# Principles of Geotechnical Engineering 7th Edition

CONVERSION FACTORS FROM ENGLISH TO SI UNITS Length: Area: Volume: 1 ft 1 ft 1 ft 1 in. 1 in. 1 in. 0.3048 m 30.48 cm 304.8 mm 0.0254 m 2.54 cm 25.4 mm 1 ft2 1 ft2 1 ft2 1 in.2 1 in.2 1 in.2 929.03 104 m2 929.03 cm2 929.03 102 mm2 6.452 104 m2 6.452 cm2 645.16 mm2 1 ft3 1 ft3 1 in.3 1 in.3 28.317 103 m3 28.317 103 cm3 16.387 106 m3 16.387 cm3 Section modulus: 1 in. 1 in.3 0.16387 10 mm 0.16387 104 m3 Hydraulic conductivity: 1 ft/min 1 ft/min 1 ft/min 1 ft/sec 1 ft/sec 1 in./min 1 in./sec 1 in./sec 0.3048 m/min 30.48 cm/min 304.8 mm/min 0.3048 m/sec 304.8 mm/sec 0.0254 m/min 2.54 cm/sec 25.4 mm/sec 3 5 Coefﬁcient of consolidation: 1 in.2/sec 1 in.2/sec 1 ft2/sec 6.452 cm2/sec 20.346 103 m2/yr 929.03 cm2/sec Force: 1 lb 1 lb 1 lb 1 kip 1 U.S. ton 1 lb 1 lb/ft 4.448 N 4.448 103 kN 0.4536 kgf 4.448 kN 8.896 kN 0.4536 103 metric ton 14.593 N/m Stress: 1 lb/ft2 1 lb/ft2 1 U.S. ton/ft2 1 kip/ft2 1 lb/in.2 47.88 N/m2 0.04788 kN/m2 95.76 kN/m2 47.88 kN/m2 6.895 kN/m2 Unit weight: 1 lb/ft3 1 lb/in.3 0.1572 kN/m3 271.43 kN/m3 Moment: 1 lb-ft 1 lb-in. 1.3558 N · m 0.11298 N · m Energy: 1 ft-lb 1.3558 J Moment of inertia: 1 in.4 1 in.4 0.4162 106 mm4 0.4162 106 m4 3 CONVERSION FACTORS FROM SI TO ENGLISH UNITS Length: Area: Volume: Force: 1m 1 cm 1 mm 1m 1 cm 1 mm 2 1m 1 cm2 1 mm2 1 m2 1 cm2 1 mm2 3.281 ft 3.281 102 ft 3.281 103 ft 39.37 in. 0.3937 in. 0.03937 in. 10.764 ft 10.764 104 ft2 10.764 106 ft2 1550 in.2 0.155 in.2 0.155 102 in.2 1 N/m2 1 kN/m2 1 kN/m2 1 kN/m2 1 kN/m2 20.885 103 lb/ft2 20.885 lb/ft2 0.01044 U.S. ton/ft2 20.885 103 kip/ft2 0.145 lb/in.2 Unit weight: 1 kN/m3 1 kN/m3 6.361 lb/ft3 0.003682 lb/in.3 Moment: 1N·m 1N·m 0.7375 lb-ft 8.851 lb-in. Energy: 1J 0.7375 ft-lb 2 1m 1 cm3 1 m3 1 cm3 35.32 ft 35.32 104 ft3 61,023.4 in.3 0.061023 in.3 1N 1 kN 1 kgf 1 kN 1 kN 1 metric ton 1 N/m 0.2248 lb 224.8 lb 2.2046 lb 0.2248 kip 0.1124 U.S. ton 2204.6 lb 0.0685 lb/ft 3 Stress: 3 Moment of inertia: 4 1 mm 1 m4 2.402 106 in.4 2.402 106 in.4 Section modulus: 1 mm3 1 m3 6.102 105 in.3 6.102 104 in.3 Hydraulic conductivity: 1 m/min 1 cm/min 1 mm/min 1 m/sec 1 mm/sec 1 m/min 1 cm/sec 1 mm/sec 3.281 ft/min 0.03281 ft/min 0.003281 ft/min 3.281 ft/sec 0.03281 ft/sec 39.37 in./min 0.3937 in./sec 0.03937 in./sec Coefﬁcient of consolidation: 1 cm2/sec 1 m2/yr 1 cm2/sec 0.155 in.2/sec 4.915 105 in.2/sec 1.0764 103 ft2/sec Principles of Geotechnical Engineering Seventh Edition BRAJA M. DAS Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Principles of Geotechnical Engineering, 7th Edition Braja M. Das Executive Director, Global Publishing Program: Chris Carson Senior Developmental Editor: Hilda Gowans Editorial Assistant: Nancy Saundercook Associate Marketing Manager: Lauren Betsos Director, Content and Media Production: Barbara Fuller Jacobsen Content Project Manager: Emily Nesheim Production Service: RPK Editorial Services Copyeditor: Shelly Gerger-Knechtl Proofreader: Nancy Benziger Indexer: Braja M. Das © 2010, 2006 Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means—graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, information storage and retrieval systems, or in any other manner—except as may be permitted by the license terms herein. 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Printed in the United States of America 1 2 3 4 5 6 7 13 12 11 10 09 To our granddaughter Elizabeth Madison This page intentionally left blank Contents Preface xiii 1 Geotechnical Engineering—A Historical Perspective 1 1.1 Geotechnical Engineering Prior to the 18th Century 1 1.2 Preclassical Period of Soil Mechanics (1700–1776) 4 1.3 Classical Soil Mechanics—Phase I (1776–1856) 4 1.4 Classical Soil Mechanics—Phase II (1856–1910) 5 1.5 Modern Soil Mechanics (1910–1927) 5 1.6 Geotechnical Engineering after 1927 7 1.7 End of an Era 10 References 12 2 Origin of Soil and Grain Size 15 2.1 Rock Cycle and the Origin of Soil 15 2.2 Soil–Particle Size 24 2.3 Clay Minerals 26 2.4 Speciﬁc Gravity (Gs ) 34 2.5 Mechanical Analysis of Soil 35 2.6 Particle–Size Distribution Curve 42 2.7 Particle Shape 46 2.8 Summary 47 Problems 47 References 50 3 Weight–Volume Relationships 51 3.1 Weight–Volume Relationships 51 3.2 Relationships among Unit Weight, Void Ratio, Moisture Content, and Speciﬁc Gravity 54 v vi Contents 3.3 Relationships among Unit Weight, Porosity, and Moisture Content 57 3.4 Various Unit-Weight Relationships 59 3.5 Relative Density 64 3.6 Comments on emax and emin 67 3.7 Summary 68 Problems 69 References 72 4 Plasticity and Structure of Soil 73 4.1 Introduction 73 4.2 Liquid Limit (LL) 74 4.3 Plastic Limit (PL) 78 4.4 Shrinkage Limit (SL) 81 4.5 Liquidity Index and Consistency Index 83 4.6 Activity 84 4.7 Plasticity Chart 87 4.8 Soil Structure 88 4.9 Summary 93 Problems 93 References 94 5 Classiﬁcation of Soil 95 5.1 5.2 5.3 5.4 5.5 Textural Classification 95 Classification by Engineering Behavior 98 AASHTO Classiﬁcation System 98 Uniﬁed Soil Classiﬁcation System 102 Summary and Comparison between the AASHTO and Uniﬁed Systems 104 Problems 112 References 113 6 Soil Compaction 114 6.1 6.2 6.3 6.4 6.5 6.6 Compaction—General Principles 114 Standard Proctor Test 115 Factors Affecting Compaction 118 Modiﬁed Proctor Test 122 Structure of Compacted Clay Soil 127 Effect of Compaction on Cohesive Soil Properties 129 Contents vii 6.7 Field Compaction 132 6.8 Speciﬁcations for Field Compaction 136 6.9 Determination of Field Unit Weight of Compaction 140 6.10 Compaction of Organic Soil and Waste Materials 144 6.11 Special Compaction Techniques 147 6.12 Summary and General Comments 155 Problems 155 References 157 7 Permeability 160 7.1 Bernoulli’s Equation 160 7.2 Darcy’s Law 162 7.3 Hydraulic Conductivity 164 7.4 Laboratory Determination of Hydraulic Conductivity 166 7.5 Relationships for Hydraulic Conductivity—Granular Soil 172 7.6 Relationships for Hydraulic Conductivity—Cohesive Soils 177 7.7 Directional Variation of Permeability 180 7.8 Equivalent Hydraulic Conductivity in Stratiﬁed Soil 182 7.9 Permeability Test in the Field by Pumping from Wells 187 7.10 In Situ Hydraulic Conductivity of Compacted Clay Soils 189 7.11 Summary and General Comments 192 Problems 193 References 196 8 Seepage 198 8.1 Laplace’s Equation of Continuity 198 8.2 Continuity Equation for Solution of Simple Flow Problems 200 8.3 Flow Nets 204 8.4 Seepage Calculation from a Flow Net 205 8.5 Flow Nets in Anisotropic Soils 209 8.6 Mathematical Solution for Seepage 211 8.7 Uplift Pressure Under Hydraulic Structures 213 8.8 Seepage Through an Earth Dam on an Impervious Base 214 8.9 L. Casagrande’s Solution for Seepage Through an Earth Dam 217 8.10 Filter Design 219 8.11 Summary 222 Problems 222 References 225 viii Contents 9 In Situ Stresses 226 9.1 Stresses in Saturated Soil without Seepage 226 9.2 Stresses in Saturated Soil with Upward Seepage 231 9.3 Stresses in Saturated Soil with Downward Seepage 233 9.4 Seepage Force 235 9.5 Heaving in Soil Due to Flow Around Sheet Piles 237 9.6 Use of Filters to Increase the Factor of Safety Against Heave 240 9.7 Effective Stress in Partially Saturated Soil 242 9.8 Capillary Rise in Soils 243 9.9 Effective Stress in the Zone of Capillary Rise 245 9.10 Summary and General Comments 248 Problems 249 References 252 10 Stresses in a Soil Mass 253 10.1 10.2 10.3 10.4 10.5 10.6 Normal and Shear Stresses on a Plane 253 The Pole Method of Finding Stresses Along a Plane 258 Stresses Caused by a Point Load 260 Vertical Stress Caused by a Line Load 262 Vertical Stress Caused by a Horizontal Line Load 264 Vertical Stress Caused by a Strip Load (Finite Width and Inﬁnite Length) 266 10.7 Vertical Stress Due to Embankment Loading 267 10.8 Vertical Stress Below the Center of a Uniformly Loaded Circular Area 273 10.9 Vertical Stress at Any Point Below a Uniformly Loaded Circular Area 275 10.10 Vertical Stress Caused by a Rectangularly Loaded Area 278 10.11 Stress Isobars 285 10.12 Inﬂuence Chart for Vertical Pressure 285 10.13 Summary and General Comments 288 Problems 289 References 293 11 Compressibility of Soil 294 11.1 11.2 11.3 11.4 Contact Pressure and Settlement Proﬁle 294 Relations for Elastic Settlement Calculation 296 Fundamentals of Consolidation 304 One-Dimensional Laboratory Consolidation Test 308 Contents 11.5 11.6 11.7 11.8 ix Void Ratio–Pressure Plots 310 Normally Consolidated and Overconsolidated Clays 313 Effect of Disturbance on Void Ratio–Pressure Relationship 316 Calculation of Settlement from One-Dimensional Primary Consolidation 317 11.9 Compression Index (Cc ) 319 11.10 Swell Index (Cs ) 320 11.11 Secondary Consolidation Settlement 326 11.12 Time Rate of Consolidation 330 11.13 Coefﬁcient of Consolidation 338 11.14 Calculation of Consolidation Settlement Under a Foundation 345 11.15 A Case History—Settlement Due to a Preload Fill for Construction of Tampa VA Hospital 347 11.16 Methods for Accelerating Consolidation Settlement 351 11.17 Precompression 354 11.18 Summary and General Comments 357 Problems 358 References 362 12 Shear Strength of Soil 365 12.1 Mohr–Coulomb Failure Criterion 365 12.2 Inclination of the Plane of Failure Caused by Shear 367 12.3 Laboratory Tests for Determination of Shear Strength Parameters 368 12.4 Direct Shear Test 369 12.5 Drained Direct Shear Test on Saturated Sand and Clay 373 12.6 General Comments on Direct Shear Test 376 12.7 Triaxial Shear Test—General 380 12.8 Consolidated-Drained Triaxial Test 381 12.9 Consolidated-Undrained Triaxial Test 389 12.10 Unconsolidated-Undrained Triaxial Test 395 12.11 Unconﬁned Compression Test on Saturated Clay 397 12.12 Empirical Relationships Between Undrained Cohesion (cu ) and Effective Overburden Pressure (soœ ) 398 12.13 Sensitivity and Thixotropy of Clay 401 12.14 Strength Anisotropy in Clay 403 12.15 Vane Shear Test 406 12.16 Other Methods for Determining Undrained Shear Strength 411 12.17 Shear Strength of Unsaturated Cohesive Soils 412 x Contents 12.18 Stress Path 414 12.19 Summary and General Comments 418 Problems 419 References 422 13 Lateral Earth Pressure: At-Rest, Rankine, and Coulomb 424 13.1 13.2 13.3 13.4 13.5 13.6 13.7 At-Rest, Active, and Passive Pressures 424 Earth Pressure At-Rest 426 Earth Pressure At-Rest for Partially Submerged Soil 429 Rankine’s Theory of Active Pressure 432 Theory of Rankine’s Passive Pressure 434 Yielding of Wall of Limited Height 436 A Generalized Case for Rankine Active and Passive Pressures—Granular Backfill 438 13.8 Diagrams for Lateral Earth-Pressure Distribution Against Retaining Walls 442 13.9 Rankine’s Pressure for c–f Soil—Inclined Backﬁll 454 13.10 Coulomb’s Active Pressure 457 13.11 Graphic Solution for Coulomb’s Active Earth Pressure 461 13.12 Coulomb’s Passive Pressure 466 13.13 Active Force on Retaining Walls with Earthquake Forces 468 13.14 Common Types of Retaining Walls in the Field 479 13.15 Summary and General Comments 482 Problems 483 References 486 14 Lateral Earth Pressure: Curved Failure Surface 488 14.1 Retaining Walls with Friction 488 14.2 Properties of a Logarithmic Spiral 490 14.3 Procedure for Determination of Passive Earth Pressure (Pp)—Cohesionless Backﬁll 492 14.4 Coefﬁcient of Passive Earth Pressure (Kp) 494 14.5 Passive Force on Walls with Earthquake Forces 498 14.6 Braced Cuts—General 499 14.7 Determination of Active Thrust on Bracing Systems of Open Cuts—Granular Soil 503 14.8 Determination of Active Thrust on Bracing Systems for Cuts—Cohesive Soil 504 Contents xi 14.9 Pressure Variation for Design of Sheetings, Struts, and Wales 505 14.10 Summary 509 Problems 509 References 511 15 Slope Stability 512 15.1 15.2 15.3 15.4 15.5 Introduction—Modes of Slope Failure 512 Factor of Safety 514 Stability of Inﬁnite Slopes 515 Finite Slopes—General 519 Analysis of Finite Slopes with Plane Failure Surfaces (Culmann’s Method) 520 15.6 Analysis of Finite Slopes with Circular Failure Surfaces—General 523 15.7 Mass Procedure—Slopes in Homogeneous Clay Soil with f 0 524 15.8 Mass Procedure—Stability of Saturated Clay Slopes (f 0 Condition) with Earthquake Forces 532 15.9 Mass Procedure—Slopes in Homogeneous c– f Soil 535 15.10 Ordinary Method of Slices 544 15.11 Bishop’s Simpliﬁed Method of Slices 548 15.12 Stability Analysis by Method of Slices for Steady-State Seepage 550 15.13 Other Solutions for Steady-State Seepage Condition 557 15.14 A Case History of Slope Failure 561 15.15 Morgenstern’s Method of Slices for Rapid Drawdown Condition 565 15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay 568 Problems 571 References 574 16 Soil-Bearing Capacity for Shallow Foundations 576 16.1 16.2 16.3 16.4 16.5 16.6 Ultimate Soil-Bearing Capacity for Shallow Foundations 577 Terzaghi’s Ultimate Bearing Capacity Equation 579 Effect of Groundwater Table 584 Factor of Safety 586 General Bearing Capacity Equation 589 A Case History for Evaluation of the Ultimate Bearing Capacity 593 xii Contents 16.7 Ultimate Load for Shallow Foundations Under Eccentric Load 597 16.8 Bearing Capacity of Sand Based on Settlement 602 16.9 Plate-Load Test 604 16.10 Summary and General Comments 607 Problems 608 References 610 17 Landﬁll Liners and Geosynthetics 611 17.1 Landﬁll Liners—Overview 611 17.2 Compaction of Clay Soil for Clay Liner Construction 612 17.3 Geosynthetics 616 17.4 Geotextiles 616 17.5 Geomembranes 619 17.6 Geonets 621 17.7 Single Clay Liner and Single Geomembrane Liner Systems 622 17.8 Recent Advances in the Liner Systems for Landﬁlls 623 17.9 Leachate Removal Systems 624 17.10 Closure of Landﬁlls 627 17.11 Summary and General Comments 628 References 628 18 Subsoil Exploration 629 18.1 Planning for Soil Exploration 629 18.2 Boring Methods 631 18.3 Common Sampling Methods 635 18.4 Sample Disturbance 639 18.5 Correlations for Standard Penetration Test 639 18.6 Other In Situ Tests 644 18.7 Rock Coring 648 18.8 Soil Exploration Report 650 Problems 652 References 653 Answers to Selected Problems 655 Index 662 Preface Principles of Geotechnical Engineering was originally published with a 1985 copyright and was intended for use as a text for the introductory course in geotechnical engineering taken by practically all civil engineering students, as well as for use as a reference book for practicing engineers. The book was revised in 1990, 1994, 1998, 2002, and 2006. This Seventh Edition is the twenty-fifth anniversary edition of the text. As in the previous editions of the book, this new edition offers an overview of soil properties and mechanics, together with coverage of field practices and basic engineering procedures without changing the basic philosophy in which the text was written originally. Unlike the Sixth Edition that had 17 chapters, this edition has 18 chapters. For better understanding and more comprehensive coverage, Weight-Volume Relationships and Plasticity and Structure of Soil are now presented in two separate chapters (Chapters 3 and 4). Most of the example and homework problems have been changed and/or modified. Other noteworthy changes for the Seventh Edition are • • • • • • New scanning electron micrographs for quartz, mica, limestone, sand grains, and clay minerals such as kaolinite and montmorillonite have been added to Chapter 2. A summary of recently published empirical relationships between liquid limit, plastic limit, plasticity index, activity, and clay-size fractions in a soil mass have been incorporated in Chapter 4. The USDA Textural Classification of Soil has now been added to Chapter 5 (Classification of Soil). Additional empirical relationships for hydraulic conductivity for granular and cohesive soils have been added, respectively, to Chapter 7 (Permeability) and Chapter 17 (Landfill Liners and Geosynthetics). The presentation of the filter design criteria has been improved in Chapter 8 (Seepage). In Chapter 11 (Compressibility of Soil), the procedure for estimating elastic settlement of foundations has been thoroughly revised with the inclusions of theories by Steinbrenner (1934) and Fox (1948). A case study related to the consolidation settlement due to a preload fill for construction of the Tampa VA Hospital is also added to this chapter. xiii xiv Preface • • • • The presentation on estimation of active force on retaining walls with earthquake forces in Chapter 13 (Lateral Earth Pressure: At-Rest, Rankine, and Coulomb) has been improved. Chapter 14 (Lateral Earth Pressure: Curved Failure Surface) now includes the procedure to estimate the passive earth pressure on retaining walls with inclined backface and horizontal granular backfill using the method of triangular slices. It also includes the relationships for passive earth pressure on retaining walls with a horizontal granular backfill and vertical backface under earthquake conditions determined by using the pseudo-static method. Chapter 15 (Slope Stability) now includes a case history of slope failure in relation to a major improvement program of Interstate Route 95 in New Hampshire. A method to calculate the ultimate bearing capacity of eccentrically loaded shallow strip foundations in granular soil using the reduction factor has been added to Chapter 16 (Soil-Bearing Capacity for Shallow Foundations). I am grateful to my wife Janice for her help in getting the manuscript ready for publication. Finally, many thanks are due to Christopher Carson, Executive Director, Global Publishing Programs; Hilda Gowans, Senior Development Editor; and the production staff of Cengage Learning (Engineering) for the final development and production of the book. Braja M. Das Henderson, Nevada 1 Geotechnical Engineering— A Historical Perspective For engineering purposes, soil is deﬁned as the uncemented aggregate of mineral grains and decayed organic matter (solid particles) with liquid and gas in the empty spaces between the solid particles. Soil is used as a construction material in various civil engineering projects, and it supports structural foundations. Thus, civil engineers must study the properties of soil, such as its origin, grain-size distribution, ability to drain water, compressibility, shear strength, and load-bearing capacity. Soil mechanics is the branch of science that deals with the study of the physical properties of soil and the behavior of soil masses subjected to various types of forces. Soils engineering is the application of the principles of soil mechanics to practical problems. Geotechnical engineering is the subdiscipline of civil engineering that involves natural materials found close to the surface of the earth. It includes the application of the principles of soil mechanics and rock mechanics to the design of foundations, retaining structures, and earth structures. 1.1 Geotechnical Engineering Prior to the 18th Century The record of a person’s ﬁrst use of soil as a construction material is lost in antiquity. In true engineering terms, the understanding of geotechnical engineering as it is known today began early in the 18th century (Skempton, 1985). For years, the art of geotechnical engineering was based on only past experiences through a succession of experimentation without any real scientiﬁc character. Based on those experimentations, many structures were built—some of which have crumbled, while others are still standing. Recorded history tells us that ancient civilizations ﬂourished along the banks of rivers, such as the Nile (Egypt), the Tigris and Euphrates (Mesopotamia), the Huang Ho (Yellow River, China), and the Indus (India). Dykes dating back to about 2000 B.C. were built in the basin of the Indus to protect the town of Mohenjo Dara (in what became Pakistan after 1947). During the Chan dynasty in China (1120 B.C. to 249 B.C.) many dykes were built for irrigation purposes. There is no evidence that measures were taken to stabilize the foundations or check erosion caused by ﬂoods (Kerisel, 1985). Ancient Greek civilization used isolated pad footings and strip-and-raft foundations for building structures. Beginning around 2750 B.C., the ﬁve most important pyramids were built in Egypt in a period of less than a century (Saqqarah, Meidum, Dahshur South and North, and Cheops). This posed formidable challenges regarding foundations, stability of slopes, and construction of 1 2 Chapter 1: Geotechnical Engineering—A Historical Perspective underground chambers. With the arrival of Buddhism in China during the Eastern Han dynasty in 68 A.D., thousands of pagodas were built. Many of these structures were constructed on silt and soft clay layers. In some cases the foundation pressure exceeded the load-bearing capacity of the soil and thereby caused extensive structural damage. One of the most famous examples of problems related to soil-bearing capacity in the construction of structures prior to the 18th century is the Leaning Tower of Pisa in Italy. (See Figure 1.1.) Construction of the tower began in 1173 A.D. when the Republic of Pisa was ﬂourishing and continued in various stages for over 200 years. The structure weighs about 15,700 metric tons and is supported by a circular base having a diameter of 20 m ( 66 ft). The tower has tilted in the past to the east, north, west and, ﬁnally, to the south. Recent investigations showed that a weak clay layer exists at a depth of about 11 m ( 36 ft) below the ground surface compression, which caused the tower to tilt. It became more than 5 m ( 16.5 ft) out of plumb with the 54 m ( 179 ft) height. The tower was closed in 1990 because it was feared that it would either fall over or collapse. It recently has been stabilized by excavating soil from under the north side of the tower. About 70 metric tons of earth were removed in 41 separate extractions that spanned the width of the tower. As the ground Figure 1.1 Leaning Tower of Pisa, Italy (Courtesy of Braja M. Das, Henderson, Nevada) 1.1 Geotechnical Engineering Prior to the 18th Century 3 Figure 1.2 Tilting of Garisenda Tower (left) and Asinelli Tower (right) in Bologna, Italy (Courtesy of Braja M. Das, Henderson, Nevada) gradually settled to fill the resulting space, the tilt of the tower eased. The tower now leans 5 degrees. The half-degree change is not noticeable, but it makes the structure considerably more stable. Figure 1.2 is an example of a similar problem. The towers shown in Figure 1.2 are located in Bologna, Italy, and they were built in the 12th century. The tower on the left is usually referred to as the Garisenda Tower. It is 48 m ( 157 ft) in height and weighs about 4210 metric tons. It has tilted about 4 degrees. The tower on the right is the Asinelli Tower, which is 97 m high and weighs 7300 metric tons. It has tilted about 1.3 degrees. After encountering several foundation-related problems during construction over centuries past, engineers and scientists began to address the properties and behaviors of soils in a more methodical manner starting in the early part of the 18th century. Based on the emphasis and the nature of study in the area of geotechnical engineering, the time span extending from 1700 to 1927 can be divided into four major periods (Skempton, 1985): 1. 2. 3. 4. Pre-classical (1700 to 1776 A.D.) Classical soil mechanics—Phase I (1776 to 1856 A.D.) Classical soil mechanics—Phase II (1856 to 1910 A.D.) Modern soil mechanics (1910 to 1927 A.D.) 4 Chapter 1: Geotechnical Engineering—A Historical Perspective Brief descriptions of some signiﬁcant developments during each of these four periods are discussed below. 1.2 Preclassical Period of Soil Mechanics (1700 –1776) This period concentrated on studies relating to natural slope and unit weights of various types of soils, as well as the semiempirical earth pressure theories. In 1717 a French royal engineer, Henri Gautier (1660–1737), studied the natural slopes of soils when tipped in a heap for formulating the design procedures of retaining walls. The natural slope is what we now refer to as the angle of repose. According to this study, the natural slope of clean dry sand and ordinary earth were 31 and 45, respectively. Also, the unit weight of clean dry sand and ordinary earth were recommended to be 18.1 kN/m3 (115 lb/ft3) and 13.4 kN/m3 (85 lb/ft3), respectively. No test results on clay were reported. In 1729, Bernard Forest de Belidor (1671–1761) published a textbook for military and civil engineers in France. In the book, he proposed a theory for lateral earth pressure on retaining walls that was a follow-up to Gautier’s (1717) original study. He also speciﬁed a soil classiﬁcation system in the manner shown in the following table. Unit weight Classiﬁcation Rock kN/m3 — lb/ft3 — Firm or hard sand Compressible sand 16.7 to 18.4 106 to 117 Ordinary earth (as found in dry locations) Soft earth (primarily silt) Clay 13.4 16.0 18.9 85 102 120 — — Peat The ﬁrst laboratory model test results on a 76-mm-high ( 3 in.) retaining wall built with sand backﬁll were reported in 1746 by a French engineer, Francois Gadroy (1705–1759), who observed the existence of slip planes in the soil at failure. Gadroy’s study was later summarized by J. J. Mayniel in 1808. 1.3 Classical Soil Mechanics—Phase I (1776 –1856) During this period, most of the developments in the area of geotechnical engineering came from engineers and scientists in France. In the preclassical period, practically all theoretical considerations used in calculating lateral earth pressure on retaining walls were based on an arbitrarily based failure surface in soil. In his famous paper presented in 1776, French scientist Charles Augustin Coulomb (1736–1806) used the principles of calculus for maxima and minima to determine the true position of the sliding surface in soil behind a retaining wall. In this analysis, Coulomb used the laws of friction and cohesion for solid bodies. In 1820, special cases of Coulomb’s work were studied by French engineer Jacques Frederic Francais (1775–1833) and by French applied mechanics professor Claude Louis Marie Henri Navier 1.5 Modern Soil Mechanics (1910 –1927) 5 (1785–1836). These special cases related to inclined backﬁlls and backﬁlls supporting surcharge. In 1840, Jean Victor Poncelet (1788–1867), an army engineer and professor of mechanics, extended Coulomb’s theory by providing a graphical method for determining the magnitude of lateral earth pressure on vertical and inclined retaining walls with arbitrarily broken polygonal ground surfaces. Poncelet was also the ﬁrst to use the symbol f for soil friction angle. He also provided the ﬁrst ultimate bearing-capacity theory for shallow foundations. In 1846 Alexandre Collin (1808–1890), an engineer, provided the details for deep slips in clay slopes, cutting, and embankments. Collin theorized that in all cases the failure takes place when the mobilized cohesion exceeds the existing cohesion of the soil. He also observed that the actual failure surfaces could be approximated as arcs of cycloids. The end of Phase I of the classical soil mechanics period is generally marked by the year (1857) of the ﬁrst publication by William John Macquorn Rankine (1820–1872), a professor of civil engineering at the University of Glasgow. This study provided a notable theory on earth pressure and equilibrium of earth masses. Rankine’s theory is a simpliﬁcation of Coulomb’s theory. 1.4 Classical Soil Mechanics—Phase II (1856 –1910) Several experimental results from laboratory tests on sand appeared in the literature in this phase. One of the earliest and most important publications is one by French engineer Henri Philibert Gaspard Darcy (1803–1858). In 1856, he published a study on the permeability of sand ﬁlters. Based on those tests, Darcy deﬁned the term coefﬁcient of permeability (or hydraulic conductivity) of soil, a very useful parameter in geotechnical engineering to this day. Sir George Howard Darwin (1845–1912), a professor of astronomy, conducted laboratory tests to determine the overturning moment on a hinged wall retaining sand in loose and dense states of compaction. Another noteworthy contribution, which was published in 1885 by Joseph Valentin Boussinesq (1842–1929), was the development of the theory of stress distribution under loaded bearing areas in a homogeneous, semiinﬁnite, elastic, and isotropic medium. In 1887, Osborne Reynolds (1842–1912) demonstrated the phenomenon of dilatency in sand. 1.5 Modern Soil Mechanics (1910 –1927) In this period, results of research conducted on clays were published in which the fundamental properties and parameters of clay were established. The most notable publications are described next. Around 1908, Albert Mauritz Atterberg (1846–1916), a Swedish chemist and soil scientist, defined clay-size fractions as the percentage by weight of particles smaller than 2 microns in size. He realized the important role of clay particles in a soil and the plasticity thereof. In 1911, he explained the consistency of cohesive soils by defining liquid, plastic, and shrinkage limits. He also defined the plasticity index as the difference between liquid limit and plastic limit (see Atterberg, 1911). In October 1909, the 17-m (56-ft) high earth dam at Charmes, France, failed. It was built between 1902–1906. A French engineer, Jean Fontard (1884–1962), carried out investigations to determine the cause of failure. In that context, he conducted undrained 6 Chapter 1: Geotechnical Engineering—A Historical Perspective double-shear tests on clay specimens (0.77 m2 in area and 200 mm thick) under constant vertical stress to determine their shear strength parameters (see Frontard, 1914). The times for failure of these specimens were between 10 to 20 minutes. Arthur Langley Bell (1874–1956), a civil engineer from England, worked on the design and construction of the outer seawall at Rosyth Dockyard. Based on his work, he developed relationships for lateral pressure and resistance in clay as well as bearing capacity of shallow foundations in clay (see Bell, 1915). He also used shear-box tests to measure the undrained shear strength of undisturbed clay specimens. Wolmar Fellenius (1876–1957), an engineer from Sweden, developed the stability analysis of saturated clay slopes (that is, 0 condition) with the assumption that the critical surface of sliding is the arc of a circle. These were elaborated upon in his papers published in 1918 and 1926. The paper published in 1926 gave correct numerical solutions for the stability numbers of circular slip surfaces passing through the toe of the slope. Karl Terzaghi (1883–1963) of Austria (Figure 1.3) developed the theory of consolidation for clays as we know today. The theory was developed when Terzaghi was teaching Image not available due to copyright restrictions 1.6 Geotechnical Engineering after 1927 7 at the American Roberts College in Istanbul, Turkey. His study spanned a five-year period from 1919 to 1924. Five different clay soils were used. The liquid limit of those soils ranged between 36 to 67, and the plasticity index was in the range of 18 to 38. The consolidation theory was published in Terzaghi’s celebrated book Erdbaumechanik in 1925. 1.6 Geotechnical Engineering after 1927 The publication of Erdbaumechanik auf Bodenphysikalisher Grundlage by Karl Terzaghi in 1925 gave birth to a new era in the development of soil mechanics. Karl Terzaghi is known as the father of modern soil mechanics, and rightfully so. Terzaghi was born on October 2, 1883 in Prague, which was then the capital of the Austrian province of Bohemia. In 1904 he graduated from the Technische Hochschule in Graz, Austria, with an undergraduate degree in mechanical engineering. After graduation he served one year in the Austrian army. Following his army service, Terzaghi studied one more year, concentrating on geological subjects. In January 1912, he received the degree of Doctor of Technical Sciences from his alma mater in Graz. In 1916, he accepted a teaching position at the Imperial School of Engineers in Istanbul. After the end of World War I, he accepted a lectureship at the American Robert College in Istanbul (1918–1925). There he began his research work on the behavior of soils and settlement of clays and on the failure due to piping in sand under dams. The publication Erdbaumechanik is primarily the result of this research. In 1925, Terzaghi accepted a visiting lectureship at Massachusetts Institute of Technology, where he worked until 1929. During that time, he became recognized as the leader of the new branch of civil engineering called soil mechanics. In October 1929, he returned to Europe to accept a professorship at the Technical University of Vienna, which soon became the nucleus for civil engineers interested in soil mechanics. In 1939, he returned to the United States to become a professor at Harvard University. The ﬁrst conference of the International Society of Soil Mechanics and Foundation Engineering (ISSMFE) was held at Harvard University in 1936 with Karl Terzaghi presiding. The conference was possible due to the conviction and efforts of Professor Arthur Casagrande of Harvard University. About 200 individuals representing 21 countries attended this conference. It was through the inspiration and guidance of Terzaghi over the preceding quarter-century that papers were brought to that conference covering a wide range of topics, such as • • • • • • • • • • • • Effective stress Shear strength Testing with Dutch cone penetrometer Consolidation Centrifuge testing Elastic theory and stress distribution Preloading for settlement control Swelling clays Frost action Earthquake and soil liquefaction Machine vibration Arching theory of earth pressure 8 Chapter 1: Geotechnical Engineering—A Historical Perspective For the next quarter-century, Terzaghi was the guiding spirit in the development of soil mechanics and geotechnical engineering throughout the world. To that effect, in 1985, Ralph Peck wrote that “few people during Terzaghi’s lifetime would have disagreed that he was not only the guiding spirit in soil mechanics, but that he was the clearing house for research and application throughout the world. Within the next few years he would be engaged on projects on every continent save Australia and Antarctica.” Peck continued with, “Hence, even today, one can hardly improve on his contemporary assessments of the state of soil mechanics as expressed in his summary papers and presidential addresses.” In 1939, Terzaghi delivered the 45th James Forrest Lecture at the Institution of Civil Engineers, London. His lecture was entitled “Soil Mechanics—A New Chapter in Engineering Science.” In it, he proclaimed that most of the foundation failures that occurred were no longer “acts of God.” Following are some highlights in the development of soil mechanics and geotechnical engineering that evolved after the ﬁrst conference of the ISSMFE in 1936: • • • • Publication of the book Theoretical Soil Mechanics by Karl Terzaghi in 1943 (Wiley, New York); Publication of the book Soil Mechanics in Engineering Practice by Karl Terzaghi and Ralph Peck in 1948 (Wiley, New York); Publication of the book Fundamentals of Soil Mechanics by Donald W. Taylor in 1948 (Wiley, New York); Start of the publication of Geotechnique, the international journal of soil mechanics in 1948 in England; After a brief interruption for World War II, the second conference of ISSMFE was held in Rotterdam, The Netherlands, in 1948. There were about 600 participants, and seven volumes of proceedings were published. In this conference, A. W. Skempton presented the landmark paper on 0 concept for clays. Following Rotterdam, ISSMFE conferences have been organized about every four years in different parts of the world. The aftermath of the Rotterdam conference saw the growth of regional conferences on geotechnical engineering, such as • • • • European Regional Conference on Stability of Earth Slopes, Stockholm (1954) First Australia-New Zealand Conference on Shear Characteristics of Soils (1952) First Pan American Conference, Mexico City (1960) Research conference on Shear Strength of Cohesive Soils, Boulder, Colorado, (1960) Two other important milestones between 1948 and 1960 are (1) the publication of A. W. Skempton’s paper on A and B pore pressure parameters which made effective stress calculations more practical for various engineering works and (2) publication of the book entitled The Measurement of Soil Properties in the Triaxial Text by A. W. Bishop and B. J. Henkel (Arnold, London) in 1957. By the early 1950’s, computer-aided finite difference and finite element solutions were applied to various types of geotechnical engineering problems. They still remain an important and useful computation tool in our profession. Since the early days, the profession of geotechnical engineering has come a long way and has matured. It is now an established branch of civil engineering, and thousands of civil engineers declare geotechnical engineering to be their preferred area of speciality. In 1997, the ISSMFE was changed to ISSMGE (International Society of Soil Mechanics and Geotechnical Engineering) to reﬂect its true scope. These international conferences have been instrumental for exchange of information regarding new 1.6 Geotechnical Engineering after 1927 9 Table 1.1 Details of ISSMFE (1936–1997) and ISSMGE (1997–present) Conferences Conference Location Year I II III IV V VI VII VIII IX X XI XII XIII XIV XV XVI XVII Harvard University, Boston, U.S.A. Rotterdam, the Netherlands Zurich, Switzerland London, England Paris, France Montreal, Canada Mexico City, Mexico Moscow, U.S.S.R. Tokyo, Japan Stockholm, Sweden San Francisco, U.S.A. Rio de Janeiro, Brazil New Delhi, India Hamburg, Germany Istanbul, Turkey Osaka, Japan Alexandria, Egypt 1936 1948 1953 1957 1961 1965 1969 1973 1977 1981 1985 1989 1994 1997 2001 2005 2009 developments and ongoing research activities in geotechnical engineering. Table 1.1 gives the location and year in which each conference of ISSMFE/ISSMGE was held, and Table 1.2 gives a list of all of the presidents of the society. In 1997, a total of 30 technical committees of ISSMGE was in place. The names of these technical committees are given in Table 1.3. Table 1.2 Presidents of ISSMFE (1936–1997) and ISSMGE (1997–present) Conferences Year President 1936–1957 1957–1961 1961–1965 1965–1969 1969–1973 1973–1977 1977–1981 1981–1985 1985–1989 1989–1994 1994–1997 1997–2001 2001–2005 2005–2009 K. Terzaghi (U. S. A.) A. W. Skempton (U. K.) A. Casagrande (U. S. A.) L. Bjerrum (Norway) R. B. Peck (U. S. A.) J. Kerisel (France) M. Fukuoka (Japan) V. F. B. deMello (Brazil) B. B. Broms (Singapore) N. R. Morgenstern (Canada) M. Jamiolkowski (Italy) K. Ishihara (Japan) W. F. Van Impe (Belgium) P. S. Sêco e Pinto (Portugal) 10 Chapter 1: Geotechnical Engineering—A Historical Perspective Table 1.3 ISSMGE Technical Committees for 1997–2001 (based on Ishihara, 1999) 1.7 Committee number Committee name TC-1 TC-2 TC-3 TC-4 TC-5 TC-6 TC-7 TC-8 TC-9 TC-10 TC-11 TC-12 TC-14 TC-15 TC-16 TC-17 TC-18 TC-19 TC-20 TC-22 TC-23 TC-24 TC-25 TC-26 TC-28 TC-29 TC-30 TC-31 TC-32 TC-33 TC-34 Instrumentation for Geotechnical Monitoring Centrifuge Testing Geotechnics of Pavements and Rail Tracks Earthquake Geotechnical Engineering Environmental Geotechnics Unsaturated Soils Tailing Dams Frost Geosynthetics and Earth Reinforcement Geophysical Site Characterization Landslides Validation of Computer Simulation Offshore Geotechnical Engineering Peat and Organic Soils Ground Property Characterization from In-situ Testing Ground Improvement Pile Foundations Preservation of Historic Sites Professional Practice Indurated Soils and Soft Rocks Limit State Design Geotechnical Engineering Soil Sampling, Evaluation and Interpretation Tropical and Residual Soils Calcareous Sediments Underground Construction in Soft Ground Stress-Strain Testing of Geomaterials in the Laboratory Coastal Geotechnical Engineering Education in Geotechnical Engineering Risk Assessment and Management Scour of Foundations Deformation of Earth Materials End of an Era In Section 1.6, a brief outline of the contributions made to modern soil mechanics by pioneers such as Karl Terzaghi, Arthur Casagrande, Donald W. Taylor, Laurits Bjerrum, and Ralph B. Peck was presented. The last of the early giants of the profession, Ralph B. Peck, passed away on February 18, 2008, at the age of 95. Professor Ralph B. Peck (Figure 1.4) was born in Winnipeg, Canada to American parents Orwin K. and Ethel H. Peck on June 23, 1912. He received B.S. and Ph.D. degrees in 1934 and 1937, respectively, from Rensselaer Polytechnic Institute, Troy, New York. During the period from 1938 to 1939, he took courses from Arthur Casagrande at Harvard University in a new subject called “soil mechanics.” From 1939 to 1943, Dr. Peck worked as an assistant to Karl Terzaghi, the “father” of modern soil mechanics, on the Chicago 1.7 End of an Era 11 Image not available due to copyright restrictions Subway Project. In 1943, he joined the University of Illinois at Champaign-Urban and was a professor of foundation engineering from 1948 until he retired in 1974. After retirement, he was active in consulting, which included major geotechnical projects in 44 states in the United States and 28 other countries on five continents. Some examples of his major consulting projects include • • • • • Rapid transit systems in Chicago, San Francisco, and Washington, D.C. Alaskan pipeline system James Bay Project in Quebec, Canada Heathrow Express Rail Project (U.K.) Dead Sea dikes His last project was the Rion-Antirion Bridge in Greece. On March 13, 2008, The Times of the United Kingdom wrote, “Ralph B. Peck was an American civil engineer who invented a controversial construction technique that would be used on some of the modern engineering wonders of the world, including the Channel Tunnel. Known as 12 Chapter 1: Geotechnical Engineering—A Historical Perspective Figure 1.5 Professor Ralph B. Peck (right) with the author, Braja Das, during his trip to attend the XVI ISSMGE Conference in Osaka, Japan—the last conference of its type that he would attend (Courtesy of Braja M. Das, Henderson, Nevada) ‘the godfather of soil mechanics,’ he was directly responsible for a succession of celebrated tunneling and earth dam projects that pushed the boundaries of what was believed to be possible.” Dr. Peck authored more than 250 highly distinguished technical publications. He was the president of the ISSMGE from 1969 to 1973. In 1974, he received the National Medal of Science from President Gerald R. Ford. Professor Peck was a teacher, mentor, friend, and counselor to generations of geotechnical engineers in every country in the world. The 16th ISSMGE Conference in Osaka, Japan (2005) would be the last major conference of its type that he would attend. During his trip to Osaka, even at the age of 93, he was intent on explaining to the author the importance of field testing and sound judgment in the decision-making process involved in the design and construction of geotechnical engineering projects (which he had done to numerous geotechnical engineers all over the world) (Figure 1.5). This is truly the end of an era. References ATTERBERG, A. M. (1911). “Über die physikalische Bodenuntersuchung, und über die Plastizität de Tone,” International Mitteilungen für Bodenkunde, Verlag für Fachliteratur. G.m.b.H. Berlin, Vol. 1, 10–43. References 13 BELIDOR, B. F. (1729). La Science des Ingenieurs dans la Conduite des Travaux de Fortiﬁcation et D’Architecture Civil, Jombert, Paris. BELL, A. L. (1915). “The Lateral Pressure and Resistance of Clay, and Supporting Power of Clay Foundations,” Min. Proceeding of Institute of Civil Engineers, Vol. 199, 233–272. BISHOP, A. W. and HENKEL, B. J. (1957). The Measurement of Soil Properties in the Triaxial Test, Arnold, London. BOUSSINESQ, J. V. (1885). Application des Potentiels â L’Etude de L’Équilibre et du Mouvement des Solides Élastiques, Gauthier-Villars, Paris. COLLIN, A. (1846). Recherches Expérimentales sur les Glissements Spontanés des Terrains Argileux Accompagnées de Considérations sur Quelques Principes de la Mécanique Terrestre, Carilian-Goeury, Paris. COULOMB, C. A. (1776). “Essai sur une Application des Règles de Maximis et Minimis à Quelques Problèmes de Statique Relatifs à L’Architecture,” Mèmoires de la Mathèmatique et de Phisique, présentés à l’Académie Royale des Sciences, par divers savans, et lûs dans sés Assemblées, De L’Imprimerie Royale, Paris, Vol. 7, Annee 1793, 343–382. DARCY, H. P. G. (1856). Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris. DARWIN, G. H. (1883). “On the Horizontal Thrust of a Mass of Sand,” Proceedings, Institute of Civil Engineers, London, Vol. 71, 350–378. FELLENIUS, W. (1918). “Kaj-och Jordrasen I Göteborg,” Teknisk Tidskrift. Vol. 48, 17–19. FRANCAIS, J. F. (1820). “Recherches sur la Poussée de Terres sur la Forme et Dimensions des Revêtments et sur la Talus D’Excavation,” Mémorial de L’Ofﬁcier du Génie, Paris, Vol. IV, 157–206. FRONTARD, J. (1914). “Notice sur L’Accident de la Digue de Charmes,” Anns. Ponts et Chaussées 9 th Ser., Vol. 23, 173–292. GADROY, F. (1746). Mémoire sur la Poussée des Terres, summarized by Mayniel, 1808. GAUTIER, H. (1717). Dissertation sur L’Epaisseur des Culées des Ponts . . . sur L’Effort et al Pesanteur des Arches . . . et sur les Proﬁles de Maconnerie qui Doivent Supporter des Chaussées, des Terrasses, et des Remparts. Cailleau, Paris. ISHIHARA, K. (1999). Personal communication. KERISEL, J. (1985). “The History of Geotechnical Engineering up until 1700,” Proceedings, XI International Conference on Soil Mechanics and Foundation Engineering, San Francisco, Golden Jubilee Volume, A. A. Balkema, 3–93. MAYNIEL, J. J. (1808). Traité Experimentale, Analytique et Pratique de la Poussé des Terres. Colas, Paris. NAVIER, C. L. M. (1839). Leçons sur L’Application de la Mécanique à L’Establissement des Constructions et des Machines, 2nd ed., Paris. PECK, R. B. (1985). “The Last Sixty Years,” Proceedings, XI International Conference on Soil Mechanics and Foundation Engineering, San Francisco, Golden Jubilee Volume, A. A. Balkema, 123–133. PONCELET, J. V. (1840). Mémoire sur la Stabilité des Revêtments et de seurs Fondations, Bachelier, Paris. RANKINE, W. J. M. (1857). “On the Stability of Loose Earth,” Philosophical Transactions, Royal Society, Vol. 147, London. REYNOLDS, O. (1887). “Experiments Showing Dilatency, a Property of Granular Material Possibly Connected to Gravitation,” Proceedings, Royal Society, London, Vol. 11, 354–363. SKEMPTON, A. W. (1948). “The f 0 Analysis of Stability and Its Theoretical Basis,” Proceedings, II International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, Vol. 1, 72–78. SKEMPTON, A. W. (1954). “The Pore Pressure Coefﬁcients A and B,” Geotechnique, Vol. 4, 143–147. SKEMPTON, A. W. (1985). “A History of Soil Properties, 1717–1927,” Proceedings, XI International Conference on Soil Mechanics and Foundation Engineering, San Francisco, Golden Jubilee Volume, A. A. Balkema, 95–121. 14 Chapter 1: Geotechnical Engineering—A Historical Perspective TAYLOR, D. W. (1948). Fundamentals of Soil Mechanics, John Wiley, New York. TERZAGHI, K. (1925). Erdbaumechanik auf Bodenphysikalisher Grundlage, Deuticke, Vienna. TERZAGHI, K. (1939). “Soil Mechanics—A New Chapter in Engineering Science,” Institute of Civil Engineers Journal, London, Vol. 12, No. 7, 106–142. TERZAGHI, K. (1943). Theoretical Soil Mechanics, John Wiley, New York. TERZAGHI, K., and PECK, R. B. (1948). Soil Mechanics in Engineering Practice, John Wiley, New York. 2 Origin of Soil and Grain Size In general, soils are formed by weathering of rocks. The physical properties of soil are dictated primarily by the minerals that constitute the soil particles and, hence, the rock from which it is derived. This chapter provides an outline of the rock cycle and the origin of soil and the grain-size distribution of particles in a soil mass. 2.1 Rock Cycle and the Origin of Soil The mineral grains that form the solid phase of a soil aggregate are the product of rock weathering. The size of the individual grains varies over a wide range. Many of the physical properties of soil are dictated by the size, shape, and chemical composition of the grains. To better understand these factors, one must be familiar with the basic types of rock that form the earth’s crust, the rock-forming minerals, and the weathering process. On the basis of their mode of origin, rocks can be divided into three basic types: igneous, sedimentary, and metamorphic. Figure 2.1 shows a diagram of the formation cycle of different types of rock and the processes associated with them. This is called the rock cycle. Brief discussions of each element of the rock cycle follow. Igneous Rock Igneous rocks are formed by the solidiﬁcation of molten magma ejected from deep within the earth’s mantle. After ejection by either ﬁssure eruption or volcanic eruption, some of the molten magma cools on the surface of the earth. Sometimes magma ceases its mobility below the earth’s surface and cools to form intrusive igneous rocks that are called plutons. Intrusive rocks formed in the past may be exposed at the surface as a result of the continuous process of erosion of the materials that once covered them. The types of igneous rock formed by the cooling of magma depend on factors such as the composition of the magma and the rate of cooling associated with it. After conducting several laboratory tests, Bowen (1922) was able to explain the relation of the rate of magma cooling to the formation of different types of rock. This explanation—known as Bowen’s reaction principle—describes the sequence by which new minerals are formed as magma cools. The mineral crystals grow larger and some of them settle. The crystals that remain suspended in the liquid react with the remaining melt to form a new mineral 15 16 Chapter 2: Origin of Soil and Grain Size Co ion act mp entation, Crystalliza tion , Ce m Sediments por a ns Tr Sedimentary rock tatio Metamorphism on, Weather in g n, Erosi Metamorphic rock Igneous rock M elti ng Magma Figure 2.1 Rock cycle Lower resistance to weathering Crystallization at higher temperature Calcium feldspar Olivine Augite e Hornblende Sodium feldspar las D ioc ro isco g ma nt pla ies gn inu us r ser es ou Biotite (black mica) o u ian s n pa nti elds se o f rie C Orthoclase s (potassium feldspar) fer Muscovite (white mica) Higher resistance to weathering Figure 2.2 Bowen’s reaction series Quartz Crystallization at lower temperature 2.1 Rock Cycle and the Origin of Soil 17 Table 2.1 Composition of Minerals Shown in Bowen’s Reaction Series Mineral Composition Biotite (black mica) calcium feldspar Plagioclase e sodium feldspar Orthoclase (potassium feldspar) Muscovite (white mica) Quartz (Mg, Fe)2SiO4 Ca, Na(Mg, Fe, Al)(Al, Si2O6) Complex ferromagnesian silicate of Ca, Na, Mg, Ti, and Al K(Mg, Fe)3AlSi3O10(OH)2 Ca1Al2Si2O82 Na1AlSi3O82 K(AlSi3O8) KAl3Si3O10(OH)2 SiO2 Olivine Augite Hornblende at a lower temperature. This process continues until the entire body of melt is solidiﬁed. Bowen classiﬁed these reactions into two groups: (1) discontinuous ferromagnesian reaction series, in which the minerals formed are different in their chemical composition and crystalline structure, and (2) continuous plagioclase feldspar reaction series, in which the minerals formed have different chemical compositions with similar crystalline structures. Figure 2.2 shows Bowen’s reaction series. The chemical compositions of the minerals are given in Table 2.1. Figure 2.3 is a scanning electron micrograph of a fractured surface of quartz showing glass-like fractures with no discrete planar cleavage. Figure 2.4 is a scanning electron micrograph that shows basal cleavage of individual mica grains. Thus, depending on the proportions of minerals available, different types of igneous rock are formed. Granite, gabbro, and basalt are some of the common types of igneous rock generally encountered in the ﬁeld. Table 2.2 shows the general composition of some igneous rocks. Table 2.2 Composition of Some Igneous Rocks Name of rock Mode of occurrence Texture Abundant minerals Intrusive Coarse Rhyolite Extrusive Fine Quartz, sodium feldspar, potassium feldspar Biotite, muscovite, hornblende Gabbro Intrusive Coarse Basalt Extrusive Fine Plagioclase, pyroxines, olivine Hornblende, biotite, magnetite Diorite Intrusive Coarse Andesite Extrusive Fine Plagioclase, hornblende Biotite, pyroxenes (quartz usually absent) Syenite Intrusive Coarse Potassium feldspar Trachyte Extrusive Fine Sodium feldspar, biotite, hornblende Peridotite Intrusive Coarse Olivine, pyroxenes Oxides of iron Granite Less abundant minerals Figure 2.3 Scanning electron micrograph of fractured surface of quartz showing glass-like fractures with no discrete planar surface (Courtesy of David J. White, Iowa State University, Ames, Iowa) Figure 2.4 Scanning electron micrograph showing basal cleavage of individual mica grains (Courtesy of David J. White, Iowa State University, Ames, Iowa) 18 2.1 Rock Cycle and the Origin of Soil 19 Weathering Weathering is the process of breaking down rocks by mechanical and chemical processes into smaller pieces. Mechanical weathering may be caused by the expansion and contraction of rocks from the continuous gain and loss of heat, which results in ultimate disintegration. Frequently, water seeps into the pores and existing cracks in rocks. As the temperature drops, the water freezes and expands. The pressure exerted by ice because of volume expansion is strong enough to break down even large rocks. Other physical agents that help disintegrate rocks are glacier ice, wind, the running water of streams and rivers, and ocean waves. It is important to realize that in mechanical weathering, large rocks are broken down into smaller pieces without any change in the chemical composition. Figure 2.5 shows several examples of mechanical erosion due to ocean waves and wind at Yehliu in Taiwan. This area is located at a long and narrow sea cape at the northwest side of Keelung, about 15 kilometers between the north coast of Chin Shan and Wanli. In chemical weathering, the original rock minerals are transformed into new minerals by chemical reaction. Water and carbon dioxide from the atmosphere form carbonic acid, which reacts with the existing rock minerals to form new minerals and soluble salts. Soluble salts present in the groundwater and organic acids formed from decayed organic matter also cause chemical weathering. An example of the chemical weathering of orthoclase to form clay minerals, silica, and soluble potassium carbonate follows: H2O CO2 S H2CO3 S H 1HCO3 2 Carbonic acid 2K1AlSi3O8 2 2H H2O S 2K 4SiO2 Al2Si2O5 1OH2 4 Orthoclase Silica Kaolinite (clay mineral) Most of the potassium ions released are carried away in solution as potassium carbonate is taken up by plants. The chemical weathering of plagioclase feldspars is similar to that of orthoclase in that it produces clay minerals, silica, and different soluble salts. Ferromagnesian minerals also form the decomposition products of clay minerals, silica, and soluble salts. Additionally, the iron and magnesium in ferromagnesian minerals result in other products such as hematite and limonite. Quartz is highly resistant to weathering and only slightly soluble in water. Figure 2.2 shows the susceptibility of rock-forming minerals to weathering. The minerals formed at higher temperatures in Bowen’s reaction series are less resistant to weathering than those formed at lower temperatures. The weathering process is not limited to igneous rocks. As shown in the rock cycle (Figure 2.1), sedimentary and metamorphic rocks also weather in a similar manner. Thus, from the preceding brief discussion, we can see how the weathering process changes solid rock masses into smaller fragments of various sizes that can range from large boulders to very small clay particles. Uncemented aggregates of these small grains in various proportions form different types of soil. The clay minerals, which are a product of chemical weathering of feldspars, ferromagnesians, and micas, give the plastic property to soils. There are three important clay minerals: (1) kaolinite, (2) illite, and (3) montmorillonite. (We discuss these clay minerals later in this chapter.) Figure 2.5 Mechanical erosion due to ocean waves and wind at Yehliu, Taiwan (Courtesy of Braja M. Das, Henderson, Nevada) 20 Figure 2.5 (Continued) 21 22 Chapter 2: Origin of Soil and Grain Size Transportation of Weathering Products The products of weathering may stay in the same place or may be moved to other places by ice, water, wind, and gravity. The soils formed by the weathered products at their place of origin are called residual soils. An important characteristic of residual soil is the gradation of particle size. Finegrained soil is found at the surface, and the grain size increases with depth. At greater depths, angular rock fragments may also be found. The transported soils may be classiﬁed into several groups, depending on their mode of transportation and deposition: 1. 2. 3. 4. 5. 6. Glacial soils—formed by transportation and deposition of glaciers Alluvial soils—transported by running water and deposited along streams Lacustrine soils—formed by deposition in quiet lakes Marine soils—formed by deposition in the seas Aeolian soils—transported and deposited by wind Colluvial soils—formed by movement of soil from its original place by gravity, such as during landslides Sedimentary Rock The deposits of gravel, sand, silt, and clay formed by weathering may become compacted by overburden pressure and cemented by agents like iron oxide, calcite, dolomite, and quartz. Cementing agents are generally carried in solution by ground-water. They ﬁll the spaces between particles and form sedimentary rock. Rocks formed in this way are called detrital sedimentary rocks. All detrital rocks have a clastic texture. The following are some examples of detrital rocks with clastic texture. Particle size Sedimentary rock Granular or larger (grain size 2 mm–4 mm or larger) Sand Silt and clay Conglomerate Sandstone Mudstone and shale In the case of conglomerates, if the particles are more angular, the rock is called breccia. In sandstone, the particle sizes may vary between 161 mm and 2 mm. When the grains in sandstone are practically all quartz, the rock is referred to as orthoquartzite. In mudstone and shale, the size of the particles are generally less than 161 mm. Mudstone has a blocky aspect; whereas, in the case of shale, the rock is split into platy slabs. Sedimentary rock also can be formed by chemical processes. Rocks of this type are classiﬁed as chemical sedimentary rock. These rocks can have clastic or nonclastic texture. The following are some examples of chemical sedimentary rock. Composition Calcite (CaCO3) Halite (NaCl) Dolomite [CaMg(CO3)] Gypsum (CaSO4 ⴢ 2H2O) Rock Limestone Rock salt Dolomite Gypsum 2.1 Rock Cycle and the Origin of Soil 23 Limestone is formed mostly of calcium carbonate deposited either by organisms or by an inorganic process. Most limestones have a clastic texture; however, nonclastic textures also are found commonly. Figure 2.6 shows the scanning electron micrograph of a fractured surface of Limestone. Individual grains of calcite show rhombohedral cleavage. Chalk is a sedimentary rock made in part from biochemically derived calcite, which are skeletal fragments of microscopic plants and animals. Dolomite is formed either by chemical deposition of mixed carbonates or by the reaction of magnesium in water with limestone. Gypsum and anhydrite result from the precipitation of soluble CaSO4 due to evaporation of ocean water. They belong to a class of rocks generally referred to as evaporites. Rock salt (NaCl) is another example of an evaporite that originates from the salt deposits of seawater. Sedimentary rock may undergo weathering to form sediments or may be subjected to the process of metamorphism to become metamorphic rock. Metamorphic Rock Metamorphism is the process of changing the composition and texture of rocks (without melting) by heat and pressure. During metamorphism, new minerals are formed, and mineral grains are sheared to give a foliated-texture to metamorphic rock. Gneiss is a metamorphic rock derived from high-grade regional metamorphism of igneous rocks, such as granite, gabbro, and diorite. Low-grade metamorphism of shales and mudstones results in slate. The clay minerals in the shale become chlorite and mica by heat; hence, slate is composed primarily of mica ﬂakes and chlorite. Phyllite is a metamorphic rock, which is derived from slate with Figure 2.6 Scanning electron micrograph of the fractured surface of limestone (Courtesy of David J. White, Iowa State University, Ames, Iowa) 24 Chapter 2: Origin of Soil and Grain Size further metamorphism being subjected to heat greater than 250 to 300C. Schist is a type of metamorphic rock derived from several igneous, sedimentary, and low-grade metamorphic rocks with a well-foliated texture and visible ﬂakes of platy and micaceous minerals. Metamorphic rock generally contains large quantities of quartz and feldspar as well. Marble is formed from calcite and dolomite by recrystallization. The mineral grains in marble are larger than those present in the original rock. Green marbles are colored by hornblends, serpentine, or talc. Black marbles contain bituminous material, and brown marbles contain iron oxide and limonite. Quartzite is a metamorphic rock formed from quartz-rich sandstones. Silica enters into the void spaces between the quartz and sand grains and acts as a cementing agent. Quartzite is one of the hardest rocks. Under extreme heat and pressure, metamorphic rocks may melt to form magma, and the cycle is repeated. 2.2 Soil-Particle Size As discussed in the preceding section, the sizes of particles that make up soil vary over a wide range. Soils generally are called gravel, sand, silt, or clay, depending on the predominant size of particles within the soil. To describe soils by their particle size, several organizations have developed particle-size classiﬁcations. Table 2.3 shows the particle-size classiﬁcations developed by the Massachusetts Institute of Technology, the U.S. Department of Agriculture, the American Association of State Highway and Transportation Ofﬁcials, and the U.S. Army Corps of Engineers and U.S. Bureau of Reclamation. In this table, the MIT system is presented for illustration purposes only. This system is important in the history of the development of the size limits of particles present in soils; however, the Uniﬁed Soil Classiﬁcation System is now almost universally accepted and has been adopted by the American Society for Testing and Materials (ASTM). Figure 2.7 shows the size limits in a graphic form. Gravels are pieces of rocks with occasional particles of quartz, feldspar, and other minerals. Sand particles are made of mostly quartz and feldspar. Other mineral grains also Table 2.3 Particle-Size Classiﬁcations Grain size (mm) Name of organization Massachusetts Institute of Technology (MIT) U.S. Department of Agriculture (USDA) American Association of State Highway and Transportation Ofﬁcials (AASHTO) Uniﬁed Soil Classiﬁcation System (U.S. Army Corps of Engineers, U.S. Bureau of Reclamation, and American Society for Testing and Materials) Gravel Sand Silt Clay 2 2 to 0.06 0.06 to 0.002 0.002 2 2 to 0.05 0.05 to 0.002 0.002 76.2 to 2 2 to 0.075 0.075 to 0.002 0.002 76.2 to 4.75 4.75 to 0.075 Fines (i.e., silts and clays) 0.075 Note: Sieve openings of 4.75 mm are found on a U.S. No. 4 sieve; 2-mm openings on a U.S. No. 10 sieve; 0.075-mm openings on a U.S. No. 200 sieve. See Table 2.5. 2.2 Soil-Particle Size 25 Massachusetts Institute of Technology U.S. Department of Agriculture American Association of State Highway and Transportation Ofﬁcials Uniﬁed Soil Classiﬁcation System 100 10 Gravel 1.0 0.1 Grain size (mm) Sand Silt 0.01 Silt and clay 0.001 Clay Figure 2.7 Soil-separate-size limits by various systems may be present at times. Figure 2.8 shows the scanning electron micrograph of some sand grains. Note that the larger grains show rounding that can occur as a result of wear during intermittent transportation by wind and/or water. Figure 2.9 is a higher magnification of the grains highlighted in Figure 2.8, and it reveals a few small clay particles adhering to larger sand grains. Silts are the microscopic soil fractions that consist of very ﬁne quartz Figure 2.8 Scanning electron micrograph of some sand grains (Courtesy of David J. White, Iowa State University, Ames, Iowa) 26 Chapter 2: Origin of Soil and Grain Size Figure 2.9 Higher magnification of the sand grains highlighted in Figure 2.8 (Courtesy of David J. White, Iowa State University, Ames, Iowa) grains and some ﬂake-shaped particles that are fragments of micaceous minerals. Clays are mostly ﬂake-shaped microscopic and submicroscopic particles of mica, clay minerals, and other minerals. As shown in Table 2.3 and Figure 2.7, clays generally are deﬁned as particles smaller than 0.002 mm. However, in some cases, particles between 0.002 and 0.005 mm in size also are referred to as clay. Particles classiﬁed as clay on the basis of their size may not necessarily contain clay minerals. Clays have been deﬁned as those particles “which develop plasticity when mixed with a limited amount of water” (Grim, 1953). (Plasticity is the putty-like property of clays that contain a certain amount of water.) Nonclay soils can contain particles of quartz, feldspar, or mica that are small enough to be within the clay classiﬁcation. Hence, it is appropriate for soil particles smaller than 2 microns (2 mm), or 5 microns (5 mm) as deﬁned under different systems, to be called clay-sized particles rather than clay. Clay particles are mostly in the colloidal size range (1 mm), and 2 mm appears to be the upper limit. 2.3 Clay Minerals Clay minerals are complex aluminum silicates composed of two basic units: (1) silica tetrahedron and (2) alumina octahedron. Each tetrahedron unit consists of four oxygen atoms surrounding a silicon atom (Figure 2.10a). The combination of tetrahedral silica units gives 2.3 Clay Minerals and and Oxygen (a) and 27 Silicon (b) Aluminum Hydroxyl (c) (d) (e) Oxygen Hydroxyl Aluminum Silicon Figure 2.10 (a) Silica tetrahedron; (b) silica sheet; (c) alumina octahedron; (d) octahedral (gibbsite) sheet; (e) elemental silica-gibbsite sheet (After Grim, 1959. With permission from ASCE.) a silica sheet (Figure 2.10b). Three oxygen atoms at the base of each tetrahedron are shared by neighboring tetrahedra. The octahedral units consist of six hydroxyls surrounding an aluminum atom (Figure 2.10c), and the combination of the octahedral aluminum hydroxyl units gives an octahedral sheet. (This also is called a gibbsite sheet—Figure 2.10d.) 28 Chapter 2: Origin of Soil and Grain Size Sometimes magnesium replaces the aluminum atoms in the octahedral units; in this case, the octahedral sheet is called a brucite sheet. In a silica sheet, each silicon atom with a positive charge of four is linked to four oxygen atoms with a total negative charge of eight. But each oxygen atom at the base of the tetrahedron is linked to two silicon atoms. This means that the top oxygen atom of each tetrahedral unit has a negative charge of one to be counterbalanced. When the silica sheet is stacked over the octahedral sheet, as shown in Figure 2.10e, these oxygen atoms replace the hydroxyls to balance their charges. Of the three important clay minerals, kaolinite consists of repeating layers of elemental silica-gibbsite sheets in a 1:1 lattice, as shown in Figures 2.11 and 2.12a. Each layer is about 7.2 Å thick. The layers are held together by hydrogen bonding. Kaolinite occurs as platelets, each with a lateral dimension of 1000 to 20,000 Å and a thickness of 100 to 1000 Å. The surface area of the kaolinite particles per unit mass is about 15 m2/g. The surface area per unit mass is deﬁned as speciﬁc surface. Figure 2.13 shows a scanning electron micrograph of a kaolinite specimen. Illite consists of a gibbsite sheet bonded to two silica sheets—one at the top and another at the bottom (Figures 2.14 and 2.12b). It is sometimes called clay mica. The illite layers are bonded by potassium ions. The negative charge to balance the potassium ions comes from the substitution of aluminum for some silicon in the tetrahedral sheets. Substitution of one element for another with no change in the crystalline form is known as Oxygen Hydroxyl Aluminum Silicon Figure 2.11 Atomic structure of montmorillonite (After Grim, 1959. With permission from ASCE.) 2.3 Clay Minerals Gibbsite sheet Silica sheet Silica sheet Gibbsite sheet Gibbsite sheet Silica sheet Silica sheet Silica sheet 10 Å 7.2 Å nH2O and exchangeable cations Potassium Silica sheet Basal spacing variable—from 9.6 Å to complete separation Silica sheet Gibbsite sheet Gibbsite sheet Silica sheet Silica sheet Silica sheet (a) (b) (c) Gibbsite sheet Silica sheet 29 Gibbsite sheet Potassium Figure 2.12 Diagram of the structures of (a) kaolinite; (b) illite; (c) montmorillonite isomorphous substitution. Illite particles generally have lateral dimensions ranging from 1000 to 5000 Å and thicknesses from 50 to 500 Å. The speciﬁc surface of the particles is about 80 m2/g. Montmorillonite has a structure similar to that of illite—that is, one gibbsite sheet sandwiched between two silica sheets. (See Figures 2.15 and 2.12c.) In montmorillonite there is isomorphous substitution of magnesium and iron for aluminum in the octahedral Figure 2.13 Scanning electron micrograph of a kaolinite specimen (Courtesy of David J. White, Iowa State University, Ames, Iowa) 30 Chapter 2: Origin of Soil and Grain Size Figure 2.14 Atomic structure of illite Oxygen Hydroxyl Aluminum Silicon Potassium sheets. Potassium ions are not present as in illite, and a large amount of water is attracted into the space between the layers. Particles of montmorillonite have lateral dimensions of 1000 to 5000 Å and thicknesses of 10 to 50 Å. The speciﬁc surface is about 800 m2/g. Figure 2.16 is a scanning electron micrograph showing the fabric of montmorillonite. Besides kaolinite, illite, and montmorillonite, other common clay minerals generally found are chlorite, halloysite, vermiculite, and attapulgite. The clay particles carry a net negative charge on their surfaces. This is the result both of isomorphous substitution and of a break in continuity of the structure at its edges. Larger negative charges are derived from larger speciﬁc surfaces. Some positively charged sites also occur at the edges of the particles. A list of the reciprocal of the average surface densities of the negative charges on the surfaces of some clay minerals follows (Yong and Warkentin, 1966): Clay mineral Reciprocal of average surface density of charge (Å2/electronic charge) Kaolinite Clay mica and chlorite Montmorillonite Vermiculite 25 50 100 75 2.3 Clay Minerals 31 Exchangeable cations nH2O Oxygen Hydroxyl Aluminum, iron, magnesium Silicon, occasionally aluminum Figure 2.15 Atomic structure of montmorillonite (After Grim, 1959. With permission from ASCE.) In dry clay, the negative charge is balanced by exchangeable cations like Ca2, Mg , Na, and K surrounding the particles being held by electrostatic attraction. When water is added to clay, these cations and a few anions ﬂoat around the clay particles. This conﬁguration is referred to as a diffuse double layer (Figure 2.17a). The cation concentration decreases with the distance from the surface of the particle (Figure 2.17b). Water molecules are polar. Hydrogen atoms are not axisymmetric around an oxygen atom; instead, they occur at a bonded angle of 105 (Figure 2.18). As a result, a water molecule has a positive charge at one side and a negative charge at the other side. It is known as a dipole. Dipolar water is attracted both by the negatively charged surface of the clay particles and by the cations in the double layer. The cations, in turn, are attracted to the soil particles. A third mechanism by which water is attracted to clay particles is hydrogen bonding, where hydrogen atoms in the water molecules are shared with oxygen atoms on the 2 32 Chapter 2: Origin of Soil and Grain Size Figure 2.16 Scanning electron micrograph showing the fabric of montmorillonite (Courtesy of David J. White, Iowa State University, Ames, Iowa) surface of the clay. Some partially hydrated cations in the pore water are also attracted to the surface of clay particles. These cations attract dipolar water molecules. All these possible mechanics of attraction of water to clay are shown in Figure 2.19. The force of attraction between water and clay decreases with distance from the surface of the particles. All Surface of clay particle (a) Figure 2.17 Diffuse double layer Concentration of ions Cations Anions Distance from the clay particle (b) 2.3 Clay Minerals 33 Oxygen Hydrogen Hydrogen 105° Figure 2.18 Dipolar character of water the water held to clay particles by force of attraction is known as double-layer water. The innermost layer of double-layer water, which is held very strongly by clay, is known as adsorbed water. This water is more viscous than free water is. Figure 2.20 shows the absorbed and double-layer water for typical montmorillonite and kaolinite particles. This orientation of water around the clay particles gives clay soils their plastic properties. It needs to be well recognized that the presence of clay minerals in a soil aggregate has a great inﬂuence on the engineering properties of the soil as a whole. When moisture is present, the engineering behavior of a soil will change greatly as the percentage of clay mineral content increases. For all practical purposes, when the clay content is about 50% or more, the sand and silt particles ﬂoat in a clay matrix, and the clay minerals primarily dictate the engineering properties of the soil. Cation Dipolar water molecule Dipolar water molecule Clay particle Figure 2.19 Attraction of dipolar molecules in diffuse double layer 34 Chapter 2: Origin of Soil and Grain Size 200 Å 10 Å 200 Å Typical montmorillonite particle, 1000 Å by 10 Å (a) 400 Å 10 Å 1000 Å 10 Å 400 Å Typical kaolinite particle, 10,000 Å by 1000 Å (b) Montmorillonite crystal Kaolinite crystal Adsorbed water Double-layer water Figure 2.20 Clay water (Redrawn after Lambe, 1958. With permission from ASCE.) 2.4 Speciﬁc Gravity (Gs ) Speciﬁc gravity is deﬁned as the ratio of the unit weight of a given material to the unit weight of water. The speciﬁc gravity of soil solids is often needed for various calculations in soil mechanics. It can be determined accurately in the laboratory. Table 2.4 shows the speciﬁc gravity of some common minerals found in soils. Most of the values fall within a range of 2.6 to 2.9. The speciﬁc gravity of solids of light-colored sand, which is mostly made of quartz, may be estimated to be about 2.65; for clayey and silty soils, it may vary from 2.6 to 2.9. 2.5 Mechanical Analysis of Soil 35 Table 2.4 Speciﬁc Gravity of Common Minerals Mineral Quartz Kaolinite Illite Montmorillonite Halloysite Potassium feldspar Sodium and calcium feldspar Chlorite Biotite Muscovite Hornblende Limonite Olivine 2.5 Speciﬁc gravity, Gs 2.65 2.6 2.8 2.65–2.80 2.0–2.55 2.57 2.62–2.76 2.6–2.9 2.8–3.2 2.76–3.1 3.0–3.47 3.6–4.0 3.27–3.7 Mechanical Analysis of Soil Mechanical analysis is the determination of the size range of particles present in a soil, expressed as a percentage of the total dry weight. Two methods generally are used to ﬁnd the particle-size distribution of soil: (1) sieve analysis—for particle sizes larger than 0.075 mm in diameter, and (2) hydrometer analysis—for particle sizes smaller than 0.075 mm in diameter. The basic principles of sieve analysis and hydrometer analysis are described brieﬂy in the following two sections. Sieve Analysis Sieve analysis consists of shaking the soil sample through a set of sieves that have progressively smaller openings. U.S. standard sieve numbers and the sizes of openings are given in Table 2.5. Table 2.5 U.S. Standard Sieve Sizes Sieve no. Opening (mm) Sieve no. Opening (mm) 4 5 6 7 8 10 12 14 16 18 20 25 30 4.75 4.00 3.35 2.80 2.36 2.00 1.70 1.40 1.18 1.00 0.850 0.710 0.600 35 40 50 60 70 80 100 120 140 170 200 270 0.500 0.425 0.355 0.250 0.212 0.180 0.150 0.125 0.106 0.090 0.075 0.053 36 Chapter 2: Origin of Soil and Grain Size The sieves used for soil analysis are generally 203 mm (8 in.) in diameter. To conduct a sieve analysis, one must ﬁrst oven-dry the soil and then break all lumps into small particles. The soil then is shaken through a stack of sieves with openings of decreasing size from top to bottom (a pan is placed below the stack). Figure 2.21 shows a set of sieves in a shaker used for conducting the test in the laboratory. The smallest-sized sieve that should be used for this type of test is the U.S. No. 200 sieve. After the soil is shaken, the mass of soil retained on each sieve is determined. When cohesive soils are analyzed, breaking the lumps into individual particles may be difﬁcult. In this case, the soil may be mixed with water to make a slurry and then washed through the sieves. Portions retained on each sieve are collected separately and oven-dried before the mass retained on each sieve is measured. 1. Determine the mass of soil retained on each sieve (i.e., M1, M2, · · · Mn) and in the pan (i.e., Mp) 2. Determine the total mass of the soil: M1 M2 · · · Mi · · · Mn Mp M 3. Determine the cumulative mass of soil retained above each sieve. For the ith sieve, it is M1 M2 · · · Mi Figure 2.21 A set of sieves for a test in the laboratory (Courtesy of Braja M. Das, Henderson, Nevada) 2.5 Mechanical Analysis of Soil 37 4. The mass of soil passing the ith sieve is M (M1 M2 · · · Mi) 5. The percent of soil passing the ith sieve (or percent ﬁner) is F M 1M1 M2 p Mi 2 M 100 Once the percent ﬁner for each sieve is calculated (step 5), the calculations are plotted on semilogarithmic graph paper (Figure 2.22) with percent ﬁner as the ordinate (arithmetic scale) and sieve opening size as the abscissa (logarithmic scale). This plot is referred to as the particle-size distribution curve. Hydrometer Analysis Hydrometer analysis is based on the principle of sedimentation of soil grains in water. When a soil specimen is dispersed in water, the particles settle at different velocities, depending on their shape, size, weight, and the viscosity of the water. For simplicity, it is assumed that all the soil particles are spheres and that the velocity of soil particles can be expressed by Stokes’ law, according to which v rs rw 2 D 18h (2.1) where v velocity rs density of soil particles rw density of water h viscosity of water D diameter of soil particles 100 Percent passing 80 60 40 20 0 10.0 5.0 1.0 0.5 Particle size (mm) — log scale Figure 2.22 Particle-size distribution curve 0.1 0.05 38 Chapter 2: Origin of Soil and Grain Size Thus, from Eq. (2.1), D where v 18h 18hv L B rs rw B rs rw B t (2.2) L Distance . Time t Note that rs Gs rw (2.3) Thus, combining Eqs. (2.2) and (2.3) gives D 18h L B 1Gs 1 2rw B t (2.4) If the units of h are (g sec)/cm2, rw is in g/cm3, L is in cm, t is in min, and D is in mm, then D1mm2 10 18h 3 1g # sec2/cm2 4 L 1cm2 B 1Gs 1 2rw 1g/cm3 2 B t 1min2 60 or D 30h L B 1Gs 1 2rw B t Assume rw to be approximately equal to 1 g/cm3, so that D 1mm2 K L 1cm2 B t 1min2 (2.5) where K 30h B 1Gs 1 2 (2.6) Note that the value of K is a function of Gs and h, which are dependent on the temperature of the test. Table 2.6 gives the variation of K with the test temperature and the speciﬁc gravity of soil solids. 2.5 Mechanical Analysis of Soil 39 Text not available due to copyright restrictions In the laboratory, the hydrometer test is conducted in a sedimentation cylinder usually with 50 g of oven-dried sample. Sometimes 100-g samples also can be used. The sedimentation cylinder is 457 mm (18 in.) high and 63.5 mm (2.5 in.) in diameter. It is marked for a volume of 1000 ml. Sodium hexametaphosphate generally is used as the dispersing agent. The volume of the dispersed soil suspension is increased to 1000 ml by adding distilled water. Figure 2.23 shows an ASTM 152H type of hydrometer. When a hydrometer is placed in the soil suspension at a time t, measured from the start of sedimentation it measures the speciﬁc gravity in the vicinity of its bulb at a depth L (Figure 2.24). The speciﬁc gravity is a function of the amount of soil particles present per unit volume of suspension at that depth. Also, at a time t, the soil particles in suspension at a depth L will have a diameter smaller than D as calculated in Eq. (2.5). The larger particles would have settled beyond the zone of measurement. Hydrometers are designed to give the amount of soil, in grams, that is still in suspension. They are calibrated for soils that have a speciﬁc gravity, Gs , of 2.65; for soils of other speciﬁc gravity, a correction must be made. By knowing the amount of soil in suspension, L, and t, we can calculate the percentage of soil by weight ﬁner than a given diameter. Note that L is the depth measured from the surface of the water to the center of gravity of the hydrometer bulb at which the density of the suspension is measured. The value of L will change with time t. Hydrometer analysis is effective for separating soil fractions down to a size of about 40 Chapter 2: Origin of Soil and Grain Size 0 L1 L 60 L2 Figure 2.23 ASTM 152H hydrometer (Courtesy of ELE International) Figure 2.24 Definition of L in hydrometer test 0.5 mm. The value of L (cm) for the ASTM 152H hydrometer can be given by the expression (see Figure 2.24) L L1 VB 1 a L2 b 2 A (2.7) where L1 distance along the stem of the hydrometer from the top of the bulb to the mark for a hydrometer reading 1cm2 L2 length of the hydrometer bulb 14 cm VB volume of the hydrometer bulb 67 cm3 A cross-sectional area of the sedimentation cylinder 27.8 cm2 The value of L1 is 10.5 cm for a reading of R 0 and 2.3 cm for a reading of R 50. Hence, for any reading R, L1 10.5 110.5 2.32 50 R 10.5 0.164R 1cm2 2.5 Mechanical Analysis of Soil 41 Table 2.7 Variation of L with Hydrometer Reading— ASTM 152H Hydrometer Hydrometer reading, R L (cm) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 16.3 16.1 16.0 15.8 15.6 15.5 15.3 15.2 15.0 14.8 14.7 14.5 14.3 14.2 14.0 13.8 13.7 13.5 13.3 13.2 13.0 12.9 12.7 12.5 12.4 12.2 12.0 11.9 11.7 11.5 11.4 Hydrometer reading, R L (cm) 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 11.2 11.1 10.9 10.7 10.6 10.4 10.2 10.1 9.9 9.7 9.6 9.4 9.2 9.1 8.9 8.8 8.6 8.4 8.3 8.1 7.9 7.8 7.6 7.4 7.3 7.1 7.0 6.8 6.6 6.5 Thus, from Eq. (2.7), L 10.5 0.164R 1 67 a 14 b 16.29 0.164R 2 27.8 (2.8) where R hydrometer reading corrected for the meniscus. On the basis of Eq. (2.8), the variations of L with the hydrometer readings R are given in Table 2.7. In many instances, the results of sieve analysis and hydrometer analysis for ﬁner fractions for a given soil are combined on one graph, such as the one shown in Figure 2.25. When these results are combined, a discontinuity generally occurs in the range where they overlap. This discontinuity occurs because soil particles are generally irregular in shape. 42 Chapter 2: Origin of Soil and Grain Size Uniﬁed classiﬁcation Silt and clay Sand Sieve analysis Sieve no. 10 16 30 40 60 100 Hydrometer analysis 200 100 Percent ﬁner 80 60 40 20 0 5 2 1 0.5 0.2 0.1 0.05 0.02 0.01 0.005 0.002 0.001 Particle diameter (mm) — log scale Sieve analysis Hydrometer analysis Figure 2.25 Particle-size distribution curve—sieve analysis and hydrometer analysis Sieve analysis gives the intermediate dimensions of a particle; hydrometer analysis gives the diameter of an equivalent sphere that would settle at the same rate as the soil particle. 2.6 Particle-Size Distribution Curve A particle-size distribution curve can be used to determine the following four parameters for a given soil (Figure 2.26): 1. Effective size (D10): This parameter is the diameter in the particle-size distribution curve corresponding to 10% ﬁner. The effective size of a granular soil is a good measure to estimate the hydraulic conductivity and drainage through soil. 2. Uniformity coefﬁcient (Cu): This parameter is deﬁned as Cu D60 D10 (2.9) where D60 diameter corresponding to 60% ﬁner. 3. Coefﬁcient of gradation (Cc ): This parameter is deﬁned as Cc D230 D60 D10 (2.10) 2.6 Particle-Size Distribution Curve 43 100 Percent ﬁner 80 75 60 40 30 25 20 10 D75 0 10.0 5.0 1.0 D60 D30 0.5 D25 D10 0.1 0.05 Particle size (mm) — log scale Figure 2.26 Definition of D75, D60, D30, D25, and D10 4. Sorting coefﬁcient (S0): This parameter is another measure of uniformity and is generally encountered in geologic works and expressed as S0 D75 B D25 (2.11) The sorting coefﬁcient is not frequently used as a parameter by geotechnical engineers. The percentages of gravel, sand, silt, and clay-size particles present in a soil can be obtained from the particle-size distribution curve. As an example, we will use the particlesize distribution curve shown in Figure 2.25 to determine the gravel, sand, silt, and clay size particles as follows (according to the Uniﬁed Soil Classiﬁcation System—see Table 2.3): Size (mm) 76.2 4.75 0.075 — Percent ﬁner 100 100 62 0 100 100 0% gravel 100 62 38% sand 62 0 62% silt and clay The particle-size distribution curve shows not only the range of particle sizes present in a soil, but also the type of distribution of various-size particles. Such types of distributions are demonstrated in Figure 2.27. Curve I represents a type of soil in which most of the soil grains are the same size. This is called poorly graded soil. Curve II represents a soil in which the particle sizes are distributed over a wide range, termed well graded. A well-graded soil has a uniformity coefﬁcient greater than about 4 for gravels and 6 for sands, and a coefﬁcient of gradation between 1 and 3 (for gravels and sands). A soil might have a combination of two or more uniformly graded fractions. Curve III represents such a soil. This type of soil is termed gap graded. 100 Percent ﬁner 80 Poorly graded 60 Well graded Gap graded 40 I II III 20 0 2.0 1.0 0.5 0.2 0.1 0.05 0.02 0.01 Particle diameter (mm) — log scale 0.005 Figure 2.27 Different types of particle-size distribution curves Example 2.1 Following are the results of a sieve analysis. Make the necessary calculations and draw a particle-size distribution curve. Mass of soil retained on each sieve (g) U.S. sieve no. 4 10 20 40 60 80 100 200 Pan 0 40 60 89 140 122 210 56 12 Solution The following table can now be prepared. a 44 U.S. sieve (1) Opening (mm) (2) 4 10 20 40 60 80 100 200 Pan 4.75 2.00 0.850 0.425 0.250 0.180 0.150 0.075 – Mass retained on each sieve (g) (3) 0 40 60 89 140 122 210 56 12 g M col. 4 729 col. 4 100 100 gM 729 Cumulative mass retained above each sieve (g) (4) Percent ﬁner a (5) 0 0 + 40 40 40 + 60 100 100 + 89 189 189 + 140 329 329 + 122 451 451 + 210 661 661 + 56 717 717 + 12 729 M 100 94.5 86.3 74.1 54.9 38.1 9.3 1.7 0 45 2.6 Particle-Size Distribution Curve The particle-size distribution curve is shown in Figure 2.28. 100 Percent ﬁner 80 60 D60 0.27 mm 40 30 20 D30 0.17 mm 10 0 10 D10 0.15 mm 5 3 1 0.5 0.3 Particle size (mm) 0.1 0.05 Figure 2.28 Particle-size distribution curve ■ Example 2.2 For the particle-size distribution curve shown in Figure 2.28 determine a. D10, D30, and D60 b. Uniformity coefficient, Cu c. Coefficient of gradation, Cz Solution Part a From Figure 2.28, D10 0.15 mm D30 0.17 mm D60 0.27 mm Part b Cu Cz D60 0.27 1.8 D10 0.15 10.17 2 2 D230 0.71 D60 D10 10.272 10.152 ■ 46 Chapter 2: Origin of Soil and Grain Size Example 2.3 For the particle-size distribution curve shown in Figure 2.28, determine the percentages of gravel, sand, silt, and clay-size particles present. Use the Unified Soil Classification System. Solution From Figure 2.28, we can prepare the following table. Size (mm) 76.2 4.75 0.075 – 2.7 Percent finer 100 100 1.7 0 100 100 0% gravel 100 1.7 98.3% sand 1.7 0 1.7% silt and clay Particle Shape The shape of particles present in a soil mass is equally as important as the particle-size distribution because it has signiﬁcant inﬂuence on the physical properties of a given soil. However, not much attention is paid to particle shape because it is more difﬁcult to measure. The particle shape generally can be divided into three major categories: 1. Bulky 2. Flaky 3. Needle shaped Bulky particles are formed mostly by mechanical weathering of rock and minerals. Geologists use such terms as angular, subangular, subrounded, and rounded to describe the shapes of bulky particles. These shapes are shown qualitatively in Figure 2.29. Small sand particles located close to their origin are generally very angular. Sand particles carried by Angular Subangular Subrounded Rounded Figure 2.29 Shape of bulky particles Problems 47 wind and water for a long distance can be subangular to rounded in shape. The shape of granular particles in a soil mass has a great inﬂuence on the physical properties of the soil, such as maximum and minimum void ratios, shear strength parameters, compressibility, etc. The angularity, A, is deﬁned as A Average radius of corners and edges Radius of the maximum inscribed sphere (2.12) The sphericity of bulky particles is deﬁned as S De Lp where De equivalent diameter of the partilce V volume of particle Lp length of particle (2.13) 3 6V B p Flaky particles have very low sphericity—usually 0.01 or less. These particles are predominantly clay minerals. Needle-shaped particles are much less common than the other two particle types. Examples of soils containing needle-shaped particles are some coral deposits and attapulgite clays. 2.8 Summary In this chapter, we discussed the rock cycle, the origin of soil by weathering, the particlesize distribution in a soil mass, the shape of particles, and clay minerals. Some important points include the following: 1. Rocks can be classiﬁed into three basic categories: (a) igneous, (b) sedimentary, and (c) metamorphic. 2. Soils are formed by chemical and mechanical weathering of rocks. 3. Based on the size of the soil particles, soil can be classiﬁed as gravel, sand, silt, or clay. 4. Clays are mostly ﬂake-shaped microscopic and submicroscopic particles of mica, clay minerals, and other minerals. 5. Clay minerals are complex aluminum silicates that develop plasticity when mixed with a limited amount of water. 6. Mechanical analysis is a process for determining the size range of particles present in a soil mass. Sieve analysis and hydrometer analysis are two tests used in the mechanical analysis of soil. Problems 2.1 2.2 For a soil, suppose that D10 0.08 mm, D30 0.22 mm, and D60 0.41 mm. Calculate the uniformity coefficient and the coefficient of gradation. Repeat Problem 2.1 with the following: D10 0.24 mm, D30 0.82 mm, and D60 1.81 mm. 48 Chapter 2: Origin of Soil and Grain Size 2.3 2.4 Repeat Problem 2.1 with the following: D10 0.18 mm, D30 0.32 mm, and D60 0.78 mm. The following are the results of a sieve analysis: U.S. sieve no. Mass of soil retained (g) 4 10 20 40 60 100 200 Pan 0 18.5 53.2 90.5 81.8 92.2 58.5 26.5 a. Determine the percent ﬁner than each sieve and plot a grain-size distribution curve. b. Determine D10 , D30 , and D60 from the grain-size distribution curve. c. Calculate the uniformity coefﬁcient, Cu . d. Calculate the coefﬁcient of gradation, Cc . 2.5 2.6 Repeat Problem 2.4 with the following: U.S. sieve no. Mass of soil retained (g) 4 10 20 40 60 80 100 200 Pan 0 44 56 82 51 106 92 85 35 Repeat Problem 2.4 with the following: U.S. sieve no. Mass of soil retained (g) 4 10 20 40 60 100 200 Pan 0 41.2 55.1 80.0 91.6 60.5 35.6 21.5 Problems 2.7 2.8 49 Repeat Problem 2.4 with the following results of a sieve analysis: U.S. sieve no. Mass of soil retained on each sieve (g) 4 6 10 20 40 60 100 200 Pan 0 0 0 9.1 249.4 179.8 22.7 15.5 23.5 The following are the results of a sieve and hydrometer analysis. Analysis Sieve number/ grain size Percent ﬁner than 40 80 170 200 0.04 mm 0.015 mm 0.008 mm 0.004 mm 0.002 mm 100 97 92 90 74 42 27 17 11 Sieve Hydrometer a. Draw the grain-size distribution curve. b. Determine the percentages of gravel, sand, silt, and clay according to the MIT system. c. Repeat part b according to the USDA system. d. Repeat part b according to the AASHTO system. 2.9 The particle-size characteristics of a soil are given in this table. Draw the particlesize distribution curve. Size (mm) 0.425 0.033 0.018 0.01 0.0062 0.0035 0.0018 0.001 Percent finer 100 90 80 70 60 50 40 35 50 Chapter 2: Origin of Soil and Grain Size Determine the percentages of gravel, sand, silt, and clay: a. According to the USDA system. b. According to the AASHTO system. 2.10 Repeat Problem 2.9 with the following data: Size (mm) 0.425 0.1 0.052 0.02 0.01 0.004 0.001 Percent finer 100 92 84 62 46 32 22 2.11 In a hydrometer test, the results are as follows: Gs 2.60, temperature of water 24 °C, and R 43 at 60 min after the start of sedimentation (see Figure 2.24). What is the diameter, D, of the smallest-size particles that have settled beyond the zone of measurement at that time (that is, t 60 min)? 2.12 Repeat Problem 2.11 with the following values: Gs 2.70, temperature 23 °C, t 120 min, and R 25. References AMERICAN SOCIETY FOR TESTING AND MATERIALS (2007). ASTM Book of Standards, Sec. 4, Vol. 04.08, West Conshohocken, Pa. BOWEN, N. L. (1922). “The Reaction Principles in Petrogenesis,” Journal of Geology, Vol. 30, 177–198. GRIM, R. E. (1953). Clay Mineralogy, McGraw-Hill, New York. GRIM, R. E. (1959). “Physico-Chemical Properties of Soils: Clay Minerals,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 85, No. SM2, 1–17. LAMBE, T. W. (1958). “The Structure of Compacted Clay,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 84, No. SM2, 1655–1 to 1655–35. YONG, R. N., and WARKENTIN, B. P. (1966). Introduction of Soil Behavior, Macmillan, New York. 3 Weight–Volume Relationships Chapter 2 presented the geologic processes by which soils are formed, the description of limits on the sizes of soil particles, and the mechanical analysis of soils. In natural occurrence, soils are three-phase systems consisting of soil solids, water, and air. This chapter discusses the weight–volume relationships of soil aggregates. 3.1 Weight–Volume Relationships Figure 3.1a shows an element of soil of volume V and weight W as it would exist in a natural state. To develop the weight–volume relationships, we must separate the three phases (that is, solid, water, and air) as shown in Figure 3.1b. Thus, the total volume of a given soil sample can be expressed as V Vs Vv Vs Vw Va (3.1) where Vs volume of soil solids Vv volume of voids Vw volume of water in the voids Va volume of air in the voids Assuming that the weight of the air is negligible, we can give the total weight of the sample as W Ws Ww (3.2) where Ws weight of soil solids Ww weight of water The volume relationships commonly used for the three phases in a soil element are void ratio, porosity, and degree of saturation. Void ratio (e) is deﬁned as the ratio of the volume of voids to the volume of solids. Thus, e Vv Vs (3.3) 51 52 Chapter 3: Weight–Volume Relationships Va V Total weight W V W Total volume V V W Ws Vs (a) (b) Air Water Solid Figure 3.1 (a) Soil element in natural state; (b) three phases of the soil element Porosity (n) is deﬁned as the ratio of the volume of voids to the total volume, or n Vv V (3.4) The degree of saturation (S) is deﬁned as the ratio of the volume of water to the volume of voids, or S Vw Vv (3.5) It is commonly expressed as a percentage. The relationship between void ratio and porosity can be derived from Eqs. (3.1), (3.3), and (3.4) as follows: Vv Vv e Vs V Vv a Vv b V n Vv 1n b 1 a V (3.6) Also, from Eq. (3.6), n e 1e (3.7) The common terms used for weight relationships are moisture content and unit weight. Moisture content (w) is also referred to as water content and is deﬁned as the ratio of the weight of water to the weight of solids in a given volume of soil: 3.1 Weight–Volume Relationships w Ww Ws 53 (3.8) Unit weight (g) is the weight of soil per unit volume. Thus, g W V (3.9) The unit weight can also be expressed in terms of the weight of soil solids, the moisture content, and the total volume. From Eqs. (3.2), (3.8), and (3.9), g Ws Ww W V V Ws c 1 a V Ww bd Ws Ws 11 w2 V (3.10) Soils engineers sometimes refer to the unit weight deﬁned by Eq. (3.9) as the moist unit weight. Often, to solve earthwork problems, one must know the weight per unit volume of soil, excluding water. This weight is referred to as the dry unit weight, gd. Thus, gd Ws V (3.11) From Eqs. (3.10) and (3.11), the relationship of unit weight, dry unit weight, and moisture content can be given as g (3.12) gd 1w Unit weight is expressed in English units (a gravitational system of measurement) as pounds per cubic foot (lb/ft3). In SI (Système International), the unit used is kilo Newtons per cubic meter (kN/m3). Because the Newton is a derived unit, working with mass densities (r) of soil may sometimes be convenient. The SI unit of mass density is kilograms per cubic meter (kg/m3). We can write the density equations [similar to Eqs. (3.9) and (3.11)] as r M V (3.13) rd Ms V (3.14) and where r density of soil 1kg/m3 2 rd dry density of soil 1kg/m3 2 M total mass of the soil sample 1kg 2 Ms mass of soil solids in the sample 1kg2 The unit of total volume, V, is m3. 54 Chapter 3: Weight–Volume Relationships The unit weight in kN/m3 can be obtained from densities in kg/m3 as g 1kN/m3 2 gr1kg/m3 2 1000 and gd 1kN/m3 2 grd 1kg/m3 2 1000 where g acceleration due to gravity 9.81 m/sec2. Note that unit weight of water (gw) is equal to 9.81 kN/m3 or 62.4 lb/ft3 or 1000 kgf/m3. 3.2 Relationships among Unit Weight, Void Ratio, Moisture Content, and Speciﬁc Gravity To obtain a relationship among unit weight (or density), void ratio, and moisture content, let us consider a volume of soil in which the volume of the soil solids is one, as shown in Figure 3.2. If the volume of the soil solids is one, then the volume of voids is numerically equal to the void ratio, e [from Eq. (3.3)]. The weights of soil solids and water can be given as Ws Gsgw Ww wWs wGsgw where Gs speciﬁc gravity of soil solids w moisture content gw unit weight of water Weight Volume V e W Gsg V Gs V 1 e W Ws Gsg Vs 1 Air Water Solid Figure 3.2 Three separate phases of a soil element with volume of soil solids equal to one 3.2 Relationships among Unit Weight, Void Ratio, Moisture Content, and Speciﬁc Gravity 55 Now, using the deﬁnitions of unit weight and dry unit weight [Eqs. (3.9) and (3.11)], we can write g 11 w2Gsgw Ws Ww Gsgw wGsgw W V V 1e 1e (3.15) and gd Ws Gsgw V 1e (3.16) Gsgw 1 gd (3.17) or e Because the weight of water for the soil element under consideration is wGsgw, the volume occupied by water is Vw Ww wGsgw wGs gw gw Hence, from the deﬁnition of degree of saturation [Eq. (3.5)], S Vw wGs e Vv or Se wGs (3.18) This equation is useful for solving problems involving three-phase relationships. If the soil sample is saturated—that is, the void spaces are completely ﬁlled with water (Figure 3.3)—the relationship for saturated unit weight (gsat) can be derived in a similar manner: gsat 1Gs e 2gw Ws Ww Gsgw egw W V V 1e 1e (3.19) Also, from Eq. (3.18) with S 1, e wGs (3.20) As mentioned before, due to the convenience of working with densities in the SI system, the following equations, similar to unit–weight relationships given in Eqs. (3.15), (3.16), and (3.19), will be useful: 56 Chapter 3: Weight–Volume Relationships Weight Volume W eg V V e V1e W Ws Gsg Vs 1 Water Figure 3.3 Saturated soil element with volume of soil solids equal to one Solid Density r 11 w2Gsrw Dry density rd Saturated density rsat 1e Gsrw 1e 1Gs e2rw 1e (3.21) (3.22) (3.23) where rw density of water 1000 kg/m3. Equation (3.21) may be derived by referring to the soil element shown in Figure 3.4, in which the volume of soil solids is equal to 1 and the volume of voids is equal to e. V e M Gs r Vs 1 Ms Gs r Air Water Solid Figure 3.4 Three separate phases of a soil element showing mass–volume relationship 3.3 Relationships among Unit Weight, Porosity, and Moisture Content 57 Hence, the mass of soil solids, Ms, is equal to Gsrw. The moisture content has been deﬁned in Eq. (3.8) as w 1mass of water2 # g Ww Ws 1mass of solid2 # g Mw Ms where Mw mass of water. Since the mass of soil in the element is equal to Gsrw, the mass of water Mw wMs wGsrw From Eq. (3.13), density r Ms Mw Gsrw wGsrw M V Vs Vv 1e 11 w2Gsrw 1e Equations (3.22) and (3.23) can be derived similarly. 3.3 Relationships among Unit Weight, Porosity, and Moisture Content The relationship among unit weight, porosity, and moisture content can be developed in a manner similar to that presented in the preceding section. Consider a soil that has a total volume equal to one, as shown in Figure 3.5. From Eq. (3.4), n Weight Vv V Volume V n W Gsg (1 n) V1 Ws Gsg (1 n) Vs 1 n Air Water Solid Figure 3.5 Soil element with total volume equal to one 58 Chapter 3: Weight–Volume Relationships If V is equal to 1, then Vv is equal to n, so Vs 1 n. The weight of soil solids (Ws) and the weight of water (Ww) can then be expressed as follows: Ws Gsgw 11 n2 (3.24) Ww wWs wGsgw 11 n2 (3.25) So, the dry unit weight equals gd Gsgw 11 n2 Ws Gsgw 11 n2 V 1 (3.26) Ws Ww Gsgw 11 n2 11 w2 V (3.27) The moist unit weight equals g Figure 3.6 shows a soil sample that is saturated and has V 1. According to this ﬁgure, gsat 11 n2Gsgw ngw Ws Ww 3 11 n2Gs n4 gw V 1 (3.28) The moisture content of a saturated soil sample can be expressed as w Ww ngw n Ws 11 n2 gwGs 11 n2Gs Weight Volume W ng V V n V1 Vs 1 n Ws Gsg (1 n) Water Solid Figure 3.6 Saturated soil element with total volume equal to one (3.29) 3.4 Various Unit-Weight Relationships 3.4 59 Various Unit-Weight Relationships In Sections 3.2 and 3.3, we derived the fundamental relationships for the moist unit weight, dry unit weight, and saturated unit weight of soil. Several other forms of relationships that can be obtained for g, gd, and gsat are given in Table 3.1. Some typical values of void ratio, moisture content in a saturated condition, and dry unit weight for soils in a natural state are given in Table 3.2. Table 3.1 Various Forms of Relationships for g, gd, and gsat Moist unit weight (G) Given w, Gs, e S, Gs, e w, Gs, S Relationship S, Gs, n Given 11 w2Gsgw g, w 1e 1Gs Se2gw Gs, e 1e 11 w2Gsgw 1 w, Gs, n Dry unit weight (Gd ) Gs, n wGs S Gs, w, S Gsgw 11 n2 11 w2 Gsgw 11 n2 nSgw e, w, S gsat, e gsat, n gsat, Gs Saturated unit weight (Gsat ) Relationship Given g 1w Gsgw 1e Relationship 1Gs e 2gw Gs, e Gsgw 11 n2 Gsgw wGs b 1 a S eSgw 11 e 2w egw gsat 1e gsat ngw 1gsat gw 2Gs 1Gs 1 2 1e Gs, n 3 11 n2 Gs n4gw Gs, wsat a 1 wsat b Gsgw 1 wsatGs e, wsat a 1 wsat e b gw ba wsat 1e n, wsat na gd, e gd a gd, n gd ngw gd, S a1 gd, wsat gd 11 wsat 2 1 wsat b gw wsat e bg 1e w 1 b g gw Gs d Table 3.2 Void Ratio, Moisture Content, and Dry Unit Weight for Some Typical Soils in a Natural State Type of soil Loose uniform sand Dense uniform sand Loose angular-grained silty sand Dense angular-grained silty sand Stiff clay Soft clay Loess Soft organic clay Glacial till Void ratio, e Natural moisture content in a saturated state (%) Dry unit weight, Gd lb/ft3 kN/m3 0.8 0.45 30 16 92 115 14.5 18 0.65 25 102 16 0.4 0.6 0.9–1.4 0.9 2.5–3.2 0.3 15 21 30–50 25 90–120 10 121 108 73–93 86 38–51 134 19 17 11.5–14.5 13.5 6–8 21 60 Chapter 3: Weight–Volume Relationships Example 3.1 For a saturated soil, show that gsat a 1w e b gw ba w 1e Solution From Eqs. (3.19) and (3.20), gsat 1Gs e 2gw 1e (a) and e wGs or Gs e w (b) Combining Eqs. (a) and (b) gives a gsat e e b gw w e 1w a b gw ba w 1e 1e Example 3.2 For a moist soil sample, the following are given. • • • • Total volume: V 1.2 m3 Total mass: M 2350 kg Moisture content: w 8.6% Speciﬁc gravity of soil solids: Gs 2.71 Determine the following. a. b. c. d. e. f. Moist density Dry density Void ratio Porosity Degree of saturation Volume of water in the soil sample Solution Part a From Eq. (3.13), r M 2350 1958.3 kg/m3 V 1.2 ■ 3.4 Various Unit-Weight Relationships Part b From Eq. (3.14), rd Ms M V 11 w2V 2350 1803.3 kg/m3 8.6 b 11.22 a1 100 Part c From Eq. (3.22), rd e Gsrw 1e 12.712 110002 Gs rw 1 0.503 1 rd 1803.3 Part d From Eq. (3.7), n 0.503 e 0.335 1e 1 0.503 Part e From Eq. (3.18), wGs S e a 8.6 b 12.71 2 100 0.463 46.3% 0.503 Part f Volume of water: Mw M Ms rw rw M 1w rw M 2350 8.6 ¢ 1 100 0.186 m3 1000 2350 ° Alternate Solution Refer to Figure 3.7. Part a r M 2350 1958.3 kg/m3 V 1.2 61 62 Chapter 3: Weight–Volume Relationships Volume (m3) Mass (kg) V 0.402 V 0.186 M 186.1 V 1.2 M 2350 Ms 2163.9 Vs 0.798 Water Air Solid Figure 3.7 Part b M 1w 2350 2163.9 kg 8.6 1 100 Ms M 2350 rd 1803.3 kg/m3 8.6 V 11 w2V a1 b 11.22 100 Ms Part c The volume of solids: Ms 2163.9 0.798 m3 Gsrw 12.71 2 110002 The volume of voids: Vv V Vs 1.2 0.798 0.402 m3 Void ratio: e Vv 0.402 0.503 Vs 0.798 Part d Porosity: n Vv 0.402 0.335 V 1.2 Part e Vw Vv Mw 186.1 Volume of water: Vw 0.186 m3 rw 1000 S 3.4 Various Unit-Weight Relationships 63 Hence, S 0.186 0.463 46.3% 0.402 Part f From Part e, Vw 0.186 m3 ■ Example 3.3 The following data are given for a soil: • Porosity: n 0.4 • Specific gravity of the soil solids: Gs 2.68 • Moisture content: w 12% Determine the mass of water to be added to 10 m3 of soil for full saturation. Solution Equation (3.27) can be rewritten in terms of density as Gsw (1 n)(1 + w) Similarly, from Eq. (3.28) sat [(1 n)Gs + n]w Thus, (2.68)(1000)(1 0.4)(1 + 0.12) 1800.96 kg/m3 sat [(1 0.4)(2.68) + 0.4] (1000) 2008 kg/m3 Mass of water needed per cubic meter equals sat 2008 1800.96 207.04 kg So, total mass of water to be added equals 207.04 10 2070.4 kg Example 3.4 A saturated soil has a dry unit weight of 103 lb/ft3. Its moisture content is 23%. Determine: a. Saturated unit weight, sat b. Specific gravity, Gs c. Void ratio, e ■ 64 Chapter 3: Weight–Volume Relationships Solution Part a: Saturated Unit Weight From Eq. (3.12), gsat gd 11 w2 1103 2 a 1 23 b 126.69 lb/ft3 126.7lb/ft3 100 Part b: Specific Gravity, Gs From Eq. (3.16), gd Gsgw 1e Also from Eq. (3.20) for saturated soils, e wGs. Thus, gd Gsgw 1 wGs So, 103 Gs 162.42 1 10.23 2 1Gs 2 or 103 23.69Gs 62.4Gs Gs 2.66 Part c: Void Ratio, e For saturated soils, e wGs (0.23)(2.66) 0.61 3.5 ■ Relative Density The term relative density is commonly used to indicate the in situ denseness or looseness of granular soil. It is deﬁned as Dr emax e emax emin (3.30) where Dr relative density, usually given as a percentage e in situ void ratio of the soil emax void ratio of the soil in the loosest state emin void ratio of the soil in the densest state The values of Dr may vary from a minimum of 0% for very loose soil to a maximum of 100% for very dense soils. Soils engineers qualitatively describe the granular soil deposits according to their relative densities, as shown in Table 3.3. In-place soils seldom 3.5 Relative Density 65 Table 3.3 uQalitative Description of Granular Soil Deposits Relative density (%) Description of soil deposit 0–15 15–50 50–70 70–85 85–100 Very loose Loose Medium Dense Very dense have relative densities less than 20 to 30%. Compacting a granular soil to a relative density greater than about 85% is difﬁcult. The relationships for relative density can also be deﬁned in terms of porosity, or emax nmax 1 nmax (3.31) emin nmin 1 nmin (3.32) n 1n (3.33) e where nmax and nmin porosity of the soil in the loosest and densest conditions, respectively. Substituting Eqs. (3.31), (3.32), and (3.33) into Eq. (3.30), we obtain Dr 11 nmin 2 1nmax n2 1nmax nmin 2 11 n2 (3.34) By using the deﬁnition of dry unit weight given in Eq. (3.16), we can express relative density in terms of maximum and minimum possible dry unit weights. Thus, 1 1 d c d gd1max2 gd gd1min2 gd1min2 gd Dr dc d c gd1max2 gd1min2 gd 1 1 c d c d gd1min2 gd1max2 c (3.35) where gd(min) dry unit weight in the loosest condition (at a void ratio of emax) gd in situ dry unit weight (at a void ratio of e) gd(max) dry unit weight in the densest condition (at a void ratio of emin) In terms of density, Eq. (3.35) can be expressed as Dr c rd rd1min2 rd1max2 rd1min2 d rd1max2 rd (3.36) 66 Chapter 3: Weight–Volume Relationships ASTM Test Designations D-4253 and D-4254 (2007) provide a procedure for determining the maximum and minimum dry unit weights of granular soils so that they can be used in Eq. (3.35) to measure the relative density of compaction in the ﬁeld. For sands, this procedure involves using a mold with a volume of 2830 cm3 (0.1 ft3). For a determination of the minimum dry unit weight, sand is poured loosely into the mold from a funnel with a 12.7 mm (12 in.) diameter spout. The average height of the fall of sand into the mold is maintained at about 25.4 mm (1 in.). The value of gd(min) then can be calculated by using the following equation gd1min2 Ws Vm (3.37) where Ws weight of sand required to ﬁll the mold Vm volume of the mold The maximum dry unit weight is determined by vibrating sand in the mold for 8 min. A surcharge of 14 kN/m2 (2 lb/in2) is added to the top of the sand in the mold. The mold is placed on a table that vibrates at a frequency of 3600 cycles/min and that has an amplitude of vibration of 0.635 mm (0.025 in.). The value of gd(max) can be determined at the end of the vibrating period with knowledge of the weight and volume of the sand. Several factors control the magnitude of gd(max): the magnitude of Example 3.5 For a given sandy soil, emax 0.75 and emin 0.4. Let Gs 2.68. In the ﬁeld, the soil is compacted to a moist density of 112 lb/f3 at a moisture content of 12%. Determine the relative density of compaction. Solution From Eq. (3.21), r 11 w2Gsgw 1e or e Gsgw 11 w2 g 1 12.682 162.42 11 0.122 112 1 0.67 From Eq. (3.30), Dr emax e 0.75 0.67 0.229 22.9% emax emin 0.75 0.4 ■ 3.6 Comments on emax and emin 67 acceleration, the surcharge load, and the geometry of acceleration. Hence, one can obtain a larger-value gd(max) than that obtained by using the ASTM standard method described earlier. Comments on emax and emin The maximum and minimum void ratios for granular soils described in Section 3.5 depend on several factors, such as • • • • Grain size Grain shape Nature of the grain-size distribution curve Fine contents, Fc (that is, fraction smaller than 0.075 mm) The amount of non-plastic ﬁnes present in a given granular soil has a great inﬂuence on emax and emin . Figure 3.8 shows a plot of the variation of emax and emin with the percentage of nonplastic ﬁnes (by volume) for Nevada 50/80 sand (Lade, et al., 1998). The ratio of 1.2 1.0 0.8 Void ratio, e 3.6 0.6 0.4 0.2 0.0 0 20 40 60 80 100 Percent ﬁnes (by volume) emax emin Figure 3.8 Variation of emax and emin (for Nevada 50/80 sand) with percentage of non-plastic ﬁnes (Redrawn from Lade et al, 1998. Copyright ASTM INTERNATIONAL. Reprinted with permission.) 