Modeling Pavement Performance Based on Data From the Swedish LTPP Database

Modeling Pavement Performance Based on Data From the Swedish LTPP Database
Modeling Pavement Performance
Based on Data From the Swedish
LTPP Database
Predicting Cracking and Rutting
Licentiate Thesis in Highway Engineering
Markus Svensson
KTH, Royal Institute of Technology
School of Architecture and the Built Environment
Department of Highway Engineering
Division of Highway and Railway Engineering
Stockholm
[email protected]
January 8, 2013
Abstract
The roads in our society are in a state of constant degradation. The reasons for
this are many, and therefore constructed to have a certain lifetime before being
reconstructed. To minimize the cost of maintaining the important transport road
network high quality prediction models are needed. This report presents new models for flexible pavement structures for initiation and propagation of fatigue cracks
in the bound layers and rutting for the whole structure.
The models are based on observations from the Swedish Long Term Pavement Performance (LTPP) database. The intention is to use them for planning maintenance
as part of a pavement management system (PMS). A statistical approach is used
for the modeling, where both cracking and rutting are related to traffic data, climate
conditions, and the subgrade characteristics as well as the pavement structure.
Keywords: performance predictions, LTPP, rut, crack, statistical modeling.
Preface
This thesis is best viewed digitally. All readers are therefore encouraged to print
this thesis only if necessary.
Thank you!
Markus Svensson, Linköping, January 8, 2013
i
Acknowledgment
There are of course many people who have contributed to this project, and for this
I am very thankful. But without the help from my fellow Ph.D. students this thesis
would not exist. The project was financed by VTI and STA, ergo. . . a big "Thanks!"
to the Swedish tax payers.
ii
List of Publications
Modeling of performance for road structure rutting and cracking based on
data from the Swedish LTPP database
Presented at the 8th International Conference on Managing Pavement, Santiago,
Chile.
MODELING PERFORMANCE PREDICTION, BASED ON RUTTING AND
CRACKING DATA
Presented at the 4th European pavement and asset management conference (EPAM),
in Malmö, Sweden.
The papers are found in Appendix F.
iii
Acronyms and Abbreviations
AADT Annual Average Daily traffic..
AADTT Annual Average Daily Truck traffic..
AADT Ty Annual Average Daily Truck Traffic volume y years after the road was
opened for traffic.
AADT f Annual Average Daily Traffic.
AADTo Annual Average Daily Traffic at the time of opening the road .
AASHO American Association of State Highway and Transportation Officials .
AASHTO American Association of State Highway and Transportation Officials.
ABD Drained Asphalt Concrete.
ABS Stone mastic asphalt concrete.
ABT Dense Asphalt conrete.
ABb AC binder layer.
AG Roadbase.
BG bitumen stabilizing gravel.
C-LTPP Canadian Long Term Pavement Performance.
C-SHRP Canadian Strategic Highway Research Program.
Ci Crack index.
DBMS Database Management System.
DBPs Deflection Basin Parameters.
DSS Decision Support System.
EER Enhanced Entity Relationship Model.
ER Entity Relationship Model.
ESALs Equivalent Single Axle Loads.
FHWA Federal Highway Administration.
FWD Falling Weight Deflectometer.
GIS Geographic Information System.
GIS Geographic Information Systems.
HABS Stone mastic Asphalt concrete AC with hard binder (pen 70 100)(SMA).
HABS Stone mastic asphalt concrete with hard binder (pen 70 100) (SMA).
HDM Highway Development and Management.
HDM-IV Highway Development and Management IV.
HIPS Highway Investment Programming System.
HMA Hot Mix Asphalt.
HVS Heavy Vehicle Simulator.
Heating/repaving a maintenance method. Heating the existing layer and placing a
new.
LTPP Long Term Pavement Performance.
iv
MAB Asphalt concrete with soft binder (pen 180 220).
MABT Dense Asphalt conrete with soft binder (pen 180 220).
MAT Mean Annual Temperature.
MEPDG Mechanistic Empirical Pavement Design Guide.
MMS Maintenance Management System.
Mr Resilient Modulus.
NVDB National Road Database.
N100Y Number of equivalent heavy vehicle axles per year.
N100 Number of equivalent heavy vehicle axles.
N100 Number of equivalent standard vehicle axles.
PCC Portland Cement Concrete.
PMS Pavement Management System.
PSI Present Serviceability Index.
PSI Present Serviceability Index.
PV Previous Value.
RDBMS Relational Database Management System.
RMS Road Management Systems.
RMSs Road Management Systems.
RST Road Surface Tester.
Remixing recycling method.
SCI300 Surface Curvature Index 300.
SHRP Strategic Highway Research Program.
SQL Structured Query Language.
STA Swedish Transport Administration.
Stabinor Adhesive layer, material created by Skanska.
TRB Transportation Research Board.
VTI The Swedish National Road and Transport Research Institute.
WIM Weigh in Motion.
vpd vehicles per day.
v
List of Symbols and Terminology
Ac - Alligator cracking
a - Parameter in the rut model, equation 2.6
b - Parameter in the rut model, equation 2.6
Ci - Crack index
Ciini - Crack index, for the crack initiation phase.
Ci propa - Crack index, for the crack propagation phase.
Lc - Longitudinal cracks
Mr - Resilient Modulus, an estimate of a materials modulus of elasticity, stress divided by strain for speedily applied loads.
Y - the average annual ESALs per lane.
N100
Cri
N100
- The total number of ESALs for specific section before crack initiation, in
2.3.
Cr
N100p - The total number of ESALs for specific section before crack propagation, in
2.3.
Cr - The total number of ESALs for specific section before cracking, in 2.3.
N100
Cr - the sum of the standard axle repetitions to crack initiations and for crack
N100
propagation respectively up to failure.
Tc - Transversal cracks
)
E((X−µX )(Y −µY ))
ρX,Y = corr(X,Y ) = cov(X,Y
- Pearson product moment correlaσX σY =
σX σY
tion coefficient.
vi
List of Figures
1.1
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
The development of the number of registered vehicles in Sweden.
[56] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stress distribution in flexible and rigid pavement.[61] . . . . . . .
Composite Pavement Design. [9] . . . . . . . . . . . . . . . . . .
The derivation of the N100 axle, used in Sweden. [59] . . . . . . .
An illustration of the common types of cracks found in flexible
pavements. [16] . . . . . . . . . . . . . . . . . . . . . . . . . . .
An illustration of the Ci division initiation and propagation. [21, 78]
Illustration of common types of deformation in HMA.[16] . . . .
Surface defects. [16] . . . . . . . . . . . . . . . . . . . . . . . .
Edge Defects. [16] . . . . . . . . . . . . . . . . . . . . . . . . .
PSI is a qualitative index, from the users perspective, that separates
structural and functional distresses. When maintenance or rehabilitation is performed the PSI value is often decreased compared to
the start value. [73] . . . . . . . . . . . . . . . . . . . . . . . . .
An illustration of the concept of RMS and PMS. . . . . . . . . . .
An example from a PMS with DSS and GIS. [54] . . . . . . . . .
The conception of the life-cycle analysis in HDM-4 [63]. . . . . .
Creep characteristic of bituminous mixtures. [57] . . . . . . . . .
The graphs show deformations of the pavement due to the cyclic
load/unload. [16] . . . . . . . . . . . . . . . . . . . . . . . . . .
Illustration of the common axle types and tire configurations. [48]
The graphs show axle loads from different types of axles.[49] . . .
The climate zones classified by the number of days when the mean
temperature is below 0◦ C in Sweden. [67] . . . . . . . . . . . . .
An illustration of performance of pavement structure factors. [53]
The RST used at VTI, and an illustration of the rut sampling and
how the lasers are placed. . . . . . . . . . . . . . . . . . . . . . .
A description of the FWD measurements. [60, 11] . . . . . . . . .
An illustration of the wire surface principle in rut depth sampling.
[20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Illustration of the variance in the rut data. The data plotted comes
from object D-RV53-2. [21] . . . . . . . . . . . . . . . . . . . .
vii
1
4
4
6
7
7
8
9
9
11
13
14
15
19
19
20
20
21
24
25
26
27
28
2.23 Example of size and development of a crack between three different inspections of the same road that is carried out at different
times. This object is 100 [m], and divided into smaller 10 [m]
sections.[77] . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.24 Ci variation in an object. [21] . . . . . . . . . . . . . . . . . . . .
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
The red dots in the map mark the geographical locations of the
objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The geographical distributions of road objects in the Swedish LTPP
database. The red dots are active objects, and the orange objects
retired from data monitoring. . . . . . . . . . . . . . . . . . . . .
Traffic estimations for object H-RV40-2. The black dots are the
traffic measured and the red lines are the estimations used in the
project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Traffic estimations for object T-2051-1. The black dots are the
traffic measured and the red lines are the estimations used in the
project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Traffic estimations for object F-RV31-1. The black dots are the
traffic measured and the red lines are the estimations used in the
project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Traffic estimations for objects H-RV34-1. The black dots are the
traffic measured and the red lines are the estimations used in the
project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Traffic estimations for objects D-RV53-2. The black dots are the
traffic measured and the red lines are the estimations used in the
project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Traffic estimations for objects C-292-1. The black dots are the
traffic measured and the red lines are the estimations used in the
project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scatter plots for the Ci factors SCI300 , Mr and N100100 , representing subgrade, structure and traffic. . . . . . . . . . . . . . . . . .
Scatter plots for the Ci factors precipitation and temperature, representing climate and age. . . . . . . . . . . . . . . . . . . . . . .
Scatter plots for the Ci factors previous value (PV). . . . . . . . .
Scatter plots for the rut factors SCI300 Mr and N100 , representing
subgrade, structure and traffic. . . . . . . . . . . . . . . . . . . .
Scatter plots for the rut factors precipitation, temperature and age.
Scatter plots for the rut factor PV and the interaction effect, between standard axles and precipitation. . . . . . . . . . . . . . . .
An illustration of how the factors in the models are expressed as
data from the LTTP database. . . . . . . . . . . . . . . . . . . . .
viii
29
30
32
33
35
36
37
38
38
39
40
41
41
42
43
44
46
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
The plots show the results for the objects D-RV53-2 and T-205-1.
The black dots are the measured data, and the red line is estimated.
The Ci is measured as a function of time. . . . . . . . . . . . . . .
The plots show the results for the objects F-RV31-1, H-RV34-1, HRV40-2 and C-292-1. The black dots are the measured data, and
the red line is estimated. The Ci is measured as a function of time.
The plot shows the results for the objects D-RV532-2, H-RV34-1.
The black dots are the measured data, and the red line is estimated.
The Ci is measured as a function of time. . . . . . . . . . . . . . .
The plot shows the results for the objects T-205-1, F-RV31-1 and
H-RV40-2. The black dots are the measured data, and the red line
is estimated. The rut is measured as a function of time. . . . . . .
The Ci models used on object G-RV23-1. The small black dots are
the measured values, the dots on the red line are the predicted values.
The Ci models used on object W-RV80-1. The small black dots are
the measured values, the dots on the red line are the predicted values.
The rut model used on object W-RV80-1. The small black dots are
the measured values, the dots on the red line are the predicted values.
The Ci models used on object Z-E45-4. The small black dots are
the measured values, the dots on the red line are the predicted values.
The rut model used on object Z-E45-4. . . . . . . . . . . . . . . .
50
51
53
54
55
56
56
57
57
A.1 An illustration of the concept of random variables. . . . . . . . . .
A.2 Image of an r.v. variable. . . . . . . . . . . . . . . . . . . . . . .
A.3 A box plot displays the differences between populations without
any assumptions of the statistical distribution, i.e. a non-parametric
method. The box plot helps identify outliers by indicating the level
of dispersion and skewness in the data.[50] . . . . . . . . . . . .
A.4 An illustration of a general controlled experiment. . . . . . . . . .
A.5 A "typical" Ci , data from the object D-RV-2. . . . . . . . . . . . .
A.6 "Typical" rut values, data from the object H-RV34-1. . . . . . . .
65
66
67
69
70
70
B.1 Relational database terminology. . . . . . . . . . . . . . . . . . .
B.2 Illustration of the normal forms. . . . . . . . . . . . . . . . . . .
73
73
C.1 The Ci and rut data presented in box plots. . . . . . . . . . . . . .
83
D.1 The red dots mark the geographical location of the validation objects. 84
D.2 The Ci and rut data presented in box plots. . . . . . . . . . . . . . 87
ix
List of Tables
2.1
2.2
2.3
2.4
The values for PSI. . . . . . . . . . . . . . . . . . . . . . . .
The periods length in days during one year, [67]. . . . . . . .
Temperature, [◦C], in the bitumen bounded layers, [67]. . . .
Deflection Basin Indexes. [11] . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
11
22
22
26
3.1 General object information. . . . . . . . . . . . . .
3.2 General construction and maintenance information.
3.3 Material information continued. . . . . . . . . . .
3.4 Material information continued. . . . . . . . . . .
3.5 Ci variable structures. . . . . . . . . . . . . . . . .
3.6 Ci variable structures. . . . . . . . . . . . . . . . .
3.7 Rut variable structures. . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
31
31
32
32
45
45
45
D.1
D.2
D.3
D.4
D.5
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
84
84
85
85
87
General info of the validation road objects. .
Material info for the validation objects. . . .
Ci validation factor data. . . . . . . . . . .
Ci validation factor data. . . . . . . . . . .
Rut validation factor data. . . . . . . . . . .
x
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Contents
PREFACE
ii
ACKNOWLEDGMENT
iv
List of PUBLICATIONS
iv
LIST OF ACRONYMS
vi
DEFINITION OF SYMBOLS AND TERMINOLOGY
vii
LIST OF FIGURES
ix
LIST OF TABLES
x
1
INTRODUCTION
1.1 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
2
LITERATURE REVIEW
2.1 Pavement Types . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Flexible Pavement Structure . . . . . . . . . . . . . . . .
2.1.2 Rigid Pavement Structure . . . . . . . . . . . . . . . . .
2.1.3 Composite Pavement Structure . . . . . . . . . . . . . . .
2.2 Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Distress Types . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Permanent Deformation . . . . . . . . . . . . . . . . . .
2.3.3 Surface Defects . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Edge Defects . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Maintenance . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Pavement Performance . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Present Serviceability Rating . . . . . . . . . . . . . . . .
2.4.2 Swedish Models . . . . . . . . . . . . . . . . . . . . . .
2.5 Road Management . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Decision Support System & Geographic Information System
xi
3
3
3
4
4
5
6
6
8
8
9
9
10
11
11
13
13
13
2.6
2.7
2.8
3
4
5
2.5.3 Highway Development and Management Model-IV
2.5.4 PMS in Sweden . . . . . . . . . . . . . . . . . . .
LTPP . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 General and Specific Pavement Studies . . . . . .
2.6.2 LTPP Database . . . . . . . . . . . . . . . . . . .
2.6.3 International LTPP . . . . . . . . . . . . . . . . .
Distress Factors . . . . . . . . . . . . . . . . . . . . . . .
2.7.1 Factor Traffic . . . . . . . . . . . . . . . . . . . .
2.7.2 Climate . . . . . . . . . . . . . . . . . . . . . . .
2.7.3 Aging . . . . . . . . . . . . . . . . . . . . . . . .
2.7.4 Material . . . . . . . . . . . . . . . . . . . . . . .
Road Measurements . . . . . . . . . . . . . . . . . . . . .
2.8.1 The Road Surface Tester . . . . . . . . . . . . . .
2.8.2 Falling Weight Deflectometer . . . . . . . . . . .
2.8.3 Rut . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.4 Crack Index . . . . . . . . . . . . . . . . . . . . .
METHOD
3.1 The Swedish LTPP . . . . . . .
3.2 Traffic Estimations . . . . . . .
3.3 Distress Factor Scatter Plots . .
3.3.1 Crack Index . . . . . . .
3.3.2 Rut . . . . . . . . . . .
3.4 Factor Modeling . . . . . . . . .
3.4.1 Crack Index . . . . . . .
3.4.2 Rut . . . . . . . . . . .
3.5 Model Structure and LTPP Data
3.5.1 Preprocessing of Data .
3.5.2 Variance . . . . . . . . .
3.5.3 Statistic Tools . . . . . .
RESULTS
4.1 Crack Index Models . . .
4.1.1 Initiation . . . .
4.1.2 Propagation . . .
4.1.3 Total Crack Index
4.2 Rut Model . . . . . . . .
4.3 Sensitivity Analysis . . .
4.4 Validation . . . . . . . .
4.4.1 G-RV23-1 . . . .
4.4.2 W-RV80-1 . . .
4.4.3 Z-E45-4 . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
CONCLUSION & DISCUSSION
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
14
16
16
17
18
18
18
18
21
22
22
25
25
25
27
29
.
.
.
.
.
.
.
.
.
.
.
.
31
33
34
39
39
42
44
44
45
46
47
47
47
.
.
.
.
.
.
.
.
.
.
49
49
49
49
50
52
54
55
55
55
56
58
xii
5.1
Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . .
References
59
60
A STATISTICS THEORY
A.1 Stochastic Variables . . . . . . .
A.1.1 Standard Deviation . . .
A.1.2 Descriptive Statistics . .
A.2 Correlation and Causality . . . .
A.2.1 Scatter Plots . . . . . .
A.3 Testing for Normality . . . . . .
A.3.1 Sample Skewness . . . .
A.3.2 Kurtosis . . . . . . . . .
A.4 Factorial Design . . . . . . . . .
A.5 Regression Analysis . . . . . . .
A.5.1 Nonlinear Regression . .
A.6 Data Preprocessing . . . . . . .
A.6.1 Min-Max Normalization
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
65
65
66
67
68
68
68
68
68
69
69
69
71
71
B THE RELATIONAL DATABASE
B.1 Enhanced Entity Relationship Model
B.1.1 Microsoft Access Database .
B.2 Terminology . . . . . . . . . . . . .
B.3 Normalization . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
72
72
72
72
73
C OBJECTS
C.1 Crack Index Initiation Factor Data . .
C.2 Crack Index Propagation Factor Data .
C.3 Rut Factor Data . . . . . . . . . . . .
C.4 Crack Index and Rut Data Ranges . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
74
74
76
78
83
D VALIDATION OBJECTS
D.1 Crack Index Factor Data . . . . . . . . . . . . . . . . . . . . . .
D.2 Rut Factor Data . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.3 Validation Data Ranges . . . . . . . . . . . . . . . . . . . . . . .
84
85
87
87
E MATLAB CODE
E.1 Data extracting Ci . . . .
E.2 Crack Index Initiation . .
E.3 Crack Index Propagation
E.4 Rut . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
88
. 88
. 95
. 99
. 105
F Publications
109
F.1 Modeling of performance for road structure rutting and cracking
based on data from the Swedish LTPP database . . . . . . . . . . 109
xiii
F.2
MODELING PERFORMANCE PREDICTION, BASED ON RUTTING AND CRACKING DATA . . . . . . . . . . . . . . . . . . 120
xiv
Chapter 1
INTRODUCTION
The value of a society’s infrastructure is difficult to measure, but the benefits of a
functioning road network are easy to understand. Our roads are in a state of constant deterioration due to factors such as climate and heavy traffic loads. Therefore
they are constructed to have a certain lifetime before being maintained or rehabilitated. The transport infrastructure in Sweden costs every year several billions, in
2010 the government decided to spend more than 480 billion SEK on infrastructure
development and maintenance over the time period 2010 to 2021. To minimize this
cost, effective models for predicting performance are needed. A Pavement Management System (PMS) is therefore used for managing the road network. The PMS
requires condition data of the network, which allows an overview to enhance the
sustainability of transportation infrastructure systems.
Road deterioration is a complex process involving numerous variables and their
interaction. Road performance is affected by external loading, and is influenced by
factors such as material properties, the environment, and construction practices. In
the past, pavement design procedures have relied mainly on empirical relationships
based on long term experience, and field tests. An estimated length of 420 000
km of asphalt road exists in Sweden out of which 100 000 km is owned by the
state, municipalities own 40 000 km and there are 280 000 km privately owned.
The number of vehicles on our road network has historically increased, and this
development is shown in Figure 1.1.
Figure 1.1: The development of the number of registered vehicles in Sweden. [56]
1
1.1. AIMS
CHAPTER 1. INTRODUCTION
A Pavement Managing System (PMS) is a tool used to plan and perform maintenance on the road network in a cost efficient manner. Usually a PMS consists of
five main parts:
◦
◦
Pavement condition studies.
Database containing all relevant
pavement condition information.
◦
◦
◦
Analysis plan.
Decision strategy.
Implementation procedures.
The analysis and decision making process require high quality distress prediction
models; and two of the most common distress types are: rutting and cracking.
[37, 33, 46, 25, 3, 56, 2]
1.1
Aims
Two of the most common distresses for flexible pavements are cracks and rut. This
project has developed performance prediction models for flexible pavement structure cracks in the bound layers and rutting for the whole pavement structure. The
performance prediction models used today are obsolete, due to reasons such as
changes in the climate, new methods and technical advancements in the construction and data gathering process. The aim for this project is to develop prediction
models for initiation and propagation of cracks and rutting, that deteriorates roads
with a gravel/bitumen superstructure. The intention is to use them for planning
of maintenance activities. The data used in this project have mainly been taken
from the Swedish Long Term Pavement Performance, (LTPP). The traffic data were
complemented with data from Swedish Road Administration.
This project investigated which variables are key factors in the deterioration process, such as type of construction, volume of heavy traffic, climate, subgrade, and
then use this information to develop new deterioration models, based on data from
the Swedish LTPP database for flexible pavements. The intention, for the models,
is to use them in planning maintenance as a part of the PMS. This will enable the
Swedish Transport Administration (STA) to make reliable distress predictions in
their pursuit to prudently manage the road network. Models for this already exist,
but they can be further optimized, or replaced with more accurate models.
Specific aims:
◦ Investigate which variables are a key factors in the deterioration process,
such as type of structure, volume of heavy traffic, climate and subgrade.
◦ Develop new deterioration models, based on LTPP data, for flexible pavements.
This requires an extensive literature study, followed by extracting and analyzing
data from the Swedish LTPP program.
2
Chapter 2
LITERATURE REVIEW
This chapter contains a review of some of the current knowledge in this area of
science.
2.1
Pavement Types
The oldest findings of engineered roads are dated from about 4000 BC. Roads
paved with stone were found at Ur, in modern day Iraq, and old timber roads preserved in a swamp have been found in Glastonbury, England. The first usage of
asphalt in a road occurred in 1824 in the Champs-Élysées in Paris. The more modern type of asphalt was created by Edward de Smedt at Columbia University in
New York City, and used in Battery Park in New York City in 1872 [4].
