/smash/get/diva2:11234/FULLTEXT01.pdf

/smash/get/diva2:11234/FULLTEXT01.pdf
Design of Concrete Pavements –
Design Criteria for Plain and Lean Concrete
Johan Söderqvist
TRITA-BKN. Bulletin 87, 2006
ISSN 1103-4270
ISRN KTH/BKN/B--87--SE
Licentiate Thesis
Preface
This licentiate thesis was carried out at the Swedish Cement and Concrete Research Institute
(CBI) and at the school of Architecture and Built Environment at the Royal Institute of
Technology (KTH), at the Division of Structural Design and Bridges.
The project is a part of the Swedish Agency for Innovation Systems’ (VINNOVA)
infrastructure programme Road-Bridge-Tunnel and was financed by Cementa AB,
VINNOVA, and KTH.
First, I thank my supervisors Prof. Johan Silfwerbrand at CBI and Dr. Erik Simonsen at
Cementa AB for all support during this thesis. I especially thank Johan for his guidance and
encouragement.
Many sincere thanks to Dr. Ali Farhang at CBI for his support and enthusiasm in this project.
Thanks to Stig Jansson at Cementa AB for motivating me in my work and always caring for
my more practical involvements in the field.
Thanks to Christer Hagert at the Swedish Road Administration (SRA) and Bengt-Åke
Hultqvist at the Swedish National Road and Transportation Institute (VTI) for good
cooperation, help, and valuable information in this project.
Thanks to Göran Olsson, Lars Melin, and Carsten Vogt at CBI for always giving me a helping
hand with logistics and material testing.
A special thank goes to my colleagues at CBI.
Stockholm, November 2006
Johan Söderqvist
I
II
Abstract
New road infrastructure projects are important and constitute of large investments that have to
serve the society for a long time. The investments have to be durable at the lowest life cycle
cost and the pavements have to sustain loads from increasing traffic intensity and heavy
traffic loads. In Sweden less than 1 ‰ of the national road network consists of concrete
pavements. In parts of Europe and in the U.S., on the other hand, concrete roads are used to a
large extent for highways as well as rural roads. To encourage the competition between
different road materials in Sweden, the tools for designing robust concrete pavements have to
be brought forward. In order to emphasize plain concrete pavements as an alternative in road
construction, the design must also be competitive.
The current Swedish design method for concrete pavements is straightforward but offers no
flexibility when designing roads with, for instance higher traffic loads. The method calculates
concrete thicknesses on the conservative side since only a limited number of parameters are
treated. Modern methods that take into account many more parameters in the design are being
developed internationally. For a new Swedish design method, these parameters have to be
established for actual conditions in Sweden. Also, the design has to be flexible and meet the
demands from contractors and clients for a wider use.
The aim of this project is to develop a new design method for plain concrete pavements that is
more flexible than today. A new design method is also intended for the Swedish Road
Administrations’ (SRA) computer based public design guide, PMS Objekt.
Information for a new design method has been assembled mainly by investigating two newly
developed design methods, VENCON2.0 in the Netherlands, and the Mechanistic-Empirical
Pavement Design Guide (MEPDG) in the USA. Comparative calculations between the
Swedish design method and the MEPDG are presented. The comparison is made on the level
of input parameters and highlights the advantageous aspects of a semi-mechanistic design
procedure where the functional properties of a concrete pavement are calculated incrementally
over the design period.
Plain and lean concrete, separately, but also the in composite beams, have been studied in
flexural fatigue testing. The results show that Tepfers’ fatigue criterion is valid for both plain
and lean concrete when subjected to flexural fatigue loading. The results also show that the
fatigue strength of composite beams of plain and lean concrete is mainly dependent on the
strength of the lean concrete but that Tepfers’ fatigue criterion is applicable. The bond
between plain in lean concrete is found to be strong and fatigue resistant, making the
composite section able to accommodate higher stresses. The bond nevertheless contributes to
the risk for reflection cracks in the plain concrete wear layer and a recommendation to focus
on stresses in the bottom of the lean concrete is formulated. Also, well distributed expansion
joints in the lean concrete are necessary.
A new project for measuring temperature gradients for use in concrete pavement design is
also presented. This is done with means of concrete prisms placed in the pavement and are
done in order to establish actual temperature gradients for various locations in Sweden. Also,
the nonlinear gradients that act in the pavement as well as the negative temperature gradients
will be analysed for the use in the design.
III
Finally, the thesis outlines a new design method for Swedish conditions. The method is
possible to develop gradually and is based on FE-analysis for fast computations. In the design,
stresses from traffic and temperature loads are calculated simultaneously in a number of
critical locations in the concrete slab. The method will also make it possible to alter design
features as slab lengths and widths, with various connections between the slabs.
IV
Sammanfattning
Långsiktiga infrastruktursatsningar inom vägbyggnad ställer höga krav på dimensionering och
kostnaderna för underhåll och reparationer måste inkluderas redan i planeringsskedet. I
Sverige utgör betongvägar mindre än 1 ‰ av det nationella vägnätet. I Europa och USA är
andelen betydligt högre, speciellt för motorvägar men även för mindre landsvägar. En
betongväg är robust och tål höga laster. Dagens svenska dimensioneringsmetod är dock
mycket förenklad och därmed inte tillräckligt flexibel. Detta innebär i sin tur att betongvägar
inte kan framställas på mest fördelaktiga sätt. En ny dimensioneringsmetod som beaktar en
rad nya parametrar i dimensioneringen är viktig för att kunna ta fram betongvägar som ett
konkurrenskraftigt alternativ för vägbyggnad i Sverige.
Detta forskningsprojekt har behandlat de första stegen i arbetet för utvecklingen av en ny
dimensioneringsmetod för betongvägar i Sverige. Den nya dimensioneringsmetoden ska vara
mer flexibel för att kunna användas för dimensionering av olika former av betongvägar.
Metoden ska också kunna implementeras i Vägverkets dimensioneringsprogram, PMS Objekt,
som utgör det program som används för statliga vägar.
Inom ramen för detta projekt har en undersökning avseende olika internationella
dimensioneringsmetoden för oarmerade betongvägar gjorts. I studien undersöktes
VENCON2.0 från Holland och Mechanistic-Empirical Pavement Design Guide (MEPDG)
från USA. Jämförande beäkningar mellan MEPDG och den svenska dimensioneringsmetoden
presenteras där de funktionella parametrarna i den amerikanska dimensioneringen har belysts
mer ingående.
I dimensioneringsmetoden för vägar ingår utmattningskriterier för olika lager som en viktig
del. Dagens kriterier för betongvägar behöver ses över och av den anledningen har
utmattningsprovning av balkar av oarmerad betong och cementbundet grus (CG) utförts. Det
har kunnat fastställas att Tepfers utmattningskriterium verkligen är tillämpbart för
böjdraghållfasthet. För CG har provning visat att ett spänningskriterium kan användas med
något reducerad utmattningshållfasthet i jämförelse med Tepfers kriterium. För
samverkansbalkar av betong och CG har det även fastställts att vidhäftningen är mycket god
och att Tepfers utmattningskriterium är applicerbart även här. Den goda vidhäftning ger dock
upphov till en risk för reflektionssprickor men den ökade bärförmågan bör ändå kunna
utnyttjas om fokus läggs på spänningsnivån i underkant CG istället för underkant betong. Täta
expansionsfogar i CG-lagret är också nödvändiga.
En annan viktig del av betongvägsdimensioneringen omfattar realistiska temperaturgradienter
för olika delar av landet och därför presenteras en ny metod för att mäta temperaturgradienter
i betong. Metoden innefattar mätningar av temperatur i betongkuber med ingjuten
mätutrustning som placerats ut invid flera vägar i Sverige. Tillsammans med tidigare utförda
mätningar kommer nya data att sammanställas för att användas i en ny dimensioneringsmetod.
Mätningarna kommer också att analyseras med anseende på negativa gradienter för att kunna
beakta laster som påverkar betongens överkant, laster som orsakar sprickor som genereras
ovanifrån s.k. top-down cracking.
Ett förslag till en ny dimensioneringsmetod för oarmerade betongvägar presenteras. Denna
metod är möjlig att utveckla successivt och behandlar både trafik- och temperaturlaster
V
samtidigt i ett FE-program. Metoden möjliggör en flexibel dimensionering där olika specifika
egenskaper ska kunna varieras beroende på förutsättningar och krav som definieras under
dimensioneringsperioden.
VI
Contents
PREFACE ..................................................................................................................................I
ABSTRACT ........................................................................................................................... III
SAMMANFATTNING ........................................................................................................... V
1.
INTRODUCTION............................................................................................................ 1
1.1
General ........................................................................................................................ 1
1.2
Background ................................................................................................................. 2
1.3
Types of Concrete Pavements ..................................................................................... 2
1.4
General Design Procedure........................................................................................... 4
1.5
Aim and Scope ............................................................................................................ 4
1.6
Outline of Thesis ......................................................................................................... 5
2.
NATIONAL AND INTERNATIONAL DESIGN METHODS.................................... 7
2.1
General ........................................................................................................................ 7
2.2
Current Design Method in Sweden ............................................................................. 8
2.2.1 Traffic Loads...................................................................................................... 8
2.2.2 Temperature Loads............................................................................................. 8
2.2.3 Fatigue and Damage Accumulation ................................................................... 9
2.3
Design Method in the USA ....................................................................................... 10
2.4
Design Method in the Netherlands............................................................................ 10
2.5
Comparison of the Design Methods.......................................................................... 13
3.
EXPERIMENTAL STUDIES ON FATIGUE............................................................. 17
3.1
General ...................................................................................................................... 17
3.2
Methodology ............................................................................................................. 18
3.3
Results ....................................................................................................................... 19
4.
TEMPERATURE MEASUREMENTS........................................................................ 21
4.1
General ...................................................................................................................... 21
4.2
Methodology ............................................................................................................. 21
VII
4.3
5.
Results ....................................................................................................................... 23
NEW DESIGN METHOD FOR SWEDISH CONDITIONS..................................... 25
5.1
General ...................................................................................................................... 25
5.2
New Method for Calculating Loads .......................................................................... 25
5.3
Fatigue Criteria and Damage Accumulation............................................................. 27
5.4
Implementation of a New Swedish Design Method ................................................. 28
6.
CONCLUSIONS............................................................................................................. 31
6.1
Concrete Pavement Design ....................................................................................... 31
6.2
Flexural Fatigue Criteria ........................................................................................... 31
7.
FURTHER RESEARCH ............................................................................................... 33
8.
REFERENCES ............................................................................................................... 35
APPENDED PAPERS (Nos. 1- 4)
VIII
Chapter 1
Introduction
1.1 General
Concrete pavement design has over the years become a more important part for the promoting
of concrete roads. A high investing cost has to be motivated, and the benefits of a pavement
with less maintenance over a longer design life have to be proved already before construction.
Efforts to avoid premature performance failing of concrete roads are, at a larger degree,
considered than for other pavement alternatives since rehabilitation techniques are expensive.
A modern design methodology has to take into account all sorts of environmental conditions
as well as future estimations on, for example traffic growth or environmental changes. The
optimisation of materials in the pavement system, demands for long-term fatigue resistance at
the lowest cost and ecologically sound choices must be considered. The understanding of the
behaviour of a concrete road is vital for the design and the performance prediction.
The current Swedish design method was developed in the 1990’s in a trial to modernise the
concrete pavement design, and create a chance for competition between road paving
alternatives in the country, Petersson (1996). The design methodology is used for highway
pavements, industrial pavements, and public pavements, accessible in Petersson (1996), SRA
(2005), Silfwerbrand (1995), AB Byggtjänst (2002), and Farhang (2004a). It is a simple
method in the way both traffic and temperature stresses are dealt with and provides a
conservative design with just a limited number of calculation steps It also simplifies both the
material and loading inputs, making it restrained for possible optimisation. Every
simplification also comes with a safety margin that can be quantified.
The development of a new design procedure consists of the quantifications of different
unknown aspects that are important for the performance of a concrete pavement. Advances in
computer modelling are tempting for a number of applications, and offer the possibility to
include many more, even new parameters in the design. Design methods with modern
computer power can be done in a whole new way and the main advantage is that even though
many more aspects are being considered, the calculations can be made very fast.
The Ph.D.-project “Design of Concrete Pavements” was started in 2004 at the Royal Institute
of Technology and the Swedish Cement & Concrete Research Institute. The project aims at
modernising the design of concrete pavements by assembling knowledge in Sweden and
abroad, to create the basis for a new computerised design method that can be integrated in the
Swedish Road Administration’s (SRA) public design tool, PMS Objekt (SRA, 2005). A new,
computerised design method has to be calibrated for Swedish conditions, where different
design aspects have to be investigated more profoundly. The design criteria for plain concrete,
actual temperatures in the field, and real traffic data are some of the main issues that have to
1
be examined. In this thesis, a proposal for a new design method is presented. The thesis also
presents three investigations on fatigue criteria as well as the description of an on-going
project dealing with field measurements on temperature gradients in concrete
1.2 Background
Concrete roads were probably first constructed in the USA in the beginning of the twentieth
century, and spread to Europe in the twenties, Williams (1986). In Sweden three main
concrete pavement epochs can be distinguished; in the twenties, the fifties, and the seventies
with the construction of plain, unjointed concrete pavements. These roads were constructed in
the south of Sweden, and apart from the highway E6 in Vellinge from 1978, only parts of
evidence of these roads remains. In the nineties, on the other hand, a new effort to develop
concrete pavements as an alternative to asphalt roads took place. A new design method was
developed by Ö. Petersson, Petersson (1996), and the construction of four new highways was
undertaken during a ten year period; a short part of a highway connecting E4 and the
international airport at Arlanda in 1990, E6 Heberg – Långås at Falkenberg in 1993, E6
Fastarp – Heberg at Falkenberg in 1996, and E20 Eskilstuna – Arphus in 1999. The latest
contribution is E4 Uppsala – Mehedeby, completed in 2006. Today, the total length of
concrete roads in Sweden is 87 km. The concrete roads constitute less than 1 ‰, and are
extremely underrepresented in comparison to the USA and several countries in Europe. For
example, Germany has 28 % of its road network consisting of concrete pavements, that is
almost as much as the total length of the primary road network in Sweden, European
Commission (1999).
1.3 Types of Concrete Pavements
There are in principle three different kinds of concrete road designs; plain jointed concrete
pavements (PJCP), continuously reinforced concrete pavements (CRCP), and jointed
reinforced concrete pavements (JRCP), see Figure 1.1. The concrete roads in Sweden are
PJCP’s and the typical Swedish pavement system consists of a wear layer of concrete, a
bound base of asphalt (100 mm) or lean concrete (150 mm), two subbase courses of
compacted unbound materials, SRA (1994, 2005), see Figure 1.2. The subgrade consists of
the virgin material (varying from rock to clay and silt), or manmade materials, e.g. fillings.
2
Figure 1.1. Different concrete pavement systems, Löfsjögård (2003).
Figure 1.2. Pavement system of a Swedish PJCP, in mm, from SRA (2003).
Four types of joints exist for concrete pavements. Transversal joints are cut at approximately
5 m distances to reduce the tensile stresses that arise from temperature expansion. These joints
are dowelled to guarantee a good load transfer between adjacent slabs. Longitudinal joints are
cut to limit the slab width for roads with more than one lane. These joints are not equally
loaded by traffic and therefore only connecting bars are placed here. Expansion or isolation
joints are used between the concrete pavement and bridges or flexible pavements.
Construction joints are used between concreting pauses, Löfsjögård (2003).
3
Figure 1.3. Slip-form paver at the construction of the PJCP at E4 Uppsala – Mehedeby in
June 2006.
A concrete road is constructed with a slip-form paver which is a set of machines that places
the concrete in one or two layers continuously. The concrete is vibrated, the dowels are put
into place, and the surface is treated by the machines automatically. In the construction of E4
Uppsala – Mehedeby 2006, the slip-form paver had an average production speed of 1 m/min,
i.e. 110 m3/h, see Figure 1.3.
1.4 General Design Procedure
The general design procedure for concrete pavements consists of the calculation of the
number of load repetitions that a pavement can resist before failure. The methodology is
iterative where a predefined pavement structure is calculated and evaluated compared to the
loading conditions applied. If the pavement system is found not to meet the loading condition,
the calculations are done all over again with increasing thicknesses or different choices of
materials in the included layers.
1.5 Aim and Scope
Future highway projects in different regions in Sweden will have to be constructed to deal
with the increasing traffic intensities from transportation. Safe, environmental friendly and
effective roads have to be built in such a way that the society at large can take advantage of
these big investments in infrastructure. Providing the tools for increased competition within
the road construction market has for long been considered essential for maintaining a good
level of competence and keeping the costs low.
4
The current design method in Sweden is not flexible and is also too simple when optimising
the design. In contrast to modern design methods it cannot take into account a more varied
climate, higher traffic loads or improved materials. The safety margins are not fully
investigated and the method is badly adopted for a computerised tool. A modern design
method ought to serve as a tool for designer and contractors to predict the future performance
of a road to be constructed.
The primary aim of this thesis is to present an overview of modern design methods that have
been developed over the last 10 – 15 years. The thesis presents the first step towards a new
design method for concrete roads in Sweden by proposing a basic model that can be enhanced
in the future. In the development of a new design procedure, many aspects of the design have
to be examined and the thesis also presents three investigations on the fatigue of plain and
lean concrete as well as measurements of temperature gradients in the field. In Sweden,
CRCP or JRCP are less prioritised for highway construction at this time and therefore this
thesis only deals with PJCP design.
1.6 Outline of Thesis
In Chapter 1, an introduction and the background of plain concrete pavement design in
Sweden is presented.
In Chapter 2, an overview on different international methods is presented and compared to the
current Swedish design.
In Chapter 3 and 4, experimental studies on fatigue are explained and measurements on
temperature gradients in Sweden are presented.
Chapter 5 provides a new design concept for possible implementation in Sweden.
Conclusions and further work are presented in Chapters 6 and 7.
This thesis includes the following appended papers, which will be referred to with their
numbers in the text.
Paper 1, “Design of Concrete Pavements – A Comparison between Swedish and U.S. Design
Methods” written by Johan Söderqvist and Johan Silfwerbrand published at the 8th
International Conference on Concrete Pavements, Colorado Springs, Colorado, August 15 –
18, 2005, pp. 1 – 17. Paper 1 describes the Swedish design method in comparison to the new
computer based Mechanistic-Empirical Pavement Design Guide in the USA. A comparison in
methodology is explained and shown by an example.
Paper 2, “Flexural Fatigue of Plain Concrete Beams” by Johan Söderqvist and Johan
Silfwerbrand, is submitted to the International Journal of Pavement Engineering. The paper is
based on flexural fatigue tests made on plain concrete beams at the Swedish Cement and
Concrete Research Institute. The objective of the paper is to investigate the fatigue criterion
that is used in Sweden and verify if the criterion is valid for tensile stresses undergoing
fatigue loading.
5
Paper 3, “Design Criteria for Lean Concrete “, by Johan Söderqvist and Johan Silfwerbrand,
was presented at the 6th International DUT-workshop on Fundamental Modelling of Design
and Performance of Concrete Pavements, Old-Tournhout, Belgium, September 15 – 16, 2006.
The paper comprises an overview of design criteria for lean concrete used as a bound base
layer under the pavement. A literature survey is presented as well as results from fatigue
testing of beams.
Paper 4, “Flexural Fatigue of Composite Beams of Plain and Lean Concrete”, by Johan
Söderqvist and Johan Silfwerbrand, submitted to the International Journal of Road Materials
and Pavement Design. The objective of Paper 4 is to analyse how the crack development and
the bond in composite beams of plain and lean concrete is influenced when subjected to
fatigue loading.
Johan Söderqvist has drawn up the proposals for methodology, independently performed the
trials, worked out the analysis and conclusions and written the papers. The co-author has
contributed with the choice of subject and his view on methodology, analysis, conclusions,
and text.
6
Chapter 2
National and International Design Methods
2.1 General
The design of concrete pavement consists of the calculations of the number of load
applications that a specific pavement system can sustain before failure, taking into account the
changes in climate, traffic, and material conditions, and summarising these effects during a set
design period. In the design, the prediction of failure relies on data from field measurements,
mechanistic, empirical, or statistical analysis. Material deterioration can be described directly
by analysing the result of instant or fatigue loading, but multiple loading combinations in
varied conditions have, to this day, been simplified in empirical methods. Nowadays,
mechanistic design has, with proper assumptions and fundamental material knowledge,
proved to be a reliable method for modelling the performance of a concrete pavement. Semimechanistic design methods, with material models that are calibrated using field data, are still
the most successful methods because they relate computer models to actual performance in
reality. These methods are possible to further develop thanks to the monitoring of roads in the
field, roads that have been in service for decades. These methods combine the safety levels
that come from an empirical approach together with powerful computational capabilities that
can be used to explain the various phenomena that affect the pavement system during years of
service.
In many countries, years of research have been put into the development of new design
methods for concrete pavements. Computer based design methods with a high level of
sophistication are introduced for highway agencies and designers with the aim of facilitating,
not only for use in the designing of more durable pavements, but also by constituting as an
economical validation tool for the optimisation of materials and calculating the effectiveness
of a certain type of construction in relation to another. Many methods are based on the
experience from road projects that have been in service for some time, making the new
methods dependent on data from empirical models.
Among the concrete pavement design methods available, two methods have been investigated
in detail, the U.S. and the Dutch one. The selection is based on two things; (i) modernity and
(ii) availability. These two methods are both new, in the international research front, computer
based, and available.
In this Chapter, the international design methods are presented and compared to the current
design method used in Sweden. The current Swedish design as well as the first international
method, the MEPDG from USA, have been investigated thoroughly in Paper 1, and are
7
therefore only presented briefly. The second method, from the Netherlands, VENCON2.0, is
described more in depth but still only outlined in this thesis. The way of calculating stresses as
well as the method’s specific aims of design is of interest when developing a new design
method for Sweden.
2.2 Current Design Method in Sweden
2.2.1 Traffic Loads
Stresses due to traffic loads are present in the interaction between the wheel of a vehicle and
the road surface. The pressure from the wheel develops stresses in the different material layers
in the road structure and, it is foremost the stresses in the concrete slab, and the strains in the
subbase that are of interest in the design. In Swedish design a 100 kN equivalent standard axle
load (ESAL) is used. Every heavy truck corresponds to 1.3 passages of an ESAL. A heavy
truck has a weight exceeding 3.5 metric tons.
Today, stresses due to traffic are calculated with an elastic multi-layer program, GIPI, van
Cauwelaert (1986). In this program, it is possible to define five layers to be calculated, each of
them given properties on thickness, module of elasticity, and Poisson’s ratio. Between the
concrete and the bound base no bond is assumed. This is a conservative assumption.
Contrarily, for industrial pavements, where a higher crack risk is accepted, bond is usually
assumed, Silfwerbrand (1995, 2001).
2.2.2 Temperature Loads
Thermal loads arise in the concrete slab when its natural tendency to expand or contract is
prevented. It is the deadweight of the concrete slab that prevents the slab to curl and in that
way stresses are generated. A positive temperature gradient corresponds to an expansion on
the top layer with tensile stresses as a consequence in the bottom layer. Curling stresses are
directly dependent on the temperature gradients.
Stresses due to thermal loads are calculated with Eisenmann’s beam equation, Eisenmann
(1979). The critical length Lcr is the decisive factor that determines the geometry of the slab.
The stresses at the bottom of a simply supported concrete beam is calculated according to the
following equations:
V temp
1.2 ˜
D ˜ 't ˜ h ˜ E § L
˜¨
2 ˜ (1 Q ) ¨© Lkr
V temp
1.2 ˜
D ˜ 't ˜ h ˜ E
2 ˜ (1 Q )
Lcr
4D'tE
h
5 ˜ (1 Q ) ˜ J
·
¸¸
¹
2
if L ” Lcr
(2.1)
if L > Lcr
(2.2)
(2.3)
where, D is the coefficient of thermal expansion of concrete, 't is the temperature gradient,
Q is the Poisson’s ratio, E is the modulus of elasticity, h is the thickness of the concrete
pavement, and J is the dead load per unit length.
8
The climate is characterised by temperature and sunshine and is described by the magnitude
and length of the thermal gradients. In Sweden, an extreme gradient of 60°C/m has been
chosen for 5 % of the year, and 40°C/m for 20 %. During the rest of the year, a thermal
gradient of 0°C/m is assumed. These values, originated from field studies in Germany, were
modified for Swedish conditions by Ö. Petersson, Petersson (1996), for the new Swedish
specifications of the 1990’s.
2.2.3 Fatigue and Damage Accumulation
Thermal and traffic loads act simultaneously with different magnitudes, hour by hour, on the
concrete pavement. The loads produce stresses and eventually, after sufficiently many
repetitions, cause fatigue damage. The number of allowable load applications for each
thickness under each part of the year is calculated with Tepfers’ fatigue equation, Tepfers
(1978, 1979a, and 1979b)
§ V ·
1 0.0685 ˜ ¨¨1 min ¸¸ ˜ log N .
© V max ¹
V max
f c, fl
(2.4)
The number of allowable load applications is also calculated with respect to the deformation
in the subgrade according to:
N
8.06 ˜ 10 8
(2.5)
H Z4
where, N is the number of load applications, Vmin is the minimum stress, i.e. temperature
stress, Vmax is the maximum stress, i.e., the traffic and temperature stress superimposed, fc,fl is
the concrete flexural strength, and Hz is the vertical compressive strain on top of the subgrade.
The stresses are only calculated in the bottom of the slab.
The accumulated fatigue damage that a certain pavement can resist is calculated with MinerPalmgren’s damage hypothesis for flexural stresses in the concrete pavement, horizontal
tensile strains in the base course or the vertical compressive strain of the subgrade. The
accumulated damage is computed by summing fatigue damage incurred during each of six
seasons of the year due to both traffic and thermal loads.
Miner-Palmgren’s damage hypothesis has the following form:
n
Di ˜ Ni
¦N
i 1
d1
(2.6)
i, allow
where, Di is the percentage of time in which damage i occurs, Ni is the total number of loads
corresponding to damage i, and Ni,allow is the number of allowable loads corresponding to
damage i.