68 Chapter 3: Weight–Volume Relationships D50 (size through which 50% of soil will pass) for the sand to that for nonplastic ﬁnes used for the tests shown in Figure 3.8 (that is, D50-sand/D50-ﬁne ) was 4.2. From Figure 3.8, it can be seen that as the percentage of ﬁnes by volume increased from zero to about 30% the magnitudes of emax and emin decreased. This is the ﬁlling-of-void phase where the ﬁnes tend to ﬁll the void spaces between the larger sand particles. There is a transition zone when the percentage of ﬁnes is between 30 to 40%. However, when the percentage of ﬁnes becomes more than about 40%, the magnitudes of emax and emin start increasing. This is the replacement-of-solids phase where the large-sized particles are pushed out and gradually are replaced by the ﬁnes. Cubrinovski and Ishihara (2002) studied the variation of emax and emin for a very larger number of soils. Based on the best-ﬁt linear-regression lines, they provided the following relationships. • Clean sand (Fc 0 to 5%) emax 0.072 1.53 emin • Sand with ﬁnes (5 Fc 15%) emax 0.25 1.37 emin • (3.39) Sand with ﬁnes and clay (15 Fc 30%; Pc 5 to 20%) emax 0.44 1.21 emin • (3.38) (3.40) Silty soils (30 Fc 70%; Pc 5 to 20%) emax 0.44 1.32 emin (3.41) where Fc ﬁne fraction for which grain size is smaller than 0.075 mm Pc clay-size fraction ( 0.005 mm) Figure 3.9 shows a plot of emax emin versus the mean grain size (D50 ) for a number of soils (Cubrinovski and Ishihara, 1999 and 2002). From this ﬁgure, the average plot for sandy and gravelly soils can be given by the relationship emax emin 0.23 3.7 0.06 D50 1mm2 (3.42) Summary In this chapter, we discussed weight–volume relations and the concept of relative density. The volume relationships are those for void ratio, porosity, and degree of saturation. The weight relationships include those for moisture content and dry, moist, and saturated unit weight. Relative density is a term to describe the denseness of a granular soil. Relative density can be expressed in terms of the maximum, minimum, and in situ unit weights/densities of a soil and is generally expressed as a percentage. Problems 69 1.0 Void ratio range, emax emin 0.8 0.6 emax emin 0.23 0.06 D50 0.4 0.2 0.0 0.1 1.0 Mean grain size, D50 (mm) 10 Clean sands (FC 0 – 5%) Sands with ﬁnes (5 FC 15%) Sands with clay (15 FC 30%, PC 5 – 20%) Silty soils (30 FC 70%, PC 5 – 20%) Gravelly sands (FC 6%, PC 17 – 36%) Gravels Figure 3.9 Plot of emax emin versus the mean grain size (Cubrinovski and Ishihara, 2002) Problems 3.1 For a given soil, show that sat d + nw 3.2 For a given soil, show that e b gw gsat gd a 1e 3.3 For a given soil, show that eSgw gd 11 e2w 3.4 A 0.4-m3 moist soil sample has the following: • Moist mass 711.2 kg • Dry mass 623.9 kg • Specific gravity of soil solids 2.68 Estimate: a. Moisture content b. Moist density 70 Chapter 3: Weight–Volume Relationships 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 c. Dry density d. Void ratio e. Porosity In its natural state, a moist soil has a volume of 0.33 ft3 and weighs 39.93 lb. The oven-dry weight of the soil is 34.54 lb. If Gs 2.67, calculate the moisture content, moist unit weight, dry unit weight, void ratio, porosity, and degree of saturation. The moist weight of 0.2 ft3 of a soil is 23 lb. The moisture content and the specific gravity of the soil solids are determined in the laboratory to be 11% and 2.7, respectively. Calculate the following: a. Moist unit weight (lb/ft3) b. Dry unit weight (lb/ft3) c. Void ratio d. Porosity e. Degree of saturation (%) f. Volume occupied by water (ft3) The saturated unit weight of a soil is 19.8 kN/m3. The moisture content of the soil is 17.1%. Determine the following: a. Dry unit weight b. Specific gravity of soil solids c. Void ratio The unit weight of a soil is 95 lb/ft3. The moisture content of this soil is 19.2% when the degree of saturation is 60%. Determine: a. Void ratio b. Specific gravity of soil solids c. Saturated unit weight For a given soil, the following are given: Gs 2.67; moist unit weight, 112 lb/ft3; and moisture content, w 10.8%. Determine: a. Dry unit weight b. Void ratio c. Porosity d. Degree of saturation Refer to Problem 3.9. Determine the weight of water, in pounds, to be added per cubic foot of soil for: a. 80% degree of saturation b. 100% degree of saturation The moist density of a soil is 1680 kg/m3. Given w 18% and Gs 2.73, determine: a. Dry density b. Porosity c. Degree of saturation d. Mass of water, in kg/m3, to be added to reach full saturation The dry density of a soil is 1780 kg/m3. Given Gs 2.68, what would be the moisture content of the soil when saturated? The porosity of a soil is 0.35. Given Gs 2.69, calculate: a. Saturated unit weight (kN/m3) b. Moisture content when moist unit weight 17.5 kN/m3 Problems 71 3.14 A saturated soil has w 23% and Gs 2.62. Determine its saturated and dry densities in kg/m3. 3.15 A soil has e 0.75, w 21.5%, and Gs 2.71. Determine: a. Moist unit weight (lb/ft3) b. Dry unit weight (lb/ft3) c. Degree of saturation (%) 3.16 A soil has w 18.2%, Gs 2.67, and S 80%. Determine the moist and dry unit weights of the soil in lb/ft3. 3.17 The moist unit weight of a soil is 112.32 lb/ft3 at a moisture content of 10%. Given Gs 2.7, determine: a. e b. Saturated unit weight 3.18 The moist unit weights and degrees of saturation of a soil are given in the table. 3.19 3.20 3.21 3.22 3.23 3.24 (lb/ft3) S (%) 105.73 112.67 50 75 Determine: a. e b. Gs Refer to Problem 3.18. Determine the weight of water, in lb, that will be in 2.5 ft3 of the soil when it is saturated. For a given sand, the maximum and minimum void ratios are 0.78 and 0.43, respectively. Given Gs 2.67, determine the dry unit weight of the soil in kN/m3 when the relative density is 65%. For a given sandy soil, emax 0.75, emin 0.46, and Gs 2.68. What will be the moist unit weight of compaction (kN/m3) in the field if Dr 78% and w 9%? For a given sandy soil, the maximum and minimum dry unit weights are 108 lb/ft3 and 92 lb/ft3, respectively. Given Gs 2.65, determine the moist unit weight of this soil when the relative density is 60% and the moisture content is 8%. The moisture content of a soil sample is 18.4%, and its dry unit weight is 100 lb/ft3. Assuming that the specific gravity of solids is 2.65, a. Calculate the degree of saturation. b. What is the maximum dry unit weight to which this soil can be compacted without change in its moisture content? A loose, uncompacted sand fill 6 ft in depth has a relative density of 40%. Laboratory tests indicated that the minimum and maximum void ratios of the sand are 0.46 and 0.90, respectively. The specific gravity of solids of the sand is 2.65. a. What is the dry unit weight of the sand? b. If the sand is compacted to a relative density of 75%, what is the decrease in thickness of the 6-ft fill? 72 Chapter 3: Weight–Volume Relationships References AMERICAN SOCIETY FOR TESTING AND MATERIALS (2007). Annual Book of ASTM Standards, Sec. 4, Vol. 04.08. West Conshohocken, Pa. CUBRINOVSKI, M., and ISHIHARA, K. (1999). “Empirical Correlation Between SPT N-Value and Relative Density for Sandy Soils,” Soils and Foundations. Vol. 39, No. 5, 61–71. CUBRINOVSKI, M., and ISHIHARA, K. (2002). “Maximum and Minimum Void Ratio Characteristics of Sands,” Soils and Foundations. Vol. 42, No. 6, 65–78. LADE, P. V., LIGGIO, C. D., and YAMAMURO, J. A. (1998). “Effects of Non-Plastic Fines on Minimum and Maximum Void Ratios of Sand,” Geotechnical Testing Journal, ASTM. Vol. 21, No. 4, 336–347. Introduction Strain Stress When clay minerals are present in ﬁne-grained soil, the soil can be remolded in the presence of some moisture without crumbling. This cohesive nature is caused by the adsorbed water surrounding the clay particles. In the early 1900s, a Swedish scientist named Atterberg developed a method to describe the consistency of ﬁne-grained soils with varying moisture contents. At a very low moisture content, soil behaves more like a solid. When the moisture content is very high, the soil and water may ﬂow like a liquid. Hence, on an arbitrary basis, depending on the moisture content, the behavior of soil can be divided into four basic states—solid, semisolid, plastic, and liquid—as shown in Figure 4.1. The moisture content, in percent, at which the transition from solid to semisolid state takes place is deﬁned as the shrinkage limit. The moisture content at the point of transition from semisolid to plastic state is the plastic limit, and from plastic to liquid state is the liquid limit. These parameters are also known as Atterberg limits. This chapter describes the procedures to determine the Atterberg limits. Also discussed in this chapter are soil structure and geotechnical parameters, such as activity and liquidity index, which are related to Atterberg limits. Stress 4.1 Stress 4 Plasticity and Structure of Soil Strain Strain Stress-strain diagrams at various states Solid Semisolid Plastic Liquid Shrinkage limit, S L Plastic limit, PL Liquid limit, L L Moisture content increasing Figure 4.1 Atterberg limits 73 74 Chapter 4: Plasticity and Structure of Soil 4.2 Liquid Limit (LL) A schematic diagram (side view) of a liquid limit device is shown in Figure 4.2a. This device consists of a brass cup and a hard rubber base. The brass cup can be dropped onto the base by a cam operated by a crank. To perform the liquid limit test, one must place a soil 46.8 mm 27 mm 54 mm Soil pat (a) 50 mm 8 mm 27 2 mm 11 mm (b) Section 11 mm 8 mm 2 mm Plan 12.7 mm (c) (d) Figure 4.2 Liquid limit test: (a) liquid limit device; (b) grooving tool; (c) soil pat before test; (d) soil pat after test 4.2 Liquid Limit (LL) 75 paste in the cup. A groove is then cut at the center of the soil pat with the standard grooving tool (Figure 4.2b). By the use of the crank-operated cam, the cup is lifted and dropped from a height of 10 mm (0.394 in.). The moisture content, in percent, required to close a distance of 12.7 mm (0.5 in.) along the bottom of the groove (see Figures 4.2c and 4.2d) after 25 blows is deﬁned as the liquid limit. It is difﬁcult to adjust the moisture content in the soil to meet the required 12.7 mm (0.5 in.) closure of the groove in the soil pat at 25 blows. Hence, at least three tests for the same soil are conducted at varying moisture contents, with the number of blows, N, required to achieve closure varying between 15 and 35. Figure 4.3 shows a photograph of a liquid limit test device and grooving tools. Figure 4.4 shows photographs of the soil pat in the liquid limit device before and after the test. The moisture content of the soil, in percent, and the corresponding number of blows are plotted on semilogarithmic graph paper (Figure 4.5). The relationship between moisture content and log N is approximated as a straight line. This line is referred to as the ﬂow curve. The moisture content corresponding to N 25, determined from the ﬂow curve, gives the liquid limit of the soil. The slope of the ﬂow line is deﬁned as the ﬂow index and may be written as IF w1 w2 N2 log a b N1 Figure 4.3 Liquid limit test device and grooving tools (Courtesy of ELE International) (4.1) (a) Figure 4.4 Photographs showing the soil pat in the liquid limit device: (a) before test; (b) after test [Note: The half-inch groove closure in (b) is marked for clarification] (Courtesy of Braja M. Das, Henderson, Nevada) (b) Moisture content (%) 50 Flow curve 45 Liquid limit 42 40 35 30 10 20 25 30 Number of blows, N (log scale) 76 40 50 Figure 4.5 Flow curve for liquid limit determination of a clayey silt 4.2 Liquid Limit (LL) 77 where IF flow index w1 moisture content of soil, in percent, corresponding to N1 blows w2 moisture content corresponding to N2 blows Note that w2 and w1 are exchanged to yield a positive value even though the slope of the ﬂow line is negative. Thus, the equation of the ﬂow line can be written in a general form as w IF log N C (4.2) where C a constant. From the analysis of hundreds of liquid limit tests, the U.S. Army Corps of Engineers (1949) at the Waterways Experiment Station in Vicksburg, Mississippi, proposed an empirical equation of the form LL wN a N tan b b 25 (4.3) where N number of blows in the liquid limit device for a 12.7 mm (0.5 in.) groove closure wN corresponding moisture content tan b 0.121 (but note that tan b is not equal to 0.121 for all soils) Equation (4.3) generally yields good results for the number of blows between 20 and 30. For routine laboratory tests, it may be used to determine the liquid limit when only one test is run for a soil. This procedure is generally referred to as the one-point method and was also adopted by ASTM under designation D-4318. The reason that the one-point method yields fairly good results is that a small range of moisture content is involved when N 20 to N 30. Another method of determining liquid limit that is popular in Europe and Asia is the fall cone method (British Standard—BS1377). In this test the liquid limit is deﬁned as the moisture content at which a standard cone of apex angle 30 and weight of 0.78 N (80 gf) will penetrate a distance d 20 mm in 5 seconds when allowed to drop from a position of point contact with the soil surface (Figure 4.6a). Due to the difﬁculty in achieving the liquid limit from a single test, four or more tests can be conducted at various moisture contents to determine the fall cone penetration, d. A semilogarithmic graph can then be plotted with moisture content (w) versus cone penetration d. The plot results in a straight line. The moisture content corresponding to d 20 mm is the liquid limit (Figure 4.6b). From Figure 4.6(b), the ﬂow index can be deﬁned as IFC w2 1% 2 w1 1% 2 log d2 log d1 where w1, w2 moisture contents at cone penetrations of d1 and d2, respectively. (4.4) 78 Chapter 4: Plasticity and Structure of Soil 30° Weight, W 0.78 N Soil d 40 mm 55 mm (a) Moisture content, (%) 50 40 Liquid limit 30 10 20 40 Penetration, d (mm) (b) 4.3 60 80 100 Figure 4.6 (a) Fall cone test (b) plot of moisture content vs. cone penetration for determination of liquid limit Plastic Limit (PL) The plastic limit is deﬁned as the moisture content in percent, at which the soil crumbles, when rolled into threads of 4.2 mm ( 18 in.) in diameter. The plastic limit is the lower limit of the plastic stage of soil. The plastic limit test is simple and is performed by repeated rollings of an ellipsoidal-sized soil mass by hand on a ground glass plate (Figure 4.7). The procedure for the plastic limit test is given by ASTM in Test Designation D-4318. As in the case of liquid limit determination, the fall cone method can be used to obtain the plastic limit. This can be achieved by using a cone of similar geometry but with a mass of 2.35 N (240 gf). Three to four tests at varying moisture contents of soil are conducted, and the corresponding cone penetrations (d) are determined. The moisture content corresponding to a cone penetration of d 20 mm is the plastic limit. Figure 4.8 shows the liquid and plastic limit determination of Cambridge Gault clay reported by Wroth and Wood (1978). 4.3 Plastic Limit (PL) 79 2 4 Figure 4.7 Rolling of soil mass on ground glass plate to determine plastic limit (Courtesy of Braja M. Das, Henderson, Nevada) W 0.78 N 70 (%) Liquid limit Moisture content, 60 Plastic limit Cone weight, W 2.35 N 50 40 1 2 5 10 20 Cone penetration, d (mm) 50 Figure 4.8 Liquid and plastic limits for Cambridge Gault clay determined by fall cone test The plasticity index (PI) is the difference between the liquid limit and the plastic limit of a soil, or PI LL PL (4.5) Table 4.1 gives the ranges of liquid limit, plastic limit, and activity (Section 4.6) of some clay minerals (Mitchell, 1976; Skempton, 1953). 80 Chapter 4: Plasticity and Structure of Soil Table 4.1 Typical Values of Liquid Limit, Plastic Limit, and Activity of Some Clay Minerals Mineral Liquid limit, LL Plastic limit, PL Activity, A Kaolinite Illite Montmorillonite Halloysite (hydrated) Halloysite (dehydrated) Attapulgite Allophane 35–100 60–120 100–900 50–70 40–55 150–250 200–250 20–40 35–60 50–100 40–60 30–45 100–125 120–150 0.3–0.5 0.5–1.2 1.5–7.0 0.1–0.2 0.4–0.6 0.4–1.3 0.4–1.3 Burmister (1949) classiﬁed the plasticity index in a qualitative manner as follows: PI Description 0 1–5 5–10 10–20 20–40 40 Nonplastic Slightly plastic Low plasticity Medium plasticity High plasticity Very high plasticity The plasticity index is important in classifying ﬁne-grained soils. It is fundamental to the Casagrande plasticity chart (presented in Section 4.7), which is currently the basis for the Uniﬁed Soil Classiﬁcation System. (See Chapter 5.) Sridharan, et al. (1999) showed that the plasticity index can be correlated to the ﬂow index as obtained from the liquid limit tests (Section 4.2). According to their study, PI 1% 2 4.12IF 1% 2 (4.6) PI 1% 2 0.74IFC 1% 2 (4.7) and In a recent study by Polidori (2007) that involved six inorganic soils and their respective mixtures with fine silica sand, it was shown that PL 0.04(LL) + 0.26(CF) + 10 (4.8) PI 0.96(LL) 0.26(CF) 10 (4.9) and where CF clay fraction (2 m) in %. The experimental results of Polidori (2007) show that the preceding relationships hold good for CF approximately equal to or greater than 30%. 4.4 Shrinkage Limit (SL) Shrinkage Limit (SL) Soil shrinks as moisture is gradually lost from it. With continuing loss of moisture, a stage of equilibrium is reached at which more loss of moisture will result in no further volume change (Figure 4.9). The moisture content, in percent, at which the volume of the soil mass ceases to change is deﬁned as the shrinkage limit. Shrinkage limit tests (ASTM Test Designation D-427) are performed in the laboratory with a porcelain dish about 44 mm (1.75 in.) in diameter and about 12.7 mm (12 in.) high. The inside of the dish is coated with petroleum jelly and is then ﬁlled completely with wet soil. Excess soil standing above the edge of the dish is struck off with a straightedge. The mass of the wet soil inside the dish is recorded. The soil pat in the dish is then oven-dried. The volume of the oven-dried soil pat is determined by the displacement of mercury. Because handling mercury may be hazardous, ASTM D-4943 describes a method of dipping the oven-dried soil pat in a melted pot of wax. The wax-coated soil pat is then cooled. Its volume is determined by submerging it in water. By reference to Figure 4.9, the shrinkage limit can be determined as SL wi 1% 2 ¢w 1 % 2 (4.10) where wi initial moisture content when the soil is placed in the shrinkage limit dish w change in moisture content (that is, between the initial moisture content and the moisture content at the shrinkage limit) However, wi 1% 2 M1 M2 100 M2 (4.11) where M1 mass of the wet soil pat in the dish at the beginning of the test (g) M2 mass of the dry soil pat (g) (see Figure 4.10) ∆ Vi Volume of soil 4.4 81 Vf Shrinkage limit Plastic limit Moisture content (%) Figure 4.9 Deﬁnition of shrinkage limit Liquid limit i 82 Chapter 4: Plasticity and Structure of Soil Soil volume Vi Soil mass M1 Soil volume Vf Soil mass M2 Porcelain dish (a) (b) Figure 4.10 Shrinkage limit test: (a) soil pat before drying; (b) soil pat after drying Also, ¢w 1% 2 1Vi Vf 2rw M2 100 (4.12) where Vi initial volume of the wet soil pat (that is, inside volume of the dish, cm3) Vf volume of the oven-dried soil pat (cm3) rw density of water (g/cm3) Finally, combining Eqs. (4.10), (4.11), and (4.12) gives SL a Vi Vf M1 M2 b 1100 2 a b 1rw 2 1100 2 M2 M2 (4.13) Another parameter that can be determined from a shrinkage limit test is the shrinkage ratio, which is the ratio of the volume change of soil as a percentage of the dry volume to the corresponding change in moisture content, or ¢V ¢V b b a M2 Vf Vf SR ¢Vrw ¢M Vfrw a b a b M2 M2 a (4.14) where V change in volume M corresponding change in the mass of moisture It can also be shown that Gs 1 1 SL a b SR 100 where Gs speciﬁc gravity of soil solids. (4.15) 4.5 Liquidity Index and Consistency Index 83 Example 4.1 Following are the results of a shrinkage limit test: • • • • Initial volume of soil in a saturated state 24.6 cm3 Final volume of soil in a dry state 15.9 cm3 Initial mass in a saturated state 44.0 g Final mass in a dry state 30.1 g Determine the shrinkage limit of the soil. Solution From Eq. (4.13), SL a Vi Vf M1 M2 b 11002 a b 1rw 2 11002 M2 M2 M1 44.0g Vi 24.6 cm3 M2 30.1g Vf 15.9 cm3 SL a rw 1 g/cm3 24.6 15.9 44.0 30.1 b 1100 2 a b 112 11002 30.1 30.1 46.18 28.9 17.28% ■ Typical values of shrinkage limit for some clay minerals are as follows (Mitchell, 1976). 4.5 Mineral Shrinkage limit Montmorillonite Illite Kaolinite 8.5–15 15–17 25–29 Liquidity Index and Consistency Index The relative consistency of a cohesive soil in the natural state can be deﬁned by a ratio called the liquidity index, which is given by LI w PL LL PL (4.16) where w in situ moisture content of soil. The in situ moisture content for a sensitive clay may be greater than the liquid limit. In this case (Figure 4.11), LI 1 These soils, when remolded, can be transformed into a viscous form to ﬂow like a liquid. 84 Chapter 4: Plasticity and Structure of Soil LI 0 LI 1 LI 1 PL Moisture content, LL PI Figure 4.11 Liquidity index Soil deposits that are heavily overconsolidated may have a natural moisture content less than the plastic limit. In this case (Figure 4.11), LI 0 Another index that is commonly used for engineering purposes is the consistency index (CI), which may be defined as CI LL w LL PI (4.17) where w in situ moisture content. If w is equal to the liquid limit, the consistency index is zero. Again, if w PI, then CI 1. Activity Because the plasticity of soil is caused by the adsorbed water that surrounds the clay particles, we can expect that the type of clay minerals and their proportional amounts in a soil will affect the liquid and plastic limits. Skempton (1953) observed that the plasticity index of a soil increases linearly with the percentage of clay-size fraction (% ﬁner than 2 mm by weight) present (Figure 4.12). The correlations of PI with the clay-size fractions for different clays plot separate lines. This difference is due to the diverse plasticity characteristics of the 100 80 Plasticity index 4.6 60 40 20 + + ++ + + 0 0 20 40 60 80 Percentage of clay-size fraction (2 mm) Shellhaven clay A 1.33 London clay A 0.95 Weald clay A 0.63 + Horten clay A 0.42 100 Figure 4.12 Activity (Based on Skempton, 1953) 4.6 Activity 85 various types of clay minerals. On the basis of these results, Skempton deﬁned a quantity called activity, which is the slope of the line correlating PI and % ﬁner than 2 mm. This activity may be expressed as A PI 1%of clay-size fraction, by weight2 (4.18) where A activity. Activity is used as an index for identifying the swelling potential of clay soils. Typical values of activities for various clay minerals are given in Table 4.1. Seed, Woodward, and Lundgren (1964a) studied the plastic property of several artiﬁcially prepared mixtures of sand and clay. They concluded that, although the relationship of the plasticity index to the percentage of clay-size fraction is linear (as observed by Skempton), it may not always pass through the origin. This is shown in Figures 4.13 and 4.14. Thus, the activity can be redeﬁned as A PI %of clay-size fraction C¿ (4.19) where C is a constant for a given soil. 500 A 5.4 Plasticity index 400 4.85 4.43 300 200 3.25 100 1.95 1.52 0.96 0.36 0 0 20 40 60 80 Percentage of clay-size fraction (2m) Commercial bentonite Bentonite/kaolinite — 4 : 1 Bentonite/kaolinite —1.5 : 1 Kaolinite/bentonite—1.5 : 1 100 Kaolinite/bentonite — 4 : 1 Kaolinite/bentonite— 9 : 1 Kaolinite/bentonite—19 : 1 Commercial kaolinite Figure 4.13 Relationship between plasticity index and clay-size fraction by weight for kaolinite/ bentonite clay mixtures (After Seed, Woodward, and Lundgren, 1964a. With permission from ASCE.) 86 Chapter 4: Plasticity and Structure of Soil 500 A 5.4 Plasticity index 400 300 3.33 200 1.78 100 1.28 0.93 0 0 20 40 60 Percentage of clay-size fraction (2m) Commercial bentonite Bentonite/illite — 4 : 1 Bentonite/illite —1.5 : 1 80 100 Illite/ bentonite —1.5 : 1 Commercial illite Figure 4.14 Relationship between plasticity index and clay-size fraction by weight for illite/ bentonite clay mixtures (After Seed, Woodward, and Lundgren, 1964a. With permission from ASCE.) Plasticity index For the experimental results shown in Figures 4.13 and 4.14, C 9. Further works of Seed, Woodward, and Lundgren (1964b) have shown that the relationship of the plasticity index to the percentage of clay-size fractions present in a soil can be represented by two straight lines. This is shown qualitatively in Figure 4.15. For clay-size fractions greater than 40%, the straight line passes through the origin when it is projected back. 0 10 40 Percentage of clay-size fraction (2 mm) Figure 4.15 Simpliﬁed relationship between plasticity index and percentage of clay-size fraction by weight (After Seed, Woodward, and Lundgren, 1964b. With permission from ASCE.) 4.7 Plasticity Chart 87 Based on Eqs. (4.8) and (4.9), Polidori (2007) provided an empirical relationship for activity as (for CF equal to or greater than 30%) A 0.961LL2 0.261CF2 10 CF (4.20) where CF is the clay fraction (2 m) Plasticity Chart Liquid and plastic limits are determined by relatively simple laboratory tests that provide information about the nature of cohesive soils. Engineers have used the tests extensively for the correlation of several physical soil parameters as well as for soil identiﬁcation. Casagrande (1932) studied the relationship of the plasticity index to the liquid limit of a wide variety of natural soils. On the basis of the test results, he proposed a plasticity chart as shown in Figure 4.16. The important feature of this chart is the empirical A-line that is given by the equation PI 0.73(LL 20). An A-line separates the inorganic clays from the inorganic silts. Inorganic clay values lie above the A-line, and values for inorganic silts lie below the A-line. Organic silts plot in the same region (below the A-line and with LL ranging from 30 to 50) as the inorganic silts of medium compressibility. Organic clays plot in the same region as inorganic silts of high compressibility (below the A-line and LL greater than 50). The information provided in the plasticity chart is of great value and is the basis for the classiﬁcation of ﬁne-grained soils in the Uniﬁed Soil Classiﬁcation System. (See Chapter 5.) 70 60 50 Plasticity index 4.7 U-Line PI 0.9(LL 8) 40 30 A-Line PI 0.73(LL 20) 20 10 0 0 20 40 60 Liquid limit 80 Cohesionless soil Inorganic clays of low plasticity Inorganic silts of low compressibility Inorganic clays of medium plasticity Inorganic silts of medium compressibility and organic silts Inorganic clays of high plasticity Inorganic silts of high compressibility and organic clays 100 Figure 4.16 Plasticity chart 88 Chapter 4: Plasticity and Structure of Soil Note that a line called the U-line lies above the A-line. The U-line is approximately the upper limit of the relationship of the plasticity index to the liquid limit for any currently known soil. The equation for the U-line can be given as PI 0.91LL 8 2 (4.21) There is another use for the A-line and the U-line. Casagrande has suggested that the shrinkage limit of a soil can be approximately determined if its plasticity index and liquid limit are known (see Holtz and Kovacs, 1981). This can be done in the following manner with reference to Figure 4.17. a. Plot the plasticity index against the liquid limit of a given soil such as point A in Figure 4.17. b. Project the A-line and the U-line downward to meet at point B. Point B will have the coordinates of LL 43.5 and PI 46.4. c. Join points B and A with a straight line. This will intersect the liquid limit axis at point C. The abscissa of point C is the estimated shrinkage limit. Plasticity index 60 40 e lin U- e lin A- A 20 C 0 B 0 8 20 40 60 Liquid limit 80 100 120 LL 43.5 PI 46.5 Figure 4.17 Estimation of shrinkage from plasticity chart (Adapted from Holtz and Kovacs, 1981) 4.8 Soil Structure Soil structure is deﬁned as the geometric arrangement of soil particles with respect to one another. Among the many factors that affect the structure are the shape, size, and mineralogical composition of soil particles, and the nature and composition of soil water. In general, soils can be placed into two groups: cohesionless and cohesive. The structures found in soils in each group are described next. 4.8 Soil Structure 89 Void Void Soil solid Soil solid (a) (b) Figure 4.18 Single-grained structure: (a) loose; (b) dense Structures in Cohesionless Soil The structures generally encountered in cohesionless soils can be divided into two major categories: single grained and honeycombed. In single-grained structures, soil particles are in stable positions, with each particle in contact with the surrounding ones. The shape and size distribution of the soil particles and their relative positions inﬂuence the denseness of packing (Figure 4.18); thus, a wide range of void ratios is possible. To get an idea of the variation of void ratios caused by the relative positions of the particles, let us consider the mode of packing of equal spheres shown in Figure 4.19. Figure 4.19a shows the case of a very loose state of packing. If we isolate a cube with each side measuring d, which is equal to the diameter of each sphere as shown in the ﬁgure, the void ratio can be calculated as e V Vs Vv Vs Vs d (a) 2d where V volume of the cube d 3 Vs volume of sphere (i.e., solid) inside the cube d 2 (b) Figure 4.19 Mode of packing of equal spheres (plan views): (a) very loose packing (e 0.91); (b) very dense packing (e 0.35) 90 Chapter 4: Plasticity and Structure of Soil Noting that V d 3 and Vs pd 3/6 yields d3 a e pd3 b 6 0.91 pd3 b a 6 Similarly, Figure 4.19b shows the case of a very dense state of packing. Figure 4.19b also shows an isolated cube, for which each side measures d 12. It can be shown that, for this case, e 0.35. Real soil differs from the equal-spheres model in that soil particles are neither equal in size nor spherical. The smaller-size particles may occupy the void spaces between the larger particles, thus the void ratio of soils is decreased compared with that for equal spheres. However, the irregularity in the particle shapes generally yields an increase in the void ratio of soils. As a result of these two factors, the void ratios encountered in real soils have approximately the same range as those obtained in equal spheres. In the honeycombed structure (Figure 4.20), relatively ﬁne sand and silt form small arches with chains of particles. Soils that exhibit a honeycombed structure have large void ratios, and they can carry an ordinary static load. However, under a heavy load or when subjected to shock loading, the structure breaks down, which results in a large amount of settlement. Structures in Cohesive Soils To understand the basic structures in cohesive soils, we need to know the types of forces that act between clay particles suspended in water. In Chapter 2, we discussed the negative charge on the surface of the clay particles and the diffuse double layer surrounding each particle. When two clay particles in suspension come close to each other, the tendency for interpenetration of the diffuse double layers results in repulsion between the particles. At the same time, an attractive force exists between the clay particles that is caused by van der Waals forces and is independent of the characteristics of water. Both repulsive and attractive forces increase with decreasing distance between the particles, but at different rates. When the spacing between the particles is very small, the force of attraction is greater than the force of repulsion. These are the forces treated by colloidal theories. Soil solid Void Figure 4.20 Honeycombed structure 4.8 Soil Structure 91 The fact that local concentrations of positive charges occur at the edges of clay particles was discussed in Chapter 2. If the clay particles are very close to each other, the positively charged edges can be attracted to the negatively charged faces of the particles. Let us consider the behavior of clay in the form of a dilute suspension. When the clay is initially dispersed in water, the particles repel one another. This repulsion occurs because with larger interparticle spacing, the forces of repulsion between the particles are greater than the forces of attraction (van der Waals forces). The force of gravity on each particle is negligible. Thus, the individual particles may settle very slowly or remain in suspension, undergoing Brownian motion (a random zigzag motion of colloidal particles in suspension). The sediment formed by the settling of the individual particles has a dispersed structure, and all particles are oriented more or less parallel to one another (Figure 4.21a). If the clay particles initially dispersed in water come close to one another during random motion in suspension, they might aggregate into visible ﬂocs with edge-to-face contact. In this instance, the particles are held together by electrostatic attraction of positively charged edges to negatively charged faces. This aggregation is known as ﬂocculation. When the ﬂocs become large, they settle under the force of gravity. The sediment formed in this manner has a ﬂocculent structure (Figure 4.21b). When salt is added to a clay-water suspension that has been initially dispersed, the ions tend to depress the double layer around the particles. This depression reduces the interparticle repulsion. The clay particles are attracted to one another to form ﬂocs and settle. The ﬂocculent structure of the sediments formed is shown in Figure 4.21c. In ﬂocculent sediment structures of the salt type, the particle orientation approaches a large degree of parallelism, which is due to van der Waals forces. (a) (b) (c) Figure 4.21 Sediment structures: (a) dispersion; (b) nonsalt ﬂocculation; (c) salt ﬂocculation (Adapted from Lambe, 1958) 92 Chapter 4: Plasticity and Structure of Soil Silt Domain Macropore Cluster Ped Micropore Silt (a) (b) Figure 4.22 Soil structure: (a) arrangement of peds and macropore spaces; (b) arrangement of domains and clusters with silt-sized particles Clays that have ﬂocculent structures are lightweight and possess high void ratios. Clay deposits formed in the sea are highly ﬂocculent. Most of the sediment deposits formed from freshwater possess an intermediate structure between dispersed and ﬂocculent. A deposit of pure clay minerals is rare in nature. When a soil has 50% or more particles with sizes of 0.002 mm or less, it is generally termed clay. Studies with scanning electron microscopes (Collins and McGown, 1974; Pusch, 1978; Yong and Sheeran, 1973) have shown that individual clay particles tend to be aggregated or ﬂocculated in submicroscopic units. These units are referred to as domains. The domains then group together, and these groups are called clusters. Clusters can be seen under a light microscope. This grouping to form clusters is caused primarily by interparticle forces. The clusters, in turn, group to form peds. Peds can be seen without a microscope. Groups of peds are macrostructural features along with joints and ﬁssures. Figure 4.22a shows the arrangement of the peds and macropore spaces. The arrangement of domains and clusters with silt-size particles is shown in Figure 4.22b. From the preceding discussion, we can see that the structure of cohesive soils is highly complex. Macrostructures have an important inﬂuence on the behavior of soils from an engineering viewpoint. The microstructure is more important from a fundamental viewpoint. Table 4.2 summarizes the macrostructures of clay soils. Table 4.2 Structure of Clay Soils Item Remarks Dispersed structures Formed by settlement of individual clay particles; more or less parallel orientation (see Figure 4.21a) Formed by settlement of ﬂocs of clay particles (see Figures 4.21b and 4.21c) Aggregated or ﬂocculated submicroscopic units of clay particles Domains group to form clusters; can be seen under light microscope Clusters group to form peds; can be seen without microscope Flocculent structures Domains Clusters Peds Problems 4.9 93 Summary This chapter discussed two major components in the study of soil mechanics. They are (a) plasticity of soil and related topics (Sections 4.2 to 4.7), and (b) structure of soil (Section 4.8). Liquid limit, plastic limit, and shrinkage limit tests of ﬁne-grained soil are indicators of the nature of its plasticity. The difference between the liquid limit and plastic limit is called the plasticity index. Liquid limit and plasticity index are required parameters for classiﬁcation of ﬁne-grained soils. The structure of cohesionless soils can be single grained or honeycombed. Honeycombed structures are encountered in relatively ﬁne sands and silts. The macrostructure of clay soils can be broadly divided into categories such as dispersed structures, ﬂocculent structures, domains, clusters, and peds. Problems 4.1 4.2 4.3 4.4 4.5 4.6 Following are the results from the liquid and plastic limit tests for a soil. Liquid limit test: Number of blows, N Moisture content (%) 16 20 28 36.5 34.1 27.0 Plastic limit test: PL 12.2% a. Draw the ﬂow curve and obtain the liquid limit. b. What is the plasticity index of the soil? Determine the liquidity index of the soil described in Problem 4.1 if win situ 31%. Following are the results from the liquid and plastic limit tests for a soil. Liquid limit test: Number of blows, N Moisture content (%) 15 20 28 42 40.8 39.1 Plastic limit test: PL 18.7% a. Draw the flow curve and obtain the liquid limit. b. What is the plasticity index of the soil? Refer to Problem 4.3. Determine the liquidity index of the soil when the in situ moisture content is 26%. A saturated soil has the following characteristics: initial volume (Vi ) 19.65 cm3, ﬁnal volume (Vf ) 13.5 cm3, mass of wet soil (Ml ) 36 g, and mass of dry soil (M2 ) 25 g. Determine the shrinkage limit and the shrinkage ratio. Repeat Problem 4.5 with the following: Vi 24.6 cm3, Vf 15.9 cm3, Ml 44 g, and M2 30.1 g. 94 Chapter 4: Plasticity and Structure of Soil References AMERICAN SOCIETY FOR TESTING AND MATERIALS (2007). Annual Book of ASTM Standards, Sec. 4, Vol. 04.08, West Conshohocken, Pa. BS:1377 (1990). British Standard Methods of Tests for Soil for Engineering Purposes, Part 2, BSI, London. BURMISTER, D. M. (1949). “Principles and Techniques of Soil Identiﬁcation,” Proceedings, Annual Highway Research Board Meeting, National Research Council, Washington, D.C., Vol. 29, 402–434. CASAGRANDE, A. (1932). “Research of Atterberg Limits of Soils,” Public Roads, Vol. 13, No. 8, 121–136. COLLINS, K., and MCGOWN, A. (1974). “The Form and Function of Microfabric Features in a Variety of Natural Soils,” Geotechnique, Vol. 24, No. 2, 223–254. HOLTZ, R. D., and KOVACS, W. D. (1981). An Introduction to Geotechnical Engineering, Prentice-Hall, Englewood Cliffs, NJ. LAMBE, T. W. (1958). “The Structure of Compacted Clay,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 85, No. SM2, 1654-1 to 1654-35. MITCHELL, J. K. (1976). Fundamentals of Soil Behavior, Wiley, New York. POLIDORI, E. (2007). “Relationship between Atterberg Limits and Clay Contents,” Soils and Foundations, Vol. 47, No. 5, 887–896. PUSCH, R. (1978). “General Report on Physico-Chemical Processes Which Affect Soil Structure and Vice Versa,” Proceedings, International Symposium on Soil Structure, Gothenburg, Sweden, Appendix, 33. SEED, H. B., WOODWARD, R. J., and LUNDGREN, R. (1964a). “Clay Mineralogical Aspects of Atterberg Limits,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 90, No. SM4, 107–131. SEED, H. B., WOODWARD, R. J., and LUNDGREN, R. (1964b). “Fundamental Aspects of the Atterberg Limits,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 90, No. SM6, 75–105. SKEMPTON, A. W. (1953). “The Colloidal Activity of Clays,” Proceedings, 3rd International Conference on Soil Mechanics and Foundation Engineering, London, Vol. 1, 57–61. SRIDHARAN, A., NAGARAJ, H. B., and PRAKASH, K. (1999). “Determination of the Plasticity Index from Flow Index,” Geotechnical Testing Journal, ASTM, Vol. 22, No. 2, 175–181. U.S. ARMY CORPS OF ENGINEERS (1949). Technical Memo 3-286, U.S. Waterways Experiment Station, Vicksburg, Miss. WROTH, C. P., and WOOD, D. M. (1978). “The Correlation of Index Properties with Some Basic Engineering Properties of Soils,” Canadian Geotechnical Journal, Vol. 15, No. 2, 137–145. YONG, R. N., and SHEERAN, D. E. (1973). “Fabric Unit Interaction and Soil Behavior,” Proceedings, International Symposium on Soil Structure, Gothenburg, Sweden, 176–184. 5 Classiﬁcation of Soil Different soils with similar properties may be classified into groups and sub-groups according to their engineering behavior. Classification systems provide a common language to concisely express the general characteristics of soils, which are infinitely varied, without detailed descriptions. Most of the soil classification systems that have been developed for engineering purposes are based on simple index properties such as particle-size distribution and plasticity. Although several classification systems are now in use, none is totally definitive of any soil for all possible applications because of the wide diversity of soil properties. 5.1 Textural Classification In a general sense, texture of soil refers to its surface appearance. Soil texture is influenced by the size of the individual particles present in it. Table 2.3 divided soils into gravel, sand, silt, and clay categories on the basis of particle size. In most cases, natural soils are mixtures of particles from several size groups. In the textural classification system, the soils are named after their principal components, such as sandy clay, silty clay, and so forth. A number of textural classification systems were developed in the past by different organizations to serve their needs, and several of those are in use today. Figure 5.1 shows the textural classification systems developed by the U.S. Department of Agriculture (USDA). This classification method is based on the particle-size limits as described under the USDA system in Table 2.3; that is • • • Sand size: 2.0 to 0.05 mm in diameter Silt size: 0.05 to 0.002 mm in diameter Clay size: smaller than 0.002 mm in diameter The use of this chart can best be demonstrated by an example. If the particle-size distribution of soil A shows 30% sand, 40% silt, and 30% clay-size particles, its textural classification can be determined by proceeding in the manner indicated by the arrows in Figure 5.1. This soil falls into the zone of clay loam. Note that this chart is based on only 95 96 Chapter 5: Classiﬁcation of Soil 0 100 10 90 20 80 30 70 lay 40 nta ge rce Sandy clay loam Soil A Silty clay loam lt Clay loam 30 60 f si Sandy clay 40 eo Silty clay tag 50 50 cen of c 60 Per Pe Clay 70 80 20 Loam Silty loam 90 10 Sand 0 100 Loamy sand 90 Sandy loam 80 70 50 40 60 Percentage of sand Silt 30 20 10 100 0 Figure 5.1 U.S. Department of Agriculture textural classification the fraction of soil that passes through the No. 10 sieve. Hence, if the particle-size distribution of a soil is such that a certain percentage of the soil particles is larger than 2 mm in diameter, a correction will be necessary. For example, if soil B has a particlesize distribution of 20% gravel, 10% sand, 30% silt, and 40% clay, the modified textural compositions are Sand size: 10 100 12.5% 100 20 Silt size: 30 100 37.5% 100 20 Clay size: 40 100 50.0% 100 20 On the basis of the preceding modified percentages, the USDA textural classification is clay. However, because of the large percentage of gravel, it may be called gravelly clay. Several other textural classification systems are also used, but they are no longer useful for civil engineering purposes. 5.1 Textural Classification 97 Example 5.1 Classify the following soils according to the USDA textural classification system. Particle-size distribution (%) Soil A B C D Gravel Sand Silt Clay 10 20 41 29 21 12 35 32 0 18 24 58 12 22 26 40 Solution Step 1. Calculate the modified percentages of sand, gravel, and silt as follows: Modified % sand %sand 100 100 %gravel Modified % silt %silt 100 100 %gravel Modified % clay %clay 100 100 %gravel Thus, the following table results: Particle-size distribution (%) Soil A B C D Sand Silt Clay 22.2 45.6 32.2 15.2 44.3 40.5 18 24 58 25 29.5 45.5 Step 2. With the modified composition calculated, refer to Figure 5.1 to determine the zone into which each soil falls. The results are as follows: Classification of soil A Gravelly clay loam B C D Gravelly silty clay Clay Gravelly clay Note: The word gravelly was added to the classification of soils A, B, and D because of the large percentage of gravel present in each. ■ 98 Chapter 5: Classiﬁcation of Soil 5.2 Classification by Engineering Behavior Although the textural classification of soil is relatively simple, it is based entirely on the particle-size distribution. The amount and type of clay minerals present in fine-grained soils dictate to a great extent their physical properties. Hence, the soils engineer must consider plasticity, which results from the presence of clay minerals, to interpret soil characteristics properly. Because textural classification systems do not take plasticity into account and are not totally indicative of many important soil properties, they are inadequate for most engineering purposes. Currently, two more elaborate classification systems are commonly used by soils engineers. Both systems take into consideration the particle-size distribution and Atterberg limits. They are the American Association of State Highway and Transportation Officials (AASHTO) classification system and the Unified Soil Classification System. The AASHTO classification system is used mostly by state and county highway departments. Geotechnical engineers generally prefer the Unified system. 5.3 AASHTO Classiﬁcation System The AASHTO system of soil classiﬁcation was developed in 1929 as the Public Road Administration classiﬁcation system. It has undergone several revisions, with the present version proposed by the Committee on Classiﬁcation of Materials for Subgrades and Granular Type Roads of the Highway Research Board in 1945 (ASTM designation D-3282; AASHTO method M145). The AASHTO classiﬁcation in present use is given in Table 5.1. According to this system, soil is classiﬁed into seven major groups: A-1 through A-7. Soils classiﬁed under groups A-1, A-2, and A-3 are granular materials of which 35% or less of the particles pass through the No. 200 sieve. Soils of which more than 35% pass through the No. 200 sieve are classiﬁed under groups A-4, A-5, A-6, and A-7. These soils are mostly silt and clay-type materials. This classiﬁcation system is based on the following criteria: 1. Grain size a. Gravel: fraction passing the 75-mm (3-in.) sieve and retained on the No. 10 (2-mm) U.S. sieve b. Sand: fraction passing the No. 10 (2-mm) U.S. sieve and retained on the No. 200 (0.075-mm) U.S. sieve c. Silt and clay: fraction passing the No. 200 U.S. sieve 2. Plasticity: The term silty is applied when the ﬁne fractions of the soil have a plasticity index of 10 or less. The term clayey is applied when the ﬁne fractions have a plasticity index of 11 or more. 3. If cobbles and boulders (size larger than 75 mm) are encountered, they are excluded from the portion of the soil sample from which classiﬁcation is made. However, the percentage of such material is recorded. To classify a soil according to Table 5.1, one must apply the test data from left to right. By process of elimination, the ﬁrst group from the left into which the test data ﬁt is the correct 99 5.3 AASHTO Classiﬁcation System Table 5.1 Classiﬁcation of Highway Subgrade Materials General classiﬁcation Granular materials (35% or less of total sample passing No. 200) A-1 Group classiﬁcation Sieve analysis (percentage passing) No. 10 No. 40 No. 200 Characteristics of fraction passing No. 40 Liquid limit Plasticity index Usual types of signiﬁcant constituent materials A-2 A-1-a A-1-b A-3 A-2-4 A-2-5 A-2-6 A-2-7 50 max. 30 max. 15 max. 50 max. 25 max. 51 min. 10 max. 35 max. 35 max. 35 max. 35 max. 6 max. Stone fragments, gravel, and sand 40 max. 10 max. NP Fine sand General subgrade rating Excellent to good General classiﬁcation Group classiﬁcation Sieve analysis (percentage passing) No. 10 No. 40 No. 200 Characteristics of fraction passing No. 40 Liquid limit Plasticity index Usual types of signiﬁcant constituent materials General subgrade rating a b 41 min. 40 max. 41 min. 10 max. 11 min. 11 min. Silty or clayey gravel and sand Silt-clay materials (more than 35% of total sample passing No. 200) A-4 A-5 A-6 36 min. 36 min. 36 min. A-7 A-7-5 a A-7-6 b 36 min. 40 max. 41 min. 40 max. 41 min. 10 max. 10 max. 11 min. 11 min. Silty soils Clayey soils Fair to poor For A-7-5, PI LL 30 For A-7-6, PI LL 30 classiﬁcation. Figure 5.2 shows a plot of the range of the liquid limit and the plasticity index for soils that fall into groups A-2, A-4, A-5, A-6, and A-7. To evaluate the quality of a soil as a highway subgrade material, one must also incorporate a number called the group index (GI) with the groups and subgroups of the soil. This index is written in parentheses after the group or subgroup designation. The group index is given by the equation GI 1F200 352 30.2 0.0051LL 402 4 0.011F200 15 2 1PI 10 2 (5.1) 100 Chapter 5: Classiﬁcation of Soil 70 60 Plasticity index 50 40 A-7-6 30 A-2-6 A-6 20 A-2-7 A-7-5 10 A-2-4 A-4 A-2-5 A-5 0 0 10 20 30 40 50 60 70 80 90 100 Liquid limit Figure 5.2 Range of liquid limit and plasticity index for soils in groups A-2, A-4, A-5, A-6, and A-7 where F200 percentage passing through the No. 200 sieve LL liquid limit PI plasticity index The ﬁrst term of Eq. (5.1)—that is, (F200 35)[0.2 + 0.005(LL 40)]—is the partial group index determined from the liquid limit. The second term—that is, 0.01(F200 15)(PI 10)—is the partial group index determined from the plasticity index. Following are some rules for determining the group index: 1. If Eq. (5.1) yields a negative value for GI, it is taken as 0. 2. The group index calculated from Eq. (5.1) is rounded off to the nearest whole number (for example, GI 3.4 is rounded off to 3; GI 3.5 is rounded off to 4). 3. There is no upper limit for the group index. 4. The group index of soils belonging to groups A-1-a, A-1-b, A-2-4, A-2-5, and A-3 is always 0. 5. When calculating the group index for soils that belong to groups A-2-6 and A-2-7, use the partial group index for PI, or GI 0.011F200 15 2 1PI 10 2 (5.2) In general, the quality of performance of a soil as a subgrade material is inversely proportional to the group index. 5.3 AASHTO Classiﬁcation System 101 Example 5.2 The results of the particle-size analysis of a soil are as follows: • Percent passing the No. 10 sieve 100 • Percent passing the No. 40 sieve 80 • Percent passing the No. 200 sieve 58 The liquid limit and plasticity index of the minus No. 40 fraction of the soil are 30 and 10, respectively. Classify the soil by the AASHTO system. Solution Using Table 5.1, since 58% of the soil is passing through the No. 200 sieve, it falls under silt-clay classifications—that is, it falls under group A-4, A-5, A-6, or A-7. Proceeding from left to right, it falls under group A-4. From Eq. (5.1) GI (F200 35)[0.2 + 0.005(LL 40)] + 0.01(F200 15)(PI 10) (58 35)[0.2 + 0.005(30 40)] + (0.01)(58 15)(10 10) 3.45 3 So, the soil will be classified as A-4(3). ■ Example 5.3 Ninety-five percent of a soil passes through the No. 200 sieve and has a liquid limit of 60 and plasticity index of 40. Classify the soil by the AASHTO system. Solution According to Table 5.1, this soil falls under group A-7 (proceed in a manner similar to Example 5.2). Since 40 60 30 c c PI LL this is an A-7-6 soil. GI (F200 35)[0.2 + 0.005(LL 40)] + 0.01(F200 15)(PI 10) (95 35)[0.2 + 0.005(60 40)] + (0.01)(95 15)(40 10) 42 So, the classification is A-7-6(42). ■ 102 Chapter 5: Classiﬁcation of Soil Example 5.4 Classify the following soil by the AASHTO Classification System: • • • • • Percentage passing No. 10 sieve 90 Percentage passing No. 40 sieve 76 Percentage passing No. 200 sieve 34 Liquid limit (–No. 40 fraction) 37 Plasticity index (–No. 40 fraction) 12 Solution The percentage passing through the No. 200 sieve is less than 35, so the soil is a granular material. From Table 5.1, we see that it is type A-2-6. From Eq. (5.2), GI 0.01(F200 15)(PI 10) For this soil, F200 34 and PI 12, so GI 0.01(34 15)(12 10) 0.38 0 Thus, the soil is type A-2-6(0). 5.4 ■ Uniﬁed Soil Classiﬁcation System The original form of this system was proposed by Casagrande in 1942 for use in the airﬁeld construction works undertaken by the Army Corps of Engineers during World War II. In cooperation with the U.S. Bureau of Reclamation, this system was revised in 1952. At present, it is used widely by engineers (ASTM Test Designation D-2487). The Uniﬁed classiﬁcation system is presented in Table 5.2. This system classiﬁes soils into two broad categories: 1. Coarse-grained soils that are gravelly and sandy in nature with less than 50% passing through the No. 200 sieve. The group symbols start with a preﬁx of G or S. G stands for gravel or gravelly soil, and S for sand or sandy soil. 2. Fine-grained soils are with 50% or more passing through the No. 200 sieve. The group symbols start with preﬁxes of M, which stands for inorganic silt, C for inorganic clay, or O for organic silts and clays. The symbol Pt is used for peat, muck, and other highly organic soils. Other symbols used for the classiﬁcation are: • • • • W—well graded P—poorly graded L—low plasticity (liquid limit less than 50) H—high plasticity (liquid limit more than 50) Table 5.2 Uniﬁed Soil Classiﬁcation System (Based on Material Passing 76.2-mm Sieve) Group symbol Criteria for assigning group symbols Coarse-grained soils More than 50% of retained on No. 200 sieve Fine-grained soils 50% or more passes No. 200 sieve Highly Organic Soils Gravels More than 50% of coarse fraction retained on No. 4 sieve Clean Gravels Less than 5% ﬁnes a Cu 4 and 1 Cc 3 c Cu 4 and/or 1 Cc 3 c GW GP Gravels with Fines More than 12% ﬁnes a,d PI 4 or plots below “A” line (Figure 5.3) PI 7 and plots on or above “A” line (Figure 5.3) GM GC Sands 50% or more of coarse fraction passes No. 4 sieve Clean Sands Less than 5% ﬁnes b Cu 6 and 1 Cc 3 c Cu 6 and/or 1 Cc 3 c SW SP Sands with Fines More than 12% ﬁnes b,d PI 4 or plots below “A” line (Figure 5.3) PI 7 and plots on or above “A” line (Figure 5.3) SM SC Inorganic PI 7 and plots on or above “A” line (Figure 5.3) e PI 4 or plots below “A” line (Figure 5.3) e CL ML Organic Liquid limit — oven dried 0.75; see Figure 5.3; OL zone Liquid limit — not dried OL Silts and clays Liquid limit less than 50 Silts and clays Liquid limit 50 or more Inorganic Organic PI plots on or above “A” line (Figure 5.3) PI plots below “A” line (Figure 5.3) Liquid limit — oven dried 0.75; see Figure 5.3; OH zone Liquid limit — not dried Primarily organic matter, dark in color, and organic odor a Gravels with 5 to 12% ﬁne require dual symbols: GW-GM, GW-GC, GP-GM, GP-GC. b Sands with 5 to 12% ﬁnes require dual symbols: SW-SM, SW-SC, SP-SM, SP-SC. 1D30 2 2 D60 ; Cc D10 D60 D10 d If 4 PI 7 and plots in the hatched area in Figure 5.3, use dual symbol GC-GM or SC-SM. c Cu e If 4 PI 7 and plots in the hatched area in Figure 5.3, use dual symbol CL-ML. CH MH OH Pt 103 104 Chapter 5: Classiﬁcation of Soil 70 60 Plasticity index 50 U- e lin PI 40 I eP l- in A CL or OL 30 20 CL-ML ML or OL 10 0. LL 9( 8) L 3(L CH or OH ) 20 0.7 MH or OH 0 0 10 16 20 30 40 50 60 70 80 90 100 Liquid limit Figure 5.3 Plasticity chart For proper classiﬁcation according to this system, some or all of the following information must be known: 1. Percent of gravel—that is, the fraction passing the 76.2-mm sieve and retained on the No. 4 sieve (4.75-mm opening) 2. Percent of sand—that is, the fraction passing the No. 4 sieve (4.75-mm opening) and retained on the No. 200 sieve (0.075-mm opening) 3. Percent of silt and clay—that is, the fraction ﬁner than the No. 200 sieve (0.075-mm opening) 4. Uniformity coefﬁcient (Cu ) and the coefﬁcient of gradation (Cc ) 5. Liquid limit and plasticity index of the portion of soil passing the No. 40 sieve The group symbols for coarse-grained gravelly soils are GW, GP, GM, GC, GC-GM, GW-GM, GW-GC, GP-GM, and GP-GC. Similarly, the group symbols for ﬁne-grained soils are CL, ML, OL, CH, MH, OH, CL-ML, and Pt. More recently, ASTM designation D-2487 created an elaborate system to assign group names to soils. These names are summarized in Figures 5.4, 5.5, and 5.6. In using these ﬁgures, one needs to remember that, in a given soil, • • • • 5.5 Fine fraction percent passing No. 200 sieve Coarse fraction percent retained on No. 200 sieve Gravel fraction percent retained on No. 4 sieve Sand fraction (percent retained on No. 200 sieve) (percent retained on No. 4 sieve) Summary and Comparison between the AASHTO and Uniﬁed Systems Both soil classiﬁcation systems, AASHTO and Uniﬁed, are based on the texture and plasticity of soil. Also, both systems divide the soils into two major categories, coarse grained and ﬁne grained, as separated by the No. 200 sieve. According to the 5.5 Summary and Comparison between the AASHTO and Uniﬁed Systems 105 Text not available due to copyright restrictions AASHTO system, a soil is considered ﬁne grained when more than 35% passes through the No. 200 sieve. According to the Uniﬁed system, a soil is considered ﬁne grained when more than 50% passes through the No. 200 sieve. A coarse-grained soil that has about 35% ﬁne grains will behave like a ﬁne-grained material. This is because enough ﬁne grains exist to ﬁll the voids between the coarse grains and hold them apart. In this respect, the AASHTO system appears to be more appropriate. In the AASHTO system, Text not available due to copyright restrictions 106 Text not available due to copyright restrictions 107 108 Chapter 5: Classiﬁcation of Soil Example 5.5 Figure 5.7 gives the grain-size distribution of two soils. The liquid and plastic limits of minus No. 40 sieve fraction of the soil are as follows: Soil A Soil B 30 22 26 20 Liquid limit Plastic limit Determine the group symbols and group names according to the Uniﬁed Soil Classiﬁcation System. No. 200 sieve 100 Percent finer 80 D60 Soil A 60 Soil B 40 D30 20 D10 0 1.0 0.1 0.01 Particle diameter (mm) — log scale 0.001 Figure 5.7 Particle-size distribution of two soils Solution Soil A The grain-size distribution curve (Figure 5.7) indicates that percent passing No. 200 sieve is 8. According to Table 5.2, it is a coarse-grained soil. Also, from Figure 5.7, the percent retained on No. 4 sieve is zero. Hence, it is a sandy soil. From Figure 5.7, D10 0.085 mm, D30 0.12 m, and D60 0.135 mm. Thus, Cu D60 0.135 1.59 6 D10 0.085 Cc 10.12 2 2 D230 1.25 1 D60 D10 10.135 2 10.0852 With LL 30 and PI 30 22 8 (which is greater than 7), it plots above the A-line in Figure 5.3. Hence, the group symbol is SP-SC. 5.5 Summary and Comparison between the AASHTO and Uniﬁed Systems 109 In order to determine the group name, we refer to Figure 5.4. Percentage of gravel 0 1which is 15% 2 So, the group name is poorly graded sand with clay. Soil B The grain-size distribution curve in Figure 5.7 shows that percent passing No. 200 sieve is 61 (50%); hence, it is a ﬁne-grained soil. Given: LL 26 and PI 26 20 6. In Figure 5.3, the PI plots in the hatched area. So, from Table 5.2, the group symbol is CL-ML. For group name (assuming that the soil is inorganic), we go to Figure 5.5 and obtain Plus No. 200 sieve 100 61 39 (which is greater than 30). Percentage of gravel 0; percentage of sand 100 61 39 Thus, because the percentage of sand is greater than the percentage of gravel, the soil is sandy silty clay. ■ Example 5.6 For a given soil, the following are known: • • • • Percentage passing No. 4 sieve 70 Percentage passing No. 200 sieve 30 Liquid limit 33 Plastic limit 12 Classify the soil using the Unified Soil Classification System. Give the group symbol and the group name. Solution Refer to Table 5.2. The percentage passing No. 200 sieve is 30%, which is less than 50%. So it is a coarse-grained soil. Thus, Coarse fraction 100 30 70% Gravel fraction percent retained on No. 4 sieve 100 70 30% Hence, more than 50% of the coarse fraction is passing No. 4 sieve. Thus, it is a sandy soil. Since more than 12% is passing No. 200 sieve, it is SM or SC. For this soil, PI 33 12 21 (which is greater than 7). With LL 33 and PI 21, it plots above the A-line in Figure 5.3. Thus, the group symbol is SC. For the group name, refer to Figure 5.4. Since the percentage of gravel is more than 15%, it is clayey sand with gravel. ■ 110 Chapter 5: Classiﬁcation of Soil the No. 10 sieve is used to separate gravel from sand; in the Uniﬁed system, the No. 4 sieve is used. From the viewpoint of soil-separated size limits, the No. 10 sieve is the more accepted upper limit for sand. This limit is used in concrete and highway basecourse technology. In the Uniﬁed system, the gravelly and sandy soils clearly are separated; in the AASHTO system, they are not. The A-2 group, in particular, contains a large variety of soils. Symbols like GW, SM, CH, and others that are used in the Uniﬁed system are more descriptive of the soil properties than the A symbols used in the AASHTO system. The classiﬁcation of organic soils, such as OL, OH, and Pt, is provided in the Uniﬁed system. Under the AASHTO system, there is no place for organic soils. Peats usually have a high moisture content, low speciﬁc gravity of soil solids, and low unit weight. Figure 5.8 shows the scanning electron micrographs of four peat samples collected in Wisconsin. Some of the properties of the peats are given in Table 5.3. Liu (1967) compared the AASHTO and Uniﬁed systems. The results of his study are presented in Tables 5.4 and 5.5. Figure 5.8 Scanning electron micrographs for four peat samples (After Dhowian and Edil, 1980. Copyright ASTM INTERNATIONAL. Reprinted with permission.) Table 5.3 Properties of the Peats Shown in Figure 5.8 Source of peat Middleton Waupaca County Portage Fond du Lac County Moisture content (%) kN/m 510 460 600 240 9.1 9.6 9.6 10.2 lb/ft Specific gravity, Gs Ash content (%) 57.9 61.1 61.1 64.9 1.41 1.68 1.72 1.94 12.0 15.0 19.5 39.8 Unit weight 3 3 Table 5.4 Comparison of the AASHTO System with the Uniﬁed System* Soil group in AASHTO system Most probable Possible A-1-a A-1-b A-3 A-2-4 A-2-5 A-2-6 A-2-7 A-4 A-5 A-6 A-7-5 A-7-6 GW, GP SW, SP, GM, SM SP GM, SM GM, SM GC, SC GM, GC, SM, SC ML, OL OH, MH, ML, OL CL OH, MH CH, CL SW, SP GP — GC, SC — GM, SM — CL, SM, SC — ML, OL, SC ML, OL, CH ML, OL, SC Comparable soil groups in Uniﬁed system Possible but improbable GM, SM — SW, GP GW, GP, SW, SP GW, GP, SW, SP GW, GP, SW, SP GW, GP, SW, SP GM, GC SM, GM GC, GM, SM GM, SM, GC, SC OH, MH, GC, GM, SM *After Liu (1967) Source: From A Review of Engineering Soil Classification Systems. In Highway Research Record 156, Highway Research Board, National Research Council, Washington, D.C., 1967, Table 5, p. 16. Reproduced with permission of the Transportation Research Board. Table 5.5 Comparison of the Uniﬁed System with the AASHTO System* Soil group in Uniﬁed system Most probable Possible Possible but improbable GW GP GM GC SW SP SM SC ML CL OL MH CH OH Pt A-1-a A-1-a A-1-b, A-2-4, A-2-5, A-2-7 A-2-6, A-2-7 A-1-b A-3, A-1-b A-1-b, A-2-4, A-2-5, A-2-7 A-2-6, A-2-7 A-4, A-5 A-6, A-7-6 A-4, A-5 A-7-5, A-5 A-7-6 A-7-5, A-5 — — A-1-b A-2-6 A-2-4 A-1-a A-1-a A-2-6, A-4 A-2-4, A-6, A-4, A-7-6 A-6, A-7-5, A-7-6 A-4 A-6, A-7-5, A-7-6 — A-7-5 — — A-2-4, A-2-5, A-2-6, A-2-7 A-3, A-2-4, A-2-5, A-2-6, A-2-7 A-4, A-5, A-6, A-7-5, A-7-6, A-1-a A-4, A-6, A-7-6, A-7-5 A-3, A-2-4, A-2-5, A-2-6, A-2-7 A-2-4, A-2-5, A-2-6, A-2-7 A-5, A-6, A-7-5, A-7-6, A-1-a A-7-5 — — — A-7-6 — A-7-6 — Comparable soil groups in AASHTO system *After Liu (1967) Source: From A Review of Engineering Soil Classification Systems. In Highway Research Record 156, Highway Research Board, National Research Council, Washington, D.C., 1967, Table 6, p. 17. Reproduced with permission of the Transportation Research Board. 111 112 Chapter 5: Classiﬁcation of Soil Problems 5.1 Classify the following soil using the U.S. Department of Agriculture textural classification chart. Particle-size distribution (%) 5.2 Soil Sand Silt Clay A B C D E 20 55 45 50 70 20 5 35 15 15 60 40 20 35 15 The sieve analysis of ten soils and the liquid and plastic limits of the fraction passing through the No. 40 sieve are given below. Classify the soils using the AASHTO classification system and give the group indexes. Soil No. 1 2 3 4 5 6 7 8 9 10 5.3 Sieve analysis (percent finer) No. 10 No. 40 No. 200 Liquid limit Plastic limit 98 100 100 85 92 48 87 90 100 92 80 92 88 55 75 28 62 76 78 74 50 80 65 45 62 6 30 34 8 32 38 56 37 28 43 — 32 37 — 44 29 23 22 20 28 NP 24 25 NP 35 Classify the following soils using the Unified soil classification system. Give group symbols and group names. Sieve analysis (percent finer) Soil No. No. 4 No. 200 Liquid limit Plasticity limit 1 2 3 4 5 6 7 94 100 100 100 92 60 99 3 77 86 45 48 40 76 — 63 55 36 30 26 60 NP 25 28 22 8 4 32 Comments Cu 4.48 and Cc 1.22 References 5.4 113 For an inorganic soil, the following grain-size analysis is given. U.S. Sieve No. Percent passing 4 10 20 40 80 200 100 90 64 38 18 13 For this soil, LL 23 and PL 19. Classify the soil by using a. AASHTO soil classiﬁcation system b. Uniﬁed soil classiﬁcation system Give group names and group symbols. References AMERICAN ASSOCIATION OF STATE HIGHWAY AND TRANSPORTATION OFFICIALS (1982). AASHTO Materials, Part I, Speciﬁcations, Washington, D.C. AMERICAN SOCIETY FOR TESTING AND MATERIALS (2007). Annual Book of ASTM Standards, Sec. 4, Vol. 04.08, West Conshohoken, Pa. CASAGRANDE, A. (1948). “Classiﬁcation and Identiﬁcation of Soils,” Transactions, ASCE, Vol. 113, 901–930. DHOWIAN, A. W., and EDIL, T. B. (1980). “Consolidation Behavior of Peats,” Geotechnical Testing Journal, ASTM, Vol. 3, No. 3, 105–114. LIU, T. K. (1967). “A Review of Engineering Soil Classiﬁcation Systems,” Highway Research Record No. 156, National Academy of Sciences, Washington, D.C., 1–22. 6 Soil Compaction In the construction of highway embankments, earth dams, and many other engineering structures, loose soils must be compacted to increase their unit weights. Compaction increases the strength characteristics of soils, which increase the bearing capacity of foundations constructed over them. Compaction also decreases the amount of undesirable settlement of structures and increases the stability of slopes of embankments. Smoothwheel rollers, sheepsfoot rollers, rubber-tired rollers, and vibratory rollers are generally used in the ﬁeld for soil compaction. Vibratory rollers are used mostly for the densiﬁcation of granular soils. Vibroﬂot devices are also used for compacting granular soil deposits to a considerable depth. Compaction of soil in this manner is known as vibroﬂotation. This chapter discusses in some detail the principles of soil compaction in the laboratory and in the ﬁeld. 6.1 Compaction— General Principles Compaction, in general, is the densiﬁcation of soil by removal of air, which requires mechanical energy. The degree of compaction of a soil is measured in terms of its dry unit weight. When water is added to the soil during compaction, it acts as a softening agent on the soil particles. The soil particles slip over each other and move into a densely packed position. The dry unit weight after compaction ﬁrst increases as the moisture content increases. (See Figure 6.1.) Note that at a moisture content w 0, the moist unit weight (g) is equal to the dry unit weight (gd), or g gd1w02 g1 When the moisture content is gradually increased and the same compactive effort is used for compaction, the weight of the soil solids in a unit volume gradually increases. For example, at w w1 , g g2 However, the dry unit weight at this moisture content is given by gd1ww12 gd1w02 ¢gd 114 6.2 Standard Proctor Test 115 Moist unit weight, g gg1 gd ( 0) g2 ∆gd 0 1 2 Moisture content, Soil solid Water Figure 6.1 Principles of compaction Beyond a certain moisture content w w2 (Figure 6.1), any increase in the moisture content tends to reduce the dry unit weight. This phenomenon occurs because the water takes up the spaces that would have been occupied by the solid particles. The moisture content at which the maximum dry unit weight is attained is generally referred to as the optimum moisture content. The laboratory test generally used to obtain the maximum dry unit weight of compaction and the optimum moisture content is called the Proctor compaction test (Proctor, 1933). The procedure for conducting this type of test is described in the following section. 6.2 Standard Proctor Test In the Proctor test, the soil is compacted in a mold that has a volume of 944 cm3 (301 ft3). The diameter of the mold is 101.6 mm (4 in.). During the laboratory test, the mold is attached to a baseplate at the bottom and to an extension at the top (Figure 6.2a). The soil is mixed with varying amounts of water and then compacted in three equal layers by a hammer (Figure 6.2b) that delivers 25 blows to each layer. The hammer has a mass of 2.5 kg (6.5 lb) and has a drop of 30.5 mm (12 in.). Figure 6.2c is a photograph of the laboratory equipment required for conducting a standard Proctor test. For each test, the moist unit weight of compaction, g, can be calculated as g W V1m2 where W weight of the compacted soil in the mold V1m2 volume of the mold 3 944 cm3 1 301 ft3 2 4 (6.1) 114.3 mm diameter (4.5 in.) Extension Drop 304.8 mm (12 in.) 116.43 mm (4.584 in.) Weight of hammer 2.5 kg (mass 5.5 lb) 101.6 mm diameter (4 in.) 50.8 mm (2 in.) (a) (b) (c) 116 Figure 6.2 Standard Proctor test equipment: (a) mold; (b) hammer; (c) photograph of laboratory equipment used for test (Courtesy of Braja M. Das, Henderson, Nevada) 6.2 Standard Proctor Test 117 For each test, the moisture content of the compacted soil is determined in the laboratory. With the known moisture content, the dry unit weight can be calculated as gd 1 g w 1% 2 (6.2) 100 where w (%) percentage of moisture content. The values of gd determined from Eq. (6.2) can be plotted against the corresponding moisture contents to obtain the maximum dry unit weight and the optimum moisture content for the soil. Figure 6.3 shows such a plot for a silty-clay soil. The procedure for the standard Proctor test is elaborated in ASTM Test Designation D-698 (ASTM, 2007) and AASHTO Test Designation T-99 (AASHTO, 1982). For a given moisture content w and degree of saturation S, the dry unit weight of compaction can be calculated as follows. From Chapter 3 [Eq. (3.16)], for any soil, gd Gsgw 1e where Gs specific gravity of soil solids gw unit weight of water e void ratio 125 19.5 19.0 Zero-air-void curve (Gs 2.69) 18.5 Maximum gd 115 18.0 17.5 Dry unit weight, gd (kN/m3 ) Dry unit weight, gd (lb/ft3 ) 120 110 17.0 Optimum moisture content 105 5 10 Moisture content, 16.5 15 (%) 18 Figure 6.3 Standard Proctor compaction test results for a silty clay 118 Chapter 6: Soil Compaction and, from Eq. (3.18), Se Gsw or e Gsw S Thus, gd Gsgw Gsw 1 S (6.3) For a given moisture content, the theoretical maximum dry unit weight is obtained when no air is in the void spaces—that is, when the degree of saturation equals 100%. Hence, the maximum dry unit weight at a given moisture content with zero air voids can be obtained by substituting S 1 into Eq. (6.3), or gzav Gsgw 1 wGs gw 1 w Gs (6.4) where gzav zero-air-void unit weight. To obtain the variation of gzav with moisture content, use the following procedure: 1. 2. 3. 4. Determine the speciﬁc gravity of soil solids. Know the unit weight of water (gw). Assume several values of w, such as 5%, 10%, 15%, and so on. Use Eq. (6.4) to calculate gzav for various values of w. Figure 6.3 also shows the variation of gzav with moisture content and its relative location with respect to the compaction curve. Under no circumstances should any part of the compaction curve lie to the right of the zero-air-void curve. 6.3 Factors Affecting Compaction The preceding section showed that moisture content has a strong inﬂuence on the degree of compaction achieved by a given soil. Besides moisture content, other important factors that affect compaction are soil type and compaction effort (energy per unit volume). The importance of each of these two factors is described in more detail in the following two sections. Effect of Soil Type The soil type—that is, grain-size distribution, shape of the soil grains, speciﬁc gravity of soil solids, and amount and type of clay minerals present—has a great inﬂuence on the maximum dry unit weight and optimum moisture content. Figure 6.4 shows typical 6.3 Factors Affecting Compaction 120 119 19.0 18.5 Sandy silt 115 Silty clay 17.5 110 Highly plastic clay 17.0 Poorly graded sand Dry unit weight, gd (kN/m3 ) Dry unit weight, gd (lb/ft3 ) 18.0 16.5 105 16.0 15.5 100 5 10 Moisture content, 15 (%) 20 Figure 6.4 Typical compaction curves for four soils (ASTM D-698) compaction curves obtained from four soils. The laboratory tests were conducted in accordance with ASTM Test Designation D-698. Note also that the bell-shaped compaction curve shown in Figure 6.3 is typical of most clayey soils. Figure 6.4 shows that for sands, the dry unit weight has a general tendency ﬁrst to decrease as moisture content increases and then to increase to a maximum value with further increase of moisture. The initial decrease of dry unit weight with increase of moisture content can be attributed to the capillary tension effect. At lower moisture contents, the capillary tension in the pore water inhibits the tendency of the soil particles to move around and be compacted densely. Lee and Suedkamp (1972) studied compaction curves for 35 soil samples. They observed that four types of compaction curves can be found. These curves are shown in Figure 6.5. The following table is a summary of the type of compaction curves encountered in various soils with reference to Figure 6.5. Type of compaction curve (Figure 6.5) Description of curve A B C Bell shaped 1-1/2 peak Double peak D Odd shaped Liquid limit Between 30 to 70 Less than 30 Less than 30 and those greater than 70 Greater than 70 120 Chapter 6: Soil Compaction Type B One and one-half peaks Dry unit weight Dry unit weight Type A Bell shaped (a) (b) Type C Double peak Type D Odd shaped Figure 6.5 Various types of compaction curves encountered in soils Dry unit weight Moisture content Dry unit weight Moisture content Moisture content Moisture content (c) (d) Effect of Compaction Effort The compaction energy per unit volume used for the standard Proctor test described in Section 6.2 can be given as Number Number Weight Height of £of blows≥ £ of ≥ £ of ≥ £ drop of ≥ per layer layers hammer hammer E Volume of mold (6.5) 6.3 Factors Affecting Compaction 121 or, in SI units, 125 2 132 a E 2.5 9.81 kN b 10.305 m2 1000 594 kN-m/m3 600 kN-m/m3 944 106 m3 In English units, E 1252 132 15.52 11 2 12,375 ft-lb/ft3 12,400 ft-lb/ft3 1 a b 30 If the compaction effort per unit volume of soil is changed, the moisture–unit weight curve also changes. This fact can be demonstrated with the aid of Figure 6.6, which shows four compaction curves for a sandy clay. The standard Proctor mold and hammer were used to obtain these compaction curves. The number of layers of soil used for compaction was three 125 19.85 Zero-air-void curve (Gs 2.7) 19.00 115 Line of optimum 4 50 blows / layer 3 30 blows / layer 18.00 110 2 25 blows / layer Dry unit weight, gd (kN/m3 ) Dry unit weight, gd (lb/ft3 ) 120 17.00 1 20 blows / layer 105 16.00 100 15.20 10 12 14 16 18 20 22 24 Moisture content, (%) Sandy clay: Liquid limit 31 Plastic limit 26 Figure 6.6 Effect of compaction energy on the compaction of a sandy clay 122 Chapter 6: Soil Compaction for all cases. However, the number of hammer blows per each layer varied from 20 to 50, which varied the energy per unit volume. From the preceding observation and Figure 6.6, we can see that 1. As the compaction effort is increased, the maximum dry unit weight of compaction is also increased. 2. As the compaction effort is increased, the optimum moisture content is decreased to some extent. The preceding statements are true for all soils. Note, however, that the degree of compaction is not directly proportional to the compaction effort. 6.4 Modiﬁed Proctor Test With the development of heavy rollers and their use in ﬁeld compaction, the standard Proctor test was modiﬁed to better represent ﬁeld conditions. This revised version sometimes is referred to as the modiﬁed Proctor test (ASTM Test Designation D-1557 and AASHTO Test Designation T-180). For conducting the modiﬁed Proctor test, the same mold is used with a volume of 944 cm3 (1/30 ft3), as in the case of the standard Proctor test. However, the soil is compacted in ﬁve layers by a hammer that has a mass of 4.54 kg (10 lb). The drop of the hammer is 457 mm (18 in.). The number of hammer blows for each layer is kept at 25 as in the case of the standard Proctor test. Figure 6.7 shows a comparison between the hammers used in standard and modified Proctor tests. The compaction energy for this type of compaction test can be calculated as 2700 kN-m/m3 (56,000 ft-lb/lb3). Because it increases the compactive effort, the modiﬁed Proctor test results in an increase in the maximum dry unit weight of the soil. The increase in the maximum dry unit weight is accompanied by a decrease in the optimum moisture content. In the preceding discussions, the speciﬁcations given for Proctor tests adopted by ASTM and AASHTO regarding the volume of the mold and the number of blows are generally those adopted for ﬁne-grained soils that pass through the U.S. No. 4 sieve. However, under each test designation, there are three suggested methods that reﬂect the mold size, the number of blows per layer, and the maximum particle size in a soil aggregate used for testing. A summary of the test methods is given in Table 6.1. Omar, et al. (2003) recently presented the results of modiﬁed Proctor compaction tests on 311 soil samples. Of these samples, 45 were gravelly soil (GP, GP-GM, GW, GWGM, and GM), 264 were sandy soil (SP, SP-SM, SW-SM, SW, SC-SM, SC, and SM), and two were clay with low plasticity (CL). All compaction tests were conducted using ASTM 1557 method C to avoid over-size correction. Based on the tests, the following correlations were developed. rd1max2 1kg/m3 2 34,804,574Gs 195.551LL2 2 156,9711R#4 2 0.5 9,527,830 4 0.5 ln1wopt 2 1.195 10 4 1LL2 2 1.964Gs 6.617 105 1R#42 7.651 (6.6) (6.7) 6.4 Modiﬁed Proctor Test 123 Figure 6.7 Comparison between standard Proctor hammer (left) and modified Proctor hammer (right) (Courtesy of Braja M. Das, Henderson, Nevada) where rd(max) maximum dry density (kg/m3) wopt optimum moisture content(%) Gs speciﬁc gravity of soil solids LL liquid limit, in percent R#4 percent retained on No. 4 sieve Gurtug and Sridharan (2004) proposed correlations for optimum moisture content and maximum dry unit weight with the plastic limit (PL) of cohesive soils. These correlations can be expressed as: wopt 1% 2 3 1.95 0.381log CE2 4 1PL2 gd 1max2 1kN/m3 2 22.68e0.0183wopt1%2 (6.8) (6.9) 124 Chapter 6: Soil Compaction Table 6.1 Summary of Standard and Modiﬁed Proctor Compaction Test Speciﬁcations (ASTM D-698 and D-1557) Description Physical data for the tests Standard Proctor test Modiﬁed Proctor test Method A Method B Method C Material Passing No. 4 sieve Passing 9.5 mm 1 38 in.2 sieve Passing 19 mm 1 34 in.2 sieve Use Used if 20% or less by weight of material is retained on No. 4 (4.75 mm) sieve Used if more than 20% by weight of material is retained on No. 4 (4.75 mm) sieve and 20% or less by weight of material is retained on 9.5 mm 1 38 in.2 sieve Used if more than 20% by weight of material is retained on 9.5 mm 1 38 in.2 sieve and less than 30% by weight of material is retained on 19 mm 1 34 in.2 sieve Mold volume 944 cm3 ( 301 ft3) 944 cm3 ( 301 ft3) 1 2124 cm3 ( 13.33 ft3) Mold diameter 101.6 mm (4 in.) 101.6 mm (4 in.) 152.4 mm (6 in.) Mold height 116.4 mm (4.584 in.) 116.4 mm (4.584 in.) 116.4 mm (4.584 in.) Weight of hammer 24.4 N (5.5 lb) 24.4 N (5.5 lb) 24.4 N (5.5 lb) Height of drop 305 mm (12 in.) 305 mm (12 in.) 305 mm (12 in.) Number of soil layers 3 3 3 Number of blows/layer 25 25 56 Weight of hammer 44.5 N (10 lb) 44.5 N (10 lb) 44.5 N (10 lb) Height of drop 457 mm (18 in.) 457 mm (18 in.) 457 mm (18 in.) Number of soil layers 5 5 5 Number of blows/layer 25 25 56 where PL plastic limit (%) CE compaction energy (kN-m/m3 ) For modiﬁed Proctor test, CE 2700 kN/m3. Hence, wopt 1% 2 0.651PL2 and gd 1max2 1kN/m3 2 22.68e0.0121PL2 6.4 Modiﬁed Proctor Test 125 Example 6.1 The laboratory test results of a standard Proctor test are given in the following table. Volume of mold (ft 3) 1 30 1 30 1 30 1 30 1 30 1 30 Weight of moist soil in mold (lb) Moisture content, w (%) 3.78 4.01 4.14 4.12 4.01 3.90 10 12 14 16 18 20 a. Determine the maximum dry unit weight of compaction and the optimum moisture content. b. Calculate and plot d versus the moisture content for degree of saturation, S 80, 90, and 100% (i.e., zav). Given: Gs 2.7. Solution Part a The following table can be prepared. Volume of mold V (ft 3) Weight of soil, W (lb) Moist unit weight, G (lb/ft 3)a Moisture content, w (%) Dry unit weight, Gd (lb/ft 3) b 3.78 4.01 4.14 4.12 4.01 3.90 113.4 120.3 124.2 123.6 120.3 117.0 10 12 14 16 18 20 103.1 107.4 108.9 106.6 101.9 97.5 1 30 1 30 1 30 1 30 1 30 1 30 W V a g b gd g w% 1 100 The plot of d versus w is shown at the bottom of Figure 6.8. From the plot, we see that the maximum dry unit weight, d(max) 109 lb/ft3 and the optimum moisture content is 14.4%. ■ 126 Chapter 6: Soil Compaction 140 Dry unit weight, gd (lb/ft3 ) gzav 130 120 110 S 100% gd (max) S 90% 100 90 S80% opt 0 8 10 12 14 16 Moisture content, (%) 18 20 Figure 6.8 Moisture content–unit weight curves Part b From Eq. (6.3), gd Gsgw Gsw 1 S The following table can be prepared. d (lb/ft3) Gs w (%) S ⴝ 80% S ⴝ 90% S ⴝ 100% 2.7 2.7 2.7 2.7 2.7 2.7 2.7 8 10 12 14 16 18 20 132.7 126.0 119.9 114.4 109.4 104.8 100.6 135.9 129.6 123.9 118.6 113.8 109.4 105.3 138.6 132.7 127.3 122.3 117.7 113.4 109.4 The plot of d versus w for the various degrees of saturation is also shown in Figure 6.8. 6.5 Structure of Compacted Clay Soil 127 Example 6.2 For a granular soil, the following are given: • Gs 2.6 • Liquid limit on the fraction passing No. 40 sieve 20 • Percent retained on No. 4 sieve 20 Using Eqs. (6.6) and (6.7), estimate the maximum dry density of compaction and the optimum moisture content based on the modified Proctor test. Solution From Eq. (6.6), d(max) (kg/m3) [4,804,574Gs 195.55(LL)2 + 156,971(R#4)0.5 9,527,830]0.5 [4,804,574(2.6) 195.55(20)2 + 156,971(20)0.5 9,527,830]0.5 1894 kg/m3 From Eq. (6.7), In(wopt) 1.195 104(LL)2 1.964Gs 6.617 105(R#4) + 7,651 1.195 104(20)2 1.964(2.6) 6.617 105(20) + 7,651 2.591 wopt 13.35% ■ 6.5 Structure of Compacted Clay Soil Lambe (1958a) studied the effect of compaction on the structure of clay soils, and the results of his study are illustrated in Figure 6.9. If clay is compacted with a moisture content on the dry side of the optimum, as represented by point A, it will possess a ﬂocculent structure. This type of structure results because, at low moisture content, the diffuse double layers of ions surrounding the clay particles cannot be fully developed; hence, the interparticle repulsion is reduced. This reduced repulsion results in a more random particle orientation and a lower dry unit weight. When the moisture content of compaction is increased, as shown by point B, the diffuse double layers around the particles expand, which increases the repulsion between the clay particles and gives a lower degree of ﬂocculation and a higher dry unit weight. A continued increase in moisture content from B to C expands the double layers more. This expansion results in a continued increase of repulsion between the particles and thus a still greater degree of particle orientation and a more or less dispersed structure. However, the dry unit weight decreases because the added water dilutes the concentration of soil solids per unit volume. At a given moisture content, higher compactive effort yields a more parallel orientation to the clay particles, which gives a more dispersed structure. The particles are closer and the soil has a higher unit weight of compaction. This phenomenon can be seen by comparing point A with point E in Figure 6.9. Figure 6.10 shows the variation in the degree of particle orientation with molding water content for compacted Boston blue clay. Works of Seed and Chan (1959) have shown similar results for compacted kaolin clay. High compactive effort E D Compacted density B C A Low compactive effort Figure 6.9 Effect of compaction on structure of clay soils (Redrawn after Lambe, 1958a. With permission from ASCE.) Molding water content Particle orientation (%) 100 a Parallel b 75 50 25 Random 0 10 12 14 16 18 20 24 22 18 114 17 106 a 102 16 b 98 15 94 90 14 10 128 12 14 16 18 20 Molding moisture content (%) Higher compaction energy 22 Lower compaction energy 24 Dry unit weight, gd (kN/m3 ) Dry unit weight, gd (lb/ft3 ) 110 Figure 6.10 Orientation against moisture content for Boston blue clay (After Lambe, 1958a. With permission from ASCE.) 6.6 Effect of Compaction on Cohesive Soil Properties Effect of Compaction on Cohesive Soil Properties Compaction induces variations in the structure of cohesive soils. Results of these structural variations include changes in hydraulic conductivity, compressibility, and strength. Figure 6.11 shows the results of permeability tests (Chapter 7) on Jamaica sandy clay. The samples used for the tests were compacted at various moisture contents by the same compactive effort. The hydraulic conductivity, which is a measure of how easily water ﬂows through soil, decreases with the increase of moisture Hydraulic conductivity, k (cm/sec) 104 105 106 107 108 12 13 14 15 16 17 18 19 20.44 130 20.00 122 19.00 118 18.00 114 110 Dry unit weight, gd (kN/m3 ) 126 Dry unit weight, gd (lb/ft3 ) 6.6 129 17.29 12 13 14 15 16 Moisture content (%) 17 18 19 Shows change in moisture and unit weight from permeation Figure 6.11 Effect of compaction on hydraulic conductivity of clayey soil (Redrawn after Lambe, 1958b. With permission from ASCE.) 130 Chapter 6: Soil Compaction content. It reaches a minimum value at approximately the optimum moisture content. Beyond the optimum moisture content, the hydraulic conductivity increases slightly. The high value of the hydraulic conductivity on the dry side of the optimum moisture content is due to the random orientation of clay particles that results in larger pore spaces. One-dimensional compressibility characteristics (Chapter 11) of clay soils compacted on the dry side of the optimum and compacted on the wet side of the optimum are shown in Figure 6.12. Under lower pressure, a soil that is compacted on the wet side of the optimum is more compressible than a soil that is compacted on the dry side of the optimum. This is shown in Figure 6.12a. Under high pressure, the trend is exactly the opposite, and this is shown in Figure 6.12b. For samples compacted on the dry side of the optimum, the pressure tends to orient the particles normal to its Void ratio, e Dry compacted or undisturbed sample Wet compacted or remolded sample Pressure (natural scale) (a) Low-pressure consolidation Void ratio, e Dry compacted or undisturbed sample Wet compacted or remolded sample Pressure (log scale) (b) High-pressure consolidation Figure 6.12 Effect of compaction on onedimensional compressibility of clayey soil (Redrawn after Lambe, 1958b. With permission from ASCE.) 131 6.6 Effect of Compaction on Cohesive Soil Properties direction of application. The space between the clay particles is also reduced at the same time. However, for samples compacted on the wet side of the optimum, pressure merely reduces the space between the clay particles. At very high pressure, it is possible to have identical structures for samples compacted on the dry and wet sides of optimum. The strength of compacted clayey soils (Chapter 12) generally decreases with the molding moisture content. This is shown in Figure 6.13, which is the result of several unconfined compression-strength tests on compacted specimens of a silty clay soil. The test specimens were prepared by kneading compaction. The insert in Figure 6.13 shows the relationship between dry unit weight and moisture content for the soil. Note that specimens A, B, and C have been compacted, respectively, on the dry side of the optimum moisture content, near optimum moisture content, and on the wet side of the optimum moisture content. The unconfined compression strength, qu, is greatly reduced for the specimen compacted on the wet side of the optimum moisture content. Some expansive clays in the field do not stay compacted, but expand upon entry of water and shrink with loss of moisture. This shrinkage and swelling of soil can cause serious distress to the foundations of structures. The nature of variation of expansion and shrinkage of expansive clay is shown in Figure 6.14. Laboratory observations such as this will help soils engineers to adopt a moisture content for compaction to minimize swelling and shrinkage. 300 16 Dry unit weight (kN/m3 ) Axial stress (kN/m2 ) qu 200 A B 14 C 12 A 10 10 qu 14 18 22 Moisture content (%) B 100 qu C 0 0 5 10 Axial strain (%) 15 Figure 6.13 Unconfined compression test on compacted specimens of a silty clay 20 26 132 Chapter 6: Soil Compaction Swelling due to wetting Volume change (%) Shrinkage due to drying Dry unit weight Molding moisture content (%) Moisture content (%) 6.7 Figure 6.14 Nature of variation of swelling and shrinkage of expansive clay Field Compaction Compaction Equipment Most of the compaction in the ﬁeld is done with rollers. The four most common types of rollers are 1. 2. 3. 4. Smooth-wheel rollers (or smooth-drum rollers) Pneumatic rubber-tired rollers Sheepsfoot rollers Vibratory rollers Smooth-wheel rollers (Figure 6.15) are suitable for proof rolling subgrades and for ﬁnishing operation of ﬁlls with sandy and clayey soils. These rollers provide 100% coverage under the wheels, with ground contact pressures as high as 310 to 380 kN/m2 (45 to 55 lb/in2). They are not suitable for producing high unit weights of compaction when used on thicker layers. Pneumatic rubber-tired rollers (Figure 6.16) are better in many respects than the smooth-wheel rollers. The former are heavily loaded with several rows of tires. These tires are closely spaced—four to six in a row. The contact pressure under the tires can range from 600 to 700 kN/m2 (85 to 100 lb/in2), and they produce about 70 to 80% coverage. Pneumatic rollers can be used for sandy and clayey soil compaction. Compaction is achieved by a combination of pressure and kneading action. Figure 6.15 Smooth-wheel roller (Ingram Compaction LLC) Figure 6.16 Pneumatic rubber-tired roller (Ingram Compaction LLC) 133 134 Chapter 6: Soil Compaction Figure 6.17 Sheepsfoot roller (SuperStock/Alamy) Sheepsfoot rollers (Figure 6.17) are drums with a large number of projections. The area of each projection may range from 25 to 85 cm2 ( 4 to 13 in2). These rollers are most effective in compacting clayey soils. The contact pressure under the projections can range from 1400 to 7000 kN/m2 (200 to 1000 lb/in2). During compaction in the ﬁeld, the initial passes compact the lower portion of a lift. Compaction at the top and middle of a lift is done at a later stage. Vibratory rollers are extremely efﬁcient in compacting granular soils. Vibrators can be attached to smooth-wheel, pneumatic rubber-tired, or sheepsfoot rollers to provide vibratory effects to the soil. Figure 6.18 demonstrates the principles of vibratory rollers. The vibration is produced by rotating off-center weights. Handheld vibrating plates can be used for effective compaction of granular soils over a limited area. Vibrating plates are also gang-mounted on machines. These plates can be used in less restricted areas. Factors Affecting Field Compaction In addition to soil type and moisture content, other factors must be considered to achieve the desired unit weight of compaction in the ﬁeld. These factors include the thickness of Vibrator Vibrator Off-center rotating weight Off-center rotating weight Figure 6.18 Principles of vibratory rollers 6.7 Field Compaction 135 lift, the intensity of pressure applied by the compacting equipment, and the area over which the pressure is applied. These factors are important because the pressure applied at the surface decreases with depth, which results in a decrease in the degree of soil compaction. During compaction, the dry unit weight of soil also is affected by the number of roller passes. Figure 6.19 shows the growth curves for a silty clay soil. The dry unit weight of a soil at a given moisture content increases to a certain point with the number of roller passes. Beyond this point, it remains approximately constant. In most cases, about 10 to 15 roller passes yield the maximum dry unit weight economically attainable. Figure 6.20a shows the variation in the unit weight of compaction with depth for a poorly graded dune sand for which compaction was achieved by a vibratory drum roller. Vibration was produced by mounting an eccentric weight on a single rotating shaft within the drum cylinder. The weight of the roller used for this compaction was 55.6 kN (12.5 kip), and the drum diameter was 1.19 m (47 in). The lifts were kept at 2.44 m (8 ft). Note that, at any given depth, the dry unit weight of compaction increases with the number of roller passes. However, the rate of increase in unit weight gradually decreases after about 15 passes. Another fact to note from Figure 6.20a is the variation of dry unit weight with depth for any given number of roller passes. The dry unit weight and hence the relative density, Dr, reach maximum values at a depth of about 0.5 m (1.5 ft) and gradually decrease at lesser depths. This decrease occurs because of the lack of conﬁning pressure toward the surface. Once the relationship between depth and relative density (or dry unit weight) for a given soil with a given number of roller passes is determined, estimating the approximate thickness of each lift is easy. This procedure is shown in Figure 6.20b (D’Appolonia, Whitman, and D’Appolonia, 1969). 120 18 Moisture content 17.8% 110 Dry unit weight, gd (lb/ft3 ) 16 100 15 90 14 13 80 12 70 11 0 8 16 24 Number of roller passes Silty clay: Liquid limit 43 Plasticity index 19 32 Dry unit weight, gd (kN/m3 ) 17 Moisture content 11.6% Figure 6.19 Growth curves for a silty clay— relationship between dry unit weight and number of passes of 84.5 kN (19 kip) three-wheel roller when the soil is compacted in 229 mm (9 in) loose layers at different moisture contents (From Full-Scale Field Tests on 3-Wheel Power Rollers. In Highway Research Bulletin 272, Highway Research Board, National Research Council, Washington, D.C., 1960, Figure 15, p. 23. Reproduced with permission of the Transportation Research Board.) 136 Chapter 6: Soil Compaction Dry unit weight, gd (lb/ft3 ) 100 104 108 2 3 1.00 1.50 2 4 5 15 5 45 1.83 6 16.50 17.00 15.72 16.00 Dry unit weight, gd (kN/m3 ) 0.00 0 Compaction after 5 roller passes 1 1 0.50 2 3 1.00 0 2 0.46 m (18 in.) 3 1.00 4 1.50 1.83 Depth (ft) 0.50 Depth (m) Depth (m) 1 Depth (ft) Number of roller passes 0.50 0.00 Depth (m) 0 Depth (ft) 0.00 4 5 1.50 6 1.83 50 60 70 80 90 Relative density, Dr (%) 5 6 50 60 70 80 90 Relative density, Dr (%) (b) (a) Figure 6.20 (a) Vibratory compaction of a sand—variation of dry unit weight with number of roller passes; thickness of lift 2.44 m (8 ft); (b) estimation of compaction lift thickness for minimum required relative density of 75% with ﬁve roller passes (After D' Appolonia, Whitman, and D' Appolonia, 1969. With permission from ASCE.) 6.8 Speciﬁcations for Field Compaction In most speciﬁcations for earthwork, the contractor is instructed to achieve a compacted ﬁeld dry unit weight of 90 to 95% of the maximum dry unit weight determined in the laboratory by either the standard or modiﬁed Proctor test. This is a speciﬁcation for relative compaction, which can be expressed as R1% 2 gd1field2 gd1max—lab2 100 (6.10) where R relative compaction. For the compaction of granular soils, speciﬁcations sometimes are written in terms of the required relative density Dr or the required relative compaction. Relative density should not be confused with relative compaction. From Chapter 3, we can write Dr c gd1field2 gd1min2 gd1max2 gd1min2 dc gd1max2 gd1field2 d (6.11) Comparing Eqs. (6.10) and (6.11), we see that R R0 1 Dr 11 R0 2 (6.12) 6.8 Speciﬁcations for Field Compaction 137 where R0 gd1min2 gd1max2 (6.13) On the basis of observation of 47 soil samples, Lee and Singh (1971) devised a correlation between R and Dr for granular soils: R 80 0.2Dr (6.14) The speciﬁcation for ﬁeld compaction based on relative compaction or on relative density is an end-product speciﬁcation. The contractor is expected to achieve a minimum dry unit weight regardless of the ﬁeld procedure adopted. The most economical compaction condition can be explained with the aid of Figure 6.21. The compaction curves A,B, and C are for the same soil with varying compactive effort. Let curve A represent the conditions of maximum compactive effort that can be obtained from the existing equipment. Let the contractor be required to achieve a minimum dry unit weight of gd(ﬁeld) Rgd(max) . To achieve this, the contractor must ensure that the moisture content w falls between w1 and w2 . As can be seen from compaction curve C, the required gd(ﬁeld) can be achieved with a lower compactive effort at a moisture content w w3 . However, for most practical conditions, a compacted ﬁeld unit weight of gd(ﬁeld) Rgd(max) cannot be achieved by the minimum compactive effort. Hence, equipment with slightly more than the minimum compactive effort should be used. The compaction curve B represents this condition. Now we can see from Figure 6.21 that the most economical moisture content is between w3 and w4 . Note that w w4 is the optimum moisture content for curve A, which is for the maximum compactive effort. The concept described in the preceding paragraph, along with Figure 6.21, is attributed historically to Seed (1964) and is elaborated on in more detail in Holtz and Kovacs (1981). Table 6.2 gives some of the requirements to achieve 95-to-100% relative compaction (based on standard Proctor maximum dry unit weight) by various ﬁeld compaction equipment (U.S. Department of Navy, 1971). Line of optimum Dry unit weight, gd gd(max) Rgd(max) B A C 1 4 3 2 Moisture content, Figure 6.21 Most economical compaction condition 138 Table 6.2 Requirements to Achieve R 95 to 100% (based on standard Proctor maximum dry unit-weight)* Requirements for compaction of 95 to 100% standard Proctor maximum dry unit weight Equipment type Sheepsfoot rollers Applicability For ﬁne-grained soils or dirty coarse-grained soils with more than 20% passing the No. 200 sieve. Not suitable for clean, coarse-grained soils. Particularly appropriate for compaction of impervious zone for earth dam or linings where bonding of lifts is important. Compacted lift thickness Passes or coverages Dimensions and weight of equipment 150 mm (6 in.) 4 to 6 passes for ﬁnegrained soil 6 to 8 passes for coarsegrained soil Foot contact area 30 to 80 cm2 (5 to 12 in2) Foot contact pressures 1700 to 3400 kN/m2 (250 to 500 psi) Fine-grained soil, PI 30 45 to 90 cm2 (7 to 14 in2) 1400 to 2800 kN/m2 (200 to 400 psi) Coarsegrained soil 65 to 90 cm2 (10 to 14 in2) 1000 to 1800 kN/m2 (150 to 250 psi) Soil type Fine-grained soil, PI 30 Efﬁcient compaction of soils on the wet side of the optimum requires less contact pressure than that required by the same soils at lower moisture contents. Rubbertired rollers For clean, coarse-grained soils with 4 to 8% passing the No. 200 sieve. 250 mm (10 in) 3 to 5 coverages Tire inﬂation pressures of 400 to 550 kN/m2 (60 to 80 psi) for clean granular material or base course and subgrade compaction. Wheel load, 80 to 110 kN (18,000 to 25,000 lb.) For ﬁne-grained soils or wellgraded, dirty, coarse-grained soils with more than 8% passing the No. 200 sieve. 150 to 200 mm (6 to 8 in) 4 to 6 coverages Tire inﬂation pressures in excess of 450 kN/m2 (65 psi) for ﬁne-grained soils of high plasticity. For uniform clean sands or silty ﬁne sands, use larger-size tires with pressures of 280 to 350 kN/m2 (40 to 50 psi.) Possible variations in equipment For earth dam, highway, and airﬁeld work, a drum of 1.5 m (60-in.) diameter, loaded to 45 to 90 kN per linear meter (1.5 to 3 tons per linear ft) of drum is generally utilized. For smaller projects, a 1 m (40-in.) diameter drum, loaded to 22 to 50 kN per linear meter (0.75 to 1.75 tons per linear ft) of drum, is used. Foot contact pressure should be regulated to avoid shearing the soil on the third or fourth pass. A wide variety of rubber-tired compaction equipment is available. For cohesive soils, light-wheel loads, such as provided by wobble-wheel equipment, may be substituted for heavy-wheel loads if lift thickness is decreased. For cohesionless soils, large-size tires are desirable to avoid shear and rutting. Appropriate for subgrade or base course compaction of well-graded sand-gravel mixtures. 200 to 300 mm (8 to 12 in.) 4 coverages Tandem-type rollers for base-course or sub-grade compaction, 90 to 135 kN (10 to 15 ton) weight, 53 to 88 kN per linear meter (300 to 500 lb per linear in.) of width of rear roller. May be used for ﬁnegrained soils other than in earth dams. Not suitable for clean, well-graded sands or silty, uniform sands. 150 to 200 mm (6 to 8 in.) 6 coverages Three-wheel roller for compaction of ﬁne-grained soil; weights from 45 to 55 kN (5 to 6 tons) for materials of low plasticity to 90 kN (10 tons) for materials of high plasticity. Vibrating baseplate compactors For coarse-grained soils with less than about 12% passing the No. 200 sieve. Best suited for materials with 4 to 8% passing the No. 200 sieve, placed thoroughly wet. 200 to 250 mm (8 to 10 in.) 3 coverages Single pads or plates should weigh no less than 0.9 kN (200 lb). May be used in tandem where working space is available. For clean, coarsegrained soil, vibration frequency should be no less than 1600 cycles per minute. Vibrating pads or plates are available, hand-propelled or self-propelled, single or in gangs, with width of coverage from 0.45 to 4.5 m (1 12 to 15 ft). Various types of vibrating-drum equipment should be considered for compaction in large areas. Crawler tractor Best suited for coarsegrained soils with less than 4 to 8% passing the No. 200 sieve, placed thoroughly wet. 250 to 300 mm (10 to 12 in.) 3 to 4 coverages No smaller than D8 tractor with blade, 153 kN (34,500 lb) weight, for high compaction. Tractor weights up to 265 kN (60,000 lb.) Power tamper or rammer For difﬁcult access, trench backﬁll. Suitable for all inorganic soils. 100 to 150 mm (4 to 6 in.) for silt or clay; 150 mm (6 in.) for coarsegrained soils 2 coverages 130 N (30 lb) minimum weight. Considerable range is tolerable, depending on materials and conditions. Weights up to 1.1 kN (250 lb); foot diameter, 100 to 250 mm (4 to 10 in.) Smoothwheel rollers * After U.S. Navy (1971). Published by U.S. Government Printing Ofﬁce Three-wheel rollers are obtainable in a wide range of sizes. Two-wheel tandem rollers are available in the 9 to 180 kN (1- to 20-ton) weight range. Three-axle tandem rollers are generally used in the 90 to 180 kN (10- to 20ton) weight range. Very heavy rollers are used for proof rolling of subgrade or base course. 139 140 Chapter 6: Soil Compaction 6.9 Determination of Field Unit Weight of Compaction When the compaction work is progressing in the ﬁeld, knowing whether the speciﬁed unit weight has been achieved is useful. The standard procedures for determining the ﬁeld unit weight of compaction include 1. Sand cone method 2. Rubber balloon method 3. Nuclear method Following is a brief description of each of these methods. Sand Cone Method (ASTM Designation D-1556) The sand cone device consists of a glass or plastic jar with a metal cone attached at its top (Figure 6.22). The jar is ﬁlled with uniform dry Ottawa sand. The combined weight of the jar, the cone, and the sand ﬁlling the jar is determined (W1). In the ﬁeld, a small hole is excavated in the area where the soil has been compacted. If the weight of the moist soil excavated from the hole (W2) is determined and the moisture content of the excavated soil is known, the dry weight of the soil can be obtained as W2 (6.15) W3 w 1% 2 1 100 where w moisture content. After excavation of the hole, the cone with the sand-ﬁlled jar attached to it is inverted and placed over the hole (Figure 6.23). Sand is allowed to ﬂow out of the jar to ﬁll the hole and the cone. After that, the combined weight of the jar, the cone, and the remaining sand in the jar is determined (W4), so Figure 6.22 Glass jar ﬁlled with Ottawa sand with sand cone attached (Courtesy of Braja M. Das, Henderson, Nevada) 6.9 Determination of Field Unit Weight of Compaction 141 Jar Ottawa sand Valve Metal plate Cone Figure 6.23 Field unit weight determined by sand cone method Hole filled with Ottawa sand W5 W1 W4 (6.16) where W5 weight of sand to ﬁll the hole and cone. The volume of the excavated hole can then be determined as V W 5 Wc gd1sand2 (6.17) where Wc weight of sand to fill the cone only gd1sand2 dry unit weight of Ottawa sand used The values of Wc and gd(sand) are determined from the calibration done in the laboratory. The dry unit weight of compaction made in the ﬁeld then can be determined as follows: gd Dry weight of the soil excavated from the hole W3 Volume of the hole V (6.18) Rubber Balloon Method (ASTM Designation D-2167) The procedure for the rubber balloon method is similar to that for the sand cone method; a test hole is made and the moist weight of soil removed from the hole and its moisture content are determined. However, the volume of the hole is determined by introducing into it a rubber balloon ﬁlled with water from a calibrated vessel, from which the volume can be read directly. The dry unit weight of the compacted soil can be determined by using Eq. (6.18). Figure 6.24 shows a calibrated vessel that would be used with a rubber balloon. Nuclear Method Nuclear density meters are often used for determining the compacted dry unit weight of soil. The density meters operate either in drilled holes or from the ground surface. It uses a radioactive isotope source. The isotope gives off Gamma rays that radiate back to the meter’s detector. Dense soil absorbs more radiation than loose soil. The instrument measures the weight of wet soil per unit volume and the weight of water present in a unit volume of soil. The dry unit weight of compacted soil can be determined by subtracting the weight of water from the moist unit weight of soil. Figure 6.25 shows a photograph of a nuclear density meter. Figure 6.24 Calibrated vessel used with rubber balloon (not shown) (Courtesy of John Hester, Carterville, Illinois) 142 Figure 6.25 Nuclear density meter (Courtesy of Braja M. Das, Henderson, Nevada) 6.9 Determination of Field Unit Weight of Compaction 143 Example 6.3 Laboratory compaction test results for a clayey silt are given in the following table. Moisture content (%) Dry unit weight (kN/m3) 6 8 9 11 12 14 14.80 17.45 18.52 18.9 18.5 16.9 Following are the results of a field unit-weight determination test performed on the same soil by means of the sand cone method: • • • • • • Calibrated dry density of Ottawa sand 1570 kg/m3 Calibrated mass of Ottawa sand to fill the cone 0.545 kg Mass of jar + cone + sand (before use) 7.59 kg Mass of jar + cone + sand (after use) 4.78 kg Mass of moist soil from hole 3.007 kg Moisture content of moist soil 10.2% Determine: a. Dry unit weight of compaction in the field b. Relative compaction in the field Solution Part a In the field, Mass of sand used to fill the hole and cone 7.59 kg 4.78 kg 2.81 kg Mass of sand used to fill the hole 2.81 kg 0.545 kg 2.265 kg Volume of the hole 1V 2 Moist density of compacted soil 2.265 kg Dry density of Ottawa sand 2.265 kg 1570 kg/m3 0.0014426 m3 Mass of moist soil Volume of hole 3.007 2.084.4 kg/m3 0.0014426 12084.42 19.812 Moist unit weight of compacted soil 20.45 kN/m3 1000 144 Chapter 6: Soil Compaction Hence, gd 1 g w1% 2 100 20.45 18.56 kN/m3 10.2 1 100 Part b The results of the laboratory compaction test are plotted in Figure 6.26. From the plot, we see that d(max) 19 kN/m3. Thus, from Eq. (6.10). 20 gd (kN/m3 ) 19 kN/m3 18 16 14 0 4 8 12 (%) R 6.10 16 20 Figure 6.26 Plot of laboratorycompaction test results gd1field2 18.56 97.7% gd1field2 19.0 ■ Compaction of Organic Soil and Waste Materials The presence of organic materials in a soil reduces its strength. In many cases, soils with a high organic content are generally discarded as ﬁll material; however, in certain economic circumstances, slightly organic soils are used for compaction. In fact, organic soils are desirable in many circumstances (e.g., for agriculture, decertiﬁcation, mitigation, and urban planning). The high costs of waste disposal have sparked an interest in the possible use of waste materials (e.g., bottom ash obtained from coal burning, copper slag, paper mill sludge, shredded waste tires mixed with inorganic soil, and so forth) in various landﬁll operations. Such use of waste materials is one of the major thrusts of present-day environmental geotechnology. Following is a discussion of the compaction characteristics of some of these materials. Organic Soil Franklin, Orozco, and Semrau (1973) conducted several laboratory tests to observe the effect of organic content on the compaction characteristics of soil. In the test program, various natural soils and soil mixtures were tested. Figure 6.27 shows the effect of organic content on the maximum dry unit weight. When the organic content exceeds 8 to 10%, 6.10 Compaction of Organic Soil and Waste Materials 145 105 16.00 100 Maximum dry unit weight (lb/ft3 ) 90 14.00 85 Oven-dried Air-dried 13.00 80 12.00 Maximum dry unit weight (kN/m3 ) 15.00 95 75 70 11.00 65 10.22 0 5 10 15 20 Organic content (%) Mixture — oven-dried Natural sample — oven-dried Mixture — air-dried 25 30 Figure 6.27 Variation of maximum dry unit weight with organic content (After Franklin, Orozco, and Semrau, 1973. With permission from ASCE.) the maximum dry unit weight of compaction decreases rapidly. Conversely, the optimum moisture content for a given compactive effort increases with an increase in organic content. This trend is shown in Figure 6.28. Likewise, the maximum unconﬁned compression strength (see Chapter 11) obtained from a compacted soil (with a given compactive effort) decreases with increasing organic content of a soil. From these facts, we can see that soils with organic contents higher than about 10% are undesirable for compaction work. Soil and Organic Material Mixtures Lancaster, et al. (1996) conducted several modiﬁed Proctor tests to determine the effect of organic content on the maximum dry unit weight and optimum moisture content of soil and organic material mixtures. The soils tested consisted of a poorly graded sandy soil (SP-SM) mixed with either shredded redwood bark, shredded rice hulls, or municipal sewage sludge. Figures 6.29 and 6.30 show the variations of maximum dry unit weight of compaction and optimum moisture content, respectively, with organic content. As in Figure 6.27, the maximum dry unit weight decreased with organic content in all cases (see Figure 6.29). Conversely, the optimum moisture content increased with organic content for soil mixed with shredded red-wood or rice hulls (see Figure 6.30), similar to the pattern shown in Figure 6.28. However, for soil and municipal sewage sludge mixtures, the optimum moisture content remained practically constant (see Figure 6.30). 146 Chapter 6: Soil Compaction Optimum moisture content (%) 35 30 25 20 15 10 0 5 10 15 Organic content (%) Oven-dried 20 Air-dried 25 Figure 6.28 Variation of optimum moisture content with organic content (After Franklin, Orozco, and Semrau, 1973. With permission from ASCE.) Bottom Ash from Coal Burning and Copper Slag Laboratory standard Proctor test results for bottom ash from coal-burning power plants and for copper slag are also available in the literature. These waste products have been shown to be environmentally safe for use as landﬁll. A summary of some of these test results is given in Table 6.3. 18 Maximum dry unit weight (kN/m3) 16 14 12 10 8 6 4 2 0 0 20 40 60 Organic content (%) Redwood bark Rice hulls 80 100 Sludge Figure 6.29 Variation of maximum dry unit weight of compaction with organic content—soil and organic material mixture. (Source: After “The Effect of Organic Content on Soil Compaction,” by J. Lancaster, R. Waco, J. Towle, and R. Chaney, 1996. In Proceedings, Third International Symposium on Environmental Geotechnology, p. 159. Used with permission of the author) 6.11 Special Compaction Techniques 147 26 Optimum moisture content (%) 24 22 20 18 16 14 12 5 10 15 20 25 Organic content (%) Redwood bark Rice hulls 30 35 Sludge Figure 6.30 Variation of optimum moisture content with organic content — soil and organic material mixtures. (Source: After “The Effect of Organic Content on Soil Compaction,” by J. Lancaster, R. Waco, J. Towle, and R. Chaney, 1996. In Proceedings, Third International Symposium on Environmental Geotechnology, p. 159. Used with permission of the author) Table 6.3 Standard Proctor Test Results of Bottom Ash and Copper Slag Maximum dry unit weight Type Bottom ash— bituminous coal (West Virginia) Bottom ash— lignite coal Copper slag 6.11 Location Fort Martin Kammer Kanawha River Mitchell Muskingham Willow Island Big Stone Power Plant, South Dakota American Smelter and Reﬁnery Company, El Paso, Texas Optimum moisture content kN/m3 lb/ft3 (%) Source 13.4 16.0 11.4 18.3 14.3 14.5 16.4 85 102 72.6 116.6 91.1 92.4 104.4 24.8 13.8 26.2 14.6 22.0 21.2 20.5 Seals, Moulton, and Ruth (1972) 19.8 126 18.8 Das, Selim, and Pfeiﬂe (1978) Das, Tarquin, and Jones (1983) Special Compaction Techniques Several special types of compaction techniques have been developed for deep compaction of in-place soils, and these techniques are used in the ﬁeld for large-scale compaction works. Among these, the popular methods are vibroﬂotation, dynamic compaction, and blasting. Details of these methods are provided in the following sections. 