Today a highway pavement is a structure consisting of superimposed layers of
processed materials above the natural soil subgrade. The purpose of a pavement
structure is to provide a surface with:
◦ adequate skid resistance.
◦ favorable light reflecting characteristics.
◦ acceptable riding quality.
◦ low noise pollution.
◦ long design life with low construction and maintenance cost.
This is done by distributing the applied vehicle loads to the subgrade, and providing
waterproofing. The body of the road is divided into:
◦ Surface layer.
◦ Base course, usually unbounded.
◦ Subgrade, reinforcement to the base course and the "naturally" occurring
soil. Normally the layer with the largest thickness.
Pavement structures can generally be categorized into two groups, flexible or rigid.
A third group; composite pavement, is a mixture of both. This type is more expensive and seldom used [27]. They are all presented in the following text.
2.1.1
Flexible Pavement Structure
The most common type of pavement structure in Sweden is flexible pavements,
i.e. flexible due to the fact that the structure bends under loading. The surface is
covered with a hard waterproof bituminous material. The top layers are designed
3
2.1. PAVEMENT TYPES
CHAPTER 2. LITERATURE REVIEW
to withstand the stresses from the vehicles and to distribute the loads to the lower
layers over a larger area. This design method modifies the stress distribution area in
such a way that less expensive weaker materials can be used [27]. This is illustrated
in Figure 2.1.
Figure 2.1: Stress distribution in flexible and rigid pavement.[61]
[27]
2.1.2
Rigid Pavement Structure
Rigid Pavements are composed of a Portland Cement Concrete (PCC) surface
course. This gives a higher elastic modulus, a stiffer construction, suitable for
areas with high traffic loads and/or heavy vehicles i.e. airports, bus stops. This
type of construction tends to distribute the load over a wider area of subgrade due
to its rigidity. (See Figure 2.1) [27].
2.1.3
Composite Pavement Structure
Composite Pavements are a combination of both flexible and rigid design. This is
the most expensive type of structure and seldom used in Sweden. There are two
types of composite pavements, one with a layer of AC over PCC or a layer of PCC
over PCC, this is further illustrated by Figure 2.2 [27].
(a) AC over PCC
(b) PCC over PCC
Figure 2.2: Composite Pavement Design. [9]
[80, 9]
As most roads in Sweden are flexible pavements this project concentrates on analyzing and predicting performance for flexible pavement structures.
4
2.2. TRAFFIC
2.2
CHAPTER 2. LITERATURE REVIEW
Traffic
One of the functions of a pavement is to protect the subgrade by distributing the
load from the traffic through the upper layers. The loads magnitudes and their
frequencies vary with time and place, which gives a highly stochastic factor. The
Annual Daily Traffic (ADT) is a measure of the average traffic flow per day, for a
certain year for a certain part of a road [67]. ADT is given as the number of vehicles
per day for a specific vehicle class per day, can be given with indices as:
◦ ADTtot - the total number of vehicles in both directions.
◦ ADTl - the traffic flow in one lane.
◦ ADTtot
- the total flow of heavy vehicles in both directions.
heavy
◦ ADTl
the flow of heavy vehicles in one direction.
heavey
ADT includes:
◦ cars.
◦ single-unit trucks and buses.
◦ multiple-unit trucks.
Annual Average Daily Traffic (AADT) is the yearly mean value of ADT. Average
Daily Truck Traffic (ADTT) is an index describing the heavy traffic. The traffic
volume is measured on a more or less regular basis, and to estimate the equivalent
heavy vehicle axles (N100 ) for other years a simple model was used, see Equation
2.1 [59].
yn
r y
N100 = AADTl ∗ 3.65 ∗ phv ∗ n̄ ∗ ∑(1 +
),
(2.1)
100
y0
where:
AADTl is the AADT in one lane.
ADT T
◦ phv is the part of heavy vehicles (
ADT ), given in %.
◦ n̄ is the equvivalent number of standard axles per heavy vehicle.
◦ y ∈ [1, 2, 3, ..., n], n ∈ N
. y is the starting year of the calculation period.
0
. y is the ending year of the calculation period.
n
◦ r is the traffic growth, in %, for heavy vehicles.
The standard axle is an imaginary axle with dual tires and a 100 kN load, homogeneously distributed over the axle. The tire/pavement contact area is circular, and
each area distributes a pressure of 800 kPa. The tires in each pair have a relative
distance of 300 mm [67]. This is further illustrated by Figure 2.3.
◦
5
2.3. DISTRESS TYPES
CHAPTER 2. LITERATURE REVIEW
Figure 2.3: The derivation of the N100 axle, used in Sweden. [59]
The Equivalent Single Axle Load (ESAL), is a concept which converts all axle
configurations and axle loads of various magnitudes and repetitions (mixed traffic)
to an equivalent number of standard or equivalent axle loads. A traffic estimation
approach, based on the idea of fixed vehicles but varying ADT. The result is a
number of repetitions of standard vehicle. The load on this axle varies internationally; in many parts of the world it is 18 kP. This allows the cumulative number of
repetitions of standard axles during the design life to be expressed as the traffic
parameter for design purpose [58, 27].
Cr
rut , N Cr , N Crini and N propa , which are appellations used in
N100 is the same as N100
100
100
100
the literature.
2.3
Distress Types
A distress in a road is not the same as a failure/fracture in many other fields of
engineering. They can have many levels, and when it becomes unacceptable a
failure occurs. Distresses can manifest in many different ways, and to avoid only
treatment of the symptoms the cause needs to be found and dealt with in the proper
manner [16]. The most common types are presented here.
2.3.1
Cracks
A fracture in the pavement leads to the formation of a crack. There are many types
of cracks identified, and here follows a more detailed description. Illustrations of
different types are found in Figure 2.4 [16].
Fatigue
When a pavement is subjected to repeated traffic loadings a set of interconnected
cracks in the initiation phase can develop. They can usually be described as manysided, sharp-angled pieces, often no more than 0.3 m on the longest side. These
cracks are usually classified as bottom-up, but top-down cracks are also reported
in the literature. The fatigue cracks are mainly generated in the wheel path. If
unattended they can lead to Crocodile cracks.
Meandering Cracks
Cracks not connected, propagating in any direction.
Transversal Cracks
Cracks similar to longitudinal cracks, mainly found in aged asphalt where the temperature differences cause the surface layer to contract and dilate.
6
2.3. DISTRESS TYPES
CHAPTER 2. LITERATURE REVIEW
Longitudinal Cracks
Cracks mainly caused by fatigue in the material.
Diagonal Cracks
Single cracks formed across the lane.
Block Cracks
Block Cracks are interlinked cracks that more often than the crocodile cracks, can
appear on areas of the road not being subjected to traffic loads. Ruts can often be
seen if crocodile cracks are present.
Crocodile (Alligator) Cracks
Are interconnected, forming a pattern that resembles the back of a Crocodile.
These are caused by fatigue failures in the asphalt layers as a result of an high
repetition of traffic loading. In pavements with thin layers the cracks normally
initiate from the bottom and propagate upward; however the opposite is true for
pavements with thicker layers. This enables moisture to infiltrate and can lead to
potholes.
Crescent shaped Cracks
Crescent shaped cracks are usually the result of lateral and shear stresses from
vehicles.
Figure 2.4: An illustration of the common types of cracks found in flexible pavements. [16]
Crack Index
The Crack Index (Ci ) is an index that describes the severity of a crack. The growth
rate for the cracks differs in the early years compared to later. Generally the lifetime
of a crack is separated into: initiation: 5 < Ci and propagation: Ci ≥ 5 (See Figure
2.5) [78].
Figure 2.5: An illustration of the Ci division initiation and propagation. [21, 78]
7
2.3. DISTRESS TYPES
2.3.2
CHAPTER 2. LITERATURE REVIEW
Permanent Deformation
Asphalt consists of a mixture composed of aggregate, binder and air voids. The
volume ratios are different in each mixture, giving different viscoelastic properties
in permanent deformation. The permanent deformation can occur in every layer,
but is only visible on the surface. An illustration of different types of deformation
is found in Figure 2.6 [16].
Rutting
Permanent surface depression in the wheel path. Sometimes the pavement along
the track can suffer uplift. This is a consequence of the compression in the HMA
and/or subgrade layers, caused mainly by heavy traffic. Rutting can also be caused
by wear from studded tires; studded tires causes abrasion of the wearing course,
which results in rutting. The studs wear the bitumen in the wearing course and
thereby the contact between aggregates in the wearing course and the tires increases. After repeated traffic passes the aggregates in the asphalt layer abrade
and this will develop rutting in the wearing course [16].
Many previous studies have shown that flexible pavement smoothness is significantly affected by rut depth variance [22].
Shoving
Shoving is horizontal plastic movement of the surface. Usually caused by accelerating or retarding of vehicles.
Depressions
Depressions are local surface areas with a negative shift in the elevations compared
to the surrounding pavement.
Corrugation
Corrugation is a form of plastic movement, seen as ripples on the surface.
Figure 2.6: Illustration of common types of deformation in HMA.[16]
2.3.3
Surface Defects
These types of distresses are mainly caused by asphalt concrete surface fatigue
[16]. An illustration of different types is found in Figure 2.7.
Pot holes
The result of disintegration in the different layers. Besides from increasing the vulnerability in the pavement structure they can cause damage to the vehicles passing.
Patches
Patches are the result of repairing smaller local areas in distress. However a patch
is considered a defect, as it generally increases the roughness which can impact the
8
2.3. DISTRESS TYPES
CHAPTER 2. LITERATURE REVIEW
fuel consumption and ride quality.
Delamination
Delamination sections of a surface layer that have come loose from the pavement.
Polishing
The result of heavy traffic polishing the aggregates in the top layer creating a
smooth surface with a low friction.
Raveling
Weakening of the bond between the aggregates and binder occurs at the surface of
the layer the result is often the dislodgement of aggregate particles.
Flushing/Bleeding
Can occur when the aggregates at the surface are totally covered by binder. Resulting in a softer surface with low friction.
Stripping
The weakening of the bond between the binder and aggregates. It initiates at the
bottom of the Hot Mix Asphalt (HMA) layer and progresses upward.
(a)
(b)
Figure 2.7: Surface defects. [16]
2.3.4
Edge Defects
Edge defects are the defaults found along a joint or the pavement edge. Usually
there are two types of classified defects edge: break and drop off [16] (see Figure
2.8).
Figure 2.8: Edge Defects. [16]
2.3.5
Maintenance
The methods needed for repairing differ depending on type of distress, but the same
strategy applies for all types of distresses;
9
2.4. PAVEMENT PERFORMANCE
CHAPTER 2. LITERATURE REVIEW
1. Identify and find the cause.
2. Rectify the design for the cause of problem.
3. Repair the symptoms.
A road can of course have several different types of distresses without being maintained. However when a road is maintained due to an unacceptable level of a
distress in one type, several of other types can also be treated at the same time
[42].
2.4
Pavement Performance
The Transportation Research Board (TRB) is a division of the American National
Research Council, which has defined pavement performance as "A function of its
relative ability to serve traffic over a period of time". The establishment of criteria
for this task is vital. At the end of the 1950’s more objective measures started
to appear [72]. This enabled the condition and performance of a pavement to be
quantified, often described in terms of:
◦
◦
Cracking
Permanent deformation
◦
◦
Surface Defects
Edge Defects
[72]
The American Association of State Highway and Transportation Officials (AASHO)
Road Test is a famous experiment carried out during the latter part of 1950. The
tests had the aim of determining the effect of traffic in highway deterioration. The
concept of Present Service Index (PSI) was derived during this study, based on data
regarding the longitudinal roughness caused by: patch work, rutting, cracking, etc
[1].
PSI is a qualitative index, from the users perspective, that separates structural and
functional distresses in an pavement. The PSI relates to the quality of the ride
from the users perspective. In order to prolong the service life of a pavement
construction it can be used as an indicator of when to perform maintenance. This
is described in Figure 2.9 [1].
10
2.4. PAVEMENT PERFORMANCE
CHAPTER 2. LITERATURE REVIEW
Figure 2.9: PSI is a qualitative index, from the users perspective, that separates
structural and functional distresses. When maintenance or rehabilitation is performed the PSI value is often decreased compared to the start value. [73]
The allowed PSI value, described in Table 2.1, before maintenance is decided
by the road owner, and usually economy
is a factor. The PMS concept is internationally recognized to be a highly cost
effective tool in planning road construction and maintenance [1].
2.4.1
Table 2.1: The values for PSI.
0-1
1-2
2-3
3-4
4-5
Very Poor
Poor
Fair
Good
Very Good
Present Serviceability Rating
The Present Serviceability Rating (PSR) is the mean of the individual rating, derived from a specific panel. The PCR is calculated as:
√
PSR ≡ 5.03 − 1.91log10 (1 + SV ) − 1.38RD − 0.01 C + P,
(2.2)
where:
◦ PSR is the Present Serviceability Rating.
◦ SV is the average slope variance.
◦ RD is the average rut depth in inches.
f t2
◦ C + P is the cracking and patching in [
].
1000 f t 2
However, this was considered unpractical since the panel was supposed to consist
of 12 trained persons [1, 55, 80, 30, 44, 38, 17, 8, 12, 16].
2.4.2
Swedish Models
The performance models used in Sweden for crack and rut predictions were created
by Wågberg and Göransson [78, 19].
Wågbergs crack model
In the crack propagation model developed by Wågberg, [78], the accumulated
heavy traffic loading is represented by equivalent standard axle, N100 , loading repCr , is given as the
etitions. The total number of N100 for a specific segment, N100
11
2.4. PAVEMENT PERFORMANCE
CHAPTER 2. LITERATURE REVIEW
sum of the standard axle repetitions to crack initiations and for crack propagation
respectively at failure as:
Cr
Crini
Cr
N100
= N100
+ N100propa .
(2.3)
Whereas the number of axle repetitions for crack initiations is given by Equation
2.4:
1
7.24−0.0052∗SCI300 −5010000∗
Crini
N100
= 10
Y
SCI300 ∗N100
,
(2.4)
and for propagation by Equation 2.5.
Cr
N100propa =
195 ∗ 105
6
4.39 + 7.1∗10
Crini
.
(2.5)
N100
◦
◦
Y is the average annual N
N100
100 per lane.
SCI300 is the surface curvature index in [µ m] based on FWD measurements
carried out at temperature 20°C on the structure recently after the structure
was built.
Göranssons rut model
A similar approach, as previously described in section 2.4.2, has been used to predict the number of accumulated standard axles to reach a certain failure rutting
depth. In the model developed by Göranssons, [19], the total number of ESAL’s
for specific segment is given by Equation 2.6:
rut
N100
=
1
rut 1
∗( )b .
−0.0209
0.9533 ∗ rut
a
(2.6)
Where:
rut
◦ N
100 is the average annual (ESALs) per lane.
◦ rut is the total rutting in [mm] on the surface used to define failure.
◦ a and b are parameters estimated from Falling Weight Deflectometer (FWD)
test as the surface curvature index SCI300 in µm measured during the autumn,
first time after the pavement structure (section) is built or rehabilitated.
The parameters a and b can be estimated as:
a = 0.0001579 ∗ SCI300 + 0.034322,
b = 0.0005695 ∗ SCI300 + 0.296.
[19].
12
2.5. ROAD MANAGEMENT
2.5
CHAPTER 2. LITERATURE REVIEW
Road Management
Road Management Systems (RMSs) are used to store and analyze road data as
a part of the infrastructure of a society. The work method is to attain the data
needed to systematically analyze and prioritize and thereafter take action needed
for maintenance and planning of everything related to the road network [74] (see
Figure 2.10).
Figure 2.10: An illustration of the concept of RMS and PMS.
A PMS can be described as "A set of tools or methods that can assist decision
making in funding cost effective strategies for providing, evaluating, and maintaining pavements in a serviceable condition". A PMS is a part of the RMS, addressing only the road structure explicitly. The Federal Highway Administration
(FHWA), U.S. Department of Transportation, uses the software Pavement Health
Track [74].
2.5.1
History
The first systems appeared in the latter part of the 1970s, and in 1985 AASHTO
published "Guidelines on Pavement Management". In 1990 the AASHO gave the
PMS guidelines: A Pavement Management System is designed to provide objective information and useful data for analysis so that highway managers can make
consistent, cost-effective, and defensible decisions related to the preservation of a
pavement network. A PMS, sometimes refereed to as a Maintenance Management
System (MMS) is a tool for planning the maintenance or rehabilitation of roads.
These strategies require prediction models, and LTPP programs are a powerful tool
when deriving them. [74]
2.5.2
Decision Support System & Geographic Information System
Decision Support System (DSS) is a software based information system that supports the decision making process, and a Geographic Information System (GIS)
is a multipurpose system designed to store and retrieve relevant information that
can be connected to a geographical position. An implementation of such system is
shown in Figure 2.11.
13
2.5. ROAD MANAGEMENT
CHAPTER 2. LITERATURE REVIEW
Figure 2.11: An example from a PMS with DSS and GIS. [54]
Using a PMS with an integrated DSS and GIS is a helpful tool in any PMS [47, 80,
16, 31].
2.5.3
Highway Development and Management Model-IV
The World Bank has developed a set of tools for highway development and management to be used in PMS. The area of use for the HDM-IV tools has expanded
beyond the traditional project appraisal and management to a potent system for
road management and investment analyzing. The gathered data for these models
are seen in Figure 2.12:
14
2.5. ROAD MANAGEMENT
CHAPTER 2. LITERATURE REVIEW
Figure 2.12: The conception of the life-cycle analysis in HDM-4 [63].
The maintenance strategy is to use a road asset as long as possible, by conducting
planned maintenance before the allowed amount of deterioration considered acceptable is reached. The allowed deterioration level is decided by the road owner;
usually economy is the most important factor [63, 64, 28, 74].
PMS Methodologies
There are generally two levels for the usage of a PMS: network and project. The
network methodology results in solutions optimized for the entire road network.
Using large quantities of combined data the optimal strategy is found, the next
15
2.6. LTPP
CHAPTER 2. LITERATURE REVIEW
step here is to target the smaller objects. The project approach solves the problem
using a fundamentally different technique, working from the low to the higher level
[63, 64, 28, 74].
2.5.4
PMS in Sweden
The Pavement Management System in Sweden started with a high focus on the
quality of road condition data in order to attain reliable and repeatable information for analysis and research. Sweden uses average values for 100 m sections,
the standard for maintenance is based on posted speed and aloud traffic classes.
The software "PMS 95" is currently used (but will be replaced by "PMS 2012")
using data from the National Road Database (NVDB). "PMS 95" focuses on the
information demand of pavement engineers, and is intended to aid the planning of
maintenance. In Sweden all the maintenance projects are outsourced. A contractor in Sweden is obliged to follow national asphalt norms. The Swedish Transport
Administration (STA) is divided, and the section Society operational area consists
of six regions [35].
2.6
LTPP
The original Long Term Pavement Performance (LTPP) was a project initiated in
the beginning of 1980 by the American TRB and AASHTO and the project began
with monitoring the deterioration of the North American highways, and storing
the results in a database. One of the outputs was the Strategic Highway Research
Program SHRP, with a focus on a LTPP monitoring program, among others. The
ambition of this original program was to:
◦ analyze existing design methods.
◦ develop improved design methods and strategies for rehabilitating existing
pavements.
◦ develop improved design equations for new and reconstructed pavements.
◦ determine the effects from:
. material variability
. loading
. construction quality
. environment
. material properties
. maintenance levels
on pavement performance and maintenance
establish a national database of pavement inventory and performance information [74].
The main purpose on any LTPP program is to prolong the life period of any pavement structure by monitoring various pavement designs and rehabilitated objects,
using different types of construction methods and materials; subjected to different
loads, environments and climate. USA. and Canada collaborates in this project, and
have a joint LTPP program, with approximately 1500 test sections. The C-LTPP
project was established in 1989. In the U.S. and Canada each State/Province Department of Transportation collects pavement condition data on a regular biannual
basis. The data is then reported to one of the six Regional LTPP Data Managers.
A pre-investigation is then performed to control the quality. The final result is then
◦
16
2.6. LTPP
CHAPTER 2. LITERATURE REVIEW
accessible by anyone in the transportation community [74].
In Canada this is executed by Transport Canada, the department responsible for
transportation, in Transport, Infrastructure and Communities Portfolio [23, 69].
2.6.1
General and Specific Pavement Studies
In the "original" LTPP program two different types of studies are preformed: General Pavement Studies (GPS) and Specific Pavement Studies (SPS). The GPS experiment targets "ordinary" structural designs of pavement versus the impacts of
climate, geology, maintenance, rehabilitation, traffic, and other factors. The SPS
has the aim of investigating the more "pavement engineering" factors. The objects
in the SPS are designed to provide a different set of design factors, enabling comparison of the performance of dissimilar design factors, both within and between
sites [74].
The GPSs were agency nominated to fit into broad categories but were paved as
part of normal paving operations within the agency. The GPS sites entered the
LTPP program with a variety of initial ages, were in a variety of conditions, and
were made of a variety of materials. Because they were paved as part of longer
paving projects, the GPS sites were believed to be more representative of standard
paving processes and ride quality than were the 500-ft test sections in the SPS program [13, 14, 74, 45, 51, 76].
Canada
The Canadian LTPP (C-LTPP) monitors 24 test sites in Canada that deals with rehabilitation of an asphalt concrete pavement. Furthermore, each test site contains
two to four adjacent test sections, each with a different rehabilitation strategy enabling a focused study on the design and optimization of overlay. The C-LTPP
started in 1989 by the C-SHRP in order to compliment the U.S. LTPP, targeting
factors more relevant to Canada. The C-SHRP monitors the construction and performance of 24 highway test sites, distributed across the major provincial system.
The focus is upon asphalt concrete overlays, constructed on existing AC pavement
having a granular base course. This gives information concerning prior condition
of the pavement. The climate and the lower (compared to the US) traffic flow were
also important factors when selecting objects suitable for monitoring. The C-LTPP
test objects are monitored throughout an entire life cycle (∼ 15 years). The aims
of this project were to:
◦ analyze the efficiency of the current rehabilitation methods and strategies.
◦ create new or calibrate performance models to suit the local conditions.
◦ institute common methods for LTPP evaluation, and build a base for future
research projects.
◦ start a database with the intent to complement the existing C-LTPP.