9
2.3 Design Method in the USA
The USA have had an old tradition of constructing roads with cement stabilised materials and
concrete and it is to no one’s surprise that the new design methods developed are very
sophisticated. Design methods for highways and airfields have been developed locally in
many states but larger project have been conducted by the Transportation Research Board
(TRB), the U.S. Military Corps of Engineers, and the Federal Airport Association (FAA). The
design method that is being launched for the public is the Mechanistic-Empirical Pavement
Design Guide (MEPDG), Transportation Research Board (2004). The software can be
downloaded from the internet and is currently being evaluated before it becomes the public
tool for designing. Since the program can be used for both rigid and flexible pavements, it is a
quite complicated program and will need a few years of investigation before all material
models and algorithms’ in the system are fully functional for the design. The program has not
been released in new versions because the risk of having people working with old versions is
considered as a problem. The software is prepared for SI-units but has not yet been released
because the lack of sponsorship. A transformation to SI-units come with a new, full
investigation to check errors in the program. The MEPDG has been presented at several
international conferences and information on the methodology used in the design is found
mainly in Transportation Research Board (2004) but also in Darter (2001) and Darter (2004),
among others.
The software calculates the degree of degradation of a concrete road by estimating the
detoriation of different functional properties as cracking (percent slabs cracked), joint faulting
(inches), and smoothness (IRI in inches/mile). The results are based on calibrated models for
fatigue development and are presented incrementally over the design life. Three different
levels of accuracy can be considered and the results are given with a percentage of
probability. The design considers both bottom-up and top-down cracking, i.e. stresses that are
generated in both the bottom and the top side of the concrete pavement. Further in-depth
analysis of the design procedure and a comparison to the current Swedish design is found in
Paper 1.
2.4 Design Method in the Netherlands
In the Netherlands, motorways have been designed with a software, VENCON 1.0, since the
middle of the 1990’s. The demands for a more user friendly software have nevertheless
emerged during the last ten years and therefore a complete upgrade of this program was
undertaken. The new software, VENCON 2.0, CROW (2004), was developed and became
available in 2005. The software is described in detail in van Leest (2006) and Houben (2006).
VENCON 2.0 is destined for plain concrete pavements and continuously reinforced concrete
pavements and has a very flexible interface. For instance, the different geometrical properties,
as the number or the width of the road lanes, can easily be changed. The design is based on
fatigue strength calculations that are made in certain locations in the concrete slab; the
longitudinal free edge, the longitudinal joints, and the transverse joint in the centre of the
wheel load.
Mainly two concrete grades are used in road construction in the Netherlands, C28/35 and
C35/45. The mean tensile strength (ffl,mean) for loading of short duration is calculated using a
safety factor of 1.2 according to
10
1.3>1.05 0.05( f ch '8)@ / 1.2
f fl,mean
(2.7)
where fch’ is the characteristic cube compressive strength at 28 days (European standard,
NEN-EN 206).
The flexural strength is also defined as a function of thickness, h (in mm), of the concrete
where
f fl,mean,h
>(1600 h) / [email protected] f fl,mean
(2.8)
Traffic loading is calculated for the total number of axles per axle group. The frequency and
average load for each axle group have been assembled from axle load measurements and can
be used in the design. Different kinds of tyres are also included in the design, where single,
double, wide base, and extra wide based tyres can be considered. The extra wide base tyre is
for instance not yet allowed but have been included for future needs. The traffic
measurements have also uncovered the number of overloaded axles that traffic the roads and
the highest average wheel load is therefore 105 kN (corresponds to an axle load of 200 – 220
kN). The traffic stresses are calculated with means of the “new” Westergaard equation,
Equation 2.9, for a circular tyre contact area, developed in Ioannides et al. (1987), at the
bottom of the slab at the three above mentioned locations. The calculation also includes the
load transfer (W) that is 60 % for construction joints and 80 % for contraction joints in a
PJCP. For comparison, the LTE is 90 % in transverse cracks in CRCP and 20 % in
longitudinal free edges on an unbound base.
For traffic stresses, the “new” Westergaard’s equation has the form
3(1 Q ) Pcal §¨ §¨ E C h 3 ·¸
a ·¸
4 1 Q
ln
1
.
84
Q
1
.
18
1
2
Q
¨ ¨
¸
l ¸¸
3
2
S (3 Q )h 2 ¨© © 100ka 4 ¹
¹
ı fl
l
Pcal
4
Eh2
121 Q k
1
(2.10)
W
200
(2.11)
where,
Vfl
Pcal
a
Ec
Q
h
k
W
(2.9)
= flexural tensile stress (N/mm2)
= wheel load, accounting for the load transfer (N)
= equivalent radius of circular contact area (mm)
= modulus of elasticity (N/mm2)
= Poisson’s ratio (-)
= thickness (mm)
= modulus of substructure reaction (N/mm3)
= load transfer that is dependent on the type of joint (%)
11
Temperature stresses are calculated along the edges of the slab. Default temperature gradients
and frequencies are used that include seven positive temperature gradient classes from 0 to
60°C/m These temperature gradients have been established from temperature measurements
in concrete pavements during the years 2000 and 2001. The calculations of temperature
induced stresses are done with respect to the magnitude of the temperature gradient. If the
temperature gradient is small, the deadweight causes the beam (the calculation consider a
beam along the edge of the slab) to remain fully supported, Equation (2.12). In case of a great
temperature gradient the beam will only be supported over certain length C at the ends
because the curling upward is greater than the effect of the deadweight, Equation (2.13) and
(2.14) for length and width, respectively. The governing equations for calculating temperature
stresses are
VT
h't
˜ DE
2
(2.12)
2
2 ·
§
¨L C¸
3 ¹
5 ©
1.8 ˜ 10 ˜
h
2
2 ·
§
W
C
¨
¸
3 ¹
1.8 ˜ 10 5 ˜ ©
h
V T, L
V T, W
C
4 .5 ˜
(2.13)
(2.14)
h
if C<<1
k'T
(2.15)
where,
VT
h
't
k
D
E
C
L
W
= flexural tensile stress due to temperature gradient (N/mm2)
= thickness of concrete (mm)
= temperature gradient (C/mm)
= modulus of substructure reaction (N/mm3)
= coefficient of linear expansion (usually 10-5 (ºC)-1)
= modulus of elasticity (N/mm2)
= supporting length (mm) in width or length of the slab
= length of the concrete slab (mm)
= width of the concrete slab (mm)
In the design, the substructure is characterised by the modulus of substructure reaction, k, that
is calculated by using known values on the modulus of subgrade reaction ko, for different
subgrades listed in the Netherlands. This k-value represents the whole substructure beneath
the concrete wear layer.
The fatigue relationship in the design is applied in all the three critical locations and
accumulated for traffic and temperature stresses with Palmgren-Miner’s damage hypothesis,
Equation (2.6).
The fatigue criterion has the following form:
12
log N i
§
V
12.903 ˜ ¨ 0.995 max
¨
f fl
©
1.000 0.7525
V min
·
¸
¸
¹ with 0.5 d V max d 0.833
f fl
(2.16)
f fl
where, Ni is the number of load applications for a specific loading condition i, ımin is the
minimum stress, ımax is the maximum stress, and ffl is the concrete flexural strength. The
fatigue equation is only valid when the maximum stress lies in the region of 50 to 83.3 % of
the maximum flexural strength. Below this region, stresses are considered harmless. Above
this region, stresses are considered to be too damaging for the pavement. The design only
considers bottom-up cracking, i.e. stresses in the bottom of the concrete pavement.
Additional checks, which include the check for robustness and the check for traffic ability at
opening of the pavement, are done as well.
2.5 Comparison of the Design Methods
The Dutch design method has many similarities with the current Swedish design method. The
equations that have been chosen, for calculating both traffic and temperature loads are newly
developed equations that originate from the same equations used in Sweden. An elastic multilayer program is, however, used for calculating traffic stresses in Sweden today. The way of
considering traffic loads in Dutch design, with different axle load frequencies for different
types of roads is more sophisticated, but differs in principle only from the Swedish method in
the number of loads, the different load magnitudes, and various tyres that are included in the
design. The high axle loads of 200 – 220 kN that is included in the design comes from new
traffic measurement in the Netherlands. Measurements conducted in Sweden have also
indicated that trucks that traffic the national roads are overloaded and should be considered in
the design, SRA (2003).
The temperature gradient distribution applied in the design is also more close to reality in
comparison to the somewhat coarse values used in Sweden. A different fatigue equation is
developed for the design which is more conservative than in Sweden, see Figure (2.1). It is
also restricted for loads exceeding 83 % of the flexural strength and no damage is accounted
for if the load is below 50 %. The main difference is nevertheless the advantageous features
that enable the possibility to alter the design. The Dutch design method makes it possible, to a
certain extent, to alter the widths and the lengths of the concrete slabs, choosing different
conditions to connect the slabs according to the type of road that is designed. The software
provides a concrete thickness with any desired information on the outcome from the
calculations.
13
1,00
0,95
Sweden - R=0,5
0,90
Sweden - R=0.2
Relative stress
0,85
Dutch - R=0,5
0,80
Dutch - R=0,2
0,75
0,70
0,65
0,60
0,55
0,50
0
2
4
6
8
10
12
14
16
LogN
Figure 2.1. Difference between the Swedish criterion, Equation 2.4, and the fatigue criterion
used in the Netherlands, Equation 2.15. The figure shows a S-N-diagram where the number of
load applications, logN, for a given relative stress, Vmax/fc,fl, and R=Vmin/Vmax is presented.
The MEPDG is also very flexible, in the same way as the Dutch method, but shifted towards a
more mechanistic design. The design aims at predicting the functional properties as joint
faulting, slab transverse cracking, and smoothness (IRI) based on the site conditions. The
detoriation of the pavement is calculated incrementally over the design period, taking into
account the various conditions that may change hourly.
Similar for all methods is the use of Miner-Palmgrens’ damage hypothesis. The hypothesis
has been investigated by Tepfers et al. (1977) among others, and is up to this day the best
method for summarising the effect of impact from more than one loading condition on
concrete fatigue. It still remains a very approximate method that is used in lack of a more
mechanistic alternative.
Critical locations that are considered in the investigated methods are presented in Figure 2.2.
Table 2.1 shows a comparison between the different design methods presented in this
Chapter. The aim of this study has been to highlight some of the features that are considered
in the different methods, compared to the current Swedish design method.
14
Longitudinal joint
Transverse joint
Connection bars
Dowels
2
5
1
4
(NL)
Traffic direction
3
(SWE)
(USA)
Figure 2.2. Critical locations for bottom-up and top-down cracking in a PJCP considered in
different design methods; Sweden (1), the Netherlands (2,3,and 4), and the USA (4 and 5).
15
Table 2.1. Comparison of design parameters for PJCP considered in the current design
method in Sweden, VENCON 2.0 in the Netherlands, and the MEPDG in the USA
Design Parameter
Sweden
The Netherlands
USA
Material properties
Subgrade moisture /temperature
No
No
Monthly variation
Material stiffness Seasonal variation
No
Monthly variation
Concrete strength over time
No*
Yes
Yes
Temperature loads
Linear temperature gradient
3 gradients
7 gradients
Daily variations
Nonlinear temperature gradient
No
No
Yes
Traffic loads
Axle loads 100 kN ESAL
10 axle groups
13 axle groups
Maximum axle load (axle type)
100 kN**
200-220 kN
100 kN /450 kN
Type of axles considered
Single
Single
Single to Quad
Axle load distribution
No*
Yes
Yes
Traffic wandering
No*
Yes
Yes
Axle spacing
Yes
No
Yes
Axle width
Yes
Yes
Yes
Tyre pressure
Yes
Yes
Yes
Dual tyre spacing
No*
Yes
Yes
Pavement properties
Joint spacing
Fixed*
Variable
Variable
Lane width
Fixed*
Variable
Variable
LTE - base layer
Variable
LTE - dowels
Variable
100%
60 or 80 %
Variable
LTE - aggregate interlock
Shoulder type
No
Yes
Yes
Traffic estimations
AADTT
Yes
Yes
Yes
Monthly adjustment factors
No
No
Yes
Vehicle class distribution
No
Yes
Yes
Hourly truck traffic distribution
No
Yes
Yes
Traffic growth factor
Yes
Yes
Yes
Types of distresses considered
No. of critical points in slab
1
3
All locations
Bottom-up cracking
Yes
Yes
Yes
Top-down cracking
No
No
Yes
Joint faulting
No
No
Yes
IRI
No
No
Yes
Interface - wear layer/road base
No bond*
No bond(?)
Variable
Note: * – Considered in the Swedish design of industrial pavements, Silfwerbrand (2001) ,
** – any load between 100 and 900 kN in the Swedish design of industrial pavements, Yes –
considered, No – not considered, Variable – can be altered, Fixed – fixed to one value in the
design.
}
}
16
Chapter 3
Experimental Studies on Fatigue
3.1 General
The fatigue criterion is the far most important factor in the design of concrete pavements. The
criterion is used to predict failure and is normally investigated through material testing of
small specimens (cubes, cylinders, or beams) in laboratories or full scale field test sections.
Small scale testing is more convenient but include various estimations on how the actual
behaviour in the field changes. Full scale testing, as full-size slabs, is rare because it needs
large testing facilities. Full-scale testing on airfield pavements have been conducted since 60
years in the U.S. and has been the main data source for the development of slab fatigue
criteria, Roesler (2004) and Smith and Roesler (2003). These tests offer real traffic loads,
environmental conditions, traffic wander, and material properties to be included in the fatigue
curve. Pavement testing, when accelerated, does not take into account all climatic variations
and soil conditions, or the beneficial factors like increasing concrete strength.
There is a vast number of different fatigue criteria available in the USA and Europe and a
comparison between different methods are presented in Figure 3.1. In Sweden, Tepfers’
fatigue criterion, Equation (2.4), has been chosen for concrete pavement design, Petersson
(1996) and SRA (2005). The equation includes the effect of minimum to maximum stress
ratio (R) and is not extremely conservative in comparison to other fatigue criteria (Figure 3.1).
The equation is developed out of compressive and splitting tests and since the fatigue strength
was equal in these two tests, it was concluded that the equation could be used for bending
stresses as well.
17
100%
95%
Relative stress, V max/f cfl
90%
85%
80%
NCHRP 1-26 (1992)
75%
Tepfers (1977) R=0
70%
Dutch VENCON2.0
(2005) R=0
65%
Darter (1990)
60%
Foxworthy (1985)
55%
50%
0
1
2
3
4
5
6
7
8
Number of load repetitions to failure, LogN
Figure 3.1. The diagram shows fatigue curves for different fatigue criteria. Darter,
Foxworthy, and NCHRP1-26 are developed in the USA, Smith (2003). VENCON2.0 comes
from the Netherlands, CROW (2004) and Tepfers from Sweden, Equation 2.4, Section 2.2.2.
Only VENCON2.0 and Tepfers consider the stress ratio (R=Vmin/Vmax).
The experimental studies conducted in this thesis are presented in Papers 2 - 4. The tests have
been done in order to verify the design criteria and are made on plain concrete, lean concrete
and composite beams of plain and lean concrete. The static system chosen for the tests was
retrieved from the preceding Swedish standard SS 13 72 12, that prescribes four point flexural
tests (Nowadays, SS-EN 12390-5-2000 is practice in Sweden. Here, three or four point
flexural tests are prescribed).
3.2 Methodology
Plain and lean concrete beams that measured 800 mm in length, 150 mm in width, and
100 mm in height were manufactured for the fatigue tests. The plain concrete beams were cast
in steel forms at the laboratory. The lean concrete beams were manufactured by compaction
of the material on a surface of 70 m2 with a thickness equivalent to the beams’ height. The
beams were extracted by sawing.
The fatigue tests were conducted at the Swedish Cement and Concrete Research Institute
(CBI) in Stockholm with the Material Test System (MTS) 810 machine, Figure 3.2. The tests
included static testing of at least three specimens to verify the static flexural strength and then
applying the selected load levels in the subsequent flexural fatigue tests using a mean value
from these tests (fc,fl in Equation (2.4)). In the fatigue flexural tests, the loading was started at
a low frequency (0.5 Hz) and increased rapidly to the maximum frequency (2.05 Hz) after a
few cycles. This method was adopted to avoid any sudden loading, or concentrated loads, that
could arise from the supports when applying the first load.
18
Figure 3.2. Composite beam of plain and lean concrete subjected to flexural fatigue loading.
All beams were stored in water and, at the time for testing, covered with a plastic folio to keep
the moisture level constant throughout the tests.
The loading caused a rough sampling of deflection and loads and the fatigue frequency was
thus limited to 2.05 Hz. Deflection data were collected through external sampling. The
sampling rate was 40 Hz. The applied loads are expected to be accurate within 5 %, but the
deviation is negligible in the context of fatigue loading where the standard deviation of the
static flexural strength is higher (up to 15 %). On the other hand, if the maximum load is
greater than 80 % of the maximum static strength, the loading accuracy and the true flexural
strength is crucial for the results. Fatigue testing is often limited to stress levels below 80 % of
the static strength. This problem is discussed in Paper 1 but also in Tepfers (1978), among
others.
3.3 Results
The fatigue test results showed that Tepfers’ fatigue criterion is applicable on the flexural
fatigue of plain concrete, but also on the flexural fatigue of lean concrete. For lean concrete
there exists a strain criterion today, developed by Örbom (1981). This criterion is very strict
compared to international criteria, see Paper 3. Since both plain and lean concrete are
cementitious materials it was of great interest to verify if a stress criterion could be applied
instead. This was proved with somewhat lower fatigue strength. The composite beams of
plain and lean concrete showed that the bond was extremely resistant and that repeated
loading resulted in all-pervading cracks. The bond has a strengthening function in the
pavement system and is likely to be advantageous if well distributed expansion joints are cut
in the lean concrete layer.
19
20
Chapter 4
Temperature Measurements
4.1 General
Data of accurate temperature variations is a key factor for the design of concrete pavements.
The temperature gradient in a concrete slab causes curling which generates stresses when the
slab is restrained. The slab can be restrained by adjoined slabs but the dead weight of a slab
also counteracts the natural movements. Thermal expansion on the upper side of a slab makes
the slab rise over the centre, but the dead load acts in the opposite direction, generating tensile
stresses in the bottom of the slab. Extreme temperature gradients, measured in for instance the
USA or Germany, have been shown to reach 90°C/m over a 200 mm thick slab, Petersson
(1996). Gradient of this magnitude can generate stresses near the ultimate tensile stress limit
of the material. The thermal conductivity of the material is low and daily temperature
variations have a pronounced effect on the concrete pavement. The temperature effects on
concrete pavements also act at the same time as heavy traffic is present, i.e. rush hour traffic.
This leads to superimposed stresses.
The temperature in the concrete is dependent on the air temperature, the solar radiation,
precipitation, and the wind conditions. The specific conditions also change with the
surrounding environmental characteristics, like shadows, hills or even the height over ocean.
In Sweden the current design only considers linear temperature gradients (presented in
Section 2.2.2). These gradients herein from measurements in Germany, but have been altered
to fit a less extreme Swedish climate by a limited number of field measurements, Petersson
(1996). New temperature measurements in concrete slabs in various parts of Sweden are
needed in a new design method to enable a more optimised design.
4.2 Methodology
Temperature measurements have been conducted within this project since 2005 with means of
concrete prisms with the dimensions 400×400 mm and the height of 250 mm. Temperature
measuring equipment has been installed inside the cubes at the time of casting and the prisms
are dug down in the asphalt road pavement so that only the top surface is exposed to the
varying environmental conditions, Figure 4.1. Four sensors measure the temperature at the
depths of 25 mm, 75 mm, 170 mm, and 200 mm, respectively. The four points over the cross
section will make it possible to investigate, not only the linear temperature gradient, but also
the duration and shape of the nonlinear temperature gradients.
21
Figure 4.1. Installation of a concrete prism for temperature measurements in Sweden. Cables
from temperature sensors are also put in the asphalt layer and connected to a data logger
beside the road. The logger is of the type ConReg 706, used for concrete maturity monitoring
in casting, CMT (2005).
The field measurements with these cubes are being conducted on three sites in Sweden; 300
km north of Stockholm in the region of Enånger, 60 km south of Stockholm in the
neighbourhood of Strängnäs, and just south of Malmö in the south of Sweden (see Figure
4.2). These locations have been chosen because of the traffic volumes at these places, making
these regions the most probable sites for concrete pavement construction in Sweden. Even
though the Swedish west coast is heavily trafficked, no measurements are conducted there in
this particular project. Here, measurements have already been done at the time for, and during
ten years after the construction of highway E4 Falkenberg by VTI, Wiman (2002). Other
measurements have also been conducted by Ö. Petersson in 1991, Petersson (1996), and A.
Farhang in 1997 - 2000, Farhang (2000), in various locations, see Figure 4.2.
22
VTI, 2006 –
Petersson, 1990
Söderqvist, 2005 –
VTI, 1996 – 2006
Farhang, 1997-2000
Figur 4.2. Locations for temperature measurements in Sweden. VTI, Petersson, and Farhang
are measurements in actual concrete pavements. Söderqvist represents measurements in
concrete prisms that are put into the existing asphalt pavement (two sites) and in existing
concrete pavement (one site).
4.3 Results
The results from the measurements will give information on the frequency and magnitude of
temperature gradients in different parts of Sweden. Measurements conducted by means of
four sensors through the cross-section of a concrete prism will also make it possible to analyse
the nonlinear temperature gradient. These gradients can reveal if stresses act in the bottom or
top of the pavement and how important this effect is for the stress level in the design.
Figure 4.3 illustrates the magnitude of a positive linear temperature gradient of 40 ºC/m, but
also a negative linear temperature gradient of 25 ºC/m. A more accurate estimation of
temperature gradients measured in the field that include positive, negative, and nonlinear
gradients, will make the design method more reliable and at the same time, possibly less
conservative.
23
T, -125 mm ( C)
T, -75 mm( C)
Linear temp.grad( C/m)
T, -200 mm( C)
T, air( C)
T, -25 mm( C)
T, air exposed(C)
50
40
Temp./Temp.gradient
30
20
10
0
00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00 06:00 12:00 18:00 00:00
-10
-20
-30
Time of the day
Figure 4.3. Temperature varying over depth in a concrete prism in Enånger, late August,
2006. A positive linear temperature gradient has its peak in the afternoon and the negative
temperature gradient has its peak in very early morning.
24
Chapter 5
New Design Method for Swedish Conditions
5.1 General
In the new proposal for the design of concrete pavements in Sweden, finite element analysis
(FE-analysis) is suggested to be utilised for the calculation of stresses in different locations in
the concrete slab. The design criteria for stresses in the concrete slab, the bound road base,
and the subbase that are included in the current design can be used in the new methodology.
Also, temperature loads with more accurate distributions can be included in the design. The
new methodology can be applied on specific projects, and later on developed to account for
true traffic loads as well as nonlinear temperature gradients.
5.2 New Method for Calculating Loads
For the calculation of traffic loads in the concrete slab, the layers under the concrete slab can
be modelled as a Wrinkler foundation, Figure 5.1. The foundation stiffness is hereby
characterised by a k-value, that is a proportionality constant between the vertical pressure and
the deflection of the slab (N/mm3). Values for the k-value exist for different materials but it is
more accurate to predict this value out of geometry and Falling Weight Deflectometer (FWD)
testing on the modulus of elasticity. The k-value is dependent on the modulus of elasticity of
the subbase but also the thickness of the concrete slab as well as the ratio in modulus of
elasticity between the concrete slab and the subbase.
Figure 5.1. Theoretical model of a Wrinkler foundation where the stress in the foundation is
proportional to the deflection of the slab in each point, from Petersson (1996).
25
In the proposed method, the calculation of stresses and strains in the concrete slab are
suggested to be done using a FE-program, for instance ISLAB2000, ARA (2004). To do this,
the software is dependent on the k-value. To estimate the k-value in the subbase FWD data
can be used, where the load and deflection is registered, or with known values on the modulus
of elasticity of the road base materials.
The k-value can also be calculated in a separate elastic multi-layer program such as GIPI or
BISAR 3.0 (1998), every time a new design is considered. The material data inputs in the
program consist of the modulus of elasticity, the poisson’s ratio, and the thickness of the
material layers. In the program, a FWD load is simulated and the k-value calculated with the
data from the program, i.e. the load and corresponding deflection under the concrete slab.
This value can be calculated for different seasons, with different temperatures and different
moisture contents in the subbase. The k-value is then used in the FE-program to calculate the
stresses in the concrete slab with varying load magnitudes and load locations, see Figure 5.2.
The use of a program like BISAR 3.0 will bring the design of concrete pavements more close
to the design of flexible pavements since the subbase is modelled in the same way. The
different design procedures will hereby be more comparable and this is advantageous for
designers and clients when evaluating various pavement alternatives.
Figure 5.2. An elastic multi-layer program is used to calculate the k-value using data on load
and deflection from the actual pavement system (1). The k-value is used in the equivalent
model in the FE-program, where stresses and strains are calculated in the concrete layer,
using model (2) or the concrete and the bound road base, using model (3).
26
The above mentioned procedure is basically the procedure that is used in the newly developed
design method in the MEPDG. ISLAB2000 is a 2.5-dimensional FE-program, specially
constructed for calculating stresses and strains in concrete slabs very fast. The program is
simple in a way that a number of adjacent slabs and pavement shoulders can be modelled with
specific interaction properties. One of the most helpful advantages is the possibility to
calculate both temperature and traffic stresses at the same time. Nonlinear temperature
gradients are also possible to apply.
In the new FE-program the calculation of stresses in the concrete slab can be utilised to
calculate the fatigue from a 100 kN standard axel load. The load can be placed in the centre of
the slab or on the edge of the slab, i.e. where the loads generate the highest stresses. The
highest stresses in the bottom of a slab is generated by a load placed in the centre or the edge
of the slab. In the top of the slab, the highest stresses are instead generated by a load placed
near the corner or the edge. For certain situations, the temperature has a crucial effect on the
total stress distribution. At a negative temperature gradient, the edges will rise, and if any load
on the edges is present, the total stress will rise as well. In a later stage, the method can be
enhanced by adding new parameters for different axle loads. These data have to be assembled
before it can be included in the design.