148 Chapter 6: Soil Compaction Vibroﬂotation Vibroﬂotation is a technique for in situ densiﬁcation of thick layers of loose granular soil deposits. It was developed in Germany in the 1930s. The ﬁrst vibroﬂotation device was used in the United States about 10 years later. The process involves the use of a Vibroﬂot unit (also called the vibrating unit), which is about 2.1 m (7 ft) long. (As shown in Figure 6.31.) This vibrating unit has an eccentric weight inside it and can develop a centrifugal force, which enables the vibrating unit to vibrate horizontally. There are openings at the bottom and top of the vibrating unit for water jets. The vibrating unit is attached to a follow-up pipe. Figure 6.31 shows the entire assembly of equipment necessary for conducting the ﬁeld compaction. Power supply Water pump Follow-up pipe Cylinder of compacted material, added from the surface to compensate for the loss of volume caused by the increase of density of the compacted soil Vibrating unit Cylinder of compacted material, produced by a single vibroflot compaction A B Figure 6.31 Vibroﬂotation unit (After Brown, 1977. With permission from ASCE.) 6.11 Special Compaction Techniques Stage 1 Stage 2 Stage 3 149 Stage 4 Figure 6.32 Compaction by vibroﬂotation process (After Brown, 1977. With permission from ASCE.) The entire vibroﬂotation compaction process in the ﬁeld can be divided into four stages (Figure 6.32): Stage 1: The jet at the bottom of the Vibroﬂot is turned on and lowered into the ground. Stage 2: The water jet creates a quick condition in the soil and it allows the vibrating unit to sink into the ground. Stage 3: Granular material is poured from the top of the hole. The water from the lower jet is transferred to the jet at the top of the vibrating unit. This water carries the granular material down the hole. Stage 4: The vibrating unit is gradually raised in about 0.3 m (1 ft) lifts and held vibrating for about 30 seconds at each lift. This process compacts the soil to the desired unit weight. The details of various types of Vibroﬂot units used in the United States are given in Table 6.4. Note that 23 kW (30-hp) electric units have been used since the latter part of the 1940s. The 75 kW (100-hp) units were introduced in the early 1970s. The zone of compaction around a single probe varies with the type of Vibroﬂot used. The cylindrical zone of compaction has a radius of about 2 m (6 ft) for a 23 kW (30-hp) unit. This radius can extend to about 3 m (10 ft) for a 75 kW (100hp) unit. Typical patterns of Vibroﬂot probe spacings are shown in Figure 6.33. Square and rectangular patterns generally are used to compact soil for isolated, shallow foundations. Equilateral triangular patterns generally are used to compact large areas. The capacity for successful densiﬁcation of in situ soil depends on several factors, the most important of which is the grain-size distribution of the soil and the type of backﬁll used to ﬁll the holes during the withdrawal period of the Vibroﬂot. The range of the grain-size distribution of in situ soil marked Zone 1 in Figure 6.34 is most suitable for compaction by vibroﬂotation. Soils that contain excessive amounts of ﬁne sand and silt-size particles are difﬁcult 150 Chapter 6: Soil Compaction Table 6.4 Types of Vibroﬂot Units* 75 kW electric and hydraulic Motor type 23 kW electric a. Vibrating tip Length Diameter Weight Maximum movement when full Centrifugal force 2.1 m (7.0 ft) 406 mm (16 in.) 17.8 kN (4000 lb) 12.5 mm (0.49 in) 160 kN (18 ton) 1.86 m (6.11 ft) 381 mm (15 in) 17.8 kN (4000 lb) 7.6 mm (0.3 in.) 89 kN (10 ton) b. Eccentric Weight Offset Length Speed 1.2 kN (260 lb) 38 mm (1.5 in) 610 mm (24 in) 1800 rpm 0.76 kN (170 lb) 32 mm (1.25 in) 390 mm (15.25 in.) 1800 rpm c. Pump Operating ﬂow rate Pressure 0–1.6 m3/min (0–400 gal/min) 700–1050 kN/m2 (100–150 lb/in2) 0–0.6 m3/min (0–150 gal/min) 700–1050 kN/m2 (100–150 lb/in2) d. Lower follow-up pipe and extensions Diameter 305 mm (12 in.) Weight 3.65 kN/m (250 lb/ft) 305 mm (12 in.) 3.65 kN/m (250 lb/ft) *After Brown 1977. With permission from ASCE. (a) (c) (b) (d) (e) Figure 6.33 Typical patterns of Vibroﬂot probe spacings for a column foundation (a, b, c, and d) and for compaction over a large area (e) 6.11 Special Compaction Techniques Unified soil classification system Coarse sand Fine sand Silts and clays Gravel 100 151 Percent finer 80 60 Zone 3 Zone 1 Zone 2 40 20 0 100.0 10.0 1.0 0.1 Grain size (mm) 0.01 0.001 Figure 6.34 Effective range of grain-size distribution of soil for vibroﬂotation to compact, and considerable effort is needed to reach the proper relative density of compaction. Zone 2 in Figure 6.34 is the approximate lower limit of grain-size distribution for which compaction by vibroﬂotation is effective. Soil deposits whose grain-size distributions fall in Zone 3 contain appreciable amounts of gravel. For these soils, the rate of probe penetration may be slow and may prove uneconomical in the long run. The grain-size distribution of the backﬁll material is an important factor that controls the rate of densiﬁcation. Brown (1977) has deﬁned a quantity called the suitability number for rating backﬁll as 3 1 1 B 1D50 2 2 1D20 2 2 1D10 2 2 SN 1.7 (6.19) where D50 , D20 , and D10 are the diameters (in mm) through which, respectively, 50, 20, and 10% of the material passes. The smaller the value of SN, the more desirable the backﬁll material. Following is a backﬁll rating system proposed by Brown: Range of SN 0–10 10–20 20–30 30–50 50 Rating as backﬁll Excellent Good Fair Poor Unsuitable Dynamic Compaction Dynamic compaction is a technique that has gained popularity in the United States for the densiﬁcation of granular soil deposits. This process consists primarily of dropping a heavy weight repeatedly on the ground at regular intervals. The weight of the hammer used varies 152 Chapter 6: Soil Compaction over a range of 80 to 360 kN (18 to 80 kip), and the height of the hammer drop varies between 7.5 and 30.5 m (25 and 100 ft). The stress waves generated by the hammer drops aid in the densiﬁcation. The degree of compaction achieved at a given site depends on the following three factors: 1. Weight of hammer 2. Height of hammer drop 3. Spacing of locations at which the hammer is dropped Leonards, Cutter, and Holtz (1980) suggested that the signiﬁcant depth of inﬂuence for compaction can be approximated by using the equation D 1 12 2 2WHh (6.20) where D significant depth of densification 1m2 WH dropping weight 1metric ton2 h height of drop 1m2 In English units, the preceding equation takes the form D 0.612WHh (6.21) where the units of D and h are ft, and the unit of WH is kip. In 1992, Poran and Rodriguez suggested a rational method for conducting dynamic compaction for granular soils in the ﬁeld. According to their method, for a hammer of width D having a weight WH and a drop h, the approximate shape of the densiﬁed area will be of the type shown in Figure 6.35 (i.e., a semiprolate spheroid). Note that in this ﬁgure b DI (where DI is the signiﬁcant depth of densiﬁcation). Figure 6.36 gives the design chart for a/D and b/D versus NWHh/Ab (D width of the hammer if not circular in cross section; A area of cross section of the hammer; and N number of required hammer drops). This method uses the following steps. Step 1: Determine the required signiﬁcant depth of densiﬁcation, DI ( b). Step 2: Determine the hammer weight (WH), height of drop (h), dimensions of the cross section, and thus, the area A and the width D. Step 3: Determine DI/D b/D. 2a a b Top view Side view Approximate shape Figure 6.35 Approximate shape of the densiﬁed area due to dynamic compaction 6.11 Special Compaction Techniques 10 153 3.5 3.0 8 6 2.5 Average a D 2.0 a D b D 4 1.5 Average b D 1.0 2 0.5 0 100 1000 NWH h (kN/m2) Ab 0.0 10,000 Figure 6.36 Poran and Rodriguez chart for a/D, b/D versus NWHh/Ab Step 4: Use Figure 6.36 and determine the magnitude of NWHh/Ab for the value of b/D obtained in Step 3. Step 5: Since the magnitudes of WH, h, A, and b are known (or assumed) from Step 2, the number of hammer drops can be estimated from the value of NWHh/Ab obtained from Step 4. Step 6: With known values of NWHh/Ab, determine a/D and thus a from Figure 6.36. Step 7: The grid spacing, Sg , for dynamic compaction may now be assumed to be equal to or somewhat less than a. (See Figure 6.37.) a Sg b Figure 6.37 Approximate grid spacing for dynamic compaction 154 Chapter 6: Soil Compaction Blasting Blasting is a technique that has been used successfully in many projects (Mitchell, 1970) for the densiﬁcation of granular soils. The general soil grain sizes suitable for compaction by blasting are the same as those for compaction by vibroﬂotation. The process involves the detonation of explosive charges, such as 60% dynamite at a certain depth below the ground surface in saturated soil. The lateral spacing of the charges varies from about 3 to 9 m (10 to 30 ft). Three to ﬁve successful detonations are usually necessary to achieve the desired compaction. Compaction (up to a relative density of about 80%) up to a depth of about 18 m (60 ft) over a large area can easily be achieved by using this process. Usually, the explosive charges are placed at a depth of about two-thirds of the thickness of the soil layer desired to be compacted. The sphere of inﬂuence of compaction by a 60% dynamite charge can be given as follows (Mitchell, 1970): r WEX B C (6.22) where r sphere of inﬂuence WEX weight of explosive—60% dynamite C 0.0122 when WEX is in kg and r is in m 0.0025 when WEX is in lb and r is in ft Figure 6.38 shows the test results of soil densiﬁcation by blasting in an area measuring 15 m by 9 m (Mitchell, 1970). For these tests, twenty 2.09-kg (4.6-lb) charges of Gelamite No. 1 (Hercules Powder Company, Wilmington, Delaware) were used. Marker No. M2 3.0 1.0 Test No. 3 0.8 2.5 2.0 0.6 4 M1 4 1.5 0.4 M1 3 Settlement (ft) Settlement (m) M2 1.0 15 m (50 ft) 0.2 9m (30 ft) M1 M2 0.5 0.0 0.0 0 5 10 15 Number of charges 20 25 Figure 6.38 Ground settlement as a function of number of explosive charges Problems 155 Example 6.4 Following are the details for the backﬁll material used in a vibroﬂotation project: • D10 0.36 mm • D20 0.52 mm • D50 1.42 mm Determine the suitability number SN . What would be its rating as a backﬁll material? Solution From Eq. (6.19), 3 1 1 B 1D50 2 2 1D20 2 2 1D10 2 2 SN 1.7 1 1 3 2 B 11.42 2 2 10.522 10.362 2 1.7 6.1 Rating: Excellent 6.12 ■ Summary and General Comments Laboratory standard and modiﬁed Proctor compaction tests described in this chapter are essentially for impact or dynamic compaction of soil; however, in the laboratory, static compaction and kneading compaction also can be used. It is important to realize that the compaction of clayey soils achieved by rollers in the ﬁeld is essentially the kneading type. The relationships of dry unit weight (gd) and moisture content (w) obtained by dynamic and kneading compaction are not the same. Proctor compaction test results obtained in the laboratory are used primarily to determine whether the roller compaction in the ﬁeld is sufﬁcient. The structures of compacted cohesive soil at a similar dry unit weight obtained by dynamic and kneading compaction may be different. This difference, in turn, affects physical properties, such as hydraulic conductivity, compressibility, and strength. For most ﬁll operations, the ﬁnal selection of the borrow site depends on such factors as the soil type and the cost of excavation and hauling. Problems 6.1 6.2 Given Gs 2.75, calculate the zero-air-void unit weight for a soil in lb/ft3 at w 5%, 8%, 10%, 12%, and 15%. Repeat Problem 6.1 with Gs 2.65. 156 Chapter 6: Soil Compaction 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Calculate the variation of dry density (kg/m3) of a soil (Gs 2.67) at w 10% and 20% for degree of saturation (S) 80%, 90%, and 100%. The results of a standard Proctor test are given below. Determine the maximum dry unit weight of compaction and the optimum moisture content. Volume of Proctor mold (ft3 ) Weight of wet soil in the mold (lb) Moisture content (%) 1/30 1/30 1/30 1/30 1/30 3.26 4.15 4.67 4.02 3.63 8.4 10.2 12.3 14.6 16.8 For the soil described in Problem 6.4, if Gs 2.72, determine the void ratio and the degree of saturation at optimum moisture content. The results of a standard Proctor test are given in the following table. Determine the maximum dry density (kg/m3) of compaction and the optimum moisture content. Volume of Proctor mold (cm3 ) Mass of wet soil in the mold (kg) Moisture content (%) 943.3 943.3 943.3 943.3 943.3 943.3 943.3 943.3 1.68 1.71 1.77 1.83 1.86 1.88 1.87 1.85 9.9 10.6 12.1 13.8 15.1 17.4 19.4 21.2 A field unit weight determination test for the soil described in Problem 6.6 yielded the following data: moisture content 10.5% and moist density 1705 kg/m3. Determine the relative compaction. The in situ moisture content of a soil is 18% and the moist unit weight is 105 lb/ft3. The specific gravity of soil solids is 2.75. This soil is to be excavated and transported to a construction site for use in a compacted fill. If the specifications call for the soil to be compacted to a minimum dry unit weight of 103.5 lb/ft3 at the same moisture content of 18%, how many cubic yards of soil from the excavation site are needed to produce 10,000 yd3 of compacted fill? How many 20-ton truckloads are needed to transport the excavated soil? A proposed embankment fill requires 8000 m3 of compacted soil. The void ratio of the compacted fill is specified as 0.7. Four borrow pits are available as described in the following table, which lists the respective void ratios of the soil and the cost per cubic meter for moving the soil to the proposed construction site. Make the References 157 necessary calculations to select the pit from which the soil should be bought to minimize the cost. Assume Gs to be the same at all pits. Borrow pit A B C D Void ratio 0.82 1.1 0.90 0.78 Cost ($/m3 ) 8 5 9 12 6.10 The maximum and minimum dry unit weights of a sand were determined in the laboratory to be 104 lb/ft3 and 93 lb/ft3, respectively. What would be the relative compaction in the field if the relative density is 78%? 6.11 The maximum and minimum dry densities of a sand were determined in the laboratory to be 1682 kg/m3 and 1510 kg/m3, respectively. In the field, if the relative density of compaction of the same sand is 70%, what are its relative compaction (%) and dry density (kg/m3)? 6.12 The relative compaction of a sand in the field is 90%. The maximum and minimum dry unit weights of the sand are 108 lb/ft3 and 93 lb/ft3, respectively. For the field condition, determine a. Dry unit weight b. Relative density of compaction c. Moist unit weight at a moisture content of 12% 6.13 Following are the results of a field unit weight determination test on a soil with the sand cone method: • Calibrated dry density of Ottawa sand 1667 kg/m3 • Calibrated mass of Ottawa sand to fill the cone 0.117 kg • Mass of jar + cone + sand (before use) 5.99 kg • Mass of jar + cone + sand (after use) 2.81 kg • Mass of moist soil from hole 3.331 kg • Moisture content of moist soil 11.6% Determine the dry unit weight of compaction in the field. 6.14 The backfill material for a vibroflotation project has the following grain sizes: • D10 0.11 mm • D20 0.19 mm • D50 1.3 mm Determine the suitability number, SN, for each. 6.15 Repeat Problem 6.14 using the following values: • D10 0.09 mm • D20 0.25 mm • D50 0.61 mm References AMERICAN ASSOCIATION OF STATE HIGHWAY AND TRANSPORTATION OFFICIALS (1982). AASHTO Materials, Part II, Washington, D.C. AMERICAN SOCIETY FOR TESTING AND MATERIALS (2007). Annual Book of ASTM Standards, Vol 04.08, West Conshohocken, Pa. 158 Chapter 6: Soil Compaction D’APPOLONIA, E. (1953). “Loose Sands—Their Compaction by Vibroﬂotation,” Special Technical Publication No. 156, ASTM, 138–154. BROWN, E. (1977). “Vibroﬂotation Compaction of Cohesionless Soils,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT12, 1437–1451. D’APPOLONIA, D. J., WHITMAN, R. V., and D’APPOLONIA, E. D. (1969). “Sand Compaction with Vibratory Rollers,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 95, No. SM1, 263–284. DAS, B. M., SELIM, A. A., and PFEIFLE, T. W. (1978). “Effective Use of Bottom Ash as a Geotechnical Material,” Proceedings, 5th Annual UMR-DNR Conference and Exposition on Energy, University of Missouri, Rolla, 342–348. DAS, B. M., TARQUIN, A. J., and JONES, A. D. (1983). “Geotechnical Properties of a Copper Slag,” Transportation Research Record No. 941, National Research Council, Washington, D.C., 1–4. FRANKLIN, A. F., OROZCO, L. F., and SEMRAU, R. (1973). “Compaction of Slightly Organic Soils,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM7, 541–557. GURTUG, Y., and SRIDHARAN, A. (2004). “Compaction Behaviour and Prediction of Its Characteristics of Fine Grained Soils with Particular Reference to Compaction Energy,” Soils and Foundations, Vol. 44, No. 5, 27–36. HOLTZ, R. D., and KOVACS, W. D. (1981). An Introduction to Geotechnical Engineering, PrenticeHall, Englewood Cliffs, N.J. JOHNSON, A. W., and SALLBERG, J. R. (1960). “Factors That Inﬂuence Field Compaction of Soil,” Highway Research Board, Bulletin No. 272. LAMBE, T. W. (1958a). “The Structure of Compacted Clay,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 84, No. SM2, 1654–1 to 1654–35. LAMBE, T. W. (1958b). “The Engineering Behavior of Compacted Clay,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 84, No. SM2, 1655–1 to 1655–35. LANCASTER, J., WACO, R., TOWLE, J., and CHANEY, R. (1996). “The Effect of Organic Content on Soil Compaction,” Proceedings, 3rd International Symposium on Environmental Geotechnology, San Diego, 152–161. LEE, K. W., and SINGH, A. (1971). “Relative Density and Relative Compaction,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. SM7, 1049–1052. LEE, P. Y., and SUEDKAMP, R. J. (1972). “Characteristics of Irregularly Shaped Compaction Curves of Soils,” Highway Research Record No. 381, National Academy of Sciences, Washington, D.C., 1–9. LEONARDS, G. A., CUTTER, W. A., and HOLTZ, R. D. (1980). “Dynamic Compaction of Granular Soils,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 106, No. GT1, 35–44. MITCHELL, J. K. (1970). “In-Place Treatment of Foundation Soils,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. SM1, 73–110. OMAR, M., ABDALLAH, S., BASMA, A., and BARAKAT, S. (2003). “Compaction Characteristics of Granular Soils in United Arab Emirates,” Geotechnical and Geological Engineering, Vol. 21, No. 3, 283–295. PORAN, C. J., and RODRIGUEZ, J. A. (1992). “Design of Dynamic Compaction,” Canadian Geotechnical Journal, Vol. 2, No. 5, 796–802. PROCTOR, R. R. (1933). “Design and Construction of Rolled Earth Dams,” Engineering News Record, Vol. 3, 245–248, 286–289, 348–351, 372–376. SEALS, R. K. MOULTON, L. K., and RUTH, E. (1972). “Bottom Ash: An Engineering Material,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 98, No. SM4, 311–325. References 159 SEED, H. B. (1964). Lecture Notes, CE 271, Seepage and Earth Dam Design, University of California, Berkeley. SEED, H. B., and CHAN, C. K. (1959). “Structure and Strength Characteristics of Compacted Clays,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 85, No. SM5, 87–128. U.S. DEPARTMENT OF NAVY (1971). “Design Manual—Soil Mechanics, Foundations, and Structures,” NAVFAC DM-7, U.S. Government Printing Ofﬁce, Washington, D.C. 7 Permeability Soils are permeable due to the existence of interconnected voids through which water can ﬂow from points of high energy to points of low energy. The study of the ﬂow of water through permeable soil media is important in soil mechanics. It is necessary for estimating the quantity of underground seepage under various hydraulic conditions, for investigating problems involving the pumping of water for underground construction, and for making stability analyses of earth dams and earth-retaining structures that are subject to seepage forces. 7.1 Bernoulli’s Equation From ﬂuid mechanics, we know that, according to Bernoulli’s equation, the total head at a point in water under motion can be given by the sum of the pressure, velocity, and elevation heads, or h u gw v2 2g Z c c c Pressure head Velocity head Elevation head (7.1) where h total head u pressure v velocity g acceleration due to gravity gw unit weight of water Note that the elevation head, Z, is the vertical distance of a given point above or below a datum plane. The pressure head is the water pressure, u, at that point divided by the unit weight of water, gw . 160 7.1 Bernoulli’s Equation 161 h uA g uB g Flow hA hB A ZA L ZB B Datum Figure 7.1 Pressure, elevation, and total heads for ﬂow of water through soil If Bernoulli’s equation is applied to the ﬂow of water through a porous soil medium, the term containing the velocity head can be neglected because the seepage velocity is small, and the total head at any point can be adequately represented by h u Z gw (7.2) Figure 7.1 shows the relationship among pressure, elevation, and total heads for the ﬂow of water through soil. Open standpipes called piezometers are installed at points A and B. The levels to which water rises in the piezometer tubes situated at points A and B are known as the piezometric levels of points A and B, respectively. The pressure head at a point is the height of the vertical column of water in the piezometer installed at that point. The loss of head between two points, A and B, can be given by ¢h hA hB a uA uB ZA b a ZB b gw gw (7.3) The head loss, h, can be expressed in a nondimensional form as i ¢h L (7.4) where i hydraulic gradient L distance between points A and B— that is, the length of flow over which the loss of head occurred In general, the variation of the velocity v with the hydraulic gradient i is as shown in Figure 7.2. This ﬁgure is divided into three zones: 1. Laminar ﬂow zone (Zone I) 2. Transition zone (Zone II) 3. Turbulent ﬂow zone (Zone III) 162 Chapter 7: Permeability Zone III Turbulent ﬂow zone Velocity, Zone II Transition zone Zone I Laminar ﬂow zone Hydraulic gradient, i Figure 7.2 Nature of variation of v with hydraulic gradient, i When the hydraulic gradient is increased gradually, the ﬂow remains laminar in oZnes I and II, and the velocity, v, bears a linear relationship to the hydraulic gradient. At a higher hydraulic gradient, the ﬂow becomes turbulent (Z one III). When the hydraulic gradient is decreased, laminar ﬂow conditions exist only in oZne I. In most soils, the ﬂow of water through the void spaces can be considered laminar; thus, v r i (7.5) In fractured rock, stones, gravels, and very coarse sands, turbulent ﬂow conditions may exist, and Eq. (7.5) may not be valid. 7.2 Darcy’s Law In 1856, Darcy published a simple equation for the discharge velocity of water through saturated soils, which may be expressed as v ki (7.6) where v discharge velocity, which is the quantity of water flowing in unit time through a unit gross cross-sectional area of soil at right angles to the direction of flow k hydraulic conductivity 1otherwise known as the coefficient of permeability2 This equation was based primarily on Darcy’s observations about the ﬂow of water through clean sands. Note that Eq. (7.6) is similar to Eq. (7.5); both are valid for laminar ﬂow conditions and applicable for a wide range of soils. In Eq. (7.6), v is the discharge velocity of water based on the gross cross-sectional area of the soil. However, the actual velocity of water (that is, the seepage velocity) through the 7.2 Darcy’s Law 163 Area of soil specimen A Flow rate, q L Area of void in the cross section A Area of soil solids in the cross section As Figure 7.3 Derivation of Eq. (7.10) void spaces is greater than v. A relationship between the discharge velocity and the seepage velocity can be derived by referring to Figure 7.3, which shows a soil of length L with a gross cross-sectional area A. If the quantity of water ﬂowing through the soil in unit time is q, then q vA Avvs (7.7) where vs seepage velocity Av area of void in the cross section of the specimen However, A Av As (7.8) where As area of soil solids in the cross section of the specimen. Combining Eqs. (7.7) and (7.8) gives q v1Av As 2 Avvs or vs v1Av As 2 Av v1Av As 2L AvL v1Vv Vs 2 Vv (7.9) where Vv volume of voids in the specimen Vs volume of soil solids in the specimen Equation (7.9) can be rewritten as vs v ≥ 1 a Vv b Vs Vv Vs ¥ v a v 1e b e n (7.10) where e void ratio n porosity Darcy’s law as deﬁned by Eq. (7.6) implies that the discharge velocity v bears a linear relationship to the hydraulic gradient i and passes through the origin as shown in Figure 7.4. Hansbo (1960), however, reported the test results for four undisturbed Discharge velocity, 164 Chapter 7: Permeability ki Clay soil Figure 7.4 Variation of discharge velocity with hydraulic gradient in clay i io Hydraulic gradient, i natural clays. On the basis of his results, a hydraulic gradient i (see Figure 7.4) appears to exist, at which (7.11) 1for i i¿ 2 v k1i i0 2 and (7.12) 1for i i¿ 2 v kim The preceding equation implies that for very low hydraulic gradients, the relationship between v and i is nonlinear. The value of m in Eq. (7.12) for four Swedish clays was about 1.5. However, several other studies refute the preceding ﬁndings. Mitchell (1976) discussed these studies in detail. Taking all points into consideration, he concluded that Darcy’s law is valid. 7.3 Hydraulic Conductivity Hydraulic conductivity is generally expressed in cm/sec or m/sec in SI units and in ft/min or ft/day in English units. The hydraulic conductivity of soils depends on several factors: ﬂuid viscosity, poresize distribution, grain-size distribution, void ratio, roughness of mineral particles, and degree of soil saturation. In clayey soils, structure plays an important role in hydraulic conductivity. Other major factors that affect the permeability of clays are the ionic concentration and the thickness of layers of water held to the clay particles. The value of hydraulic conductivity (k) varies widely for different soils. Some typical values for saturated soils are given in Table 7.1. The hydraulic conductivity of unsaturated soils is lower and increases rapidly with the degree of saturation. Table 7.1 Typical Values of Hydraulic Conductivity of Saturated Soils k Soil type cm /sec ft /min Clean gravel Coarse sand Fine sand Silty clay Clay 1001.0 1.00.01 0.010.001 0.0010.00001 0.000001 2002.0 2.00.02 0.020.002 0.0020.00002 0.000002 7.3 Hydraulic Conductivity 165 The hydraulic conductivity of a soil is also related to the properties of the ﬂuid ﬂowing through it by the equation k gw K h (7.13) where gw unit weight of water h viscosity of water K absolute permeability The absolute permeability K is expressed in units of L2 (that is, cm2, ft2, and so forth). Equation (7.13) showed that hydraulic conductivity is a function of the unit weight and the viscosity of water, which is in turn a function of the temperature at which the test is conducted. So, from Eq. (7.13), kT1 kT2 a hT2 hT1 ba gw1T12 gw1T22 b (7.14) where kT1 , kT2 hydraulic conductivity at temperatures T1 and T2 , respectively hT1 , hT2 viscosity of water at temperatures T1 and T2 , respectively gw1T12 , gw1T22 unit weight of water at temperatures T1 and T2 , respectively It is conventional to express the value of k at a temperature of 20C. Within the range of test temperatures, we can assume that gw1T12 gw1T22 . So, from Eq. (7.14) k20°C a hT°C b kT°C h20°C (7.15) The variation of hTC /h20C with the test temperature T varying from 15 to 30C is given in Table 7.2. Table 7.2 Variation of hTC/h20C Temperature, T (ⴗC) HTⴗC/H20ⴗC Temperature, T (ⴗC) HTⴗC/H20ⴗC 15 16 17 18 19 20 21 22 1.135 1.106 1.077 1.051 1.025 1.000 0.976 0.953 23 24 25 26 27 28 29 30 0.931 0.910 0.889 0.869 0.850 0.832 0.814 0.797 166 Chapter 7: Permeability 7.4 Laboratory Determination of Hydraulic Conductivity Two standard laboratory tests are used to determine the hydraulic conductivity of soil— the constant-head test and the falling-head test. A brief description of each follows. Constant-Head Test A typical arrangement of the constant-head permeability test is shown in Figure 7.5. In this type of laboratory setup, the water supply at the inlet is adjusted in such a way that the difference of head between the inlet and the outlet remains constant during the test period. After a constant ﬂow rate is established, water is collected in a graduated ﬂask for a known duration. The total volume of water collected may be expressed as Q Avt A1ki2t (7.16) where Q volume of water collected A area of cross section of the soil specimen t duration of water collection h L Graduated ﬂask Porous stone Soil specimen Figure 7.5 Constant-head permeability test 7.4 Laboratory Determination of Hydraulic Conductivity 167 And because i h L (7.17) where L length of the specimen, Eq. (7.17) can be substituted into Eq. (7.16) to yield h Q Aak b t L (7.18) or k QL Aht (7.19) Falling-Head Test A typical arrangement of the falling-head permeability test is shown in Figure 7.6. Water from a standpipe ﬂows through the soil. The initial head difference h1 at time t 0 is recorded, and water is allowed to ﬂow through the soil specimen such that the ﬁnal head difference at time t t2 is h2 . dh Stand pipe h1 h h2 Porous stone Soil specimen Figure 7.6 Falling-head permeability test 168 Chapter 7: Permeability The rate of ﬂow of the water through the specimen at any time t can be given by qk dh h A a L dt (7.20) where q flow rate a cross-sectional area of the standpipe A cross-sectional area of the soil specimen Rearrangement of Eq. (7.20) gives dt aL dh a b Ak h (7.21) Integration of the left side of Eq. (7.21) with limits of time from 0 to t and the right side with limits of head difference from h1 to h2 gives t h1 aL loge Ak h2 or k 2.303 h1 aL log10 At h2 (7.22) Example 7.1 Refer to the constant-head permeability test arrangement shown in Figure 7.5. A test gives these values: • • • • L 30 cm A area of the specimen 177 cm2 Constant-head difference, h 50 cm Water collected in a period of 5 min 350 cm3 Calculate the hydraulic conductivity in cm/sec. Solution From Eq. (7.19), QL Aht Given Q 350 cm3, L 30 cm, A 177 cm2, h 50 cm, and t 5 min, we have k k 1350 2 1302 1177 2 150 2 152 160 2 3.95 103 cm/sec ■ 7.4 Laboratory Determination of Hydraulic Conductivity 169 Example 7.2 For a falling-head permeability test, the following values are given: • • • • • Length of specimen 8 in. Area of soil specimen 1.6 in.2 Area of standpipe 0.06 in.2 Head difference at time t 0 20 in. Head difference at time t 180 sec 12 in. Determine the hydraulic conductivity of the soil in in./sec. Solution From Eq. (7.22), h1 aL log10 a b At h2 We are given a 0.06 in.2, L 8 in., A 1.6 in.2, t 180 sec, h1 20 in., and h2 300 mm, k 2.303 k 2.303 10.06 2 182 11.6 2 11802 log10 a 20 b 12 8.52 : 10ⴚ4 in./sec ■ Example 7.3 The hydraulic conductivity of a clayey soil is 3 107 cm/sec. The viscosity of water at 25°C is 0.0911 104 g # sec/cm2. Calculate the absolute permeability K of the soil. Solution From Eq. (7.13), k gw K 3 107 cm/sec h so 3 107 a 1 g/cm3 0.0911 104 bK K 0.2733 : 10ⴚ11 cm2 ■ 170 Chapter 7: Permeability Example 7.4 A permeable soil layer is underlain by an impervious layer, as shown in Figure 7.7a. With k 5.3 105 m/sec for the permeable layer, calculate the rate of seepage through it in m3/hr/m width if H 3 m and a 8°. Solution From Figure 7.7b, i L tan a head loss sin a L length a b cos a q kiA 1k 2 1sin a2 13 cos a2 11 2 k 5.3 105 m/sec q 15.3 10 5 2 1sin 8°2 13 cos 8°2 136002 0.0789 m3/hr/m c To change to m/hr a Groundwater table (free surface) H Direction of seepage a Impervious layer (a) Permeable layer Ground surface h L tan a 3 cos a (m) L cos a 8 a L (b) Figure 7.7 ■ 7.4 Laboratory Determination of Hydraulic Conductivity 171 Example 7.5 Find the ﬂow rate in m3/sec/m length (at right angles to the cross section shown) through the permeable soil layer shown in Figure 7.8 given H 8 m, H1 3 m, h 4 m, L 50 m, a 8, and k 0.08 cm/sec. h H H1 a Directio n of ﬂow L Impervious layer Permeable layer Figure 7.8 Flow through permeable layer Solution Hydraulic gradient 1i2 h L cos a From Eqs. (7.17) and (7.18), q kiA k a h cos a b 1H1 cos a 12 L 10.08 102 m/sec2 a 0.19 103 m3/sec/m 4 cos 8° b 13 cos 8° 1 2 50 ■ 172 Chapter 7: Permeability 7.5 Relationships for Hydraulic Conductivity— Granular Soil For fairly uniform sand (that is, sand with a small uniformity coefﬁcient), Hazen (1930) proposed an empirical relationship for hydraulic conductivity in the form k 1cm/sec2 cD210 (7.23) where c a constant that varies from 1.0 to 1.5 D10 the effective size, in mm Equation (7.23) is based primarily on Hazen’s observations of loose, clean, ﬁlter sands. A small quantity of silts and clays, when present in a sandy soil, may change the hydraulic conductivity substantially. Over the last several years, experimental observations have shown that the magnitude of c for various types of granular soils may vary by three orders of magnitude (Carrier, 2003) and, hence, is not very reliable. Another form of equation that gives fairly good results in estimating the hydraulic conductivity of sandy soils is based on the Kozeny-Carman equation (Kozeny, 1927; Carman, 1938, 1956). The derivation of this equation is not presented here. Interested readers are referred to any advanced soil mechanics book. According to the KozenyCarman equation k gw e3 1 CsS 2s T2 h 1 e (7.24) where Cs shape factor, which is a function of the shape of ﬂow channels Ss speciﬁc surface area per unit volume of particles T tortuosity of ﬂow channels gw unit weight of water h viscosity of permeant e void ratio For practical use, Carrier (2003) has modiﬁed Eq. (7.24) in the following manner. At 1 20°C, gw /h for water is about 9.93 104 a b . Also, (CsT 2 ) is approximately cm # s equal to 5. Substituting these values in Eq. (7.24), we obtain k 1.99 104 a 1 2 e3 b Ss 1 e (7.25) Again, Ss SF 1 b a Deff cm (7.26) 7.5 Relationships for Hydraulic Conductivity— Granular Soil 173 with Deff 100% fi ga b D1av2i (7.27) fi fraction of particles between two sieve sizes, in percent (Note: larger sieve, l; smaller sieve, s) D (av)i (cm) [Dli (cm)]0.5 [Dsi(cm)]0.5 SF shape factor where (7.28) Combining Eqs. (7.25), (7.26), (7.27), and (7.28), k 1.99 104 £ g 2 100% fi § a 1 2 e3 b a b SF 1e (7.29) 0.5 D0.5 li Dsi The magnitude of SF may vary between 6 to 8, depending on the angularity of the soil particles. Carrier (2003) further suggested a slight modiﬁcation to Eq. (7.29), which can be written as k 1.99 104 £ g 2 100% fi § a 1 2 e3 b a b SF 1e (7.30) D0.404 D0.595 li si Equation (7.30) suggests that k r e3 1e (7.31) The author recommends the use of Eqs. (7.30) and (7.31). It is important to note that Eqs. (7.23) and (7.31) assume that laminar ﬂow condition does exist. More recently, Chapuis (2004) proposed an empirical relationship for k in conjunction with Eq. (7.31) as k1cm/s2 2.4622 c D210 0.7825 e3 d 11 e2 (7.32) where D10 effective size (mm). The preceding equation is valid for natural, uniform sand and gravel to predict k that is in the range of 101 to 103 cm/s. This can be extended to natural, silty sands without plasticity. It is not valid for crushed materials or silty soils with some plasticity. Based on laboratory experimental results, Amer and Awad (1974) proposed the following relationship for k in granular soil: rw e3 2.32 k 3.5 10 4 a b C0.6 b u D10 a h 1e (7.33) 174 Chapter 7: Permeability where k is in cm/sec Cu uniformity coefficient D10 effective size (mm) w density of water (g/cm3) viscosity (g s/cm2) At 20 C ° , w 1 g/cm3 and 0.1 104 g s/cm2. So k 3.5 10 4 a e3 1 2.32 b C0.6 b u D10 a 1e 0.1 10 4 or k1cm/sec2 35 a e3 2.32 b C0.6 u 1D10 2 1e (7.34) Mention was made at the end of Section 7.1 that turbulent ﬂow conditions may exist in very coarse sands and gravels and that Darcy’s law may not be valid for these materials. However, under a low hydraulic gradient, laminar ﬂow conditions usually exist. Kenney, Lau, and Ofoegbu (1984) conducted laboratory tests on granular soils in which the particle sizes in various specimens ranged from 0.074 to 25.4 mm. The uniformity coefﬁcients, Cu , of these specimens ranged from 1.04 to 12. All permeability tests were conducted at a relative density of 80% or more. These tests showed that for laminar ﬂow conditions, K1mm2 2 10.05 to 12D25 (7.35) where D5 diameter (mm) through which 5% of soil passes. On the basis of laboratory experiments, the U.S. Department of Navy (1971) provided an empirical correlation between k (ft/min) and D10 (mm) for granular soils with the uniformity coefﬁcient varying between 2 and 12 and D10 /D5 1.4. This correlation is shown in Figure 7.9. Example 7.6 The hydraulic conductivity of a sand at a void ratio of 0.5 is 0.02 cm/sec. Estimate its hydraulic conductivity at a void ratio of 0.65. Solution From Eq. (7.31), e31 k1 1 e1 e32 k2 1 e2 10.5 2 3 1 0.5 0.02 10.652 3 k2 1 0.65 k2 0.04 cm/sec ■ 7.5 Relationships for Hydraulic Conductivity— Granular Soil 175 10 8 6 4 Hydraulic conductivity, k (ft /min) 2 1 0.8 0.6 Void ratio, e 0.7 0.6 0.5 0.4 0.3 0.4 0.2 D10 1.4 D5 0.1 0.08 0.06 Cu 2 to 12 0.04 0.02 0.01 0.1 0.2 0.4 0.6 0.8 1 D10 (mm) 2 4 6 8 10 Figure 7.9 Hydraulic conductivity of granular soils (After U.S. Department of Navy, 1971) Example 7.7 The grain-size distribution curve for a sand is shown in Figure 7.10. Estimate the hydraulic conductivity using Eq. (7.30). Given: the void ratio of the sand is 0.6. Use SF 7. Solution From Figure 7.10, the following table can be prepared. Sieve no. Sieve opening (cm) Percent passing 30 40 60 100 200 0.06 0.0425 0.02 0.015 0.0075 100 96 84 50 0 Fraction of particles between two consecutive sieves (%) 4 12 34 50 For fraction between Nos. 30 and 40 sieves; fi D0.404 li D0.595 si 10.06 2 0.404 4 81.62 10.04252 0.595 176 Chapter 7: Permeability 100 Percent passing 80 60 40 20 0 1.0 0.4 0.2 0.1 Grain size (mm) 0.04 0.02 0.01 Figure 7.10 For fraction between Nos. 40 and 60 sieves; fi D0.404 li D0.595 si 10.0425 2 12 440.76 10.022 0.595 0.404 Similarly, for fraction between Nos. 60 and 100 sieves; fi D0.404 D0.595 li si 34 2009.5 10.02 2 0.404 10.0152 0.595 And, for between Nos. 100 and 200 sieves; fi D0.404 li g D0.595 si 100% fi 50 5013.8 10.00752 0.595 10.015 2 100 0.0133 81.62 440.76 2009.5 5013.8 0.404 D0.404 D0.595 li si From Eq. (7.30), 1 2 0.63 b 0.0097 cm/s k 11.99 104 2 10.0133 2 2 a b a 7 1 0.6 ■ 177 7.6 Relationships for Hydraulic Conductivity— Cohesive Soils Example 7.8 Solve Example 7.7 using Eq. (7.32). Solution From Fig. 7.10, D10 0.09 mm. From Eq. (7.32), k 2.4622 c D210 0.7825 0.7825 e3 0.63 2.4622 c 10.092 2 0.0119 cm/sec d d 1e 1 0.6 ■ Example 7.9 Solve Example 7.7 using Eq. (7.34). Solution From Figure 7.10, D60 0.16 mm and D10 0.09 mm. Thus, Cu D60 0.16 1.78 D10 0.09 From Eq. (7.34), k 35 a 7.6 e3 0.63 2.32 b C0.6 b 11.782 0.6 10.09 2 2.32 0.025 cm/sec 35 a u 1D10 2 1e 1 0.6 ■ Relationships for Hydraulic Conductivity— Cohesive Soils The Kozeny-Carman equation [Eq. (7.24)] has been used in the past to see if it will hold good for cohesive soil. Olsen (1961) conducted hydraulic conductivity tests on sodium illite and compared the results with Eq. (7.24). This comparison is shown in Figure 7.11. The marked degrees of variation between the theoretical and experimental values arise from several factors, including deviations from Darcy’s law, high viscosity of the pore water, and unequal pore sizes. Taylor (1948) proposed a linear relationship between the logarithm of k and the void ratio as log k log ko eo e Ck (7.36) where ko in situ hydraulic conductivity at a void ratio eo k hydraulic conductivity at a void ratio e Ck hydraulic conductivity change index The preceding equation is a good correlation for eo less than about 2.5. In this equation, the value of Ck may be taken to be about 0.5eo. 178 Chapter 7: Permeability 105 104 Kaolinite 5 10 Hydraulic conductivity k (mm/sec) k (mm/sec) 106 107 108 Eq. (6.24) 106 Illite 107 108 109 Montmorillonite 1010 109 0.2 0.4 0.6 Porosity, n Sodium illite 101 N–NaCl 0.8 Sodium illite 104 N–NaCl Figure 7.11 Coefﬁcient of permeability for sodium illite (Based on Olsen, 1961) 1011 0.3 1.0 3.0 Void ratio, e 10 30 Figure 7.12 Variation of hydraulic conductivity of sodium clay minerals (Based on Mesri and Olson, 1971) For a wide range of void ratio, Mesri and Olson (1971) suggested the use of a linear relationship between log k and log e in the form log k A¿log e B¿ (7.37) Figure 7.12 shows the plot of log k versus log e obtained in the laboratory based on which Eq. (7.37) was proposed. Samarasinghe, et al. (1982) conducted laboratory tests on New Liskeard clay and proposed that, for normally consolidated clays, k Ca en b 1e (7.38) where C and n are constants to be determined experimentally. Tavenas, et al. (1983) also gave a correlation between the void ratio and the hydraulic conductivity of clayey soil. This correlation is shown in Figure 7.13. An important point to note, however, is that in Figure 7.13, PI, the plasticity index, and CF, the clay-size fraction in the soil, are in fraction (decimal) form. One should keep in mind, however, that any empirical relationship of this type is for estimation only, because the magnitude of k is a highly variable parameter and depends on several factors. 7.6 Relationships for Hydraulic Conductivity— Cohesive Soils 179 PI CF 1.25 2.8 1.0 2.4 Void ratio, e 2.0 0.75 1.6 1.2 0.5 0.8 0.4 1011 1010 109 5 109 k (m /sec) Figure 7.13 Variation of void ratio with hydraulic conductivity of clayey soils (Based on Tavenas, et al, 1983) Example 7.10 For a normally consolidated clay soil, the following values are given: Void ratio k (cm /sec) 1.1 0.9 0.302 107 0.12 107 Estimate the hydraulic conductivity of the clay at a void ratio of 0.75. Use Eq. (7.38). Solution From Eq. (7.38), kCa en b 1e en1 b k1 1 e1 en2 k2 a b 1 e2 a 11.12 n 0.302 107 1 1.1 7 10.9 2 n 0.12 10 1 0.9 1.9 1.1 n 2.517 a ba b 2.1 0.9 180 Chapter 7: Permeability 2.782 11.2222 n n log 12.782 2 log 11.2222 0.444 5.1 0.087 so kCa To ﬁnd C, 0.302 107 C c C e5.1 b 1e 11.1 2 5.1 1 1.1 d a 1.626 bC 2.1 10.302 107 2 12.12 1.626 0.39 107 Hence, k 10.39 107 cm/sec2 a en b 1e At a void ratio of 0.75, k 10.39 107 2 a ■ Directional Variation of Permeability Most soils are not isotropic with respect to permeability. In a given soil deposit, the magnitude of k changes with respect to the direction of ﬂow. Figure 7.14 shows a soil layer through which water ﬂows in a direction inclined at an angle with the vertical. Let the hydraulic conductivity in the vertical (a 0) and horizontal (a 90) directions be kV and kH, respectively. The magnitudes of kV and kH in a given soil depend on several factors, including the method of deposition in the ﬁeld. Figure 7.15 shows the laboratory test results obtained by Fukushima and Ishii (1986) related to kV and kH for compacted Masa-do soil (weathered granite). The soil a Soil layer Fl dir ow ec tio n 7.7 0.755.1 b 0.514 : 10ⴚ8 cm/sec 1 0.75 Figure 7.14 Directional variation of permeability Dry unit weight (kN/m3) 7.7 Directional Variation of Permeability 181 17.2 Compaction curve 16.7 16.2 102 k (cm/s) 103 Confining pressure (kN/m2) 104 20 105 200 kV 106 20 kH 107 12 200 16 20 Molding moisture content (%) 24 Figure 7.15 Variation of kV and kH for Masa-do soil compacted in the laboratory (Based on the results of Fukushima and Ishii, 1986) specimens were initially compacted at a certain moisture content, and the hydraulic conductivity was determined at 100% saturation. Note that, for any given molding moisture content and confining pressure, kH is larger than kV. There are several published results for fine-grained soils that show the ratio of kH/kV may vary over a wide range. Table 7.3 provided a summary of some of those studies. Table 7.3 kH/kV for Fine-Grained Soils—Summary of Several Studies Soil type kH/kV Reference Organic silt with peat Plastic marine clay Soft clay Varved clay Varved clay Varved clay Varved clay 1.2 to 1.7 1.2 1.5 1.5 to 1.7 1.5 3 to 15 4 to 40 Tsien (1955) Lumb and Holt (1968) Basett and Brodie (1961) Chan and Kenney (1973) Kenney and Chan (1973) Wu, et al. (1978) Casagrande and Poulos (1969) 182 Chapter 7: Permeability 7.8 Equivalent Hydraulic Conductivity in Stratiﬁed Soil In a stratiﬁed soil deposit where the hydraulic conductivity for ﬂow in a given direction changes from layer to layer, an equivalent hydraulic conductivity can be computed to simplify calculations. The following derivations relate to the equivalent hydraulic conductivities for ﬂow in vertical and horizontal directions through multilayered soils with horizontal stratiﬁcation. Figure 7.16 shows n layers of soil with ﬂow in the horizontal direction. Let us consider a cross section of unit length passing through the n layer and perpendicular to the direction of ﬂow. The total ﬂow through the cross section in unit time can be written as qv#1#H v1 # 1 # H1 v2 # 1 # H2 v3 # 1 # H3 p vn # 1 # Hn (7.39) v average discharge velocity where v1 , v2 , v3 , . . . , vn discharge velocities of flow in layers denoted by the subscripts If kH1, kH2, kH3, . . ., kHn are the hydraulic conductivities of the individual layers in the horizontal direction and kH(eq) is the equivalent hydraulic conductivity in the horizontal direction, then, from Darcy’s law, v kH1eq2ieq ; v1 kH1i1 ; v2 kH2i2 ; v3 kH3i3 ; . . . vn kHnin Substituting the preceding relations for velocities into Eq. (7.39) and noting that ieq i1 i2 i3 . . . in results in kH1eq2 kV1 kV2 Direction of ﬂow kV3 kVn 1 1kH1H1 kH2H2 kH3H3 p kHnHn 2 H kH1 kH2 kH3 kHn (7.40) H1 H2 H3 Hn H Figure 7.16 Equivalent hydraulic conductivity determination—horizontal ﬂow in stratiﬁed soil 7.8 Equivalent Hydraulic Conductivity in Stratiﬁed Soil 183 h3 h h2 h1 kV1 H2 kV2 H H3 kV3 Hn kVn kH1 kH2 kH3 kHn Direction of ﬂow H1 Figure 7.17 Equivalent hydraulic conductivity determination—vertical ﬂow in stratiﬁed soil Figure 7.17 shows n layers of soil with ﬂow in the vertical direction. In this case, the velocity of ﬂow through all the layers is the same. However, the total head loss, h, is equal to the sum of the head losses in all layers. Thus, v v1 v2 v3 p vn (7.41) h h1 h2 h3 p hn (7.42) and Using Darcy’s law, we can rewrite Eq. (7.41) as kV1eq2 a h b kV1i1 kV2i2 kV3i3 p kVnin H (7.43) where kV1, kV2, kV3, . . . , kVn are the hydraulic conductivities of the individual layers in the vertical direction and kV(eq) is the equivalent hydraulic conductivity. Again, from Eq. (7.42), h H1i1 H2i2 H3i3 p Hnin (7.44) Solving Eqs. (7.43) and (7.44) gives kV1eq2 H H1 H2 H3 Hn a b a b a b p a b kV1 kV2 kV3 kVn (7.45) 184 Chapter 7: Permeability Grain size 4 3 2 m 1m 1 0 20 30 40 50 60 Moisture content (%) 70 6m Boring no. 1 Sample no. 14 Depth, 53.0 ft Elevation, 570.5 ft m 2m 0 0.0 m 0 0.0 0.00 Distance along sample (in.) Coarse Transition Fine Coarse Trans. Fine Coarse Moisture content 80 0 50 100 Percent less than Figure 7.18 Variation of moisture content and grain-size distribution in New Liskeard varved soil. (Source: After “Laboratory Investigation of Permeability Ratio of New Liskeard Varved Clay,” by H. T. Chan and T. C. Kenney, 1973, Canadian Geotechnical Journal, 10(3), p. 453–472. ©2008 NRC Canada or its licensors. Reproduced with permission.) An excellent example of naturally deposited layered soil is varved soil, which is a rhythmically layered sediment of coarse and ﬁne minerals. Varved soils result from annual seasonal ﬂuctuation of sediment conditions in glacial lakes. Figure 7.18 shows the variation of moisture content and grain-size distribution in New Liskeard, Canada, varved soil. Each varve is about 41 to 51 mm (1.6 to 2.0 in.) thick and consists of two homogeneous layers of soil— one coarse and oneﬁne—with a transition layer between. Example 7.11 A layered soil is shown in Figure 7.19. Given: • H1 2 m • H2 3 m • H3 4 m k1 104 cm/sec k2 3.2 102 cm/sec k3 4.1 105 cm/sec Estimate the ratio of equivalent hydraulic conductivity, kH1eq2 kV1eq2 7.8 Equivalent Hydraulic Conductivity in Stratiﬁed Soil H1 k1 H2 k2 H3 k3 185 Figure 7.19 A layered soil proﬁle Solution From Eq. (7.40), kH1eq2 1 1k H kH2H2 kH3H3 2 H H1 1 1 3 1104 2 122 13.2 102 2 132 14.1 105 2 142 4 12 3 42 107.07 104 cm/sec Again, from Eq. (7.45), kV1eq2 H H1 H2 H3 a b a b a b kV1 kV2 kV3 234 2 3 4 a 4 b a b a b 10 3.2 102 4.1 105 0.765 104 cm/sec Hence, kH1eq2 kV1eq2 107.07 104 139.96 0.765 104 ■ 186 Chapter 7: Permeability Example 7.12 Figure 7.20 shows three layers of soil in a tube that is 100 mm 100 mm in cross section. Water is supplied to maintain a constant-head difference of 300 mm across the sample. The hydraulic conductivities of the soils in the direction of ﬂow through them are as follows: Soil k (cm /sec) A B C 102 3 103 4.9 104 Find the rate of water supply in cm3/hr. Solution From Eq. (7.45), kV1eq2 H 450 H1 H2 H3 150 150 150 a ba ba b a 2 b a ba b 3 k1 k2 k3 10 3 10 4.9 104 0.001213 cm/sec q kV1eq2iA 10.0012132 a 300 100 100 ba b 450 10 10 0.0809 cm3/sec 291.24 cm3/hr Water supply Constant-head difference 300 mm hA hB A B C 150 mm 150 mm 150 mm Figure 7.20 Three layers of soil in a tube 100 mm 100 mm in cross section ■ 7.9 Permeability Test in the Field by Pumping from Wells 7.9 187 Permeability Test in the Field by Pumping from Wells In the ﬁeld, the average hydraulic conductivity of a soil deposit in the direction of ﬂow can be determined by performing pumping tests from wells. Figure 7.21 shows a case where the top permeable layer, whose hydraulic conductivity has to be determined, is unconﬁned and underlain by an impermeable layer. During the test, water is pumped out at a constant rate from a test well that has a perforated casing. Several observation wells at various radial distances are made around the test well. Continuous observations of the water level in the test well and in the observation wells are made after the start of pumping, until a steady state is reached. The steady state is established when the water level in the test and observation wells becomes constant. The expression for the rate of ﬂow of groundwater into the well, which is equal to the rate of discharge from pumping, can be written as q ka dh b 2prh dr (7.46) or r1 r2 2pk dr a b r q h1 h dh h2 Thus, 2.303q log10 a r1 b r2 p1h21 h22 2 k r dr (7.47) r1 r2 Water table before pumping Draw-down curve during pumping dh h2 h1 h Impermeable layer Test well Observation wells Figure 7.21 Pumping test from a well in an unconﬁned permeable layer underlain by an impermeable stratum. 188 Chapter 7: Permeability From ﬁeld measurements, if q, r1 , r2 , h1 , and h2 are known, the hydraulic conductivity can be calculated from the simple relationship presented in Eq. (7.47). This equation also can be written as 2.303q log10 a r1 b r2 k 1cm/sec2 14.7p1h21 h22 2 (7.48) where q is in gpm and h1 and h2 are in ft. The average hydraulic conductivity for a conﬁned aquifer can also be determined by conducting a pumping test from a well with a perforated casing that penetrates the full depth of the aquifer and by observing the piezometric level in a number of observation wells at various radial distances (Figure 7.22). Pumping is continued at a uniform rate q until a steady state is reached. Because water can enter the test well only from the aquifer of thickness H, the steady state of discharge is qk a dh b 2prH dr (7.49) or r1 r2 dr r h1 h2 2pkH dh q r r1 r2 dr Piezometric level before pumping Piezometric level during pumping h dh h2 h1 H Impermeable layer Conﬁned aquifer Test well Observation wells Figure 7.22 Pumping test from a well penetrating the full depth in a conﬁned aquifer 7.10 In Situ Hydraulic Conductivity of Compacted Clay Soils 189 This gives the hydraulic conductivity in the direction of ﬂow as q log10 a r1 b r2 k 2.727H1h1 h2 2 7.10 (7.50) In Situ Hydraulic Conductivity of Compacted Clay Soils Daniel (1989) provided an excellent review of nine methods to estimate the in situ hydraulic conductivity of compacted clay layers. Three of these methods are described. Boutwell Permeameter A schematic diagram of the Boutwell permeameter is shown in Figure 7.23. A hole is ﬁrst drilled and a casing is placed in it (Figure 7.23a). The casing is ﬁlled with water and a falling-head permeability test is conducted. Based on the test results, the hydraulic conductivity k1 is calculated as k1 h1 pd2 ln a b pD1t2 t1 2 h2 d d D D (7.51) h L (a) (b) Compacted clay Grout Casing Figure 7.23 Permeability test with Boutwell permeameter h 190 Chapter 7: Permeability where d diameter of the standpipe D diameter of the casing h1 head at time t1 h2 head at time t2 After the hydraulic conductivity is determined, the hole is deepened by augering, and the permeameter is reassembled as shown in Figure 7.23b. A falling-head hydraulic conductivity test is conducted again. The hydraulic conductivity is calculated as k2 h1 A¿ ln a b B¿ h2 (7.52) where A¿ d2 e ln c B¿ 8D L¿ L¿ 2 1 a b df D B D L¿ L¿ 1t2 t1 2 e 1 0.562 exp c 1.57 a b d f D D (7.53) (7.54) The anisotropy with respect to hydraulic conductivity is determined by referring to Figure 7.24, which is a plot of k2 /k1 versus m (m 1kH/kV ) for various values of L/D. Figure 7.24 can be used to determine m using the experimental values of k2 /k1 and L/D. The plots in this ﬁgure are determined from ln3 1L¿/D2 21 1L¿/D2 2 4 k2 m k1 ln3 1mL¿/D2 21 1mL¿/D2 2 4 (7.55) Once m is determined, we can calculate kH mk1 (7.56) and kV LD 1.0 12 k1 m (7.57) 1.5 2.0 m 8 4 0 1 2 3 k2/k1 4 Figure 7.24 Variation of k2/k1 with m [Eq. (7.55)] 7.10 In Situ Hydraulic Conductivity of Compacted Clay Soils 191 Constant-Head Borehole Permeameter Figure 7.25 shows a constant-head borehole permeameter. In this arrangement a constant head h is maintained by supplying water, and the rate of ﬂow q is measured. The hydraulic conductivity can be calculated as q k r 2R 1 3F1 1F2/A– 2 4 R h r 2 2 (7.58) where F1 (7.59) 4.11711 R2 2 ln1R 2R2 1 2 31 11/R2 2 4 0.5 4.280 F2 ln1R 2R2 1 2 A– 1 ar 2 (7.60) (7.61) (7.62) Typical values of a range from 0.002 to 0.01 cm1 for ﬁne-grained soil. Porous Probes Porous probes (Figure 7.26) are pushed or driven into the soil. Constant- or falling-head permeability tests are performed. The hydraulic conductivity is calculated as follows: Stand pipe h 2r Granular backﬁll Perforated casing Compacted clay Seal Figure 7.25 Borehole test with constant water level 192 Chapter 7: Permeability d d h Seal h Seal L1 L1 D D (a) (b) Figure 7.26 Porous probe: (a) test with permeable base; (b) test with impermeable base The constant head is given by q Fh (7.63) h1 pd2/4 ln a b F1t2 t1 2 h2 (7.64) k The falling head is given by k For probes with permeable bases (Figure 7.26a), F 2pL1 ln3 1L1/D2 21 1L1/D2 2 4 (7.65) For probes with impermeable bases (Figure 7.26b), F 7.11 2pL1 ln3 1L1/D2 21 1L1/D2 2 4 2.8D (7.66) Summary and General Comments In this chapter, we discussed Darcy’s law, the deﬁnition of hydraulic conductivity, laboratory determinations of hydraulic conductivity and the empirical relations for it, and ﬁeld determinations of hydraulic conductivity of various types of soil. Hydraulic conductivity of various soil layers is highly variable. The empirical relations for hydraulic conductivity should be used as a general guide for all practical considerations. The accuracy of the values of k determined in the laboratory depends on several factors: 1. Temperature of the ﬂuid 2. Viscosity of the ﬂuid Problems 3. 4. 5. 6. 193 Trapped air bubbles present in the soil specimen Degree of saturation of the soil specimen Migration of ﬁnes during testing Duplication of ﬁeld conditions in the laboratory The hydraulic conductivity of saturated cohesive soils also can be determined by laboratory consolidation tests. The actual value of the hydraulic conductivity in the ﬁeld also may be somewhat different than that obtained in the laboratory because of the nonhomogeneity of the soil. Hence, proper care should be taken in assessing the order of the magnitude of k for all design considerations. Problems 7.1 7.2 7.3 7.4 7.5 Refer to the constant-head arrangement shown in Figure 7.5. For a test, the following are given: • L 18 in. • A area of the specimen 3.5 in.2 • Constant-head difference h 28 in. • Water collected in 3 min 21.58 in.3 Calculate the hydraulic conductivity (in./sec). Refer to Figure 7.5. For a constant-head permeability test in a sand, the following are given: • L 300 mm • A 175 cm2 • h 500 mm • Water collected in 3 min 620 cm3 • Void ratio of sand 0.58 Determine a. Hydraulic conductivity, k (cm/sec) b. Seepage velocity In a constant-head permeability test in the laboratory, the following are given: L 12 in. and A 15 in.2. If the value of k 0.006 in./sec and a ﬂow rate of 450 in.3/hr must be maintained through the soil, what is the head difference, h, across the specimen?Also, determine the discharge velocity under the test conditions. For a falling-head permeability test, the following are given: • Length of the soil specimen 20 in. • Area of the soil specimen 4 in.2 • Area of the standpipe 0.2 in.2 • Head difference at time t 0 is 30 in. • Head difference at time t 10 min is 12 in. a. Determine the hydraulic conductivity of the soil (in./min). b. What was the head difference at time t 5 min? For a falling-head permeability test, the following are given: length of specimen 380 mm; area of specimen 6.5 cm2; k 0.175 cm/min. What should be the area of the standpipe for the head to drop from 650 cm to 300 cm in 8 min? 194 Chapter 7: Permeability 7.6 For a falling-head permeability test, the following are given: • Length of soil specimen 700 mm • Area of the soil specimen 20 cm2 • Area of the standpipe 1.05 cm2 • Head difference at time t 0 is 800 mm • Head difference at time t 8 min is 500 mm a. Determine the absolute permeability of the soil. b. What is the head difference at time t 6 min? Assume that the test was conducted at 20C, and at 20C, gw 9.789 kN/m3 and h 1.005 103 N # s/m2. A sand layer of the cross-sectional area shown in Fig. 7.27 has been determined to exist for a 800-m length of the levee. The hydraulic conductivity of the sand layer is 2.8 m/day. Determine the quantity of water which ﬂows into the ditch in m3/min. A permeable soil layer is underlain by an impervious layer, as shown in Figure 7.28. With k 5.2 104 cm/sec for the permeable layer, calculate the rate of seepage through it in m3/hr/m length if H 3.8 m and a 8. 7.7 7.8 Elv. 160 m Levee Elv. 145 m 1.6 m Ditch 180 m Impervious Sand Figure 7.27 a Groundwater table (free surface) H Direction of seepage a Impervious layer Figure 7.28 Ground surface Problems h H H1 a Directio n of ﬂo w L Impervious layer Figure 7.29 7.9 7.10 7.11 7.12 7.13 7.14 Refer to Figure 7.29. Find the ﬂow rate in m3/sec/m length (at right angles to the cross section shown) through the permeable soil layer. Given: H 5 m, H1 2.8 m, h 3.1 m, L 60 m, a 5, k 0.05 cm/sec. The hydraulic conductivity of a sand at a void ratio of 0.5 is 0.022 cm/sec. Estimate its hydraulic conductivity at a void ratio of 0.7. Use Eq. (7.31). For a sand, the following are given: porosity, n 0.31 and k 0.2 ft/min. Determine k when n 0.4. Use Eq. (7.31). The maximum dry density determined in the laboratory for a quartz sand is 1800 kg/m3. In the ﬁeld, if the relative compaction is 80%, determine the hydraulic conductivity of the sand in the ﬁeld-compaction condition (given that D10 0.15 mm, Cu 2.2 and Gs 2.66). Use Eq. (7.34). For a sandy soil, the following are given: • Maximum void ratio 0.7 • Minimum void ratio 0.46 • D10 0.2 mm Determine the hydraulic conductivity of the sand at a relative density of 60%. Use Eq. (7.32). The sieve analysis for a sand is given in the following table. Estimate the hydraulic conductivity of the sand at a void ratio of 0.5. Use Eq. (7.30) and SF 6.5. U.S. Sieve No. Percent passing 30 40 60 100 200 100 80 68 28 0 195 196 Chapter 7: Permeability 7.15 For a normally consolidated clay, the following are given: Void ratio, e k (cm /sec) 0.8 1.4 1.2 106 3.6 106 Estimate the hydraulic conductivity of the clay at a void ratio, e 0.9. Use Eq. (7.38). 7.16 The in situ void ratio of a soft clay deposit is 2.1 and the hydraulic conductivity of the clay at this void ratio is 0.91 106 cm/sec. What is the hydraulic conductivity if the soil is compressed to have a void ratio of 1.1. Use Eq. (7.36). 7.17 A layered soil is shown in Figure 7.30. Given that • H1 1.5 m • k1 105 cm/sec • H2 2.5 m • k2 3.0 103 cm/sec • H3 3.0 m • k3 3.5 105 cm/sec Estimate the ratio of equivalent hydraulic conductivity, kH(eq) /kV(eq) . 7.18 A layered soil is shown in Figure 7.31. Estimate the ratio of equivalent hydraulic conductivity, kH(eq) /kV(eq) . 1.5 m 1m H1 1.5 m H2 1m H3 k 2 10 3 cm/sec (top layer) k 2 10 4 cm/sec k1 Figure 7.30 k2 k 10 4 cm/sec k 3 10 4 cm/sec (bottom layer) k3 Figure 7.31 References AMER, A. M., and AWAD, A. A. (1974). “Permeability of Cohesionless Soils,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 100, No. GT12, 1309–1316. BASETT, D. J., and BRODIE, A. F. (1961). “A Study of Matabitchuan Varved Clay,” Ontario Hydroelectric Research News, Vol. 13, No. 4, 1–6. CARMAN, P. C. (1938). “The Determination of the Speciﬁc Surface of Powders.” J. Soc. Chem. Ind. Trans., Vol. 57, 225. References 197 CARMAN, P. C. (1956). Flow of Gases through Porous Media, Butterworths Scientiﬁc Publications, London. CARRIER III, W. D. (2003). “Goodbye, Hazen; Hello, Kozeny-Carman,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 129, No. 11, 1054–1056. CASAGRANDE, L., and POULOS, S. J. (1969). “On the Effectiveness of Sand Drains,” Canadian Geotechnical Journal, Vol. 6, No. 3, 287–326. CHAN, H. T., and KENNEY, T. C. (1973). “Laboratory Investigation of Permeability Ratio of New Liskeard Varved Soil,” Canadian Geotechnical Journal, Vol. 10, No. 3, 453–472. CHAPUIS, R. P. (2004). “Predicting the Saturated Hydraulic Conductivity of Sand and Gravel Using Effective Diameter and Void Ratio,” Canadian Geotechnical Journal, Vol. 41, No. 5, 787–795. CHAPUIS, R. P., GILL, D. E., and BAASS, K. (1989). “Laboratory Permeability Tests on Sand: Inﬂuence of Compaction Method on Anisotropy,” Canadian Geotechnical Journal, Vol. 26, 614–622. DANIEL, D. E. (1989). “In Situ Hydraulic Conductivity Tests for Compacted Clay,” Journal of Geotechnical Engineering, ASCE, Vol. 115, No. 9, 1205–1226. DARCY, H. (1856). Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris. FUKUSHIMA, S., and ISHII, T. (1986). “An Experimental Study of the Influence of Confining Pressure on Permeability Coefficients of Filldam Core Materials,” Soils and Foundations, Vol. 26, No. 4, 32–46 HANSBO, S. (1960). “Consolidation of Clay with Special Reference to Inﬂuence of Vertical Sand Drains,” Swedish Geotechnical Institute, Proc. No. 18, 41–61. HAZEN , A. (1930). “Water Supply,” in American Civil Engineers Handbook, Wiley, New York. KENNEY, T. C., and CHAN, H. T. (1973). “Field Investigation of Permeability Ratio of New Liskeard Varved Soil,” Canadian Geotechnical Journal, Vol. 10, No. 3, 473–488. KENNEY, T. C., LAU, D., and OFOEGBU, G. I. (1984). “Permeability of Compacted Granular Materials,” Canadian Geotechnical Journal, Vol. 21, No. 4, 726–729. KOZENY , J. (1927). “Ueber kapillare Leitung des Wassers im Boden.” Wien, Akad. Wiss., Vol. 136, No. 2a, 271. KRUMBEIN, W. C., and MONK, G. D. (1943). “Permeability as a Function of the Size Parameters of Unconsolidated Sand,” Transactions, AIMME (Petroleum Division), Vol. 151, 153–163. LUMB, P., and HOLT, J. K. (1968). “The Undrained Shear Strength of a Soft Marine Clay from Hong Kong,” Geotechnique, Vol. 18, 25–36. MESRI, G., and OLSON, R. E. (1971). “Mechanism Controlling the Permeability of Clays,” Clay and Clay Minerals, Vol. 19, 151–158. MITCHELL, J. K. (1976). Fundamentals of Soil Behavior, Wiley, New York. OLSEN, H. W. (1961). “Hydraulic Flow Through Saturated Clay,” Sc.D. Thesis, Massachusetts Institute of Technology. SAMARASINGHE, A. M., HUANG, Y. H., and DRNEVICH, V. P. (1982). “Permeability and Consolidation of Normally Consolidated Soils,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 108, No. GT6, 835–850. TAVENAS, F., JEAN, P., LEBLOND, F. T. P., and LEROUEIL, S. (1983). “The Permeability of Natural Soft Clays. Part II: Permeability Characteristics,” Canadian Geotechnical Journal, Vol. 20, No. 4, 645–660. TSIEN, S. I. (1955). “Stabilization of Marsh Deposit,” Highway Research Record, Bulletin 115, 15–43. U.S. DEPARTMENT OF NAVY (1971). “Design Manual—Soil Mechanics, Foundations, and Earth Structures,” NAVFAC DM-7, U.S. Government Printing Ofﬁce, Washington, D.C. Wu, T. H., Chang, N. Y., AND Ali, E. M. (1978). “Consolidation and Strength Properties of a Clay,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 104, No. GT7, 899–905. 8 Seepage In the preceding chapter, we considered some simple cases for which direct application of Darcy’s law was required to calculate the ﬂow of water through soil. In many instances, the ﬂow of water through soil is not in one direction only, nor is it uniform over the entire area perpendicular to the ﬂow. In such cases, the groundwater ﬂow is generally calculated by the use of graphs referred to as ﬂow nets. The concept of the ﬂow net is based on Laplace’s equation of continuity, which governs the steady ﬂow condition for a given point in the soil mass. In the following sections of this chapter, the derivation of Laplace’s equation of continuity will be presented along with its application to seepage problems. 8.1 Laplace’s Equation of Continuity To derive the Laplace differential equation of continuity, let us consider a single row of sheet piles that have been driven into a permeable soil layer, as shown in Figure 8.1a. The row of sheet piles is assumed to be impervious. The steady state ﬂow of water from the upstream to the downstream side through the permeable layer is a two-dimensional ﬂow. For ﬂow at a point A, we consider an elemental soil block. The block has dimensions dx, dy, and dz (length dy is perpendicular to the plane of the paper); it is shown in an enlarged scale in Figure 8.1b. Let vx and vz be the components of the discharge velocity in the horizontal and vertical directions, respectively. The rate of ﬂow of water into the elemental block in the horizontal direction is equal to vx dz dy, and in the vertical direction it is vz dx dy. The rates of outﬂow from the block in the horizontal and vertical directions are, respectively, a vx 0vx dx b dz dy 0x a vz 0vz and 198 0z dz b dx dy 8.1 Laplace’s Equation of Continuity 199 Sheet pile H1 h H2 A dz Impermeable layer dx (a) ( z z z dz) dx dy dy x ( x dz dy x x dx) dz dy dz dx z dx dy (b) Figure 8.1 (a) Single-row sheet piles driven into permeable layer; (b) ﬂow at A Assuming that water is incompressible and that no volume change in the soil mass occurs, we know that the total rate of inﬂow should equal the total rate of outﬂow. Thus, c a vx 0vz 0vx dx b dz dy a vz dz b dx dy d 3vx dz dy vz dx dy4 0 0x 0z or 0vz 0vx 0 0x 0z (8.1) 200 Chapter 8: Seepage With Darcy’s law, the discharge velocities can be expressed as vx kxix kx 0h 0x (8.2) vz kziz kz 0h 0z (8.3) and where kx and kz are the hydraulic conductivities in the horizontal and vertical directions, respectively. From Eqs. (8.1), (8.2), and (8.3), we can write kx 0 2h 0 2h kz 2 0 2 0x 0z (8.4) If the soil is isotropic with respect to the hydraulic conductivity—that is, kx kz—the preceding continuity equation for two-dimensional ﬂow simpliﬁes to 0 2h 0 2h 0 0x2 0z2 8.2 (8.5) Continuity Equation for Solution of Simple Flow Problems The continuity equation given in Eq. (8.5) can be used in solving some simple ﬂow problems. To illustrate this, let us consider a one-dimensional ﬂow problem, as shown in Figure 8.2, in which a constant head is maintained across a two-layered soil for the ﬂow of water. The head difference between the top of soil layer no. 1 and the bottom of soil layer no. 2 is h1 . Because the ﬂow is in only the z direction, the continuity equation [Eq. (8.5)] is simpliﬁed to the form 0 2h 0 0z2 (8.6) h A1z A2 (8.7) or where A1 and A2 are constants. To obtain A1 and A2 for ﬂow through soil layer no. 1, we must know the boundary conditions, which are as follows: Condition 1: At z 0, h h1 . Condition 2: At z H1 , h h2 . 8.2 Continuity Equation for Solution of Simple Flow Problems 201 Water supply h1 h z H1 h2 H2 Soil 1 (k1) Soil 2 (k 2 ) Figure 8.2 Flow through a two-layered soil Combining Eq. (8.7) and Condition 1 gives A2 h1 (8.8) Similarly, combining Eq. (8.7) and Condition 2 with Eq. (8.8) gives h2 A1H1 h1 or A1 a h1 h2 b H1 (8.9) Combining Eqs. (8.7), (8.8), and (8.9), we obtain h a h1 h2 b z h1 H1 1for 0 z H1 2 (8.10) For ﬂow through soil layer no. 2, the boundary conditions are Condition 1: At z H1 , h h2 . Condition 2: At z H1 + H2 , h 0. From Condition 1 and Eq. (8.7), A2 h2 A1H1 (8.11) 202 Chapter 8: Seepage Also, from Condition 2 and Eqs. (8.7) and (8.11), 0 A1 1H1 H2 2 1h2 A1H1 2 A1H1 A1H2 h2 A1H1 0 or A1 h2 H2 (8.12) So, from Eqs. (8.7), (8.11), and (8.12), h a H1 h2 b z h2 a 1 b H2 H2 1for H1 z H1 H2 2 (8.13) At any given time, ﬂow through soil layer no. 1 equals ﬂow through soil layer no. 2, so q k1 a h1 h2 h2 0 b A k2 a bA H1 H2 where A area of cross section of the soil k1 hydraulic conductivity of soil layer no. 1 k2 hydraulic conductivity of soil layer no. 2 or h2 h1k1 k1 k2 H1 a b H1 H2 (8.14) Substituting Eq. (8.14) into Eq. (8.10), we obtain h h1 a 1 k2z b k1H2 k2H1 1for 0 z H1 2 (8.15) Similarly, combining Eqs. (8.13) and (8.14) gives h h1 c a k1 b 1H1 H2 z2 d k1H2 k2H1 1for H1 z H1 H2 2 (8.16) 8.2 Continuity Equation for Solution of Simple Flow Problems 203 Example 8.1 Refer to Figure 8.2. Given: H1 12 in., H2 20 in., h1 24 in., h 20 in., z 8 in., k1 0.026 in./sec, and diameter of the soil specimen is D 3 in. Determine the rate of flow of water through the two-layered soil (in.3/hr). Solution Since z 8 in. is located in soil layer 1, Eq. (8.15) is valid. Thus, h h1 a 1 20 24D1 k2z b h1D1 k1H2 k2H1 8 k1 a b 20 12 k2 z T k1 a b H2 H1 k2 T k1 1.795 1.8 k2 Given that k1 0.026 in./sec, k2 k1 0.026 0.0144 in./sec 1.8 1.8 The rate of flow is q keq iA i A keq h1 24 0.75 H1 H2 12 20 p p 2 D 132 2 7.069 in.2 4 4 H1 H2 12 20 0.0173 in./sec 62.28 in./hr H1 H2 12 20 k1 k2 0.026 0.0144 Thus, q keq iA (62.28)(0.75)(7.069) 330.19 in.3/hr ■ 204 Chapter 8: Seepage 8.3 Flow Nets The continuity equation [Eq. (8.5)] in an isotropic medium represents two orthogonal families of curves—that is, the ﬂow lines and the equipotential lines. A ﬂow line is a line along which a water particle will travel from upstream to the downstream side in the permeable soil medium. An equipotential line is a line along which the potential head at all points is equal. Thus, if piezometers are placed at different points along an equipotential line, the water level will rise to the same elevation in all of them. Figure 8.3a demonstrates the Sheet pile H1 H2 Flow line kx kz k Equipotential line Impervious layer (a) Sheet pile Water level H1 Water level b d a e H2 c kx kz k Nf 4 Nd 6 f g Impervious layer (b) Figure 8.3 (a) Deﬁnition of ﬂow lines and equipotential lines; (b) completed ﬂow net 205 8.4 Seepage Calculation from a Flow Net deﬁnition of ﬂow and equipotential lines for ﬂow in the permeable soil layer around the row of sheet piles shown in Figure 8.1 (for kx kz k). A combination of a number of ﬂow lines and equipotential lines is called a ﬂow net. As mentioned in the introduction, ﬂow nets are constructed for the calculation of groundwater ﬂow and the evaluation of heads in the media. To complete the graphic construction of a ﬂow net, one must draw the ﬂow and equipotential lines in such a way that 1. The equipotential lines intersect the ﬂow lines at right angles. 2. The ﬂow elements formed are approximate squares. Figure 8.3b shows an example of a completed ﬂow net. One more example of ﬂow net in isotropic permeable layer are given in Figure 8.4. In these ﬁgures, Nf is the number of ﬂow channels in the ﬂow net, and Nd is the number of potential drops (deﬁned later in this chapter). Drawing a ﬂow net takes several trials. While constructing the ﬂow net, keep the boundary conditions in mind. For the ﬂow net shown in Figure 8.3b, the following four boundary conditions apply: Condition 1: The upstream and downstream surfaces of the permeable layer (lines ab and de) are equipotential lines. Condition 2: Because ab and de are equipotential lines, all the ﬂow lines intersect them at right angles. Condition 3: The boundary of the impervious layer—that is, line fg—is a ﬂow line, and so is the surface of the impervious sheet pile, line acd. Condition 4: The equipotential lines intersect acd and fg at right angles. H H1 H2 Toe ﬁlter kx kz k Nf 5 Nd 9 Figure 8.4 Flow net under a dam with toe ﬁlter 8.4 Seepage Calculation from a Flow Net In any ﬂow net, the strip between any two adjacent ﬂow lines is called a ﬂow channel. Figure 8.5 shows a ﬂow channel with the equipotential lines forming square elements. Let h1 , h2 , h3 , h4 , . . ., hn be the piezometric levels corresponding to the equipotential lines. The rate of seepage through the ﬂow channel per unit length (perpendicular to the vertical section through the permeable layer) can be calculated as follows. Because there is no ﬂow across the ﬂow lines, ¢q1 ¢q2 ¢q3 p ¢q (8.17) 206 Chapter 8: Seepage h1 q h2 l1 h3 l1 q1 l2 h4 l2 l3 q2 l3 q q3 Figure 8.5 Seepage through a ﬂow channel with square elements From Darcy’s law, the ﬂow rate is equal to kiA. Thus, Eq. (8.17) can be written as ¢q k a h1 h2 h2 h3 h3 h4 b l1 k a b l2 k a b l3 p l1 l2 l3 (8.18) Eq. (8.18) shows that if the ﬂow elements are drawn as approximate squares, the drop in the piezometric level between any two adjacent equipotential lines is the same. This is called the potential drop. Thus, h1 h2 h2 h3 h3 h4 p H Nd (8.19) and ¢q k H Nd (8.20) where H head difference between the upstream and downstream sides Nd number of potential drops In Figure 8.3b, for any ﬂow channel, H H1 H2 and Nd 6. If the number of ﬂow channels in a ﬂow net is equal to Nf , the total rate of ﬂow through all the channels per unit length can be given by qk HNf Nd (8.21) Although drawing square elements for a ﬂow net is convenient, it is not always necessary. Alternatively, one can draw a rectangular mesh for a ﬂow channel, as shown in Figure 8.6, provided that the width-to-length ratios for all the rectangular elements in the ﬂow net are the same. In this case, Eq. (8.18) for rate of ﬂow through the channel can be modiﬁed to h2 h3 h3 h4 h1 h2 (8.22) b b1 k a b b2 k a b b3 p l1 l2 l3 If b1/l1 b2/l2 b3/l3 p n (i.e., the elements are not square), Eqs. (8.20) and (8.21) can be modiﬁed to ¢q k a ¢q kH a n b Nd (8.23) 8.4 Seepage Calculation from a Flow Net 207 h1 q h2 l1 h3 b1 l2 q1 h4 b2 l3 b3 q q2 q3 Figure 8.6 Seepage through a ﬂow channel with rectangular elements and q kH a Nf Nd bn (8.24) Figure 8.7 shows a ﬂow net for seepage around a single row of sheet piles. Note that ﬂow channels 1 and 2 have square elements. Hence, the rate of ﬂow through these two channels can be obtained from Eq. (8.20): ¢q1 ¢q2 k k 2kH H H Nd Nd Nd However, ﬂow channel 3 has rectangular elements. These elements have a width-to-length ratio of about 0.38; hence, from Eq. (8.23) ¢q3 k H10.38 2 Nd Water level H 5.6 m Water table 2.2 m Ground surface b d a c 4.1 m e Flow channel 1 l 1 b Flow channel 2 l 1 b Flow channel 3 l 1 b 0.38 Scale 5m Impervious layer Figure 8.7 Flow net for seepage around a single row of sheet piles 208 Chapter 8: Seepage So, the total rate of seepage can be given as q ¢q1 ¢q2 ¢q3 2.38 kH Nd (8.25) Example 8.2 A ﬂow net for ﬂow around a single row of sheet piles in a permeable soil layer is shown in Figure 8.7. Given that kx kz k 5 103 cm/sec, determine a. How high (above the ground surface) the water will rise if piezometers are placed at points a and b. b. The total rate of seepage through the permeable layer per unit length c. The approximate average hydraulic gradient at c. Solution Part a From Figure 8.7, we have Nd 6, H1 5.6 m, and H2 2.2 m. So the head loss of each potential drop is ¢H H1 H2 5.6 2.2 0.567 m Nd 6 At point a, we have gone through one potential drop. So the water in the piezometer will rise to an elevation of (5.6 0.567) 5.033 m above the ground surface At point b, we have five potential drops. So the water in the piezometer will rise to an elevation of [5.6 (5)(0.567)] 2.765 m above the ground surface Part b From Eq. (8.25), q 2.38 k1H1 H2 2 Nd 12.38 2 15 10 5 m/sec2 15.6 2.22 6.74 10 5 6 m /sec/m 3 Part c The average hydraulic gradient at c can be given as i head loss ¢H 0.567 m 0.138 average length of flow between d and e ¢L 4.1 m (Note: The average length of flow has been scaled.) ■ 8.5 Flow Nets in Anisotropic Soil 8.5 209 Flow Nets in Anisotropic Soil The ﬂow-net construction described thus far and the derived Eqs. (8.21) and (8.24) for seepage calculation have been based on the assumption that the soil is isotropic. However, in nature, most soils exhibit some degree of anisotropy. To account for soil anisotropy with respect to hydraulic conductivity, we must modify the ﬂow net construction. The differential equation of continuity for a two-dimensional ﬂow [Eq. (8.4)] is kx 0 2h 0 2h k 0 z 0x2 0z2 For anisotropic soils, kx kz . In this case, the equation represents two families of curves that do not meet at 90. However, we can rewrite the preceding equation as 0 2h 0 2h 2 0 2 1kz /kx 2 0x 0z (8.26) Substituting x¿ 1kz /kx x, we can express Eq. (8.26) as 0 2h 0 2h 0 0x¿ 2 0z2 (8.27) Now Eq. (8.27) is in a form similar to that of Eq. (8.5), with x replaced by x, which is the new transformed coordinate. To construct the ﬂow net, use the following procedure: Step 1: Adopt a vertical scale (that is, z axis) for drawing the cross section. Step 2: Adopt a horizontal scale (that is, x axis) such that horizontal scale 1kz/kx vertical scale. Step 3: With scales adopted as in Steps 1 and 2, plot the vertical section through the permeable layer parallel to the direction of ﬂow. Step 4: Draw the ﬂow net for the permeable layer on the section obtained from Step 3, with ﬂow lines intersecting equipotential lines at right angles and the elements as approximate squares. The rate of seepage per unit length can be calculated by modifying Eq. (8.21) to q 2kxkz where HNf Nd (8.28) H total head loss Nf and Nd number of flow channels and potential drops, respectively 1from flow net drawn in Step 42 Note that when ﬂow nets are drawn in transformed sections (in anisotropic soils), the ﬂow lines and the equipotential lines are orthogonal. However, when they are redrawn in a true section, these lines are not at right angles to each other. This fact is shown in Figure 8.8. In this ﬁgure, it is assumed that kx 6kz . Figure 8.8a shows a ﬂow element in a transformed section. The ﬂow element has been redrawn in a true section in Figure 8.8b. 210 Chapter 8: Seepage kz 1 kx 6 Vertical scale 20 ft Horizontal scale 20(√6) 49 ft (a) Scale 20 ft Figure 8.8 A ﬂow element in anisotropic soil: (a) in transformed section; (b) in true section (b) Example 8.3 A dam section is shown in Figure 8.9a. The hydraulic conductivity of the permeable layer in the vertical and horizontal directions are 2 10 2 mm/s and 4 102 mm/s, respectively. Draw a ﬂow net and calculate the seepage loss of the dam in ft3/day/ft Solution From the given data, kz 2 102 mm/s 5.67 ft/day kx 4 102 mm/s 11.34 ft/day 8.6 Mathematical Solution for Seepage 211 20 ft (a) 25 ft 20 ft 1.0 1.0 0.5 Horizontal scale 25 √2 35.36 ft Vertical scale 25 ft Permeable layer (b) Impermeable layer Figure 8.9 and h 20 ft. For drawing the ﬂow net, 2 102 1vertical scale2 B 4 102 1 1vertical scale2 12 Horizontal scale On the basis of this, the dam section is replotted, and the ﬂow net drawn as in Figure 8.9b. The rate of seepage is given by q 1kxkz H(Nf /Nd). From Figure 8.9b, Nd 8 and Nf 2.5 (the lowermost ﬂow channel has a width-to-length ratio of 0.5). So, q 115.672 111.342 1202 12.5/82 50.12 ft3/day/ft 8.6 ■ Mathematical Solution for Seepage The seepage under several simple hydraulic structures can be solved mathematically. Harr (1962) has analyzed many such conditions. Figure 8.10 shows a nondimensional plot for the rate of seepage around a single row of sheet piles. In a similar manner, Figure 8.11 is a nondimensional plot for the rate of seepage under a dam. In Figure 8.10, the depth of penetration of the sheet pile is S, and the thickness of the permeable soil layer is T. 212 Chapter 8: Seepage Text not available due to copyright restrictions Example 8.4 Refer to Figure 8.11. Given; the width of the dam, B 6 m; length of the dam, L 120 m; S 3 m; T ¿ 6 m; x 2.4 m; and H1 H2 5 m. If the hydraulic conductivity of the permeable layer is 0.008 cm/sec, estimate the seepage under the dam (Q) in m3/day/m. Solution Given that B 6 m, T ¿ 6 m, and S 3 m, so b B/2 3 m. 3 b 0.5 T¿ 6 3 S 0.5 T¿ 6 2.4 x 0.8 b 3 From Figure 8.11, for b/T ¿ 0.5, S/T ' 0.5, and x/b 0.8, the value of q/kH 0.378. Thus, Q q L 0.378 k HL (0.378)(0.008 102 60 60 24 m/day)(5)(120) 1567.64 m3/day ■ 8.7 Uplift Pressure Under Hydraulic Structures 213 Text not available due to copyright restrictions 8.7 Uplift Pressure Under Hydraulic Structures Flow nets can be used to determine the uplift pressure at the base of a hydraulic structure. This general concept can be demonstrated by a simple example. Figure 8.12a shows a weir, the base of which is 2 m below the ground surface. The necessary ﬂow net also has been drawn (assuming that kx kz k). The pressure distribution diagram at the base of the weir can be obtained from the equipotential lines as follows. There are seven equipotential drops (Nd) in the ﬂow net, and the difference in the water levels between the upstream and downstream sides is H 7 m. The head loss for each potential drop is H/7 7/7 1 m. The uplift pressure at a 1left corner of the base2 1Pressure head at a2 1gw 2 3 17 22 14 gw 8gw 214 Chapter 8: Seepage 14 m 7m 8m 2m a b c d e f 10 m kx kz k Impermeable layer (a) 14 m a b c d e f 3g kN/m2 4g kN/m2 5g kN/m2 6g kN/m2 7g kN/m2 8g kN/m2 (b) Figure 8.12 (a) A weir; (b) uplift force under a hydraulic structure Similarly, the uplift pressure at b 39 122 11 2 4gw 7gw and at f 39 16 2 11 2 4gw 3gw The uplift pressures have been plotted in Figure 8.12b. The uplift force per unit length measured along the axis of the weir can be calculated by ﬁnding the area of the pressure diagram. 8.8 Seepage Through an Earth Dam on an Impervious Base Figure 8.13 shows a homogeneous earth dam resting on an impervious base. Let the hydraulic conductivity of the compacted material of which the earth dam is made be equal to k. The free surface of the water passing through the dam is given by abcd. It is assumed 8.8 Seepage Through an Earth Dam on an Impervious Base 215 z d B 0.3 a Water level a dz H b b c dx f x z L k e a d Impervious layer Figure 8.13 Flow through an earth dam constructed over an impervious base that abc is parabolic. The slope of the free surface can be assumed to be equal to the hydraulic gradient. It also is assumed that, because this hydraulic gradient is constant with depth (Dupuit, 1863), i dz dx (8.29) Considering the triangle cde, we can give the rate of seepage per unit length of the dam (at right angles to the cross section shown in Figure 8.13) as q kiA i dz tan a dx A 1ce2 112 L sin a So q k1tan a2 1L sin a2 kL tan a sin a (8.30) Again, the rate of seepage (per unit length of the dam) through the section bf is q kiA k a dz dz b 1z 1 2 kz dx dx (8.31) 216 Chapter 8: Seepage For continuous ﬂow, qEq. 18.302 qEq. 18.312 or kz dz kL tan a sin a dx or zH kz dz zL sin a 1 2 2 1H xd 1kL tan a sin a2 dx xL cos a L2 sin2 a2 L tan a sin a1d L cos a2 L2 sin2 a sin2 a H2 b L2 sin2 a Ld a cos a 2 2 L2 cos a H2 cos a Ld L2 cos a 2 2 2 sin a or L2 cos a 2Ld H2 cos a 0 sin2 a So, L d2 d H2 cos a B cos2 a sin2 a (8.32) Following is a step-by-step procedure to obtain the seepage rate q (per unit length of the dam): Step 1: Step 2: Step 3: Step 4: Step 5: Obtain a. Calculate (see Figure 8.13) and then 0.3. Calculate d. With known values of a and d, calculate L from Eq. (8.32). With known value of L, calculate q from Eq. (8.30). The preceding solution generally is referred to as Schaffernak’s solution (1917) with Casagrande’s correction, since Casagrande experimentally showed that the parabolic free surface starts from a, not a (Figure 8.13). 8.9 L. Casagrande’s Solution for Seepage Through an Earth Dam 217 Example 8.5 Refer to the earth dam shown in Figure 8.13. Given that 45°, 30°, B 10 ft, H 20 ft, height of dam 25 ft, and k 2 104 ft/min, calculate the seepage rate, q, in ft3/day/ft length. Solution We know that 45° and 30°. Thus, ¢ 20 H 20 ft tan b tan 45° d 0.3¢ 6 125 202 tan b 125 202 tan 45° B 10 0.3¢ 10.32 120 2 6 ft 25 tan a 25 64.3 ft tan 30 From Eq. (8.32), L H2 d2 d cos a B cos 2a sin 2a 2 2 64.3 20 64.3 a b a b 11.7 ft cos 30 B cos 30 sin 30 From Eq. (8.30) q kL tan sin (2 104)(11.7)(tan 30)(sin 30) 6.754 104 ft3/min/ft 0.973 ft3/day/ft 8.9 ■ L. Casagrande’s Solution for Seepage Through an Earth Dam Equation (8.32) is derived on the basis of Dupuit’s assumption (i.e., i dz/dx). It was shown by Casagrande (1932) that, when the downstream slope angle a in Figure 8.13 becomes greater than 30, deviations from Dupuit’s assumption become more noticeable. Thus (see Figure 8.13), L. Casagrande (1932) suggested that i where ds 2dx2 dz2 dz sin a ds (8.33) 218 Chapter 8: Seepage So Eq. (8.30) can now be modiﬁed as q kiA k sin a1L sin a 2 kL sin2 a (8.34) Again, q kiA k a dz b 11 z2 ds (8.35) Combining Eqs. (8.34) and (8.35) yields H z dz L sin a s L sin2 a ds (8.36) L where s length of curve abc 1 2 1H L2 sin2 a2 L sin2 a1s L 2 2 or Ls B s2 H2 sin2 a (8.37) With about 4 to 5% error, we can write s 2d2 H2 (8.38) Combining Eqs. (8.37) and (8.38) yields L 2d2 H2 2d2 H2 cot2 a (8.39) Once the magnitude of L is known, the rate of seepage can be calculated from Eq. (8.34) as q kL sin2 a In order to avoid the approximation introduced in Eqs. (8.38) and (8.39), a solution was provided by Gilboy (1934). This is shown in a graphical form in Figure 8.14. Note, in this graph, m L sin a H In order to use the graph, Step 1: Determine d/H. Step 2: For a given d/H and a, determine m. mH Step 3: Calculate L . sin a Step 4: Calculate kL sin2 a. (8.40) 8.10 Filter Design 100 0.4 0.3 0.2 0.15 2 3 219 m 0.1 80 60 a (deg) 40 0.5 0.6 0.7 20 0.8 0 0 1 4 5 6 7 8 9 d /H Figure 8.14 Chart for solution by L. Casagrande’s method based on Gilboy’s solution 8.10 Filter Design When seepage water flows from a soil with relatively fine grains into a coarser material, there is danger that the fine soil particles may wash away into the coarse material. Over a period of time, this process may clog the void spaces in the coarser material. Hence, the grain-size distribution of the coarse material should be properly manipulated to avoid this situation. A properly designed coarser material is called a filter. Figure 8.15 shows the steady-state seepage condition in an earth dam which has a toe filter. For proper selection of the filter material, two conditions should be kept in mind: Condition 1: The size of the voids in the filter material should be small enough to hold the larger particles of the protected material in place. Condition 2: The filter material should have a high hydraulic conductivity to prevent buildup of large seepage forces and hydrostatic pressures in the filters. Toe filter Seepage Impermeable layer Figure 8.15 Steady-state seepage in an earth dam with a toe filter 220 Chapter 8: Seepage It can be shows that, if three perfect spheres have diameters greater than 6.5 times the diameter of a smaller sphere, the small sphere can move through the void spaces of the larger ones (Figure 8.16a). Generally speaking, in a given soil, the sizes of the grains vary over a wide range. If the pore spaces in a filter are small enough to hold D85 of the soil to be protected, then the finer soil particles also will be protected (Figure 8.16b). This means that the effective diameter of the pore spaces in the filter should be less than D85 of the soil to be protected. The effective pore diameter is about 15 D15 of the filter. With this in mind and based on the experimental investigation of filters, Terzaghi and Peck (1948) provided the following criteria to satisfy Condition 1: D151F2 D851S2 4to 5 1to satisfy Condition 12 (8.41) In order to satisfy Condition 2, they suggested that D151F2 D151S2 4to 5 1to satisfy Condition 22 (8.42) Large sphere Small sphere (a) Seepage of water Soil to be protected Filter D85(S) D15(F) (b) Figure 8.16 (a) Large spheres with diameters of 6.5 times the diameter of the small sphere; (b) boundary between a filter and the soil to be protected 8.10 Filter Design 221 where D15(F) diameter through which 15% of ﬁlter material will pass D15(S) diameter through which 15% of soil to be protected will pass D85(S) diameter through which 85% of soil to be protected will pass The proper use of Eqs. (8.41) and (8.42) to determine the grain-size distribution of soils used as ﬁlters is shown in Figure 8.17. Consider the soil used for the construction of the earth dam shown in Figure 8.15. Let the grain-size distribution of this soil be given by curve a in Figure 8.17. We can now determine 5D85(S) and 5D15(S) and plot them as shown in Figure 8.17. The acceptable grain-size distribution of the ﬁlter material will have to lie in the shaded zone. (Note: The shape of curves b and c are approximately the same as curve a.) The U.S. Navy (1971) requires the following conditions for the design of ﬁlters. Condition 1: For avoiding the movement of the particles of the protected soil: D151F2 D851S2 D50 1F2 D50 1S2 D151F2 D151S2 Condition 2: Curve c 5 25 20 If the uniformity coefﬁcient Cu of the protected soil is less than 1.5, D15(F) /D85(S) may be increased to 6. Also, if Cu of the protected soil is greater than 4, D15(F) /D15(S) may be increased to 40. For avoiding buildup of large seepage force in the ﬁlter: D151F2 4 D151S2 Curve a Curve b (soil to be protected) 100 5D85(S) 0.55 mm D85(S) 0.11 mm Percent ﬁner 80 60 Range of good ﬁlter 40 5D15(S) 0.045 mm D 15(S) 0.009 mm 20 0 10.0 5.0 1.0 0.5 0.1 0.05 Grain-size distribution (mm) 0.01 0.005 0.002 Figure 8.17 Determination of grain-size distribution of ﬁlter using Eqs. (8.41) and (8.42) 222 Chapter 8: Seepage Condition 3: Condition 4: Condition 5: The ﬁlter material should not have grain sizes greater than 76.2 mm (3 in.). (This is to avoid segregation of particles in the ﬁlter.) To avoid internal movement of ﬁnes in the ﬁlter, it should have no more than 5% passing a No. 200 sieve. When perforated pipes are used for collecting seepage water, ﬁlters also are used around the pipes to protect the ﬁne-grained soil from being washed into the pipes. To avoid the movement of the ﬁlter material into the drain-pipe perforations, the following additional conditions should be met: D851F2 slot width D851F2 hole diameter 8.11 1.2 to 1.4 1.0 to 1.2 Summary In this chapter, we studied Laplace’s equation of continuity and its application in solving problems related to seepage calculation. The continuity equation is the fundamental basis on which the concept of drawing ﬂow nets is derived. Flow nets are very powerful tools for calculation of seepage as well as uplift pressure under various hydraulic structures. Also discussed in this chapter (Sections 8.8 and 8.9) is the procedure to calculate seepage through an earth dam constructed over an impervious base. Section 8.8 derives the relationship for seepage based on Dupuit’s assumption that the hydraulic gradient is constant with depth. An improved procedure (L. Casagrande’s solution) for seepage calculation is provided in Section 8.9. The principles of designing a protective ﬁlter are given in Section 8.10. Problems 8.1 8.2 8.3 Refer to the constant-head permeability test arrangement in a two-layered soil as shown in Figure 8.2. During the test, it was seen that when a constant head of h1 200 mm was maintained, the magnitude of h2 was 80 mm. If k1 is 0.004 cm/sec, determine the value of k2 given H1 100 mm and H2 150 mm. Refer to Figure 8.18. Given: • H1 6 m • D3m • H2 1.5 m • D1 6 m draw a ﬂow net. Calculate the seepage loss per meter length of the sheet pile (at a right angle to the cross section shown). Draw a ﬂow net for the single row of sheet piles driven into a permeable layer as shown in Figure 8.18. Given: • H1 3 m • D 1.5 m • H2 0.5 m • D1 3.75 m calculate the seepage loss per meter length of the sheet pile (at right angles to the cross section shown). Problems 223 Sheet pile H1 H2 D D1 k 4 104 cm /sec Impermeable layer Figure 8.18 8.4 8.5 8.6 Refer to Figure 8.18. Given: • H1 4 m • D1 6 m • H2 1.5 m • D 3.6 m calculate the seepage loss in m3/day per meter length of the sheet pile (at right angles to the cross section shown). Use Figure 8.10. For the hydraulic structure shown in Figure 8.19, draw a ﬂow net for ﬂow through the permeable layer and calculate the seepage loss in m3/day/m. Refer to Problem 8.5. Using the ﬂow net drawn, calculate the hydraulic uplift force at the base of the hydraulic structure per meter length (measured along the axis of the structure). 25 m 10 m 3.34 m 1.67 m 20 m k 0.002 cm /sec 1.67 m Permeable layer Figure 8.19 1.67 m Impermeable layer 224 Chapter 8: Seepage 37 m 10 m 1.5 m 3m Sheet pile 24 m 9m k 103 cm /sec Impermeable layer Figure 8.20 8.7 Draw a ﬂow net for the weir shown in Figure 8.20. Calculate the rate of seepage under the weir. 8.8 For the weir shown in Figure 8.21, calculate the seepage in the permeable layer in m3/day/m for (a) x 1 m and (b) x 2 m. Use Figure 8.11. 8.9 An earth dam is shown in Figure 8.22. Determine the seepage rate, q, in m3/day/m length. Given: a1 35, a2 40, L 1 5 m, H 7 m, H1 10 m, and k 3104 cm/sec. Use Schaffernak’s solution. 8.10 Repeat Problem 8.9 using L. Casagrande’s method. 8m 6m 4m Sheet pile 8m k 103 cm /sec x Permeable layer Figure 8.21 Impermeable layer References 225 L1 Water level k H1 H a1 a2 Impervious base Figure 8.22 References CASAGRANDE, L. (1932). “Naeherungsmethoden zur Bestimmurg von Art und Menge der Sickerung durch geschuettete Daemme,” Thesis, Technische Hochschule, Vienna. DUPUIT, J. (1863). Etudes Theoriques et Practiques sur le Mouvement des Eaux dans les Canaux Decouverts et a Travers les Terrains Permeables, Dunod, Paris. GILBOY, G. (1934). “Mechanics of Hydraulic Fill Dams,” in Contributions to Soil Mechanics 1925–1940, Boston Society of Civil Engineers, Boston. HARR, M. E. (1962). Ground Water and Seepage, McGraw-Hill, New York. SCHAFFERNAK, F. (1917). “Über die Standicherheit durchlaessiger geschuetteter Dämme,” Allgem. Bauzeitung. TERZAGHI, K., and PECK, R. B. (1948). Soil Mechanics in Engineering Practice, Wiley, New York. U.S. DEPARTMENT OF THE NAVY, Naval Facilities Engineering Command (1971). “Design Manual—Soil Mechanics, Foundations, and Earth Structures,” NAVFAC DM-7, Washington, D.C. 9 In Situ Stresses As described in Chapter 3, soils are multiphase systems. In a given volume of soil, the solid particles are distributed randomly with void spaces between. The void spaces are continuous and are occupied by water and/or air. To analyze problems (such as compressibility of soils, bearing capacity of foundations, stability of embankments, and lateral pressure on earth-retaining structures), we need to know the nature of the distribution of stress along a given cross section of the soil proﬁle. We can begin the analysis by considering a saturated soil with no seepage. 9.1 Stresses in Saturated Soil without Seepage Figure 9.1a shows a column of saturated soil mass with no seepage of water in any direction. The total stress at the elevation of point A can be obtained from the saturated unit weight of the soil and the unit weight of water above it. Thus, s Hgw 1HA H2 gsat (9.1) where s total stress at the elevation of point A gw unit weight of water gsat saturated unit weight of the soil H height of water table from the top of the soil column HA distance between point A and the water table The total stress, s, given by Eq. (9.1) can be divided into two parts: 1. A portion is carried by water in the continuous void spaces. This portion acts with equal intensity in all directions. 2. The rest of the total stress is carried by the soil solids at their points of contact. The sum of the vertical components of the forces developed at the points of contact of the solid particles per unit cross-sectional area of the soil mass is called the effective stress. This can be seen by drawing a wavy line, a–a, through point A that passes only through the points of contact of the solid particles. Let P1 , P2 , P3 , . . . , Pn be the forces that act at the points of contact of the soil particles (Figure 9.1b). The sum of the vertical 226 9.1 Stresses in Saturated Soil without Seepage 227 H HA Pore water a Solid particle A a Cross-sectional area A (a) a1 P1 a2 a4 a3 P4 P2 P3 Cross-sectional area A (b) Figure 9.1 (a) Effective stress consideration for a saturated soil column without seepage; (b) forces acting at the points of contact of soil particles at the level of point A components of all such forces over the unit cross-sectional area is equal to the effective stress s, or P11v2 P21v2 P31v2 p Pn1v2 (9.2) s¿ A where P1(v) , P2(v) , P3(v) , . . ., Pn(v) are the vertical components of P1 , P2 , P3 , . . ., Pn, respectively, and A is the cross-sectional area of the soil mass under consideration. Again, if as is the cross-sectional area occupied by solid-to-solid contacts (that is, as a1 a2 a3 p an), then the space occupied by water equals (A as). So we can write s s¿ u1A as 2 A s¿ u11 asœ 2 (9.3) 228 Chapter 9: In Situ Stresses where u HAgw pore water pressure 1that is, the hydrostatic pressure at A2 asœ as /A fraction of unit cross-sectional area of the soil mass occupied by solid-to-solid contacts The value of asœ is extremely small and can be neglected for pressure ranges generally encountered in practical problems. Thus, Eq. (9.3) can be approximated by s s¿ u (9.4) where u is also referred to as neutral stress. Substitution of Eq. (9.1) for s in Eq. (9.4) gives s¿ 3Hgw 1HA H2gsat 4 HAgw 1HA H2 1gsat gw 2 1Height of the soil column2 g¿ (9.5) where g gsat gw equals the submerged unit weight of soil. Thus, we can see that the effective stress at any point A is independent of the depth of water, H, above the submerged soil. Figure 9.2a shows a layer of submerged soil in a tank where there is no seepage. Figures 9.2b through 9.2d show plots of the variations of the total stress, pore water pressure, and effective stress, respectively, with depth for a submerged layer of soil placed in a tank with no seepage. The principle of effective stress [Eq. (9.4)] was ﬁrst developed by Terzaghi (1925, 1936). Skempton (1960) extended the work of Terzaghi and proposed the relationship between total and effective stress in the form of Eq. (9.3). In summary, effective stress is approximately the force per unit area carried by the soil skeleton. The effective stress in a soil mass controls its volume change and strength. Increasing the effective stress induces soil to move into a denser state of packing. The effective stress principle is probably the most important concept in geotechnical engineering. The compressibility and shearing resistance of a soil depend to a great extent on the effective stress. Thus, the concept of effective stress is signiﬁcant in solving geotechnical engineering problems, such as the lateral earth pressure on retaining structures, the load-bearing capacity and settlement of foundations, and the stability of earth slopes. In Eq. (9.2), the effective stress, s, is deﬁned as the sum of the vertical components of all intergranular contact forces over a unit gross cross-sectional area. This deﬁnition is mostly true for granular soils; however, for ﬁne-grained soils, intergranular contact may not physically be there, because the clay particles are surrounded by tightly held water ﬁlm. In a more general sense, Eq. (9.3) can be rewritten as s sig u11 a¿s 2 A¿ R¿ where sig intergranular stress A¿ electrical attractive force per unit cross-sectional area of soil R¿ electrical repulsive force per unit cross-sectional area of soil (9.6) 9.1 Stresses in Saturated Soil without Seepage H1 229 A z C H2 B Valve (closed) (a) 0 H1 Total stress, s H 1g H1g H1g H2gsat Depth Effective stress, s 0 zg (H1 H2)g Depth (b) 0 (H1 z)g H1g zgsat H1 z H1 H2 Pore water pressure, u 0 H2g Depth (c) (d) Figure 9.2 (a) Layer of soil in a tank where there is no seepage; Variation of (b) total stress, (c) pore water pressure, and (d) effective stress with depth for a submerged soil layer without seepage For granular soils, silts, and clays of low plasticity, the magnitudes of A and R are small. Hence, for all practical purposes, sig s¿ s u However, if A R is large, then sig Z s. Such situations can be encountered in highly plastic, dispersed clay. Many interpretations have been made in the past to distinguish between the intergranular stress and effective stress. In any case, the effective stress principle is an excellent approximation used in solving engineering problems. 230 Chapter 9: In Situ Stresses Example 9.1 A soil proﬁle is shown in Figure 9.3. Calculate the total stress, pore water pressure, and effective stress at points A, B, and C. A 6m Dry sand gdry 16.5 kN/m3 z B Groundwater table Saturated sand gsat 19.25 kN/m3 13 m C Dry sand Saturated sand Clay Figure 9.3 Soil proﬁle Solution At Point A, Total stress: A 0 Pore water pressure: uA 0 Effective stress: 'A 0 At Point B, B 6dry(sand) 6 16.5 99 kN/m2 uB 0 kN/m2 'B 99 0 99 kN/m2 At Point C, C 6dry(sand) + 13sat(clay) 6 16.5 + 13 19.25 99 + 250.25 349.25 kN/m2 uC 13w 13 9.81 127.53 kN/m2 'C 349.25 127.53 221.72 kN/m2 ■ 9.2 Stresses in Saturated Soil with Upward Seepage 9.2 231 Stresses in Saturated Soil with Upward Seepage If water is seeping, the effective stress at any point in a soil mass will differ from that in the static case. It will increase or decrease, depending on the direction of seepage. Figure 9.4a shows a layer of granular soil in a tank where upward seepage is caused by adding water through the valve at the bottom of the tank. The rate of water supply is kept constant. The loss of head caused by upward seepage between the levels of A and B is h. Keeping in mind that the total stress at any point in the soil mass is due solely to the h H1 Hh2 z A z H2 C B Valve (open) Inflow (a) 0 H1 Total stress, s H1g 0 H1g H1g H2gsat Depth 0 (H1 H2 h)g Depth (b) Effective stress, s (H1 z iz)g H1g zgsat H1 z H1 H2 Pore water pressure, u 0 zg izg H2g hg Depth (c) (d) Figure 9.4 (a) Layer of soil in a tank with upward seepage. Variation of (b) total stress; (c) pore water pressure; and (d) effective stress with depth for a soil layer with upward seepage 232 Chapter 9: In Situ Stresses weight of soil and water above it, we ﬁnd that the effective stress calculations at points A and B are as follows: At A, • • • Total stress: sA H1gw Pore water pressure: uA H1gw œ Effective stress: sA sA uA 0 At B, • • • Total stress: sB H1gw H2gsat Pore water pressure: uB (H1 H2 h)gw Effective stress: sBœ sB uB H2 1gsat gw 2 hgw H2g¿ hgw Similarly, the effective stress at a point C located at a depth z below the top of the soil surface can be calculated as follows: At C, • Total stress: sC H1gw zgsat • Pore water pressure: uC a H1 z • Effective stress: sCœ sC uC z1gsat gw 2 zg¿ h z b gw H2 h zgw H2 h zgw H2 Note that h/H2 is the hydraulic gradient i caused by the ﬂow, and therefore, sCœ zg¿ izgw (9.7) The variations of total stress, pore water pressure, and effective stress with depth are plotted in Figures 9.4b through 9.4d, respectively. A comparison of Figures 9.2d and 9.3d shows that the effective stress at a point located at a depth z measured from the surface of a soil layer is reduced by an amount izgw because of upward seepage of water. If the rate of seepage and thereby the hydraulic gradient gradually are increased, a limiting condition will be reached, at which point sCœ zg¿ icr zgw 0 (9.8) where icr critical hydraulic gradient (for zero effective stress). Under such a situation, soil stability is lost. This situation generally is referred to as boiling, or a quick condition. From Eq. (9.8), icr g¿ gw For most soils, the value of icr varies from 0.9 to 1.1, with an average of 1. (9.9) 9.3 Stresses in Saturated Soil with Downward Seepage 233 Example 9.2 A 20-ft thick layer of stiff saturated clay is underlain by a layer of sand (Figure 9.5). The sand is under artesian pressure. Calculate the maximum depth of cut H that can be made in the clay. Solution Due to excavation, there will be unloading of the overburden pressure. Let the depth of the cut be H, at which point the bottom will heave. Let us consider the stability of point A at that time: sA 120 H2 gsat1clay2 uA 12gw For heave to occur, sA should be 0. So sA uA 120 H2gsat1clay2 12gw or 120 H2120 112 262.4 0 H 1202120 1122 62.4 120 13.76 ft gsat 120 lb/ft3 H 20 ft 12 ft A 4 ft gsat 108 lb/ft3 Saturated clay Sand Figure 9.5 9.3 ■ Stresses in Saturated Soil with Downward Seepage The condition of downward seepage is shown in Figure 9.6a on the next page. The water level in the soil tank is held constant by adjusting the supply from the top and the outﬂow at the bottom. 234 Chapter 9: In Situ Stresses Inflow H1 h A Hh2 z z C H2 B Valve (open) Outflow (a) 0 H1 H1 z H1 H2 Total stress, s Pore water pressure, u 0 H1gw 0 H 1g w Depth (H1 H2 h)gw Depth (b) 0 (H1 z iz)gw H1gw zgsat H1gw H2gsat Effective stress, s zg izgw H2g hgw Depth (c) (d) Figure 9.6 (a) Layer of soil in a tank with downward seepage; variation of (b) total stress; (c) pore water pressure; (d) effective stress with depth for a soil layer with downward seepage The hydraulic gradient caused by the downward seepage equals i h/H2 . The total stress, pore water pressure, and effective stress at any point C are, respectively, sC H1gw zgsat uC 1H1 z iz2gw sCœ 1H1gw zgsat 2 1H1 z iz2gw zg¿ izgw The variations of total stress, pore water pressure, and effective stress with depth also are shown graphically in Figures 9.6b through 9.6d. 9.4 Seepage Force 9.4 235 Seepage Force The preceding section showed that the effect of seepage is to increase or decrease the effective stress at a point in a layer of soil. Often, expressing the seepage force per unit volume of soil is convenient. In Figure 9.2, it was shown that, with no seepage, the effective stress at a depth z measured from the surface of the soil layer in the tank is equal to zg. Thus, the effective force on an area A is (9.10) P1œ zg¿A (The direction of the force P1 is shown in Figure 9.7a.) Again, if there is an upward seepage of water in the vertical direction through the same soil layer (Figure 9.4), the effective force on an area A at a depth z can be given by P2œ 1zg¿ izgw 2A (9.11) Hence, the decrease in the total force because of seepage is P1œ P2œ izgwA (9.12) The volume of the soil contributing to the effective force equals zA, so the seepage force per unit volume of soil is P1œ P2œ izgwA igw 1Volume of soil2 zA (9.13) Volume of soil z A z zgA (a) Volume of soil z A z (zg izg ) A zgA izg A seepage force (b) z (zg izg ) A zgA izg A seepage force (c) Figure 9.7 Force due to (a) no seepage; (b) upward seepage; (c) downward seepage on a volume of soil 236 Chapter 9: In Situ Stresses The force per unit volume, igw , for this case acts in the upward direction—that is, in the direction of ﬂow. This upward force is demonstrated in Figure 9.7b. Similarly, for downward seepage, it can be shown that the seepage force in the downward direction per unit volume of soil is igw (Figure 9.7c). From the preceding discussions, we can conclude that the seepage force per unit volume of soil is equal to igw , and in isotropic soils the force acts in the same direction as the direction of ﬂow. This statement is true for ﬂow in any direction. Flow nets can be used to ﬁnd the hydraulic gradient at any point and, thus, the seepage force per unit volume of soil. Example 9.3 Consider the upward ﬂow of water through a layer of sand in a tank as shown in Figure 9.8. For the sand, the following are given: void ratio (e) 0.52 and speciﬁc gravity of solids 2.67. a. Calculate the total stress, pore water pressure, and effective stress at points A and B. b. What is the upward seepage force per unit volume of soil? Solution Part a The saturated unit weight of sand is calculated as follows: gsat 1Gs e 2gw 1e 12.67 0.5229.81 1 0.52 20.59 kN/m3 1.5 m 0.7 m 1m 2m A B Valve (open) Inflow Sand Figure 9.8 Upward ﬂow of water through a layer of sand in a tank 9.5 Heaving in Soil Due to Flow Around Sheet Piles 237 Now, the following table can be prepared: Point Total stress, S (kN/m2) A 0.7gw 1gsat 10.72 19.81 2 112 120.592 27.46 B 0.7gw 2gsat 10.72 19.81 2 122 120.592 48.05 Pore water pressure, u (kN/m2) Effective stress, Sⴕ ⴝ S ⴚ u (kN/m2) 1.5 b 112 d gw 2 12.45 2 19.812 24.03 c 11 0.7 2 a 3.43 12 0.7 1.52 gw 14.2 2 19.81 2 41.2 6.85 Part b Hydraulic gradient (i) 1.5/2 0.75. Thus, the seepage force per unit volume can be calculated as igw 10.752 19.812 7.36 kN/m3 9.5 ■ Heaving in Soil Due to Flow Around Sheet Piles Seepage force per unit volume of soil can be used for checking possible failure of sheetpile structures where underground seepage may cause heaving of soil on the downstream side (Figure 9.9a). After conducting several model tests, Terzaghi (1922) concluded that heaving generally occurs within a distance of D/2 from the sheet piles (when D equals depth of embedment of sheet piles into the permeable layer). Therefore, we need to Sheet pile D 2 H1 H2 D D T W D 2 U (a) Heave zone (b) Impermeable layer Figure 9.9 (a) Check for heaving on the downstream side for a row of sheet piles driven into a permeable layer; (b) enlargement of heave zone 238 Chapter 9: In Situ Stresses investigate the stability of soil in a zone measuring D by D/2 in cross-section as shown in Figure 9.9b. The factor of safety against heaving can be given by FS W¿ U (9.14) where FS factor of safety W submerged weight of soil in the heave zone per unit length of sheet pile D1D/2 2 1gsat gw 2 1 12 2D2g¿ U uplifting force caused by seepage on the same volume of soil From Eq. (9.13), U 1Soil volume2 1iavgw 2 12 D2iavgw where iav average hydraulic gradient at the bottom of the block of soil (see Example 9.4). Substituting the values of W and U in Eq. (9.14), we can write FS g¿ iavgw (9.15) For the case of ﬂow around a sheet pile in a homogeneous soil, as shown in Figure 9.9, it can be demonstrated that U Co 0.5gwD1H1 H2 2 (9.16) where Co is a function of D/T (see Table 9.1). Hence, from Eq. (9.14), FS 0.5D2g¿ Dg¿ W¿ U 0.5CogwD1H1 H2 2 Cogw 1H1 H2 2 Table 9.1 Variation of Co with D/T D/T Co 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.385 0.365 0.359 0.353 0.347 0.339 0.327 0.309 0.274 (9.17) 9.5 Heaving in Soil Due to Flow Around Sheet Piles 239 Example 9.4 Figure 9.10 shows the ﬂow net for seepage of water around a single row of sheet piles driven into a permeable layer. Calculate the factor of safety against downstream heave, given that gsat for the permeable layer 112 lb/ft3. (Note: thickness of permeable layer T 60 ft) Sheet pile H1 30 ft H2 5 ft 20 ft gsat 112 lb/ft3 Heave zone Impermeable layer Figure 9.10 Flow net for seepage of water around sheet piles driven into permeable layer Solution From the dimensions given in Figure 9.10, the soil prism to be considered is 20 ft 10 ft in cross section. The soil prism is drawn to an enlarged scale in Figure 9.11. By use of the ﬂow net, we can calculate the head loss through the prism as follows: • At b, the driving head 36 1H1 H2 2 • At c, the driving head 1.6 6 1H1 H2 2 Similarly, for other intermediate points along bc, the approximate driving heads have been calculated and are shown in Figure 9.11. The average value of the head loss in the prism is 0.36(H1 H2), and the average hydraulic gradient is iav 0.361H1 H2 2 D 240 Chapter 9: In Situ Stresses a 10 ft Soil prism d 20 ft c b Driving head (H1 H2 ) 0 Average 0.36 0.5 Figure 9.11 Soil prism—enlarged scale Thus, the factor of safety [Eq. (9.15)] is FS 1112 62.42 20 g¿D g¿ 1.77 iavgw 0.361H1 H2 2gw 0.36130 5 2 62.4 Alternate Solution For this case, D/T 1/3. From Table 9.1, for D/T 1/3, the value of Co 0.357. Thus, from Eq. (9.17), FS 9.6 1202 1112 62.42 Dg¿ 1.78 Cogw 1H1 H2 2 10.3572 162.42 130 5 2 ■ Use of Filters to Increase the Factor of Safety against Heave The factor of safety against heave as calculated in Example 9.4 is low. In practice, a minimum factor of safety of about 4 to 5 is required for the safety of the structure. Such a high factor of safety is recommended primarily because of the inaccuracies inherent in the analysis. One way to increase the factor of safety against heave is to use a ﬁlter in the downstream side of the sheet-pile structure (Figure 9.12a). A ﬁlter is a granular material with openings small enough to prevent the movement of the soil particles upon which it is placed and, at the same time, is pervious enough to offer little resistance to seepage through it (see Section 8.10). In Figure 9.12a, the thickness of the ﬁlter material is D1 . In this case, the factor of safety against heave can be calculated as follows (Figure 9.12b). 