17
2.7. DISTRESS FACTORS
CHAPTER 2. LITERATURE REVIEW
The highway agencies in Canada were given technical guidelines containing detailed information in how to collect and report different types of performance data
[7, 23].
2.6.2
LTPP Database
Gathering and storing the data require several millions of $ in funding, and this
demonstrates the importance of transport. One of the most common ways to store
the LTPP data is in a relationship database, i.e. a multipurpose storage concept. The
data is simply stored in tables (row/column) with different types of relations to each
other. This can be a very powerful tool, and at the same time highly user-friendly
with a good Relational Database Management System (RDBMS) creating SQL
queries combined with a logical designed Enhanced Entity Relationship Model
(EER) [74]. More information about the EER database can be found in appendix
B.
2.6.3
International LTPP
The interest in this project rapidly spread throughout the world. Some of the implementations are further described here. [74]
The Nordic Countries
The original SHRP LTPP program was not designed for the cold climate in the
north (Finland, Norway, Sweden); therefore a different set of data samples are
used. "Cold Climate" can be defined by the Freezing Index (the number of days
times degrees below freezing point) at least 100. Frost is an extremely important
design factor resulting in thin overlays and thick unbound layers [35].
New Zealand and Australia
The New Zealand LTPP project started in 2000, and after performing studies on
several existing LTPP projects the aim was to:
◦ establish a representative sample of LTPP sections across New Zealand, relating to the interaction effects of climate and sub-soil moisture sensitivity.
◦ reinforce the current road data collection program.
◦ collect data at the precision level needed for the models used.
Australia’s Austroads (Australia’s peak road agency organization), funded an LTPP
study in 1994/95 with the aim of improving the existing pavement performance
prediction models [26, 24].
2.7
Distress Factors
In this section the main factors causing the pavement distresses are presented,
though there are many more.
2.7.1
Factor Traffic
Based on the data from [56] the number of vehicles is increasing and highways
are often constructed to deal with this. The vehicles give a dynamic load/unload
stress, that cumulative gives permanent strain shown in Figure 2.13. Here the strain
18
2.7. DISTRESS FACTORS
CHAPTER 2. LITERATURE REVIEW
increases differently depending on how many axle loads the site has been subjected
to [57].
Figure 2.13: Creep characteristic of bituminous mixtures. [57]
The results from a dynamic repeated load test are shown in Figure 2.14.
Figure 2.14: The graphs show deformations of the pavement due to the cyclic
load/unload. [16]
The geometrical placement of the wheel/pavement interface differs for different
axle types; the most common types and tire configurations are shown by Figure
2.15. The current rules in the EU are a maximum length of 18.75 m and a maximum
weight of 40 tons, however in Sweden 60 tons are allowed. The method used to
sample vehicle weight is often static weigh stations, which produce data with a
19
2.7. DISTRESS FACTORS
CHAPTER 2. LITERATURE REVIEW
very high bias [62].
Figure 2.15: Illustration of the common axle types and tire configurations. [48]
This can be avoided by using Weigh in Motion (WIM) technique, which enables
the weight of the axles to be measured when the vehicle is in motion so that the
traffic can be divided into an axle load spectrum. This results in a spectrum of
load pulses with different frequencies. Each vehicle type contributes to the distress
of a pavement; the load from the wheels creates a unique stress wave that travels
through the pavement (see Figure 2.18) [49].
Figure 2.16: The graphs show axle loads from different types of axles.[49]
In Sweden the following axle loads are allowed:
◦ 11.5 ton on a steering axle.
◦ 19 tons on a driving tandem axle.
◦ 10 ton axle load.
◦ 60 ton total weight.
◦ 18 tons on a tandem axle.
[70]
20
2.7. DISTRESS FACTORS
CHAPTER 2. LITERATURE REVIEW
Tire/Pavement interface
The shape of the contact area between tire/pavement depends on the tire pressure;
this is one of the elements in the pavement response. According to the rules in
Sweden studded tires are required from the first of November to the end of April.
The studded tire can give a mechanical dislodging of aggregates, due to wear, similar to the pavement deformation caused by heavy loading [27, 70, 20]. The rut
given by the LTTP database does not give the source [20].
2.7.2
Climate
The climate is a key factor in road deterioration, and several different types of data
are available, such as the number of days of frost, per year and rain amounts etc.
These factors are strongly dependent on their geographical placement, as seen in
Table 2.2 and Figure 2.17. The climate has also a significant impact on the design
of pavements. The strength, durability and load bearing capacity of the pavement
are affected, adding traffic and complex interaction effects [36, 67].
Figure 2.17: The climate zones classified by the number of days when the mean
temperature is below 0◦ C in Sweden. [67]
The climate plays an important part in the selection of materials and for the thickness dimensioning of the layers. Table 2.3 gives the temperatures. The thickness
and material are selected to protect the road against frost heave/thaw.
21
2.7. DISTRESS FACTORS
CHAPTER 2. LITERATURE REVIEW
Table 2.2: The periods length in days during one year, [67].
Climate zone :
Winter :
Frost heave :
Frost thaw :
Spring :
Summer :
Fall :
1
49
10
15
46
153
92
2
80
10
31
15
153
76
3
121
4
151
5
166
45
61
91
123
76
77
76
47
61
Table 2.3: Temperature, [◦C], in the bitumen bounded layers, [67].
Climate zone :
Winter :
Frost heave :
Frost thaw :
Spring :
Summer :
Fall :
1
-1.9
1
1
4
19.8
6.9
2
-1.9
1
2.3
3
18.1
3.8
3
-3.6
4.5
17.2
3.8
4
-5.1
6.5
18.1
3.8
5
-7
7.5
16.4
3.2
To represent the climate the Mean Annual Temperature (MAT) in ◦ C, and the perception, measured in [mm], was used. The values are calculated as the mean value
over a longer period. The climate is believed to be an important factor, however the
lack of control over the observations is believed to be the cause for the unexpected
low correlation values.
2.7.3
Aging
This term describes the effects of long term exposure to the failing factors and how
this changes the material properties by altering the bond strength in and between
molecules, the chemical structure of the bitumen and the settings for the exposure.
This is measured in years from opening [60, 18].
2.7.4
Material
The Pavement body is the engineered part, where material and geometrics can be
designed; the subgrade is the natural occurring soil. The properties and the settings
change with time and the subgrade can be composed of a wide range of different
materials; some are less desirable when it comes to road construction.
Structure
An important explanatory variable for crack initiation of flexible pavements is the
Surface Curvature Index (SCI300 ), used as a value of the strength of the pavement
structure. SCI300 is explained in the FWD section 2.8.2.
22
2.7. DISTRESS FACTORS
CHAPTER 2. LITERATURE REVIEW
Subgrade
The quality of a subgrade can be described by four highly interconnected factors:
1. Load bearing capacity.
2. Moisture content.
3. Frost heave\thaw
4. Shrinkage and\or swelling.
The load bearing capacity
A "good" subgrade is able to withstand the forces transmitted by the pavement
structure without deforming. This parameter can be described as the stiffness of
the subgrade. The Resilient Modulus, (Mr ), is required when for determining the
stresses, strains, and deflections in pavement design.
Mr ≡
qd
,
εr
(2.7)
where
◦
◦
◦
Mr is the Resilient Modulus.
qd is the applied deviatoric stress.
εr is the resilient strain.
[27]
The deviator stress is a result when there is a difference in the principal stresses σ1
and σ3 , qd = |σ1 − σ3 | [32].
According to the STA [60] the Mr can be estimated as:
Mr =
5200
D900
(2.8)
where
D900 is the deflection measured at 900 mm from an FWD measurement,
descried in section 2.8.2.
The SCI300 and Mr values are to be regarded as constants, for both phases in the Ci
and rut model.
◦
Climate and Material
For most of the highways the design is, among other factors, based on typical historic climatic patterns, reflecting the local climate and incorporating assumptions
about a reasonable range of temperatures and precipitation levels. Regions with
cold climate can be subjected to frost and freezing of the roadbed. Certain subgrade soils are particularly susceptible to frost action, though generally frost heave
is limited to areas with silty soils. Frost heave is the upward motion of the subgrade, caused by the expansion of the moisture in the soil when it freezes. If the
enthalpy increases the moisture changes state from solid to liquid, resulting in a reduced bearing capacity. The temperature in the pavement also affects the oxidation
process that can increase the viscosity properties. In the presence of moisture, one
of the mechanisms for the deterioration of asphalt is the debonding effect. This
23
2.7. DISTRESS FACTORS
CHAPTER 2. LITERATURE REVIEW
can cause durability problems such as stripping, release of stones, and cracking.
While relatively unimportant for loose aggregate, aggregate chemical properties
are important in a pavement material. In HMA, aggregate surface chemistry can
determine how well an asphalt cement binder will adhere to an aggregate surface.
Poor adherence, commonly referred to as stripping, can cause premature structural
failure. The materials used in road construction are often described in terms of:
◦
◦
◦
Gradation and size
Toughness and abrasion resistance
Durability and soundness
◦
◦
◦
Particle shape and surface texture
Specific gravity
Cleanliness and deleterious materials
Previous Ci /Rut value
The state of the object is highly dependent on the Previous Value (PV). The PV
acts as the "starting point" when making predictions.
Pavement Deterioration
The reasons for a road to fail are many, and the distress a road has to withstand
varies over time with frequency and severeness. The Figure 2.18 illustrates the
complexity of road performance.
Figure 2.18: An illustration of performance of pavement structure factors. [53]
Pavement deterioration is a negative change in performance or condition of the
pavement, i.e, an increase in distress or decrease in serviceability. Pavements are
often designed to maintain functionality in whichever environment they are built.
However material properties vary with the seasons and temperature changes. The
climate’s effect on a road also depends on the environment in which it is built, some
microhealing has been noticed for pavements [74].
24
2.8. ROAD MEASUREMENTS
2.8
CHAPTER 2. LITERATURE REVIEW
Road Measurements
The objects in the LTPP are inspected on a yearly basis and are given a Ci according
to the principles described in: [77]. Here follows a description of some of the data
found in the LTPP and how they are measured [20].
2.8.1
The Road Surface Tester
The RST measures the state of a road, and the data is stored in the LTPP database.
The sampling is done automatically when the car is driven, the RST is illustrated
by Figure 2.19 [75].
Figure 2.19: The RST used at VTI, and an illustration of the rut sampling and how
the lasers are placed.
The figure also gives an illustration of the sampling process. Sampling with 17
lasers is the current standard, a measuring distance of 3.2 m is reached, however
the Road Surface Tester (RST) owned by VTI carries 19 laser units and that gives
3.6 m. A sampling occurs every 10 cm and using the data the rut depth is calculated.
The placement of the lasers is shown in Figure 2.19. More information regarding
these tests can be found in: [71].
2.8.2
Falling Weight Deflectometer
The purpose of a Falling Weight Deflectometer (FWD) is to simulate the load pulse
from a heavy vehicle. This is done with a load plate of 30 [cm] being dropped
25
2.8. ROAD MEASUREMENTS
CHAPTER 2. LITERATURE REVIEW
from a height so that a force of approximate 50 [kN] is reached when it impacts the
pavement surface. This is usually done three times, and the deflection is measured
at the impact center and at 0, 200, 300, 450, 600, 900 and 1200 mm from the center.
The FWD are spot measurements representing longer sections, and the road needs
to be closed while measuring. The aim here is, of course, that this type of data can
be attained without disturbing the traffic. The Figure 2.20 shows the placements of
the geophones, and Table 2.4 states how the results are used [11].
Figure 2.20: A description of the FWD measurements. [60, 11]
Table 2.4: Deflection Basin Indexes. [11]
Sensor no., Di
D1 D2
D3
D4
Distance from the center [mm], ri 0 200 300 450
26
D5
600
D6
900
D7
1200
2.8. ROAD MEASUREMENTS
CHAPTER 2. LITERATURE REVIEW
Maximum deflection recorded - D1 - Indirect proportional to the overall stiffness
of an elastic half-space.
Surface Curvature Index - SCI300 = D1 − D3 - Curvature of the inner portion of
the basin. Indicates the stiffness of the top part of the pavement.
Base Curvature Index - BCI = D7 − D6 - Curvature of the outer part of the basin.
Indicates the stiffness of the bottom part of the pavement or the top part of the subgrade soil.
Base Damage Index - BDI = D3 − D5 - Curvature of the middle part of the basin.
Indicates the stiffness of the pavement.
Basin Area - Area = D10 ∑Ni=0 ((Di−1 + Di ) ∗ (ri − ri−1 )) - Considered a good indicator of overall pavement strength during spring thaw.
2
− Dx )2 + a2 )
Radius of Curvature of the center of the basin - R = (D12(D
≈ 2(D1a− Dx ) 1 − Dx )
Dx is the deflection measured at the sensor just outside the loading plate and a is
radius of the plate.
h1
Tensile Strain at the bottom of the AC layer - ε = 2R
- h1 is the thickness of the
asphalt bound layer [mm].
D
Subgrade Strength Index - SSI = D55t - SSI indicates a measurement during thaw
s
and SSI indicates measurement after thaw recovery.
The deflections are measured by accelerometers that measure the vertical displacement speed of the surface. The influence of passing traffic is eliminated by having
a very short sampling time when the weight is dropped [52, 60, 11, 40].
2.8.3
Rut
The rut data is sampled by the RST, which measures unevenness in the horizontal plane. In the LTPP database rut depth, given in mm, is measured by a RST,
equipped with 17 Laser sensors, the data are measured in both directions using the
wire surface principle, illustrated by Figure 2.21, and the average value from all of
the sections in an object are used in this study [21].
Figure 2.21: An illustration of the wire surface principle in rut depth sampling.
[20]
The rut data stored in LTPP are the ruts in the left and right track, and the maximum
depth for the entire sampling width.
Rut variance
The rut values in the LTPP database are given in mm, and from the objects investigated they seem to have a profile with a small value of variance (see the plots in
Figures 2.22) [21] .
27
2.8. ROAD MEASUREMENTS
CHAPTER 2. LITERATURE REVIEW
Figure 2.22: Illustration of the variance in the rut data. The data plotted comes
from object D-RV53-2. [21]
The data plotted come from the object D-RV53-2, and shows how the rut values
from all sections in an object have been transformed into one value representing the
whole object. The sampling is done automatically several times for each section
and gives deviations each time the rut is measured. When needed, the available rut
data were used to interpolate new values, in order to replace missing samples in an
interval.
28
2.8. ROAD MEASUREMENTS
2.8.4
CHAPTER 2. LITERATURE REVIEW
Crack Index
Crack index (Ci ), is an index derived in the EU project "PARIS", [15]. Every
section in a road object is manually inspected and given an Ci , that includes all
types of cracks. A crack is classified and given a weight in the calculation, the
magnitude of the weight is dependent on the classification and size [77].
Figure 2.23: Example of size and development of a crack between three different
inspections of the same road that is carried out at different times. This object is
100 [m], and divided into smaller 10 [m] sections.[77]
The index increases with the level of severity and spreading, but also with the type
of crack. The crack index is empirical, based on a visual survey, and is calculated
as Equation 2.9:
Ci = 2Ac + Lc + Tc ,
(2.9)
where:
Ac : Alligator cracking; Ac low [m] + 1.5 ∗ Ac average [m] + 2 ∗ Ac bad [m]
Lc : Longitudinal cracks; Lc low [m] + 1.5 ∗ Lc average [m] + 2 ∗ Lc bad [m]
Tc : Transversalcracks; Tc low (no) + 1.5 ∗ Tc average (no) + 2 ∗ Tc bad (no.)
Low, average and bad are weights defined in [77], and no. stands for the number
of cracks. Cracks shorter than 1[m] are assigned a length of 1[m] [78].
The Ci used in this project is the mean value from the sections in an object. However the magnitude of the Ci increases the standard deviation in the prediction. As
further described by Figure 2.23 [21].
29
2.8. ROAD MEASUREMENTS
CHAPTER 2. LITERATURE REVIEW
Figure 2.24: Ci variation in an object. [21]
The figure shows that the standard deviation is low in the first time period, but
once cracks appears in different sections they grows with a speed that increases
with time. Resulting in large variances between the sections in an object. The Ci
can be divided in to two phases: initiation (when Ci < 5) and propagation (when
Ci ≥ 5). This is due the different growth rate in the phases and creates the need of
two models.
30
Chapter 3
METHOD
The prediction models created are empirical-mechanistic models, based on data
from a database that include the structural information, traffic volume, and condition data for each "homogeneous" section of a road. The road objects used are
described by Table 3.1, 3.3, 3.4.
Table 3.1: General object information.
Observation Length [year]
Object
Name
Opened [year] Sections no. Speed [ km
h ]
C-292-1
Gimo
1994
9
90
16
D-RV53-2 Nyköping 1987
10
90
22
F-RV31-1 Nässjö
1988
11
90
20
H-RV34-1 Målilla
1987
10
90
16
H-RV40-2 Vimmerby 1980
12
90
30
T-205-1
Laxå
1994
8
80
16
Table 3.2: General construction and maintenance information.
Name
Surface layer
∆Thickness [mm] Maintenance
Gimo
AG, Stabinor
85
1994
ABb, ABS
43
2009
D-RV53-2 Nyköping MABT
40
1987
HABS
28
1993
ABS
43
2008
F-RV31-1 Nässjö
AG
85
1988
Remixing, ABS 24
2007
Remixing, ABS 24
2007
H-RV34-1 Målilla
AG, MAB
75
1987
HABS, Heating 28
2002
HABS, Heating 28
2002
H-RV40-2 Vimmerby BG, MABT
75
1980
ABT
13
2008
ABT
12
2008
ABT
8
2008
Remixing, ABS 28
2009
Remixing, ABS 28
2009
T-205-1
Laxå
AG, MABT
130
1994
Object
C-292-1
31
CHAPTER 3. METHOD
The materials in the surface layer are further described by [68].
Object
C-292-1
D-RV53-2
F-RV31-1
H-RV34-1
H-RV40-2
T-205-1
Table 3.3: Material information continued.
Subgrade
Type Subbase
clay
3
rock
silty clay / silt
3
gravel and sand
silty moraine
6
sand
1
gravel and sand
s moraine/s moraine on rock 6
blockr Si Mn
6
crushed/uncrushed mtrl
Object
C-292-1
D-RV53-2
F-RV31-1
H-RV34-1
H-RV40-2
T-205-1
Table 3.4: Material information continued.
Base Course
Thickness [mm]
crushed rock
160
gravel
160
gravel
115
gravel
125
gravel
80
crushed soil/rock 150
Figure 3.1: The red dots in the map mark the geographical locations of the objects.
The subgrade is classified in the same way as in the EU project PARIS;
1. sand
2. silty sand
3. clay
4. peat
5. bedrock
6. other
[15]
More data are given in the Appendix D and by [21].
32
3.1. THE SWEDISH LTPP
3.1
CHAPTER 3. METHOD
The Swedish LTPP
The Swedish LTPP project started 1984, and has continued to grow ever since. It
began with a limited number of objects but has expanded with time. The LTPP
database contains relevant pavement information with focus on road deterioration
caused by heavy traffic and exposure to climate. Today it contains data from over
650 sections, distributed in over 65 road objects in the stately road network, the
object are represented as dots the map in Figure 3.2. The objects are selected from
the national road network to ensure that they are constructed according to national
standards. Each object is divided into smaller, 100 m sections but the number of
sections in an object varies. The performance monitoring is mainly focused on
road deterioration caused by heavy traffic.
Figure 3.2: The geographical distributions of road objects in the Swedish LTPP
database. The red dots are active objects, and the orange objects retired from data
monitoring.
Most of the road objects in the LTPP database are concentrated toward the southern
part of the country as most of the traffic is there. The Swedish LTPP has a database
containing almost all relevant pavement information. The Swedish Transport Administration (STA), has given the National Road and Transport Research Institute
(VTI) the assignment to collect a large amount of data concerning the state of
several objects in Sweden. The main objective for this project is to create road deterioration models for a PMS, used for conducting road maintenance. An overview
of the stored data in the Swedish LTPP database is given by the list:
road construction and structure
◦ climate
material properties
◦ traffic
◦ road state
◦ pavement distresses
The Swedish LTPP database is accessible to the public and has been used in various research projects for pavement performance analysis and management decisions. The Swedish pavement design software "PMS Objekt" has been verified
◦
◦
33
3.2. TRAFFIC ESTIMATIONS
CHAPTER 3. METHOD
with the LTPP database. The database can also be used for calibrating and validating models, that require detailed data regarding local conditions of pavement
structures, bound and unbound material characterization, environmental conditions, traffic loading, and distress data, such as rutting, cracking, that stretches
over the lives of pavement structures. The database is stored in a Microsoft Access
format. [65, 20, 60, 49]
3.2
Traffic Estimations
From the measured ADT and ADTT the change in the ADT and ADTT has been
has been estimated with models given by equations 3.1 and 3.2:
ADTi+1 = ADTi ∗
(1 + r)Y − 1
,
r
ADT Ti+1 = ADT Ti ∗
(1 + r)Y − 1
,
r
(3.1)
(3.2)
where:
◦ ADTi - is the measured ADT for year i.
◦ ADTi+1 - is the estimated ADT for the next year.
◦ ADT Ti - is the measured ADTT for year i.
◦ ADT Ti+1 - is the estimated ADTT for the next year.
◦ Y - is the number of years from the measured.
◦ r - is the traffic volume growth, estimated to 3%.
◦ i ∈ [0, 1, 2, ..., n], n ∈ N
The ADT data used in this project from data comes from [66], and the models from
[27].
Results are shown in the plots in Figures 3.3, 3.4, 3.5, 3.7 and 3.8. The traffic
volume is plotted as a function of time in the ADT and ADT T estimations. In
all the plots are the black dots are the traffic measured; the red lines gives the
estimations used in the project.
34
3.2. TRAFFIC ESTIMATIONS
CHAPTER 3. METHOD
Figure 3.3: Traffic estimations for object H-RV40-2. The black dots are the traffic
measured and the red lines are the estimations used in the project.
35
3.2. TRAFFIC ESTIMATIONS
CHAPTER 3. METHOD
Figure 3.4: Traffic estimations for object T-2051-1. The black dots are the traffic
measured and the red lines are the estimations used in the project.
36
3.2. TRAFFIC ESTIMATIONS
CHAPTER 3. METHOD
Figure 3.5: Traffic estimations for object F-RV31-1. The black dots are the traffic
measured and the red lines are the estimations used in the project.