To gain control over different extreme situations at least four different load locations should
be considered for fatigue analysis; two load locations on top and two load locations in the
bottom of the concrete slab. This approach would complete the current design method by
considering the combination of loads near the edge of the slab when a negative temperature
gradient is present.
The temperature is very important when designing concrete pavements. New temperature
measurements in Sweden indicate that the temperature gradients are more complex and can be
more accurately estimated compared to the current gradients. For instance, research data have
shown that a negative temperature gradient is present during a short period of time (Farhang,
2004b). This is not accounted for today but will be included in the new design method. The
use of linear temperature gradients is also a simplification of reality.
5.3 Fatigue Criteria and Damage Accumulation
A fatigue criterion that considers two load levels, as Tepfers’ fatigue criterion, or the Dutch
correspondent equation, is valuable in a design that accounts for both temperature and traffic
loads. A new fatigue criterion can be developed but for the time being, it is suggested that
Tepfers’ criterion remains in Swedish design.
Damage is influenced by the loading frequency, the time at resting, and the magnitude of the
loading, Hsu (1981). Calculating the accumulated damage is still generally done by means of
Miner-Palmgren’s damage hypothesis in pavement design, Equation (2.6). The method is
criticised by many because its linear illustration of damage simplifies the influence of damage
from multiple loads. Any new or better method has, however, yet not been presented. The
damage hypothesis works as an engineering tool more than a refined tool for accumulating
different loads. In modern design methods, the damage hypothesis is used in combination
with a calibrated damage model where the damage is varied statistically, as in the MEPDG.
27
5.4 Implementation of a New Swedish Design Method
The general idea of the proposed design method is presented in Figure 5.3. The development
of a new method consists of the gathering of parameters for traffic and temperature loads that
are unique in Sweden, Figure 5.3 (A and B). Different material properties and design criteria
for new failure modes have to be investigated, Figure 5.3 (C and D) and new calculation
methods have to be analysed in comparison to existing ones to guarantee the reliability of the
method. The more practical implementation of a computer based, interactive program for the
design is much dependent on the amount of parameters that are required in the design, i.e. the
level of sophistication. The method proposed is meant to be developed gradually so that it
replaces the current method without any too dramatically changes for constructors or clients.
Also, the level of simplicity is a key factor to reach a wider use among future concrete
pavement designers.
28
4. Fatigue analysis
Design criteria for critical
locations in concrete, bound
road base, and subbase.
2. Elastic multi-layer
analysis
Program:
GIPI or BISAR 3.0
Input:
- Equivalent load
- Material data
- Geometry
Output:
- Substructure reaction, k-value
(N/mm3)
- Strains in substructure
5. Miner-Palmgren’s
Damage Hypothesis
Damage accumulation
3. FE-analysis
Program:
ISLAB2000
Input:
- Traffic loads
- Temperature loads
- Concrete properties
- k-value (from 2)
- Load transfer efficiency (LTE)
Output:
- Stresses in various locations,
bottom and top of pavement
Does the design meet the
design criteria
throughout the design
period?
1. Design data
A. Climate data
Temperature :
- Linear gradient
- Nonlinear
gradient
D. Design Criteria
- Design criteria
- Design period
- Accepted distresses
B. Traffic data
- Axle loads
- Types of
vehicles
- Traffic growth
C. Material data
Suggested pavement
system.
- Geometry
- Thicknesses of
layers
- Material properties
(Modulus of elasticity
poisson’s ratio)
NO
New design
properties and
new
calculation.
Figure 5.3. Proposed procedure for the design of PJCP in Sweden.
29
YES
Design
approved!
30
Chapter 6
Conclusions
6.1 Concrete Pavement Design
The design of plain concrete pavements involves a large number of parameters. The current
Swedish design method was developed 15 years ago. The method is simple and limited since
many parameters are neglected and therefore the safety margins applied are important. In
different design methods found, new information on traffic and temperature distributions are
incorporated into the design. These methods are more accurate because it is possible to
quantify the safety margins. These methods are also computerised to a large extent, making
them faster and more flexible.
This thesis presents a basis of a new design method that could be easily implemented in
Sweden. The method can be developed gradually and is based on FE-analysis for fast
calculation. In the process of validating the new method it is recommended that it is compared
to some of the international design methods available and the current Swedish design method.
6.2 Flexural Fatigue Criteria
The laboratory tests conducted in this project have aimed at investigating the fatigue criterion
in Swedish design. The tests involved beams of plain concrete, lean concrete, and composite
beams of plain and lean concrete.
Tepfers’ fatigue criterion was originally developed out of compression and splitting tests and
the test results for the plain concrete beams showed that the criterion is valid also for flexural
fatigue.
The fatigue of lean concrete is, in Swedish pavement design, calculated using a strain
criterion that is very strict compared to international standards. Since lean concrete is a
cementitious material, the tests were done in order to analyse the possibility to apply a stress
criterion instead, as done for plain concrete. The tests showed that a stress criterion is
applicable with a small reduction in fatigue strength. The reduction compared to Tepfers’
fatigue equation can be illustrated by a reduction constant C, that reduces the impact of
amplitude between the load levels. A stress criterion for lean concrete would be less strict and
consequently make lean concrete more competitive than today.
31
The composite beams were investigated to verify the cracking behaviour and the fatigue
strength properties of the bond. In testing, all-pervading cracks were difficult to avoid. The
cracking is explained by (1) the full bond between the materials that allowed stresses to pass
through the interface, and (2) the concentrated stresses that are assembled at the crack
opening, caused by the decreasing cross-section. The phenomenon of entire cracking of the
cross-section is referred to as reflection cracks, and challenges the question on whether the
bond really is desirable or not, i.e. strength versus risk of cracking. The bond might be
detrimental because it ruins the whole section when the capacity of the lean concrete is
surpassed and the cracking of the concrete wear layer is consequently difficult to control.
However, if the design focuses on the fatigue and cracking in the bottom of the lean concrete
instead of the bottom of the plain concrete, a much stiffer pavement can be obtained and
stresses further down in the substructure are limited. Of course, cracking due to temperature
expansion and shrinkage in the lean concrete has to be limited by cutting expansion joints in
the material at short distances.
32
Chapter 7
Further Research
In order to establish a modern design method for plain concrete pavements in Sweden,
research in various areas within concrete pavement design is required. Material testing for
ascertaining the materials degradation parameters and how temperature or moisture affects the
different materials properties is important. These are issues that have to be addressed in the
development of durable and cost effective concrete pavements. Future research includes:
x
Extended laboratory testing and/or road pavement testing regarding the fatigue criteria for
plain and lean concrete to incorporate the most accurate and best fitting design criteria in
Swedish design
x
Further research on the benefits or disadvantages with a bond between plain and lean
concrete.
x
Traffic monitoring to improve traffic estimations in the design. Measurements on traffic
have to contain load axle weights, frequencies, and type of vehicles or tyre.
Measurements have been conducted in Sweden but have to be extended, and assembled.
x
Continuous temperature measurements for the establishment of real temperature gradients
for various parts of Sweden.
x
Further investigations for the development of a new method to replace Miner-Palmgrens’
damage hypothesis. The solution is regarded to be found in fracture mechanics.
x
Further development of the proposed Swedish design method and verifications by
performing comparison studies with international design methods, the old Swedish
method, and long-term performance studies on real concrete roads in Sweden.
33
34
Chapter 8
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Ave., Champaigh, IL 61820, USA.
AB Svensk Byggtjänst and Cementa AB (2002), Handbok Betong på mark – Platsgjutna
lösningar (Handbook Concrete Pavements – In-situ Casting Solutions). AB Svensk
Byggtjänst and Cementa AB, Stockholm, Sweden, 256 pp, in Swedish.
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Swedish).
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CROW, Ede, the Netherlands, in Dutch.
Darter M., Khazanovich, L., Snyder, M., Rao, S., and Hallin, J. (2001), Development and
Calibration of a Mechanistic Design Procedure for Jointed Plain Concrete Pavements.
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September, 9-13, 2001, pp.113-131.
Darter M., Khazanovich, L., Yu, T., Selezneva, O., Hallin, J., Titus-Glover, L., and
Larson, G. (2004), Design of Jointed Concrete Pavement Using Incremental Damage.
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Performance of Concrete Pavements, Istanbul, Turkey, March 1, 2004, CROW Technology
Centre, The Netherlands.
Eisenmann, J. (1979), Betonfahrbahnen, Handbuch für Beton-, Stahlbeton und
Spannbetonbau. Verlag von Wilhelm Ernst & Sohn, Berlin, München, Düsseldorf, 305 pp. (in
German).
European Commission, Directorate General Transport (1999), COST 333: Development of
New Bituminous Pavement Design Method. Office for Official Publications of the European
Communities, Luxembourg, 373 pp.
35
Farhang A. (2000), Concrete Structures Subjected to Combined Mechanical and Thermal
Loading. Bulletin 60, Doctoral thesis, Royal Institute of Technology, Department of Civil and
Architectural Engineering, Stockholm, Sweden, 294 pp.
Farhang A. (2004a), Dimensionering av platsgjutna betongbeläggningar för kommunala ytor
(Design of In-situ Concrete Pavements for Public Spaces). Bulletin 67. Royal Institute of
Technology, Department of Structural Engineering, Stockholm, Sweden, 49 pp. (in Swedish).
Farhang A. (2004b). Calculation of Thermal Stresses for Jointed Plain Concrete Pavements
Based on Non-linear Temperature Gradients Using Modified Eisenmann’s Equation and
Modified Silfwerbrand’s Plate Method. Proceedings, 5th International CROW-workshop on
Fundamental Modelling of the Design and Performance of Concrete Pavements, Istanbul,
Turkey, March 1, 2004, CROW Technology Centre, The Netherlands, 27 pp.
Houben, L.J.M., Braam, C.R, van Leest, A.J., Stet, M., Fréney, J.W., and Bouquet, C. (2006),
Backgrounds of VENCON2.0 Software for the Structural Design of Plain and Continuously
Reinforced Pavements. Proceedings, 10th International DUT-Workshop on Fundamental
Modelling of Design and Performance of Concrete Pavements. Old-Turnhout, Belgium,
September 15-16, 2006, pp. 1-20.
Hsu, T. C. (1981), Fatigue of Plain Concrete. ACI Journal, Proceedings V. 78, No.4, JulyAugust, 1981, pp. 294-304.
Ioannides, A.M., Thompson, M.R. and Barenberg, E.J. (1987), The Westergaard Solutions
Reconsidered. Proceedings, Workshop on Theoretical Design of Concrete Pavements, June 56, 1987, Epen. Record 1, CROW, Ede, The Netherlands.
Löfsjögård, M. (2003). Functional Properties of Concrete Roads. Bulletin 73, Doctoral thesis,
Royal Institute of Technology, Department of Civil and Architectural Engineering,
Stockholm, Sweden.
Petersson, Ö. (1996), Svensk metod för dimensionering av betongvägar (Swedish Method for
the Design of Concrete Pavements). Bulletin 16, Licentiate thesis, Royal Institute of
Technology, Department of Civil and Architectural Engineering, Stockholm, Sweden (In
Swedish).
Roesler, J. (2006), Fatigue Resistance of Concrete Pavements. Proceedings, 10th International
DUT-Workshop on Fundamental Modelling of Design and Performance of Concrete
Pavements, Old-Turnhout, Belgium, September 15-16, 2006, 19 pp.
Silfwerbrand, J. (1995), Dimensionering av betongbeläggningar (Design of Concrete
Pavements). Report No. 9, 2nd Edition., Royal Institute of Technology, Department of
Structural Engineering, Stockholm, Sweden, 91 pp. (in Swedish).
Silfwerbrand, J. (2001), Swedish Design of Industrial Pavements. Proceedings, 7th
International Conference on Concrete Pavements, ISCP, Orlando, USA, September, 9-13,
2001, pp. 791-806.
36
Smith, K.D and Roesler, J.R. (2003), Review of Fatigue Models for Concrete Airfield
Pavement Design. Proceedings of the ASCE Specialty Conference, September, 21-24, 2003,
Las Vegas, Nevada, USA, 25 pp.
SRA (1994), Allmän teknisk beskrivning för vägkonstruktioner VÄG 94 (Technical Guidelines
for Road Structures VÄG 94). Part 1-10. Report Nos.1994:21 to 1994:30, Swedish Road
Administration, Borlänge, Sweden (in Swedish).
SRA (2003), BWIM-mätningar 2002 och 2003 (Inspection and measurements of heavy traffic
loads on Swedish roads 2002 and 2003). Final report., Publ.2003:165., Swedish Road
Administration, Borlänge, Sweden (in Swedish).
SRA (2005), Allmän teknisk beskrivning för vägkonstruktioner ATB VÄG 2005 (Technical
Guidelines for Road Structures ATB VÄG 2005). Part A-K. Report No. 2005:112, Swedish
Road Administration, Borlänge, Sweden (in Swedish).
Tepfers, R., Fridén, and C., Georgsson, L. (1977), A Study of the Applicability to the Fatigue
of Concrete of the Palmgren-Miner Partial Damage Hypothesis. Magazine of Concrete
Research, Vol. 29, No. 100, September, 1977.
Tepfers, R. (1978), En undersökning av betongens utmattningshållfasthet (An examination of
the fatigue properties of concrete). Byggforskningen, Report R86:1978, 1978, 121 pp. (in
Swedish).
Tepfers, R. (1979a), Tensile Fatigue Strength of Plain Concrete. ACI Journal, Proceedings
V.76, No. 8, August, 1979, pp. 919-933.
Tepfers, R., & Kutti, T. (1979b), Fatigue Strength of Plain, Ordinary, and Lightweight
Concrete. ACI Journal, Proceedings V. 76, No. 5, May, 1979, pp. 635-652.
Transportation Research Board (2004). Guide for Mechanistic-Empirical Design of New and
Rehabilitated Pavement Structures, Part 1-3+App. AA, JJ, QQ, Final Report/Document,
NCHRP 1-37A, Washington, D.C., USA.
van Cauwelaert, F. (1986), Computer Programs for the Determination of Stresses and
Displacements in Four Layer Systems with Fixed Bottom. Centre de Recherches de l'Institut
Superieur Industriel Catholique du Hainaut, Mons, Belgium, 61 pp.
van Leest, A., Stet, A.J., and Fréney, J.W. (2005), VENCON 2.0: A Fast and Reliable Tool for
Concrete Road Pavements (Jointed and Continuously Reinforced Applications). Proceedings,
8th International Conference on Concrete Pavements. Colorado Springs, Colorado, USA,
August 14-18, 2005, Vol. III, pp. 1303-1321.
Williams, R.I.T. (1986), Cement-Treated Pavements: Materials, Design and Construction.
Elsevier Applied Science Publishers Ltd., New York, NY, USA.
Wiman, L. and Carlsson, H. (2002), Prov med olika överbyggnadstyper: observationssträckor
på väg E6, Fastarp – Heberg. Resultatrapport efter 5 års uppföljning, 1996 – 2001 (Tests of
different wear layers: observation roads on E4, Fastarp – Heberg. Results from 5 years of
37
monitoring, 1996 – 2001). Statens väg- och transportinstitut (VTI), VTI notat 52-2002,
Linköping, Sweden, 75 pp, (in Swedish).
Örbom B. (1981), Trafikbetingade förändringar hos bärlager av cementstabiliserade,
välgraderade material enligt undersökning vid Pennsylvania Transportation Institute samt
härpå grundade dimensioneringsanvisningar (Traffic dependent changes of cement stabilised
base layer, continuous grading materials, according to investigations at Pennsylvania
Transportation Institute and on these supported design guidelines). Study Report, Swedish
National Road and Transport Research Institute (VTI), Linköping, Sweden (in Swedish).
38
Design of Concrete Pavements: A Comparison between
Swedish and U.S. Methods
Johan Söderqvist1, Member ISCP
Johan Silfwerbrand2, Member ISCP
Abstract
The Swedish regulations for roads have now been converted into a computerised
design guide that will provide engineers with the necessary, up to date, tool for the
design of roads, no matter the surfacing material. The design of concrete roads is,
however, based on old tables that offer little change in the concrete thickness when
altering parameters like climate, traffic, and material properties. In an ongoing
project a new computer-based design method for concrete roads is being developed.
The aim of the project is to facilitate the design method and establish a method that
treats the concrete pavement with the same ease, accuracy and safety level as current
asphalt pavement design methods. In the Ph. D. project an inventory of various
design methods is conducted. The project has a focus on investigating and comparing
the mechanistic-based design procedure available in the United States to the current
Swedish aspects of design.
Introduction
In Sweden, less than one percent of the national roads are paved with concrete
(European Commission 1999). The concrete roads of the 1960’s were constructed
with long slab lengths on unbound road bases and had severe joint faulting problems.
The unbound granular base under the slab eroded and cracks appeared as the loading
capacity decreased. New attempts in the 1970’s resulted in a few roads that have
been performing well but have now reached a critical point where they are in need of
rehabilitation. The Swedish National Road Administration (SNRA) has during the
last 15 years regained interest in concrete roads because of the growing problems
with rutting on heavily trafficked asphalt roads and the need for competition between
different road surfacing materials (Löfsjögård 2003).
The current design method for concrete roads in Sweden herein from the
investigation conducted by Petersson (Petersson 1990) in the 1990’s. A thorough
analysis of different design methods for concrete roads was performed and several
methods for calculating traffic and temperature loads were considered. This work
resulted in a new design method, later implemented in the Swedish regulations for
roads. The methodology was evaluated with the design of two highways in the south
of Sweden.
The modern concrete roads that have been constructed since 1990 are plain
jointed concrete pavements (PJCP) on a cement bound or asphalt bound road base
1
Ph.D. student, Swedish Cement & Concrete Research Institute, SE 100 44 Stockholm,
Sweden, email: [email protected]
2
Professor, President, Swedish Cement & Concrete Research Institute, SE 100 44 Stockholm,
Sweden, email: [email protected]
1
and the results from the follow-ups show a high reliability and an excellent long-term
performance (Löfsjögård 2003). The Swedish model for the design of concrete roads
is based on Westergaard’s, by Eisenmann, improved equations. It is a design method
used in several countries in Europe.
Nowadays, new computerised tools are available that can handle many more
parameters in the design and make calculations instantly. Overcoming the difficulties
of how to estimate traffic, variations in climate, and different material types, it would
be possible to optimise and improve the accuracy in pavement design with these
tools.
The Swedish regulations for roads have been converted to a computerised
design guide that will provide engineers with the necessary, up to date, tool for the
design of roads, no matter the surfacing material. This change in design methodology
has, however, left the design of concrete pavements behind. The old tables for
calculating pavement thicknesses are still used in new design, probably resulting in
less optimised design.
The thicker the pavement is the less attractive it becomes and, therefore, a
new more flexible design method is needed in Sweden, a method that can account for
a greater variation of inputs and have a high level of reliability.
The new mechanistic-empirical design procedure, the Design Guide, that has
been developed in the U.S. during the last ten years is based on mechanistic concepts
but calibrated to observed data (Darter 2004, Transportation Research Board 2004).
This design concept can be applied to both flexible and rigid pavements and the
procedure is based on damage calculation accumulated incrementally over the entire
analysis period. The mechanistic-based design procedure enables the functional
properties, i.e., cracking, joint faulting and smoothness, to be evaluated throughout
the pavement’s designated lifetime. This is necessary when estimating the total cost
of a highway construction project and makes it possible to compare different
surfacing materials. The design method is more attractive both for the designers and
the contractors, as both economical and practical issues become possible to analyse.
Comparing this procedure to the Swedish aspect of design many questions
arise on weather it would be possible to apply some of its features on projects in
Sweden. It is of course fundamental that the design procedure is validated for the
actual construction site and the actual materials as well as to the type of traffic and
climate that the pavement will endure. In the U.S., it has been feasible to validate the
design process to a huge bank of collected data and is, therefore, in some way
optimised for these specific conditions. Taking this data to a new environment and
using other materials in layers beneath the concrete, the parameters for particular
parts of the design can be put faulty if not examined well. It is also courageous to
replace design methods that differ too much in inputs, but also in redefining design
criteria. The specifications for the design are always validated to test sections and
reference projects. Specifications for all sorts of structures are always improved to
new materials and new construction methods but follow the same principles.
The objective of this paper is to describe the differences between the new
mechanistic-empirical design procedure for plain joint concrete pavements
developed in the U.S., and the current Swedish design method. The paper will
highlight the Swedish aspects of design and analyse the corresponding methodology
in the Design Guide. The Design Guide software has been available for evaluation
through the Transportation Research Board’s homepage and a limited investigation
of the program has been done. The aim of this work is to investigate how new tools
2
and new modelling techniques could be used in Sweden and how these methods are
developed.
A New Design Criteria
The new design procedure for plain joint concrete pavements in the U.S. is based on
mechanistic concepts. In the development of the procedure many new parameters for
the design are taken into account. The intention is to predict the functional properties
as joint faulting, slab transverse cracking, and smoothness (IRI) based on the site
conditions. These performance parameters are considered the most important in
pavement design since they directly affect the riding comfort for the people
travelling on the roads. Modern methods with high capacity for calculations and
reliability prediction based on observed data have made it possible to consider
further parameters in the design. It is nevertheless difficult to understand exactly
which parameters to investigate and how these particular parameters change over
time in ever changing conditions.
The Swedish design method
For pavement design, the procedure in Sweden is similar to the methods used in
many European countries (Silfwerbrand 2001). The procedure consists of the
determination of the acceptable number of load applications for a selected pavement
type. Taking seasonal effects into account, the Miner-Palmgren’s hypothesis is used
to determine the required pavement structure for each of the traffic loading and
subgrade bearing capacity combinations. In the design, the diurnal combinations of
thermal loads and traffic loads are critical for determining the bearing capacity of the
road. For this it is necessary to calculate tensile stresses in the concrete pavement and
the road base but also the vertical compressive strain on top of the subgrade. Tensile
stresses in the road base are, however, often neglected for a cement bound base since
it is considered cracked under the concrete pavement. The procedure is simple in a
way that stresses are calculated assuming linear elastic properties of the materials.
The likelihood of the method to lead to unsuitable thicknesses of the pavement is
imminent since specific conditions are difficult to take into account, i.e. the true
properties of the materials. It is believed that in most cases the method results in too
thick pavements, taking into account the safety margins needed and the observed
performance of roads constructed in Sweden.
Traffic load calculations in Swedish method. In traffic estimation the heavy trucks
are represented by vehicles with a total load exceeding 3,5 metric tons and a
wheelbase exceeding 3.3 meters. The load from these vehicles is substituted by an
equivalent single axle load (ESAL) of 100 kN. The heavy trucks are given a factor of
0.9 to 2.0 ESAL’s depending on the type of pavement. Estimating the traffic, and
consequently the number of load applications that the concrete slab has to withstand,
it is also important to take into account the traffic wandering over the lanes, the
design period, and the growth factor.
Stresses due to traffic were first calculated with Westergaard’s equations,
improved by Eisenmann (Eisenmann 1979). These equations are used to calculate the
stresses in the centre, on the edge, and in the corner of the concrete slab. Today,
stresses due to traffic are calculated with an elastic multi-layer program, GIPI (van
Cauwelaert 1986). In this program, it is possible to define five layers to be
3
calculated, each of them given properties on thickness, module of elasticity, and
Poisson’s ratio. Between the concrete and the bound base no bond is assumed. This
is not always the case (contrarily, bond is assumed in industrial pavements) and a
higher flexural stress is hereby calculated in the concrete (Silfwerbrand 2001). The
design is because of this rather conservative providing additional safety to the design.
Calculations have to be made for each season of the year to capture the differences in
strength of the materials and the depth of frozen material with the changing
temperatures. The linear elastic layer program is infinite in the horizontal plane and
stresses on the edges are therefore not possible to obtain. The case of loads near the
edge is either avoided by markings on the road or by thickening the edge according
to a distress twice as high as the computed stresses for internal loading. The last
option is more common for industrial pavements. This practice is adopted from
results from analysis with the two-dimensional finite element program ILLISLAB
that indicated a 100% increase of stresses when approaching the edge of the slab
(Silfwerbrand 1995). The load efficiency factor between doweled joints for concrete
slabs on bound bases is considered to be 100 percent.
Traffic load calculations in U.S. Design Guide. The U.S. design Guide considers a
broad variety of axle loads and axle configurations (Transportation Research Board
2004). The number of heavy trucks is calculated in the same manner as in Sweden
but instead of one single axle, several different types of wheel loads can be modelled
simultaneously. Mechanistically, this is interesting because the impact of more than
one wheel on the concrete slab generates other types of distresses and damages on
the pavement, e.g., top down cracking. The possibility to change anything from axle
configuration to traffic distribution hour by hour is developed in a way that the traffic
can be modelled with such accuracy that the main error lies within the traffic
estimations. Table 1 shows the inputs for the Swedish design in comparison to the
U.S. Design Guide.
Three main types of traffic related distresses for plain jointed concrete
pavements are observed for analysis with the Design Guide; stresses that develop
top-down cracking, bottom-up cracking, and joint faulting damage. The loads
generating these distresses are identified at three critical locations on the concrete
slab where the impact of the traffic loads is the most important (Transportation
Research Board 2004, Darter 2001). For bottom-up cracking the critical point is
situated at mid span on the edge of the slab. The same location is critical for topdown cracking but is produced by two axles acting simultaneously, developing
tensile stresses on top of the slab between the axles. Faulting occurs when the loads
are concentrated, and causing deflections on either side of a joint. The critical point
regarding faulting is the corner of the slab.
For transverse joints, the total load transfer efficiency (LTE) includes the
contribution of three major mechanisms of transfer; by concrete aggregates, by joint
dowels, and by the base course (Transportation Research Board 2004).
Calibrated mechanistic based models have been developed to predict faulting,
cracking, and IRI. The faulting is determined incrementally and considers the effect
of previous maximum faulting, current faulting level, and differential energy. IRI
depends on the initial smoothness, the change in distress, i.e., cracking and faulting,
and age, subgrade type, and climate. The model predicts smoothness incrementally
over the design life (Darter 2004, Transportation Research Board 2004).
4
Stresses due to both traffic and thermal loads are calculated with a finite
element program, ISLAB2000. The procedure is explained further in the section
describing thermal loads calculations.