9.6 Use of Filters to Increase the Factor of Safety against Heave 241 D/2 WF D1 W D1 Filter D D U (a) (b) Figure 9.12 Factor of safety against heave, with a ﬁlter The submerged weight of the soil and the ﬁlter in the heave zone per unit length of sheet pile W WF, where W¿ 1D2 a D 1 b 1gsat gw 2 D2g¿ 2 2 WFœ 1D1 2 a D 1 b 1gFœ 2 D1DgFœ 2 2 in which gF effective unit weight of the ﬁlter. The uplifting force caused by seepage on the same volume of soil is given by U 12 D2iavgw The preceding relationship was derived in Section 9.5. The factor of safety against heave is thus D1 œ 1 1 2 D g¿ D1DgFœ b gF g¿ a W¿ WFœ 2 2 D FS U 1 2 iavgw D iavgw 2 (9.18) The principles for selection of ﬁlter materials were given in Section 8.10. If Eq. (9.16) is used, 1 2 1 D g¿ D1DgFœ Dg¿ D1gFœ 2 2 FS 0.5Cogw D1H1 H2 2 Cogw 1H1 H2 2 The value of Co is given in Table 9.1. (9.19) 242 Chapter 9: In Situ Stresses 9.7 Effective Stress in Partially Saturated Soil In partially saturated soil, water in the void spaces is not continuous, and it is a three-phase system—that is, solid, pore water, and pore air (Figure 9.13). Hence, the total stress at any point in a soil proﬁle consists of intergranular, pore air, and pore water pressures. From laboratory test results, Bishop, et al. (1960) gave the following equation for effective stress in partially saturated soils sœ s ua x1ua uw 2 (9.20) where s effective stress s total stress ua pore air pressure uw pore water pressure In Eq. (9.20), x represents the fraction of a unit cross-sectional area of the soil occupied by water. For dry soil x 0, and for saturated soil x 1. Pore air Solid particle Pore water Figure 9.13 Partially saturated soil Bishop, et al. have pointed out that the intermediate values of x will depend primarily on the degree of saturation S. However, these values also will be inﬂuenced by factors such as soil structure. The nature of variation of x with the degree of saturation for a silt is shown in Figure 9.14. 9.8 Capillary Rise in Soils 243 1.0 0.8 0.6 x Theory 0.4 0.2 0.0 0 20 40 60 Degree of saturation, S (%) 80 100 Drained test Figure 9.14 Relationship between the parameter x and the degree of saturation for Bearhead silt (After Bishop et al, 1960. With permission from ASCE.) 9.8 Capillary Rise in Soils The continuous void spaces in soil can behave as bundles of capillary tubes of variable cross section. Because of surface tension force, water may rise above the phreatic surface. Figure 9.15 shows the fundamental concept of the height of rise in a capillary tube. The height of rise of water in the capillary tube can be given by summing the forces in the vertical direction, or a p 2 d b hcgw pdT cos a 4 4T cos a hc dgw (9.21) where T surface tension 1force/length2 a angle of contact d diameter of capillary tube gw unit weight of water For pure water and clean glass, a 0. Thus, Eq. (9.21) becomes hc 4T dgw (9.22) 244 Chapter 9: In Situ Stresses T T d a a Atmospheric pressure hcg Capillary tube hc Free water surface Pressure h hg (a) (b) Figure 9.15 (a) Rise of water in the capillary tube; (b) pressure within the height of rise in the capillary tube (atmospheric pressure taken as datum) For water, T 72 mN/m. From Eq. (9.22), we see that the height of capillary rise hc r 1 d (9.23) Thus, the smaller the capillary tube diameter, the larger the capillary rise. Although the concept of capillary rise as demonstrated for an ideal capillary tube can be applied to soils, one must realize that the capillary tubes formed in soils because of the continuity of voids have variable cross sections. The results of the nonuniformity on capillary rise can be seen when a dry column of sandy soil is placed in contact with water (Figure 9.16). After the lapse of a given amount of time, the variation of the degree of saturation with the height of the soil column caused by capillary rise is approximately as shown in Figure 9.16b. The degree of saturation is about 100% up to a height of h2 , and this corresponds to the largest voids. Beyond the height h2 , water can occupy only the smaller voids; hence, the degree of saturation is less than 100%. The maximum height of capillary rise corresponds to the smallest voids. Hazen (1930) gave a formula for the approximation of the height of capillary rise in the form. h1 1mm2 C eD10 (9.24) where D10 effective size 1mm2 e void ratio C a constant that varies from 10 to 50 mm2 Equation (9.24) has an approach similar to that of Eq. (9.23). With the decrease of D10 , the pore size in soil decreases, which causes higher capillary rise. Table 9.2 shows the approximate range of capillary rise that is encountered in various types of soils. 9.9 Effective Stress in the Zone of Capillary Rise 245 h h1 h2 0 100 Degree of saturation (%) Screen (b) (a) Sandy soil Water Figure 9.16 Capillary effect in sandy soil: (a) a soil column in contact with water; (b) variation of degree of saturation in the soil column Table 9.2 Approximate Range of Capillary Rise in Soils Range of capillary rise Soil type m ft Coarse sand Fine sand Silt Clay 0.1–0.2 0.3–1.2 0.75–7.5 7.5–23 0.3–0.6 1–4 2.5–25 25–75 Capillary rise is important in the formation of some types of soils such as caliche, which can be found in the desert Southwest of the United States. Caliche is a mixture of sand, silt, and gravel bonded by calcareous deposits. These deposits are brought to the surface by a net upward migration of water by capillary action. The water evaporates in the high local temperature. Because of sparse rainfall, the carbonates are not washed out of the top soil layer. 9.9 Effective Stress in the Zone of Capillary Rise The general relationship among total stress, effective stress, and pore water pressure was given in Eq. (9.4) as s s¿ u The pore water pressure u at a point in a layer of soil fully saturated by capillary rise is equal to gw h (h height of the point under consideration measured from the 246 Chapter 9: In Situ Stresses groundwater table) with the atmospheric pressure taken as datum. If partial saturation is caused by capillary action, it can be approximated as u a S bg h 100 w (9.25) where S degree of saturation, in percent. Example 9.5 A soil proﬁle is shown in Figure 9.17. Given: H1 6 ft, H2 3 ft, H3 6 ft. Plot the variation of s, u, and s with depth. Solution Determination of Unit Weight Dry sand: gd1sand2 12.65 2 162.4 2 Gsgw 110.24 lb/ft 3 1e 1 0.5 A Gs 2.65 e 0.5 H1 B H2 C Zone of capillary rise Gs 2.65; e 0.5 Degree of saturation S 50% 42% (moisture content) Gs 2.71 H3 D Sand Figure 9.17 Ground water table Saturated clay Rock 9.9 Effective Stress in the Zone of Capillary Rise Moist sand: gsand 1Gs Se2gw 1e 32.65 10.5 2 10.52 4 62.4 1 0.5 120.64 lb/ft3 Saturated clay: 12.71 2 10.42 2 Gsw 1.1382 S 1.0 1Gs e2gw 12.71 1.1382262.4 112.3 lb/ft3 1e 1 1.1382 e gsat1clay2 Calculation of Stress At the ground surface (i.e., point A): s0 u0 sœ s u 0 At depth H1 (i.e., point B): d(sand)(6) (110.24)(6) 661.44 lb/ft2 u 0 (immediately above) u (Sw H2) (0.5)(62.4)(3) ⴚ93.6 lb/ft2 (immediately below) ' 661.44 0 661.44 lb/ft2 (immediately above) ' 661.44 (93.6) 755.04 lb/ft2 (immediately below) At depth H1 + H2 (i.e., at point C): (110.24)(6) + (120.64)(3) 1023.36 lb/ft2 u0 ' 1023.36 0 1023.36 lb/ft2 At depth H1 + H2 + H3 (i.e., at point D): 1023.36 + (112.3)(6) 1697.17 lb/ft2 u 6w (6)(62.4) 374.4 lb/ft2 ' 1697.17 374.4 1322.77 lb/ft2 The plot of the stress variation is shown in Figure 9.18. 247 248 Chapter 9: In Situ Stresses 0 0 s (lb/ft2) 500 1000 6 1500 661.44 0 93.6 0 u (l/ft2) 500 6 0 0 s (lb/ft2) 500 1000 6 1500 755.04 661.44 9 15 Depth (ft) 1023.36 1697.17 9 15 Depth (ft) 9 374.4 1023.36 15 Figure 9.18 9.10 1322.77 Depth (ft) ■ Summary and General Comments The effective stress principle is probably the most important concept in geotechnical engineering. The compressibility and shearing resistance of a soil depend to a great extent on the effective stress. Thus, the concept of effective stress is signiﬁcant in solving geotechnical engineering problems, such as the lateral earth pressure on retaining structures, the load-bearing capacity and settlement of foundations, and the stability of earth slopes. In Eq. (9.2), the effective stress s is deﬁned as the sum of the vertical components of all intergranular contact forces over a unit gross cross-sectional area. This deﬁnition is mostly true for granular soils; however, for ﬁne-grained soils, intergranular contact may not be there physically, because the clay particles are surrounded by tightly held water ﬁlm. In a more general sense, Eq. (9.3) can be rewritten as s sig u11 asœ 2 A¿ R¿ (9.26) where sig intergranular stress A¿ electrical attractive force per unit cross-sectional area of soil R¿ electrical repulsive force per unit cross-sectional area of soil For granular soils, silts, and clays of low plasticity, the magnitudes of A and R are small. Hence, for all practical purposes, sig s¿ s u However, if A R is large, then sig Z s. Such situations can be encountered in highly plastic, dispersed clay. Many interpretations have been made in the past to distinguish Problems 249 between the intergranular stress and effective stress. In any case, the effective stress principle is an excellent approximation used in solving engineering problems. Problems 9.1 through 9.5 Refer to Figure 9.19. Calculate s, u, and s at A, B, C, and D for the following cases and plot the variations with depth. (Note: e void ratio, w moisture content, Gs speciﬁc gravity of soil solids, gd dry unit weight, and gsat saturated unit weight.) Details of soil layer Problem I II III 9.1 H1 5 ft gd 112 lb/ft3 H2 6 ft gsat 120 lb/ft3 H3 8 ft gsat 125 lb/ft3 9.2 H1 5 ft gd 100 lb/ft3 H2 10 ft gsat 116 lb/ft3 H3 9 ft gsat 122 lb/ft3 9.3 H1 3 m gd 15 kN/m3 H2 4 m gsat 16 kN/m3 H3 5 m gsat 18 kN/m3 9.4 H1 4 m e 0.4 Gs 2.62 H2 5 m e 0.6 Gs 2.68 H3 3 m e 0.81 Gs 2.73 9.5 H1 4 m e 0.6 Gs 2.65 H2 3 m e 0.52 Gs 2.68 H3 1.5 m w 40% e 1.1 A Layer I H1 B Groundwater table Layer II H2 C Layer III H3 D Dry sand Figure 9.19 Sand Clay Rock 250 Chapter 9: In Situ Stresses e 0.52 Gs 2.66 5m Groundwater table e 1.0 Gs 2.75 4m Dry Sand 9.6 9.7 9.8 9.9 Figure 9.20 Clay Refer to the soil proﬁle shown in Figure 9.20. a. Calculate the variation of s, u, and s with depth. b. If the water table rises to the top of the ground surface, what is the change in the effective stress at the bottom of the clay layer? c. How many meters must the groundwater table rise to decrease the effective stress by 15 kN/m2 at the bottom of the clay layer? An exploratory drill hole was made in a stiff saturated clay (see Figure 9.21). The sand layer underlying the clay was observed to be under artesian pressure. Water in the drill hole rose to a height of 12 ft above the top of the sand layer. If an open excavation is to be made in the clay, how deep can the excavation proceed before the bottom heaves? A cut is made in a stiff saturated clay that is underlain by a layer of sand. (See Figure 9.22.) What should be the height of the water, h, in the cut so that the stability of the saturated clay is not lost? Refer to Figure 9.4a. If H1 1.5 m, H2 2.5 m, h 1.5 m, gsat 18.6 kN/m3, hydraulic conductivity of sand (k) 0.12 cm/sec, and area of tank 0.45 m2, what is the rate of upward seepage of water (m3/min)? Exploratory drill hole Gs 2.72 Moisture content 35% 20 ft 12 ft 12 ft Saturated clay Sand Figure 9.21 Problems Saturated clay psat 1900 kg/m3 H 5 m 7m 251 h 3.0 m A Sand rsat 1800 kg/m3 2m Figure 9.22 9.10 Refer to Figure 9.4a. Given: H1 1 m, H2 2 m, h 1.2 m, void ratio of sand (e) 0.55, speciﬁc gravity of soil solids (Gs) 2.68, area of the tank 0.5 m2, and hydraulic conductivity of sand 0.1 cm/sec. a. What is the rate of upward seepage? b. If h 1.2 m, will boiling occur? Why? c. What should be the value of h to cause boiling? 9.11 Find the factor of safety against heave on the downstream side of the single-row sheet pile structure shown in Figure 8.7. (Note: Thickness of permeable layer 10 m and depth of penetration of sheet pile 6 m.) Assume gsat 19 kN/m2. 9.12 through 9.13 Refer to Figure 9.23. For the following variables, calculate and plot s, u, and s with depth. Problem H1 H2 H3 Degree of saturation in capillary rise zone, S (%) 9.12 9.13 8 ft 5m 4 ft 3m 7 ft 3.5 m 50 65 H1 Gs 2.66 e 0.5 H2 Gs 2.71; e 0.75 Groundwater table Gs 2.72 e 0.95 H3 Dry sand Clay; zone of capillary rise Clay Rock Figure 9.23 252 Chapter 9: In Situ Stresses References BISHOP, A. W., ALPAN, I., BLIGHT, G. E., and DONALD, I. B. (1960). “Factors Controlling the Strength of Partially Saturated Cohesive Soils,” Proceedings, Research Conference on Shear Strength of Cohesive Soils, ASCE, 502–532. HAZEN, A. (1930). “Water Supply,” in American Civil Engineering Handbook, Wiley, New York. SKEMPTON, A. W. (1960). “Correspondence,” Geotechnique, Vol. 10, No. 4, 186. TERZAGHI, K. (1922). “Der Grundbruch an Stauwerken und seine Verhütung,” Die Wasserkraft, Vol. 17, 445–449. TERZAGHI, K. (1925). Erdbaumechanik auf Bodenphysikalischer Grundlage, Dueticke, Vienna. TERZAGHI, K. (1936). “Relation Between Soil Mechanics and Foundation Engineering: Presidential Address,” Proceedings, First International Conference on Soil Mechanics and Foundation Engineering, Boston, Vol. 3, 13–18. TERZAGHI, K., and PECK, R. B. (1948). Soil Mechanics in Engineering Practice, Wiley, New York. 10 Stresses in a Soil Mass Construction of a foundation causes changes in the stress, usually a net increase. The net stress increase in the soil depends on the load per unit area to which the foundation is subjected, the depth below the foundation at which the stress estimation is desired, and other factors. It is necessary to estimate the net increase of vertical stress in soil that occurs as a result of the construction of a foundation so that settlement can be calculated. The settlement calculation procedure is discussed in more detail in Chapter 11. This chapter discusses the principles of estimation of vertical stress increase in soil caused by various types of loading, based on the theory of elasticity. Although natural soil deposits, in most cases, are not fully elastic, isotropic, or homogeneous materials, calculations for estimating increases in vertical stress yield fairly good results for practical work. 10.1 Normal and Shear Stresses on a Plane Students in a soil mechanics course are familiar with the fundamental principles of the mechanics of deformable solids. This section is a brief review of the basic concepts of normal and shear stresses on a plane that can be found in any course on the mechanics of materials. Figure 10.1a shows a two-dimensional soil element that is being subjected to normal and shear stresses (sy sx). To determine the normal stress and the shear stress on a plane EF that makes an angle u with the plane AB, we need to consider the free body diagram of EFB shown in Figure 10.1b. Let sn and tn be the normal stress and the shear stress, respectively, on the plane EF. From geometry, we know that EB EF cos u (10.1) FB EF sin u (10.2) and Summing the components of forces that act on the element in the direction of N and T, we have sn 1EF2 sx 1EF2 sin2 u sy 1EF2 cos2 u 2txy 1EF2 sin u cos u 253 254 Chapter 10: Stresses in a Soil Mass sy tx y N D C F sn t n F sx sx tx y A E u T E u B B tx y tx y sy (a) (b) Figure 10.1 (a) A soil element with normal and shear stresses acting on it; (b) free body diagram of EFB as shown in (a) or sn sx sin2 u sy cos2 u 2txy sin u cos u or sn sy sx 2 sy sx 2 cos 2u txy sin 2u (10.3) Again, tn 1EF2 sx 1EF2 sin u cos u sy 1EF2 sin u cos u txy 1EF2 cos2 u txy 1EF2 sin2 u or tn sy sin u cos u sx sin u cos u txy 1cos2 u sin2 u2 or tn sy sx 2 sin 2u txy cos 2u (10.4) From Eq. (10.4), we can see that we can choose the value of u in such a way that tn will be equal to zero. Substituting tn 0, we get tan 2u 2txy sy sx (10.5) 10.1 Normal and Shear Stresses on a Plane 255 For given values of txy , sx , and sy , Eq. (10.5) will give two values of u that are 90 apart. This means that there are two planes that are at right angles to each other on which the shear stress is zero. Such planes are called principal planes. The normal stresses that act on the principal planes are referred to as principal stresses. The values of principal stresses can be found by substituting Eq. (10.5) into Eq. (10.3), which yields Major principal stress: sn s1 sy sx 2 B c 1sy sx 2 2 d t2xy (10.6) d t2xy (10.7) 2 Minor principal stress: sn s3 sy sx 2 B c 1sy sx 2 2 2 The normal stress and shear stress that act on any plane can also be determined by plotting a Mohr’s circle, as shown in Figure 10.2. The following sign conventions are used in Mohr’s circles: compressive normal stresses are taken as positive, and shear stresses are considered positive if they act on opposite faces of the element in such a way that they tend to produce a counterclockwise rotation. For plane AD of the soil element shown in Figure 10.1a, normal stress equals sx and shear stress equals txy . For plane AB, normal stress equals sy and shear stress equals txy . The points R and M in Figure 10.2 represent the stress conditions on planes AD and AB, respectively. O is the point of intersection of the normal stress axis with the line RM. The circle MNQRS drawn with O as the center and OR as the radius is the Shear stress, t sx sy 2 R √ sx sy 2 t 2 xy 2 (sx , tx y ) (sn, tn ) Q O S s3 2u (sy,tx y ) s1 N M Figure 10.2 Principles of the Mohr’s circle Normal stress, s 256 Chapter 10: Stresses in a Soil Mass Mohr’s circle for the stress conditions considered. The radius of the Mohr’s circle is equal to B c 1sy sx 2 2 d t2xy 2 The stress on plane EF can be determined by moving an angle 2u (which is twice the angle that the plane EF makes in a counterclockwise direction with plane AB in Figure 10.1a) in a counterclockwise direction from point M along the circumference of the Mohr’s circle to reach point Q. The abscissa and ordinate of point Q, respectively, give the normal stress sn and the shear stress tn on plane EF. Because the ordinates (that is, the shear stresses) of points N and S are zero, they represent the stresses on the principal planes. The abscissa of point N is equal to s1 [Eq. (10.6)], and the abscissa for point S is s3 [Eq. (10.7)]. As a special case, if the planes AB and AD were major and minor principal planes, the normal stress and the shear stress on plane EF could be found by substituting txy 0. Equations (10.3) and (10.4) show that sy s1 and sx s3 (Figure 10.3a). Thus, sn s1 s3 s1 s3 cos 2u 2 2 (10.8) tn s1 s3 sin 2u 2 (10.9) The Mohr’s circle for such stress conditions is shown in Figure 10.3b. The abscissa and the ordinate of point Q give the normal stress and the shear stress, respectively, on the plane EF. s1 D C F s3 s3 A E Shear stress Q(sn, tn ) s3 S 2u O s1 N u B s1 (a) (b) Figure 10.3 (a) Soil element with AB and AD as major and minor principal planes; (b) Mohr’s circle for soil element shown in (a) Normal stress 10.1 Normal and Shear Stresses on a Plane 257 Example 10.1 A soil element is shown in Figure 10.4. The magnitudes of stresses are sx 2000 lb/ft2, t 800 lb/ft2, sy 2500 lb/ft2, and u 20. Determine a. Magnitudes of the principal stresses b. Normal and shear stresses on plane AB. Use Eqs. (10.3), (10.4), (10.6), and (10.7). sy t B t sx sx t u A t sy Figure 10.4 Soil element with stresses acting on it Solution Part a From Eqs. (10.6) and (10.7), sy sx sy sx 2 s3 f c d t2xy s1 2 B 2 2500 2000 2500 2000 2 c d 1800 2 2 2 B 2 s1 3088.15 lb/ft2 s3 1411.85 lb/ft2 Part b From Eq. (10.3), sn sy sx 2 sy sx 2 cos 2u t sin 2u 2500 2000 2500 2000 cos 12 20 2 18002 sin 12 20 2 2 2 1927.28 lb/ft2 258 Chapter 10: Stresses in a Soil Mass From Eq. (10.4), tn sy sx 2 sin 2u t cos 2u 2500 2000 sin 12 202 1800 2 cos 12 20 2 2 773.5 lb/ft2 The Pole Method of Finding Stresses Along a Plane Another important technique of ﬁnding stresses along a plane from a Mohr’s circle is the pole method, or the method of origin of planes. This is demonstrated in Figure 10.5. Figure 10.5a is the same stress element that is shown in Figure 10.1a; Figure 10.5b is the Mohr’s circle for the stress conditions indicated. According to the pole method, we draw a line from a known point on the Mohr’s circle parallel to the plane on which the state of stress acts. The point of intersection of this line with the Mohr’s circle is called the pole. This is a unique point for the state of stress under consideration. For example, the point M on the Mohr’s circle in Figure 10.5b represents the stresses on the plane AB. The line MP is drawn parallel to AB. So point P is the pole (origin of planes) in this case. If we need to ﬁnd the stresses on a plane EF, we draw a line from the pole parallel to EF. The point of intersection of this line with the Mohr’s circle is Q. The coordinates of Q give the stresses on the plane EF. (Note: From geometry, angle QOM is twice the angle QPM.) sy tx y D C F sx tx y A E Shear stress 10.2 ■ R s3 2u s1 S u Q(sn , tn ) (sx ,tx y ) N O u B M(sy ,tx y ) P (a) Normal stress (b) Figure 10.5 (a) Soil element with normal and shear stresses acting on it; (b) use of pole method to ﬁnd the stresses along a plane 10.2 The Pole Method of Finding Stresses Along a Plane 259 Example 10.2 For the stressed soil element shown in Figure 10.6a, determine a. Major principal stress b. Minor principal stress c. Normal and shear stresses on the plane AE Use the pole method. Shear stress (lb/in.2) 150 lb/in.2 60 lb/in.2 D N(150, 60) P C E Q (60, 30) 90 lb/in.2 Normal stress (lb/in.2) 60 lb/in.2 45 A B M(90, 60) (a) (b) Figure 10.6 (a) Soil element with stresses acting on it; (b) Mohr’s circle Solution On plane AD: Normal stress 90 lb/in.2 Shear stress 60 lb/in.2 On plane AB: Normal stress 150 lb/in.2 Shear stress 60 lb/in.2 The Mohr’s circle is plotted in Figure 10.6b. From the plot, Part a Major principal stress 187.1 lb/in.2 Part b Minor principal stress 52.9 lb/in.2 260 Chapter 10: Stresses in a Soil Mass Part c NP is the line drawn parallel to the plane CD. P is the pole. PQ is drawn parallel to AE (see Figure 10.6a). The coordinates of point Q give the stresses on the plane AE. Thus, Normal stress 60 lb/in.2 Shear stress 30 lb/in.2 10.3 ■ Stresses Caused by a Point Load Boussinesq (1883) solved the problem of stresses produced at any point in a homogeneous, elastic, and isotropic medium as the result of a point load applied on the surface of an inﬁnitely large half-space. According to Figure 10.7, Boussinesq’s solution for normal stresses at a point caused by the point load P is ¢sx x2 y2 y 2z P 3x2z 3 2d f e 5 11 2m2 c 2 2p L Lr 1L z2 Lr (10.10) ¢sy y2 x2 P 2y2z x2z 3 2d f e 5 11 2m2 c 2 2p L Lr 1L z2 Lr (10.11) 3P z3 3P z3 2p L5 2p 1r2 z2 2 5/2 (10.12) and ¢sz P x r y x y L sz z sx z sy Figure 10.7 Stresses in an elastic medium caused by a point load 10.3 Stresses Caused by a Point Load 261 where r 2x2 y2 L 2x2 y2 z2 2r2 z2 m Poisson’s ratio Note that Eqs. (10.10) and (10.11), which are the expressions for horizontal normal stresses, depend on the Poisson’s ratio of the medium. However, the relationship for the vertical normal stress, sz , as given by Eq. (10.12), is independent of Poisson’s ratio. The relationship for sz can be rewritten as ¢sz 1 P 3 P e f 2 I1 z2 2p 3 1r/z2 2 1 4 5/2 z (10.13) where I1 1 3 2p 3 1r/z2 2 1 4 5/2 (10.14) The variation of I1 for various values of r/z is given in Table 10.1. Table 10.1 Variation of I1 for Various Values of r/z [Eq. (10.14)] r/z I1 r/z I1 r/z I1 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.4775 0.4770 0.4765 0.4723 0.4699 0.4657 0.4607 0.4548 0.4482 0.4409 0.4329 0.4242 0.4151 0.4050 0.3954 0.3849 0.3742 0.3632 0.36 0.38 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.20 1.40 1.60 0.3521 0.3408 0.3294 0.3011 0.2733 0.2466 0.2214 0.1978 0.1762 0.1565 0.1386 0.1226 0.1083 0.0956 0.0844 0.0513 0.0317 0.0200 1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 4.20 4.40 4.60 4.80 5.00 0.0129 0.0085 0.0058 0.0040 0.0029 0.0021 0.0015 0.0011 0.00085 0.00066 0.00051 0.00040 0.00032 0.00026 0.00021 0.00017 0.00014 262 Chapter 10: Stresses in a Soil Mass Example 10.3 Consider a point load P 5 kN (Figure 10.7). Calculate the vertical stress increase (z) at z 0, 2 m, 4 m, 6 m, 10 m, and 20 m. Given x 3 m and y 4 m. Solution r 2x2 y2 232 42 5 m The following table can now be prepared. 10.4 ⌬Sz ⴝ a r (m) z (m) r z I1 5 0 2 4 6 10 20 q 2.5 1.25 0.83 0.5 0.25 0 0.0034 0.0424 0.1295 0.2733 0.4103 P b I1 z2 (kN/m2) 0 0.0043 0.0133 0.0180 0.0137 0.0051 ■ Vertical Stress Caused by a Vertical Line Load Figure 10.8 shows a vertical ﬂexible line load of inﬁnite length that has an intensity q/unit length on the surface of a semi-inﬁnite soil mass. The vertical stress increase, sz , inside the soil mass can be determined by using the principles of the theory of elasticity, or ¢sz 2qz3 p1x2 z2 2 2 (10.15) This equation can be rewritten as ¢sz 2q pz3 1x/z2 2 1 4 2 or ¢sz 1q/z2 2 p 3 1x/z2 2 1 4 2 (10.16) Note that Eq. (10.16) is in a nondimensional form. Using this equation, we can calculate the variation of sz/(q/z) with x/z. This is given in Table 10.2. The value of sz calculated by using Eq. (10.16) is the additional stress on soil caused by the line load. The value of sz does not include the overburden pressure of the soil above point A. 10.4 Vertical Stress Caused by a Vertical Line Load 263 q/ Unit length x sz z A x z Figure 10.8 Line load over the surface of a semi-inﬁnite soil mass Table 10.2 Variation of sz /(q/z) with x/z [Eq. (10.16)] x/z ⌬Sz /(q/z) x/z ⌬Sz /(q/z) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.637 0.624 0.589 0.536 0.473 0.407 0.344 0.287 0.237 0.194 0.159 0.130 0.107 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0 0.088 0.073 0.060 0.050 0.042 0.035 0.030 0.025 0.019 0.014 0.011 0.008 0.006 Example 10.4 Figure 10.9a shows two line loads on the ground surface. Determine the increase of stress at point A. Solution Refer to Figure 10.9b. The total stress at A is ¢sz ¢sz112 ¢sz122 ¢sz112 2q1z3 p1x21 z2 2 2 122 1500 2 142 3 p152 42 2 2 12.12 lb/ft2 264 Chapter 10: Stresses in a Soil Mass ¢sz122 2q2z3 p1x22 z2 2 2 122 11000 2 142 3 p1102 42 2 2 3.03 lb/ft2 ¢sz 12.12 3.03 15.15 lb/ft2 q2 1000 lb/ft q1 500 lb/ft 5 ft 5 ft 4 ft A (a) 500 lb/ft 1000 lb/ft x1 x2 4 ft sz(1) sz(2) A 5 ft 4 ft A 10 ft z z (b) Figure 10.9 (a) Two line loads on the ground surface; (b) use of superposition principle to obtain stress at point A 10.5 ■ Vertical Stress Caused by a Horizontal Line Load Figure 10.10 shows a horizontal ﬂexible line load on the surface of a semi-inﬁnite soil mass. The vertical stress increase at point A in the soil mass can be given as ¢sz 2q xz2 p 1x2 z2 2 2 Table 10.3 gives the variation of sz /(q/z) with x/z. (10.17) 10.5 Vertical Stress Caused by a Horizontal Line Load 265 q/ Unit length x sz Table 10.3 Variation of sz /(q/z) with x/z z A x z Figure 10.10 Horizontal line load over the surface of a semi-inﬁnite soil mass x /z ⌬Sz /(q/z) x /z ⌬Sz /(q/z) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.062 0.118 0.161 0.189 0.204 0.207 0.7 0.8 0.9 1.0 1.5 2.0 3.0 0.201 0.189 0.175 0.159 0.090 0.051 0.019 Example 10.5 An inclined line load with a magnitude of 1000 lb/ft is shown in Figure 10.11. Determine the increase of vertical stress sz at point A due to the line load. Solution The vertical component of the inclined load qV 1000 cos 20 939.7 lb/ft, and the horizontal component qH 1000 sin 20 342 lb/ft. For point A, x/z 5/4 1.25. Using Table 10.2, the vertical stress increase at point A due to qV is ¢sz1V2 0.098 qv azb qv 939.7 b 23 lb/ft2 ¢sz1V2 10.0982 a z b 10.0982 a 4 1000 lb/ft 20 5 ft x 4 ft A z Figure 10.11 266 Chapter 10: Stresses in a Soil Mass Similarly, using Table 10.3, the vertical stress increase at point A due to qH is ¢sz1H2 0.125 qH b a z ¢sz1V2 10.1252 a 342 b 10.7 lb/ft2 4 Thus, the total is ¢sz ¢sz1V2 ¢sz1H2 23 10.7 33.7 lb/ft2 10.6 ■ Vertical Stress Caused by a Vertical Strip Load (Finite Width and Inﬁnite Length) The fundamental equation for the vertical stress increase at a point in a soil mass as the result of a line load (Section 10.4) can be used to determine the vertical stress at a point caused by a ﬂexible strip load of width B. (See Figure 10.12.) Let the load per unit area of the strip shown in Figure 10.12 be equal to q. If we consider an elemental strip of width dr, B q Load per unit area x r dr xr z sz x A z Figure 10.12 Vertical stress caused by a ﬂexible strip load 10.7 Vertical Stress Due to Embankment Loading 267 the load per unit length of this strip is equal to q dr. This elemental strip can be treated as a line load. Equation (10.15) gives the vertical stress increase dsz at point A inside the soil mass caused by this elemental strip load. To calculate the vertical stress increase, we need to substitute q dr for q and (x r) for x. So, dsz 21q dr2z3 p 3 1x r2 2 z2 4 2 (10.18) The total increase in the vertical stress (sz) at point A caused by the entire strip load of width B can be determined by integration of Eq. (10.18) with limits of r from B/2 to B/2, or ¢sz dsz B/2 a B/2 2q z3 f dr be p 3 1x r2 2 z2 4 2 q z z e tan1 c d tan1 c d p x 1B/22 x 1B/2 2 Bz3x2 z2 1B2/42 4 3x2 z2 1B2/42 4 2 B2z2 (10.19) f Table 10.4 shows the variation of sz /q with 2z/B for 2x/B. This table can be used conveniently for the calculation of vertical stress at a point caused by a ﬂexible strip load. 10.7 Vertical Stress Due to Embankment Loading Figure 10.14 shows the cross section of an embankment of height H. For this twodimensional loading condition the vertical stress increase may be expressed as ¢sz qo B1 B2 B1 b 1a1 a2 2 1a2 2 d ca p B2 B2 (10.20) where qo gH g unit weight of the embankment soil H height of the embankment a1 1radians2 tan1 a a2 tan1 a B1 B2 B1 b tan1 a b z z (10.21) B1 b z (10.22) Table 10.4 Variation of sz /q with 2z/B and 2x/B [Eq. (10.19)] 2z/B 0.0 0.1 0.2 0.3 0.4 2x /B 0.5 0.6 0.7 0.8 0.9 1.0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 1.000 1.000 0.997 0.990 0.977 0.959 0.937 0.910 0.881 0.850 0.818 0.787 0.755 0.725 0.696 0.668 0.642 0.617 0.593 0.571 0.550 0.530 0.511 0.494 0.477 0.462 0.447 0.433 0.420 0.408 0.396 0.385 0.374 0.364 0.354 0.345 0.337 0.328 0.320 0.313 0.306 0.299 0.292 0.286 0.280 0.274 0.268 0.263 0.258 0.253 0.248 1.000 1.000 0.997 0.989 0.976 0.958 0.935 0.908 0.878 0.847 0.815 0.783 0.752 0.722 0.693 0.666 0.639 0.615 0.591 0.569 0.548 0.529 0.510 0.493 0.476 0.461 0.446 0.432 0.419 0.407 0.395 0.384 0.373 0.363 0.354 0.345 0.336 0.328 0.320 0.313 0.305 0.299 0.292 0.286 0.280 0.274 0.268 0.263 0.258 0.253 0.248 1.000 0.999 0.996 0.987 0.973 0.953 0.928 0.899 0.869 0.837 0.805 0.774 0.743 0.714 0.685 0.658 0.633 0.608 0.585 0.564 0.543 0.524 0.506 0.489 0.473 0.458 0.443 0.430 0.417 0.405 0.393 0.382 0.372 0.362 0.352 0.343 0.335 0.327 0.319 0.312 0.304 0.298 0.291 0.285 0.279 0.273 0.268 0.262 0.257 0.252 0.247 1.000 0.999 0.995 0.984 0.966 0.943 0.915 0.885 0.853 0.821 0.789 0.758 0.728 0.699 0.672 0.646 0.621 0.598 0.576 0.555 0.535 0.517 0.499 0.483 0.467 0.452 0.439 0.425 0.413 0.401 0.390 0.379 0.369 0.359 0.350 0.341 0.333 0.325 0.317 0.310 0.303 0.296 0.290 0.283 0.278 0.272 0.266 0.261 0.256 0.251 0.246 1.000 0.999 0.992 0.978 0.955 0.927 0.896 0.863 0.829 0.797 0.766 0.735 0.707 0.679 0.653 0.629 0.605 0.583 0.563 0.543 0.524 0.507 0.490 0.474 0.460 0.445 0.432 0.419 0.407 0.396 0.385 0.375 0.365 0.355 0.346 0.338 0.330 0.322 0.315 0.307 0.301 0.294 0.288 0.282 0.276 0.270 0.265 0.260 0.255 0.250 0.245 1.000 0.998 0.988 0.967 0.937 0.902 0.866 0.831 0.797 0.765 0.735 0.706 0.679 0.654 0.630 0.607 0.586 0.565 0.546 0.528 0.510 0.494 0.479 0.464 0.450 0.436 0.424 0.412 0.400 0.389 0.379 0.369 0.360 0.351 0.342 0.334 0.326 0.318 0.311 0.304 0.298 0.291 0.285 0.279 0.274 0.268 0.263 0.258 0.253 0.248 0.244 1.000 0.997 0.979 0.947 0.906 0.864 0.825 0.788 0.755 0.724 0.696 0.670 0.646 0.623 0.602 0.581 0.562 0.544 0.526 0.510 0.494 0.479 0.465 0.451 0.438 0.426 0.414 0.403 0.392 0.382 0.372 0.363 0.354 0.345 0.337 0.329 0.321 0.314 0.307 0.301 0.294 0.288 0.282 0.276 0.271 0.266 0.260 0.255 0.251 0.246 0.242 1.000 0.993 0.959 0.908 0.855 0.808 0.767 0.732 0.701 0.675 0.650 0.628 0.607 0.588 0.569 0.552 0.535 0.519 0.504 0.489 0.475 0.462 0.449 0.437 0.425 0.414 0.403 0.393 0.383 0.373 0.364 0.355 0.347 0.339 0.331 0.323 0.316 0.309 0.303 0.296 0.290 0.284 0.278 0.273 0.268 0.263 0.258 0.253 0.248 0.244 0.239 1.000 0.980 0.909 0.833 0.773 0.727 0.691 0.662 0.638 0.617 0.598 0.580 0.564 0.548 0.534 0.519 0.506 0.492 0.479 0.467 0.455 0.443 0.432 0.421 0.410 0.400 0.390 0.381 0.372 0.363 0.355 0.347 0.339 0.331 0.324 0.317 0.310 0.304 0.297 0.291 0.285 0.280 0.274 0.269 0.264 0.259 0.254 0.250 0.245 0.241 0.237 1.000 0.909 0.775 0.697 0.651 0.620 0.598 0.581 0.566 0.552 0.540 0.529 0.517 0.506 0.495 0.484 0.474 0.463 0.453 0.443 0.433 0.423 0.413 0.404 0.395 0.386 0.377 0.369 0.360 0.352 0.345 0.337 0.330 0.323 0.316 0.310 0.304 0.298 0.292 0.286 0.280 0.275 0.270 0.265 0.260 0.255 0.251 0.246 0.242 0.238 0.234 0.000 0.500 0.500 0.499 0.498 0.497 0.495 0.492 0.489 0.485 0.480 0.474 0.468 0.462 0.455 0.448 0.440 0.433 0.425 0.417 0.409 0.401 0.393 0.385 0.378 0.370 0.363 0.355 0.348 0.341 0.334 0.327 0.321 0.315 0.308 0.302 0.297 0.291 0.285 0.280 0.275 0.270 0.265 0.260 0.256 0.251 0.247 0.243 0.239 0.235 0.231 268 Table 10.4 (continued) 2z/B 1.1 1.2 1.3 1.4 1.5 2x /B 1.6 1.7 1.8 1.9 2.0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 0.000 0.091 0.225 0.301 0.346 0.373 0.391 0.403 0.411 0.416 0.419 0.420 0.419 0.417 0.414 0.411 0.407 0.402 0.396 0.391 0.385 0.379 0.373 0.366 0.360 0.354 0.347 0.341 0.335 0.329 0.323 0.317 0.311 0.305 0.300 0.294 0.289 0.284 0.279 0.274 0.269 0.264 0.260 0.255 0.251 0.247 0.243 0.239 0.235 0.231 0.227 0.000 0.020 0.091 0.165 0.224 0.267 0.298 0.321 0.338 0.351 0.360 0.366 0.371 0.373 0.374 0.374 0.373 0.370 0.368 0.364 0.360 0.356 0.352 0.347 0.342 0.337 0.332 0.327 0.321 0.316 0.311 0.306 0.301 0.296 0.291 0.286 0.281 0.276 0.272 0.267 0.263 0.258 0.254 0.250 0.246 0.242 0.238 0.235 0.231 0.227 0.224 0.000 0.007 0.040 0.090 0.141 0.185 0.222 0.250 0.273 0.291 0.305 0.316 0.325 0.331 0.335 0.338 0.339 0.339 0.339 0.338 0.336 0.333 0.330 0.327 0.323 0.320 0.316 0.312 0.307 0.303 0.299 0.294 0.290 0.286 0.281 0.277 0.273 0.268 0.264 0.260 0.256 0.252 0.248 0.244 0.241 0.237 0.234 0.230 0.227 0.223 0.220 0.000 0.003 0.020 0.052 0.090 0.128 0.163 0.193 0.218 0.239 0.256 0.271 0.282 0.291 0.298 0.303 0.307 0.309 0.311 0.312 0.311 0.311 0.309 0.307 0.305 0.302 0.299 0.296 0.293 0.290 0.286 0.283 0.279 0.275 0.271 0.268 0.264 0.260 0.256 0.253 0.249 0.246 0.242 0.239 0.235 0.232 0.229 0.225 0.222 0.219 0.216 0.000 0.002 0.011 0.031 0.059 0.089 0.120 0.148 0.173 0.195 0.214 0.230 0.243 0.254 0.263 0.271 0.276 0.281 0.284 0.286 0.288 0.288 0.288 0.288 0.287 0.285 0.283 0.281 0.279 0.276 0.274 0.271 0.268 0.265 0.261 0.258 0.255 0.252 0.249 0.245 0.242 0.239 0.236 0.233 0.229 0.226 0.223 0.220 0.217 0.215 0.212 0.000 0.001 0.007 0.020 0.040 0.063 0.088 0.113 0.137 0.158 0.177 0.194 0.209 0.221 0.232 0.240 0.248 0.254 0.258 0.262 0.265 0.267 0.268 0.268 0.268 0.268 0.267 0.266 0.265 0.263 0.261 0.259 0.256 0.254 0.251 0.249 0.246 0.243 0.240 0.238 0.235 0.232 0.229 0.226 0.224 0.221 0.218 0.215 0.213 0.210 0.207 0.000 0.001 0.004 0.013 0.027 0.046 0.066 0.087 0.108 0.128 0.147 0.164 0.178 0.191 0.203 0.213 0.221 0.228 0.234 0.239 0.243 0.246 0.248 0.250 0.251 0.251 0.251 0.251 0.250 0.249 0.248 0.247 0.245 0.243 0.241 0.239 0.237 0.235 0.232 0.230 0.227 0.225 0.222 0.220 0.217 0.215 0.212 0.210 0.208 0.205 0.203 0.000 0.000 0.003 0.009 0.020 0.034 0.050 0.068 0.086 0.104 0.122 0.138 0.152 0.166 0.177 0.188 0.197 0.205 0.212 0.217 0.222 0.226 0.229 0.232 0.234 0.235 0.236 0.236 0.236 0.236 0.236 0.235 0.234 0.232 0.231 0.229 0.228 0.226 0.224 0.222 0.220 0.218 0.216 0.213 0.211 0.209 0.207 0.205 0.202 0.200 0.198 0.000 0.000 0.002 0.007 0.014 0.025 0.038 0.053 0.069 0.085 0.101 0.116 0.130 0.143 0.155 0.165 0.175 0.183 0.191 0.197 0.203 0.208 0.212 0.215 0.217 0.220 0.221 0.222 0.223 0.223 0.223 0.223 0.223 0.222 0.221 0.220 0.218 0.217 0.216 0.214 0.212 0.211 0.209 0.207 0.205 0.203 0.201 0.199 0.197 0.195 0.193 0.000 0.000 0.002 0.005 0.011 0.019 0.030 0.042 0.056 0.070 0.084 0.098 0.111 0.123 0.135 0.146 0.155 0.164 0.172 0.179 0.185 0.190 0.195 0.199 0.202 0.205 0.207 0.208 0.210 0.211 0.211 0.212 0.212 0.211 0.211 0.210 0.209 0.208 0.207 0.206 0.205 0.203 0.202 0.200 0.199 0.197 0.195 0.194 0.192 0.190 0.188 269 270 Chapter 10: Stresses in a Soil Mass Example 10.6 With reference to Figure 10.12, we are given q 200 kN/m2, B 6 m, and z 3 m. Determine the vertical stress increase at x 9, 6, 3, and 0 m. Plot a graph of ¢sz against x. Solution The following table can be made: a b x(m) 2x/B 2z/B ⌬Sz /q a ⌬Sz b (kN/m2) 9 6 3 0 3 2 1 0 1 1 1 1 0.017 0.084 0.480 0.818 3.4 16.8 96.0 163.6 From Table 10.4 q 200 kN/m2 The plot of sz against x is given in Figure 10.13. 200 sz (kN/m2 ) 160 120 80 40 0 10 8 6 4 2 0 x (m) 2 4 6 Figure 10.13 Plot of sz against distance x 8 10 ■ For a detailed derivation of the equation, see Das (2008). A simpliﬁed form of Eq. (10.20) is ¢sz qoI2 (10.23) where I2 a function of B1 /z and B2 /z. The variation of I2 with B1/z and B2 /z is shown in Figure 10.15 (Osterberg, 1957). B2 B1 H qo gH a2 a1 Figure 10.14 Embankment loading 0.50 z 3.0 2.0 1.6 1.4 1.2 0.45 1.0 0.9 0.40 0.8 0.7 0.35 0.6 0.5 0.30 I2 0.4 0.25 0.3 0.20 0.2 0.15 0.1 0.10 0.05 Figure 10.15 Osterberg’s chart for determination of vertical stress due to embankment loading B1/z 0 0.00 0.01 0.1 1.0 10.0 B2 /z 271 272 Chapter 10: Stresses in a Soil Mass Example 10.7 An embankment is shown in Figure 10.16a. Determine the stress increase under the embankment at point A. Solution Refer to Figures 10.16b and c. At point A: z z(1) z(2) qo(1) qo(2) (10)(18) 180 kN/m2 From Fig. 10.16b, B1 21 m and B2 20 m. So, B1 21 3.5 z 6 B2 20 3.33 z 6 6m 10 m IV:1.5H IV:2H g 18 kN/m3 6m sz A (a) 15 m 6m 20 m 10 m qo(1) A sz(1) (b) 15 m 10 m qo(2) A sz(2) (c) Figure 10.16 10.8 Vertical Stress Below the Center of a Uniformly Loaded Circular Area 273 From Fig. 10.15, I2(1) 0.495. Again, from Fig. 10.16c, B1 0 and B2 15 m. So, B1 0 0 z 6 B2 15 2.5 z 6 From Fig. 10.15, I2(2) 0.39. Therefore, z z(1) z(2) qo(1)I2(1) qo(2)I2(2) 180[0.490 0.39] 18.9 kN/m2 10.8 ■ Vertical Stress Below the Center of a Uniformly Loaded Circular Area Using Boussinesq’s solution for vertical stress sz caused by a point load [Eq. (10.12)], one also can develop an expression for the vertical stress below the center of a uniformly loaded ﬂexible circular area. From Figure 10.17, let the intensity of pressure on the circular area of radius R be equal to q. The total load on the elemental area (shaded in the ﬁgure) is equal to qr dr da. The vertical stress, dsz , at point A caused by the load on the elemental area (which may be assumed to be a concentrated load) can be obtained from Eq. (10.12): dsz 31qr dr da2 2p z3 1r2 z2 2 5/2 (10.24) Load per unit area q r R dr da z sz A Figure 10.17 Vertical stress below the center of a uniformly loaded ﬂexible circular area 274 Chapter 10: Stresses in a Soil Mass The increase in the stress at point A caused by the entire loaded area can be found by integrating Eq. (10.24): ¢sz dsz a2p a0 rR r0 3q z3r dr da 2p 1r2 z2 2 5/2 So, ¢sz q e 1 1 f 3 1R/z2 2 1 4 3/2 (10.25) The variation of sz /q with z/R as obtained from Eq. (10.25) is given in Table 10.5. A plot of this also is shown in Figure 10.18. The value of sz decreases rapidly with depth, and at z 5R, it is about 6% of q, which is the intensity of pressure at the ground surface. Table 10.5 Variation of sz /q with z/R [Eq. (10.25)] z /R ⌬Sz /q z /R ⌬Sz /q 0 0.02 0.05 0.10 0.2 0.4 0.5 0.8 1 0.9999 0.9998 0.9990 0.9925 0.9488 0.9106 0.7562 1.0 1.5 2.0 2.5 3.0 4.0 5.0 0.6465 0.4240 0.2845 0.1996 0.1436 0.0869 0.0571 0 1 2 z 3 R 4 5 6 0.0 0.2 0.4 0.6 sz q 0.8 1.0 Figure 10.18 Stress under the center of a uniformly loaded ﬂexible circular area 10.9 Vertical Stress at any Point Below a Uniformly Loaded Circular Area 10.9 275 Vertical Stress at Any Point Below a Uniformly Loaded Circular Area A detailed tabulation for calculation of vertical stress below a uniformly loaded ﬂexible circular area was given by Ahlvin and Ulery (1962). Referring to Figure 10.19, we ﬁnd that sz at any point A located at a depth z at any distance r from the center of the loaded area can be given as ¢sz q1A¿ B¿ 2 (10.26) where A and B are functions of z/R and r/R. (See Tables 10.6 and 10.7.) q R r z z r sz A sz Figure 10.19 Vertical stress at any point below a uniformly loaded circular area Example 10.8 Consider a uniformly loaded flexible circular area on the ground surface, as shown in Fig. 10.19. Given: R 3 m and uniform load q 100 kN/m2. Calculate the increase in vertical stress at depths of 1.5 m, 3 m, 4.5 m, 6 m, and 12 m below the ground surface for points at (a) r 0 and (b) r 4.5 m. Solution From Eq. (10.26), ¢sz q 1A¿ B¿ 2 Given R 3 m and q 100 kN/m2. (continued on page 278) Table 10.6 Variation of A with z/R and r/R * r/R z/R 0 0.2 0.4 0.6 0.8 1 1.2 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.2 1.5 2 2.5 3 4 5 6 7 8 9 10 1.0 0.90050 0.80388 0.71265 0.62861 0.55279 0.48550 0.42654 0.37531 0.33104 0.29289 0.23178 0.16795 0.10557 0.07152 0.05132 0.02986 0.01942 0.01361 0.01005 0.00772 0.00612 1.0 0.89748 0.79824 0.70518 0.62015 0.54403 0.47691 0.41874 0.36832 0.32492 0.28763 0.22795 0.16552 0.10453 0.07098 0.05101 0.02976 0.01938 1.0 0.88679 0.77884 0.68316 0.59241 0.51622 0.45078 0.39491 0.34729 0.30669 0.27005 0.21662 0.15877 0.10140 0.06947 0.05022 0.02907 1.0 0.86126 0.73483 0.62690 0.53767 0.46448 0.40427 0.35428 0.31243 0.27707 0.24697 0.19890 0.14804 0.09647 0.06698 0.04886 0.02802 1.0 0.78797 0.63014 0.52081 0.44329 0.38390 0.33676 0.29833 0.26581 0.23832 0.21468 0.17626 0.13436 0.09011 0.06373 0.04707 0.02832 0.5 0.43015 0.38269 0.34375 0.31048 0.28156 0.25588 0.21727 0.21297 0.19488 0.17868 0.15101 0.11892 0.08269 0.05974 0.04487 0.02749 0.01835 0.01307 0.00976 0.00755 0.00600 0 0.09645 0.15433 0.17964 0.18709 0.18556 0.17952 0.17124 0.16206 0.15253 0.14329 0.12570 0.10296 0.07471 0.05555 0.04241 0.02651 0 0.02787 0.05251 0.07199 0.08593 0.09499 0.10010 0.10228 0.10236 0.10094 0.09849 0.09192 0.08048 0.06275 0.04880 0.03839 0.02490 0 0.00856 0.01680 0.02440 0.03118 0.03701 0.00477 0.04558 0.05185 0.05260 0.05116 0.04496 0.03787 0.03150 0.02193 0.01573 0.01168 0.00894 0.00703 0.00566 0.00465 * Source: From Ahlvin, R. G., and H. H. Ulery. Tabulated Values for Determining the Complete Pattern of Stresses, Strains, and Deflections Beneath a Uniform Circular Load on a Homogeneous Half Space. In Highway Research Bulletin 342, Highway Research Board, National Research Council, Washington, D.C., 1962, Tables 1 and 2, p. 3. Reproduced with permission of the Transportation Research Board. Table 10.7 Variation of B with z/R and r/R * r/R z/R 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.2 1.5 2 2.5 3 4 5 6 7 8 9 10 0 0.09852 0.18857 0.26362 0.32016 0.35777 0.37831 0.38487 0.38091 0.36962 0.35355 0.31485 0.25602 0.17889 0.12807 0.09487 0.05707 0.03772 0.02666 0.01980 0.01526 0.01212 0 0.10140 0.19306 0.26787 0.32259 0.35752 0.37531 0.37962 0.37408 0.36275 0.34553 0.30730 0.25025 0.18144 0.12633 0.09394 0.05666 0.03760 0 0.11138 0.20772 0.28018 0.32748 0.35323 0.36308 0.36072 0.35133 0.33734 0.32075 0.28481 0.23338 0.16644 0.12126 0.09099 0.05562 0 0.13424 0.23524 0.29483 0.32273 0.33106 0.32822 0.31929 0.30699 0.29299 0.27819 0.24836 0.20694 0.15198 0.11327 0.08635 0.05383 0 0.18796 0.25983 0.27257 0.26925 0.26236 0.25411 0.24638 0.23779 0.22891 0.21978 0.20113 0.17368 0.13375 0.10298 0.08033 0.05145 0 0.05388 0.08513 0.10757 0.12404 0.13591 0.14440 0.14986 0.15292 0.15404 0.15355 0.14915 0.13732 0.11331 0.09130 0.07325 0.04773 0.03384 0.02468 0.01868 0.01459 0.01170 0 0.07899 0.07759 0.04316 0.00766 0.02165 0.04457 0.06209 0.07530 0.08507 0.09210 0.10002 0.10193 0.09254 0.07869 0.06551 0.04532 * 276 1.5 0 0.02672 0.04448 0.04999 0.04535 0.03455 0.02101 0.00702 0.00614 0.01795 0.02814 0.04378 0.05745 0.06371 0.06022 0.05354 0.03995 0.00924 2 0 0.00845 0.01593 0.02166 0.02522 0.02651 0.02329 0.01005 0.00023 0.01385 0.02836 0.03429 0.03511 0.03066 0.02474 0.01968 0.01577 0.01279 0.01054 0.00879 Source: From Ahlvin, R. G., and H. H. Ulery. Tabulated Values for Determining the Complete Pattern of Stresses, Strains, and Deflections Beneath a Uniform Circular Load on a Homogeneous Half Space. In Highway Research Bulletin 342, Highway Research Board, National Research Council, Washington, D.C., 1962, Tables 1 and 2, p. 3. Reproduced with permission of the Transportation Research Board. Table 10.6 (continued) 3 4 5 0 0.00211 0.00419 0.00622 0 0.00084 0.00167 0.00250 0 0.00042 0.00083 0 6 0 7 0 8 0.00048 0.00030 0.00020 0.01013 0.00407 0.00209 0.00118 0.00071 0.01742 0.01935 0.02142 0.02221 0.02143 0.01980 0.01592 0.01249 0.00983 0.00784 0.00635 0.00520 0.00438 0.00761 0.00871 0.01013 0.01160 0.01221 0.01220 0.01109 0.00949 0.00795 0.00661 0.00554 0.00466 0.00397 0.00393 0.00459 0.00548 0.00659 0.00732 0.00770 0.00768 0.00708 0.00628 0.00548 0.00472 0.00409 0.00352 0.00226 0.00269 0.00325 0.00399 0.00463 0.00505 0.00536 0.00527 0.00492 0.00445 0.00398 0.00353 0.00326 5 6 10 12 14 0 0 0 0.00053 0.00025 0.00014 0.00009 0.00143 0.00171 0.00210 0.00264 0.00308 0.00346 0.00384 0.00394 0.00384 0.00360 0.00332 0.00301 0.00273 0.00097 0.00115 0.00141 0.00180 0.00214 0.00242 0.00282 0.00298 0.00299 0.00291 0.00276 0.00256 0.00241 0.00050 0.00029 0.00018 0.00073 0.00094 0.00115 0.00132 0.00160 0.00179 0.00188 0.00193 0.00189 0.00184 0.00043 0.00056 0.00068 0.00079 0.00099 0.00113 0.00124 0.00130 0.00134 0.00133 0.00027 0.00036 0.00043 0.00051 0.00065 0.00075 0.00084 0.00091 0.00094 0.00096 7 8 10 12 14 Table 10.7 (continued) 3 4 0 0.00210 0.00412 0.00599 0 0.00084 0.00166 0.00245 0 0.00042 0.00083 0 0 0 0 0 0 0.00024 0.00015 0.00010 0.00991 0.00388 0.00199 0.00116 0.00073 0.00049 0.00025 0.00014 0.00009 0.01115 0.00995 0.00669 0.00028 0.00661 0.01112 0.01515 0.01522 0.01380 0.01204 0.01034 0.00888 0.00764 0.00608 0.00632 0.00600 0.00410 0.00130 0.00157 0.00595 0.00810 0.00867 0.00842 0.00779 0.00705 0.00631 0.00344 0.00378 0.00401 0.00371 0.00271 0.00134 0.00155 0.00371 0.00496 0.00547 0.00554 0.00533 0.00501 0.00210 0.00236 0.00265 0.00278 0.00250 0.00192 0.00029 0.00132 0.00254 0.00332 0.00372 0.00386 0.00382 0.00135 0.00156 0.00181 0.00202 0.00201 0.00179 0.00094 0.00013 0.00110 0.00185 0.00236 0.00265 0.00281 0.00092 0.00107 0.00126 0.00148 0.00156 0.00151 0.00109 0.00043 0.00028 0.00093 0.00141 0.00178 0.00199 0.00048 0.00028 0.00018 0.00068 0.00084 0.00094 0.00099 0.00094 0.00070 0.00037 0.00002 0.00035 0.00066 0.00040 0.00050 0.00059 0.00065 0.00068 0.00061 0.00047 0.00029 0.00008 0.00012 0.00026 0.00033 0.00039 0.00046 0.00050 0.00049 0.00045 0.00037 0.00025 0.00012 277 278 Chapter 10: Stresses in a Soil Mass Part (a) We can prepare the following table: (Note: r/R 0. A and B values are from Tables 10.6 and 10.7.) Depth, z (m) z /R A B ⌬z (kN/m2) 1.5 3 4.5 6 12 0.5 1.0 1.5 2.0 4.0 0.553 0.293 0.168 0.106 0.03 0.358 0.354 0.256 0.179 0.057 91.1 64.7 42.4 28.5 8.7 Part (b) r/R 4.5/3 1.5 10.10 Depth, z (m) z /R A 1.5 3 4.5 6 12 0.5 1.0 1.5 2.0 4.0 0.095 0.098 0.08 0.063 0.025 B 0.035 0.028 0.057 0.064 0.04 ⌬z (kN/m2) 6.0 12.6 13.7 12.7 6.5 ■ Vertical Stress Caused by a Rectangularly Loaded Area Boussinesq’s solution also can be used to calculate the vertical stress increase below a ﬂexible rectangular loaded area, as shown in Figure 10.20. The loaded area is located at the ground surface and has length L and width B. The uniformly distributed load per unit area is equal to q. To determine the increase in the vertical stress (sz) at point A, which is located at depth z below the corner of the rectangular area, we need to consider a small elemental area dx dy of the rectangle. (This is shown in Figure 10.20.) The load on this elemental area can be given by dq q dx dy (10.27) The increase in the stress (dsz) at point A caused by the load dq can be determined by using Eq. (10.12). However, we need to replace P with dq q dx dy and r2 with x2 + y2. Thus, dsz 3q dx dy z3 2p1x2 y2 z2 2 5/2 (10.28) 10.10 Vertical Stress Caused by a Rectangularly Loaded Area 279 q x dy dx B L y sz A sz Figure 10.20 Vertical stress below the corner of a uniformly loaded ﬂexible rectangular area z The increase in the stress, at point A caused by the entire loaded area can now be determined by integrating the preceding equation. We obtain ¢sz dsz B L y0 x0 3qz3 1dx dy2 2p1x2 y2 z2 2 5/2 qI3 (10.29) where I3 1 2mn2m2 n2 1 m2 n2 2 2mn2m2 n2 1 1 a a b tan bd c 2 4p m n2 m2n2 1 m2 n2 1 m2 n2 m2n2 1 (10.30) m B z (10.31) n L z (10.32) The variation of I3 with m and n is shown in Table 10.8 and Figure 10.21. 280 Chapter 10: Stresses in a Soil Mass Table 10.8 Variation of I3 with m and n [Eq. (10.30)] m n 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 4.0 5.0 6.0 0.0047 0.0092 0.0132 0.0168 0.0198 0.0222 0.0242 0.0258 0.0270 0.0279 0.0293 0.0301 0.0306 0.0309 0.0311 0.0314 0.0315 0.0316 0.0316 0.0316 0.0092 0.0179 0.0259 0.0328 0.0387 0.0435 0.0474 0.0504 0.0528 0.0547 0.0573 0.0589 0.0599 0.0606 0.0610 0.0616 0.0618 0.0619 0.0620 0.0620 0.0132 0.0259 0.0374 0.0474 0.0559 0.0629 0.0686 0.0731 0.0766 0.0794 0.0832 0.0856 0.0871 0.0880 0.0887 0.0895 0.0898 0.0901 0.0901 0.0902 0.0168 0.0328 0.0474 0.0602 0.0711 0.0801 0.0873 0.0931 0.0977 0.1013 0.1063 0.1094 0.1114 0.1126 0.1134 0.1145 0.1150 0.1153 0.1154 0.1154 0.0198 0.0387 0.0559 0.0711 0.0840 0.0947 0.1034 0.1104 0.1158 0.1202 0.1263 0.1300 0.1324 0.1340 0.1350 0.1363 0.1368 0.1372 0.1374 0.1374 0.0222 0.0435 0.0629 0.0801 0.0947 0.1069 0.1169 0.1247 0.1311 0.1361 0.1431 0.1475 0.1503 0.1521 0.1533 0.1548 0.1555 0.1560 0.1561 0.1562 0.0242 0.0474 0.0686 0.0873 0.1034 0.1168 0.1277 0.1365 0.1436 0.1491 0.1570 0.1620 0.1652 0.1672 0.1686 0.1704 0.1711 0.1717 0.1719 0.1719 0.0258 0.0504 0.0731 0.0931 0.1104 0.1247 0.1365 0.1461 0.1537 0.1598 0.1684 0.1739 0.1774 0.1797 0.1812 0.1832 0.1841 0.1847 0.1849 0.1850 0.0270 0.0528 0.0766 0.0977 0.1158 0.1311 0.1436 0.1537 0.1619 0.1684 0.1777 0.1836 0.1874 0.1899 0.1915 0.1938 0.1947 0.1954 0.1956 0.1957 0.0279 0.0547 0.0794 0.1013 0.1202 0.1361 0.1491 0.1598 0.1684 0.1752 0.1851 0.1914 0.1955 0.1981 0.1999 0.2024 0.2034 0.2042 0.2044 0.2045 The increase in the stress at any point below a rectangularly loaded area can be found by using Eq. (10.29). This can be explained by reference to Figure 10.22. Let us determine the stress at a point below point A at depth z. The loaded area can be divided into four rectangles as shown. The point A is the corner common to all four rectangles. The increase in the stress at depth z below point A due to each rectangular area can now be calculated by using Eq. (10.29). The total stress increase caused by the entire loaded area can be given by ¢sz q 3I3112 I3122 I3132 I3142 4 (10.33) where I3(1) , I3(2) , I3(3) , and I3(4) values of I3 for rectangles 1, 2, 3, and 4, respectively. In most cases the vertical stress increase below the center of a rectangular area (Figure 10.23) is important. This stress increase can be given by the relationship ¢sz qI4 (10.34) 10.10 Vertical Stress Caused by a Rectangularly Loaded Area 281 Table 10.8 (continued) 1.2 1.4 1.6 1.8 2.0 2.5 3.0 4.0 5.0 6.0 0.0293 0.0573 0.0832 0.1063 0.1263 0.1431 0.1570 0.1684 0.1777 0.1851 0.1958 0.2028 0.2073 0.2103 0.2124 0.2151 0.2163 0.2172 0.2175 0.2176 0.0301 0.0589 0.0856 0.1094 0.1300 0.1475 0.1620 0.1739 0.1836 0.1914 0.2028 0.2102 0.2151 0.2183 0.2206 0.2236 0.2250 0.2260 0.2263 0.2264 0.0306 0.0599 0.0871 0.1114 0.1324 0.1503 0.1652 0.1774 0.1874 0.1955 0.2073 0.2151 0.2203 0.2237 0.2261 0.2294 0.2309 0.2320 0.2324 0.2325 0.0309 0.0606 0.0880 0.1126 0.1340 0.1521 0.1672 0.1797 0.1899 0.1981 0.2103 0.2184 0.2237 0.2274 0.2299 0.2333 0.2350 0.2362 0.2366 0.2367 0.0311 0.0610 0.0887 0.1134 0.1350 0.1533 0.1686 0.1812 0.1915 0.1999 0.2124 0.2206 0.2261 0.2299 0.2325 0.2361 0.2378 0.2391 0.2395 0.2397 0.0314 0.0616 0.0895 0.1145 0.1363 0.1548 0.1704 0.1832 0.1938 0.2024 0.2151 0.2236 0.2294 0.2333 0.2361 0.2401 0.2420 0.2434 0.2439 0.2441 0.0315 0.0618 0.0898 0.1150 0.1368 0.1555 0.1711 0.1841 0.1947 0.2034 0.2163 0.2250 0.2309 0.2350 0.2378 0.2420 0.2439 0.2455 0.2460 0.2463 0.0316 0.0619 0.0901 0.1153 0.1372 0.1560 0.1717 0.1847 0.1954 0.2042 0.2172 0.2260 0.2320 0.2362 0.2391 0.2434 0.2455 0.2472 0.2479 0.2482 0.0316 0.0620 0.0901 0.1154 0.1374 0.1561 0.1719 0.1849 0.1956 0.2044 0.2175 0.2263 0.2323 0.2366 0.2395 0.2439 0.2461 0.2479 0.2486 0.2489 0.0316 0.0620 0.0902 0.1154 0.1374 0.1562 0.1719 0.1850 0.1957 0.2045 0.2176 0.2264 0.2325 0.2367 0.2397 0.2441 0.2463 0.2481 0.2489 0.2492 where I4 1 m21 2n21 m1 m1n1 2 sin1 d (10.35) c 2 2 2 2 2 2 p 21 m n 11 n1 2 1m1 n1 2 2m1 n21 21 n21 1 1 m1 L B (10.36) n1 z b (10.37) b B 2 (10.38) The variation of I4 with m1 and n1 is given in Table 10.9. 0.26 0.24 2.0 1.8 1.4 0.22 1.2 1.0 0.20 0.9 0.8 0.18 0.7 0.6 0.14 0.5 I3 0.16 0.12 0.4 0.10 0.3 0.08 0.2 0.06 0.04 n 0.1 0.02 0.00 1.0 m 10.0 0.1 Figure 10.21 Variation of I3 with m and n 1 B 3 A 4 2 L 282 Figure 10.22 Increase of stress at any point below a rectangularly loaded ﬂexible area 10.10 Vertical Stress Caused by a Rectangularly Loaded Area 283 L q B x LB y sz A sz z Figure 10.23 Vertical stress below the center of a uniformly loaded ﬂexible rectangular area Table 10.9 Variation of I4 with m1 and n1 [Eq. (10.35)] m1 n1 1 2 3 4 5 6 7 8 9 10 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 0.994 0.960 0.892 0.800 0.701 0.606 0.522 0.449 0.388 0.336 0.179 0.108 0.072 0.051 0.038 0.029 0.023 0.019 0.997 0.976 0.932 0.870 0.800 0.727 0.658 0.593 0.534 0.481 0.293 0.190 0.131 0.095 0.072 0.056 0.045 0.037 0.997 0.977 0.936 0.878 0.814 0.748 0.685 0.627 0.573 0.525 0.348 0.241 0.174 0.130 0.100 0.079 0.064 0.053 0.997 0.977 0.936 0.880 0.817 0.753 0.692 0.636 0.585 0.540 0.373 0.269 0.202 0.155 0.122 0.098 0.081 0.067 0.997 0.977 0.937 0.881 0.818 0.754 0.694 0.639 0.590 0.545 0.384 0.285 0.219 0.172 0.139 0.113 0.094 0.079 0.997 0.977 0.937 0.881 0.818 0.755 0.695 0.640 0.591 0.547 0.389 0.293 0.229 0.184 0.150 0.125 0.105 0.089 0.997 0.977 0.937 0.881 0.818 0.755 0.695 0.641 0.592 0.548 0.392 0.298 0.236 0.192 0.158 0.133 0.113 0.097 0.997 0.977 0.937 0.881 0.818 0.755 0.696 0.641 0.592 0.549 0.393 0.301 0.240 0.197 0.164 0.139 0.119 0.103 0.997 0.977 0.937 0.881 0.818 0.755 0.696 0.641 0.593 0.549 0.394 0.302 0.242 0.200 0.168 0.144 0.124 0.108 0.997 0.977 0.937 0.881 0.818 0.755 0.696 0.642 0.593 0.549 0.395 0.303 0.244 0.202 0.171 0.147 0.128 0.112 284 Chapter 10: Stresses in a Soil Mass Example 10.9 The plan of a uniformly loaded rectangular area is shown in Figure 10.24a. Determine the vertical stress increase ¢sz below point A¿ at a depth of z 4 m. 3m 2 m q 150 kN/m2 1m A (a) 2m q 150 kN/m2 q 150 kN/m2 A 2m A 4m 1m (b) (c) Figure 10.24 Solution The stress increase ¢sz can be written as sz sz(1) sz(2) where sz(1) stress increase due to the loaded area shown in Figure 10.24b sz(2) stress increase due to the loaded area shown in Figure 10.24c For the loaded area shown in Figure 10.24b: m 2 B 0.5 z 4 n 4 L 1 z 4 From Figure 10.21 for m 0.5 and n 1, the value of I3 0.1225. So sz(1) qI3 (150)(0.1225) 18.38 kN/m2 Similarly, for the loaded area shown in Figure 10.24c: m 1 B 0.25 z 4 n 2 L 0.5 z 4 Thus, I3 0.0473. Hence sz(2) (150)(0.0473) 7.1 kN/m2 So sz sz(1) sz(2) 18.38 7.1 11.28 kN/m2 ■ 10.12 Inﬂuence Chart for Vertical Pressure B B Flexible strip load a 285 B a a a Uniformly loaded square area (flexible) Plan Plan q q 0 0.9 sz q 0.7 0.5 sz 0.9 q 0.7 0.5 B B 2B 2B 0.3 0.2 0.06 0.08 0.1 0.3 0.1 0.08 Graph 3B sz 0.2 q Graph 0.06 3B 0.04 0 B 2B 4B 4B 5B 5B 2.5B Figure 10.25 Vertical pressure isobars under a ﬂexible strip load (Note: Isobars are for line a–a as shown on the plan) 10.11 0.02 2.5B 2B B sz q 0 Figure 10.26 Vertical pressure isobars under a uniformly loaded square area (Note: Isobars are for line a–a as shown on the plan) Stress Isobars In Section 10.6, we developed the relationship to estimate sz at any point due to a vertical strip loading. Also, Section 10.10 provides the relationships to calculate sz at any point due to a vertically and uniformly loaded rectangular area. These relationships for sz can be used to calculate the stress increase at various grid points below the loaded area. Based on those calculated stress increases, stress isobars can be plotted. Figures 10.25 and 10.26 show such stress isobars under uniformly loaded (vertically) strip and square areas. 10.12 Inﬂuence Chart for Vertical Pressure Equation (10.25) can be rearranged and written in the form ¢sz 2/3 R 1 a1 b z q B (10.39) 286 Chapter 10: Stresses in a Soil Mass Table 10.10 Values of R/z for Various Pressure Ratios [Eq. (10.39)] ⌬SZ/q R/z ⌬SZ/q R/z 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0 0.1865 0.2698 0.3383 0.4005 0.4598 0.5181 0.5768 0.6370 0.6997 0.7664 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 0.8384 0.9176 1.0067 1.1097 1.2328 1.3871 1.5943 1.9084 2.5232 q Note that R/z and sz /q in this equation are nondimensional quantities. The values of R/z that correspond to various pressure ratios are given in Table 10.10. Using the values of R/z obtained from Eq. (10.39) for various pressure ratios, Newmark (1942) presented an inﬂuence chart that can be used to determine the vertical pressure at any point below a uniformly loaded ﬂexible area of any shape. Figure 10.27 shows an inﬂuence chart that has been constructed by drawing concentric circles. The radii of the circles are equal to the R/z values corresponding to sz /q 0, 0.1, 0.2, . . . , 1. (Note: For sz /q 0, R/z 0, and for sz /q 1, R/z q, so nine A B Influence value 0.005 Figure 10.27 Inﬂuence chart for vertical pressure based on Boussinesq’s theory (Bulletin No. 338. Influence Charts for Computation of Stresses in Elastic. Foundations, by Nathan M. Newmark. University of Illinois, 1942.) 10.12 Inﬂuence Chart for Vertical Pressure 287 circles are shown.) The unit length for plotting the circles is AB. The circles are divided by several equally spaced radial lines. The inﬂuence value of the chart is given by 1/N, where N is equal to the number of elements in the chart. In Figure 10.27, there are 200 elements; hence, the inﬂuence value is 0.005. The procedure for obtaining vertical pressure at any point below a loaded area is as follows: 1. Determine the depth z below the uniformly loaded area at which the stress increase is required. 2. Plot the plan of the loaded area with a scale of z equal to the unit length of the chart (AB). 3. Place the plan (plotted in step 2) on the inﬂuence chart in such a way that the point below which the stress is to be determined is located at the center of the chart. 4. Count the number of elements (M) of the chart enclosed by the plan of the loaded area. The increase in the pressure at the point under consideration is given by ¢sz 1IV2 qM (10.40) where IV influence value q pressure on the loaded area Example 10.10 The cross section and plan of a column footing are shown in Figure 10.28. Find the increase in vertical stress produced by the column footing at point A. 660 kN 1.5 m Footing size 3m3m 3m A 1.5 m 3m A 3m Figure 10.28 Cross section and plan of a column footing 288 Chapter 10: Stresses in a Soil Mass A A B Influence value 0.005 Figure 10.29 Determination of stress at a point by use of Newmark’s inﬂuence chart Solution Point A is located at à depth 3 m below the bottom of the footing. The plan of the square footing has been replotted to a scale of AB 3 m and placed on the inﬂuence chart (Figure 10.29) in such a way that point A on the plan falls directly over the center of the chart. The number of elements inside the outline of the plan is about 48.5. Hence, ¢sz 1IV2qM 0.005 a 10.13 660 b 48.5 17.78 kN/m2 33 ■ Summary and General Comments This chapter presents the relationships for determining vertical stress at a point due to the application of various types of loading on the surface of a soil mass. The types of loading considered here are point, line, strip, embankment, circular, and rectangular. These relationships are derived by integration of Boussinesq’s equation for a point load. The equations and graphs presented in this chapter are based entirely on the principles of the theory of elasticity; however, one must realize the limitations of these theories when they are applied to a soil medium. This is because soil deposits, in general, are not homogeneous, perfectly elastic, and isotropic. Hence, some deviations from the theoretical stress calculations can be expected in the ﬁeld. Only a limited number of Problems 289 ﬁeld observations are available in the literature for comparision purposes. On the basis of these results, it appears that one could expect a difference of 25 to 30% between theoretical estimates and actual ﬁeld values. Problems 10.1 A soil element is shown in Figure 10.30. Determine the following: a. Maximum and minimum principal stresses b. Normal and shear stresses on the plane AB Use Eqs. (10.3), (10.4), (10.6), and (10.7). 10.2 Repeat Problem 10.1 for the element shown in Figure 10.31. 10.3 Using the principles of Mohr’s circles for the soil element shown in Figure 10.32, determine the following: a. Maximum and minimum principal stresses b. Normal and shear stresses on the plane AB 10.4 Repeat Problem 10.3 for the soil element shown in Figure 10.33. 10.5 A soil element is shown in Figure 10.34. Determine the following: a. Maximum and minimum principal stresses b. Normal and shear stresses on the plane AB Use the pole method. 100 kN/m2 45 400 lb/ft2 kN/m2 300 lb/ft2 A B 30 45 kN/m2 60 kN/m2 750 lb/ft2 B 300 lb/ft2 45 A Figure 10.30 Figure 10.31 80 kN/m2 125 lb/ft2 30 kN/m2 55 lb/ft2 A B 20 30 kN/m2 B 150 kN/m2 50 55 lb/ft2 A Figure 10.32 300 lb/ft2 Figure 10.33 290 Chapter 10: Stresses in a Soil Mass 75 kN/m2 90 lb/ft2 30 kN/m2 35 lb/ft2 B B 50 kN/m2 45 lb/ft2 30 kN/m2 35 lb/ft2 50 40 A A Figure 10.34 Figure 10.35 10.6 Repeat Problem 10.5 for the soil element shown in Figure 10.35. 10.7 Point loads of magnitude 2000, 4000, and 6000 lb act at A, B, and C, respectively (Figure 10.36). Determine the increase in vertical stress at a depth of 10 ft below point D. Use Boussinesq’s equation. 10.8 Refer to Figure 10.37. Determine the vertical stress increase, sz, at point A with the following values: • q1 75 kN/m • x1 2 m • q2 300 kN/m • x2 3 m • z2m 10.9 Repeat Problem 10.8 with the following data: • q1 300 kN/m • x1 4 m • q2 260 kN/m • x2 3 m • z3m 10.10 Refer to Figure 10.37. Given: q1 750 lb/ft, x1 8 ft, x2 4 ft, and z 3 ft. If the vertical stress increase at point A due to the loading is 35 lb/ft2, determine the magnitude of q2. B 10 ft A Line load q1 Line load q2 x1 10 ft sz A C 5 ft Figure 10.36 x2 D Figure 10.37 z Problems 291 q1 250 kN/m q2 45 2m 3m 2m sz A Figure 10.38 10.11 Refer to Figure 10.38. Due to the application of line loads q1 and q2, the vertical stress increase, sz , at A is 30 kN/m2. Determine the magnitude of q2. 10.12 Refer to Figure 10.12. Given: B 12 ft, q 350 lb/ft2, x 9 ft, and z 5 ft. Determine the vertical stress increase, sz, at point A. 10.13 Repeat Problem 10.12 for B 3 m, q 60 kN/m2, x 1.5 m, and z 3 m. 10.14 An earth embankment diagram is shown in Figure 10.39. Determine the stress increase at point A due to the embankment load. 10.15 Figure 10.40 shows an embankment load for a silty clay soil layer. Determine the vertical stress increase at points A, B, and C. 10.16 Consider a circularly loaded ﬂexible area on the ground surface. Given that the radius of the circular area R 4 m and that the uniformly distributed load q 200 kN/m2, calculate the vertical stress increase, sz , at points 1.5, 3, 6, 9, and 12 m below the ground surface (immediately below the center of the circular area). 10.17 Figure 10.19 shows a ﬂexible circular area of radius R 10 ft. The uniformly distributed load on the circular area is 2000 lb/ft2. Calculate the vertical stress increase at r 0, 2, 4, 8, and 12 ft, and z 5 ft. 9m Unit weight g 16.5 kN/m3 1 1 8m 1.5 1.5 6m A Figure 10.39 292 Chapter 10: Stresses in a Soil Mass 10 ft 10 ft Unit weight g 120 lb/ft3 1 1 40 ft 2 2 20 ft C B A Figure 10.40 10.18 Refer to Figure 10.41. The circular ﬂexible area is uniformly loaded. Given q 300 kN/m2 and using Newmark’s chart, determine the vertical stress increase sz at point A. 10.19 The plan of a ﬂexible rectangular loaded area is shown in Figure 10.42. The uniformly distributed load on the ﬂexible area, q, is 100 kN/m2. Determine the increase in the vertical stress, sz , at a depth of z 2 m below a. Point A b. Point B c. Point C 4m A Plan sz 4m A Cross section Figure 10.41 References 293 4m 1.6 m B 2m 0.8 m q 100 kN/m2 C A 1.2 m Figure 10.42 10.20 Refer to the ﬂexible loaded rectangular area shown in Figure 10.42. Using Eq. (10.34), determine the vertical stress increase below the center of the area at a depth of 3.5 m. References AHLVIN, R. G., and ULERY, H. H. (1962). “Tabulated Values for Determining the Complete Pattern of Stresses, Strains, and Deﬂections Beneath a Uniform Circular Load on a Homogeneous Half Space,” in Highway Research Bulletin 342, Transportation Research Board, National Research Council, Washington, D.C., 1–13. BOUSSINESQ, J. (1883). Application des Potentials à L’Etude de L’Equilibre et du Mouvement des Solides Elastiques, Gauthier-Villars, Paris. DAS, B. (2008). Advanced Soil Mechanics, 3rd ed., Taylor and Francis, London. NEWMARK, N. M. (1942). “Inﬂuence Charts for Computation of Stresses in Elastic Soil,” University of Illinois Engineering Experiment Station, Bulletin No. 338. OSTERBERG, J. O. (1957). “Inﬂuence Values for Vertical Stresses in Semi-Inﬁnite Mass Due to Embankment Loading,” Proceedings, Fourth International Conference on Soil Mechanics and Foundation Engineering, London, Vol. 1, 393–396. 11 Compressibility of Soil A stress increase caused by the construction of foundations or other loads compresses soil layers. The compression is caused by (a) deformation of soil particles, (b) relocations of soil particles, and (c) expulsion of water or air from the void spaces. In general, the soil settlement caused by loads may be divided into three broad categories: 1. Elastic settlement (or immediate settlement), which is caused by the elastic deformation of dry soil and of moist and saturated soils without any change in the moisture content. Elastic settlement calculations generally are based on equations derived from the theory of elasticity. 2. Primary consolidation settlement, which is the result of a volume change in saturated cohesive soils because of expulsion of the water that occupies the void spaces. 3. Secondary consolidation settlement, which is observed in saturated cohesive soils and is the result of the plastic adjustment of soil fabrics. It is an additional form of compression that occurs at constant effective stress. This chapter presents the fundamental principles for estimating the elastic and consolidation settlements of soil layers under superimposed loadings. The total settlement of a foundation can then be given as ST Sc Ss Se where ST total settlement Sc primary consolidation settlement Ss secondary consolidation settlement Se elastic settlement When foundations are constructed on very compressible clays, the consolidation settlement can be several times greater than the elastic settlement. ELASTIC SETTLEMENT 11.1 Contact Pressure and Settlement Proﬁle Elastic, or immediate, settlement of foundations (Se) occurs directly after the application of a load without a change in the moisture content of the soil. The magnitude of the contact 294 11.1 Contact Pressure and Settlement Proﬁle 295 Contact pressure distribution Settlement proﬁle (a) Contact pressure distribution Settlement proﬁle (b) Figure 11.1 Elastic settlement proﬁle and contact pressure in clay: (a) ﬂexible foundation; (b) rigid foundation settlement will depend on the ﬂexibility of the foundation and the type of material on which it is resting. In Chapter 10, the relationships for determining the increase in stress (which causes elastic settlement) due to the application of line load, strip load, embankment load, circular load, and rectangular load were based on the following assumptions: • • • The load is applied at the ground surface. The loaded area is ﬂexible. The soil medium is homogeneous, elastic, isotropic, and extends to a great depth. In general, foundations are not perfectly ﬂexible and are embedded at a certain depth below the ground surface. It is instructive, however, to evaluate the distribution of the contact pressure under a foundation along with the settlement proﬁle under idealized conditions. Figure 11.1a shows a perfectly ﬂexible foundation resting on an elastic material such as saturated clay. If the foundation is subjected to a uniformly distributed load, the contact pressure will be uniform and the foundation will experience a sagging proﬁle. On the other hand, if we consider a perfectly rigid foundation resting on the ground surface subjected to a uniformly distributed load, the contact pressure and foundation settlement proﬁle will be as shown in Figure 11.1b: the foundation will undergo a uniform settlement and the contact pressure will be redistributed. The settlement proﬁle and contact pressure distribution described are true for soils in which the modulus of elasticity is fairly constant with depth. In the case of cohesionless sand, the modulus of elasticity increases with depth. Additionally, there is a lack of lateral conﬁnement on the edge of the foundation at the ground surface. The sand at the edge of a ﬂexible foundation is pushed outward, and the deﬂection curve of the foundation takes a concave downward shape. The distributions of contact pressure and the settlement proﬁles of a ﬂexible and a rigid foundation resting on sand and subjected to uniform loading are shown in Figures 11.2a and 11.2b, respectively. 296 Chapter 11: Compressibility of Soil Contact pressure distribution Settlement proﬁle (a) Contact pressure distribution Settlement proﬁle (b) 11.2 Figure 11.2 Elastic settlement proﬁle and contact pressure in sand: (a) ﬂexible foundation; (b) rigid foundation Relations for Elastic Settlement Calculation Figure 11.3 shows a shallow foundation subjected to a net force per unit area equal to . Let the Poisson’s ratio and the modulus of elasticity of the soil supporting it be s and Es, respectively. Theoretically, if the foundation is perfectly flexible, the settlement may be expressed as Se ¢s1aB¿ 2 Foundation BL ∆s Rigid foundation settlement 1 m2s IsIf Es (11.1) Df Flexible foundation settlement H ms Poisson’s ratio Es modulus of elasticity Soil Rock Figure 11.3 Elastic settlement of flexible and rigid foundations 11.2 Relations for Elastic Settlement Calculation 297 where net applied pressure on the foundation s Poisson’s ratio of soil Es average modulus of elasticity of the soil under the foundation measured from z 0 to about z 4B B' B/2 for center of foundation B for corner of foundation Is shape factor (Steinbrenner, 1934) F1 1 2ms F2 1 ms (11.2) F1 1 1A 0 A 1 2 p (11.3) F2 n¿ tan 2p (11.4) A0 m¿ ln A1 ln A2 1 A2 11 2m¿ 2 1 2 2m¿ 2 n¿ 2 (11.5) m¿11 2m¿ 2 n¿ 2 1 2 1m¿ 2m¿ 2 1 2 21 n¿ 2 (11.6) m¿ 2m¿ 2 n¿ 2 1 m¿ (11.7) n¿ 2m¿ 2 n¿ 2 1 If depth factor 1Fox, 19482 f a Df B , ms, and L b B (11.8) factor that depends on the location on the foundation where settlement is being calculated • For calculation of settlement at the center of the foundation: a4 L m¿ B n¿ • H B a b 2 For calculation of settlement at a corner of the foundation: a1 L m¿ B H n¿ B The variations of F1 and F2 [Eqs. (11.3) and (11.4)] with m' and n' given in Tables 11.1 and 11.2. Also the variation of If with Df/B and s is given in Table 11.3. Note that when Df 0, the value of If 1 in all cases. 298 Chapter 11: Compressibility of Soil Table 11.1 Variation of F1 with m' and n' m’ n' 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 20.00 50.00 100.00 0.014 0.049 0.095 0.142 0.186 0.224 0.257 0.285 0.309 0.330 0.348 0.363 0.376 0.388 0.399 0.408 0.417 0.424 0.431 0.437 0.443 0.448 0.453 0.457 0.461 0.465 0.468 0.471 0.474 0.477 0.480 0.482 0.485 0.487 0.489 0.491 0.493 0.495 0.496 0.498 0.529 0.548 0.555 0.013 0.046 0.090 0.138 0.183 0.224 0.259 0.290 0.317 0.341 0.361 0.379 0.394 0.408 0.420 0.431 0.440 0.450 0.458 0.465 0.472 0.478 0.483 0.489 0.493 0.498 0.502 0.506 0.509 0.513 0.516 0.519 0.522 0.524 0.527 0.529 0.531 0.533 0.536 0.537 0.575 0.598 0.605 0.012 0.044 0.087 0.134 0.179 0.222 0.259 0.292 0.321 0.347 0.369 0.389 0.406 0.422 0.436 0.448 0.458 0.469 0.478 0.487 0.494 0.501 0.508 0.514 0.519 0.524 0.529 0.533 0.538 0.541 0.545 0.549 0.552 0.555 0.558 0.560 0.563 0.565 0.568 0.570 0.614 0.640 0.649 0.011 0.042 0.084 0.130 0.176 0.219 0.258 0.292 0.323 0.350 0.374 0.396 0.415 0.431 0.447 0.460 0.472 0.484 0.494 0.503 0.512 0.520 0.527 0.534 0.540 0.546 0.551 0.556 0.561 0.565 0.569 0.573 0.577 0.580 0.583 0.587 0.589 0.592 0.595 0.597 0.647 0.678 0.688 0.011 0.041 0.082 0.127 0.173 0.216 0.255 0.291 0.323 0.351 0.377 0.400 0.420 0.438 0.454 0.469 0.481 0.495 0.506 0.516 0.526 0.534 0.542 0.550 0.557 0.563 0.569 0.575 0.580 0.585 0.589 0.594 0.598 0.601 0.605 0.609 0.612 0.615 0.618 0.621 0.677 0.711 0.722 0.011 0.040 0.080 0.125 0.170 0.213 0.253 0.289 0.322 0.351 0.378 0.402 0.423 0.442 0.460 0.476 0.484 0.503 0.515 0.526 0.537 0.546 0.555 0.563 0.570 0.577 0.584 0.590 0.596 0.601 0.606 0.611 0.615 0.619 0.623 0.627 0.631 0.634 0.638 0.641 0.702 0.740 0.753 0.010 0.038 0.077 0.121 0.165 0.207 0.247 0.284 0.317 0.348 0.377 0.402 0.426 0.447 0.467 0.484 0.495 0.516 0.530 0.543 0.555 0.566 0.576 0.585 0.594 0.603 0.610 0.618 0.625 0.631 0.637 0.643 0.648 0.653 0.658 0.663 0.667 0.671 0.675 0.679 0.756 0.803 0.819 0.010 0.038 0.076 0.118 0.161 0.203 0.242 0.279 0.313 0.344 0.373 0.400 0.424 0.447 0.458 0.487 0.514 0.521 0.536 0.551 0.564 0.576 0.588 0.598 0.609 0.618 0.627 0.635 0.643 0.650 0.658 0.664 0.670 0.676 0.682 0.687 0.693 0.697 0.702 0.707 0.797 0.853 0.872 0.010 0.037 0.074 0.116 0.158 0.199 0.238 0.275 0.308 0.340 0.369 0.396 0.421 0.444 0.466 0.486 0.515 0.522 0.539 0.554 0.568 0.581 0.594 0.606 0.617 0.627 0.637 0.646 0.655 0.663 0.671 0.678 0.685 0.692 0.698 0.705 0.710 0.716 0.721 0.726 0.830 0.895 0.918 0.010 0.037 0.074 0.115 0.157 0.197 0.235 0.271 0.305 0.336 0.365 0.392 0.418 0.441 0.464 0.484 0.515 0.522 0.539 0.554 0.569 0.584 0.597 0.609 0.621 0.632 0.643 0.653 0.662 0.671 0.680 0.688 0.695 0.703 0.710 0.716 0.723 0.719 0.735 0.740 0.858 0.931 0.956 11.2 Relations for Elastic Settlement Calculation 299 Table 11.1 (continued) m’ n' 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 20.00 50.00 100.00 4.5 5.0 6.0 7.0 8.0 9.0 0.010 0.036 0.073 0.114 0.155 0.195 0.233 0.269 0.302 0.333 0.362 0.389 0.415 0.438 0.461 0.482 0.516 0.520 0.537 0.554 0.569 0.584 0.597 0.611 0.623 0.635 0.646 0.656 0.666 0.676 0.685 0.694 0.702 0.710 0.717 0.725 0.731 0.738 0.744 0.750 0.878 0.962 0.990 0.010 0.036 0.073 0.113 0.154 0.194 0.232 0.267 0.300 0.331 0.359 0.386 0.412 0.435 0.458 0.479 0.496 0.517 0.535 0.552 0.568 0.583 0.597 0.610 0.623 0.635 0.647 0.658 0.669 0.679 0.688 0.697 0.706 0.714 0.722 0.730 0.737 0.744 0.751 0.758 0.896 0.989 1.020 0.010 0.036 0.072 0.112 0.153 0.192 0.229 0.264 0.296 0.327 0.355 0.382 0.407 0.430 0.453 0.474 0.484 0.513 0.530 0.548 0.564 0.579 0.594 0.608 0.621 0.634 0.646 0.658 0.669 0.680 0.690 0.700 0.710 0.719 0.727 0.736 0.744 0.752 0.759 0.766 0.925 1.034 1.072 0.010 0.036 0.072 0.112 0.152 0.191 0.228 0.262 0.294 0.324 0.352 0.378 0.403 0.427 0.449 0.470 0.473 0.508 0.526 0.543 0.560 0.575 0.590 0.604 0.618 0.631 0.644 0.656 0.668 0.679 0.689 0.700 0.710 0.719 0.728 0.737 0.746 0.754 0.762 0.770 0.945 1.070 1.114 0.010 0.036 0.072 0.112 0.152 0.190 0.227 0.261 0.293 0.322 0.350 0.376 0.401 0.424 0.446 0.466 0.471 0.505 0.523 0.540 0.556 0.571 0.586 0.601 0.615 0.628 0.641 0.653 0.665 0.676 0.687 0.698 0.708 0.718 0.727 0.736 0.745 0.754 0.762 0.770 0.959 1.100 1.150 0.010 0.036 0.072 0.111 0.151 0.190 0.226 0.260 0.291 0.321 0.348 0.374 0.399 0.421 0.443 0.464 0.471 0.502 0.519 0.536 0.553 0.568 0.583 0.598 0.611 0.625 0.637 0.650 0.662 0.673 0.684 0.695 0.705 0.715 0.725 0.735 0.744 0.753 0.761 0.770 0.969 1.125 1.182 10.0 25.0 50.0 0.010 0.036 0.071 0.111 0.151 0.189 0.225 0.259 0.291 0.320 0.347 0.373 0.397 0.420 0.441 0.462 0.470 0.499 0.517 0.534 0.550 0.585 0.580 0.595 0.608 0.622 0.634 0.647 0.659 0.670 0.681 0.692 0.703 0.713 0.723 0.732 0.742 0.751 0.759 0.768 0.977 1.146 1.209 0.010 0.036 0.071 0.110 0.150 0.188 0.223 0.257 0.287 0.316 0.343 0.368 0.391 0.413 0.433 0.453 0.468 0.489 0.506 0.522 0.537 0.551 0.565 0.579 0.592 0.605 0.617 0.628 0.640 0.651 0.661 0.672 0.682 0.692 0.701 0.710 0.719 0.728 0.737 0.745 0.982 1.265 1.408 0.010 0.036 0.071 0.110 0.150 0.188 0.223 0.256 0.287 0.315 0.342 0.367 0.390 0.412 0.432 0.451 0.462 0.487 0.504 0.519 0.534 0.549 0.583 0.576 0.589 0.601 0.613 0.624 0.635 0.646 0.656 0.666 0.676 0.686 0.695 0.704 0.713 0.721 0.729 0.738 0.965 1.279 1.489 100.0 0.010 0.036 0.071 0.110 0.150 0.188 0.223 0.256 0.287 0.315 0.342 0.367 0.390 0.411 0.432 0.451 0.460 0.487 0.503 0.519 0.534 0.548 0.562 0.575 0.588 0.600 0.612 0.623 0.634 0.645 0.655 0.665 0.675 0.684 0.693 0.702 0.711 0.719 0.727 0.735 0.957 1.261 1.499 300 Chapter 11: Compressibility of Soil Table 11.2 Variation of F2 with m' and n' m’ n' 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 20.00 50.00 100.00 1.0 1.2 1.4 1.6 0.049 0.074 0.083 0.083 0.080 0.075 0.069 0.064 0.059 0.055 0.051 0.048 0.045 0.042 0.040 0.037 0.036 0.034 0.032 0.031 0.029 0.028 0.027 0.026 0.025 0.024 0.023 0.022 0.022 0.021 0.020 0.020 0.019 0.018 0.018 0.017 0.017 0.017 0.016 0.016 0.008 0.003 0.002 0.050 0.077 0.089 0.091 0.089 0.084 0.079 0.074 0.069 0.064 0.060 0.056 0.053 0.050 0.047 0.044 0.042 0.040 0.038 0.036 0.035 0.033 0.032 0.031 0.030 0.029 0.028 0.027 0.026 0.025 0.024 0.023 0.023 0.022 0.021 0.021 0.020 0.020 0.019 0.019 0.010 0.004 0.002 0.051 0.080 0.093 0.098 0.096 0.093 0.088 0.083 0.077 0.073 0.068 0.064 0.060 0.057 0.054 0.051 0.049 0.046 0.044 0.042 0.040 0.039 0.037 0.036 0.034 0.033 0.032 0.031 0.030 0.029 0.028 0.027 0.026 0.026 0.025 0.024 0.024 0.023 0.023 0.022 0.011 0.004 0.002 0.051 0.081 0.097 0.102 0.102 0.099 0.095 0.090 0.085 0.080 0.076 0.071 0.067 0.064 0.060 0.057 0.055 0.052 0.050 0.048 0.046 0.044 0.042 0.040 0.039 0.038 0.036 0.035 0.034 0.033 0.032 0.031 0.030 0.029 0.028 0.028 0.027 0.026 0.026 0.025 0.013 0.005 0.003 1.8 0.051 0.083 0.099 0.106 0.107 0.105 0.101 0.097 0.092 0.087 0.082 0.078 0.074 0.070 0.067 0.063 0.061 0.058 0.055 0.053 0.051 0.049 0.047 0.045 0.044 0.042 0.041 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.032 0.031 0.030 0.029 0.029 0.028 0.014 0.006 0.003 2.0 2.5 3.0 3.5 4.0 0.052 0.084 0.101 0.109 0.111 0.110 0.107 0.102 0.098 0.093 0.089 0.084 0.080 0.076 0.073 0.069 0.066 0.063 0.061 0.058 0.056 0.054 0.052 0.050 0.048 0.046 0.045 0.043 0.042 0.041 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.033 0.032 0.031 0.016 0.006 0.003 0.052 0.086 0.104 0.114 0.118 0.118 0.117 0.114 0.110 0.106 0.102 0.097 0.093 0.089 0.086 0.082 0.079 0.076 0.073 0.070 0.067 0.065 0.063 0.060 0.058 0.056 0.055 0.053 0.051 0.050 0.048 0.047 0.046 0.045 0.043 0.042 0.041 0.040 0.039 0.038 0.020 0.008 0.004 0.052 0.086 0.106 0.117 0.122 0.124 0.123 0.121 0.119 0.115 0.111 0.108 0.104 0.100 0.096 0.093 0.090 0.086 0.083 0.080 0.078 0.075 0.073 0.070 0.068 0.066 0.064 0.062 0.060 0.059 0.057 0.055 0.054 0.053 0.051 0.050 0.049 0.048 0.047 0.046 0.024 0.010 0.005 0.052 0.0878 0.107 0.119 0.125 0.128 0.128 0.127 0.125 0.122 0.119 0.116 0.112 0.109 0.105 0.102 0.099 0.096 0.093 0.090 0.087 0.084 0.082 0.079 0.077 0.075 0.073 0.071 0.069 0.067 0.065 0.063 0.062 0.060 0.059 0.057 0.056 0.055 0.054 0.052 0.027 0.011 0.006 0.052 0.087 0.108 0.120 0.127 0.130 0.131 0.131 0.130 0.127 0.125 0.122 0.119 0.116 0.113 0.110 0.107 0.104 0.101 0.098 0.095 0.092 0.090 0.087 0.085 0.083 0.080 0.078 0.076 0.074 0.072 0.071 0.069 0.067 0.066 0.064 0.063 0.061 0.060 0.059 0.031 0.013 0.006 11.2 Relations for Elastic Settlement Calculation 301 Table 11.2 (continued) m’ n' 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 20.00 50.00 100.00 4.5 5.0 6.0 7.0 0.053 0.087 0.109 0.121 0.128 0.132 0.134 0.134 0.133 0.132 0.130 0.127 0.125 0.122 0.119 0.116 0.113 0.110 0.107 0.105 0.102 0.099 0.097 0.094 0.092 0.090 0.087 0.085 0.083 0.081 0.079 0.077 0.076 0.074 0.072 0.071 0.069 0.068 0.066 0.065 0.035 0.014 0.007 0.053 0.087 0.109 0.122 0.130 0.134 0.136 0.136 0.136 0.135 0.133 0.131 0.129 0.126 0.124 0.121 0.119 0.116 0.113 0.111 0.108 0.106 0.103 0.101 0.098 0.096 0.094 0.092 0.090 0.088 0.086 0.084 0.082 0.080 0.078 0.077 0.075 0.074 0.072 0.071 0.039 0.016 0.008 0.053 0.088 0.109 0.123 0.131 0.136 0.138 0.139 0.140 0.139 0.138 0.137 0.135 0.133 0.131 0.129 0.127 0.125 0.123 0.120 0.118 0.116 0.113 0.111 0.109 0.107 0.105 0.103 0.101 0.099 0.097 0.095 0.093 0.091 0.089 0.088 0.086 0.085 0.083 0.082 0.046 0.019 0.010 0.053 0.088 0.110 0.123 0.132 0.137 0.140 0.141 0.142 0.142 0.142 0.141 0.140 0.138 0.137 0.135 0.133 0.131 0.130 0.128 0.126 0.124 0.122 0.120 0.118 0.116 0.114 0.112 0.110 0.108 0.106 0.104 0.102 0.101 0.099 0.097 0.096 0.094 0.092 0.091 0.053 0.022 0.011 8.0 0.053 0.088 0.110 0.124 0.132 0.138 0.141 0.143 0.144 0.144 0.144 0.144 0.143 0.142 0.141 0.139 0.138 0.136 0.135 0.133 0.131 0.130 0.128 0.126 0.124 0.122 0.121 0.119 0.117 0.115 0.114 0.112 0.110 0.108 0.107 0.105 0.104 0.102 0.100 0.099 0.059 0.025 0.013 9.0 0.053 0.088 0.110 0.124 0.133 0.138 0.142 0.144 0.145 0.146 0.146 0.145 0.145 0.144 0.143 0.142 0.141 0.140 0.139 0.137 0.136 0.134 0.133 0.131 0.129 0.128 0.126 0.125 0.123 0.121 0.120 0.118 0.117 0.115 0.114 0.112 0.110 0.109 0.107 0.106 0.065 0.028 0.014 10.0 25.0 50.0 100.0 0.053 0.088 0.110 0.124 0.133 0.139 0.142 0.145 0.146 0.147 0.147 0.147 0.147 0.146 0.145 0.145 0.144 0.143 0.142 0.140 0.139 0.138 0.136 0.135 0.134 0.132 0.131 0.129 0.128 0.126 0.125 0.124 0.122 0.121 0.119 0.118 0.116 0.115 0.113 0.112 0.071 0.031 0.016 0.053 0.088 0.111 0.125 0.134 0.140 0.144 0.147 0.149 0.151 0.152 0.152 0.153 0.153 0.154 0.154 0.154 0.154 0.154 0.154 0.154 0.154 0.154 0.153 0.153 0.153 0.153 0.152 0.152 0.152 0.151 0.151 0.150 0.150 0.150 0.149 0.149 0.148 0.148 0.147 0.124 0.071 0.039 0.053 0.088 0.111 0.125 0.134 0.140 0.144 0.147 0.150 0.151 0.152 0.153 0.154 0.155 0.155 0.155 0.156 0.156 0.156 0.156 0.156 0.156 0.157 0.157 0.157 0.157 0.157 0.157 0.157 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.148 0.113 0.071 0.053 0.088 0.111 0.125 0.134 0.140 0.145 0.148 0.150 0.151 0.153 0.154 0.154 0.155 0.155 0.156 0.156 0.156 0.157 0.157 0.157 0.157 0.157 0.157 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.156 0.142 0.113 302 Chapter 11: Compressibility of Soil Table 11.3 Variation of If with L/B and Df /B If L /B Df/B s ⴝ 0.3 s ⴝ 0.4 s ⴝ 0.5 1 0.5 0.75 1 0.5 0.75 1 0.5 0.75 1 0.77 0.69 0.65 0.82 0.75 0.71 0.87 0.81 0.78 0.82 0.74 0.69 0.86 0.79 0.75 0.91 0.86 0.82 0.85 0.77 0.72 0.89 0.83 0.79 0.93 0.89 0.85 2 5 The elastic settlement of a rigid foundation can be estimated as Se1rigid2 0.93Se1flexible, center2 (11.9) Due to the nonhomogeneous nature of soil deposits, the magnitude of Es may vary with depth. For that reason, Bowles (1987) recommended using a weighted average value of Es in Eq. (11.1) or Es ©Es1i2 ¢z (11.10) z Where Es(i) soil modulus of elasticity within a depth z – z H or 5B, whichever is smaller Representative values of the modulus of elasticity and Poisson’s ratio for different types of soils are given in Tables 11.4 and 11.5, respectively. Table 11.4 Representative Values of the Modulus of Elasticity of Soil Es Soil type Soft clay Hard clay Loose sand Dense sand kN/m2 lb/in.2 1,800–3,500 6,000–14,000 10,000–28,000 35,000–70,000 250–500 850–2,000 1,500–4,000 5,000–10,000 11.2 Relations for Elastic Settlement Calculation 303 Table 11.5 Representative Values of Poisson’s Ratio Type of soil Loose sand Medium sand Dense sand Silty sand Soft clay Medium clay Poisson’s ratio, Ms 0.2–0.4 0.25–0.4 0.3–0.45 0.2–0.4 0.15–0.25 0.2–0.5 Example 11.1 A rigid shallow foundation 1 m 2 m is shown in Figure 11.4. Calculate the elastic settlement at the center of the foundation. Figure 11.4 304 Chapter 11: Compressibility of Soil Solution Given: B 1 m and L 2 m. Note that z 5 m 5B. From Eq. (11.10), Es ©Es1i2 ¢z z 110,0002 122 18,000 2 11 2 112,000 2 122 5 10,400 kN/m2 For the center of the foundation, a4 L 2 2 m¿ B 1 H 5 n¿ 10 B 1 b a a b 2 2 From Tables 11.1 and 11.2, F1 0.641 and F2 0.031. From Eq. (11.2). 2 ms F2 1 ms 2 0.3 10.0312 0.716 0.641 1 0.3 Is F1 Again, Df B 1 L 1, 2, ms 0.3. From Table 11.3, If 0.71. Hence, 1 B 1 m2s IsIf Es 1 0.32 1 ba b 10.716 2 10.71 2 0.0133 m 13.3 mm 11502 a 4 2 10,400 Se1flexible2 ¢s1aB¿ 2 Since the foundation is rigid, from Eq. (11.9), Se(rigid) (0.93)(13.3) 12.4 mm ■ CONSOLIDATION SETTLEMENT 11.3 Fundamentals of Consolidation When a saturated soil layer is subjected to a stress increase, the pore water pressure is increased suddenly. In sandy soils that are highly permeable, the drainage caused by the increase in the pore water pressure is completed immediately. Pore water drainage is accompanied by a reduction in the volume of the soil mass, which results in settlement. Because of rapid drainage of the pore water in sandy soils, elastic settlement and consolidation occur simultaneously. When a saturated compressible clay layer is subjected to a stress increase, elastic settlement occurs immediately. Because the hydraulic conductivity of clay is signiﬁcantly 11.3 Fundamentals of Consolidation 305 smaller than that of sand, the excess pore water pressure generated by loading gradually dissipates over a long period. Thus, the associated volume change (that is, the consolidation) in the clay may continue long after the elastic settlement. The settlement caused by consolidation in clay may be several times greater than the elastic settlement. The time-dependent deformation of saturated clayey soil best can be understood by considering a simple model that consists of a cylinder with a spring at its center. Let the inside area of the cross section of the cylinder be equal to A. The cylinder is ﬁlled with water and has a frictionless watertight piston and valve as shown in Figure 11.5a. At this time, if we place a load P on the piston (Figure 11.5b) and keep the valve closed, the entire load will be taken by the water in the cylinder because water is incompressible. The spring will not go through any deformation. The excess hydrostatic pressure at this time can be given as ¢u P A (11.11) This value can be observed in the pressure gauge attached to the cylinder. In general, we can write P Ps Pw (11.12) P Valve closed Valve closed u P A u 0 (b) (a) P Valve open P Valve open u u P A u 0 (c) Figure 11.5 Spring-cylinder model (d) 306 Chapter 11: Compressibility of Soil where Ps load carried by the spring and Pw load carried by the water. From the preceding discussion, we can see that when the valve is closed after the placement of the load P, Ps 0 and Pw P Now, if the valve is opened, the water will ﬂow outward (Figure 11.5c). This ﬂow will be accompanied by a reduction of the excess hydrostatic pressure and an increase in the compression of the spring. So, at this time, Eq. (11.12) will hold. However, Ps 0 and Pw P 1that is, ¢u P/A2 After some time, the excess hydrostatic pressure will become zero and the system will reach a state of equilibrium, as shown in Figure 11.5d. Now we can write Ps P and Pw 0 and P Ps Pw With this in mind, we can analyze the strain of a saturated clay layer subjected to a stress increase (Figure 11.6a). Consider the case where a layer of saturated clay of thickness H that is conﬁned between two layers of sand is being subjected to an instantaneous increase of total stress of s. This incremental total stress will be transmitted to the pore water and the soil solids. This means that the total stress, s, will be divided in some proportion between effective stress and pore water pressure. The behavior of the effective stress change will be similar to that of the spring in Figure 11.5, and the behavior of the pore water pressure change will be similar to that of the excess hydrostatic pressure in Figure 11.5. From the principle of effective stress (Chapter 9), it follows that ¢s ¢s¿ ¢u (11.13) where ¢s¿ increase in the effective stress ¢u increase in the pore water pressure Because clay has a very low hydraulic conductivity and water is incompressible as compared with the soil skeleton, at time t 0, the entire incremental stress, s, will be carried by water (s u) at all depths (Figure 11.6b). None will be carried by the soil skeleton—that is, incremental effective stress (s) 0. After the application of incremental stress, s, to the clay layer, the water in the void spaces will start to be squeezed out and will drain in both directions into the sand layers. By this process, the excess pore water pressure at any depth in the clay layer s Groundwater table H Depth Sand Clay (a) Total stress increase Effective stress increase Pore water pressure increase H s Depth s 0 u s Depth Depth (b) At time t 0 Total stress increase Effective stress increase Pore water pressure increase u s H s Depth s 0 s Depth s Depth (c) At time 0 t Total stress increase Effective stress increase Pore water pressure increase H u 0 s Depth Depth s s Depth (d) At time t Figure 11.6 Variation of total stress, pore water pressure, and effective stress in a clay layer drained at top and bottom as the result of an added stress, s 307 308 Chapter 11: Compressibility of Soil gradually will decrease, and the stress carried by the soil solids (effective stress) will increase. Thus, at time 0 t q, ¢s ¢s¿ ¢u 1¢s¿ 0 and ¢u ¢s2 However, the magnitudes of s and u at various depths will change (Figure 11.6c), depending on the minimum distance of the drainage path to either the top or bottom sand layer. Theoretically, at time t q, the entire excess pore water pressure would be dissipated by drainage from all points of the clay layer; thus, u 0. Now the total stress increase, s, will be carried by the soil structure (Figure 11.6d). Hence, ¢s ¢s¿ This gradual process of drainage under an additional load application and the associated transfer of excess pore water pressure to effective stress cause the time-dependent settlement in the clay soil layer. 11.4 One-Dimensional Laboratory Consolidation Test The one-dimensional consolidation testing procedure was ﬁrst suggested by Terzaghi. This test is performed in a consolidometer (sometimes referred to as an oedometer). The schematic diagram of a consolidometer is shown in Figure 11.7a. Figure 11.7b. shows a photograph of a consolidometer. The soil specimen is placed inside a metal ring with two porous stones, one at the top of the specimen and another at the bottom. The specimens are usually 64 mm ( 2.5 in.) in diameter and 25 mm. ( 1 in.) thick. The load on the specimen is applied through a lever arm, and compression is measured by a micrometer dial gauge. The specimen is kept under water during the test. Each load usually is kept for 24 hours. After that, the load usually is doubled, which doubles the pressure on the specimen, and the compression measurement is continued. At the end of the test, the dry weight of the test specimen is determined. Figure 11.7c shows a consolidation test in progress (righthand side). The general shape of the plot of deformation of the specimen against time for a given load increment is shown in Figure 11.8. From the plot, we can observe three distinct stages, which may be described as follows: Stage I: Stage II: Initial compression, which is caused mostly by preloading. Primary consolidation, during which excess pore water pressure gradually is transferred into effective stress because of the expulsion of pore water. Stage III: Secondary consolidation, which occurs after complete dissipation of the excess pore water pressure, when some deformation of the specimen takes place because of the plastic readjustment of soil fabric. Load Dial gauge Porous stone Soil specimen (a) (b) Specimen ring Figure 11.7 (a) Schematic diagram of a consolidometer; (b) photograph of a consolidometer; (c) a consolidation test in progress (right-hand side) (Courtesy of Braja M. Das, Henderson, Nevada) (c) 309 310 Chapter 11: Compressibility of Soil Deformation Stage I Stage II Stage III Time (log scale) Figure 11.8 Time–deformation plot during consolidation for a given load increment Stage I: Initial compression Stage II: Primary consolidation Stage III: Secondary consolidation 11.5 Void Ratio–Pressure Plots After the time–deformation plots for various loadings are obtained in the laboratory, it is necessary to study the change in the void ratio of the specimen with pressure. Following is a step-by-step procedure for doing so: Step 1: Calculate the height of solids, Hs, in the soil specimen (Figure 11.9) using the equation Hs Ws Ms AGsgw AGs rw (11.14) where Ws dry weight of the specimen Ms dry mass of the specimen A area of the specimen Gs specific gravity of soil solids gw unit weight of water rw density of water Step 2: Calculate the initial height of voids as Hv H Hs where H initial height of the specimen. (11.15) 11.5 Void Ratio–Pressure Plots 311 H2 H1 Initial height of specimen H H H Hs Hs Specimen area A Void Ws AGsg Solid Figure 11.9 Change of height of specimen in one-dimensional consolidation test Step 3: Calculate the initial void ratio, eO, of the specimen, using the equation eO Vv Hv A Hv Vs Hs A Hs (11.16) Step 4: For the ﬁrst incremental loading, s1 (total load/unit area of specimen), which causes a deformation H1 , calculate the change in the void ratio as ¢e1 ¢H1 Hs (11.17) (H1 is obtained from the initial and the ﬁnal dial readings for the loading). It is important to note that, at the end of consolidation, total stress s1 is equal to effective stress s1. Step 5: Calculate the new void ratio after consolidation caused by the pressure increment as e1 eO ¢e1 (11.18) For the next loading, s2 (note: s2 equals the cumulative load per unit area of specimen), which causes additional deformation H2 , the void ratio at the end of consolidation can be calculated as e2 e1 ¢H2 Hs (11.19) At this time, s2 effective stress, s2. Proceeding in a similar manner, one can obtain the void ratios at the end of the consolidation for all load increments. The effective stress s and the corresponding void ratios (e) at the end of consolidation are plotted on semilogarithmic graph paper. The typical shape of such a plot is shown in Figure 11.10. 312 Chapter 11: Compressibility of Soil e0 e1 Void ratio, e e2 s1 s2 Effective pressure, s (log scale) Figure 11.10 Typical plot of e against log s Example 11.2 Following are the results of a laboratory consolidation test on a soil specimen obtained from the field: Dry mass of specimen 128 g, height of specimen at the beginning of the test 2.54 cm, Gs 2.75, and area of the specimen 30.68 cm2. Effective pressure, ' (ton /ft2) Final height of specimen at the end of consolidation (cm) 0 0.5 1 2 4 8 16 32 2.540 2.488 2.465 2.431 2.389 2.324 2.225 2.115 Make necessary calculations and draw an e versus log ' curve. Solution From Eq. (11.14), Hs 128 g Ws Ms 1.52 cm 2 AGsgw AGs rw 130.68 cm 2 12.75 2 11 g/cm3 2 11.6 Normally Consolidated and Overconsolidated Clays 313 Now the following table can be prepared. Effective pressure, ' (ton /ft2) Height at the end of consolidation, H (cm) Hv ⴝ H ⴚ Hs (cm) e ⴝ Hv/Hs 0 0.5 1 2 4 8 16 32 2.540 2.488 2.465 2.431 2.389 2.324 2.225 2.115 1.02 0.968 0.945 0.911 0.869 0.804 0.705 0.595 0.671 0.637 0.622 0.599 0.572 0.529 0.464 0.390 The e versus log ´ plot is shown in Figure 11.11 0.8 0.7 Void ratio, e 0.6 0.5 0.4 0.3 0.2 0.1 11.6 0.3 1 3 10 30 Effective pressure, s (ton/ft2)–log scale 100 Figure 11.11 Variation of void ratio with effective pressure ■ Normally Consolidated and Overconsolidated Clays Figure 11.10 shows that the upper part of the e–log s plot is somewhat curved with a ﬂat slope, followed by a linear relationship for the void ratio with log s having a steeper slope. This phenomenon can be explained in the following manner: A soil in the ﬁeld at some depth has been subjected to a certain maximum effective past pressure in its geologic history. This maximum effective past pressure may be equal to or less than the existing effective overburden pressure at the time of sampling. The reduction of effective pressure in the ﬁeld may be caused by natural geologic processes or human processes. During the soil sampling, the existing effective overburden pressure is also released, which results in some expansion. When this specimen is subjected to a 314 Chapter 11: Compressibility of Soil consolidation test, a small amount of compression (that is, a small change in void ratio) will occur when the effective pressure applied is less than the maximum effective overburden pressure in the ﬁeld to which the soil has been subjected in the past. When the effective pressure on the specimen becomes greater than the maximum effective past pressure, the change in the void ratio is much larger, and the e–log s relationship is practically linear with a steeper slope. This relationship can be veriﬁed in the laboratory by loading the specimen to exceed the maximum effective overburden pressure, and then unloading and reloading again. The e–log s plot for such cases is shown in Figure 11.12, in which cd represents unloading and dfg represents the reloading process. This leads us to the two basic deﬁnitions of clay based on stress history: 1. Normally consolidated, whose present effective overburden pressure is the maximum pressure that the soil was subjected to in the past. 2. Overconsolidated, whose present effective overburden pressure is less than that which the soil experienced in the past. The maximum effective past pressure is called the preconsolidation pressure. Casagrande (1936) suggested a simple graphic construction to determine the preconsolidation pressure sc from the laboratory e–log s plot. The procedure is as follows (see Figure 11.13): 1. By visual observation, establish point a, at which the e–log s plot has a minimum radius of curvature. 2. Draw a horizontal line ab. 3. Draw the line ac tangent at a. a b d Void ratio, e Void ratio, e a c f g Figure 11.12 Plot of e against log s showing loading, unloading, and reloading branches a a d c h g Effective pressure, s (log scale) b f sc Pressure, s (log scale) Figure 11.13 Graphic procedure for determining preconsolidation pressure 11.6 Normally Consolidated and Overconsolidated Clays 315 4. Draw the line ad, which is the bisector of the angle bac. 5. Project the straight-line portion gh of the e–log s plot back to intersect line ad at f. The abscissa of point f is the preconsolidation pressure, sc . The overconsolidation ratio (OCR) for a soil can now be deﬁned as s¿c s¿ OCR (11.20) where scœ preconsolidation pressure of a specimen sœ present effective vertical pressure In the literature, some empirical relationships are available to predict the preconsolidation pressure. Some examples are given next. • Nagaraj and Murty (1985): eo b 0.0463so¿ eL 0.188 1.112 a log sc¿ where eo in situ void ratio eL void ratio at liquid limit c LL1 % 2 (11.21) d Gs (11.22) sc¿ 10 31.111.621LI24 pa (11.23) 100 Gs specific gravity of soil solids ´o in situ effective overburden pressure (Note: 'c and 'o are in kN/m2) • Stas and Kulhawy (1984): where pa atmospheric pressure (100 kN/m2) LI liquidity index • Hansbo (1957) ´c (VST) Cu(VST) where (VST) an empirical coefficient (11.24) 222 LL1% 2 Cu(VST) undrained shear strength obtained from vane shear test (Chapter 12) In any case, these above relationships may change from soil to soil. They may be taken as an initial approximation. 316 Chapter 11: Compressibility of Soil Effect of Disturbance on Void Ratio–Pressure Relationship A soil specimen will be remolded when it is subjected to some degree of disturbance. This remolding will result in some deviation of the e –log s plot as observed in the laboratory from the actual behavior in the ﬁeld. The ﬁeld e –log s plot can be reconstructed from the laboratory test results in the manner described in this section (Terzaghi and Peck, 1967). Normally Consolidated Clay of Low to Medium Plasticity (Figure 11.14) 1. In Figure 11.14, curve 2 is the laboratory e–log s plot. From this plot, determine the preconsolidation pressure (sc) sO (that is, the present effective overburden pressure). Knowing where sc sO , draw vertical line ab. 2. Calculate the void ratio in the ﬁeld, eO [Section 11.5, Eq. (11.16)]. Draw horizontal line cd. 3. Calculate 0.4eO and draw line ef. (Note: f is the point of intersection of the line with curve 2.) 4. Join points f and g. Note that g is the point of intersection of lines ab and cd. This is the virgin compression curve. It is important to point out that if a soil is remolded completely, the general position of the e–log s plot will be as represented by curve 3. a g eO c 1 3 0.4eO Laboratory consolidation curve Consolidation curve for remolded specimen e d Virgin consolidation curve; slope = Cc 2 Void ratio, e 11.7 b f s O sc Pressure, s (log scale) Figure 11.14 Consolidation characteristics of normally consolidated clay of low to medium sensitivity 11.8 Calculation of Settlement from One-Dimensional Primary Consolidation c a f g eO h i j 1 2 Virgin consolidation curve; slope Cc compression index Void ratio, e 0.4eO 317 Laboratory consolidation curve 3 Laboratory rebound curve; slope Cs swell index e k d s O b sc Pressure, s (log scale) Figure 11.15 Consolidation characteristics of overconsolidated clay of low to medium sensitivity Overconsolidated Clay of Low to Medium Plasticity (Figure 11.15) 1. In Figure 11.15, curve 2 is the laboratory e–log s plot (loading), and curve 3 is the laboratory unloading, or rebound, curve. From curve 2, determine the preconsolidation pressure sc . Draw the vertical line ab. 2. Determine the ﬁeld effective overburden pressure sO . Draw vertical line cd. 3. Determine the void ratio in the ﬁeld, eO. Draw the horizontal line fg. The point of intersection of lines fg and cd is h. 4. Draw a line hi, which is parallel to curve 3 (which is practically a straight line). The point of intersection of lines hi and ab is j. 5. Join points j and k. Point k is on curve 2, and its ordinate is 0.4eO. The ﬁeld consolidation plot will take a path hjk. The recompression path in the ﬁeld is hj and is parallel to the laboratory rebound curve (Schmertmann, 1953). 11.8 Calculation of Settlement from One-Dimensional Primary Consolidation With the knowledge gained from the analysis of consolidation test results, we can now proceed to calculate the probable settlement caused by primary consolidation in the ﬁeld, assuming one-dimensional consolidation. Let us consider a saturated clay layer of thickness H and cross-sectional area A under an existing average effective overburden pressure, sO. Because of an increase of effective pressure, s, let the primary settlement be Sc. Thus, the change in volume (Figure 11.16) can be given by ¢V V0 V1 HA 1H Sc 2A ScA (11.25) 318 Chapter 11: Compressibility of Soil Cross-sectional area A Cross-sectional area A Volume Volume V V Height Height Sc V Sc V0 V1 0 V1 H H Soil Void Vs Solid Figure 11.16 Settlement caused by one-dimensional consolidation where V0 and V1 are the initial and ﬁnal volumes, respectively. However, the change in the total volume is equal to the change in the volume of voids, Vv. Hence, ¢V ScA Vv0 Vv1 ¢Vv (11.26) where Vv and Vv are the initial and ﬁnal void volumes, respectively. From the deﬁnition of void ratio, it follows that 0 1 ¢Vv ¢eVs (11.27) V0 AH 1 eo 1 eo (11.28) where e change of void ratio. But Vs where eO initial void ratio at volume V0 . Thus, from Eqs. (11.25) through (11.28), ¢V ScA ¢eVs AH ¢e 1 eo or Sc H ¢e 1 eo (11.29) For normally consolidated clays that exhibit a linear e–log s relationship (see Figure 11.14), ¢e Cc 3 log1s¿o ¢s¿ 2 log s¿o 4 (11.30) where Cc slope of the e–log s plot and is deﬁned as the compression index. Substitution of Eq. (11.30) into Eq. (11.29) gives Sc CcH s¿o ¢s¿ log a b 1 eo s¿o (11.31) 11.9 Compression Index (Cc) 319 In overconsolidated clays (see Figure 11.15), for sO + s sc , ﬁeld e–log s variation will be along the line hj, the slope of which will be approximately equal to that for the laboratory rebound curve. The slope of the rebound curve Cs is referred to as the swell index; so ¢e Cs 3log1s¿o ¢s¿ 2 log s¿o 4 (11.32) From Eqs. (11.29) and (11.32), we obtain CsH s¿o ¢s¿ log a b 1 eo s¿o (11.33) CsH s¿c CcH s¿o ¢s¿ log log a b 1 eo s¿o 1 eo s¿c (11.34) Sc If sO s sc , then Sc However, if the e–log s curve is given, one can simply pick e off the plot for the appropriate range of pressures. This number may be substituted into Eq. (11.29) for the calculation of settlement, Sc. 11.9 Compression Index (Cc ) The compression index for the calculation of ﬁeld settlement caused by consolidation can be determined by graphic construction (as shown in Figure 11.14) after one obtains the laboratory test results for void ratio and pressure. Skempton (1944) suggested the following empirical expression for the compression index for undisturbed clays: Cc 0.0091LL 10 2 (11.35) where LL liquid limit. Several other correlations for the compression index are also available. They have been developed by tests on various clays. Some of these correlations are given in Table 11.6. On the basis of observations on several natural clays, Rendon-Herrero (1983) gave the relationship for the compression index in the form Cc 0.141G1.2 s a 1 eo 2.38 b Gs (11.36) 320 Chapter 11: Compressibility of Soil Table 11.6 Correlations for Compression Index, Cc * Equation Reference Region of applicability Cc 0.007(LL 7) Cc 0.01wN Cc 1.15(eO 0.27) Cc 0.30(eO 0.27) Cc 0.0115wN Cc 0.0046(LL 9) Cc 0.75(eO 0.5) Cc 0.208eO 0.0083 Cc 0.156eO 0.0107 Skempton (1944) Remolded clays Chicago clays All clays Inorganic cohesive soil: silt, silty clay, clay Organic soils, peats, organic silt, and clay Brazilian clays Soils with low plasticity Chicago clays All clays Nishida (1956) Hough (1957) *After Rendon-Herrero, 1980. With permission from ASCE. Note: eO in situ void ratio; wN in situ water content. Nagaraj and Murty (1985) expressed the compression index as Cc 0.2343 c LL 1% 2 100 d Gs (11.37) Based on the modified Cam clay model, Wroth and Wood (1978) have shown that Cc 0.5GS 3PI1% 2 4 100 (11.38) where PI plasticity index. If an average value of Gs is taken to be about 2.7 (Kulhawy and Mayne, 1990) Cc PI 74 (11.39) More recently, Park and Koumoto (2004) expressed the compression index by the following relationship. Cc no 371.747 4.275no (11.40) where no in situ porosity of the soil 11.10 Swell Index (Cs ) The swell index is appreciably smaller in magnitude than the compression index and generally can be determined from laboratory tests. In most cases, Cs 15 to 101 Cc 11.10 Swell Index (Cs) 321 The swell index was expressed by Nagaraj and Murty (1985) as Cs 0.0463 c LL1% 2 100 d Gs (11.41) Based on the modified Cam clay model, Kulhawy and Mayne (1990) have shown that Cc PI 370 (11.42) Typical values of the liquid limit, plastic limit, virgin compression index, and swell index for some natural soils are given in Table 11.7. Table 11.7 Compression and Swell of Natural Soils Soil Boston blue clay Chicago clay Ft. Gordon clay, Georgia New Orleans clay Montana clay Liquid limit Plastic limit Compression index, Cc Swell index, Cs 41 60 51 80 60 20 20 26 25 28 0.35 0.4 0.12 0.3 0.21 0.07 0.07 — 0.05 0.05 Example 11.3 The following are the results of a laboratory consolidation test: Pressure, Sⴕ (ton /ft 2) Void ratio, e 0.25 0.5 1.0 2.0 4.0 1.03 1.02 0.98 0.91 0.79 Remarks Loading Pressure, Sⴕ (ton /ft 2) Void ratio, e 8.0 16.0 8 4 2 0.71 0.62 0.635 0.655 0.67 Remarks Loading Unloading a. Draw an e-log sOœ graph and determine the preconsolidation pressure, scœ b. Calculate the compression index and the ratio of Cs /Cc c. On the basis of the average e-log s plot, calculate the void ratio at sOœ 12 ton/ft2 322 Chapter 11: Compressibility of Soil 1.1 sc 1.2 ton /ft 2 1.0 b a 0.9 Void ratio, e d c 0.8 0.7 0.6 0.5 0.1 0.2 0.5 1.0 2 4 5 Pressure s (ton /ft 2) 10 20 Figure 11.17 Plot of e versus log s Solution Part a The e versus log s plot is shown in Figure 11.17. Casagrande’s graphic procedure is used to determine the preconsolidation pressure: sOœ 1.2 ton/ft2 Part b From the average e-log s plot, for the loading and unloading branches, the following values can be determined: Branch Loading Unloading e SⴕO (ton /ft 2) 0.9 0.8 0.67 0.655 2 4 2 4 From the loading branch, Cc e1 e 2 0.9 0.8 0.33 s2œ 4 log œ log a b s1 2 11.10 Swell Index (Cs) 323 From the unloading branch, Cs e1 e 2 0.67 0.655 0.05 œ s2 4 log œ log a b s1 2 Cs 0.05 0.15 Cc 0.33 Part c CC e1 e3 s3œ log œ s1 We know that e1 0.9 at s1œ 2 ton/ft2 and that Cc 0.33 [part (b)]. Let s3œ 12 ton/ft2. So, 0.33 0.9 e3 12 log a b 2 e3 0.9 0.33 log a 12 b 0.64 2 ■ Example 11.4 A soil proﬁle is shown in Figure 11.18. If a uniformly distributed load, s, is applied at the ground surface, what is the settlement of the clay layer caused by primary consolidation if a. The clay is normally consolidated b. The preconsolidation pressure (scœ ) 190 kN/m2 c. scœ 170 kN/m2 Use Cs 16 Cc. Solution Part a The average effective stress at the middle of the clay layer is soœ 2gdry 43gsat1sand2 gw 4 42 3 gsat1clay2 gw 4 soœ 122 1142 4118 9.81 2 2119 9.812 79.14 kN/m2 From Eq. (11.31), Sc CcH soœ ¢s¿ log a b 1 eo soœ 324 Chapter 11: Compressibility of Soil s 100 kN/m2 gdry 14 kN/m3 2m Groundwater table gsat 18 kN/m3 4m gsat 19 kN/m3 Void ratio, e 0.8 4m LL 40 Clay Sand Figure 11.18 From Eq. (11.35), Cc 0.0091LL 102 0.009140 10 2 0.27 So, Sc 10.272 142 1 0.8 log a 79.14 100 b 0.213 m 213 mm 79.14 Part b sOœ ¢s¿ 79.14 100 179.14 kN/m2 scœ 190 kN/m2 Because sOœ ¢sœ scœ , use Eq. (11.33): Sc CsH sOœ ¢s¿ log a b 1 eO sOœ Cs Cc 0.27 0.045 6 6 Sc 10.0452 142 1 0.8 log a 79.14 100 b 0.036 m 36 mm 79.14 Part c soœ 79.14 kN/m2 soœ ¢s¿ 179.14 kN/m2 scœ 170 kN/m2 11.10 Swell Index (Cs) 325 Because soœ scœ soœ ¢sœ, use Eq. (11.34) Sc CsH scœ CcH soœ ¢s¿ log œ log a b 1 eo so 1 eo scœ 10.0452 142 1.8 log a 10.27 2 142 170 179.14 b log a b 79.14 1.8 170 0.0468 m 46.8 mm ■ Example 11.5 A soil proﬁle is shown in Figure 11.19a. Laboratory consolidation tests were conducted on a specimen collected from the middle of the clay layer. The ﬁeld consolidation curve interpolated from the laboratory test results is shown in Figure 11.19b. Calculate the settlement in the ﬁeld caused by primary consolidation for a surcharge of 48 kN/m2 applied at the ground surface. Solution sOœ 15 2 1gsat gw 2 5118.0 9.81 2 40.95 kN/m2 eO 1.1 ¢s¿ 48 kN/m2 sOœ ¢s¿ 40.95 48 88.95 kN/m2 48 kN/m2 Groundwater table eO 1.1 gsat 18 kN/m3 10 m Rock Clay (a) Figure 11.19 (a) Soil proﬁle 326 Chapter 11: Compressibility of Soil 1.12 1.10 Void ratio, e 1.08 1.076 e 1.06 1.045 1.04 88.95 1.02 1.00 40.95 70 Pressure sO (log scale) (kN/m2) 100 Figure 11.19 (b) ﬁeld consolidation curve (b) The void ratio corresponding to 88.95 kN/m2 (see Figure 11.19b) is 1.045. Hence, e 1.1 1.045 0.055. We have Settlement 1Sc 2 H so, Sc 10 11.11 10.055 2 1 1.1 ¢e 1 eo 0.262 m 262 mm [Eq. (11.29)] ■ Secondary Consolidation Settlement Section 11.4 showed that at the end of primary consolidation (that is, after complete dissipation of excess pore water pressure) some settlement is observed because of the plastic adjustment of soil fabrics. This stage of consolidation is called secondary consolidation. During secondary consolidation the plot of deformation against the log of time is practically linear (see Figure 11.8). The variation of the void ratio, e, with time t for a given load increment will be similar to that shown in Figure 11.8. This variation is shown in Figure 11.20. From Figure 11.20, the secondary compression index can be deﬁned as Ca ¢e ¢e log t2 log t1 log1t2/t1 2 (11.43) Void ratio, e 11.11 Secondary Consolidation Settlement ep Ca e t log t2 1 327 e t2 t1 Time, t (log scale) Figure 11.20 Variation of e with log t under a given load increment and deﬁnition of secondary consolidation index where Ca secondary compression index ¢e change of void ratio t1 , t2 time The magnitude of the secondary consolidation can be calculated as Ss C aœ H log a t2 b t1 (11.44) and C aœ Ca 1 ep (11.45) where ep void ratio at the end of primary consolidation 1see Figure 10.202 H thickness of clay layer The general magnitudes of Ca as observed in various natural deposits (also see Figure 11.21) are as follows: • • • Overconsolidated clays 0.001 or less Normally consolidated clays 0.005 to 0.03 Organic soil 0.04 or more 328 Chapter 11: Compressibility of Soil 100 6 10 2 1 Ca (%) 7 3 8 5 4 1.0 11 10 9 0.1 10 100 1000 3000 Natural water content (%) 1—Whangamarino clay 2 — Mexico City clay 3—Calcareous organic silt 4 — Leda clay 5 — Norwegian plastic clay 6 — Amorphous and ﬁbrous peat 7 — Canadian muskeg 8 — Organic marine deposits 9 — Boston blue clay 10 — Chicago blue clay 11 — Organic silty clay — Organic silt, etc. Figure 11.21 Ca for natural soil deposits (After Mesri, 1973. With permission from ASCE.) Mersri and Godlewski (1977) compiled the ratio of Ca/Cc for a number of natural clays. From this study, it appears that Ca/Cc for • • • Inorganic clays and silts 0.04 0.01 Organic clays and silts 0.05 0.01 Peats 0.075 0.01 Secondary consolidation settlement is more important than primary consolidation in organic and highly compressible inorganic soils. In overconsolidated inorganic clays, the secondary compression index is very small and of less practical signiﬁcance. 