37
3.2. TRAFFIC ESTIMATIONS
CHAPTER 3. METHOD
Figure 3.6: Traffic estimations for objects H-RV34-1. The black dots are the traffic
measured and the red lines are the estimations used in the project.
Figure 3.7: Traffic estimations for objects D-RV53-2. The black dots are the traffic
measured and the red lines are the estimations used in the project.
38
3.3. DISTRESS FACTOR SCATTER PLOTS
CHAPTER 3. METHOD
Figure 3.8: Traffic estimations for objects C-292-1. The black dots are the traffic
measured and the red lines are the estimations used in the project.
3.3
Distress Factor Scatter Plots
Here are the "key" factors used in the models described by scatter plots.
3.3.1
Crack Index
A scatter plot can suggest the type of relation between variables, but does not give
more. Here are the scatter plots for the factors in the Ci model found (see Figures
3.9, 3.10 and 3.11).
39
3.3. DISTRESS FACTOR SCATTER PLOTS
CHAPTER 3. METHOD
Figure 3.9: Scatter plots for the Ci factors SCI300 , Mr and N100100 , representing
subgrade, structure and traffic.
40
3.3. DISTRESS FACTOR SCATTER PLOTS
CHAPTER 3. METHOD
Figure 3.10: Scatter plots for the Ci factors precipitation and temperature, representing climate and age.
Figure 3.11: Scatter plots for the Ci factors previous value (PV).
41
3.3. DISTRESS FACTOR SCATTER PLOTS
3.3.2
CHAPTER 3. METHOD
Rut
The scatter plots for the rut factors are shown in the Figures .
Figure 3.12: Scatter plots for the rut factors SCI300 Mr and N100 , representing subgrade, structure and traffic.
42
3.3. DISTRESS FACTOR SCATTER PLOTS
CHAPTER 3. METHOD
Figure 3.13: Scatter plots for the rut factors precipitation, temperature and age.
43
3.4. FACTOR MODELING
CHAPTER 3. METHOD
Figure 3.14: Scatter plots for the rut factor PV and the interaction effect, between
standard axles and precipitation.
3.4
Factor Modeling
Both the crack index and rut have the same factors, but with different parameters and structures. Using the results from the scatter plots and hypotheses from
the literature the structures were derived and can be found here. β represents the
different coefficients for the factors.
3.4.1
Crack Index
The Ci models share the same structure, but the β coefficients in the initiation and
propagation phases are different.
Ci initiation
44
3.4. FACTOR MODELING
CHAPTER 3. METHOD
Table 3.5: Ci variable structures.
factor
SCI300
MR
N1 00
Precipitation
MAT
Age
PV
model
β2 x1
β3 x2
β4x3
β5 x4
β6 x5
β7 x62
β8 x7
If the Ci in the initiation phase is estimated as: Ci is < 0, then Ci ≡ 0.
If the Ci in the propagation phase is estimated as: Ci is < 5, then Ci ≡ 5.
Ci propagation
Table 3.6: Ci variable structures.
factor
SCI300
MR
N1 00
Precipitation
MAT
Age
PV
3.4.2
model
β2 x1
β3 x2
β4x3
β5 x4
β6 x5
β7 x62
β8 x7
Rut
Table 3.7: Rut variable structures.
factor
SCI300
MR
N1 00
Precipitation
MAT
Age
PV
x3 ∗ x5
model
β2 x1
β3 x2
β4 log10 (x3 )
β5 x4
β6 x5
β7 x62
β8 x7
β9 x8
If the estimated rut is < 0, then rut ≡ 0. The interaction between the two factors
standard axles and temperature, x3 ∗x5 , has been shown in the results of [79].
45
3.5. MODEL STRUCTURE AND LTPP DATA
3.5
CHAPTER 3. METHOD
Model Structure and LTPP Data
The models in this study are intentionally of only valid for objects that are of virginal form. The models have been developed under the assumption that all the
objects used had no cracks and that the rut depth started from zero when no measured data were available. The models are to be used in the maintenance planning
phase for roads in cold climate regions. After a review of the various types of
prediction models, it was concluded that an empirical-mechanistic model is best
suited, with a systematic database that includes the structural information, traffic
volume, and condition data for each section of the road. Figure 3.13 shows an
illustration of how the factors in the models are expressed as data from the LTTP
database.
Figure 3.15: An illustration of how the factors in the models are expressed as data
from the LTTP database.
The data in the LTPP database that represent the input factors:
◦ Accumulated traffic, represented with N100 (ESALs).
◦ Climate, represented by:
. mean annual temperature, MAT, in [oC] measured over a time period
of 30 years.
. mean precipitation in [mm] over a time period of 30 years.
◦ Aging, in the meaning of years from construction date.
◦ Pavement Structure, represented with SCI300 .
◦ The present Ci value.
◦ Subgrade stiffness, represented with Mr .
The SCI300 , from FWD measurements, and Mr , the resilient modulus, should be
the value measured one or two years after construction, or estimated from the measure nearest in time, and those values are to be used during the entire prediction
time.
46
3.5. MODEL STRUCTURE AND LTPP DATA
3.5.1
CHAPTER 3. METHOD
Preprocessing of Data
The input for all models should be normalized, with the Min-Max technique, and
the Ci and rut values from the LTPP database should be processed by Equation 3.3
before being used for model construction.
i f y(i − 1) > y(i) then
(3.3)
y(i) ≡ y(i − 1),
where:
◦ y(i) is the current Ci /rut sample.
◦ y(i − 1) is prior sample to y(i).
The models assumes that the Ciinitiation and the rut value start at "0", and that Ci propagation
start with the value "5" or higher.
Min-Max normalization This is a linear transformation of the data
xi −xmin
xmax −xmin
xinormalized =
0 |i f xmax − xmin = 0|
(3.4)
Where:
◦ xi
is the i:th sample, normalized.
normalized
◦ xi is the i:th sample, unnormalized.
◦ xmax is the max value sample in the current unit dimension.
◦ xmin is the min value sample in the current unit dimension.
[43]
3.5.2
Variance
The variance for the models is estimated as:
s2 =
SSRES
,
n−k−1
(3.5)
where:
◦
SSRES - The variation in the result
that the models were not able to
predict.
◦
◦
n - the number of sampled data.
k - number of explanation factors.
For the initiation model the Ci models gives a s2 = 0.86, and for propagation 0.89.
For the rut model the s2 is 0.0006.
3.5.3
Statistic Tools
In Appendix A are some of the tools used in the creation of the model structures
discussed. The chosen structures have been selected after analyzing the results
from the scatter plots in section 3.3, combined with educated guesses.
47
3.5. MODEL STRUCTURE AND LTPP DATA
CHAPTER 3. METHOD
In chapter 4 are the results from the models presented. The data is plotted against
the time in years from when the object was opened for traffic. The structures of
the models are based on the observed factor data’s relationship to Ci and rut values.
Several different structures has been tested and the result is based on the variance
and R2 for the models.
48
Chapter 4
RESULTS
This chapter gives the results from the models.
4.1
Crack Index Models
The Ci values show a large difference in growth rate in the two phases; hence there
is a need for two models. When the measured Ci is below 5 then the initiation model
should be applied, until the Ci reaches 5. Then should the propagation model be
uses.
4.1.1
Initiation
The initiation model is given as Equation 4.1:
Cii = − 0.847 − 10100x1 + 413x2 + 4.95x3
− 637x4 + 216000x5 − 1180000000x62
+ −1180000000x7 + ε
i f Ci < 0, then Ci ≡ 0.
(4.1)
Where:
◦
◦
◦
◦
4.1.2
x4 , precipitation [mm].
x5 , temperature [o C].
◦ x6 , age [years].
◦ x7 , previous value [-].
ε is an independent random variable with the expectation value = 0 and variance = σ 2 .
x1 , SCI300 [µm].
x2 , Mr [Mpa].
x3 , N100 [standard axle].
◦
◦
Propagation
Equation 4.2 gives the propagation model:
Ci p =31.9 + 247000x1 − 43800x2
+ 20.8x3 − 46100x4 − 3780000x5
+ (0.571x6 )2 + 1260000x7 + ε
49
4.1. CRACK INDEX MODELS
CHAPTER 4. RESULTS
i f Ci < 5, then Ci ≡ 5.
(4.2)
Where:
◦ Ci is the predicted Ci values.
.
.
.
◦
4.1.3
.
x1 , SCI300 [µm].
x2 , Mr [Mpa].
x3 , N100 [standard axle].
.
.
.
x4 , precipitation [mm].
x5 , temperature [o C].
x6 , age [years].
x7 , previous value [-].
ε is an independent random variable with the expectation value = 0 and variance = σ 2 .
Total Crack Index
The total crack index (Ci ) is estimated by:
Cii , i f ,Ci < 5
Ci =
Ci p , i f ,Ci ≥ 5
(4.3)
The results are shown by the plots in Figures 4.1 and 4.2. When used on a road
newly built the ingoing Ci is to be set to zero. The initiation model is to be used
when Ci < 5, and the propagation when Ci ≥ 5.
Figure 4.1: The plots show the results for the objects D-RV53-2 and T-205-1. The
black dots are the measured data, and the red line is estimated. The Ci is measured
as a function of time.
50
4.1. CRACK INDEX MODELS
CHAPTER 4. RESULTS
Figure 4.2: The plots show the results for the objects F-RV31-1, H-RV34-1, HRV40-2 and C-292-1. The black dots are the measured data, and the red line is
estimated. The Ci is measured as a function of time.
The R2 values range from [0.93,0.99];
51
4.2. RUT MODEL
4.2
CHAPTER 4. RESULTS
Rut Model
For the rutting the model is given by Equation 4.4:
rut =11.682 + (−1.2562 ∗ 108 x1 + 4.3704 ∗ 106 x2 + 46278log10 (x3 )
− 5.7504 ∗ 107 x4 + 1.1234 ∗ 108 x5 + 3.3172 ∗ 109 x7
+ 1.3567 ∗ 109 (x5 x3 )) ∗ ln(1.0001 + x6 ) + ε.
(4.4)
where:
◦ rut is the predicted rut value in [mm].
.
.
.
.
.
x1 , SCI300 [µm].
x2 , Mr [Mpa].
x3 , N100 [standard axle].
x4 , precipitation [mm].
.
.
.
x5 , temperature [o C].
x6 , age [years].
x7 , previous value [mm].
x3 ∗ x5 , interaction effect [standard axle o C].
ε is an independent random variable with the expectation value = 0 and variance = σ 2
The model was created using data from five different objects, and the test results
shown by the plots in Figures 4.5 and 4.6.
◦
52
4.2. RUT MODEL
CHAPTER 4. RESULTS
Figure 4.3: The plot shows the results for the objects D-RV532-2, H-RV34-1. The
black dots are the measured data, and the red line is estimated. The Ci is measured
as a function of time.
53
4.3. SENSITIVITY ANALYSIS
CHAPTER 4. RESULTS
Figure 4.4: The plot shows the results for the objects T-205-1, F-RV31-1 and HRV40-2. The black dots are the measured data, and the red line is estimated. The
rut is measured as a function of time.
The R2 values ranged between [0.96,0.99]. The bumps in the plotted data are believed to have been caused by the change in the number of lasers used in the sampling process.
4.3
Sensitivity Analysis
All the data in the Swedish LTPP database are of course measured on a finite number of occasions that are meant to be representative of a longer time period, but
54
4.4. VALIDATION
CHAPTER 4. RESULTS
some measurements are more sensitive to errors than others, sensitivity tests were
performed on each model and the results were:
◦ Ci initiation model: the traffic is the most sensitive factors for errors.
◦ Ci propagation model: the traffic is the most sensitive factors for errors.
◦ rut model: the age and previous rut value seem important.
4.4
Validation
A low number of samples can only give an indication for assessing the models
quality. The models created have been validated on objects G-RV23-1, W-RV80-1
and Z-E45-4. The objects are selected based on the date that they opened for traffic,
where the aim is to test them on "new" objects.
4.4.1
G-RV23-1
The results for crack index can be seen in Figure 4.7.
Figure 4.5: The Ci models used on object G-RV23-1. The small black dots are the
measured values, the dots on the red line are the predicted values.
The total R2 is 0.99, although the result is from a very small sample set. Samples
needed for the rut model was not found in the LTPP.
4.4.2
W-RV80-1
The result for crack index and rut can be seen in Figures 4.8, 4.9.
55
4.4. VALIDATION
CHAPTER 4. RESULTS
Figure 4.6: The Ci models used on object W-RV80-1. The small black dots are the
measured values, the dots on the red line are the predicted values.
The total R2 is 0.64; although the result indicates a need for calibration. The result
for the rut models is seen in Figure 4.9:
Figure 4.7: The rut model used on object W-RV80-1. The small black dots are the
measured values, the dots on the red line are the predicted values.
The R2 is 0.98, however the number of available samples is very low and that
causes high predicted values.
4.4.3
Z-E45-4
The crack index models tested on the object Z-E45-4, Figure 4.10:
56
4.4. VALIDATION
CHAPTER 4. RESULTS
Figure 4.8: The Ci models used on object Z-E45-4. The small black dots are the
measured values, the dots on the red line are the predicted values.
The rut model tested on the object Z-E45-4, Figure 4.11:
Figure 4.9: The rut model used on object Z-E45-4.
The R2 is 0.99; however the number of available samples is very low.
57
Chapter 5
CONCLUSION &
DISCUSSION
Prediction models have been developed based on observation from the Swedish
LTPP database, with the intention of aiding planning of maintenance activities as
a part a PMS. A strength of the models is that they are able to connect the factors
described in chapter 2.
These new models will allow the Swedish National Transport Administration to
prudently manage the road network, in a cost-effective manner. The crack index
is based on visual inspections, although cracks often initiates in the lower layers
before being visible. The present crack index has a large impact in the propagation
phase, the assumed explanation is that vulnerability for damage increases when a
crack is present. This effect doesn’t seem to be as big in the rut deformation.
A large part of the road network in Sweden consists of roads designed according
to older standards giving some variation in pavement thickness and support layer
material. This can cause problems when evaluating the performance models, and
calibration of the factor parameters is needed before they are applied. Road objects
with other large deviations in the distress data are less suitable for these models,
and recalibration is not always a guarantee.
By constructing performance models the understanding of the factors increases; the
quality of construction and maintenance can thereby increase. This can decrease
the costs of the infrastructure.
The models seem to be able to predict performance adequately, however the models needs further testing and calibration before they can be used. To increase the
models reliability larger dataset are needed. The performed validation can only
give indications due to the small set of data and the high number of factors. The
factor data concerning one individual object in the LTTP database are often given
in different time series that are not synchronized, thus resulting in small datasets
when working with models that uses several factors.
58
5.1. RECOMMENDATIONS
5.1
CHAPTER 5. CONCLUSION & DISCUSSION
Recommendations
Here follows a few recommendations of what can be done in the future:
The design of the database and the plan for gathering of data need to be changed.
The data in the LTPP tables needs a higher level of quality control and assurance
procedures, and the design of the database should be redone and follow stricter
rules. This can result in higher accuracy from future models.
The plan for which data to collect, and how, can be updated. For example the input
parameters for the HDM − 4 are more detailed when it comes to traffic flow pattern, vehicle fleet, traffic growth, climate etc. [64]
A better design of the database will simplify the future creation of high quality
prediction models, and together with an updated plan for data gathering lower the
cost of our roads. [6]
59
Bibliography
[1] AASHO. http://www.transportation.org/Pages/default.aspx.
[2] M. Andersson, J. Nyström, K. Odolinski, L. Wieweg, and W. Åsa. Strategi
för utveckling av en samhällsekonomisk analysmodell för drift, underhåll och
reinvestering av väg- och järnvägsinfrastruktur. Technical report, VTI, 2011.
VTI report 706.
[3] Asfaltskolan. www.asfaltskolan.se, 2011 09.
[4] M. Bellis. History of roads. http://inventors.about.com, 10 2011.
[5] G. Blom. Sannolihetsteori och statistikteori med tillämpningar. Studentlitteratur, 2001. 4th ed.
[6] C. Brown. U.S. Representative for Florida’s 3rd congressional district.
[7] Canadian Strategic Highway Research Program. http://www.cshrp.org, 08
2011.
[8] S. Coleman and B. Diefenderfer. Analysis of Virginia specific traffic data
for use with mechanistic empirical pavement design guide. Transportation
Research Record: Journal of the Transportation Research Board, 11 2010.
[9] M. Darter and T. Derek. Composite pavements and design, construction, and
benefits. http://www.dot.state.mn.us, 12 2011. University of Minnesota.
[10] A. C. Davison. Statistical Models. Cambridge University Press„ 12 2003.
Swiss Federal Institute of Technology, Department of Mathematics.
[11] G. Dore and H. Zubeak. Cold Region Pavement Engineering. McGraw Hill
Professional, 2008.
[12] M. El-Basyouny and M. G. J. Jeong. Probabilistic performance-related
specifications methodology based on mechanistic-empirical pavement design
guide. Journal of the Transportation Research Board, issue 2151, 2151, 2010.
[13] G. Elkins, S. Peter, T. Travis, and S. Amy. Longterm pavement performance
information management system pavement performance database user reference guide. FHWA, 2003.
[14] R. Elmasri and N. Shamkant. Database Systems. Pearson, 2010. 6th ed.
[15] EU. Paris, performance analysis of road infrastructure. http://europa.eu.int,
1999. ISBN 92-828-7827-9.
[16] F. T. Fang, editor. The Handbook of Highway Engineering. Taylor and Francis, 2006.
[17] FHWA. Distress identification manual for the long term pavement performance program. Technical report, Federal Highway Administration, 2003.
60
BIBLIOGRAPHY
BIBLIOGRAPHY
FHWA-RD-03-031.
[18] K. George, A. Rajagopal, and L. Lim. Models for predicting pavement deterioration. Transportation Research Board, Issue Number: 1215, 1989. ISSN:
0361-1981.
[19] N. G. Göransson. Prognosmodell för spårutveckling orsakad av tung trafik.
VTI notat 2-2007, 2007. http://www.vti.se/sv/publikationer.
[20] N. G. Göransson. Den Svenska Nationella LTPP databasen. VTI, 12 2009.
http://www.vti.se/.
[21] N. G. Göransson. Long term pavement performance database. http://www.
vti.se, 2009. Microsoft Acess database.
[22] M. Gustafsson, C. Berglund, and B. e. Forsberg. Effekter av vinterdäck - en
kunskapsöversikt. Technical report, VTI, 2006. VTI Rapport 543.
[23] R. Haas, N. Li, and S. Tighe. ROUGHNESS TRENDS AT C-SHRP LTPP
SITES. Ottawa, Ont.: Canadian Strategic Highway Research Program, 03
1999.
[24] T. Henning, R. Dunn, S. Costello, G. Hart, C. Parkman, and G. Burgess.
Long-term pavement performance (ltpp) studies in New Zealand lessons the
challenges and the way ahead. http://pavementmanagement.org, 07 2011.
[25] F. Holt and W. Grambling. Pavement management implementation. American
Society for Testing and Materials, 1992. Editors: Holt and Grambling.
[26] Z. Hoque and T. Martin. Status of Australia’s long term pavement performance maintenance (ltppm) study. In 6th International Conference on Managing Pavements, 2004.
[27] Y. Huang. Pavement Analysis and Design. Pearson, Prentice Hall, 2 edition,
2004.
[28] A. Ihs and L. Sjögren. An overview of HDM-4 and the Swedish pavement
management system (PMS). Technical report, Infrastructure maintenance,
VTI, 2003.
[29] Indiana University. Data mining. http://www.soic.indiana.edu/informatics/,
01 2005. Data Mining,.
[30] B. Kallas. Pavement maintenance and rehabilitation, 1985. Symposium.
[31] P. Keenan. Using a GIS as a DSS generator. http://mis.ucd.ie/staff/pkeenan/
gis_as_a_dss.html, 2011. University College Dublin.
[32] KTH. Handbook of Solid Mechanics. KTH, 2010. Department of Solid
Mechanics, editor Sundström Bengt.
[33] KTH, Royal Institute of Technology. Road construction and maintenance.
http://www.kth.se/student/kurser/kurs/AF2901, 2010.
[34] M. Kutner, C. Nachtsheim, J. Neter, and L. William. Applied Linear Statistical Models. IRWIN, 1996. 4th ed.
[35] J. Lang. Comparison of pavement management in the nordic countries. In
The 4th EPAM, European pavement and asset management conference, 2012.
[36] L. S. Li, Qiang and Mills. The implications of climate change on pavement performance and design. Technical report, University of Delaware University, Transportation Center (UD-UTC), 2011. http://www.ce.udel.edu.
61
BIBLIOGRAPHY
BIBLIOGRAPHY
[37] F. Lynn, K. S, H. Ronald, and H. Ralph. Long cost term cost benefit analysis
of pavement management system implementation. Transportation Research
Board, 1994. 3rd International Conference on Managing Pavements.
[38] T. McDonald. Guide to Pavement Maintenance. iUniverse Publishing, 2010.
[39] Microsoft Corporation. Defining relationships between tables in a Microsoft
Access database. http://support.microsoft.com/.
[40] A. Molenaar. Structural design of pavements. Technical report, Delft University of Technology, Faculty of Civil Engineering and Geosciences, 09 2009.
http://www.citg.tudelft.nl/.
[41] D. Montgomery. Design and Analysis of Experiments. Wiley, 2009.
[42] G. Mukesh. Road Rehabilitation and Maintenance Strategy, in Solomon Islands. Chief Technical Adviser.
[43] K. Nandakumar, A. K. Jain, and A. A. Ross. Score normalization in multimodal biometric systems. http://biometrics.cse.mse.edu, 08 2012. Michigan
State University,West Virginia University,.
[44] National Research Council (U.S.). Transportation Research Board. Pavement
analysis, design, rehabilitation, and evironmental factors, 1991. Transportation Research Board, National Research Council, 1991.
[45] NHCHR.
Mechanistic empirical pavement design guide.
http://onlinepubs.trb.org/, 07 2011. National Cooperative Highway Research Program.
[46] O. Nima Kargah, S. Stoffels, and N. Tabatabaee. Network level pavement
roughness prediction model for rehabilitation recommendations. Transportation Research Record Journal of the Transportation Research Board, 2, 2010.
Issue 2155.
[47] T. Parsons and C. Ogden. Recommended guidelines for the collection and
use of geospatially referenced data airfield pavement management. Technical
report, ACRP, 2010.
[48] Pavement Interactive. Loads. http://www.pavementinteractive.org/article/
loads/, 02 2012.