Table 1. Comparison of design inputs for traffic calculations. The Swedish
method against U.S. Design Guide
Traffic Estimation Parameters
AADTT
Monthly adjustment factors
Vehicle class distribution
Hourly truck traffic distribution
AADTT distribution by vehicle class
Traffic growth factor
Axle load distribution factor
Wheel location
Traffic wandering
Lane width
Axle spacing
Axle width
Tyre spacing
Tyre pressure
Dual tyre spacing
Sweden
X
X
X
-
USA
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Discussion. The U.S. Design Guide is, compared to the Swedish design method,
more sophisticated considering the amount of inputs that are taken into account, see
Table 1. The Swedish method does not take more than one single axle into
consideration in comparison to the different axle spectra that defines traffic loads in
the Design Guide. Introducing multiples axles into the design enables to consider the
effect of tensile stresses on top of the concrete slab.
The reliability in traffic estimation also has a significant effect on the traffic
calculations, and the Swedish design method is sensitive to the number of ESAL’s
per heavy vehicle, e.g., increasing the factor from 1 to 2 equals a theoretical decrease
in pavement design life from 20 to 10 years (SNRA 2003).
Thermal load calculations with Swedish method. Stresses due to thermal loads are
calculated with Eisenmann’s beam equation (Eisenmann 1979). The critical length
Lcr is the decisive factor that determines the geometry of the slab. The stresses at the
bottom of a simply supported concrete beam is calculated according to the following
equations:
V temp
1.2 ˜
V temp
D ˜ 't ˜ h ˜ E § L
˜¨
2 ˜ (1 Q ) ¨© Lkr
1.2 ˜
D ˜ 't ˜ h ˜ E
2 ˜ (1 Q )
5
·
¸¸
¹
2
if L d Lcr
(1)
if L ! Lcr
(2)
Lcr
4D'tE
h
5 ˜ (1 Q ) ˜ J
(3)
where, D is the coefficient of thermal expansion of concrete, 't is the temperature
gradient, Q is the Poisson’s ratio, E is the modulus of elasticity, h is the thickness
of the beam, and J is the dead load per unit length.
The climate is characterised by temperature and sunshine and is described by
the magnitude and length of the thermal gradients. In Sweden an extreme gradient of
60°C/m has been chosen for 5 percent of the year, and 40°C/m for 20 percent.
During the rest of the year, a thermal gradient of 0°C/m is assumed. These values,
originated from field studies in Germany, were modified for Swedish conditions by
Petersson (Petersson 1990) for the new Swedish specifications of the 1990’s.
Thermal load calculations with the U.S. Design Guide. In the Design Guide, the
thermal loads in the concrete slab are calculated with a two-dimensional finite
element program, ISLAB2000. The temperature variations in the Design Guide are
treated by computerised means where the daily variations are assembled through
weather station data, obtained from the National Climatic Data Centre database.
(Transportation Research Board 2004)
The climatic information is given by providing the pavement location
(longitude and latitude) and the elevation. It is recommended that the data be chosen
from several weather stations to compensate for missing data in any one of the
weather stations. The hourly inputs contain air temperature, precipitation, wind
speed, percentage sunshine, and ambient relative humidity. Inputs provided from site
such as seasonal or constant water table depth give the climate model sufficient
information to monthly recalculate material properties over the entire design period.
The temperatures directly affect the concrete slab properties with transient
hourly negative and positive nonlinear temperature differences between top and
bottom, caused by solar radiation and computed using the Enhanced Integrated
Climate Module (EICM). The temperature and other climate properties also have an
indirect affect on the slab by changing the subgrade properties, i.e., subgrade strength
and stiffness.
The analysis requires several input parameters described in Table 2. The
structural model for stress computations only considers the concrete slab and the
base course. For all layers beneath these two layers a dynamic k-value (psi/in) is
assigned through calculations with a linear elastic layer program. In this program, a
Falling Weight Deflectometer (FWD) load is simulated. The computed deflection
that is produced by the FWD is then used to backcalculate the appropriate k-value.
The k-value is recalculated for every month to reflect changes in material properties
due to climate changes and is used directly for computations of stresses and
deflections. (Transportation Research Board 2004)
To enable rapid solutions neural networks (NN) have been developed based
on results from ISLAB2000. This approach is needed to deal with the large numbers
of calculations that are required (Transportation Research Board 2004).
The thermal loads are calculated from a temperature profile of 11 point
through the slab. The thermal gradients are generated from input data on
temperatures in these points, creating nonlinear gradients.
6
Table 2. Parameters considered for calculation of distresses in the
Comparison between the Swedish method and the U.S. Design Guide.
Parameter
Sweden
Material
Thickness
X
Properties
Modulus of elasticity
X
Coefficient of thermal expansion X
Unit weight
X
Subgrade properties
X
Loads
Nonlinear temperature gradient
Linear temperature gradient
X
Axle type, weight, and position
Single axle
Design features
concrete slab.
USA
X
X
X
X
X
Hourly input
Several
combinations
Bond/no bond X
X
Variable
X
Variable
X
X
100%
X
X
Interface conditions
Joint spacing
Lane width
LTE - base layer
LTE - dowels
LTE - aggregate interlock
Shoulder type
}
Discussion. Eisenmann’s equation, used in the Swedish method, is based on the
assumption that the thermal gradient is linear and the chosen gradients are all
positive, i.e., during daytime conditions. Negative gradients are not considered in the
design since they are less significant and are counterbalanced by the traffic loads.
The Design Guide, on the other hand, automatically computes hourly variations of
both positive and negative nonlinear gradients.
Farhang investigated how the stresses are calculated and compared
Eisenmann’s beam equation to results obtained by ISLAB2000 (Farhang 2004). The
results showed that Eisenmann’s method produces almost twice as high stresses
assuming linear, positive gradients. Farhang also showed that by introducing a
nonlinear positive temperature gradient (based on measurements) the tensile stresses
are reduced further.
Damage from stresses and fatigue
The concrete slab is, as mentioned above, affected by both thermal and traffic loads.
These loads act simultaneously with different magnitudes, hour by hour. The loads
produce stresses in the concrete and eventually, after sufficiently many repetitions,
cause fatigue damage.
Fatigue damage analysis for concrete in Sweden. The number of allowable load
applications for each thickness under each part of the year is calculated with Tepfers
fatigue equation (Tepfers 1979)
V max
f ct
§ V
1 0.0685 ˜ ¨¨1 min
© V max
7
·
¸¸ ˜ log N .
¹
(5)
The number of allowable load applications are also calculated with respect to
the deformation in the subgrade according to:
N
8.06 ˜ 10 8
(6)
H Z4
where, N is the number of load applications, V min is the minimum stress, i.e.
temperature stress, V max is the maximum stress, i.e., the traffic and thermal stress
superimposed, f ct is the concrete flexural strength, and H Z is the vertical
compressive strain on top of the subgrade. The stresses are only calculated in the
bottom of the slab.
The accumulated fatigue damage that a certain pavement can resist is
calculated with Miners-Palmgren’s damage hypothesis for flexural stresses in the
concrete pavement, horizontal tensile strains in the base course or the vertical
compressive strain of the subgrade. The accumulated damage is computed by
summing fatigue damage incurred during each of the 6 seasons of the year due to
both traffic and thermal loads.
The accumulated damage is calculated with the following equation:
n
Di ˜ Ni
¦N
i 1
d1
(7)
i, allow
where, D i is the percentage of time in which damage i occurs, Ni is the total number
of loads corresponding to damage i, and Ni,allow is the number of allowable loads
corresponding to damage i.
Damage analysis in the Design Guide. The Design Guide contains prediction
models for calculating joint faulting, slab transverse cracking, and IRI. The models
are calibrated with data from JPCP sections in the field.
The maximum bending stresses and bending strength are used to compute the
number of allowable axle load applications and aggregate interlock wear due to each
wheel load for each time increment using the fatigue relationship (Darter 2004,
Transportation Research Board 2004):
logN i , j , k ,l , m, n § MRi
C1 ˜ ¨
¨V
© i , j , k ,l , m , n
·
¸
¸
¹
C2
1
(10)
where, MRi is the modulus of rupture at age i for concrete, Ni,j,… is the allowable
number of loads applications at specified condition, ıi,j,k,…is the applied stress at
specified condition, and C1 and C2 are calibration constants.
The fatigue damage, for all traffic load increments over the design period is
calculated in the same way as in Sweden with Miner-Palmgren’s hypothesis. The
fatigue is calculated both on the top and bottom of the slab and the conversion to
physical pavement distress, i.e., bottom-up and top-down transverse cracking, is
related to observed cracking in the field through calibration.
8
Comparing Swedish and U.S. Design through a Case Study
In a trial to analyse the Design Guide, a section of a Swedish road is evaluated with
the program. The particular section is a highway north of Stockholm, Sweden, that
will be constructed as a plain jointed concrete pavement in 2006.
The following calculations with the Design Guide are not exact because no
laboratory testing for this purpose have been done prior to this investigation. Many
of the needed parameters are not directly considered in Swedish design and
calculations are due to these reasons made on a level 3 basis; with a minimum of
inputs. Materials parameters are, for instance, chosen from recommended values in
the Guide. Traffic distribution by vehicle class and axle load values are also taken
directly from the Guide.
The objective is to evaluate the impact of some of the parameters in the
design of new concrete pavements. The inputs are changed one by one and the effect
in performance at the end of the design life is evaluated. Different improvements
may affect performance on faulting but have less effect on the cracking ability. It is
beneficial to know what parameters are the most significant in the design and know
under which circumstances they are valid, i.e., different climate or traffic conditions.
It is also essential to scrutinize the most important parameters and develop
new criteria to be able to judge performance in future Swedish design.
General Information. The highway, E4 Uppsala, is situated north of Stockholm. A
23 km long section will consist of a plain joint concrete pavement with 2 lanes in
each direction. Traffic is estimated to 12 500 AADT from which 17% are heavy
vehicles, i.e., vehicles exceeding 3,5 metric tons. Construction starts in July 2006 and
the highway will open to traffic in October 2007.
Material Data. The pavement (Figure 1) consists of a wear layer of concrete, a
bound base of asphalt, two subbase courses of compacted unbound materials, and a
subbase of crushed gravel. The subgrade consists of a typical Swedish, low strength
and silty, material.
Figure 1. Pavement system according to the Swedish specifications.
9
Climate conditions. For each climate zone, the number of accumulated negative
degrees per day is given. The site is situated in climate zone 2 in Sweden and the
climate properties are given for six seasons with a corresponding number of days, see
Table 4. For climate zone 2, 300-600 accumulated negative degrees Celsius per day
is estimated. This means, for instance, that 300 negative degrees per day equals 60
days with an average day temperature of –5ºC. The maximal frost depth is 2.0
meters.
Traffic Data. The Average Annual Daily Traffic (AADT) for one lane, in one
direction, is estimated to 12 500 vehicles of which 17 % are heavy trucks. Traffic is
also estimated to increase annually with a growth factor of 2.6%. This value is not
really reliable considering a design life of 40 years but should be accounted for over
the first 15-20 years.
Design with the Swedish method
Material data. The pavements system is from start chosen from a standard design.
Properties, e.g., elastic modulus and Poisson’s ratio for the different layers and
different seasons, are retrieved from the Swedish specifications (SNRA 2004), see
Table 3.
Traditionally, concrete pavements for highways are constructed with concrete
with a flexural strength of 6 MPa and an elastic modulus of 36 000 MPa.
Traffic loads. For the design of this road section a factor of 2.45 standard axles per
heavy vehicle and a growth factor over 40 years of 2.6 % is estimated. The total
number of estimated ESAL’s are shown in Table 3.
Table 3. Traffic calculation for E4 Uppsala
AADT
Percent of trucks, A
Number of axles per truck, B
Traffic growth, n
Design life, years
Number of ESAL’s, Nekv
12500
17 %
2,45
2.6 % (compound)
40
134 million
The traffic stresses are calculated for a 100 kN standard axle with a tyre
pressure of 0.8 MPa. The stresses induced by this axle are computed with an elastic
multi-layer program and no bond between the concrete and the road base is assumed.
The load transfer efficiency is considered to be 100 percent in dowelled joints
between two slabs on bound bases. The calculations are based on a slab with 5 m
joint distance.
Thermal loads. The pavement is affected by a positive thermal gradient of 40°C/m
under 20 % of the year and 60°C/m under 5 % (SNRA 2004). These gradients take
place under the summer period. Thermal stresses are calculated with Eisenmann’s
beam model, Eq. 1-3.
10
Table 4. Material and climate data (SNRA 2004). (Note: 200 mm = 7.9 in, 36 GPa =
5.2 Mpsi)
Material
Concrete
Asphalt base
Unbound base
Unbound
subbase
Crushed rock
Subgrade
Climate period
Freeze- Spring
Winter
Spring Summer Autumn
thaw
thaw
80 days 10 days 31days 15days 153 days 76 days
Thickness
[mm]
200
100
80
36 000 36 000
11500 10000
1000
150
36 000
9000
300
36 000 36 000
8500
2500
450
450
36 000
8000
450
0.2
0.4
0.35
220
450
450
450
450
450
450
0.35
850
1000
1000
1000
1000
70
10
85
20
100
45
100
45
0.35
0.35
Poisson’s
ratio
Modulus of elasticity [MPa]
Results. The limitation for the pavement system, is the vertical compressive strain of
the subgrade, see Table 5. The requirement for traffic class 7, i.e., traffic exceeding
19·106 standard axles, is nevertheless fulfilled. The thickness of the pavement
structure is satisfactory against frost heave.
Table 5. Number of permissible axle loads for the pavement system calculated with
the Swedish method. The subgrade is the weakest component in the system.
Pavement layer
Concrete
Asphalt base
Subgrade
Allowable ESAL’s
2.65·1012
7.93·108
2.28·108
Evaluation of the Design Guide
Materials. The materials for the structure are chosen to correspond to the actual
material properties in the Swedish specifications. The Design Guide is most accurate
after affecting thorough analysis of the materials in the laboratory. Because of this,
the input values for the unbound materials are, without exception, chosen from the
default values provided by the Guide. The common rule for the choice of elastic
modulus, or corresponding strength properties, is that the deeper under the slab the
material is situated, the lower the modulus. No stress-dependent properties are
accounted for and the climate model is restrained from altering the modulus over the
months, i.e., specified modulus values are given for each month.
Traffic. The Design Guide inputs for traffic is the number of AADT and the number
of heavy vehicles. These parameters are computed in the same way as in the Swedish
method. The inputs concerning the traffic are then chosen through the help of default
values recommended in the guide as showed in Table 6.
11
Table 6. Inputs for traffic calculation in the Design Guide.
Parameter
Operational Speed
Volume Adjustment factors
Vehicle class distribution
Hourly truck traffic distribution
Traffic growth
Axle load distribution factors
Mean wheel location
Traffic wander std deviation
Number of axles per Truck
Axle Configuration
Input
60 mph (100 km/h)
Default, 100% for all vehicle classes
TTC 8 "High percentage of single-trailer
truck with some single-unit trucks."
Default, maximum of 6% per hour
around noon
2.6% (compound)
Default, level 3
18 in (457 mm)
10
Default for single-, tandem-, tridem-, and
quad axles
Default
Climate. The New York climate was chosen to represent the climate of this particular
design because it is well defined and does not bring too extreme temperature
variations to the calculations.
Design features – Structure. The dowels, the slab size, and the joint spacing are the
constructive features that have been altered in the design, see Table 7. Case 4
corresponds to Swedish practice regarding these features.
Table 7. Five different design features analysed with the Design Guide.
(Note: 5.0 m = 16.4 in. 25 mm = 1 in.)
Case 0 Case 1 Case 2 Case 3 Case 4
Joint spacing
5.0 m
4.6 m
4.6 m
4.6 m
5.0 m
Dowel diameter 25 mm 25 mm 38 mm 38 mm 25 mm
Slab width
3.7 m
3.7 m
3.7 m
4.3 m
4.3 m
Results. The results show that the design features, e.g., joint design and edge support,
have significant importance in crack development and damage accumulation under
the design period.
The most effective parameters to avoid premature cracking, faulting and rapid
damage accumulation is done by the use of widened slabs, decreased joint spacing,
increased dowel diameter, increased thickness of the concrete slab, and higher
strength of the concrete. The faulting is also affected by the capability of erosion of
the base and how the slab shoulder is tied to the adjoining pavement. (Darter 2004,
Transportation Research Board 2004)
The first model, Case 0, with the same parameters for joint spacing and dowel
diameter as in Swedish practise shows the most damage. The slab width is, however,
the same as the lane width, contrarily to Swedish practice where the slab is more or
less square with a maximum width of 4.5 m. Because of the fact that the properties
are not correct regarding materials we only consider this model as the basic model,
the model from where we evaluate the following ones.
12
The model in Case 1 has shorter joint spacing. The percentage of cracking is
decreased and illustrated by decreased bottom-up and top-down cracking. The LTE
is about the same as for Case 0.
In Case 2, the joint spacing is shorter, the dowel diameter is enlarged, and the
slab width is increased. In this case, the LTE is not affected at all over the entire
design period, see Figure 2.
Shortening the joint spacing, increasing dowel diameter, and widening the
slab yields the best result in comparison to the basic model. This is done in Case 3.
Bottom-up cracking is significantly decreased.
In Case 4, the basic model is simply modified by a widened slab. The result
shows on a significant improvement in cracking development. Faulting is, however,
considerably increased after 20 years as shown in Figure 6 and the LTE is,
consequentially reduced, see Figure 2.
The main results from this limited study show that faulting is significantly
depending on the dowel size, and that both top-down and bottom-up cracking can be
controlled by slab size. The bottom-up cracking is more dependent of the width of
the slab. IRI and LTE can be related to cracking and faulting and are, consequently,
dependent of the same parameters, see Figure 6 and Figure 7.
100
Case 2
Case 3
90
Case 4
80
Load transfer efficiency, %
70
Case 0
Case 1
60
50
40
30
20
10
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
Pavement age, years
Figure 2. Load transfer efficiency during design life for cases 0-4. Cases 2
and 3 show the beneficial influence of enlarged dowel diameter.
13
1,0
0,9
Case 1
Case 2
0,8
Case 0
Cumulative damage
0,7
0,6
0,5
0,4
0,3
0,2
Case 4
0,1
Case 3
0,0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
Pavement age, years
Figure 3. Bottom-up cracking over the design life for cases 0-4. Case 3 and
case 4 show the efficiency of widened slabs to avoid bottom-up cracking.
1,0
0,9
0,8
Cumulative damage
0,7
0,6
0,5
Case 0
0,4
0,3
0,2
Case 1
Case 2
0,1
Case 4
Case 3
0,0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
Pavement age, years
Figure 4. Top-down cracking over the design life for Cases 0-4. Cases 3 and
4 show the efficiency of widened slabs in the development of top-down cracking.
Case 1 and 2 have shorter joint spacing compared to Case 0.
14
100
90
80
Percent slabs cracked, %
70
Case 0
60
50
Case 1
40
30
20
10
Case 4
Case 2 Case 3
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
Pavement age, years
Figure 5. Cases 0-4. Predicted slabs cracked over the design life, considering
the effect from both bottom-up and top-down cracking. The damage is predicted on a
50% reliability level.
0,20
Case 0
0,18
Case 1
0,16
0,14
Faulting, in
0,12
Case 4
0,10
0,08
0,06
Case 2
0,04
0,02
Case 3
0,00
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
Pavement age, years
Figure 6. Faulting development over the design life for cases 0-4. The most
effective parameters are dowel diameter and joint spacing.
15
600
540
480
420
IRI, in/mile
360
300
240
Case 0
Case 1
180
Case 2
120
Case 4
Case 3
60
0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
Pavement age, years
Figure 7. IRI growth for cases 0-4. Similarly to faulting, the IRI is controlled
by dowel diameter and joint spacing.
Discussion and Conclusions
The design of concrete pavements in Sweden is based on specifications established
in the 1990’s. The procedure offers little change when altering parameters like
climate, traffic, and material properties. In a first step towards the development of a
new and more flexible method, an inventory of the U.S. Design Guide has been
conducted. This paper focuses on the differences in traffic and thermal load
calculations, as well as design criteria and damage analysis. Apart from the direct
comparison of defined parts of the design procedure, a quantitative study of a
highway project in Sweden has been performed with both methods.
The Swedish method considers distresses in the concrete slab, the bound base
course, and the subgrade. The study shows, for instance, that the subgrade is decisive
for the design. In the U.S. Design Guide the distresses in the pavement itself, i.e.,
cracking, smoothness, and joint faulting, are decisive in the design
The Swedish method accounts for bottom-up cracking and calculations with
the Design Guide show that bottom-up cracking is critical in the design. Nonetheless,
both top-down cracking and the load transfer efficiency contributes to the over all
performance. The Design Guide is flexible in such a way that, by changing
parameters for joint design, it can demonstrate how cracking or faulting can be
decreased.
A level of damage of approximate 50%, can in the Swedish method, in this
particular case study, be formulated in the subgrade after 40 years, i.e., by dividing
the total number of ESAL’s with the allowable number ESAL’s calculated for the
subgrade. The U.S. Design Guide reveals instead, on a 50% reliability level, a
predicted 5 % damage in the concrete slab (Figure 5).
The level of distress used as design criteria in the U.S. Design Guide is to a
certain extent dependent on the type of road that is designed. The designer
16
establishes the allowable limit of cracking and expresses the criteria in terms of
percentage of cracked slabs, percentage of LTE, or inches of faulting.
The calculations performed with the Design Guide are not accurate for the
reason that the many input parameters, especially for materials in the subbase and
subgrade, have not been evaluated regarding U.S. standards. Most inputs are taken
directly from the recommended values provided by the Guide, i.e., level 3 model
with the least possible inputs. The aim is, however, to model the Swedish pavement
system and investigate in how to interpret the results.
The criteria for road construction in Sweden are different in that manner that
it is difficult to put a number on the effective, accumulated damage. In the method,
the safety margins are achieved by underestimating mechanistic properties.
The performance criteria must, nevertheless, be measurable in terms of
relating percentages of distress to established criteria in preceding methods. Cracking
is for instance, in Sweden, related to the flexural strength of concrete beams. The
safety lies within the assumption that a slab is more resistant to stresses than a beam
due to bi-axial load bearing, and that the stresses are calculated for the characteristic
strength of the material, i.e., 28-days flexural strength (Silfwerbrand 1995, SNRA
2004).
The performance criteria in the Design Guide are developed with the intention
that designers can decide what level of performance is acceptable for the actual
design. The design can be weighed to the cost, i.e., using existent or available
materials, chose simpler joint design features etc.. The modelling example
considered in this paper could, for instance, be further improved by varying materials
in the pavement system or choosing a stronger base course. The way of proceeding to
achieve the best possible pavement depends much on the resources available and the
engineering experience. The U.S. design procedure is different to methods applied in
Sweden. Attempting to adopt a similar method requires new procedures to determine
specific environmental effects, integrating new material characterisation methods,
and assembling data for regional calibration of damage models.
Calibration data for a mechanistic based design, similar to the data used for
the U.S. Design Guide, is for Swedish conditions very limited since less than 1 % of
the national roads consist of concrete pavements but many of the necessary tools to
establish a new methodology in Sweden are already at hand. Climate data are, as an
example, collected continuously for weather predictions and could be retrieved and
applied in the same way as in the U.S. Design Guide.
New methods to estimate traffic are also being developed in Sweden. Here,
information on actual loads and actual axle configurations that are trafficking the
roads in Sweden are being assembled. Traffic is divided in traffic classes in the same
manner as in the Design Guide and these results will be applicable in future design
(SNRA 2002). Material properties in Sweden originate from FWD tests conducted
by the Swedish National Road and Transport Research Institute. Typical values for
elastic modulus for different materials have been established in accordance with the
SNRA. New methods to estimate nonlinear material behaviour will have to be
utilised to get necessary information for pavement design if a more optimised
method is chosen.
Further Research
This Ph.D. project is a part in the development of a new design method for concrete
roads in Sweden.
17
The project will involve further investigation in international methods for
design of concrete pavements and especially a deeper understanding of the
methodology applied in the U.S. Design Guide. Further analysis of the Design Guide
will include a profound examination of the material characterisation methods, an
analysis of the calibration procedure, and an evaluation of the procedure in Swedish
conditions.
Furthermore, laboratory tests on concrete and lean concrete will be conducted
in order to verify and analyse the existing fatigue criteria in Sweden, and parameters
for climate affects will also be analysed within this project.
Together with the Swedish National Road and Transport Research Institute
and the Swedish National Road Administration a new computer-based design
method for concrete pavements will be chosen, modified for Swedish conditions, and
implemented in a computerised design guide.
Acknowledgment
The financial support provided by Cementa AB, the Royal Institute of Technology,
and the Swedish Agency for Innovation Systems is gratefully acknowledged.
18
References
Darter M., et al. (2001). “Development and calibration of a mechanistic design
procedure for jointed plain concrete pavements.” 7th Int. Conf. on Concrete
Pavements, ISCP, Orlando, USA, Sept. 9-13, 113-131
Darter M., et al. (2004). “Design of jointed concrete pavement using incremental
damage”. 5th Int. CROW-workshop on Fund. Modelling of the Design and
Performance of Concrete Pavements, Istanbul, Turkey, March 1, CROW Technology
Centre, The Netherlands
Eisenmann, J. (1979). “Betonfahrbahnen.“ Handbuch für Beton-, Stahlbeton und
Spannbetonbau. Verlag von Wilhelm Ernst & Sohn, Berlin, München, Düsseldorf,
305 pp. (In German).
European Commission, Directorate General Transport (1999). COST 333:
Development of New Bituminous Pavement Design Method., Office for Official
Publications of the European Communities, Luxembourg, 373 pp.
Farhang A. (2004). “Calculation of thermal stresses for jointed plain concrete
pavements based on non-linear temperature gradients using modified Eisenmann’s
equation and modified Silfwerbrand’s plate method.” 5th Int. CROW-workshop on
Fund. Modelling of the Design and Performance of Concrete Pavements, Istanbul,
Turkey, March 1, CROW Technology Centre, The Netherlands, 27pp.
Löfsjögård, M. (2003). Functional Properties of Concrete Roads, PhD thesis., Royal
Inst. of Tech., Dep. of Civ. and Arch. Eng., Stockholm, Sweden.
Petersson, Ö. (1996). Swedish Method for the Design of Concrete Pavements,
Licentiate thesis., Royal Inst. of Tech., Dep. of Civ. and Arch. Eng., Stockholm,
Sweden. (In Swedish)
Silfwerbrand, J. (1995). Design of Concrete Pavements (Dimensionering av
betongbeläggningar), Report No. 9, 2nd Edition., Royal Inst. of Tech., Dept. of
Struct. Eng., Stockholm, 91 pp. (In Swedish).
Silfwerbrand, J. (2001). “Swedish design of industrial pavements”. 7th Int. Conf. on
Concrete Pavements, ISCP, Orlando, USA, Sept. 9-13, 791-806
SNRA (2003). Inspection and measurements of heavy traffic loads on Swedish roads
2002 and 2003(BWIM-mätningar 2002 och 2003), Final report., Publ.2003:165.,
Swedish National Road Adm., Borlänge, Sweden, (In Swedish).
SNRA (2004). Technical Guidelines for Road Structures. Part 1 – General
Prerequisite and Part 3 – Structural Design of the Pavement System. Publ. 2004:111,
Swedish National Road Adm., Borlänge, Sweden, 2004, 39+80pp (In Swedish).
Tepfers, R. (1979). “Tensile Fatigue Strength of Plain Concrete.” ACI Journal, proc.
76(8), Aug., 919-933.
19
Transportation Research Board (2004). Guide for Mechanistic-Empirical Design of
New and Rehabilitated Pavement Structures, Part 1-3+App. AA, JJ, QQ, Final
Report/Document., NCHRP 1-37A, Washington, D.C.
van Cauwelaert, F. (1986). “Computer Programs for the Determination of Stresses
and Displacements in Four Layer Systems with Fixed Bottom.” Centre de
Recherches de l'Institut Superieur Industriel Catholique du Hainaut, Mons, Belgium,
61 pp.
20
Flexural Fatigue of Plain Concrete Beams
J. Söderqvist & J.Silfwerbrand
Swedish Cement & Concrete Research Institute, Stockholm, Sweden
Abstract
This paper investigates the applicability of Tepfers’ fatigue equation to flexural
strength of plain concrete. The flexural fatigue is an important property in pavement
design and given that Tepfers’ fatigue equation, originally, was developed out of
compression and splitting tests, the relationship to flexural tests has not been fully
investigated. In this study twelve plain concrete beams have been subjected to
flexural loading and the results have, with good agreement, been compared to other
research results. Palmgren-Miner’s partial damage hypothesis has also been analysed
with flexural tests with sound deviation compared to theory.
1 Introduction
The fatigue strength is an important property, which has to be taken into account in
the design of various structures requiring long fatigue life. Although the major part
of research on fatigue of plain concrete is devoted to compressive stresses, it is also
recognized that tensile fatigue in bending is an important factor in the design of
certain types of structures, such as roads and airfields. In Swedish pavement design,
Tepfers’ fatigue equation is commonly used. The fatigue principle is based on
compressive and splitting tests and, since the material can be described with the same
equation for both compressive and tensile stresses, the same equation has also been
used, tentatively, for tensile stresses in bending, i.e. flexural stresses.
This paper investigates fatigue strength of plain concrete beams subjected to
flexural loading. First, twelve plain concrete beams were tested to determine the
fatigue strength and, second, the results were compared with other tests performed on
concrete beams, prisms, cylinders, and columns over the years. Tepfers’ fatigue
equation is analysed and modified by the introduction of an additional constant in the
formula. The Palmgren-Miner partial damage hypothesis is examined as well.
1.1 Previous Research
The fatigue strength of concrete is normally defined as a fraction of the static
strength that the material can support for a given number of stress cycles, N. The
most common way of illustrating this is by Wöhler curves, also called S-N-curves,
see Figure 1. The logarithm of the number of load cycles to failure, N, at a specific
maximum stress level, Vmax, is plotted in the diagram. For each curve the relationship
between the minimum and the maximum stresses is held constant.
1
1
Relative Load
0,8
0,6
R=0,5
R=0,4
R=0,3
R=0,2
R=0,1
R=0
0,4
0,2
0
1,E+00
1,E+01
1,E+02 1,E+03 1,E+04
Number of load repetitions, N
1,E+05
1,E+06
Figure 1. Wöhler diagram with constant values on R.
The principal form of the current fatigue formula has its origin in the work
conducted by Aas-Jacobsen in 1970 (Aas-Jacobsen, 1970). Aas-Jacobsen performed
a series of tests on prisms, prestressed beams and eccentrically loaded columns, and
formulated
V max
f cfl
§ V
·
1 E ˜ ¨1 min ¸ ˜ log N
¨ V
¸
max ¹
©
(1)
where,
N
= the number of load applications
= the minimum stress
= the maximum stress
f cfl
= the concrete strength
E= a material constant
V min
V max
R
V min
V max
The equation considers a linear relationship between the minimum and maximum
stress ratio making the equation independent of the concrete strength. Aas-Jacobsen’s
tests showed that failure will not occur due to fatigue if the load is less than 60 % of
the static stress limit, that repeated loading has little effect on the ultimate load if the
2
specimen does not fail due to fatigue, and that the deflection at 1 million load cycles
is 40 - 100 % larger than the deflection after the first load cycle. In Aas-Jacobsen’s
investigation the material constant ȕ = 0.064.
Tepfers (Tepfers & Kutti, 1979, Tepfers, 1979, and Tepfers, 1978) examined
the fatigue of concrete with compressive and splitting tests on concrete cubes. The
tests were also compared with other research results from compressive tests on
ordinary and lightweight concrete (data are presented in (Tepfers, 1978)). Tepfers
concluded that Aas-Jacobsen’s fatigue equation (Equation (1)) is valid for plain
concrete but suggested ȕ = 0.0685.
Plain concrete in flexure has been recognized as an important factor in the
design of roads and airfields and therefore, and as the methods to test concrete in
flexure are growing more accurate, this has been done in many countries. This type
of testing is associated with inexactness because small imperfections in the
specimens can cause large discrepancies (Tepfers & Kutti, 1979). Also, fatigue
testing is time-consuming, especially in the high-cycle region, i.e. more than 1 000
load cycles.
Over the years, a number of studies have been conducted to investigate the
fatigue of plain concrete. These studies have different approaches and the most
significant ones are listed in this section.
Hsu (Hsu, 1981) has tested beams to investigate the three-variable
relationship of f, N and R with the aim of developing a four-variable relationship f, N,
T, and R, where T is the period of repetitive loads. Hsu suggests two different fatigue
equations, one for the low-cycle region, and one for the high-cycle region. For the
high-cycle region Equation (2) is suggested.
V max
f cc
§
·
V
1 0.0662 ˜ ¨1 0.556 min ¸ ˜ log N 0.0294 log T (2)
¨
V max ¸¹
©
In comparison to Tepfers’ fatigue equation the R-value is decreased (by a
factor of 0.556) and the loading time, T, is introduced. ȕ is still very near what
Tepfers’ suggested. The resulting R-value reduces the impact of the amplitude,
especially noticeable in case of high R-values.
Also, completely empirical models have been and are being developed to
predict the fatigue life of concrete structures. These models are specifically
developed from long-term testing on real structures as airfields and roads and are
often formed as
V max
AN B
MR
where A and B are experimental coefficients and MR is the modulus of rupture, i.e.
f cfl
Shi has investigated flexural fatigue strength with 78 plain concrete beams in
1994 (Shi et al., 1993). Shi uses a mathematical approach to modify the power
formula with statistical tools to account for the stress ratio in Equation 2.
Oh (Oh, 1986) uses statistical tools, i.e. a Weibull distribution, to more
convincingly describe the fatigue behavior of concrete as this kind of testing exhibits
larger scattering than static tests. This approach makes it possible to calculate a
probability of failure for a given level of maximum load.
3
Other research has aimed at investigating different conditions as material
properties, time related aspect on loading, and stress reversals.
1.2 Fatigue Equation
In Sweden, Tepfers’ fatigue equation (i.e. Equation (1) with ȕ = 0.0685) is
commonly used in the design of concrete roads, airfields and industrial pavements
where flexural stresses are dominant (Silfwerbrand, 1995, SRA, 2005). The
compilation of various test results conducted by Tepfers showed that the fatigue
equation, with satisfactory results, could be used for both compression and tension
(i.e. splitting), and therefore, the same equation has been proposed and used for
flexural stresses. This hypothesis is generally accepted but has not been fully
investigated in Sweden. Tepfers also concluded that the deviation between measured
and calculated results is significant when testing specimens with a stress ratio R >
0.75. Improvements in the development of measuring equipment and more complex
arrangements in testing are considered to be useful in new testing to overcome these
problems.
2 Laboratory Tests
2.1 Test Specimens
The concrete mixture is presented in Table 1. Test specimens were cast in four
batches, each of which included eight 800 × 150 × 100 mm beams, three 150 mm
high cylinders with diameter 100 mm, and six 150 × 150 × 150 mm cubes. The
specimens were cured for four days in water and then transferred to conditioning at
20°C with a relative humidity of 50 % for 60 days. This was done to avoid uneven
moisture content during testing.
Table 1. Concrete mixture.
Swedish Concrete Grade K60
Quantity
[kg/m3]
CEM I, Std Degerhamn
Swedish cement for civil engineering
structures
Aggregate 0-8 mm, Underås
375
737
Aggregate 8-16 mm, Underås
1018
Water
149.4
Silica
26.3
Plasticiser, 92M
8.6
Air Entraining Agent, L14
300
w/c
0.40
w/b
0.37
4
2.2 Test Procedure
The flexural strength was tested in a Material Test System (MTS) 810 machine
(Figure 2). For each batch three beams were used to determine the mean flexural
strength. The beams were simply supported with a span of 700 mm, and tested in
four-point flexural loading according to Swedish standard SS 13 72 12. A total of
twelve tests were made on three different values of Vmin, corresponding to 5, 20 and
40 percent of the maximum static load. On each level four different values of Vmax
were tested. In addition, cylinders and cubes were cast to determine the elastic
modulus and the compressive strength. The fatigue tests were conducted in load
control with a sinus waveform at 0.05 Hz in the beginning and 2.05 Hz after 100
cycles. Two Linear Variable Differential Transformers (LVDT) were also mounted
on every beam to measure the midspan deflection.
Figure 2. Flexural fatigue testing with a MTS 810 machine. LVDT’s are placed on
both sides of the concrete beam, as shown.
Table 2. Intended test scheme.
Type
fcfl
Vmin/ fcfl Vmax/ fcfl
6 Static load
1
Fatigue
0.05
0.6
Fatigue
0.05
0.7
Fatigue
0.05
0.8
Fatigue
0.05
0.9
6 Static load
1
Fatigue
0.2
0.6
Fatigue
0.2
0.7
Fatigue
0.2
0.8
Fatigue
0.2
0.9
6 Static load
1
Fatigue
0.4
0.6
Fatigue
0.4
0.7
Fatigue
0.4
0.8
Fatigue
0.4
0.9
R =Vmin/Vmax n
0
3
0.083
1
0.071
1
0.063
1
0.056
1
0
3
0.333
1
0.286
1
0.250
1
0.222
1
0
3
0.667
1
0.571
1
0.500
1
0.444
1
Sum 21
5
3 Tests Results
3.1 Static Tests
Cubes were cast to determine the compressive strength of the concrete during the test
period. The compressive strength was tested at 28 days, and at the start and end of
the fatigue test period. The cubes were stored similarly to the concrete beams at 50 %
relative humidity and 20°C.
For each batch the flexural strength of the concrete was tested statically
with three beams. The mean values from these test series were used to find the
appropriate load level in the subsequent fatigue tests.
The modulus of elasticity, Ec, was determined at the start of the tests
with 150 mm cylinders with radius 100 mm. The cylinders were stored in water until
testing according to Swedish standard.
Also, the modulus of elasticity in bending, Efl, was determined from
static tests on three beams using an equation, developed for four-point flexural
loading, with the form
Efl
131^F2 F1`l 3
686^G 2 G1`bh 3
where,
l = length
b
= width
h
= height
F1
= 1/4.Fcr
F2
= 3/4.Fcr
Fcr
= failure load
G1,2
= midspan deflections corresponding to F1 and F2
All static material strengths are shown in Table 3.
3.2 Fatigue Tests
The fatigue tests were carried out according to the method described in previous
sections. Results are presented in Table 4, Figure 3, and Figure 4. Specimens that
exceeded 2 million loading cycles do not appear here. These tests were instead used
to verify the partial damage accumulation and were subjected to one, or even two,
higher load levels.
Since the fatigue tests were limited to ten beams and the discrepancies
were substantial, the relationship between the expected and the measured number of
loads can be used to visualize the fatigue strength (Figure 3). Before testing each
beam was checked for voids and other irregularities and even though no voids
appeared on the surface, some were found inside the beams after failure. It is difficult
to predict the impact of these imperfections. By including research results from other
tests it is possible to put the test results into perspective. Test results from
compression and splitting tests are put together in Figure 5.
6
Table 3. Static material strength values (mean values out of three tests).
Batch Age
fcc
Ec
Efl
fcfl
(Cubes) (Cylinders)
(Beams)
(Beams)
Id
Days MPa
MPa
MPa
MPa
1
2
3
1
2
1
2
2
3
3
3
28
28
28
60
158
161
204
231
232
294
425
75
77
78
78
86
79
84
86
86
-
36200
33300
35200
-
22100
19500
21000
5.51
5.53
5.81
Table 4. Summary of flexural fatigue tests
Id
(1)
139B
149E
139E
149H
139C
149G
149F
992e
149D
139A
Vmax/ fcfl Vmin/ fcfl
(2)
0.90
0.90
0.80
0.80
0.80
0.90
0.90
0.70
0.80
0.70
(3)
0.20
0.40
0.05
0.05
0.20
0.60
0.60
0.05
0.40
0.20
R
logNc
logNm
(4)
0.222
0.444
0.063
0.063
0.250
0.667
0.667
0.071
0.500
0.286
(5)
1.88
2.63
3.11
3.11
3.89
4.38
4.38
4.72
5.84
6.13
(6)
2.43
2.41
2.34
4.41
3.82
2.60
2.27
4.06
5.42
5.67
ȕ
(C = 1)
(7)
0.0530
0.0745
0.0912
0.0484
0.0697
0.1155
0.1322
0.0797
0.0738
0.0740
10
0.0812
ȕ
(C = 0.7556)
(8)
0.0495
0.0623
0.0897
0.0476
0.0645
0.0776
0.0888
0.0782
0.0593
0.0674
10
0.0685
n
Mean
Standard
0.0259
0.0148
Deviation
Variance
0.0007
0.0002
Note. Nc is the calculated number, and Nm is the measured number of load
repetitions. The 7th column show Evalues for Nm when C = 1 in Equation (3). The
values of the 8th column show E values for Nm when C is selected to to fit Eaverage =
0.0685.
7
Shi, Fwa & Tan, 1993 - Beams
Söderqvist, 2005 - Beams
Linear (Shi, Fwa, Tan, 1993 - Beams)
Linear (Söderqvist, 2005 - Beams)
9,000
y = 0,9389x
2
R = 0,9361
8,000
7,000
logNm
6,000
5,000
4,000
y = 0,8668x
R2 = 0,4924
3,000
2,000
1,000
0,000
0,000
2,000
4,000
6,000
8,000
10,000
logNc
Figure 3. Correlation between the calculated number of load applications, logNc,
and the measured number of load applications, logNc, according to Equation (1).
logNm/logNc, C=1
smin/ fcfl
smax/ fcfl
mean value, C=1
160%
Damage / Load level
140%
120%
100%
80%
60%
40%
20%
0%
139A 992e 149D 139C 139E 149H 149G 149F 149E 139B
Test specimen
Figure 4. Damage as a percentage of the calculated number of load applications,
Nm/Nc, and the load level that each beam is subjected to. 100 % is the fatigue limit
according to Equation (1) with C = 1.
8
Graf-Brenner, 1934
Graf-Brenner, 1936
Antrim-McLaughlin, 1959
Assimacopoulos, Warner, Ekberg, 1959
Söderqvist, 2005 (beams)
Gaede, 1962
Bennet & Muir, 1967
Tepfers, Fridén, Georgsson, 1977
Shi, Fwa & Tan, 1993 (beams)
Optimal
12
10
logNm
8
6
4
2
0
0
2
4
6
8
10
12
logNc
Figure 5. Correlation between the calculated number of load applications, logNc, and
the measured number of load applications, logNc. Results from previous research
consisting of both compression, splitting and flexural tests (data obtained from (Shi
et al., 1993, and Tepfers, 1978)).
3.3 Modified Fatigue Equation
In this study, the tests are slightly weaker than predicted with Equation (1), with a
mean damage of 92 %. The series with the higher total load can be distinguished
with a somewhat lower fatigue capacity, see Figure 5. This could indicate that, with
an increasing minimum load, the discrepancies would increase. This could be
explained by the type of testing which is influenced by cracks and irregularities in
the specimens. The supports are also inelastic and could lock ordinary movements
and produce additional shear. To reduce the influence of the amplitude in Tepfers’
fatigue equation, a factor, C, is introduced into Equation 1 according to:
V max
f cfl
§
·
V
1 0.0685 ˜ ¨1 C ˜ min ¸ ˜ log N
¨
V max ¸¹
©
where C = 0.7556 fits the test results in this particular study. These results are
presented in Table 4 and Figure 6.
Performing the same procedure on previous results from both compression,
splitting and flexural tests found in literature, the constant C = 0.8035, see Figure 7.
9
(3)
logNm/logNc, C=0,7556
smax/ fcfl
smin/ fcfl
mean value, C=0,7556
160%
Damage / Load level
140%
120%
100%
80%
60%
40%
20%
0%
139A 992e 149D 139C 139E 149H 149G 149F 149E 139B
Test specimen
Figure 6. Damage as a percentage of the calculated number of load applications, Nm/Nc,
and the load level that each beam is subjected to. 100 % is the fatigue limit according to
Equation (1) with C = 0.7556.
logNc, C=1
logNc, C=0,8035
Linear, C=1
Linear, C=0,8035
18
16
y = 0,8016x
14
logNm
12
10
y = 1,0539x
8
6
4
2
0
0
5
10
logNc
15
Figure 7. Correlation between the calculated number of load applications, logNc, and
the measured number of load applications, logNc. Results from previous research
consisting of both compression, splitting and flexural tests (also presented in Figure 4).
The results are recalculated using Equation (3) with C = 0.8035.
10
149_D
992f
149E
139_A
149F
139_B
149G
139_C
149H
139_D
992e
139_E
Increase from first deflection (%)
140%
120%
100%
80%
60%
40%
20%
0%
0
1
2
3
4
5
6
7
Number of load applications, logNc
Figure 8. Deflection increase as a percentage of the deflection after the first loading
cycle. The last deflection measured is not necessarily the deflection immediately
before failure. Beam 992f is subjected to a higher load for the last 22 497 cycles,
hence the steep curve at the end. Loading properties are presented in Table 4.
3.4 Measured Deflections
The deflections are measured during testing with two LVDT’s. In previous research
it has been suggested that there is a linear relation between the range of deflection
and the logarithm of the number of load applications. The measurements in this
particular study indicate that the deflection indeed grows in a linear way and that the
deflection increases just before failure. The deflection during testing is presented for
each beam in Figure 8.
4 Miner-Palmgren’s Partial Damage Hypothesis
The accumulated fatigue damage can be calculated with Miners-Palmgren’s partial
damage hypothesis. This formula takes into account the damage from each specific
loading circumstance and accumulates the amount of fractional damage to predict the
total damage. In pavement design, the accumulated damage is computed by summing
fatigue damage incurred during each season of the year due to both traffic and
thermal loads.
The basic shape of Miners-Palmgren’s damage hypothesis is
n
¦
i
D i ˜ Ni
1 Ni, allow
11
d1
where, Įi is the percentage of time in which damage i occurs, Ni is the total number
of loads corresponding to damage i, and Ni,allow is the number of allowable loads
corresponding to damage i.
Two beams were used to verify Palmgren-Miner’s damage hypothesis. If a
beam has reached over 2 million load repetitions without failing, the maximum load
has been increased and the accumulated number of load repetitions is counted until
failure. The partial sum according to
n
¦
Ni
i
1 Ni, allow
n
¦
log Ni
log
Ni, allow
1
(4)
and,
i
(5)
are calculated and analysed. The results show that the partial sum of load repetitions
reaches 167 and 141 %, respectively, and that the calculated sum for the logarithm of
repetitive loads are higher (see Table 5). The accumulated damage is decreased when
using Tepfers’ modified fatigue equation with C = 0.7556. The accumulated damage
is shown in Figure 9.
12
Table 5. Number of load applications in each load level and the accumulated damage
for each beam.
Nc
Nc
Id
Nm
Vmin/ fcfl
Vmax/ fcfl
(C = 1)
(C = 0,7556)
139D
0.2
0.7
2.462.106
5.742.108
3.911.1011
.
5
.
6
0.2
0.8
3.808 10
1.353 10
1.302.108
.
4
.
3
0.2
0.9
1.080 10
7.815 10
1.420.105
3
Ni
¦
1.67
0.07
i 1 Ni, allow
992f
0.05
0.6
2.289.106
2.346.106
2.696.108
.
4
.
4
0.05
0.7
2.250 10
5.205 10
1.746.106
2
Ni
¦
1.41
0.02
i 1 Ni, allow
logNc
logNc
logNm
(C = 1)
(C = 0.7556)
139D
0.2
0.7
6.39
8.76
11.59
0.2
0.8
5.58
6.13
8.11
0.2
0.9
4.03
3.89
5.15
3
log Ni
¦
2.68
2.02
i 1 log Ni, allow
992f
0.05
0.6
6.36
6.37
8.43
0.05
0.7
4.35
4.72
6.24
2
log Ni
¦
1.92
1.45
i 1 log Ni, allow
13
logNm - 139D, C=1
logNm - 992f, C=1
logNm - 139D, C=0,7556
logNm - 992f, C=0,7556
300%
250%
Damage
200%
150%
100%
50%
0%
0
5
10
15
Number of load applications, logN m
20
Figure 9. Damage accumulation, calculated with Miners-Palmgren’s damage
hypothesis with logarithmic values (Table 5). Comparison between Tepfers’ fatigue
equation, Equation (1), and Tepfers’ modified equation, Equation (3). The fatigue
limit is 100 %.
5 Discussion
This investigation must be incorporated in the context of other research. In flexural
testing, the static flexural strength, fcfl, is determined on separately tested beams and
is therefore related to inexactness to some extent. However, this problem exists for
all sorts of fatigue testing and a mean value is used instead. By using a minimum of
three beams to determine the static strength, reliability is increased. Also, every beam
was manually manufactured and small imperfections in each specimen have
influence on the results. Deviations in fcfl occur and the fatigue is very sensitive to
this. A 5 % deviation between assumed and true fcfl is a fact for these tests and,
according to Tepfers, this has an increasing effect, especially on higher R-values, i.e.
R > 0.8. Tepfers also considered the long-term strength of concrete and concluded
that Vmax/ fcfl exceeding the long-term strength of concrete is sensitive to the loading
frequency because of creeping effects. For these reasons, no tests were performed for
R-values in this region.
By introducing a constant, C < 1, in Tepfers’ fatigue equation, the influence
of the amplitude is decreased. This approach can be compared to Equation (2),
formulated by Hsu (Equation (2)). In this present study, C = 0,7556 correlates with
the actual tests performed, and C = 0,8035 correlates with previous tests from both
compression, splitting, and flexural tests found in literature. A new constant related
to the R-value could indicate steeper curves in the Wöhler diagram (Figure 1) due to
strength reducing factors from the set-up of the tests. Since the R-values in this study
14
varies from 0.071 to 0.667, it has not been possible to examine the possibility that the
curves should be other than linear.
Measured deflections confirm the findings by Aas-Jacobsen that the
deformations during fatigue loading increase with 40 – 100 % from the first
measured deflection after one load cycle (Aas-Jacobsen, 1970).
Palmgren-Miner’s partial damage hypothesis is generally used to accumulate
the influence of multiple loading. Even though discrepancies of more than a factor of
ten have been generally accepted, the two examples above show a 140 – 160 %
deviation. This indicates satisfactory results, in agreement with previously performed
tests, and in particular (Tepfers et al., 1977).
6 Conclusions
Tepfers’ fatigue equation is, as a general rule, used in concrete pavement design in
Sweden. In the design, the flexural strength is dominant. Since this equation
originally was developed from compression and splitting tests, it is of great interest
to investigate how well the equation is suited for flexural fatigue. The following
conclusions may be drawn:
x
x
x
x
x
The flexural fatigue can be described with Tepfers’ fatigue equation and is
compatible with results from a number of previously performed compressive
and splitting tests.
The flexural fatigue can also be described with Tepfers’ fatigue equation by
the introduction of a constant to decrease the impact of the R-value. In this
study it was found that C = 0.756 can be fitted to the test results. For previous
tests C = 0.804 is matched with the same procedure.
For a calculated number of load applications exceeding 108, the measured
number of load applications is generally lower than predicted for both
compressive and splitting fatigue tests made in the past. This can directly be
derived from Figure 3.
It is difficult to determine the damage if a specimen has been subjected to a
small number of loads compared to the number of maximum load
applications that the specimen can withstand. The Palmgren-Miner partial
damage hypothesis is through this investigation still applicable to the fatigue
of plain concrete, especially in the view of previous research in this area.
Repeated loading increases the deflection at midspan of a beam and the
deflection grows increasingly just before failure.
Plain concrete in flexure has so far been described with Tepfers’ fatigue equation by
means of the assumption that the material can be described with the same equation
for both compressive and tensile stresses, and consequently, also for flexural stresses.
On the basis of this present study, this assumption is not contradicted, but rather
confirmed. However, a larger scale investigation may reveal differences that were
not possible to point out here.
15
References
Aas-Jacobsen, K., Fatigue of concrete beams and columns, NTH Institutt for
Betonkonstruksjoner, September 1970. Bulletin No. 70-1, 1970.
Hsu, T. C., Fatigue of Plain Concrete, ACI Journal, Proceedings V. 78, No.4, JulyAugust, 1981.
Oh, B. H., Fatigue Analysis of Plain Concrete in Flexure’, Journal of Structural
Engineering, Vol. 112, No. 2, February, 1986.
Shi, X. P., Fwa T. F., Tan, S. A., Flexural Fatigue Strength of Plain Concrete, ACI
Journal, Proceedings V. 90, No.5, September, 1993.
Silfwerbrand, J., Design of Concrete Pavements (Dimensionering av
betongbeläggningar), Report No. 9, 2nd Edition., Royal Inst. of Tech., Dept. of
Struct. Eng., Stockholm, 91 pp. 1995 (In Swedish).
SRA, ATB VÄG 2005 (Technical Guidelines for Road Structures). Part A-K. Report
No. 2005:112, Swedish National Road Administration, Borlänge, Sweden, 2005. (In
Swedish).
Tepfers, R., Fridén, C., Georgsson, L., A study of the applicability to the fatigue of
concrete of the Palmgran-Miner partial damage hypothesis. Magazine of Concrete
Research, Vol. 29, No. 100, September 1977.
Tepfers, R., An examination of the fatigue properties of concrete (En undersökning
av betongens utmattningshållfasthet). Byggforskningen, Report R86:1978, 121 pp,
1978.
Tepfers, R., & Kutti, T., Fatigue Strength of Plain, Ordinary, and Lightweight
Concrete, ACI Journal,Proceedings V. 76, nr 5, pp. 635-652, May 1979.
Tepfers, R., Tensile Fatigue Strength of Plain Concrete, ACI Journal, Proceedings V.
76, No.8, pp. 919-933, August, 1979.
16
Design Criteria for Lean Concrete
Mr. J. Söderqvist and Prof. J. Silfwerbrand
Swedish Cement & Concrete Research Institute, Stockholm, Sweden
ABSTRACT: In Sweden, the fatigue of lean concrete has, up to today, been described with a
strain criterion. By comparing different design criteria found in the literature, the Swedish
design criterion becomes very rigorous, especially for high numbers of load repetitions. The
lean concrete is due to this reason thick in the Swedish pavement system and, therefore, not
as favourable as an asphalt base. In this paper, an international comparison of design
criteria for lean concrete is done. Flexural tests on lean concrete beams, conducted in this
study, show the unmotivated rigorousness of the Swedish fatigue criterion, and a new
criterion based on tensile stresses is suggested.
KEY WORDS: Pavement, road base, lean concrete, flexural strength, fatigue.
1. INTRODUCTION
Swedish concrete pavement systems containing lean concrete tend to be thick. One example
is the previous Swedish pavement design guide (SRA, 1994) where 150 mm of lean concrete
is considered to be equivalent to a 100 mm asphalt stabilised base. In the current design
guide (SRA, 2005), the design tables are replaced by a computer program. Its solution
shows, however, the same result. For a person not familiar with Swedish pavement design,
this fact might be surprising. Lean concrete is always (for all seasons) stiffer than asphalt
and provides, thus, a better base in both asphalt and concrete pavement systems. The
assumed superiority of the asphalt base is based on three reasons: (i) the Swedish design
criterion for lean concrete is more rigorous than the corresponding one for asphalt, (ii) the
asphalt base is more suitable for construction traffic than the lean concrete base, and (iii)
several Swedish pavement engineers have the opinion that pavement systems containing
asphalt base perform better than those containing lean concrete. The authors’ opinion is that
the third reason lacks scientific support and that the used Swedish design criterion for lean
concrete is too rigorous. This will be shown both in an international comparison and by
analysing new fatigue test data.
2. STRAIN CRITERIA FOR LEAN CONCRETE
The Swedish design of lean concrete was originally developed by Björn Örbom (Örbom,
1981). It was based on field test results in Pennsylvania and adjusted to Swedish conditions
by including measuring results from test roads of the Swedish National Road and Transport
Research Institute (VTI). The design criterion for the Pennsylvania tests may be defined by
the following expressions:
N
4.092 ˜ 10 6 ˜ İ y 2.597 ; 10000 d N d 3.5 ˜ 10 7
(1a)
Hy
8.42 ˜ 10 3 ˜ N 0,385
(1b)
where N is the allowable number of load repetitions and Hy is the horizontal tensile strain in
the lowest fibre of the lean concrete layer. The quantity is non-dimensional.
The Swedish design criterion was originally only shown as a curve in a diagram. VTI has,
however, later developed an equation that is fitted to the diagram curve. It has the following
expression:
N
1.06 ˜ 10 10
(2a)
H 3y.86
or
Hy
2.61 ˜ 10 3
(2b)
N 0.259
The relationship (2a) can be found in both the previous and the current version of the
Swedish pavement design guide, VÄG 94 (SRA, 1994) and ATB VÄG 2005 (SRA, 2005),
respectively.
The Finnish researcher Józef Judycki (Judycki, 1991) has investigated how cementitious
materials are characterized and, among other things, rewritten a number of fatigue criteria.
The Belgian researchers De Henau & Verstraeten (Henau & Verstraeten, 1971) proposed the
following equation:
Hy
H y1 ˜ N 0.025 ; 10 4 d N d 108
(3a)
or rewritten
N
§ Hy ·
¨
¸
¨ H y1 ¸
©
¹
40
(3b)
The American highway engineers Pretorius & Monismith (Pretorius & Monismith,1972)
proposed a curve that can be recalculated to the following equations:
N 1.288 ˜ 109 ˜ 10
57800 ˜ H y
(5a)
H y 1.576 ˜ 10 4 1.73 ˜ 10 5 ˜ log N
(5b)
In South Africa, Otte (Otte, 1978) proposed the following alternative relationships:
Hy
H y1 ˜ (1 0.11 ˜ log N )
(6a)
2
N
§
¨
¨1 ¨
10©
·
¸
¸ / 0,11
H y1 ¸
¹
Hy
(6b)
H y H y1 ˜ N 0.079
N
§ H ·
¨ y ¸
¨¨ H ¸¸
© y1 ¹
(7a)
12.66
(7b)
In the Netherlands, Ros et al. (Ros et al., 1982) proposed a curve that can be expressed with
the following equation:
N
13,5
§¨ 2.0 ˜ 10 4 / H ·¸
y¹
©
(10a)
Hy
2.0 ˜ 104 ˜ N 0,074
(10b)
Figures Nos. 1-5 show comparisons between the various fatigue criteria. The first three
curves show how the allowable strain declines with increasing number of load repetitions. In
Figures Nos. 2 and 3, also four Swedish field test results have been included. All figures
contain a horizontal axis drawn logarithmically. Figure 2 has a linear vertical axis showing the
strain in Pstrain whereas Figure 3 has a logarithmic vertical axis. We see that the results are
well gathered close to the Swedish design criterion, especially in the log-log diagram (Figure
3). This is the diagram Örbom used when he established the Swedish design criterion. If he
had used a diagram with linear vertical axis, he might instead have defined a curve closer to
the curve proposed by Ros et al. In this sense, the Swedish design criterion for lean concrete
is a coincidence.
Figure 4 shows how allowable strain declines in relation to allowable strain at a single load.
Since a couple of the equations are not valid for small number of load repetitions (N < 10
000), Figure 5 shows how allowable strain declines in relation to allowable strain at N = 10
000. As shown in Figure 1, the Swedish design criterion gives large allowable strain for small
numbers of load repetitions, i.e., for N < 1 million. For higher numbers, it is more
conservative than the criterion proposed by Ros et al. If we study how allowable strain
declines in relation to allowable strain at single load instead, we observe that the Swedish
criterion as its origin in Pennsylvania lies substantially below the other criteria. The fatigue is,
consequently, assumed to occur much more dramatically.
3
Strain ( P strain)
1000
900
VÄG 94
Pennsylvania
800
700
600
Pretorius
Ros
500
400
300
200
100
0
1.E+00
1.E+02
1.E+04
1.E+06
1.E+08
Number of load repetitions
Figure 1: Allowable horizontal tensile strain as a function of number of load
repetitions.
250
VÄG 94
Pennsylvania
Strain ( P strain)
200
Pretorius
Ros
150
VTI
100
50
0
1.E+00
1.E+02
1.E+04
1.E+06
1.E+08
Number of load repetitions
Figure 2: Allowable horizontal tensile strain as a function of number of load
repetitions. Figure 1 has been supplemented with measuring data provided from
Swedish National Road and Transport Research Institute (VTI).
4
VÄG 94
10000
Pennsylvania
Strain ( P strain)
Pretorius
1000
Ros
VTI
100
10
1
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
Number of load repetitions
Relative strain
Figure 3: Allowable horizontal tensile strain as a function of number of load
repetitions. Please, note that the figure shows a log-log diagram.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.E+00
VÄG 94
Pennsylvania
Pretorius
Ros
Verstraeten
Otte log
Otte exp
1.E+02
1.E+04
1.E+06
1.E+08
Number f load repetitions
Figure 4: Allowable horizontal tensile strain in relation to allowable strain at single
load.
5
Relative strain
1
0.9
VÄG 94
Pennsylvania
0.8
0.7
0.6
Pretorius
Ros
Verstraeten
0.5
0.4
0.3
0.2
0.1
0
1.E+04
Otte log
Otte exp
1.E+05
1.E+06
1.E+07
1.E+08
Number of load repetitions
Figure 5: Allowable horizontal tensile strain in relation to allowable strain at N =
10 000 load repetition.
3. STRESS CRITERIA FOR LEAN CONCRETE
Some criteria do not deal with tensile strain but tensile stress. These two groups of criteria
are not that simple to compare. According to the theory of elasticity, stresses might be
recalculated as strains. The following relationship is valid:
Hy
1
˜ V y Q ˜ V x V z E
^
`
(11)
where, Vx, Vy, andVz are stresses, E is the modulus of elasticity, and Q is Poisson’s ratio. In
the bottom fibre of the lean concrete layer, the traffic stresses in two perpendicular
directions Vx and Vy are approximately equal while the vertical stressVz might be neglected.
Equation (11) might be rewritten as follows:
İy
1
˜ 1 Ȟ ˜ ı y
E
(12)
E
˜İ
1 Ȟ y
(13)
or
ıy
In Swedish concrete pavement design, Tepfers’ fatigue criterion (Tepfers, 1979) is used:
V max
f ct
§ V
·
1 0.0685 ˜ ¨1 min ¸ ˜ log N
¨ V
¸
max ¹
©
(14)
where, Vmin and Vmax are the minimum and maximum stress in the fatigue cycle, respectively,
and fct is the flexural tensile strength of concrete. Usually Vmin is the thermal stress and Vmax
6
is the sum of the thermal and traffic stresses (Söderqvist & Silfwerbrand, 2005). If we
neglect the thermal stress, Equation 14 may be simplified as follows:
Vy
f ct
1 0.0685 ˜ log N
(15a)
1V y / f ct / 0.0685
(15b)
or
N
10
If we assume that lean concrete has flexural tensile strength fct = 2,0 MPa, modulus of
elasticity E = 17 000 MPa, and Poisson’s ratio Q= 0,25, Equations (12) and (13) give us the
following expression:
8.82 ˜ 10 5 ˜ (1 0.0685 ˜ log N )
Hy
(16)
At a concrete block pavement workshop organized in August 2000 on Lidingö island in
Sweden, Shackel, Wellner and Huurman presented design criteria for lean concrete or other
cement treated materials. Wellner (Wellner, 2000) assumes that the flexural tensile strength
of a certain material equals 1/6 of its compressive strength and that the modulus of elasticity
is 15 000 MPa if the material is uncracked and 5000 MPa if it is cracked. He uses the
following criterion:
Vy
f ct ˜
84 5 ˜ log N
100
(17)
Huurman (Huurman, 2000) stresses that the modulus of elasticity varies substantially
between cement stabilised materials (4100 MPa), cement treated sand (5000 – 12 000 MPa)
and lean concrete (10 000 – 20 000 MPa) dependent on constituting materials and age at
testing. He states that the flexural tensile strength is 20 percent of the compressive strength
and uses the following equation:
log N
11.872 12.12 ˜
Vy
(18a)
fct
which may be rewritten as follows:
ıy
f ct ˜ 0.979 0.0825 ˜ logN (18b)
If we for the sake of comparison reuse fct = 2,0 MPa, E = 17 000 MPa and Q= 0,25, we
arrive at the following equations:
Hy
8.82 ˜ 105 ˜
84 5 ˜ log N
100
(19)
and
7
H y 8.82 ˜ 105 ˜ 0.979 0.0825 ˜ log N (20)
for Wellner and Huurman’s criteria, respectively.
Figure 6 shows a comparison between the Swedish concrete design criterion and Wellner
and Huurman’s criteria. They are all also compared with the Swedish design criterion for lean
concrete. As shown, the traditional Swedish design criterion for lean concrete is the most
favourable for low numbers of load repetitions, i.e., for N < 3 million. For large numbers, it is
the opposite. Figure 7 shows that the Swedish lean concrete criterion implies a more rapid
fatigue than any of the others. Wellner’s criterion is the mildest and Huurman’s the most
rigorous.
Strain ( P strain)
1000
900
VÄG 94
Tepfers
800
700
600
Wellner
Huurman
500
400
300
200
100
0
1,E+00
1,E+02
1,E+04
1,E+06
1,E+08
Number of load repetitions
Figure 6: Allowable horizontal tensile strain as a function of the number of load
repetitions according to the Swedish (or, e.g., Tepfers’) concrete criterion, Wellner’s
criterion, and Huurman’s criterion after recalculation from stresses to strains. The
Swedish lean concrete criterion according to VÄG 94 is shown for comparison.
VÄG 94
Relative stain
1
0,9
Tepfers
Wellner
0,8
0,7
0,6
Huurman
0,5
0,4
0,3
0,2
0,1
0
1,E+00
1,E+02
1,E+04
1,E+06
1,E+08
Number of load repetitions
Figure 7: Allowable horizontal tensile strain in relation to allowable strain at single
load according to the Swedish (or, e.g., Tepfers’) concrete criterion, Wellner’s
criterion, and Huurman’s criterion after recalculation from stresses to strains. The
Swedish lean concrete criterion according to VÄG 94 is shown for comparison.
8
4. LABORATORY TESTS
The laboratory tests conducted in this study are done in order to (i) evaluate the design
criterion in Sweden and (ii) analyse a stress related approach on the fatigue of lean
concrete. For plain concrete, Tepfers’ fatigue equation, Equation 14, is widely used, and the
fatigue of lean concrete is compared to this equation through this study. The equation
considers two stress levels and has been chosen in this study because of the similarities in
material properties between plain and lean concrete. By introducing a correlation factor in
this equation, a new criterion for lean concrete fatigue is suggested.
4.1
Test specimens
The lean concrete mixture chosen for the fatigue tests is retrieved from the Swedish design
guide and fabricated in a concrete plant in Täby, Stockholm, Sweden. The plant was situated
at a transportation distance of 30 min from the production site in Upplands-Väsby, north of
Stockholm. Since the project took place in November 2004, with an air temperature of
approximately 5 – 10 °C, the concrete was transported in open trucks. Data for the concrete
mixture are presented in Table 1. The production site was prepared with a 70 m2 base of
compacted gravel. The lean concrete was tipped on the surface and compacted 3 – 4 times
using a CA15 roller to a target thickness of 150 mm, see Figure 8. The level of compaction
and the relative moisture content were monitored with a nuclear density gauge to guarantee
a 97 % compaction level. After seven days, 13 beams were cut out of the surface. The
specimens were stored in water for 12 –15 months before testing.
Table 1: Typical Swedish lean concrete mixture used in this study.
Quantity (kg/m3)
Swedish Lean Concrete
CEM I, Std Degerhamn, Swedish cement for civ. eng. struct.
110
Aggregate 0-8 mm
1576
Aggregate 8-16 mm
674
Water
115
w/c
1.05
Density
2476
9
Figure 8: A CA15 roller was used for the compaction of the lean concrete. Three to four
passages were needed to reach a good compaction level.
4.2
Test procedure
The flexural strength and fatigue were tested in a Material Test System (MTS) 810 machine.
At first, three beams were tested to retrieve the static flexural strength. The static strength
properties were then used to pinpoint the load levels in the following fatigue tests, taking
into account each beam’s individual cross-section. The beams were simply supported with a
span of 700 mm, and tested in four point flexural loading according to Swedish standard SS
13 72 12, see Figures 9 and 10.
F/2
F/2
300 mm
200 mm
200 mm
Figure 9: Static system for flexural and fatigue tests.
10
Figure 10: MTS 810 machine for flexural testing at the Swedish Cement and Concrete
Research Institute, Stockholm, Sweden.
A total of 10 fatigue tests were made on two different values of Vmin, corresponding to
approximately 5 and 20 percent of the maximum static strength, fcfl. On each level different
values of Vmax were tested while the ratio R = Vmin/Vmax was held constant on two levels. The
test program is presented in Table 2. In addition, cylinders and cubes were tested to
determine the modulus of elasticity, Efl, and the compressive strength, fcc. The fatigue tests
were conducted in load control with a sinus waveform at 0.05 Hz in the beginning and 2.05
Hz from 100 cycles to failure. The beams were also covered with a plastic folio to keep the
moisture level constant throughout the test. Two Linear Variable Differential Transformers
(LVDT) were mounted on every beam to measure the midspan deflection.
Table 2: Program for fatigue testing
concrete.
Id
Vmax/fcfl (%) Vmin/fcfl (%)
1
62
5.2
2
64
5.3
3
70
5.8
4
75
6.3
5
80
6.7
6
70
20.0
7
75
21.4
8
80
22.9
9
83
23.5
10
85
24.3
4.3
of lean
R
0.0833
0.0833
0.0833
0.0833
0.0833
0.2857
0.2857
0.2857
0.2857
0.2857
Static test results
Three 150 mm cubes were cast and tested in compression after 28 days according to
Swedish practice. These cubes were compacted with a vibrating plate in three layers to
guarantee a 97 % compaction level.
11
Cylinders drilled from the lean concrete beams were used to determine the compressive
strength, fcc, at the end of the fatigue test period. The modulus of elasticity, Ec, was also
determined with cylinders cut out of the original lean concrete surface. All cylinders were
stored at 100 % relative humidity and 20°C as the lean concrete beams. The results are
shown in Table 3.
Before the fatigue testing begun the flexural strength, fcfl, of the lean concrete was tested
statically with three beams, see Table 3. The mean value from this test series was used to
find the appropriate load level in the subsequent fatigue tests. The static tests were also
used to find the elastic modulus in flexure, Efl.
The modulus of elasticity is different for cylinders and beams. Cylinders are often used for
establishing this value but in the design, when the flexural strength determines the material’s
capacity it is relevant to calculate the actual modulus of elasticity in flexure, Efl. The static
tests provided Efl = 16 000 MPa, approximately the same as suggested by Huurman and
Wellner, see Section 3. The reason to the difference in E between cylinders and beams has
not been investigated in this study. It might be a result of micro cracking, not visible, but
possibly present during flexural loading.
Table 3 - Static material strengths. The late age is due to delayed start of testing.
Type
Strength
Age at test
No. of tests
Type of strength
(mm)
(MPa)
(days)
(-)
fcc 28d
Cubes (150×150)
16.3
28
3
fcc 400d
Cylinders (Ø100×100)
13.7
400
6
fcfl 440d
Beams (800×150×100)
2.1
440
3
Ec 500d
Cylinders (Ø100×200)
28 000
500
6
Efl 440d
Beams (800×150×100)
16 000
440
3
Differences between compressive strengths are probably due to the fact that the 28-days
compressive strength comes from the fabricated cubes that were compacted manually. The
400-day compressive strength comes from cylinders from the lean concrete surface, where a
roller was used for the compaction of the material.
The flexural strength is in agreement with other research results, in the region of 1/3 to 1/2
of the flexural strength of plain concrete. In a previous study, plain concrete beam had a
flexural strength of approximately 5.5 MPa (Söderqvist & Silfwerbrand, 2006).
4.4
Fatigue test results
The fatigue tests were carried out according to the method described in the previous
sections. The specimens were carefully measured before testing to be able to apply the most
accurate stresses with regard to the height and base of each individual specimen. Two
different values on R = Vmin/Vmax corresponding to 0.2857 and 0.0833 were considered.
Before each fatigue test, the number of loads, referred to as logNc, that the current beam
could withstand was calculated according to Equation 14. This number was then evaluated
compared to the number of loads that the current beam could take before failure, logNm.
The strain criterion is translated to a stress criterion and evaluated compared to results
achieved in this study in Section 4.6. In Equation 14, Tepfers came to the conclusion that ǃ
= 0.0685 was valid for plain concrete (Tepfers, 1979). The evaluation of the fatigue tests in
this study is mainly done by investigating the constant ǃ, but also by introducing a new
constant, C, in a modified fatigue equation, see Section 4.7. Equation 14 is presented in a SN-diagram in Figure 11 and basic results from the fatigue tests are shown in Table 4.
12
In Figure 12, the results are also presented in relation to the predicted number of loads for
each specimen according to Equation 14.
Table 4: Summary of flexural fatigue tests with results on fatigue life and different values
on ǃ with the correlation factor C.
Id
cg9
cg10
cg6
cg7
cg8
Vmax/ fcfl
0.62
0.64
0.70
0.75
0.70
Vmin/ fcfl
0.05
0.05
0.06
0.06
0.20
R
logNc
logNm
ȕ, C = 1
ȕ, C=0,28
0.0833
6.05
6.04
0.0686
0.0644
0.0833
5.73
5.12
0.0767
0.0720
0.0833
4.78
4.42
0.0741
0.0696
0.0833
3.98
3.96
0.0689
0.0646
0.2857
6.13
3.49*1
0.1204
0.0935
0.0918
0.0713
2
0.20
0.21
0.21
0.23
0.2857
6.13
4.58
0.2857
5.11
4.00*1
0.0875
0.0679
0.2857
5.11
4.302
0.0813
0.0631
cg4
0.70
0.75
0.75
0.80
0.2857
4.09
3.25
0.0862
0.0670
cg11
0.85
0.24
0.2857
3.07
3.18
0.0660
0.0512
N
10
10
0.0685
cg12
cg5
cg13
Mean
0.0821
Standard Deviation
0.0160
0.0106
Variance
0.0003
0.0001
R = 0,5
R = 0,4
R = 0,3
R = 0,2
R = 0,1
R=0
1
0,9
0,8
Relative stress
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
1,0E+00
1,0E+01
1,0E+02
1,0E+03
1,0E+04
1,0E+05
1,0E+06
1,0E+07
Number of load applications, logN
Figure 11: Tepfers’ fatigue equation, Equation 14. The number of allowable load applications
in relation to the relative stress and different ratios on R.
13
logNm/logNc
smax / fcfl
mean value
smin / fcfl
120%
Damage / Load level
100%
80%
60%
40%
20%
0%
cg9 cg10 cg6
cg7
cg8 cg12 cg5 cg13 cg4 cg11
Test specimen
Figure 12: Damage as a percentage of the calculated number of load applications,
logNm/logNc, and the load level that each beam is subjected to, Vmin/ fcfl and Vmin/ fcfl. 100 %
is the fatigue limit according to Equation 14. The mean value is 86 %.
4.5
Deflections due to repeated loading
The deflections were measured with two LVDT’s mounted on each side of the beams. The
deflections were recorded at the beginning of the test and then every 50 to 10 000 cycles
depending on the predicted number of loads to failure. In some rare cases the maximum
deflection before failure was captured. All beams exceeded a 30 % increase in deflections
compared to the deflection in the first loading cycle. Aas-Jacobsen found a 40 – 100 %
increase in deflection for plain concrete (Aas-Jacobsen, 1970), and this was confirmed in the
preceding study on plain concrete beams (Söderqvist & Silfwerbrand, 2006). The final
deflection for lean concrete could reach a 180 % increase. Deflections are presented in
Figure 13.
14
cg8
cg5
cg4
cg11
cg13
cg12
cg10
cg9
cg7
cg6
200%
180%
Deflection (%)
160%
140%
120%
100%
80%
60%
40%
20%
0%
1,E+00
1,E+01
1,E+02
1,E+03
1,E+04
1,E+05
1,E+06
1,E+07
Number of load applications, LogNm
Figure 13: Deflections for beams subjected to fatigue loading. Since the deflections were
recorded manually, the final deflection is not necessarily the maximum deflection before
failure.
4.6
The Swedish strain criterion
The tensile stresses in the road base are often neglected for a cement bound base since it is
considered cracked under the concrete pavement. In Sweden, the use of a strain criterion
has been chosen instead. This criteria is discussed in Section 2, i.e. Equation (2a) and (2b).
From Table 3, values on the modulus of elasticity, Efl, and the flexural strength, fcfl, is used
together with Equation (13) and a Poisson’s ratio Q= 0,25 to rewrite the strain criterion to a
stress criterion. The produced equation has the following form:
Vy
55.68 ˜ N 0.259
(21)
The equation is plotted in Figure 17 together with results from the fatigue tests.
The obtained results from testing are spreading over a range of 10 000 to 1 million load
applications but are clearly more resistible than predicted by the strain criterion. It can
hereby, once again, be shown how strict the Swedish criterion is, not only in comparison to
other international criteria, as discussed in previous sections, but also to new test data.
15
Väg94-strain to stress
Test result R=0,2857
Test result R=0,0833
1
0,9
Relative stresses
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
1,E+00
1,E+02
1,E+04
1,E+06
1,E+08
1,E+10
Number of load applications, N
Figure 14: Allowable relative horizontal stress in relation to allowable stress at single load.
The Swedish strain criterion is translated to a stress criterion. Here, compared to tests
results from the fatigue tests.
4.7
Tepfers’ modified fatigue equation
Equation 14 is, as a general rule, used in pavement design in Sweden for plain concrete. In
the design, the flexural strength is dominant for both plain and lean concrete. In a previous
study, plain concrete beams were tested in order to verify the fatigue criteria in flexure
(Söderqvist & Silfwerbrand, 2006). In this study the same test system was used and the
design criterion was in agreement with the flexural tests results. Since Tepfers’ fatigue
criterion is consistent with ordinary concrete, it is not too unrealistic to think that it could be
used for lean concrete, implying the material is uncracked.
In this study, the number of tests is limited but there is a tendency of coherence between
the test results. All tests are slightly weaker than predicted with Equation 14, but the series
with the higher minimum load can be distinguished with a somewhat lower fatigue capacity,
see Figure 11. This could indicate that, with an increasing minimum load, the discrepancies
would increase. This hypothesis can be shown by introducing a correlation factor, C, to
Equation 14 according to:
V max
f ct
§
·
V
1 0.0685 ˜ ¨1 C ˜ min ¸ ˜ log N
¨
V max ¸¹
©
(22)
where C = 0,28 fits the test results in this particular study. These results are presented in
Table 4 and Figure Nos. 15, 16 and 17.
16
R = 0,0833
Test result R=0,2857
Test result R=0,0833
R = 0,2857
1,0E+00
9,0E-01
Relative stresses
8,0E-01
7,0E-01
6,0E-01
5,0E-01
4,0E-01
3,0E-01
2,0E-01
1,0E-01
0,0E+00
1,E+00
1,E+02
1,E+04
1,E+06
1,E+08
1,E+10
Number of load applications, N
Figure 15: Tests results and the correlation to Tepfers’ fatigue equation, Equation 14.
R = 0,2857, C=0,28
R = 0,0833, C=0,28
Test result R=0,2857
Test result R=0,0833
1,0E+00
9,0E-01
Relative stresses
8,0E-01
7,0E-01
6,0E-01
5,0E-01
4,0E-01
3,0E-01
2,0E-01
1,0E-01
0,0E+00
1,E+00
1,E+02
1,E+04
1,E+06
1,E+08
1,E+10
Number of load applications, N
Figure 16: Tests results and the correlation to Tepfers’ modified fatigue equation, Equation
(21), with C=0.28.
17
160%
logNm/logNc (C = 0,28)
mean (C = 0.28)
smax / fcfl
smin / fcfl
Damage / Load level
140%
120%
100%
80%
60%
40%
20%
0%
cg9 cg10 cg6
cg7
cg8 cg12 cg5 cg13 cg4 cg11
Test specimen
Figure 17: Damage as a percentage of the calculated number of load applications
considering a constant C = 0.28 in Equation 22. The mean value is in this case 102 %.
5. DISCUSSION
The strain criterion in Sweden is developed out of field test result from Pennsylvania, USA. A
discussion weather this criterion is fair, especially in relation to international criteria, has
been presented. The strain criterion has also been translated into a stress criterion with
actual material properties, to consider tensile stresses. In comparison to Tepfer’s fatigue
criterion and new fatigue test data, the strain criterion is found to be discriminatory and this
study may even raise a few question marks on whether this criterion really is not too strict.
Tepfers’ fatigue equation is generally applied to plain concrete in the Swedish design. The
flexural fatigue of lean concrete, estimated through flexural tests in this study, is found to be
in agreement with Tepfers’ fatigue equation, possibly with some modifications. The results
are also compatible with results from previously performed fatigue tests on plain concrete. A
larger scale investigation may, however, reveal the exact shape of a new approach that was
not possible to obtain here. Since the fatigue of plain concrete is based on compression and
splitting tests, it would also be valuable to perform these types of tests on lean concrete to
more directly compare fatigue properties for these two materials.
6. ACKNOWLEDGEMENTS
The financial support provided by Cementa AB, the Royal Institute of Technology, and the
Swedish Agency for Innovation Systems is gratefully acknowledged. A special thank goes to
Peter Glinning at Betong Industri, Stig Jansson at Cementa AB, and helpful personnel at the
Swedish Road Administration in Upplands-Väsby, Stockholm. Their support and practical
knowledge made this project possible.
18
REFERENCES
Aas-Jacobsen, K., 1970. Fatigue of concrete beams and columns. NTH Institutt for
Betonkonstruksjoner, September. Bulletin No. 70-1.
De Henau A. & Verstraeten J., 1971. Fatigue in Lean Concrete, Application to the Design of
Bases. Chapter 2.2.3 in the Belgian Report to the XIVth World Congress, Prague, Check
Republic, Question II, pp. 16-18.
Huurman R., 2000. The Analytical Structural Design of Industrial Concrete Block Pavements.
Proceedings, International Workshop on Design of Industrial Concrete Block Pavements,
Lidingö, Stockholm, August 24-26. Report 61, Structural Design and Bridges, Department of
Structural Engineering, Royal Institute of Technology, Stockholm, pp. 107-128.
Judycki J., 1991. Structural Characterization of Road Base Materials Treated with Hydraulic
Binders. Publication No. 12, Road and Transport Laboratory, University of Oulu, Oulo,
Finland.
Otte E., 1978. Factors Affecting the Behaviour of Cement-Treated Layers in Pavements. The
9th Australian Road Research Board Conference, University of Queensland, Brisbane, Vol. 9,
pp. 191-202.
Örbom B., 1981. Traffic dependent changes of cement stabilised base layer, continuous
grading materials, according to investigations at Pennsylvania Transportation Institute and
on these supported design guidelines. Study Report, Swedish National Road and Transport
Research Institute (VTI), Linköping, 1981. (In Swedish).
Pretorius P S & Monismith C. L., 1972. Fatigue Cracks Formation and Propagation in
Pavements Containing Sili-Cement Bases. HRR, No. 407, pp. 102-115.
Ros J., Pronk A. C. & Eikelbloom J., 1982). The Performance of Highway Pavements in the
Neterlands and the Applicability ot Linear Elastic Theory of Pavement Design. 5th
International Conference on Structural Design of Asphalt Pavements, Delft, The Netherlands,
Vol. 1, pp. 285-302.
Shackel B., 2001. Pavement Design Studies for the Port of Helsingborg. Proceedings,
International Workshop on Design of Industrial Concrete Block Pavements, Lidingö,
Stockholm, August 24-26. Report 61, Structural Design and Bridges, Department of
Structural Engineering, Royal Institute of Technology, Stockholm, pp. 21-36.
Söderqvist, J., & Silfwerbrand, J., 2005. Design of Concrete Pavements: A Comparison
between Swedish and U.S. Methods. Proceedings, 8th International Conference on Concrete
Pavements. Colorado Springs, Colorado, USA, August 14-18, 2005, Vol. I, pp. 1-20.
Söderqvist, J., & Silfwerbrand, J., 2006. Flexural Fatigue of Plain Concrete Beams. Materials
& Structures. Submitted march 2006.
SRA, 1994. VÄG 94 (Technical Guidelines for Road Structures). Part 1-10. Report Nos.
1994:21 to 1994:30, Swedish National Road Administration, Borlänge, Sweden, 1994. (In
Swedish).
19
SRA, 2005. ATB VÄG 2005 (Technical Guidelines for Road Structures). Part A-K. Report No.
2005:112, Swedish National Road Administration, Borlänge, Sweden, 2005. (In Swedish).
Tepfers, R., 1979. Tensile Fatigue Strength of Plain Concrete. ACI Journal, Proceedings V.
76, No. 8, August, pp. 919-933.
Wellner F., 2001. Design of Block Pavings. Proceedings, International Workshop on Design of
Industrial Concrete Block Pavements, Lidingö, Stockholm, August 24-26. Report 61,
Structural Design and Bridges, Department of Structural Engineering, Royal Institute of
Technology, Stockholm, pp. 49-65
20
Flexural Fatigue of Composite Beams of Plain and
Lean Concrete
J. Söderqvist and J. Silfwerbrand
Swedish Cement and Concrete Research Institute, Stockholm, Sweden
ABSTRACT
Lean concrete is often used in concrete or asphalt pavement design because it supplies
additional strength and stiffness to the pavement system at a low cost. In concrete pavement
design the lean concrete is mostly used for heavily trafficked roads, or industrial pavements
where very high loads are present. The lean concrete contributes with strength and stiffness,
and makes it possible to design pavements without dramatically increasing the overlaying
concrete thickness. In this study, composite beams of plain and lean concrete have been
manufactured and subjected to flexural fatigue testing. First, the flexural static strength was
determined with four beams. Eight beams where subjected to flexural fatigue loading. The
results are analysed using Tepfers’ fatigue equation, a fatigue criterion used for plain
concrete. The study was conducted in order to analyse how the composite beam behaviour,
the crack development, and bond between the materials were affected by cyclic loading.
INTRODUCTION
Lean concrete is generally described as a compacted, hydraulically bound material with a
relative low cement and water content. It is widely used as a stabilised road base for highways
as well as airports and industrial areas. The most common applications are roads with high
traffic intensity or industrial pavements trafficked with very high loads. As an alternative to
an asphalt road base, the lean concrete is much stiffer during all seasons of the year. It is of
course affected by the temperature changes but with a minimal effect on strength. The lower
content of cement in lean concrete makes it economically and environmentally advantageous
to use it under the concrete pavement.
Lean concrete is used to distribute stresses and to minimise the overlaying concrete thickness.
It has a strengthening function in the overall pavement system by contributing with additional
strength in the material itself, but also by attaching to the concrete overlay. The bond is
mechanistically desirable to reduce stresses but is thought to possibly be time dependent and
diminish some time after construction. One disadvantage with bonding is the hypothesis that
cracks that develop in the lean concrete can propagate through the overlaying pavement
(reflection cracks).
In this study the interaction between plain and lean concrete is investigated with composite
beams subjected to flexural stresses. The bonding between plain and lean concrete is
investigated with static as well as fatigue flexural loading and pull-off tests. In the fatigue
tests, the bonding is analysed by monitoring the deflection but also by visual inspections. The
1
main objective is to analyse the nature of the crack development in the composite beams and
investigate how well Tepfers’ fatigue criterion is suited for the prediction of fatigue life. The
load levels in the fatigue testing are chosen in relation to the material strength to capture the
crack development in each material.
FLEXURAL FATIGUE OF PLAIN AND LEAN CONCRETE
The fatigue strength of plain concrete is defined as the fraction of the static stress that the
material can support for a given number of stress cycles, N. A fatigue criterion that accounts
for the variation in loading magnitude, originally developed by Aas-Jacobsen (Aas-Jacobsen,
1970) is:
ı max
f cfl
§ ı
1 ȕ ˜ ¨¨1 min
© ı max
·
¸¸ ˜ log N
¹
(1)
where,
N
Vmin
Vmax
fc,fl
E
R
= the number of load applications
= the minimum stress
= the maximum stress
= the concrete flexural strength
= a material constant
=Vmin /Vmax
A common way of illustrating the fatigue strength is in a Wöhler diagram, with so called S-Ncurves. In the Wöhler diagram, the logarithm of the number of load cycles to failure, logN, at
a specific maximum stress level, is plotted. For each curve the relationship between the
minimum and the maximum stresses, R, is held constant, see Figure 1.
2
1
Relative Stress
0,8
0,6
R=0,5
R=0,4
R=0,3
R=0,2
R=0,1
R=0
0,4
0,2
0
1,E+00
1,E+01
1,E+02
1,E+03
1,E+04
1,E+05
1,E+06
Number of load repetitions, logN
Figure 1. Wöhler diagram with constant values on R=Vmin/Vmax.
In Equation (1), theE-value is a material constant that is determined experimentally. Tepfers
(Tepfers & Kutti, 1979, Tepfers, 1979, and Tepfers, 1978) examined the fatigue of concrete
with compressive and splitting tests. The tests were also compared with other research results
from compressive tests on ordinary and lightweight concrete (data are presented in (Tepfers,
1978)). Tepfers concluded that Aas-Jacobsen’s fatigue equation is valid for plain concrete and
suggested ȕ = 0.0685.
The fatigue of lean concrete, on the other hand, is often described with a strain criterion. In
two preceding studies, lean concrete and plain concrete beams have been tested in bending
and compared to Tepfers’ fatigue criterion (Söderqvist & Silfwerbrand, 2005 and Söderqvist
& Silfwerbrand, 2006). The hypothesis is that both lean and plain concrete are cementitiuous
materials and that Tepfers’ fatigue equation ought to be used for both. The tests were done in
order to examine the hypothesis. The results show that the fatigue of lean concrete can be
described with a stress criterion and therefore the fatigue life for a composite beam is chosen
to be analysed with Tepfers’ fatigue equation in this study.
BENDING STRESSES IN A COMPOSITE BEAM
Fatigue testing requires the capability to calculate stresses in a specimen to conduct more
effective testing. To reach a certain load level in a specific specimen, the stresses have to be
calculated in advance with regard to the effective height, width, and material strength and
stiffness of each layer. For a two layered composite beam with plain concrete on top and lean
concrete in the bottom, and assuming full bonding, the stress distribution can be calculated by
transforming the beam into one equivalent section, a T-beam, in one material, see Figure 2.
The stresses can then be obtained by the simple theory of elasticity, using equation
3
V PCC
M ˜ y max
I PCC(equiv)
(2)
for the plain concrete layer and the relationship
V LC
E PCC
˜ V PCC
E LC
(3)
for the stresses in the lean concrete, where the stresses, Vthe modulus of elasticity, E, and the
moment of inertia I, are denoted with PCC for the plain concrete and LC for the lean concrete.
IPCC(equiv) is the moment of inertia of the equivalent section, the T-beam.
a
b
Figure 2. Cross-section of a beam. The composite section with two layers of different materials, on
the left, is transformed into an equivalent section of one material with a reduced width, a T-beam, on
the right.
Attention has to be made to the height relation between the two layers since the modulus of
elasticity of both materials are fairly equal. If the height of both plain and lean concrete is the
same, see Figure 3a, the failure load for the lean concrete layer will result in a load exceeding
the failure load for the plain concrete when it has to carry the entire load. In this case, a brittle
fracture through the whole section will occur. The stress level in the interface between the
materials is also small since the neutral axis lies near the middle of the beam, and the interface
is therefore unstressed or nearly unstressed. In this study, this has been tested with four
beams.
In four beams, the thickness of the lean concrete is reduced by sawing to approximately half
the plain concrete thickness, see Figure 3b. This has been done in order to try to reach the
fatigue limit for the lean concrete while keeping the load level lower than the failure load for
the plain concrete. By doing this, the bonding between the materials should be affected by the
fatigue loading. In this case it would be interesting to see weather a crack that develop due to
the repeated loading travels along or through the interface between the two materials.
4
PCC
-
-
1.0
+
1.0
+
LC
+
a) Bond
1.0
PCC
-
+
0.5
LC
+
+
b) Bond
1.0
PCC
-
+
0.5
LC
c) No bond
+
+
Figure 3. Schematic stress distribution in a composite beam with a higher modulus of elasticity in top
layer. For a and b, full bond generates lower relative stresses in bottom of top layer. In c, less or no
bond generates higher tensile stresses in both layers. On the right, the resultant stress distribution after
that the bottom layer (lean concrete) has failed is shown.
The benefits of a composite pavement system with plain and lean concrete can be illustrated
by making an example with the calculation method mentioned above. Taking lean concrete
and plain concrete with a flexural strength of 2.60 MPa and 5.50 MPa, respectively, and
assuming full bonding, a 200 mm plain concrete layer with 150 mm lean concrete can take a
55 % higher load compared to a 200 mm plain concrete layer. Decreasing the thickness of the
plain concrete layer to 180 mm with a remained thickness of 150 mm of lean concrete, the
load can be increased with 38 %, see Table 1.
5
Table 1. Example of changing load capacity of
different composite pavement systems. Flexural
strength of lean and plain concrete is 2.60 MPa
and 5.50 MPa, respectively.
Increased load capacity
PCC
LC
compared to reference
(mm) (mm)
(%)
200
0
O
200
150
55.0
180
150
38.0
150
150
14.0
200
100
16.0
180
100
0.3
100
100
-50.0
SCOPE OF THE LABORATORY TESTS
The laboratory tests are conducted in order to analyse the interaction between layers in a
composite beams of plain and lean concrete subjected to fatigue loading. The fatigue strength
is evaluated with Equation 1. The equation considers two stress levels and has been chosen in
this study because of the similarities in material properties between plain and lean concrete.
The objective of the tests is to investigate the fatigue strength of a composite beam but also to
study the bond between the materials. The crack development is of great interest and here, the
question is if the crack will propagate through both the lean and plain concrete, i.e. a so called
reflection crack, or if the crack travels between the two materials, horizontally.
TEST SPECIMENS
Twelve composite beams with plain concrete on top and lean concrete in the bottom were
used in this study. The beam dimensions (L × B × H) were approximately 800 × 150 ×
150 mm (six beams) and 800 × 150 × 120 mm (six beams). All beams had from the start
equal height of plain and lean concrete. Four beams were later cut to have a lean concrete
layer that was approximately half the thickness of the plain concrete. Beam dimensions are
presented in Table 2.
6
Table 2. Dimension of the cross-section of the tested beams, and the type of testing the
beams were subjected to.
B
Type of
HLC HPCC
(mm) (mm) (mm) testing
Balk 1-1
Balk 1-2
Balk 2-5
Balk 2-2
Balk 1-3
Balk 1-4
Balk 1-5
Balk 1-6
Balk 2-1
Balk 2-3
Balk 2-4
Balk 2-6
67
64
91
96
93
95
96
90
67
65
64
62
71
67
90
95
86
85
81
88
35
40
36
41
156
160
156
155
157
149
151
150
145
153
153
158
Comment
Static
“
“
“
Fatigue
“
“
“
Specimen failed at first loading
“
Subjected to three load levels, static test at the end
“
“
“
The lean concrete mixture was retrieved from the Swedish design guide (SRA, 2005) and
fabricated in a concrete plant. Mixtures are presented in Table 4. The production site was
prepared with a 70 m2 base of compacted gravel. The lean concrete was tipped on the surface
and compacted 3 – 4 times using a CA15 roller in two equal layers with the total thickness of
150 mm (Figure 4). The level of compaction and the relative moisture content were monitored
with a nuclear density gauge to guarantee a 97 % compaction level. After seven days, twelve
beams were cut out of the surface. The specimens were stored in water for 18 months before
casting plain concrete on the beams. After casting the specimens were cured in water until
testing. The surface of the lean concrete was only washed with water before applying the
plain concrete.
Figure 4. Preparation of lean concrete layer. The material was compacted to a 97 % compaction level.
7
Table 3. Activities over time for fatigue testing of composite beams
Activity
Time
Production of lean concrete on site
November 2004
Extraction of beams from surface, storage in 100 % RH November 2004
Casting of plain concrete on lean concrete beams
June 2006
Static tests of composite beams
June –July 2006
Fatigue testing of composite beams
Sept – Oct 2006
Age (Days)
1
14
550
600
630
Table 4. Typical Swedish plain and lean concrete mixtures used in this
study.
Quantity
Swedish Lean Concrete
CEM I, Std Degerhamn*
110 kg/m3
Aggregate 0-8 mm
1576
“
Aggregate 8-16 mm
674
“
Water
115
“
w/c
1.05
Density
2476 kg/m3
Swedish Concrete Grade K60
CEM I, Std Degerhamn*
375 kg/m3
Aggregate 0-8 mm, Underås
737
“
Aggregate 8-16 mm, Underås
1018
“
Water
149.4
“
Silica
26.3
“
Plasticiser, 92M
8.6
“
Air Entraining Agent, L14
0.3
“
w/c
0.40
w/b
0.37
Density
2400 kg/m3
* Swedish cement for civil engineering structures
TEST PROCEDURE
The flexural strength and fatigue were tested in a Material Test System (MTS) 810 machine
(Figure 5). At first, four beams were tested to retrieve the static flexural strength. Here, both
Linear Variable Differential Transformers (LVDT) and strain gauges were used to monitor
the tests. The static strength properties were then used to pinpoint the load levels in the
following fatigue tests, taking into account each beam’s individual cross-section. The beams
were simply supported with a span of 700 mm, and tested in four point flexural loading
according to Swedish standard SS 13 72 12, see Figure 6.
8
Figure 5. Material Test System 810 machine at the CBI was used for the fatigue testing of composite
beams in this study.
F/2
F/2
300 mm
200 mm
200 mm
Figure 6. Static system for flexural and fatigue tests.
STATIC TEST RESULTS
The static tests in this study are additional to tests made in an earlier study (Söderqvist &
Silfwerbrand, 2006), where the material from the same lean concrete surface was tested. For
the plain concrete, the same mixture was used as in an earlier study where plain concrete
beams were subjected to fatigue loading (Söderqvist & Silfwerbrand, 2005).
The compressive strength of the lean concrete as well as the plain concrete was
determined by means of cubes and cylinders. For the lean concrete, cubes were cast and tested
in compression after 28 days according to Swedish practice. These cubes were compacted
with a vibrating plate in three layers to guarantee a 97 % compaction level. Plain concrete
9
cubes were cast at the same time as the overlaying concrete was cast on the lean concrete
beams, and tested at 28 days.
The bond strength between plain and lean concrete is determined by pull-off tests made on
cylinders drilled out of tested beams.
Four beams with equal height on both plain and lean concrete were tested statically to
determine the static flexural strength. Strain gauges were mounted on both sides of the
boundaries between the lean and plain concrete to monitor the failure load for the two
materials respectively.
The modulus of elasticity in bending for both plain and lean concrete was also determined
with the measured deflection through static tests, on specimens tested in an earlier study
(Söderqvist & Silfwerbrand, 2005 and Söderqvist & Silfwerbrand, 2006 ) with the following
equation:
Efl
131^F2 F1`l 3
686^G 2 G1`bh 3
(4)
where,
l
b
h
F1
F2
Fcr
G1,2
= length
= width
= height
= 1/4.Fcr
= 3/4.Fcr
= failure load
= midspan deflections corresponding to F1 and F2
The static modulus of elasticity is also determined with Ø100×200 mm cylinders for both
plain and lean concrete. Results from the static tests are presented in Tables Nos. 5 - 8.
Table 5. Static material strengths for lean concrete. The late age is due to delayed start of
testing. The test is made on lean concrete beams in an earlier study. The materiel is
nevertheless from the same batch and production site.
Age at testing
No. of tests
Strength
Type of
Type
(days)
(-)
and
strength
(mm)
Stiffness
cubes (heigth×width×length)
(MPa)
cylinders (diameter×heigth)
beams (length×width×height)
fcc 28d
Cubes (150×150×150)
16.3
28
3
fcc 400d
Cylinders (Ø100×100)
13.7
400
6
fcfl 440d
Beams (800×150×100)
2.1
440
3
Ec 500d
Cylinders (Ø100×200)
28 000
500
6
Efl 440d
Beams (800×150×100)
16 000
440
3
10
Table 6. Static material strengths for plain concrete.
Type of strength
Type
Strength and
(mm)
stiffness
(MPa)
fcc 28d
Cubes (150×150)
86.3
fcc 146d
Cylinders (Ø100×100)
97.5
fcfl 200d
Beams (800×150×100)
5.5*
Ec 200d
Cylinders (Ø100×200)
35 000*
Efl 200d
Beams (800×150×100)
21 000*
Age at test
(days)
No. of tests
(-)
28
146
200
200
200
3
5
3
6
3
*Tests made on different beams in an earlier study, where the same mixture was utilised.
Table 7. Pull-off tests on cylinders from tested beams. All failures occurred in the lean
concrete.
Dry Specimens
Type of
Type*
Strength
Age at test
No. of tests
strength
(mm)
(MPa)
(days)
(-)
balk 1-1
Cylinders (Ø72×100)
0.87
450
3
balk 1-2
“
0.81
“
3
balk 2-5
“
0.24
“
2
Wet Specimens
balk 2-5
“
0.16
“
1
balk 2-6
“
0.52
“
3
balk 2-3
“
0.56
“
2
balk 2-4
“
0.71
“
2
Table 8. Static flexural testing of
composite beams.
Fmax
fcfl,LC
Id
(MPa)
(kN)
Balk 1-1
15.00
2.84
Balk 1-2
12.58
2.57
Balk 2-5
24.19
2.66
Balk 2-2
25.20
2.50
Mean value:
2.64
Standard deviation:
0.14
Balk 2-3*
9.25
3.06
* Beam was first subjected to 2 million loading
cycles before conducting the static test
FATIGUE TEST RESULTS
A total of seven beams were tested in fatigue loading according to the tests procedure
described in the previous section (one beam cracked at first loading due to mistreatment).
First, four beams with equal height of plain and lean concrete were tested. All these beams
showed an excellent bond between the two materials and failed in one single vertical crack, in
a brittle behaviour. There were no crack development and as soon as the lean concrete failed,
the plain concrete also failed due to the high load, see section Bending Stresses in a
Composite Beam. In the next test series, the lean concrete layer was cut horizontally to
11
approximately half the thickness of the plain concrete. This procedure is adopted so that a
failure load of the lean concrete would not exceed the failure load of the plain concrete,
making it possible to analyse the bonding effect and a possible crack development in the lean
concrete layer in the test specimens.
The specimens were carefully measured before testing to apply a load as close as possible to
the desired load level with regard to the height and width of each individual specimen. A
constant value of R = Vmin/Vmax corresponding to 0.22 was considered. Before each fatigue
test, the number of loads, referred to as logNc, that the current beam could withstand in the
bottom of both the plain and lean concrete layer was calculated according to Tepfers’ fatigue
equation and the calculation method described in the section Bending Stresses in a Composite
Beam. This number was then evaluated compared to the number of loads that the current
beam could take before failure, logNm, for each of the two layers. The beams were covered
with a plastic folio to keep the moisture level constant throughout the test. Two LVDT’s were
mounted on every beam to measure the midspan deflection.
EVALUATION OF FATIGUE TEST RESULTS
Each beam was measured after testing, and the dimensions of the crack cross-section of the
beams were used to calculate a more accurate relative stress level. The load level that each
beam was subjected to is calculated using the relation between the presumed maximum load,
i.e. flexural static strength, and the applied load. Since the static test results had a large scatter
on the static flexural strength, a 14 % deviation (i.e. the standard deviation from the static
tests, see Table 8) from the mean value is considered in the evaluation of the fatigue tests.
This means that if the flexural strength is varied, consequentially, the load levels and the
number of predicted load applications will vary. For high load levels, when the maximum
load generates stresses near the static stress limit, small changes in the static strength have a
large impact in the number of predicted load applications. Since no crack development was
observed during testing, calculations on fatigue strength is based only on the flexural strength
of the lean concrete. Results are presented in Table 9 and Figure 7.
Table 9. Flexural fatigue test results. Results are evaluated using a 14 % variation of the
calculated mean static flexural strength fcfl, mean = 2.64 MPa for the lean concrete.
HPCC/
Id
logNm logNm/logNc
logNc
R
Vmax/ fcfl
HLC
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Balk 1-3
Balk 1-4
Balk 1-5
Balk 1-6
Balk 2-4
Balk 2-6
1.08
1.12
1.19
1.03
1.75
1.51
0.81
0.76
0.69
0.69
0.88
0.98
0.61
0.57
0.52
0.52
0.66
0.74
0.22
0.22
0.22
0.22
0.22
0.22
3.61
4.58
5.78
5.84
2.33
0.46
7.33
8.06
8.96
9.00
6.36
4.94
3.70
4.09
5.46
5.38
5.72
5.37
1.02
0.89
0.94
0.92
2.45
11.76
0.52
0.38
0.34
0.32
1.55
10.67
Note: Columns 3, 6 and 9 show values when calculating with a static flexural strength that is decreased with
14 % and columns 4, 7, and 10 correspond to calculations done with a static flexural strength increased with
14 % from the mean value.
12
LogNm/logNc (fcfl,assumed=1,14*fcfl, mean)
LogNm/logNc (fcfl,assumed= 0.86*fcfl, mean)
sigma_max/(0.86*fcfl, mean)
sigma_max/(1.14*fcfl, mean)
200
Damage / V max/f cfl [%]
175
150
125
100
75
50
25
0
Balk 1-3
Balk 1-4
Balk 1-5
Balk 1-6
Balk 2-4
Balk 2-6
Specimen [ID]
Figure 7. Fatigue test results. The diagram shows the damage (logNm/logNc) and maximum load
(Vmax/fcfl). The columns represent the damage of each beam in relation to the predicted number of load
applications. 100 % is the fatigue limit according to Equation 1. The markings represent the relative
maximum load that each beam was subjected to. Specimens Balk 2-4 and Balk 2-6 are extremely
persistent and therefore cut in this diagram, see values in Table 9.
The beams with a lean concrete layer of approximately half the thickness of the plain concrete
were superior in fatigue loading. One possible explanation is the prestress in the specimens
that is generated from the autogenous shrinkage of the plain concrete. This effect will be more
pronounced in a thin lean concrete layer with less ability to counteract the prestressing.
One beam was also subjected to several load levels and the accumulated damage was
calculated, see Table 10 and Figure 8. The beam did not fail due to the fatigue loading and
was therefore tested statically at the end, see Table 10.
Table 10. Accumulated damage for three load levels. Results are evaluated using a 14 %
variation of the calculated mean static flexural strength fcfl, LC = 2.64 MPa.
Vmax/ fcfl
(1)
(2)
(3)
R
(4)
Balk 2-3, 1st
Balk 2-3, 2nd
Balk 2-3, 3rd
0.54
0.57
0.61
0.66
0.70
0.74
0.22 6.42
0.22 5.65
0.22 4.88
Id
logNc
(5)
(6)
8.62
7.99
7.36
logNm
(7)
5.97
5.18
6.08
6logNm/logNc
(8)
(9)
0.4
0.7
16.5
0.0
0.0
0.1
Note: Columns 2, 5, and 8 show values when calculating with a static flexural strength that is decreased with
14 % and columns 3, 6, and 9 correspond to calculations done with a static flexural strength increased with 14 %
from the mean value.
13
balk 2-3, Fcfl=2.38
balk 2-3, Fcfl=2.90
100
90
80
Damage (%)
70
60
50
40
30
20
10
0
0
500000
1000000
1500000
2000000
2500000
Load applications (N)
Figure 8. Beam subjected to three different load levels. Diagram showing the accumulated damage
calculated with Miner-Palmgren’s damage hypothesis. In reality, the beam withstood all three load
levels and finally failed under a static load of F = 9.25 kN equal to a flexural strength of
fc,fl LC = 3.06 MPa (Table 10). The two lines correspond to two different values on the assumed static
flexural strength (fcfl=2.38 MPa and fcfl=2.90 MPa).
DEFLECTIONS DUE TO REPEATED LOADING
The deflections were measured with two LVDT’s mounted on each side of the beams. The
deflection was also used to determine if cracking occurred in the lean concrete layer. The
deflections were recorded at the beginning of the test and then every 50 to 10 000 cycles
depending on the predicted number of loads to failure. The maximum deflection before failure
is difficult to capture since it is impossible to predict the exact time of failure. Deflections are
presented in Figure 9. The deflection grows linearly versus the logN, with an accelerated
growth just before failure.
14
balk2-4
balk2-6
balk2-3
balk 1-6
balk 1-3
balk1-4
balk1-5
400
Increase deflection [%]
350
300
250
200
150
100
50
0
2
3
3
4
4
5
5
6
6
7
7
Load Applications [LogN]
Figure 9. The deflection growth as a percentage of the first deflection. The final deflection is not
necessarily the maximum deflection before failure. The last measured deflection is nevertheless
recorded less than 10 000 cycles before failure.
DISCUSSION
In concrete pavement design, a lean concrete road base is used to reduce the stresses in the
concrete overlay and the vertical strain in the subgrade, making it possible to design
pavements for high traffic loads without increasing the concrete pavement thickness
dramatically.
The tests show a very good bond between the materials. All cracks were inspected and pulloff tests were conducted. The ratio between the material strength and the modulus of elasticity
is low and as the tensile stress reaches the stress strength limit of the material, a crack goes
through the whole section. In these tests, peak stresses in the bottom of the lean concrete in
the composite beam were investigated. The crack development did not stop in the interface
between the plain and lean concrete, instead the whole beam cracked as soon as the stresses
reached the fatigue limit in the lean concrete. Since the bonding was extremely good for all
beams, the cracking of the beams can be explained by the natural behaviour of a crack
development; the crack develops at the smallest cross-section, in the most stressed region, and
opens perpendicularly to the stress direction. The crack develops because there is a
concentration of stresses at the end of the crack, see Figure Nos. 10 - 11. Stresses and strains
have to be taken care of by a smaller and smaller cross-section, and these stresses are
multiplied instantly at the end of the crack. The bond between the lean and plain concrete
allows strains and stresses to pass through the interface, and a stress concentration at the
interface is therefore possible.
15
The tests also show that the deflections grow linearly in a logarithmic diagram. This also has
been observed when testing plain and lean concrete separately. The deflection did not show
any signs of intermediate cracking either.
Figure 10. Finite element model (FE-model) showing the stress concentration over the crack of a
partly cracked beam with plain concrete on top and lean concrete in the bottom. Full bond is
considered. The model constitutes of shell elements and is modelled in the finite element analysis
(FEA) software LUSAS Bridge (LUSAS, 2006).
0,1
FE-model uncracked F=5,00 kN
0,09
Equation (2), F=6,65 kN uncracked
H [m] (LC up to 0.03 m)
0,08
FE-model cracked (120*120 mesh)
PCC F=6,65 kN
0,07
0,06
0,05
0,04
0,03
0,02
0,01
0
-8,00E+06
-6,00E+06
-4,00E+06
-2,00E+06
0,00E+00
2,00E+06
4,00E+06
6,00E+06
8,00E+06
1,00E+07
Stresses [MPa]
Figure 11. Stress distribution over the cross-section at the crack calculated with Equation (2), and the
FEA software, LUSAS Bridge. Stresses are concentrated in the bottom of the plain concrete layer
above the crack in the lean concrete. The stresses (8 MPa) exceed the strength limit (5.5 MPa) for the
plain concrete, which results in reflection cracks. The bond is set to 100 % in the FE-model.
16
CONCLUSIONS
In the flexural fatigue tests, the first four beams had the same height of both plain and lean
concrete. These four beams showed consistent fatigue properties; full bond and no crack
development were observed. In the tests where the height of the lean concrete was
approximately half of the plain concrete, the fatigue was superior compared to Tepfers’
fatigue criterion. These beams were tested in a trial to achieve partial cracking in the
specimen. The load levels were chosen in a way that, even though the lean concrete would
fail, the plain concrete would be sufficiently thick to could carry the applied load. However,
all beams cracked entirely and no partial cracking was observed. The cracking is explained by
(1) the full bond between the materials that allowed stresses to pass through the interface, and
(2) the concentrated stresses that are assembled at the crack opening, caused by the decreasing
cross-section.
The phenomenon of entire cracking of the cross-section is referred to as reflection cracks, and
challenges the question on whether the bond really is desirable or not, i.e. strength versus risk
of cracking. The conclusion is that bond may be detrimental because it ruins the whole section
and is difficult to control.
The higher flexural fatigue strength of the beams with a relatively thinner lean concrete layer
is explained by a possible prestress from the shrinkage of the plain concrete on top. It is likely
that a thinner lean concrete layer is more influenced by this prestress.
The deflection increased steadily throughout every test, and as the specimen come near the
fatigue limit the deflections grow more rapidly. The increasing deflections work as a
forewarning that the beam is about to crack. In this study, the average deflections increased
with 100 % or more from the first deflection, compared to the deflections of observed for
plain concrete beams (Söderqvist & Silfwerbrand, 2005), that just reached 60 – 100 %.
The conclusions from the study can be summarised as follows:
1. The bond between the plain and lean concrete is remarkably good still after 1 million
loading cycles. The bond has been obtained without any additional treatment made on
the surfaces of the specimens.
2. The static flexural tests did not indicate any bond loss. Neither did the flexural fatigue
tests.
3. The static tests did not show any partial cracking in the lean concrete, the crack at the
failure load always went through the entire cross-section.
4. The deflection increases more steadily versus the logarithm of loading cycles for the
composite beams than for plain concrete beams.
5. The fatigue strength of composite beams of plain and lean concrete was equal or better
than predicted with Tepfers’ fatigue criterion, Equation (1).
6. The bond is may be detrimental because, even though it brings more strength to the
structure, overloading or fatigue loading causes all-pervading cracks.
In concrete pavement design the bond can be beneficial because a more stiff construction
reduces stresses and strains in the subgrade. To account for a higher loading capacity, a bond
could be considered but it would in that case be more important to focus on the stresses in the
bottom of the lean concrete instead of the plain concrete. Furthermore, the thermal cracking of
the lean concrete has to be avoided by sawing at short distances.
17
References
Aas-Jacobsen, K., (1970), Fatigue of concrete beams and columns. NTH Institutt for
Betonkonstruksjoner, September 1970. Bulletin No. 70-1.
LUSAS, (2006), LUSAS Bridge finite element software, version 13.8. FEA Ltd, Forge House,
66 High Street, Kingston upon Thames, Surrey, KT1 1HN, United Kingdom
Silfwerbrand, J., (1995), Design of Concrete Pavements (Dimensionering av
betongbeläggningar), Report No. 9, 2nd Edition., Royal Inst. of Tech., Dept. of Struct. Eng.,
Stockholm, 91 pp. 1995 (In Swedish).
SRA, (2005), ATB VÄG 2005 (Technical Guidelines for Road Structures). Part A-K. Report
No. 2005:112, Swedish National Road Administration, Borlänge, Sweden, 2005. (In
Swedish).
Söderqvist, J., & Silfwerbrand, J., (2005), Design of Concrete Pavements: A Comparison
between Swedish and U.S. Methods. Proceedings, 8th International Conference on Concrete
Pavements. Colorado Springs, Colorado, USA, August 14-18, 2005, Vol. I, pp. 1-20.
Söderqvist, J., & Silfwerbrand, J., (2005), Flexural Fatigue of Plain Concrete Beams.
International Journal of Pavement Engineering. Submitted 2006.
Söderqvist, J., & Silfwerbrand, J., (2006), Design Criteria for Lean Concrete. Proceedings,
10th International DUT-Workshop on Fundamental Modelling of Design and Performance of
Concrete Pavements. Old-Turnhout, Belgium, September 15-16, 2006, pp. 1-20.
Tepfers, R., Fridén, C., Georgsson, L., (1977). A study of the applicability to the fatigue of
concrete of the Palmgran-Miner partial damage hypothesis. Magazine of Concrete Research,
Vol. 29, No. 100, September 1977.
Tepfers, R., (1978), An examination of the fatigue properties of concrete (En undersökning av
betongens utmattningshållfasthet). Byggforskningen, Report R86:1978, 1978, 121 pp.
Tepfers, R., & Kutti, T., (1979), Fatigue Strength of Plain, Ordinary, and Lightweight
Concrete. ACI Journal,Proceedings V. 76, nr 5, May 1979, pp. 635-652.
Tepfers, R., (1979), Tensile Fatigue Strength of Plain Concrete. ACI Journal, Proceedings V.
76, No.8, August, 1979, pp. 919-933.
18
List of Bulletins from the Department of Structural Engineering,
The Royal Institute of Technology, Stockholm
TRITA-BKN. Bulletin
Pacoste, C., On the Application of Catastrophe Theory to Stability Analyses of Elastic Structures.
Doctoral Thesis, 1993. Bulletin 1.
Stenmark, A-K., Dämpning av 13 m lång stålbalk "Ullevibalken". Utprovning av dämpmassor och
fastsättning av motbalk samt experimentell bestämning av modformer och förlustfaktorer. Vibration tests
of full-scale steel girder to determine optimum passive control.
Licentiatavhandling, 1993. Bulletin 2.
Silfwerbrand, J., Renovering av asfaltgolv med cementbundna plastmodifierade avjämningsmassor.
1993. Bulletin 3.
Norlin, B., Two-Layered Composite Beams with Nonlinear Connectors and Geometry Tests and
Theory.
Doctoral Thesis, 1993. Bulletin 4.
Habtezion, T., On the Behaviour of Equilibrium Near Critical States.
Licentiate Thesis, 1993. Bulletin 5.
Krus, J., Hållfasthet hos frostnedbruten betong.
Licentiatavhandling, 1993. Bulletin 6.
Wiberg, U., Material Characterization and Defect Detection by Quantitative Ultrasonics.
Doctoral Thesis, 1993. Bulletin 7.
Lidström, T., Finite Element Modelling Supported by Object Oriented Methods.
Licentiate Thesis, 1993. Bulletin 8.
Hallgren, M., Flexural and Shear Capacity of Reinforced High Strength Concrete Beams without
Stirrups. Licentiate Thesis, 1994. Bulletin 9.
Krus, J., Betongbalkars lastkapacitet efter miljöbelastning.
1994. Bulletin 10.
Sandahl, P., Analysis Sensitivity for Wind-related Fatigue in Lattice Structures.
Licentiate Thesis, 1994. Bulletin 11.
Sanne, L., Information Transfer Analysis and Modelling of the Structural Steel Construction Process.
Licentiate Thesis, 1994. Bulletin 12.
Zhitao, H., Influence of Web Buckling on Fatigue Life of Thin-Walled Columns.
Doctoral Thesis, 1994. Bulletin 13.
Kjörling, M., Dynamic response of railway track components. Measurements during train passage and
dynamic laboratory loading.
Licentiate Thesis, 1995. Bulletin 14.
Yang, L., On Analysis Methods for Reinforced Concrete Structures.
Doctoral Thesis, 1995. Bulletin 15.
Petersson, Ö., Svensk metod för dimensionering av betongvägar.
Licentiatavhandling, 1996. Bulletin 16.
Lidström, T., Computational Methods for Finite Element Instability Analyses.
Doctoral Thesis, 1996. Bulletin 17.
Krus, J., Environment- and Function-induced Degradation of Concrete Structures.
Doctoral Thesis, 1996. Bulletin 18.
Editor, Silfwerbrand, J., Structural Loadings in the 21st Century.
Sven Sahlin Workshop, June 1996. Proceedings. Bulletin 19.
Ansell, A., Frequency Dependent Matrices for Dynamic Analysis of Frame Type Structures.
Licentiate Thesis, 1996. Bulletin 20.
Troive, S., Optimering av åtgärder för ökad livslängd hos infrastrukturkonstruktioner.
Licentiatavhandling, 1996. Bulletin 21.
Karoumi, R., Dynamic Response of Cable-Stayed Bridges Subjected to Moving Vehicles.
Licentiate Thesis, 1996. Bulletin 22.
Hallgren, M., Punching Shear Capacity of Reinforced High Strength Concrete Slabs.
Doctoral Thesis, 1996. Bulletin 23.
Hellgren, M., Strength of Bolt-Channel and Screw-Groove Joints in Aluminium Extrusions.
Licentiate Thesis, 1996. Bulletin 24.
Yagi, T., Wind-induced Instabilities of Structures.
Doctoral Thesis, 1997. Bulletin 25.
Eriksson, A., and Sandberg, G., (editors), Engineering Structures and Extreme Events proceedings
from a symposium, May 1997. Bulletin 26.
Paulsson, J., Effects of Repairs on the Remaining Life of Concrete Bridge Decks.
Licentiate Thesis, 1997. Bulletin 27.
Olsson, A., Object-oriented finite element algorithms.
Licentiate Thesis, 1997. Bulletin 28.
Yunhua, L., On Shear Locking in Finite Elements.
Licentiate Thesis, 1997. Bulletin 29.
Ekman, M., Sprickor i betongkonstruktioner och dess inverkan på beständigheten.
Licentiate Thesis, 1997. Bulletin 30.
Karawajczyk, E., Finite Element Approach to the Mechanics of Track-Deck Systems.
Licentiate Thesis, 1997. Bulletin 31.
Fransson, H., Rotation Capacity of Reinforced High Strength Concrete Beams.
Licentiate Thesis, 1997. Bulletin 32.
Edlund, S., Arbitrary Thin-Walled Cross Sections. Theory and Computer Implementation.
Licentiate Thesis, 1997. Bulletin 33.
Forsell, K., Dynamic analyses of static instability phenomena.
Licentiate Thesis, 1997. Bulletin 34.
Ikäheimonen, J., Construction Loads on Shores and Stability of Horizontal Formworks.
Doctoral Thesis, 1997. Bulletin 35.
Racutanu, G., Konstbyggnaders reella livslängd.
Licentiatavhandling, 1997. Bulletin 36.
Appelqvist, I., Sammanbyggnad. Datastrukturer och utveckling av ett IT-stöd för byggprocessen.
Licentiatavhandling, 1997. Bulletin 37.
Alavizadeh-Farhang, A., Plain and Steel Fibre Reinforced Concrete Beams Subjected to Combined
Mechanical and Thermal Loading.
Licentiate Thesis, 1998. Bulletin 38.
Eriksson, A. and Pacoste, C., (editors), Proceedings of the NSCM-11: Nordic Seminar on Computational
Mechanics, October 1998. Bulletin 39.
Luo, Y., On some Finite Element Formulations in Structural Mechanics.
Doctoral Thesis, 1998. Bulletin 40.
Troive, S., Structural LCC Design of Concrete Bridges.
Doctoral Thesis, 1998. Bulletin 41.
Tärno, I., Effects of Contour Ellipticity upon Structural Behaviour of Hyparform Suspended Roofs.
Licentiate Thesis, 1998. Bulletin 42.
Hassanzadeh, G., Betongplattor på pelare. Förstärkningsmetoder och dimensioneringsmetoder för plattor
med icke vidhäftande spännarmering.
Licentiatavhandling, 1998. Bulletin 43.
Karoumi, R., Response of Cable-Stayed and Suspension Bridges to Moving Vehicles. Analysis methods
and practical modeling techniques.
Doctoral Thesis, 1998. Bulletin 44.
Johnson, R., Progression of the Dynamic Properties of Large Suspension Bridges during Construction
A Case Study of the Höga Kusten Bridge.
Licentiate Thesis, 1999. Bulletin 45.
Tibert, G., Numerical Analyses of Cable Roof Structures.
Licentiate Thesis, 1999. Bulletin 46.
Ahlenius, E., Explosionslaster och infrastrukturkonstruktioner - Risker, värderingar och kostnader.
Licentiatavhandling, 1999. Bulletin 47.
Battini, J-M., Plastic instability of plane frames using a co-rotational approach.
Licentiate Thesis, 1999. Bulletin 48.
Ay, L., Using Steel Fiber Reinforced High Performance Concrete in the Industrialization of Bridge
Structures.
Licentiate Thesis, 1999. Bulletin 49.
Paulsson-Tralla, J., Service Life of Repaired Concrete Bridge Decks.
Doctoral Thesis, 1999. Bulletin 50.
Billberg, P., Some rheology aspects on fine mortar part of concrete.
Licentiate Thesis, 1999. Bulletin 51.
Ansell, A., Dynamically Loaded Rock Reinforcement.
Doctoral Thesis, 1999. Bulletin 52.
Forsell, K., Instability analyses of structures under dynamic loads.
Doctoral Thesis, 2000. Bulletin 53.
Edlund, S., Buckling of T-Section Beam-Columns in Aluminium with or without Transverse Welds.
Doctoral Thesis, 2000. Bulletin 54.
Löfsjögård, M., Functional Properties of Concrete Roads í General Interrelationships and Studies on
Pavement Brightness and Sawcutting Times for Joints.
Licentiate Thesis, 2000. Bulletin 55.
Nilsson, U., Load bearing capacity of steel fibree reinforced shotcrete linings.
Licentiate Thesis, 2000. Bulletin 56.
Silfwerbrand, J. and Hassanzadeh, G., (editors), International Workshop on Punching Shear Capacity of
RC Slabs í Proceedings. Dedicated to Professor Sven Kinnunen. Stockholm June 7-9, 2000. Bulletin 57.
Wiberg, A., Strengthening and repair of structural concrete with advanced, cementitious composites.
Licentiate Thesis, 2000. Bulletin 58.
Racutanu, G., The Real Service Life of Swedish Road Bridges - A case study.
Doctoral Thesis, 2000. Bulletin 59.
Alavizadeh-Farhang, A., Concrete Structures Subjected to Combined Mechanical and Thermal Loading.
Doctoral Thesis, 2000. Bulletin 60.
Wäppling, M., Behaviour of Concrete Block Pavements - Field Tests and Surveys.
Licentiate Thesis, 2000. Bulletin 61.
Getachew, A., Trafiklaster på broar. Analys av insamlade och Monte Carlo genererade fordonsdata.
Licentiatavhandling, 2000. Bulletin 62.
James, G., Raising Allowable Axle Loads on Railway Bridges using Simulation and Field Data.
Licentiate Thesis, 2001. Bulletin 63.
Karawajczyk, E., Finite Elements Simulations of Integral Bridge Behaviour.
Doctoral Thesis, 2001. Bulletin 64.
Thöyrä, T., Strength of Slotted Steel Studs.
Licentiate Thesis, 2001. Bulletin 65.
Tranvik, P., Dynamic Behaviour under Wind Loading of a 90 m Steel Chimney.
Licentiate Thesis, 2001. Bulletin 66.
Ullman, R., Buckling of Aluminium Girders with Corrugated Webs.
Licentiate Thesis, 2002. Bulletin 67.
Getachew, A., Traffic Load Effects on Bridges. Statistical Analysis of Collected and Monte Carlo
Simulated Vehicle Data.
Doctoral Thesis, 2003. Bulletin 68.
Quilligan, M., Bridge Weigh-in-Motion. Development of a 2-D Multi-Vehicle Algorithm.
Licentiate Thesis, 2003. Bulletin 69.
James, G., Analysis of Traffic Load Effects on Railway Bridges.
Doctoral Thesis 2003. Bulletin 70.
Nilsson, U., Structural behaviour of fibre reinforced sprayed concrete anchored in rock.
Doctoral Thesis 2003. Bulletin 71.
Wiberg, A., Strengthening of Concrete Beams Using Cementitious Carbon Fibre Composites.
Doctoral Thesis 2003. Bulletin 72.
Löfsjögård, M., Functional Properties of Concrete Roads – Development of an Optimisation Model and
Studies on Road Lighting Design and Joint Performance.
Doctoral Thesis 2003. Bulletin 73.
Bayoglu-Flener, E., Soil-Structure Interaction for Integral Bridges and Culverts.
Licentiate Thesis 2004. Bulletin 74.
Lutfi, A., Steel Fibrous Cement Based Composites. Part one: Material and mechanical properties.
Part two: Behaviour in the anchorage zones of prestressed bridges.
Doctoral Thesis 2004. Bulletin 75.
Johansson, U., Fatigue Tests and Analysis of Reinforced Concrete Bridge Deck Models.
Licentiate Thesis 2004. Bulletin 76.
Roth, T., Langzeitverhalten von Spannstählen in Betonkonstruktionen.
Licentitate Thesis 2004. Bulletin 77.
Hedebratt, J., Integrerad projektering och produktion av industrigolv – Metoder för att förbättra
kvaliteten.
Licentiatavhandling, 2004. Bulletin 78.
Österberg, E., Revealing of age-related deterioration of prestressed reinforced concrete containments in
nuclear power plants – Requirements and NDT methods.
Licentiate Thesis 2004. Bulletin 79.
Broms, C.E., Concrete flat slabs and footings New design method for punching and detailing for
ductility.
Doctoral Thesis 2005. Bulletin 80.
Wiberg, J., Bridge Monitoring to Allow for Reliable Dynamic FE Modelling - A Case Study of the New
Årsta Railway Bridge.
Licentiate Thesis 2006. Bulletin 81.
Mattsson, H-Å., Funktionsentreprenad Brounderhåll – En pilotstudie i Uppsala län.
Licentiate Thesis 2006. Bulletin 82.
Masanja, D. P, Foam concrete as a structural material.
Doctoral Thesis 2006. Bulletin 83.
Johansson, A., Impregnation of Concrete Structures – Transportation and Fixation of Moisture in Water
Repellent Treated Concrete.
Licentiate Thesis 2006. Bulletin 84.
Billberg, P., Form Pressure Generated by Self-Compacting Concrete – Influence of Thixotropy and
Structural Behaviour at Rest.
Doctoral Thesis 2006. Bulletin 85.
Enckell, M., Structural Health Monitoring using Modern Sensor Technology – Long-term Monitoring of
the New Årsta Railway Bridge.
Licentiate Thesis 2006. Bulletin 86.
The bulletins enumerated above, with the exception for those which are out of print, may be purchased
from the Department of Civil and Architectural Engineering, The Royal Institute of Technology, SE-100
44 Stockholm, Sweden.
The department also publishes other series. For full information see our homepage http://www.byv.kth.se
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