11.11 Secondary Consolidation Settlement 329 Example 11.6 For a normally consolidated clay layer in the ﬁeld, the following values are given: • • • • • • Thickness of clay layer 8.5 ft Void ratio (eO) 0.8 Compression index (Cc) 0.28 Average effective pressure on the clay layer 1soœ 2 2650 lb/ft2 s 970 lb/ft2 Secondary compression index (Ca) 0.02 What is the total consolidation settlement of the clay layer ﬁve years after the completion of primary consolidation settlement? (Note: Time for completion of primary settlement 1.5 years.) Solution From Eq. (11.45), Caœ Ca 1 ep The value of ep can be calculated as ep eo ¢eprimary Combining Eqs. (11.29) and (11.30), we ﬁnd that ¢e Cc log a soœ ¢s¿ 2650 970 b b 0.28 log a œ so 2650 0.038 Primary consolidation, Sc 10.038 2 18.5 12 2 ¢eH 2.15 in. 1 eo 1 0.8 It is given that eO 0.8, and thus, ep 0.8 0.038 0.762 Hence, Caœ 0.02 0.011 1 0.762 From Eq. (11.44), Ss Caœ H log a t2 5 b 0.59 in. b 10.011 2 18.5 12 2 log a t1 1.5 Total consolidation settlement primary consolidation (Sc) secondary settlement (Ss). So Total consolidation settlement 2.15 0.59 2.74 in. ■ 330 Chapter 11: Compressibility of Soil 11.12 Time Rate of Consolidation The total settlement caused by primary consolidation resulting from an increase in the stress on a soil layer can be calculated by the use of one of the three equations—(11.31), (11.33), or (11.34)—given in Section 11.8. However, they do not provide any information regarding the rate of primary consolidation. Terzaghi (1925) proposed the ﬁrst theory to consider the rate of one-dimensional consolidation for saturated clay soils. The mathematical derivations are based on the following six assumptions (also see Taylor, 1948): 1. 2. 3. 4. 5. 6. The clay–water system is homogeneous. Saturation is complete. Compressibility of water is negligible. Compressibility of soil grains is negligible (but soil grains rearrange). The ﬂow of water is in one direction only (that is, in the direction of compression). Darcy’s law is valid. Figure 11.22a shows a layer of clay of thickness 2Hdr (Note: Hdr length of maximum drainage path) that is located between two highly permeable sand layers. If the clay layer is subjected to an increased pressure of s, the pore water pressure at any point A in the clay layer will increase. For one-dimensional consolidation, water will be squeezed out in the vertical direction toward the sand layer. Figure 11.22b shows the ﬂow of water through a prismatic element at A. For the soil element shown, Rate of outflow Rate of inflow Rate of of water of water volume change Thus, a vz 0vz 0z dz b dx dy vz dx dy 0V 0t where V volume of the soil element vz velocity of flow in z direction or 0vz 0z dx dy dz 0V 0t (11.46) Using Darcy’s law, we have vz ki k 0h k 0u gw 0z 0z (11.47) where u excess pore water pressure caused by the increase of stress. From Eqs. (11.46) and (11.47), k 0 2u 1 0V gw 0z2 dx dy dz 0t (11.48) 11.12 Time Rate of Consolidation 331 z s h gu Groundwater table A 2Hdr z Sand Clay (a) ( z z z dz) dx dy dz dy z dx dy dx (b) Figure 11.22 (a) Clay layer undergoing consolidation; (b) ﬂow of water at A during consolidation During consolidation, the rate of change in the volume of the soil element is equal to the rate of change in the volume of voids. Thus, 01Vs eVs 2 0Vv 0Vs 0Vs 0e 0V Vs e 0t 0t 0t 0t 0t 0t where Vs volume of soil solids Vv volume of voids (11.49) 332 Chapter 11: Compressibility of Soil But (assuming that soil solids are incompressible) 0Vs 0 0t and Vs dx dy dz V 1 eo 1 eo Substitution for 0Vs /0t and Vs in Eq. (11.49) yields dx dy dz 0e 0V 0t 1 eo 0t (11.50) where eO initial void ratio. Combining Eqs. (11.48) and (11.50) gives 0e 1 k 0 2u 2 gw 0z 1 eo 0t (11.51) The change in the void ratio is caused by the increase of effective stress (i.e., a decrease of excess pore water pressure). Assuming that they are related linearly, we have 0e av 01¢s¿ 2 av 0u (11.52) where 01¢s¿ 2 change in effective pressure av coefficient of compressibility 1av can be considered constant for a narrow range of pressure increase2 Combining Eqs. (11.51) and (11.52) gives av 0u 0u k 0 2u mv 2 gw 0z 1 eo 0t 0t where mv coefficient of volume compressibility av /11 eo 2 (11.53) 0 2u 0u cv 2 0t 0z (11.54) or, where cv coefficient of consolidation k/1gwmv 2 (11.55) Thus, cv k gwmv k av gw a b 1 eo (11.56) 11.12 Time Rate of Consolidation 333 Eq. (11.54) is the basic differential equation of Terzaghi’s consolidation theory and can be solved with the following boundary conditions: z 0, u 0 z 2Hdr, t 0, u0 u uO The solution yields mq 2uo Mz 2 sin a u a c b d eM Tv M Hdr m0 (11.57) where m an integer M 1p/2 2 12m 12 uO initial excess pore water pressure Tv cvt H2dr time factor (11.58) The time factor is a nondimensional number. Because consolidation progresses by the dissipation of excess pore water pressure, the degree of consolidation at a distance z at any time t is Uz uo uz uo uz 1 (11.59) uo where uz excess pore water pressure at time t. Equations (11.57) and (11.59) can be combined to obtain the degree of consolidation at any depth z. This is shown in Figure 11.23. The average degree of consolidation for the entire depth of the clay layer at any time t can be written from Eq. (11.59) as U Sc1t2 Sc a 1 1 b 2Hdr 2Hdr uz dz 0 uo (11.60) where U average degree of consolidation Sc1t2 settlement of the layer at time t Sc ultimate settlement of the layer from primary consolidation Substitution of the expression for excess pore water pressure uz given in Eq. (11.57) into Eq. (11.60) gives mq 2 M2Tv U1 a e 2 m0 M (11.61) 334 Chapter 11: Compressibility of Soil 2.0 0.1 T 0 1.5 0.2 0.7 0.3 0.4 0.8 0.5 z 1.0 Hdr 0.6 0.9 0.5 Figure 11.23 Variation of Uz with Tv and z/Hdr 0.0 0.0 0.1 0.2 0.3 0.4 0.7 0.5 0.6 Degree of consolidation, Uz 0.8 0.9 1.0 The variation in the average degree of consolidation with the nondimensional time factor, Tv , is given in Figure 11.24, which represents the case where uO is the same for the entire depth of the consolidating layer. The values of the time factor and their corresponding average degrees of consolidation for the case presented in Figure 11.24 may also be approximated by the following simple relationship: For U 0 to 60%, Tv p U% 2 a b 4 100 For U 60%, Tv 1.781 0.933 log1100 U% 2 (11.62) (11.63) Table 11.8 gives the variation of Tv with U on the basis of Eqs. (11.62) and (11.63). Sivaram and Swamee (1977) gave the following equation for U varying from 0 to 100%: 14Tv /p2 U% 100 31 14Tv /p2 2.8 4 0.179 0.5 or Tv 1p/4 2 1U%/1002 2 31 1U%/1002 5.6 4 0.357 (11.64) (11.65) Equations (11.64) and (11.65) give an error in Tv of less than 1% for 0% U 90% and less than 3% for 90% U 100%. 2Hdr uO Hdr uO Average degree of consolidation, U(%) Two-way drainage One-way drainage One-way drainage uO 0 20 40 60 80 100 Hdr 0.0 0.2 0.4 Time factor, T 0.6 0.8 0.9 Different types of drainage with uo constant Figure 11.24 Variation of average degree of consolidation with time factor, Tv (uO constant with depth) Table 11.8 Variation of Tv with U U (%) Tv U (%) Tv U (%) Tv U (%) Tv 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 0 0.00008 0.0003 0.00071 0.00126 0.00196 0.00283 0.00385 0.00502 0.00636 0.00785 0.0095 0.0113 0.0133 0.0154 0.0177 0.0201 0.0227 0.0254 0.0283 0.0314 0.0346 0.0380 0.0415 0.0452 0.0491 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 0.0531 0.0572 0.0615 0.0660 0.0707 0.0754 0.0803 0.0855 0.0907 0.0962 0.102 0.107 0.113 0.119 0.126 0.132 0.138 0.145 0.152 0.159 0.166 0.173 0.181 0.188 0.197 0.204 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 0.212 0.221 0.230 0.239 0.248 0.257 0.267 0.276 0.286 0.297 0.307 0.318 0.329 0.304 0.352 0.364 0.377 0.390 0.403 0.417 0.431 0.446 0.461 0.477 0.493 0.511 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 0.529 0.547 0.567 0.588 0.610 0.633 0.658 0.684 0.712 0.742 0.774 0.809 0.848 0.891 0.938 0.993 1.055 1.129 1.219 1.336 1.500 1.781 q 335 336 Chapter 11: Compressibility of Soil Example 11.7 The time required for 50% consolidation of a 25-mm-thick clay layer (drained at both top and bottom) in the laboratory is 2 min. 20 sec. How long (in days) will it take for a 3-m-thick clay layer of the same clay in the field under the same pressure increment to reach 50% consolidation? In the field, there is a rock layer at the bottom of the clay. Solution T50 cvtlab 2 Hdr1lab2 cvtfield 2 Hdr1field2 or tlab 2 Hdr1lab2 tfield 2 Hdr1field2 tfield 140sec 2 13 m2 2 0.025m b a 2 tfield 8,064,000 sec 93.33 days ■ Example 11.8 Refer to Example 11.7. How long (in days) will it take in the field for 30% primary consolidation to occur? Use Eq. (11.62). Solution From Eq. (11.62) cvtfield Hdr2 1field2 Tv r U2 So t r U2 t1 U21 2 t2 U2 or 93.33 days 502 t2 302 t2 33.6 days ■ 11.12 Time Rate of Consolidation 337 Example 11.9 A 3-m-thick layer (double drainage) of saturated clay under a surcharge loading underwent 90% primary consolidation in 75 days. Find the coefﬁcient of consolidation of clay for the pressure range. Solution T90 cvt90 H2dr Because the clay layer has two-way drainage, Hdr 3 m/2 1.5 m. Also, T90 0.848 (see Table 11.8). So, 0.848 cv cv 175 24 60 60 2 11.5 100 2 2 0.848 2.25 104 0.00294 cm2/sec 75 24 60 60 ■ Example 11.10 For a normally consolidated laboratory clay specimen drained on both sides, the following are given: • • • • 'O 3000 lb/ft2, e eO 1.1 'O + ' 6000 lb/ft2, e 0.9 Thickness of clay specimen 1 in. Time for 50% consolidation 2 min a. Determine the hydraulic conductivity (ft/min) of the clay for the loading range. b. How long (in days) will it take for a 6-ft clay layer in the field (drained on one side) to reach 60% consolidation? Solution Part a The coefficient of compressibility is av mv 1 eav ¢e b ¢sœ 1 eav a ¢e 1.1 0.9 0.2 ¢sœ 6000 3000 3000 lb/ft2 1.1 0.9 eav 1.0 2 338 Chapter 11: Compressibility of Soil So 0.2 3000 3.33 10 5ft2/lb mv 1 1.0 From Table 11.8, for U 50%, Tv 0.197; thus, 10.1972 a cv 2 1 b 2 12 1.71 10 4ft2/min 2 k cvmvgw 11.71 10 4ft2/min 2 13.33 10 5ft2/lb 2 162.4 lb/ft3 2 3.55 10 7 ft/min Part b T60 t60 cvt60 H2dr T60H2dr cv From Table 11.8, for U 60%, Tv 0.286, t60 11.13 10.2862 16 2 2 1.71 10 4 60,211 min 41.8 days ■ Coefﬁcient of Consolidation The coefﬁcient of consolidation cv generally decreases as the liquid limit of soil increases. The range of variation of cv for a given liquid limit of soil is wide. For a given load increment on a specimen, two graphical methods commonly are used for determining cv from laboratory one-dimensional consolidation tests. The ﬁrst is the logarithm-of-time method proposed by Casagrande and Fadum (1940), and the other is the square-root-of-time method given by Taylor (1942). More recently, at least two other methods were proposed. They are the hyperbola method (Sridharan and Prakash, 1985) and the early stage log-t method (Robinson and Allam, 1996). The general procedures for obtaining cv by these methods are described in this section. 11.13 Coefﬁcient of Consolidation Deformation (increasing) d0 D B 339 E x C x d 0 d100 2 F d50 d100 A t1 t2 Figure 11.25 Logarithm-of-time method for determining coefﬁcient of consolidation t50 Time (log scale) Logarithm-of-Time Method For a given incremental loading of the laboratory test, the specimen deformation against log-of-time plot is shown in Figure 11.25. The following constructions are needed to determine cv . Step 1: Extend the straight-line portions of primary and secondary consolidations to intersect at A. The ordinate of A is represented by d100 —that is, the deformation at the end of 100% primary consolidation. Step 2: The initial curved portion of the plot of deformation versus log t is approximated to be a parabola on the natural scale. Select times t1 and t2 on the curved portion such that t2 4t1 . Let the difference of specimen deformation during time (t2 t1) be equal to x. Step 3: Draw a horizontal line DE such that the vertical distance BD is equal to x. The deformation corresponding to the line DE is d0 (that is, deformation at 0% consolidation). Step 4: The ordinate of point F on the consolidation curve represents the deformation at 50% primary consolidation, and its abscissa represents the corresponding time (t50). Step 5: For 50% average degree of consolidation, Tv 0.197 (see Table 11.8), so, T50 cv t50 H2dr or cv 0.197Hdr2 t50 where Hdr average longest drainage path during consolidation. (11.66) 340 Chapter 11: Compressibility of Soil For specimens drained at both top and bottom, Hdr equals one-half the average height of the specimen during consolidation. For specimens drained on only one side, Hdr equals the average height of the specimen during consolidation. Square-Root-of-Time Method In the square-root-of-time method, a plot of deformation against the square root of time is made for the incremental loading (Figure 11.26). Other graphic constructions required are as follows: Step 1: Draw a line AB through the early portion of the curve. Step 2: Draw a line AC such that OC 1.15OB. The abscissa of point D, which is the intersection of AC and the consolidation curve, gives the square root of time for 90% consolidation ( 1t90). Step 3: For 90% consolidation, T90 0.848 (see Table 11.8), so T90 0.848 cv t90 H2dr or cv 0.848H2dr t90 Hdr in Eq. (11.67) is determined in a manner similar to that in the logarithm-of-time method. Deformation (increasing) A √t90 D O B C √Time, t Figure 11.26 Square-root-of-time ﬁtting method (11.67) 11.3 Coefﬁcient of Consolidation 341 t H c m 1 b d D a Time, t Figure 11.27 Hyperbola method for determination of cv Hyperbola Method In the hyperbola method, the following procedure is recommended for the determination of cv . Step 1: Obtain the time t and the specimen deformation (H) from the laboratory consolidation test. Step 2: Plot the graph of t/H against t as shown in Figure 11.27. Step 3: Identify the straight-line portion bc and project it back to point d. Determine the intercept D. Step 4: Determine the slope m of the line bc. Step 5: Calculate cv as cv 0.3 a mHdr2 b D (11.68) Note that because the unit of D is time/length and the unit of m is (time/length)/time 1/length, the unit of cv is a 1 b 1length2 2 1length2 2 length time time a b length The hyperbola method is fairly simple to use, and it gives good results for U 60 to 90%. 342 Chapter 11: Compressibility of Soil D G E Deformation (increasing) d0 F t 22.14 Time, t (log scale) Figure 11.28 Early stage log-t method Early Stage log-t Method The early stage log-t method, an extension of the logarithm-of-time method, is based on specimen deformation against log-of-time plot as shown in Figure 11.28. According to this method, follow Steps 2 and 3 described for the logarithm-of-time method to determine d0 . Draw a horizontal line DE through d0 . Then draw a tangent through the point of inﬂection, F. The tangent intersects line DE at point G. Determine the time t corresponding to G, which is the time at U 22.14%. So cv 0.0385Hdr2 t22.14 (11.69) In most cases, for a given soil and pressure range, the magnitude of cv determined by using the logarithm-of-time method provides lowest value. The highest value is obtained from the early stage log-t method. The primary reason is because the early stage log-t method uses the earlier part of the consolidation curve, whereas the logarithm-of-time method uses the lower portion of the consolidation curve. When the lower portion of the consolidation curve is taken into account, the effect of secondary consolidation plays a role in the magnitude of cv. This fact is demonstrated for several soils in Table 11.9. Several investigators also have reported that the cv value obtained from the ﬁeld is substantially higher than that obtained from laboratory tests conducted by using conventional testing methods (that is, logarithm-of-time and square-root-of-time methods). This ﬁnding is shown in Table 11.10 (Leroueil, 1988). Hence, the early stage log-t method may provide a more realistic value of ﬁeldwork. Table 11.9 Comparison of cv Obtained from Various Methods* cv ⴛ 10 4 cm 2/sec Soil Red earth Brown soil Black cotton soil Illite Bentonite Chicago clay (Taylor, 1948) Range of pressure Sⴕ (kN/m2) Logarithmof-time method Square-rootof-time method Early stage log-t method 25–50 50–100 100–200 200–400 400–800 25–50 50–100 100–200 200–400 400–800 25–50 50–100 100–200 200–400 400–800 25–50 50–100 100–200 200–400 400–800 25–50 50–100 100–200 200–400 400–800 12.5–25 25–50 50–100 100–200 200–400 400–800 800–1600 4.63 6.43 7.32 8.14 8.10 3.81 3.02 2.86 2.09 1.30 5.07 3.06 2.00 1.15 0.56 1.66 1.34 2.20 3.15 4.15 0.063 0.046 0.044 0.021 0.015 25.10 20.10 13.70 3.18 4.56 6.05 7.09 5.45 7.98 9.99 10.90 11.99 4.45 3.77 3.40 2.21 1.45 6.55 3.69 2.50 1.57 0.64 2.25 3.13 3.18 4.59 5.82 0.130 0.100 0.052 0.022 0.017 45.50 23.90 17.40 4.71 4.40 6.44 8.62 6.12 9.00 11.43 12.56 12.80 5.42 3.80 3.52 2.74 1.36 9.73 4.78 3.45 2.03 0.79 2.50 3.32 3.65 5.14 6.45 0.162 0.130 0.081 0.040 0.022 46.00 31.50 20.20 4.97 4.91 7.41 9.09 * After a table from "Determination of Coefficient of Consolidation from Early Stage of Log t Plot," by R.G. Robinson and M.M Allam, 1996, Geotechnical Testing Journal, 19(3) pp. 316-320. Copyright ASTM INTERNATIONAL. Reprinted with permission. 343 Table 11.10 Comparison Between the Coefﬁcients of Consolidation Determined in the Laboratory and Those Deduced from Embankment Settlement Analysis, as Observed by Leroueil (1988) c v (lab) c v (in situ ) Site (m2/sec) (m2/sec) c v (in situ )/c v (lab) Reference Ska-Edeby IV Oxford (1) Donnington Oxford (2) Avonmouth Tickton Over causeway Melbourne Penang Cubzac B Cubzac C A-64 Saint-Alban R-7 Matagami Berthierville 5.0 109 1.0 107 1.6 108 2.0 108 1.4 108 7.5 108 1.0 108 6.0 109 8.0 109 1.1 106 2.0 107 4.3 107 2.0 106 8.0 108 2.8 107 8.5 108 4.0 108 20 4–57 4–7 3–36 6–47 7–47 3–12 200 70 10 31 27 8 47 10 3–10 Holtz and Broms (1972) Lewis, Murray, and Symons (1975) Lewis, Murray, and Symons (1975) Lewis, Murray, and Symons (1975) Lewis, Murray, and Symons (1975) Lewis, Murray, and Symons (1975) Lewis, Murray, and Symons (1975) Walker and Morgan (1977) Adachi and Todo (1979) Magnan, et al. (1983) Leroueil, Magnan, and Tavenas (1985) Leroueil, Magnan, and Tavenas (1985) Leroueil, Magnan, and Tavenas (1985) Leroueil, Magnan, and Tavenas (1985) Leroueil, Magnan, and Tavenas (1985) Kabbaj (1985) Example 11.11 During a laboratory consolidation test, the time and dial gauge readings obtained from an increase of pressure on the specimen from 50 kN/m2 to 100 kN/m2 are given here. Time (min) Dial gauge reading (cm ⴛ 104) Time (min) Dial gauge reading (cm ⴛ 104) 0 0.1 0.25 0.5 1.0 2.0 4.0 8.0 3975 4082 4102 4128 4166 4224 4298 4420 16.0 30.0 60.0 120.0 240.0 480.0 960.0 1440.0 4572 4737 4923 5080 5207 5283 5334 5364 Using the logarithm-of-time method, determine cv. The average height of the specimen during consolidation was 2.24 cm, and it was drained at the top and bottom. Solution The semi-logarithmic plot of dial reading versus time is shown in Figure 11.29. For this, t1 0.1 min, t2 0.4 min to determine d0. Following the procedure outlined in Figure 11.25, t50 19 min. From Eq. (11.66) Cv 344 0.197H2dr t50 2.24 2 b 2 0.013 cm2/min 2.17 10 4 cm2/sec 19 0.197 a 11.14 Calculation of Consolidation Settlement Under a Foundation 0.40 345 d0 0.42 Dial reading (cm) 0.44 d50 0.46 0.48 0.50 d100 0.52 t50 = 19 min. 0.54 0.1 0.3 0.4 1.0 3 10 30 100 300 1000 Time (min.)—log scale Figure 11.29 11.14 ■ Calculation of Consolidation Settlement Under a Foundation Chapter 10 showed that the increase in the vertical stress in soil caused by a load applied over a limited area decreases with depth z measured from the ground surface downward. Hence to estimate the one-dimensional settlement of a foundation, we can use Eq. (11.31), (11.33), or (11.34). However, the increase of effective stress, s, in these equations should be the average increase in the pressure below the center of the foundation. The values can be determined by using the procedure described in Chapter 10. Assuming that the pressure increase varies parabolically, using Simpson’s rule, we œ can estimate the value of ¢sav as œ ¢sav œ ¢sbœ ¢stœ 4¢sm 6 (11.70) œ where ¢stœ , ¢sm , and ¢sbœ represent the increase in the effective pressure at the top, middle, and bottom of the layer, respectively. 346 Chapter 11: Compressibility of Soil Example 11.12 Calculate the settlement of the 10-ft-thick clay layer (Figure 11.30) that will result from the load carried by a 5-ft-square footing. The clay is normally consolidated. Use the weighted average method [Eq. (11.70)] to calculate the average increase of effective pressure in the clay layer. 200 kips Footing size 5 ft 5 ft 5 ft gdry 100 pcf 10 ft Groundwater table 10 ft gsat 120 pcf 10 ft gsat 110 pcf eO 1.0 LL 40 Dry sand Sand Clay Figure 11.30 Solution For normally consolidated clay, from Eq. (11.31), Se œ CcH soœ ¢sav log œ 1 eo so where Cc 0.0091LL 102 0.009140 10 2 0.27 H 10 12 120 in. eo 1.0 11.15 A Case History—Settlement Due to a Preload Fill for Construction of Tampa VA Hospital 347 10 3 gsat1clay2 62.4 4 2 sOœ 10 ft gdry1sand2 10 ft3gsat1sand2 62.4 4 10 100 101120 62.42 51110 62.4 2 1814 lb/ft2 From Eq. (11.70), œ ¢sav œ ¢sbœ ¢stœ 4¢sm 6 œ ¢stœ , ¢sm , and ¢sbœ below the center of the footing can be obtained from Eq. (10.34). Now we can prepare the following table (Note: L/B 5/5 1): b ⴝ B/2 (ft) n1 ⴝ z /b m1 z (ft) 1 15 2.5 6 1 1 20 25 2.5 2.5 8 10 q (kip/ft 2) 200 8 55 8 8 I4 ⌬Sⴕ ⴝ qI4 (kip/ft 2) 0.051 0.408 ¢stœ 0.029 0.019 œ 0.232 ¢sm 0.152 ¢sbœ So, œ ¢sav 0.408 142 10.2322 0.152 6 0.248 kip/ft2 248 lb/ft2 Hence, Sc 111.15 10.272 11202 11 log 1814 248 0.9 in. 1814 ■ A Case History—Settlement Due to a Preload Fill for Construction of Tampa VA Hospital Wheeless and Sowers (1972) have presented the field measurements of settlement due to a preload fill used for the construction of Tampa Veterans Administration Hospital. Figure 11.31 shows the simplified general subsoil conditions at the building site. In general, the subsoil consisted of 15 to 20 ft of subangular quartz sand at the top followed by clayey soil of varying thicknesses. The void ratio of the clayey soil varied from 0.7 to 1.4. The silt and clay content of the clayey soil varied from 5 to 75%. The Tampa limestone underlying the clay layer is a complex assortment of chalky, poorly 348 Chapter 11: Compressibility of Soil Firm fine sand Fine sandy clay—clayey fine sand Peat, organic silt Soft porous limestone—Tampa formation Surcharge fill 100 0 8-story hospital West Ground surface East Fill Mat 25 0 25 50 75 Figure 11.31 Simplified general subsoil conditions at the site of Tampa VA Hospital (After Wheelen and Sowers, 1972. With permission from ASCE.) consolidated calcareous deposits. The groundwater table was located at a depth of about 15 ft below the ground surface (elevation +25 ft). Figure 11.32 shows the consolidation curves obtained in the laboratory for clayey sand and sandy clay samples collected from various depths at the site. The plan of the hospital building is shown in Figure 11.33 (broken lines). Figure 11.31 also shows the cross section of the building. For a number of reasons, it was decided that the hospital should be built with a mat foundation. As can be seen from Figure 11.31, some soil had to be excavated to build the mat. As reported by Wheeless and Sowers, preliminary calculations indicated that the average building load in the eight-story area would be equal to the weight of the soil to be excavated for the construction of the mat. In that case, the consolidation settlement of the clay layer under the building would be rather small. However, the grading plan required a permanent fill of 16 ft over the original ground surface to provide access to the main floor on the east side. This is also shown in Figure 11.31. Preliminary calculations indicated that the weight of this fill could be expected to produce a soil settlement of about 4 in. near the east side of the building. This settlement would produce undue bending and overstressing of the mat foundation. For that reason, it was decided to build a temporary fill that was 26 ft high and limited to the front area of the proposed building. The fill area is shown in Figures 11.31 and 11.33. This temporary fill was built because the vertical stress that it induced in the clay layer would be greater than the stress induced by the permanent fill of 16 ft as required by the grading plan. This would 11.15 A Case History—Settlement Due to a Preload Fill for Construction of Tampa VA Hospital 349 1.3 B29—54 ft 1.2 7B4—24 ft Void ratio, e 1.1 1.0 7B3—38 ft B25—26 ft 0.9 2B2—24 ft 2B14—24 ft 0.8 B26—57 ft 0.7 B16—26 ft 100 1000 Pressure, s (log scale) (lb/ft2) 10,000 Figure 11.32 Consolidation curves of clayey sands and sandy clays (After Wheelen and Sowers, 1972. With permission from ASCE.) produce faster consolidation settlement. In a period of about four months, the settlement would be approximately 4 in., which is the magnitude of maximum settlement expected from the required permanent fill of 16 ft. At that time, if the excess fill material is removed and the building constructed, the settlement of the mat on the east side will be negligible. This technique of achieving the probable settlement of soil before construction is referred to as preloading. Figure 11.33 shows the locations of eight settlement plates placed on the original ground surface before the construction of the temporary fill. Figure 11.34 shows the 350 Chapter 11: Compressibility of Soil N 3-story 8-story Location of settlement plates 3-story Surcharge fill area Plan 0 100 Figure 11.33 Plan of the Tampa VA Hospital (After Wheelen and Sowers, 1972. With permission from ASCE.) time-settlement records beneath the surcharge fill area as observed from the settlement plates. Following is a comparison of total estimated and observed consolidation settlement due to preloading. Settlement plate location Observed settlement (in.) Estimated consolidation settlement (in.) 3 4 6 7 2.6 2.5 2.9 3.4 2.9 2.9 3.0 3.8 From the preceding comparisons of observed and estimated settlements given by Wheeless and Sowers and Figure 11.34, the following conclusions can be drawn: 1. In all cases, the estimated settlement slightly exceeded the observed settlement. 2. Most of the settlement was complete in a period of about 90 days. 3. The difference between the estimated and observed settlement varied from 3 to 16%, with an average of 13%. 4. Two-thirds to four-fifths of the total observed settlement was completed during the period of fill construction. The rate of consolidation was much faster than anticipated. Wheeless and Sowers have suggested that the accelerated rate of consolidation may be due primarily to irregular sandy seams within the clay stratum. In Section 11.12, it was shown that the average degree of consolidation is related to the time factor, Tv. Also t TvH 2dr cv 11.16 Methods for Accelerating Consolidation Settlement 351 Load 100% 26-ft sand fill 0 Surcharge period Fill Remove Permanent 0 1 Settlement plate no. Settlement (in.) 1 2 8 2 5 4 3 6 3 7 4 0 20 40 60 80 100 Time in days from beginning surcharge 120 140 Figure 11.34 Settlement-time curves beneath the surcharge fill area for the construction of Tampa VA Hospital (After Wheelen and Sowers, 1972. With permission from ASCE.) For similar values of Tv (or average degree of consolidation) and cv, the time t will be less if the maximum length of the drainage path (Hdr) is less. The presence of irregular sandy seams in the clay layer tends to reduce the magnitude of Hdr. This is the reason why a faster rate of consolidation was attained in this area. The structural load of the VA hospital was completed in the early part of 1970. No noticeable foundation movement has occurred. 11.16 Methods for Accelerating Consolidation Settlement In many instances, sand drains and prefabricated vertical drains are used in the ﬁeld to accelerate consolidation settlement in soft, normally consolidated clay layers and to achieve precompression before the construction of a desired foundation. Sand drains are constructed by drilling holes through the clay layer(s) in the ﬁeld at regular intervals. The holes then are backﬁlled with sand. This can be achieved by several means, such as (a) rotary drilling and then backﬁlling with sand; (b) drilling by continuous ﬂight auger with a hollow stem and backﬁlling with sand (through the 352 Chapter 11: Compressibility of Soil Surcharge Groundwater table Sand drain; radius r Vertical drainage Sand drain; radius r de Sand drain He Radial drainage Radial drainage Vertical drainage Sand Clay layer (a) Section (b) Plan Figure 11.35 Sand drains hollow stem); and (c) driving hollow steel piles. The soil inside the pile is then jetted out, and the hole is backﬁlled with sand. Figure 11.35 shows a schematic diagram of sand drains. After backﬁlling the drill holes with sand, a surcharge is applied at the ground surface. This surcharge will increase the pore water pressure in the clay. The excess pore water pressure in the clay will be dissipated by drainage—both vertically and radially to the sand drains—which accelerates settlement of the clay layer. In Figure 11.35a, note that the radius of the sand drains is rw . Figure 11.35b shows the plan of the layout of the sand drains. The effective zone from which the radial drainage will be directed toward a given sand drain is approximately cylindrical, with a diameter of de . The surcharge that needs to be applied at the ground surface and the length of time it has to be maintained to achieve the desired degree of consolidation will be a function of rw, de, and other soil parameters. Figure 11.36 shows a sand drain installation in progress. Prefabricated vertical drains (PVDs), which also are referred to as wick or strip drains, originally were developed as a substitute for the commonly used sand drain. With the advent of materials science, these drains are manufactured from synthetic polymers such as polypropylene and high-density polyethylene. PVDs normally are manufactured with a corrugated or channeled synthetic core enclosed by a geotextile ﬁlter, as shown schematically in Figure 11.37. Installation rates reported in the literature are on the order of 0.1 to 0.3 m/s, excluding equipment mobilization and setup time. PVDs have been used extensively in the past for expedient consolidation of low permeability soils under surface surcharge. The main advantage of PVDs over sand drains is that they do not require drilling and, thus, installation is much faster. Figure 11.38 shows the installation of PVDs in the ﬁeld. Figure 11.36 Sand drain installation in progress (Courtesy of E.C. Shin, University of Inchon, South Korea) Polypropylene core Geotextile fabric Figure 11.37 Prefabricated vertical drain (PVD) 353 354 Chapter 11: Compressibility of Soil Figure 11.38 Installation of PVDs in progress (Courtesy of E.C. Shin, University of Inchon, South Korea) 11.17 Precompression When highly compressible, normally consolidated clayey soil layers lie at a limited depth and large consolidation settlements are expected as a result of the construction of large buildings, highway embankments, or earth dams, precompression of soil may be used to minimize postconstruction settlement. The principles of precompression are best explained by referring to Figure 11.39. Here, the proposed structural load per unit area is s( p) and the thickness of the clay layer undergoing consolidation is H. The maximum primary consolidation settlement caused by the structural load, Sc S( p ) , then is Sc S1p2 soœ ¢s1p2 CcH log 1 eo soœ (11.71) Note that at the end of consolidation, s s( p) . The settlement–time relationship under the structural load will be like that shown in Figure 11.39b. However, if a surcharge of s( p) s( f ) is placed on the ground, then the primary consolidation settlement will be Sc S1pf2 soœ 3 ¢s1p2 ¢s1 f2 4 CcH log 1 eo soœ (11.72) 11.17 Precompression 355 Surcharge per unit area Surcharge Groundwater table s(p) s(f) s(p) t2 H Time t1 Time S(p) Sand Clay S(pf) Settlement (a) (b) Figure 11.39 Principles of precompression Note that at the end of consolidation, s s( p) s( f ) The settlement–time relationship under a surcharge of s( p) s( f ) is also shown in Figure 11.39b. Note that a total settlement of S( p) would occur at a time t2 , which is much shorter than t1 . So, if a temporary total surcharge of s( f ) s( p) is applied on the ground surface for time t2 , the settlement will equal S( p) . At that time, if the surcharge is removed and a structure with a permanent load per unit area of s( p) is built, no appreciable settlement will occur. The procedure just described is precompression. The total surcharge, s( p) s( f ) , can be applied by using temporary ﬁlls. Derivation of Equations to Obtain ⌬S ( f ) and t 2 Figure 11.39b shows that, under a surcharge of s( p) s( f ) , the degree of consolidation at time t2 after load application is U S1p2 (11.73) S1pf2 Substitution of Eqs. (11.71) and (11.72) into Eq. (11.73) yields log c U log c soœ ¢s1p2 soœ d soœ ¢s1p2 ¢s1 f2 soœ log c 1 d log e 1 ¢s1p2 soœ ¢s1p2 soœ c1 d ¢s1 f2 ¢s1p2 df (11.74) 356 Chapter 11: Compressibility of Soil From Eq. (11.61), we know that U f(Tv ) (11.75) where Tv time factor cv t2 /H 2dr cv coefﬁcient of consolidation t2 time Hdr maximum drainage path (H/2 for two-way drainage and H for one-way drainage) The variation of U with Tv is shown in Table 11.8. Procedure for Obtaining Precompression Parameters Engineers may encounter two problems during precompression work in the ﬁeld: œ 1. The value of s( f ) is known, but t2 must be obtained. In such case, obtain so and s( p) and solve for U using Eq. (11.74). For this value of U, obtain Tv from Table 11.8. Then, t2 TvH 2dr cv (11.76) 2. For a speciﬁed value of t2 , s( f ) must be obtained. In such case, calculate Tv . Then refer to Table 11.8 to obtain the degree of consolidation, U. With the estimated value of U, go to Eq. (11.74) to ﬁnd the required s( f ) /s( p) and then calculate s( f ) . The case history given in Section 11.15 is an example of precompression. Example 11.13 Refer to Figure 11.39. During the construction of a highway bridge, the average permanent load on the clay layer is expected to increase by about 115 kN/m2. The average effective overburden pressure at the middle of the clay layer is 210 kN/m2. Here, H 6 m, Cc 0.28, eo 0.9, and cv 0.36 m2/mo. The clay is normally consolidated. a. Determine the total primary consolidation settlement of the bridge without precompression. b. What is the surcharge, s( f ) , needed to eliminate by precompression the entire primary consolidation settlement in nine months? Solution Part a The total primary consolidation settlement may be calculated from Eq. (11.71): Sc S1p2 soœ ¢s1p2 10.28 2 162 CcH 210 115 log c d log c d œ 1 eo so 1 0.9 210 0.1677 m 167.7 mm 11.18 Summary and General Comments 357 Part b Tv cvt2 H2dr cv 0.36 m2/mo. Hdr 3 m 1two-way drainage2 t2 9 mo. Hence, Tv 10.36 2 192 32 0.36 According to Table 11.8, for Tv 0.36, the value of U is 67%. Now ¢s1p2 115 kN/m2 soœ 210 kN/m2 So, ¢s1p2 soœ Thus, from Eq. (11.74), U 0.67 ¢s1f2 ¢s1p2 115 0.548 210 log11 0.548 2 ¢s1f2 log e 1 0.548 c 1 df ¢s1p2 0.677 So, s( f ) (0.677)(115) 78 kN/m2 11.18 ■ Summary and General Comments In this chapter, we discussed the fundamental concepts and theories for estimating elastic and consolidation (primary and secondary) settlement. Elastic settlement of a foundation is primarily a function of the size and rigidity of the foundation, the modulus of elasticity, the Poisson’s ratio of the soil, and the intensity of load on the foundation. Consolidation is a time-dependent process of settlement of saturated clay layers located below the ground water table by extrusion of excess water pressure generated by application of load on the foundation. Total consolidation settlement of a foundation on clay 358 Chapter 11: Compressibility of Soil is a function of compression index (Cc), swell index (Cs), initial void ratio, (eO) and the average stress increase in the clay layer. The degree of consolidation for a given soil layer at a certain time after the load application depends on its coefﬁcient of consolidation (cv) and also on the length of the minimum drainage path. Installation of sand drains and wick drains helps reduce the time for accomplishing the desired degree of consolidation for a given construction project. There are several case histories in the literature for which the fundamental principles of soil compressibility have been used to predict and compare the actual total settlement and the time rate of settlement of soil proﬁles under superimposed loading. In some cases, the actual and predicted maximum settlements agree remarkably; in many others, the predicted settlements deviate to a large extent from the actual settlements observed. The disagreement in the latter cases may have several causes: 1. Improper evaluation of soil properties 2. Nonhomogeneity and irregularity of soil proﬁles 3. Error in the evaluation of the net stress increase with depth, which induces settlement The variation between the predicted and observed time rate of settlement may also be due to • • Improper evaluation of cv (see Section 11.13) Presence of irregular sandy seams within the clay layer, which reduces the length of the maximum drainage path, Hdr Problems 11.1 Refer to Figure 11.3 for the rigid foundation. Given: B 3 ft; L 6 ft; Df 3 ft; H 15 ft; Es 140 ton/ft2; s 0.4; and net increase of pressure on the foundation, 4000 lb/ft2. Estimate the elastic settlement. 11.2 Refer to Figure 11.3 for a square rigid foundation measuring 3 m 3 m in plan supported by a layer of sand. Given that Df 1.5 m, Es 16,000 kN/m2, s 0.3; H 20 m, and 100 kN/m2, calculate the elastic settlement. 11.3 The following are the results of a consolidation test. e Pressure, Sⴕ (ton /ft2) 1.1 1.085 1.055 1.01 0.94 0.79 0.63 0.25 0.50 1.00 2.00 4.00 8.00 16.00 a. Plot the e-log s curve b. Using Casagrande’s method, determine the preconsolidation pressure c. Calculate the compression index, Cc , from the laboratory e-log s curve Problems 359 11.4 Repeat Problem 11.3 using the following values. e 1.21 1.195 1.15 1.06 0.98 0.925 Pressure, Sⴕ (kN/m2) 25 50 100 200 400 500 11.5 The results of a laboratory consolidation test on a clay specimen are the following. Pressure, Sⴕ (lb/ft 2) Total height of specimen at end of consolidation (in.) 500 1,000 2,000 4,000 8,000 16,000 0.6947 0.6850 0.6705 0.6520 0.6358 0.6252 Given the initial height of specimen 0.748 in., Gs 2.68, mass of dry specimen 95.2 g, and area of specimen 4.91 in.2: a. Plot the e-log s curve b. Determine the preconsolidation pressure c. Calculate the compression index, Cc 11.6 Figure 11.40 shows a soil proﬁle. The uniformly distributed load on the ground surface is s. Given: s 1000 lb/ft2, H1 8 ft, H2 15 ft, and H3 17 ft. Also, • Sand: gdry 110 lb/ft3, gsat 115 lb/ft3 • Clay: gsat 120 lb/ft3, LL 50, e 0.9 Estimate the primary consolidation settlement if a. The clay is normally consolidated 1 b. The preconsolidation pressure is 2600 lb/ft2 (Cs 6 Cc) 11.7 Refer to Figure 11.40. Given: H1 2.5 m, H2 2.5 m, H3 3 m, and s 100 kN/m2. Also, • Sand: e 0.64, Gs 2.65 • Clay: e 0.9, Gs 2.75, LL 55 Estimate the primary consolidation settlement of the clay layer assuming that it is normally consolidated. 11.8 Refer to Figure 11.40. Given: H1 5 ft, H2 7 ft, H3 6 ft, and s 3000 lb/ft2. Also, • Clay: e 1.1, Gs 2.72, LL 45 • Sand: e 0.58, Gs 2.65 360 Chapter 11: Compressibility of Soil s H1 Groundwater table H2 Void ratio e H3 Sand Clay Figure 11.40 11.9 11.10 11.11 11.12 Estimate the primary consolidation settlement if the preconsolidation pressure is 3500 lb/ft2. Assume CS 15 Cc . The coordinates of two points on a virgin compression curve are as follows: • e1 0.82 • s1œ 2500 lb/ft2 • e2 0.70 • s2œ 4000 lb/ft2 Determine the void ratio that corresponds to a pressure of 6000 lb/ft2. The laboratory consolidation data for an undisturbed clay specimen are as follows: • e1 1.1 • s1œ 1 ton/ft2 • e2 0.9 • s2œ 3 ton/ft2 What is the void ratio for a pressure of 3.5 ton/ft2? (Note: scœ 0.8 ton/ft2.) Refer to Problem 11.7. Given: cv 2.8 106m2/min. How long will it take for 60% consolidation to occur? Following are the relationships of e and s for a clay soil: e Sⴕ (ton /ft 2) 1.0 0.97 0.85 0.75 0.2 0.5 1.8 3.2 For this clay soil in the ﬁeld, the following values are given: H 4.5 ft, soœ 0.7 ton/ft2, and soœ s 2.0 ton/ft2. Calculate the expected settlement caused by primary consolidation. Problems 361 11.13 The coordinates of two points on a virgin compression curve are as follows: • e1 1.7 • s1œ 150 kN/m2 • e2 1.48 • s2œ 400 kN/m2 a. Determine the coefﬁcient of volume compressibility for the pressure range stated. b. Given that cv 0.002 cm2/sec, determine k in cm/sec corresponding to the average void ratio. 11.14 The time for 50% consolidation of a 1-in. thick clay layer (drained at top and bottom) in the laboratory is 2 min, 20 sec. How long (in days) will it take for a 8-ft thick layer of the same clay in the ﬁeld (under the same pressure increment) to reach 30% consolidation? In the ﬁeld, there is a rock layer at the bottom of the clay. 11.15 For a normally consolidated clay, the following are given: • soœ 2 ton/ft2 • e eo 1.21 2 œ • so s 4 ton/ft • e 0.96 The hydraulic conductivity k of the clay for the preceding loading range is 1.8 104 ft/day. a. How long (in days) will it take for a 9 ft thick clay layer (drained on one side) in the ﬁeld to reach 60% consolidation? b. What is the settlement at that time (that is, at 60% consolidation)? 11.16 For a laboratory consolidation test on a clay specimen (drained on both sides), the following were obtained: • Thickness of the clay layer 25 mm • s1œ 200 kN/m2 • e1 0.73 • s2œ 400 kN/m2 • e2 0.61 • Time for 50% consolidation (t50) 2.8 min Determine the hydraulic conductivity of the clay for the loading range. 11.17 The time for 50% consolidation of a 25-mm thick clay layer (drained at top and bottom) in the laboratory is 225 sec. How long (in days) will it take for a 2-m thick layer of the same clay in the ﬁeld (under the same pressure increment) to reach 50% consolidation? There is a rock layer at the bottom of the clay in the ﬁeld. 11.18 A 3-m thick layer of saturated clay (two-way drainage) under a surcharge loading underwent 90% primary consolidation in 100 days. The laboratory test’s specimen will have two-way drainage. a. Find the coefﬁcient of consolidation of clay for the pressure range. b. For a 25 mm. thick undisturbed clay specimen, how long will it take to undergo 80% consolidation in the laboratory for a similar pressure range? 11.19 A normally consolidated clay layer is 3 m thick (one-way drainage). From the application of a given pressure, the total anticipated primary consolidation settlement will be 80 mm. a. What is the average degree of consolidation for the clay layer when the settlement is 25 mm? b. If the average value of cv for the pressure range is 0.002 cm2/sec, how long will it take for 50% settlement to occur? c. How long will it take for 50% consolidation to occur if the clay layer is drained at both top and bottom? 11.20 Refer to Figure 11.41. Given that B 1 m, L 3 m, and Q 110 kN, calculate the primary consolidation settlement of the foundation. 362 Chapter 11: Compressibility of Soil Load Q BL g 15 kN/m3 1.5 m Groundwater table 1.5 m gsat 18 kN/m3 2.5 m 35% Gs 2.7, LL 38 Sand Clay (normally consolidated) Figure 11.41 References ADACHI, K., and TODO, H. (1979). “A Case Study on Settlement of Soil Clay in Penang,” Proceedings, 6th Asian Regional Conference on Soil Mechanics and Foundation Engineering, Vol. 1, 117–120. BOWLES, J. E. (1987). “Elastic Foundation Settlement on Sand Deposits,” Journal of Geotechnical Engineering, ASCE, Vol. 113, No. 8, 846–860. CASAGRANDE, A. (1936). “Determination of the Preconsolidation Load and Its Practical Signiﬁcance,” Proceedings, 1st International Conference on Soil Mechanics and Foundation Engineering, Cambridge, Mass., Vol. 3, 60–64. CASAGRANDE, A., and FADUM, R. E. 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Manual of Estimating Soil Properties for Foundation Design, Final Report (EL-6800) submitted to Electric Power Research Institute (EPRI), Palo Alto, CA. LEROUEIL, S. (1988). “Tenth Canadian Geotechnical Colloquium: Recent Developments in Consolidation of Natural Clays,” Canadian Geotechnical Journal, Vol. 25, No. 1, 85–107. LEROUEIL, S., MAGNAN, J. P., and TAVENAS, F. (1985). Remblais sur Argiles Molles. Technique et Documentation, Paris, France. LEWIS, W. A., MURRAY, R. T., and SYMONS, I. F. (1975). “Settlement and Stability of Embankments Constructed on Soft Alluvial Soils,” Proceedings, The Institute of Civil Engineers, Vol. 59, 571–593. MAGNAN, J. P., MIEUSSENS, C., and QUEYROI, D. (1983). Research Report LPC 127, Étude d’un Remblai sur Sols Compressibles: le Remblai B du Site Expérimental de Cubzac-les-Ponts. Laboratorie Central des Ponts et Chaussées, Paris. MESRI, G. (1973). “Coefﬁcient of Secondary Compression,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM1, 122–137. M ESRI , G., AND G ODLEWSKI , P. M. (1977). “Time and Stress—Compressiblity Interrelationship,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT5, 417–430. NAGARAJ, T., and MURTY, B. R. S. (1985). “Prediction of the Preconsolidation Pressure and Recompression Index of Soils,” Geotechnical Testing Journal, Vol. 8, No. 4, 199–202. NISHIDA, Y. (1956). “A Brief Note on Compression Index of Soils,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 82, No. SM3, 1027-1–1027-14. PARK, J. H., and KOUMOTO, T. (2004). “New Compression Index Equation,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 130, No. 2, 223–226. RENDON-HERRERO, O. (1983). “Universal Compression Index Equation,” Discussion, Journal of Geotechnical Engineering, ASCE, Vol. 109, No. 10, 1349. RENDON-HERRERO, O. (1980). “Universal Compression Index Equation,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 106, No. GT11, 1179–1200. ROBINSON, R. G., and ALLAM, M. M. (1996). “Determination of Coefﬁcient of Consolidation from Early Stage of log t Plot,” Geotechnical Testing Journal, ASTM, Vol. 19, No. 3, 316–320. SCHMERTMANN, J. H. (1953). “Undisturbed Consolidation Behavior of Clay,” Transactions, ASCE, Vol. 120, 1201. SIVARAM, B., and SWAMEE, A. (1977). “A Computational Method for Consolidation Coefﬁcient,” Soils and Foundations, Vol. 17, No. 2, 48–52. SKEMPTON, A. W. (1944). “Notes on the Compressibility of Clays,” Quarterly Journal of the Geological Society of London, Vol. 100, 119–135. SRIDHARAN, A., and PRAKASH, K. (1985). “Improved Rectangular Hyperbola Method for the Determination of Coefﬁcient of Consolidation,” Geotechnical Testing Journal, ASTM, Vol. 8, No. 1, 37–40. STAS, C.V., AND KULHAWY, F. H. (1984). “Critical Evaluation of Design Methods for Foundations under Axial Uplift and Compression Loading,” Report EL-3771, Electric Power Research Institute (EPRI), Palo Alto, CA. STEINBRENNER, W. (1934). “Tafeln zur Setzungsberechnung,” Die Strasse, Vol. 1, 121–124. TAYLOR, D. W. (1942). “Research on Consolidation of Clays,” Serial No. 82, Department of Civil and Sanitary Engineering, Massachusetts Institute of Technology, Cambridge, Mass. TAYLOR, D. W. (1948). Fundamentals of Soil Mechanics, Wiley, New York. TERZAGHI, K. (1925). Erdbaumechanik auf Bodenphysikalischer Grundlager, Deuticke, Vienna. TERZAGHI, K., and PECK, R. B. (1967). Soil Mechanics in Engineering Practice, 2nd ed., Wiley, New York. KULHAWY, F. H., 364 References WALKER, L. K. and MORGAN, J. R. (1977). “Field Performance of a Firm Silty Clay,” Proceedings, 9th International Conference on Soil Mechanics and Foundation Engineering, Tokyo, Vol. 1, 341–346. WHEELEN, L. D., AND SOWERS, G. F. (1972). “Mat Foundation and Preload Fill, VA Hospital, Tampa,” Proceedings, Specialty Conference on Performance of Earth and Earth-Supported Structures, ASCE, Vol. 1, Part 2, 939–951. WROTH, C. P., AND WOOD, D. M. (1978). “The Correlation of Index Properties with Some Basic Engineering Properties of Soils,” Canadian Geotechnical Journal, Vol. 15, No. 2, 137–145. 12 Shear Strength of Soil The shear strength of a soil mass is the internal resistance per unit area that the soil mass can offer to resist failure and sliding along any plane inside it. One must understand the nature of shearing resistance in order to analyze soil stability problems, such as bearing capacity, slope stability, and lateral pressure on earth-retaining structures. 12.1 Mohr–Coulomb Failure Criterion Mohr (1900) presented a theory for rupture in materials that contended that a material fails because of a critical combination of normal stress and shearing stress and not from either maximum normal or shear stress alone. Thus, the functional relationship between normal stress and shear stress on a failure plane can be expressed in the following form: tf f1s 2 (12.1) The failure envelope deﬁned by Eq. (12.1) is a curved line. For most soil mechanics problems, it is sufﬁcient to approximate the shear stress on the failure plane as a linear function of the normal stress (Coulomb, 1776). This linear function can be written as tf c s tan f (12.2) where c cohesion f angle of internal friction s normal stress on the failure plane tf shear strength The preceding equation is called the Mohr–Coulomb failure criterion. In saturated soil, the total normal stress at a point is the sum of the effective stress (s) and pore water pressure (u), or s s¿ u 365 366 Chapter 12: Shear Strength of Soil The effective stress s is carried by the soil solids. The Mohr–Coulomb failure criterion, expressed in terms of effective stress, will be of the form tf c¿ s¿ tan f¿ (12.3) where c cohesion and f friction angle, based on effective stress. Thus, Eqs. (12.2) and (12.3) are expressions of shear strength based on total stress and effective stress. The value of c for sand and inorganic silt is 0. For normally consolidated clays, c can be approximated at 0. Overconsolidated clays have values of c that are greater than 0. The angle of friction, f, is sometimes referred to as the drained angle of friction. Typical values of f for some granular soils are given in Table 12.1. The signiﬁcance of Eq. (12.3) can be explained by referring to Fig. 12.1, which shows an elemental soil mass. Let the effective normal stress and the shear stress on the Table 12.1 Typical Values of Drained Angle of Friction for Sands and Silts Soil type Fⴕ (deg) Sand: Rounded grains Loose Medium Dense 27–30 30–35 35–38 Sand: Angular grains Loose Medium Dense 30–35 35–40 40–45 Gravel with some sand 34–48 Silts 26–35 C sx Shear stress, t t b s t t sy B f a A Mohr– Coulomb failure criteria c Effective normal stress, s (a) Figure 12.1 Mohr–Coulomb failure criterion (b) 367 12.2 Inclination of the Plane of Failure Caused by Shear plane ab be s and t, respectively. Figure 12.1b shows the plot of the failure envelope deﬁned by Eq. (12.3). If the magnitudes of s and t on plane ab are such that they plot as point A in Figure 12.1b, shear failure will not occur along the plane. If the effective normal stress and the shear stress on plane ab plot as point B (which falls on the failure envelope), shear failure will occur along that plane. A state of stress on a plane represented by point C cannot exist, because it plots above the failure envelope, and shear failure in a soil would have occurred already. 12.2 Inclination of the Plane of Failure Caused by Shear As stated by the Mohr–Coulomb failure criterion, failure from shear will occur when the shear stress on a plane reaches a value given by Eq. (12.3). To determine the inclination of the failure plane with the major principal plane, refer to Figure 12.2, where s1œ and s3œ are, respectively, the major and minor effective principal stresses. The failure plane EF makes an angle u with the major principal plane. To determine the angle u and the relationship between s1œ and s3œ , refer to Figure 12.3, which is a plot of the Mohr’s circle for the state of stress shown in Figure 12.2 (see Chapter 10). In Figure 12.3, fgh is the failure envelope deﬁned by the relationship tf c s tan f. The radial line ab deﬁnes the major principal plane (CD in Figure 12.2), and the radial line ad deﬁnes the failure plane (EF in Figure 12.2). It can be shown that ⬔bad 2u 90 f, or u 45 f¿ 2 (12.4) Again, from Figure 12.3, ad sin f¿ fa (12.5) fa fO Oa c¿ cot f¿ s1œ s3œ 2 (12.6a) s1 A B F Shear stress h s3 u D C E g f s1 s3 Figure 12.2 Inclination of failure plane in soil with major principal plane f O tf c s tan f d 2u c e a s3 Effective normal stress b s1 Figure 12.3 Mohr’s circle and failure envelope 368 Chapter 12: Shear Strength of Soil Also, ad s1œ s3œ 2 (12.6b) Substituting Eqs. (12.6a) and (12.6b) into Eq. (12.5), we obtain s1œ s3œ 2 sin f¿ s1œ s3œ c¿ cot f¿ 2 or s1œ s3œ a cos f¿ 1 sin f¿ b 2c a b 1 sin f¿ 1 sin f¿ (12.7) However, f¿ 1 sin f¿ tan2 a 45 b 1 sin f¿ 2 and f¿ cos f¿ tan a 45 b 1 sin f¿ 2 Thus, s1œ s3œ tan2 a 45 f¿ f¿ b 2c¿ tan a 45 b 2 2 (12.8) An expression similar to Eq. (12.8) could also be derived using Eq. (12.2) (that is, total stress parameters c and f), or s1 s3 tan2 a 45 12.3 f f b 2c tan a 45 b 2 2 (12.9) Laboratory Test for Determination of Shear Strength Parameters There are several laboratory methods now available to determine the shear strength parameters (i.e., c, f, c, f) of various soil specimens in the laboratory. They are as follows: • • • • • Direct shear test Triaxial test Direct simple shear test Plane strain triaxial test Torsional ring shear test 12.4 Direct Shear Test 369 The direct shear test and the triaxial test are the two commonly used techniques for determining the shear strength parameters. These two tests will be described in detail in the sections that follow. 12.4 Direct Shear Test The direct shear test is the oldest and simplest form of shear test arrangement. A diagram of the direct shear test apparatus is shown in Figure 12.4. The test equipment consists of a metal shear box in which the soil specimen is placed. The soil specimens may be square or circular in plan. The size of the specimens generally used is about 51 mm 51 mm or 102 mm 102 mm (2 in. 2 in. or 4 in. 4 in.) across and about 25 mm (1 in.) high. The box is split horizontally into halves. Normal force on the specimen is applied from the top of the shear box. The normal stress on the specimens can be as great as 1050 kN/m2 (150 lb/in.2). Shear force is applied by moving one-half of the box relative to the other to cause failure in the soil specimen. Depending on the equipment, the shear test can be either stress controlled or strain controlled. In stress-controlled tests, the shear force is applied in equal increments until the specimen fails. The failure occurs along the plane of split of the shear box. After the application of each incremental load, the shear displacement of the top half of the box is measured by a horizontal dial gauge. The change in the height of the specimen (and thus the volume change of the specimen) during the test can be obtained from the readings of a dial gauge that measures the vertical movement of the upper loading plate. In strain-controlled tests, a constant rate of shear displacement is applied to onehalf of the box by a motor that acts through gears. The constant rate of shear displacement is measured by a horizontal dial gauge. The resisting shear force of the soil corresponding to any shear displacement can be measured by a horizontal proving ring or load cell. The volume change of the specimen during the test is obtained in a manner Normal force Shear force t t Shear box Shear force Loading plate Porous stone Figure 12.4 Diagram of direct shear test arrangement 370 Chapter 12: Shear Strength of Soil Figure 12.5 Strain-controlled direct shear equipment (Courtesy of Braja M. Das, Henderson, Nevada) similar to that in the stress-controlled tests. Figure 12.5 shows a photograph of straincontrolled direct shear test equipment. Figure 12.6 shows a photograph taken from the top of the direct shear test equipment with the dial gages and proving ring in place. The advantage of the strain-controlled tests is that in the case of dense sand, peak shear resistance (that is, at failure) as well as lesser shear resistance (that is, at a point after failure called ultimate strength) can be observed and plotted. In stress-controlled tests, only the peak shear resistance can be observed and plotted. Note that the peak shear resistance in stress-controlled tests can be only approximated because failure occurs at a stress level somewhere between the prefailure load increment and the failure load increment. Nevertheless, compared with strain-controlled tests, stress-controlled tests probably model real ﬁeld situations better. For a given test, the normal stress can be calculated as s Normal stress Normal force Cross-sectional area of the specimen (12.10) The resisting shear stress for any shear displacement can be calculated as t Shear stress Resisting shear force Cross-sectional area of the specimen (12.11) 12.4 Direct Shear Test 371 Figure 12.6 A photograph showing the dial gauges and proving ring in place (Courtesy of Braja M. Das, Henderson, Nevada) Figure 12.7 shows a typical plot of shear stress and change in the height of the specimen against shear displacement for dry loose and dense sands. These observations were obtained from a strain-controlled test. The following generalizations can be developed from Figure 12.7 regarding the variation of resisting shear stress with shear displacement: 1. In loose sand, the resisting shear stress increases with shear displacement until a failure shear stress of tf is reached. After that, the shear resistance remains approximately constant for any further increase in the shear displacement. 2. In dense sand, the resisting shear stress increases with shear displacement until it reaches a failure stress of tf . This tf is called the peak shear strength. After failure stress is attained, the resisting shear stress gradually decreases as shear displacement increases until it ﬁnally reaches a constant value called the ultimate shear strength. Since the height of the specimen changes during the application of the shear force (as shown in Figure 12.7), it is obvious that the void ratio of the sand changes (at least in the vicinity of the split of the shear box). Figure 12.8 shows the nature of variation of the void ratio for loose and dense sands with shear displacement. At large shear displacement, the void ratios of loose and dense sands become practically the same, and this is termed the critical void ratio. It is important to note that, in dry sand, s s¿ and c¿ 0 372 Chapter 12: Shear Strength of Soil Peak shear strength Dense sand Shear stress Ultimate shear strength tult tf Loose sand tf s constant Loose sand Dense sand Void ratio Change in height of specimen Compression Expansion Shear displacement Ultimate strength Peak strength Shear displacement Dense sand Loose sand Figure 12.7 Plot of shear stress and change in height of specimen against shear displacement for loose and dense dry sand (direct shear test) Shearing displacement Figure 12.8 Nature of variation of void ratio with shearing displacement Direct shear tests are repeated on similar specimens at various normal stresses. The normal stresses and the corresponding values of tf obtained from a number of tests are plotted on a graph from which the shear strength parameters are determined. Figure 12.9 shows such a plot for tests on a dry sand. The equation for the average line obtained from experimental results is tf s¿ tan f¿ (12.12) So, the friction angle can be determined as follows: f¿ tan1 a tf s¿ b (12.13) It is important to note that in situ cemented sands may show a c intercept. If the variation of the ultimate shear strength (tult) with normal stress is known, it can be plotted as shown in Figure 12.9. The average plot can be expressed as tult s tan fult (12.14) 12.5 Drained Direct Shear Test on Saturated Sand and Clay 0 10 s (lb/in2) 20 30 40 200 30 150 20 t (lb/in2) Shear stress, t (kN/m2) 373 100 10 50 fult 29 f 42 0 0 0 50 100 150 200 Effective normal stress, s (kN/m2) 250 300 Figure 12.9 Determination of shear strength parameters for a dry sand using the results of direct shear tests or fœult tan 1 a 12.5 tult b s¿ (12.15) Drained Direct Shear Test on Saturated Sand and Clay In the direct shear test arrangement, the shear box that contains the soil specimen is generally kept inside a container that can be ﬁlled with water to saturate the specimen. A drained test is made on a saturated soil specimen by keeping the rate of loading slow enough so that the excess pore water pressure generated in the soil is dissipated completely by drainage. Pore water from the specimen is drained through two porous stones. (See Figure 12.4.) Because the hydraulic conductivity of sand is high, the excess pore water pressure generated due to loading (normal and shear) is dissipated quickly. Hence, for an ordinary loading rate, essentially full drainage conditions exist. The friction angle, f, obtained from a drained direct shear test of saturated sand will be the same as that for a similar specimen of dry sand. The hydraulic conductivity of clay is very small compared with that of sand. When a normal load is applied to a clay soil specimen, a sufﬁcient length of time must elapse for full consolidation—that is, for dissipation of excess pore water pressure. For this reason, the shearing load must be applied very slowly. The test may last from two to ﬁve days. Figure 12.10 shows the results of a drained direct shear test on an overconsolidated clay. Figure 12.11 shows the plot of tf against s obtained from a number of drained direct shear tests on a normally consolidated clay and an overconsolidated clay. Note that the value of c¿ 0 for a normally consolidated clay. 374 Chapter 12: Shear Strength of Soil Peak shear strength 600 60 400 40 200 Shear stress (lb/in2) Shear stress (kN/m2) 80 20 Residual shear strength 0 0 0 1 2 3 4 Horizontal deformation (%) Dimension of specimen in the direction of shear force Figure 12.10 Results of a drained direct shear test on an overconsolidated clay [Note: Residual shear strength in clay is similar to ultimate shear strength in sand (see Figure 12.7)] Similar to the ultimate shear strength in the case of sand (Figure 12.8), at large shearing displacements, we can obtain the residual shear strength of clay (tr) in a drained test. This is shown in Figure 12.10. Figure 12.11 shows the plot of tr versus s. The average plot will pass through the origin and can be expressed as Shear stress at failure tr s tan fr f c f Effective normal stress, s Overconsolidated clay tf c s tan f (c 0) Normally consolidated clay tf s tan f (c 0) Residual strength plot tr s tan fr Figure 12.11 Failure envelope for clay obtained from drained direct shear tests 12.5 Drained Direct Shear Test on Saturated Sand and Clay 375 or fœr tan 1 a tr b s¿ (12.16) The drained angle of friction, f, of normally consolidated clays generally decreases with the plasticity index of soil. This fact is illustrated in Figure 12.12 for a number of clays from data reported by Kenney (1959). Although the data are scattered considerably, the general pattern seems to hold. Skempton (1964) provided the results of the variation of the residual angle of friction, fr, of a number of clayey soils with the clay-size fraction (2 m) present. The following table shows a summary of these results. Soil Selset Wiener Tegel Jackfield Oxford clay Jari London clay Walton’s Wood Weser-Elbe Little Belt Biotite Clay-size fraction (%) Residual friction angle, fr (deg) 17.7 22.8 35.4 41.9 46.5 54.9 67 63.2 77.2 100 29.8 25.1 19.1 16.3 18.6 16.3 13.2 9.3 11.2 7.5 1.0 0.8 Sin f 0.6 0.4 0.2 0.0 5 10 15 20 30 40 Plasticity index, PI (%) Undisturbed soil 60 80 100 150 Remolded soil Figure 12.12 Variation of sin with plasticity index for a number of soils (After Kenney, 1959. With permission from ASCE.) 376 Chapter 12: Shear Strength of Soil 12.6 General Comments on Direct Shear Test The direct shear test is simple to perform, but it has some inherent shortcomings. The reliability of the results may be questioned because the soil is not allowed to fail along the weakest plane but is forced to fail along the plane of split of the shear box. Also, the shear stress distribution over the shear surface of the specimen is not uniform. Despite these shortcomings, the direct shear test is the simplest and most economical for a dry or saturated sandy soil. In many foundation design problems, one must determine the angle of friction between the soil and the material in which the foundation is constructed (Figure 12.13). The foundation material may be concrete, steel, or wood. The shear strength along the surface of contact of the soil and the foundation can be given as tf caœ s¿ tan dœ (12.17) where caœ adhesion dœ effective angle of friction between the soil and the foundation material Note that the preceding equation is similar in form to Eq. (12.3). The shear strength parameters between a soil and a foundation material can be conveniently determined by a direct shear test. This is a great advantage of the direct shear test. The foundation material can be placed in the bottom part of the direct shear test box and then the soil can be placed above it (that is, in the top part of the box), as shown in Figure 12.14, and the test can be conducted in the usual manner. Figure 12.15 shows the results of direct shear tests conducted in this manner with a quartz sand and concrete, wood, and steel as foundation materials, with s 100 kN/m2 (14.5 lb/in.2). It was mentioned brieﬂy in Section 12.1 [related to Eq. (12.1)] that Mohr’s failure envelope is curvilinear in nature, and Eq. (12.2) is only an approximation. This fact should be kept in mind when considering problems at higher conﬁning pressures. Figure 12.16 shows the decrease of f and d with the increase of normal stress (s) for the same materials discussed in Figure 12.15. This can be explained by referring to Figure 12.17, which shows a curved Mohr’s failure envelope. If a direct shear test is conducted with s sœ112, the shear strength will be tf (1) . So, d1œ tan1 c tf112 sœ112 d t Interface t Foundation material Soil Figure 12.13 Interface of a foundation material and soil Normal force Shear force Soil Foundation material Figure 12.14 Direct shear test to determine interface friction angle Relative density, Dr (%) 75 50 25 100 1.2 0 Normal stress s 100 kN/m2 (14.5 lb/in2) 50 1.0 40 Friction angle, d or f () tan (f or d) 0.8 0.6 0.4 30 20 10 0.2 emin 0.51 emax 0.716 0.0 0 2.0 1.9 1.8 1.7 1.6 1.5 1.4 1.3 Sand Concrete 0.5 1.0 2.0 4.0 6.0 8.0 10 Normal stress, s ( 10 2 kN/m2) 1/e Wood Steel Figure 12.15 Variation of tan f and tan d with 1/e [Note: e void ratio, s 100 kN/m2 (14.5 lb/in.2), quartz sand] (After Acar, Durgunoglu, and Tumay, 1982. With permission from ASCE.) Sand Concrete Wood Steel Figure 12.16 Variation of f and d with s (Note: Relative density 45%; quartz sand) (After Acar, Durgunoglu, and Tumay, 1982. With permission from ASCE.) 377 378 Chapter 12: Shear Strength of Soil Mohrs fail ure envelope in sand Shear stress, t [tf (2) , s(2 )] )] [tf(1), s(1 d1 d2 Figure 12.17 Curvilinear nature of Mohr’s failure envelope in sand Normal stress, s This is shown in Figure 12.17. In a similar manner, if the test is conducted with s sœ122, then d¿ d2œ tan1 c tf122 sœ122 d œ As can be seen from Figure 12.17, d2œ d1œ since s2œ s112. Keeping this in mind, it must be realized that the values of f given in Table 12.1 are only the average values. Example 12.1 Following are the results of four drained direct shear tests on an overconsolidated clay: • Diameter of specimen 50 mm • Height of specimen 25 mm Test no. Normal force, N (N) Shear force at failure, Speak (N) Residual shear force, Sresidual (N) 1 2 3 4 150 250 350 550 157.5 199.9 257.6 363.4 44.2 56.6 102.9 144.5 Determine the relationships for peak shear strength (tf) and residual shear strength (tr). 12.6 General Comments on Direct Shear Test 379 Solution 50 2 b 0.0019634 m2. Now the following Area of the specimen 1A2 1p/4 2 a 1000 table can be prepared. Test no. Normal force, N (N) Normal stress, Sⴕ (kN/m2) Peak shear force, Speak (N) 1 2 3 4 150 250 350 550 76.4 127.3 178.3 280.1 157.5 199.9 257.6 363.4 A (kN/m2) Residual shear force, Sresidual (N) 80.2 101.8 131.2 185.1 44.2 56.6 102.9 144.5 Tf ⴝ Speak Tr ⴝ Sresidual A (kN/m2) 22.5 28.8 52.4 73.6 The variations of tf and tr with s are plotted in Figure 12.18. From the plots, we ﬁnd that Peak strength: tf 1kN/m2 2 40 ⴙ Sⴕ tan 27 Residual strength: tr 1kN/m2 2 Sⴕ tan 14.6 (Note: For all overconsolidated clays, the residual shear strength can be expressed as tr s¿ tan frœ where frœ effective residual friction angle.) 300 Shear stress, t (kN/m2) 250 200 150 tf versus s 100 27 f 50 tr versus s c 40 kN/m2 fr 14.6 0 0 50 100 150 200 250 Effective normal stress, s (kN/m2) 300 350 Figure 12.18 Variations of tf and tr with s ■ 380 Chapter 12: Shear Strength of Soil 12.7 Triaxial Shear Test (General) The triaxial shear test is one of the most reliable methods available for determining shear strength parameters. It is used widely for research and conventional testing. A diagram of the triaxial test layout is shown in Figure 12.19. In this test, a soil specimen about 36 mm (1.4 in.) in diameter and 76 mm (3 in.) long generally is used. The specimen is encased by a thin rubber membrane and placed inside a plastic cylindrical chamber that usually is ﬁlled with water or glycerine. The specimen is subjected to a conﬁning pressure by compression of the ﬂuid in the chamber. (Note: Air is sometimes used as a compression medium.) To cause shear failure in the specimen, one must apply axial stress through a vertical loading ram (sometimes called deviator stress). This stress can be applied in one of two ways: 1. Application of dead weights or hydraulic pressure in equal increments until the specimen fails. (Axial deformation of the specimen resulting from the load applied through the ram is measured by a dial gauge.) 2. Application of axial deformation at a constant rate by means of a geared or hydraulic loading press. This is a strain-controlled test. The axial load applied by the loading ram corresponding to a given axial deformation is measured by a proving ring or load cell attached to the ram. Axial load Loading ram Air release valve Air Rubber ring Air Top cap Flexible tube Pressure gauge Sealing ring To cell pressure control Connections for drainage or pore pressure measurement Water Porous disc Specimen enclosed in a rubber membrane Figure 12.19 Diagram of triaxial test equipment (After Bishop and Bjerrum, 1960. With permission from ASCE.) 12.8 Consolidated-Drained Triaxial Test 381 Connections to measure drainage into or out of the specimen, or to measure pressure in the pore water (as per the test conditions), also are provided. The following three standard types of triaxial tests generally are conducted: 1. Consolidated-drained test or drained test (CD test) 2. Consolidated-undrained test (CU test) 3. Unconsolidated-undrained test or undrained test (UU test) The general procedures and implications for each of the tests in saturated soils are described in the following sections. 12.8 Consolidated-Drained Triaxial Test In the CD test, the saturated specimen ﬁrst is subjected to an all around conﬁning pressure, s3 , by compression of the chamber ﬂuid (Figure 12.20a). As conﬁning pressure is applied, the pore water pressure of the specimen increases by uc (if drainage is prevented). This increase in the pore water pressure can be expressed as a nondimensional parameter in the form B uc s3 (12.18) where B Skempton’s pore pressure parameter (Skempton, 1954). For saturated soft soils, B is approximately equal to 1; however, for saturated stiff soils, the magnitude of B can be less than 1. Black and Lee (1973) gave the theoretical values of B for various soils at complete saturation. These values are listed in Table 12.2. Now, if the connection to drainage is opened, dissipation of the excess pore water pressure, and thus consolidation, will occur. With time, uc will become equal to 0. In saturated soil, the change in the volume of the specimen (Vc) that takes place during consolidation can be obtained from the volume of pore water drained (Figure 12.21a). Next, the deviator stress, sd , on the specimen is increased very slowly (Figure 12.20b). The drainage connection is kept open, and the slow rate of deviator stress application allows complete dissipation of any pore water pressure that developed as a result (ud 0). sd s3 s3 s3 uc 0 s3 s3 s3 ud 0 s3 sd (a) (b) s3 Figure 12.20 Consolidated-drained triaxial test: (a) specimen under chamberconﬁning pressure; (b) deviator stress application Table 12.2 Theoretical Values of B at Complete Saturation Theoretical value Type of soil 0.9998 0.9988 0.9877 0.9130 Compression Expansion Vc Normally consolidated soft clay Lightly overconsolidated soft clays and silts Overconsolidated stiff clays and sands Very dense sands and very stiff clays at high conﬁning pressures Time sd (sd )f (sd )f Axial strain (b) (c) Vd Compression Expansion Axial strain Compression Expansion Vd sd (a) Axial strain Axial strain (d) (e) Figure 12.21 Consolidated-drained triaxial test: (a) volume change of specimen caused by chamber-conﬁning pressure; (b) plot of deviator stress against strain in the vertical direction for loose sand and normally consolidated clay; (c) plot of deviator stress against strain in the vertical direction for dense sand and overconsolidated clay; (d) volume change in loose sand and normally consolidated clay during deviator stress application; (e) volume change in dense sand and overconsolidated clay during deviator stress application 382 12.8 Consolidated-Drained Triaxial Test 383 A typical plot of the variation of deviator stress against strain in loose sand and normally consolidated clay is shown in Figure 12.21b. Figure 12.21c shows a similar plot for dense sand and overconsolidated clay. The volume change, Vd, of specimens that occurs because of the application of deviator stress in various soils is also shown in Figures 12.21d and 12.21e. Because the pore water pressure developed during the test is completely dissipated, we have Total and effective confining stress s3 s3œ and Total and effective axial stress at failure s3 1¢sd 2 f s1 s1œ In a triaxial test, s1œ is the major principal effective stress at failure and s3œ is the minor principal effective stress at failure. Several tests on similar specimens can be conducted by varying the conﬁning pressure. With the major and minor principal stresses at failure for each test the Mohr’s circles can be drawn and the failure envelopes can be obtained. Figure 12.22 shows the type of effective stress failure envelope obtained for tests on sand and normally consolidated clay. The coordinates of the point of tangency of the failure envelope with a Mohr’s circle (that is, point A) give the stresses (normal and shear) on the failure plane of that test specimen. For normally consolidated clay, referring to Figure 12.22 sin f¿ s1 u 45 f 2 Effective stress failure envelope tf s tan f u s3 f Shear stress s3 AO¿ OO¿ B s1 A 2u O s3 s3 2u O s1 s1 Normal stress (sd )f (sd )f Figure 12.22 Effective stress failure envelope from drained tests on sand and normally consolidated clay 384 Chapter 12: Shear Strength of Soil or sœ1 sœ3 b 2 sin f¿ sœ1 sœ3 b a 2 a f¿ sin 1 a sœ1 sœ3 b sœ1 sœ3 (12.19) Also, the failure plane will be inclined at an angle of u 45 f/2 to the major principal plane, as shown in Figure 12.22. Overconsolidation results when a clay initially is consolidated under an all-around chamber pressure of sc ( scœ ) and is allowed to swell by reducing the chamber pressure to s3 ( s3œ ). The failure envelope obtained from drained triaxial tests of such overconsolidated clay specimens shows two distinct branches (ab and bc in Figure 12.23). The portion ab has a ﬂatter slope with a cohesion intercept, and the shear strength equation for this branch can be written as tf c¿ s¿ tan f1œ (12.20) The portion bc of the failure envelope represents a normally consolidated stage of soil and follows the equation tf s tan f. If the triaxial test results of two overconsolidated soil specimens are known, the magnitudes of f1 and c can be determined as follows. From Eq. (12.8), for Specimen 1: s1(1) s3(1) tan2 (45 + f1/2) + 2c tan(45 + f1/2) (12.21) And, for Specimen 2: s1(2) s3(2) tan2 (45 + f1/2) + 2c tan(45 + f1/2) (12.22) or s1(1) s1(2) [s3(1) s3(2)] tan2 (45 + f1/2) Overconsolidated c f Shear stress a Normally consolidated b f1 c s3 s3 s1 s1 sc Normal stress Figure 12.23 Effective stress failure envelope for overconsolidated clay 12.8 Consolidated-Drained Triaxial Test 385 Hence, f1œ 2 e tan 1 c œ œ s1122 s1112 œ s3112 œ s3122 d 0.5 45° f (12.23) Once the value of f1 is known, we can obtain c as sœ1112 sœ3112 tan2 a 45 c¿ 2 tan a 45 fœ1 b 2 fœ1 b 2 (12.24) A consolidated-drained triaxial test on a clayey soil may take several days to complete. This amount of time is required because deviator stress must be applied very slowly to ensure full drainage from the soil specimen. For this reason, the CD type of triaxial test is uncommon. Example 12.2 A consolidated-drained triaxial test was conducted on a normally consolidated clay. The results are as follows: • s3 16 lb/in.2 • (sd )f 25 lb/in.2 Determine a. Angle of friction, f b. Angle u that the failure plane makes with the major principal plane Solution For normally consolidated soil, the failure envelope equation is tf s¿ tan f¿ 1because c¿ 0 2 For the triaxial test, the effective major and minor principal stresses at failure are as follows: s1œ s1 s3 1¢sd 2 f 16 25 41 lb/in.2 and s3œ s3 16 lb/in.2 Part a The Mohr’s circle and the failure envelope are shown in Figure 12.24. From Eq. (12.19), sin fœ s1œ s3œ 41 16 0.438 œ œ s1 s3 41 16 or f¿ 26ⴗ 386 Chapter 12: Shear Strength of Soil s1 Shear stress u s3 s3 Effective stress failure envelope f B s1 2u O s3 16 lb/in2 s1 41 lb/in2 A Normal stress Figure 12.24 Mohr’s circle and failure envelope for a normally consolidated clay Part b From Eq. (12.4), u 45 f¿ 26° 45° 58ⴗ 2 2 ■ Example 12.3 Refer to Example 12.2. a. Find the normal stress s and the shear stress tf on the failure plane. b. Determine the effective normal stress on the plane of maximum shear stress. Solution Part a From Eqs. (10.8) and (10.9), s¿ 1on the failure plane2 s1œ s3œ s1œ s3œ cos 2u 2 2 and tf sœ1 sœ3 sin 2u 2 Substituting the values of s1 41 lb/in.2, s3 16 lb/in.2, and u 58° into the preceding equations, we get s¿ 41 16 41 16 cos 12 58 2 23.0 lb/in.2 2 2 12.8 Consolidated-Drained Triaxial Test 387 and tf 41 16 sin 12 582 11.2 lb/in.2 2 Part b From Eq. (10.9), it can be seen that the maximum shear stress will occur on the plane with u 45°. From Eq. (10.8), s1œ s3œ s1œ s3œ cos 2u 2 2 Substituting u 45° into the preceding equation gives s¿ s¿ 41 16 41 16 cos 90 28.5 lb/in.2 2 2 ■ Example 12.4 The equation of the effective stress failure envelope for normally consolidated clayey soil is tf s tan 30. A drained triaxial test was conducted with the same soil at a chamberconﬁning pressure of 10 lb/in.2 Calculate the deviator stress at failure. Solution For normally consolidated clay, c 0. Thus, from Eq. (12.8), s1œ s3œ tan2 a 45 f¿ b 2 f¿ 30° s1œ 10 tan2 a 45 30 b 30 lb/in.2 2 So, 1¢sd 2 f s1œ s3œ 30 10 20 lb/in.2 Example 12.5 The results of two drained triaxial tests on a saturated clay follow: Specimen I: s3 70 kN/m2 1¢sd 2 f 130 kN/m2 ■ 388 Chapter 12: Shear Strength of Soil Specimen II: s3 160 kN/m2 1¢sd 2 f 223.5 kN/m2 Determine the shear strength parameters. Solution Refer to Figure 12.25. For Specimen I, the principal stresses at failure are s3œ s3 70 kN/m2 and s1œ s1 s3 1 ¢sd 2 f 70 130 200 kN/m2 Similarly, the principal stresses at failure for Specimen II are s3œ s3 160 kN/m2 and s1œ s1 s3 1¢sd 2 f 160 223.5 383.5 kN/m2 Now, from Eq. (12.23), 2 e tan 1 c œ œ s11II2 s11I2 œ œ s31I2 s31II2 d 0.5 45° f 2 e tan1 c 200 383.5 0.5 d 45° f 20° 70 160 Shear stress (kN/m2 ) f1œ f c 70 160 200 383.5 Normal stress, s (kN/m2 ) Figure 12.25 Effective stress failure envelope and Mohr’s circles for Specimens I and II 12.9 Consolidated-Undrained Triaxial Test 389 Again, from Eq. (12.24), œ œ s31I2 tan2 a 45 s11I2 c¿ 12.9 f1œ b 2 tan a 45 2 f1œ b 2 200 70 tan2 a 45 20 b 2 20 2 tan a 45 b 2 20 kN/m2 ■ Consolidated-Undrained Triaxial Test The consolidated-undrained test is the most common type of triaxial test. In this test, the saturated soil specimen is ﬁrst consolidated by an all-around chamber ﬂuid pressure, s3 , that results in drainage (Figures 12.26a and 12.26b). After the pore water pressure generated by the application of conﬁning pressure is dissipated, the deviator stress, sd , on the specimen is increased to cause shear failure (Figure 12.26c). During this phase of the test, the drainage line from the specimen is kept closed. Because drainage is not permitted, the pore water pressure, ud , will increase. During the test, simultaneous measurements of sd and ud are made. The increase in the pore water pressure, ud , can be expressed in a nondimensional form as A ¢ud ¢sd (12.25) where A Skempton’s pore pressure parameter (Skempton, 1954). The general patterns of variation of sd and ud with axial strain for sand and clay soils are shown in Figures 12.26d through 12.26g. In loose sand and normally consolidated clay, the pore water pressure increases with strain. In dense sand and overconsolidated clay, the pore water pressure increases with strain to a certain limit, beyond which it decreases and becomes negative (with respect to the atmospheric pressure). This decrease is because of a tendency of the soil to dilate. Unlike the consolidated-drained test, the total and effective principal stresses are not the same in the consolidated-undrained test. Because the pore water pressure at failure is measured in this test, the principal stresses may be analyzed as follows: • • • Major principal stress at failure 1total2: s3 1¢sd 2 f s1 Minor principal stress at failure 1total2: s3 Major principal stress at failure 1effective2 : s1 1¢ud 2 f s1œ œ • Minor principal stress at failure 1effective2: s3 1¢ud 2 f s3 In these equations, (ud)f pore water pressure at failure. The preceding derivations show that s1 s3 s1œ s3œ 390 Chapter 12: Shear Strength of Soil sd s3 s3 uc 0 (sd )f s3 Axial strain (d) (a) sd Compression Expansion Vc s3 (sd )f Axial strain (e) Time (b) ud sd s3 Axial strain (f) s3 s3 sd (c) ud s3 Axial strain (g) Figure 12.26 Consolidated undrained test: (a) specimen under chamber conﬁning pressure; (b) volume change in specimen caused by conﬁning pressure; (c) deviator stress application; (d) deviator stress against axial strain for loose sand and normally consolidated clay; (e) deviator stress against axial strain for dense sand and overconsolidated clay; (f) variation of pore water pressure with axial strain for loose sand and normally consolidated clay; (g) variation of pore water pressure with axial strain for dense sand and overconsolidated clay Tests on several similar specimens with varying conﬁning pressures may be conducted to determine the shear strength parameters. Figure 12.27 shows the total and effective stress Mohr’s circles at failure obtained from consolidated-undrained triaxial tests in sand and normally consolidated clay. Note that A and B are two total stress Mohr’s circles obtained from two tests. C and D are the effective stress Mohr’s circles corresponding to total stress circles A and B, respectively. The diameters of circles A and C are the same; similarly, the diameters of circles B and D are the same. 12.9 Consolidated-Undrained Triaxial Test 391 Effective stress failure envelope tf s tan f f Total stress failure envelope tf s tan f Shear stress f s3 s3 (ud )f C s1 A s1 D B Normal stress (ud )f Figure 12.27 Total and effective stress failure envelopes for consolidated undrained triaxial tests. (Note: The ﬁgure assumes that no back pressure is applied.) In Figure 12.27, the total stress failure envelope can be obtained by drawing a line that touches all the total stress Mohr’s circles. For sand and normally consolidated clays, this will be approximately a straight line passing through the origin and may be expressed by the equation tf s tan f (12.26) where s total stress f the angle that the total stress failure envelope makes with the normal stress axis, also known as the consolidated-undrained angle of shearing resistance Equation (12.26) is seldom used for practical considerations. Similar to Eq. (12.19), for sand and normally consolidated clay, we can write f sin 1 a s1 s3 b s1 s3 (12.27) and f¿ sin 1 a sin 1 e sin 1 c s1œ s3œ b s1œ s3œ 3s1 1 ¢ud 2 f 4 3 s3 1¢ud 2 f 4 3s1 1 ¢ud 2 f 4 3 s3 1¢ud 2 f 4 s1 s3 d s1 s3 21¢ud 2 f f (12.28) Again referring to Figure 12.27, we see that the failure envelope that is tangent to all the effective stress Mohr’s circles can be represented by the equation tf s tan f, which is the same as that obtained from consolidated-drained tests (see Figure 12.22). 392 Chapter 12: Shear Strength of Soil Shear stress c tf s tan f f tf c s tan f1 b a f1 c s3 s1 Normal stress Figure 12.28 Total stress failure envelope obtained from consolidated-undrained tests in over-consolidated clay In overconsolidated clays, the total stress failure envelope obtained from consolidated-undrained tests will take the shape shown in Figure 12.28. The straight line ab is represented by the equation tf c s tan f1 (12.29) and the straight line bc follows the relationship given by Eq. (12.26). The effective stress failure envelope drawn from the effective stress Mohr’s circles will be similar to that shown in Figure 12.23. Consolidated-drained tests on clay soils take considerable time. For this reason, consolidated-undrained tests can be conducted on such soils with pore pressure measurements to obtain the drained shear strength parameters. Because drainage is not allowed in these tests during the application of deviator stress, they can be performed quickly. Skempton’s pore water pressure parameter A was deﬁned in Eq. (12.25). At failure, the parameter A can be written as A Af 1 ¢ud 2 f 1¢sd 2 f (12.30) The general range of Af values in most clay soils is as follows: • • Normally consolidated clays: 0.5 to 1 Overconsolidated clays: 0.5 to 0 Table 12.3 gives the values of Af for some normally consolidated clays as obtained by the Norwegian Geotechnical Institute. Laboratory triaxial tests of Simons (1960) on Oslo clay, Weald clay, and London clay showed that Af becomes approximately zero at an overconsolidation value of about 3 or 4 (Figure 12.29). Table 12.3 Triaxial Test Results for Some Normally Consolidated Clays Obtained by the Norwegian Geotechnical Institute* Location Seven Sisters, Canada Sarpborg Lilla Edet, Sweden Fredrikstad Fredrikstad Lilla Edet, Sweden Gö ta River, Sweden Gö ta River, Sweden Oslo Trondheim Drammen a Plastic limit Liquidity index Sensitivity 127 69 68 59 57 63 60 60 48 36 33 35 28 30 22 22 30 27 30 25 20 18 0.28 0.68 1.32 0.58 0.63 1.58 1.30 1.50 0.87 0.50 1.08 5 50 5 6 50 12 40 4 2 8 a Drained friction angle, F (deg) 19 25.5 26 28.5 27 23 28.5 24 31.5 34 28 Af 0.72 1.03 1.10 0.87 1.00 1.02 1.05 1.05 1.00 0.75 1.18 After Bjerrum and Simons, 1960. With permission from ASCE. See Section 12.13 for the deﬁnition of sensitivity 1.4 1.2 0.8 0.4 Oslo clay Af * Liquid limit Weald clay 0.0 Oslo and London clay 0.4 0.8 1 2 4 6 8 10 20 40 60 80 100 Overconsolidation ratio, OCR Figure 12.29 Variation of Af with overconsolidation ratio for three clays (Based on Simon, 1960) 393 394 Chapter 12: Shear Strength of Soil Example 12.6 A specimen of saturated sand was consolidated under an all-around pressure of 12 lb/in.2 The axial stress was then increased and drainage was prevented. The specimen failed when the axial deviator stress reached 9.1 lb/in.2 The pore water pressure at failure was 6.8 lb/in.2 Determine a. Consolidated-undrained angle of shearing resistance, f b. Drained friction angle, f Solution Part a For this case, s3 12 lb/in.2, s1 12 + 9.1 21.1 lb/in.2, and (ud)f 6.8 lb/in.2. The total and effective stress failure envelopes are shown in Figure 12.30. From Eq. (12.27), f sin 1 a s1 s3 21.1 12 b 16° b sin 1 a s1 s3 21.1 12 Part b From Eq. (12.28), f¿ sin 1 c s1 s3 21.1 12 d 27.8° d sin 1 c s1 s3 21¢ud 2 f 21.1 12 12 2 16.82 f Shear stress (lb/in2) Effective stress failure envelope Total stress failure envelope f B B 5.2 A 12 14.3 A 21.1 Normal stress (lb/in2) Figure 12.30 Failure envelopes and Mohr’s circles for a saturated sand ■ 12.10 Unconsolidated-Undrained Triaxial Test 395 Example 12.7 Refer to the soil specimen described in Example 12.6. What would be the deviator stress at failure, (sd )f , if a drained test was conducted with the same chamber allaround pressure (that is, 12 lb/in.2 )? Solution From Eq. (12.8) (with c 0), s1œ s3œ tan2 a 45 f¿ b 2 s3œ 12 lb/in.2 and f 27.8° (from Example 12.6). So, s1œ 12 tan2 a 45 27.8 b 33 lb/in.2 2 1¢sd 2 f s1œ s3œ 33 12 21 lb/in.2 12.10 ■ Unconsolidated-Undrained Triaxial Test In unconsolidated-undrained tests, drainage from the soil specimen is not permitted during the application of chamber pressure s3 . The test specimen is sheared to failure by the application of deviator stress, sd , and drainage is prevented. Because drainage is not allowed at any stage, the test can be performed quickly. Because of the application of chamber conﬁning pressure s3 , the pore water pressure in the soil specimen will increase by uc . A further increase in the pore water pressure (ud) will occur because of the deviator stress application. Hence, the total pore water pressure u in the specimen at any stage of deviator stress application can be given as u uc ¢ud (12.31) From Eqs. (12.18) and (12.25), uc Bs3 and ud Asd , so u Bs3 A¢sd Bs3 A1s1 s3 2 (12.32) This test usually is conducted on clay specimens and depends on a very important strength concept for cohesive soils if the soil is fully saturated. The added axial stress at failure (sd)f is practically the same regardless of the chamber conﬁning pressure. This property is shown in Figure 12.31. The failure envelope for the total stress Mohr’s circles becomes a horizontal line and hence is called a f 0 condition. From Eq. (12.9) with f 0, we get tf c cu (12.33) where cu is the undrained shear strength and is equal to the radius of the Mohr’s circles. Note that the f 0 concept is applicable to only saturated clays and silts. Shear stress 396 Chapter 12: Shear Strength of Soil Total stress Mohr’s circles at failure Failure envelope f 0 cu s3 s3 s1 s3 Normal stress s1 s1 Figure 12.31 Total stress Mohr’s circles and failure envelope (f 0) obtained from unconsolidated-undrained triaxial tests on fully saturated cohesive soil The reason for obtaining the same added axial stress (sd)f regardless of the conﬁning pressure can be explained as follows. If a clay specimen (No. I) is consolidated at a chamber pressure s3 and then sheared to failure without drainage, the total stress conditions at failure can be represented by the Mohr’s circle P in Figure 12.32. The pore pressure developed in the specimen at failure is equal to (ud)f . Thus, the major and minor principal effective stresses at failure are, respectively, s1œ 3s3 1¢sd 2 f 4 1¢ud 2 f s1 1¢ud 2 f and s3œ s3 1 ¢ud 2 f Q is the effective stress Mohr’s circle drawn with the preceding principal stresses. Note that the diameters of circles P and Q are the same. Shear stress f Total stress Mohr’s circle at failure P Q s3 s1 s3 (sd )f (ud )f Figure 12.32 The f 0 concept f R s1 Normal stress (sd )f (sd )f s3 uc 12.11 Unconﬁned Compression Test on Saturated Clay 397 Now let us consider another similar clay specimen (No. II) that has been consolidated under a chamber pressure s3 with initial pore pressure equal to zero. If the chamber pressure is increased by s3 without drainage, the pore water pressure will increase by an amount uc . For saturated soils under isotropic stresses, the pore water pressure increase is equal to the total stress increase, so uc s3 (B 1). At this time, the effective conﬁning pressure is equal to s3 s3 uc s3 s3 s3 s3 . This is the same as the effective conﬁning pressure of Specimen I before the application of deviator stress. Hence, if Specimen II is sheared to failure by increasing the axial stress, it should fail at the same deviator stress (sd)f that was obtained for Specimen I. The total stress Mohr’s circle at failure will be R (see Figure 12.32). The added pore pressure increase caused by the application of (sd)f will be (ud)f . At failure, the minor principal effective stress is 3 1s3 ¢s3 2 4 3 ¢uc 1¢ud 2 f 4 s3 1¢ud 2 f s3œ and the major principal effective stress is 3s3 ¢s3 1¢sd 2 f 4 3 ¢uc 1¢ud 2 f 4 3 s3 1¢sd 2 f 4 1¢ud 2 f s1 1¢ud 2 f s1œ Thus, the effective stress Mohr’s circle will still be Q because strength is a function of effective stress. Note that the diameters of circles P, Q, and R are all the same. Any value of s3 could have been chosen for testing Specimen II. In any case, the deviator stress (sd)f to cause failure would have been the same as long as the soil was fully saturated and fully undrained during both stages of the test. 12.11 Unconﬁned Compression Test on Saturated Clay The unconﬁned compression test is a special type of unconsolidated-undrained test that is commonly used for clay specimens. In this test, the conﬁning pressure s3 is 0. An axial load is rapidly applied to the specimen to cause failure. At failure, the total minor principal stress is zero and the total major principal stress is s1 (Figure 12.33). Because the undrained shear strength is independent of the conﬁning pressure as long as the soil is fully saturated and fully undrained, we have tf qu s1 cu 2 2 (12.34) where qu is the unconﬁned compression strength. Table 12.4 gives the approximate consistencies of clays on the basis of their unconﬁned compression strength. A photograph of 398 Chapter 12: Shear Strength of Soil Shear stress s1 s1 f0 Total stress Mohr’s circle at failure cu s1 qu s3 0 Normal stress Figure 12.33 Unconﬁned compression test Table 12.4 General Relationship of Consistency and Unconﬁned Compression Strength of Clays qu Consistency Very soft Soft Medium Stiff Very stiff Hard 2 kN/m ton /ft2 0–25 25–50 50–100 100–200 200–400 400 0–0.25 0.25–0.5 0.5–1 1–2 2–4 4 unconﬁned compression test equipment is shown in Figure 12.34. Figures 12.35 and 12.36 show the failure in two specimens—one by shear and one by bulging—at the end of unconﬁned compression tests. Theoretically, for similar saturated clay specimens, the unconﬁned compression tests and the unconsolidated-undrained triaxial tests should yield the same values of cu . In practice, however, unconﬁned compression tests on saturated clays yield slightly lower values of cu than those obtained from unconsolidated-undrained tests. 12.12 Empirical Relationships Between Undrained Cohesion (cu) and Effective Overburden Pressure (SOœ ) Several empirical relationships have been proposed between cu and the effective overburden pressure so. The most commonly cited relationship is that given by Skempton (1957) which can be expressed as cu1VST2 soœ 0.11 0.00371PI2 1for normally consolidated clay2 (12.35) 12.12 Empirical Relationships Between Undrained Cohesion & Effective Overburden Pressure 399 Figure 12.35 Failure by shear of an unconfined compression test specimen (Courtesy of Braja M. Das, Henderson, Nevada) Figure 12.34 Unconﬁned compression test equipment (Courtesy of ELE International) where cu(VST) undrained shear strength from vane shear test (see Section 12.15) PI plasticity index (%) Chandler (1988) suggested that the preceding relationship will hold good for overconsolidated soil with an accuracy of 25%. This does not include sensitive and fissured clays. Ladd, et al. (1977) proposed that a cu b soœ overconsolidated cu a œb so normally consolidated where OCR overconsolidation ratio. 1OCR2 0.8 (12.36) 400 Chapter 12: Shear Strength of Soil Figure 12.36 Failure by bulging of an unconfined compression test specimen (Courtesy of Braja M. Das, Henderson, Nevada) Example 12.8 An overconsolidated clay deposit located below the groundwater table has the following: • Average present effective overburden pressure 160 kN/m2 • Overconsolidation ratio 3.2 • Plasticity index 28 Estimate the average undrained shear strength of the clay [that is, cu(VST)]. Solution From Eq. (12.35), c cu1VST2 soœ d 0.11 0.00371PI2 0.11 10.0037 2 128 2 0.2136 normally consolidated 12.13 Sensitivity and Thixotropy of Clay 401 From Eq. (12.36), c cu1VST2 soœ d 1OCR2 0.8 c overconsolidated cu1VST2 soœ d 13.22 0.8 10.21362 0.542 normally consolidated Thus, cu(VST) 0.542o (0.542)(160) 86.7 kN/m2 Sensitivity and Thixotropy of Clay For many naturally deposited clay soils, the unconﬁned compression strength is reduced greatly when the soils are tested after remolding without any change in the moisture content, as shown in Figure 12.37. This property of clay soils is called sensitivity. The degree of sensitivity may be deﬁned as the ratio of the unconﬁned compression strength in an undisturbed state to that in a remolded state, or St qu1undisturbed2 (12.37) qu1remolded2 The sensitivity ratio of most clays ranges from about 1 to 8; however, highly ﬂocculent marine clay deposits may have sensitivity ratios ranging from about 10 to 80. Some clays turn to viscous ﬂuids upon remolding. These clays are found mostly in the previously glaciated areas of North America and Scandinavia. Such clays are referred to as quick clays. Rosenqvist (1953) classiﬁed clays on the basis of their sensitivity. This general classiﬁcation is shown in Figure 12.38. s1 12.13 ■ Undisturbed Remolded qu qu Axial strain Figure 12.37 Unconﬁned compression strength for undisturbed and remolded clay 402 Chapter 12: Shear Strength of Soil Extra quick 64 Very quick Sensitivity, St (log scale) 32 Medium quick 16 Slightly quick 8 Very sensitive 4 Medium sensitive 2 Slightly sensitive 1 Insensitive Figure 12.38 Classiﬁcation of clays based on sensitivity The loss of strength of clay soils from remolding is caused primarily by the destruction of the clay particle structure that was developed during the original process of sedimentation. If, however, after remolding, a soil specimen is kept in an undisturbed state (that is, without any change in the moisture content), it will continue to gain strength with time. This phenomenon is referred to as thixotropy. Thixotropy is a time-dependent, reversible process in which materials under constant composition and volume soften when remolded. This loss of strength is gradually regained with time when the materials are allowed to rest. This phenomenon is illustrated in Figure 12.39a. Most soils, however, are partially thixotropic—that is, part of the strength loss caused by remolding is never regained with time. The nature of the strength-time variation for partially thixotropic materials is shown in Figure 12.39b. For soils, the difference between the undisturbed strength and the strength after thixotropic hardening can be attributed to the destruction of the clay-particle structure that was developed during the original process of sedimentation. Seed and Chan (1959) conducted several tests on three compacted clays with a water content near or below the plastic limit to study the thixotropic strength regain 12.14 Strength Anisotropy in Clay St qu (undisturbed) qu (remolded) en in g Ha rd Remolding en in g Ha rd g Remolding Initial undisturbed strength Ha r de ni n Remolding qu (undisturbed) Strength 403 Remolded strength qu (remolded) Time (a) rde ni n g g Ha Remolding Ha r de ni n g Remolding Ha r de ni n Strength after thixotropic hardening Remolding Strength Initial undisturbed strength Remolded strength Time (b) Figure 12.39 Behavior of (a) thixotropic material; (b) partially thixotropic material characteristics of the clays. The results of these tests are shown in Figure 12.40. Note that in Figure 12.40, Thixotropic strength ratio 12.14 cu 1at time t after compaction2 cu 1at time t 0 after compaction2 (12.38) Strength Anisotropy in Clay The unconsolidated-undrained shear strength of some saturated clays can vary, depending on the direction of load application; this variation is referred to as anisotropy with respect to strength. Anisotropy is caused primarily by the nature of the deposition of the cohesive soils, and subsequent consolidation makes the clay particles orient perpendicular to the 404 Chapter 12: Shear Strength of Soil 1.6 Thixotropic strength ratio 1.5 1.4 1.3 1.2 1.1 1.0 10 100 1,000 Time (min) 10,000 100,000 Vicksburg silty clay PL 23; 19.5% Pittsburgh sandy clay PL 20; 17.4% Friant-Kern clay PL 35; 22% Figure 12.40 Thixotropic strength increase with time for three clays (Based on Seed and Chan, 1959) direction of the major principal stress. Parallel orientation of the clay particles can cause the strength of clay to vary with direction. Figure 12.41 shows an element of saturated clay in a deposit with the major principal stress making an angle a with respect to the horizontal. For anisotropic clays, the magnitude of cu is a function of a. s1 a s3 Saturated clay Figure 12.41 Strength anisotropy in clay 405 12.14 Strength Anisotropy in Clay As an example, the variation of cu with a for undisturbed specimens of Winnipeg Upper Brown clay (Loh and Holt, 1974) is shown in Figure 12.42. Based on several laboratory test results, Casagrande and Carrillo (1944) proposed the following relationship for the directional variation of undrained shear strength: cu 1a2 cu 1a0°2 3cu 1a90°2 cu 1a0°2 4 sin2 a (12.39) 56 8 42 6 28 4 14 2 cu(a) (lb/in2) cu(a) (kN/m2) For normally consolidated clays, cu(a90) cu(a0) ; for overconsolidated clays, cu(a90) cu(a0) . Figure 12.43 shows the directional variation for cu(a) based on Eq. (12.39). The anisotropy with respect to strength for clays can have an important effect on various stability calculations. 0 0 0 10 20 30 40 50 60 70 80 90 a (deg) Figure 12.42 Directional variation of cu for undisturbed Winnipeg Upper Brown clay (Based on Loh and Holt, 1974) Normally consolidated anisotropic clay cu(a 90) Overconsolidated anisotropic clay Isotropic clay cu(a) cu(a 0) Figure 12.43 Graphical representation of Eq. (12.39) 406 Chapter 12: Shear Strength of Soil Vane Shear Test Fairly reliable results for the undrained shear strength, cu (f 0 concept), of very soft to medium cohesive soils may be obtained directly from vane shear tests. The shear vane usually consists of four thin, equal-sized steel plates welded to a steel torque rod (Figure 12.44). First, the vane is pushed into the soil. Then torque is applied at the top of the torque rod to rotate the vane at a uniform speed. A cylinder of soil of height h and diameter d will resist the torque until the soil fails. The undrained shear strength of the soil can be calculated as follows. If T is the maximum torque applied at the head of the torque rod to cause failure, it should be equal to the sum of the resisting moment of the shear force along the side surface of the soil cylinder (Ms) and the resisting moment of the shear force at each end (Me) (Figure 12.45): (12.40) ⎧ ⎪ ⎨ ⎪ ⎩ T Ms Me Me Two ends The resisting moment can be given as 1d/22 ⎧ ⎨ ⎩ Ms 1pdh2cu ⎧ ⎪ ⎨ ⎪ ⎩ 12.15 Surface area Moment arm (12.41) where d diameter of the shear vane h height of the shear vane For the calculation of Me, investigators have assumed several types of distribution of shear strength mobilization at the ends of the soil cylinder: 1. Triangular. Shear strength mobilization is cu at the periphery of the soil cylinder and decreases linearly to zero at the center. 2. Uniform. Shear strength mobilization is constant (that is, cu) from the periphery to the center of the soil cylinder. 3. Parabolic. Shear strength mobilization is cu at the periphery of the soil cylinder and decreases parabolically to zero at the center. T h d Figure 12.44 Diagram of vane shear test equipment 12.15 Vane Shear Test 407 Triangular mobilization of shear strength cu d 2 d 2 Me cu h Uniform mobilization of shear strength d 2 Ms d 2 Parabolic form of mobilization of shear strength cu Me d 2 d d 2 (b) (a) Figure 12.45 Derivation of Eq. (12.43): (a) resisting moment of shear force; (b) variations in shear strength-mobilization These variations in shear strength mobilization are shown in Figure 12.45b. In general, the torque, T, at failure can be expressed as T pcu c d2h d3 b d 2 4 (12.42) or cu T d3 dh b d pc 2 4 2 (12.43) where b 12 for triangular mobilization of undrained shear strength b 23 for uniform mobilization of undrained shear strength b 35 for parabolic mobilization of undrained shear strength Note that Eq. (12.43) usually is referred to as Calding’s equation. Vane shear tests can be conducted in the laboratory and in the ﬁeld during soil exploration. The laboratory shear vane has dimensions of about 13 mm (12 in.) in diameter and 25 mm (1 in.) in height. Figure 12.46 shows a photograph of laboratory vane shear test equipment. Figure 12.47 shows the ﬁeld vanes recommended by ASTM (2004). Table 12.5 gives the ASTM recommended dimensions of ﬁeld vanes. Figure 12.46 Laboratory vane shear test device (Courtesy of ELE International) Text not available due to copyright restrictions 408 12.15 Vane Shear Test 409 Text not available due to copyright restrictions According to ASTM (2004), if h/d 2, then cu 1kN/m2 2 T 1N # m2 1366 108 2d3 c (12.44) 1cm2 and cu 1lb/ft2 2 T 1lb # ft2 0.0021d3 c (12.45) 1in.2 In the ﬁeld, where considerable variation in the undrained shear strength can be found with depth, vane shear tests are extremely useful. In a short period, one can establish a reasonable pattern of the change of cu with depth. However, if the clay deposit at a given site is more or less uniform, a few unconsolidated-undrained triaxial tests on undisturbed specimens will allow a reasonable estimation of soil parameters for design work. Vane shear tests also are limited by the strength of soils in which they can be used. The undrained shear strength obtained from a vane shear test also depends on the rate of application of torque T. Figure 12.48 shows a comparison of the variation of cu with the depth obtained from ﬁeld vane shear tests, unconﬁned compression tests, and unconsolidated-undrained triaxial tests for Morgan City recent alluvium (Arman, et al., 1975). It can be seen that the vane shear test values are higher compared to the others. Bjerrum (1974) also showed that, as the plasticity of soils increases, cu obtained from vane shear tests may give results that are unsafe for foundation design. For this reason, he suggested the correction cu1design2 lcu 1vane shear2 (12.46) cu (lb/ft2) 0 500 1000 1500 2000 2500 0 0 6 2 12 4 Depth (ft) Depth (m) 18 6 24 8 30 10 36 12 0 20 40 60 80 Undrained shear strength, cu (kN/m2) Unconﬁned compression Undrained triaxial 100 120 Field vane shear Figure 12.48 Variation of cu with depth obtained from various tests for Morgan City recent alluvium (Drawn from the test results of Arman et al, 1975) 410 12.16 Other Methods for Determining Undrained Shear Strength 411 where l correction factor 1.7 0.54 log1PI2 (12.47) PI plasticity index 12.16 Other Methods for Determining Undrained Shear Strength A modiﬁed form of the vane shear test apparatus is the Torvane (Figure 12.49), which is a handheld device with a calibrated spring. This instrument can be used for determining cu for tube specimens collected from the ﬁeld during soil exploration, and it can be used in the ﬁeld. The Torvane is pushed into the soil and then rotated until the soil fails. The undrained shear strength can be read at the top of the calibrated dial. Figure 12.50 shows a pocket penetrometer, which is pushed directly into the soil. The unconﬁned compression strength (qu) is measured by a calibrated spring. This device can be used both in the laboratory and in the ﬁeld. Figure 12.49 Torvane (Courtesy of ELE International) 412 Chapter 12: Shear Strength of Soil Figure 12.50 Pocket penetrometer (Courtesy of ELE International) 12.17 Shear Strength of Unsaturated Cohesive Soils The equation relating total stress, effective stress, and pore water pressure for unsaturated soils, can be expressed as s¿ s ua x1ua uw 2 (12.48) where s¿ effective stress s total stress ua pore air pressure uw pore water pressure When the expression for s is substituted into the shear strength equation [Eq. (12.3)], which is based on effective stress parameters, we get tf c¿ 3s ua x1ua uw 2 4 tan f¿ (12.49) The values of x depend primarily on the degree of saturation. With ordinary triaxial equipment used for laboratory testing, it is not possible to determine accurately the effective stresses in unsaturated soil specimens, so the common practice is to conduct undrained triaxial tests on unsaturated specimens and measure only the total stress. Figure 12.51 shows a total stress failure envelope obtained from a number of undrained triaxial tests conducted with a given initial degree of saturation. The failure envelope is generally curved. Higher conﬁning pressure causes higher compression of the air in void spaces; thus, the solubility of void air in void water is increased. For design purposes, the curved envelope is sometimes approximated as a straight line, as shown in Figure 12.51, with an equation as follows: tf c s tan f (Note: c and f in the preceding equation are empirical constants.) (12.50) 12.17 Shear Strength of Unsaturated Cohesive Soils 413 tf c s tan f Shear stress f c Normal stress (total) Figure 12.51 Total stress failure envelope for unsaturated cohesive soils Figure 12.52 shows the variation of the total stress envelopes with change of the initial degree of saturation obtained from undrained tests on an inorganic clay. Note that for these tests the specimens were prepared with approximately the same initial dry unit weight of about 16.7 kN/m3 (106 lb/ft3). For a given total normal stress, the shear stress needed to cause failure decreases as the degree of saturation increases. When the degree of saturation reaches 100%, the total stress failure envelope becomes a horizontal line that is the same as with the f 0 concept. In practical cases where a cohesive soil deposit may become saturated because of rainfall or a rise in the groundwater table, the strength of partially saturated clay should not be used for design considerations. Instead, the unsaturated soil specimens collected from the ﬁeld must be saturated in the laboratory and the undrained strength determined. s (lb/in2) 0 160 240 320 160 1000 Inorganic clay (CL) 800 Degree of saturation 61% 600 120 80 400 74.8% 200 t (lb/in2) Shear stress, t (kN/m2) 80 40 86.8% 88.8% 0 0 500 1000 1500 Total normal stress, s (kN/m2) 0 2000 2200 Figure 12.52 Variation of the total stress failure envelope with change of initial degree of saturation obtained from undrained tests of an inorganic clay (After Casagrande and Hirschfeld, 1960. With permission from ASCE.) 414 Chapter 12: Shear Strength of Soil Stress Path Results of triaxial tests can be represented by diagrams called stress paths. A stress path is a line that connects a series of points, each of which represents a successive stress state experienced by a soil specimen during the progress of a test. There are several ways in which a stress path can be drawn. This section covers one of them. Lambe (1964) suggested a type of stress path representation that plots q against p (where p and q are the coordinates of the top of the Mohr’s circle). Thus, relationships for p and q are as follows: p¿ s1œ s3œ 2 (12.51) q¿ s1œ s3œ 2 (12.52) This type of stress path plot can be explained with the aid of Figure 12.53. Let us consider a normally consolidated clay specimen subjected to an isotropically consolidated-drained triaxial test. At the beginning of the application of deviator stress, s1œ s3œ s3 , so p¿ s3œ s3œ s3œ s3 2 (12.53) s3œ s3œ 0 2 (12.54) and q¿ F f F' a Shear stress, or q 12.18 D D B D s3 s3 O I 45 s1 s1' A s, s, or p sd (sd)f Figure 12.53 Stress path—plot of q against p for a consolidated-drained triaxial test on a normally consolidated clay 12.18 Stress Path 415 For this condition, p and q will plot as a point (that is, I in Figure 12.53). At some other time during deviator stress application, s1œ s3œ sd s3 sd ; s3œ s3 . The Mohr’s circle marked A in Figure 12.53 corresponds to this state of stress on the soil specimen. The values of p and q for this stress condition are p¿ 1s3œ ¢sd 2 s3œ ¢sd ¢sd s1œ s3œ s3œ s3 2 2 2 2 (12.55) and q¿ 1s3œ ¢sd 2 s3œ 2 ¢sd 2 (12.56) If these values of p and q were plotted in Figure 12.53, they would be represented by point D at the top of the Mohr’s circle. So, if the values of p and q at various stages of the deviator stress application are plotted and these points are joined, a straight line like ID will result. The straight line ID is referred to as the stress path in a q-p plot for a consolidated-drained triaxial test. Note that the line ID makes an angle of 45 with the horizontal. Point D represents the failure condition of the soil specimen in the test. Also, we can see that Mohr’s circle B represents the failure stress condition. For normally consolidated clays, the failure envelope can be given by tf s tan f. This is the line OF in Figure 12.53. (See also Figure 12.22.) A modiﬁed failure envelope now can be deﬁned by line OF. This modiﬁed line commonly is called the Kf line. The equation of the Kf line can be expressed as q¿ p¿ tan a (12.57) where a the angle that the modiﬁed failure envelope makes with the horizontal. The relationship between the angles f and a can be determined by referring to Figure 12.54, in which, for clarity, the Mohr’s circle at failure (that is, circle B) and lines Shear stress, or q F f F O a C D s3 3 O' (sd )f Figure 12.54 Relationship between f and a s1 s1 s, s, or p 416 Chapter 12: Shear Strength of Soil OF and OF, as shown in Figure 12.53, have been redrawn. Note that O is the center of the Mohr’s circle at failure. Now, DO¿ tan a OO¿ and thus, we obtain s1œ s3œ s1œ s3œ 2 tan a œ œ s1 s3 s1œ s3œ 2 (12.58) Again, CO¿ sin f¿ OO¿ or s1œ s3œ s1œ s3œ 2 sin f¿ œ œ s1 s3 s1œ s3œ 2 (12.59) Comparing Eqs. (12.58) and (12.59), we see that sin f¿ tan a (12.60) f¿ sin1 1tan a2 (12.61) or Figure 12.55 shows a q– p plot for a normally consolidated clay specimen subjected to an isotropically consolidated-undrained triaxial test. At the beginning of the application of deviator stress, s1œ s3œ s3 . Hence, p s3œ and q 0. This relationship is represented by point I. At some other stage of the deviator stress application, s1œ s3 ¢sd ¢ud and s3œ s3 ¢ud So, p¿ s1œ s3œ ¢sd s3 ¢ud 2 2 (12.62) 12.18 Stress Path 417 Shear stress, or q F Effective stress Mohr’s circle U Total stress Mohr’s circle U U a O 1 s3 s3 I s1 s, s, or p ud Figure 12.55 Stress path—plot of q against p for a consolidated-undrained triaxial test on a normally consolidated clay and q¿ s1œ s3œ ¢sd 2 2 (12.63) The preceding values of p and q will plot as point U in Figure 12.55. Points such as U represent values of p and q as the test progresses. At failure of the soil specimen, p¿ s3 1¢sd 2 f 2 1¢ud 2 f (12.64) and q¿ 1¢sd 2 f 2 (12.65) The values of p and q given by Eqs. (12.64) and (12.65) will plot as point U. Hence, the effective stress path for a consolidated-undrained test can be given by the curve IU U. Note that point U will fall on the modiﬁed failure envelope, OF (see Figure 12.54), which is inclined at an angle a to the horizontal. Lambe (1964) proposed a technique to evaluate the elastic and consolidation settlements of foundations on clay soils by using the stress paths determined in this manner. Example 12.9 For a normally consolidated clay, the failure envelope is given by the equation tf s tan f. The corresponding modiﬁed failure envelope (q-p plot) is given by Eq. (12.57) as q p tan a. In a similar manner, if the failure envelope is tf c s tan f, the corresponding modiﬁed failure envelope is a q-p plot that can be expressed as q m p tan a. Express a as a function of f, and give m as a function of c and f. 418 Chapter 12: Shear Strength of Soil Solution From Figure 12.56, s1œ s3œ b 2 AB AB sin f¿ s1œ s3œ AC CO OA c¿ cot f¿ a b 2 a So, s1œ s3œ s1œ s3œ c¿ cos f¿ a b sin f¿ 2 2 (a) q¿ m p¿ tan a (b) or Comparing Eqs. (a) and (b), we ﬁnd that m cⴕ cos Fⴕ and tan a sin f¿ or a tan1 1sin Fⴕ2 f Shear stress tf c s' tan f B s1' s3' 2 c C O c cot f s3 s1 s3 2 A s1 Normal stress Figure 12.56 Derivation of a as a function of f and m as a function of c and f 12.19 ■ Summary and General Comments In this chapter, the shear strengths of granular and cohesive soils were examined. Laboratory procedures for determining the shear strength parameters were described. In textbooks, determination of the shear strength parameters of cohesive soils appears to be fairly simple. However, in practice, the proper choice of these parameters for design and stability checks of various earth, earth-retaining, and earth-supported structures Problems 419 is very difﬁcult and requires experience and an appropriate theoretical background in geotechnical engineering. In this chapter, three types of strength parameters (consolidateddrained, consolidated-undrained, and unconsolidated-undrained) were introduced. Their use depends on drainage conditions. Consolidated-drained strength parameters can be used to determine the long-term stability of structures such as earth embankments and cut slopes. Consolidated-undrained shear strength parameters can be used to study stability problems relating to cases where the soil initially is fully consolidated and then there is rapid loading. An excellent example of this is the stability of slopes of earth dams after rapid drawdown. The unconsolidatedundrained shear strength of clays can be used to evaluate the end-of-construction stability of saturated cohesive soils with the assumption that the load caused by construction has been applied rapidly and there has been little time for drainage to take place. The bearing capacity of foundations on soft saturated clays and the stability of the base of embankments on soft clays are examples of this condition. The unconsolidated-undrained shear strength of some saturated clays can vary depending on the direction of load application; this is referred to as anisotropy with respect to strength. Anisotropy is caused primarily by the nature of the deposition of the cohesive soils, and subsequent consolidation makes the clay particles orient perpendicular to the direction of the major principal stress. Parallel orientation of the clay particles can cause the strength of clay to vary with direction. The anisotropy with respect to strength for clays can have an important effect on the load-bearing capacity of foundations and the stability of earth embankments because the direction of the major principal stress along the potential failure surfaces changes. The sensitivity of clays was discussed in Section 12.13. It is imperative that sensitive clay deposits are properly identiﬁed. For instance, when machine foundations (which are subjected to vibratory loading) are constructed over sensitive clays, the clay may lose its load-bearing capacity substantially, and failure may occur. Problems 12.1 For a direct shear test on a dry sand, the following are given: • Specimen size: 75 mm 75 mm 30 mm (height) • Normal stress: 200 kN/m2 • Shear stress at failure: 175 kN/m2 a. Determine the angle of friction, f b. For a normal stress of 150 kN/m2, what shear force is required to cause failure in the specimen? 12.2 For a dry sand specimen in a direct shear test box, the following are given: • Angle of friction: 38 • Size of specimen: 2 in. 2 in. 1.2 in. (height) • Normal stress: 20 lb/in.2 Determine the shear force required to cause failure. 12.3 The following are the results of four drained, direct shear tests on a normally consolidated clay. Given: • Size of specimen 60 mm 60 mm • Height of specimen 30 mm 420 Chapter 12: Shear Strength of Soil Test no. Normal force (N) Shear force at failure (N) 1 2 3 4 200 300 400 500 155 230 310 385 Draw a graph for the shear stress at failure against the normal stress, and determine the drained angle of friction from the graph. 12.4 Repeat Problem 12.3 with the following data. Given specimen size: • Diameter 2 in. • Height 1 in. Test no. Normal force (lb) Shear force at failure (lb) 1 2 3 4 60 90 110 125 37.5 55 70 80 12.5 The equation of the effective stress failure envelope for a loose, sandy soil was obtained from a direct shear test at tf s tan 30. A drained triaxial test was conducted with the same soil at a chamber conﬁning pressure of 10 lb/in.2. Calculate the deviator stress at failure. 12.6 For the triaxial test described in Problem 12.5: a. Estimate the angle that the failure plane makes with the major principal plane. b. Determine the normal stress and shear stress (when the specimen failed) on a plane that makes an angle of 30 with the major principal plane. Also, explain why the specimen did not fail along the plane during the test. 12.7 The relationship between the relative density, Dr , and the angle of friction, f, of a sand can be given as f 25 0.18Dr (Dr is in %). A drained triaxial test on the same sand was conducted with a chamber-conﬁning pressure of 18 lb/in.2. The relative density of compaction was 60%. Calculate the major principal stress at failure. 12.8 For a normally consolidated clay, the results of a drained triaxial test are as follows. • Chamber conﬁning pressure: 15 lb/in.2 • Deviator stress at failure: 34 lb/in.2 Determine the soil friction angle, f. 12.9 For a normally consolidated clay, f 24. In a drained triaxial test, the specimen failed at a deviator stress of 175 kN/m2. What was the chamber conﬁning pressure, s3œ ? 12.10 For a normally consolidated clay, f 28. In a drained triaxial test, the specimen failed at a deviator stress of 30 lb/in.2. What was the chamber conﬁning pressure, s3œ ? Problems 421 12.11 A consolidated-drained triaxial test was conducted on a normally consolidated clay. The results were as follows: s3 250 kN/m2 1¢sd 2 f 275 kN/m2 Determine: a. Angle of friction, f b. Angle u that the failure plane makes with the major principal plane c. Normal stress, s, and shear stress, tf , on the failure plane 12.12 The results of two drained triaxial tests on a saturated clay are given next: Specimen I: Chamber conﬁning pressure 15 lb/in.2 Deviator stress at failure 31.4 lb/in.2 Specimen II: Chamber-conﬁning pressure 25 lb/in.2 Deviator stress at failure 47 lb/in.2 Calculate the shear strength parameters of the soil. 12.13 If the clay specimen described in Problem 12.12 is tested in a triaxial apparatus with a chamber-conﬁning pressure of 25 lb/in.2, what is the major principal stress at failure? 12.14 A sandy soil has a drained angle of friction of 38. In a drained triaxial test on the same soil, the deviator stress at failure is 175 kN/m2. What is the chamberconﬁning pressure? 12.15 A consolidated-undrained test on a normally consolidated clay yielded the following results: • s3 15 lb/in.2 • Deviator stress: (sd)f 11 lb/in.2 Pore pressure: (ud )f 7.2 lb/in.2 Calculate the consolidated-undrained friction angle and the drained friction angle. 12.16 Repeat Problem 12.15 with the following: s3 140 kN/m2 1¢sd 2 f 125 kN/m2 1¢ud 2 f 75 kN/m2 12.17 The shear strength of a normally consolidated clay can be given by the equation tf s tan 31. A consolidated-undrained triaxial test was conducted on the clay. Following are the results of the test: • Chamber conﬁning pressure 112 kN/m2 • Deviator stress at failure 100 kN/m2 Determine: a. Consolidated-undrained friction angle b. Pore water pressure developed in the clay specimen at failure 12.18 For the clay specimen described in Problem 12.17, what would have been the deviator stress at failure if a drained test had been conducted with the same chamber-conﬁning pressure (that is, s3 112 kN/m2 )? 12.19 For a normally consolidated clay soil, f 32 and f 22. A consolidatedundrained triaxial test was conducted on this clay soil with a chamber-conﬁning pressure of 15 lb/in.2. Determine the deviator stress and the pore water pressure at failure. 12.20 The friction angle, f, of a normally consolidated clay specimen collected during ﬁeld exploration was determined from drained triaxial tests to be 25. The unconﬁned compression strength, qu , of a similar specimen was found to be 100 kN/m2. Determine the pore water pressure at failure for the unconﬁned compression test. 422 Chapter 12: Shear Strength of Soil 12.21 Repeat Problem 12.20 using the following values. f¿ 23° qu 120 kN/m2 12.22 The results of two consolidated-drained triaxial tests on a clayey soil are as follows. Test no. S3œ (lb/in.2) œ S11failure2 (lb/in.2) 1 2 27 12 73 48 Use the failure envelope equation given in Example 12.9—that is, q m p tan a. (Do not plot the graph.) a. Find m and a b. Find c and f 12.23 A 15-m thick normally consolidated clay layer is shown in Figure 12.57. The plasticity index of the clay is 18. Estimate the undrained cohesion as would be determined from a vane shear test at a depth of 8 m below the ground surface. Use Eq. (12.35). g 16 kN/m3 3m Groundwater table 15 m gsat 18.6 kN/m3 Dry sand Clay Rock Figure 12.57 References ACAR, Y. B., DURGUNOGLU, H. T., and TUMAY, M. T. (1982). “Interface Properties of Sand,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 108, No. GT4, 648–654. ARMAN, A., POPLIN, J. K., and AHMAD, N. (1975). “Study of Vane Shear,” Proceedings, Conference on In Situ Measurement and Soil Properties, ASCE, Vol. 1, 93–120. AMERICAN SOCIETY FOR TESTING AND MATERIALS (2004). Annual Book of ASTM Standards, Vol. 04.08, Philadelphia, Pa. BISHOP, A. W., and BJERRUM, L. (1960). “The Relevance of the Triaxial Test to the Solution of Stability Problems,” Proceedings, Research Conference on Shear Strength of Cohesive Soils, ASCE, 437–501. BJERRUM, L. (1974). “Problems of Soil Mechanics and Construction on Soft Clays,” Norwegian Geotechnical Institute, Publication No. 110, Oslo. References 423 BLACK, D. K., and LEE, K. L. (1973). “Saturating Laboratory Samples by Back Pressure,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM1, 75–93. CASAGRANDE, A., and CARRILLO, N. (1944). “Shear Failure of Anisotropic Materials,” in Contribution to Soil Mechanics 1941–1953, Boston Society of Civil Engineers, Boston. CASAGRANDE, A., and HIRSCHFELD, R. C. (1960). “Stress Deformation and Strength Characteristics of a Clay Compacted to a Constant Dry Unit Weight,” Proceedings, Research Conference on Shear Strength of Cohesive Soils, ASCE, 359–417. CHANDLER, R. J. (1988). “The in situ Measurement of the Undrained Shear Strength of Clays Using the Field Vane,” STP 1014, Vane Shear Strength Testing in Soils: Field and Laboratory Studies, ASTM, 13–44. COULOMB, C. A. (1776). “Essai sur une application des regles de Maximums et Minimis á quelques Problèmes de Statique, relatifs ál’Architecture,” Memoires de Mathematique et de Physique, Présentés, ál’Academie Royale des Sciences, Paris, Vol. 3, 38. KENNEY, T. C. (1959). “Discussion,” Proceedings, ASCE, Vol. 85, No. SM3, 67–79. LADD, C. C., FOOTE, R., ISHIHARA, K., SCHLOSSER, F., and POULOS, H. G. (1977). “Stress Deformation and Strength Characteristics,” Proceedings, 9th International Conference on Soil Mechanics and Foundation Engineering, Tokyo, Vol. 2, 421–494. LAMBE, T. W. (1964). “Methods of Estimating Settlement,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 90, No. SM5, 47–74. LOH, A. K., and HOLT, R. T. (1974). “Directional Variation in Undrained Shear Strength and Fabric of Winnipeg Upper Brown Clay,” Canadian Geotechnical Journal, Vol. 11, No. 3, 430–437. MOHR, O. (1900). “Welche Umstände Bedingen die Elastizitätsgrenze und den Bruch eines Materiales?” Zeitschrift des Vereines Deutscher Ingenieure, Vol. 44, 1524–1530, 1572– 1577. ROSENQ VIST , I. TH. (1953). “Considerations on the Sensitivity of Norwegian u Qick Clays,” Geotechnique, Vol. 3, No. 5, 195–200. SEED, H. B., and CHAN, C. K. (1959). “Thixotropic Characteristics of Compacted Clays,” Transactions, ASCE, Vol. 124, 894–916. SIMONS, N. E. (1960). “The Effect of Overconsolidation on the Shear Strength Characteristics of an Undisturbed Oslo Clay,” Proceedings, Research Conference on Shear Strength of Cohesive Soils, ASCE, 747–763. SKEMPTON, A. W. (1954). “The Pore Water Coefﬁcients A and B,” Geotechnique, Vol. 4, 143–147. SKEMPTON, A. W. (1957). “Discussion: The Planning and Design of New Hong Kong Airport,” Proceedings, Institute of Civil Engineers, London, Vol. 7, 305–307. SKEMPTON, A. W. (1964). “Long-Term Stability of Clay Slopes,” Geotechnique, Vol. 14, 77. 13 Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Retaining structures such as retaining walls, basement walls, and bulkheads commonly are encountered in foundation engineering as they support slopes of earth masses. Proper design and construction of these structures require a thorough knowledge of the lateral forces that act between the retaining structures and the soil masses being retained. These lateral forces are caused by lateral earth pressure. This chapter is devoted to the study of the various earth pressure theories. 13.1 At-Rest, Active, and Passive Pressures Consider a mass of soil shown in Figure. 13.1a. The mass is bounded by a frictionless wall of height AB. A soil element located at a depth z is subjected to a vertical effective pressure, soœ , and a horizontal effective pressure, shœ . There are no shear stresses on the vertical and horizontal planes of the soil element. Let us deﬁne the ratio of shœ to soœ as a nondimensional quantity K, or K shœ soœ (13.1) Now, three possible cases may arise concerning the retaining wall: and they are described Case 1 If the wall AB is static—that is, if it does not move either to the right or to the left of its initial position—the soil mass will be in a state of static equilibrium. In that case, shœ is referred to as the at-rest earth pressure, or K Ko shœ soœ (13.2) where Ko at-rest earth pressure coefﬁcient. 424 Case 2 If the frictionless wall rotates sufﬁciently about its bottom to a position of AB (Figure 13.1b), then a triangular soil mass ABC adjacent to the wall will reach a state of 425 13.1 At-Rest, Active, and Passive Pressures A A La At-rest pressure A so Active pressure so z z K a so sh K o so sh H C tf c s tan f tf c s tan f B B (b) (a) A A Lp Passive pressure C z Kp so sh so tf c s tan f B (c) Figure 13.1 Deﬁnition of at-rest, active, and passive pressures (Note: Wall AB is frictionless) plastic equilibrium and will fail sliding down the plane BC. At this time, the horizontal effective stress, shœ saœ , will be referred to as active pressure. Now, K Ka shœ saœ soœ soœ (13.3) where Ka active earth pressure coefﬁcient. Case 3 If the frictionless wall rotates sufﬁciently about its bottom to a position AB (Figure 13.1c), then a triangular soil mass ABC will reach a state of plastic 426 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Earth pressure, sh Passive pressure, sp At-rest pressure, sh Active pressure, sa Wall tilt La H Lp H Wall tilt Figure 13.2 Variation of the magnitude of lateral earth pressure with wall tilt Table 13.1 Typical Values of La /H and Lp/H Soil type Loose sand Dense sand Soft clay Stiff clay ⌬La /H ⌬Lp /H 0.001–0.002 0.0005–0.001 0.02 0.01 0.01 0.005 0.04 0.02 equilibrium and will fail sliding upward along the plane BC. The horizontal effective stress at this time will be shœ spœ , the so-called passive pressure. In this case, K Kp spœ shœ soœ soœ (13.4) where Kp passive earth pressure coefﬁcient Figure 13.2 shows the nature of variation of lateral earth pressure with the wall tilt. Typical values of La /H (La AA in Figure 13.1b) and Lp /H (Lp AA in Figure 13.1c) for attaining the active and passive states in various soils are given in Table 13.1. AT-REST LATERAL EARTH PRESSURE 13.2 Earth Pressure At-Rest The fundamental concept of earth pressure at rest was discussed in the preceding section. In order to deﬁne the earth pressure coefﬁcient Ko at rest, we refer to Figure 13.3, 13.2 Earth Pressure At-Rest 427 A so gz z sh K o gz H tf c s tan f Figure 13.3 Earth pressure at rest B which shows a wall AB retaining a dry soil with a unit weight of g. The wall is static. At a depth z, Vertical effective stress soœ gz Horizontal effective stress shœ Kogz So, Ko shœ at-rest earth pressure coefficient soœ For coarse-grained soils, the coefﬁcient of earth pressure at rest can be estimated by using the empirical relationship (Jaky, 1944) Ko 1 sin f¿ (13.5) where f drained friction angle. While designing a wall that may be subjected to lateral earth pressure at rest, one must take care in evaluating the value of Ko . Sherif, Fang, and Sherif (1984), on the basis of their laboratory tests, showed that Jaky’s equation for Ko [Eq. (13.5)] gives good results when the backﬁll is loose sand. However, for a dense, compacted sand backﬁll, Eq. (13.5) may grossly underestimate the lateral earth pressure at rest. This underestimation results because of the process of compaction of backﬁll. For this reason, they recommended the design relationship Ko 11 sin f2 c where gd 1 d 5.5 gd1min2 gd actual compacted dry unit weight of the sand behind the wall gd1min2 dry unit weight of the sand in the loosest state 1Chapter 32 (13.6) 428 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb The increase of Ko observed from Eq. (13.6) compared to Eq. (13.5) is due to overconsolidation. For that reason, Mayne and Kulhawy (1982), after evaluating 171 soils, recommended a modiﬁcation to Eq. (13.5). Or Ko 11 sin f¿ 2 1OCR2 sinf¿ (13.7) where OCR overconsolidation ratio preconsolidation pressure, scœ present effective overburden pressure, soœ Equation (13.7) is valid for soils ranging from clay to gravel. For ﬁne-grained, normally consolidated soils, Massarsch (1979) suggested the following equation for Ko : Ko 0.44 0.42 c PI 1% 2 100 d (13.8) For overconsolidated clays, the coefﬁcient of earth pressure at rest can be approximated as Ko 1overconsolidated2 Ko 1normally consolidated2 1OCR (13.9) Figure 13.4 shows the distribution of lateral earth pressure at rest on a wall of height H retaining a dry soil having a unit weight of g. The total force per unit length of the wall, Po , is equal to the area of the pressure diagram, so Po 12 KogH2 tf c s tan f Unit weight g H H 3 KogH Figure 13.4 Distribution of lateral earth pressure at-rest on a wall (13.10) 13.3 Earth Pressure At-Rest for Partially Submerged Soil 13.3 429 Earth Pressure At-Rest for Partially Submerged Soil Figure 13.5a shows a wall of height H. The groundwater table is located at a depth H1 below the ground surface, and there is no compensating water on the other side of the wall. For z H1 , the lateral earth pressure at rest can be given as shœ Kogz. The variation of shœ with depth is shown by triangle ACE in Figure 13.5a. However, for z H1 (i.e., below the groundwater table), the pressure on the wall is found from the effective stress and pore water pressure components via the equation Effective vertical pressure soœ gH1 g¿1z H1 2 (13.11) where g gsat gw the effective unit weight of soil. So, the effective lateral pressure at rest is shœ Kosoœ Ko 3 gH1 g¿1z H1 2 4 (13.12) The variation of shœ with depth is shown by CEGB in Figure 13.5a. Again, the lateral pressure from pore water is u gw 1z H1 2 (13.13) The variation of u with depth is shown in Figure 13.5b. A Unit weight of soil g H1 z C H Groundwater table I E KogH1 H2 sh Saturated unit weight of soil gsat u F J G B Ko(gH1 gH2 ) K g H2 (a) (b) H1 KogH1 H2 sh Ko(gH1 gH2 ) g H2 (c) Figure 13.5 Distribution of earth pressure at rest for partially submerged soil 430 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Hence, the total lateral pressure from earth and water at any depth z H1 is equal to sh shœ u Ko 3gH1 g¿1z H1 2 4 gw 1z H1 2 (13.14) The force per unit length of the wall can be found from the sum of the areas of the pressure diagrams in Figures 13.5a and 13.5b and is equal to (Figure 13.5c) ⎧ ⎪ ⎨ ⎪ ⎩ ⎧ ⎪ ⎨ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Po 12 KogH21 KogH1H2 12 1Kog¿ gw 2H22 Area ACE Area CEFB Areas EFG and IJK (13.15) Example 13.1 Figure 13.6a shows a 15-ft-high retaining wall. The wall is restrained from yielding. Calculate the lateral force Po per unit length of the wall. Also, determine the location of the resultant force. Assume that for sand OCR 2. Solution Ko 11 sin f¿ 2 1OCR2 sinf¿ 11 sin 302 12 2 sin30 0.707 At z 0: soœ 0; shœ 0; u 0 At z 10 ft: soœ 110 2 11002 1000 lb/ft2 shœ Kosoœ 10.7072 110002 707 lb/ft2 u0 At z 15 ft: soœ 110 2 11002 15 2 1122.4 62.4 2 1300 lb/ft2 shœ Kosoœ 10.7072 113002 919.1 lb/ft2 u 152 1gw 2 152 162.42 312 lb/ft2 The variations of shœ and u with depth are shown in Figures 13.6b and 13.6c. Lateral force Po Area 1 Area 2 Area 3 Area 4 or 1 1 1 Po a b 1102 17072 152 17072 a b 152 1212.12 a b 152 13122 2 2 2 3535 3535 530.3 780 8380.3 lb/ft 13.3 Earth Pressure At-Rest for Partially Submerged Soil 431 A c 0 f 30 g 100 lb/ft3 z 10 ft B Groundwater table c 0 f 30 gsat 122.4 lb/ft3 5 ft C Sand (a) sh (lb/ft2) u (lb/ft2) 1 10 10 707 2 3 15 z (ft) 4 919.1 15 z (ft) (b) 312 (c) Figure 13.6 The location of the resultant, measured from the bottom of the wall, is z g moment of pressure diagram about C Po or 135352 a 5 z 5 5 5 10 b 13535 2 a b 1530.32 a b 17802 a b 3 2 3 3 4.83 ft 8380.3 ■ 432 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb RANKINE’S LATERAL EARTH PRESSURE 13.4 Rankine’s Theory of Active Pressure The phrase plastic equilibrium in soil refers to the condition where every point in a soil mass is on the verge of failure. Rankine (1857) investigated the stress conditions in soil at a state of plastic equilibrium. In this section and in Section 13.5, we deal with Rankine’s theory of earth pressure. Figure 13.7a shows a soil mass that is bounded by a frictionless wall, AB, that extends to an inﬁnite depth. The vertical and horizontal effective principal stresses on a soil element at a depth z are soœ and shœ , respectively. As we saw in Section 13.2, if the wall AB is not allowed to move, then shœ Kosoœ . The stress condition in the soil element can be L A A 2c √K a so z 2ctan 45 f g 2 sh z Unit weight of soil g tf c s tan f gzKa 2c√Ka B B (c) (a) Shear stress tf c + s tan f c A 45 D f 2 45 b sa f f O C so K o so Normal stress a D (b) Figure 13.7 Rankine’s active earth pressure (d) f 2 13.4 Rankine’s Theory of Active Pressure 433 represented by the Mohr’s circle a in Figure 13.7b. However, if the wall AB is allowed to move away from the soil mass gradually, the horizontal principal stress will decrease. Ultimately a state will be reached when the stress condition in the soil element can be represented by the Mohr’s circle b, the state of plastic equilibrium and failure of the soil will occur. This situation represents Rankine’s active state, and the effective pressure saœ on the vertical plane (which is a principal plane) is Rankine’s active earth pressure. We next derive saœ in terms of g, z, c, and f from Figure 13.7b sin f¿ CD CD AC AO OC But CD radius of the failure circle soœ saœ 2 AO c¿ cot f¿ and OC soœ saœ 2 So, soœ saœ 2 sin f¿ soœ saœ c¿ cot f¿ 2 or c¿ cos f¿ soœ saœ soœ saœ sin f¿ 2 2 or saœ soœ 1 sin f¿ cos f¿ 2c¿ 1 sin f¿ 1 sin f¿ But soœ vertical effective overburden pressure gz f¿ 1 sin f¿ tan2 a 45 b 1 sin f¿ 2 and cos f¿ f¿ tan a 45 b 1 sin f¿ 2 (13.16) 434 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Substituting the preceding values into Eq. (13.16), we get saœ gz tan2 a 45 f¿ f¿ b 2c¿ tan a 45 b 2 2 (13.17) The variation of saœ with depth is shown in Figure 13.7c. For cohesionless soils, c 0 and f¿ saœ soœ tan2 a 45 b (13.18) 2 The ratio of saœ to soœ is called the coefﬁcient of Rankine’s active earth pressure and is given by Ka saœ f¿ 2 b œ tan a 45 so 2 (13.19) Again, from Figure 13.7b we can see that the failure planes in the soil make (45 f/2)-degree angles with the direction of the major principal plane—that is, the horizontal. These are called potential slip planes and are shown in Figure 13.7d. It is important to realize that a similar equation for sa could be derived based on the total stress shear strength parameters—that is, tf c s tan f. For this case, sa gz tan2 a 45 13.5 f f b 2c tan a 45 b 2 2 (13.20) Theory of Rankine’s Passive Pressure Rankine’s passive state can be explained with the aid of Figure 13.8. AB is a frictionless wall that extends to an inﬁnite depth (Figure 13.8a). The initial stress condition on a soil element is represented by the Mohr’s circle a in Figure 13.8b. If the wall gradually is pushed into the soil mass, the effective principal stress sh will increase. Ultimately, the wall will reach a situation where the stress condition for the soil element can be expressed by the Mohr’s circle b. At this time, failure of the soil will occur. This situation is referred to as Rankine’s passive state. The lateral earth pressure spœ , which is the major principal stress, is called Rankine’s passive earth pressure. From Figure 13.8b, it can be shown that spœ soœ tan2 a 45 f¿ f¿ b 2c¿ tan a 45 b 2 2 gz tan2 a 45 f¿ f¿ b 2c¿ tan a 45 b 2 2 (13.21) 13.5 Theory of Rankine’s Passive Pressure 435 The derivation is similar to that for Rankine’s active state. Figure 13.8c shows the variation of passive pressure with depth. For cohesionless soils (c 0), f¿ b spœ soœ tan2 a 45 2 or spœ soœ Kp tan2 a 45 f¿ b 2 (13.22) Kp (the ratio of effective stresses) in the preceding equation is referred to as the coefﬁcient of Rankine’s passive earth pressure. L A A so z sh Unit weight of soil g tf c s tan f B B (a) Shear stress tf c + s tan f D b f A O f a Ko so C so sp Normal stress D (b) Figure 13.8 Rankine’s passive earth pressure 436 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb z gzKp 2c√Kp (c) 45 f 2 45 f 2 (d) Figure 13.8 (continued) The points D and D on the failure circle (see Figure 13.8b) correspond to the slip planes in the soil. For Rankine’s passive state, the slip planes make (45 f/2)-degree angles with the direction of the minor principal plane—that is, in the horizontal direction. Figure 13.8d shows the distribution of slip planes in the soil mass. 13.6 Yielding of Wall of Limited Height We learned in the preceding discussion that sufﬁcient movement of a frictionless wall extending to an inﬁnite depth is necessary to achieve a state of plastic equilibrium. However, the distribution of lateral pressure against a wall of limited height is inﬂuenced very much by the manner in which the wall actually yields. In most retaining walls of limited height, movement may occur by simple translation or, more frequently, by rotation about the bottom. 13.6 Yielding of Wall of Limited Height 437 For preliminary theoretical analysis, let us consider a frictionless retaining wall represented by a plane AB as shown in Figure 13.9a. If the wall AB rotates sufﬁciently about its bottom to a position AB, then a triangular soil mass ABC adjacent to the wall will reach Rankine’s active state. Because the slip planes in Rankine’s active state make angles of (45 f/2) degrees with the major principal plane, the soil mass in the state of plastic equilibrium is bounded by the plane BC, which makes an angle of (45 f/2) degrees with the horizontal. The soil inside the zone ABC undergoes the same unit deformation in the horizontal direction everywhere, which is equal to La /La . The lateral earth pressure on the wall at any depth z from the ground surface can be calculated by using Eq. (13.17). In a similar manner, if the frictionless wall AB (Figure 13.9b) rotates sufﬁciently into the soil mass to a position AB, then the triangular mass of soil ABC will reach Rankine’s La La f 45 2 A A C 45 f 2 z H B (a) Lp Lp A A f 45 2 45 f 2 C z H 45 f 2 B (b) Figure 13.9 Rotation of frictionless wall about the bottom 438 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb passive state. The slip plane BC bounding the soil wedge that is at a state of plastic equilibrium will make an angle of (45 f/2) degrees with the horizontal. Every point of the soil in the triangular zone ABC will undergo the same unit deformation in the horizontal direction, which is equal to Lp /Lp . The passive pressure on the wall at any depth z can be evaluated by using Eq. (13.21). 13.7 A Generalized Case for Rankine Active and Passive Pressure— Granular Backfill In Sections 13.4, 13.5, and 13.6, we discussed the Rankine active and passive pressure cases for a frictionless wall with a vertical back and a horizontal backﬁll of granular soil. This can be extended to general cases of frictionless wall with inclined backﬁll (granular soil) as shown in Figure 13.10 (Chu, 1991). a a z u u H sa or sp b Frictionless wall b Figure 13.10 General case for Rankine active and passive pressures 13.7 A Generalized Case for Rankine Active and Passive Pressure— Granular Backfill 439 Rankine Active Case For the Rankine active case, the lateral earth pressure (saœ ) at a depth z can be given as saœ gz cos a21 sin2 f¿ 2 sin f¿ cos ca cos a 2sin2 f¿ sin2 a (13.23) where ca sin1 a sin a b a 2u (13.24) sin f¿ The pressure saœ will be inclined at an angle b with the plane drawn at right angle to the backface of the wall, and b tan1 a sin f¿ sin ca b 1 sin f¿ cos ca (13.25) The active force Pa for unit length of the wall can then be calculated as Pa 1 gH2Ka 1R2 2 (13.26) where Ka1R2 cos1a u 2 21 sin2 f¿ 2 sin f¿ cos ca cos2 u1cos a 2sin2 f¿ sin2 a 2 Rankine active earth-pressure coefﬁcient for generalized case (13.27) The location and direction of the resultant force Pa is shown in Figure 13.11a. Also shown in this ﬁgure is the failure wedge, ABC. Note that BC will be inclined at an angle h. Or h f¿ p sin a a 1 sin1 a b 4 2 2 2 sin f¿ (13.28) As a special case, for a vertical backface of the wall (that is, u 0) as shown in Figure 13.12, Eqs. (13.26) and (13.27) simplify to the following Pa 1 Ka1R2gH2 2 where Ka1R2 cos a cos a 2cos2 a cos2 f¿ cos a 2cos2 a cos2 f¿ (13.29) 440 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb C Active case C a A Passive case a a A Failure wedge Failure wedge a Pp Pa u u H u b u H b b H 3 h p 4 f a 2 2 b H 3 h B 1 sin1 sin a 2 sin f h p 4 (a) f a 2 2 h B 1 sin1 sin a 2 sin f (b) Figure 13.11 Location and direction of resultant Rankine force a Frictionless wall g c f z H a sa Figure 13.12 Frictionless vertical retaining wall with sloping backﬁll Also note that, for this case, the angle b will be equal to a. The variation of Ka(R) given in Eq. (13.29) with a and f is given in Table 13.2. Rankine Passive Case Similar to the active case, for the Rankine passive case, we can obtain the following relationships. 13.7 A Generalized Case for Rankine Active and Passive Pressure— Granular Backfill 441 Table 13.2 Values of Ka(R) [Eq. (13.29)] Fⴕ (deg) S T A (deg) 28 30 32 34 36 38 40 0 5 10 15 20 25 0.361 0.366 0.380 0.409 0.461 0.573 0.333 0.337 0.350 0.373 0.414 0.494 0.307 0.311 0.321 0.341 0.374 0.434 0.283 0.286 0.294 0.311 0.338 0.385 0.260 0.262 0.270 0.283 0.306 0.343 0.238 0.240 0.246 0.258 0.277 0.307 0.217 0.219 0.225 0.235 0.250 0.275 spœ gz cos a21 sin2 f¿ 2 sin f¿ cos cp cos a 2sin2 f¿ sin2 a (13.30) where cp sin1 a sin a b a 2u sin f¿ (13.31) The inclination b of spœ , as shown in Figure 13.10, is b tan1 a sin f¿ sin cp 1 sin f¿ cos cp b (13.32) The passive force per unit length of the wall is Pp 1 gH2Kp1R2 2 where Kp1R2 cos1a u 2 21 sin2 f¿ 2sin f¿ cos cp cos2 u1cos a 2sin2 f¿ sin2 a2 (13.33) The location and direction of Pa along with the failure wedge is shown in Figure 13.11b. For walls with vertical backface, u 0, Pp where Kp1R2 cos a 1 Kp1R2gH2 2 cos a 2cos2 a cos2 f¿ cos a 2cos2 a cos2 f¿ (13.34) 442 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Table 13.3 Passive Earth Pressure Coefﬁcient, Kp(R) [Eq. (13.34)] Fⴕ (deg) S T A (deg) 28 30 32 34 36 38 40 0 5 10 15 20 25 2.770 2.715 2.551 2.284 1.918 1.434 3.000 2.943 2.775 2.502 2.132 1.664 3.255 3.196 3.022 2.740 2.362 1.894 3.537 3.476 3.295 3.003 2.612 2.135 3.852 3.788 3.598 3.293 2.886 2.394 4.204 4.136 3.937 3.615 3.189 2.676 4.599 4.527 4.316 3.977 3.526 2.987 The variation of Kp(R) with a and f, as expressed by Eq. (13.34), is given in Table 13.3. Again, for this special case, the angle of Pp with the normal drawn to the back of the wall will be equal to a (that is, b a). 13.8 Diagrams for Lateral Earth-Pressure Distribution Against Retaining Walls Backﬁll— Cohesionless Soil with Horizontal Ground Surface Active Case Figure 13.13a shows a retaining wall with cohensionless soil backﬁll that has a horizontal ground surface. The unit weight and the angle of friction of the soil are g and f, respectively. For Rankine’s active state, the earth pressure at any depth against the retaining wall can be given by Eq. (13.17): saœ Kagz 1Note: c¿ 02 Note that saœ increases linearly with depth, and at the bottom of the wall, it is saœ KagH (13.35) The total force per unit length of the wall is equal to the area of the pressure diagram, so Pa 12 KagH2 (13.36) Passive Case The lateral pressure distribution against a retaining wall of height H for Rankine’s passive state is shown in Figure 13.13b. The lateral earth pressure at any depth z [Eq. (13.22), c 0] is spœ KpgH (13.37) The total force per unit length of the wall is Pp 12 KpgH2 (13.38) 13.8 Diagrams for Lateral Earth-Pressure Distribution Against Retaining Walls 45 443 f 2 Failure wedge H g f c 0 H sa Pa H 3 KagH (a) 45 f 2 Failure wedge H g f c 0 sp H Pp H 3 KpgH (b) Figure 13.13 Pressure distribution against a retaining wall for cohensionless soil backﬁll with horizontal ground surface: (a) Rankine’s active state; (b) Rankine’s passive state Backﬁll—Partially Submerged Cohensionless Soil Supporting a Surcharge Active Case Figure 13.14a shows a frictionless retaining wall of height H and a backﬁll of cohensionless soil. The groundwater table is located at a depth of H1 below the ground surface, and the backﬁll is supporting a surcharge pressure of q per unit area. From Eq. (13.19), the effective active earth pressure at any depth can be given by saœ Kasoœ (13.39) where soœ and saœ the effective vertical pressure and lateral pressure, respectively. At z 0, so soœ q (13.40) saœ Kaq (13.41) and 444 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Surcharge q f 45 2 g f Groundwater table Failure wedge H1 H H2 gsat f z (a) qKa qKa H1 sa KagH1 qKa g H2 Ka(q gH1) KagH2 g H2 (c) (d) Figure 13.14 Rankine’s active earth-pressure distribution against a retaining wall with partially submerged cohesionless soil backﬁll supporting a surcharge soœ 1q gH1 2 (13.42) saœ Ka 1q gH1 2 (13.43) soœ 1q gH1 g¿H2 2 (13.44) H2 u Ka(q gH1 gH2 ) (b) At depth z H1 , and At depth z H, and sa saœ Ka 1q gH1 g¿H2 2 where g gsat gw . The variation of saœ with depth is shown in Figure 13.14b. (13.45) 13.8 Diagrams for Lateral Earth-Pressure Distribution Against Retaining Walls 445 The lateral pressure on the wall from the pore water between z 0 and H1 is 0, and for z H1 , it increases linearly with depth (Figure 13.14c). At z H, u gwH2 The total lateral-pressure diagram (Figure 13.14d) is the sum of the pressure diagrams shown in Figures 13.14b and 13.14c. The total active force per unit length of the wall is the area of the total pressure diagram. Thus, Pa KaqH 12 KagH21 KagH1H2 12 1Kag¿ gw 2H22 (13.46) Passive Case Figure 13.15a shows the same retaining wall as was shown in Figure 13.14a. Rankine’s passive pressure at any depth against the wall can be given by Eq. (13.22): spœ Kpsoœ Surcharge q 45 H1 f 2 g f Groundwater table Failure wedge H gsat f H2 z (a) H1 Kp(gH1 q) H2 sp qKp Kp(gH1 gH2 ) (b) sp u g H2 (c) Kp(q gH1) KpgH2 g H2 (d) Figure 13.15 Rankine’s passive earth-pressure distribution against a retaining wall with partially submerged cohesionless soil backﬁll supporting a surcharge 446 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Using the preceding equation, we can determine the variation of spœ with depth, as shown in Figure 13.15b. The variation of the pressure on the wall from water with depth is shown in Figure 13.15c. Figure 13.15d shows the distribution of the total pressure sp with depth. The total lateral passive force per unit length of the wall is the area of the diagram given in Figure 13.15d, or Pp KpqH 12 KpgH21 KpgH1H2 12 1Kpg¿ gw 2H22 (13.47) Backﬁll— Cohesive Soil with Horizontal Backﬁll Active Case Figure 13.16a shows a frictionless retaining wall with a cohesive soil backﬁll. The active pressure against the wall at any depth below the ground surface can be expressed as [Eq. (13.17)] saœ Kagz 2 1Kac¿ The variation of Kagz with depth is shown in Figure 13.16b, and the variation of 21Kac¿ with depth is shown in Figure 13.16c. Note that 2 1Kac¿ is not a function of z; hence, Figure 13.16c is a rectangle. The variation of the net value of saœ with depth is plotted in Figure 13.16d. Also note that, because of the effect of cohesion, saœ is negative in the upper part of the retaining wall. The depth zo at which the active pressure becomes equal to 0 can be found from Eq. (13.17) as Kagzo 2 1Kac¿ 0 or zo 2c¿ g1Ka (13.48) For the undrained condition—that is, f 0, Ka tan2 45 1, and c cu (undrained cohesion)—from Eq. (13.20), zo 2cu g (13.49) So, with time, tensile cracks at the soil–wall interface will develop up to a depth zo . The total active force per unit length of the wall can be found from the area of the total pressure diagram (Figure 13.16d), or Pa 12 KagH2 2 1Kac¿H (13.50) Pa 12 gH2 2cuH (13.51) For the f 0 condition, For calculation of the total active force, common practice is to take the tensile cracks into account. Because no contact exists between the soil and the wall up to a depth of zo after the development of tensile cracks, only the active pressure distribution 13.8 Diagrams for Lateral Earth-Pressure Distribution Against Retaining Walls 45 447 f 2 Failure wedge z H (a) 2c√Ka zo H H zo Kag H (b) 2c√Ka (c) Kag H − 2c√Ka (d) Figure 13.16 Rankine’s active earth-pressure distribution against a retaining wall with cohesive soil backﬁll against the wall between z 2c/(g 1Ka) and H (Figure 13.16d) is considered. In this case, Pa 12 1KagH 21Kac¿ 2 a H 12 KagH2 2 1Kac¿H 2 2c¿ b g1Ka c¿ 2 g (13.52) 448 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb For the f 0 condition, Pa 12 gH2 2cuH 2 c2u g (13.53) Passive Case Figure 13.17a shows the same retaining wall with backﬁll similar to that considered in Figure 13.16a. Rankine’s passive pressure against the wall at depth z can be given by [Eq. (13.21)] spœ Kpgz 2 1Kpc¿ At z 0, spœ 2 1Kpc¿ (13.54) spœ KpgH 2 1Kpc¿ (13.55) and at z H, The variation of spœ with depth is shown in Figure 13.17b. The passive force per unit length of the wall can be found from the area of the pressure diagrams as Pp 12 KpgH2 2 1Kpc¿H (13.56) For the f 0 condition, Kp 1 and Pp 12 gH2 2cuH 45 (13.57) f 2 Failure wedge H z sp KpgH 2c√Kp (a) (b) Figure 13.17 Rankine’s passive earth-pressure distribution against a retaining wall with cohesive soil backﬁll 13.8 Diagrams for Lateral Earth-Pressure Distribution Against Retaining Walls 449 Example 13.2 An 6 m high retaining wall is shown in Figure 13.18a. Determine a. Rankine active force per unit length of the wall and the location of the resultant b. Rankine passive force per unit length of the wall and the location of the resultant Solution Part a Because c 0, to determine the active force we can use from Eq. (13.19). saœ Kasoœ Kagz Ka 1 sin f¿ 1 sin 36 0.26 1 sin f¿ 1 sin 36 At z 0, saœ 0; at z 6 m, saœ 10.26 2 1162 16 2 24.96 kN/m2 The pressure-distribution diagram is shown in Figure 13.18b. The active force per unit length of the wall is as follows: Pa 12 162 124.962 74.88 kN/m Also, z2m Part b To determine the passive force, we are given that c 0. So, from Eq. (13.22), spœ Kpsoœ Kpgz Kp 1 sin f¿ 1 sin 36 3.85 1 sin f¿ 1 sin 36 At z 0, spœ 0; at z 6 m, spœ 13.852 116 2 162 369.6 kN/m2 The pressure-distribution diagram is shown in Figure 13.18c. The passive force per unit length of the wall is as follows. Pp 12 16 2 1369.6 2 1108.8 kN/m2 450 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Normally consolidated sand (OCR 1) g 16 kN/m3 f 36 c 0 6m 6m 74.88 kN/m z̄ 2 m 24.96 kN/m2 (a) (b) 6m 1108.8 kN/m z̄ 2 m 369.6 kN/m2 (c) Figure 13.18 Diagrams for determining active and passive forces ■ Example 13.3 Refer to Figure 13.10. Given: H 12 ft, +20°, and +20°. For the granular backfill, it is given that 115 lb/ft3 and 30°. Determine the active force Pa per unit length of the wall as well as the location and direction of the resultant. Solution From Eq. (13.24), ca sin 1 a sin a sin 20 b a 2u sin1 a b 20°122 120°2 sin fœ sin 30 43.16° 20° 40° 63.16° 13.8 Diagrams for Lateral Earth-Pressure Distribution Against Retaining Walls 451 From Eq. (13.27), Ka1R2 cos1a u2 21 sin 2 fœ 2 sin fœ # cos ca cos2 u 1 cos a 2 sin 2 fœ sin 2 a 2 cos120 202 21 sin 2 30 12 2 1sin 302 1cos 63.162 cos2 20 1cos 20 2 sin 2 30 sin 2 20 2 From Eq. (13.26), Pa 1 2 gH2Ka1R2 1 2 0.776 11152 1122 2 10.7762 6425 lb/ft From Eq. (13.25), b tan 1 a tan 1 c sin f¿ sin ca b 1 sin f¿ cos ca 1sin 302 1sin 63.162 1 1sin 302 1cos 63.162 d 29.95° 30° The resultant will act a distance of 12/3 4 ft above the bottom of the wall with 30°. ■ Example 13.4 For the retaining wall shown in Figure 13.19a, determine the force per unit length of the wall for Rankine’s active state. Also ﬁnd the location of the resultant. g 16 kN/m3 f 30 c 0 3m Groundwater table gsat 18 kN/m3 f 35 c 0 z 3m (a) 3m 16 16 3m 13.0 19.67 (b) 29.43 (c) Pa 117.15 kN/m 1.78 m 13.0 36.1 (d) Figure 13.19 Retaining wall and pressure diagrams for determining Rankine’s active earth pressure. (Note: The units of pressure in (b), (c), and (d) are kN/m2) 452 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Solution Given that c 0, we known that saœ Kasoœ . For the upper layer of the soil, Rankine’s active earth-pressure coefﬁcient is Ka Ka 112 1 sin 30° 1 1 sin 30° 3 For the lower layer, Ka Ka122 1 sin 35° 0.271 1 sin 35° œ At z 0, so 0. At z 3 m (just inside the bottom of the upper layer), soœ 3 16 48 kN/m2. So saœ Ka112soœ 13 48 16 kN/m2 Again, at z 3 m (in the lower layer), soœ 3 16 48 kN/m2, and saœ Ka122soœ 10.2712 1482 13.0 kN/m2 At z 6 m, soœ 3 16 3118 9.812 72.57 kN/m2 c gw and saœ Ka122soœ 10.2712 172.57 2 19.67 kN/m2 The variation of saœ with depth is shown in Figure 13.19b. The lateral pressures due to the pore water are as follows. At z 0: u0 At z 3 m: u 0 At z 6 m: u 3 gw 3 9.81 29.43 kN/m2 The variation of u with depth is shown in Figure 13.19c, and that for sa (total active pressure) is shown in Figure 13.19d. Thus, Pa 1 12 2 13 2 1162 3113.02 1 12 2 132 136.12 24 39.0 54.15 117.15 kN/m The location of the resultant can be found by taking the moment about the bottom of the wall: 24 a 3 z 1.78 m 3 3 3 b 39.0 a b 54.15 a b 3 2 3 117.15 ■ 13.8 Diagrams for Lateral Earth-Pressure Distribution Against Retaining Walls 453 Example 13.5 A retaining wall that has a soft, saturated clay backﬁll is shown in Figure 13.20a. For the undrained condition (f 0) of the backﬁll, determine a. Maximum depth of the tensile crack b. Pa before the tensile crack occurs c. Pa after the tensile crack occurs 20 kN/m2 1.21 m Soft saturated clay 6m g 16.5 kN/m3 f0 cu 10 kN/m2 4.79 m 79 kN/m2 (a) (b) Figure 13.20 Rankine active pressure due to a soft, saturated clay backﬁll Solution For f 0, Ka tan2 45 1 and c cu . From Eq. (13.20), sa gz 2cu At z 0, sa 2cu 12 2 110 2 20 kN/m2 At z 6 m, sa 116.5 2 162 12 2 110 2 79 kN/m2 The variation of sa with depth is shown in Figure 13.20b. Part a From Eq. (13.49), the depth of the tensile crack equals zo 122 1102 2cu 1.21 m g 16.5 454 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Part b Before the tensile crack occurs [Eq. (13.51)], Pa 12 gH2 2cuH or Pa 12 116.52 16 2 2 2110 2 162 177 kN/m Part c After the tensile crack occurs, Pa 12 16 1.212 1792 189.2 kN/m [Note: The preceding Pa can also be obtained by substituting the proper values into Eq. (13.53).] ■ 13.9 Rankine Pressure for cⴕ– Fⴕ Soil—Inclined Backfill Figure 13.21 shows a retaining wall with a vertical back with an inclined backﬁll which is a c–f soil. The analysis for determining the active and passive Rankine earth pressure for this condition has been provided by Mazindrani and Ganjali (1997). According to this analysis, œ Active pressure: saœ gzKa1R2 gzKa1R2 cos a where Ka1R2 Rankline active earth-pressure coefﬁcient Ka1R2 œ Ka1R2 cos a a z H g f c Frictionless wall Figure 13.21 Rankine active and passive pressures with an inclined backﬁll of c–f soil (13.58) (13.59) 13.9 Rankine Pressure for cⴕ–F ⴕ Soil—Inclined Backfill 455 Similarly, œ Passive pressure: spœ gzKp1R2 gzKp1R2 cos a where Kp1R2 Rankine passive earth-pressure coefﬁcient Kp1R2 œ Kp1R2 cos a (13.60) (13.61) Also, œ œ , Kp1R2 Ka1R2 2 cos2 a 2 a µ 1 cos2 f¿ c¿ b cos f¿ sin f¿ gz ∂ 1 c¿ 2 2 c¿ 2 2 2 2 B c 4 cos a1cos a cos f¿ 2 4 a gz b cos f¿ 8 a gz b cos a sin f¿ cos f¿ d (13.62) œ œ Tables 13.4 and 13.5 give the variation of Ka1R2 and Kp1R2 with a, c/gz, and f. Table 13.4 Variation of Ka(R) with , c/z, and c/␥z ␣ (deg) (deg) 0 0.025 0.050 0.100 0 15 20 25 30 35 40 0.589 0.490 0.406 0.333 0.271 0.217 0.550 0.455 0.374 0.305 0.245 0.194 0.512 0.420 0.342 0.276 0.219 0.171 0.435 0.351 0.278 0.218 0.167 0.124 5 15 20 25 30 35 40 0.607 0.502 0.413 0.339 0.275 0.220 0.566 0.465 0.381 0.309 0.248 0.196 0.525 0.429 0.348 0.280 0.222 0.173 0.445 0.357 0.283 0.221 0.169 0.126 10 15 20 25 30 35 40 0.674 0.539 0.438 0.355 0.286 0.228 0.621 0.497 0.402 0.323 0.258 0.203 0.571 0.456 0.366 0.292 0.230 0.179 0.477 0.377 0.296 0.230 0.175 0.130 15 15 20 25 30 35 40 1.000 0.624 0.486 0.386 0.307 0.243 0.776 0.567 0.443 0.350 0.276 0.216 0.683 0.514 0.401 0.315 0.246 0.190 0.546 0.417 0.321 0.246 0.186 0.337 456 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Table 13.5 Variation of Kp(R) with , c/z, and c/␥z (deg) (deg) 0 0.025 0.050 0.100 0 15 20 25 30 35 40 1.698 2.040 2.464 3.000 3.690 4.599 1.764 2.111 2.542 3.087 3.786 4.706 1.829 2.182 2.621 3.173 3.882 4.813 1.959 2.325 2.778 3.346 4.074 5.028 5 15 20 25 30 35 40 1.674 1.994 2.420 2.954 3.641 5.545 1.716 2.067 2.499 3.042 3.738 4.652 1.783 2.140 2.578 3.129 3.834 4.760 1.916 2.285 2.737 3.303 4.027 4.975 10 15 20 25 30 35 40 1.484 1.854 2.285 2.818 3.495 4.383 1.564 1.932 2.368 2.907 3.593 4.491 1.641 2.010 2.450 2.996 3.691 4.600 1.788 2.162 2.614 3.174 3.887 4.817 15 15 20 25 30 35 40 1.000 1.602 2.058 2.500 3.255 4.117 1.251 1.696 2.147 2.684 3.356 4.228 1.370 1.786 2.236 2.777 3.456 4.338 1.561 1.956 2.409 2.961 3.656 4.558 For the active case, the depth of the tensile crack can be given as zo 2c¿ 1 sin f¿ g B 1 sin f¿ (13.63) COULOMB’S EARTH PRESSURE THEORY More than 200 years ago, Coulomb (1776) presented a theory for active and passive earth pressures against retaining walls. In this theory, Coulomb assumed that the failure Example 13.6 Refer to Figure 13.21. Given: H 15 ft, a 10, g 118 lb/ft3, f 20, and c 250 lb/ft2. Determine the Rankine active force Pa on the retaining wall after the tensile crack occurs. 13.10 Coulomb’s Active Pressure 457 Solution From Eq. (13.63), the depth of tensile crack is zo 12 2 12502 1 sin 20 2c¿ 1 sin f¿ 6.05 ft g B 1 sin f¿ 118 B 1 sin 20 So At z 0: saœ 0 œ cos a At z 15 ft: saœ gzKa1R2 250 c¿ 0.14 gz 11182 1152 œ From Table 13.4, for a 10 and c/gz 0.14, the magnitude of Ka1R2 0.3. So saœ 11182 115 2 10.32 1cos 10°2 522.9 lb/ft 2 Hence, Pa 1 1 1H zo 2 1522.9 2 115 6.052 1522.9 2 2340.16 lb/ft 2 2 ■ surface is a plane. The wall friction was taken into consideration. The following sections discuss the general principles of the derivation of Coulomb’s earthpressure theory for a cohesionless backﬁll (shear strength deﬁned by the equation tf s tan f). 13.10 Coulomb’s Active Pressure Let AB (Figure 13.22a) be the back face of a retaining wall supporting a granular soil; the surface of which is constantly sloping at an angle a with the horizontal. BC is a trial failure surface. In the stability consideration of the probable failure wedge ABC, the following forces are involved (per unit length of the wall): 1. W—the weight of the soil wedge. 2. F—the resultant of the shear and normal forces on the surface of failure, BC. This is inclined at an angle of f to the normal drawn to the plane BC. 3. Pa —the active force per unit length of the wall. The direction of Pa is inclined at an angle d to the normal drawn to the face of the wall that supports the soil. d is the angle of friction between the soil and the wall. The force triangle for the wedge is shown in Figure 13.22b. From the law of sines, we have Pa W sin190 u d¿ b f¿ 2 sin1b f¿ 2 (13.64) 458 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb C a ba A Pa 90 u a 90 u d 90 u d b f W u F D H 90 u b W b f dʹ Pa f F b B (a) (b) Figure 13.22 Coulomb’s active pressure: (a) trial failure wedge; (b) force polygon or Pa sin1b f¿ 2 sin190 u d¿ b f¿ 2 W (13.65) The preceding equation can be written in the form cos1u b2cos1u a2 sin1b f¿ 2 1 d Pa gH2 c 2 2 cos u sin1b a2sin190 u d¿ b f¿ 2 (13.66) where g unit weight of the backﬁll. The values of g, H, u, a, f, and d are constants, and b is the only variable. To determine the critical value of b for maximum Pa , we have dPa 0 db (13.67) After solving Eq. (13.67), when the relationship of b is substituted into Eq. (13.66), we obtain Coulomb’s active earth pressure as Pa 12 KagH2 (13.68) where Ka is Coulomb’s active earth-pressure coefﬁcient and is given by Ka cos2 1f¿ u 2 cos u cos1d¿ u2 c 1 sin1d¿ f¿ 2sin1f¿ a2 d B cos1d¿ u 2cos1u a2 2 2 (13.69) 13.10 Coulomb’s Active Pressure 459 Table 13.6 Values of Ka [Eq. (13.69)] for u 0, a 0 Dⴕ (deg) S T Fⴕ (deg) 0 5 10 15 20 25 28 30 32 34 36 38 40 42 0.3610 0.3333 0.3073 0.2827 0.2596 0.2379 0.2174 0.1982 0.3448 0.3189 0.2945 0.2714 0.2497 0.2292 0.2089 0.1916 0.3330 0.3085 0.2853 0.2633 0.2426 0.2230 0.2045 0.1870 0.3251 0.3014 0.2791 0.2579 0.2379 0.2190 0.2011 0.1841 0.3203 0.2973 0.2755 0.2549 0.2354 0.2169 0.1994 0.1828 0.3186 0.2956 0.2745 0.2542 0.2350 0.2167 0.1995 0.1831 Note that when a 0, u 0, and d 0, Coulomb’s active earth-pressure coefﬁcient becomes equal to (1 sin f)/(1 sin f), which is the same as Rankine’s earth-pressure coefﬁcient given earlier in this chapter. The variation of the values of Ka for retaining walls with a vertical back (u 0) and horizontal backﬁll (a 0) is given in Table 13.6. From this table, note that for a given value of f, the effect of wall friction is to reduce somewhat the active earth-pressure coefﬁcient. Tables 13.7 and 13.8 also give the variation of Ka [Eq. (13.69)] for various values of a, f, u, and d (d 23 f in Table 13.7 and d 12 f in Table 13.8). Example 13.7 Refer to Figure 13.22. Given: a 10; u 5; H 4 m; unit weight of soil, g 15 kN/m3; soil friction angle, f 30; and d 15. Estimate the active force, Pa , per unit length of the wall. Also, state the direction and location of the resultant force, Pa . Solution From Eq. (13.68), Pa For f 30; d 15—that is, 1 gH2Ka 2 15 1 d¿ ; a 10; and u 5, the magnitude f¿ 30 2 of Ka is 0.3872 (Table 13.8). So, Pa 1 115 2 14 2 2 10.38722 46.46 kN/m 2 The resultant will act at a vertical distance equal to H/3 4/3 1.33 m above the bottom of the wall and will be inclined at an angle of 15 ( d) to the back face of the wall. ■ Table 13.7 Values of Ka [Eq. (13.69)] (Note: d 23 f¿ ) Fⴕ (deg) 0 5 10 15 20 25 0 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0.3213 0.3091 0.2973 0.2860 0.2750 0.2645 0.2543 0.2444 0.2349 0.2257 0.2168 0.2082 0.1998 0.1918 0.1840 0.3431 0.3295 0.3165 0.3039 0.2919 0.2803 0.2691 0.2583 0.2479 0.2379 0.2282 0.2188 0.2098 0.2011 0.1927 0.3702 0.3548 0.3400 0.3259 0.3123 0.2993 0.2868 0.2748 0.2633 0.2522 0.2415 0.2313 0.2214 0.2119 0.2027 0.3588 0.3467 0.3349 0.3235 0.3125 0.3019 0.2916 0.2816 0.2719 0.2626 0.2535 0.2447 0.2361 0.2278 0.2197 0.3845 0.3709 0.3578 0.3451 0.3329 0.3211 0.3097 0.2987 0.2881 0.2778 0.2679 0.2582 0.2489 0.2398 0.2311 0.4164 0.4007 0.3857 0.3713 0.3575 0.3442 0.3314 0.3190 0.3072 0.2957 0.2846 0.2740 0.2636 0.2537 0.2441 0.4007 0.3886 0.3769 0.3655 0.3545 0.3439 0.3335 0.3235 0.3137 0.3042 0.2950 0.2861 0.2774 0.2689 0.2606 0.4311 0.4175 0.4043 0.3916 0.3792 0.3673 0.3558 0.3446 0.3338 0.3233 0.3131 0.3033 0.2937 0.2844 0.2753 0.4686 0.4528 0.4376 0.4230 0.4089 0.3953 0.3822 0.3696 0.3574 0.3456 0.3342 0.3231 0.3125 0.3021 0.2921 0.4481 0.4362 0.4245 0.4133 0.4023 0.3917 0.3813 0.3713 0.3615 0.3520 0.3427 0.3337 0.3249 0.3164 0.3080 0.4843 0.4707 0.4575 0.4447 0.4324 0.4204 0.4088 0.3975 0.3866 0.3759 0.3656 0.3556 0.3458 0.3363 0.3271 0.5287 0.5128 0.4974 0.4826 0.4683 0.4545 0.4412 0.4283 0.4158 0.4037 0.3920 0.3807 0.3697 0.3590 0.3487 0.5026 0.4908 0.4794 0.4682 0.4574 0.4469 0.4367 0.4267 0.4170 0.4075 0.3983 0.3894 0.3806 0.3721 0.3637 0.5461 0.5325 0.5194 0.5067 0.4943 0.4823 0.4707 0.4594 0.4484 0.4377 0.4273 0.4172 0.4074 0.3978 0.3884 0.5992 0.5831 0.5676 0.5526 0.5382 0.5242 0.5107 0.4976 0.4849 0.4726 0.4607 0.4491 0.4379 0.4270 0.4164 0.5662 0.5547 0.5435 0.5326 0.5220 0.5117 0.5017 0.4919 0.4824 0.4732 0.4641 0.4553 0.4468 0.4384 0.4302 0.6190 0.6056 0.5926 0.5800 0.5677 0.5558 0.5443 0.5330 0.5221 0.5115 0.5012 0.4911 0.4813 0.4718 0.4625 0.6834 0.6672 0.6516 0.6365 0.6219 0.6078 0.5942 0.5810 0.5682 0.5558 0.5437 0.5321 0.5207 0.5097 0.4990 5 10 460 U (deg) A (deg) 13.11 Graphic Solution for Coulomb’s Active Earth Pressure 461 Table 13.7 (continued) Fⴕ (deg) 0 5 10 15 20 25 15 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0.4065 0.3881 0.3707 0.3541 0.3384 0.3234 0.3091 0.2954 0.2823 0.2698 0.2578 0.2463 0.2353 0.2247 0.2146 0.4602 0.4364 0.4142 0.3935 0.3742 0.3559 0.3388 0.3225 0.3071 0.2925 0.2787 0.2654 0.2529 0.2408 0.2294 0.4585 0.4397 0.4219 0.4049 0.3887 0.3732 0.3583 0.3442 0.3306 0.3175 0.3050 0.2929 0.2813 0.2702 0.2594 0.5205 0.4958 0.4728 0.4513 0.4311 0.4121 0.3941 0.3771 0.3609 0.3455 0.3308 0.3168 0.3034 0.2906 0.2784 0.5179 0.4987 0.4804 0.4629 0.4462 0.4303 0.4150 0.4003 0.3862 0.3726 0.3595 0.3470 0.3348 0.3231 0.3118 0.5900 0.5642 0.5403 0.5179 0.4968 0.4769 0.4581 0.4402 0.4233 0.4071 0.3916 0.3768 0.3626 0.3490 0.3360 0.5868 0.5672 0.5484 0.5305 0.5133 0.4969 0.4811 0.4659 0.4513 0.4373 0.4237 0.4106 0.3980 0.3858 0.3740 0.6714 0.6445 0.6195 0.5961 0.5741 0.5532 0.5335 0.5148 0.4969 0.4799 0.4636 0.4480 0.4331 0.4187 0.4049 0.6685 0.6483 0.6291 0.6106 0.5930 0.5761 0.5598 0.5442 0.5291 0.5146 0.5006 0.4871 0.4740 0.4613 0.4491 0.7689 0.7406 0.7144 0.6898 0.6666 0.6448 0.6241 0.6044 0.5856 0.5677 0.5506 0.5342 0.5185 0.5033 0.4888 0.7670 0.7463 0.7265 0.7076 0.6895 0.6721 0.6554 0.6393 0.6238 0.6089 0.5945 0.5805 0.5671 0.5541 0.5415 0.8880 0.8581 0.8303 0.8043 0.7799 0.7569 0.7351 0.7144 0.6947 0.6759 0.6579 0.6407 0.6242 0.6083 0.5930 20 13.11 U (deg) A (deg) Graphic Solution for Coulomb’s Active Earth Pressure An expedient method for creating a graphic solution of Coulomb’s earth-pressure theory was given by Culmann (1875). Culmann’s solution can be used for any wall friction, regardless of irregularity of backﬁll and surcharges. Hence, it provides a powerful technique for estimating lateral earth pressure. The steps in Culmann’s solution of active pressure with granular backﬁll (c 0) are described next, with reference to Figure 13.23a: Step 1: Draw the features of the retaining wall and the backﬁll to a convenient scale. Step 2: Determine the value of c (degrees) 90 u d, where u the inclination of the back face of the retaining wall with the vertical, and d angle of wall friction. Table 13.8 Values of Ka [Eq. (13.69)] (Note: d f/2) Fⴕ (deg) 0 5 10 15 20 25 0 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0.3264 0.3137 0.3014 0.2896 0.2782 0.2671 0.2564 0.2461 0.2362 0.2265 0.2172 0.2081 0.1994 0.1909 0.1828 0.3477 0.3337 0.3202 0.3072 0.2946 0.2825 0.2709 0.2596 0.2488 0.2383 0.2282 0.2185 0.2090 0.1999 0.1911 0.3743 0.3584 0.3432 0.3286 0.3145 0.3011 0.2881 0.2757 0.2637 0.2522 0.2412 0.2305 0.2202 0.2103 0.2007 0.3629 0.3502 0.3379 0.3260 0.3145 0.3033 0.2925 0.2820 0.2718 0.2620 0.2524 0.2431 0.2341 0.2253 0.2168 0.3879 0.3737 0.3601 0.3470 0.3342 0.3219 0.3101 0.2986 0.2874 0.2767 0.2662 0.2561 0.2463 0.2368 0.2276 0.4187 0.4026 0.3872 0.3723 0.3580 0.3442 0.3309 0.3181 0.3058 0.2938 0.2823 0.2712 0.2604 0.2500 0.2400 0.4034 0.3907 0.3784 0.3665 0.3549 0.3436 0.3327 0.3221 0.3118 0.3017 0.2920 0.2825 0.2732 0.2642 0.2554 0.4327 0.4185 0.4048 0.3915 0.3787 0.3662 0.3541 0.3424 0.3310 0.3199 0.3092 0.2988 0.2887 0.2788 0.2693 0.4688 0.4525 0.4368 0.4217 0.4071 0.3930 0.3793 0.3662 0.3534 0.3411 0.3292 0.3176 0.3064 0.2956 0.2850 0.4490 0.4363 0.4241 0.4121 0.4005 0.3892 0.3782 0.3675 0.3571 0.3469 0.3370 0.3273 0.3179 0.3087 0.2997 0.4837 0.4694 0.4556 0.4422 0.4292 0.4166 0.4043 0.3924 0.3808 0.3695 0.3585 0.3478 0.3374 0.3273 0.3174 0.5261 0.5096 0.4936 0.4782 0.4633 0.4489 0.4350 0.4215 0.4084 0.3957 0.3833 0.3714 0.3597 0.3484 0.3375 0.5011 0.4886 0.4764 0.4645 0.4529 0.4415 0.4305 0.4197 0.4092 0.3990 0.3890 0.3792 0.3696 0.3602 0.3511 0.5425 0.5282 0.5144 0.5009 0.4878 0.4750 0.4626 0.4505 0.4387 0.4272 0.4160 0.4050 0.3944 0.3840 0.3738 0.5928 0.5761 0.5599 0.5442 0.5290 0.5143 0.5000 0.4862 0.4727 0.4597 0.4470 0.4346 0.4226 0.4109 0.3995 0.5616 0.5492 0.5371 0.5253 0.5137 0.5025 0.4915 0.4807 0.4702 0.4599 0.4498 0.4400 0.4304 0.4209 0.4117 0.6115 0.5972 0.5833 0.5698 0.5566 0.5437 0.5312 0.5190 0.5070 0.4954 0.4840 0.4729 0.4620 0.4514 0.4410 0.6719 0.6549 0.6385 0.6225 0.6071 0.5920 0.5775 0.5633 0.5495 0.5361 0.5230 0.5103 0.4979 0.4858 0.4740 5 10 462 U (deg) A (deg) 13.11 Graphic Solution for Coulomb’s Active Earth Pressure 463 Table 13.8 (continued) U (deg) A (deg) Fⴕ (deg) 0 5 10 15 20 25 15 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0.4095 0.3908 0.3730 0.3560 0.3398 0.3244 0.3097 0.2956 0.2821 0.2692 0.2569 0.2450 0.2336 0.2227 0.2122 0.4614 0.4374 0.4150 0.3941 0.3744 0.3559 0.3384 0.3218 0.3061 0.2911 0.2769 0.2633 0.2504 0.2381 0.2263 0.4594 0.4402 0.4220 0.4046 0.3880 0.3721 0.3568 0.3422 0.3282 0.3147 0.3017 0.2893 0.2773 0.2657 0.2546 0.5188 0.4940 0.4708 0.4491 0.4286 0.4093 0.3910 0.3736 0.3571 0.3413 0.3263 0.3120 0.2982 0.2851 0.2725 0.5159 0.4964 0.4777 0.4598 0.4427 0.4262 0.4105 0.3953 0.3807 0.3667 0.3531 0.3401 0.3275 0.3153 0.3035 0.5844 0.5586 0.5345 0.5119 0.4906 0.4704 0.4513 0.4331 0.4157 0.3991 0.3833 0.3681 0.3535 0.3395 0.3261 0.5812 0.5611 0.5419 0.5235 0.5059 0.4889 0.4726 0.4569 0.4417 0.4271 0.4130 0.3993 0.3861 0.3733 0.3609 0.6608 0.6339 0.6087 0.5851 0.5628 0.5417 0.5216 0.5025 0.4842 0.4668 0.4500 0.4340 0.4185 0.4037 0.3894 0.6579 0.6373 0.6175 0.5985 0.5803 0.5627 0.5458 0.5295 0.5138 0.4985 0.4838 0.4695 0.4557 0.4423 0.4293 0.7514 0.7232 0.6968 0.6720 0.6486 0.6264 0.6052 0.5851 0.5658 0.5474 0.5297 0.5127 0.4963 0.4805 0.4653 0.7498 0.7284 0.7080 0.6884 0.6695 0.6513 0.6338 0.6168 0.6004 0.5846 0.5692 0.5543 0.5399 0.5258 0.5122 0.8613 0.8313 0.8034 0.7772 0.7524 0.7289 0.7066 0.6853 0.6649 0.6453 0.6266 0.6085 0.5912 0.5744 0.5582 20 Step 3: Step 4: Step 5: Step 6: Step 7: Draw a line BD that makes an angle f with the horizontal. Draw a line BE that makes an angle c with line BD. To consider some trial failure wedges, draw lines BC1 , BC2 , BC3 , . . ., BCn . Find the areas of ABC1 , ABC2 , ABC3 , . . . , ABCn . Determine the weight of soil, W, per unit length of the retaining wall in each of the trial failure wedges as follows: W1 (Area of ABC1) (g) (1) W2 (Area of ABC2) (g) (1) W3 (Area of ABC3) (g) (1) o Wn (Area of ABCn) (g) (1) 464 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Step 8: Adopt a convenient load scale and plot the weights W1 , W2 , W3 , . . . , Wn determined from step 7 on line BD. (Note: Bc1 W1 , Bc2 W2 , Bc3 W3 , . . . , Bcn Wn .) Step 9: Draw c1c1œ , c2c2œ , c3c3œ , . . . , cncnœ parallel to the line BE. (Note: c1œ , c2œ , c3œ , . . . , cnœ are located on lines BC1 , BC2 , BC3 , . . . , BCn , respectively.) Step 10: Draw a smooth curve through points c1œ , c2œ , c3œ , . . . , cnœ . This curve is called the Culmann line. Step 11: Draw a tangent BD to the smooth curve drawn in Step 10. BD is parallel to line BD. Let caœ be the point of tangency. Step 12: Draw a line cacaœ parallel to the line BE. Step 13: Determine the active force per unit length of wall as Pa 1Length of cacaœ 2 1Load scale2 Step 14: Draw a line Bcaœ Ca . ABCa is the desired failure wedge. Note that the construction procedure entails, in essence, drawing a number of force polygons for a number of trial wedges and ﬁnding the maximum value of the active force that the wall can be subjected to. For example, Figure 13.23b shows the force polygon for the failure wedge ABCa (similar to that in Figure 13.22b), in which W weight of the failure wedge of soil ABCa Pa active force on the wall F the resultant of the shear and normal forces acting along BCa b ⬔CaBF 1the angle that the failure wedge makes with the horizontal2 The force triangle (Figure 13.23b) is simply rotated in Figure 13.23a and is represented by the triangle Bcacaœ . Similarly, the force triangles Bc1c1œ , Bc2c2œ , Bc3c3œ , . . . , Bcncnœ correspond to the trial wedges ABC1 , ABC2 , ABC3 , . . . , ABCn . Ca C2 C1 C4 C3 A D Pa c D ca u c2 W c3 B c1 dʹ Pa c4 c3 c2 c c4 ca F b f c1 f' B F c E (a) (b) Figure 13.23 Culmann’s solution for active earth pressure 13.11 Graphic Solution for Coulomb’s Active Earth Pressure 465 C A O d O Figure 13.24 Approximate method for ﬁnding the point of application of the resultant active force Pa B The preceding graphic procedure is given in a step-by-step manner only to facilitate basic understanding. These problems can be easily and effectively solved by the use of computer programs. The Culmann solution provides us with only the magnitude of the active force per unit length of the retaining wall—not with the point of application of the resultant. The analytic procedure used to ﬁnd the point of application of the resultant can be tedious. For this reason, an approximate method, which does not sacriﬁce much accuracy, can be used. This method is demonstrated in Figure 13.24, in which ABC is the failure wedge determined by Culmann’s method. O is the center of gravity of the wedge ABC. If a line OO is drawn parallel to the surface of sliding, BC, the point of intersection of this line with the back face of the wall will give the point of application of Pa . Thus, Pa acts at O inclined at angle d with the normal drawn to the back face of the wall. Example 13.8 A 15-ft-high retaining wall with a granular soil backﬁll is shown in Figure 13.25. Given that g 100 lb/ft3, f 35, u 5, and d 10, determine the active thrust per foot length of the wall. Solution For this problem, c 90 u d 90 5 10 75. The graphic construction is shown in Figure 13.25. The weights of the wedges considered are as follows. Wedge Weight (lb/ft) ABC1 ABC2 ABC3 ABC4 ABC5 1 2 (4.38)(17.88)(100) 3,916 3,916 [ 12 (2.36)(18.56)](100) 6,106 6,106 [ 12 (2.24)(19.54)](100) 8,295 8,295 [ 21 (2.11)(20.77)](100) 10,486 10,486 [ 12 (1.97)(22.22)](100) 12,675 466 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb C3 2.1 1f t ft 4 ft ft 19. 5 18.5 6 4.38 ft 17.88 ft A 2.2 4 2.5 ft 2.5 ft C4 1.9 7f C5 t 22 .22 ft C2 2.3 6 ft C1 2.5 ft 20 .77 ft 2.5 ft 5 ft b 5l ,67 2 1 b P a 15 ft 6 lb 16 9 , 3 d 10 ,4 10 4,0 90 lb/ ft b l c3 6 ,10 l 86 c5 c4 17.5 ft 8,295 lb c2 c1 f 35 B c 90 u d 75 Length (ft) Weight (1000 lb) 0 1 2 3 4 5 0 1 2 3 4 5 u 5 Figure 13.25 Culmann’s solution for determining active thrust per unit length of wall In Figure 13.25, Bc1 3,916 lb Bc2 6,106 lb Bc3 8,295 lb Bc4 10,486 lb Bc5 12,675 lb The active thrust per unit length of the wall is 4,090 lb/ft. 13.12 ■ Coulomb’s Passive Pressure Figure 13.26a shows a retaining wall with a sloping cohensionless backﬁll similar to that considered in Figure 13.22a. The force polygon for equilibrium of the wedge ABC for the passive state is shown in Figure 13.26b. Pp is the notation for the passive force. Other 13.12 Coulomb’s Passive Pressure C 467 a A 90 u a W u H Pp d 90 u b f F b B (a) [180 (90 u d) ( b f)] Pp 90 u d F W b f (b) Figure 13.26 Coulomb’s passive pressure: (a) trial failure wedge; (b) force polygon notations used are the same as those for the active case (Section 13.10). In a procedure similar to the one that we followed in the active case [Eq. (13.68)], we get Pp 12 KpgH2 (13.70) where Kp Coulomb’s passive earth-pressure coefﬁcient, or Kp cos2 1f¿ u2 cos u cos1d¿ u2 c 1 2 sin1f¿ d¿ 2sin1f¿ a2 B cos1d¿ u2cos1a u 2 d 2 (13.71) 468 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Table 13.9 Values of Kp [Eq. 13.71] for u 0, a 0 Dⴕ (deg) S T Fⴕ (deg) 0 5 10 15 20 15 20 25 30 35 40 1.698 2.040 2.464 3.000 3.690 4.600 1.900 2.313 2.830 3.506 4.390 5.590 2.130 2.636 3.286 4.143 5.310 6.946 2.405 3.030 3.855 4.977 6.854 8.870 2.735 3.525 4.597 6.105 8.324 11.772 For a frictionless wall with the vertical back face supporting granular soil backﬁll with a horizontal surface (that is, u 0, a 0, and d 0), Eq. (13.71) yields Kp 1 sin f¿ f¿ tan2 a 45 b 1 sin f¿ 2 This relationship is the same as that obtained for the passive earth-pressure coefﬁcient in Rankine’s case, given by Eq. (13.22). The variation of Kp with f and d (for u 0 and a 0) is given in Table 13.9. We can see from this table that for given value of f, the value of Kp increases with the wall friction. 13.13 Active Force on Retaining Walls with Earthquake Forces Active Case (Granular Backﬁll) Coulomb’s analysis for active force on retaining walls discussed in Section 13.10 can be conveniently extended to include earthquake forces. To do so, let us consider a retaining wall of height H with a sloping granular backﬁll, as shown in Figure 13.27a. Let the unit weight and the friction angle of the granular soil retained by the wall be equal to g and f, respectively. Also, let d be the angle of friction between the soil and the wall. ABC is a trial failure wedge. The forces acting on the wedge are as follows: 1. 2. 3. 4. 5. Weight of the soil in the wedge, W Resultant of the shear and normal forces on the failure surface BC, F Active force per unit length of the wall, Pae Horizontal inertial force, khW Vertical inertial force, kvW Note that Horizontal component of earthquake acceleration g Vertical component of earthquake acceleration kv g kh where g acceleration due to gravity. (13.72) (13.73) 13.13 Active and Passive Forces on Retaining Walls with Earthquake Forces C 469 a A Pae k W g f d u khW H kW W W k hW d f Pae h F F B (a) (b) Figure 13.27 Active force on a retaining wall with earthquake forces The force polygon demonstrating these forces is shown in Figure 13.27b. The dynamic active force on the wall is the maximum value of Pae exerted by any wedge. This value can be expressed as Pae 12 gH2 11 kv 2Kaﬂ (13.74) where Kaﬂ cos2 1f¿ u b2 sin1d¿ f¿ 2sin1f¿ a b2 1/2 2 cos2 u cos b cos1d¿ u b2 e 1 c d f cos1d¿ u b2cos1u a2 (13.75) and b tan1 a kh b 1 kv (13.76) Note that with no inertia forces from earthquakes, b is equal to 0. Hence, Kaﬂ Ka as given in Eq. (13.69). Equations (13.74) and (13.75) generally are referred to as the MononobeOkabe equations (Mononobe, 1929; Okabe, 1926). The variation of Kaﬂ with u 0 and kv 0 is given in Table 13.10. Table 13.10 Values of Kaﬂ [Eq. (13.75)] with u 0 and kv 0 Fⴕ (deg) kh Dⴕ (deg) 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 28 30 35 40 45 0 0.427 0.508 0.611 0.753 1.005 0.397 0.473 0.569 0.697 0.890 0.328 0.396 0.478 0.581 0.716 0.268 0.382 0.400 0.488 0.596 0.217 0.270 0.334 0.409 0.500 0 5 0.457 0.554 0.690 0.942 — 0.423 0.514 0.635 0.825 — 0.347 0.424 0.522 0.653 0.855 0.282 0.349 0.431 0.535 0.673 0.227 0.285 0.356 0.442 0.551 0.1 0.2 0.3 0.4 0.5 0 10 0.497 0.623 0.856 — — 0.457 0.570 0.748 — — 0.371 0.461 0.585 0.780 — 0.299 0.375 0.472 0.604 0.809 0.238 0.303 0.383 0.486 0.624 0.1 0.2 0.3 0.4 0.5 f/2 0 0.396 0.485 0.604 0.778 1.115 0.368 0.452 0.563 0.718 0.972 0.306 0.380 0.474 0.599 0.774 0.253 0.319 0.402 0.508 0.648 0.207 0.267 0.340 0.433 0.552 0.1 0.2 0.3 0.4 0.5 f/2 5 0.428 0.537 0.699 1.025 — 0.396 0.497 0.640 0.881 — 0.326 0.412 0.526 0.690 0.962 0.268 0.342 0.438 0.568 0.752 0.218 0.283 0.367 0.475 0.620 0.1 0.2 0.3 0.4 0.5 f/2 10 0.472 0.616 0.908 — — 0.433 0.562 0.780 — — 0.352 0.454 0.602 0.857 — 0.285 0.371 0.487 0.656 0.944 0.230 0.303 0.400 0.531 0.722 2 f 3 0 0.393 0.366 0.306 0.256 0.212 0.486 0.612 0.801 1.177 0.454 0.572 0.740 1.023 0.384 0.486 0.622 0.819 0.326 0.416 0.533 0.693 0.276 0.357 0.462 0.600 0.427 0.395 0.327 0.271 0.224 0.541 0.714 1.073 — 0.501 0.655 0.921 — 0.418 0.541 0.722 1.034 0.350 0.455 0.600 0.812 0.294 0.386 0.509 0.679 0.472 0.434 0.354 0.290 0.237 0.625 0.942 — — 0.570 0.807 — — 0.463 0.624 0.909 — 0.381 0.509 0.699 1.037 0.317 0.423 0.573 0.800 0.1 A (deg) 0.2 0.3 0.4 0.5 0.1 2 f 3 5 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 470 2 f 3 10 13.13 Active and Passive Forces on Retaining Walls with Earthquake Forces 471 Seed and Whitman (1970) provided a simple procedure to obtain the value of Ka from the standard charts of Ka [see Eq. (13.69)]. This procedure is explained next. Referring to Eqs. (13.68) and (13.69), we can write Pa 1 1 1 gH 2 Ka a gH 2b a b 1A c 2 2 2 cos2 u (13.77) where cos2 1fœ u 2 A c Ka cos2 u œ œ sin 1 d f 2 sin 1f œ a 2 2 cos 1d u 2 c 1 d œ B cos 1d u2 cos 1u a2 (13.78) œ 2 Now, referring to Eqs. (13.74) and (13.75), we can write Pae 1 1 1 œœ b 1A m 2 gH 2 11 kn 2Ka a gH 2 b 11 kn 2 a 2 2 2 cos u cos b (13.79) where œœ A m Ka cos2 u cos b cos 2 1f u b 2 œ 2 sin1d f 2 sin 1f a b 2 cos 1d u b 2 c 1 d œ B cos 1d u b2 cos 1u a2 œ œ œ œ (13.80) Now, let us define u* u b (13.81) a* a b (13.82) and Substituting Eqs. (13.81) and (13.82) into the relationship for Am [that is, Eq. (13.80)], we obtain cos2 1f u* 2 œ Am cos 1d u 2 c 1 œ * sin 1d f¿ 2 sin 1f¿ a* 2 œ B cos1d œ u* 2 cos 1u* a* 2 d 2 (13.83) Comparing Eqs. (13.78) and (13.83), we have Am Ac(*, *) Ka(*, *)cos2 * (13.84) 472 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Hence, Pae 1 1 Ka 1u*, a* 2 cos2 u* gH 2 11 kn 2 2 cos2 u cos b or Pae 3Pa 1u*, a* 2 4 11 kn 2 a cos 2 u* b cos2 u cos b 3 Pa 1u*, a* 2 4 11 kn 2 c cos 2 1 u b 2 d cos 2 u cos b (13.85) The term Pa(*, * ) in Eq. (13.85) is the active earth pressure on an imaginary retaining wall with a wall inclination of * and backfill slope inclination of * . The value of Ka can be obtained from standard charts or tables, such as Tables 13.7 and 13.8. Considering the active force relation given by Eqs. (13.74) through (13.76), we ﬁnd that the term sin (f a ) in Eq. (13.75) has two important implications. First, if f a b 0 (i.e., negative), no real solution of Kaﬂ is possible. Physically, this implies that an equilibrium condition will not exist. Hence, for stability, the limiting slope of the backﬁll may be given as a f¿ b (13.86) For no earthquake condition, b 0; for stability, Eq. (13.86) gives the familiar relation a f¿ (13.87) Second, for horizontal backﬁll, a 0; for stability, b f¿ (13.88) Because b tan1[kh/(1 kv)], for stability, combining Eqs. (13.76) and (13.88) results in kh 11 kv 2tan f¿ (13.89) Hence, the critical value of the horizontal acceleration can be deﬁned as kh 1cr2 11 kv 2tan f¿ where kh(cr) critical of horizontal acceleration (Figure 13.28). (13.90) 13.13 Active and Passive Forces on Retaining Walls with Earthquake Forces k 0 0.5 0.2 473 0.4 0.4 kh(cr) 0.3 0.2 0.1 0.0 0 10 20 Soil friction angle, f (deg) 30 40 Figure 13.28 Critical values of horizontal acceleration (Eq. 13.90) Location of Line of Action of Resultant Force, Pae Seed and Whitman (1970) proposed a simple procedure to determine the location of the line of action of the resultant, Pae . Their method is as follows: 1. Let Pae Pa ¢Pae (13.91) where Pa Coulomb’s active force as determined from Eq. 113.682 ¢Pae additional active force caused by the earthquake effect 2. 3. 4. 5. Calculate Pa [Eq. (13.68)]. Calculate Pae [Eq. (13.74)]. Calculate Pae Pae Pa . According to Figure 13.29, Pa will act at a distance of H/3 from the base of the wall. Also, Pae will act at a distance of 0.6H from the base of the wall. 6. Calculate the location of Pae as Pa a z H b ¢Pae 10.6H2 3 Pae (13.92) where z distance of the line of action of Pae from the base of the wall. Note that the line of action of Pae will be inclined at an angle of d to the normal drawn to the back face of the retaining wall. It is very important to realize that this method of determining Pae is approximate and does not actually model the soil dynamics. 474 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Pae Pae d Pa d H d 0.6 H H 3 Figure 13.29 Location of the line of action of Pae Active Case (c ⴕ– Fⴕ Backﬁll) The Mononobe–Okabe equation for estimating Pae for cohesionless backﬁll also can be extended to c–f soil (Prakash and Saran, 1966; Saran and Prakash, 1968). Figure 13.30 shows a retaining wall of height H with a horizontal c– f backﬁll. The depth of tensile crack that may develop in a c– f soil was given in Eq. (13.48) as zo 2c¿ g1Ka where Ka tan2(45 f/2). A E zo B D khW u C g f c C H H zo W u d i f Pae F C Figure 13.30 Trial failure wedge behind a retaining wall with a c– f backﬁll 13.13 Active and Passive Forces on Retaining Walls with Earthquake Forces 475 Referring to Figure 13.30, the forces acting on the soil wedge (per unit length of the wall) are as follows: • • • • • • The weight of the wedge ABCDE, W Resultant of the shear and normal forces on the failure surface CD, F Active force, Pae Horizontal inertia force, khW Cohesive force along CD, C c( CD ) Adhesive force along BC, C c( BC ) It is important to realize that the following two assumptions have been made: 1. The vertical inertia force (kvW) has been taken to be zero. 2. The unit adhesion along the soil–wall interface (BC) has been taken to be equal to the cohesion (c) of the soil. Considering these forces, we can show that œ œ c¿1H zo 2Nac Pae g1H zo 2 2Nag (13.93) where œ Nac œ Nag cos h¿ sec u cos f¿ sec i sin1h¿ d¿ 2 (13.94) 31n 0.52 1tan u tan i2 n2 tan u4 3cos1i f¿2 kh sin1i f¿24 sin1h¿ d¿ 2 (13.95) in which h¿ u i f¿ zo n H zo (13.96) (13.97) œ œ The values of Nac and Nag can be determined by optimizing each coefﬁcient separately. Thus, Eq. (13.93) gives the upper bound of Pae . For the static condition, kh 0. Thus, Pae g1H zo 2 2Nag c¿1H zo 2Nac (13.98) The relationships for Nac and Nag can be determined by substituting kh 0 into Eqs. (13.94) and (13.95). Hence, œ Nac Nac Nag œ Nag l cos h¿ sec u cos f¿ sec i sin1h¿ d¿ 2 3 1n 0.5 2 1tan u tan i2 n2 tan u 4 cos1i f¿ 2 sin1h¿ d¿ 2 (13.99) (13.100) The variations of Nac , Nag , and l with f and u are shown in Figures 13.31 through 13.34. 4.5 4.5 4.0 4.0 20 u Earth-pressure coefﬁcient, Nac 3.5 3.5 15 10 3.0 3.0 5 u 0 2.5 2.5 2.0 2.0 1.5 1.5 u 5 10 1.0 15 1.0 u 20 0.5 0.5 0 5 10 15 20 25 f (deg) 30 35 40 45 œ Figure 13.31 Variation of Nac N ac with f and u (Based on Prakash and Saran, 1966, and Saran and Prakash, 1968) 1.0 1.0 Earth-pressure coefﬁcient, Nag n 0.2 0.8 u 20 0.8 15 10 5 0.6 0.6 0 0.4 0.4 u 5 0.2 10 0.2 15 0.0 0 5 10 15 20 0.0 20 25 f (deg) 30 35 40 45 Figure 13.32 Variation of Nag with f and u (n 0.2) (Based on Prakash and Saran, 1966, and Saran and Prakash, 1968) 476 1.0 1.0 n0 0.8 Earth-pressure coefﬁcient, Nag 0.8 0.6 0.6 u 20 15 10 5 0.4 0.4 0 u 5 10 15 20 0.2 0.2 0.0 0.0 0 5 10 15 20 25 f (deg) 30 35 40 45 Figure 13.33 Variation of Nag with f and u (n 0) (Based on Prakash and Saran, 1966, and Saran and Prakash, 1968) 2.0 1.9 u 0 1.8 10 20 0 1.7 1.6 10 20 0 10 20 0 10 20 0 10 20 l 1.5 1.4 1.3 1.2 1.1 1.0 0 kh : 10 20 30 40 Angle of internal friction, f 0.25 0.10 0.20 0.05 50 0.15 Figure 13.34 Variation of l with kh , f, and u (Based on Prakash and Saran, 1966, and Saran and Prakash, 1968) 477 478 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Example 13.9 For a retaining wall with a cohesionless soil backﬁll, g 15.5 kN/m3, f 30, d 15, u 0, a 0, H 4 m, kv 0, and kh 0.2. Determine Pae . Also determine the location of the resultant line of action of Pae—that is, z. Solution From Eqs. (13.81) and (13.82), u* u b a* a b b tan 1 a kh 0.2 b tan1 a b 11.3° 1 kn 10 So, * 0 + 11.3° 11.3° * 0 + 11.3° 11.3° From Eq. (13.85), Pae 3Pa 1u*, a* 2 4 11 kn 2 a cos2 u* b cos2 u cos2 b cos2 1u b 2 3Pa 1u*, a* 2 4 11 kn 2 c d cos u cos2 b Pa 1u*, a* 2 12 gH2Ka where Ka is a function of * and *. Since / 15/30 0.5, we will use Table 13.8. For * * 11.3°, the value of Ka 0.452. Thus, Pae 12 115.52 142 2 10.452 2 11 02 c cos 2 10 11.32 1 cos 02 1 cos 20 2 d 56.05 kN/m [Note: We can also get the same values of Pae using Eq. (13.75) and Kae from Table 13.10.] We now locate the resultant line of action. From Eq. (13.68), Pa 12 KagH 2 For f 30 and d 15, Ka 0.3014 (Table 13.8), so Pa 12 10.30142 115.5 2 14 2 2 37.37 kN/m Hence, Pae 56.05 37.37 18.68 kN/m. From Eq. (13.92), Pa a z 4 H 137.37 2 a b 118.682 12.42 b ¢Pae 10.6H2 3 3 1.69 m Pae 56.05 ■ 13.14 Common Types of Retaining Walls in the Field 479 Example 13.10 For a retaining wall, the following are given. • H 28 ft • u 10° • f 20° • c 210 lb/ft2 • g 118 lb/ft3 • kh 0.1 Determine the magnitude of the active force, Pae . Solution From Eq. (13.48), zo 2c¿ g1Ka 2c¿ f¿ b g tan a 45 2 122 1210 2 1118 2tan a 45 20 b 2 5.08 ft From Eq. (13.97), n zo 5.08 0.22 0.2 H zo 28 5.08 From Eqs. (13.93), (13.99), and (13.100), Pae g1H zo 2 2 1lNag 2 c¿1H zo 2Nac For u 10, f 20, kh 0.1, and n 0.2, Nac 1.60 Nag 0.375 l 1.17 1Figure 13.312 1Figure 13.322 1Figure 13.342 Thus, Pae 11182 128 5.082 2 11.17 0.375 2 12102 128 5.082 11.602 19,496 lb/ft 13.14 ■ Common Types of Retaining Walls in the Field The preceding sections present the theoretical concepts for estimating the lateral earth pressure for retaining walls. In practice, the common types of retaining walls constructed can be divided into two major categories: rigid retaining walls and mechanically stabilized earth (MSE) walls. The following is a brief overview of the various types of retaining walls constructed in the ﬁeld. Rigid Retaining Walls Under this category, the wall may be subdivided to four categories. They are: 1. Gravity retaining walls 2. Semigravity retaining walls 480 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb 3. Cantilever retaining walls 4. Counterfort retaining walls Gravity retaining walls (Figure 13.35a) are constructed with plain concrete or stone masonry. They depend on their own weight and any soil resting on the masonry for stability. This type of construction is not economical for high walls. In many cases, a small amount of steel may be used for the construction of gravity walls, thereby minimizing the size of wall sections. Such walls generally are referred to as semigravity walls (Figure 13.35b). Cantilever retaining walls (Figure 13.35c) are made of reinforced concrete that consists of a thin stem and a base slab. This type of wall is economical to a height of about 8 m (25 ft). Plain concrete or stone masonry Reinforcement (a) Gravity wall (b) Semigravity wall Reinforcement (c) Cantilever wall Counterfort (d) Counterfort wall Figure 13.35 Types of retaining wall 13.14 Common Types of Retaining Walls in the Field 481 Counterfort retaining walls (Figure 13.35d) are similar to cantilever walls. At regular intervals, however, they have thin, vertical concrete slabs known as counterforts that tie the wall and the base slab together. The purpose of the counterforts is to reduce the shear and the bending moments. Mechanically Stabilized Earth (MSE) Walls Mechanically stabilized earth walls are ﬂexible walls, and they are becoming more common nowadays. The main components of these types of walls are • • • Backﬁll—which is granular soil Reinforcement in the backﬁll A cover (or skin) on the front face The reinforcement can be thin galvanized steel strips, geogrid, or geotextile (see Chapter 17 for descriptions of geogrid and geotextile). In most cases, precast concrete slabs are used as skin. The slabs are grooved to ﬁt into each other so that soil cannot ﬂow between the joints. Thin galvanized steel also can be used as skin when the reinforcements are metallic strips. When metal skins are used, they are bolted together, and reinforcing strips are placed between the skins. Figure 13.36 shows an MSE wall with metallic strips as reinforcement along with a metal skin. Figure 13.37 shows some typical MSE walls with geogrid reinforcement in the backﬁll. The retaining walls are designed using various earth-pressure theories described in this chapter. For actual wall design, refer to any foundation engineering book (for instance, Das, 2007). Skin SH Tie S Soil Figure 13.36 MSE wall with metallic strip reinforcement and metallic skin 482 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb Geogrids—biaxial Geogrids—uniaxial Geogrids Pinned connection (a) Precast concrete panel Gabion facing Geogrids Leveling pad (c) (b) Figure 13.37 Typical schematic diagrams of retaining walls with geogrid reinforcement: (a) geogrid wraparound wall; (b) wall with gabion facing; (c) concrete-panel-faced wall 13.15 Summary and General Comments This chapter covers the general topics of lateral earth pressure, including the following: 1. 2. 3. 4. At-rest earth pressure Active earth pressure—Rankine’s and Coulomb’s Passive earth pressure—Rankine’s and Coulomb’s Active earth pressure, which includes earthquake forces. This is an extension of Coulomb’s theory For design, it is important to realize that the lateral active pressure on a retaining wall can be calculated using Rankine’s theory only when the wall moves sufﬁciently outward by rotation about the toe of the footing or by deﬂection of the wall. If sufﬁcient wall movement cannot occur (or is not allowed to occur), then the lateral earth pressure will be greater than the Rankine active pressure and sometimes may be closer to the at-rest earth pressure. Hence, proper selection of the lateral earth-pressure coefﬁcient is crucial for safe Problems 483 and proper design. It is a general practice to assume a value for the soil friction angle (f) of the backﬁll in order to calculate the Rankine active pressure distribution, ignoring the contribution of the cohesion (c). The general range of f used for the design of retaining walls is given in the following table. Soil friction angle, Fⴕ (deg) Soil type Soft clay Compacted clay Dry sand and gravel Silty sand 0–15 20–30 30–40 20–30 In Section 13.8, we saw that the lateral earth pressure on a retaining wall is increased greatly in the presence of a water table above the base of the wall. Most retaining walls are not designed to withstand full hydrostatic pressure; hence, it is important that adequate drainage facilities are provided to ensure that the backﬁll soil does not become fully saturated. This can be achieved by providing weepholes at regular intervals along the length of the wall. Problems 13.1 through 13.4 Figure 13.38 shows a retaining wall that is restrained from yielding. For each problem, determine the magnitude of the lateral earth force per unit length of the wall. Also, state the location of the resultant, z, measured from the bottom of the wall. Problem H F (deg) G Overconsolidation ratio, OCR 13.1 13.2 13.3 13.4 7m 15 ft 6m 20 ft 38 32 35 30 17 kN/m3 110 lb/ft3 18 kN/m3 105 lb/ft3 2.5 2 2 1 Sand H Figure 13.38 Unit weight g (or density r) f c 0 d (angle of wall friction) 0 484 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb 13.5 through 13.8 Assume that the retaining wall shown in Figure 13.38 is frictionless. For each problem, determine the Rankine active force per unit length of the wall, the variation of active earth pressure with depth, and the location of the resultant. Problem H Fⴕ (deg) G 13.5 13.6 13.7 13.8 15 ft 18 ft 4m 5m 30 32 36 40 105 lb/ft3 100 lb/ft3 18 kN/m3 17 kN/m3 13.9 through 13.12 Assume that the retaining wall shown in Figure 13.38 is frictionless. For each problem, determine the Rankine passive force per unit length of the wall, the variation of lateral earth pressure with depth, and the location of the resultant. Problem H Fⴕ (deg) G 13.9 13.10 13.11 13.12 8 ft 10 ft 5m 4m 34 36 35 30 110 lb/ft3 105 lb/ft3 14 kN/m3 15 kN/m3 13.13 through 13.15 A retaining wall is shown in Figure 13.39. For each problem, determine the Rankine active force, Pa , per unit length of the wall and the location of the resultant. F1œ F2œ (deg) (deg) Problem H H1 G1 G2 13.13 13.14 13.15 10 ft 20 ft 6m 5 ft 6 ft 3m 105 lb/ft3 110 lb/ft3 15.5 kN/m3 122 lb/ft3 126 lb/ft3 19.0 kN/m3 30 34 30 Surcharge q H1 Sand g1 f1 c1 0 Groundwater table H Sand g2 (saturated unit weight) f2 c2 0 Frictionless wall Figure 13.39 30 34 36 q 0 300 lb/ft3 15 kN/m3 Problems 485 13.16 Refer to the frictionless wall shown in Figure 13.10. Given: H 5 m, u 10, and a 15. For the granular backﬁll, f 35 and g 15 kN/m3. a. Determine the magnitude of active earth pressure, saœ , at the bottom of the wall. Also, state the direction of application of saœ . b. Determine Pa per meter length of the wall and its location and direction. 13.17 Refer to the retaining wall shown in Figure 13.40. Given: g 16.5 kN/m3, H 4 m, and d 0. Determine the magnitude of the passive force per unit length of the wall, its location, and its direction of the line of action. 13.18 For the retaining wall described in Problem 13.13, determine the Rankine passive force per unit length of the wall and the location of the resultant. 13.19 A 15 ft high retaining wall with a vertical back face retains a homogeneous saturated soft clay. The saturated unit weight of the clay is 122 lb/ft3. Laboratory tests showed that the undrained shear strength, cu , of the clay is equal to 350 lb/ft2. a. Make the necessary calculations and draw the variation of Rankine’s active pressure on the wall with depth. b. Find the depth up to which a tensile crack can occur. c. Determine the total active force per unit length of the wall before the tensile crack occurs. d. Determine the total active force per unit length of the wall after the tensile crack occurs. Also, ﬁnd the location of the resultant. 13.20 Redo Problem 13.19 assuming that the backﬁll is supported by a surcharge of 200 lb/ft2. 13.21 A 5-m-high retaining wall with a vertical back face has a c f soil for backﬁll. For the backﬁll, g 19 kN/m3, c 26 kN/m2, and f 16. Considering the existence of the tensile crack, determine the active force, Pa , on the wall for Rankine’s active state. 13.22 Consider a 5-m-high frictionless retaining wall with a vertical back and inclined backﬁll. The inclination of the backﬁll with the horizontal, a, is 10. For the backﬁll, given: g 18 kN/m3, f 25, and c 5 kN/m3. Determine the Rankine active force, Pa , per unit length of the wall after the occurrence of the tensile crack. 13.23 Consider the retaining wall shown in Figure 13.40. The height of the wall is 5 m, and the unit weight of the sand backﬁll is 18 kN/m3. Using Coulomb’s equation, calculate the active force, Pa , on the wall for the following values of the angle of wall friction. a. d 15 b. d 20 Comment on the direction and location of the resultant. Sand u 10 H Unit weight g (or density r) c 0 f 30 d (wall friction) Figure 13.40 486 Chapter 13: Lateral Earth Pressure: At-Rest, Rankine, and Coulomb 1 n Cohesionless soil Unit weight g (or density r) c 0 f d (angle of wall friction) H H1 b Figure 13.41 13.24 Referring to Figure 13.41, determine Coulomb’s active force, Pa , per unit length of the wall for the following cases. a. H 15 ft, b 85, n 1, H1 20 ft, g 128 lb/ft3, f 38, d 20 b. H 5.5 m, b 80, n 1, H1 6.5 m, r 1680 kg/m3, f 30, d 30 Use Culmann’s graphic construction procedure. 13.25 Refer to Figure 13.27. Given: H 4 m, u 10, a 10, g 15 kN/m3, f 35, d 23 f, kh 0.15, and kv 0. Determine the active force, Pae , per unit length of the retaining wall. 13.26 Refer to Figure 13.30. Given: H 6 m, u 10, f 15, c 20 kN/m2, g 19 kN/m3, and kh 0.15. Using the method described in Section 13.13, determine Pae . Assume that the depth of tensile crack is zero. References CHU, S. C. (1991). “Rankine Analysis of Active and Passive Pressures on Dry Sand,” Soils and Foundations, Vol. 31, No. 4, 115–120. COULOMB, C. A. (1776). “Essai sur une Application des Règles de Maximis et Minimis a quelques Problèmes de Statique, relatifs a l’Architecture,” Mem. Roy. des Sciences, Paris, Vol. 3, 38. CULMANN, C. (1875). Die graphische Statik, Meyer and Zeller, Zurich. JAKY, J. (1944). “The Coefﬁcient of Earth Pressure at Rest,” Journal of the Society of Hungarian Architects and Engineers, Vol. 7, 355–358. MASSARSCH, K. R. (1979). “Lateral Earth Pressure in Normally Consolidated Clay,” Proceedings of the Seventh European Conference on Soil Mechanics and Foundation Engineering, Brighton, England, Vol. 2, 245–250. MAYNE, P. W., and KULHAWY, F. H. (1982). “Ko—OCR Relationships in Soil,” Journal of the Geotechnical Division, ASCE, Vol. 108, No. 6, 851–872. MAZINDRANI, Z. H., and GANJALI, M. H. (1997). “Lateral Earth Pressure Problem of Cohesive Backﬁll with Inclined Surface,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 123, No. 2, 110–112. OKABE, S. (1926). “General Theory of Earth Pressure,” Journal of the Japanese Society of Civil Engineers, Tokyo, Vol. 12, No. 1. References 487 PRAKASH, S., and SARAN, S. (1966). “Static and Dynamic Earth Pressure Behind Retaining Walls,” Proceedings, 3rd Symposium on Earthquake Engineering, Roorkee, India, Vol. 1, 277–288. RANKINE, W. M. J. (1857). “On Stability on Loose Earth,” Philosophic Transactions of Royal Society, London, Part I, 9–27. SARAN, S., and PRAKASH, S. (1968). “Dimensionless Parameters for Static and Dynamic Earth Pressure for Retaining Walls,” Indian Geotechnical Journal, Vol. 7, No. 3, 295–310. SEED, H. B., and WHITMAN, R. V. (1970). “Design of Earth Retaining Structures for Dynamic Loads,” Proceedings, Specialty Conference on Lateral Stresses in the Ground and Design of Earth Retaining Structures, ASCE, 103–147. SHERIF, M. A., FANG, Y. S., and SHERIF, R. I. (1984). “KA and KO Behind Rotating and NonYielding Walls,” Journal of Geotechnical Engineering, ASCE, Vol. 110, No. GT1, 41–56. 14 Lateral Earth Pressure: Curved Failure Surface In Chapter 13, we considered Coulomb’s earth pressure theory, in which the retaining wall was considered to be rough. The potential failure surfaces in the backﬁll were considered to be planes. In reality, most failure surfaces in soil are curved. There are several instances where the assumption of plane failure surfaces in soil may provide unsafe results. Examples of these cases are the estimation of passive pressure and braced cuts. This chapter describes procedures by which passive earth pressure and lateral earth pressure on braced cuts can be estimated using curved failure surfaces in the soil. 14.1 Retaining Walls with Friction In reality, retaining walls are rough, and shear forces develop between the face of the wall and the backﬁll. To understand the effect of wall friction on the failure surface, let us consider a rough retaining wall AB with a horizontal granular backﬁll as shown in Figure 14.1. In the active case (Figure 14.1a), when the wall AB moves to a position AB, the soil mass in the active zone will be stretched outward. This will cause a downward motion of the soil relative to the wall. This motion causes a downward shear on the wall (Figure 14.1b), and it is called a positive wall friction in the active case. If d is the angle of friction between the wall and the backﬁll, then the resultant active force Pa will be inclined at an angle d to the normal drawn to the back face of the retaining wall. Advanced studies show that the failure surface in the backﬁll can be represented by BCD, as shown in Figure 14.1a. The portion BC is curved, and the portion CD of the failure surface is a straight line. Rankine’s active state exists in the zone ACD. Under certain conditions, if the wall shown in Figure 14.1a is forced downward with reference to the backﬁll, the direction of the active force, Pa , will change as shown in Figure 14.1c. This is a situation of negative wall friction (d) in the active case. Figure 14.1c also shows the nature of the failure surface in the backﬁll. The effect of wall friction for the passive state is shown in Figures 14.1d and e. When the wall AB is pushed to a position AB (Figure 14.1d), the soil in the passive zone will be 488 45 f 2 A f 45 2 D A H H d Pa 3 C Soil friction angle f B (a) Active case (d) 45 f 2 A (b) 45 A Pa f 2 H Soil friction angle f d H 3 B (c) Active case (d) 45 f 2 A A 45 D Pp f 2 H d C H 3 Soil friction angle f B (e) (d) Passive case (d) 45 f 2 45 A A' f 2 H Soil friction angle f H 3 d Pp B (f) Passive case (d) Figure 14.1 Effect of wall friction on failure surface 489 490 Chapter 14: Lateral Earth Pressure: Curved Failure Surface compressed. The result is an upward motion relative to the wall. The upward motion of the soil will cause an upward shear on the retaining wall (Figure 14.1e). This is referred to as positive wall friction in the passive case. The resultant passive force, Pp , will be inclined at an angle d to the normal drawn to the back face of the wall. The failure surface in the soil has a curved lower portion BC and a straight upper portion CD. Rankine’s passive state exists in the zone ACD. If the wall shown in Figure 14.1d is forced upward relative to the backﬁll by a force, then the direction of the passive force Pp will change as shown in Figure 14.1f. This is negative wall friction in the passive case (d). Figure 14.1f also shows the nature of the failure surface in the backﬁll under such a condition. For practical considerations, in the case of loose granular backﬁll, the angle of wall friction d is taken to be equal to the angle of friction of soil, f. For dense granular backﬁlls, d is smaller than f and is in the range of f/2 d (2/3)f. The assumption of plane failure surface gives reasonably good results while calculating active earth pressure. However, the assumption that the failure surface is a plane in Coulomb’s theory grossly overestimates the passive resistance of walls, particularly for d f/2. 14.2 Properties of a Logarithmic Spiral The case of passive pressure shown in Figure 14.1d (case of d) is the most common one encountered in design and construction. Also, the curved failure surface represented by BC in Figure 14.1d is assumed most commonly to be the arc of a logarithmic spiral. In a similar manner, the failure surface in soil in the case of braced cuts (Sections 14.8 to 14.9) also is assumed to be the arc of a logarithmic spiral. Hence, some useful ideas concerning the properties of a logarithmic spiral are described in this section. The equation of the logarithmic spiral generally used in solving problems in soil mechanics is of the form r ro eu tan f¿ (14.1) where r radius of the spiral ro starting radius at u 0 f¿ angle of friction of soil u angle between r and ro The basic parameters of a logarithmic spiral are shown in Figure 14.2, in which O is the center of the spiral. The area of the sector OAB is given by A 0 u 1 r1r du 2 2 (14.2) 14.2 Properties of a Logarithmic Spiral 491 O u r0 r1 r 90 90 n m A B f Figure 14.2 General parameters of a logarithmic spiral Substituting the values of r from Eq. (14.1) into Eq. (14.2), we get A u1 0 1 2 2u tan f¿ roe du 2 r21 r2o 4 tan f¿ (14.3) The location of the centroid can be deﬁned by the distances m and n (Figure 14.2), measured from OA and OB, respectively, and can be given by the following equations (Hijab, 1956): tan f¿ 4 ≥ m ro 3 19 tan2 f¿ 1 2 tan f¿ 4 n ro ≥ 3 19 tan2 f¿ 1 2 a a r1 3 b 13 tan f¿ sin u cos u 2 1 ro ¥ r1 2 a b 1 ro (14.4) r1 3 b 3 tan f¿ sin u cos u ro ¥ r1 2 a b 1 ro (14.5) Another important property of the logarithmic spiral deﬁned by Eq. (14.1) is that any radial line makes an angle f with the normal to the curve drawn at the point where the 492 Chapter 14: Lateral Earth Pressure: Curved Failure Surface radial line and the spiral intersect. This basic property is useful particularly in solving problems related to lateral earth pressure. PASSIVE EARTH PRESSURE 14.3 Procedure for Determination of Passive Earth Pressure (Pp )— Cohesionless Backﬁll Figure 14.1d shows the curved failure surface in the granular backﬁll of a retaining wall of height H. The shear strength of the granular backﬁll is expressed as tf s¿ tan f¿ (14.6) The curved lower portion BC of the failure wedge is an arc of a logarithmic spiral deﬁned by Eq. (14.1). The center of the log spiral lies on the line CA (not necessarily within the limits of points C and A). The upper portion CD is a straight line that makes an angle of (45 f/2) degrees with the horizontal. The soil in the zone ACD is in Rankine’s passive state. Figure 14.3 shows the procedure for evaluating the passive resistance by trial wedges (Terzaghi and Peck, 1967). The retaining wall is ﬁrst drawn to scale as shown in Figure 14.3a. The line C1A is drawn in such a way that it makes an angle of (45 f/2) degrees with the surface of the backﬁll. BC1D1 is a trial wedge in which BC1 is the arc of a logarithmic spiral. According to the equation r1 roeu tan f, O1 is the center of the spiral. (Note: O1B ro and O1C1 r1 and ⬔BO1C1 u1 ; refer to Figure 14.2.) Now let us consider the stability of the soil mass ABC1C1 (Figure 14.3b). For equilibrium, the following forces per unit length of the wall are to be considered: 1. Weight of the soil in zone ABC1C1œ W1 (g)(Area of ABC1C1œ )(1). 2. The vertical face, C1C1œ , is in the zone of Rankine’s passive state; hence, the force acting on this face is f¿ 1 b Pd112 g1d1 2 2 tan2 a 45 2 2 (14.7) where d1 C1C1œ . Pd(1) acts horizontally at a distance of d1 /3 measured vertically upward from C1 . 3. F1 is the resultant of the shear and normal forces that act along the surface of sliding, BC1 . At any point on the curve, according to the property of the logarithmic spiral, a radial line makes an angle f with the normal. Because the resultant, F1 , makes an angle f with the normal to the spiral at its point of application, its line of application will coincide with a radial line and will pass through the point O1 . 4. P1 is the passive force per unit length of the wall. It acts at a distance of H/3 measured vertically from the bottom of the wall. The direction of the force P1 is inclined at an angle d with the normal drawn to the back face of the wall. O1 C 1 A D1 f 45 2 H f 45 2 g f c 0 u d Granular soil C1 H 3 B (a) O1 A l W(1) C 1 lP(1) l1 d1 d1 3 P1 d W1 B Pd(1) C1 F1 f (b) Pp O1 A H 2 31 P1 P2 P3 C 1 Pp d Trial 2 Trial 1 C1 B Trial 3 (c) Figure 14.3 Passive earth pressure against retaining wall with curved failure surface 493 494 Chapter 14: Lateral Earth Pressure: Curved Failure Surface Now, taking the moments of W1 , Pd(1) , F1 , and P1 about the point O1 , for equilibrium, we have W1 3lW112 4 Pd112 3l1 4 F1 3 04 P1 3 lP112 4 (14.8) or P1 1 lP112 3W1lW112 Pd112l1 4 (14.9) where lW(1) , l1 , and lP(1) are moment arms for the forces W1 , Pd(1) , and P1 , respectively. The preceding procedure for ﬁnding the trial passive force per unit length of the wall is repeated for several trial wedges such as those shown in Figure 14.3c. Let P1 , P2 , P3 , . . . , Pn be the forces that correspond to trial wedges 1, 2, 3, . . . , n, respectively. The forces are plotted to some scale as shown in the upper part of the ﬁgure. A smooth curve is plotted through the points 1, 2, 3, . . . , n. The lowest point of the smooth curve deﬁnes the actual passive force, Pp, per unit length of the wall. 14.4 Coefﬁcient of Passive Earth Pressure (Kp ) Referring to the retaining wall with a granular backfill (c 0) shown in Fig. 14.3, the passive earth pressure Kp can be expressed as 1 Pp KpgH2 2 (14.10) or Kp Pp 0.5gH2 (14.11) Following is a summary of results obtained by several investigators. Procedure of Terzaghi and Peck Using the procedure of Terzaghi and Peck (1967) described in Section 14.3, the passive earth-pressure coefficient can be evaluated for various combinations of u, d, and f. Figure 14.4 shows the variation of Kp for f 30° and 40° (for u 0) with d. Solution by the Method of Slices Shields and Tolunay (1973) improved the trail wedge solutions described in Section 14.3 using the method of slices to consider the stability of the trial soil wedge such as ABC1 C1 in Fig. 14.3a. The details of the analysis are beyond the scope of this text. However the values of Kp (passive earth-pressure coefficient) obtained by this method are given in Fig. 14.5. Note that the values of Kp shown in Fig. 14.5 are for retaining walls with a vertical back (that is, u 0 in Fig. 14.3) supporting a granular backfill with a horizontal ground surface. 18 16 d 1 f 16 14 14 0.8 0.6 12 12 10 Kp 0.4 Kp f 40 10 8 8 0.2 6 6 0 f 30 4 4 2 2 0 0 0 10 20 30 40 d (deg) 495 Figure 14.4 Variation of Kp with f and d based on the procedure of Terzaghi and Peck (1967) (Note: u 0) (Based on Terzaghi and Peck, 1967) 20 25 30 35 40 Soil friction angle, f (deg) Figure 14.5 Kp based on Shields and Tolunay’s analysis (Note u 0) (Based on Shields and Tolunay's analysis) 45 496 Chapter 14: Lateral Earth Pressure: Curved Failure Surface Solution by the Method of Triangular Slices Zhu and Qian (2000) used the method of triangular slices (such as in the zone of ABC1 in Fig. 14.3a) to obtain the variation of Kp. According to this analysis Kp Kp(d 0)R (14.12) where Kp passive earth pressure coefficient for a given value of u, d, and f Kp(d 0) Kp for a given value of u, f, with d 0 R modification factor which is a function of f, u, d/f The variations of Kp(d 0) are given in Table 14.1. The interpolated values of R are given in Table 14.2. Table 14.1 Variation of Kp(d 0) [see Eq. (14.12) and Figure 14.3a]* u (deg) Fⴕ (deg) 30 25 20 15 10 5 0 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1.70 1.74 1.77 1.81 1.84 1.88 1.91 1.95 1.99 2.03 2.07 2.11 2.15 2.20 2.24 2.29 2.33 2.38 2.43 2.48 2.53 2.59 2.64 2.70 2.76 2.82 1.69 1.73 1.77 1.81 1.85 1.89 1.93 1.98 2.02 2.07 2.11 2.16 2.21 2.26 2.32 2.37 2.43 2.49 2.55 2.61 2.67 2.74 2.80 2.88 2.94 3.02 1.72 1.76 1.80 1.85 1.90 1.95 1.99 2.05 2.10 2.15 2.21 2.27 2.33 2.39 2.45 2.52 2.59 2.66 2.73 2.81 2.89 2.97 3.05 3.14 3.23 3.32 1.77 1.81 1.87 1.92 1.97 2.03 2.09 2.15 2.21 2.27 2.34 2.41 2.48 2.56 2.64 2.72 2.80 2.89 2.98 3.07 3.17 3.27 3.38 3.49 3.61 3.73 1.83 1.89 1.95 2.01 2.07 2.14 2.21 2.28 2.35 2.43 2.51 2.60 2.68 2.77 2.87 2.97 3.07 3.18 3.29 3.41 3.53 3.66 3.80 3.94 4.09 4.25 1.92 1.99 2.06 2.13 2.21 2.28 2.36 2.45 2.54 2.63 2.73 2.83 2.93 3.04 3.16 3.28 3.41 3.55 3.69 3.84 4.00 4.16 4.34 4.52 4.72 4.92 2.04 2.12 2.20 2.28 2.37 2.46 2.56 2.66 2.77 2.88 3.00 3.12 3.25 3.39 3.53 3.68 3.84 4.01 4.19 4.38 4.59 4.80 5.03 5.27 5.53 5.80 *Based on Zhu and Qian, 2000 14.4 Coefﬁcient of Passive Earth Pressure (Kp) 497 Table 14.2 Variation of R [Eq. (14.12)] R for f (deg) u (deg) d/f 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 5 10 15 20 30 35 40 45 1.2 1.4 1.65 1.95 2.2 1.2 1.4 1.6 1.9 2.15 1.15 1.35 1.6 1.8 2.05 1.15 1.35 1.55 1.8 2.0 1.15 1.35 1.5 1.8 1.9 1.28 1.6 1.95 2.4 2.85 1.25 1.6 1.9 2.35 2.8 1.2 1.5 1.85 2.25 2.65 1.2 1.5 1.8 2.2 2.6 1.2 1.45 1.8 2.1 2.4 1.35 1.8 2.4 3.15 3.95 1.32 1.8 2.35 3.05 3.8 1.3 1.7 2.25 2.9 3.6 1.3 1.65 2.2 2.8 3.4 1.3 1.65 2.1 2.6 3.2 1.45 2.2 3.2 4.45 6.1 1.4 2.1 3.0 4.3 5.7 1.4 2.0 2.9 4.0 5.3 1.35 1.95 2.7 3.8 4.95 1.35 1.9 2.6 3.55 4.8 Example 14.1 Consider a 3-m-high (H) retaining wall with a vertical back (u 0) and a horizontal granular backﬁll. Given: g 15.7 kN/m3, d 15, and f 30. Estimate the passive force, Pp, by using a. b. c. d. Coulomb’s theory Terzaghi and Peck’s wedge theory Shields and Tolunay’s solution (method of slices) Zhu and Qian’s solution (method of triangular slices) Solution Part a From Eq. (13.70), Pp 12 KpgH2 498 Chapter 14: Lateral Earth Pressure: Curved Failure Surface From Table 13.9, for f 30 and d 15, the value of Kp is 4.977. Thus, P (12 )(4.977)(15.7)(3)2 351.6 kN/m Part b From Fig. 14.4, for f 30° and d 15°, the value of Kp is about 4.53. Thus, Pp 1 12 2 14.53 2 115.7 2 13 2 2 320 kN/m Part c 1 Pp KpgH2 2 From Figure 14.5, for f 30 and d 15 (i.e d¿ 0.5) the value of Kp is 4.13. Hence, f¿ 1 Pp a b 14.132 115.72 13 2 2 292 kN/m 2 Part d From Eq. (14.12), Kp Kp(d 0)R For f 30° and u 0, Kp(d 0) is equal to 3.0 (Table 14.1). Again, for u 0 and d/f 0.5, the value of R is about 1.52. Thus, Kp (3)(1.52) 4.56. Pp 1 12 2 14.56115.7 2 132 2 322 kN/m 14.5 ■ Passive Force on Walls with Earthquake Forces The relationship for passive earth pressure on a retaining wall with a granular horizontal backfill and vertical back face under earthquake conditions was evaluated by Subba Rao and Choudhury (2005) using the pseudo-static approach to the method of limited equilibrium. The failure surface in soil assumed in the analysis was similar to that shown in Fig. 14.3 (with u 0; that is, vertical back face). The notations used in the analysis were H height of retaining wall Ppe passive force per unit length of the wall f soil friction angle d angle of wall friction kh horizontal component of earthquake acceleration acceleration due to gravity,g kv vertical component of earthquake acceleration acceleration due to gravity,g 14.6 Braced Cuts— General 15 10 k 0 k kh f 40 12 9 6 40 4 30 Kpg(e) Kpg(e) 40 6 f 40 8 k 0 k kh 30 30 30 3 20 2 20 10 499 20 10 10 20 10 d 1 f 0 d .5 f 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 kh kh (a) (b) 0.4 0.5 d¿ Figure 14.6 Variation of Kpg(e): (a) d/f 1; (b) ¿ 0.5 f Based on this analysis, the passive force Ppe can be expressed as Ppe c 1 1 gH2Kpg1e2 d 2 cos d¿ (14.13) where Kpg(e) passive earth-pressure coefficient in the normal direction to the wall. Kpg(e) is a function of kh and kv. The variations of Kpg(e) for d/f 0.5 and 1 are shown in Figure 14.6. The passive pressure Ppe will be inclined at an angle d to the back face of the wall and will act at a distance of H/3 above the bottom of the wall. BRACED CUTS 14.6 Braced Cuts— General Frequently during the construction of foundations or utilities (such as sewers), open trenches with vertical soil slopes are excavated. Although most of these trenches are temporary, the sides of the cuts must be supported by proper bracing systems. Figure 14.7 shows one of several bracing systems commonly adopted in construction practice. The bracing consists of sheet piles, wales, and struts. 500 Chapter 14: Lateral Earth Pressure: Curved Failure Surface Strut A Wale Sheet pile Sheet pile Strut Wale X B X C (a) (b) Figure 14.7 Braced cut: (a) cross section; (b) plan (section at X–X) Proper design of these elements requires a knowledge of the lateral earth pressure exerted on the braced walls. The magnitude of the lateral earth pressure at various depths of the cut is very much inﬂuenced by the deformation condition of the sheeting. To understand the nature of the deformation of the braced walls, one needs to follow the sequence of construction. Construction of the unit begins with driving the sheetings. The top row of the wales and struts (marked A in Figure 14.7a) is emplaced immediately after a small cut is made. This emplacement must be done immediately so that the soil mass outside the cut has no time to deform and cause the sheetings to yield. As the sequence of driving the sheetings, excavating the soil, and placing rows of wales and struts (see B and C in Figure 14.7) continues, the sheetings move inward at greater depths. This action is caused by greater earth pressure exerted by the soil outside the cut. The deformation of the braced walls is shown by the broken lines in Figure 14.7a. Essentially, the problem models a condition where the walls are rotating about the level of the top row of struts. A photograph of braced cuts made for subway construction in Chicago is shown in Figure 14.8a. Figures 14.8b and 14.8c are photographs of two braced cuts—one in Seoul, South Korea, and the other in Taiwan. The deformation of a braced wall differs from the deformation condition of a retaining wall in that, in a braced wall, the rotation is about the top. For this reason, neither Coulomb’s nor Rankine’s theory will give the actual earth-pressure distribution. This fact is illustrated in Figure 14.9, in which AB is a frictionless wall with a granular soil backﬁll. When the wall deforms to position AB, failure surface BC develops. Because the upper portion of the soil mass in the zone ABC does not undergo sufﬁcient deformation, it does not pass into Rankine’s active state. The sliding surface BC intersects the ground surface almost at 90. The corresponding earth pressure will be somewhat parabolic, like acb shown in Figure 14.9b. With this type of pressure distribution, the point of application of the resultant active thrust, Pa , will be at a height of naH measured from the bottom of the wall, with na 13 (for triangular pressure distribution na 13 ). Theoretical evaluation and ﬁeld measurements have shown that na could be as high as 0.55. Image not available due to copyright restrictions 501 Image not available due to copyright restrictions a A C 90 Sand P H na H Logarithmic spiral b B B (a) c Kag H Kog H (b) Figure 14.9 Earth-pressure distribution against a wall with rotation about the top 502 14.7 Determination of Active Thrust on Bracing Systems of Open Cuts— Granular Soil 14.7 503 Determination of Active Thrust on Bracing Systems of Open Cuts— Granular Soil The active thrust on the bracing system of open cuts can be estimated theoretically by using trial wedges and Terzaghi’s general wedge theory (1941). The basic procedure for determination of the active thrust are described in this section. Figure 14.10a shows a braced wall AB of height H that deforms by rotating about its top. The wall is assumed to be rough, with the angle of wall friction equal to d. The point of application of the active thrust (that is, naH) is assumed to be known. The curve of sliding is b1 lW(1) O1 lP(1) A 90 f b1 H g f c 0 W1 d P1 na H f F1 B B (a) b2 b1 b3 P2 P1 P3 Pa O1 A b2 H b1 b3 d Pa Trial 3 na H Trial 1 Trial 2 B B (b) Figure 14.10 Determination of active force on bracing system of open cut in cohesionless soil 504 Chapter 14: Lateral Earth Pressure: Curved Failure Surface assumed to be an arc of a logarithmic spiral. As we discussed in the preceding section, the curve of sliding intersects the horizontal ground surface at 90. To proceed with the trial wedge solution, let us select a point b1 . From b1 , a line b1 b1 that makes an angle f with the ground surface is drawn. (Note that f effective angle of friction of the soil.) The arc of the logarithmic spiral, b1B, which deﬁnes the curve of sliding for this trial, can now be drawn, with the center of the spiral (point O1) located on the line b1b1œ . Note that the equation for the logarithmic spiral is given by r1 roeu1 tan f¿ and, in this case, O1b1 ro and O1B r1 . Also, it is interesting to see that the horizontal line that represents the ground surface is the normal to the curve of sliding at the point b1 , and that O1b1 is a radial line. The angle between them is equal to f, which agrees with the property of the spiral. To look at the equilibrium of the failure wedge, let us consider the following forces per unit length of the braced wall: • • • W1 the weight of the wedge ABb1 (Area of ABb1) (g) (1). P1 the active thrust acting at a point naH measured vertically upward from the bottom of the cut and inclined at an angle d with the horizontal. F1 the resultant of the shear and normal forces that act along the trial failure surface. The line of action of the force F1 will pass through the point O1 . Now, taking the moments of these forces about O1 , we have W1 3 lW112 4 F1 102 P1 3 lP 112 4 0 or P1 W1lW112 lP 112 (14.14) where lW(1) and lP (1) are the moment arms for the forces W1 and P1, respectively. This procedure of ﬁnding the active thrust can now be repeated for several wedges such as ABb2 , ABb3 , . . . , ABbn (Figure 14.10b). Note that the centers of the logarithmic-spiral arcs will lie on lines b2b2œ , b3b3œ , . . . , bnbnœ , respectively. The active thrusts P1 , P2 , P3 , . . . , Pn derived from the trial wedges are plotted to some scale in the upper portion of Figure 14.10b. The maximum point of the smooth curve drawn through these points will yield the desired maximum active thrust, Pa, on the braced wall. Kim and Preber (1969) determined the values of Pa /0.5gH2 for braced excavations for various values of f, d, and na. These values are given in Table 14.3. In general, the average magnitude of Pa is about 10% greater when the wall rotation is about the top as compared with the value obtained by Coulomb’s active earth-pressure theory. 14.8 Determination of Active Thrust on Bracing Systems for Cuts— Cohesive Soil Using the principles of the general wedge theory, we also can determine the active thrust on bracing systems for cuts made in c– f soil. Table 14.4 gives the variation of Pa in a nondimensional form for various values of f, d, na, and c/gH. 14.9 Pressure Variation for Design of Sheetings, Struts, and Wales 505 Table 14.3 Pa /0.5gH2 Against f, d, and na (c 0) for Braced Cuts* Pa /0.5GH 2 Fⴕ (deg) Dⴕ (deg) na ⴝ 0.3 na ⴝ 0.4 na ⴝ 0.5 na ⴝ 0.6 10 0 5 10 0.653 0.623 0.610 0.734 0.700 0.685 0.840 0.799 0.783 0.983 0.933 0.916 15 0 5 10 15 0.542 0.518 0.505 0.499 0.602 0.575 0.559 0.554 0.679 0.646 0.629 0.623 0.778 0.739 0.719 0.714 20 0 5 10 15 20 0.499 0.430 0.419 0.413 0.413 0.495 0.473 0.460 0.454 0.454 0.551 0.526 0.511 0.504 0.504 0.622 0.593 0.575 0.568 0.569 25 0 5 10 15 20 25 0.371 0.356 0.347 0.342 0.341 0.344 0.405 0.389 0.378 0.373 0.372 0.375 0.447 0.428 0.416 0.410 0.409 0.413 0.499 0.477 0.464 0.457 0.456 0.461 0 5 10 15 20 25 30 0.304 0.293 0.286 0.282 0.281 0.284 0.289 0.330 0.318 0.310 0.306 0.305 0.307 0.313 0.361 0.347 0.339 0.334 0.332 0.335 0.341 0.400 0.384 0.374 0.368 0.367 0.370 0.377 30 * Pa /0.5GH 2 Fⴕ (deg) Dⴕ (deg) na ⴝ 0.3 na ⴝ 0.4 na ⴝ 0.5 na ⴝ 0.6 35 0 5 10 15 20 25 30 35 0.247 0.239 0.234 0.231 0.231 0.232 0.236 0.243 0.267 0.258 0.252 0.249 0.248 0.250 0.254 0.262 0.290 0.280 0.273 0.270 0.269 0.271 0.276 0.284 0.318 0.318 0.300 0.296 0.295 0.297 0.302 0.312 40 0 5 10 15 20 25 30 35 40 0.198 0.192 0.189 0.187 0.187 0.188 0.192 0.197 0.205 0.213 0.206 0.202 0.200 0.200 0.202 0.205 0.211 0.220 0.230 0.223 0.219 0.216 0.216 0.218 0.222 0.228 0.237 0.252 0.244 0.238 0.236 0.235 0.237 0.241 0.248 0.259 45 0 5 10 15 20 25 30 35 40 45 0.156 0.152 0.150 0.148 0.149 0.150 0.153 0.158 0.164 0.173 0.167 0.163 0.160 0.159 0.159 0.160 0.164 0.168 0.175 0.184 0.180 0.175 0.172 0.171 0.171 0.173 0.176 0.181 0.188 0.198 0.196 0.190 0.187 0.185 0.185 0.187 0.190 0.196 0.204 0.213 After Kim and Preber, 1969. With permission from ASCE. 14.9 Pressure Variation for Design of Sheetings, Struts, and Wales The active thrust against sheeting in a braced cut, calculated by using the general wedge theory, does not explain the variation of the earth pressure with depth that is necessary for design work. An important difference between bracings in open cuts and retaining walls is that retaining walls fail as single units, whereas bracings in an open cut undergo progressive failure where one or more struts fail at one time. Empirical lateral pressure diagrams against sheetings for the design of bracing systems have been given by Peck (1969). These pressure diagrams for cuts in sand, soft to medium clay, and stiff clay are given in Figure 14.11. Strut loads may be determined by assuming that the vertical members are hinged at each strut level except the topmost and bottommost ones (Figure 14.12). Example 14.2 illustrates the procedure for the calculation of strut loads. Table 14.4 Values of Pa /0.5gH 2 for Cuts in a c-f Soil with the Assumption caœ c(tan d/tan f)* Dⴕ (deg) na ⴝ 0.3 and c ⴕ/GH ⴝ 0.1 na ⴝ 0.4 and c ⴕ/GH ⴝ 0.1 na ⴝ 0.5 and c ⴕ/GH ⴝ 0.1 0.254 0.214 0.187 0.169 0.285 0.240 0.210 0.191 0.322 0.270 0.238 0.218 0.191 0.160 0.140 0.122 0.113 0.210 0.179 0.156 0.127 0.124 0.236 0.200 0.173 0.154 0.140 0.138 0.116 0.099 0.085 0.074 0.065 0.150 0.128 0.110 0.095 0.083 0.074 0.167 0.141 0.122 0.106 0.093 0.083 0.093 0.078 0.066 0.056 0.047 0.036 0.029 0.103 0.086 0.073 0.060 0.051 0.042 0.033 0.113 0.094 0.080 0.067 0.056 0.047 0.038 f 15 0 5 10 15 f 20 0 5 10 15 20 f 25 0 5 10 15 20 25 f 30 0 5 10 15 20 25 30 * After Kim and Preber, 1969. With permission from ASCE. f sa 0.65 gH tan2 45 2 sa gH 1 4cu gH gH for 4 cu 0.25H sa 0.2 gH to 0.4gH gH for 4 cu 0.25H H 0.5H 0.75H 0.25H sa sa (a) sa (b) Sand Soft to medium clay (c) Stiff clay Figure 14.11 Peck’s pressure diagrams for design of bracing systems 506 14.9 Pressure Variation for Design of Sheetings, Struts, and Wales 507 A A B BC C DE D F E sa Figure 14.12 Determination of strut loads from empirical lateral pressure diagrams F Example 14.2 A 7-m-deep braced cut in sand is shown in Figure 14.13. In the plan, the struts are placed at s 2 m center to center. Using Peck’s empirical pressure diagram, calculate the design strut loads. 5m 1m A Sand f 30 g 16 kN/m3 2m B 2m C 2m Bottom of cut Figure 14.13 Braced cut in sand Solution Refer to Figure 14.11a. For the lateral earth pressure diagram, sa 0.65gH tan 2 a 45 f¿ 30 b 10.65 2 116 2 17 2 tan 2 a 45 b 24.27 kN/m2 2 2 508 Chapter 14: Lateral Earth Pressure: Curved Failure Surface B2 1m 24.27 kN/m2 1m A C 24.27 kN/m2 2m 2m B1 Figure 14.14 Calculation of strut loads from pressure envelope Assume that the sheeting is hinged at strut level B. Now refer to the diagram in Figure 14.14 We need to find reactions at A, B1, B2, and C. Taking the moment about B1, we have 2A 124.27 2 132 a 3 b ; A 54.61 kN/m 2 Hence, B1 (24.27)(3) 54.61 18.2 kN/m Again, taking the moment about B2, we have 2C 124.272 142 a 4 b 2 C 97.08 kN/m So B2 (24.27)(4) 97.08 0 The strut loads are as follows: At level A: (A)(s) (54.61)(2) 109.22 kN At level B: (B1 + B2)(s) (18.2 + 0)(2) 36.4 kN At level C: (C)(s) (97.08)(2) 194.16 kN ■ Problems 14.10 509 Summary This chapter covers two major topics: • • Estimation of passive pressure using curved failure surface in soil Lateral earth pressure on braced cuts using the general wedge theory and pressure envelopes for design of struts, wales, and sheet piles. Passive pressure calculations using curved failure surface are essential for the case in which d f/2, since plane-failure surface assumption provides results on the unsafe side for design. In the case of braced cuts, although the general wedge theory provides the force per unit length of the cut, it does not provide the nature of distribution of earth pressure with depth. For that reason, pressure envelopes are necessary for practical design. Problems 14.1 Refer to the retaining wall shown in Figure 14.15. Given: u 10, a 0, g 15.5 kN/m3, f 35, d 21, and H 6 m. Estimate the passive force, Pp , per unit length of the wall. Use Tables 14.1 and 14.2. 14.2 Refer to Figure 14.15. Given: H 15 ft, g 100 lb/ft3, f 30, d 23 f¿ , a 0, and u 0. Calculate the passive force per unit length of the wall using Figure 14.5. 14.3 Solve Problem 14.2 using Tables 14.1 and 14.2. 14.4 Refer to Figure 14.15. Given: u 0, a 0, H 2.5 m, f 30, d 20, and g 14.8 kN/m3. Estimate the passive force per unit length of the wall using Figure 14.4. a Granular backﬁll H u Figure 14.15 g f c 0 d 510 Chapter 14: Lateral Earth Pressure: Curved Failure Surface 14.5 Refer to Figure 14.3a. Given: H 5 m, g 16 kN/m3, f 30, d 15, kv 0, and kh 0.3. Calculate Ppe for the retaining wall (Section 14.5). 14.6 A braced wall is shown in Figure 14.16. Given: H 5 m, naH 2 m, f 35, d 20, g 16 kN/m3, and c 0. Determine the active thrust, Pa , on the wall using the general wedge theory. 14.7 Repeat Problem 14.6 with the following given: H 15.6 m, naH 4.68 m, g 18 kN/m3, f 20, d 15, and c 28 kN/m2. Assume caœ tan d¿ . c¿ tan f¿ 14.8 The elevation and plan of a bracing system for an open cut in sand are shown in Figure 14.17. Assuming gsand 105 lb/ft3 and f 38°, determine the strut loads. g c f d H d Pa na H Figure 14.16 2 ft Struts—8 ft center to center A 5 ft B 5 ft Sand 22 ft 8 ft g f C 5 ft D 8 ft 5 ft (a) Section Figure 14.17 (b) Plan References 511 References HIJAB, W. (1956). “A Note on the Centroid of a Logarithmic Spiral Sector,” Geotechnique, Vol. 4, No. 2, 96–99. KIM, J. S., and PREBER, T. (1969). “Earth Pressure Against Braced Excavations,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 95, No. SM6, 1581–1584. PECK, R. B. (1969). “Deep Excavation and Tunneling in Soft Ground,” Proceedings, 7th International Conference on Soil Mechanics and Foundation Engineering, Mexico City, Stateof-the-Art Vol., 225–290. SUBBA RAO, K.S., AND CHOUDHURY, D. (2005). “Seismic Passive Earth Pressures in Soil,” Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Vol. 131, No. 1, 131–135. SHIELDS, D. H., and TOLUNAY, A. Z. (1973). “Passive Pressure Coefﬁcients by Method of Slices,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM12, 1043–1053. TERZAGHI, K. (1941). “General Wedge Theory of Earth Pressure,” Transactions, ASCE, Vol. 106, 68–97. TERZAGHI, K., and PECK, R. B. (1967). Soil Mechanics in Engineering Practice, 2nd ed., Wiley, New York. ZHU, D. Y., AND QIAN, Q. (2000). “Determination of Passive Earth Pressure Coefficient by the Method of Triangular Slices,” Canadian Geotechnical Journal, Vol. 37, No. 2, 485–491. 15 15.1 Slope Stability Introduction—Modes of Slope Failure An exposed ground surface that stands at an angle with the horizontal is called an unrestrained slope. The slope can be natural or man-made. It can fail in various modes. Cruden and Varnes (1996) classified the slope failures into the following five major categories. They are 1. Fall. This is the detachment of soil and/or rock fragments that fall down a slope (Figure 15.1). Figure 15.2 shows a fall in which a large amount of soil mass has slid down a slope. 2. Topple. This is a forward rotation of soil and/or rock mass about an axis below the center of gravity of mass being displaced (Figure 15.3). 3. Slide. This is the downward movement of a soil mass occurring on a surface of rupture (Figure 15.4). 4. Spread. This is a form of slide (Figure 15.5) by translation. It occurs by “sudden movement of water-bearing seams of sands or silts overlain by clays or loaded by fills” (Cruden and Varnes, 1996). 5. Flow. This is a downward movement of soil mass similar to a viscous fluid (Figure 15.6). This chapter primarily relates to the quantitative analysis that fall under the category of slide. 512 Figure 15.1 “Fall” type of landslide Figure 15.2 Soil and rock “fall” in a slope (Courtesy of E.C. Shin, University of Inchon, South Korea) Figure 15.3 Slope failure by “toppling” Figure 15.4 Slope failure by “sliding” 513 514 Chapter 15: Slope Stability Figure 15.5 Slope failure by lateral “spreading” Figure 15.6 Slope failure by “flowing” 15.2 Factor of Safety The task of the engineer charged with analyzing slope stability is to determine the factor of safety. Generally, the factor of safety is deﬁned as tf (15.1) Fs td where Fs factor of safety with respect to strength tf average shear strength of the soil td average shear stress developed along the potential failure surface The shear strength of a soil consists of two components, cohesion and friction, and may be written as tf c¿ s¿ tan f¿ where c¿ cohesion f¿ angle of friction s¿ normal stress on the potential failure surface (15.2) 15.3 Stability of Inﬁnite Slopes 515 In a similar manner, we can write td cdœ s¿ tan fdœ (15.3) where cdœ and fdœ are, respectively, the cohesion and the angle of friction that develop along the potential failure surface. Substituting Eqs. (15.2) and (15.3) into Eq. (15.1), we get Fs c¿ s¿ tan f¿ cdœ s¿ tan fdœ (15.4) Now we can introduce some other aspects of the factor of safety—that is, the factor of safety with respect to cohesion, Fc¿ , and the factor of safety with respect to friction, Ff¿ . They are deﬁned as c¿ cdœ (15.5) tan f¿ tan fdœ (15.6) Fc¿ and Ff¿ When we compare Eqs. (15.4) through (15.6), we can see that when Fc¿ becomes equal to Ff¿ , it gives the factor of safety with respect to strength. Or, if tan f¿ c¿ cdœ tan fdœ then we can write Fs Fc¿ Ff¿ (15.7) When Fs is equal to 1, the slope is in a state of impending failure. Generally, a value of 1.5 for the factor of safety with respect to strength is acceptable for the design of a stable slope. 15.3 Stability of Inﬁnite Slopes In considering the problem of slope stability, let us start with the case of an inﬁnite slope as shown in Figure 15.7. The shear strength of the soil may be given by Eq. (15.2): tf c¿ s¿ tan f¿ Assuming that the pore water pressure is zero, we will evaluate the factor of safety against a possible slope failure along a plane AB located at a depth H below the ground surface. The slope failure can occur by the movement of soil above the plane AB from right to left. 516 Chapter 15: Slope Stability L d a F b W Na b B F Ta H b c Tr b A b Nr R Figure 15.7 Analysis of inﬁnite slope (without seepage) Let us consider a slope element abcd that has a unit length perpendicular to the plane of the section shown. The forces, F, that act on the faces ab and cd are equal and opposite and may be ignored. The weight of the soil element is W 1Volume of soil element2 1Unit weight of soil2 gLH (15.8) The weight W can be resolved into two components: 1. Force perpendicular to the plane AB Na W cos b gLH cos b. 2. Force parallel to the plane AB Ta W sin b gLH sin b. Note that this is the force that tends to cause the slip along the plane. Thus, the effective normal stress and the shear stress at the base of the slope element can be given, respectively, as s¿ Na gLH cos b gH cos2 b Area of base L b a cos b (15.9) and t gLH sin b Ta gH cos b sin b Area of base L b a cos b (15.10) The reaction to the weight W is an equal and opposite force R. The normal and tangential components of R with respect to the plane AB are Nr R cos b W cos b (15.11) 15.3 Stability of Inﬁnite Slopes 517 and Tr R sin b W sin b (15.12) For equilibrium, the resistive shear stress that develops at the base of the element is equal to (Tr)/(Area of base) gH sin b cos b. The resistive shear stress also may be written in the same form as Eq. (15.3): td cdœ s¿ tan fdœ The value of the normal stress is given by Eq. (15.9). Substitution of Eq. (15.9) into Eq. (15.3) yields td cdœ gH cos2 b tan fdœ (15.13) Thus, gH sin b cos b cdœ gH cos2 b tan fdœ or cdœ sin b cos b cos2 b tan fdœ gH cos2 b1tanb tan fdœ 2 (15.14) The factor of safety with respect to strength has been deﬁned in Eq. (15.7), from which we get tan fdœ tan f¿ Fs and cdœ c¿ Fs Substituting the preceding relationships into Eq. (15.14), we obtain Fs tan f¿ c¿ 2 tan b gH cos b tan b (15.15) For granular soils, c 0, and the factor of safety, Fs , becomes equal to (tan f)/ (tan b). This indicates that in an inﬁnite slope in sand, the value of Fs is independent of the height H and the slope is stable as long as b f. If a soil possesses cohesion and friction, the depth of the plane along which critical equilibrium occurs may be determined by substituting Fs 1 and H Hcr into Eq. (15.15). Thus, Hcr 1 c¿ 2 g cos b1tan b tan f¿ 2 (15.16) 518 Chapter 15: Slope Stability L d a b of tion age seep c Dire Na W H B Ta c b Tr b R Nr A Figure 15.8 Infinite slope with steadystate seepage If there is steady state seepage through the soil and the ground water table coincides with the ground surface, as shown in Figure 15.8, the factor of safety against sliding can be determined as Fs gœ tan fœ cœ gsat tan b gsatH cos 2 b tan b (15.17) where sat saturated unit weight of soil sat w effective unit weight of soil Example 15.1 For the inﬁnite slope with a steady state seepage shown in Figure 15.9, determine: a. The factor of safety against sliding along the soil-rock interface. b. The height, H, that will give a factor of safety (Fs) of 2 against sliding along the soil-rock interface. Solution Part a From Eq. (15.17), Fs g œ tan f œ cœ gsat tan b gsatH cos 2 b tan b gsat 17.8 kN/m3 15.4 Finite Slopes— General 519 b 25 gsat 17.8 kN/m3 c 10 kN/m2 f 20 age H6m Seep b 15 Rock Figure 15.9 gœ gsat gw 17.8 9.81 7.99 kN/m3 7.99 tan 20 10 0.375 0.61 0.985 Fs 17.8 tan 15 117.82 162 1cos 152 2 1tan 152 Part b g œ tan f œ cœ 2 gsat tan b gsatH cos b tan b 7.99 tan 20 2.247 10 0.61 2 17.8 tan 15 H 117.8 2 1H2 1 cos 152 2 1 tan 152 2.247 1.62 m H 2 0.61 Fs 15.4 ■ Finite Slopes— General When the value of Hcr approaches the height of the slope, the slope generally may be considered ﬁnite. For simplicity, when analyzing the stability of a ﬁnite slope in a homogeneous soil, we need to make an assumption about the general shape of the surface of potential failure. Although considerable evidence suggests that slope failures usually occur on curved failure surfaces, Culmann (1875) approximated the surface of potential failure as a plane. The factor of safety, Fs , calculated by using Culmann’s approximation, gives fairly good results for near-vertical slopes only. After extensive investigation of slope failures in the 1920s, a Swedish geotechnical commission recommended that the actual surface of sliding may be approximated to be circularly cylindrical. 520 Chapter 15: Slope Stability Since that time, most conventional stability analyses of slopes have been made by assuming that the curve of potential sliding is an arc of a circle. However, in many circumstances (for example, zoned dams and foundations on weak strata), stability analysis using plane failure of sliding is more appropriate and yields excellent results. 15.5 Analysis of Finite Slopes with Plane Failure Surfaces (Culmann’s Method) Culmann’s analysis is based on the assumption that the failure of a slope occurs along a plane when the average shearing stress tending to cause the slip is more than the shear strength of the soil. Also, the most critical plane is the one that has a minimum ratio of the average shearing stress that tends to cause failure to the shear strength of soil. Figure 15.10 shows a slope of height H. The slope rises at an angle b with the horizontal. AC is a trial failure plane. If we consider a unit length perpendicular to the section of the slope, we ﬁnd that the weight of the wedge ABC is equal to W 12 1H2 1BC2 11 2 1g2 12 H1H cot u H cot b2 g sin1b u2 1 d gH 2 c 2 sin b sin u (15.18) The normal and tangential components of W with respect to the plane AC are as follows. sin1b u 2 1 d cos u Na normal component W cos u gH 2 c 2 sin b sin u sin1b u2 1 Ta tangential component W sin u gH 2 c d sin u 2 sin b sin u (15.19) (15.20) C B W Na Ta Tr Nr H R b A u tf c s tan f Unit weight of soil g Figure 15.10 Finite slope analysis— Culmann’s method 15.5 Analysis of Finite Slopes with Plane Failure Surfaces (Culmann’s Method) 521 The average effective normal stress and the average shear stress on the plane AC are, respectively, s¿ Na 1AC2 112 Na H b a sin u sin1b u2 1 d cos u sin u gH c 2 sin b sin u (15.21) and t Ta 1AC2 112 Ta H a b sin u sin1b u 2 1 gH c d sin2 u 2 sin b sin u (15.22) The average resistive shearing stress developed along the plane AC also may be expressed as td cdœ s¿ tan fdœ sin1b u2 1 cdœ gH c d cos u sin u tan fdœ 2 sin b sin u (15.23) Now, from Eqs. (15.22) and (15.23), sin1b u2 sin1b u2 1 1 gH c d sin2 u cdœ gH c d cos u sin u tan fdœ 2 sin b sin u 2 sin b sin u (15.24) or sin1b u 2 1sin u cos u tan fdœ 2 1 cd gH c d 2 sin b (15.25) The expression in Eq. (15.25) is derived for the trial failure plane AC. In an effort to determine the critical failure plane, we must use the principle of maxima and minima (for a given value of fdœ ) to ﬁnd the angle u where the developed cohesion would be maximum. Thus, the ﬁrst derivative of cd with respect to u is set equal to zero, or 0cdœ 0 0u (15.26) Because g, H, and b are constants in Eq. (15.25), we have 0 3sin1b u2 1sin u cos u tan fdœ 2 4 0 0u (15.27) 522 Chapter 15: Slope Stability Solving Eq. (15.27) gives the critical value of u, or ucr b fdœ 2 (15.28) Substitution of the value of u ucr into Eq. (15.25) yields cdœ œ gH 1 cos1b fd 2 c d 4 sin b cos fdœ (15.29) The preceding equation also can be written as 1 cos1b fdœ 2 cdœ m gH 4 sin b cos fdœ (15.30) where m stability number. The maximum height of the slope for which critical equilibrium occurs can be obtained by substituting cdœ c and fdœ f into Eq. (15.29). Thus, Hcr sin b cos f¿ 4c¿ d c g 1 cos1b f¿ 2 (15.31) Example 15.2 A cut is to be made in a soil having g 105 lb/ft3, c 600 lb/ft2, and f 15. The side of the cut slope will make an angle of 45 with the horizontal. What should be the depth of the cut slope that will have a factor of safety (Fs) of 3? Solution Given: f 15; c 600 lb/ft2. If Fs 3, then Fcœ and Ffœ should both be equal to 3. Fc¿ c¿ cdœ or cdœ c¿ c¿ 600 200 lb/ft2 Fcœ Fs 3 Similarly, Ffœ tan fdœ tan f¿ tan fdœ tan f¿ tan f¿ tan 15 Ff œ Fs 3 15.6 Analysis of Finite Slopes with Circular Failure Surfaces— General 523 or fdœ tan1 c tan 15 d 5.1° 3 Substituting the preceding values of cdœ and fdœ in Eq. (15.29) H sin b # cos fdœ 4cdœ d c g 1 cos1b fdœ 2 sin 45 # cos 5.1 4 200 c d 105 1 cos145 5.12 23.05 ft 15.6 ■ Analysis of Finite Slopes with Circular Failure Surfaces— General Modes of Failure In general, ﬁnite slope failure occurs in one of the following modes (Figure 15.11): 1. When the failure occurs in such a way that the surface of sliding intersects the slope at or above its toe, it is called a slope failure (Figure 15.11a). The failure circle is referred to as a toe circle if it passes through the toe of the slope and as a slope circle if it passes above the toe of the slope. Under certain circumstances, a shallow slope failure can occur, as shown in Figure 15.11b. 2. When the failure occurs in such a way that the surface of sliding passes at some distance below the toe of the slope, it is called a base failure (Figure 15.11c). The failure circle in the case of base failure is called a midpoint circle. Types of Stability Analysis Procedures Various procedures of stability analysis may, in general, be divided into two major classes: 1. Mass procedure: In this case, the mass of the soil above the surface of sliding is taken as a unit. This procedure is useful when the soil that forms the slope is assumed to be homogeneous, although this is not the case in most natural slopes. 2. Method of slices: In this procedure, the soil above the surface of sliding is divided into a number of vertical parallel slices. The stability of each slice is calculated separately. This is a versatile technique in which the nonhomogeneity of the soils and pore water pressure can be taken into consideration. It also accounts for the variation of the normal stress along the potential failure surface. The fundamentals of the analysis of slope stability by mass procedure and method of slices are given in the following sections. 524 Chapter 15: Slope Stability O Toe circle (b) L O L O Midpoint circle Slope circle (a) (c) Figure 15.11 Modes of failure of ﬁnite slope: (a) slope failure; (b) shallow slope failure; (c) base failure 15.7 Mass Procedure—Slopes in Homogeneous Clay Soil with F ⴝ 0 Figure 15.12 shows a slope in a homogeneous soil. The undrained shear strength of the soil is assumed to be constant with depth and may be given by tf cu . To perform the stability analysis, we choose a trial potential curve of sliding, AED, which is an arc of a circle that has a radius r. The center of the circle is located at O. Considering a unit length perpendicular to the section of the slope, we can give the weight of the soil above the curve AED as W W1 W2 , where W1 1Area of FCDEF2 1g2 and W2 1Area of ABFEA2 1g2 15.7 Mass Procedure—Slopes in Homogeneous Clay Soil with 0 525 O D C u Radius r H l2 A l1 F W2 cd W1 B E cd Unit weight of soil g tf cu cd Nr (normal reaction) Figure 15.12 Stability analysis of slope in homogeneous saturated clay soil (f 0) Failure of the slope may occur by sliding of the soil mass. The moment of the driving force about O to cause slope instability is Md W1l1 W2 l2 (15.32) where l1 and l2 are the moment arms. The resistance to sliding is derived from the cohesion that acts along the potential surface of sliding. If cd is the cohesion that needs to be developed, the moment of the resisting forces about O is MR cd 1AED2 11 2 1r2 cdr2u (15.33) For equilibrium, MR Md ; thus, cdr2u W1l1 W2 l2 or cd W1l1 W2 l2 r2u (15.34) The factor of safety against sliding may now be found. Fs tf cd cu cd (15.35) 526 Chapter 15: Slope Stability Note that the potential curve of sliding, AED, was chosen arbitrarily. The critical surface is that for which the ratio of cu to cd is a minimum. In other words, cd is maximum. To ﬁnd the critical surface for sliding, one must make a number of trials for different trial circles. The minimum value of the factor of safety thus obtained is the factor of safety against sliding for the slope, and the corresponding circle is the critical circle. Stability problems of this type have been solved analytically by Fellenius (1927) and Taylor (1937). For the case of critical circles, the developed cohesion can be expressed by the relationship cd gHm or cd m gH (15.36) Note that the term m on the right-hand side of the preceding equation is nondimensional and is referred to as the stability number. The critical height (i.e., Fs 1) of the slope can be evaluated by substituting H Hcr and cd cu (full mobilization of the undrained shear strength) into the preceding equation. Thus, Hcr cu gm (15.37) Values of the stability number, m, for various slope angles, b, are given in Figure 15.13. Terzaghi used the term gH/cd , the reciprocal of m and called it the stability factor. Readers should be careful in using Figure 15.13 and note that it is valid for slopes of saturated clay and is applicable to only undrained conditions (f 0). In reference to Figure 15.13, the following must be pointed out: 1. For a slope angle b greater than 53, the critical circle is always a toe circle. The location of the center of the critical toe circle may be found with the aid of Figure 15.14. 2. For b 53, the critical circle may be a toe, slope, or midpoint circle, depending on the location of the ﬁrm base under the slope. This is called the depth function, which is deﬁned as D Vertical distance from top of slope to firm base Height of slope (15.38) 15.7 Mass Procedure—Slopes in Homogeneous Clay Soil with 0 527 For b 53: All circles are toe circles. For b 53: H DH Toe circle nH Midpoint circle Slope circle (a) Stability number, m 0.3 b 53 0.2 D 4.0 0.1 1.2 2.0 1.0 1.5 0.0 90 80 70 60 50 40 30 20 10 0 Slope angle, b (deg) (b) Figure 15.13 (a) Deﬁnition of parameters for midpoint circle type of failure; (b) plot of stability number against slope angle (Adapted from Terzaghi and Peck, 1967. With permission of John Wiley & Sons, Inc.) 3. When the critical circle is a midpoint circle (i.e., the failure surface is tangent to the ﬁrm base), its position can be determined with the aid of Figure 15.15. 4. The maximum possible value of the stability number for failure as a midpoint circle is 0.181. Fellenius (1927) also investigated the case of critical toe circles for slopes with 53°. The location of these can be determined with the use of Figure 15.16 and Table 15.1 on the next page. Note that these critical toe circles are not necessarily the most critical circles that exist. O 80 u 70 b a a and u (deg) 60 u 50 40 a 30 50 60 70 80 90 b (deg) Figure 15.14 Location of the center of critical circles for b 53° Table 15.1 Location of the Center of Critical Toe Circles ( 53°) n (deg) 1 (deg) 2 (deg) 1.0 1.5 2.0 3.0 5.0 45 33.68 26.57 18.43 11.32 28 26 25 25 25 37 35 35 35 37 Note: for notations of n, , 1, and 2, see Figure 15.16 528 15.7 Mass Procedure—Slopes in Homogeneous Clay Soil with 0 529 Figure 15.15 Location of midpoint circle (Based on Fellenius, 1927; and Terzaghi and Peck, 1967) O a2 1 n a1 b Figure 15.16 Location of the center of critical toe circles for 53° Example 15.3 A cut slope is to be made in a soft saturated clay with its sides rising at an angle of 60 to the horizontal (Figure 15.17). Given: cu 40 kN/m2 and g 17.5 kN/m3. a. Determine the maximum depth up to which the excavation can be carried out. b. Find the radius, r, of the critical circle when the factor of safety is equal to 1 (Part a). c. Find the distance BC. 530 Chapter 15: Slope Stability O r C B u Radius r Hcr D a A 60 F E Figure 15.17 Solution Part a Since the slope angle b 60 53, the critical circle is a toe circle. From Figure 15.13, for b 60, the stability number 0.195. Hcr cu 40 11.72 m gm 17.5 0.195 Part b From Figure 15.17, r DC u sin 2 But DC AC 2 a Hcr b sin a 2 so, Hcr r 2 sin a sin u 2 From Figure 15.14, for b 60, a 35 and u 72.5. Substituting these values into the equation for r we get r Hcr 2 sin a sin u 2 11.72 17.28 m 21sin 352 1sin 36.252 15.7 Mass Procedure—Slopes in Homogeneous Clay Soil with 0 531 Part c BC EF AF AE Hcr 1cot a cot 75°2 11.721cot 35 cot 602 9.97 m ■ Example 15.4 A cut slope was excavated in a saturated clay. The slope made an angle of 40 with the horizontal. Slope failure occurred when the cut reached a depth of 6.1 m. Previous soil explorations showed that a rock layer was located at a depth of 9.15 m below the ground surface. Assuming an undrained condition and gsat 17.29 kN/m3, ﬁnd the following. a. Determine the undrained cohesion of the clay (Figure 15.13). b. What was the nature of the critical circle? c. With reference to the toe of the slope, at what distance did the surface of sliding intersect the bottom of the excavation? Solution Part a Referring to Figure 15.13, 9.15 1.5 6.1 gsat 17.29 kN/m3 cu Hcr gm D From Figure 15.13, for b 40 and D 1.5, m 0.175. So, cu 1Hcr 2 1g2 1m 2 16.12 117.292 10.1752 18.46 kN/m2 Part b Midpoint circle. Part c Again, from Figure 15.15, for D 1.5, b 40; n 0.9. So, Distance 1n2 1Hcr 2 10.92 16.12 5.49 m ■ 532 Chapter 15: Slope Stability T 15.8 Mass Procedure—Stability of Saturated Clay Slope (F ⴝ 0 Condition) with Earthquake Forces The stability of saturated clay slopes (f 0 condition) with earthquake forces has been analyzed by Koppula (1984). Figure 15.18 shows a clay slope with a potential curve of sliding AED, which is an arc of a circle that has radius r. The center of the circle is located at O. Taking a unit length perpendicular to the slope, we consider these forces for stability analysis: 1. Weight of the soil wedge: W 1area of ABCDEA2 1g2 2. Horizontal inertia force, khW: horizontal component of earthquake acceleration g kh (g acceleration from gravity) 3. Cohesive force along the surface of sliding, which will have a magnitude of u. 1AED2c The moment of the driving forces about O can now be given as Md Wl1 khWl2 (15.39) O Radius r C D l2 H DH l1 khW cu W b A cu B E cu cu cu g f0 Figure 15.18 Stability analysis of slope in homogeneous saturated clay with earthquake forces (f 0 condition) 15.8 Mass Procedure—Stability of Saturated Clay Slope ( 0 Condition) 533 Similarly, the moment of the resisting forces about O is 1cu 2r Mr 1AED2 (15.40) Thus, the factor of safety against sliding is Fs 1AED2 1cu 2 1r2 Mr cu M Md Wl1 khWl2 gH (15.41) where M stability factor. The variations of the stability factor M with slope angle b and kh based on Koppula’s (1984) analysis are given in Figures 15.19 and 15.20. b 10 20 16 b 20 12 M b 30 8 D D 1.0 1.5 2.0 1.0 1.5 2.0 4.0 4.0 4 0 0.0 0.1 6 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 D 1.0 1.5 2.0 4.0 0.0 0.1 0.2 0.3 kh kh kh b 40 b 45 b 50 0.5 D D D 1 1 1 2 2 2 4 4 4 M 4 0.4 2 0 0.0 0.1 0.2 0.3 kh 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 kh Figure 15.19 Variation of M with kh and b based on Koppula’s analysis 0.1 0.2 0.3 kh 0.4 0.5 534 Chapter 15: Slope Stability 6 kh 0.00 0.05 5 0.10 0.15 4 0.20 0.25 M 0.30 3 0.35 0.40 0.45 2 1 0 50 60 70 80 90 b (deg) Figure 15.20 Variation of M with kh based on Koppula’s analysis (for b 55) Example 15.5 A cut slope in saturated clay (Figure 15.21) makes an angle of 56° with the horizontal. Assume kh 0.25. a. Determine the maximum depth up to which the cut could be made. b. How deep should the cut be made if a factor of safety of 2 against sliding is required? Solution Part a For the critical height of the slopes, Fs 1. So, from Eq. (15.41), Hcr cuM g 15.9 Mass Procedure—Slopes in Homogeneous c– Soil H 535 g 100 lb/ft3 cu 500 lb/ft2 f0 b 56 Figure 15.21 From Figure 15.20, for b 56 and kh 0.25, M 3.66. Thus, Hcr 15002 13.662 100 18.3 ft Part b From Eq. (15.41), H 15.9 1500 2 13.662 cuM 9.15 ft gFs 1100 2 122 ■ Mass Procedure—Slopes in Homogeneous c ⴕ– Fⴕ Soil A slope in a homogeneous soil is shown in Figure 15.22a. The shear strength of the soil is given by tf c¿ s¿ tan f¿ is a trial circular arc that passes through The pore water pressure is assumed to be zero. AC the toe of the slope, and O is the center of the circle. Considering a unit length perpendicular to the section of the slope, we ﬁnd Weight of soil wedge ABC W (Area of ABC)(g) 536 Chapter 15: Slope Stability r sin f O r u C B a tf c s tan f r W H Cd b a f A F (a) Cd W Cd F d Cd (b) (c) Figure 15.22 Stability analysis of slope in homogeneous c–f soil For equilibrium, the following other forces are acting on the wedge: • Cd—resultant of the cohesive force that is equal to the cohesion per unit area developed times the length of the cord AC. The magnitude of Cd is given by the following (Figure 15.22b). Cd cdœ 1AC2 (15.42) Cd acts in a direction parallel to the cord AC (see Figure 15.22 b) and at a distance a from the center of the circle O such that r Cd 1a2 cdœ 1AC2 15.9 Mass Procedure—Slopes in Homogeneous c– Soil 537 or a • r cdœ 1AC2 Cd AC r AC (15.43) F—the resultant of the normal and frictional forces along the surface of sliding. For equilibrium, the line of action of F will pass through the point of intersection of the line of action of W and Cd . Now, if we assume that full friction is mobilized (fd f or Ff 1), the line of action of F will make an angle of f with a normal to the arc and thus will be a tangent to a circle with its center at O and having a radius of r sin f. This circle is called the friction circle. Actually, the radius of the friction circle is a little larger than r sin f. Because the directions of W, Cd , and F are known and the magnitude of W is known, a force polygon, as shown in Figure 15.22c, can be plotted. The magnitude of Cd can be determined from the force polygon. So the cohesion per unit area developed can be found. cdœ Cd AC Determination of the magnitude of cdœ described previously is based on a trial surface of sliding. Several trials must be made to obtain the most critical sliding surface, along which the developed cohesion is a maximum. Thus, we can express the maximum cohesion developed along the critical surface as cdœ gH3 f1a, b, u, f¿ 2 4 (15.44) For critical equilibrium—that is, Fc¿ Ff¿ Fs 1—we can substitute H Hcr and cdœ c into Eq. (15.44) and write c¿ gHcr 3 f1a, b, u, f¿ 2 4 or c¿ f1a, b, u, f¿ 2 m gHcr (15.45) where m stability number. The values of m for various values of f and b are given in Figure 15.23, which is based on Taylor (1937). This can be used to determine the factor of safety, Fs, of the homogeneous slope. The procedure to do the analysis is given as Step 1: Determine c, , , and H. Step 2: Assume several values of d (Note: d , such as d(1), d(2) . . . . (Column 1 of Table 15.2). 538 Chapter 15: Slope Stability Table 15.2 Determination of Fs by Frication Circle Method d (1) F¿ tan œ tan dœ (2) m cd (3) (4) œ Fc' (5) œ d(1) tan f œ tan fd112 m1 m1H cd(1) c Fc¿112 œ cd112 d(2) tan f¿ tan f¿d122 m2 m2H cd(2) cœ Fc¿122 œ cd122 Step 3: Determine F , for each assumed value of d as (Column 2, Table 15.2) FSf¿112 FSf¿122 tan f¿ ¿ tan fd112 tan f¿ ¿ tan fd122 Step 4: For each assumed value of d and , determine m (that is, m1, m2, m3, . . .) from Figure 15.23 (Column 3, Table 15.2). Step 5: Determine the developed cohesion for each value of m as (Column 4, Table 15.2) cd(1) m1H cd(2) m2H Step 6: Calculate Fc' for each value of cd (Column 5, Table 15.2), or Fcœ112 cœ œ cd112 Fcœ122 cœ œ cd122 Step 7: Plot a graph of F versus the corresponding Fc (Figure 15.24) and determine Fs F Fc. An example of determining FSs using the procedure just described is given in Example 15.6. Using Taylor’s friction circle method of slope stability (as shown in Example 15.6) Singh (1970) provided graphs of equal factors of safety, FSs, for various slopes. This is shown in Figure 15.25. Calculations have shown that for f 3, the critical circles are all toe circles. More recently, Michalowski (2002) made a stability analysis of simple slopes using the kinematic approach of limit analysis applied to a rigid rotational collapse mechanism. The failure surface in soil assumed in this study is an arc of a logarithmic spiral (Figure 15.26). The results of this study are summarized in Figure 15.27, from which Fs can be obtained directly. Figure 15.23 Taylor’s stability number Ff Fs Fs 45 Fc Figure 15.24 Plot of F versus Fc to determine Fs 539 0.6 0.6 0.5 0.5 0.4 0.4 Fs 2.5 c′ 0.3 gH 2.0 0.2 Fs 3 c′ gH 0.3 0.2 2.5 1.6 0.1 0.8 1.0 0 1.4 1.2 1.1 0.6 10 20 30 ′ (deg) 0.6 50 40 2.0 1.8 1.6 1.2 1.4 0.8 1.0 0.1 0 10 20 30 ′ (deg) (a) 40 50 (b) 0.6 0.6 0.5 0.5 0.4 0.4 c′ gH 0.3 c′ gH 0.3 Fs 3 0.2 0.1 Fs 3.0 0.2 1.2 1.0 0.6 0 10 40 20 30 ′ (deg) 2.0 1.4 1.8 1.6 1.0 0.8 1.2 2.5 2.0 1.8 1.6 1.4 0.8 2.5 0.1 0.6 0 50 10 20 30 ′ (deg) 0.6 0.6 0.6 0.5 0.5 0.5 Fs 3 0.4 2.5 c′ 0.3 gH c′ 0.3 gH 1.6 1.0 0.2 1.4 0.2 0.8 0.6 10 20 30 ′ (deg) (e) 40 Fs 3 c′ 0.3 gH 2.0 1.8 0.2 1.6 1.4 1.2 0.1 0.8 0.6 0.1 1.2 0 0.4 2.5 2.0 1.8 50 50 (d) (c) 0.4 40 0 1.0 0.1 Fs 3 2.5 1.8 1.2 1.0 2.0 0.8 1.6 1.4 0.6 10 20 30 ′ (deg) (f) 40 50 0 10 20 30 ′ (deg) 40 50 (g) Figure 15.25 Contours of equal factors of safety: (a) slope – 1 vertical to 0.5 horizontal; (b) slope – 1 vertical to 0.75 horizontal; (c) slope – 1 vertical to 1 horizontal; (d) slope – 1 vertical to 1.5 horizontal; (e) slope – 1 vertical to 2 horizontal; (f) slope – 1 vertical to 2.5 horizontal; (g) slope – 1 vertical to 3 horizontal (After Singh, 1970. With permission from ASCE.) 540 15.9 Mass Procedure—Slopes in Homogeneous c– Soil 541 v O rO E A V H Log-spiral b D C f 5 b 15 30 Figure 15.26 Stability analysis using rotational collapse mechanism V V B 45 14 60 12 75 10 90 8 b 15 30 45 60 75 4 Fs tan f 3 Fs tan f 90 6 2 4 1 2 0 0 0.0 0.1 0.2 0.3 0.4 0.5 c gH tan f 0.0 0.5 1.0 1.5 c gH tan f 2.0 2.5 3.0 Figure 15.27 Michalowski’s analysis for stability of simple slopes Example 15.6 A slope with 45° is to be constructed with a soil that has 20° and c 24 kN/m2. The unit weight of the compacted soil will be 18.9 kN/m3. a. Find the critical height of the slope. b. If the height of the slope is 10 m, determine the factor of safety with respect to strength. 542 Chapter 15: Slope Stability Solution Part a We have m cœ gHcr From Figure 15.23, for 45° and 20°, m 0.06. So Hcr 24 cœ 21.1 m gm 118.9 2 10.062 Part b If we assume that full friction is mobilized, then, referring to Figure 15.23 (for 45° and d 20°), we have m 0.06 cdœ gH or cd (0.06)(18.9)(10) 11.34 kN/m2 Thus, Ffœ tan fœ tan 20 1 œ tan fd tan 20 and Fcœ cœ 24 2.12 œ cd 11.34 Since FSc FS, this is not the factor of safety with respect to strength. Now we can make another trial. Let the developed angle of friction, d, be equal to 15°. For 45° and the friction angle equal to 15°, we find from Figure 15.23. m 0.083 cdœ gH or cd (0.083)(18.9)(10) 15.69 kN/m2 For this trial, Ffœ tan fœ tan 20 1.36 œ tan fd tan 15 15.9 Mass Procedure—Slopes in Homogeneous c– Soil 543 and Fcœ cœ 24 1.53 œ cd 15.69 Similar calculations of F and Fc for various assumed values of d are given in the following table. d tan d F m 20 15 10 5 0.364 0.268 0.176 0.0875 1.0 1.36 2.07 4.16 0.06 0.083 0.105 0.136 cd (kN/m2) 11.34 15.69 19.85 25.70 Fc 2.12 1.53 1.21 0.93 The values of F are plotted against their corresponding values of Fc in Figure 15.28, from which we find FSc FS FSs 1.42 Note: We could have found the value of FSs from Figure 15.25c. Since 45°, it is a slope of 1V:1H. For this slope cœ 24 0.127 gH 118.9 2 110 2 From Figure 15.25c, for’ c/H 0.127, the value of FSs 1.4 5 4 Ff 3 2 Fs 1 Fs 45 0 0 1 2 Fc 3 4 5 Figure 15.28 ■ 544 Chapter 15: Slope Stability Example 15.7 Solve Example 15.6 using Michalowski’s solution. Solution Part a For critical height (Hcr), FSs 1. Thus, 3.49 24 cœ gH tan fœ 118.92 1Hcr 2 1tan 202 Hcr FSs 1 2.747 tan fœ tan 20 b 45° From Figure 15.27, for 45° and Fs /tan 2.747, the value of c/H tan 0.17. So 3.49 0.17; Hcr 20.5 m Hcr Part b cœ 24 0.349 gH tan fœ 118.9 2 1102 1tan 202 b 45° From Figure 15.27, Fs /tan 4. Fs 4 tan (4)(tan 20) 1.46 15.10 ■ Ordinary Method of Slices Stability analysis by using the method of slices can be explained with the use of Figure 15.29a, in which AC is an arc of a circle representing the trial failure surface. The soil above the trial failure surface is divided into several vertical slices. The width of each slice need not be the same. Considering a unit length perpendicular to the cross section shown, the forces that act on a typical slice (nth slice) are shown in Figure 15.29b. Wn is the weight of the slice. The forces Nr and Tr, respectively, are the normal and tangential components of the reaction R. Pn and Pn1 are the normal forces that act on the sides of the slice. Similarly, the shearing forces that act on the sides of the slice are Tn and Tn1. For simplicity, the pore water pressure is assumed to be zero. The forces Pn, Pn1, Tn, and Tn1 are difﬁcult to determine. However, we can make an approximate assumption that the resultants of Pn and Tn are equal in magnitude to the resultants of Pn1 and Tn1 and that their lines of action coincide. 15.10 Ordinary Method of Slices 545 r sin an O r C bn B 1 2 r n H r Wn A tf c s tan f p an (a) Tn Wn Pn Pn1 Tn1 an Tr Nr R Wn an Ln Figure 15.29 Stability analysis by ordinary method of slices: (a) trial failure surface; (b) forces acting on nth slice (b) For equilibrium consideration, Nr Wn cos an The resisting shear force can be expressed as Tr td 1¢Ln 2 tf 1¢Ln 2 Fs 1 3 c¿ s¿ tan f¿ 4 ¢Ln Fs (15.46) 546 Chapter 15: Slope Stability The normal stress, s, in Eq. (15.46) is equal to Wn cos an Nr ¢Ln ¢Ln For equilibrium of the trial wedge ABC, the moment of the driving force about O equals the moment of the resisting force about O, or np np Wn cos an 1 W r sin a tan f¿ b 1¢Ln 2 1r2 n n a F a c¿ a ¢Ln n1 n1 s or np a 1c¿¢Ln Wn cos an tan f¿ 2 Fs n1 (15.47) np a Wn sin an n1 [Note: Ln in Eq. (15.47) is approximately equal to (bn)/(cos an), where bn the width of the nth slice.] Note that the value of an may be either positive or negative. The value of an is positive when the slope of the arc is in the same quadrant as the ground slope. To ﬁnd the minimum factor of safety—that is, the factor of safety for the critical circle—one must make several trials by changing the center of the trial circle. This method generally is referred to as the ordinary method of slices. For convenience, a slope in a homogeneous soil is shown in Figure 15.29. However, the method of slices can be extended to slopes with layered soil, as shown in Figure 15.30. The general procedure of stability analysis is the same. However, some minor points should be kept in mind. When Eq. (15.47) is used for the factor of safety calculation, the values of f and c will not be the same for all slices. For example, for slice No. 3 (see Figure 15.30), we have to use a friction angle of f f3œ and cohesion c c3œ ; similarly, for slice No. 2, f f2œ and c c2œ . C B 1 g1, f 1, c 1 2 4 3 g2 , f 2, c 2 5 A 6 g3 , f 3, c 3 Figure 15.30 Stability analysis, by ordinary method of slices, for slope in layered soils 15.10 Ordinary Method of Slices 547 It is of interest to note that if total shear strength parameters (that is, tf c tan f) were used, Eq. (15.47) would take the form np a 1c¢Ln Wn cos an tan f2 Fs n1 (15.48) np a Wn sin an n1 Example 15.8 For the slope shown in Figure 15.31, ﬁnd the factor of safety against sliding for the trial slip surface AC. Use the ordinary method of slices. 18 m O 5m B C 1 2 70 3 W1 4 g 16 kN/m 3 c 20 kN/m2 f 20 5 A 7 54 W2 6 30 24 W3 8 0 W6 W5 W7 14 m 12 38 W4 Figure 15.31 Stability analysis of a slope by ordinary method of slices Solution The sliding wedge is divided into seven slices. Now the following table can be prepared: Slice no. (1) W (kN/m) (2) An (deg) (3) sin An (4) cos An (5) ⌬Ln (m) (6) Wn sin An (kN/m) (7) Wn cos An (kN/m) (8) 1 2 3 4 5 6 7 22.4 294.4 435.2 435.2 390.4 268.8 66.58 70 54 38 24 12 0 8 0.94 0.81 0.616 0.407 0.208 0 0.139 0.342 0.588 0.788 0.914 0.978 1 0.990 2.924 6.803 5.076 4.376 4.09 4 3.232 Col. 6 30.501 m 21.1 238.5 268.1 177.1 81.2 0 9.25 Col. 7 776.75 kN/m 7.66 173.1 342.94 397.8 381.8 268.8 65.9 Col. 8 1638 kN/m 548 Chapter 15: Slope Stability Fs 15.11 1© Col. 62 1c¿ 2 1© Col. 82tan f¿ © Col. 7 130.5012 1202 116382 1tan 202 776.75 1.55 ■ Bishop’s Simpliﬁed Method of Slices In 1955, Bishop proposed a more reﬁned solution to the ordinary method of slices. In this method, the effect of forces on the sides of each slice are accounted for to some degree. We can study this method by referring to the slope analysis presented in Figure 15.29. The forces that act on the nth slice shown in Figure 15.29b have been redrawn in Figure 15.32a. Now, let Pn Pn1 P and Tn Tn1 T. Also, we can write Tr Nr 1tan fdœ 2 cdœ ¢Ln Nr a tan f¿ c¿¢Ln b Fs Fs (15.49) Figure 15.32b shows the force polygon for equilibrium of the nth slice. Summing the forces in the vertical direction gives Wn ¢T Nr cos an c Nr tan f¿ c¿¢Ln d sin an Fs Fs Tn Wn Tn1 Pn Nr tan f Fs cLn Fs Pn1 Wn Nr fd an Tr R Wn an an N r T Ln (a) P (b) Figure 15.32 Bishop’s simpliﬁed method of slices: (a) forces acting on the nth slice; (b) force polygon for equilibrium 15.11 Bishop’s Simpliﬁed Method of Slices 549 or c¿¢Ln sin an Fs tan f¿ sin an cos an Fs Wn ¢T Nr (15.50) For equilibrium of the wedge ABC (Figure 15.29a), taking the moment about O gives np np n1 n1 a Wnr sin an a Tr r (15.51) where Tr 1 1c¿ s¿ tan f¿ 2 ¢Ln Fs 1 1c¿¢Ln Nr tan f¿ 2 Fs (15.52) Substitution of Eqs. (15.50) and (15.52) into Eq. (15.51) gives np Fs 1 a 1c¿bn Wn tan f¿ ¢T tan f¿ 2 m a1n2 n1 np (15.53) a Wn sin an n1 where ma1n2 cos an tan f¿ sin an Fs (15.54) Figure 15.33 shows the variation of ma(n) with an and tan f/Fs. Figure 15.33 Variation of m(n) with n and tan /Fs [Eq. (15.54)] 550 Chapter 15: Slope Stability For simplicity, if we let T 0, Eq. (15.53) becomes np Fs 1 a 1c¿bn Wn tan f¿ 2 m a1n2 n1 np (15.55) a Wn sin an n1 Note that the term Fs is present on both sides of Eq. (15.55). Hence, we must adopt a trial-and-error procedure to ﬁnd the value of Fs . As in the method of ordinary slices, a number of failure surfaces must be investigated so that we can ﬁnd the critical surface that provides the minimum factor of safety. Bishop’s simpliﬁed method is probably the most widely used. When incorporated into computer programs, it yields satisfactory results in most cases. The ordinary method of slices is presented in this chapter as a learning tool only. It is used rarely now because it is too conservative. 15.12 Stability Analysis by Method of Slices for Steady-State Seepage The fundamentals of the ordinary method of slices and Bishop’s simpliﬁed method of slices were presented in Sections 15.10 and 15.11, respectively, and we assumed the pore water pressure to be zero. However, for steady-state seepage through slopes, as is the situation in many practical cases, the pore water pressure must be considered when effective shear strength parameters are used. So we need to modify Eqs. (15.47) and (15.55) slightly. Figure 15.34 shows a slope through which there is steady-state seepage. For the nth slice, the average pore water pressure at the bottom of the slice is equal to un hngw . The total force caused by the pore water pressure at the bottom of the nth slice is equal to unLn . z Phreatic surface h Seepage H b Figure 15.34 Stability analysis of slope with steady-state seepage 15.12 Stability Analysis by Method of Slices for Steady-State Seepage 551 Thus, Eq. (15.47) for the ordinary method of slices will be modiﬁed to read as follows. np a 1c¿¢Ln 1Wn cos an un ¢Ln 2 4 tan f¿ Fs n1 (15.56) np a Wn sin an n1 Similarly, Eq. (15.55) for Bishop’s simpliﬁed method of slices will be modiﬁed to the form np Fs 1 a 3c¿bn 1Wn unbn 2tan f¿ 4 m 1a2n n1 np (15.57) a Wn sin an n1 Note that Wn in Eqs. (15.56) and (15.57) is the total weight of the slice. Using Eq. (15.57), Bishop and Morgenstern (1960) developed tables for the calculation of Fs for simple slopes. The principles of these developments can be explained as follows. In Eq. (15.57), Wn total weight of the nth slice gbnzn (15.58) where zn average height of the nth slice. Also in Eq. (15.57), un hngw So, we can let ru1n2 un hngw gzn gzn (15.59) Note that ru(n) is a nondimensional quantity. Substituting Eqs. (15.58) and (15.59) into Eq. (15.57) and simplifying, we obtain bn zn c¿ bn 1 np 31 ru1n2 4 tan f¿ b gH H H H z § a• ¶ (15.60) Fs £ n n sin a n n1 a HH m a1n2 n1 np For a steady-state seepage condition, a weighted average value of ru(n) can be taken, which is a constant. Let the weighted averaged value of ru(n) be ru . For most practical cases, the value of ru may range up to 0.5. Thus, Fs £ bn zn 1 c¿ bn np 11 ru 2tan f¿ d c bn zn HH § a • gH H ¶ n1 a H H sin an ma1n2 n1 np (15.61) 552 Chapter 15: Slope Stability The factor of safety based on the preceding equation can be solved and expressed in the form Fs m¿ n¿ru (15.62) where m and n stability coefﬁcients. Table 15.3 gives the values of m and n for various combinations of c/gH, D, f, and b. Table 15.3 Values of m and n [Eq. (15.62)] a. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 10.0 12.5 15.0 17.5 0.353 0.443 0.536 0.631 0.441 0.554 0.670 0.789 0.529 0.665 0.804 0.946 0.588 0.739 0.893 1.051 0.705 0.887 1.072 1.261 0.749 0.943 1.139 1.340 0.882 1.109 1.340 1.577 0.917 1.153 1.393 1.639 20.0 22.5 25.0 27.5 0.728 0.828 0.933 1.041 0.910 1.035 1.166 1.301 1.092 1.243 1.399 1.562 1.213 1.381 1.554 1.736 1.456 1.657 1.865 2.082 1.547 1.761 1.982 2.213 1.820 2.071 2.332 2.603 1.892 2.153 2.424 2.706 30.0 32.5 35.0 37.5 1.155 1.274 1.400 1.535 1.444 1.593 1.750 1.919 1.732 1.911 2.101 2.302 1.924 2.123 2.334 2.558 2.309 2.548 2.801 3.069 2.454 2.708 2.977 3.261 2.887 3.185 3.501 3.837 3.001 3.311 3.639 3.989 40.0 1.678 2.098 2.517 2.797 3.356 3.566 4.196 4.362 b. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.025 and D ⴝ 1.00 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 10.0 12.5 15.0 17.5 0.678 0.790 0.901 1.012 0.534 0.655 0.776 0.898 0.906 1.066 1.224 1.380 0.683 0.849 1.014 1.179 1.130 1.337 1.544 1.751 0.846 1.061 1.273 1.485 1.367 1.620 1.868 2.121 1.031 1.282 1.534 1.789 20.0 22.5 25.0 27.5 1.124 1.239 1.356 1.478 1.022 1.150 1.282 1.421 1.542 1.705 1.875 2.050 1.347 1.518 1.696 1.882 1.962 2.177 2.400 2.631 1.698 1.916 2.141 2.375 2.380 2.646 2.921 3.207 2.050 2.317 2.596 2.886 30.0 32.5 35.0 37.5 1.606 1.739 1.880 2.030 1.567 1.721 1.885 2.060 2.235 2.431 2.635 2.855 2.078 2.285 2.505 2.741 2.873 3.127 3.396 3.681 2.622 2.883 3.160 3.458 3.508 3.823 4.156 4.510 3.191 3.511 3.849 4.209 40.0 2.190 2.247 3.090 2.993 3.984 3.778 4.885 4.592 (continued) Table 15.3 (continued) c. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.025 and D ⴝ 1.25 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 10.0 12.5 15.0 17.5 0.737 0.878 1.019 1.162 0.614 0.759 0.907 1.059 0.901 1.076 1.253 1.433 0.726 0.908 1.093 1.282 1.085 1.299 1.515 1.736 0.867 1.098 1.311 1.541 1.285 1.543 1.803 2.065 1.014 1.278 1.545 1.814 20.0 22.5 25.0 27.5 1.309 1.461 1.619 1.783 1.216 1.379 1.547 1.728 1.618 1.808 2.007 2.213 1.478 1.680 1.891 2.111 1.961 2.194 2.437 2.689 1.775 2.017 2.269 2.531 2.334 2.610 2.879 3.196 2.090 2.373 2.669 2.976 30.0 32.5 35.0 37.5 1.956 2.139 2.331 2.536 1.915 2.112 2.321 2.541 2.431 2.659 2.901 3.158 2.342 2.686 2.841 3.112 2.953 3.231 3.524 3.835 2.806 3.095 3.400 3.723 3.511 3.841 4.191 4.563 3.299 3.638 3.998 4.379 40.0 2.753 2.775 3.431 3.399 4.164 4.064 4.958 4.784 d. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.05 and D ⴝ 1.00 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 10.0 12.5 15.0 17.5 0.913 1.030 1.145 1.262 0.563 0.690 0.816 0.942 1.181 1.343 1.506 1.671 0.717 0.878 1.043 1.212 1.469 1.688 1.904 2.117 0.910 1.136 1.353 1.565 1.733 1.995 2.256 2.517 1.069 1.316 1.567 1.825 20.0 22.5 25.0 27.5 1.380 1.500 1.624 1.753 1.071 1.202 1.338 1.480 1.840 2.014 2.193 1.380 1.387 1.568 1.757 1.952 2.333 2.551 2.778 3.013 1.776 1.989 2.211 2.444 2.783 3.055 3.336 3.628 2.091 2.365 2.651 2.948 30.0 32.5 35.0 37.5 1.888 2.029 2.178 2.336 1.630 1.789 1.958 2.138 2.574 2.777 2.990 3.215 2.157 2.370 2.592 2.826 3.261 3.523 3.803 4.103 2.693 2.961 3.253 3.574 3.934 4.256 4.597 4.959 3.259 3.585 3.927 4.288 40.0 2.505 2.332 3.451 3.071 4.425 3.926 5.344 4.668 (continued) 553 Table 15.3 (continued) e. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.05 and D ⴝ 1.25 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 10.0 12.5 15.0 17.5 0.919 1.065 1.211 1.359 0.633 0.792 0.950 1.108 1.119 1.294 1.471 1.650 0.766 0.941 1.119 1.303 1.344 1.563 1.782 2.004 0.886 1.112 1.338 1.567 1.594 1.850 2.109 2.373 1.042 1.300 1.562 1.831 20.0 22.5 25.0 27.5 1.509 1.663 1.822 1.988 1.266 1.428 1.595 1.769 1.834 2.024 2.222 2.428 1.493 1.690 1.897 2.113 2.230 2.463 2.705 2.957 1.799 2.038 2.287 2.546 2.643 2.921 3.211 3.513 2.107 2.392 2.690 2.999 30.0 32.5 35.0 37.5 2.161 2.343 2.535 2.738 1.950 2.141 2.344 2.560 2.645 2.873 3.114 3.370 2.342 2.583 2.839 3.111 3.221 3.500 3.795 4.109 2.819 3.107 3.413 3.740 3.829 4.161 4.511 4.881 3.324 3.665 4.025 4.405 40.0 2.953 2.791 3.642 3.400 4.442 4.090 5.273 4.806 f. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.05 and D ⴝ 1.50 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 10.0 12.5 15.0 17.5 1.022 1.202 1.383 1.565 0.751 0.936 1.122 1.309 1.170 1.376 1.583 1.795 0.828 1.043 1.260 1.480 1.343 1.589 1.835 2.084 0.974 1.227 1.480 1.734 1.547 1.829 2.112 2.398 1.108 1.399 1.690 1.983 20.0 22.5 25.0 27.5 1.752 1.943 2.143 2.350 1.501 1.698 1.903 2.117 2.011 2.234 2.467 2.709 1.705 1.937 2.179 2.431 2.337 2.597 2.867 3.148 1.993 2.258 2.534 2.820 2.690 2.990 3.302 3.626 2.280 2.585 2.902 3.231 30.0 32.5 35.0 37.5 2.568 2.798 3.041 3.299 2.342 2.580 2.832 3.102 2.964 3.232 3.515 3.817 2.696 2.975 3.269 3.583 3.443 3.753 4.082 4.431 3.120 3.436 3.771 4.128 3.967 4.326 4.707 5.112 3.577 3.940 4.325 4.735 40.0 3.574 3.389 4.136 3.915 4.803 4.507 5.543 5.171 (continued) 554 Table 15.3 (continued) g. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.075 and toe circles Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 20 25 30 35 40 1.593 1.853 2.133 2.433 2.773 1.158 1.430 1.730 2.058 2.430 2.055 2.426 2.826 3.253 3.737 1.516 1.888 2.288 2.730 3.231 2.498 2.980 3.496 4.055 4.680 1.903 2.361 2.888 3.445 4.061 2.934 3.520 4.150 4.846 5.609 2.301 2.861 3.461 4.159 4.918 h. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.075 and D ⴝ 1.00 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 20 25 30 35 40 1.610 1.872 2.142 2.443 2.772 1.100 1.386 1.686 2.030 2.386 2.141 2.502 2.884 3.306 3.775 1.443 1.815 2.201 2.659 3.145 2.664 3.126 3.623 4.177 4.785 1.801 2.259 2.758 3.331 3.945 3.173 3.742 4.357 5.024 5.776 2.130 2.715 3.331 4.001 4.759 i. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.075 and D ⴝ 1.25 Stability coefﬁcients for earth slopes Slope 2:1 Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 20 25 30 35 40 1.688 2.004 2.352 2.728 3.154 1.285 1.641 2.015 2.385 2.841 2.071 2.469 2.888 3.357 3.889 1.543 1.957 2.385 2.870 3.428 2.492 2.972 3.499 4.079 4.729 1.815 2.315 2.857 3.457 4.128 2.954 3.523 4.149 4.831 5.603 2.173 2.730 3.357 4.043 4.830 j. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.075 and D ⴝ 1.50 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 20 25 30 35 40 1.918 2.308 2.735 3.211 3.742 1.514 1.914 2.355 2.854 3.397 2.199 2.660 3.158 3.708 4.332 1.728 2.200 2.714 3.285 3.926 2.548 3.083 3.659 4.302 5.026 1.985 2.530 3.128 3.786 4.527 2.931 3.552 4.128 4.961 5.788 2.272 2.915 3.585 4.343 5.185 (continued) 555 Table 15.3 (continued) k. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.100 and toe circles Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 20 25 30 35 40 1.804 2.076 2.362 2.673 3.012 2.101 1.488 1.786 2.130 2.486 2.286 2.665 3.076 3.518 4.008 1.588 1.945 2.359 2.803 3.303 2.748 3.246 3.770 4.339 4.984 1.974 2.459 2.961 3.518 4.173 3.190 3.796 4.442 5.146 5.923 2.361 2.959 3.576 4.249 5.019 l. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.100 and D ⴝ 1.00 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 20 25 30 35 40 1.841 2.102 2.378 2.692 3.025 1.143 1.430 1.714 2.086 2.445 2.421 2.785 3.183 3.612 4.103 1.472 1.845 2.258 2.715 3.230 2.982 3.458 3.973 4.516 5.144 1.815 2.303 2.830 3.359 4.001 3.549 4.131 4.751 5.426 6.187 2.157 2.743 3.372 4.059 4.831 m. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.100 and D ⴝ 1.25 Slope 2:1 Stability coefﬁcients for earth slopes Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 20 25 30 35 40 1.874 2.197 2.540 2.922 3.345 1.301 1.642 2.000 2.415 2.855 2.283 2.681 3.112 3.588 4.119 1.558 1.972 2.415 2.914 3.457 2.751 3.233 3.753 4.333 4.987 1.843 2.330 2.858 3.458 4.142 3.253 3.833 4.451 5.141 5.921 2.158 2.758 3.372 4.072 4.872 n. Stability coefﬁcients m ⴕ and n ⴕ for c ⴕ/GH ⴝ 0.100 and D ⴝ 1.50 Stability coefﬁcients for earth slopes Slope 2:1 556 Slope 3:1 Slope 4:1 Slope 5:1 Fⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ mⴕ nⴕ 20 25 30 35 40 2.079 2.477 2.908 3.385 3.924 1.528 1.942 2.385 2.884 3.441 2.387 2.852 3.349 3.900 4.524 1.742 2.215 2.728 3.300 3.941 2.768 3.297 3.881 4.520 5.247 2.014 2.542 3.143 3.800 4.542 3.158 3.796 4.468 5.211 6.040 2.285 2.927 3.614 4.372 5.200 15.13 Other Solutions for Steady-State Seepage Condition 557 To determine Fs from Table 15.3, we must use the following step-by-step procedure: Step 1: Obtain f, b, and c/gH. Step 2: Obtain ru (weighted average value). Step 3: From Table 15.3, obtain the values of m and n for D 1, 1.25, and 1.5 (for the required parameters f, b, ru , and c/gH). Step 4: Determine Fs , using the values of m and n for each value of D. Step 5: The required value of Fs is the smallest one obtained in Step 4. 15.13 Other Solutions for Steady-State Seepage Condition Spencer’s Solution Bishop’s simpliﬁed method of slices described in Sections 15.11 and 15.12 satisﬁes the equations of equilibrium with respect to the moment but not with respect to the forces. Spencer (1967) has provided a method to determine the factor of safety (Fs ) by taking into account the interslice forces (Pn , Tn , Pn1 , Tn1 , as shown in Figure 15.32), which does satisfy the equations of equilibrium with respect to moment and forces. The details of this method of analysis are beyond the scope of this text; however, the ﬁnal results of Spencer’s work are summarized in this section in Figure 15.35. Note that ru , as shown in Figure 15.35, is the same as that deﬁned by Eq. (15.61). In order to use the charts given in Figure 15.35 and to determine the required value of Fs , the following step-by-step procedure needs to be used. Step 1: Determine c, g, H, b, f, and ru for the given slope. Step 2: Assume a value of Fs . Step 3: Calculate c/[Fs (assumed) gH]. c Step 2 Step 4: With the value of c/FsgH calculated in Step 3 and the slope angle b, enter the proper chart in Figure 15.35 to obtain fdœ . Note that Figures 15.35 a, b, and c, are, respectively, for ru of 0, 0.25, and 0.5, respectively. Step 5: Calculate Fs tan f/tan fdœ . c Step 4 Step 6: If the values of Fs as assumed in Step 2 are not the same as those calculated in Step 5, repeat Steps 2, 3, 4, and 5 until they are the same. Michalowski’s Solution Michalowski (2002) used the kinematic approach of limit analysis similar to that shown in Figures 15.26 and 15.27 to analyze slopes with steady-state seepage. The results of this analysis are summarized in Figure 15.36 for ru 0.25 and ru 0.5. Note that Figure 15.27 is applicable for the ru 0 condition. 0.12 0.10 c/FsgH 0.08 fd 10 ru 0 12 14 16 18 20 22 24 26 28 30 0.06 0.04 0.02 0.00 4 6 10 14 18 22 Slope angle b (deg) 26 30 34 (a) 0.12 0.10 fd 10 ru 0.25 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 c/FsgH 0.08 0.06 0.04 0.02 0.00 4 6 10 14 18 22 Slope angle b (deg) 26 30 34 (b) fd 10 0.12 0.10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 ru 0.5 c/FsgH 0.08 0.06 0.04 0.02 0.00 4 6 10 14 18 22 26 Slope angle b (deg) (c) Figure 15.35 Spencer’s solution—plot of c/Fs H versus b 558 30 34 b 15 14 30 45 60 75 b 15 30 45 14 60 75 90 12 10 10 8 8 6 6 4 4 90 Fs tan f 12 2 2 ru 0.50 ru 0.25 0 0 0.0 0.5 1.0 1.5 2.0 b 15 5 2.5 3.0 30 0.0 0.5 1.0 1.5 2.0 2.5 3.0 b 15 5 30 45 4 60 75 Fs tan f 3 4 45 60 3 75 90 2 2 90 1 1 ru 0.50 ru 0.25 0 0 0.0 0.1 0.2 0.3 c gH tan f 0.4 0.5 0.0 0.1 0.2 0.3 c gH tan f 0.4 0.5 Figure 15.36 Michalowski’s solution for steady-state seepage condition 559 560 Chapter 15: Slope Stability Example 15.9 A given slope under steady-state seepage has the following: H 21.62 m, f 25, slope: 2H:1V, c 20 kN/m2, g 18.5 kN/m3, ru 0.25. Determine the factor of safety, Fs . Use Table 15.3. Solution 1 b tan1 a b 26.57° 2 20 c¿ 0.05 gH 118.5 2 121.622 Now the following table can be prepared. Fs ⴝ mⴕ ⴚ nⴕru c B (deg) Fⴕ (deg) cⴕ/GH D mⴕ a nⴕ b 26.57 26.57 26.57 25 25 25 0.05 0.05 0.05 1.00 1.25 1.5 1.624 d 1.822 e 2.143 f 1.338 d 1.595 e 1.903 f 1.29 1.423 1.667 a From Table 15.3 From Table 15.3 c Eq. (15.62); ru 0.25 d Table 15.3 e Table 15.3 f Table 15.3 b So, Fs 1.29 ■ Example 15.10 Solve Example 15.9 using Spencer’s solution (Figure 15.35). Solution Given: H 21.62 m, b 26.57, c 20 kN/m2, g 18.5 kN/m3, f 25, and ru 0.25. Now the following table can be prepared. a B (deg) Fs (assumed) 26.57 26.57 26.57 26.57 1.1 1.2 1.3 1.4 From Figure 15.35b c¿ Fs 1assumed2GH 0.0455 0.0417 0.0385 0.0357 Fdœ a (deg) 18 19 20 21 Fs (calculated) ⴝ tan F¿ tan Fdœ 1.435 1.354 1.281 1.215 15.14 A Case History of Slope Failure 561 Figure 15.37 shows a plot of Fs(assumed) against Fs(calculated) , from which Fs 1.3. Fs(calculated) 1.6 1.4 1.3 1.2 45 1.0 1.0 1.2 1.4 Fs(assumed) 1.6 Figure 15.37 ■ Example 15.11 Solve Example 15.9 using Michalowski’s solution (Figure 15.36). Solution 20 c¿ 0.107 gH tan f¿ 118.5 2 121.622 1tan 252 Fs For ru 0.25, from Figure 15.36, 3.1 So, tan f¿ Fs 13.1 2 1tan 252 1.45 15.14 ■ A Case History of Slope Failure Ladd (1972) reported the results of a study of the failure of a slope that had been constructed over a sensitive clay. The study was conducted in relation to a major improvement program of Interstate Route 95 in Portsmouth, New Hampshire, which is located 50 miles north of Boston on the coast. To study the stability of the slope, a test embankment was built to failure during the spring of 1968. The test embankment was heavily instrumented. The general subsoil condition at the test site, the section of the test embankment, and the instruments placed to monitor the performance of the test section are shown in Figure 15.38. The ground water level at the test section was at an elevation of +20 ft (mean sea level). The general physical properties of the soft to very soft gray silty clay layer as shown in Figure 15.38 are as follows: Natural moisture content 50 5% Undrained shear strength as obtained from field vane shear tests 250 50 lb/ft2 562 70 60 Settlement platform Elevation (fr—mean sea level) 50 40 El. 41.5 ft Settlement platform with Borros point Alignment stake Vibrating wire Piezometer Casagrande piezometer 30 1 4 1 Sand fill 20 Slope indicator 8 Medium–stiff clay 10 Soft to very soft gray silty clay 0 10 Loose fine sand–silt Soft clay Sandy till Rock 20 30 40 W100 80 60 40 20 0 20 Distance (ft) 40 60 80 100 120 140 Figure 15.38 Cross-section through the centerline of the experimental test section looking north (After Ladd, 1972. With permission from ASCE.) 160E 15.14 A Case History of Slope Failure 563 Remolded shear strength 25 5 lb/ft2 Liquid limit 35 5 Plastic limit 20 2 During construction of the test embankment, fill was placed at a fairly uniform rate within a period of about one month. Failure of the slope (1 vertical: 4 horizontal) occurred on June 6, 1968, at night. The height of the embankment at failure was 21.5 ft. Figure 15.39 shows the actual failure surface of the slope. The rotated section shown in Figure 15.39 is the “before failure” section rotated through an angle of 13 degrees about a point W 45 ft, El. 51 ft. Ladd (1972) reported the total stress ( 0 concept) stability analysis of the slope that failed by using Bishop’s simplified method (Section 15.11). The variation of the undrained shear strengths (cu) used for the stability analysis is given below. Note that these values have not been corrected via Eq. (12.46). Elevation (ft—mean sea level) cu as obtained from vane shear strength tests (lb/ft2) 20 to 15 15 to 10 10 to 5 5 to 0 0 to 2.5 2 to 5 5 to 10 10 to 13 1000 400 240 250 300 235 265 300 60 Centerline Elevation (fr—mean sea level) 50 Before 40 Fill Rotated 30 After Rotated 20 Medium clay Rotated Slope indicator, before 10 0 Original ground surface After Soft clay LEGEND Before failure After failure Rotated section 10 Sand–Silt 20 120 110 100 90 80 40 60 50 40 30 Distance (ft) 20 10 0 10 Figure 15.39 Cross-section of the experimental test section before and after failure (After Ladd, 1972. With permission from ASCE.) 20 30 564 Chapter 15: Slope Stability The factor of safety (Fs) as obtained from the stability analysis for the critical circle of sliding was 0.88. The critical circle of sliding is shown in Figure 15.40. The factor of safety for actual surface of sliding as obtained by using Bishop’s simplified method was 0.92. For comparison purposes, the actual surface of sliding is also shown in Figure 15.40. Note that the bottom of the actual failure surface is about 3 ft (0.91 m) above the theoretically determined critical failure surface. Ladd (1972) also reported the stability analysis of the slope based on the average undrained shear strength variation of the clay layer as determined by using the Stress History And Normalized Soil Engineering Properties (SHANSEP). The details of obtaining cu by this procedure are beyond the scope of this text. However, the final results are given in the following table. Elevation (ft—mean sea level) Average cu as obtained from SHANSEP (lb/ft2) 20 to 15 15 to 10 10 to 5 5 to 0 0 to 2.5 2.5 to 5 5 to 10 10 to 13 1000 335 230 260 300 320 355 400 80 Critical failure surface (from vane shear test) Actual failure surface Critical failure surface from SHANSEP Elevation (ft—mean sea level) 60 g 110 lb/ft3 f 30 El. 41.5 ft 40 Ground water table g 115 lb/ft3 f 40 Sand fill 20 Medium clay Soft clay 0 20 120 Sand-silt; g 130 lb/ft3; f 30 100 80 60 40 Distance (ft) 20 0 Figure 15.40 Results of total stress stability analysis (After Ladd, 1972. With permission from ASCE.) (Note: SHANSEP Stress History And Normalized Soil Engineering Properties) 15.15 Morgenstern’s Method of Slices for Rapid Drawdown Condition 565 Using the preceding average value of cu, Bishop’s simplified method of stability analysis yields the following results: Failure surface Factor of safety, Fs actual failure surface critical failure surface 1.02 1.01 Figure 15.40 also shows the critical failure surface as determined by using the values of cu obtained from SHANSEP. Based on the preceding results, we can draw the following conclusions: a. The actual failure surface of a slope with limited height is an arc of a circle. b. The disagreement between the predicted critical failure surface and the actual failure surface is primarily due to the shear strength assumptions. The cu values obtained from SHANSEP give an Fs 1, and the critical failure surface is practically the same as the actual failure surface. This case study is another example that demonstrates the importance of proper evaluation of soil parameters for prediction of the stability of various structures. 15.15 Morgenstern’s Method of Slices for Rapid Drawdown Condition Morgenstern (1963) used Bishop’s method of slices (Section 15.11) to determine the factor of safety, Fs , during rapid drawdown. In preparing the solution, Morgenstern used the following notation (Figure 15.41): • • • L height of drawdown H height of embankment b angle that the slope makes with the horizontal Water table before drawdown L Water table after drawdown H b Impervious base Figure 15.41 Stability analysis for rapid drawdown condition 566 Chapter 15: Slope Stability Morgenstern also assumed that 1. The embankment is made of homogeneous material and rests on an impervious base. 2. Initially, the water level coincides with the top of the embankment. 3. During drawdown, pore water pressure does not dissipate. 4. The unit weight of saturated soil (gsat) 2gw (gw unit weight of water). Figures 15.42 through 15.44 provide the drawdown stability charts developed by Morgenstern. 1.0 tan b 0.8 1 2 tan b f 20 f 20 0.6 f 30 L/H f 30 f 40 f 40 0.4 1 4 0.2 0.0 0 1 2 Fs 3 0 1 2 3 4 Fs 1.0 tan b 0.8 1 3 tan b f 20 f 20 0.6 f 30 L /H f 30 f 40 f 40 0.4 0.2 0.0 0 1 2 Fs 3 1 5 0 1 2 3 Fs Figure 15.42 Morgenstern’s drawdown stability chart for c/gH 0.0125 4 1.0 tan b 1 2 1 4 tan b 1.0 tan b 0.8 1 2 tan b 1 4 0.8 f 20 f 20 f 30 f 30 f 40 f 40 0.4 f 20 f 20 0.6 f 30 f 30 L/H L/H 0.6 f 40 f 40 0.4 0.2 0.2 0 0 1 2 Fs 3 4 0 1 2 3 4 0 Fs 0 1 1.0 tan b 1 3 tan b 1 5 2 Fs 1.0 L/H f 40 f 40 0.4 1 2 3 1 3 1 5 f 20 f 30 f 30 0.6 4 tan b f 20 f 30 f 30 0 0.8 f 20 L/H 0.6 4 Fs tan b 0.8 f 20 3 f 40 f 40 0.4 0.2 0.2 0 0 1 2 Fs 3 4 0 1 2 3 Fs 567 Figure 15.43 Morgenstern’s drawdown stability chart for c/gH 0.025 4 0 0 1 2 Fs 3 4 0 1 2 3 4 Fs Figure 15.44 Morgenstern’s drawdown stability chart for c/gH 0.05 568 Chapter 15: Slope Stability 15.16 Fluctuation of Factor of Safety of Slopes in Clay Embankment on Saturated Clay Figure 15.45a shows a clay embankment constructed on a saturated soft clay. Let P be a point on a potential failure surface APB that is an arc of a circle. Before construction of the embankment, the pore water pressure at P can be expressed as u hgw (15.63) Under ideal conditions, let us assume that the height of the ﬁll needed for the construction of the embankment is placed uniformly, as shown in Figure 15.45b. At time t t1 , the embankment height is equal to H, and it remains constant thereafter (that is, t t1). The average shear stress increase, t, on the potential failure surface caused by the construction of the embankment also is shown in Figure 15.45b. The value of t will increase linearly with time up to time t t1 and remain constant thereafter. The pore water pressure at point P (Figure 15.45a) will continue to increase as construction of the embankment progresses, as shown in Figure 15.45c. At time t t1 , u u1 hgw . This is because of the slow rate of drainage from the clay layer. However, after construction of the embankment is completed (that is, t t1), the pore water pressure gradually will decrease with time as the drainage (thus consolidation) progresses. At time t t2 , u hgw For simplicity, if we assume that the embankment construction is rapid and that practically no drainage occurs during the construction period, the average shear strength of the clay will remain constant from t 0 to t t1 , or tf cu (undrained shear strength). This is shown in Figure 15.45d. For time t t1 , as consolidation progresses, the magnitude of the shear strength, tf , will gradually increase. At time t t2—that is, after consolidation is completed—the average shear strength of the clay will be equal to tf c s tan f (drained shear strength) (Figure 15.45d). The factor of safety of the embankment along the potential surface of sliding can be given as Fs Average shear strength of clay, tf, along sliding surface 1Figure 14.45d2 Average shear stress, t, along sliding surface 1Figure 14.45b2 (15.64) The general nature of the variation of the factor of safety, Fs , with time is shown in Figure 15.45e. As we can see from this ﬁgure, the magnitude of Fs initially decreases with time. At the end of construction (time t t1), the value of the factor of safety is a minimum. Beyond this point, the value of Fs continues to increase with drainage up to time t t2 . Cuts in Saturated Clay Figure 15.46a on page 570 shows a cut slope in a saturated soft clay in which APB is a circular potential failure surface. During advancement of the cut, the average shear stress, t, on the potential failure surface passing through P will increase. The maximum value of the average shear stress, t, will be attained at the end of construction—that is, at time t t1 . This property is shown in Figure 15.46b on page 570. A Embankment B Groundwater table h P (a) Height of ﬁll Average shear stress, t, on a given surface through P Pore water pressure, u t1 u1 (b) Time t2 u excess pore water pressure due to the stress imposed hg t1 t2 Time Shear strength of soil, tf (c) t1 t2 Time Factor of safety, Fs (d) t1 t2 (e) Time Figure 15.45 Factor of safety variation with time for embankment on soft clay (Redrawn after Bishop and Bjerrum, 1960. With permission from ASCE.) 569 A Groundwater table B P Shear stress, t (a) Average shear stress, t, on a surface through P t1 (b) t2 Time Excess pore water pressure, u (positive) t1 t2 Shear strength of soil, tf (negative) Time (c) t1 t2 Time Factor of safety, Fs (d) t2 t1 (e) 570 Time Figure 15.46 Variation of factor of safety for cut slope in soft clay (Redrawn after Bishop and Bjerrum, 1960. With permission from ASCE.) Problems 571 Because of excavation of the soil, the effective overburden pressure at point P will decrease, which will induce a reduction in the pore water pressure. The variation of the net change of pore water pressure, u, is shown in Figure 15.46c. After excavation is complete (time t t1), the net negative excess pore water pressure will gradually dissipate. At time t t2 , the magnitude of u will be equal to 0. The variation of the average shear strength, tf , of the clay with time is shown in Figure 15.46d. Note that the shear strength of the soil after excavation gradually decreases. This decrease occurs because of dissipation of the negative excess pore water pressure. If the factor of safety of the cut slope, Fs , along the potential failure surface is deﬁned by Eq. (15.64), its variation will be as shown in Figure 15.46e. Note that the magnitude of Fs decreases with time, and its minimum value is obtained at time t t2 . Problems 15.1 Refer to Figure 15.47. Given: b 30, g 15.5 kN/m3, f 20, and c 15 kN/m2. Find the height, H, that will have a factor of safety, Fs , of 2 against sliding along the soil-rock interface. 15.2 For the slope shown in Figure 15.47, ﬁnd the height, H, for critical equilibrium. Given: b 22, g 100 lb/ft3, f 15, and c 200 lb/ft2. 15.3 Refer to Figure 15.48. If there were seepage through the soil and the groundwater table coincided with the ground surface, what would be the value of Fs ? Use H 8 m, rsat (saturated density of soil) 1900 kg/m3, b 20, c 18 kN/m2, and f 25. 15.4 For the inﬁnite slope shown in Figure 15.48, ﬁnd the factor of safety against sliding along the plane AB, given that H 25 ft, Gs 2.6, e 0.5, f 22, and c 600 lb/ft2. Note that there is seepage through the soil and that the groundwater table coincides with the ground surface. 15.5 For a ﬁnite slope such as that shown in Figure 15.10, assume that the slope failure would occur along a plane (Culmann’s assumption). Find the height of the slope for critical equilibrium. Given: f 25, c 400 lb/ft2, g 115 lb/ft3, and b 50. b b g f c H Rock Figure 15.47 572 Chapter 15: Slope Stability Groundwater table ion rect ge eepa of s Di 20 Gs e f c B H 20 A Figure 15.48 15.6 Refer to Figure 15.10. Using the soil parameters given in Problem 15.5, ﬁnd the height of the slope, H, that will have a factor of safety of 2 against sliding. Assume that the critical surface for sliding is a plane. 15.7 Refer to Figure 15.10. Given that f 15, c 200 lb/ft2, g 115 lb/ft3, b 50, and H 12 ft, determine the factor of safety with respect to sliding. Assume that the critical surface for sliding is a plane. 15.8 Determine the height of a ﬁnite slope (1 vertical to 2 horizontal) that should have a factor of safety of 2 against sliding. For the soil, the following values are given: c 18 kN/m2, f 20, and r 1700 kg/m3. Assume the critical surface for sliding to be a plane. 15.9 A cut slope is to be made in a saturated clay. Given: cu 30 kN/m2 (f 0 condition) and g 17 kN/m3. The slope makes an angle, b, of 60 with the horizontal. Determine the maximum depth up to which the cut could be made. Assume that the critical surface for sliding is circular. What is the nature of the critical circle (that is, toe, slope, or midpoint)? 15.10 For the cut slope described in Problem 15.9, if we need a factor of safety of 2.0 against sliding, how deep should the cut be made? 15.11 Using the graph given in Figure 15.13, determine the height of a slope (1 vertical to 1 horizontal) in saturated clay having an undrained shear strength of 25 kN/m2. The desired factor of safety against sliding is 2.5 Given: g 18 kN/m3 and D 1.20. 15.12 Refer to Problem 15.11. What should be the critical height of the slope? What is the nature of the critical circle? 15.13 A cut slope was excavated in a saturated clay. The slope angle, b, is equal to 40 with the horizontal. Slope failure occurred when the cut reached a depth of 8.5 m. Previous soil explorations showed that a rock layer was located at a depth of 12 m below the ground surface. Assuming an undrained condition and that g 18.5 kN/m3: a. Determine the undrained cohesion of the clay (Figure 15.13). b. What was the nature of the critical circle? c. With reference to the top of the slope, at what distance did the surface of sliding intersect the bottom of the excavation? Problems 573 15.14 A clay slope is built over a layer of rock. Determine the factor of safety if kh 0.4 for the slope with values • Height, H 16 m • Slope angle, b 30 • Saturated unit weight of soil, gsat 17 kN/m3 • Undrained shear strength, cu 50 kN/m2 15.15 For a slope in clay where H 50 ft, g 115 lb/ft3, b 60, and cu 1000 lb/ft2, determine the factor of safety for kh 0.3. 15.16 Refer to Figure 15.49. Use Figure 15.27 (f 0) to solve the following. a. If n 2, f 20, c 20 kN/m2, and g 16 kN/m3, ﬁnd the critical height of the slope. b. If n 1.5, f 25, c 750 lb/ft2, and g 110 lb/ft3, ﬁnd the critical height of the slope. 15.17 Refer to Figure 15.49. Using Figure 15.23, ﬁnd the factor of safety, Fs , with respect to sliding for a slope with the following. • Slope: 2H:1V • g 110 lb/ft3 • f 10 • H 50 ft • c 700 lb/ft2 15.18 Repeat Problem 15.17 with the following. • Slope: 1H:1V • g 115 lb/ft3 • f 20 • H 30 ft • c 400 lb/ft2 15.19 Repeat Problem 15.17 with the following. • Slope: 2.5H:1V • g 16.5 kN/m3 • f 12 • H 12 m • c 24 kN/m2 15.20 Referring to Figure 15.50 and using the ordinary method of slices, ﬁnd the factor of safety with respect to sliding for the following trial cases. a. b 45, f 20, c 400 lb/ft2, g 115 lb/ft3, H 40 ft, a 30, and u 70 b. b 45, f 15, c 18 kN/m2, g 17.1 kN/m3, H 5 m, a 30, and u 80 15.21 Determine the minimum factor of safety of a slope with the following parameters: H 25 ft, b 26.57, f 20, c 300 lb/ft2, g 120 lb/ft3, and ru 0.5. Use Bishop and Morgenstern’s method. 1 H n g f c Figure 15.49 574 Chapter 15: Slope Stability u B C n 1 H b g c f a A Figure 15.50 15.22 Determine the minimum factor of safety of a slope with the following parameters: H 6 m, b 18.43, f 20, c 6 kN/m2, g 20 kN/m3, and ru 0.5. Use Bishop and Morgenstern’s method. 15.23 Use Spencer’s chart to determine the value of Fs for the given slope: b 20, H 15 m, f 15, c 20 kN/m2, g 17.5 kN/m3, and ru 0.5. 15.24 For a slope, given: • Slope: 3H:1V • c 12 kN/m2 • H 12.63 m • g 19 kN/m3 • f 25 • ru 0.25 Using Spencer’s chart, determine the factor of safety, Fs . References BISHOP, A. W. (1955). “The Use of Slip Circle in the Stability Analysis of Earth Slopes,” Geotechnique, Vol. 5, No. 1, 7–17. BISHOP, A. W., and BJERRUM, L. (1960). “The Relevance of the Triaxial Test to the Solution of Stability Problems,” Proceedings, Research Conference on Shear Strength of Cohesive Soils, ASCE, 437–501. BISHOP, A. W., and MORGENSTERN, N. R. (1960). “Stability Coefﬁcients for Earth Slopes,” Geotechnique, Vol. 10, No. 4, 129–147. CRUDEN, D. M., and VARNES, D. J. (1996). “Landslide Types and Processes,” Special Report 247, Transportation Research Board, 36–75. CULMANN, C. (1875). Die Graphische Statik, Meyer and Zeller, Zurich. FELLENIUS, W. (1927). Erdstatische Berechnungen, rev. ed., W. Ernst u. Sons, Berlin. KOPPULA, S. D. (1984). “On Stability of Slopes on Clays with Linearly Increasing Strength,” Canadian Geotechnical Journal, Vol. 21, No. 3, 557–581. LADD, C. C. (1972). “Test Embankment on Sensitive Clay,” Proceedings, Conference on Performance of Earth and Earth-Supported Structures, ASCE, Vol. 1, Part 1, 101–128. MICHALOWSKI, R. L. (2002). “Stability Charts for Uniform Slopes,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 128, No. 4, 351–355. MORGENSTERN, N. R. (1963). “Stability Charts for Earth Slopes During Rapid Drawdown,” Geotechnique, Vol. 13, No. 2, 121–133. References 575 SINGH, A. (1970). “Shear Strength and Stability of Man-Made Slopes,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. SM6, 1879–1892. SPENCER, E. (1967). “A Method of Analysis of the Stability of Embankments Assuming Parallel Inter-Slice Forces,” Geotechnique, Vol. 17, No. 1, 11–26. TAYLOR, D. W. (1937). “Stability of Earth Slopes,” Journal of the Boston Society of Civil Engineers, Vol. 24, 197–246. TERZAGHI, K., and PECK, R. B. (1967). Soil Mechanics in Engineering Practice, 2nd ed., Wiley, New York. 16 Soil-Bearing Capacity for Shallow Foundations The lowest part of a structure generally is referred to as the foundation. Its function is to transfer the load of the structure to the soil on which it is resting. A properly designed foundation transfers the load throughout the soil without overstressing the soil. Overstressing the soil can result in either excessive settlement or shear failure of the soil, both of which cause damage to the structure. Thus, geotechnical and structural engineers who design foundations must evaluate the bearing capacity of soils. Depending on the structure and soil encountered, various types of foundations are used. Figure 16.1 shows the most common types of foundations. A spread footing is simply an enlargement of a load-bearing wall or column that makes it possible to spread the load of the structure over a larger area of the soil. In soil with low load-bearing capacity, the size of the spread footings required is impracticably large. In that case, it is more economical to construct the entire structure over a concrete pad. This is called a mat foundation. Pile and drilled shaft foundations are used for heavier structures when great depth is required for supporting the load. Piles are structural members made of timber, concrete, or steel that transmit the load of the superstructure to the lower layers of the soil. According to how they transmit their load into the subsoil, piles can be divided into two categories: friction piles and end-bearing piles. In the case of friction piles, the superstructure load is resisted by the shear stresses generated along the surface of the pile. In the end-bearing pile, the load carried by the pile is transmitted at its tip to a ﬁrm stratum. In the case of drilled shafts, a shaft is drilled into the subsoil and then is ﬁlled with concrete. A metal casing may be used while the shaft is being drilled. The casing may be left in place or may be withdrawn during the placing of concrete. Generally, the diameter of a drilled shaft is much larger than that of a pile. The distinction between piles and drilled shafts becomes hazy at an approximate diameter of 1 m (3 ft), and the deﬁnitions and nomenclature are inaccurate. Spread footings and mat foundations generally are referred to as shallow foundations, whereas pile and drilled-shaft foundations are classiﬁed as deep foundations. In a more general sense, shallow foundations are foundations that have a depth-of-embedmentto-width ratio of approximately less than four. When the depth-of-embedment-to-width ratio of a foundation is greater than four, it may be classiﬁed as a deep foundation. 576 16.1 Ultimate Soil-Bearing Capacity for Shallow Foundations (a) 577 (b) Pile (c) (d) Figure 16.1 Common types of foundations: (a) spread footing; (b) mat foundation; (c) pile foundation; (d) drilled shaft foundation In this chapter, we discuss the soil-bearing capacity for shallow foundations. As mentioned before, for a foundation to function properly, (1) the settlement of soil caused by the load must be within the tolerable limit, and (2) shear failure of the soil supporting the foundation must not occur. Compressibility of soil—consolidation and elasticity theory—was introduced in Chapter 10. This chapter introduces the load-carrying capacity of shallow foundations based on the criteria of shear failure in soil. 16.1 Ultimate Soil-Bearing Capacity for Shallow Foundations To understand the concept of the ultimate soil-bearing capacity and the mode of shear failure in soil, let us consider the case of a long rectangular footing of width B located at the surface of a dense sand layer (or stiff soil) shown in Figure 16.2a. When a uniformly distributed load of q per unit area is applied to the footing, it settles. If the uniformly distributed load (q) is increased, the settlement of the footing gradually increases. When the value of q qu is reached (Figure 16.2b), bearing capacity failure occurs; the footing undergoes a very large settlement without any further increase of q. The soil on one or both sides of 578 Chapter 16: Soil-Bearing Capacity for Shallow Foundations qu qu Load per unit area Settlement B q II I General shear failure Local shear failure (a) (b) Figure 16.2 Ultimate soil-bearing capacity for shallow foundation: (a) model footing; (b) load settlement relationship the foundation bulges, and the slip surface extends to the ground surface. The loadsettlement relationship is like Curve I shown in Figure 16.2b. In this case, qu is deﬁned as the ultimate bearing capacity of soil. The bearing capacity failure just described is called a general shear failure and can be explained with reference to Figure 16.3a. When the foundation settles under the application of a load, a triangular wedge-shaped zone of soil (marked I) is pushed down, Original surface of soil B qu III III I II II (a) Original surface of soil B qu I II II (b) Figure 16.3 Modes of bearing capacity failure in soil: (a) general shear failure of soil; (b) local shear failure of soil 16.2 Terzaghi’s Ultimate Bearing Capacity Equation 579 and, in turn, it presses the zones marked II and III sideways and then upward. At the ultimate pressure, qu , the soil passes into a state of plastic equilibrium and failure occurs by sliding. If the footing test is conducted instead in a loose-to-medium dense sand, the loadsettlement relationship is like Curve II in Figure 16.2b. Beyond a certain value of q quœ , the load-settlement relationship becomes a steep, inclined straight line. In this case, quœ is deﬁned as the ultimate bearing capacity of soil. This type of soil failure is referred to as local shear failure and is shown in Figure 16.3b. The triangular wedge-shaped zone (marked I) below the footing moves downward, but unlike general shear failure, the slip surfaces end somewhere inside the soil. Some signs of soil bulging are seen, however. 16.2 Terzaghi’s Ultimate Bearing Capacity Equation In 1921, Prandtl published the results of his study on the penetration of hard bodies (such as metal punches) into a softer material. Terzaghi (1943) extended the plastic failure theory of Prandtl to evaluate the bearing capacity of soils for shallow strip footings. For practical considerations, a long wall footing (length-to-width ratio more than about ﬁve) may be called a strip footing. According to Terzaghi, a foundation may be deﬁned as a shallow foundation if the depth Df is less than or equal to its width B (Figure 16.4). He also assumed that, for ultimate soil-bearing capacity calculations, the weight of soil above the base of the footing may be replaced by a uniform surcharge, q gDf . The failure mechanism assumed by Terzaghi for determining the ultimate soilbearing capacity (general shear failure) for a rough strip footing located at a depth Df measured from the ground surface is shown in Figure 16.5a. The soil wedge ABJ (Zone I) is an elastic zone. Both AJ and BJ make an angle f with the horizontal. Zones marked II (AJE and BJD) are the radial shear zones, and zones marked III are the Rankine passive zones. The rupture lines JD and JE are arcs of a logarithmic spiral, and DF and EG are straight lines. AE, BD, EG, and DF make angles of Df Unit weight of soil g q gDf Df B B Figure 16.4 Shallow strip footing 580 Chapter 16: Soil-Bearing Capacity for Shallow Foundations B qu f f G 45 f 45 f 2 2 III A C PP E I J B 45 C PP II Df q gDf f 45 f 2 2 III D II F Soil g c f (a) B 2b qu A f f W C c(AJ ) cb cos f f PP J B C c(BJ) cb cos f f PP (b) Figure 16.5 Terzaghi’s bearing capacity analysis 45f/2 degrees with the horizontal. The equation of the arcs of the logarithmic spirals JD and JE may be given as r ro eu tan f¿ If the load per unit area, qu , is applied to the footing and general shear failure occurs, the passive force Pp is acting on each of the faces of the soil wedge ABJ. This concept is easy to conceive of if we imagine that AJ and BJ are two walls that are pushing the soil wedges AJEG and BJDF, respectively, to cause passive failure. Pp should be inclined at an angle d (which is the angle of wall friction) to the perpendicular drawn to the wedge faces (that is, AJ and BJ). In this case, d should be equal to the angle of friction of soil, f. Because AJ and BJ are inclined at an angle f to the horizontal, the direction of Pp should be vertical. Now let us consider the free-body diagram of the wedge ABJ as shown in Figure 16.5b. Considering the unit length of the footing, we have, for equilibrium, 1qu 2 12b 2 11 2 W 2C sin f¿ 2Pp (16.1) where b B/2 W weight of soil wedge ABJ gb2 tan f¿ C cohesive force acting along each face , AJ and BJ, that is equal to the unit cohesion times the length of each face c¿b/1cos f¿ 2 Thus, 2bqu 2Pp 2bc¿ tan f¿ gb2 tan f¿ (16.2) 16.2 Terzaghi’s Ultimate Bearing Capacity Equation 581 or qu Pp b c¿ tan f¿ gb tan f¿ 2 (16.3) The passive pressure in Eq. (16.2) is the sum of the contribution of the weight of soil g, cohesion c, and surcharge q and can be expressed as Pp 12 g1b tan f¿ 2 2Kg c¿1b tan f¿ 2Kc q1b tan f¿ 2 Kq (16.4) where Kg , Kc , and Kq are earth-pressure coefﬁcients that are functions of the soil friction angle, f. Combining Eqs. (16.3) and (16.4), we obtain 1 qu c¿Nc qNq gBNg 2 where Nc tan f¿1Kc 1 2 (16.5) Nq Kq tan f¿ (16.6) Ng 12 tan f¿1Kg tan f¿ 1 2 (16.7) The terms Nc , Nq , and Ng are, respectively, the contributions of cohesion, surcharge, and unit weight of soil to the ultimate load-bearing capacity. It is extremely tedious to evaluate Kc , Kq , and Kg . For this reason, Terzaghi used an approximate method to determine the ultimate bearing capacity, qu . The principles of this approximation are the following. 1. If c 0 and surcharge (q) 0 (that is, Df 0), then qu qg 12 gBNg (16.8) 2. If g 0 (that is, weightless soil) and q 0, then qu qc c¿Nc (16.9) 3. If g 0 (weightless soil) and c 0, then qu qq qNq (16.10) By the method of superimposition, when the effects of the unit weight of soil, cohesion, and surcharge are considered, we have qu qc qq qg c¿Nc qNq 12 gBNg (16.11) Equation (16.11) is referred to as Terzaghi’s bearing capacity equation. The terms Nc , Nq , and Ng are called the bearing capacity factors. The values of these factors are given in Table 16.1. 582 Chapter 16: Soil-Bearing Capacity for Shallow Foundations Table 16.1 Terzaghi’s Bearing Capacity Factors—Nc , Nq and Ng—Eqs. (16.11), (16.12), and (16.13), respectively a Fⴕ (deg) Nc Nq NG 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 5.70 6.00 6.30 6.62 6.97 7.34 7.73 8.15 8.60 9.09 9.61 10.16 10.76 11.41 12.11 12.86 13.68 14.60 15.12 16.56 17.69 18.92 20.27 21.75 23.36 25.13 1.00 1.10 1.22 1.35 1.49 1.64 1.81 2.00 2.21 2.44 2.69 2.98 3.29 3.63 4.02 4.45 4.92 5.45 6.04 6.70 7.44 8.26 9.19 10.23 11.40 12.72 0.00 0.01 0.04 0.06 0.10 0.14 0.20 0.27 0.35 0.44 0.56 0.69 0.85 1.04 1.26 1.52 1.82 2.18 2.59 3.07 3.64 4.31 5.09 6.00 7.08 8.34 a Fⴕ (deg) Nc Nq NG 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 27.09 29.24 31.61 34.24 37.16 40.41 44.04 48.09 52.64 57.75 63.53 70.01 77.50 85.97 95.66 106.81 119.67 134.58 161.95 172.28 196.22 224.55 258.28 298.71 347.50 14.21 16.90 17.81 19.98 22.46 25.28 28.52 32.23 36.50 41.44 47.16 53.80 61.55 70.61 81.27 93.85 108.75 126.50 147.74 173.28 204.19 241.80 287.85 344.63 416.14 9.84 11.60 13.70 16.18 19.13 22.65 26.87 31.94 38.04 45.41 54.36 65.27 78.61 95.03 116.31 140.51 171.99 211.56 261.60 325.34 407.11 512.84 650.67 831.99 1072.80 a N values from Kumbhojkar (1993) For square and circular footings, Terzaghi suggested the following equations for ultimate soil-bearing capacity: The square footing is qu 1.3c¿Nc qNq 0.4gBNg (16.12) The circular footing is qu 1.3c¿Nc qNq 0.3gBNg where B diameter of the footing. (16.13) 16.2 Terzaghi’s Ultimate Bearing Capacity Equation 583 Krizek (1965) gave the following approximations for the values of Nc, Nq, and N with a maximum deviation of 15%. Nc 228 4.3f¿ Nq 40 f¿ 40 5f¿ 40 f¿ (16.14) (16.15) and Ng 6f¿ 40 f¿ (16.16) where soil friction angle, in degrees. Equations (16.14) through (16.16) are valid for 0° to 35°. Equation (16.11) was derived on the assumption that the bearing capacity failure of soil takes place by general shear failure. In the case of local shear failure, we may assume that c¿ 23 c¿ (16.17) tan f¿ 23 tan f¿ (16.18) and The ultimate bearing capacity of soil for a strip footing may be given by quœ c¿Ncœ qNqœ 12 gBNgœ (16.19) The modiﬁed bearing capacity factors Ncœ , Nqœ , and Ngœ are calculated by using the same general equation as that for Nc , Nq , and Ng , but by substituting f¿ tan1 ( 32 tan f) for f. The values of the bearing capacity factors for a local shear failure are given in Table 16.2. The ultimate soil-bearing capacity for square and circular footings for the local shear failure case now may be given as follows [similar to Eqs. (16.12) and (16.13)]: The square footing is quœ 1.3c¿Ncœ qNqœ 0.4gBNgœ (16.20) quœ 1.3c¿Ncœ qNqœ 0.3gBNgœ (16.21) The circular footing is For an undrained condition with f 0 and tf cu , the bearing capacity factors are Ng 0 and Nq 1. Also, Nc 5.7. In that case, Eqs. (16.11), (16.12), and (16.13) (which are the cases for general shear failure) take the forms 584 Chapter 16: Soil-Bearing Capacity for Shallow Foundations Table 16.2 Terzaghi’s Modiﬁed Bearing Capacity Factors—Nc , Nq , and Ng—Eqs. (16.19), (16.20), and (16.21), respectively Fⴕ (deg) N cœ N qœ N Gœ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 5.70 5.90 6.10 6.30 6.51 6.74 6.97 7.22 7.47 7.74 8.02 8.32 8.63 8.96 9.31 9.67 10.06 10.47 10.90 11.36 11.85 12.37 12.92 13.51 14.14 14.80 1.00 1.07 1.14 1.22 1.30 1.39 1.49 1.59 1.70 1.82 1.94 2.08 2.22 2.38 2.55 2.73 2.92 3.13 3.36 3.61 3.88 4.17 4.48 4.82 5.20 5.60 0.00 0.005 0.02 0.04 0.055 0.074 0.10 0.128 0.16 0.20 0.24 0.30 0.35 0.42 0.48 0.57 0.67 0.76 0.88 1.03 1.12 1.35 1.55 1.74 1.97 2.25 Fⴕ (deg) N cœ N qœ N Gœ 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 16.53 16.30 17.13 18.03 18.99 20.03 21.16 22.39 23.72 25.18 26.77 28.51 30.43 32.53 34.87 37.45 40.33 43.54 47.13 51.17 55.73 60.91 66.80 73.55 81.31 6.05 6.54 7.07 7.66 8.31 9.03 9.82 10.69 11.67 12.75 13.97 16.32 16.85 18.56 20.50 22.70 25.21 28.06 31.34 35.11 39.48 44.54 50.46 57.41 65.60 2.59 2.88 3.29 3.76 4.39 4.83 5.51 6.32 7.22 8.35 9.41 10.90 12.75 14.71 17.22 19.75 22.50 26.25 30.40 36.00 41.70 49.30 59.25 71.45 85.75 qu 5.7cu q 1strip footing2 (16.22) and qu 11.32 15.72cu q 7.41cu q 1square and circular footing2 (16.23) 16.3 Effect of Groundwater Table In developing the bearing capacity equations given in the preceding section we assumed that the groundwater table is located at a depth much greater than the width, B of the footing. However, if the groundwater table is close to the footing, some changes are required in the second and third terms of Eqs. (16.11) to (16.13), and Eqs. (16.19) to (16.21). Three different conditions can arise regarding the location of the groundwater table with respect to the bottom of the foundation. They are shown in Figure 16.6. Each of these conditions is brieﬂy described next. 16.3 Effect of Groundwater Table g Groundwater table Df g 585 Df D gsat Groundwater table gsat B B (a) (b) Df g D Groundwater table gsat B (c) Figure 16.6 Effect of the location of groundwater table on the bearing capacity of shallow foundations: (a) Case I; (b) Case II; (c) Case III Case I (Figure 16.6a) If the groundwater table is located at a distance D above the bottom of the foundation, the magnitude of q in the second term of the bearing capacity equation should be calculated as q g1Df D2 g¿D (16.24) where g gsat gw effective unit weight of soil. Also, the unit weight of soil, g, that appears in the third term of the bearing capacity equations should be replaced by g. Case II (Figure 16.6b) If the groundwater table coincides with the bottom of the foundation, the magnitude of q is equal to gDf . However, the unit weight, g, in the third term of the bearing capacity equations should be replaced by g. Case III (Figure 16.6c) When the groundwater table is at a depth D below the bottom of the foundation, q gDf . The magnitude of g in the third term of the bearing capacity equations should be replaced by gav. gav 1 3gD g¿1B D2 4 B gav g 1for D B2 1for D B2 (16.25a) (16.25b) 586 Chapter 16: Soil-Bearing Capacity for Shallow Foundations 16.4 Factor of Safety Generally, a factor of safety, Fs , of about 3 or more is applied to the ultimate soil-bearing capacity to arrive at the value of the allowable bearing capacity. An Fs of 3 or more is not considered too conservative. In nature, soils are neither homogeneous nor isotropic. Much uncertainty is involved in evaluating the basic shear strength parameters of soil. There are two basic deﬁnitions of the allowable bearing capacity of shallow foundations. They are gross allowable bearing capacity, and net allowable bearing capacity. The gross allowable bearing capacity can be calculated as qall qu Fs (16.26) As deﬁned by Eq. (16.26) qall is the allowable load per unit area to which the soil under the foundation should be subjected to avoid any chance of bearing capacity failure. It includes the contribution (Figure 16.7) of (a) the dead and live loads above the ground surface, W(DL ) ; (b) the self-weight of the foundation, WF ; and (c) the weight of the soil located immediately above foundation, WS . Thus, qall W1DL2 WF WS qu c d Fs A (16.27) where A area of the foundation. The net allowable bearing capacity is the allowable load per unit area of the foundation in excess of the existing vertical effective stress at the level of the foundation. The vertical effective stress at the foundation level is equal to q gDf . So, the net ultimate load is qu1net2 qu q (16.28) Hence, qall1net2 W(D L) WS 2 WF WS 2 Figure 16.7 Contributions to qall qu1net2 Fs qu q Fs (16.29) 16.4 Factor of Safety 587 If we assume that the weight of the soil and the weight of the concrete from which the foundation is made are approximately the same, then q gDf WS W F A Hence, qall1net2 W1DL2 A qu q Fs (16.30) Example 16.1 A continuous foundation is shown in Figure 16.8. Using Terzaghi’s bearing capacity factors, determine the gross allowable load per unit area (qall) that the foundation can carry. Given: • g 110 lb/ft3 • c 200 lb/ft2 • f 20 • Df 3 ft • B 4 ft • Factor of safety 3 Assume general shear failure. 3 ft g 110 lb/ft3 f 20 c 200 lb/ft2 4 ft Figure 16.8 Solution From Eq. (16.11), qu c¿Nc qNq 12 gBNg From Table 16.1, for f 20, Nc 17.69 Nq 7.44 Ng 3.64 Also, q gDf 11102 132 330 lb/ft2 588 Chapter 16: Soil-Bearing Capacity for Shallow Foundations So, qu 12002 117.692 13302 17.442 1 12 2 11102 142 13.64 2 6794 lb/ft2 From Eq. (16.26), qall qu 6794 2265 lb/ft2 Fs 3 ■ Example 16.2 A square foundation is shown in Figure 16.9. The footing will carry a gross mass of 30,000 kg. Using a factor of safety of 3, determine the size of the footing—that is, the size of B. Use Eq. (16.12). 30,000 kg r 1850 kg/m3 f 35 c 0 1m B Figure 16.9 Solution It is given that soil density 1850 kg/m3. So g 1850 9.81 18.15 kN/m3 1000 Total gross load to be supported by the footing is 130,0002 9.81 1000 294.3 kN Qall From Eq. (16.12) qu 1.3cNc + qNq + 0.4BN 16.5 General Bearing Capacity Equation 589 With a factor of safety of 3, qall qu 1 11.3c¿Nc qNq 0.4gBNg 2 3 3 (a) Also, qall Qall 2 B 294.3 B2 (b) From Eqs. (a) and (b), 1 294.3 11.3c¿Nc qNq 0.4gBNg 2 2 3 B (c) From Table 16.1, for 35°, Nc 57.75, Nq 41.44, and N 45.41. Substituting these values into Eq. (c) yields 1 294.3 3 11.3 2 102 157.752 118.15 1 2 141.442 0.4118.15 2 1B2 145.412 4 2 3 B or 294.3 250.7 109.9 B2 The preceding equation may now be solved by trial and error, and from that we get B 0.95 m 16.5 ■ General Bearing Capacity Equation After the development of Terzaghi’s bearing capacity equation, several investigators worked in this area and reﬁned the solution (that is, Meyerhof, 1951 and 1963; Lundgren and Mortensen, 1953; Balla, 1962; Vesic, 1973; and Hansen, 1970). Different solutions show that the bearing capacity factors Nc and Nq do not change much. However, for a given value of f, the values of Ng obtained by different investigators vary widely. This difference is because of the variation of the assumption of the wedge shape of soil located directly below the footing, as explained in the following paragraph. While deriving the bearing capacity equation for a strip footing, Terzaghi used the case of a rough footing and assumed that the sides AJ and BJ of the soil wedge ABJ (see Figure 16.5a) make an angle f with the horizontal. Later model tests (for example, DeBeer and Vesic, 1958) showed that Terzaghi’s assumption of the general nature of the rupture surface in soil for bearing capacity failure is correct. However, tests have shown that the sides AJ and BJ of the soil wedge ABJ make angles of about 45 f/2 degrees (instead of f) with the horizontal. This type of failure mechanism is shown in Figure 16.10. It consists of a Rankine active zone ABJ (Zone I), two radial shear zones (Zones II), and two Rankine passive zones (Zones III). The curves JD and JE are arcs of a logarithmic spiral. 590 Chapter 16: Soil-Bearing Capacity for Shallow Foundations B qu q gDf Df 45 f/2 G 45 f/2 45 f/2 45 f/2 45 f/2 45 f/2 B A J E Zone I Zone II F D Zone III Figure 16.10 Soil-bearing capacity calculation—general shear failure On the basis of this type of failure mechanism, the ultimate bearing capacity of a strip footing may be evaluated by the approximate method of superimposition described in Section 16.2 as qu qc qq qg (16.31) where qc , qq , and qg are the contributions of cohesion, surcharge, and unit weight of soil, respectively. Reissner (1924) expressed qq as qq qNq (16.32) where Nq ep tan f¿ tan2 a 45 f¿ b 2 (16.33) Prandtl (1921) showed that qc c¿Nc (16.34) where Nc 1Nq 1 2cot f¿ c Eq. 116.332 (16.35) qg 12 BgNg (16.36) Meyerhof (1963) expressed qg as where Ng 1Nq 1 2tan11.4f¿ 2 c Eq. 116.332 (16.37) 16.5 General Bearing Capacity Equation 591 Combining Eqs. (16.31), (16.32), (16.34), and (16.36), we obtain qu c¿Nc qNq 12 gBNg (16.38) This equation is in the same general form as that given by Terzaghi [Eq. (16.11)]; however, the values of the bearing capacity factors are not the same. The values of Nq , Nc , and Ng , deﬁned by Eqs. (16.33), (16.35), and (16.37), are given in Table 16.3. But for all practical purposes, Terzaghi’s bearing capacity factors will yield good results. Differences in bearing capacity factors are usually minor compared with the unknown soil parameters. The soil-bearing capacity equation for a strip footing given by Eq. (16.38) can be modiﬁed for general use by incorporating the following factors: Depth factor: To account for the shearing resistance developed along the failure surface in soil above the base of the footing. To determine the bearing capacity of rectangular and circular footings. Shape factor: Table 16.3 Bearing Capacity Factors Nc, Nq, and N [Eqs. (16.33), (16.35) and (16.37)] ⴕ (deg) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Nc Nq N␥ (deg) 5.14 5.38 5.63 5.90 6.19 6.49 6.81 7.16 7.53 7.92 8.35 8.80 9.28 9.81 10.37 10.98 11.63 12.34 13.10 13.93 14.83 15.82 16.88 18.05 19.32 20.72 1.00 1.09 1.20 1.31 1.43 1.57 1.72 1.88 2.06 2.25 2.47 2.71 2.97 3.26 3.59 3.94 4.34 4.77 5.26 5.80 6.40 7.07 7.82 8.66 9.60 10.66 0 0.002 0.010 0.023 0.042 0.070 0.106 0.152 0.209 0.280 0.367 0.471 0.596 0.744 0.921 1.129 1.375 1.664 2.009 2.403 2.871 3.421 4.066 4.824 5.716 6.765 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Nc 22.25 23.94 25.80 27.86 30.14 32.67 35.49 38.64 42.16 46.12 50.59 55.63 61.35 67.87 75.31 83.86 93.71 105.11 118.37 133.88 152.10 173.64 199.26 229.93 266.89 Nq 11.85 13.20 14.72 16.44 18.40 20.63 23.18 26.09 29.44 33.30 37.75 42.92 48.93 55.96 64.20 73.90 85.38 99.02 115.31 134.88 158.51 187.21 222.31 265.51 319.07 N␥ 8.002 9.463 11.190 13.236 15.668 18.564 22.022 26.166 31.145 37.152 44.426 53.270 64.073 77.332 93.690 113.985 139.316 171.141 211.406 262.739 328.728 414.322 526.444 674.908 873.843 592 Chapter 16: Soil-Bearing Capacity for Shallow Foundations Inclination factor: To determine the bearing capacity of a footing on which the direction of load application is inclined at a certain angle to the vertical. Thus, the modiﬁed general ultimate bearing capacity equation can be written as qu c¿lcslcdlciNc qlqslqdlqiNq 12 lgslgdlgigBNg (16.39) where lcs , lqs , and lgs shape factors lcd , lqd , and lgd depth factors lci , lqi , and lgi inclination factors The approximate values of these shape, depth, and inclination factors recommended by Meyerhof are given in Table 16.4. Table 16.4 Meyerhof’s Shape, Depth, and Inclination Factors for a Rectangular Footing a Shape factors For f 0: lcs 1 0.2 a B b L lqs 1 lgs 1 For f 10: f¿ B b tan2 a 45 b L 2 f¿ B b lqs lgs 1 0.1 a b tan2 a 45 L 2 lcs 1 0.2 a Depth factors For f 0: lcd 1 0.2 a Df B b lqd lgd 1 For f 10: lcd 1 0.2 a Df B b tan a 45 lqd lgd 1 0.1 a Df B f¿ b 2 b tan a 45 Inclination factors lci a 1 a° 2 b 90° a° 2 lqi a 1 b 90° a° 2 b lgi a 1 f¿° a B width of footing; L length of footing a f¿ b 2 16.6 A Case History for Evaluation of the Ultimate Bearing Capacity 593 For undrained condition, if the footing is subjected to vertical loading (that is, a 0°), then f0 c cu Ng 0 Nq 1 Nc 1 lci lqi lgi 1 So Eq. (16.39) transforms to qu 5.14cu c 1 0.2 a 16.6 Df B b d c 1 0.2 a bd q L B (16.40) A Case History for Evaluation of the Ultimate Bearing Capacity Several documented cases of large-scale ﬁeld-load tests used to determine the ultimate bearing capacity of shallow foundations are presently available. One of these ﬁeld-load tests is discussed in this section. The results of this test are compared with the theories presented in this chapter. Skempton (1942) reported a ﬁeld-load test in clay for a large foundation with B 2.44 m (8 ft) and L 2.74 m (9 ft). This test also was reported by Bishop and Bjerrum (1960). Figure 16.11 shows a diagram of the foundation and the soil proﬁle. The variation of the undrained cohesion (cu) of the soil proﬁle also is shown in Figure 16.11. The average moisture content, liquid limit, and plastic limit of the clay underlying the foundation were 50, 70, and 28%, respectively. The foundation was loaded to failure immediately after contruction. The net ultimate bearing capacity was determined to be 119.79 kN/m2 (2500 lb/ft2). The net ultimate bearing capacity was deﬁned in Eq. (16.28) as qu1net2 qu q From Eq. (16.39), for the vertical loading condition and f 0 (note: Nq 1, Ng 0, lqs 1, and lqd 1), qu culcslcdNc q So, qu1net2 1culcslcdNc q2 q culcslcdNc 594 Chapter 16: Soil-Bearing Capacity for Shallow Foundations Load Undrained shear strength (lb/ft2) 0 400 800 1200 0.00 2 0.61 4 1.22 6 1.83 8 2.44 10 3.05 (ft) 1.68 m (5 ft 6 in.) 0 (m) Stanchion 3.66 0 19 38 ( kN/m2) 57 Net foundation pressure at failure 119.79 kN/m2 (2500 lb/ft2) Hard core Peaty clay Top soil Firm brown clay Firm sandy clay Mottled clay Soft blue clay Figure 16.11 Skempton’s ﬁeld-load test on a foundation supported by a saturated clay (After Bishop and Bjerrum, 1960. With permission from ASCE.) For the case under consideration, cu 16.8 kN/m2 (350 lb/ft2) (see Figure 16.11) and Nc 5.14 (see Table 16.3). From Table 16.4, lcs 1 0.2 a 2.44 B b 1 0.2 a b 1.2 L 2.74 lcd 1 0.2 a Df B b 1 0.2 a 1.68 b 1.138 2.44 Hence, qu1net2 116.82 11.22 11.138 2 15.142 117.9 kN/m2 So, for this ﬁeld-load test, qu1net-theory2 qu1net-actual2 117.9 0.98 119.79 16.6 A Case History for Evaluation of the Ultimate Bearing Capacity 595 Thus, the agreement between the theoretical estimate and the ﬁeld-load test result is fairly good. The slight variation between them may be because of the estimation of the average value of cu . Bishop and Bjerrum (1960) cited several end-of-construction failures of footings on saturated clay. The data for these failures are given in Table 16.5. We can see from this table that in all cases qu(net-theory)/qu(net-actual) is approximately 1. This finding confirms that the design of shallow foundations based on the net ultimate bearing capacity is a reliable technique. Table 16.5 End-of-Construction Failures of Footings—Saturated Clay Foundation: 0 Condition* Data of clay Locality Loading test, Marmorerá Kensal Green Silo, Transcona Kippen Screw pile, Lock Ryan Screw pile, Newport Oil tank, Fredrikstad Oil tank A, Shellhaven Oil tank B, Shellhaven Silo, U.S.A. Loading test, Moss Loading test, Hagalund Loading test, Torp Loading test, Rygge w (%) LL PL PI w PL PI 10 — 50 50 — — 45 70 — 40 9 68 27 45 35 — 110 70 — — 55 87 — — — 55 24 37 25 — 30 28 — — 25 25 — — — 20 16 19 20 — 80 42 — — 30 62 — — — 35 8 18 0.25 — 0.25 0.52 — — 0.67 0.73 — — — 1.37 1.39 1.44 qu1nettheory2 0.92 1.02 1.09 0.95 1.05 1.07 1.08 1.03 1.05 0.98 1.10 0.93 0.96 0.95 qu1netactual2 * After Bishop and Bjerrum (1960). With permission from ASCE. Note: w moisture content, LL liquid limit; PL plastic limit; PI plasticity index Example 16.3 A square footing is shown in Figure 16.12. Determine the safe gross load (factor of safety of 3) that the footing can carry. Use Eq. (16.39). Solution From Eq. (16.39), qu c¿lcslcdNc qlqslqdNq 12 g¿lgslgdBNg 596 Chapter 16: Soil-Bearing Capacity for Shallow Foundations g 16 kN/m3 c 0 f 32 0.5 m Groundwater table 0.5 m gsat 19.5 kN/m3 1.2 m Figure 16.12 (Note: lci , lqi , and lgi are all equal to 1 because the load is vertical.) Because c 0, qu qlqslqdNq 12 g¿lgslgdBNg From Table 16.3 for f 32, Nq 23.18 and Ng 22.02. From Table 16.4, lqs lgs 1 0.1 a 1 0.1 a 32 1.2 b tan2 a 45 b 1.325 1.2 2 lqd lgd 1 0.1 a 1 0.1 a f¿ B b tan2 a 45 b L 2 Df B b tan a 45 f¿ b 2 32 1 b tan a 45 b 1.15 1.2 2 The groundwater table is located above the bottom of the foundation, so, from Eq. (16.23), q (0.5)(16) (0.5)(19.5 9.81) 12.845 kN/m2 Thus, qu 112.8452 11.3252 11.152 123.182 1 12 2 119.5 9.812 11.3252 11.152 11.2 2 122.022 453.7 195.1 648.8 kN/m2 qu 648.8 216.3 kN/m2 qall 3 3 Hence, the gross load is as follows: Q qall(B2) 216.3(1.2)2 311.5 kN ■ 16.7 Ultimate Load for Shallow Foundations Under Eccentric Load Ultimate Load for Shallow Foundations Under Eccentric Load One-Way Eccentricity To calculate the bearing capacity of shallow foundations with eccentric loading, Meyerhof (1953) introduced the concept of effective area. This concept can be explained with reference to Figure 16.13, in which a footing of length L and width B is subjected to an eccentric load, Qu . If Qu is the ultimate load on the footing, it may be approximated as follows: 1. Referring to Figures 16.13b and 16.13c, calculate the effective dimensions of the foundation. If the eccentricity (e) is in the x direction (Figure 16.13b), the effective dimensions are X B 2e and YL However, if the eccentricity is in the y direction (Figure 16.13c), the effective dimensions are Y L 2e Qu e Df BL (a) Section y y A A e x X B 2e B B (b) Plan L Y L L 2e L Y L 16.7 597 e x X B B (c) Plan Figure 16.13 Ultimate load for shallow foundation under eccentric load 598 Chapter 16: Soil-Bearing Capacity for Shallow Foundations and XB 2. The lower of the two effective dimensions calculated in Step 1 is the effective width (B) and the other is the effective length (L). Thus, B X or Y, whichever is smaller L X or Y, whichever is larger 3. So the effective area is equal to B times L. Now, using the effective width, we can rewrite Eq. (16.39) as qu c¿lcslcdNc qlqslqdNq 12 lgslgdgB¿Ng (16.41) Note that the preceding equation is obtained by substituting B for B in Eq. (16.39). While computing the shape and depth factors, one should use B for B and L for L. 4. Once the value of qu is calculated from Eq. (16.41), we can obtain the total gross ultimate load as follows: Qu qu 1B¿L¿ 2 quA¿ (16.42) where A effective area. Purkayastha and Char (1977) carried out stability analysis of eccentrically loaded continuous foundations on granular soil (i.e., c 0) using the method of slices. Based on that analysis, they proposed that Rk 1 qu1eccentric2 qu1centric2 (16.43) where Rk reduction factor qu(eccentric) ultimate bearing capacity of eccentrically loaded continuous foundations Qu1eccentric2 B qu(centric) ultimate bearing capacity of centrally loaded continuous foundations Qu1centric2 B 599 16.7 Ultimate Load for Shallow Foundations Under Eccentric Load The magnitude of Rk can be expressed as Rk a a e k b B (16.44) where a and k are functions of the embedment ratio Df/B (Table 16.6). Hence, combining Eqs. (16.43) and (16.44) gives Qu1eccentric2 Qu1centric2 c 1 a a e k b d B (16.45) where Qu(eccentric) and Qu(centric) ultimate load per unit length, respectively, for eccentrically and centrically loaded foundations. Two-Way Eccentricity When foundations are subjected to loads with two-way eccentricity, as shown in Figure 16.14, the effective area is determined such that the centroid coincides with the load. The procedure y y B B A A ex ex ey L L ey L x x L B B (a) (b) y B A Table 16.6 Variations of a and k [Eq. (16.44)] Df /B a k 0 0.25 0.5 1.0 1.862 1.811 1.754 1.820 0.73 0.785 0.80 0.888 ex x L (c) Figure 16.14 Foundation subjected to two-way eccentricity 600 Chapter 16: Soil-Bearing Capacity for Shallow Foundations for ﬁnding the effective dimensions, B and L, are beyond the scope of this text and readers may refer to Das (2007). Once B and L are determined, Eqs. (16.41) and (16.42) may be used to determine the ultimate load. Example 16.4 A rectangular footing 1.5 m 1 m is shown in Figure 16.15. Determine the magnitude of the gross ultimate load applied eccentrically for bearing capacity failure in soil. Qu e 0.1 m 1m g 18 kN/m3 c 0 f 30 1m y x 1.5 m 0.1 m 1m Figure 16.15 Solution From Figures 16.13b and 16.15, X B 2e 1 2e 1 122 10.1 2 0.8 m Y L 1.5 m 16.7 Ultimate Load for Shallow Foundations Under Eccentric Load 601 So, effective width (B) 0.8 m and effective length (L) 1.5 m. From Eq. (16.39), qu qlqslqdNq 12 lgslgdgB¿Ng From Table 16.3 for f 30, Nq 18.4 and Ng 15.668. From Table 16.4, lqs lgs 1 0.1 a 1 0.1 a 30 0.8 b tan2 a 45 b 1.16 1.5 2 lqd lgd 1 0.1 a 1 0.1 a f¿ B¿ b tan2 a 45 b L¿ 2 Df B¿ b tan a 45 f¿ b 2 30 1 b tan a 45 b 1.217 0.8 2 So qu 11 182 11.16 2 11.2172 118.4 2 1 12 2 11.162 11.2172 1182 10.8 2 115.6682 627 kN/m 2 Hence, from Eq. (16.42), Qu qu(BL) (627) (0.8) (1.5) 752 kN ■ Example 16.5 Consider an eccentrically loaded continuous foundation supported by a granular soil. Given: B 1.5 m, Df 0.75 m, load eccentricity e/B 0.1, 17.5 kN/m3, 30°, and c 0. Use the reduction factor method [Eq. (16.45)] and determine the gross ultimate load per unit length that the foundation can carry. Solution From Eq. (16.45), Qu1eccentric2 Qu1centric2 c 1 a a e k b d B 602 Chapter 16: Soil-Bearing Capacity for Shallow Foundations Also, Df /B 0.75/1.5 0.5. From Table 16.6, a 1.754 and k 0.8. Thus, Qu1centric2 qNq lqd 1 lgdBNg 2 (Note: The shape factors are all equal to one, since it is a continuous foundation.) q Df (17.5)(0.75) 13.125 kN/m2 From Table 16.3 for ' 30°, Nq 18.4 and N 15.668. Therefore, lqd lgd 1 0.1 a 1 0.1 a Df B b tan a45 f¿ b 2 30 0.75 b tan a45 b 1.087 1.5 2 Qu(centric) (13.125)(18.4)(1.087) + (0.5)(1.087)(1.5)(15.668) 275.28 kN/m2 Hence, Qu(eccentric) (275.28)[1 (1.754)(0.1)0.8] 198.76 kN 199 kN 16.8 ■ Bearing Capacity of Sand Based on Settlement Obtaining undisturbed specimens of cohesionless sand during a soil exploration program is usually difﬁcult. For this reason, the results of standard penetration tests (SPTs) performed during subsurface exploration are commonly used to predict the allowable soil-bearing capacity of foundations on sand. (The procedure for conducting SPTs is discussed in detail in Chapter 18.) Meyerhof (1956) proposed correlations for the net allowable bearing capacity (qnet) based on settlement (elastic). It was further revised by Meyerhof (1965) based on the field performance of foundations. The correlations can be expressed as follows. SI Units qnet 1kN/m2 2 Se 1mm2 N60 Fd c d 0.05 25 1forB 1.22 m2 (16.46) 16.8 Bearing Capacity of Sand Based on Settlement qnet 1kN/m2 2 Se 1mm2 N60 B 0.3 2 a b Fd c d 0.08 B 25 1forB 1.22 m2 603 (16.47) where B foundation width (m) Se settlement English Units qnet 1kip/ft2 2 qnet 1kip/ft2 2 N60 Fd 3Se 1in.2 4 2.5 1forB 4 ft2 N60 B 1 2 a b Fd 3Se 1in.2 4 4 B 1forB 4 ft2 (16.48) (16.49) where B foundation width (ft) In Eqs. (16.46) through (16.49), N60 field standard penetration number based on 60% average energy ratio Se allowable settlement (elastic) Fd depth factor 1 0.33 a Df B b 1.33 (16.50) The N60 values referred to in Eqs. (16.46) through (16.49) are the average values between the bottom of the foundation and 2B below the bottom. Comparison with Field Settlement Observation Meyerhof (1965) compiled the observed maximum settlement (Se) for several mat foundations constructed on sand and gravel. These are shown in Table 16.7 (Column 5) along with the values of B, qnet, and N60. From Eq. (16.49), we can write Se 1in.2 qnet N60 B1 2 a ba b Fd 4 B (16.51) As can be seen from Table 16.7, the widths B for the mats are large. Hence B1 2 b 1 a B Fd 1 0.33 Df B 1 So Se 1in.2 qnet 0.25N60 (16.52) 604 Chapter 16: Soil-Bearing Capacity for Shallow Foundations Table 16.7 Observed and Calculated Maximum Settlement of Mat Foundations on Sand and Gravel Structure (1) T. Edison Sao Paulo, Brazil Banco de Brasil Sao Paulo, Brazil Iparanga Sao Paulo, Brazil C.B.I., Esplanada Sao Paulo, Brazil Riscala Sao Paulo, Brazil Thyssen Dusseldorf, Germany Ministry Dusseldorf, Germany Chimney Cologne, Germany Se(observed) (in.) (5)a Se(predicted) (in.) (6)b Se1predicted2 B (ft) (2)a N60 (3)a qnet (kip/ft2) (4)a 60 15 4.8 0.6 1.28 2.1 75 18 5.0 1.1 1.11 1.0 30 9 6.4 1.4 2.84 2.03 48 22 8.0 1.1 1.45 1.32 13 20 4.8 0.5 0.96 1.92 74 25 5.0 0.95 0.8 0.84 52 20 4.6 0.85 0.92 1.08 67 10 3.6 0.4 1.44 3.6 Se1observed2 (7) a Compiled from Meyerhof (1965) From Eq. (16.52) b Using the actual values of qnet and N60 given in Table 16.7, the magnitudes of Se have been calculated via Eq. (16.52). These are shown in Column 6 of Table 16.7 as Se(predicted). The ratio of Se(predicted)/Se(observed) is shown in Column 7. This ratio varies from 0.84 to 3.6. Hence, it can be concluded that the allowable net bearing capacity for a given allowable settlement calculated using the empirical relation is safe and conservative. 16.9 Plate-Load Test In some cases, conducting ﬁeld-load tests to determine the soil-bearing capacity of foundations is desirable. The standard method for a ﬁeld-load test is given by the American Society for Testing and Materials (ASTM) under Designation D-1194 (ASTM, 1997). Circular steel bearing plates 162 to 760 mm (6 to 30 in.) in diameter and 305 mm 305 mm (1 ft 1 ft) square plates are used for this type of test. A diagram of the load test is shown in Figure 16.16. To conduct the test, one must have a pit of depth Df excavated. The width of the test pit should be at least four times the width of the bearing plate to be used for the test. The bearing plate is placed on the soil at the bottom of the pit, and an incremental load on the bearing plate is applied. After the application of an incremental load, enough time is allowed for settlement to occur. When the settlement of the bearing plate becomes negligible, another incremental load is applied. In this manner, a load-settlement plot can be obtained, as shown in Figure 16.17. 16.9 Plate-Load Test 605 Df Jack W 4B B W Reaction beam Test plate Figure 16.16 Diagram of plate load test Anchor pile From the results of ﬁeld load tests, the ultimate soil-bearing capacity of actual footings can be approximated as follows: For clays, (16.53) qu 1footing2 qu 1plate2 For sandy soils, qu 1footing2 qu 1plate2 B1footing2 B1plate2 (16.54) For a given intensity of load q, the settlement of the actual footing also can be approximated from the following equations: In clay, Se 1footing2 Se 1plate2 B1footing2 B1plate2 (16.55) Settlement, Se Load per unit area, q Figure 16.17 Typical load-settlement curve obtained from plate load test 606 Chapter 16: Soil-Bearing Capacity for Shallow Foundations In sandy soil, Se 1footing2 Se 1plate2 c 2B1footing2 B1footing2 B1plate2 d 2 (16.56) Example 16.6 The ultimate bearing capacity of a 700-mm diameter plate as determined from ﬁeld load tests is 280 kN/m2. Estimate the ultimate bearing capacity of a circular footing with a diameter of 1.5 m. The soil is sandy. Solution From Eq. (16.54), qu 1footing2 qu 1plate2 B1footing2 B1plate2 280 a 1.5 b 0.7 680 kN/m2 ■ Example 16.7 The results of a plate-load test in a sandy soil are shown in Figure 16.18. The size of the plate is 0.305 m 0.305 m. Determine the size of a square column foundation that should carry a load of 2500 kN with a maximum settlement of 25 mm. 0 Load/unit area (kN/m2) 200 400 600 800 10 Settlement (mm) 20 30 40 50 60 70 Figure 16.18 16.10 Summary and General Comments 607 Solution The problem has to be solved by trial and error using the following table and Eq. (16.56). Q (kN) (1) Assumed width, BF (m) (2) 2500 2500 2500 4.0 3.0 3.2 qⴝ 1kN/m 2 B2F (3) Q 2 156.25 277.80 244.10 Se(plate) corresponding to q in Column 3 from Fig. 16.18 (mm) (4) Se(footing) using Eq. (16.56) (mm) (5) 4.0 8.0 6.8 13.81 26.37 22.67 So a column footing with dimensions of 3.2 m 3.2 m will be appropriate. 16.10 ■ Summary and General Comments In this chapter, theories for estimating the ultimate and allowable bearing capacities of shallow foundations were presented. Procedures for ﬁeld-load tests and estimation of the allowable bearing capacity of granular soil based on limited settlement criteria were discussed brieﬂy. Several building codes now used in the United States and elsewhere provide presumptive bearing capacities for various types of soil. It is extremely important to realize that they are approximate values only. The bearing capacity of foundations depends on several factors: 1. 2. 3. 4. 5. 6. 7. Subsoil stratiﬁcation Shear strength parameters of the subsoil Location of the groundwater table Environmental factors Building size and weight Depth of excavation Type of structure Hence, it is important that the allowable bearing capacity at a given site be determined based on the ﬁndings of soil exploration at that site, past experience of foundation construction, and fundamentals of the geotechnical engineering theories for bearing capacity. The allowable bearing capacity relationships based on settlement considerations such as those given in Section 16.9 do not take into account the settlement caused by consolidation of the clay layers. Excessive settlement usually causes the building to crack, which ultimately may lead to structural failure. Uniform settlement of a structure does not produce cracking; on the other hand, differential settlement may produce cracks and damage to a building. 608 Chapter 16: Soil-Bearing Capacity for Shallow Foundations Problems 16.1 A continuous footing is shown in Figure 16.19. Using Terzaghi’s bearing capacity factors, determine the gross allowable load per unit area (qall ) that the footing can carry. Assume general shear failure. Given: g 115 lb/ft3, c 600 lb/ft2, f 25, Df 3.5 ft, B 4 ft, and factor of safety 3. 16.2 Repeat Problem 16.1 with the following: g 17.5 kN/m3, c 14 kN/m2, f 20, Df 1.0 m, B 1.2 m, and factor of safety 3. 16.3 Repeat Problem 16.1 with the following: g 17.7 kN/m3, cu 48 kN/m2, f 0, Df 0.6 m, B 0.8 m, and factor of safety 4. 16.4 Repeat Problem 16.1 using Eq. (16.39). 16.5 Repeat Problem 16.2 using Eq. (16.39). 16.6 Repeat Problem 16.3 using Eq. (16.39). 16.7 A square footing is shown in Figure 16.20. Determine the gross allowable load, Qall , that the footing can carry. Use Terzaghi’s equation for general shear failure (Fs 3). Given: g 105 lb/ft3, gsat 118 lb/ft3, c 0, f 35, B 5 ft, Df 4 ft, and h 2 ft. 16.8 Repeat Problem 16.7 with the following: density of soil above the groundwater table, r 1800 kg/m3; saturated soil density below the groundwater table, r sat 1980 kg/m3; c 23.94 kN/m2; f 25; B 1.8 m; Df 1.2 m; and h 2 m. 16.9 A square footing (B B) must carry a gross allowable load of 42,260 lb. The base of the footing is to be located at a depth of 3 ft below the ground surface. For the soil, we are given that g 110 lb/ft3, c 200 lb/ft2, and f 20. If the required factor of safety is 3, determine the size of the footing. Use Terzaghi’s bearing capacity factors and general shear failure of soil. 16.10 Repeat Problem 16.9 with the following: gross allowable load 92.5 kip; Df 2 ft; g 115 lb/ft3; c 0; f 35; and required factor of safety 3. 16.11 Repeat Problem 16.7 using Eq. (16.39). 16.12 A square footing is shown in Figure 16.21. The footing is subjected to an eccentric load. For the following cases, determine the gross allowable load that the footing could carry. Use Meyerhof’s procedure and Fs 4. qall Df Unit weight of soil g c f B Figure 16.19 Problems h 609 Unit weight of soil g c f Qall Df Groundwater table gsat B Figure 16.20 Qult x Df Unit weight of soil g (or density r) c f B x y B B Figure 16.21 a. g 110 lb/ft3, c 0, f 35, B 5 ft, Df 3.5 ft, x 0.6 ft, y 0 b. g 120 lb/ft3, c 400 lb/ft2, f 25, B 6 ft, Df 4.5 ft, x 0, y 0.5 ft c. r 1950 kg/m3, c 0, f 40, B 3 m, Df 1.4 m, x 0.3 m, y 0 16.13 A plate-load test was conducted in a sandy soil in which the size of the bearing plate was 1 ft 1 ft. The ultimate load per unit area (qu ) for the test was found to be 3850 lb/ft2. Estimate the total allowable load (Qall ) for a footing of size 6 ft 6 ft. Use a factor of safety of 4. 16.14 A plate load test (bearing plate of 762 mm diameter) was conducted in clay. The ultimate load per unit area (qu ) for the test was found to be 248.9 kN/m2. What should be the total allowable load (Qall ) for a column footing with a diameter of 2 m? Use a factor of safety of 3. 610 Chapter 16: Soil-Bearing Capacity for Shallow Foundations References AMERICAN SOCIETY FOR TESTING AND MATERIALS (1997). Annual Book of Standards, ASTM, Vol. 04.08, West Conshohocken, Pa. BALLA, A. (1962). “Bearing Capacity of Foundations,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 89, No. SM5, 12–34. BISHOP, A. W., and BJERRUM, L. (1960). “The Relevance of the Triaxial Test to the Solution of Stability Problems,” Proceedings, Research Conference on Shear Strength of Cohesive Soils, ASCE, 437–501. DAS, B. M. (2007), Principles of Foundation Engineering, 6th ed., Thomson, Toronto, Canada. HANSEN, J. B. (1970). “A Revised and Extended Formula for Bearing Capacity,” Danish Geotechnical Institute, Bulletin No. 28, Copenhagen. KUMBHOJKAR, A. S. (1993). “Numerical Evaluation of Terzaghi’s Ng ,” Journal of Geotechnical Engineering, ASCE, Vol. 119, No. GT3, 598–607. LUNDGREN, H., and MORTENSEN, K. (1953). “Determination by the Theory of Elasticity of the Bearing Capacity of Continuous Footing on Sand,” Proceedings, 3rd International Conference on Soil Mechanics and Foundation Engineering, Vol. I, 409–412. MEYERHOF, G. G. (1951). “The Ultimate Bearing Capacity of Foundations,” Geotechnique, Vol. 2, No. 4, 301–331. MEYERHOF, G. G. (1953). “The Bearing Capacity of Foundations Under Eccentric and Inclined Loads,” Proceedings, 3rd International Conference on Soil Mechanics and Foundation Engineering, Vol. 1, 440–445. MEYERHOF, G. G. (1956). “Penetration Tests and Bearing Capacity of Cohesionless Soils,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 82, No. SM1, pp. 1–19. MEYERHOF, G. G. (1963). “Some Recent Research on the Bearing Capacity of Foundations,” Canadian Geotechnical Journal, Vol. 1, 16–26. MEYERHOF, G. G. (1965). “Shallow Foundations,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 91, No. SM2, pp. 21–31. PRANDTL, L. (1921). “Über die Eindringungsfestigkeit (Harte) plasticher Baustoffe und die Festigkeit von Schneiden,” Zeitschrift für Angewandte Mathematik und Mechanik, Basel, Switzerland, Vol. 1, No. 1, 15–20. PURKAYASTHA, R. D., and CHAR, R. A. N. (1977). “Stability Analysis for Eccentrically Loaded Footings,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. 6, 647–651. REISSNER, H. (1924). “Zum Erddruckproblem,” Proceedings, 1st International Congress of Applied Mechanics, 295–311. SKEMPTON, A. W. (1942). “An Investigation of the Bearing Capacity of a Soft Clay Soil,” Journal of the Institute of Civil Engineers, London, Vol. 18, 307–321. TERZAGHI, K. (1943). Theoretical Soil Mechanics, Wiley, New York. VESIC, A. S. (1973). “Analysis of Ultimate Loads of Shallow Foundations,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM1, 45–73. 17 Landﬁll Liners and Geosynthetics Enormous amounts of solid waste are generated every year in the United States and other industrialized countries. These waste materials, in general, can be classiﬁed into four major categories: (1) municipal waste, (2) industrial waste, (3) hazardous waste, and (4) lowlevel radioactive waste. Table 17.1 lists the waste material generated in 1984 in the United States in these four categories (Koerner, 1994). The waste materials generally are placed in landﬁlls. The landﬁll materials interact with moisture received from rainfall and snow to form a liquid called leachate. The chemical composition of leachates varies widely, depending on the waste material involved. Leachates are a main source of groundwater pollution; therefore, they must be contained properly in all landﬁlls, surface impoundments, and waste piles within some type of liner system. In the following sections of this chapter, various types of liner systems and the materials used in them are discussed. 17.1 Landﬁll Liners— Overview Until about 1982, the predominant liner material used in landﬁlls was clay. Proper clay liners have a hydraulic conductivity of about 107 cm/sec or less. In 1984, the U.S. Environmental Protection Agency’s minimum technological requirements for hazardous-waste landﬁll design and construction were introduced by the U.S. Congress in Hazardous and Solid Waste amendments. In these amendments, Congress stipulated that all new landﬁlls should have double liners and systems for leachate collection and removal. Table 17.1 Waste Material Generation in the United States Waste type Municipal Industrial (building debris, degradable waste, nondegradable waste, and near hazardous) Hazardous Low-level radioactive Approximate quantity in 1984 (millions of metric tons) 300 600 150 15 611 612 Chapter 17: Landﬁll Liners and Geosynthetics To understand the construction and functioning of the double-liner system, we must review the general properties of the component materials involved in the system—that is, clay soil and geosynthetics (such as geotextiles, geomembranes, and geonets). Section 17.2 gives the details for the compaction of clay soils in the ﬁeld for liner construction. A brief review of the essential properties of geosynthetics is given in Sections 17.3 through 17.6. 17.2 Compaction of Clay Soil for Clay Liner Construction It was shown in Chapter 6 (Section 6.5) that, when a clay is compacted at a lower moisture content, it possesses a ﬂocculent structure. Approximately at the optimum moisture content of compaction, the clay particles have a lower degree of ﬂocculation. A further increase in the moisture content at compaction provides a greater degree of particle orientation; however, the dry unit weight decreases, because the added water dilutes the concentration of soil solids per unit volume. Figure 17.1 shows the results of laboratory compaction tests on a clay soil as well as the variation of hydraulic conductivity on the compacted clay specimens. From the laboratory test results shown, the following observations can be made: 1. For a given compaction effort, the hydraulic conductivity, k, decreases with the increase in molding moisture content, reaching a minimum value at about the optimum moisture content (that is, approximately where the soil has a higher unit weight with the clay particles having a lower degree of ﬂocculation). Beyond the optimum moisture content, the hydraulic conductivity increases slightly. 2. For similar compaction effort and dry unit weight, a soil will have a lower hydraulic conductivity when it is compacted on the wet side of the optimum moisture content. Benson and Daniel (1990) conducted laboratory compaction tests by varying the size of clods of moist clayey soil. These tests show that, for similar compaction effort and molding moisture content, the magnitude of k decreases with the decrease in clod size. In some compaction work in clayey soils, the compaction must be done in a manner so that a certain speciﬁed upper level of hydraulic conductivity of the soil is achieved. Examples of such works are compaction of the core of an earth dam and installation of clay liners in solid-waste disposal sites. To prevent groundwater pollution from leachates generated from solid-waste disposal sites, the U.S. Environmental Protection Agency (EPA) requires that clay liners have a hydraulic conductivity of 107 cm/sec or less. To achieve this value, the contractor must ensure that the soil meets the following criteria (Environmental Protection Agency, 1989): 1. The soil should have at least 20% ﬁnes (ﬁne silt and clay-sized particles). 2. The plasticity index (PI) should be greater than 10. Soils that have a PI greater than about 30 are difﬁcult to work with in the ﬁeld. 3. The soil should not include more than 10% gravel-sized particles. 4. The soil should not contain any particles or chunks of rock that are larger than 25 to 50 mm (1 to 2 in.). In many instances, the soil found at the construction site may be somewhat nonplastic. Such soil may be blended with imported clay minerals (like sodium bentonite) to 613 17.2 Compaction of Clay Soil for Clay Liner Construction 2000 Dry density (kg/m3 ) 1900 1800 1700 1600 1500 8 10 12 14 16 18 20 Moisture content (%) (a) Hydraulic conductivity (cm /s) 106 107 108 109 8 10 12 14 16 18 Moisture content (%) (b) 20 Figure 17.1 Tests on a clay soil: (a) modiﬁed Proctor compaction curve; (b) variation of k with molding moisture content achieve the desired range of hydraulic conductivity. In addition, during ﬁeld compaction, a heavy sheepsfoot roller can introduce larger shear strains during compaction that create a more dispersed structure in the soil. This type of compacted soil will have an even lower hydraulic conductivity. Small lifts should be used during compaction so that the feet of the compactor can penetrate the full depth of the lift. There are a few published studies on the variation of hydraulic conductivity of mixtures of nonplastic soils and Bentonite. Sivapullaiah, et al. (2000) evaluated the hydraulic conductivity of Bentonite-sand and Bentonite-silt mixtures in the laboratory. Based on this study the following correlation was developed log k e 0.05351LL2 5.286 0.00631LL2 0.2516 where k hydraulic conductivity of Bentonite-nonplastic soil mixture (m/sec) e void ratio of compacted mixture LL liquid limit of mixture (%) Equation (17.1) is valid for mixtures having a liquid limit greater than 50%. (17.1) 614 Chapter 17: Landﬁll Liners and Geosynthetics The size of the clay clods has a strong inﬂuence on the hydraulic conductivity of a compacted clay. Hence, during compaction, the clods must be broken down mechanically to as small as possible. A very heavy roller used for compaction helps to break them down. Bonding between successive lifts is also an important factor; otherwise, permeant can move through a vertical crack in the compacted clay and then travel along the interface between two lifts until it ﬁnds another crack, as is shown schematically in Figure 17.2. Poor bonding can increase substantially the overall hydraulic conductivity of a compacted clay. An example of poor bonding was seen in a trial pad construction in Houston in 1986. The trial pad was 0.91 m (3 ft) thick and built in six, 15.2 mm (6 in.) lifts. The results of the hydraulic conductivity tests for the compact soil from the trial pad are given in Table 17.2. Note that, although the laboratory-determined values of k for various lifts are on the order of 107 to 109 cm/sec, the actual overall value of k increased to the order of 104. For this reason, scariﬁcation and control of the moisture content after compaction of each lift are extremely important in achieving the desired hydraulic conductivity. In the construction of clay liners for solid-waste disposal sites where it is required that k 107 cm/sec, it is important to establish the moisture content–unit weight criteria in the laboratory for the soil to be used in ﬁeld construction. This helps in the development of proper speciﬁcations. Lift 4 Lift 3 Lift 2 Lift 1 Figure 17.2 Pattern of ﬂow through a compacted clay with improper bonding between lifts (After U.S. Environmental Protection Agency, 1989) Table 17.2 Hydraulic Conductivity from Houston Liner Tests* Location Sample Laboratory k (cm /sec) Lower lift Upper lift Lift interface Lower lift Upper lift 76 mm (3 in.) tube 76 mm (3 in.) tube 76 mm (3 in.) tube Block Block 4 109 1 109 1 107 8 105 1 108 Actual overall k 1 104 cm/sec * After U.S. Environmental Protection Agency, 1989 17.2 Compaction of Clay Soil for Clay Liner Construction 615 Daniel and Benson (1990) developed a procedure to establish the moisture content–unit weight criteria for clayey soils to meet the hydraulic conductivity requirement. The following is a step-by-step procedure to develop the criteria. Step 1: Conduct Proctor tests to establish the dry unit weight versus molding moisture content relationships (Figure 17.3a). Modiﬁed Proctor Dry unit weight Standard Proctor Hydraulic conductivity, k Moisture content (a) Maximum allowable k, kall Moisture content (b) Dry unit weight Acceptable zone Moisture content (c) Figure 17.3 (a) Proctor curves; (b) variation of hydraulic conductivity of compacted specimens; (c) determination of acceptable zone 616 Chapter 17: Landﬁll Liners and Geosynthetics Step 2: Conduct permeability tests on the compacted soil specimens (from Step 1), and plot the results, as shown in Figure 17.3b. In this ﬁgure, also plot the maximum allowable value of k (that is, kall). Step 3: Replot the dry unit weight–moisture content points (Figure 17.3c) with different symbols to represent the compacted specimens with k kall and k kall. Step 4: Plot the acceptable zone for which k is less than or equal to kall (Figure 17.3c). 17.3 Geosynthetics In general, geosynthetics are fabric-like material made from polymers such as polyester, polyethylene, polypropylene, polyvinyl chloride (PVC), nylon, chlorinated polyethylene, and others. The term geosynthetics includes the following: • • • • • Geotextiles Geomembranes Geogrids Geonets Geocomposites Each type of geosynthetic performs one or more of the following ﬁve major functions: 1. 2. 3. 4. 5. Separation Reinforcement Filtration Drainage Moisture barrier Geosynthetics have been used in civil engineering construction since the late 1970s, and their use currently is growing rapidly. In this chapter, it is not possible to provide detailed descriptions of manufacturing procedures, properties, and uses of all types of geosynthetics. However, an overview of geotextiles, geomembranes, and geonets is given. For further information, refer to a geosynthetics text, such as that by Koerner (1994). 17.4 Geotextiles Geotextiles are textiles in the traditional sense; however, the fabrics usually are made from petroleum products such as polyester, polyethylene, and polypropylene. They also may be made from ﬁberglass. Geotextiles are not prepared from natural fabrics, which decay too quickly. They may be woven, knitted, or nonwoven. Woven geotextiles are made of two sets of parallel ﬁlaments or strands of yarn systematically interlaced to form a planar structure. Knitted geotextiles are formed by interlocking a series of loops of one or more ﬁlaments or strands of yarn to form a planar 17.4 Geotextiles 617 structure. Nonwoven geotextiles are formed from ﬁlaments or short ﬁbers arranged in an oriented or a random pattern in a planar structure. These ﬁlaments or short ﬁbers ﬁrst are arranged into a loose web. They then are bonded by using one or a combination of the following processes: • • • Chemical bonding—by glue, rubber, latex, cellulose derivative, and so forth Thermal bonding—by heat for partial melting of ﬁlaments Mechanical bonding—by needle punching The needle-punched nonwoven geotextiles are thick and have high in-plane hydraulic conductivity. Geotextiles have four major uses: 1. Drainage: The fabrics can channel water rapidly from soil to various outlets. 2. Filtration: When placed between two soil layers, one coarse grained and the other ﬁne grained, the fabric allows free seepage of water from one layer to the other. At the same time, it protects the ﬁne-grained soil from being washed into the coarsegrained soil. 3. Separation: Geotextiles help keep various soil layers separate after construction. For example, in the construction of highways, a clayey subgrade can be kept separate from a granular base course. 4. Reinforcement: The tensile strength of geotextiles increases the load-bearing capacity of the soil. Geotextiles currently available commercially have thicknesses that vary from about 0.25 to 7.6 mm (0.01 to 0.3 in.). The mass per unit area of these geotextiles ranges from about 150 to 700 g/cm2. One of the major functions of geotextiles is ﬁltration. For this purpose, water must be able to ﬂow freely through the fabric of the geotextile (Figure 17.4). Hence, the cross-plane hydraulic conductivity is an important parameter for design purposes. It should be realized that geotextile fabrics are compressible, however, and their thickness may change depending on the effective normal stress to which they are being subjected. The change in thickness under normal stress also changes the cross-plane hydraulic Direction of ﬂow Geotextile Figure 17.4 Cross-plane ﬂow through geotextile 618 Chapter 17: Landﬁll Liners and Geosynthetics conductivity of a geotextile. Thus, the cross-plane capability is generally expressed in terms of a quantity called permittivity, or P kn t (17.2) where P permittivity kn hydraulic conductivity for cross-plane flow t thickness of the geotextile In a similar manner, to perform the function of drainage satisfactorily, geotextiles must possess excellent in-plane permeability. For reasons stated previously, the in-plane hydraulic conductivity also depends on the compressibility, and, hence, the thickness of the geotextile. The in-plane drainage capability can thus be expressed in terms of a quantity called transmissivity, or T kpt (17.3) where T transmissivity kp hydraulic conductivity for in-plane flow 1Figure 17.52 The units of kn and kp are cm/sec or ft/min; however, the unit of permittivity, P, is sec1 or min1. In a similar manner, the unit of transmissivity, T, is m3/sec # m or ft3/min # ft. Depending on the type of geotextile, kn and P, and kp and T can vary widely. The following are some typical values for kn, P, kp, and T. • • • • Hydraulic conductivity, kn : 1 103 to 2.5 101 cm/sec Permittivity, P: 2 102 to 2.0 sec1 Hydraulic conductivity, kp : Nonwoven: 1 103 to 5 102 cm/sec Woven: 2 103 to 4 103 cm/sec Transmissivity T: Nonwoven: 2 106 to 2 109 m3/sec # m Woven: 1.5 108 to 2 108 m3/sec # m Geotextile Direction of ﬂow Figure 17.5 In-plane ﬂow in geotextile 17.5 Geomembranes 619 When a geotextile is being considered for use in the design and construction of landﬁll liners, certain properties must be measured by tests on the geotextile to determine its applicability. A partial list of these tests follows. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 17.5 Mass per unit area Percentage of open area Equivalent opening size Thickness Ultraviolet resistivity Permittivity Transmissivity Puncture resistance Resistance to abrasion Compressibility Tensile strength and elongation properties Chemical resistance Geomembranes Geomembranes are impermeable liquid or vapor barriers made primarily from continuous polymeric sheets that are ﬂexible. The type of polymeric material used for geomembranes may be thermoplastic or thermoset. The thermoplastic polymers include PVC, polyethylene, chlorinated polyethylene, and polyamide. The thermoset polymers include ethylene vinyl acetate, polychloroprene, and isoprene-isobutylene. Although geomembranes are thought to be impermeable, they are not. Water vapor transmission tests show that the hydraulic conductivity of geomembranes is in the range of 1010 to 1013 cm/sec; hence, they are only “essentially impermeable.” Many scrim-reinforced geomembranes manufactured in single piles have thicknesses that range from 0.25 to about 0.4 mm (0.01 to 0.016 in.). These single piles of geomembranes can be laminated together to make thicker geomembranes. Some geomembranes made from PVC and polyethylene may be as thick as 4.5 to 5 mm (0.18 to 0.2 in.). The following is a partial list of tests that should be conducted on geomembranes when they are to be used as landﬁll liners. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Density Mass per unit area Water vapor transmission capacity Tensile behavior Tear resistance Resistance to impact Puncture resistance Stress cracking Chemical resistance Ultraviolet light resistance Thermal properties Behavior of seams 620 Chapter 17: Landﬁll Liners and Geosynthetics The most important aspect of construction with geomembranes is the preparation of seams. Otherwise, the basic reason for using geomembranes as a liquid or vapor barrier will be defeated. Geomembrane sheets generally are seamed together in the factory to prepare larger sheets. These larger sheets are ﬁeld seamed into their ﬁnal position. There are several types of seams, some of which are described brieﬂy. Lap Seam with Adhesive • A solvent adhesive is used for this type of seam (Figure 17.6a). After application of the solvent, the two sheets of geomembrane are overlapped, then roller pressure is applied. Lap Seam with Gum Tape • This type of seam (Figure 17.6b) is used mostly in dense thermoset material, such as isoprene-isobutylene. Tongue-and-Groove Splice • A schematic diagram of the tongue-and-groove splice is shown in Figure 17.6c. The tapes used for the splice are double sided. (a) (b) Factory vulcanized (c) (d) (e) (f) Adhesive Gum tape Figure 17.6 Conﬁgurations of ﬁeld geomembrane seams: (a) lap seam; (b) lap seam with gum tape; (c) tongueand-groove splice; (d) extrusion weld lap seam; (e) ﬁllet weld lap seam; (f) double hot air or wedge seam (After U.S. Environmental Protection Agency, 1989) 17.6 Geonets 621 Extrusion Weld Lap Seam • Extrusion or fusion welding is done on geomembranes made from polyethylene. A ribbon of molten polymer is extruded between the two surfaces to be joined (Figure 17.6d). Fillet Weld Lap Seam • This seam is similar to an extrusion weld lap seam; however, for ﬁllet welding, the extrudate is placed over the edge of the seam (Figure 17.6e). Double Hot Air or Wedge Seam • In the hot air seam, hot air is blown to melt the two opposing surfaces. For melting, the temperatures should rise to about 500F or more. After the opposite surfaces are melted, pressure is applied to form the seam (Figure 17.6f). For hot wedge seams, an electrically heated element like a blade is passed between the two opposing surfaces of the geomembrane. The heated element helps to melt the geomembrane, after which pressure is applied by a roller to form the seam. 17.6 Geonets Geonets are formed by the continuous extrusion of polymeric ribs at acute angles to each other. They have large openings in a netlike conﬁguration. The primary function of geonets is drainage. Figure 17.7 is a photograph of a typical piece of geonet. Most geonets currently available are made of medium-density and high-density polyethylene. They are available in rolls with widths of 1.8 to 2.1 m (6 to 7 ft) and lengths of 30 to 90 m (100 to 300 ft). The approximate aperture sizes vary from 30 mm 30 mm (1.2 in. 1.2 in.) to about 6 mm 6 mm (0.25 in. 2.5 in.). The thickness of geonets available commercially can vary from 3.8 to 7.6 mm (0.15 to 0.3 in.). Seaming of geonets is somewhat more difﬁcult. For this purpose, staples, threaded loops, and wire sometimes are used. Figure 17.7 Geonet (Courtesy of Braja M. Das, Henderson, Nevada) 622 Chapter 17: Landﬁll Liners and Geosynthetics 17.7 Single Clay Liner and Single Geomembrane Liner Systems Until about 1982—that is, before the guidelines for the minimum technological requirements for hazardous-waste landﬁll design and construction were mandated by the U.S. Environmental Protection Agency—most landﬁll liners were single clay liners. Figure 17.8 shows the cross section of a single clay liner system for a landﬁll. It consists primarily of a compacted clay liner over the native foundation soil. The thickness of the compacted clay liner varies between 0.9 and 1.8 m (3 and 6 ft). The maximum required hydraulic conductivity, k, is 107 cm/sec. Over the clay liner is a layer of gravel with perforated pipes for leachate collection and removal. Over the gravel layer is a layer of ﬁlter soil. The ﬁlter is used to protect the holes in the perforated pipes against the movement of ﬁne soil particles. In most cases, the ﬁlter is medium coarse to ﬁne sandy soil. It is important to note that this system does not have any leak-detection capability. Around 1982, single layers of geomembranes also were used as a liner material for landﬁll sites. As shown in Figure 17.9, the geomembrane is laid over native foundation soil. Over the geomembrane is a layer of gravel with perforated pipes for leachate collection and removal. A layer of ﬁlter soil is placed between the solid waste material and the gravel. As in the single clay liner system, no provision is made for leak detection. Native foundation soil Waste Filter soil Gravel Clay liner Perforated pipe Figure 17.8 Cross section of single clay liner system for a landﬁll Native foundation soil Waste Filter soil Gravel Perforated pipe Geomembrane Figure 17.9 Cross section of single geomembrane liner system for a landﬁll 17.8 Recent Advances in the Liner Systems for Landﬁlls 17.8 623 Recent Advances in the Liner Systems for Landﬁlls Since 1984, most landﬁlls developed for solid and hazardous wastes have double liners. The two liners are an upper primary liner and a lower secondary liner. Above the top liner is a primary leachate collection and removal system. In general, the primary leachate collection system must be able to maintain a leachate head of 0.3 m (12 in.) or less. Between the primary and secondary liners is a system for leak detection, collection, and removal (LDCR) of leachates. The general guidelines for the primary leachate collection system and the LDCR system are as follows: 1. It can be a granular drainage layer or a geosynthetic drainage material, such as a geonet. 2. If a granular drainage layer is used, it should have a minimum thickness of 0.3 m (12 in.) 3. The granular drainage layer (or the geosynthetic) should have a hydraulic conductivity, k, greater than 102 cm/sec. 4. If a granular drainage layer is used, it should have a granular ﬁlter or a layer of geotextile over it to prevent clogging. A layer of geotextile also is required over the geonet when it is used as the drainage layer. 5. The granular drainage layer, when used, must be chemically resistant to the waste material and the leachate that are produced. It also should have a network of perforated pipes to collect the leachate effectively and efﬁciently. In the design of the liner systems, the compacted clay layers should be at least 1 m (3 ft) thick, with k 107 cm/sec. Figures 17.10 and 17.11 show schematic diagrams of two double-liner systems. In Figure 17.10, the primary leachate collection system is made of a granular material with perforated pipes and a ﬁlter system above it. The primary liner is a geomembrane. The LDCR system is made of a geonet. The secondary liner Native foundation soil Waste Filter soil Gravel Perforated pipe Geomembrane (primary liner) Geonet (leak detection and leachate collection) Geomembrane (secondary composite liner) Clay liner Figure 17.10 Cross section of double-liner system (note the secondary composite liner) 624 Chapter 17: Landﬁll Liners and Geosynthetics Native foundation soil Perforated pipe Waste Filter soil Gravel Primary composite liner (geomembrane) Primary composite liner (clay liner) Geotextile Geonet Secondary composite liner (geomembrane) Secondary composite liner (clay liner) Figure 17.11 Cross section of double-liner system (note the primary and secondary composite liners) is a composite liner made of a geomembrane with a compacted clay layer below it. In Figure 17.11, the primary leachate collection system is similar to that shown in Figure 17.10; however, the primary and secondary liners are both composite liners (geomembrane-clay). The LDCR system is a geonet with a layer of geotextile over it. The layer of geotextile acts as a ﬁlter and separator. The geomembranes used for landﬁll lining must have a minimum thickness of 0.76 mm (0.03 in.); however, all geomembranes that have a thickness of 0.76 mm (0.03 in.) may not be suitable in all situations. In practice, most geomembranes used as liners have thicknesses ranging from 1.8 to 2.54 mm (0.7 to 0.1 in.). 17.9 Leachate Removal Systems The bottom of a landﬁll must be graded properly so that the leachate collected from the primary collection system and the LDCR system will ﬂow to a low point by gravity. Usually, a grade of 2% or more is provided for large landﬁll sites. The low point of the leachate collection system ends at a sump. For a primary leachate collection, a manhole is located at the sump, which rises through the waste material. Figure 17.12 shows a schematic diagram of the leachate removal system with a low-volume sump. A typical leachate removal system for high-volume sumps (for primary collection) is shown in Figure 17.13. Leachate can be removed from the LDCR system by means of pumping, as shown in Figure 17.14, or by gravity monitoring, as shown in Figure 17.15. When leachate is removed by pumping, the plastic pipe used for removal must penetrate the primary liner. On the other hand, if gravity monitoring is used, the pipe will penetrate the secondary liner. Concrete base 1–1.2 m (36 – 48 in.) Reinforced concrete pipe (RCP) Primary leachate Solid waste Sand Gravel Geomembrane Steel plate Figure 17.12 Primary leachate removal system with a low-volume sump (After U.S. Environmental Protection Agency, 1989). Standpipe Air space RCP pipe Operational cover 2% minimum Sand Geomembrane Compacted clay Figure 17.13 Primary leachate removal system with a high-volume sump (After U.S. Environmental Protection Agency, 1989) 625 Leachate removal Plastic pipe Submersible pump Leak detection and collection system Native foundation soil Waste Primary geomembrane Secondary geomembrane Clay liner Figure 17.14 Secondary leak detection, collection, and removal (LDCR) system—by means of pumping. (Note: The plastic pipe penetrates the primary geomembrane) Leak detection system Pipe Sump Native foundation soil Waste Clay liner Primary geomembrane Secondary geomembrane Figure 17.15 Secondary LDCR system, by means of gravity monitoring. (Note: The plastic pipe penetrates the secondary geomembrane) 626 17.10 Closure of Landﬁlls Closure of Landﬁlls When the landﬁll is complete and no more waste can be placed into it, a cap must be put on it (Figure 17.16). This cap will reduce and ultimately eliminate leachate generation. A schematic diagram of the layering system recommended by the U.S. Environmental Protection Agency (1979, 1986) and Koerner (1994) for hazardous-waste landﬁlls is shown in Figure 17.17. Essentially, it consists of a compacted clay cap over the solid waste, a geomembrane liner, a drainage layer, and a cover of topsoil. The manhole used for leachate collection penetrates the landﬁll cover. Leachate removal continues until its generation is stopped. For hazardous-waste landﬁll sites, the EPA (1989) recommends this period to be about 30 years. Cap (prevents inﬁltration) Liner (prevents migration of leachates) Waste 600 mm (24 in.) Figure 17.16 Landﬁll with liner and cap k 10 2 cm/sec Geomembrane [minimum thickness 0.5 mm (0.02 in.)] 600 mm (24 in.) 17.10 627 k 10 7 cm/sec Cover topsoil Drainage layer Compacted clay cap Waste Figure 17.17 Schematic diagram of the layering system for landﬁll cap 628 Chapter 17: Landﬁll Liners and Geosynthetics 17.11 Summary and General Comments This chapter provided a brief overview of the problems associated with solid- and hazardous-waste landﬁlls. The general concepts for the construction of landﬁll liners using compacted clayey soil and geosynthetics (that is, geotextiles, geomembranes, and geonets) were discussed. Several areas were not addressed, however, because they are beyond the scope of the text. Areas not discussed include the following: 1. Selection of material. The chemicals contained in leachates generated from hazardous and nonhazardous waste may interact with the liner materials. For this reason, it is essential that representative leachates are used to test the chemical compatibility so that the liner material remains intact during the periods of landﬁll operation and closure, and possibly longer. Selection of the proper leacheates becomes difﬁcult because of the extreme variations encountered in the ﬁeld. The mechanical properties of geomembranes also are important. Properties such as workability, creep, stress cracking, and the thermal coefﬁcient of expansion should be investigated thoroughly. 2. Stability of side-slope liner. The stability and slippage checks of the side-slope liners of a landﬁll site are important and complicated because of the variation of the frictional characters of the composite materials involved in liner construction. For a detailed treatment of this topic, refer to any book on geosynthetics (e.g., Koerner, 1994). 3. Leak-response action plan. It is extremely important that any leaks or clogging of the drainage layer(s) in a given waste-disposal site be detected as quickly as possible. Leaks or cloggings are likelihoods at a site even with good construction quality control. Each waste-disposal facility should have a leak-response action plan. References BENSON, C. H. and DANIEL, D. E. (1990). “Inﬂuence of Clods on Hydraulic Conductivity of Compacted Clay,” Journal of Geotechnical Engineering, ASCE, Vol. 116, No. 8, 1231–1248. DANIEL, D. E. and BENSON, C. H. (1990). “Water Content-Density Criteria for Compacted Soil Liners,” Journal of Geotechnical Engineering, ASCE, Vol. 116, No. 12, 1811–1830. KOERNER, R. M. (1994). Designing with Geosynthetics, 3rd ed., Prentice-Hall, Englewood Cliffs, N.J. SIVAPULLAIAH, P. V., SRIDHARAN, A., and STALIN, V. K. (2000). “Hydraulic Conductivity of Bentonite-Sand Mixtures,” Canadian Geotechnical Journal, Vol. 37, No. 2, 406–413. U.S. ENVIRONMENTAL PROTECTION AGENCY (1979). Design and Construction for Solid Waste Landﬁlls, Publication No. EPA-600/2-79-165, Cincinnati, Ohio. U.S. ENVIRONMENTAL PROTECTION AGENCY (1986). Cover for Uncontrolled Hazardous Waste Sites, Publication No. EPA-540/2-85-002, Cincinnati, Ohio. U.S. ENVIRONMENTAL PROTECTION AGENCY (1989). Requirements for Hazardous Waste Landﬁll Design, Construction, and Closure, Publication No. EPA-625/4-89-022, Cincinnati, Ohio. 18 Subsoil Exploration The preceding chapters reviewed the fundamental properties of soils and their behavior under stress and strain in idealized conditions. In practice, natural soil deposits are not homogeneous, elastic, or isotropic. In some places, the stratiﬁcation of soil deposits even may change greatly within a horizontal distance of 15 to 30 m (50 to 100 ft). For foundation design and construction work, one must know the actual soil stratiﬁcation at a given site, the laboratory test results of the soil samples obtained from various depths, and the observations made during the construction of other structures built under similar conditions. For most major structures, adequate subsoil exploration at the construction site must be conducted. The purposes of subsoil exploration include the following: 1. Determining the nature of soil at the site and its stratiﬁcation 2. Obtaining disturbed and undisturbed soil samples for visual identiﬁcation and appropriate laboratory tests 3. Determining the depth and nature of bedrock, if and when encountered 4. Performing some in situ ﬁeld tests, such as permeability tests (Chapter 7), vane shear tests (Chapter 12), and standard penetration tests 5. Observing drainage conditions from and into the site 6. Assessing any special construction problems with respect to the existing structure(s) nearby 7. Determining the position of the water table This chapter brieﬂy summarizes subsoil exploration techniques. For additional information, refer to the Manual of Foundation Investigations of the American Association of State Highway and Transportation Ofﬁcials (1967). 18.1 Planning for Soil Exploration A soil exploration program for a given structure can be divided broadly into four phases: 1. Compilation of the existing information regarding the structure: This phase includes gathering information such as the type of structure to be constructed and its future use, the requirements of local building codes, and the column and load-bearing wall 629 630 Chapter 18: Subsoil Exploration loads. If the exploration is for the construction of a bridge foundation, one must have an idea of the length of the span and the anticipated loads to which the piers and abutments will be subjected. 2. Collection of existing information for the subsoil condition: Considerable savings in the exploration program sometimes can be realized if the geotechnical engineer in charge of the project thoroughly reviews the existing information regarding the subsoil conditions at the site under consideration. Useful information can be obtained from the following sources: a. Geologic survey maps b. County soil survey maps prepared by the U.S. Department of Agriculture and the Soil Conservation Service c. Soil manuals published by the state highway departments d. Existing soil exploration reports prepared for the construction of nearby structures Information gathered from the preceding sources provides insight into the type of soil and problems that might be encountered during actual drilling operations. 3. Reconnaissance of the proposed construction site: The engineer visually should inspect the site and the surrounding area. In many cases, the information gathered from such a trip is invaluable for future planning. The type of vegetation at a site, in some instances, may indicate the type of subsoil that will be encountered. The accessibility of a site and the nature of drainage into and from it also can be determined. Open cuts near the site provide an indication about the subsoil stratiﬁcation. Cracks in the walls of nearby structure(s) may indicate settlement from the possible existence of soft clay layers or the presence of expansive clay soils. 4. Detailed site investigation: This phase consists of making several test borings at the site and collecting disturbed and undisturbed soil samples from various depths for visual observation and for laboratory tests. No hard-and-fast rule exists for determining the number of borings or the depth to which the test borings are to be advanced. For most buildings, at least one boring at each corner and one at the center should provide a start. Depending on the uniformity of the subsoil, additional test borings may be made. Table 18.1 gives guidelines for initial planning of borehole spacing. The test borings should extend through unsuitable foundation materials to ﬁrm soil layers. Sowers and Sowers (1970) provided a rough estimate of the minimum depth of Table 18.1 Spacing of Borings Boring spacings Project One-story buildings Multistory buildings Highways Earth dams Residential subdivision planning m ft 25–30 15–25 250–300 25–50 60–100 75–100 50–75 750–1000 75–150 200–300 18.2 Boring Methods 631 borings (unless bedrock is encountered) for multistory buildings. They can be given by the following equations, applicable to light steel or narrow concrete buildings: zb 1m2 3S0.7 zb 1ft2 10S0.7 (18.1a) (18.1b) or to heavy steel or wide concrete buildings: zb 1m2 6S0.7 zb 1ft2 20S0.7 (18.2a) (18.2b) In Eqs. (18.1) and (18.2), zb is the approximate depth of boring and S is the number of stories. The American Society of Civil Engineers (1972) recommended the following rules of thumb for estimating the boring depths for buildings. 1. Estimate the variation of the net effective stress increase, s, that will result from the construction of the proposed structure with depth. This variation can be estimated by using the principles outlined in Chapter 10. Determine the depth D1 at which the value of s is equal to 10% of the average load per unit area of the structure. œ 2. Plot the variation of the effective vertical stress, so , in the soil layer with depth. Compare this with the net stress increase variation, s, with depth as determined in Step 1. Determine the depth D2 at which s 0.05soœ . 3. The smaller of the two depths, D1 and D2 , is the approximate minimum depth of the boring. When the soil exploration is for the construction of dams and embankments, the depth of boring may range from one-half to two times the embankment height. The general techniques used for advancing test borings in the ﬁeld and the procedure for the collection of soil samples are described in the following sections. 18.2 Boring Methods The test boring can be advanced in the ﬁeld by several methods. The simplest is the use of augers. Figure 18.1 shows two types of hand augers that can be used for making boreholes up to a depth of about 3 to 5 m (10 to 15 ft). They can be used for soil exploration work for highways and small structures. Information regarding the types of soil present at various depths is obtained by noting the soil that holds to the auger. The soil samples collected in this manner are disturbed, but they can be used to conduct laboratory tests such as grainsize determination and Atterberg limits. When the boreholes are to be advanced to greater depths, the most common method is to use continuous-ﬂight augers, which are power operated. The power for drilling is delivered by truck- or tractor-mounted drilling rigs. Continuous-ﬂight augers are available commercially in 1 to 1.5 m (3 to 5 ft) sections. During the drilling operation, section after section of auger can be added and the hole extended downward. Continuous-ﬂight augers 632 Chapter 18: Subsoil Exploration Figure 18.1 Hand augers: (a) Iwan auger; (b) slip auger (Courtesy of Braja M. Das, Henderson, Nevada) can be solid stem or hollow stem. Some of the commonly used solid-stem augers have outside diameters of 67 mm 12 58 in.2 , 83 mm 13 14 in.2 , 102 mm (4 in.), and 114 mm 14 12 in.2 . The inside and outside diameters of some hollow-stem augers are given in Table 18.2. Flight augers bring the loose soil from the bottom of the hole to the surface. The driller can detect the change in soil type encountered by the change of speed and the sound of drilling. Figure 18.2 shows a drilling operation with ﬂight augers. When solid-stem augers are used, the auger must be withdrawn at regular intervals to obtain soil samples and to conduct other operations such as standard penetration tests. Hollow-stem augers have a distinct advantage in this respect—they do not have to be removed at frequent intervals for sampling or other tests. As shown in Figure 18.3, the outside of the auger acts like a casing. A removable plug is attached to the bottom of the auger by means of a center rod. Table 18.2 Dimensions of Commonly Used Hollow-Stem Augers Inside diameter Outside diameter mm in. mm in. 63.5 69.85 76.2 88.9 101.6 2.5 2.75 3.0 3.5 4.0 158.75 187.8 203.2 228.6 254.0 6.25 7.0 8.0 9.0 10.0 18.2 Boring Methods 633 Figure 18.2 Drilling with ﬂight augers (Courtesy of Danny R. Anderson, PE, of Professional Service Industries, Inc., El Paso, Texas) Center rod Hollow-stem auger Removable plug Figure 18.3 Schematic diagram of hollow-stem auger with removable plug During the drilling, the plug can be pulled out with the auger in place, and soil sampling and standard penetration tests can be performed. When hollow-stem augers are used in sandy soils below the groundwater table, the sand might be pushed several feet into the stem of the auger by excess hydrostatic pressure immediately after the removal of the plug. In such conditions, the plug should not be used. Instead, water inside the hollow stem should be maintained at a higher level than the groundwater table. 634 Chapter 18: Subsoil Exploration Rotary drilling is a procedure by which rapidly rotating drilling bits attached to the bottom of drilling rods cut and grind the soil and advance the borehole down. Several types of drilling bits are available for such work. Rotary drilling can be used in sand, clay, and rock (unless badly ﬁssured). Water or drilling mud is forced down the drilling rods to the bits, and the return ﬂow forces the cuttings to the surface. Drilling mud is a slurry prepared by mixing bentonite and water (bentonite is a montmorillonite clay formed by the weathering of volcanic ash). Boreholes with diameters ranging from 50 to 200 mm (2 to 8 in.) can be made easily by using this technique. Wash boring is another method of advancing boreholes. In this method, a casing about 2 to 3 m (6 to 10 ft) long is driven into the ground. The soil inside the casing then is removed by means of a chopping bit that is attached to a drilling rod. Water is forced through the drilling rod, and it goes out at a very high velocity through the holes at the bottom of the chopping bit (Figure 18.4). The water and the chopped soil particles rise upward in the drill hole and overﬂow at the top of the casing through a T-connection. The wash water then is collected in a container. The casing can be extended with additional pieces as the borehole progresses; however, such extension is not necessary if the borehole will stay open without caving in. Percussion drilling is an alternative method of advancing a borehole, particularly through hard soil and rock. In this technique, a heavy drilling bit is raised and lowered to chop the hard soil. Casing for this type of drilling may be required. The chopped soil particles are brought up by the circulation of water. Derrick Rope Pressure hose Suction hose Wash-water tub Engine Drill rod Casing Chopping bit Driving shoe Water jet at high velocity Figure 18.4 Wash boring 18.3 Common Sampling Methods 18.3 635 Common Sampling Methods During the advancement of the boreholes, soil samples are collected at various depths for further analysis. This section brieﬂy discusses some of the methods of sample collection. Sampling by Standard Split Spoon Figure 18.5 shows a diagram of a split-spoon sampler. It consists of a tool-steel driving shoe at the bottom, a steel tube (that is split longitudinally into halves) in the middle, and a coupling at the top. The steel tube in the middle has inside and outside diameters of 34.9 mm 11 38 in.2 and 50.8 mm (2 in.), respectively. Figure 18.6 shows a photograph of an unassembled split-spoon sampler. When the borehole is advanced to a desired depth, the drilling tools are removed. The split-spoon sampler is attached to the drilling rod and then lowered to the bottom of the borehole (Figure 18.7). The sampler is driven into the soil at the bottom of the borehole by means of hammer blows. The hammer blows occur at the top of the drilling rod. The hammer weighs 623 N (140 lb). For each blow, the hammer drops a distance of 0.762 m (30 in.). The number of blows required for driving the sampler through three 152.4 mm (6 in.) intervals is recorded. The sum of the number of blows required for Drill rod Head Water port Pin Ball valve Split barrel 457.2 mm (18 in.) Driving shoe (138 in.) 76.2 mm (3 in.) 34.9 mm 1 in. 1.59 mm (16 ) 50.8 mm (2 in.) Figure 18.5 Diagram of standard splitspoon sampler Figure 18.6 Split-spoon sampler, unassembled (Courtesy of ELE International) 636 Chapter 18: Subsoil Exploration Figure 18.7 Drilling rod with split-spoon sampler lowered to the bottom of the borehole (Courtesy of Braja M. Das, Henderson, Nevada) driving the last two 152.4 mm (6 in.) intervals is referred to as the standard penetration number, N. It also commonly is called the blow count. The interpretation of the standard penetration number is given in Section 18.5. After driving is completed, the sampler is withdrawn and the shoe and coupling are removed. The soil sample collected inside the split tube then is removed and transported to the laboratory in small glass jars. Determination of the standard penetration number and collection of split-spoon samples usually are done at 1.5 m (5 ft) intervals. At this point, it is important to point out that there are several factors that will contribute to the variation of the standard penetration number, N, at a given depth for similar soil proﬁles. These factors include SPT hammer efﬁciency, borehole diameter, sampling method, and rod length factor (Skempton, 1986; Seed, et al., 1985). The two most common types of SPT hammers used in the ﬁeld are the safety hammer and donut hammer. They commonly are dropped by a rope with two wraps around a pulley. The SPT hammer energy efficiency can be expressed as Er 1% 2 actual hammer energy to the sampler 100 input energy Theoretical input energy Wh (18.3) (18.4) where W weight of the hammer 0.623 kN (140 lb) h height of drop 0.76 mm (30 in.) So, Wh (0.623)(0.76) 0.474 kN-m (4200 in.-lb) In the field, the magnitude of Er can vary from 30 to 90%. The standard practice now in the U.S. is to express the N-value to an average energy ratio of 60% ( N60). Thus, correcting for field procedures and on the basis of field observations, it appears reasonable to 18.3 Common Sampling Methods 637 standardize the field penetration number as a function of the input driving energy and its dissipation around the sampler into the surrounding soil, or N60 NhHhBhShR 60 (18.5) where N60 standard penetration number corrected for ﬁeld conditions N measured penetration number hH hammer efﬁciency (%) hB correction for borehole diameter hS sampler correction hR correction for rod length Based on the recommendations of Seed, et al. (1985) and Skempton (1986), the variations of hH, hB, hS, and hR are summarized in Table 18.3. Table 18.3 Variations of hH, hB, hS, and hR [Eq. (18.5)] 1. Variation of HH Country Hammer type Hammer release HH (%) Japan Donut Donut Free fall Rope and pulley 78 67 United States Safety Donut Rope and pulley Rope and pulley 60 45 Argentina China Donut Donut Donut Rope and pulley Free fall Rope and pulley 45 60 50 2. Variation of HB Diameter mm in. HB 60–120 150 200 2.4–4.7 6 8 1 1.05 1.15 3. Variation of HS Variable Standard sampler With liner for dense sand and clay With liner for loose sand 4. Variation of HR Rod length (m) HR 10 6–10 4–6 0–4 1.0 0.95 0.85 0.75 HS 1.0 0.8 0.9 638 Chapter 18: Subsoil Exploration Sampling by Thin-Wall Tube Sampling by thin-wall tube is used for obtaining fairly undisturbed soil samples. The thinwall tubes are made of seamless, thin tubes and commonly are referred to as Shelby tubes (Figure 18.8). To collect samples at a given depth in a borehole, one ﬁrst must remove the drilling tools. The sampler is attached to a drilling rod and lowered to the bottom of the borehole. After this, it is pushed hydraulically into the soil. It then is spun to shear off the base and is pulled out. The sampler with the soil inside is sealed and taken to the laboratory for testing. Most commonly used thin-wall tube samplers have outside diameters of 76.2 mm (3 in.). Sampling by Piston Sampler Piston samplers are particularly useful when highly undisturbed samples are required. The cost of recovering such samples is, of course, higher. Several types of piston samplers can be used; however, the sampler proposed by Osterberg (1952) is the most advantageous (Figure 18.9). It consists of a thin-wall tube with a piston. Initially, the piston closes the end of the thin-wall tube. The sampler ﬁrst is lowered to the bottom of the borehole (Figure 18.9a), then the thin-wall tube is pushed into the soil hydraulically—past the piston. After this, the pressure is released through a hole in the piston rod (Figure 18.9b). The presence of the piston prevents distortion in the sample by neither letting the soil squeeze into the sampling tube very fast nor admitting excess soil. Samples obtained in this manner consequently are disturbed less than those obtained by Shelby tubes. Drill rod Vent Water (in) Water (out) Drill rod Thin wall tube Piston Di Sample (a) Do Figure 18.8 Thin-wall tube sampler (b) Figure 18.9 Piston sampler: (a) sampler lowered to bottom of borehole; (b) pressure released through hole in piston rod 18.5 Correlations for Standard Penetration Test 18.4 639 Sample Disturbance The degree of disturbance of the sample collected by various methods can be expressed by a term called the area ratio, which is given by Ar 1 % 2 D2o D2i D2i 100 (18.6) where Do outside diameter of the sampler Di inside diameter of the sampler A soil sample generally can be considered undisturbed if the area ratio is less than or equal to 10%. The following is a calculation of Ar for a standard split-spoon sampler and a 50.8 mm (2 in.) Shelby tube: For the standard spit-spoon sampler, Di 1.38 in. and Do 2 in. Hence, Ar 1% 2 12 2 2 11.382 2 11.38 2 2 100 110% For the Shelby-tube sampler (2-in. diameter), Di 1.875 in. and Do 2 in. Hence, Ar 1% 2 12 2 2 11.8752 2 11.875 2 2 100 13.8% The preceding calculation indicates that the sample collected by split spoons is highly disturbed. The area ratio (Ar) of the 50.8 mm (2 in.) diameter Shelby tube samples is slightly higher than the 10% limit stated previously. For practical purpose, however, it can be treated as an undisturbed sample. The disturbed but representative soil samples recovered by split-spoon samplers can be used for laboratory tests, such as grain-size distribution, liquid limit, plastic limit, and shrinkage limit. However, undisturbed soil samples are necessary for performing tests such as consolidation, triaxial compression, and unconﬁned compression. 18.5 Correlations for Standard Penetration Test The procedure for conducting standard penetration tests was outlined in Section 18.3. The standard penetration number, N60, is commonly used to correlate several useful physical parameters of soil. Some of these are briefly described next. Cohesive Soil The consistency of clay soils can be estimated from the standard penetration number N60. In order to achieve that, Szechy and Vargi (1978) calculated the consistency index (CI) as CI LL w LL PL (18.7) 640 Chapter 18: Subsoil Exploration where w natural moisture content LL liquid limit PL plastic limit The approximate correlation between CI, N60, and the unconfined compression strength (qu) is given in Table 18.4. It is important to point out that the correlation between N60 and qu given in Table 18.4 is approximate. The sensitivity, St , of clay soil also plays an important role in the actual N60 value obtained in the ﬁeld. In any case, for clays of a given geology, a reasonable correlation between N60 and qu can be obtained, as shown in Figure 18.10. In this ﬁgure, the notation pa is the atmospheric pressure (in the same unit as qu). For the data shown in Figure 18.10, the reported regression is given by (Kulhawy and Mayne, 1990). qu 0.58N0.72 60 pa (18.8) Table 18.4 Approximate Correlation between CI, N60, and qu Unconﬁned compression strength, qu Standard penetration number, N60 Consistency CI kN/m2 lb/ft2 2 2 to 8 8 to 15 15 to 30 30 Very soft Soft to medium Stiff Very stiff Hard 0.5 0.5 to 0.75 0.75 to 1.0 1.0 to 1.5 1.5 25 25 to 80 80 to 150 150 to 400 400 500 500 to 1700 1700 to 3100 3100 to 8400 8400 25 low er bo un d ck r Pe ld e nd Go ann) a i m h g ert r za ) hm lde Ter m PI (Sc y u Go ay); i cla lay) l ed go c y c s (m ica (silty r nd nd e h a C rat s bou w an t( per So e l a p Jap u r g PI ) oess); San gle w n o l ( (l Sa ers ois lay ) Sow ); Illin ) c ann R y a India (cl SB rtm (U hme erat l n g c o n (S Sa ust ia Ho Yugoslav (hi gh PI ) 20 N60 So we rs 15 10 5 0 0 1 2 qu pa 3 Figure 18.10 Correlation between N60 and qu /pa (Based on Djoenaidi, 1985) 4 18.5 Correlations for Standard Penetration Test 641 Granular Soil In granular soils, the standard penetration number is highly dependent on the effective overburden pressure, soœ . A number of empirical relationships have been proposed to convert the ﬁeldstandard penetration number N60 to a standard effective overburden pressure, soœ , of 96 kN/m2 (2000 lb/ft2). The general form is 1N1 2 60 CNN60 (18.9) Several correlations have been developed over the years for the correction factor, CN . In this text, two of these are recommended for use. Correlation of Liao and Whitman (1986) In SI units, CN 9.78 1 B soœ (18.10) where soœ effective overburden pressure in kN/m2. In English units, CN 1 B soœ (18.11) where soœ effective overburden pressure in U.S. ton/ft2 (1 U.S. ton 2000 lb). Correlation of Skempton (1986) In SI units, CN 2 1 0.01soœ where soœ effective overburden pressure in kN/m2. In English units, 2 CN 1 soœ (18.12) (18.13) where soœ effective overburden pressure in U.S. ton/ft2. Table 18.5 shows approximate correlations for the standard penetration number, (N1)60 , and relative density, Dr . Table 18.5 Approximate Relationship Between Corrected Standard Penetration Number and Relative Density of Sand Corrected standard penetration number, (N 1)60 0–5 5–10 10–30 30–50 Relative density, Dr (%) 0–5 5–30 30–60 60–95 642 Chapter 18: Subsoil Exploration Cubrinovski and Ishihara (1999) proposed a correlation between N60 and the relative density of granular soils, Dr , in the form Dr 1% 2 £ N60 a 0.23 0.06 1.7 b D50 9 0.5 a 98 § b soœ 1100 2 (18.14) soœ effective overburden pressure in kN/m2 where D50 sieve size through which 50% of soil will pass 1mm2 The drained angle of friction of granular soils, f, also has been correlated to the standard penetration number. Peck, Hanson, and Thornburn (1974) gave a correlation between (N1)60 and f in a graphic form, which can be approximated as (Wolff, 1989) f¿ 1deg2 27.1 0.31N1 2 60 0.000541N1 2 260 (18.15) soœ . After Kulhawy and Schmertmann (1975) also provided a correlation for N60 versus Mayne (1990), this correlation can be approximated as 0.34 1 f¿ tan £ N60 soœ 12.2 20.3 a b pa § (18.16) where pa atmospheric pressure (same unit as soœ ). The standard penetration number is a useful guideline in soil exploration and the assessment of subsoil conditions, provided that the results are interpreted correctly. Note that all equations and correlations relating to the standard penetration numbers are approximate. Because soil is not homogeneous, a wide variation in the N60 value may be obtained in the ﬁeld. For soil deposits that contain large boulders and gravel, the standard penetration numbers may be erratic. Example 18.1 Following are the results of a standard penetration test in sand. Determine the corrected standard penetration numbers, (N1)60 , at various depths. Note that the water table was not observed within a depth of 35 ft below the ground surface. Assume that the average unit weight of sand is 110 lb/ft3. Use Eq. (18.11). Depth, z (ft) N 60 5 10 15 20 25 8 7 12 14 13 18.5 Correlations for Standard Penetration Test 643 Solution From Eq. (18.11), CN (sO)0.5. Now the following table can be prepared: Depth, z (ft) S Oœ ⴝ zG (ton /ft2) CN N 60 (N1)60 5 10 15 20 25 0.275 0.55 0.825 1.1 1.375 1.91 1.35 1.1 0.95 0.853 8 7 12 14 13 15 10 13 13 11 ■ Example 18.2 Solve Example 18.1 using Eq. (18.13). Solution From Eq. (18.13), CN 2 1 soœ Now we can prepare the following table. Depth, z (ft) S Oœ (ton /ft2) CN N 60 (N1)60 5 10 15 20 25 0.275 0.55 0.825 1.1 1.375 1.57 1.29 1.1 0.95 0.84 8 7 12 14 13 13 9 13 13 11 ■ Example 18.3 Refer to Example 18.1. Using Eq. (18.16), estimate the average soil friction angle, f, from z 0 to z 25 ft. Solution From Eq. (18.16), 0.34 1 f¿ tan £ N60 sOœ 12.2 20.3 a b pa § pa 14.7 lb/in.2 2115 lb/ft2 1 ton/ft2 644 Chapter 18: Subsoil Exploration Now the following table can be prepared. Depth, z (ft) S Oœ (ton /ft2) N60 Fⴕ (deg) [Eq. (18.16)] 5 10 15 20 25 0.275 0.55 0.825 1.1 1.375 8 7 12 14 13 37.3 33.6 36.5 36.3 34.3 Average f 36° 18.6 ■ Other In Situ Tests Depending on the type of project and the complexity of the subsoil, several types of in situ tests can be conducted during the exploration period. In many cases, the soil properties evaluated from the in situ tests yield more representative values. This better accuracy results primarily because the sample disturbance during soil exploration is eliminated. Following are some of the common tests that can be conducted in the ﬁeld. Vane Shear Test The principles and the application of the vane shear test were discussed in Chapter 12. When soft clay is encountered during the advancement of a borehole, the undrained shear strength of clay, cu , can be determined by conducting a vane shear test in the borehole. This test provides valuable information about the strength in undisturbed clay. Borehole Pressuremeter Test The pressuremeter is a device that originally was developed by Menard in 1965 for in situ measurement of the stress–strain modulus. This device basically consists of a pressure cell and two guard cells (Figure 18.11). The test involves expanding the pressure cell inside a borehole and measuring the expansion of its volume. The test data are interpreted on the basis of the theory of expansion of an inﬁnitely thick cylinder of soil. Figure 18.12 shows the variation of the pressure-cell volume with changes in the cell pressure. In this ﬁgure, Zone I represents the reloading portion, during which the soil around the borehole is pushed back to its initial state—that is, the state it was in before drilling. Zone II represents a pseudoelastic zone, in which the cell volume versus cell pressure is practically linear. The zone marked III is the plastic zone. For the pseudoelastic zone, Es 211 ms 2Vo ¢p ¢V where Es modulus of elasticity of soil ms Poisson’s ratio of soil Vo cell volume corresponding to pressure po 1that is, the cell pressure corresponding to the beginning of Zone II2 ¢p/¢V Slope of straight-line plot of Zone II (18.17) 18.6 Other In Situ Tests 645 Measuring cell volume, V Water pressure (for expansion of main cell) V p Vo po Measuring cell pressure, p Zone I Reloading Zone II Pseudoelastic zone Zone III Plastic zone Gas pressure (for expansion of guard cell) Guard cell Measuring cell Figure 18.11 Schematic diagram for pressuremeter test Figure 18.12 Relationship between measuring pressure and measuring volume for Menard pressuremeter Menard recommended a value of ms 0.33 for use in Eq. 18.17, but other values can be used. With ms 0.33, Es 2.66 Vo ¢p ¢V (18.18) From the theory of elasticity, the relationship between the modulus of elasticity and the shear modulus can be given as Es 211 ms 2Gs (18.19) where Gs shear modulus of soil. Hence, combining Eqs. (18.17) and (18.19) gives G Vo ¢p ¢V (18.20) Pressuremeter test results can be used to determine the at-rest earth-pressure coefﬁcient, Ko (Chapter 13). This coefﬁcient can be obtained from the ratio of po and soœ (soœ effective vertical stress at the depth of the test), or Ko po soœ (18.21) Note that po (see Figure 18.12) represents the in situ lateral pressure. The pressuremeter tests are very sensitive to the conditions of a borehole before the test. 646 Chapter 18: Subsoil Exploration Cone Penetration Test The Dutch cone penetrometer is a device by which a 60 cone with a base area of 10 cm2 (1.54 in2) (Figure 18.13) is pushed into the soil, and the cone end resistance, qc , to penetration is measured. Most cone penetrometers that are used commonly have friction sleeves that follow the point. This allows independent determination of the cone resistance (qc) and the frictional resistance ( fc) of the soil above it. The friction sleeves have an exposed surface area of about 150 cm2 (23 in2). The penetrometer shown in Figure 18.13 is a mechanical-friction cone penetrometer. At the present time, electrical-friction cone penetrometers also are used for ﬁeld investigation. 35.7 mm 15 mm 15 mm 12.5 mm 30 mm 47 mm 52.5 mm 187 mm 20 mm 45 mm 11.5 mm 133.5 mm 35.7 mm 25 mm 387 mm 266 mm 69 mm 33.5 mm 23 mm 32.5 mm 146 mm 35.7 mm 30 35 mm mm 60 Collapsed Extended Figure 18.13 Dutch cone penetrometer with friction sleeve (From Annual Book of ASTM Standards, 04.08. Copyright ASTM INTERNATIONAL. Reprinted with permission.) 18.6 Other In Situ Tests 647 One of the major advantages of the cone penetration test is that boreholes are not necessary to conduct the test. Unlike the standard penetration test, however, soil samples cannot be recovered for visual observation and laboratory tests. Robertson and Campanella (1983) provided correlations between the vertical effecœ tive stress 1so 2, drained soil friction angle (f), and qc for sand. The relationship between œ so , f, and qcœ can be approximated (after Kulhawy and Mayne, 1990) as f¿ tan1 c 0.1 0.38 log a qc bd soœ (18.22) The cone penetration resistance also has been correlated with the equivalent modulus of elasticity, Es , of soils by various investigators. Schmertmann (1970) gave a simple correlation for sand as Es 2qc (18.23) Troﬁmenkov (1974) also gave the following correlations for the modulus of elasticity in sand and clay: Es 3qc Es 7qc 1for sands2 1for clays 2 (18.24) (18.25) Correlations such as Eqs. (18.23) through (18.25) can be used in the calculation of the elastic settlement of foundations (Chapter 11). Anagnostopoulos, et al. (2003) have provided several correlations based on a large number of ﬁeld-test results conducted on a wide variety of soils. The correlations obtained from this study are summarized below. Case 1. Correlation between Undrained Shear Strength (cu) and qc. cu qc so Nk (18.26) where o total vertical stress NK bearing capacity factor ( 18.3 for all cones) Consistent units need to be used in Eq. (18.26). The values of cu in the field tests were equal to or less than about 250 kN/m2. Case 2. Correlation between Sleeve-Frictional Resistance (fc) and cu. For mechanical cones, fc 1.26cu (18.27) fc cu (18.28) fc 1.21cu (18.29) For electric cones, Average for all cones 648 Chapter 18: Subsoil Exploration Case 3. Correlation Between qc and N60. Based on a large number of field test results on various types of soil with mean grain size (D50) varying from 0.001 mm to about 7 to 8 mm, Anagnostopoulos, et al. (2003) provided the following correlation between qc, N60, and D50. a qc b pa 7.64D0.26 50 N60 (18.30) where pa atmospheric pressure ( 100 kN/m2 or 1 ton/ft2) D50 mean grain size, in mm Case 4. Correlation Between Friction Ratio (Rf) and D50. The friction ratio can be defined as Rf fc qc (18.31) Based on the soils investigated (with D50 ranging from 0.001 mm to about 7 to 8 mm), the correlation between Rf and D50 can be given as. Rf 1% 2 1.45 1.36 log1D50 2 1electric cone2 (18.32 and Rf 1% 2 0.7811 1.611 log1D50 2 1mechanical cone2 (18.33) where D50 is in mm. 18.7 Rock Coring It may be necessary to core rock if bedrock is encountered at a certain depth during drilling. It is always desirable that coring be done for at least 3 m (10 ft). If the bedrock is weathered or irregular, the coring may need to be extended to a greater depth. For coring, a core barrel is attached to the drilling rod. A coring bit is attached to the bottom of the core barrel. The cutting element in the bit may be diamond, tungsten, or carbide. The coring is advanced by rotary drilling. Water is circulated through the drilling rod during coring, and the cuttings are washed out. Figure 18.14a shows a diagram of rock coring by the use of a single-tube core barrel. Rock cores obtained by such barrels can be fractured because of torsion. To avoid this problem, one can use double-tube core barrels (Figure 18.14b). Table 18.6 gives the details of various types of casings and core barrels, diameters of core barrel bits, and diameters of core samples obtained. The core samples smaller than the BX size tend to break away during coring. 18.7 Rock Coring 649 Drill rod Inner barrel Core barrel Rock Outer barrel Rock core Rock core Core lifter Core lifter Coring bit Coring bit (a) Figure 18.14 Rock coring: (a) single-tube core barrel; (b) double-tube core barrel (b) Rock Table 18.6 Details of Core Barrel Designations, Bits, and Core Samples Casing and core barrel designation EX AX BX NX Outside diameter of core barrel bit, mm (in.) 36.5 47.6 58.7 74.6 11167 2 1178 2 12 165 2 12 15 16 2 Diameter of core sample, mm (in.) 22.2 28.6 41.3 54.0 1 78 2 1118 2 1158 2 12 18 2 On the basis of the length of the rock core obtained from each run, the following quantities can be obtained for evaluation of the quality of rock. Recovery ratio Rock quality designation 1RQD2 Length of rock core recovered Length of coring g Length of rock pieces recovered having lengths of 101.6 mm 14 in.2 or more Length of coring (18.34) (18.35) 650 Chapter 18: Subsoil Exploration Table 18.7 Qualitative Description of Rocks Based on RQD RQD 1–0.9 0.9–0.75 0.75–0.5 0.5–0.25 0.25–0 Rock quality Excellent Good Fair Poor Very poor A recovery ratio equal to 1 indicates intact rock. However, highly fractured rocks have a recovery ratio of 0.5 or less. Deere (1963) proposed the classiﬁcation system in Table 18.7 for in situ rocks on the basis of their RQD. 18.8 Soil Exploration Report At the end of the soil exploration program, the soil and rock samples collected from the ﬁeld are subjected to visual observation and laboratory tests. Then, a soil exploration report is prepared for use by the planning and design ofﬁce. Any soil exploration report should contain the following information: 1. Scope of investigation 2. General description of the proposed structure for which the exploration has been conducted 3. Geologic conditions of the site 4. Drainage facilities at the site 5. Details of boring 6. Description of subsoil conditions as determined from the soil and rock samples collected 7. Groundwater table as observed from the boreholes 8. Details of foundation recommendations and alternatives 9. Any anticipated construction problems 10. Limitations of the investigation The following graphic presentations also need to be attached to the soil exploration report: 1. 2. 3. 4. 5. Site location map Location of borings with respect to the proposed structure Boring logs (Figure 18.15) Laboratory test results Other special presentations The boring log is the graphic presentation of the details gathered from each borehole. Figure 18.15 shows a typical boring log. BORING LOG Shopping center PROJECT TITLE Intersection LOCATION Hill Street and Miner Street BORING NUMBER 4 DESCRIPTION OF SOIL TYPE OF BORING June 7, 1997 GROUND ELEVATION 132.2 ft STANDARD PENETRATION NUMBER, N60 MOISTURE CONTENT, w (%) COMMENTS 13 11 5 24 Hollow-stem auger DEPTH (ft) AND SAMPLE NUMBER Tan sandy silt DATE 1 2 3 4 Light brown silty clay (CL) 5 SS-1 6 Liquid limit = 32 PI = 9 7 8 9 Groundwater table June 14, 1997 10 SS-2 11 12 13 14 Soft clay (CL) 15 ST-1 6 16 17 28 Liquid limit = 44 PI = 26 qu = unconﬁned compression strength = 850 lb/ft2 18 19 Compact sand and gravel End of boring @ 22 ft 20 SS-3 32 21 22 Figure 18.15 Typical boring log (Note: SS split-spoon sample; ST Shelby tube sample) 651 652 Chapter 18: Subsoil Exploration Problems 18.1 Determine the area ratio of a Shelby tube sampler that has outside and inside diameters of 3.5 in. and 3.375 in., respectively. 18.2 Repeat Problem 18.1 with outside diameter 114 mm. and inside diameter 111 mm. 18.3 The following are the results of a standard penetration test in sand. Determine the corrected standard penetration numbers, (N1 )60 , at the various depths given. Note that the water table was not observed within a depth of 12 m below the ground surface. Assume that the average unit weight of sand is 15.5 kN/m3. Use Eq. (18.10). Depth (m) N 60 1.5 3 4.5 6 7.5 8 7 12 14 13 18.4 Using the values of N60 given in Problem 18.3 and Eq. (18.16), estimate the average soil friction angle, f. Use pa 100 kN/m2. 18.5 The following are the results of a standard penetration test in dry sand. 18.6 18.7 18.8 18.9 Depth (m) N 60 1.5 3 4.5 6 7.5 6 12 17 21 23 For the sand deposit, assume the mean grain size, D50 , to be 0.26 mm and the unit weight of sand to be 15.5 kN/m3. Estimate the variation of relative density with depth. Use Eq. (18.14). A boring log in a sandy soil is shown in Figure 18.16. Determine the corrected standard penetration numbers from Eq. (18.13). a. From the results of Problem 18.3, estimate a design value of (N1 )60 (corrected standard penetration number) for the construction of a shallow foundation. b. Refer to Eq. (16.47). For a 2-m-square column foundation in plan, what allowable load could the column carry? The bottom of the foundation is to be located at a depth of 1.0 m below the ground surface. The maximum tolerable settlement is 25 mm. For the soil proﬁle described in Problem 18.5, estimate the variation of the cone penetration resistance, qc , with depth. Use Eq. (18.30). In a layer of saturated clay, a cone penetration was conducted. At a depth of 10 m below the ground surface, qc was determined to be 1400 kN/m2. Estimate the undrained shear strength of the clay. Given: unit weight of saturated clay 18 kN/m2. References Depth (ft) 17.5 ft 653 N60 5 7 10 6 15 9 20 10 25 12 gsat = 118 lb/ft3 30 14 g = 115 lb/ft3 Sand Figure 18.16 18.10 The average cone penetration resistance at a certain depth in a sandy soil is 205 kN/m2. Estimate the modulus of elasticity of the soil at that depth. 18.11 During a ﬁeld exploration program, rock was cored for a length of 4 ft. and the length of the rock core recovered was 3 ft. Determine the discovery ratio. References AMERICAN ASSOCIATION OF STATE HIGHWAY AND TRANSPORTATION OFFICIALS (1967). Manual of Foundation Investigations, National Press Building, Washington, D.C. AMERICAN SOCIETY OF CIVIL ENGINEERS (1972). “Subsurface Investigation for Design and Construction of Foundations of Buildings, Part I,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 98, No. SM5, 481–490. AMERICAN SOCIETY FOR TESTING AND MATERIALS (1991). Annual Book of ASTM Standards, Vol. 04.08. Philadelphia, Pa. ANAGNOSTOPOULOS, A., KOUKIS, G., SABATAKAKIS, N., and TSIAMBAOS, G. (2003). “Empirical Correlation of Soil Parameters Based on Cone Penetration Tests (CPT) for Greek Soils,” Geotechnical and Geological Engineering, Vol. 21, No. 4, 377–387. CUBRINOVSKI, M., and ISHIHARA, K. (1999). “Empirical Correlation Between SPT N–Value and Relative Density for Sandy Soils,” Soils and Foundations, Vol. 39, No. 5, 61–92. DEERE, D. U. (1963). “Technical Description of Rock Cores for Engineering Purposes,” Felsmechanik und Ingenieurgeologie, Vol. 1, No. 1, 16–22. DJOENAIDI, W. J. (1985). “A Compendium of Soil Properties and Correlations.” Master’s thesis, University of Sydney, Australia. KULHAWY, F. H., and MAYNE, P. W. (1990). Manual on Estimating Soil Properties for Foundation Design, Final Report (EL-6800) submitted to Electric Power Research Institute (EPRI), Palo Alto, Calif. LIAO, S., and WHITMAN, R. V. (1986). “Overburden Correction Factor for SPT in Sand,” Journal of Geotechnical Engineering, ASCE, Vol. 112, No. 3, 373–377. 654 Chapter 18: Subsoil Exploration MENARD, L. (1965). “Rules for Calculation of Bearing Capacity and Foundation Settlement Based on Pressuremeter Tests,” Proceedings, 6th International Conference on Soil Mechanics and Foundation Engineering, Montreal, Canada, Vol. 2, 295–299. OSTERBERG, J. O. (1952). “New Piston-Type Sampler,” Engineering News Record, April 24. PECK, R. B., HANSON, W. E., and Thornburn, T. H. (1974). Foundation Engineering, 2nd ed., Wiley, New York. ROBERTSON, P. K., and CAMPANELLA, R. G. (1983). “Interpretation of Cone Penetration Tests. Part I: Sand,” Canadian Geotechnical Journal, Vol. 20, No. 4, 718–733. SCHMERTMANN, J. H. (1970). “Static Cone to Compute Static Settlement Over Sand,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. SM3, 1011–1043. SCHMERTMANN, J. H. (1975). “Measurement of in situ Shear Strength,” Proceedings, Specialty Conference on in situ Measurement of Soil Properties, ASCE, Vol. 2, 57–138. SEED, H. B., TOKIMATSU, K., HARDER, L. F., and CHUNG, R. M. (1985). “Inﬂuence of SPT Procedures in Soil Liquefaction Resistance Evaluations,” Journal of Geotechnical Engineering, ASCE, Vol. 111, No. 12, 1426–1445. SKEMPTON, A. W. (1986). “Standard Penetration Test Procedures and the Effect in Sands of Overburden Pressure, Relative Density, Particle Size, Aging and Overconsolidation,” Geotechnique, Vol. 36, No. 3, 425–447. SOWERS, G. B., and SOWERS, G. F. (1970). Introductory Soil Mechanics and Foundations, Macmillan, New York. SZECHY, K., and VARGA, L., (1978). Foundation Engineering—Soil Exploration and Spread Foundtion, Akademiai Kiado, Hungary. TROFIMENKOV, J. G. (1974). “General Reports: Eastern Europe,” Proceedings, European Symposium of Penetration Testing, Stockholm, Sweden, Vol. 2.1, 24–39. WOLFF, T. F. (1989). “Pile Capacity Prediction Using Parameter Functions,” in Predicted and Observed Axial Behavior of Piles, Results of a Pile Prediction Symposium, sponsored by Geotechnical Engineering Division, ASCE, Evanston, Ill., June 1989, ASCE Geotechnical Special Publication No. 23, 96–106. Answers to Selected Problems Chapter 2 2.1 2.3 2.5 2.7 2.9 2.11 Cu 5.13; Cc 1.48 Cu 4.33; Cc 0.73 b. D10 0.11 mm; D30 0.17 mm; D60 0.3 mm c. 2.73 d. 0.88 b. D10 0.23 mm; D30 0.33 mm; D60 0.48 mm c. 2.09 d. 0.99 Gravel 0%; Sand 6%; Silt 52%; Clay 42% 0.0052 mm Chapter 3 3.5 3.7 3.9 3.11 3.13 w 15.6% 121 lb/ft3 d 104.7 lb/ft3 e 0.59 n 0.37 S 70.6% a. 16.91 kN/m3 b. 2.44 c. 0.417 a. 101.1 lb/ft3 b. 0.648 c. 0.39 d. 44.5% a. 1423.7 kg/m3 b. 0.479 c. 53.5% d. 222 kg/m3 a. 20.59 kN/m3 b. 1.99% 655 656 Answers to Selected Problems 3.15 3.17 3.19 3.21 3.23 a. 117.4 lb/ft3 b. 96.6 lb/ft3 c. 77.7% a. 0.65 b. 126.7 lb/ft3 69.75 lb 18.8 kN/m3 a. 74.6% b. 111.1 lb/ft3 Chapter 4 4.1 4.3 4.5 a. 28.5 b. 16.3 a. 39.7 b. 21 SL 19.4% SR 1.85 Chapter 5 5.1 5.3 Soil Classification A B C D E Clay Sandy clay Loam Sandy clay and sandy clay loam (borderline) Sandy Loam Soil Symbol Group name 1 2 3 4 5 6 7 SP MH CH SC SC GM-GC CH Poorly graded sand Elastic silt with sand Fat clay Clayey sand Clayey sand Silty clayey gravel with sand Fat clay with sand Chapter 6 6.1 w (%) ␥zav (lb/ft3) 5 8 10 12 15 150.9 140.7 134.6 129.0 121.5 Answers to Selected Problems 6.3 w (%) 10 20 6.5 6.7 6.9 6.11 6.13 6.15 d @ S (%) (kg/m3) 80 90 100 2002 1601 2059 1676 2107 1741 e 0.358 S 94% 89.9% Pit B d 1626.4 kg/m3 R 96.7% 15.94 kN/m3 20.65; Rating: Fair Chapter 7 7.1 7.3 7.5 7.7 7.9 7.11 7.13 7.15 7.17 0.022 in./sec h 16.67 in. v 0.0083 in./sec 0.31 cm2 0.207 m3/min 7.18 105 m3/sec/m 0.57 ft/min 0.354 cm/sec 1.52 106 cm/sec 36.8 Chapter 8 8.1 8.3 8.5 8.7 8.9 0.009 cm/sec 0.518 m3/m/day 7.2 m3/m/day 2.1 m3/m/day 0.271 m3/m/day Chapter 9 9.1 lb/ft2 Point u ⴕ A B C D 0 560 1280 2280 0 0 374.4 873.6 0 560 905.6 1406.4 657 658 Answers to Selected Problems 9.3 kN/m2 Point u ⴕ A B C D 0 45 109 199 0 0 39.24 88.29 0 45 69.76 110.71 9.5 9.7 9.9 9.11 9.13 kN/m2 Point u A B C D 0 65 126.95 153.94 0 0 29.43 44.158 0 65 97.52 109.79 13.62 ft 0.019 m3/min 4.88 kN/m2 Depth (m) 0 5 8 11.5 0 87 140.76 205.37 u 0 0 19.13 0 34.34 Chapter 10 1 129.24 kN/m2 3 30.76 kN/m2 n 51.03 kN/m2 n 39.82 kN/m2 10.3 1 161.1 kN/m2 3 68.9 kN/m2 n 138.5 kN/m2 n 39.7 kN/m2 10.5 1 95 kN/m2 3 30 kN/m2 n 94.2 kN/m2 n 7.1 kN/m2 10.7 20.15 lb/ft2 10.9 15.32 kN/m2 10.11 674 kN/m 10.1 ⴕ ⴕ 0 87 106.13 140.76 171.03 Answers to Selected Problems 10.13 24.54 kN/m2 10.15 @ A: 4492.8 lb/ft2 @ B: 4320 lb/ft2 @ C: 384 lb/ft2 10.17 r (ft) ⌬z (lb/ft2) 0 2 4 8 12 1821 1803 1739 1293 414 10.19 a. b. c. 20 kN/m2 46.72 kN/m2 6.59 kN/m2 Chapter 11 11.1 11.3 11.5 11.7 11.9 11.11 11.13 11.15 11.17 11.19 0.393 in. b. 3.1 ton/ft2 c. 0.53 b. 940 lb/ft2 c. 0.133 229 mm 0.596 159.6 days a. 0.00034 m2/kN b. 6.67 108 cm/sec a. 240.8 days b. 7.33 in. 66.7 days a. 31.25% b. 102.6 days c. 25.65 days Chapter 12 12.1 a. 41.2° b. 0.739 kN 12.3 37.5° 12.5 20 lb/in.2 12.7 68.7 lb/in.2 12.9 127.7 kN/m2 12.11 a. 20.8° b. 55.4° c. ' 338.7 kN/m2 128.7 kN/m2 659 660 Answers to Selected Problems 12.13 72 lb/in.2 12.15 15.6° 24.4° 12.17 a. 18° b. 64.9 kN/m2 12.19 d(f) 17.97 lb/in.2 ud (f) 7.01 lb/in.2 12.21 93.5 kN/m2 12.23 16.24 kN/m2 Chapter 13 13.1 13.3 13.5 13.7 13.9 13.11 13.13 13.15 13.17 13.19 13.21 13.23 13.25 Po 281.55 kN/m; –z 2.33 m Po 205.7 kN/m; –z 2 m Pa 3933.6 lb/ft; –z 5 ft Pa 37.44 kN/m; –z 1.33 m Pp 12,450 lb/ft; –z 2.67 ft Pp 645.8 kN/m; –z 1.67 m Pa 2340.8 lb/ft; –z 2.91 ft Pa 141.1 kN/m; –z 2.04 m 402.6 kN/m. Pp acts at 1.33 m from the bottom of the wall inclined at an angle 6.64°. b. 5.74 ft c. 3225 lb/ft d. Pa 5233 lb/ft; –z 3.09 ft 10.02 kN/m Part 1: 85.14 kN/m Part 2: 84.80 kN/m 68.3 kN/m Chapter 14 14.1 14.3 14.5 14.7 1533 kN/m 59,063 lb/ft 710.2 kN/m 267.2 kN/m Chapter 15 15.1 15.3 15.5 15.7 15.9 15.11 15.13 1.63 m 0.98 103.1 ft 1.76 9.19 m; toe failure 3.86 m a. 27.5 kN/m2 b. Mid-point circle c. 5.95 m Answers to Selected Problems 15.15 15.17 15.19 15.21 15.23 0.57 1.5 1.75 0.75 1.09 Chapter 16 16.1 16.3 16.5 16.7 16.9 16.11 16.13 7372 lb/ft2 71 kN/m2 138.7 kN/m2 153 kip 4.1 ft 208.8 kip 207.9 kip Chapter 18 18.1 18.3 18.5 7.54% Depth (m) (N1)60 1.5 3.0 4.5 6.0 7.5 16 10 14 14 12 Depth (m) Dr (%) 1.5 3.0 4.5 6.0 7.5 87 87 84 81 76 18.7 211.8 kN/m2 18.9 66.7 kN/m2 18.11 75% 661 Index A AASHTO classification, 98–100 Absolute permeability, 165 Active pressure: braced cut, 503–506 Coulomb, 457–459 Rankine, 432–434 Active thrust, braced cut, 503–506 Activity, 84–87 Adhesion, 376 Adsorbed water, 33 Aeolian soil, 22 A-line, 87 Alluvial soil, 22 Alumina octahedron, 26–27 Angle of friction: consolidated-undrained, 391 definition of, 365 drained, 366 drained, clay, 374 foundation material and soil, 376–378 residual, clay, 374–375 typical values for, 366 Angularity, 47 Anisotropic soil, flow net, 209–210 Anisotropy, clay, 403–405 Anisotropy ratio, 181 A parameter, 392 Area ratio, 639 Atterberg limit, 73 662 Auger, 631–632 Average degree of consolidation, 333–334 B Bearing capacity, shallow foundation: based on settlement, 602–604 depth factor, 592 eccentric load, 597–600 effective area, 598 effective width, 598 effect of ground water table, 584–585 factor of safety, 586–687 factors, general, 590–591 factors, Terzaghi, 582, 584 general equation, 592 gross allowable, 586 inclination factor, 592 net allowable, 586 shape factor, 592 Terzaghi’s equation, 581–582 Blasting, compaction, 154 Boiling, 232 Bottom ash, compaction, 146 Boussinesq’s solution, 260–261 Bowen’s reaction principle, 15–16 B parameter: definition of, 381 typical values for, 382 Braced cut: active thrust, cohesive soil, 504, 506 active thrust, granular soil, 503–505 general, 499–502 Brownian motion, 91 Brucite sheet, 28 C Calding’s equation, 407 Capillary rise, 243–245 Chemical sedimentary rock, 22 Chemical weathering, 19 Chopping bit, 634 Classification, particle size, 24–26 Clay, 26 Clay liner: compaction, 612–616 double, 623–624 single, 622 single, geomembrane, 622 Clay mica, 28 Clay mineral, 26–34 Cluster, structure, 92 Coefficient: active pressure with earthquake, 469 compressibility, 332 consolidation, 332 Coulomb’s active pressure, 458 Index Coulomb’s passive pressure, 467 earth pressure at rest, 427–428 gradation, 42 Rankine active pressure, 432–434 Rankine passive pressure, 434–436 sorting, 43 volume compressibility, 332 Cohesion, definition of, 365 Colluvial soil, 22 Compaction: blasting, 154 bottom ash, 146 compaction effort, 120–122 copper slag, 146 dynamic, 151–153 effect of soil type, 118–120 effect on hydraulic conductivity, 129 general principles, 114–115 modified Proctor test, 122–124 maximum dry unit weight, 115 optimum moisture content, 115 organic soil, 144–145 relative, 136 soil-organic material mixture, 145–146 specifications for, 136–137 standard Proctor test, 115–118 zero-air-void unit weight, 118 Compression index, 319–320 Cone penetration test, 646–648 Consistency, 72 Consolidated-drained triaxial test, 381–385 Consolidated-undrained triaxial test, 389–393 Consolidation: coefficient of, 332 degree of, 333 effect of sample disturbance, 316–317 laboratory test, 308–312 logarithm-of-time method, 339–340 overconsolidation ratio, 315 preconsolidation pressure, 314–315 secondary, 326–328 settlement calculation, 317–319 spring-cylinder model, 305 square-root-of-time method, 340 time rate of, 330–335 void ratio-pressure plot, 310–312 Constant head test, 166–167 Contact pressure, 294–296 Continuity equation, Laplace, 198–200 Continuous plagioclase reaction series, 17 Copper slag, compaction, 146, 147 Coulomb’s earth pressure: active, 457–459 graphical solution, 461, 463–465 passive, 466–468 Criteria filter, 219–222 Critical hydraulic gradient, 232 Cross-plane hydraulic conductivity, 617–618 Culmann’s solution, 461, 463–465 D Darcy’s law, 162 Degree of consolidation, 333 Degree of saturation, 52 Density: definition of, 53 relative, 64–66 Depth of tensile crack, 446 Detrital sedimentary rock, 22 Diffuse double layer, 31 Dipole, 31 Direct shear test, 369–375 Discharge velocity, 162 663 Discontinuous ferromagnesian reaction series, 17 Dispersing agent, 39 Disturbance, effect on consolidation, 316–317 Domain, structure, 92 Double layer water, 33 Drained angle of friction, 366 Drilled shaft foundation, 576 Dry density, 53 Dry unit weight, 53 Dynamic compaction, 151–152 Dynamic earth pressure: c′−φ′ soil, 474–477 E Earth dam, seepage, 214–216 Earth pressure at rest: coefficient of, 427–428 normally consolidated clay, 428 overconsolidated clay, 428 Effective size, 42 Effective stress: definition of, 226 Elastic settlement, 296–303 Elasticity modulus, 302 Elevation head, 160 Empirical relations, hydraulic conductivity, 172–174, 177–179 Equipotential line, 204 Equivalent hydraulic conductivity, 182–184 Evaporite, 23 F Factor of safety, slope: clay embankment, 568–571 cohesion, 515 friction, 515 strength, 514–515 Failure criteria, MohrCoulomb, 365 Falling head test, 167–168 664 Index Field unit weight: nuclear method, 141 rubber balloon method, 141 sand cone method, 140–141 Field vanes, 408–409 Filter: criteria, 219–222 definition of, 219 Finite slope, 519–520 Fissure eruption, 15 Flight auger, 631–632 Flocculation, 91 Flow channel, 205 Flow index, 75 Flow line, 204 Flow net, 204–205 Foundation material, friction angle, 376–378 Friction circle, 537 G Gap-graded soil, 43 Geomembrane, 619–621 Geonet, 621 Geotextile, 616–619 Gibbsite sheet, 27 Glacial soil, 22 Gradation. coefficient of, 42 Gravel, 24 Group index, classification, 99 H Hazen’s equation, 172 Hazen’s formula, capillary rise, 243–245 Head: elevation, 160 pressure, 160 velocity, 106 Heaving, factor of safety, 240–241 Honeycombed structure, 90 Hydraulic conductivity: bonding between lifts, effect of, 614 definition of, 162 directional variation of, 180–181 effect of temperature, 165 empirical relations for, 172–175, 177–179 equivalent, 182–184 size of clay clods, 612 typical values, 165 Hydraulic gradient, 161 Hydrogen bonding, 31 Hydrometer analysis, 37–42 I Igneous rock, 15–17 Illite, 28 Immediate settlement, 294–303 Index: compression, 319–320 consistency, 84 liquidity, 83–84 plasticity, 79 swell, 320–321 Influence chart, 285–298 Influence value, stress, 297 Isomorphous substitution, 29 K Kaolinite, 28 Kozeny-Carman equation, 172 L Laboratory test, consolidation, 308–312 Lacustrine soil, 22 Laminar flow, 161 Laplace’s equation of continuity, 198–200 Leachate removal system, 624–626 Line load, stress, 262–265 Liquidity index, 83 Liquid limit: definition, 73 one point method, 77 typical values for, 80 Logarithmic spiral, 490–492 Logarithm-of-time method, consolidation, 339–340 M Magma, 15 Major principal stress, 255 Marine soil, 22 Mat foundation, 576 Maximum dry unit weight, compaction, 115 Mechanical weathering, 19 Metamorphic rock, 23–24 Mid-point circle, 523 Minor principal stress, 255 Modified Proctor test, 122–124 Mohr-Coulomb failure criteria, 365 Mohr’s circle, 255–256 Moist unit weight, 53 Moisture content, 52–53 Mononobe-Okabe solution: active pressure coefficient, 469 equation for, 469 line of action, active force, 473–474 Montmorillonite, 29 N Needle punched nonwoven geotextile, 617 Neutral stress, 228 Nonwoven geotextile, 617 Normally consolidated clay, 314 Normal stress. plane, 254 Nuclear method, compaction, 141 O Octahedral sheet, 27 Oedometer, 308 One point method, liquid limit, 77 Optimum moisture content, 115 Ordinary method of slices, slope, 544–547 Organic soil, compaction, 144–145 Overconsolidated clay, 314 Overconsolidation ratio: definition of, 315 variation of Af, 392–393 P Partially saturated soil, effective stress, 245–246 Particle shape, 46–47 Index Particle size distribution curve, 37 Passive pressure: Coulomb, 466–468 curved failure surface, 492–499 Rankine, 434–436 wall friction, 488–490 Peak shear strength, 371 Peat, 110 Peds, 92 Percent finer, 37 Percussion drilling, 634 Permeability test: constant head, 166–167 falling head, 167–168 Permeability test pumping from wells, 187–189 Permittivity, 618 Piezometer, 161 Pile, 576 Piston sampler, 638 Plane, principal, 255 Plasticity chart, 86–88 Plasticity index, 79 Plastic limit, 78 Plate load test, 604–606 Pluton, 15 Pneumatic roller, 132 Pocket penetrometer, 411 Point load, stress, 260–261 Poisson’s ratio, 303 Poorly graded soil, 43 Pore pressure parameter: A, 389, 393 B, 381–382 Pore water pressure: definition of, 228 in zone of capillary rise, 246 Porosity, 52 Potential drop, 206 Preconsolidation pressure: definition of, 314 graphical construction for, 314 Pressure head, 160 Pressuremeter test, 644–645 Primary leachate collection system, 624–625 Principal plane, 255 Principal stress, 255 Q Quartzite, 24 Quick condition, 232 R Rankine active state, 433 Rankine theory: active pressure, 432–434 coefficient of active pressure, 434 coefficient of passive pressure, 435 depth of tensile crack, 446 passive pressure, 434 – 436 Reaction principle, Bowen, 15 Rectangular loaded area, stress, 278–283 Relative compaction, 136 Relative density, 64–66 Residual friction angle, clay, 374, 375 Residual soil, 22 Retaining wall: cantilever, 480 counterfort, 480–481 gravity, 480 mechanically stabilized earth, 481–482 Rock coring, 648–650 Rock cycle, 18 Rock quality designation, 649–650 Roller: pneumatic, 132 sheepsfoot, 134 smooth-wheel, 132 vibratory, 134 Rotary drilling, 634 Rubber balloon method, field unit test, 141 S Sand, 24 Sand cone method, 140–141 Saturation, degree of, 52 Secondary compression index, 326 665 Secondary consolidation, 326–328 Sedimentary rock, 22–23 Seepage: force, 235–236 through earth dam, 214–219 velocity, 163 Sensitivity, 401 Settlement calculation, consolidation, 317–319 Shallow foundation: general shear failure, 578 local shear failure, 579 Shape, particle, 46–47 Shear stress, plane, 254 Sheepsfoot roller, 134 Shelby tube, 638 Shrinkage limit, 81–84 Shrinkage ratio, 82 Sieve analysis, 35–37 Sieve size, 35 Silica tetrahedron, 26–27 Silt, 24 Single-grained structure, 89 Slip plane, 438 Slope stability: base failure, 523 Bishop’s simplified method, 548–550 Culmann’s method, 520–522 friction circle, 537 infinite slope, without seepage, 515–517 infinite slope, with seepage, 518 Michalowski’s solution, 538, 541, 557, 559 ordinary method of slices, 544–547 rapid drawdown, 565–567 slope failure, 512–514 Spencer’s solution, 557–558 stability number, slope, 526 Smooth-wheel roller, 132 Sorting coefficient, 43 Specific gravity: definition, 34 typical values for, 35 666 Index Specific surface, 28 Spiral, logarithmic, 490–492 Spring-cylinder model, consolidation, 304–306 Square-root-of-time method, 340 Standard penetration number, 636–637 Standard Proctor test, 115–118 Standard split spoon, 635 Stoke’s law, 37 Stress: influence chart for, 285–287 line load, 262–265 Mohr’s circle for, 255–256 path, 414–417 point load, 260–261 principal, 255 rectangularly loaded area, 278–283 strip load, 266–267 uniformly loaded circular area, 273–275 Structure, compacted clay, 127–128 Surface tension, 243 Swell index, 320–321 T Tensile crack, 446 Textural classification, 95–96 Thixotropy, definition of, 402 Time factor, 333 Time rate of consolidation, 330–335 Torvane, 411 Total stress, 226 Transmissivity, 618 Triaxial test: consolidated-drained, 381–385 consolidated-undrained, 389–393 general, 380–381 unconsolidated-undrained, 395–397 Turbulent flow, 161 U U-line, 88 Ultimate strength, 371 Unconfined compression strength, 397 Unconfined compression test, 397–398 Unconsolidated-undrained test, 395–397 Undrained shear strength: definition of, 395 empirical relations for, 398–399 Unified classification system, 102–104 Uniformity coefficient, 42 Uniformly loaded circular area, stress, 273–277 Unit weight: definition of, 53 dry, 53 moist, 53 relationship for, 59 Uplift pressure, 213–214 V Vane shear test: correlation for, 409, 411 procedure for, 406–409 Varved soil, 184 Velocity: discharge, 162 head, 160 seepage, 163 Vibratory roller, 134 Vibroflot, 148, 150 Vibroflotation, 148–151 Virgin compression curve, 316 Void ratio, 51 Void ratio-pressure plot, 310-312 Volcanic eruption, 15 Volume compressibility, coefficient of, 332 W Wall friction, passive pressure, 488-490 Wall yielding, earth pressure, 436-438 Wash boring, 634 Weathering, 19-21 Well graded, 43 Woven geotextile, 616 Z Zero-air-void unit weight, 118

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