[49] M. Quilligan. Bridge weigh in motion. Royal Institute of Technology, 2003.
ISSN 1103-4270.
[50] L. Råde and B. Westergren. Mathematics Handbook. Studentlitteratur, fifth
edition, 2004.
[51] R. Roos, M. Stöcker, and A. Grossman. Implementation of a communal PMS
in Germany -state-of-the-art-. In The 3rd European Pavement and Asset Management EPAM3, 07 2008.
[52] P. Rosengren and S. Davor. Model for classification of areas for selective reinforcement measures. documents.vsect.chalmers.se, 10 2010. Examensarbete
2007:15.
[53] R. G. Saba. Pavement performance prediction models, 11 2011. Norwegian
Public Roads Administration, Road Technology Division.
[54] SATELLITE IMAGING CORPORATION.
http://www.ucancode.net/
Gis-Source-Code.htm, 03 2012.
62
BIBLIOGRAPHY
BIBLIOGRAPHY
[55] M. Shahin. Pavement management for airports, roads, and parking lots.
Springer-Verlag New York, 2 edition, September 2009.
[56] Statistiska centralbyrån. Transport and communications. http://www.scb.se,
09 2011. Sweden’s Statistics Agency.
[57] Z. Suo and W. G. Wong. Nonlinear properties analysis on rutting behavior
of bituminous materials with different air void contents. Construction and
Building Materials, July 2009. www.elsevier.com/locate/conbuildmat.
[58] Swedish National Road Administration. Publikation väg 94. www.vv.se, 5
2010.
[59] Swedish
National
Road
Administration.
VVTK
VÄG.
http://publikationswebbutik.vv.se, 02 2012. PUBLIKATION 2008:78.
[60] Swedish Transport Administration.
Bearbetning av deflektionsmätdata,erhållna vid provbelasning av väg med fwd-apparat.
http://publikationswebbutik.vv.se, 2000.
Metodbeskrivning 114:2000,
publikation 2000:29, BYA20A 99:3235.
[61] The South African National Roads Agency Limited SANRAL.
http://www.nra.co.za, 12 2011.
[62] The
Swedish
Transport
Agency.
Transportstyrelsen.
http://www.transportstyrelsen.se.
[63] The World Bank. http://web.worldbank.org, 2011.
[64] The World Bank. HDM-4 manual. http://www.worldbank.org/, 02 2012.
[65] Trafikverket. ATB VÄG 2005. http://www.trafikverket.se, 2011.
[66] Trafikverket. Klickbara kartan. http://www.trafikverket.se/Foretag/Trafikeraoch-transportera/, 2011.
[67] Trafikverket. TRVK väg. Technical Report TRV 2011:072, Trafikverket,
2011.
[68] Trafikverket. Vägunderhåll 2000, 2012. http://www.trafikverket.se.
[69] Transportation Association of Canada.
Canadian long-term pavement performance (c-ltpp) database users guide.
Technical report, Canadian Strategic Highway Research Program C-SHRP, 1997.
http://www.cshrp.org/products/userg2.pdf.
[70] Transportstyrelsen. Lasta lagligt vikt- och dimensionsbestämmelser för tunga
fordon 2010. Technical report, http://www.transportstyrelsen.se/, 2010.
[71] Transportverket. Vvmb 122 vägytemätning med mätbil objektmätning. publikationswebbutik.vv.se, 07 2010.
[72] TRB. The transportation research board. http://www.trb.org. Division of the
American National Research Council.
[73] University of Washington.
Pavement guide interactive.
http://training.ce.washington.edu, 02 2012.
[74] US Department of Transportation.
Federal highway administration.
http://www.fhwa.dot.gov, 07 2011.
[75] VTI, the Swedish National Road and Transport Research Institute.
Linköping, head office, Olaus Magnus väg 35, [email protected] http://www.vti.se.
[76] A. Weinger and S. P. The new Austrian Method for the Structural Assessment
63
BIBLIOGRAPHY
[77]
[78]
[79]
[80]
BIBLIOGRAPHY
of Pavement Constructions for PMS Purposes. In The 3rd European Pavement
and Asset Management EPAM3, 07 2008.
L. G. Wågberg. Bära eller brista,. Svenska kommunförbundet, 1991.
L. G. Wågberg. Utveckling av nedbrytningsmodeller sprickinitiering och
sprickpropagering. Kommunikationsforskningsberedningen, 2001. VTI meddelande 9161.
L. G. Wiman. Accelererad provning av vägkonstruktioner, Med och utan
polymermodifierat bitumen i bindlager. Technical report, VTI, 2010.
E. Yoder and M. Witczak. Principles of pavement design. Wiley, 1975.
64
Appendix A
STATISTICS THEORY
Here follows a brief description of some of the tools used in this project. 1 "Statistics is an aid, not a substitute for common sense". 2 A scientific model is an
abstraction of the reality, a simplification. In science statistics is a tool based on
the study of:
likelihood and probability
for occurring events, based on information known and deducted from samples.
Statistics allows educated guesses, estimates, to be made from a small amount
samples; that may be expensive or difficult to obtain. Statistical Models are constructed from:
◦
◦
◦
A.1
numerical data.
estimating the probabilistic future
behavior of a system based on
past behavior.
extrapolation or interpolation of
◦
◦
data based on some best-fit.
error estimates of observations.
spectral analysis of data or model
generated output.
Stochastic Variables
A stochastic/random variable (r.v.) is defined as a function defined for a sample
space Ω. For real numbers, ℜ, one can say that the r.v. X is a function from Ω to
R , where µ is a sample in Ω. This is further explained by the illustration in Figure
A.1.
Figure A.1: An illustration of the concept of random variables.
[5]
An unbiased estimation of the "true value" of a stochastic variable is given by a
point estimation:
1 Using
the notation as in [50]
Sealy Gosset
2 William
65
A.1. STOCHASTIC VARIABLES
APPENDIX A. STATISTICS THEORY
E(X) = µ,
(A.1)
where the function E(X) gives the expected value of X. There are three kinds of
X;
 
 ∗
x1
x1
x2 
x∗ 
 
 2
the true values, X =  . , the observed values X∗ =  .  and the estimated values
 .. 
 .. 
xn∗
xn
 
xb1
xb2 

b =
X
 .. , estimated from the X∗ .
.
xbn
xn∗ are the random variables describing how X∗ varies for different observations.
E(X) is the expected value of X. The empirical estimate can be calculated as the
arithmetic mean of the repeated measure observations of the variable. This estimates the true expected value in an unbiased manner, given that this value exists.
From the law of large numbers it can be concluded that, if the size of the sample
set gets larger, the variance of this estimate gets smaller; an illustration is found in
Figure A.2.
Figure A.2: Image of an r.v. variable.
The true value can never be sampled, but it can be estimated if n → ∞. There are
often many ways to derive a point estimation (see Figure A.2). The effectiveness
of estimation is measured by the variance, V [X], which describes how far values lie
from the mean. If V (Xb1 ) < Var(Xb2 ) then Xb1 is the most effective estimate.
A.1.1
Standard Deviation
We can only sample the variable as:
66
A.1. STOCHASTIC VARIABLES
APPENDIX A. STATISTICS THEORY
X∗ = µ + σ .
(A.2)
From these values the mean value, X̂ can be calculated,
pwhere σ is the standard
deviation, the square root of the expected value of σ = E[X−µ]2 . The standard
deviation is given by:
Var(X) = σ 2 = X2 − E[X]2 .
(A.3)
The standard deviation can be estimated as: s2 =
A.1.2
1
n−1
∑nj=1 (X j − X̄)2 .
Descriptive Statistics
The box plot is a method of visualizing data invented by J. Tukey. An American mathematician who also developed the Cooley−Tukey Fast Fourier Transform
(FFT) algorithm. This method displays differences between populations without
assumptions of the statistical distribution, i.e. a non-parametric method. The box
plot helps identify outliers by indicating the level of dispersion and skewness in the
data. An example is shown in Figure A.3. [50]
Figure A.3: A box plot displays the differences between populations without any
assumptions of the statistical distribution, i.e. a non-parametric method. The box
plot helps identify outliers by indicating the level of dispersion and skewness in the
data.[50]
67
A.2. CORRELATION AND CAUSALITY
APPENDIX A. STATISTICS THEORY
A.2
Correlation and Causality
The Pearson correlation coefficient gives a measurement of the strength of a linear
relationship between two variables. The value does not characterize their relationship. The dictum that "correlation does not imply causation" signifies that correlation cannot be used to infer a causal relationship between the variables. Correlation
is sometimes used as evidence of a possible causal relationship, but correlation can
only "hint" a causal relationship. The coefficient of determination generalizes the
correlation coefficient for relationships beyond linear regression.
A.2.1
Scatter Plots
A scatter plot is a graph made by plotting ordered pairs in a coordinate plane to
show the correlation and trends between two sets of data. A scatter plot can show
the relationship between to variables, however it does not by itself prove that one
variable causes the other. [34]
A.3
Testing for Normality
The histogram can give you a general idea of the shape of a measure, but two
numerical measures of shape give a more precise evaluation: skewness tells you
the amount and direction of skew (departure from horizontal symmetry), and kurtosis tells you how tall and sharp the central peak is, relative to a standard bell
curve.
Why do we care? One application is testing for normality: many Statistics inferences often require that a distribution be normal or nearly normal. A normal
distribution has skewness and excess kurtosis of 0, so if your distribution is close
to those values then it is probably close to normal.
A.3.1
Sample Skewness
By calculating the skewness, which is a measure of the asymmetry of the probability distribution of a random variable (r.v), it is possible to make inferences
regarding the samples. The value can be positive or negative, or undefined. The
skewness value can be positive or negative, or even undefined. Qualitatively it can
give indications regarding the probability density function (pdf).
◦ A negative value suggests that the tail on the left side of the pdf is longer than
the right side and the majority of the samples lie to the right of the mean.
◦ A positive result implies a tail on the right side is longer than the left side
and the bulk of the values lie to the left of the mean.
◦ A zero value indicates that the samples are equally distributed around the
mean, suggesting a symmetric distribution.
[10]
A.3.2
Kurtosis
Kurtosis is a measure of the height and sharpness of the peak relative to the rest of
the data. A normal distribution for instance has a kurtosis of 3. [10]
68
A.4. FACTORIAL DESIGN
A.4
APPENDIX A. STATISTICS THEORY
Factorial Design
In general an experiment consists of controllable and uncontrollable factors, as
shown in Figure A.4.
Figure A.4: An illustration of a general controlled experiment.
Many factors vary with the seasons and the traffic amount varies over time. The
climate’s effect on a road also depends on the environment in which it is built.
Our data come from an "experiment" where almost all factors are uncontrolled.
This means that the magnitude of the factors and interaction factors are unknown.
Nonetheless different tools from mathematical science combined with reasoning
can be used in finding the best structure for the prediction models, and values for
the parameters. [41]
A.5
Regression Analysis
This is a way of modeling, i.e. an estimation technique, the relationship between
one or more variables. The problems that can be solved using statistics are:
◦ How to find the mathematical relationship, describing a line, that fits "best"?
◦ How to describe deviation from this line?
◦ How do new data affect the created model?
  
   
y1
1 x11 . . . x1k
β1
ε1
y2  1 x21 . . . x2k  β2  ε2 
  
   
Model: Y = Xβ + ε =  .  =  .
  ..  +  .. .
..
.
.
 .  .
 .   . 
.
yn
1 xn1 . . . xnk
βk
εn
A.5.1
Nonlinear Regression
However, in many cases the relationship between the data is not linear. "Nonlinear regression" is a type of regression in which the observed data are modeled by
a nonlinear function model parameters, and depends on one or more independent
variables. The measured data is comprised of explanatory variables "xn " and response variables "yn ". The measured yn is considered as a stochastic (random)
variable with a mean resulting in a nonlinear function f (x, β ). A "closed-form expression" that provides a "prefect" answer for the best-fitting parameters is seldom
found. Instead numerical optimization algorithms are often used. This is due to
the fact that there may be many local minima of the function to be optimized and
69
A.5. REGRESSION ANALYSIS
APPENDIX A. STATISTICS THEORY
even the global minimum may produce a biased estimate. These estimations can be
found using the "Levenberg-Marquardt algorithm". This is an iterative method that
requires an initial "guess" for the parameter vector, β . The resulting models are
always estimations of an often more complex relationship than in the "real" world.
The Figures A.5 and A.6 give an example of typical Ci and rut values. [34]
Figure A.5: A "typical" Ci , data from the object D-RV-2.
The rut values were in this study inter- and extrapolated to increase the number of
samples for creating the models.
Figure A.6: "Typical" rut values, data from the object H-RV34-1.
70
A.6. DATA PREPROCESSING
A.6
APPENDIX A. STATISTICS THEORY
Data Preprocessing
All data stored should be free from noise before being used to ensure quality. When
making models a normalization technique should be used to make the numerically
data comparable when different units are used. In the following section the MinMax technique is described (see Section A.5.1).
A.6.1
Min-Max Normalization
This process is done in order to take data measured in the relevant engineering units
and transform them into a value in the interval [0,1] by using Equation A.4.
xi −xmin
xmax −xmin
xinormalized =
(A.4)
0 |i f xmax − xmin = 0|
Where:
◦ xi
is the i:th sample, normalized.
normalized
◦ xi is the i:th sample, unnormalized.
◦ xmax is the max value sample in the current unit dimension.
◦ xmin is the min value sample in the current unit dimension.
Min-max normalization performs a linear transformation on the original data values. When the min-max normalization is applied, each feature will lie within the
new range of values will remain the same. The min-max normalization has the
advantage of preserving exactly all relationships in the data.
[43, 29]
71
Appendix B
THE RELATIONAL
DATABASE
The concept "relational database model" was developed in 1969 by Edgar Frank
Codd, a British computer scientist working for IBM. The model is based on mathematical set theory and predicate logic, and constructed by tables, relations, that
can be "managed" using commands that return tables. [14]
B.1
Enhanced Entity Relationship Model
An Enhanced Entity Relationship (EER) Model is a more advanced way of modeling, which includes all the functionality of an Entity Relationship Model (ER) and
in addition the concepts of:
◦
◦
◦
subclass
superclass
specialization
◦
◦
generalization
union types
As well as the properties and constraints that help to organize and structure the data,
in addition it leads to forcing the database to conform to a set of requirements, as
required by complex software systems and GIS. [14]
B.1.1
Microsoft Access Database
The database used here considers only relationships between pairs of tables, which
can only be related as:
◦
one-to-one
◦
one-to-many
◦
many-to-many
[14, 39]
B.2
Terminology
The theory of relational databases employs a set of mathematical terms. The most
important ones, and their SQL database equivalents, are given in the list below:
◦
◦
relation - table
derived relation - view
◦
◦
tuple - row
attribute -column
A relation consists of a set of tuples, all sharing the same attributes. The tuple
can be used to represent a physical object and its information. The tuples form a
72
B.3. NORMALIZATION
APPENDIX B. THE RELATIONAL DATABASE
table, organized into rows and columns. All data in an attribute belongs to the same
dimension, having the same requirements. This is illustrated in Figure B.1.
Figure B.1: Relational database terminology.
B.3
Normalization
When an EER is translated into tables certain procedures can be used to eliminate
nonsimple domains and redundant information. This offers protection against data
loss of integrity in the data and manipulation anomalies. An illustration of this is
found in Figure B.2.
Figure B.2: Illustration of the normal forms.
Normalisation is a set of simple rules that increases the "quality" of the architectural design, thus enabling a faster DBMS. To address the data in the tables specific
types of "keys" are used when addressing the specific data wanted. [14]
73
Appendix C
OBJECTS
Here follows a description of the factors for the road objects from which the data
in both models were collected.
C.1
Crack Index Initiation Factor Data
C-292-1
SCI300 [µm]
113
113
113
113
113
113
113
113
113
Mr [MPa]
220
220
220
220
220
220
220
220
220
N100
33117.34
67228.19
102362.37
138550.58
175824.43
214216.50
253760.33
294490.47
336442.512
Mr [MPa]
43
43
43
43
43
43
43
N100
64586.80
131111.20
199631.33
270207.07
342900.08
417773.89
494893.90
Precip. [mm]
590
590
590
590
590
590
590
594
594
Temp. [◦ C]
5
5
5
5
5
5
5
5
5
Age [year]
0
1
2
3
4
5
6
7
8
H-RV34-1
SCI300 [µm]
134.94
134.94
134.94
134.94
134.94
134.94
134.94
Precip. [mm]
566
566
566
566
576
576
576
74
Temp. [◦ C]
6.20
6.20
6.20
6.20
6
6
6
Age [year]
0
1
2
3
4
5
6
C.1. CRACK INDEX INITIATION FACTOR DATA APPENDIX C. OBJECTS
D-RV53-2
SCI300 [µm]
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Mr [MPa]
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
N100
58433.04
118619.07
180610.68
244462.04
310228.95
377968.85
447740.96
519606.22
593627.45
669869.31
748398.43
829283.42
912594.97
998405.85
1086791.07
H-RV40-2
SCI300 [µm]
145
145
145
145
145
145
145
145
145
145
145
145
145
Mr [MPa]
165
165
165
165
165
165
165
165
165
165
165
165
165
N100
26792.74
54389.25
82813.67
112090.81
142246.27
173306.40
205298.33
238250.01
272190.25
307148.69
343155.89
380243.30
418443.33
Precip. [mm]
666
666
666
688
688
688
688
688
688
688
688
688
688
688
688
Precip. [mm]
506
506
506
506
506
506
506
506
507
506
506
532
532
75
Temp. [◦ C]
5.30
5.30
5.30
5.20
5.20
5.20
5.20
5.20
5.2
5.2
5.2
5.2
5.2
5.2
5.2
Temp. [◦ C]
6.90
6.90
6.90
6.90
6.90
6.90
6.61
6.60
6.60
6.60
6.56
6.50
6.50
Age [year]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Age [year]
0
1
2
3
4
5
6
7
8
9
10
11
12
C.2. CRACK INDEX PROPAGATION FACTOR DATA
APPENDIX C. OBJECTS
T-205-1
SCI300 [µm]
145
145
145
145
145
145
145
145
145
145
145
145
145
C.2
Mr [MPa]
165
165
165
165
165
165
165
165
165
165
165
165
165
Precip. [mm]
506
506
506
506
506
506
506
506
507
506
506
532
532
N100
26792.73
54389.25
82813.67
112090.81
142246.28
173306.30
205298.32
238250.01
272190.25
307148.69
343155.89
380243.30
418443.33
Temp. [◦ C]
6.90
6.90
6.90
6.90
6.90
6.90
6.60
6.60
6.60
6.60
6.60
6.60
6.50
Age [year]
0
1
2
3
4
5
6
7
8
9
10
11
12
Crack Index Propagation Factor Data
C-292-1
SCI300 [µm]
108
108
108
108
108
108
108
Mr [MPa]
205
205
205
205
205
205
205
N100
379653.13
424160.06
470002.19
517219.59
565853.52
615946.46
667542.18
Precip. [mm]
594
594
594
594
594
594
594
Temp. [◦ C]
5
5
5
5
5
5
5
Age [year]
9
10
11
12
13
14
15
Mr [MPa]
65
65
65
65
65
65
65
N100
337652.61
369401.93
402103.73
435786.59
470479.93
506214.07
543020.24
Precip. [mm]
508
508
508
508
470
470
470
Temp. [◦ C]
6.3
6.3
6.3
6.3
6.3
6.3
6.3
Age [year]
12
13
14
15
16
17
18
D-RV53-2
SCI300 [µm]
110
110
110
110
110
110
110
76
C.2. CRACK INDEX PROPAGATION FACTOR DATA
APPENDIX C. OBJECTS
F-RV31-1
SCI300 [µm]
100
100
100
100
100
Mr [MPa]
80
80
80
80
80
N100
1177827.84
1271595.72
1368176.62
1467654.97
1570117.65
Precip. [mm]
688
688
688
688
688
Temp. [◦ C]
52
52
52
52
52
Age [year]
15
16
17
18
19
Mr [MPa]
43
43
43
43
43
43
43
43
43
N100
574327.51
656144.13
740415.26
827214.52
916617.75
1008703.08
1103550.97
1201244.30
1301868.42
Precip. [mm]
576
576
576
576
576
576
570
569
569
Temp. [◦ C]
6
6
6
6
6
6
6
6
6
Age [year]
7
8
9
10
11
12
13
14
15
Mr [MPa]
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
N100
457789.37
498315.78
540058.00
583052.47
627336.78
672949.62
719930.85
768321.50
818163.88
869501.54
922379.32
976843.43
1032941.47
1090722.45
1150236.86
1211536.70
1274675.54
Precip. [mm]
532
532
601
601
601
601
601
601
601
601
601
601
601
601
601
601
601
Temp. [◦ C]
6.5
6.5
6.5
6.4
6.4
6.4
6.4
6.4
6.4
6.4
6.4
6.4
6.4
6.4
6.4
6.4
6.4
Age [year]
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
H-RV34-1
SCI300 [µm]
134.94
134.94
134.94
134.94
134.94
134.94
134.94
134.94
134.94
H-RV40-2
SCI300 [µm]
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
77
C.3. RUT FACTOR DATA
APPENDIX C. OBJECTS
T-205-1
SCI300 [µm]
88
88
88
88
88
88
88
88
88
88
88
88
88
C.3
Mr [MPa]
103
103
103
103
103
103
103
103
103
103
103
103
103
N100
198135.61
251842.74
307161.09
364138.99
422826.23
483274.09
545535.38
609664.51
675717.51
743752.11
813827.74
886005.64
960348.87
Precip. [mm]
687
687
687
687
687
687
687
687
687
687
687
687
687
Temp. [◦ C]
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
Age [year]
3
4
5
6
7
8
9
10
11
12
13
14
15
Precip. [mm]
507
507
507
507
524
524
524
524
524
524
489
508
508
508
508
508
470
470
470
470
470
470
Temp. [◦ C]
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
6.30
Age [year]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Rut Factor Data
D-RV53-2
SCI300 [µm]
140
140
140
140
140
140
140
140
140
140
140
140
110
110
110
110
110
110
110
110
110
110
Mr [MPa]
64
64
64
64
64
64
64
64
64
64
64
64
65
65
65
65
65
65
65
65
65
65
N100
21619.75
43888.08
66824.43
90448.94
114782.15
139845.36
165660.46
192250.02
219637.26
247846.12
276901.25
306828.03
337652.61
369401.93
402103.73
435786.59
470479.93
506214.07
543020.23
580930.58
619978.24
660197.33
78
C.3. RUT FACTOR DATA
APPENDIX C. OBJECTS
F-RV31-1
SCI300 [µm]
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Mr [MPa]
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
N100
58433.04
118619.07
180610.68
244462.04
310228.94
377968.85
447740.96
519606.22
593627.45
669869.31
748398.43
829283.42
912594.97
998405.85
1086791.07
1177827.84
1271595.72
1368176.63
1467654.97
1570117.65
79
Precip. [mm]
666
666
666
688
688
688
688
688
688
688
688
688
688
688
688
688
688
688
688
688
Temp. [◦ C]
5.30 0
5.30 1
5.30 2
5.20 3
5.20 4
5.20 5
5.20 6
5.20 7
5.20 8
5.20 9
5.20 10
5.20 11
5.20 12
5.20 13
5.20 14
5.20 15
5.20 16
5.20 17
5.20 18
5.20 19
Age [year]
C.3. RUT FACTOR DATA
APPENDIX C. OBJECTS
H-RV34-1
SCI300 [µm]
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Mr [MPa]
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
N100
58433.04
118619.07
180610.68
244462.04
310228.94
377968.85
447740.96
519606.22
593627.45
669869.31
748398.43
829283.42
912594.97
998405.85
1086791.07
1177827.84
1271595.72
1368176.63
1467654.97
1570117.65
80
Precip. [mm]
666
666
666
688
688
688
688
688
688
688
688
688
688
688
688
688
688
688
688
688
Temp. [◦ C]
5.30
5.20
5.20
5.20
5.20
5.20
5.20
5.20
5.2
5.2
5.2
5.2
5.2
5.2
5.2
5.2
5.2
5.2
5.2
5.2
Age [year]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
C.3. RUT FACTOR DATA
APPENDIX C. OBJECTS
H-RV40-2
SCI300 [µm]
145
145
145
145
145
145
145
145
145
145
145
145
145
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
135
Mr [MPa]
165
165
165
165
165
165
165
165
165
165
165
165
165
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
N100
26792.74
54389.25
82813.67
112090.81
142246.27
173306.40
205298.32
238250.01
272190.25
307148.69
343155.89
380243.30
418443.33
457789.37
498315.78
540057.99
583052.47
627336.78
672949.62
719930.84
768321.50
818163.88
869501.54
922379.32
976843.43
1032941.47
1090722.45
1150236.86
1211536.70
1274675.54
81
Precip. [mm]
506
506
506
506
506
506
506
506
507
506
506
532
532
532
532
601
601
601
601
601
601
601
601
601
601
601
601
601
601
601
Temp. [◦ C]
6.90
6.90
6.90
6.90
6.90
6.90
6.60
6.60
6.60
6.60
6.60
6.50
6.50
6.50
6.50
6.50
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
6.40
Age [year]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
C.3. RUT FACTOR DATA
APPENDIX C. OBJECTS
T-205-1
SCI300 [µm]
88
88
88
88
88
88
88
88
88
88
88
88
88
88
88
88
Mr [MPa]
103
103
103
103
103
103
103
103
103
103
103
103
103
103
103
103
Precip. [mm]
687
687
687
687
687
687
687
687
687
687
687
687
687
687
687
687
N100
47233.09
95883.18
145992.75
198135.61
251842.74
307161.09
364138.99
422826.23
483274.09
545535.38
609664.51
675717.51
743752.10
813827.74
886005.64
960348.88
82
Temp. [◦ C]
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
5.40
Age [year]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
C.4. CRACK INDEX AND RUT DATA RANGES
C.4
APPENDIX C. OBJECTS
Crack Index and Rut Data Ranges
The Ci and rut data are described by the box plots in Figure C.1.
Figure C.1: The Ci and rut data presented in box plots.
83
Appendix D
VALIDATION OBJECTS
Here follows a description of the validation road objects W-RV80-1 and Z-E454.
Table D.1: General info of the validation road objects.
Object
Name
Opened
Sections no. Speed [ km
h ]
G-RV23-1 Älmhult
1986-07-01 11
90
W-RV80-1 Falun
1989-07-01 10
90
Z-E45-4
Härjedalen 2002-07-01 8
90
The map in, Figure D.1, marks the geographical location of the objects.
Figure D.1: The red dots mark the geographical location of the validation objects.
Table D.2: Material info for the validation objects.
Object
Subgrade
Type
Subbase
Thickness [mm]
Roadbase
Thickness [mm]
G-RV23-1
sand moraine
6
sand and gravel
300
gravel
125
W-RV80-1
gravel, sand till
6
Material A
500
gravel
120
Z-E45-4
sand, gravel till
6
crushed VÄG 94
420
crushed VÄG 94
80
The subgrade is classified as in the EU project PARIS;
1. sand
2. silty sand
3. clay
4. peat
5. bedrock
6. other
84
D.1. CRACK INDEX FACTOR DATAAPPENDIX D. VALIDATION OBJECTS
[15]
D.1
Crack Index Factor Data
G-RV23-1
Table D.3: Ci validation factor data.
SCI300 [µm]
119
119
119
Mr [MPa]
78.37
78.37
78.37
N100
52342
106250
161780
Precip. [mm]
816
816
816
Temp. [◦ C]
5.7
5.7
5.7
Age
6
7
8
W-RV80-1
Table D.4: Ci validation factor data.
SCI300 [µm]
138.64
138.64
138.64
138.64
191.66
191.66
191.66
191.66
191.66
191.66
191.66
Mr [MPa]
79.262
79.262
79.262
79.262
72.166
72.166
72.166
72.166
72.166
72.166
72.166
N100
40098
81399
123940
167760
212890
259370
307250
356570
407360
459680
513570
Precip. [mm]
579
579
617
617
617
617
617
617
617
617
617
Temp. [◦ C]
4.6
4.6
4.2
4.2
4.2
4.2
4.2
4.2
4.2
4.2
4.2
Age
0
1
2
3
4
5
6
7
8
9
10
Mr [MPa]
101.61
101.61
101.61
101.61
101.61
101.61
101.61
101.61
N100
441.44
947.77
1528.5
2194.7
2958.7
3835.1
4840.3
5993.3
Precip. [mm]
603
603
603
603
603
603
603
603
Temp. [◦ C]
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
Age
0
1
2
3
4
5
6
7
W-RV80-1
SCI300 [µm]
110.76
110.76
110.76
110.76
110.76
110.76
110.76
110.76
Z-E45-4
85
D.1. CRACK INDEX FACTOR DATAAPPENDIX D. VALIDATION OBJECTS
SCI300 [µm]
110.76
110.76
110.76
110.76
110.76
110.76
110.76
110.76
Mr [MPa]
101.61
101.61
101.61
101.61
101.61
101.61
101.61
101.61
N100
38363
82366
132840
190730
257130
333290
420650
520840
Precip. [mm]
603
603
603
603
603
603
603
603
86
Temp. [◦ C]
2.2
2.2
2.2
2.2
2.2
2.2
2.2
2.2
Age
0
1
2
3
4
5
6
7
D.2. RUT FACTOR DATA
D.2
APPENDIX D. VALIDATION OBJECTS
Rut Factor Data
W-RV80-1
Table D.5: Rut validation factor data.
SCI300 [µm]
138.64
191.66
191.66
191.66
Mr [MPa]
79.262
72.166
72.166
72.166
N100
40098
407360
459680
513570
Precip. [mm]
579
617
617
617
Temp. [◦ C]
4.6
4.2
4.2
4.2
Age
0
8
9
10
Mr [MPa]
101.61
101.61
101.61
101.61
101.61
N100
38363
82366
140260
216420
316620
Precip. [mm]
603
603
603
603
603
Temp. [◦ C]
2.2
2.2
2.2
2.2
2.2
Age
0
1
2
3
4
Z-E45-4
SCI300 [µm]
110
110
110
110
110
D.3
Validation Data Ranges
The Ci and rut data are described by the box plots in Figure D.2.
Figure D.2: The Ci and rut data presented in box plots.
87
Appendix E
MATLAB CODE
The software Matlab was used for creating the models, and the code can be found
in this section.
E.1
Data extracting Ci
% FWD %%%%%%%%%%%%%%%%%%%%%%%%%%%
l o a d periodInit_T2051 ;
l o a d periodPropa_T2051 ;
%%
SCI_InitT2051 = 88 * ones ( l e n g t h ( periodInit_T2051 ) , 1 ) ;
SCI_PropaT2051 = 88 * ones ( l e n g t h ( periodPropa_T2051 ) , 1 ) ;
MrInit_T2051 = 103 * ones ( l e n g t h ( periodInit_T2051 ) , 1 ) ;
MrPropa_T2051 = 103 * ones ( l e n g t h ( periodPropa_T2051 ) , 1 ) ;
save ( ' SCI_InitT2051 ' , ' SCI_InitT2051 ' )
s a v e ( ' SCI_PropaT2051 ' , ' SCI_PropaT2051 ' )
save ( ' MrInit_T2051 ' , ' MrInit_T2051 ' )
s a v e ( ' MrPropa_T2051 ' , ' MrPropa_T2051 ' )
precep6190Init_T2051 = [ 6 8 7
687
687];
precep6190Propa_T2051 = 687 * ones ( l e n g t h ( [ 1 9 9 7 : 2 0 0 9 ] ) , 1 ) ;
save ( ' precep6190Init_T2051 ' , ' precep6190Init_T2051 ' )
save ( ' precep6190Propa_T2051 ' , ' precep6190Propa_T2051 ' )
temp6190Init_T2051 = [ 5 . 4 0 0 0 0 0 0 9 5
5.400000095
5.400000095];
temp6190Propa_T2051 = 5 . 4 0 0 0 0 0 0 9 5 * ones ( l e n g t h ( [ 1 9 9 7 : 2 0 0 9 ] ) , 1 ) ;
save ( ' temp6190Init_T2051 ' , ' temp6190Init_T2051 ' )
s a v e ( ' temp6190Propa_T2051 ' , ' temp6190Propa_T2051 ' )
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot (211)
p l o t ( [ periodInit_T2051 ; periodPropa_T2051 ] , [ temp6190Init_T2051 ; ←temp6190Propa_T2051 ] , '−−bo ' , ' l i n e w i d t h ' , 3 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' g ' , ' MarkerSize ' ,4)
g r i d on
subplot (212)
p l o t ( [ periodInit_T2051 ; periodPropa_T2051 ] , [ precep6190Init_T2051 ; ←precep6190Propa_T2051 ] , '−−r o ' , ' l i n e w i d t h ' , 3 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' g ' , ' MarkerSize ' ,4)
g r i d on
88
E.1. DATA EXTRACTING CI
APPENDIX E. MATLAB CODE
% T r a f f i c %%%%%%%%%%%%%%%%%%%%%%%%%%%
clc
clear
close all
l o a d periodInit_T2051
l o a d periodPropa_T2051
%%
openD = 1 9 9 4 ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% measure d i r e c t i o n : 0
measureTraffic_T2051 = [ 1 9 9 4
1997
2001
2005
2009];
save ( ' measureTraffic_T2051 ' , ' measureTraffic_T2051 ' )
AADT_T2051 = [ 1 3 2 0
1380
1490
1510
1530];
s a v e ( ' AADT_T2051 ' , ' AADT_T2051 ' )
AADTTrucks_T2051 = [ 1 8 0
220
210
260
250];
s a v e ( ' AADTTrucks_T2051 ' , ' AADTTrucks_T2051 ' )
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% K l i c k b a r a k a r t a n s ä g e r open M ä t t 9 3 , LTPP s ä g e r 9 4 , k ö r på 9 4 ? ? ? ? ? ? ? ?
% f1i = inline ( '1320./((1+0.03) .^ x) ') ;
% f11i = inline ( '180./((1+0.03) .^ x) ') ;
% xx1i = s o r t ( [ 1 : 1 ] , ' descend ' ) ; % 1983:1994 7 sampels
% AADT01= f 1 i ( x x 1 i ' ) ;
% AADTT01= f 1 1 i ( x x 1 i ' ) ;
f1p = inline ( ' 1 3 2 0 . * ( 1 + 0 . 0 3 ) . ^ x ' ) ;
f11p=inline ( ' 1 8 0 . * ( 1 + 0 . 0 3 ) . ^ x ' ) ;
xx1p = [ 0 : 1 5 ] ' ; % 1 9 9 4 : 2 0 0 9 15 s a m p l e s
AADT1=f1p ( xx1p ) ;
AADTT1=f11p ( xx1p ) ;
% 97
f2i = inline ( ' 1 4 9 0 . / ( ( 1 + 0 . 0 3 ) . ^ x ) ' ) ;
f22i = inline ( ' 2 1 0 . / ( ( 1 + 0 . 0 3 ) . ^ x ) ' ) ;
xx2i= s o r t ( [ 1 : l e n g t h ( 1 9 9 4 : 1 9 9 6 ) ] , ' d e s c e n d ' ) ; % 1 9 9 8 : 2 0 0 0 7 s a m p e l s
AADT02=f2i ( xx2i ' ) ;
AADTT02=f22i ( xx2i ' ) ;
f2p = inline ( ' 1 3 8 0 . * ( 1 + 0 . 0 3 ) . ^ x ' ) ;
f22p = inline ( ' 2 2 0 . * ( 1 + 0 . 0 3 ) . ^ x ' ) ;
xx2p= [ 0 : l e n g t h ( [ 1 9 9 8 : 2 0 0 9 ] ) ] ' ; % 1 9 9 7 : 2 0 0 9 12 s a m p l e s
AADT2=f2p ( xx2p ) ;
AADTT2=f22p ( xx2p ) ;
% 01
f3i = inline ( ' 1 4 9 0 . / ( ( 1 + 0 . 0 3 ) . ^ x ) ' ) ;
f33i = inline ( ' 2 1 0 . / ( ( 1 + 0 . 0 3 ) . ^ x ) ' ) ;
89
E.1. DATA EXTRACTING CI
APPENDIX E. MATLAB CODE
xx3i= s o r t ( [ 1 : l e n g t h ( 1 9 9 4 : 2 0 0 0 ) ] , ' d e s c e n d ' ) ; % 1 9 9 4 : 2 0 0 0 7 s a m p e l s
AADT03=f3i ( xx3i ' ) ;
AADTT03=f33i ( xx3i ' ) ;
f3p = inline ( ' 1 4 9 0 . * ( 1 + 0 . 0 3 ) . ^ x ' ) ;
f33p = inline ( ' 2 1 0 . * ( 1 + 0 . 0 3 ) . ^ x ' ) ;
xx3p= [ 0 : l e n g t h ( [ 2 0 0 2 : 2 0 0 9 ] ) ] ' ; % 2 0 0 1 : 2 0 0 9 9 s a m p l e s
AADT3=f3p ( xx3p ) ;
AADTT3=f33p ( xx3p ) ;
% 05
f4i = inline ( ' 1 5 1 0 . / ( ( 1 + 0 . 0 3 ) . ^ x ) ' ) ;
f44i = inline ( ' 2 6 0 . / ( ( 1 + 0 . 0 3 ) . ^ x ) ' ) ;
xx4i= s o r t ( [ 1 : 1 1 ] , ' d e s c e n d ' ) ; % 1 9 9 4 : 2 0 0 4 11 s a m p e l s
AADT04=f4i ( xx4i ' ) ;
AADTT04=f44i ( xx4i ' ) ;
f4p = inline ( ' 1 5 1 0 . * ( 1 + 0 . 0 3 ) . ^ x ' ) ;
f44p = inline ( ' 2 6 0 . * ( 1 + 0 . 0 3 ) . ^ x ' ) ;
xx4p= [ 0 : l e n g t h ( [ 2 0 0 6 : 2 0 0 9 ] ) ] ' ; % 2 0 0 5 : 2 0 0 9 s a m p l e s
AADT4=f4p ( xx4p ) ;
AADTT4=f44p ( xx4p ) ;
% 09
f9i = inline ( ' 1 5 3 0 . / ( ( 1 + 0 . 0 3 ) . ^ x ) ' ) ;
f99i = inline ( ' 2 5 0 . / ( ( 1 + 0 . 0 3 ) . ^ x ) ' ) ;
xx9i= s o r t ( [ 1 : l e n g t h ( 1 9 9 4 : 2 0 0 8 ) ] , ' d e s c e n d ' ) ; % 1 9 9 4 : 2 0 0 8 s a m p e l s
AADT09=f9i ( xx9i ' ) ;
AADTT09=f99i ( xx9i ' ) ;
AADT9 = 1 5 3 0 ;
AADTT9 = 2 5 0 ;
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure (1)
subplot (211)
p l o t ( measureTraffic_T2051 , AADT_T2051 , ' ko ' , ' L i n e W i d t h ' , 3 , ' M a r k e r E d g e C o l o r ' ←, ' k ' , ' MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,9)
g r i d on
% h o l d on
% p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , AADT1 , ' rx ' )
% h o l d on
% p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , [ AADT02 ; AADT2 ] , ' rx ' )
% h o l d on
% p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , [ AADT03 ; AADT3 ] , ' rx ' )
% h o l d on
% p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , [ AADT04 ; AADT4 ] , ' rx ' )
% h o l d on
% p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , [ AADT09 ; AADT9 ] , ' rx ' )
h o l d on
%%
ADTmean = mean ( [ [ AADT1 ] [ AADT02 ; AADT2 ] [ AADT03 ; AADT3 ] [ AADT04 ; AADT4 ] [ ←AADT09 ; AADT9 ] ] , 2 ) ; % [ AADT04 ; AADT4 ] ] , 2 ) ;
%AADTstd = s t d ( [ [ AADT01 ; AADT1] [ AADT02 ; AADT2] [ AADT03 ; AADT3 ] ] , 0 , 2 ) ;
subplot (211)
p l o t ( [ 1 9 9 4 : 2 0 0 9 ] ' , ADTmean , '−−r ' , ' l i n e w i d t h ' , 3 , ' M a r k e r E d g e C o l o r ' , ' r ' , ' ←MarkerFaceColor ' , ' r ' , ' MarkerSize ' ,4)
% t i t l e ( ' T−205−1 ADT' , ' F o n t S i z e ' , 1 2 , ' f o n t w e i g h t ' , ' b o l d ' )
XLABEL ( ' Year ' , ' F o n t S i z e ' , 2 0 , ' f o n t w e i g h t ' , ' b o l d ' )
YLABEL ( 'AADT ' , ' F o n t S i z e ' , 2 0 , ' f o n t w e i g h t ' , ' b o l d ' )
s e t ( gca , ' F o n t S i z e ' , 1 4 )
90
E.1. DATA EXTRACTING CI
APPENDIX E. MATLAB CODE
%h _ l e g e n d = l e g e n d ( ' measured ' , ' e s t i m a t e d 1 9 9 4 ' , ' 9 8 ' , ' 0 2 ' , ' 0 6 ' , ' mean ←e s t i m a t e s ' , ' l o c a t i o n ' , ' NorthWest ' )
%s e t ( h _ l e g e n d , ' F o n t S i z e ' , 1 2 ) ;
g r i d on
% h= g c f ;
% s a v e a s ( g c f , ' C : \ U s e r s \ MarkusS \ Dropbox \ l i c T \ _ l i c R e p o r t \ I m a g e s \ ADT_T2051 . ←png ' )
% s a v e a s ( g c f , ' T2051_ADT . f i g ' )
% s a v e a s ( g c f , ' T2051_ADT . t i f f ' )
% s a v e a s ( g c f , ' T2051_ADT . pdf ' )
% AADT_T2051 = 365 * ADTmean ;
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot (212)
p l o t ( measureTraffic_T2051 , AADTTrucks_T2051 , ' ko ' , ' L i n e W i d t h ' , 3 , ' ←MarkerEdgeColor ' , ' k ' , ' MarkerFaceColor ' , ' k ' , ' MarkerSize ' , 9 )
h o l d on
g r i d on
% p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , [ AADTT1 ] , ' rx ' )
% h o l d on
% p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , [ AADTT02 ; AADTT2 ] , ' rx ' )
% h o l d on
% p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , [ AADTT03 ; AADTT3 ] , ' rx ' )
% h o l d on
% p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , [ AADTT04 ; AADTT4 ] , ' rx ' )
h o l d on
ADTTmean = mean ( [ [ AADTT1 ] [ AADTT02 ; AADTT2 ] [ AADTT03 ; AADTT3 ] [ AADTT04 ; ←AADTT4 ] [ AADTT09 ; AADTT9 ] ] , 2 ) ;% [ AADTT04 ; AADTT4 ] ] , 2 ) ;
%AADTstd = s t d ( [ [ AADT01 ; AADT1] [ AADT02 ; AADT2] [ AADT03 ; AADT3 ] ] , 0 , 2 ) ;
subplot (212)
p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , ADTTmean , '−−r ' , ' l i n e w i d t h ' , 3 , ' M a r k e r E d g e C o l o r ' , ' r ' , ' ←MarkerFaceColor ' , ' r ' , ' MarkerSize ' ,4)
%h _ l e g e n d = l e g e n d ( ' measured ' , ' e s t i m a t e d 1 9 9 4 ' , ' 9 7 ' , ' 0 1 ' , ' 0 5 ' , ' 0 9 ' , ' mean ←e s t i m a t e s ' , ' l o c a t i o n ' , ' NorthWest ' )
% t i t l e ( ' T−205−1 ADTT ' , ' F o n t S i z e ' , 1 2 , ' f o n t w e i g h t ' , ' b o l d ' )
XLABEL ( ' Year ' , ' F o n t S i z e ' , 2 0 , ' f o n t w e i g h t ' , ' b o l d ' )
YLABEL ( 'AADTT ' , ' F o n t S i z e ' , 2 0 , ' f o n t w e i g h t ' , ' b o l d ' )
s e t ( gca , ' F o n t S i z e ' , 1 4 )
%s e t ( h _ l e g e n d , ' F o n t S i z e ' , 1 2 ) ;
g r i d on
h= g c f ;
saveas ( g c f , ' C : \ U s e r s \ MarkusS \ D e s k t o p \ c i B i l d e r \ ADTT2_T2051 . png ' )
saveas ( g c f , ' T2051_ADTT . f i g ' )
saveas ( g c f , ' T2051_ADTT . t i f f ' )
saveas ( g c f , ' T2051_ADTT . p d f ' )
AADTT_T2051 = 365 * ADTTmean ;
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%N100
n_T2051 = 4 . 4 ;
% Medelantal axlar för l a s t b i l a r
s a v e ( ' n_T2051 ' , ' n_T2051 ' )
pHV_T2051 = AADTT_T2051 . / AADT_T2051 * 1 0 0 ;
N100_T2051 = AADT_T2051 . * ( pHV_T2051 / 1 0 0 ) * ( 1 / 2 ) * n_T2051 * 0 . 3 3 ;
s a v e ( ' N100_T2051 ' , ' N100_T2051 ' )
%%
figure (22)
91
E.1. DATA EXTRACTING CI
APPENDIX E. MATLAB CODE
p l o t ( [ 1 9 9 4 : 2 0 0 9 ] , N100_T2051 , '−−g ' , ' l i n e w i d t h ' , 3 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' g ' , ' MarkerSize ' ,4)
N100CumSum_T2051 = cumsum ( N100_T2051 ) ;
N100CumSumInit_T2051 = N100CumSum_T2051 ( 1 : l e n g t h ( periodInit_T2051 ) ) ;
N100CumSumPropa_T2051 = N100CumSum_T2051 ( l e n g t h ( periodInit_T2051 ) + 1 : end ) ←;
s a v e ( ' N100CumSum_T2051 ' , ' N100CumSum_T2051 ' )
s a v e ( ' N100CumSumInit_T2051 ' , ' N100CumSumInit_T2051 ' )
s a v e ( ' N100CumSumPropa_T2051 ' , ' N100CumSumPropa_T2051 ' )
h o l d on
p l o t ( periodInit_T2051 , N100CumSumInit_T2051 , ' r o ' , ' l i n e w i d t h ' , 3 , ' ←MarkerEdgeColor ' , ' r ' , ' MarkerFaceColor ' , ' g ' , ' MarkerSize ' , 4 )
h o l d on
p l o t ( periodPropa_T2051 , N100CumSumPropa_T2051 , ' bo ' , ' l i n e w i d t h ' , 3 , ' ←MarkerEdgeColor ' , ' k ' , ' MarkerFaceColor ' , ' g ' , ' MarkerSize ' , 4 )
h_legend = l e g e n d ( ' N_{100} ' , ' c u m m u l a t i v e sum I n i t i t i o n ' , ' P r o p a g a t i o n ' , ' ←l o c a t i o n ' , ' NorthWest ' )
t i t l e ( ' N_{100} & N_{100} c u m m u l a t i v e ' , ' F o n t S i z e ' , 1 2 , ' f o n t w e i g h t ' , ' b o l d ' )
s e t ( h_legend , ' F o n t S i z e ' , 1 2 ) ;
g r i d on
h= g c f ;
saveas ( g c f , ' T2051_N100C . f i g ' )
saveas ( g c f , ' T2051_N100C . t i f f ' )
saveas ( g c f , ' T2051_N100C . p d f ' )
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
periodInit_T2051 = [ 1 9 9 4 : 1 9 9 6 ] ' ;
periodPropa_T2051 = [ 1 9 9 7 : 2 0 0 9 ] ' ;
CiInit_T2051 = [ 0
2.2667
4.5333];
%%%%%%%%%%%%%%%%
CiPropa_T2051 = [ 5 % 1997
10.5625
18.25
24.5625
69.125
74.75
48.25
64.25
79
94.9375
118.3125
158.625
182.9375
];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
save ( ' periodInit_T2051 ' , ' periodInit_T2051 ' ) ;
save ( ' periodPropa_T2051 ' , ' periodPropa_T2051 ' ) ;
save ( ' CiInit_T2051 ' , ' CiInit_T2051 ' ) ;
save ( ' CiPropa_T2051 ' , ' CiPropa_T2051 ' ) ;
%%%%%%%%%%%%%%%%%%%%%%% % m e a s u r e d r u t
mRUT = [ 0
3.66875001788139
92
E.1. DATA EXTRACTING CI
APPENDIX E. MATLAB CODE
4.03125
4.21875001490116
4.56874999403953
5.2749999910593
6.41249999403953
7.54375004768371
8.37187498807907
9.2
];
% d a t a m e a s u r e d RUT
dmRUT = [ 1 9 9 4
1997
1998
1999
2000
2002
2004
2006
2008
2009 % n o t m e a s u r e d
];
periodRUT_T2051 = [ 1 9 9 4 : 2 0 0 9 ] ' ;
s a v e ( ' periodRUT_T2051 ' , ' periodRUT_T2051 ' )
RUTi_T205= i n t e r p 1 ( dmRUT , mRUT , periodRUT_T2051 ) ;
s a v e ( ' RUTi_T205 ' , ' RUTi_T205 ' )
y = RUT_T2051 ;
x = T2051_Measure ;
T2051_period = periodInit_T2051 ( 1 ) : periodPropa_T2051 ( end ) ; % u p d a t e ←!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
xi = T2051_period ' ;
T2051_RUTperiod=T2051_period ' ;
RUTi_T2051 = i n t e r p 1 ( x , y , xi , ' l i n e a r ' ) ;
l o a d CiInit_T2051 . mat
l o a d CiPropa_T2051 . mat
Yinit_T2051 = CiInit_T2051 ;
Ypropa_T2051 = CiPropa_T2051 ;
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
l o a d N100CumSumPropa_T2051 . mat
l o a d N100CumSumInit_T2051 . mat
l o a d precep6190Init_T2051 . mat
l o a d precep6190Propa_T2051 . mat
l o a d temp6190Init_T2051 . mat
l o a d temp6190Propa_T2051 . mat
l o a d SCI_InitT2051 . mat
l o a d SCI_PropaT2051 . mat
l o a d MrInit_T2051 . mat
l o a d MrPropa_T2051 . mat
l o a d periodInit_T2051 . mat
l o a d periodPropa_T2051 . mat
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x1Init = SCI_InitT2051 ;
x1Propa = SCI_PropaT2051 ;
x2Init = MrInit_T2051 ;
x2Propa = MrPropa_T2051 ;
x3Init = N100CumSumInit_T2051 ;
93
E.1. DATA EXTRACTING CI
APPENDIX E. MATLAB CODE
x3Propa = N100CumSumPropa_T2051 ;
x4Init = precep6190Init_T2051 ;
x4Propa = precep6190Propa_T2051 ;
x5Init = temp6190Init_T2051 ;
x5Propa = temp6190Propa_T2051 ;
%%%% Age %%%%%%%%%%%%
ageInit_T2051 = [ 0 : l e n g t h ( periodInit_T2051 ) − 1 ] ' ;
startAgePropa = l e n g t h ( periodInit_T2051 ) ;
endAgePropa = ( l e n g t h ( periodInit_T2051 ) + l e n g t h ( periodPropa_T2051 ) ) ;
agePropa_T2051 = [ startAgePropa : ( endAgePropa−1) ] ' ;
lifeAgePeriod_T2051 = [ ( periodInit_T2051 ( 1 ) ) : ( periodPropa_T2051 ( end ) ) ] ' ;
lifeAgeYear_T2051 = [ 0 : ( endAgePropa−1) ] ' ;
save ( ' ageInit_T2051 ' , ' ageInit_T2051 ' )
save ( ' agePropa_T2051 ' , ' agePropa_T2051 ' )
save ( ' lifeAgePeriod_T2051 ' , ' lifeAgePeriod_T2051 ' )
save ( ' lifeAgeYear_T2051 ' , ' lifeAgeYear_T2051 ' )
x6Init = ageInit_T2051 ;
x6Propa = agePropa_T2051 ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Xinit_T2051 = [ x1Init x2Init x3Init x4Init x5Init x6Init ] ;
Xpropa_T2051 = [ x1Propa x2Propa x3Propa x4Propa x5Propa x6Propa ] ;
save (
save (
save (
save (
' Yinit_T2051 '
' Ypropa_T2051
' Xinit_T2051 '
' Xpropa_T2051
,
'
,
'
' Yinit_T2051 ' )
, ' Ypropa_T2051 ' )
' Xinit_T2051 ' )
, ' Xpropa_T2051 ' )
l o a d periodRUT_T2051
l o a d RUTi_T2051 . mat
Y = RUTi_T2051 ;
Yrut_T2051 = Y ;
s a v e ( ' Yrut_T2051 ' , ' Yrut_T2051 ' )
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
l o a d N100CumSumPropa_T2051 . mat
l o a d N100CumSumInit_T2051 . mat
l o a d precep6190Init_T2051 . mat
l o a d precep6190Propa_T2051 . mat
l o a d temp6190Init_T2051 . mat
l o a d temp6190Propa_T2051 . mat
l o a d SCI_InitT2051 . mat
l o a d SCI_PropaT2051 . mat
l o a d MrInit_T2051 . mat
l o a d MrPropa_T2051 . mat
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
l o a d periodInit_T2051 . mat
l o a d periodPropa_T2051 . mat
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x1Init = SCI_InitT2051 ;
x1Propa = SCI_PropaT2051 ;
x2Init = MrInit_T2051 ;
x2Propa = MrPropa_T2051 ;
x3Init = N100CumSumInit_T2051 ;
x3Propa = N100CumSumPropa_T2051 ;
x4Init = precep6190Init_T2051 ;
x4Propa = precep6190Propa_T2051 ;
x5Init = temp6190Init_T2051 ;
x5Propa = temp6190Propa_T2051 ;
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
94
E.2. CRACK INDEX INITIATION
x1
x2
x3
x4
x5
=
=
=
=
=
APPENDIX E. MATLAB CODE
[ x1Init ; x1Propa ] ;
[ x2Init ; x2Propa ] ;
l o g 1 0 ( [ x3Init ; x3Propa ] ) ; % t a k e l o g o f ! !
[ x4Init ; x4Propa ] ;
[ x5Init ; x5Propa ] ;
%% %% Age %%%%%%%%%%%%
ageInit_T2051 = [ 0 : l e n g t h ( periodInit_T2051 ) − 1 ] ' ;
startAgePropa = l e n g t h ( periodInit_T2051 ) ;
endAgePropa = ( l e n g t h ( periodInit_T2051 ) + l e n g t h ( periodPropa_T2051 ) ) ;
agePropa_T2051 = [ startAgePropa : ( endAgePropa−1) ] ' ;
lifeAgePeriod_T2051 = [ ( periodInit_T2051 ( 1 ) ) : ( periodPropa_T2051 ( end ) ) ] ' ;
lifeAgeYear_T2051 = [ 0 : ( endAgePropa−1) ] ' ;
save ( ' ageInit_T2051 ' , ' ageInit_T2051 ' )
save ( ' agePropa_T2051 ' , ' agePropa_T2051 ' )
save ( ' lifeAgePeriod_T2051 ' , ' lifeAgePeriod_T2051 ' )
save ( ' lifeAgeYear_T2051 ' , ' lifeAgeYear_T2051 ' )
x6Init = ageInit_T2051 ;
x6Propa = agePropa_T2051 ;
x6 = [ x6Init ; x6Propa ] ;
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
x0 = x1 . / x1 ;
X = [ x0 x1 x2 x3 x4 x5 x6 ] ;
Xrut_T2051 = X ;
s a v e ( ' Xrut_T2051 ' , ' Xrut_T2051 ' )
[ b , bint , r , rint , stats ] = regress ( Y , X ) ;
stats = regstats ( Y , X , ' l i n e a r ' )
Ymodel1 = X * b ;
Ymodel1 ( f i n d ( Ymodel1 < 0 ) ) = 0 ;
% c o e f ? c i e n t o f d e t e r m i n a t i o n R2
SSE=sum ( ( Y−Ymodel1 ) . ^ 2 ) ;
SST=sum ( ( Y−mean ( Y ) ) . ^ 2 ) ;
R2=(1 −(SSE / SST ) ) ;
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%
E.2
Crack Index Initiation
clc
close all
clear
f o r m a t long g
%%
l o a d Xinit_HRV341 . mat
l o a d Xinit_HRV402 . mat
l o a d Xinit_FRV311 . mat
l o a d Xinit_DRV532 . mat
l o a d Xinit_C2921 . mat
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
l o a d Yinit_HRV341 . mat % model1
l o a d Yinit_FRV311 . mat % model3
l o a d Yinit_HRV402 . mat % model2
l o a d Yinit_DRV532 . mat % model4
l o a d Yinit_C2921 . mat % model5
%% p u t i n o l d Ci
95
E.2. CRACK INDEX INITIATION
APPENDIX E. MATLAB CODE
Xinit_HRV341 = [ Xinit_HRV341 [ 0 ; Yinit_HRV341 ( 1 : ( end −1) )
Xinit_HRV402 = [ Xinit_HRV402 [ 0 ; Yinit_HRV402 ( 1 : ( end −1) )
Xinit_FRV311 = [ Xinit_FRV311 [ 0 ; Yinit_FRV311 ( 1 : ( end −1) )
Xinit_DRV532 = [ Xinit_DRV532 [ 0 ; Yinit_DRV532 ( 1 : ( end −1) )
Xinit_C2921 = [ Xinit_C2921 [ 0 ; Yinit_C2921 ( 1 : ( end −1) ) ] ] ;
%% c l e a n t h e d a t a
% Yinit_DRV532
f o r i = 2 : ( l e n g t h ( Yinit_DRV532 ) )
i f Yinit_DRV532 ( i−1) > Yinit_DRV532 ( i )
Yinit_DRV532 ( i ) = Yinit_DRV532 ( i−1) ;
end
end
]];
]];
]];
]];
% Yinit_HRV341
f o r i = 2 : ( l e n g t h ( Yinit_HRV341 ) )
i f Yinit_HRV341 ( i−1) > Yinit_HRV341 ( i )
Yinit_HRV341 ( i ) = Yinit_HRV341 ( i−1) ;
end
end
% Yinit_FRV311
f o r i = 2 : ( l e n g t h ( Yinit_FRV311 ) )
i f Yinit_FRV311 ( i−1) > Yinit_FRV311 ( i )
Yinit_FRV311 ( i ) = Yinit_FRV311 ( i−1) ;
end
end
% Yinit_HRV402
f o r i = 2 : ( l e n g t h ( Yinit_HRV402 ) )
i f Yinit_HRV402 ( i−1) > Yinit_HRV402 ( i )
Yinit_HRV402 ( i ) = Yinit_HRV402 ( i−1) ;
end
end
% Yinit_C2921
f o r i = 2 : ( l e n g t h ( Yinit_C2921 ) )
i f Yinit_C2921 ( i−1) > Yinit_C2921 ( i )
Yinit_C2921 ( i ) = Yinit_C2921 ( i−1) ;
end
end
%% n o r m a l i s e r a %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
minXinit_DRV532 = min ( min ( Xinit_DRV532 ) ) ;
maxXinit_DRV532 = max ( max ( Xinit_DRV532 ) ) ;
f o r i = 1 : s i z e ( Xinit_DRV532 , 1 )
f o r j = 1 : s i z e ( Xinit_DRV532 , 2 )
Xinit_DRV532 ( i , j ) = Xinit_DRV532 ( i , j ) / ( maxXinit_DRV532−minXinit_DRV532←);
end
end
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
minXinit_FRV311 = min ( min ( Xinit_FRV311 ) ) ;
maxXinit_FRV311 = max ( max ( Xinit_FRV311 ) ) ;
f o r i = 1 : s i z e ( Xinit_FRV311 , 1 )
f o r j = 1 : s i z e ( Xinit_FRV311 , 2 )
Xinit_FRV311 ( i , j ) = Xinit_FRV311 ( i , j ) / ( maxXinit_FRV311−minXinit_FRV311←);
end
end
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
minXinit_HRV341 = min ( min ( Xinit_HRV341 ) ) ;
maxXinit_HRV341 = max ( max ( Xinit_HRV341 ) ) ;
f o r i = 1 : s i z e ( Xinit_HRV341 , 1 )
f o r j = 1 : s i z e ( Xinit_HRV341 , 2 )
Xinit_HRV341 ( i , j ) = Xinit_HRV341 ( i , j ) / ( maxXinit_HRV341−minXinit_HRV341←);
end
96
E.2. CRACK INDEX INITIATION
APPENDIX E. MATLAB CODE
end
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
minXinit_HRV402 = min ( min ( Xinit_HRV402 ) ) ;
maxXinit_HRV402 = max ( max ( Xinit_HRV402 ) ) ;
f o r i = 1 : s i z e ( Xinit_HRV402 , 1 )
f o r j = 1 : s i z e ( Xinit_HRV402 , 2 )
Xinit_HRV402 ( i , j ) = Xinit_HRV402 ( i , j ) / ( maxXinit_HRV402−minXinit_HRV402←);
end
end
%%
minXinit_C2921 = min ( min ( Xinit_C2921 ) ) ;
maxXinit_C2921 = max ( max ( Xinit_C2921 ) ) ;
f o r i = 1 : s i z e ( Xinit_C2921 , 1 )
f o r j = 1 : s i z e ( Xinit_C2921 , 2 )
Xinit_C2921 ( i , j ) = Xinit_C2921 ( i , j ) / ( maxXinit_C2921−minXinit_C2921 ) ;
end
end
x2 = [ Xinit_HRV341 ; Xinit_HRV402 ; Xinit_FRV311 ; Xinit_DRV532 ; Xinit_C2921 ] ;
y = [ Yinit_HRV341 ; Yinit_HRV402 ; Yinit_FRV311 ; Yinit_DRV532 ; Yinit_C2921 ] ;
%%
%%
b02 = [ 1 1 1 1 1 1 1 1 ] ' ;%
OPTIONS = STATSET ( ' M a x I t e r ' , 1 0 0 0 0 , ' TolFun ' , 1 e−50 , ' D i s p l a y ' , ' i t e r ' ) ;
beta2 = nlinfit ( x2 , y , @modelCiI_2 , b02 , OPTIONS ) ;
betaI2 = beta2 ;
model_2 = modelCiI_2 ( betaI2 , x2 )
f o r i = 1 : l e n g t h ( model_2 )
i f model_2 ( i ) < 0
model_2 ( i ) = 0 ;
end
end
R2_A = corr ( model_2 , y ) ^2
%%
model1 = modelCiI_2 ( beta2 , Xinit_HRV341 ) ;
model2 = modelCiI_2 ( beta2 , Xinit_HRV402 ) ;
model3 = modelCiI_2 ( beta2 , Xinit_FRV311 ) ;
model4 = modelCiI_2 ( beta2 , Xinit_DRV532 ) ;
model5 = modelCiI_2 ( beta2 , Xinit_C2921 ) ;
f o r i = 1 : l e n g t h ( model1 )
i f model1 ( i ) < 0
model1 ( i ) = 0 ;
end
end
f o r i = 1 : l e n g t h ( model2 )
i f model2 ( i ) < 0
model2 ( i ) = 0 ;
end
end
f o r i = 1 : l e n g t h ( model3 )
i f model3 ( i ) < 0
model3 ( i ) = 0 ;
end
end
f o r i = 1 : l e n g t h ( model4 )
i f model4 ( i ) < 0
model4 ( i ) = 0 ;
end
end
f o r i = 1 : l e n g t h ( model5 )
i f model5 ( i ) < 0
97
E.2. CRACK INDEX INITIATION
APPENDIX E. MATLAB CODE
model5 ( i ) = 0 ;
end
end
%%
figure (1)
subplot (211)
p l o t ( Yinit_HRV341 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , . . .
' MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model1 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' ←,...
' k ' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' H−RV34−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
R2_1 = corr ( model1 , Yinit_HRV341 ) ^2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot (212)
p l o t ( Yinit_HRV402 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , . . .
' MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model2 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , . . .
' MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' H−RV40−2 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
R2_2 = corr ( model2 , Yinit_HRV402 ) ^2
figure (3)
subplot (211)
p l o t ( Yinit_FRV311 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model3 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' F−RV31−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
R2_3 = corr ( model3 , Yinit_FRV311 ) ^2
subplot (212)
p l o t ( Yinit_DRV532 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model4 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' D−RV53−2 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
R2_4 = corr ( model4 , Yinit_DRV532 ) ^2
figure (5)
subplot (211)
p l o t ( Yinit_C2921 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
98
E.3. CRACK INDEX PROPAGATION
APPENDIX E. MATLAB CODE
p l o t ( model5 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' C−292−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
R2_5 = corr ( model5 , Yinit_C2921 ) ^2
E.3
Crack Index Propagation
close all
clear
clc
f o r m a t short g
%%
l o a d Xpropa_DRV532 . mat
l o a d Xpropa_T2051 . mat
l o a d Xpropa_FRV311 . mat
l o a d Xpropa_HRV341 . mat
l o a d Xpropa_HRV402 . mat
l o a d Xpropa_C2921 . mat
%%
l o a d Ypropa_DRV532 . mat
l o a d Ypropa_HRV341 . mat
l o a d Ypropa_FRV311 . mat
l o a d Ypropa_HRV402 . mat
l o a d Ypropa_T2051 . mat
l o a d Ypropa_C2921 . mat
%%
l o a d Yinit_HRV341 . mat
l o a d Yinit_FRV311 . mat
l o a d Yinit_HRV402 . mat
l o a d Yinit_DRV532 . mat
l o a d Yinit_C2921 . mat
l o a d Yinit_T2051 . mat
%% X w i t h o l d Y
Xpropa_DRV532 = [ Xpropa_DRV532 [ Yinit_DRV532 ( end ) ; Ypropa_DRV532 ( 1 : end −1)←]];
Xpropa_T2051 = [ Xpropa_T2051 [ Yinit_T2051 ( end ) ; Ypropa_T2051 ( 1 : end −1) ] ] ;
Xpropa_FRV311 = [ Xpropa_FRV311 [ Yinit_FRV311 ( end ) ; Ypropa_FRV311 ( 1 : end −1)←]];
Xpropa_HRV341 = [ Xpropa_HRV341 [ Yinit_HRV341 ( end ) ; Ypropa_HRV341 ( 1 : end −1)←]];
Xpropa_HRV402 = [ Xpropa_HRV402 [ Yinit_HRV402 ( end ) ; Ypropa_HRV402 ( 1 : end −1)←]];
Xpropa_C2921 = [ Xpropa_C2921 [ Yinit_C2921 ( end ) ; Ypropa_C2921 ( 1 : end −1) ] ] ;
%% n o r m a l i s e r a med max i v a r j e o b j e k t
minXpropa_DRV532 = min ( min ( Xpropa_DRV532 ) ) ;
maxXpropa_DRV532 = max ( max ( Xpropa_DRV532 ) ) ;
f o r i = 1 : s i z e ( Xpropa_DRV532 , 1 )
f o r j = 1 : s i z e ( Xpropa_DRV532 , 2 )
Xpropa_DRV532 ( i , j ) = Xpropa_DRV532 ( i , j ) / ( maxXpropa_DRV532−←minXpropa_DRV532 ) ;
end
end
99
E.3. CRACK INDEX PROPAGATION
APPENDIX E. MATLAB CODE
%
%%
minXpropa_T2051 = min ( min ( Xpropa_T2051 ) ) ;
maxXpropa_T2051 = max ( max ( Xpropa_T2051 ) ) ;
f o r i = 1 : s i z e ( Xpropa_T2051 , 1 )
f o r j = 1 : s i z e ( Xpropa_T2051 , 2 )
Xpropa_T2051 ( i , j ) = Xpropa_T2051 ( i , j ) / ( maxXpropa_T2051−←minXpropa_T2051 ) ;
end
end
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
minXpropa_FRV311 = min ( min ( Xpropa_FRV311 ) ) ;
maxXpropa_FRV311 = max ( max ( Xpropa_FRV311 ) ) ;
f o r i = 1 : s i z e ( Xpropa_FRV311 , 1 )
f o r j = 1 : s i z e ( Xpropa_FRV311 , 2 )
Xpropa_FRV311 ( i , j ) = Xpropa_FRV311 ( i , j ) / ( maxXpropa_FRV311−←minXpropa_FRV311 ) ;
end
end
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
minXpropa_HRV341 = min ( min ( Xpropa_HRV341 ) ) ;
maxXpropa_HRV341 = max ( max ( Xpropa_HRV341 ) ) ;
f o r i = 1 : s i z e ( Xpropa_HRV341 , 1 )
f o r j = 1 : s i z e ( Xpropa_HRV341 , 2 )
Xpropa_HRV341 ( i , j ) = Xpropa_HRV341 ( i , j ) / ( maxXpropa_HRV341−←minXpropa_HRV341 ) ;
end
end
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
minXpropa_HRV402 = min ( min ( Xpropa_HRV402 ) ) ;
maxXpropa_HRV402 = max ( max ( Xpropa_HRV402 ) ) ;
f o r i = 1 : s i z e ( Xpropa_HRV402 , 1 )
f o r j = 1 : s i z e ( Xpropa_HRV402 , 2 )
Xpropa_HRV402 ( i , j ) = Xpropa_HRV402 ( i , j ) / ( maxXpropa_HRV402−←minXpropa_HRV402 ) ;
end
end
%%
minXpropa_C2921 = min ( min ( Xpropa_C2921 ) ) ;
maxXpropa_C2921 = max ( max ( Xpropa_C2921 ) ) ;
f o r i = 1 : s i z e ( Xpropa_C2921 , 1 )
f o r j = 1 : s i z e ( Xpropa_C2921 , 2 )
Xpropa_C2921 ( i , j ) = Xpropa_C2921 ( i , j ) / ( maxXpropa_C2921−←minXpropa_C2921 ) ;
end
end
%% c l e a n t h e d a t a
% Ypropa_DRV532
f o r i = 2 : ( l e n g t h ( Ypropa_DRV532 ) )
i f Ypropa_DRV532 ( i−1) > Ypropa_DRV532 ( i )
Ypropa_DRV532 ( i ) = Ypropa_DRV532 ( i−1) ;
end
end
100
E.3. CRACK INDEX PROPAGATION
APPENDIX E. MATLAB CODE
% Ypropa_HRV341
f o r i = 2 : ( l e n g t h ( Ypropa_HRV341 ) )
i f Ypropa_HRV341 ( i−1) > Ypropa_HRV341 ( i )
Ypropa_HRV341 ( i ) = Ypropa_HRV341 ( i−1) ;
end
end
% Ypropa_FRV311
f o r i = 2 : ( l e n g t h ( Ypropa_FRV311 ) )
i f Ypropa_FRV311 ( i−1) > Ypropa_FRV311 ( i )
Ypropa_FRV311 ( i ) = Ypropa_FRV311 ( i−1) ;
end
end
% Ypropa_HRV402
f o r i = 2 : ( l e n g t h ( Ypropa_HRV402 ) )
i f Ypropa_HRV402 ( i−1) > Ypropa_HRV402 ( i )
Ypropa_HRV402 ( i ) = Ypropa_HRV402 ( i−1) ;
end
end
% Ypropa_T2051
f o r i = 2 : ( l e n g t h ( Ypropa_T2051 ) )
i f Ypropa_T2051 ( i−1) > Ypropa_T2051 ( i )
Ypropa_T2051 ( i ) = Ypropa_T2051 ( i−1) ;
end
end
% Ypropa_C2921
f o r i = 2 : ( l e n g t h ( Ypropa_C2921 ) )
i f Ypropa_C2921 ( i−1) > Ypropa_C2921 ( i )
Ypropa_C2921 ( i ) = Ypropa_C2921 ( i−1) ;
end
end
%% s k a p a bigX och bigY
x = [ Xpropa_DRV532 ; Xpropa_T2051 ; Xpropa_FRV311 ; Xpropa_HRV341 ; Xpropa_HRV402←; Xpropa_C2921 ] ;
y = [ Ypropa_DRV532 ; Ypropa_T2051 ; Ypropa_FRV311 ; Ypropa_HRV341 ; Ypropa_HRV402←; Ypropa_C2921 ] ;
%%
b0 = [ 1 1 1 1 1 1 1 1 ] ' ; %
OPTIONS = STATSET ( ' M a x I t e r ' , 1 0 0 0 0 , ' TolFun ' , 1 e−50 , ' D i s p l a y ' , ' i t e r ' ) ;
beta2 = nlinfit ( x , y , @modelCiPropa_10 , b0 , OPTIONS ) ;
betaCP = beta2 ;
save ( ' betaCP ' , ' betaCP ' )
%%
model_2 = modelCiPropa_10 ( beta2 , x ) ;
f o r i = 1 : l e n g t h ( model_2 )
i f model_2 ( i ) < 5
model_2 ( i ) = 5 ;
end
end
R22 = corr ( model_2 , y ) ^2
%%
figure (1)
p l o t ( y , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k ' , ' ←MarkerSize ' ,4)
h o l d on
101
E.3. CRACK INDEX PROPAGATION
APPENDIX E. MATLAB CODE
% p l o t ( model1 , ' − rx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' r ' , ' M a r k e r F a c e C o l o r ←' , ' r ' , ' MarkerSize ' , 4 )
% h o l d on
p l o t ( model_2 , '−gx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' r ' , ' M a r k e r F a c e C o l o r ' , ←' r ' , ' MarkerSize ' ,4)
g r i d on
l e g e n d ( ' m e a s u r e d ' , ' model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' F o n t W e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' FontWeight ' , ' bold ' )
%% c l e a n i n p a r t 1
%
model1 = ( modelCiPropa_10 ( beta2 , Xpropa_DRV532 ) ) ;
f o r i = 1 : l e n g t h ( model1 )
i f model1 ( i ) < 5
model1 ( i ) = 5 ;
end
end
model2 = ( modelCiPropa_10 ( beta2 , Xpropa_T2051 ) ) ;
f o r i = 1 : l e n g t h ( model2 )
i f model2 ( i ) < 5
model2 ( i ) = 5 ;
end
end
model3 = ( modelCiPropa_10 ( beta2 , Xpropa_FRV311 ) ) ;
f o r i = 1 : l e n g t h ( model3 )
i f model3 ( i ) < 5
model3 ( i ) = 5 ;
end
end
model4 = ( modelCiPropa_10 ( beta2 , Xpropa_HRV341 ) ) ;
f o r i = 1 : l e n g t h ( model4 )
i f model4 ( i ) < 5
model4 ( i ) = 5 ;
end
end
model5 = ( modelCiPropa_10 ( beta2 , Xpropa_HRV402 ) ) ;
f o r i = 1 : l e n g t h ( model5 )
i f model5 ( i ) < 5
model5 ( i ) = 5 ;
end
end
model6 = ( modelCiPropa_10 ( beta2 , Xpropa_C2921 ) ) ;
f o r i = 1 : l e n g t h ( model6 )
i f model6 ( i ) < 5
model6 ( i ) = 5 ;
end
end
%% model c l e a n i n g p a r t 2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% model1 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model1 ) )
i f model1 ( i−1) > model1 ( i )
model1 ( i ) = model1 ( i−1) ;
end
end
% model2 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model2 ) )
i f model2 ( i−1) > model2 ( i )
model2 ( i ) = model2 ( i−1) ;
end
102
E.3. CRACK INDEX PROPAGATION
APPENDIX E. MATLAB CODE
end
% model3 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model3 ) )
i f model3 ( i−1) > model3 ( i )
model3 ( i ) = model3 ( i−1) ;
end
end
% model4 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model4 ) )
i f model4 ( i−1) > model4 ( i )
model4 ( i ) = model4 ( i−1) ;
end
end
% model5 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model5 ) )
i f model5 ( i−1) > model5 ( i )
model5 ( i ) = model5 ( i−1) ;
end
end
% model6 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model6 ) )
i f model6 ( i−1) > model6 ( i )
model6 ( i ) = model6 ( i−1) ;
end
end
%%
figure (2)
subplot (211)
p l o t ( Ypropa_DRV532 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model1 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' DRV532 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
t i t l e ( ' D−RV53−2 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr1 = corr ( Ypropa_DRV532 , model1 ) ^2
subplot (212)
p l o t ( Ypropa_T2051 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model2 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' T2051 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
t i t l e ( ' T−205−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr2 = corr ( Ypropa_T2051 , model2 ) ^2
h= g c f ;
saveas ( g c f , ' C : \ U s e r s \ MarkusS \ Dropbox \ l i c T \ _ l i c R e p o r t \ I m a g e s \ CiP_Sub1 . png ' ←)
figure (3)
subplot (211)
p l o t ( Ypropa_FRV311 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
103
E.3. CRACK INDEX PROPAGATION
APPENDIX E. MATLAB CODE
h o l d on
p l o t ( model3 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' FRV311 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
t i t l e ( ' F−RV31−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr3 = corr ( Ypropa_FRV311 , model3 ) ^2
subplot (212)
p l o t ( Ypropa_HRV341 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model4 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' HRV341 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
t i t l e ( ' H−RV34−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr4 = corr ( Ypropa_HRV341 , model4 ) ^2
h= g c f ;
saveas ( g c f , ' C : \ U s e r s \ MarkusS \ Dropbox \ l i c T \ _ l i c R e p o r t \ I m a g e s \ CiP_Sub2 . png ' ←)
figure (4)
subplot (211)
p l o t ( Ypropa_HRV402 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model5 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' HRV402 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
t i t l e ( ' H−RV40−2 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr5 = corr ( Ypropa_HRV402 , model5 ) ^2
subplot (212)
p l o t ( Ypropa_C2921 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model6 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' C2921 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
y l a b e l ( ' C_i [ −] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' year ' , ' FontSize ' ,24 , ' fontweight ' , ' bold ' )
t i t l e ( ' C−292−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr6 = corr ( Ypropa_C2921 , model6 ) ^2
h= g c f ;
saveas ( g c f , ' C : \ U s e r s \ MarkusS \ Dropbox \ l i c T \ _ l i c R e p o r t \ I m a g e s \ CiP_Sub3 . png ' ←)
%
%
R2A = [ corr1 corr2 corr3 corr4 corr5 corr6 ] ;
104
E.4. RUT
E.4
APPENDIX E. MATLAB CODE
Rut
clc
close all
clear
f o r m a t short g
%% l o a d i n d a t a
l o a d Xinit_HRV341 ;
l o a d Xinit_HRV402 ;
l o a d Xinit_FRV311 ;
l o a d Xinit_T2051 ;
l o a d Xinit_DRV532 . mat
l o a d Xinit_C2921 . mat
l o a d Xpropa_HRV341 ;
l o a d Xpropa_HRV402 ;
l o a d Xpropa_FRV311 ;
l o a d Xpropa_T2051 ;
l o a d Xpropa_DRV532 . mat
l o a d Xpropa_C2921 . mat
l o a d Yrut_DRV532 ;
l o a d Yrut_HRV341 ;
l o a d Yrut_T2051 ;
l o a d Yrut_FRV311 ;
l o a d Yrut_HRV402 . mat
%% c r e a t e f a c t o r p r e v i o u s v a l u e
Xrut_DRV532 = [ Xinit_DRV532 ; Xpropa_DRV532 ] ;
Xrut_HRV341 = [ Xinit_HRV341 ; Xpropa_HRV341 ] ;
Xrut_T2051 = [ Xinit_T2051 ; Xpropa_T2051 ] ;
Xrut_FRV311 = [ Xinit_FRV311 ; Xpropa_FRV311 ] ;
Xrut_HRV402 = [ Xinit_HRV402 ; Xpropa_HRV402 ] ;
Xrut_DRV532 = [ Xrut_DRV532 , [ 0 ; Yrut_DRV532 ( 1 : ( end −1) ) ] ] ;
Xrut_HRV341 = [ Xrut_HRV341 , [ 0 ; Yrut_HRV341 ( 1 : ( end −1) ) ] ] ;
Xrut_T2051 = [ Xrut_T2051 , [ 0 ; Yrut_T2051 ( 1 : ( end −1) ) ] ] ;
Xrut_FRV311 = [ Xrut_FRV311 , [ 0 ; Yrut_FRV311 ( 1 : ( end −1) ) ] ] ;
Xrut_HRV402 = [ Xrut_HRV402 , [ 0 ; Yrut_HRV402 ( 1 : ( end −1) ) ] ] ;
%% n o r m a l i z e MaxMin
minXrut_FRV311 = min ( min ( Xrut_FRV311 ) ) ;
maxXrut_FRV311 = max ( max ( Xrut_FRV311 ) ) ;
f o r i = 1 : s i z e ( Xrut_FRV311 , 1 )
f o r j = 1 : s i z e ( Xrut_FRV311 , 2 )
Xrut_FRV311 ( i , j ) = Xrut_FRV311 ( i , j ) / ( maxXrut_FRV311−minXrut_FRV311 ) ;
end
end
%
minXrut_HRV341 = min ( min ( Xrut_HRV341 ) ) ;
maxXrut_HRV341 = max ( max ( Xrut_HRV341 ) ) ;
f o r i = 1 : s i z e ( Xrut_HRV341 , 1 )
f o r j = 1 : s i z e ( Xrut_HRV341 , 2 )
Xrut_HRV341 ( i , j ) = Xrut_HRV341 ( i , j ) / ( maxXrut_HRV341−minXrut_HRV341 ) ;
end
end
minXrut_HRV402 = min ( min ( Xrut_HRV402 ) ) ;
maxXrut_HRV402 = max ( max ( Xrut_HRV402 ) ) ;
f o r i = 1 : s i z e ( Xrut_HRV402 , 1 )
f o r j = 1 : s i z e ( Xrut_HRV402 , 2 )
Xrut_HRV402 ( i , j ) = Xrut_HRV402 ( i , j ) / ( maxXrut_HRV402−minXrut_HRV402 ) ;
end
end
105
E.4. RUT
APPENDIX E. MATLAB CODE
%% c l e a n t h e d a t a
% Yinit_DRV532
f o r i = 2 : ( l e n g t h ( Yrut_DRV532 ) )
i f Yrut_DRV532 ( i−1) > Yrut_DRV532 ( i )
Yrut_DRV532 ( i ) = Yrut_DRV532 ( i−1) ;
end
end
% Yrut_HRV341
f o r i = 2 : ( l e n g t h ( Yrut_HRV341 ) )
i f Yrut_HRV341 ( i−1) > Yrut_HRV341 ( i )
Yrut_HRV341 ( i ) = Yrut_HRV341 ( i−1) ;
end
end
% Yrut_FRV311
f o r i = 2 : ( l e n g t h ( Yrut_FRV311 ) )
i f Yrut_FRV311 ( i−1) > Yrut_FRV311 ( i )
Yrut_FRV311 ( i ) = Yrut_FRV311 ( i−1) ;
end
end
% Yrut_HRV402
f o r i = 2 : ( l e n g t h ( Yrut_HRV402 ) )
i f Yrut_HRV402 ( i−1) > Yrut_HRV402 ( i )
Yrut_HRV402 ( i ) = Yrut_HRV402 ( i−1) ;
end
end
%% c r e a t e x and y
x = [ Xrut_DRV532 ; Xrut_HRV341 ; Xrut_T2051 ; Xrut_FRV311 ; Xrut_HRV402 ] ;
y = [ Yrut_DRV532 ; Yrut_HRV341 ; Yrut_T2051 ; Yrut_FRV311 ; Yrut_HRV402 ] ;
%% c r e a t e model n l i n f i t
b0 = [ 1 1 1 1 1 1 1 1 ] ;
OPTIONS = STATSET ( ' M a x I t e r ' , 1 0 0 0 0 , ' TolFun ' , 1 e−50 , ' D i s p l a y ' , ' i t e r ' ) ;
b e t a = nlinfit ( x , y , @modelrut_02 , b0 , OPTIONS ) ;
model = modelrut_02 ( b e t a , x ) ;
save ( ' beta ' , ' beta ' )
%% u s e b e t a and model
model1 = ( modelrut_02 ( b e t a , Xrut_DRV532 ) ) ;
f o r i = 1 : l e n g t h ( model1 )
i f model1 ( i ) < 0
model1 ( i ) = 0 ;
end
end
model2 = ( modelrut_02 ( b e t a , Xrut_HRV341 ) ) ;
f o r i = 1 : l e n g t h ( model2 )
i f model2 ( i ) < 0
model2 ( i ) = 0 ;
end
end
model3 = ( modelrut_02 ( b e t a , Xrut_T2051 ) ) ;
f o r i = 1 : l e n g t h ( model3 )
i f model3 ( i ) < 0
model3 ( i ) = 0 ;
end
end
model4 = ( modelrut_02 ( b e t a , Xrut_FRV311 ) ) ;
f o r i = 1 : l e n g t h ( model4 )
i f model4 ( i ) < 0
model4 ( i ) = 0 ;
end
end
model5 = ( modelrut_02 ( b e t a , Xrut_HRV402 ) ) ;
f o r i = 1 : l e n g t h ( model5 )
106
E.4. RUT
APPENDIX E. MATLAB CODE
i f model5 ( i ) < 0
model5 ( i ) = 0 ;
end
end
%% c l e a n t h e m o d e l s
f o r i = 2 : ( l e n g t h ( model1 ) )
i f model1 ( i−1) > model1 ( i )
model1 ( i ) = model1 ( i−1) ;
end
end
% model2 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model2 ) )
i f model2 ( i−1) > model2 ( i )
model2 ( i ) = model2 ( i−1) ;
end
end
% model3 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model3 ) )
i f model3 ( i−1) > model3 ( i )
model3 ( i ) = model3 ( i−1) ;
end
end
% model4 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model4 ) )
i f model4 ( i−1) > model4 ( i )
model4 ( i ) = model4 ( i−1) ;
end
end
% model5 ( 1 ) = 5 ;
f o r i = 2 : ( l e n g t h ( model5 ) )
i f model5 ( i−1) > model5 ( i )
model5 ( i ) = model5 ( i−1) ;
end
end
%% p l o t r e s u l t
figure (2)
subplot (211)
p l o t ( Yrut_DRV532 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
y l a b e l ( ' r u t [mm] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' age ' , ' F o n t S i z e ' ,24 , ' f o n t w e i g h t ' , ' bold ' )
p l o t ( model1 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
l e g e n d ( ' m e a s u r e d ' , ' DRV532 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' D−RV53−2 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr1 = ( corr ( Yrut_DRV532 , model1 ) ) ^ 2 ;
subplot (212)
p l o t ( Yrut_HRV341 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model2 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
y l a b e l ( ' r u t [mm] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' age ' , ' F o n t S i z e ' ,24 , ' f o n t w e i g h t ' , ' bold ' )
l e g e n d ( ' m e a s u r e d ' , ' HRV341 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' H−RV34−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr2 = ( corr ( Yrut_HRV341 , model2 ) ) ^ 2 ;
107
E.4. RUT
APPENDIX E. MATLAB CODE
h= g c f ;
%s a v e a s ( g c f , ' C : \ U s e r s \ MarkusS \ Dropbox \ l i c T \ _ l i c R e p o r t \ I m a g e s \ r u t _ S u b 1 . png ←')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure (3)
subplot (211)
p l o t ( Yrut_T2051 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ←' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model3 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
y l a b e l ( ' r u t [mm] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' age ' , ' F o n t S i z e ' ,24 , ' f o n t w e i g h t ' , ' bold ' )
l e g e n d ( ' m e a s u r e d ' , ' T2051 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' T−205−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr3 = ( corr ( Yrut_T2051 , model3 ) ) ^ 2 ;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
subplot (212)
p l o t ( Yrut_FRV311 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model4 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
y l a b e l ( ' r u t [mm] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' age ' , ' F o n t S i z e ' ,24 , ' f o n t w e i g h t ' , ' bold ' )
l e g e n d ( ' m e a s u r e d ' , ' FRV311 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' F−RV31−1 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr4 = ( corr ( Yrut_FRV311 , model4 ) ) ^ 2 ;
h= g c f ;
%s a v e a s ( g c f , ' C : \ U s e r s \ MarkusS \ Dropbox \ l i c T \ _ l i c R e p o r t \ I m a g e s \ r u t _ S u b 2 . png ←')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure (4)
subplot (211)
p l o t ( Yrut_HRV402 , ' kx ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' ←MarkerFaceColor ' , ' k ' , ' MarkerSize ' ,4)
g r i d on
h o l d on
p l o t ( model5 , '−r ' , ' l i n e w i d t h ' , 2 , ' M a r k e r E d g e C o l o r ' , ' k ' , ' M a r k e r F a c e C o l o r ' , ' k←' , ' MarkerSize ' ,4)
y l a b e l ( ' r u t [mm] ' , ' F o n t S i z e ' , 2 4 , ' f o n t w e i g h t ' , ' b o l d ' )
x l a b e l ( ' age ' , ' F o n t S i z e ' ,24 , ' f o n t w e i g h t ' , ' bold ' )
l e g e n d ( ' m e a s u r e d ' , ' HRV402 model ' , ' L o c a t i o n ' , ' N o r t h W e s t ' )
t i t l e ( ' H−RV40−2 ' , ' F o n t S i z e ' , 1 5 , ' f o n t w e i g h t ' , ' b o l d ' )
corr5 = ( corr ( Yrut_HRV402 , model5 ) ) ^ 2 ;
h= g c f ;
saveas ( g c f , ' C : \ U s e r s \ MarkusS \ Dropbox \ l i c T \ _ l i c R e p o r t \ I m a g e s \ r u t _ S u b 3 . png ' ←)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
108
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement