THE CHARACTERIZATION OF A PLAIN CONCRETE EQUIVALENT ELASTIC
FATIGUE CRACK RESISTANCE CURVE UNDER VARIOUS LOADING REGIMES
By
Nicholas Andres Brake

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Civil Engineering
2012

ABSTRACT
THE CHARACTERIZATION OF A PLAIN CONCRETE EQUIVALENT ELASTIC
FATIGUE CRACK RESISTANCE CURVE UNDER VARIOUS LOADING REGIMES
By
Nicholas Andres Brake
Concrete is a quasi-brittle material that exhibits a large residual bridging stress zone in the wake
of a dominant crack tip. These stresses often influence the ‘size effect’ observed in standard
strength tests. In fatigue, they often cause a ‘load history’ effect that alters the propagation rate
by mitigating the stress intensities located at the crack tip. This mitigation often leads to the
formation of two distinct crack rate regions known as the deceleration and acceleration region,
respectively.
The cyclically induced residual bridging stresses also influence the ‘size effect’ that is
manifested through the logC in the Paris Law. In this study, it is shown that with the use of size
dependent equivalent cyclic crack resistance curve, one may obtain a unique and sizeindependent set of Paris parameters.
A total of 48 three point bending single-edge notch specimens were tested. Two different
sizes were studied under both quasi-static and fatigue loading. The fatigue tests were conducted
using three different loading regimes: constant, variable, and random amplitude loading.
Under quasi-static loading, a new method to determine an equivalent cyclic crack
resistance curve is proposed. It is a hybrid experimental technique driven by two governing beam
equilibrium conditions and a ‘corrected’ crack length criterion. The proposed technique backcalculates 4 bridging parameters that govern an assumed exponential stress distribution. A
weight function was then used to determine the equivalent resistance curve as a function of crack

extension and applied load. The behavior of the cyclic equivalent resistance curve was then
parameterized. It is then concluded that the back-calculated bridging stress distribution could be
used to determine the capacity of the structure with a moment equilibrium condition and a
resistance that could be used for fatigue loading scenarios.
Under constant amplitude fatigue loading, it is shown that the equivalent cyclic crack
resistance curve is directly related to the crack propagation rate and can be obtained if the
following two conditions are satisfied: i) the crack resistance starts at zero, and ii) the post-peak
slope is defined. It is then shown that if these conditions are satisfied, a unique 3 parameter
equivalent resistance curve is obtainable using only experimental crack rate and stress intensity
data.
Fatigue tests were then carried out under constant, variable, and random amplitude
loading. The results suggest that the proposed functional form of the equivalent crack resistance
curve under quasi-static loading is adequate in describing the equivalent fatigue resistance under
the three fatigue loading regimes. In addition, it is also shown that if a size dependent fatigue
resistance curve is inserted into the Paris law, logC and n become unique.
Finally, the fatigue damage under variable and random amplitude loading is simulated
using the average values for the larger size specimens. The simulated fatigue fracture prediction
is compared to the prediction using a linear damage rule (LDR). The error is shown in terms of
number of cycles to failure, N, and depends on the loading sequence. The adequacy of the LDR
is assessed under a random concrete pavement stress distribution and is shown to over-predict
damage by nearly 30 % if the LDR is calibrated with constant amplitude loading tests.

Copyright by
Nicholas Andres Brake
2012

ACKNOWLEDGMENTS
I would like to acknowledge that if it were not for the love, enthusiasm, guidance, and support
from my family, adviser, and closest friends, I would have never completed this work. I would
like to extend a personal thanks to my parents, Mariana Alvarez Brake and John Hugh Brake for
their inspiration and dedication to my education. Without them, this would not be possible. I
would like to thank my brothers Jake and Patrick Brake for their charisma and joviality. I would
like to thank my adviser Dr. Karim Chatti for his invaluable knowledge and experience. Without
him, this would not be possible. I would also like to thank Dr. Nizar Lajnef for allowing me to
use his groups’ COD gage and MTS machine. Obviously, without his generosity, this would
have never been possible. I would also like to thank Dr.’s Buch and Burgueno for giving me the
opportunity to be the lead instructor of CE 305 in spring of 2012. I would also like to
acknowledge the help of Siavosh, the lab manager at the civil infrastructure lab. Without his
knowledge and guidance of concrete experimental methods, this would not have been possible.
Finally, I would like to thank my girlfriend and future wife, Lindsay Dubbs. Without her, this
would not be possible. Once again, thank you all.

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TABLE OF CONTENTS

CHAPTER I .................................................................................................................................... 1
INTRODUCTION .......................................................................................................................... 1
Background ................................................................................................................................. 1
Problem Identification ................................................................................................................ 2
Research Objectives .................................................................................................................... 4
Chapter Overview ....................................................................................................................... 5
CHAPTER II................................................................................................................................... 8
LITERATURE REVIEW ............................................................................................................... 8
Classical pavement stress solutions ............................................................................................ 8
Mechanistic-Empirical Design ................................................................................................. 12
Rapid Stress Calculations ......................................................................................................... 21
Linear Damage Accumulation .................................................................................................. 22
Classic Linear Damage Rule..................................................................................................... 25
Fatigue Damage ........................................................................................................................ 26
Linear Elastic Fracture Mechanics ........................................................................................... 28
Fracture Mechanics in Concrete ............................................................................................... 30
Crack Resistance Curves .......................................................................................................... 31
Weight Function ....................................................................................................................... 34
Bridging Stresses ...................................................................................................................... 36
Concrete Fatigue Fracture ......................................................................................................... 38
CHAPTER III ............................................................................................................................... 44
METHODS AND MATERIALS .................................................................................................. 44
Testing Equipment .................................................................................................................... 44
COD Gage Attachment ............................................................................................................. 44
Equipment Accessories ............................................................................................................. 45
Concrete Mix Properties ........................................................................................................... 46
Specimen Geometry .................................................................................................................. 47
Loading Regimes ...................................................................................................................... 47
Crack length measurements ...................................................................................................... 48
Equivalent Elastic Crack ........................................................................................................... 49
Jenq-Shah Method .................................................................................................................... 49
S
Specific crack resistance (KR ) curve ....................................................................................... 51
CHAPTER IV ............................................................................................................................... 55
A METHOD TO DETERMINE AN EQUIVALENT CYCLIC CRACK RESISTANCE CURVE
UNDER QUASI-STATIC LOADING ......................................................................................... 55
Load and Equivalent Crack Data .............................................................................................. 59
CMOD condition ...................................................................................................................... 62
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Force and Moment Equilibrium ................................................................................................ 65
S
KR and bridging stress parameters ........................................................................................... 71
ini

Comparison of KI and KR curves with inclusion of KIC ...................................................... 77
Determination of the equivalent crack resistance curve ........................................................... 80
Application to fatigue loading .................................................................................................. 83
Results ....................................................................................................................................... 88
CHAPTER V ................................................................................................................................ 94
THE DETERMINATION OF AN EQUIVALENT CYCLIC CRACK RESISTANCE CURVE
UNDER CONSTANT AMPLITUDE LOADING ....................................................................... 94
Experimental test setup and mix characteristics ....................................................................... 95
Crack length measurements ...................................................................................................... 96
Fatigue crack resistance curves................................................................................................. 97
Experimental Results .............................................................................................................. 103
S
Comparison between quasi-static specific KR curve and cyclic resistance curve ................. 104
Comparison to previously published results ........................................................................... 108
Fatigue prediction using the cyclic resistance curves ............................................................. 110
Discussion ............................................................................................................................... 112
SIZE DEPENDENT FATIGUE CRACK RESISTANCE CURVES UNDER VARIOUS
LOADING REGIMES ................................................................................................................ 114
Variable load effects ............................................................................................................... 114
Size effect in fatigue ............................................................................................................... 118
Calibration using the cumulative crack propagation .............................................................. 118
Determining the fatigue parameters ........................................................................................ 120
Results ..................................................................................................................................... 122
Description of proposed model ............................................................................................... 137
Summary ................................................................................................................................. 138
CHAPTER VII ............................................................................................................................ 140
THE EVALUATION OF THE LINEAR DAMAGE RULE UNDER RANDOM PAVEMENT
EDGE STRESSES ...................................................................................................................... 140
S-N Calibration ....................................................................................................................... 146
MEPDG and Monte Carlo Simulation .................................................................................... 148
Monte Carlo Simulation.......................................................................................................... 149
Truck Traffic Randomization ................................................................................................. 149
Stress-time history .................................................................................................................. 153
Rapid Stress Interpolation scheme .......................................................................................... 160
Interpolation Databases........................................................................................................... 164
Pavement edge stress simulation ............................................................................................ 167
Damage Prediction .................................................................................................................. 169
Calibration of Percent Slabs Cracked ..................................................................................... 171
CHAPTER VIII .......................................................................................................................... 175
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CONCLUSION ........................................................................................................................... 175
Summary of findings .............................................................................................................. 175
Recommendations for future research .................................................................................... 181
APPENDICES ............................................................................................................................ 183
APPENDIX A: Quasi-static Loading Test Results................................................................. 184
APPENDIX B: Fatigue Loading Test Results ........................................................................ 200
APPENDIX C: Pavement Edge Stress Results....................................................................... 236
BIBLIOGRAPHY ....................................................................................................................... 245

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LIST OF TABLES
Table 1: Different truck and axle Configurations ......................................................................... 13
Table 2: Experimental Program .................................................................................................... 48
S

Table 3: KR curve and bridging stress parameters ..................................................................... 75
Table 4: Fracture characteristics under quasi-static loading ......................................................... 76
Table 5: Initial fracture toughness using acorr for 100 mm specimens under quasi-static loading
....................................................................................................................................................... 79
Table 6: Equivalent crack resistance curve parameters ................................................................ 89
Table 7: Summary of constant amplitude fatigue results............................................................ 104
Table 8: Summary of constant amplitude fatigue results using data from Subramaniam et al.
(2000) .......................................................................................................................................... 109
Table 9: Summary of random, variable, and constant amplitude fatigue results using data for the
large specimens, 100 mm ............................................................................................................ 123
Table 10: Summary of random, variable, and constant amplitude fatigue results using data for the
small specimens, 50 mm ............................................................................................................. 124
Table 11: Summary of random, variable, and constant amplitude fatigue results using data for the
large specimens, 100 mm-only transient/acceleration stage ....................................................... 125
Table 12: Summary of random, variable, and constant amplitude fatigue results using data for the
small specimens, 50 mm-only transient/acceleration stage ........................................................ 126
Table 13: p-values generated for the Paris parameters log C and n and the slope variant
coefficient, only transient/acceleration stage .............................................................................. 134
Table 14: p-values generated for the Paris parameters log C and n, aeq,u, m, and 2 ............. 134

x

Table 15: Average values for the equivalent fatigue crack resistance curve and Paris parametersonly transient/acceleration stage ................................................................................................. 136
Table 16: Average values for the equivalent fatigue crack resistance curve and Paris parameters
..................................................................................................................................................... 136
Table 17: Summary of random fatigue crack parameters using a zero slope post peak ............. 143
Table 18: Quasi-static specimen geometry (mm) ....................................................................... 199
Table 19: Beam geometry/properties and number of cycles to failure ...................................... 235

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LIST OF FIGURES
Figure 1: Critical stress locations (for interpretation of the references to color in this and all other
figures, the reader is referred to the electronic version of this dissertation) ................................... 9
Figure 2: (a) positive thermal gradient and (b) negative thermal gradient ................................... 12
Figure 3: Induced stresses at the mid-slab edge under various axle types.................................... 14
Figure 4: Concrete pavement design architecture ......................................................................... 23
Figure 5: S-N Curve ...................................................................................................................... 25
Figure 6: Mode I Fracture ............................................................................................................. 29
Figure 7: Relationship between strength and size......................................................................... 31
Figure 8: Crack resistance curve for two different size concrete specimens ................................ 33
Figure 9: Common bridging stress distributions used to predict concrete fracture ...................... 37
Figure 10: (a) Transient Stage Crack Propagation, (b) Steady-state Crack Propagation.............. 40
Figure 11: COD Gage ................................................................................................................... 45
Figure 12: MTS machine .............................................................................................................. 46
Figure 13: Load v. Crack Mouth Opening Displacement ............................................................. 49
Figure 14: KR and KI curves ........................................................................................................ 53
Figure 15: (a) P-CMOD curve and (b) P-crack curve................................................................... 54
Figure 16: Comparison of compliance techniques: Linear asymptotic assumption (secant) and
load-unload (tangent) .................................................................................................................... 57
Figure 17: Crack opening displacement fields for a crack that is traction-less and one with
tractions ......................................................................................................................................... 58
Figure 18: Construction of P-aeq curve ...................................................................................... 61
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Figure 19: Point on the P-aeq curve ............................................................................................ 65
Figure 20: Bending and bridging stresses at mid-span ................................................................. 67
Figure 21: External and internal moment equilibrium .................................................................. 70
Figure 22: Equivalent and corrected crack lengths ....................................................................... 70
Figure 23: KI v. KR curve ............................................................................................................ 78
Figure 24: Bridging stresses acting over the equivalent crack domain ........................................ 82
Figure 25: Equivalent and corrected crack resistance curve ......................................................... 83
Figure 26: Equivalent crack resistance under varying loads ........................................................ 85
Figure 27: Change in the slope of the post-peak crack resistance curve as a function of load ..... 88
Figure 28: Relationship between total fracture toughness and equivalent fracture toughness ..... 90
Figure 29: Relationship between the equivalent critical crack extension and the critical crack
opening .......................................................................................................................................... 91
Figure 30: Relationship between the initial post peak slope and the ratio between the tensile
strength and specimen size............................................................................................................ 92
Figure 31: Relationship between the applied load-crack resistance slope sensitivity and nondimensional size ............................................................................................................................ 93
Figure 32: Smoothened crack rate data versus crack length ......................................................... 97
f

Figure 33: Satisfying condition 3: dKR /daeq is zero beyond aeq,c ......................................... 101
f

Figure 34: KR shown as a function of aeq,c............................................................................. 102
s

f

Figure 35: KR and KR resistance curves and KI ...................................................................... 107
Figure 36: Simulation of crack rate under stress ratios 0.85 and 0.95 ........................................ 111
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Figure 37: Variation of log C1 and n1 as a function of stress ratio ............................................. 113
Figure 38: Differences between transition zone for (a) random loading and (b) constant
amplitude loading........................................................................................................................ 117
Figure 39: Results for specimen P22: Larger beam, 100 mm depth, (a) equivalent crack rate v.
crack extension, and (b) equivalent crack extension v. Number of Cycles ................................ 128
Figure 40: Results for specimen B05: Larger beam, 100 mm depth, (a) applied load histogram,
(b) equivalent crack rate v. crack extension (c) equivalent crack extension v. Number of Cycles
..................................................................................................................................................... 130
Figure 41: Random load distribution: (a) uniform distribution, specimen P07 (b) normal
distribution, specimen S24 .......................................................................................................... 132
Figure 42: (a) Large, Crack extension and number of cycles, (b) Large, Crack rate and crack
extension, (c) Small, Crack extension and number of cycles, (b) Small, Crack rate and crack
extension ..................................................................................................................................... 144
Figure 43: Calibration of S-N curve against MEPDG ................................................................ 147
Figure 44: Pavement cross-section used in the simulation ......................................................... 148
Figure 45: Axle load placement for a class 11 truck that has 5 single axles .............................. 156
Figure 46: Truck traffic and thermal loading induced stresses ................................................... 157
Figure 47: Inter- and Intra-axle loads for a tandem axle ............................................................ 158
Figure 48: Procedure for calculating the pavement edge stress .................................................. 159
Figure 49: Comparison of stress output from EVERFE and the interpolation scheme .............. 167
Figure 50: (a) Bi-variate histogram of peak stress and stress range, (b) histogram of stress ratios,
(c) histogram of R – ratios .......................................................................................................... 168
Figure 51: (a) Histogram of modulus of subgrade reaction, k for Miami and Lansing, (b)
Histogram of Equivalent Linear Temperature Gradients for Miami and Lansing Simulations . 171

xiv

Figure 52: (a) Lansing: linear and non-linear damage, (b) Lansing: % slabs cracked, linear and
non-linear damage (c) Miami: linear and non-linear damage, (d) Miami: % slabs cracked, linear
and non-linear damage ................................................................................................................ 173
S

Figure 53: Specimen L1: (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 185
S

Figure 54: Specimen L2: (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 186
S

Figure 55: Specimen L3 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 187
S

Figure 56: Specimen L4 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 188
S

Figure 57: Specimen L5 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 189
S

Figure 58: Specimen L6 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 190
S

Figure 59: Specimen S1 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 191
S

Figure 60: Specimen S2 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 192
S

Figure 61: Specimen S3 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 193
S

Figure 62: Specimen S4 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 194
S

Figure 63: Specimen S5 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 195

xv

S

Figure 64: Specimen S6 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 196
S

Figure 65: Specimen S7 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 197
S

Figure 66: Specimen S8 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and
corrected crack lengths ............................................................................................................... 198
Figure 67: Specimen P07 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 201
Figure 68: Specimen P09 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 202
Figure 69: Specimen P11 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 203
Figure 70: Specimen B02 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 204
Figure 71: Specimen B04 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 205
Figure 72: Specimen B05 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 206
Figure 73: Specimen B06 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 207
Figure 74: Specimen B10 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 208
Figure 75: Specimen P02 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 209
Figure 76: Specimen P08 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 210
Figure 77: Specimen P15 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 211
xvi

Figure 78: Specimen P17 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 212
Figure 79: Specimen B01 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 213
Figure 80: Specimen B08 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 214
Figure 81: Specimen P10 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 215
Figure 82: Specimen P19 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 216
Figure 83: Specimen P22 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 217
Figure 84: Specimen S21 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 218
Figure 85: Specimen S24 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 219
Figure 86: Specimen S25 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 220
Figure 87: Specimen S04 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 221
Figure 88: Specimen S10 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 222
Figure 89: Specimen S11 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 223
Figure 90: Specimen S12 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 224
Figure 91: Specimen S13 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 225
xvii

Figure 92: Specimen S14 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 226
Figure 93: Specimen S15 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 227
Figure 94: Specimen S16 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 228
Figure 95: Specimen S17 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 229
Figure 96: Specimen S18 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 230
Figure 97: Specimen S01 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 231
Figure 98: Specimen S03 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 232
Figure 99: Specimen S05 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 233
Figure 100: Specimen S08 (a) Load History, (b) Experimental and model eq. crack rate v. eq.
crack extension, (c) Experimental and model eq. crack extension v. N ..................................... 234
Figure 101: Stress Interpolation for (a) single axle, positive temperature gradient, (b) single axle,
negative thermal gradient, (c) tandem axle (front axle at mid-slab), positive thermal gradient (d)
tandem axle, (front and back axle equidistant from mid-slab), positive thermal gradient ......... 237
Figure 102: Stress Interpolation for (a) tandem axle (front axle at mid-slab), negative thermal
gradient, (b) tandem axle, (front and back axle equidistant from mid-slab), negative thermal
gradient (c) tridem axle, (middle axle at mid-slab), positive thermal gradient, (d) tridem axle,
(front and second axle equidistant from mid-slab), positive thermal gradient ........................... 238
Figure 103: Pavement Edge Stress Results for Lansing, MI (a) peak stress and stress range
histogram..................................................................................................................................... 239
Figure 104: Pavement Edge Stress Results for Miami, FL (a) peak stress and stress range
histogram..................................................................................................................................... 242
xviii

CHAPTER I

INTRODUCTION

This dissertation focuses on the characterization of crack wake bridging stresses under different
forms of loading in plain Portland Cement Concrete (PCC). The bridging stresses are interpreted
through a cyclic crack resistance curve. A large emphasis is placed on defining the cyclic crack
resistance curve under fatigue, developing new methods to obtain it, and comparing it to quasistatic behavior. Since quasi-static fracture tests are significantly easier and less time consuming
to conduct, a large emphasis of this research is placed on comparing fatigue and quasi-static
results. The study is then concluded by highlighting the advantages and disadvantages of
predicting transverse fatigue cracking in concrete pavements using a linear damage rule (LDR).
Background
Transverse fatigue cracking is a common concrete pavement distress and is triggered by a
combination of truck traffic, thermal loads, pre-existing flaws, and built-in moisture and thermal
gradients. Moreover, the fatigue damage process is complex because the material is
heterogeneous and suffers from a variety of pre-loading flaws such as air voids and shrinkage
cracks. The current design method for transverse cracking is a mechanistic-empirical approach
that uses a calibrated, field-inspired, LDR to predict damage. The LDR is calibrated based on the
percentage of slabs cracked over a given stretch of roadway. It is argued that a mechanisticempirical approach is necessary due to the myriad of unknown, stochastic processes that
influence pavement performance. The argument is that many of the unknowns can be
1

circumvented by this ‘brute force’ approach and produce a reliable design. The disadvantage
however, is that it relies on regional calibration factors that limits the broadness of the model.
The LDR used in the design method is founded on Miner’s hypothesis and relates stress
ratio to the number of cycles to failure. It is not sensitive to load sequence effects and will incur
some prediction error under variable and random amplitude loading scenarios. Since pavement
stresses depend on a collection of random processes, the stresses tend to follow a random
distribution and thus, the current prediction model should incur some error. This error caused by
load sequence effects has yet to be explained.
The current design method can be strengthened if the damage mechanism is described in
terms of fracture mechanics, which can account for load history effects, R-ratio, and stress ratio
using a modified Paris law. The load sequence effect can be assessed by inserting a pavement
random stress sequence into a modified Paris law and comparing it to a LDR output over the
same sequence. The difference of the results can be used as a design reliability measure and used
to explain a portion of the total prediction error observed at the distress level (percent slabs
cracked).
Problem Identification
The investigation of the load history effects shown in this dissertation was carried out with the
use of a stress ratio based LDR and a modified Paris fatigue fracture model. There is a
considerably large library of linear elastic fracture mechanics fatigue models for metallic
materials. For concrete materials however, the library is quite small. Although there has been a
considerable amount of research dedicated to concrete fatigue, most explain the phenomenon
within the context Miner’s hypothesis (or an LDR).
2

Very little research in fatigue has been conducted using fracture mechanics and for the
studies that do exist, they are limited to simple Paris laws. The major limitation of the Paris law
is that it cannot account for the residual bridging stresses along the wake of a dominant crack.
These stresses strongly influence the fatigue life and are one of the variables that create the peak
stress ‘size effect’. The effect of these stresses in fatigue is manifested by two distinct crack rate
stages: a deceleration stage and an acceleration stage. Many of the fatigue studies neglect the
deceleration stage altogether because the Paris law alone cannot explain this stage.
The residual stresses may be influenced by the type of concrete, nominal maximum
aggregate size, water to cement ratio, boundary conditions, and curing conditions. It is also
influenced by a crack closure effect that is governed by an inelastic zone ahead of the crack tip.
In concrete, the behavior above the crack tip will dictate the R-ratio effect and control the
permanent deformation along the crack wake, which in turn influences the crack shielding.
Before assessing the load history effect in a pavement system, a fatigue fracture model
that can account for residual stresses in the crack wake and front (FPZ) needs to be developed.
This means that it is necessary to determine the evolution of these residual stresses under
different loading regimes and in different structural sizes. It is also important to understand the
similarities of these residual bridging stresses between quasi-static and fatigue loading. Quasistatic tests can be executed more rapidly than high-cycle fatigue tests and thus it would be useful
to identify common characteristics. If they are similar, a quasi-static test can be used in lieu of
the high cycle fatigue test.

3

Research Objectives
This research has two major objectives: 1) quantify the bridging residual stresses under quasistatic loading and determine an equivalent crack resistance curve. Next, assess whether it is
possible to use the functional form of the found equivalent crack resistance curve (under quasistatic loading) to predict fatigue loading for two different sized, three point bending, single edge
notch specimens (TPBSEN); 2) Determine the load history prediction error associated with using
a LDR under a random load distribution.
More specifically, the objectives of this dissertation are the following:


Objective 1: For different size specimens,
o Determine the equivalent crack resistance curve under quasi-static loading.
o Develop a fatigue fracture model that can account for the bridging stresses.
o Calibrate the fatigue equivalent crack resistance curve using the functional form
obtained from quasi-static loading.
o Compare the fatigue and quasi-static cracking parameters and assess the statistical
significance between the two.
o Compare the Paris coefficients for each fatigue loading scenario.
o Compare the crack resistance curve for each loading scenario.



Objective 2:
o Develop a pavement stress model that outputs a random distribution of stresses at
the critical, mid-slab edge location.
o Identify the variables that govern the load history effect in the fracture model.
o Assess the load history effect under a two-load sequence.
4

o Compare the number of cycles to failure prediction between the LDR and the
fracture model using the distribution from the pavement stress model.
Chapter Overview
This dissertation describes some new methods to characterize the effect of cyclically induced
bridging stresses under quasi-static and fatigue loading. An overview of each chapter is shown
below.
Chapter II is a review of the literature pertinent to the area of fatigue fracture in concrete
with an application in pavements. The review covers the state of the art in fatigue fracture in
concrete materials, focusing on linear elastic fracture mechanics techniques. There is also a brief
synopsis of cohesive zone models that encompasses both monotonic and fatigue loading cases.
Finally, there is a brief review on pavement stress analysis that covers the classical works by
Westergaard and newer techniques using finite elements. In addition, the fatigue cracking
distress as interpreted in the new pavement mechanistic design guide is outlined.
Chapter III covers the experimental setup, procedure, and mix characteristics for all 48
test specimens conducted under quasi-static and fatigue loading. The details shown therein cover
the experimental matrix, the equipment used for the test, and the mix time characteristics of the
concrete used.
Chapter IV covers a new method to determine cyclically induced bridging stresses in
concrete under quasi-static loading. It is shown that bridging stresses can be back-calculated by
imposing a force, moment, and displacement equilibrium condition at the crack location. The
crack resistance curve is then obtained with the use of a weight function. The equivalent crack
concept is discussed, and a new method to determine the equivalent crack resistance is proposed.
5

The equivalent crack resistance curve is then discussed within the context of fatigue and is
shown to depend on the level of applied load. Finally, the quasi-static test results are presented
and discussed.
Chapter V covers a new procedure to determine an equivalent cyclic crack resistance
curve under constant amplitude loading. The equivalent crack propagation rate is directly related
to the equivalent cyclic crack resistance curve. The solution is considered unique under the
conditions that the initial crack resistance is zero and the slope of the curve after post-critical
crack extension is pre-defined. The constant amplitude fatigue results are then compared to the
quasi-static results.
Chapter VI presents and summarizes the modified Paris Law used to characterize fatigue
crack propagation under constant, variable, and random amplitude loading. The fatigue fracture
law uses an equivalent crack resistance curve that has the same functional form obtained from
quasi-static loading and satisfies the uniqueness condition specified in chapter V. Next, a
summary of all quasi-static and fatigue results are shown. A statistical analysis is then conducted
to compare the results between different loading regimes and sizes. It includes 14 quasi-static, 8
constant, 20 variable, and 6 random amplitude loading tests. The chapter concludes by showing
that the Paris law coefficients, log C and n, are not size dependent for the two sizes tested.
Chapter VII presents the results of an analysis that compares the damage prediction
between a linear damage rule and a fatigue fracture model under a random distribution of
pavement edge stresses. The chapter begins by showing the procedure used to calibrate the
modified Paris fatigue fracture model and then describes in detail the procedure for rapidly
predicting the pavement edge stresses. The results indicate that the LDR under-predicts fatigue
6

damage by nearly 30 % if the linear damage rule was calibrated under constant amplitude
loading.
Chapter VIII presents the summary of findings, some conclusions, and recommendations
for future research.

7

CHAPTER II

LITERATURE REVIEW

The motivation of this research was due in large part the desire to improve performance
prediction models for concrete pavement design.

Concrete pavements are subjected to a

complex combination of environmental and traffic loads creating a unique distribution of peak
stress and stress ranges at the critical mid-slab edge. Moreover, the fatigue damage accumulation
process is complex because the material is heterogeneous and suffers from a variety of preloading flaws such as air voids and shrinkage cracks leading to a substantial variability in fatigue
performance.
Classical pavement stress solutions
The cornerstone of any reliable pavement cross-section design lies with the ability to first predict
the stresses induced by truck traffic and environmental loads. In the 1920’s, Westergaard (1926)
developed solutions for a plate supported by a Winkler (liquid) foundation at three different
locations: Edge, Interior, and Corner as shown in Figure 1. The governing equation is shown in
equation (1).

(

)

(

)
(1)

Where

8

(

)

( )

k = stiffness of the Winkler foundation
u = displacement.
E = elastic modulus
h = thickness of concrete layer
= poisson’s ratio
= Laplace operator
F(x) = applied force

Interior

Edge
Corner

Figure 1: Critical stress locations (for interpretation of the references to color in this and
all other figures, the reader is referred to the electronic version of this dissertation)

The solutions are valid under the assumptions that the plate is infinitely long in the xdirection, and no gap between the plate and the foundation exists. Westergaard asserted that the
critical stress location is located at the mid-slab edge where the tensile portion is located at the
bottom of the plate. He also noted that at the corner, although the stresses are smaller in
9

magnitude compared to the edge, the critical tensile stresses are located at the top of the plate.
Note that concrete is much weaker in tension than in compression, so for unreinforced sections,
the critical location will always be located where the tensile stresses are maximum.
Over the last few decades, specific concrete pavement finite element software has been
developed to predict the stresses under more complicated structural and loading scenarios. The
most common are ILSL2, WESLIQID, J-SLAB, KENSLAB, and EVERFE. The differences
between them are mainly in the interpretation of subgrade and temperature gradients. It is
asserted in a NCHRP report (2004) that ILSL2 (Khazanovich, L., Yu, T., 1998) is technically
superior to the other models because of its ability to implement a variety of complex sub-grade
models and account for varying linear temperature gradients for different layers.
EVERFE (Davids, W.G., Wang, Z.M., Turkiyyah, G., Mahoney, J. and Bush, D., 2003)
is a three dimensional finite element program (compared to the other FE programs that use 2-D
plate elements) that can account for multiple sub-surface layers, multiple slabs, non-linear
temperature gradients, non-linear joint transfer behavior, and a tensionless foundation. The
limitation, as stated in the NCHRP report, is that the computational time is large relative to the
other FEM software, especially when considering elastic layers (rather than a Winkler
foundation). However, it is the program of choice for this study, because the computational time
of the FEM runs was not of the essence.
Ioannides et al. (1985) revisited the Westergaard solutions using FEM and stated that the
analytical stress and displacement solutions agreed with the numerical solutions when the ratio
between the slab length, L, and the radius of relative stiffness, , was large. The radius of relative
stiffness is defined in equation (2):
10

√

(

)
(2)

Iaonnides asserted that for interior stresses, L/ must be larger than 3.5; for edge stresses,
L/ must be greater than 5; and for corner deflections, L/ must be greater than 8.Thus, it was
concluded that the infinite slab length assumption will lead to erroneous predictions when the
ratio between radius of relative stiffness and slab length is small. This work was a major impetus
in shedding light on the importance of using numerical solutions to predict stresses, especially
under complex subgrade conditions, and thermal gradients.
More recently, several studies have investigated the location of the critical stresses using
FEM (Vongchusiri, 2005; Hiller, J.E and Roesler, J.R., 2005). The proposition was that the
critical stress location originally found by Westergaard may change under certain complex
combinations of thermal and wheel loads. Hiller and Roesler (2005) concluded that the location
of the critical tensile stresses will change depending on the type of thermal gradient: positive or
negative. A positive thermal gradient is defined when the temperature is greater at the top
(relative to the bottom) of the pavement. A negative gradient is the opposite; where the
temperature of the pavement is greater at the bottom.

11

Critical Locations

Critical
Location

(b)

(a)

Figure 2: (a) positive thermal gradient and (b) negative thermal gradient

Figure 2a shows that there are two critical locations for the negative thermal gradient case. One
of the critical locations is located along the longitudinal edge and the other is located along the
transverse edge. The studies show that that the longitudinal stresses (x-direction) in general are
still greater than the transverse stresses (y-direction). Note that the critical location along the
longitudinal direction in Figure 2a is not located in the same position as in Figure 2b
Mechanistic-Empirical Design
The current design methodology (NCHRP, 2004) for transverse cracking uses a hybrid
mechanistic-empirical analysis. The stresses of an un-cracked pavement are predicted via an
artificial neural network (ANN) that is trained by a stress database. The database is made up of
numerous finite element runs, each of which considers a different scenario, e.g. thickness, tire
pressure, thermal gradient, etc.
Generation of pavement stresses
Stresses are generated in a pavement when it is subjected to both traffic and
environmentally induced loads. In the design guide, traffic loads are established through user12

defined hourly truck traffic distributions and axle load spectra. Truck traffic is divided into ten
classes, each of which is distributed according to the hourly distribution factor. Each truck class
then has a specific collection of different axle types (single, tandem, tridem, quad, etc.). Table 1
shows an example of several different trucks and the different axle types associated with them.
Figure 3 shows the mid-slab edge stresses induced by several of the shown axle types under zero
temperature gradient. The axle loads are then finally determined from an axle load spectrum
(distribution of axle loads for a given axle type) and positioned a certain distance away from the
slab edge depending on the user-defined wheel wander distribution.

Table 1: Different truck and axle Configurations

Axle/truck

Example truck
configurations

Single
Tandem
Tridem
Quad
Five
Six
Eight

13

Axle configurations

Stress

Time
Figure 3: Induced stresses at the mid-slab edge under various axle types

The environmentally induced stresses are generated from both temperature and moisture
gradients. The design guide uses an equivalent linear gradient concept (Ioannides, A.M and
Khazanovich, L., 1998) which essentially constructs an equivalent linear temperature gradient
from a thermally induced non-linear stress distribution. Equation (3) shows the mathematical
representation of the equivalency concept. Next, moisture induced gradients are added to the
thermal gradients and form the total equivalent gradient used in the pavement stress simulation.

( )

∫

( ) ( ) [ ( )

( )

∫
(3)

Where
z = location along the pavement cross-section
14

( )

]

h = pavement thickness
TL = linear thermal gradient

T0 = reference temperature
= coefficient of thermal expansion
E = elastic modulus
The design guide also accounts for changes in PCC elastic modulus and foundation
support. Both the stiffness of the elastic base, Ebase, and the stiffness of the subgrade, k, will
vary depending on monthly sub-surface conditions. As stated previously, the stresses induced by
truck traffic, thermal and moisture gradients, and changes in subsurface conditions are then
calculated with the ANN.
The NCHRP report (2004) states that there are many variables that can affect the
magnitude of the pavement edge stresses, and creating an ANN database taking into account
each variable individually is not feasible. The report states with the use of an equivalent
thickness concept, the number of stress influencing variable can be reduced. In the report, it is
argued that a reasonably sized stress database can be obtained using this concept.
Equivalency Concept
The equivalency concept is used to consolidate the total number of variables affecting the
pavement edge stresses to a select few, non-dimensional, variables. It is founded on the nondimensional plate-elastic foundation solutions determined by Koronev (1962) as stated in the
NCHRP report. The non-dimensional stress function is shown in equation (4).
15

( )

(

)

(4)
Where
= unit weight of the elastic layer
= radius of relative stiffness
h = thickness of elastic layer
L = length of plate
=non-dimensional linear temperature gradient
=non-dimensional radial co-ordinate (r/L)
P = applied load
Q = self-weight of the elastic layer
M* = non-dimensional moment distribution
The equivalency concept states that if the non-dimensional variables are constant for two plate
systems, the stress between the first and second system is related by the following expression
shown in equation (5).

16

( )

( )
(5)

This means that the stress database can be constructed using less variables because, for instance,
a ‘reference system’ which uses a constant elastic modulus, poisson’s ratio, modulus of subgrade
reaction, and unit weight, (the radius of relative stiffness in this case will only vary as a function
of layer thickness) can be related to the actual system (which may have a different elastic
modulus, poisson’s ratio, etc.) by knowing the ratio between the thickness, radius of relative
stiffness, and unit weight of the two systems.
Note that the procedure shown in the report is separated into two classes: bonded and unbonded. In this case, un-bonded signifies that the strain at the bottom of the concrete layer is not
compatible with the elastic layer directly below it. In this study, the un-bonded case is used, so
the calculations pertaining to such cases are shown here.
The first step delineated in the report is to convert the multi-layer pavement system that
consists of two elastic layers and a subgrade layer to a system of one elastic and subgrade layer.
This is accomplished by determining an effective concrete thickness and is shown in equation
(6).

√
(6)
Where
17

Ebase = elastic modulus of elastic soil layer

Epcc = elastic modulus of concrete layer

hbase = thickness of elastic soil layer

hpcc = thickness of concrete layer
The next step is to calculate the effective unit weight of the concrete, which can be obtained from
equation (7).

(7)
Where
= concrete unit weight

Next, one must calculate the radius of relative stiffness which is shown in equation (2) and
substitute h for heff. Next, the effective, linear temperature gradient is calculated using equation
(3) and substituting h for heff. It is then stated that a non-dimensional temperature gradient can be
determined. The expression is shown in equation (8). The report asserts that the stresses
generated in two different pavement systems have the same non-dimensional thermal gradient,
the stresses will be the same between the two systems.

18

(

)

(8)
Where
= effective linear thermal gradient (difference between TL at top and TL at bottom of

effective thickness)
The next step is to determine the non-dimensional applied load/pavement weight ratio, and is
shown in equation (9).

(9)
Where
L = length of the slab
W = width of the slab
P = applied load
One of the last steps is to convert the effective concrete thickness to an equivalent
thickness. This is shown in equation (10). It is stated that in developing the stress database, the
PCC elastic modulus, poisson’s ratio, and subgrade stiffness can be held constant since the stress
between one system and another is relatable by equation (5). Note that the value inside of the
denominator in equation (10) is constant. This is because the reference system uses a constant
19

6

elastic modulus and poisson’s ratio of 4X10 psi and 0.15, respectively, and is resting on a
Winkler foundation with a stiffness, k, of 100 psi/in as shown in the NCHRP report (2004). Note
that equation (10) simply determines the height of the pavement layer in the reference system
needed to equal to the radius of relative stiffness in the effective system.

√
(10)
The equivalent stresses from the reference system are then calculated using the nondimensional variables shown in equation (4). The stresses are related back to the effective system
with equation (5) and then related to the actual system with equation (11).

(11)
Finally, a non-linear stress term (that is induced by the non-linear thermal gradient) is
added to the stress from equation (11). The contribution of the non-linear stress gradient can be
determined by subtracting the equivalent linear temperature gradient from the actual temperature
gradient and converting the temperature to stress by multiplying by the elastic modulus and
coefficient of thermal expansion. Recently, Hiller and Roesler (2010) developed a simple method
to determine the non-linear stresses contribution by characterizing the temperature gradient with
a quadratic distribution. The researchers developed a new term called the non-linear area
(NOLA), defined as the area between the quadratic temperature distribution and equivalent linear
20

distribution. The non-linear stress and the mathematical description of NOLA are shown in
equations (12) and (13). The authors assert that the addition of the non-linear stress term will
tend to increase the tension at the top of the pavement and lower it at the bottom.

(12)

∫ [ ( )

]

(13)

Note that the influence of load transfer efficiency at the transverse and longitudinal joints
is not mentioned here. The NCHRP report (2004) should be consulted for a more detailed
description of the equivalency concept, the neural network architecture, the effect of a bonded
sub-base, and load transfer efficiency.
Rapid Stress Calculations
As mentioned previously, the method of choice to rapidly predict the stresses from the FEM
constructed database are the ANN. However, other methods have been developed to rapidly
predict pavement stresses. Vongchusiri (2005) developed a scheme that uses a network of onedimensional piecewise interpolations to predict the mid slab edge stresses and stress influence
surface under various temperature gradients and axle configurations.

21

Linear Damage Accumulation
In the MEPDG design methodology, the stresses are inserted into a field inspired calibrated
mechanistic-empirical performance model that uses a linear damage rule (LDR) in the spirit of a
classical Miner’s rule, to predict damage. However, the strength reference is established from
laboratory strength tests (Modulus of Rupture, MR). A graphic of the methodology is shown in
Error! Reference source not found..
Damage is defined as the inverse of number of cycles to failure, 1/Nf (Miner, 1945).
Equation (14) shows the damage algorithm used to predict fatigue cracking in concrete
pavements. The indices i-m, represent each of the possible loading combinations (i = age, j =
month, k = axle type, l = load level, m = equivalent temperature gradient, n = traffic path) that
the pavement can be subjected to; where  is the applied stress.

∑

∑
(
(14)

22

)

Figure 4: Concrete pavement design architecture

23

Figure 4 (cont’d)

One of the advantages of using this type of damage model is its computational efficiency,
since it can rapidly account for, process and convert millions of load repetitions to damage. This
allows for multiple designs to be considered within minutes.

Some of the disadvantages

however, are that it is insufficient in determining the in-situ state of damage because no
information is given on the state of the material itself (no information on the stress-strain
behavior and the reduction of the elastic modulus). In addition, it cannot account for size effect,
load history effect, and variable amplitude loading without using some empirical calibration
factors. Thus, there is a need for the development of a concrete fatigue model that can account
for all three of the aforementioned effects and be able to maintain a comparable level of
computational efficiency to the S-N approach.

24

Classic Linear Damage Rule
There are several ways of predicting fatigue damage in concrete. The classical method is
to use a linear damage rule (LDR) accompanied with an experimentally generated log-log S-N
curve relating applied stress ratio (SR) to number of cycles to failure, Nf. (Miner, 1945). Note
that SR is the ratio between the applied peak stress and the maximum allowable stress (/max).
Figure 5 displays a series of S-N curves that show Nf increases as the stress ratio decreases
(Chatti et al.). The solid lines represent a least squares linear regression which is described
mathematically using equation (15)
A LDR can be used to determine the cumulative damage by using equation (16), where,
as stated previously, damage (D) is defined as 1/Nf , and failure occurs when D = 1.

1.0

Stress Ratio

0.9
0.8
0.7
0.6
0.5
0.4
1

10

100

1000

10000

100000

1000000

Nf
Single
Six
Log. (Tridem)

Tandem
Tridem
Figure 5: S-N Curve (Single)
Eight
Log.
Log. (Quad) 25
Log. (Eight)

Quad
Log. (Tandem)
Log. (Six)

(

)

(

)

(15)

∑
(16)
Fatigue Damage
Over the last few decades, a number of concrete fatigue studies have been conducted for various
mix types and loading scenarios (Murdock, J.W., and Kesler, C.E., 1959; Tepfers, R., Kutti,T.,
1979; Raithby, 1979; Hsu, Fatigue of Plain Concrete, 1981; Zhang, B., Phillips, D.V., Wu, K.,
1996). In summary, the studies show that fatigue life is dependent on strength, age, drying
shrinkage, loading frequency, SR, and R ratio (ratio of applied valley and peak load). In the
metal industry, Goodman diagrams (or constant life diagrams) have been used extensively as a
means to establish the relationship between the endurance limit (stress level at which fatigue
damage becomes negligible), peak stress and stress range (Sendeckyj, 2001). Other researchers,
as a means to acknowledge the large variability in fatigue, used a reliability approach to predict
the probability of fatigue damage failure (Oh, 1991; Park, 1990; Singh, S.P., Kaushik, S.K.,
2001). In general, this was accomplished by inserting an experimentally observed Nf probability
density function (pdf) into the LDR to express Nf in terms of a probability.

26

Variable Amplitude and Load History Effects
In metallic structures, variable amplitude and load history effects have been investigated
thoroughly over the last few decades. There is a collection of works that focus on the variable
amplitude (or load interaction) and load history effect. The purpose of these studies was to
address the inadequacies of a LDR to predict fatigue under varying loading conditions. Sufficient
experimental evidence suggests that fatigue life predictions using a LDR can both over-predict
and under-predict damage depending on the sequence of loading (Fatemi, 1998). In a low to high
(L-H) loading sequence, the LDR’s will typically over-predict damage and in a high to low
loading sequence (H-L), the LDR’s will typical under-predict damage.
Several researchers proposed modifications to the original LDR approach. The double
linear damage rule (DLDR) was introduced by Manson (1966) and it separated the S-N curve
into two regions to represent the differences in damage accumulation rates within the crack
initiation and propagation stages. This method was then refined to include a damage curve
function (Manson, S.S., Halford, G.R., 1981) that allowed damage to be expressed as a nonlinear power function dependent on load-level. Oh (1991) developed a non-linear damage rule
for concrete that expresses the damage (D) as a load-level dependent cubic polynomial, and was
able to predict the LDR error for both L-H and H-L loading sequences. More recently, a one
dimensional continuum damage mechanics model was used to predict the load sequence effect
(Lemaitre, 1992; Xiao, Y.C., Li, S., Gao, Z., 1998). These models associate damage directly to
elastic modulus degradation and are founded on thermodynamic principles in which the damage
induced strain energy release rate, ̇ , is the driving force for damage.
Under random amplitude loading, frequency-based techniques are often used to count the
level of load repetitions. The rainflow counting method is one example where a random load
27

time history is transformed into an equivalent constant amplitude load histogram allowing
classical LDR’s to be applicable (Anthes, R.J., 1997). A more recent technique uses power
spectral density (PSD) functions to transform random stress time histories to the frequency
domain for the purpose of correlating Gaussian or non-Gaussian distribution parameters to
damage using LDR’s (Benasciutti, D., Tovo, R., 2005).
Linear Elastic Fracture Mechanics
Linear Elastic Fracture Mechanics (LEFM) is a field that focuses on the study of cracks in elastic
media. The field was introduced in the late 1940’s and was first used to describe the near
instantaneous crack extension observed after a critical applied force was reached. Initially,
engineers measured the fracture energy,

(strain energy release per crack extension) to

assess a structures ability to resist unstable crack propagation, but they did not have any
information on the stress fields promoting the unstable behavior. A decade later, the stress field
solutions around a crack tip were developed and it was discovered that crack growth was
governed by the intensity of the stress field around the crack. It was shown that the stress
intensity was solvable for three different modes. Mode I fracture (as shown in Figure 6 it is
associated with the crack opening response caused by normal forces along the crack face. Modes
II and III facture correspond to crack opening response caused by in plane and out of plane shear
stress, respectively. Equation (17) shows the function of the stress field around a crack tip in
mode I. Equation (17) shows that the stress is singular when the ordinate is zero (at the crack
tip). It also shows that it is governed by the stress intensity, KI. (Tada, H., Paris, P.C. and G.R.
Irwin, 2000).

28

Figure 6: Mode I Fracture

(𝜃

)

𝐾
√ 𝜋𝑟

𝜃
cos (

𝜃
𝜃
sin sin )

(17)
Where
r = radial distance away from the crack tip
𝜃 = angle between the crack plane and the point of interest
The stress intensity in mode I fracture is defined in equation (18). It is a function of the far-field
stress and the crack geometry.

29

𝐾

lim √ 𝜋𝑟

(𝑟 𝜃

)

(18)

Griffith (Tada, H., Paris, P.C. and G.R. Irwin, 2000) later proved that the fracture energy was
related to stress intensity through equation (19).

𝐾
(19)

Fracture Mechanics in Concrete
Concrete is a quasi-brittle material that exhibits a large residual zone along the wake of a
dominant crack and exhibits a size dependent R-curve behavior, which limits the applicability of
classical LEFM techniques/concepts to predict material response (Bazant, Z.P., Planas, J., 1998).
Figure 7 shows a typical plot of strength versus size in concrete. The figure highlights the issue
that strength and LEFM theories are only applicable for very small and large sizes, respectively.
The transition between the two is defined as the non-linear fracture mechanics regime. Within
this regime, failure is strongly influenced by the residual bridging stresses. This means that the
typical failure criterion in LEFM: KI > KIC, i.e. unstable fracture occurs when stress intensity is
greater than fracture toughness, is somewhat more complicated because the fracture toughness
becomes a function of specimen size, D, and crack extension, a.

30

Strength

Non-Linear

LEFM

Figure 7: Relationship between strength and size

Crack Resistance Curves
The R-curve or crack resistance curve, describes the fracture toughening mechanism that is
commonly observed in concrete as the crack length increases. The curves are typically described
in terms of fracture energy, or stress intensity and can be used to decipher stable versus unstable
crack growth. The crack growth becomes unstable when the applied stress intensity is tangent to
the crack resistance. These conditions are shown mathematically in equations (20-21).

31

𝐾

𝐾
(20)

𝐾

𝐾

(21)

Figure 8 shows the typical shape of an R-curve in concrete and highlights two important aspects
of this phenomenon. The first aspect is that unstable fracture occurs at a distinct point along the
crack resistance curve. Failure occurs at the tangent point between the applied stress intensity
and the crack resistance curve. The second aspect is that the bridging stresses that are holding the
crack faces together are actually directly related to the crack resistance. As the crack grows,
more bridging stresses will develop, which will lead to a greater resistance to future fracture.

32

Figure 8: Crack resistance curve for two different size concrete specimens

Over the years, there have been several methods proposed to determine the crack
resistance curve in concrete. Bazant showed that the general, size independent R-curve could be
determined by the size effect law (1990) shown in equation (22).

𝐾
√

(
(

Where

= nominal stress at failure
( )=geometric function
33

)
)

= non-dimensional crack length (a/D)
cf = limiting crack extension at failure as
The study showed that by using the size effect law, one can arrive at an expression that
describes the R-curve in terms of cf, and the non-dimensional crack length at failure. In order to
obtain the limiting specific fracture energy, Gf, and the limiting crack extension, the nominal
strength at failure for three different size specimens needs to be obtained.
Other researchers have used the R-curve approach to describe the failure mechanism in
concrete and other quasi-brittle materials (Bazant Z.P., Cedolin L., 1984; Bazant, Z.P., Jirasek,
M., 1993; Mai, 2002; Wecharatana, M., Shah, S.P., 1983). More recently, Xu and Reinhardt
(1999a) developed a double fracture resistance criterion that de-couples the initial crack
resistance and the unstable crack resistance. This work was then extended by Kumar and Barai
(2009) who developed closed form solutions for the bridging/cohesive/residual stresses using a
weight function.
Weight Function
Several researchers have quantified the R-curve with the use of a weight function (Foote,
R.M.L., Mai, Y.W., Cotterell, B., 1986; Mai, 2002; Fett, T., Munz, D., Geraghty, R.D, White,
K.W., 2000; S. Funfschilling, T. Fett, S.E. Gallops, J.J. Kruzic, R. Oberacker, M.J. Hoffman,
2010; Kumar, S., Barai, S.V., 2009; Li, V.C., Matsumoto, T., 1998). The concept of the weight
function was first proposed by Bueckner (1970; 1971). In this case, the weight function (Tada,
H., Paris, P.C. and G.R. Irwin, 2000) is a device used to correlate both stress and displacement to

34

stress intensity, as shown in equation (23) and is deemed to be a geometric property, independent
of loading.

(

(

)

)
𝐾( )

(23)

Where
x = arbitrary point along the crack path
a = crack length
u = crack opening displacement (COD)
With the use of a weight function, the stress intensity, KI, can be described for any
arbitrary stress, as shown in equation (24).

𝐾( )

(

∫

) ( )

(24)
Where
( )=arbitrary stress function
The weight function approach has also been used to determine the crack opening
displacement, and the relationship between the two is shown in equation (25).

35

(

𝛿( )

)

∫

(

)∫

(

) ( )

x≤

(25)

Primarily, the weight function can be used to determine the influence of the bridging stresses on
fracture resistance (or net stress intensity) and the crack opening displacement.

𝐾

( )

∫

(

)[

( 𝛿( ))]

(26)

𝛿( )

(

)

∫

(

)∫

(

)[

( )

(𝛿( ))]

x≤

(27)
The crack opening displacement can be solved for by using an iterative algorithm that satisfies
the double integral shown in equation (27).
Bridging Stresses
For concrete, there are three common functions that are used to describe the bridging stress
distribution: linear, bi-linear, and exponential (Bazant, Z.P., Planas, J., 1998). Bazant and Li
(1995) used a linear distribution to model the size effect of the modulus of rupture for unnotched specimens. Guinea et al. (1994) and Petersson (1981) developed a bi-linear softening
curve for concrete. More recently, Roesler et al. (2007) used a bi-linear curve and determined the
kink-point by using the specific and the total fracture energy. Figure 9 shows the three different
distributions. The variable f’t is the material tensile strength and wc is the critical crack opening
36

displacement, at which point no bridging stresses are transferred from one crack face to the other.
Note that the area under the softening curves represents the total fracture energy, GF. This value
has been shown to be greater than the specific fracture energy, Gf, which is defined as the energy
released at the onset of unstable fracture. It is for this reason that some researchers believe that a
bi-linear or exponential bridging stress distribution is superior to that of a linear distribution (for
failure prediction) because these distributions can be separated into two separate portions, one of
which describes the specific fracture energy.

f’t

f’t

f’t

GF

GF

GF

Linear

wc

wc

wc

Exponential

Bi-Linear

Figure 9: Common bridging stress distributions used to predict concrete fracture

37

Equation (28) shows a common exponential function used to describe the bridging stresses that
was originally proposed by Reinhardt (Kumar, S., Barai, S.V., 2010; Reinhardt, H.W.,
Cornilesson, H.A.W., Hordijk, D.A., 1986).

(𝛿( ))

𝑓 {[

(

)exp(

𝛿( )
) ] exp (
𝑤

𝛿( )
)
𝑤

(

𝛿( )
)(
𝑤

)}
(28)

Where

f’t = tensile strength
𝑤 = critical crack opening displacement at which no stresses are transmitted from one crack face
to the other
= curve shape factor
= curve shape factor
Concrete Fatigue Fracture
In plain portland cement concrete (PCC), the fatigue cracking process is similar to other quasibrittle materials in that two distinct stages are observed: a transient stage where the crack growth
rate is decreasing and a steady state stage where the rate is increasing (Kruzic, 2005).
Subramaniam et al. (2000), Perdikaris and Calomino (1987), Bazant and Xu (1991) have all
shown that concrete fatigue fracture in the steady state range follows the well-known Paris Law
38

shown in equation (29). The fatigue crack propagation rate (da/dN, where a is the crack length
and N is the number of cycles) follows a power law in which stress intensity, KI, is the argument.
The coefficients, C and n, are considered to be material properties. However, equation (29) can
only be used to describe the steady stage region.

( 𝐾)
(29)

Subramaniam et al. (2000) modeled the transient stage separately as a function of crack
extension (a). The two regions were separated at a unique point called abend, which they
found to be equal to the critical crack extension at failure, ac, in the quasi-static crack
resistance curve at peak load under quasi-static loading (see Figure 10).

39

(a)

(b)
Figure 10: (a) Transient Stage Crack Propagation, (b) Steady-state Crack Propagation

The crack propagation law in the transient stage is shown in equation (30) from
Subramaniam et al. (2000).
40

(

)

(30)

This means that four fatigue parameters would need to be defined: C and n in equation
(29) and C1, and n1 in equation (30). The major limitation of this approach, however, is that it
cannot predict crack growth under variable amplitude loading since it cannot account for the
change in stress in the transient stage. This is because the argument in equation (30) is crack
extension as opposed to stress intensity. Another limitation is that because the bridging stresses
(which are considered to be a material property) are not explicitly determined, the information
gathered from experimental tests (that use a simple geometry) would not be applicable for other
geometries commonly used in the field (Gallops, S., Fett, T., Ager III, J.W., Kruzic, J.J., 2011).
The reason is because the influence that bridging stresses have on fatigue may be different from
one geometry to another. Bazant and Xu (1991) showed that there is a distinct size effect in
concrete fatigue. The results of the study show that the C parameter in the Paris law tends to be
smaller for larger sized structures. This is a clear indication that the bridging stresses (although
they are a constant material property) affect the fatigue behavior depending on the relative sizes
and/or geometry. However, if the bridging stresses are known, their influence on a specific
geometry can be accounted for, so the size effect will not be observed on the Paris parameters.
One way of accounting for the variable stresses and the bridging stresses in both
cracking regions, is to include a crack resistance term in the argument of equation (29). This can
be done using the concept of effective stress intensity as defined in equation (31) from Ritchie et
al. (1987).
41

𝐾

𝐾

𝐾

(31)

The term KI is the stress intensity and Kbr is the crack shielding contribution from the
residual stress near the crack tip. Li and Matsumoto (1998) proposed a fatigue model in which
the fatigue crack propagation followed a unique relationship dependent on the combination of
both the applied stress intensity, KI and the contribution of the bridging stresses behind the crack
tip, Kbr. The modified Paris Law is shown in equation (32).

[𝐾

𝐾 (𝛿 )]

(32)

They argued that the bridging stresses were governed by the crack opening displacement
along the length of the crack, x. However, their model used a bridging stress degradation
function that is dependent on the number of cycles N, which makes it difficult to use under
variable amplitude loading.
Several researchers recently have published experimental data for other quasi-brittle
materials such as Alumina, and have reported cyclic threshold stress intensity (Kth) curves that
have a similar shape to that of a quasi-static resistance curve (KR); i.e., crack resistance rate
decreasing with increasing crack extension.

42

Under variable amplitude loading, the number of overall fatigue fracture models for
concrete is limited. Slowik et al. (1996) developed a concrete fatigue fracture model similar to
the Forman equation in metals (Beden, 2009) that accounts for the effect of stress intensity range
(KI), peak stress intensity, load history, and overloads (F) as shown in equation (33). This
version of the Paris Law works well in predicting concrete fatigue cracking under variable
amplitude loading in the steady state cracking stage; however, no information on the crack
propagation rates is given for the transient stage, which makes it difficult to predict.

𝐾 ( 𝐾)
(𝐾
)
𝐾
(33)

43

(

)

CHAPTER III

METHODS AND MATERIALS

This chapter covers the experimental procedures used to quantify the equivalent cyclic crack
resistance under four different loading regimes: quasi-static, constant, variable, and random
amplitude loading. The specimen dimensions, mix characteristics, and testing are also discussed
in this chapter.
Testing Equipment
The notched beam specimens were tested on a 25 kN/100 Hz capacity, servo-hydraulic MTS
Landshark. A MTS crack opening displacement (COD) gage, was used to measure the crack
mouth opening displacement.

COD Gage Attachment
The COD gage was mounted to the specimens by a pair of 3 mm thick steel knife edges. The
knife edges were glued directly to the bottom of the specimen with a fast drying epoxy resin
which was allowed to harden for 24 hours. The distance between the tip of the knife edges
(where the COD gage was attached) was determined with the use of a MTS manufactured,
aluminum separator. Figure 11 shows a photograph of the mounted COD gage.

44

Figure 11: COD Gage

Equipment Accessories
The specimens were tested under three point bending. The loads were transferred from the MTS
to the specimens with an aluminum I-beam which was attached to the uniaxial hydraulic loading
ram. The I-beam was designed such that the vertical deflection ratio between the concrete beam
and aluminum beam was approximately 1 %. Roller supports were attached to the aluminum
beam and were the designated load transfer point to the concrete beams. Figure 12 shows a
photograph of the test setup, which includes the MTS machine, the COD gage, a notched
concrete beam, the aluminum beam, and the roller supports.

45

Figure 12: MTS machine

Concrete Mix Properties
The concrete mix used in this research consisted of ASTMC-150 Type I cement, a natural sand,
and a limestone coarse aggregate (nominal maximum size of 25 mm). The water to cement ratio
3

was 0.45 and the air content was 6.5%. The unit weight was 2274.62 kg/m .
The average 28 day Modulus of Rupture (MOR) and the split tensile strength, f’t, were
5.23 and 2.89 MPa, respectively. The 28 day compressive strength was 25 MPa. The specimens
were cured for one year inside of a humidity room and then placed in ambient temperature for
46

one more month to ensure minimal strength gain during fatigue testing. The average split tensile
strength and compressive strength at the time of testing was 3.79 and 40.62 MPa, respectively.
Specimen Geometry
A total of 48 plain PCC three-point single edge notched beam specimens (TPB-SEN) were tested
under both cyclic quasi-static and random amplitude fatigue loading. Two different beam sizes
were tested. The larger specimens had a span of 400 mm, a depth of 100 mm (S/D=4), and a
width of 100 mm and the smaller specimens had a span of 200 mm, a depth of 50 mm, and a
width of 50 mm. Three different notch to depth ratios () were used in the larger size specimens.
For the smaller size specimens, only a notch to depth ratio of 0.15 was used due to the limited
space between the tip of the notch and the top of the beam. Note that the smaller beams were
sawn cut into 4 equal parts from the larger beams with the dimensions mentioned above. This
was done in lieu of casting small beams altogether to ensure that the large 1” aggregates would
not dominate a local region and influence. Casting larger beams should allow for a larger random
distribution of aggregates; if a large aggregate did dominate the critical location where the notch
was located, it would be simply saw cut and its effect would be minimized.
Loading Regimes
A total of fourteen beams were tested under cyclic, quasi-static loading. For the larger
specimens, the COD controlled loading rate was 0.0005 mm/s was. For the smaller size
specimens, a COD controlled loading rate of 0.00025 mm/s was used. The loading rates were
determined in a manner that the peak load was reached no sooner than one minute after the
beginning of loading (on average). The remaining 34 specimens were tested in fatigue. Each
specimen was subjected to a 2 Hz cyclical load. For the constant amplitude tests, an SR (stress
ratio, max load/peak load) of 0.95 and 0.85 and an R-ratio of 0.05 was used.
47

For the variable amplitude tests, the SR’s were changed abruptly after a noticeable
amount of damage had been detected with the COD gage (change in compliance). The SR’s were
varied between 0.95 and 0.75. In some tests, the R-ratios were also changed abruptly after
significant damage was observed. The R-ratios used in the tests ranged between 0.05 and 0.5.
Under random loading, both a uniform and random distribution were used. The SR’s
ranged from 0.5 to 0.9 and R ratio ranged from 0.5 to 0.05.

Table 2 shows the entire

experimental program used in this study.
Table 2: Experimental Program
Loading

Size

Quasistatic

100

Constant

50
100
50

Variable

100
50

Random

100
50

Total

Geometry Replicates
0.15
3
0.35
3
0.15
8
0.15
1
0.35
2
0.5
2
0.15
4
0.15
2
0.35
2
0.5
5
0.15
10
0.15
2
0.35
1
0.15
1
48

Crack length measurements
Direct determination of effective crack lengths in both quasi-static and fatigue loading was done
using the Jenq-Shah compliance technique (1985). The compliance is defined as the inverse of
the tangent modulus (slope of the applied load and the crack mouth opening displacement),

48

cmod/P. Figure 13 shows a P-CMOD curve under quasi-static loading which highlights the
idealized compliance at cycles i and i+1.

Figure 13: Load v. Crack Mouth Opening Displacement

Equivalent Elastic Crack
The equivalent crack concept is commonly used in fracture mechanics and it is an idealization of
the cracking mechanism, which assumes that the crack is traction-less. Using this assumption, it
allows one to determine the crack length with the information gathered only from compliance
measurements. The Jenq-Shah method is described below.
Jenq-Shah Method
The Jenq-Shah method is a compliance technique that uses the analytical displacement solution
at the crack mouth to back calculate the crack length. The method has been used to determine the

49

equivalent crack under quasi-static loading and was extended to fatigue loading, as shown by
Subramaniam (2000) .
The method uses the compliance information gathered in the initial loading cycle to
determine the elastic modulus. In subsequent cycles, if compliance increases, the method equates
this change to an increase in crack length, assuming the elastic modulus remains constant. The
equivalent crack length is calculated using equations (34) and (35), and a simple NewtonRaphson technique. In this study, a three-point bending single edge notch specimen was used.
For such specimens that also have a span to depth ratio (S/D) of 4, the relationship between the
compliance and the crack length is shown in equation (35).

( )
(34)

( )

(
(35)

Where
Ci = current compliance;

ai = current effective crack length,

i =current non-dimensional effective crack
b = beam width,
D = beam depth
50

)

P = applied load at mid-span.
s

Specific crack resistance (KR ) curve
The stress intensity for a three point bending, single edge notch specimen can be determined with
equation (36) after the effective crack length has been determined (Tada, H., Paris, P.C. and G.R.
s

Irwin, 2000). Under quasi-static cyclic loading, the KR curve is quantified by determining the
peak stress intensity at every loading cycle, which can be calculated by knowing the equivalent
crack length and peak load over the cycle. The superscript s denotes that this type of resistance
curve is specific to the size and geometry of the structure. It is expected that if the geometry
s

and/or size of the specimen were to change, so would the shape of the KR curve, so thus it is
said to be a specific crack resistance curve.

𝐾

√

( )

(36)

(37)

( )

√ [

(
(

)(
)(
(38)

51

]
)

s

The critical parameters governing the crack resistance curve are KRc , aeq,c and m. One
way of describing the resistance curves, as shown by Morel (2007), is through equation (39).
s

The parameters KRc and aeq,c are the specific quasi-static crack resistance and the equivalent
critical crack extension, respectively. These parameters can be determined either graphically or
mathematically. The two points are tangent to the KI curve at peak load. Figure 14 shows the
s

two curves, KR , and KI, under quasi-static loading. The figure highlights the point of tangency
between the curves and shows where the two parameters are located.

𝐾

𝐾

(

(39)
Where

𝐾

= critical crack resistance
= equivalent critical crack extension

m = shape parameter

52

)

KRC

s

, aeq,c

Figure 14: KR and KI curves

Load-Crack Data Smoothening
This method of determining the crack resistance curve can also be used as a means to smoothen
the experimental load-crack data. Once the KR

s

curve has been determined from the

experimental data, the load can be determined by rearranging equation (36) to solve for stress,

N, and using the relationship between load and stress shown in equation (37). The end result
s

leads to equation (40). Figure 14 shows the smooth KR function fitted onto discrete data points
s

obtained from experimental tests. The smooth KR function can then be used to determine a
smooth load-crack curve as shown in Figure 15.

53

𝐾 (
√

)
( )

(40)

(a)

(b)
Figure 15: (a) P-CMOD curve and (b) P-crack curve

54

CHAPTER IV

A METHOD TO DETERMINE AN EQUIVALENT CYCLIC CRACK RESISTANCE
CURVE UNDER QUASI-STATIC LOADING

The current global economic downturn has forced many governments to change their strategy on
infrastructure rehabilitation. An onus has been placed on engineers to design and build longer
lasting, eco-friendly, and more sustainable infrastructure. A design that incorporates
sophisticated damage modeling is becoming a necessity to meet the design targets specified by
the transportation agencies. These mechanistic damage models are quite different from the
classical SN and LDR approaches proposed by Miner (1945) because damage is physically
quantifiable and is usually related to some measure of strain energy release (Lemaitre, 1992).
The benefit of these mechanistic models is that they can also be used in conjunction with nondestructive tests needed to assess the current state of the infrastructure.
Prior to the launching of any damage design protocol, the fracture mechanism, especially
in concrete materials, needs to be better understood by scientists and practicing engineers alike.
Concrete is a unique material that exhibits ‘small crack’ behavior, where the residual stresses
strongly influence fracture characteristics like fatigue crack propagation rate, fracture toughness,
and critical crack length. One of the many interesting aspects of concrete is that the material
constituency is made up of relatively large aggregates (compared to the grain size of alumina for
example) which engenders a proportionally large region of residual stress. This softening region
is often denoted as the bridging stress zone (Cox, B.N., Marshall, D.B., 1991) and is responsible,
in large part, for the size effect, R-curve behavior (Fett, T., Munz, D., Geraghty, R.D, White,
55

K.W., 2000), crack deceleration region in fatigue (Li, V.C., Matsumoto, T., 1998), and also
influences the endurance limit, as espoused by Kruzic et al. (2005).
Fracture characterization in concrete under quasi-static and monotonic loading has been
of particular interest to researchers. Many studies have been dedicated to understanding the
relationship between stress softening behavior and size effect (Bazant, Z.P., Kazemi, M.T., 1990;
Morel, 2007; Roesler J., Paulino, G.H., Park, K., Gaedicke, C., 2007), and the impact it has on
the R-curve (Foote, R.M.L., Mai, Y.W., Cotterell, B., 1986; Mai, 2002; Gallops, S. et al., 2011;
Bazant Z.P., Cedolin L., 1984). Recently, Xu and Reinhardt (1999a; 1999b; 1998) proposed a
double K criterion that uses a linear asymptotic (or secant modulus) compliance technique to
determine the equivalent crack length and the contribution of the bridging stresses. The
researchers assert that the total stress intensity at failure is defined as the sum of the initial
fracture toughness, KIC
coh

stresses, KI

ini
,

and the resistance generated by the contribution of the bridging

, as shown in equation (41). More recently, Kumar and Barai (2010) used a

simplified weight function to describe the bridging stress zone and their results showed good
agreement between predicted and modeled P-CMOD behavior.

𝐾 (

)

𝐾
(

𝐾
)

The linear asymptotic assumption used in the formulation of the double K criterion developed by
the researchers is a simple but powerful way of interpreting the non-linear behavior in the
concrete. The assumptions are 1) a change in compliance is proportional to a change in crack
length and 2) the non-linearity observed in the response of the material is caused only by
56

cracking. This simplification is made in order to develop a simple closed form solution between
the P-CMOD behavior and crack length. However, it is known that some of the non-linearity can
be attributed to plastic deformation and not cracking. As a result, the linear asymptotic
assumption will predict larger crack lengths than peak to peak compliance techniques, e.g. the
Jenq-Shah compliance technique. Peak to peak compliance (or tangent modulus) methods
eliminate the plastic deformation when calculating equivalent crack lengths. The peak to peak
compliance methods, however, seem to under-predict CMOD. Figure 16 shows the difference
between the two compliance techniques.

Ci
P

Ci
Ci+1

Ci+1

P

Ci+2

Ci+2

Ci+3

CMOD

Ci+3

CMOD

Linear Asymptotic

Peak to Peak

Figure 16: Comparison of compliance techniques: Linear asymptotic assumption (secant)
and load-unload (tangent)

Note that, in both methods, the crack is assumed to be traction-less and is denoted as an
equivalent crack. It is a convenient assumption that allows one to directly relate crack length to a
change in compliance or CMOD. However, because there are bridging stresses along the crack
57

face, the equivalent cracks will be smaller than the actual crack length. Figure 17 shows an
illustration of the two disparate lengths, but shows how both share the same CMOD, which is
measured experimentally.

Figure 17: Crack opening displacement fields for a crack that is traction-less and one with
tractions

In general, compliance techniques are computationally efficient but are limited by their
inability to determine the actual crack lengths and/or bridging stresses. The methods have been
used, however, in conjunction with optical devices, to determine the bridging stresses. Mai
(2002) showed that one can determine the bridging stresses if both the actual crack length and
equivalent crack length are known. Kruzic et al. (2005) and Gallops et al. (2011) showed that
one can determine the fatigue bridging stresses generated using a successive approximation
iterative algorithm that used a CMOD boundary condition and an initial fracture toughness
58

condition. They also developed a simple first and second order approximation scheme to
determine the bridging stresses with only the use of weight functions and the KR curve.
Funcshcilling et. al (2010) showed that one can obtain the actual crack length by satisfying a
series of integral equations simultaneously and using an energetic equilibrium condition. They
also showed that at crack initiation, compliance techniques are more sensitive in identifying
changes in crack length than optical techniques because the surface cracks are not visible until
the center portion of the crack has elongated substantially.
In this chapter, a new method to determine the bridging parameters, f’t, wc, c1, and c2,
from equation (28), and an equivalent fracture resistance curve under a quasi-static loading test
for a three point bending specimen is proposed. The method uses a weight function and a
successive approximation scheme to calculate the crack opening displacement field and bridging
stresses. The properties governing the bridging stresses are back-calculated with an optimization
scheme, programmed in MATLAB that satisfies displacement equilibrium at the crack mouth
and global force and moment equilibrium along the depth of the beam where the notch is located.
Load and Equivalent Crack Data
The equivalent crack lengths were obtained using the Jenq-Shah compliance technique (using
tangent compliance). The crack lengths were measured after the completion of each loading
cycle. Using the peak load and crack length at each cycle, a specific quasi-static crack resistance
s

curve, KR , was generated; it describes the variation of the peak stress intensity after each cycle
and identifies the critical stress intensity and crack extension at failure (or unstable cracking).
The word specific is attached to resistance curve to signify its dependence on size.

59

s

A smooth KR function in the spirit of the one proposed by Morel (2007) was fit over the
s

data. The function, shown in equation (39), has three governing parameters: KRc , aeq,c and m.
s

The parameters KRc and aeq,c correspond to the stress intensity (or specific fracture toughness)
and critical crack extension at peak load, respectively. The parameter m was then obtained using
a least squares, non-linear regression technique that fit the function over the experimental data.
The smooth P-aeq function shown in Figure 18c was obtained using the smooth KR

s

curve with the two critical fracture parameters and the power m from least squares regression.
The load was determined by inserting the nominal stress term, N, from equation (36) into (37)
s

and by substituting KI for KR shown in equation (39). The smooth P-aeq function is described
mathematically in equation (42) and its behavior is shown in Figure 18.

(

)

𝐾

(
(42)

60

)

√

( )

(a)

(b)

(c)
Figure 18: Construction of P-aeq curve
61

The next two subsections in this chapter describe the successive approximation scheme
used to determine the crack opening displacement field and the double loop optimization scheme
that solves for the ‘corrected’ crack length and the four bridging stress parameters. The
parameters were solved for by satisfying two conditions under a given load from the smooth Paeq curve: 1) CMOD for the corrected and equivalent cracks must be equal, and 2) the applied
moment and internal moment at mid-span must be equal. A total of 8 points were chosen along
the P-aeq curve. Note, four or more points along the curve must be chosen to ensure enough
equations exist to solve for the unknown parameters. Initial seed values were assigned to the four
bridging parameters before starting the optimization scheme (f’t = 3.5 MPa, wc = 0.25 mm, c1 =
3 and c2 = 8). Finally, a double loop optimization scheme using a least squares objective function
was executed.
CMOD condition
The crack opening displacement field was determined using a weight function g(x,a), which is
shown in equation (27).
Where

𝛿(

)

𝛿 [(

)

(

) (

(43)

62

( ) ) ]

( )

𝛿
(44)

The variable  shown in equation (43) is the crack opening displacement and shown in
equationis the crack opening at the mouth of the crack, or CMOD. The variable x is the
ordinate along the length of the crack and  is the non-dimensional crack length.
The crack opening displacement field 𝛿( ) was calculated using equation (27) and was
solved iteratively using successive approximation. In the first iteration, the bridging stresses had
a value of zero and only the contribution of app was evaluated. The ensuing displacement field
was then inserted into the bridging stress function,

(𝛿( )) shown in equation (28) and the

integral was re-evaluated.
This procedure was repeated until adequate convergence was observed. GaussChebyshev quadrature was used to evaluate the integral because a singularity exists in the weight
function at x=a. This type of quadrature is useful in evaluating functions with singularities and as
such, yielded more accurate results than with a normal Gauss integration scheme.
The corrected crack length was determined by satisfying the condition that the peak to
peak CMOD for the traction-less crack, 0,TL must be equal to the peak to peak CMOD for a
crack with tractions, 0,T under the same applied load. Note that the peak to peak and not the
total crack opening displacements were used since the author is determining an equivalent cyclic
crack resistance curve. The reason for this is that the author wanted to compare the equivalent
63

cyclic crack resistance curve under quasi-static and fatigue loading cases. Moreover, since plastic
deformations are larger in fatigue, as shown in Subramaniam et. al (2000), the author decided to
neglect the plastic deformations and use the peak to peak (or tangent) approach to evaluate crack
lengths in order to provide a more reasonable comparison.
First, the P-aeq curve was evaluated at a given point (Papp*, aeq*) as shown in Figure
19. Next, the crack mouth opening displacement 0,TL, for the given applied load, Papp* and
the equivalent crack length, a0 +aeq*, was evaluated with equation (31). The crack mouth
opening displacement with the presence of bridging stresses, 0,T, was then evaluated. In the
first loop, 0,T < 0,TL, because the bridging stresses were resisting crack opening. In the next
iteration, the length of the crack with the tractions was increased and CMOD was re-evaluated.
An optimization scheme in MATLAB was used to determine the corrected crack length, acorr,
that satisfied the condition 0,T = 0,TL, under Papp*.
illustration of this condition.

64

Refer to Figure 17 to see an

Papp*, aeq*

Figure 19: Point on the P-aeq curve

Force and Moment Equilibrium
The second loop in the optimization scheme dealt with the task of satisfying internal force and
moment equilibrium at the mid-span point along the beam. This type of equilibrium condition
was also used in the works of Iyengar et. al (2002), Zhang et. al (2001), and Wu et. al (2006).
Iyengar et. al proposed a solution that used a moment-rotation formulation to determine the
relationship between the power softening curve and ‘size effect’. Zhang et. al used a hybrid
LEFM solution, similar to the one proposed here (although without a successive approximation
scheme) that determined the crack opening displacements with fracture mechanics equations and
evaluated the capacity of the beams in fatigue with a moment equilibrium condition. Wu et. al
proposed an analytical solution using LEFM to determine the capacity of a three point bending
beam with a moment equilibrium condition, under the assumption that the crack opening
displacement profile was linear.
65

Under quasi-static loading, the stress distribution was assumed to follow a linear stress
distribution above the corrected crack length, denoted as 2, and follow a stress softening
relationship shown in equation (28) between the tip of corrected crack length and the top of the
notch or CTOD, denoted as 1. The mathematical description of the stress over the entire midspan beam depth is shown in equation (45), and Figure 20 shows an illustration of the
distribution.

≤
(𝛿( ))
{

≤

𝑓 (

≤

)
(

≤

≤ }

)

The centroid, C, (or zero-stress location above the notch) was found using an
optimization function in MATLAB. The initial value for C was

½ (acorr+ D), where D is the

depth of the beam. It was solved by integrating the stress distribution caused by the applied load,
Papp*. Equation (46) shows the force equilibrium condition from which the Centroid, C, was
determined.

∫ (

)
(

)

66

FR,C

2,C

Centroid, C

d
FR,T

Beam

2
2,T

Depth

1

Corrected
Crack
Notch

Figure 20: Bending and bridging stresses at mid-span

The stresses were then partitioned at the centroid, C, into two separate parts: the
compressive stresses above C and the tensile stresses below C. The resultant force for each part
was determined with equation (47). Note that both the resultant forces, FR,C and FR,T, should be
equal since equation (46) states that the total net force acting along the cross-section of the beam
is zero. The centroid for the tension and compression sides were determined with equations (48)
and (49). The moment arm, d, was then determined by subtracting the distance CC from CT as
shown in equation (50).

∫
(47)

67

∫

(

)

∫

(

)

(

)

Finally, the internal moment, M, was calculated by multiplying either FR,T or FR,C by the
moment arm, d, as shown in equation (51). The internal moment was then converted to a point
load, denoted as Pint through equation (52). The conversion was made in order to compare the
internal forces to the external forces. Note that the moment at mid-span of a simply supported
beam is PL/4. A comparison could have easily been made between external and internal
moments; however, it seemed more reasonable to present the results in these terms so as to be
consistent with common practice in fracture mechanics and express the results in terms of load.
Global equilibrium was established when Papp* = Pint. Note that the global equilibrium
condition was evaluated at each of the eight points obtained from the smooth P-aeq curve. The
error between the points was established with a least squares expression and is shown in equation
(53). The double loop optimization scheme stopped after the correct combination of f’t, wc, c1,
68

and c2 were selected and the minimum error between the two was achieved. Figure 21 shows an
example of a converged global equilibrium solution. Figure 22 shows how the equivalent crack
length deviates from the corrected crack length as the crack moves forward. It should be noted
that the shape of the deviation between crack lengths is similar to the one obtained by
Funfschilling et. al. (2010), for a quasi-brittle material containing a portion of Al203 and Y203.
Figure 22 shows that at the onset of cracking, the two crack lengths do not substantially differ
from each other. However, as the crack moves forward, the deviation grows and eventually
becomes steady.

(

)

(

)

∑(
(

)
)

69

Figure 21: External and internal moment equilibrium

Figure 22: Equivalent and corrected crack lengths
70

One should note that under the premise that the bridging stresses can be back-calculated, the fact
that the method depends on P-aeq is a limitation because the curve may be affected by the rate
of loading. Therefore, only slowly loaded quasi-static tests would be applicable for this particular
method.
s

KR and bridging stress parameters
A total 14 specimens were tested under quasi-static loading using CMOD control, with the
loading rates specified in chapter II. The larger beams (100 mm depth) were tested with two
initial notch to depth ratios, 0: 0.15 and 0.35. The smaller beam sizes (50 mm) had only one
notch to depth ratio of 0.15 to ensure that there was enough clearance between the tip of the
notch and the top surface of the beam. Tables 3 and 4 show the results for all the specimens. The
first column shows the specimen number. The larger specimens are labeled with an ‘L’ and the
smaller specimens with an ‘S’. The second and third column shows the beam depth dimension
and the initial notch size, respectively. Columns 4-6 show the three parameters that govern the
s

specific quasi-static crack resistance curve, KR curve obtained from the cyclic P-CMOD data.
S

The parameters KRC and aeq,c represent the specific fracture toughness and equivalent critical
crack extension at peak load, respectively. Both parameters were found to be size dependent. The
bridging stress parameters, f’t, wc , c1, and c2 are shown in columns 7-10. The tensile strength,
f’t was not found to be size dependent. The critical crack opening displacement, wc, was found to
be size dependent. The two shape parameters, c1 and c2 were not found to be size dependent.
71

Note that the average shape parameters back-calculated here, c1= 4.45 and c2= 25.58, are greater
than the values reported by Xu and Reinhardt (1999a) and Kumar and Barai (2009), c1= 3, c2= 7.
The reason for this is that plastic deformations were neglected from cycle to cycle during the
back-calculation procedure, forcing the shape parameters to be greater than the previously
published results since the COD field would be smaller in magnitude. Therefore, to satisfy
moment equilibrium under a smaller COD field, the shape parameters must be greater, as shown
here.
In Table 4, column 3 shows the elastic modulus, E; it was obtained with the Jenq-Shah
compliance technique over the first cycle and also knowing the initial crack length, a0. Refer to
equation (34) in chapter III for the mathematical formulation. Surprisingly, this property was
found to be size dependent. However, the elastic modulus of the larger beams agreed well with
measurements made separately with cube specimens. The smaller elastic modulus may be
attributed to the small un-notched ligament length which can be influenced by the relatively large
isolated aggregates. For the larger beams, the aggregates are more dispersed because of the larger
un-notched area.
Total and cyclic fracture energy
The last seven columns in Table 4 show the fracture properties for each specimen. It
T

T

includes the total fracture energy, GF , and fracture toughness, KIC , the cyclic fracture energy,
c

c

GF , and fracture toughness, KIC , the ratio between the two toughness’s, and the nondimensional specimen size. The total fracture toughness was determined simply by dividing the
72

total area under the P-CMOD curve by the specimen width, b, and the un-notched ligament
length (D-a0). The cyclic fracture energy was determined by integrating the bridging stress
function (with the back-calculated f,t, wc, c1, and c2 values obtained with the technique proposed
in this chapter), as shown in equation (55). The total and cyclic fracture toughness’s were
determined with equation (54). Note that equation (54) shows the equation for the cyclic fracture
toughness, however, if one replaces the cyclic fracture toughness with the total, the total fracture
toughness can be obtained. The total non-dimensional size is shown in the second to last column
and the cyclic non-dimensional size is shown in the last column. The difference between the two
is the characteristic length lch. The total non-dimensional size uses a characteristic length with
the total fracture toughness.

The cyclic non-dimensional size uses a characteristic length

determined by using the cyclic fracture toughness, as shown in equation (56). The variable D is
c

the specimen depth and lch is the cyclic characteristic length of the material with units of length,
and is defined as the square of the quotient between cyclic fracture toughness and the backcalculated tensile strength (for each individual specimen; therefore subject to change), as shown
in equation (56).
As expected, the total fracture energy is larger for the larger size specimens because the
critical crack opening displacement, wc, is size dependent. The cyclic fracture energy is smaller
than the total fracture toughness, which was also expected since the bridging stress distribution
obtained in the proposed technique neglected plastic deformation. The ratio between the cyclic
and total fracture energy, on average, was between 0.72-0.81. In addition, the total non-

73

dimensional size is smaller than the cyclic non-dimensional size, which was expected because
the total fracture energy is larger than the cyclic.

√

𝐾
(

)

( 𝛿) 𝛿

∫
(

)

(

𝐾
𝑓

(56)

74

)

s

Table 3: KR curve and bridging stress parameters
KRC

s

Specimen

Size
(mm)

Notch
Ratio

L1
L2
L3

100
100
100

0.15
0.15
0.15

31.31
30.63
29.08

Average
L4
L5
L6

100
100
100
100

0.15
0.35
0.35
0.35

30.34
34.44
25.10*
32.97

Average

100

0.35

33.70

S1
S2
S3
S4
S5
S6
S7
S8
Average
Average
p-value
*outlier

50
50
50
50
50
50
50
50
50
100

0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15,0.35

24.14
22.78
22.99
24.02
20.68
15.24
23.75
16.95
21.32
31.69
0.001

1.5

(N/mm

)

aeq,c
(mm)

mc

f't
(MPa)

wc
(mm)

c1

c2

7.36
6.65
6.71

0.30
0.21
0.23

4.31
4.57
4.28

0.56*
0.33
0.31

4.49
4.46
4.50

25.40
25.41
25.42

6.91 0.25
7.23
0.19
7.06
0.21
13.95* 0.29

4.39
5.13
3.69
3.70

0.32
0.29
0.28
0.42

4.48 25.41
4.54 25.44
6.69* 26.26
4.26 25.54

0.23

4.17

0.33

4.40

4.47
0.32
2.80
0.18
10.39* 0.31
9.39* 0.22
3.81
0.13
4.56
0.27
3.69
0.16
2.23
0.20
3.59 0.22
7.00 0.24
0.086 0.65

4.39
5.09
3.23
4.00
4.60
2.84
5.12
3.87
4.14
4.28
0.714

0.23
0.18
0.30
0.21
0.16
0.28
0.19
0.20
0.22
0.32
0.019

4.91 25.08
4.98 25.02
5.01 25.09
4.96 25.09
5.02 25.06
2.50* 24.70
5.00 25.05
4.99 25.11
4.98 25.02
4.45 25.58
0.76 0.009

7.14

75

25.7

Table 4: Fracture characteristics under quasi-static loading

T

T

1.5

Specimen Size (mm) E (MPa) GF (N/mm) KIC (N/mm
L1
L2
L3
Average
L4
L5
L6

100
100
100
100
100
100
100

30646
33039
36781
33489

Average
S1
S2
S3
S4
S5
S6
S7
S8
Average
Average
p-value

c

c

1.5

) GF (N/mm) KIC (N/mm

c

) KIC /KIC

T

T

c

D/lch

D/lch

n/a

n/a

0.098

54.75

n/a

n/a

0.619

0.096

56.37

0.061

44.73

0.79

0.66

1.05

0.079

53.75

0.053

44.35

0.83

0.63

0.93

0.087

55.06

0.071

47.94

0.81

0.65

0.87

33122
28967

n/a
0.075

n/a
46.46

0.060
0.043

44.66
35.10

n/a
0.76

n/a
0.63

1.32
1.10

30831

0.086

51.62

0.062

43.77

0.85

0.51

0.71

100
50
50
50
50
50
50
50
50

30973
36774
26217
21013
22966
19608
16935
22380
21171

0.080

49.0

0.055

41.18

0.80

0.57

1.05

0.091
0.060
0.072

58.00
39.65
38.94

0.043
0.038
0.041

39.73
31.74
29.34

0.69
0.80
0.75

0.29
0.83
0.34

0.61
1.29
0.61

0.068

39.53

0.035

28.39

0.72

0.51

0.99

0.063
0.061
0.080
0.063

35.07
32.14
42.41
36.41

0.030
0.032
0.041
0.032

24.20
23.24
30.11
25.88

0.69
0.72
0.71
0.71

0.86
0.39
0.73
0.56

1.81*
0.75
1.45
1.12

50
100

23383
32231
0.006

0.070
0.084

40.27
52.05

0.036
0.063

29.08
44.56

0.72
0.81

0.56
0.61

0.97
0.96

0.049

0.007

0.001

0.002

0.004

0.608

0.552

76

Comparison of KI and KR curves with inclusion of KIC

ini

Following the double K criterion proposed by Xu and Reinhardt (1999a), the KI and KR curves
were compared. KI and KR were determined with equations (57) and (58)

𝐾( )

(

∫

)

( )

(57)

𝐾 ( )

(

∫

)

( )

(58)
The KIC

ini

term, which describes the initial fracture toughness of the concrete, was

determined by simply subtracting the difference between KI and KR at the critical corrected
crack extension, ac,corr, which corresponds to the peak load. When this subtraction is made, the
KI and KR curves overlap, suggesting equilibrium is maintained as the crack grows under
displacement control. These results are in slight contrast to the ones published by Xu and
Reinhardt in that KI and KR shown here are tangent at multiple locations and do not just intersect
at the peak load. However, the results shown here imply that stable fracture is maintained for
most of the crack growth and that KIC

ini

does not seem to vary significantly (until well beyond

the critical crack extension, as shown in Figure 23), a condition that has been maintained by
77

researchers investigating other types of quasi-brittle materials (Gallops, S., Fett, T., Ager III,
J.W., Kruzic, J.J., 2011; Kruzic, J.J., Cannon, R.M., Ager III, J. W. ,Ritchie, R. O., 2005; S.
Funfschilling, T. Fett, S.E. Gallops, J.J. Kruzic, R. Oberacker, M.J. Hoffman, 2010).

Figure 23: KI v. KR curve

Table 5 shows the initial fracture toughness for all six 100 mm depth quasi-static
specimens. The average was 0.40 MPa-m

0.5

for the large beams and 0.271 MPa-m

0.5

. These

values are lower than the published values shown in Kumar and Barai (2010), which can be
explained with the following rationale: 1) the concrete mix used here exhibits lower fracture
toughness, 2) the calculated critical crack lengths at failure using the linear asymptotic
assumption are longer (since a secant compliance is used) which translates to larger fracture
toughness and 3) since the crack opening displacement field is determined using an iterative
78

successive approximation, the contribution of the bridging stresses (and thus KR) will tend to be
larger than when the bridging stresses are simply superposed onto a traction-less crack opening
displacement field and thus, necessitate the condition that KIC

ini

be larger to satisfy the

requirement that the applied stress intensity and crack resistance be equal at peak load. In
addition, KIC

ini

is shown to be size dependent, which is consistent with the results found for the

specific fracture toughness, KIC

s
.

Table 5: Initial fracture toughness using acorr for 100 mm specimens under quasi-static
loading
Specimen

KICini

Specimen

KICini

units

L1

11.46

S1

8.73

N/mm1.5

L2

12.81

S2

10.42

N/mm1.5

L3

11.52

S3

6.36

N/mm1.5

L4

16.99

S4

7.94

N/mm1.5

L5

11.81

S5

10.80

N/mm1.5

L6

11.99

S6

5.39

N/mm1.5

n/a

n/a

S7

10.79

N/mm1.6

n/a

n/a

S8

8.21

N/mm1.7

Average

12.76

8.58

N/mm1.5

Average

0.404

0.271

Mpa-m.5

Moreover, Kumar and Barai (2012) recently showed that the double K-criterion can yield
size and geometry dependent results. The researchers showed that for a varying, non-dimensional
T

size, D/lch, the ratio KIC /KIC

ini

can vary significantly. The results shown here indicate that the
79

T

ratio KIC /KIC

ini

for both sizes is approximately 3.33, a similar result to what is shown by

Kumar and Barai (they showed a ratio of approximately 4) for a non-dimensional size of around
1 (similar to the size shown here).

One should note however, that the method of determining KIC

ini

shown here is less

sophisticated (Zhang, X., Xu, S., 2011), but it was done to give the reader some context as to the
appropriateness of the entire methodology described herein and to identify similarities.
Determination of the equivalent crack resistance curve
The use of LEFM equivalent crack lengths has been shown to be an efficient way of
characterizing the state of damage in a structure. In fatigue for example, Subramaniam et. al
(2000) used the Jenq-Shah compliance method to determine the equivalent crack lengths until
unstable failure and showed that the critical equivalent crack extension under quasi-static loading
can be identified in the log(daeq/dN) curve. In addition, the researchers showed that the fatigue
specimens failed at the same fracture toughness as the quasi-static specimens. The implications
of these results suggest that the cyclic bridging stresses in fatigue must be similar and/or
relatable to that under quasi-static loading. Morel et. al (2010) recently showed that the bridging
stress parameters: f’t, wc, and Gf/GF (ratio of specific and total fracture energy) govern the early
shape of the equivalent R-curve, critical crack extension, and peak load, respectively. Therefore,
if wc governs the equivalent critical crack extension, aeq,c, which happens to be the crack
extension corresponding to the bending point in the log(daeq /dN) in fatigue (refer to Figure 10),
it leads to a conclusion that the fatigue bridging stresses have similar characteristics to that

80

observed under quasi-static loading. Moreover, under constant amplitude loading, in order for
the log(daeq/dN) v. aeq curve to have zero slope, it implies that at this equivalent crack length,
the rate of change between the applied stress intensity and the crack resistance must be equal.
This is a similar trait observed at failure under quasi-static loading. Therefore, it seems plausible
that the shape of fatigue and quasi-static R-curve is similar, and hence, so are the bridging
stresses. Kruzic et al. (2005) measured the cyclic resistance curve for Alumina and showed that it
also shared similar characteristics to quasi-static resistance curves in that there is a pre-peak
decreasing slope region, and a post-peak steady state region.
The purpose of this sub-section is to show the steps necessary to obtain the equivalent
cyclic crack resistance curve. Using the found bridging law parameters, the equivalent cyclic
crack resistance curve was obtained with equation (59). This equation states that the equivalent
crack resistance can be obtained by evaluating the contribution of the bridging stresses with a
weight function, at a given equilibrium position (Papp*=Pint at aeq), between the bounds of the
initial crack, a0, and the equivalent crack aeq (obtained from the P-aeq curve). Also note that
the equivalent resistant curve was evaluated at every equilibrium point along the P-aeq curve.
The term f’teq represents the tensile stress at the location x = aeq. This value will change
depending on the difference between the corrected crack and the equivalent crack length. For
simplification, an average value of f’teq over the crack range 0.15*D (or 0.35*D, depending on
the initial crack geometry) to 0.65*D was used and this average value will be denoted henceforth
simply as f’teq. The upper bound of 0.65*D was chosen because this was the upper bound of the

81

weight function. Table 6 shows f’teq for all specimens and the ratio between f’t and f’teq. On
average, the ratio was approximately 0.75. Successive approximation was not used to determine
the equivalent crack resistance here, meaning the solution was assumed to converge on the first
iteration. The reason this was done was to satisfy the condition that over the equivalent crack
domain, the CMOD need be equal to the CMOD from the corrected crack domain over the same
applied load. In summary, the equivalent bridging stresses (with parameters f’teq, wc, c1,and c2)
were evaluated over the traction-less crack opening displacement field that exists over the crack
domain of a0 and aeq.

𝐾

(

)

∫

(

)

(𝛿( ≤

≤

))

(59)

f't
f'teq

acorr

2

GFc,eq=1/E*KRc,eq

aeq

Figure 24: Bridging stresses acting over the equivalent crack domain

82

Figure 24 shows the integration bounds between the initial crack length and the
equivalent crack length. Figure 25 shows how the corrected and equivalent crack resistances
differ as the crack propagates forward. This result is expected since there was a substantial
deviation between the two crack lengths as shown in Figure 22. Note that here, the initial fracture
toughness was assumed to be zero in both curves.

Figure 25: Equivalent and corrected crack resistance curve

Application to fatigue loading
Since compliance techniques are easily used (and preferred by many researchers) to determine
crack lengths in fatigue, it has created a need to quantify crack resistance in the equivalent
domain. In this last section, a new method to determine the equivalent crack resistance with
moment equilibrium and combined use of the corrected and equivalent crack length was shown.
83

In this section, a method to determine the equivalent crack resistance curve under a fatigue
loading scenario is described. The main difference here is that instead of evaluating the
resistance curve at the equilibrium loads from the P-aeq curve, it is evaluated under a set of
loads ranging from 0 to Pmax (the peak load sustained by a given specimen).
The procedure for obtaining the fatigue equivalent crack resistance curve is the following: 1)
determine the bridging stress parameters using moment equilibrium and the corrected crack
length, 2) fit a smooth function (spline interpolation was chosen here) over the aeq (acorr)
curve and 3) evaluate the integral shown in equation (59) under a set of loads ranging from 0 to
Pmax.. A total of 10 equally spaced loads were evaluated at 25 equally spaced aeq (acorr)
points; combining for a total of 250 cases. Note the variable aeq (acorr) denotes the equivalent
crack extension is a function of the corrected crack extension. Note that as the crack extension
approaches values near 0.3*D, the equivalent resistance starts to become non-linear. Therefore,
only values between a0 and 0.3*D were used in the parameterization process.

84

Figure 26: Equivalent crack resistance under varying loads

The result for specimen L1 is shown in Figure 26. The Figure shows during the early
stages of cracking, a change in the load does not change the trajectory of the resistance curve
significantly. The reason is because the bridging stresses overwhelm the applied stress and do
not allow the crack extension to open. As the crack elongates, the applied stress begins to
overcome the bridging stresses, and force the crack open, which spawns a change in slope. This
behavior was described mathematically with the function shown in equation (60). It is similar to
85

the power function shown in Bazant and Planas (1998) used to describe the R-curve. The original
function however, had a slope of zero beyond the critical crack extension, ac, and was governed
by three parameters: the shape parameter, m, fracture toughness at peak load, KRc, and ac. The
f

f

function here, denoted as KR (fatigue resistance curve), is governed by 5 parameters: KRc ,

aeq,c, m, 1, and 2. A modification was made to the original function to accommodate the
change in slope beyond the pivot point, as shown in Figure 26.

𝐾

[

(

≤

) ]

𝐾
{

𝐾

(

)
(

≤

}

)

Where

(

(

)

𝐾

)

(

𝐾

𝐾

𝐾

)

[

(
(

86

)

) ]

[

(
The variable

]

)
f

is the crack length where the first and second segments of KR meet

and is obtained by satisfying the condition that the slopes at this point must be equal. Note that
that m must be greater than 1, the ratio

number, and
the equivalent crack,

must be less than 1 to yield a positive-real
f

is a function of applied load. The variable Kc is the fracture toughness at
. The subscript  denotes that the variable varies as a function of

the post peak slope.
The slope in the post-peak region was governed by equation (63). The equation states that
the slope will change as a function of applied load. The relationship between the applied load
and the post-peak slope, , is shown in Figure 27 along with the smooth fit obtained with
equation (63). The variable 1 is the initial slope corresponding to the scenario where applied
load is zero. The variable 2 characterizes the relationship between the load and post-peak slope.

87

Figure 27: Change in the slope of the post-peak crack resistance curve as a function of load

The five parameters were obtained with a non-linear regression scheme in MATLAB.
Fitting the behavior with equations (60-63) yielded adequate results, as shown in Figure 26. The
f

solid lines represent the smooth fit provided by the KR function.
Results
Table 6 shows the results from the 14 specimens. Four of the five parameters are size dependent:
f

KRc , aeq,c, 1, and 2. The equivalent critical crack extensions obtained through non-linear
regression agreed well with the ones obtained from experiments: 100 mm: (7.00, 6.69) and 50
mm: (3.59, 3.47). This means that the functional form governing the equivalent crack resistance
curve shown in equation (60) seems to be adequate in describing the behavior under varying
loads and in identifying the critical crack extension.
88

Table 6: Equivalent crack resistance curve parameters
f

Specimen

Size
(mm)

Notch
Ratio

KRC
(N/mm1.5)

aeq,c
(mm)

m

1

2

f'teq

f’teq/f’t

L1
L2
L3

100
100
100

0.15
0.15
0.15

12.18
10.98
10.69

6.82
4.96
5.46

2.39
1.82
2.01

1.53
1.52
1.45

-0.86
-1.39
-1.25

3.28
3.22
3.07

0.76
0.70
0.72

Average
L4
L5
L6

100
100
100
100

0.15
0.35
0.35
0.35

11.28
10.94
8.20
9.89

5.75
3.55
4.10
5.56

2.08
2.05
2.35
2.47

1.50
1.91
1.39
1.49

-1.17
-1.35
-1.21
-0.83

3.19
3.92
2.85
3.05

0.73
0.76
0.77
0.83

Average

100

0.35

9.68

4.41

2.29

1.60

-1.13

3.28

0.79

S1
S2
S3
S4
S5
S6
S7
S8
Average
Average
p-value

50
50
50
50
50
50
50
50
50
100

0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15,0.35

8.10
7.42
6.29
6.69
3.18
5.43
7.49
6.12
6.34
10.48
0.000

2.86
1.91
3.22
2.53
0.97
3.37
1.87
2.17
2.36
5.08
0.001

2.45
1.87
2.40
2.08
1.26
2.52
1.41
1.74
1.97
2.18
0.341

2.31
2.39
1.64
1.91
1.29
1.42
2.28
1.77
1.88
1.55
0.077

-0.72
-1.40
-0.81
-1.23
-3.11*
-0.96
-1.86
-1.38
-1.19
-1.15
0.811

3.43
3.54
2.49
2.86
1.94*
2.13
3.42
2.70
2.94
3.23
0.287

0.78
0.69
0.77
0.71
0.75
0.67
0.70
0.73
0.76
0.205

Figures 28-31 show the relationship between the bridging parameters, specimen size, and
equivalent fatigue crack resistance curve parameters. In summary, the total fracture toughness,
T

f

T2

KIC , is related to the equivalent fatigue fracture toughness, KRc , and the ratio, (wcE/KIC )

(units of length), is linearly related to the equivalent critical crack extension, aeq,c. Morel et al.
(2010) showed the non-dimensional equivalent critical crack extension is related to the square of
the non-dimensional crack opening displacement and non-dimensional size. In addition, using
the Jenq-Shah model, one may prove that the critical equivalent crack is proportional the ratio
89

T2

(wcE/KIC ) , as shown in Bazant and Planas (1998). In general, the results shown here agree well
with those findings.

Figure 28: Relationship between total fracture toughness and equivalent fracture toughness
in fatigue

90

Figure 29: Relationship between the equivalent critical crack extension and the critical
crack opening

The two parameters governing the shape of the post-peak slope, 1 and 2, are linearly
c

related to the ratio f’t/D, and the non-dimensional size, D/lch , respectively. The relationship
between 1 and f’t/D seems reasonable since the initial slope is governed by the bridging stresses
and for small sizes (or smaller crack lengths) they overwhelm the applied stresses, keeping the
crack opening displacement at a minimum, which in turn leads to a steeper post-peak slope. The
c

relationship between 2 and D/ lch can be explained using the same rationale: The larger size
specimens (or larger crack lengths) will begin to overcome the bridging stresses because the
stress intensity will increase, and thus will create a situation where the post-peak slope change is
more sensitive to changes in applied load.
91

2

The regression error shown in Figures 28-31 (R = 0.94, 0.89, 0.82, 0.88) can be
explained by the following rationale: 1) error was incurred in the non-linear fitting process and
2) as shown in Morel (2010), there may be more than one variable related to a given bridging
stress parameter.

Figure 30: Relationship between the initial post peak slope and the ratio between the tensile
strength and specimen size

92

Figure 31: Relationship between the applied load-crack resistance slope sensitivity and
non-dimensional size

93

CHAPTER V

THE DETERMINATION OF AN EQUIVALENT CYCLIC CRACK RESISTANCE
CURVE UNDER CONSTANT AMPLITUDE LOADING

Fatigue damage is a time, moisture, temperature, and loading dependent process and is a
major factor in the loss of material structural integrity. For concrete structures, fatigue damage is
typically observed in bridges, off-shore structures, and pavements. These particular structures
are continuously subjected to cyclical forces and changes in environmental conditions which
cause incremental damage over time and can eventually lead to failure. To prevent this, damage
must be evaluated periodically and repaired prior to unstable crack growth. This means that
damage must be defined physically in order to be assessed and contained in the field.
Traditionally, fatigue damage has been quantified using Miner’s Law that defines damage
through the Number of cycles to failure, D = 1/Nf, which is a function of the applied stress ratio
(/MR) under constant amplitude loading (Miner, 1945; Okamoto, 1999; Oh, 1991) One of the
main advantages of using this type of damage model is its computational efficiency. Some
performance based design codes (e.g. concrete pavement design) use the S-N approach because
currently, it is the only feasible way of rapidly accounting for, processing, and converting
millions of load repetitions to damage (Guide for Mechanistic-Empirical Design of New and
Rehabilitated Pavement Structures, 2004). This allows for multiple designs to be considered
within minutes. Some of the disadvantages however, are that it is insufficient in determining the
in-situ state of damage because no information is given on the state of the material itself (no
94

information on the stress-strain behavior and the reduction of the Elastic Modulus). In addition,
it cannot account for size effect, load history effect, and variable amplitude loading without using
some empirical calibration factors. Thus, there is a need for the development of a concrete
fatigue model that can account for all three of the aforementioned effects and in addition, be able
to maintain a comparable level of computational efficiency to the S-N approach.
The objectives of this chapter are to present a new method of determining a fatigue crack
resistance curve under cyclic loading. The results at the end of the chapter show that if the postini

peak slope, , and the initial fracture toughness, KIC , are pre-defined, one arrives at a unique
expression for the fatigue crack resistance curve. The purpose of using the crack resistance curve
in the formulation is that it allows for a continuous prediction of fatigue, which is necessary if
variable amplitude loading scenarios are considered and it could also be used to explain the size
effect.

Experimental test setup and mix characteristics
A total of 10 large beam specimens were tested under constant amplitude fatigue loading. The
notch to depth ratio () for each specimen was 0.35.

Each specimen was subjected to a 2 Hz

cyclical load with an R-ratio (min load/max load) of 0.05. Half of the fatigue specimens (5)
were subjected to a stress ratio (max load/peak load) of 0.85 and the other half to a stress ratio of
0.95.

95

Crack length measurements
Peak to peak load and CMOD measurements were recorded after each cycle. After a test was
completed, compliance calculations (using a tangent modulus, or peak to peak response; plastic
deformation was neglected) were made. Next, the Jenq-Shah compliance technique was used to
determine the crack length after each cycle. Please refer to chapter II for a detailed description
of the technique. The crack data was then filtered by trimming the size so there was only one
data point per each 0.01 mm increase in crack length. From here, the data was converted to crack
rate by dividing each crack increment, aeq, by the number of cycles, N, needed to produce the
new crack increment. The crack rate data was then smoothened by a cubic piecewise regression
function. The smooth crack rate data was then used to derive the fatigue crack resistance curve.
Figure 32 shows an example of the difference between the trimmed down crack rate data and the
cubic piecewise regression fit.

96

Figure 32: Smoothened crack rate data versus crack length
Fatigue crack resistance curves
f

In this section, a new method to determine the fatigue crack resistance curve (KR ) is proposed.
f

The KR curve can be obtained if the crack propagation rate (daeq/dN), the equivalent crack
extension (aeq), and the stress intensity (KI) are known. There is a direct relationship between
the decreasing crack propagation rate (transient stage) and the crack resistance curve. A unique
f

description of the KR curve can be obtained by satisfying the following three conditions: 1) An
f

intrinsic linear relationship between log(KI -KR ) and log(daeq/dN) exists (Li, V.C., Matsumoto,
T., 1998), 2) The initial fatigue cracking resistance is zero, and 3) The fracture resistance in the
97

post peak region (after aeq,c has been reached) should be a constant (zero slope), as observed
by Kruzic (2005) for Alumina under fatigue loading.
f

The derivation of the KR curve is shown in equations (64-66) Equation (64) shows the
fatigue cracking equation, which is simply a Paris Law equation that was modified by
f

subtracting the stress intensity term, KI with the KR term which satisfies the first condition.
f

Rearranging equation (64) and solving for KR yields equation (65) where f(aeq) is shown in
equation (66).

(

(

))

(𝐾 (

)

𝐾 (

))

(64)

𝐾 (

𝐾(

)

)

𝑓(

)

(65)

𝑓(

)

(

(

(

))

( ))

(66)

Note that the critical crack extension, aeq,c corresponds to the aeq value at which

𝑓

=0

(see Figure 33). To elaborate, recall that for quasi-static loading, the definition of the R-curve
states that at the critical crack extension, the KI and KR curves must be tangent (recall equations
98

f

20 and 21). For fatigue loading, by definition, the KR curve must be below the KI curve.
Therefore only equation (21) applies. Taking the derivative of equation (65) leads to equation
(67).

𝐾

𝐾

𝑓

(67)

For equation (21) to be satisfied,

𝑓

, must be equal to zero. Next, using condition 2, shown in

equation (68), an expression for C can be derived, as shown in equation (69).

𝐾 (

)
(68)
( )

(

( ))

(69)
Condition 3 states that the derivative of equation (65) is zero at some arbitrary crack extension
beyond aeq,c:

𝐾

(70)

99

This leads to the condition shown in equation (71).

𝐾

𝑓

(71)
The fatigue parameters C and n can then be solved for simultaneously by trial and error until
f

equation (71) is satisfied. This is shown graphically in Figure 33. The resulting KR values as a
f

function of aeq are shown in Figure 34 as individual data points. Note that KR =0 at a =0, per
condition 2.

100

f

Figure 33: Satisfying condition 3: dKR /daeq is zero beyond aeq,c

101

f

Figure 34: KR shown as a function of aeq,c

Figure 34 also shows the corresponding fit using equation (72) which has the same form as
equation (39) except that it is only valid for

≤

f

, where

is the crack

f

extension at which the KR function becomes constant (matching the KR plateau). A non-linear
least squares regression technique in MATLAB was used to determine the values of each of the
parameters aeq,c and m. The parameter aeq,u can be determined by adding a shift factor
obtained from fatigue tests, toaeq,c ;i.e., aeq,u = aeq,u+ 

102

𝐾
{

𝐾

(

𝐾

)

≤

𝐾

}
(72)

Experimental Results
Ten specimens were tested under fatigue loading, the results of which are shown in Table 7. The
average Paris fatigue coefficients log C and n are -23.46 and 16.85, respectively. The average
f

maximum fatigue resistance, KRc , and the critical crack length are 13.25 N/mm3/2 and 5.80 mm,
respectively. Table 7 shows the p-values from an independent t-test between the fatigue
parameters at 0.95 and 0.85 stress ratios. The smaller the p-value, the greater the probability the
two populations have different means (A p-value of 0.05 would mean that there is 95% chance
the two means are different).
The two stress ratio population means (0.95 and 0.85 stress ratios) are statistically
different (using a 95 % confidence interval). However, the smallest p-values occur for the aeq,f
f

m and KRc populations. This could be explained as follows: 1) aeq,f is larger for the lower
stress ratio because the crack must elongate further before failure in order to reach the fracture
toughness in fatigue; 2) The m parameter is 0.95 at the higher stress ratio as compared to 0.69 for
the lower stress ratio because of the bridging stresses deteriorating more rapidly; 3) KRcf is lower
at the higher stress ratio because of a smaller maximum bridging stress, which is consistent with
the cyclic cohesive zone model proposed by Roe and Siegmund (2003).

103

Table 7: Summary of constant amplitude fatigue results
Stress
Ratio
0.95
0.95
0.95
0.95
0.95
Average
0.85
0.85
0.85
0.85
0.85
Average
p-value
Combined
Average

f

KRc

3/2

aeq,u

m

f

aeq,c

aeq,f

KIC

(mm)
8.58
8.42
15.13
17.20
11.89
12.24
15.20
10.85
19.14
18.74
34.03
19.59
0.20

(N/mm )
32.04
31.90
39.22
44.36
38.60
37.22
38.90
33.97
38.22
37.65
43.50
38.45
0.68

log C

n

-19.16
-31.58
-23.24
-22.84
-26.18
-24.60
-20.07
-28.74
-27.25
-14.66
-20.88
-22.32
0.51

13.50
23.00
16.00
16.00
18.00
17.30
15.00
22.00
20.00
10.00
15.00
16.40
0.74

15.92

37.84

-23.46

16.85

(N/mm )
7.19
9.35
13.14
17.94
13.10
12.14
17.59
12.59
14.55
9.98
17.04
14.35
0.370

(mm)
4.04
7.01
5.07
8.59
11.75
7.29
11.36
4.55
10.12
7.14
9.20
8.47
0.534

0.80
0.91
0.81
1.66
0.54
0.94
0.50
0.91
0.83
0.66
0.58
0.70
0.26

(mm)
3.82
6.25
4.75
9.25
4.07
5.63
8.12
4.32
7.15
5.62
4.61
5.96
0.79

13.25

7.88

0.82

5.80

3/2

s

Comparison between quasi-static specific KR curve and cyclic resistance curve
In this section, we compare the important crack resistance parameters under quasi-static and
fatigue loading. Referring to Table 3 (quasi-static) and Table 7 (constant amplitude fatigue), we
can make the following observations:
The average critical crack extension, aeq,c for the fatigue crack resistance curve (5.80
mm) was not statistically different from the observed in quasi-static resistance curve (7 mm). An
independent t-test was used to compare the mean difference between the populations and it
yielded a p-value of 0.532. This corroborates one of the conclusions made by Subramaniam et al.

104

(2000) which stated that the crack extension at aeq,bend in fatigue is equivalent to aeq,c in
quasi-static loading.
f

The average value for the fracture toughness in fatigue, KIC , at failure, is 37.8 N/mm3/2.
s

This value is greater than the value recorded under quasi-static loading, KIC , which was 30.59
N/mm

3/2

. Statistically, the means are different (p=0.001). The reason for this is that there is an

increasing quasi-static crack resistance beyond the critical crack extension. Furthermore, the
average crack extension at failure in fatigue, (aeq,f=15.92 mm) is greater than the crack
extension at peak load under quasi-static loading (aeq,c=7.00 mm). This is expected since the
fracture toughness in fatigue is higher than that under quasi-static loading and the applied stress
in fatigue is lower than the peak stress under quasi-static testing.
f

The average values for m and KRc , were 0.82 and 13.25 N/mm

3/2

, respectively, which

s

are 3.56 and 0.44 (respectively) times the average value for m and KIC observed in the quasistatic fracture resistance curves (0.24 and 30.59 N/mm3/2, respectively). The reason is because
there is a build-up of damage within the bridging stress zone which will reduce the magnitude of
the peak bridging stresses and therefore reduce the initial slope of the cyclic resistance curve.
This is consistent with the cyclic cohesive damage model proposed by Roe and Siegmund (2003)
and later extended to multiple-dimensions by Jiang et al. (2009) which states that the peak
bridging stress will erode depending on the level of damage found within the zone and will be
governed by the magnitude of the displacement (of the cohesive bridging zone) occurring within
each cycle. Equation (73) shows how the magnitude of the bridging stresses will change as a
105

function of a damage variable D, which varies from 0 to 1 depending on the level of sustained
damage within the bridging zone.

(

)

(73)
s

f

Figure 35 shows the average KR and KR curves (using the average parameters from and Table 7
as well as constant load KI curve and relevant crack extension and fracture parameters. The KR

s

f

and KR curves can be thought of as the upper and lower bounds for fatigue damage. The KI
f

curve is the forcing function; when it is above the KR curve, fatigue damage occurs, and when it
s

reaches the KR curve, failure occurs.
One interesting conclusion that can be drawn, besides the critical crack extensions being
f

f

statistically equal, is that the maximum value of the KR curve (denoted as KRc, in Figure 35) is
s

on average (using all fatigue specimens), approximately 44% of the average KRc value from the
f

quasi-static tests. In addition, the power m for the KR curve is approximately 3.56 times the
f

magnitude of the power m under quasi-static loading. So in theory, one may relate the KR curve
s

to the KR curve using the expression shown in equation (74). For this concrete mix, 1 = 0.44
and 2=3.56. Recall that aeq,u = aeq,c +

106

s

f

Figure 35: KR and KR resistance curves and KI
107



𝐾

𝐾

(

)

(74)

Comparison to previously published results
Using data from for quasi-static fracture tests reported by Subramaniam (1999) five fracture
resistance curves were developed. This was done by using the peak load and the critical crack
length along with the maximum loads and corresponding crack extensions for the various
loading/unloading post peak cycles. The plain concrete beams had similar dimensions to the ones
tested in this study. The notch to depth ratio () was 0.35, the depth was 100 mm, the width was
50 mm and the span was 400 mm. The average critical crack extension aeq,c was 9.97 mm.
s

s

The critical stress intensity at failure (KIC =KRc ), and the power m, was 40.48 N/mm

3/2

, and

0.26, respectively.
In fatigue, two stress ratios were analyzed: SR = 0.94 and 0.74. The results are
f

summarized in. The average KRc, , aeq,u and m were 15.85 N/mm1.5, 11.67 mm and 0.92,
respectively. The average log C and n values were -18.67 and 11.87, respectively. Similar to

f

what was observed in the current tests, the KRc, , values for the lower stress ratio were larger
than those for the higher stress ratio, while the reverse was observed for the power m (which is
consistent with our results). The 1 and 2 parameters of equation (74) were determined to be
108

0.39, and 3.54, respectively. These numbers are very similar to what was observed in the current
tests (0.44 and 3.56). This seems to confirm that at least for this geometry (=0.35), the cyclic
resistance curve seems to follow a similar trend in which the following conclusions can be made:
1) The cyclic resistance curve seems to vary as a function of applied stress level, although the
difference is not statistically significant given the current limited data; (2) averaging the results
f

from different stress ratios produces consistent results with scaling factors 1 of the KRc, to
s

KRc of about 0.4, and 2 of the power m under fatigue loading to the one under quasi-static
loading of about 3.5.

Table 8: Summary of constant amplitude fatigue results using data from Subramaniam et
al. (2000)
f

Stress
Ratio
0.94
0.94
0.94
Mean
0.74
0.74
0.74
Mean
Mean
Total

KRc

3/2

aeq,u

(N/mm )
16.62
15.13
10.03
13.93
17.82
16.20
19.32
17.78

m
1.03
1.00
1.39
1.14
0.56
0.99
0.54
0.69

(mm)
10.00
6.50
6.50
7.67
16.00
12.00
19.00
15.67

log C
-21.71
-9.15
-33.37
-21.41
-15.71
-15.10
-16.97
-15.93

n
14.00
5.50
22.00
13.83
10.00
8.70
11.00
9.90

15.85

0.92

11.67

-18.67

11.87

109

Fatigue prediction using the cyclic resistance curves
f

For fatigue prediction, a better mathematical form to describe the KR, curve is:

𝐾

𝐾

[

(

) ]

(75)
f

In equation (75), aeq,u= aeq,c+ The formulation for KR, in equation (72) was useful
s

for the purpose of comparing the fracture toughness under quasi-static loading (KRc =KIC) to the
f

KRc (the maximum value of the fatigue crack resistance curve) and the power m for both loading
condition. In this section, two separate fatigue simulations were carried out using the results from
two specimens at 0.95 and 0.85 stress ratios. The R-ratio for both specimens was 0.05. Figure 36
f

shows the predicted crack propagation rate using the fitted KR curve (equation 75).

110

There is a good fit between the experimental and simulated results. Equation (76) was then used
to predict the number of cycles to failure. The predicted average number of cycles to failure (Nf)
for 0.85 and 0.95 stress ratios were 818 and 63 respectively. The actual Nf for 0.85 and 0.95
were 944 and 60, respectively.

∫

(𝐾

𝐾 )

(76)

Figure 36: Simulation of crack rate under stress ratios 0.85 and 0.95

111

Discussion
In order to accurately predict the entire fatigue life of plain concrete, it is important to account
for both the transient and steady state stages. The transient stage makes up for approximately 4050% of the concrete’s fatigue lifespan (under constant amplitude loading). In the past, the
transient stage was accounted for by a separate Paris-type fatigue law that only used the crack
extension as its argument. In doing so, one will encounter a situation where the Paris parameters
log C1 and n1 will not be unique. As can be seen from Figure 36, as the applied stress decreases,
the rate of propagation will also decrease as a function of crack extension. This means that in
order to accommodate the reduction in crack rate, the log C1 and n1 values must be altered to
provide lower values over the same crack domain. A second series of simulations was conducted
to show this effect. The average cyclic threshold curve and fatigue parameters log C and n were
used to predict the fatigue crack propagation in the concrete specimens tested by Subramaniam et
al. (2000). Using the crack rates from the simulation, one may see from Figure 37 how log C1
and n1 can vary as a function of applied stress. This matches the researchers’ observations. The
individual points represent the actual data obtained in their experiments and the lines represent
the results from the fatigue simulations.

112

Figure 37: Variation of log C1 and n1 as a function of stress ratio

113

CHAPTER VI

SIZE DEPENDENT FATIGUE CRACK RESISTANCE CURVES UNDER VARIOUS
LOADING REGIMES

In the last chapter, it was shown that an equivalent fatigue crack resistance curve can be obtained
if the post-peak slope and the initial fracture toughness are defined. In this chapter, the influence
of size effect and variable amplitude loading are discussed. A new method that uses the
cumulative crack, rather than crack rate data to determine the Paris parameters, log C and n
under variable and random amplitude loading is proposed. It is consistent with the findings
discussed in the last chapter, that both the initial fracture toughness and post-peak are predefined. It is then shown that with the use of a non-linear regression scheme, if those two
f

conditions are upheld, 2 of the 5 parameters governing the KR curve, aeq,u and 2 may be
f

obtained. The other 3 parameters: KRc , m, and 1, were held constant, using the average values
obtained for each beam size in chapter IV. Finally, the 4 equivalent fatigue crack resistance
parameters and the Paris parameters are compared for different beam sizes and loading regimes.
It is shown that for random loading scenarios, the bending point in the fatigue curve, aeq,bend is
longer than that observed under constant amplitude loading. The reason for this is then discussed.
Variable load effects
Up to this point, it was shown that one may determine an equivalent crack resistance curve from
stress intensity and crack rate data. In addition, it was also shown that one may obtain an
equivalent crack resistance curve under varying loads from an inverse analysis of quasi-static
114

flexural specimens. Note that the differences in shape between two crack resistance curves
mainly comes from the behavior in the post-peak region (beyond aeq,c). One curve was shown
to have a constant post-peak and the other a varying post-peak that was load dependent. In
chapters II and V it was shown that under fatigue loading, concrete exhibits a crack deceleration
and acceleration which creates a ‘v’ shaped crack rate, da/dN, curve. To model this behavior it
was shown that a fatigue crack resistance term must be inserted into the Paris law to describe the
competition between the evolving stress intensities and crack resistance. If the equivalent crack
resistance curve was pre-defined to have zero initial fracture toughness and zero post peak slope,
then a unique expression for log C and n could be obtained under constant amplitude loading.
The question is however: will this post-peak assumption be appropriate to explain a variable or
random amplitude loading case?
As mentioned in chapter III, an extensive series of fatigue experiments was conducted: a
total of 44 fatigue tests, 34 of which are shown in this chapter. The other ten were shown in
chapter V. Of the 34 specimens discussed in this chapter, 6 specimens were subjected to random
amplitude loading, 19 to variable amplitude loading and the remaining 9 to constant amplitude
loading. After reviewing the data, an interesting result was observed under random amplitude
loading. The transition between the deceleration and acceleration region in the ‘v’ shaped curve
for the random amplitude loading scenarios was substantially longer than that under constant
amplitude loading. This means, aside from the fact that there are varying peak loads and R-ratio,
which immediately calls into question the applicability of the original Paris law, if the constant
post-peak slope assumption is used under this type of loading, the ultimate equivalent crack
extension, aeq,u, would also need to be larger. Figure 38
115

shows the differences in the

transitional region between constant amplitude and random amplitude loading tests. As can be
seen, the length of the transition zone for the random amplitude case is approximately 8 mm in
length while it is 3 mm for the constant amplitude loading case..
In chapter V, it was stated that the transition point, or aeq,bend in the ‘v’ shape curve
occurs when the change in stress intensity is equal to the change in crack resistance (with respect
to crack length). Under constant amplitude loading, this occurs at a unique point and could easily
be found graphically as shown in Figure 33. This point on the figure was denoted as aeq,c,
which corresponded to the critical crack extension under quasi-static loading.
Under random amplitude loading, a unique transition point does not exist. The reasons
could be explained by the load-dependent equivalent crack resistance obtained in chapter IV. If
the post-peak slope changes as function of applied load, then presumably so will the transitional
point, or aeq,bend. If the assertion that this point will occur when the rate of the stress intensity
is equal to that of the crack resistance is held, and if this resistance is changing due to changes in
load, the obvious result should then lead to a change in aeq,bend.. Of course, the transition will
cease after the crack has elongated substantially and the steepest post-peak crack resistance slope
is smaller in magnitude than the average rate of the stress intensity under random amplitude
loading. Therefore, in general, using an equivalent crack resistance curve that is dependent on
load is more powerful than using a constant post-peak region when attempting to predict the
cracking mechanism under random and/or variable amplitude loading.

116

(a)

(b)
Figure 38: Differences between transition zone for (a) random loading and (b) constant
amplitude loading

117

Size effect in fatigue
Size effect in fatigue has been studied by several researchers. Bazant and Xu (1991) showed that
the logC Paris parameter will decrease as the specimen size is decreased. The researchers also
determined a fatigue characteristic length using the size effect law and showed that if the applied
stress intensity is normalized, the size dependent crack rate curves will consolidate and form one
unique curve. Sain and Kishen (2007) showed that the C term in the Paris law follows a linear
relationship with non-dimensional specimen size, D/lch and also, they determined the moment
capacity, and fracture resistance with a Greens function. Zhang et al. (2001) used a moment
equilibrium condition and LEFM solutions to show the flexural capacity of beams of various size
depths under fatigue loading.
The works mentioned above tried to predict the influence of size by considering the
presence of bridging stresses. In this chapter, this is done by using an equivalent fatigue crack
resistance curve.
Calibration using the cumulative crack propagation
It was mentioned previously that crack length measurements were made with the Jenq-Shah
compliance technique at every cycle spanning an entire fatigue test. It was mentioned that the
crack data was filtered by a trimming process and then smoothened with a cubic piecewise
regression function. The crack length data was originally filtered because on a per-cycle basis the
changes in compliance were not large enough to overcome the noise level of the measuring
device. Therefore, the original crack data vector was trimmed and smoothened so that the crack
rate was decipherable. For constant amplitude loading tests, this data filtration process works

118

reasonably well since the applied load is constant and the relationship between stress intensity
and crack rate is easily identified.
Under random amplitude loading, the filtration process described above is insufficient
because the load changes after each cycle. This means that if the data is trimmed, vital load
history information will be lost and the calibrated model will not be explaining the true behavior
of the cracking mechanism under random/variable amplitude loading. Therefore, a different
method based on cumulative crack data was used and the corresponding steps are described
below:


Equivalent crack lengths were recorded at each cycle in a given fatigue test.



Stress intensities were then determined with equations (36-38) for each crack length
and corresponding applied load.



The R-ratio (Pmin/Pmax) was recorded for each cycle.



A second equivalent crack data vector was recorded; it uses only selected points
along the original crack data vector: the specific data point was chosen at every 0.25
mm increase in crack length. This data vector consisted of j elements.



The second equivalent crack data vector was then converted to an equivalent crack
exp

extension, aeq,j

by subtracting the initial crack length, aeq0 and it consisted of j

elements.


A cycle increment data vector, Nj, with the same number of elements as aeq, j

exp

was then constructed. Each element within the data vector showed the number of

119

cycles necessary to produce a crack increment of 0.25 mm, e.g., one element in Nj
would correspond to the number of cycles needed to go from aj-1 to aj-1 + aj.


The Paris and equivalent crack resistance parameters were then determined using
equation (77-78) and a non-linear least squares approach.

∑

∑ (𝐾

𝐾 )

(77)

∑ (

𝑟𝑟

)
̅̅̅̅̅̅̅̅̅̅̅̅)

∑ (
(78)

Where
The total number of cycles to failure was the following:

∑
(79)
Determining the fatigue parameters
f

The fatigue parameters: log C, n, KRc , aeq,u, m, and 2, were obtained for each of the 34
beams tested under constant, variable and random amplitude loading using a non-linear
regression scheme. Recall from chapter V that to arrive at a unique solution for log C and n, the
120

post-peak slope and the initial fatigue fracture toughness must be defined. Here, the initial
toughness was also defined to be zero, but the initial post peak slope was not. The slope was
defined to have the average value 1 obtained from chapter IV to account for both size effect and
variable amplitude loading effects. Using equations (77-78) a non-linear regression scheme was
used to determine the four parameters, and the procedure was the following:


Define the initial post-peak slope, 1 for the given specimen:
o Large specimens: 1 = 1.55
o Small specimens: 1 = 1.88



f

Define the initial seed values for the fatigue crack resistance parameters: KRu , m, and

aeq,u
f

o Large specimens: KRc =10.48 N/mm1.5 ,aeq,u = 5.08, m = 2
f

o Small specimens: KRc =6.34 N/mm1.5 , aeq,u = 2.36, m = 2
o Define the bounds for log C and n; they were [-100,0] and [0,75], respectively
o Define the bounds for 2; they were [-6 ]


2

Define the objective function; here it was defined as 1-R .
o The solution was deemed to be found if the error was below 0.01;



Run the optimization scheme.
f

o By trial and error, modify the values for the three fatigue parameters: KRu ,

aeq,u, and m if the error exceeded 0.01.
121



The parameter m was modified first (while holding the other variables
constant) until the error was minimized. If the error still exceeded the
threshold, then:



aeq,u was modified (while holding the other variables constant) until
the error was minimized. If the error still exceeded the threshold, then:



f

KRc was modified (while holding the other variables constant) until the
error was minimized.



For most of the specimens, the model error was below the 0.01 threshold (29 out of 34)

Results
The following section presents the results for all 34 fatigue specimens. Tables 9-10 show the
Paris parameters and equivalent fatigue crack resistance parameters for all 34 specimens,
including the p-values for the two independent size populations. Tables 11-12 show the results
for all 34 specimens without the use of the resistance curve. This means that the deceleration
region (or transient stage) was truncated. One column in Tables 11 and 12 is entitled ‘%
transient’. This is the percentage of time (or cycles) spent within the deceleration/transient stage
relative to the total time (or cycles) to reach failure. Note that for the variable amplitude tests,
‘n/a’ is written in the cells. This is because under variable amplitude loading, the % transient
may be skewed depending on the level of applied loading.

122

Table 9: Summary of random, variable, and constant amplitude fatigue results using data for the large specimens, 100 mm
Specimen

D

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100

Notch
to
Depth
0.15
0.15
0.15
0.50
0.50
0.50
0.50
0.50
0.15
0.15
0.35
0.35
0.50
0.50
0.15
0.35
0.35

ID

Loading Type

logC

n

f
KRC

aeq,u

m

1

2

2
1-R

aeq,fail

KIC

P07
P09
P11
B02
B04
B05
B06
B10
P02
P08
P15
P17
B01
B08
P10
P19
P22

Uniform Random
Uniform Random
Uniform Random
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Const-0.95
Const-0.95
Const-0.95
Const-0.95
Const-0.90

-23.98
-15.24
-39.97
-23.29
-14.77
-15.44
-28.26
-17.45
-15.48
-29.19
-26.37
-46.67
-28.15
-14.85
-34.41
-26.90
-32.89

17.98
11.48
29.94
17.58
10.85
11.86
21.72
13.89
11.46
21.27
18.33
33.41
20.51
10.83
24.60
18.89
24.07

10.48
10.48
10.48
9.48
10.48
12.48
10.48
8.48
10.48
12.48
11.48
10.48
6.48
10.48
6.48
7.48
10.48

5.08
5.00
5.04
6.08
5.08
4.58
4.14
3.36
4.47
6.51
6.30
8.02
5.07
7.08
1.49*
6.08
6.08

2
2
2
2
2
2
1
2
2
2
1.5
1
2
1.3
1.35
2
2

1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.55
1.76
1.55
1.55

-1.38
-1.16
-1.17
-3.00
-5.03
-0.99
-2.53
-5.99*
-1.64
-5.71
-3.25
-0.58
-1.15
-1.15
-2.21
-1.15
-1.15

7.0E-03
8.8E-03
4.1E-03
6.3E-03
2.0E-02
9.7E-03
4.2E-03
9.9E-03
6.7E-03
2.8E-02
1.1E-03
9.5E-03
7.1E-03
9.5E-03
4.3E-02
7.1E-03
6.7E-03

20.74
19.31
23.86
15.16
3.92
15.43
11.31
4.87
19.27
34.40
15.72
17.54
8.93
8.75
18.34
12.62
15.66

42.20
40.42
46.07
31.74
22.97
36.52
32.43
22.96
36.70
41.48
38.20
43.13
31.19
31.03
37.47
37.94
36.52

123

f

Table 10: Summary of random, variable, and constant amplitude fatigue results using data for the small specimens, 50 mm
Specimen

D

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50
50

Notch
to
Depth
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15
0.15

ID

Loading Type

logC

S21
S24
S25
S04
S10
S11
S12
S13
S14
S15
S16
S17
S18
S01
S03
S05
S08

Uniform Random
Uniform Random
Uniform Random
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Const-0.90
Const-0.80
Const-0.95
Const-0.85

-25.57
-31.43
-20.34
-29.27
-26.59
-26.97
-27.68
-6.72
-34.93
-42.68
-34.34
-24.67
-37.29
-21.32
-20.79
-9.62
-13.27

n

f
KRC

20.61 7.34
25.65 6.34
16.10 7.78
23.24 9.34
20.81 8.34
21.81 9.34
22.01 7.34
4.70
7.34
30.38 3.34
32.81 3.34
29.75 6.34
19.27 8.34
31.91 8.34
16.92 6.34
15.48 6.34
7.32
6.34
9.83 14.34*

124

aeq,u

m

1

2

2
1-R

aeq,fail

KIC

f

1.95
1.91
3.03
1.08
2.02
3.05
0.89
1.90
2.89
2.27
0.63
2.88
4.13
3.15
4.35
1.28
7.35*

2
2
2.35
3
2
1
2
2
2
2
2
1
1.5
2
1.5
2
1.5

1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88
1.88

-0.78
-0.66
-1.78
-0.44
-0.000033*
-1.03
-1.31
-2.37
-1.07
-1.18
-0.27
-0.82
-0.90
-1.43
-1.43
-1.43
-1.43

9.7E-03
9.2E-03
7.9E-03
7.9E-03
8.7E-03
2.9E-03
3.6E-02
2.0E-03
4.2E-03
3.0E-02
3.5E-03
1.2E-02
8.9E-03
3.9E-03
6.2E-03
5.5E-03
4.6E-03

10.63
13.90
9.14
9.70
3.77
11.04
13.79
7.70
9.29
9.78
11.41
8.82
11.50
8.97
7.79
8.65
14.41

31.77
37.11
27.75
36.97
25.27
32.87
35.40
27.86
22.69
29.34
35.51
28.00
29.51
27.42
27.95
28.86
36.52

Table 11: Summary of random, variable, and constant amplitude fatigue results using data for the large specimens, 100 mmonly transient/acceleration stage
Specimen

D

1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
12.00
13.00
14.00
15.00
16.00
17.00

100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00

Notch to
Depth
15.00
15.00
15.00
50.00
50.00
50.00
50.00
50.00
15.00
15.00
35.00
35.00
50.00
50.00
15.00
35.00
35.00

ID

Loading Type

logC

n

% Transient

2
1-R

P07
P09
P11
B02
B04
B05
B06
B10
P02
P08
P15
P17
B01
B08
P10
P19
P22

Uniform Random
Uniform Random
Uniform Random
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Const-0.95
Const-0.95
Const-0.95
Const-0.95
Const-0.9

-32.88
-20.18
-41.23
-64.79
-6.69
-29.87
-49.98
-39.50
-20.96
-4.37
-41.42
-41.73
-33.07
-25.66
-27.56
-22.37
-40.58

20.62
12.47
25.14
42.49
3.36
18.83
33.45
28.31
13.32
2.07
25.89
25.27
22.11
16.80
17.77
14.32
25.40

73.28%
58.69%
64.78%
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
51.88%
63.79%
98.27%
40.24%
81.55%

4.90E-03
1.79E-03
9.91E-03
4.69E-03
2.63E-02
2.37E-03
1.68E-02
1.57E-03
2.63E-03
8.79E-03
2.35E-03
3.31E-03
1.71E-03
3.00E-03
5.17E-03
4.89E-03
7.87E-03

125

Table 12: Summary of random, variable, and constant amplitude fatigue results using data for the small specimens, 50 mmonly transient/acceleration stage
Specimen

D

Notch to
Depth

ID

Loading Type

logC

n

% Transient

2
1-R

1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
12.00
13.00
14.00
15.00
16.00
17.00

50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00

15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00
15.00

S21
S24
S25
S04
S10
S11
S12
S13
S14
S15
S16
S17
S18
S01
S03
S05
S08

Uniform Random
Uniform Random
Uniform Random
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Variable
Const-0.9
Const-0.8
Const-0.95
Const-0.85

-26.10
-22.40
-18.97
-21.69
-14.24
-25.39
-20.38
-9.48
-22.45
-28.76
-61.54*
-34.82
-37.31
-22.65
-28.24
-17.21
-27.54

16.94
14.09
12.58
12.92
7.67
16.67
13.33
6.10
16.10
19.79
39.33*
27.51
25.43
15.69
19.18
11.84
17.23

78.36%
42.29%
72.58%
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
n/a
44.86%
73.09%
64.82%
68.84%

5.50E-03
7.65E-03
8.85E-03
2.64E-03
4.92E-02
6.00E-03
7.24E-03
2.11E-03
5.48E-03
3.15E-03
4.65E-03
1.50E+00
5.87E-03
6.55E-03
1.90E-03
6.98E-03
1.80E-03

126

Constant Amplitude Loading
In the constant amplitude loading tests, the slope variant parameter was held constant (-2.21 and
-2.59 for the 100 and 50 mm specimens, respectively). Therefore, only the ultimate crack
extension and the Paris parameters were obtained via the regression scheme. Figure 39 shows the
relationship between log(daeq/dN) and equivalent crack extension, aeq.
For the smaller specimens, the results show that the Paris parameters are quite different
than for random and variable loading. The reason is because some of the variable amplitude tests
were originally tested under constant amplitude loading. However, those specimens did not show
any signs of degradation over several hours so the load level was increased. Therefore, the beams
that failed without having to increase the load were inherently less tough and more prone to
fatigue failure. This is the reason that the Paris parameters are lower than the rest of the
population. This problem was not encountered in the larger size specimens. Even for specimens
that took several hours to reach failure under a stress ratio of 0.9 for example, it was obvious
from the oscilloscope (output from the voltage signal from the COD gage) that some degradation
was taking place. Therefore, for the larger size specimens, the constant amplitude results were
more aligned with the rest of the population.

127

(a)

(b)
Figure 39: Results for specimen P22: Larger beam, 100 mm depth, (a) equivalent crack
rate v. crack extension, and (b) equivalent crack extension v. Number of Cycles

Variable Amplitude Loading
The variable amplitude loading cases consisted of varying segments of constant
amplitude loads, e.g., for a given number of cycles, the peak load amplitude and the R-ratio
would remain constant and then at a pre-defined time, was changed and held constant over

128

another period. Figure 40 shows the results for specimen B05. Figure 40a shows how the load
and R-ratio change abruptly at approximately 3,000 and 4,500 cycles, respectively.

129

(a)

(b)

(c)

Figure 40: Results for specimen B05: Larger beam, 100 mm depth, (a) applied load histogram, (b) equivalent crack rate v.
crack extension (c) equivalent crack extension v. Number of Cycles
130

Random Amplitude Loading
Two different types of random loading distributions were used in this study: uniform and normal.
Figure 41 shows the histogram for the two types of loading used in this study. Specimens P07,
P09, and P11 were subjected to uniform random loading distribution and specimens S21, S24,
and S25 were subjected to a normal random loading distribution. The mean stress ratio and Rratio in the normal distributions were 0.8 and 0.1, respectively, with a standard deviation of 0.07
for both. The range of stress ratios and R-ratios in the uniform distribution was 0.9-0.75 and 0.50.05, respectively. Refer to the appendix for more detail on each specimen.

131

(a)

(b)
Figure 41: Random load distribution: (a) uniform distribution, specimen P07 (b) normal
distribution, specimen S24

132

Fatigue crack resistance parameters
Recall, the fatigue crack resistance parameters were found with a non-linear regression scheme
2

that minimized the 1-R error between the fit and the cumulative crack data. From Table 9 and
Table 10, it can be seen that the average ultimate equivalent crack extension, aeq,u, is
approximately the same as the average critical equivalent crack extension under quasi-static
loading shown in chapter IV (D=100: 5.08, 5.50, and D=50: 2.36, 2.34 mm). The same could be
f

said for the critical fatigue fracture toughness, KRc , (D=100: 10.48, 9.95, and D=50: 6.34, 6.99
N/mm

1.5

). The m parameter was also similar (D=100: 2.18, 1.77 and D=50: 1.97, 1.87). On

average, the 2 parameter seems to be also similar to the quasi-static results (D=100: -1.15, -2.08
and D=50: -1.19, -2.15).
Size Effect
Size effect in concrete fatigue has been shown to manifest through the log C term. Recall, several
researchers have shown that log C will decrease as the non-dimensional structural size increases
(Sain, T., Kishen, J.M., 2007). However, most of these observations have come from data that
has only included the transient cracking region. Moreover, the Paris model used in these studies
considers only the stress intensity, KI, and does not include any contribution from the crack
resistance. Table 13 shows the results for when the transient cracking stage is not accounted for,
and a simple form of the Paris law (no crack resistance) is used. Note that similar results are
obtained to previously published results; log C is significantly larger for the larger size
specimens. The log C term for the larger size specimens was found to be -31.93 and for the
smaller size specimens was found to be -23.60, with a p-value of 0.052. The n term, however,
133

was not found to be size dependent (20.44 and 15.81) for the two different sizes although the pvalue was still quite low (0.112).
Table 13: p-values generated for the Paris parameters log C and n and the slope variant
coefficient, only transient/acceleration stage
log C
-31.93
-23.60
0.052

, All Fatigue,100
, All Fatigue,50
p-value

n
20.44
15.81
0.112

When the bridging stress contributions are included into the fatigue formulation, and the
model is calibrated with steps shown in the beginning of this chapter, very promising results are
obtained. Table 14 shows that on average, the logC and n terms are not size dependent. The
average logC for the large specimens was -25.49 and for the small specimens it was -25.50; with
a p-value of 0.990. The average n for the large specimens was 18.75 and for the small specimens
it was 20.5; with a p-value of 0.502. These results suggest that logC size effect is essentially a
product of the influence that the bridging stresses have on a given sized structure. However, if
the relative contribution is considered, then the Paris parameters become unique and are no
longer a function of size.

Table 14: p-values generated for the Paris parameters log C and n, aeq,u, m, and 2
logC

, All Fatigue, 100
, All Fatigue, 50
p-value

-25.49
-25.50
0.990

f
KRC
9.95
6.99
0.000

n
18.75
20.5
0.502

aeq,u

m

2

5.59
2.34
0.000

1.77
1.87
0.502

-2.08
-1.14
0.030

The size dependency in this case comes from the fatigue crack resistance curve. The
f

1.5

average critical fatigue fracture toughness, KRc , for the large size specimens was 9.95 N/mm
134

and for the smaller sizes it was 6.99 N/mm

1.5

; with a p-value of 0.000. The crack extension

dividing the pre- and post-peak cracking regions, aeq,u, was also size dependent (p-value of
0.000), where the average for the large specimens was 5.59 mm and for the smaller specimens
was 2.34 mm.
Loading Regime
As shown in Tables 15-16, the type of loading regime does not significantly influence the fatigue
cracking characteristics in the larger size specimens. The reason for this is because the stress
range effect is accounted for by the two post-peak slope parameters and . If these two
parameters were not included in the formulation, it would be expected that the parameter aeq,u
would be different under these three loading regimes since it was shown that under random
loading, the transition zone between the acceleration and deceleration stage is much larger than
under constant amplitude loading. For the smaller size specimens, the results show that the Paris
parameters were significantly different under constant amplitude loading. The reason for this was
previously explained.

135

Table 15: Average values for the equivalent fatigue crack resistance curve and Paris
parameters-only transient/acceleration stage
Loading Type

logC

n

% Transient

fval

Average,100 R
Average,100 V
Average,100 C
Average,100
Average,50 R
Average,50 V
Average,50 C
Average,50

-31.43
-33.26
-29.85
-31.93
-22.49
-23.84
-23.91
-23.60

19.41
21.44
19.28
20.45
14.54
16.17
15.98
15.82

65.58%

5.53E-03
7.64E-03
4.53E-03
6.36E-03
7.33E-03
1.59E-01
4.31E-03
9.56E-02

Average

-27.89

18.20

67.15%
64.41%
62.90%

5.10E-02

Table 16: Average values for the equivalent fatigue crack resistance curve and Paris
parameters
Loading Type

logC

n

Average,100 R
Average,100 V
Average,100 C
Average,100
Average,50 R
Average,50 V
Average,50 C
Average,50
Average

-26.40
-24.10
-27.44
-25.49
-25.78
-29.11
-16.25
-25.50
-25.49

19.80
17.82
19.78
18.75
20.79
23.67
12.39
20.50
19.63

f
KRC

aeq,u

m

1

2

2
1-R

aeq,fail

KIC

f

10.48
10.70
8.28
9.95
7.15
7.14
6.34
6.99

5.04
5.39
6.08
5.50
2.30
2.17
2.93
2.34

2.00
1.72
1.73
1.77
2.12
1.85
1.75
1.87
1.82

1.55
1.55
1.59
1.56
1.88
1.88
1.88
1.88
1.72

-1.24
-2.84
-1.36
-2.08
-1.07
-1.04
-1.43
-1.15
-1.61

0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01

21.30
15.29
12.86
15.64
11.23
9.68
9.96
10.02
12.83

42.89
34.01
34.83
35.82
32.21
30.34
30.19
30.63
33.23

Fracture toughness at failure
f

The fracture toughness at fatigue failure, KIC , was defined as the maximum equivalent stress
intensity applied to the beam right before unstable fracture, i.e., the last recorded stress intensity.
The last two columns in Tables 9-10 show the crack extension at failure, aeq,f and the fracture
toughness for each of the 34 beams tested in fatigue. The average crack extension at failure for
136

the large and small sizes was 15.64 and 10.02 mm, respectively. The corresponding stress
intensities at failure were 35.73 and 30.63 N/mm

1.5
,

respectively. These fracture toughness

values were then compared to the average quasi-static fracture toughness at those respective
crack extensions and they were 36.41 and 29.20 N/mm

1.5
,

respectively; with p-values of 0.667

and 0.2, respectively, and thus considered to be statistically equivalent.
These results are similar to the ones obtained in chapter V. The ramifications of this are
twofold: 1) the equivalent stress intensity may be used as a measure to characterize the failure
point in both quasi-static and fatigue loading, which means 2) the moment equilibrium and
corrected crack formulation proposed in chapter IV can also be used to predict the capacity
and/or failure point in fatigue if the cyclic bridging stress parameters are known.
Description of proposed model
The results from this chapter indicate that the functional form of the equivalent crack resistance
curve obtained from the cyclic quasi-static tests can be used in the fatigue formulation to predict
crack resistance under constant, variable, and random amplitude loading. Moreover, the same
bridging stresses that govern the fatigue crack resistance curve can also be used to predict the
moment capacity of the beam in fatigue. The benefit of using such a formulation is that one
escapes the need for the excessive computational step of introducing degrading bridging law into
the fatigue formulation. The relationship between the cyclic bridging stresses and the cyclic
resistance curve has been explained in detail in chapter IV and here, its usefulness in fatigue
prediction has been confirmed. The entire formulation from start to finish is the following:


Determine the cyclic bridging stress parameters from a cyclic quasi-static test containing
P-CMOD data
137



Once the bridging stresses are known, the Pint-acorr curve is determined by satisfying
the condition Pint(acorr)=Papp(acorr)



Determine the equivalent crack length, aeq, by satisfying the condition TL(aeq) =

T(acorr)


s

Determine the equivalent KR curve by using equations (36-38) to establish the fatigue
failure envelope in terms of stress intensity.



Determine the load and crack dependent fatigue resistance curve by evaluating the
contribution of the bridging stresses over a series of applied loads, Papp, ranging from 0
to Pmax over a series of crack extensions, ranging from 0 to 0.3*D



Use the unique Paris parameters, fatigue resistance curve, and the failure envelope to
predict the fatigue crack growth and number of cycles to failure, Nf.

Summary
In this chapter, it was shown that the equivalent fatigue crack resistance parameters can be
obtained with a non-linear regression scheme that used the average results from chapter IV to
define the initial seed values. On average, the fatigue crack parameters found in this chapter
agree well with the ones obtained in chapter IV.

It was also shown that by including a size

dependent fatigue crack resistance curve, the Paris parameters will no longer depend on the size
of the structure.
The fatigue parameters did not vary substantially as a function of the loading regime. The
reason for this is that the fatigue crack resistance curve included two terms and  that
138

account for the load range effect. If this effect is not considered in the formulation, then under
variable and/or random amplitude loading, the crack extension parameter would have to be much
larger than under constant amplitude loading to accommodate the longer transition zone between
the deceleration and acceleration stages.
The equivalent fracture toughness at failure in fatigue, on average, agreed well with that
under quasi-static loading. This means that the capacity of the beam can be predicted in fatigue
with the moment equilibrium and corrected crack length formulation proposed in chapter IV.

139

CHAPTER VII

THE EVALUATION OF THE LINEAR DAMAGE RULE UNDER RANDOM
PAVEMENT EDGE STRESSES

Concrete pavements are subjected to a complex combination of environmental and traffic loads
which produce a unique distribution of peak stress and stress ranges at the critical mid-slab edge.
Moreover, the fatigue damage accumulation process is complex because the material is
heterogeneous and suffers from a variety of pre-loading flaws such as air voids and shrinkage
cracks leading to a substantial variability in fatigue performance.
There are several different ways of predicting fatigue damage in concrete. The classical
method is to use a linear damage rule (LDR) accompanied with an experimentally generated loglog S-N curve that relates applied stress ratio (SR) to the number of cycles to failure in fatigue,
Nf (Miner, 1945).
The current design methodology (NCHRP, 2004) for transverse cracking uses a hybrid
mechanical and empirical analysis. The stresses of an un-cracked pavement are predicted via an
artificial neural network that is trained by an extensive Finite Element response database.
Damage is accounted for by inserting the stresses into the field calibrated S-N performance
model and is defined as the inverse of the number of cycles to failure, 1/Nf (Miner, 1945).
Equation (14) shows the damage algorithm that is used to predict fatigue cracking in concrete
pavements.
140

One of the advantages of using this type of damage model is its computational efficiency,
since it can rapidly account for, process and convert millions of load repetitions to damage. This
allows for multiple designs to be considered within minutes.

Some of the disadvantages

however, are that it is insufficient in determining the in-situ state of damage because no
information is given on the state of the material itself (no information on the stress-strain
behavior and the reduction of the elastic modulus). In addition, it cannot account for size effect,
load history effect, and variable amplitude loading without using some empirical calibration
factors. Thus, there is a need for the development of a concrete fatigue model that can account
for all three of the aforementioned effects and at the same time, be able to maintain a comparable
level of computational efficiency to the S-N approach.
The objective of this chapter is to compare the fatigue prediction using a LDR and a
fracture mechanics model under a randomly distributed mid-slab edge pavement stress history.
Studies have shown that LDR’s under-predict damage for H-L loading sequences, but this effect
may be lessened when the stress amplitudes have large R ratios. Therefore, it is important to
know the dominant R ratio frequency and determine whether it counteracts the load history
effect. It is also important to determine the prediction error generated by the LDR under a
random stress distribution. A modified Slowik fatigue fracture model (refer to equation 33) was
used for the comparison. The reason for this is that the model directly accounts for the R-ratio
effect and an overload effect. The original model, however, was slightly modified by replacing
f

the original KI term with KIeff, which is defined as the difference between KI and KR . The
parameter KImax is defined as the maximum stress intensity ever applied to the crack. Under
141

random loading, it would be expected that this value would increase at a faster rate than K I,
which means that this model would predict more damage accumulation under such loadings than
one without KImax. Note that the model used in this chapter differs from the ones in the previous
chapters in that there is a term specifically accounting for the R-ratio effect and load history
effect. In the future, the two model predictions will be compared.
In total, there were 4 fatigue and 3 crack resistance parameters governing the fatigue
process. Equation (80) shows the modified Slowik Model and equation (81) shows the function
describing fatigue crack resistance curve.

(𝐾

𝐾 ) (
(𝐾
𝐾

)
)

(80)

𝐾

[

(

) ]

≤

𝐾
{

𝐾

}
(81)

The modified Paris law was calibrated using the 6 random amplitude loaded specimens.
This method is exactly the same as the one proposed in chapter VI; except that the Paris law is
different. Equation (82) shows the modeled cumulative crack extension that was used to
2

determine the (1-R ) error in the same way as shown in equation (78).

142

( )

∑

∑

𝐾

(

)
𝐾

(𝐾

)

𝑓
|

(82)
The average coefficients in equation (82) were: log C = -7.05; n = 5.14; p = 6.25 and q = 0.22. These parameters were not found to be size dependent or geometry dependent.
Figure 42 shows the experimental results and fit from two different specimen sizes. The
fatigue crack parameters were found to be size dependent; the results are shown in Table 17.
As discussed previously, if a zero slope post peak crack resistance curve is used, the
ultimate crack extensions are larger to accommodate the larger transition zone, under such
loadings.
Table 17: Summary of random fatigue crack parameters using a zero slope post peak

D
100
100
100
Average
50
50
50
Average

f

1.5

KRc (N/mm

)

aeq,u (mm)

m

13
17
12
14.00
5
8
3
5.33

3
2.8
2.5
2.77
2.8
2.6
2.6
2.67

15
18
18
17.00
16
16
13
15.00

143

1.5

KIC (N/mm
41.99
39.43
45.7
42.37
31.77
37.11
27.74
32.21

)

0
-0.5

log(da/dN)

a (mm)

-1
-1.5
-2
-2.5
-3
-3.5
-4
0

Experimental
Model
5

a (mm)
Da (mm)

10

(b)

(a)

Figure 42: (a) Large, Crack extension and number of cycles, (b) Large, Crack rate and crack extension, (c) Small, Crack
extension and number of cycles, (b) Small, Crack rate and crack extension

144

15

Figure 42 (cont’d)

-0.4

log(da/dN)

a (mm)

-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
0
(c)

Experimental
Model
5

10
a (mm)
Da (mm)
(d)

145

15

20

S-N Calibration
In this study, the fatigue fracture model was used to construct the S-N curve by simulating
fatigue crack growth under various constant stress ratios. The R ratio was constant (0.1) for all
stress ratios. The peak stress under monotonic loading was 5.23 MPa. The width and depth of the
beam was 100 mm and the span was 400 mm. The initial notch was 6 mm. This notch length was
chosen so that the nominal stress at failure (far field beam stress) would match the modulus of
rupture found experimentally. Next, the simply supported fracture model was calibrated to match
the S-N curve used in MEPDG to predict pavement slab fracture. This rationale was used
because (1) in MEPDG the transverse cracking model also uses beam modulus of rupture as the
strength reference, and (2) damage is limited to the edge of the pavement (meaning that only one
dimensional crack propagation can exist, similar to a beam). The author does acknowledge
however, that the MEPDG S-N curve is a field calibrated model and would expect a handful of
unknowns to influence both the crack propagation rate and direction.
The beam model was calibrated to explain the fracture phenomenon of a fully supported
slab that is larger in cross-section (size effect), exhibits out of plane fracture (vertical and
transverse direction), is subjected to frictional forces at the concrete/foundation interface, and
exhibits mixed mode fracture. The calibration constants were strategically placed onto three
f

terms: log C, KR , and n. The rationale is the following: log C should be calibrated to account for
f

out of plane fracture, KR should be calibrated to account for size effect (increasing R-curve
resistance as a function of size), and n should also be calibrated to account for the effect of the
foundation induced frictional resistance (should decrease because friction will reduce the crack
propagation rate). Figure 43 shows the S-N curve before and after calibration. The calibration

146

factors 1, 2, and 3 were 0.713, 0.977, and 0.879, respectively. Equation (83) shows the
calibrated relationship between Nf and stress intensities and was used to construct the calibrated
S-N curve.

𝑓

∫ ((

)

(𝐾

𝐾 )
(𝐾
𝐾

(

)
)

)

(83)

Figure 43: Calibration of S-N curve against MEPDG

147

𝐾

𝐾

MEPDG and Monte Carlo Simulation
Two simulations were conducted to assess the effect of load history. Two different climatic
regions were chosen: Lansing, MI and Miami, FL. These regions were selected because of their
annual mean temperature difference. The structural geometry and the traffic were identical for
both simulations. The pavement cross-section consisted of a 200 mm PCC slab overlying a 150
mm A-1-A granular base and a A-2-7 subgrade. The length of the slab was 4.5 m. The 28 day
modulus of rupture was 5.23 MPa. The 28 day elastic modulus was 29.6 GPa. For the purpose of
analysis, the strength and elastic modulus were assumed to remain constant because no data for
f

f

KRc and KIC as a function of age was available to the author. An annual average daily truck
traffic (AADTT) of 185 with no growth was used to reduce simulation time. MEPDG default
truck distributions and axle spectra’s were used. The mean wander was 450 mm with a standard
deviation of 250 mm. Medium spacing between axle configurations was used (4.5 m). Default
axle spacing for tandem and tridem axles was also used (1.3 m). A zero built in temperature curl
was used in the simulation to accelerate damage growth. Figure 44 shows the pavement crosssection used in the simulation.

Figure 44: Pavement cross-section used in the simulation
148

Monte Carlo Simulation
The randomized truck traffic simulation was created in MATLAB. The random environmental
conditions were generated using output from the Enhanced Integrated Climatic Model (EICM) in
MEPDG: cross-sectional pavement temperature, monthly modulus of subgrade reaction k, and
the elastic modulus of the granular base Ebase. The foundation properties varied monthly, and the
truck traffic varied hourly. The linear temperature gradients were calculated as shown in
Appendix QQ in the MEPDG Guide using the output cross-sectional temperatures from EICM.
In addition, the non-linear temperature gradients were calculated based on the method proposed
by Hiller and Roesler (2010) using the non-linear area (NOLA) concept. The temperature also
varied hourly. The critical stresses were interpolated from a stress database (70,690 runs) that
was compiled with EVERFE (Davids, W.G., Wang, Z.M., Turkiyyah, G., Mahoney, J. and Bush,
D., 2003). The equivalent layer concept was used to calculate the stresses, as shown in Appendix
QQ. The adequacy of the interpolation scheme is shown in Figure 49. The interpolation scheme
output and EVERFE output are compared using 150 random runs using different pavement
thickness, elastic modulus, traffic wander, and load for a tandem axle. Reasonable results were
obtained.

Truck Traffic Randomization
Every hour, the total truck volume was established by the AADTT and the hourly truck traffic
distribution. The hourly volume was multiplied by the truck class distribution and rounded. The
result yielded a finite number of ‘possible’ truck repetitions for a given hour and truck class. A
finite truck population was then created corresponding the number of ‘possible’ truck repetitions
for each class, e.g., if three class 9 trucks and two class 13 trucks were ‘possible’, the truck
149

population vector would be [9 9 9 13 13]. Using the randperm function in MATLAB, the vector
can be re-arranged randomly to look like [13 9 9 13 9].
The axle population corresponding to each of the truck classes was created in a similar
fashion. Each axle load spectra was multiplied by 1000 and rounded to yield a finite number of
possible axle repetitions corresponding to each truck class, axle load, and month. The population
was then indexed according to the axle load. Thus, for a class 9 truck in January for example, if
there are total of three 178 kN and two 169 kN ‘possible’ single axle loads, the load vector
would look like [178 178 178 169 169] and after randomization, the re-arranged vector would
look like [169 178 178 169 178]. Please note that there would be a similar vector of possible
loads for every truck class and month, and the vector should have a length of 1000 (not 5; the
vector shown here is for the purpose of example only).
Next, the total number of ‘possible’ axle load repetitions per truck was determined from
the default probability matrix given in MEPDG. The population was arbitrarily chosen to be
large (1000). The population was then conditioned by requiring the population average to equal
the value shown in the MEPD probability matrix. For example, if a population of five was
chosen and class 4 number of single axles per truck was 1.6, the population vector would look
like [1 1 2 2 2] and after randomization would look like [2 1 2 2 1] (The average of the vector is
1.6).
In summary, trucks were selected at random and the number of corresponding ‘possible’
single, tandem, and tridem axles was established and used to select the number of random load
repetitions. The axle load was then placed a random distance away from edge. The distance was
dependent on the normal wander distribution. The stresses at the mid-slab edge were then
calculated for every random axle load repetition. In summary, the stresses were calculated using
150

hourly linear and non-linear temperature gradients, hourly random truck traffic data, and monthly
foundation properties. A step by step procedure is shown below:


Obtain an hourly truck traffic random truck population by multiplying the AADTT by
the truck class distribution and the hour distribution.


AADTT*TruckClassDist*HourlyDist = round(Number of trucks per
class)



Create 10 truck vectors, each having a length of the number of trucks
per class and each element within the vector has the same number (413) that corresponds to the specific truck distribution.



Concatenate all 10 vectors and randomize their positions

o Select one truck
o Determine the number of single, tandem, and tridem axles pertaining to the
truck


Create a vector that represents the average axle repetition per truck


If the number of axle repetitions per truck falls between 1 and
2, then create a vector of ones and twos that has a length of
1000 and has an average equal the average axle repetition per
truck. This step should be repeated for each axle type and for
each truck class.



For the truck, select an element from each of the 3 vectors.
These elements represents the number of axle loads per
configuration will be subjected to the pavement.

151

o Determine a population of axle loads that corresponds to a particular truck
class and month


This can be done by multiplying 1000 by the axle load spectra. This
results in a matrix that shows the number of ‘possible’ repetitions that
correspond to each truck class and month.



Round the matrix.



Sum the elements in each row: this number should equal 1000.



Create a vector of zeros with a length of 1000 elements



Fill the vector with the total axle load population for the given month
and axle load, e.g. in January for a class 4 truck, if 10 axles equaling
100 kN exist (after multiplying and rounding the matrix), then the first
10 slots in the zero vector will be filled with a value of 100 kN. This
step is repeated for all the ‘possible’ axle loads. The total length of the
vector should be 1000.



Randomize this vector



Repeat for every truck class and month

o For the given truck, month, and known number of single, tandem, and tridem
axle load repetitions, fetch the load from the axle load population.


Record the axle loads in this order: single, tandem, and tridem. This means that for a
given truck, the single axles will always come first, the tandem axles second, and the
tridem axles third.

152

Stress-time history
In this section, the procedure for determining the critical mid-slab edge stresses for a given truck
is presented. The example shown here is for a class 11 truck that has 5 single axles, as shown in
Figure 46. The axles were partitioned into two categories: inter-axle and intra-axle contributions.
As first axle enters the slab, and the backside of the wheel is positioned along the edge of the
joint, the first stress is recorded. The next recorded stress corresponds to the scenario where axle
1 (A1) is positioned at mid-slab. The third stress is then recorded and corresponds to the scenario
where A1 is placed along the edge of the left joint and A2 is place along the edge of the right
joint. The procedure is repeated until all five axles have passed over the pavement. Note that this
event considers both the contribution of the individual single axle, the interaction between the
axles, and thermal stresses. This was done to so that both peak and valley stresses could be
determined.
The intra-axle contributions for the tandem and tridem axles were also considered.

An

intra-axle stress cycle for a tandem axle would look like the following: thermal stress + axle load
(first axle at the edge of the right joint; second axle has not affect; total axle load is

½*Tandem

axle)  thermal stress + axle load (first axle at mid-slab, second axle placed 1.3 m to the right)
 thermal stress + axle load (first and second axle placed equidistant from mid-slab),  thermal
stress + axle load induced stress (second axle over mid-slab)  initial thermal stress + axle load
(second axle at the edge of the left joint; total axle load is

½*Tandem axle). Figure 46 shows a

stress time history caused by a thermally and truck traffic induced loads and Figure 47 shows an
example of how the stresses are counted for a tandem axle. The intra-axle stress cycle for a
tridem axle would look like the following: thermal stress + axle load (first axle near the joint) 
153

thermal stress + axle load (first axle at mid-slab, second axle placed away at a distance equal to
the default axle spacing, third axle has no affect because it is still off the slab; this means that the
tandem loading database was used with a load of 2/3*Tridem axle)  thermal stress + axle load
(first and second axle placed equidistant from mid-slab, third axle now appears and is spaced at a
distance of 1.3 m),  thermal stress + axle load induced stress (first axle left of the mid-slab by
1.3 m, second axle over mid-slab, third axle right of the of the second axle)  initial thermal
stress + axle load (first and second axle placed equidistant from mid-slab, third axle now appears
and is spaced at a distance of 1.3 m, the second loading case was simply repeated)  initial
thermal stress + axle load (third axle at mid-slab, second axle 1.3 m left of third axle, first axle
has no affect because it is still off the slab; this means that the tandem loading database was used
with a load of 2/3*Tridem axle)  initial thermal stress + axle load (third axle at the edge of the
left joint; the axle interaction database is used with a load of 1/3*Tridem axle). Note that the
stress repetitions were only counted when there was an axle load present, meaning that isolated
thermal cycles were not accounted for. The stresses were counted using the following procedure:
The stress at the ith axle load repetition,i, was counted as ‘peak stress’ when the condition i-1
<i >i+1 was identified. Similarly, a ‘valley stress’ was counted when the condition i-1 >i
<i+1 was identified.
The inter-axle contributions between single-tandem, single-tridem, and tandem-tridem
were also considered when necessary. Under these circumstances only one axle (out two for the
tandem and three for the tridem) was considered to effect the pavement edge stress. This means
that under the circumstance that a tandem axle is leaving the slab and a tridem axle is
approaching it, the inter-axle scenario would be simulated by placing the rear axle of the tandem
154

along the edge of the left joint and the front axle of the tridem along the edge of the right joint.
The magnitude of the axle loads would be 1/2 of the total tandem load and 1/3 of the total tridem
load.

155

1

2

5

6

9
Critical location

3

7

10
# Stress Calculation

11

Wheel load

Figure 45: Axle load placement for a class 11 truck that has 5 single axles
156

4

8

Figure 46: Truck traffic and thermal loading induced stresses
157

Figure 47: Inter- and Intra-axle loads for a tandem axle
158

Figure 48: Procedure for calculating the pavement edge stress

159

Rapid Stress Interpolation scheme
In order to rapidly predict the mid-slab edge stresses induced by the axle and thermal
loads, a multi-level quartic interpolation scheme was devised. Please refer to Vongchusiri (2005)
for details on the interpolation scheme. Please note however, the interpolation functions used
here are different than the ones proposed in Vongchusiri (2005).
The interpolation scheme used herein was a function of 7 variables: 1) axle type, 2) axle
placement (along longitudinal direction of the slab), 3) axle load 4) wander (distance from the
edge of the pavement), 5) concrete density, 6) equivalent linear thermal gradient, and 7) concrete
thickness. The scheme interpolated between 70,690 anchor points. Each point corresponded to
one pavement simulation in EVERFE. For the single axle simulations for example, there were a
total of 3,125 runs (5 variables each having 5 data points for total combination of 5 5) under
positive thermal gradient and another 3,125 under a negative gradient. The database was
partitioned into two separate parts: 1) positive temperature curl and 2) negative temperature curl.
The inputs used in EVERFE are provided below:


Single Axle
o Positive Gradient: [0, 6.94, 13.89, 20.83, 27.78] 0C


Load: [0, 48.95, 97.9, 14.7, 18.7] kN



Wander: [0, 229, 457, 686, 914] mm



Concrete density: [1931, 2207, 2483, 2759, 3035] kg/m



Concrete thickness: [152.4, 190.5, 241.3, 292.1, 330.2] mm

3

o Negative Gradient: [0, -6.94, -13.89, -20.83, -27.78] 0C


Wander, Load, density, and thickness were all the same as the positive.
160



Tandem Axle
o Positive Gradient: [0, 6.94, 13.89, 20.83, 27.78] 0C


Load: [26.7, 93.5, 186.9, 280.4, 364.9] kN



Wander, thickness, density are the same as single axle, positive gradient

o Negative Gradient: [0, -6.94, -13.89, -20.83, -27.78] 0C



Wander, Load, density, and thickness were all the same as the positive.

Tridem Axle
o Positive Gradient: [0, 6.94, 13.89, 20.83, 27.78] 0C


Load: [53.4, 153.5, 253.7, 353.8, 453.9] kN



Wander, thickness, density are the same as single axle, positive gradient

o Negative Gradient: [0, -6.94, -13.89, -20.83, -27.78] 0C



Wander, Load, density, and thickness were all the same as the positive.

Axle Interaction (single axle placed on the edge of both left and right joints)
o Positive Gradient: [0, 9.44, 18.89, 27.78] 0C
o Negative Gradient: [0, -9.44, -18.89, -27.78] 0C
o Load @ left joint: [0, 62.3, 124.6, 186.9] kN
o Load @ right joint: [0, 62.3, 124.6, 186.9] kN
o Wander: [0, 204.8, 635 914.4] mm
o Concrete Density: [1931, 2345, 2759, 3034] kg/m

3

o Concrete thickness: [152.4, 203.2, 254, 330.2] mm
Please note that the individual axle interpolation functions were quartic polynomials and
the inter-axle interaction interpolation functions were cubic polynomials; the scheme works in

161

the following way: suppose there are two input variables, x and y that affect the behavior of z.
The objective is to develop a cubic polynomial network interpolation function that can describe
the behavior of z(x,y). First, a database must be constructed so the interpolation scheme may
operate on the ‘anchor’ points. This is done by running a computer simulation of the behavior of
z as a function of the two inputs. Since a cubic polynomial will be used for both input variables x
and y, there should be a total of 16 output data points (equal to the total number of possible
combinations between the x and y inputs, 42). Therefore, the database should be 27 X 2 matrix.
Let’s say the objective is to interpolate a value for z at inputs x* and y*. The network
interpolation would then operate on the database anchor points in the following way:


First, determine the one-dimensional cubic interpolation functions for the x and y inputs
with equation (84). The interpolation function for the variable is shown below

[

]

[

]
(84)

Where

[

]
(85)

xi= x input put into the computer simulation

162



Second, multiply the matrix Nij with the field variable, zi over the region i = 1 to 4
(Note that

≤ ≤

) to determine the coefficients governing the cubic function over

this region.

{

}

[

]

(86)


Third, multiply the coefficient vector j, by the xj* input to arrive at the interpolated
result.

{

}

{

}

(87)


Repeat this process three more times over the region
≤ ≤

≤ ≤

,

≤ ≤

, and

.



Now, there should be 4 interpolated field variables: {zi*, z2*, z3*, z4*} = zi*



Next, using the y cubic interpolation function, determine the final interpolated field
variable using equation (88), which is just combining equations (86-87) but using the y
inputs instead.

[

]
(88)

163

Interpolation Databases
Two databases were constructed for the single axle: one for the positive gradient and one for the
negative gradient. For the tandem case, four databases were constructed: one with the front axle
at the mid-slab and one where the front and rear axle is placed at an equal distance from the midslab; these two cases are run for both a positive and negative thermal gradient. For the tridem
case, there are a total of four databases: one with the front axle over the mid-slab and one with
the front and middle axle

Equivalent slab system
In chapter II the equivalent slab concept was discussed. The concept states, since the stresses in a
fully supported plate can be described non-dimensionally with equation (4), it is possible to
relate the stresses from one system to another using equation (5). The benefit of using this
concept is that it reduces the amount of inputs needed to model a pavement system. For example,
in the stress database generated with EVERFE, it was not necessary to include the elastic
modulus of the soil, Esoil or the concrete, Epcc, the modulus of subgrade reaction k, the
coefficient of thermal expansion, CTE, or poisson’s ratio. The reason is because according to
equation (5) if the non-dimensional inputs in one fully supported plate system are the same as in
another fully supported plate system, then the stress in the two systems are relatable. The
pavement system analyzed in this chapter consisted of an elastic concrete layer, and elastic soil
layer and a subgrade. The enhanced integrative climatic module in the MEPDG was used to
convert the soil properties to a composite elastic soil layer, Ebase, and a liquid foundation with a
modulus of subgrade reaction of k. In the simulation, these two properties change monthly.
164

Using equation (6), this composite layered system was transformed to an effective layer system
having only an elastic concrete layer and a subgrade modulus. The effective thickness was then
converted to an equivalent thickness using equation (89) that can now be used to generate a
stress output with the interpolation scheme. The equation satisfies the condition where the radius
of relative stiffness between the two systems must be equal so the stress output between the two
systems is relatable.

√

(

)
(
(

)
)

The linear thermal gradient was calculated with equation (90), which is the numerical
form of equation (3) as shown in the MEPDG (2004).

(∑ (

)

∑ (

)

)

(90)

The linear thermal gradient was then converted to an equivalent linear thermal gradient with
equation (91).

165

(91)

Next, the axle load was converted to an equivalent load with equation (92)

(92)

Finally, the equivalent thickness, temperature gradient, and load were input into the interpolation
scheme. The output was an equivalent stress. This stress was then converted back to an effective
stress with equation (5) and then converted back to the actual system with equation (11). The
accuracy of the interpolation scheme was then assessed by comparing it to 150 EVERFE runs
with random inputs of axle load, wander, modulus of subgrade reaction, and positive thermal
gradients. Figure 49 shows that the accuracy of the interpolation scheme is quite good.

166

Figure 49: Comparison of stress output from EVERFE and the interpolation scheme

Pavement edge stress simulation
Random mid-slab edge stresses were generated for the Lansing, MI and the Miami, FL cases
using the randomized truck traffic and the interpolation scheme described in the previous
sections. Figure 50a is a bi-variate histogram that shows the distribution of peak stresses and
stress ranges. Figure 50b and Figure 50c show that the dominant R and stress ratio frequency
falls between 0 and 0.1 and the dominant stress ratio is also between 0 and 0.1. Note that the
negative R-ratios were changed to zero because equation (80) was not calibrated to account for
stress reversals, which explains why there is a dominant occurrence of zero.

167

(a)

(b)
Figure 50: (a) Bi-variate histogram of peak stress and stress range, (b) histogram of stress
ratios, (c) histogram of R – ratios

168

Figure 50 (cont’d)

(c)

Damage Prediction
The stresses and R ratios calculated in the previous section were used to predict fatigue damage
with the fracture mechanics model and the calibrated S-N curve. The Lansing simulation
produced more damage because there was a higher stress ratio frequency between 0.7 and 0.9
(1,948 for Lansing compared to 334 for Miami), meaning the pavement should fail faster. The
reason for the higher stresses can be caused by either a more severe thermal gradient or a more
flexible subgrade. Figure 51a shows the comparison between the k histograms for Miami and
Lansing. The left tail of the Lansing histogram has smaller k values compared to Miami. This
means that during the spring period, the Lansing foundations are significantly more flexible
169

which leads to higher pavement edge stresses. Moreover, the positive mean equivalent thermal
gradient is higher in Lansing which will produce higher bottom up stresses.
Figure 52 shows the damage accumulation from MEPDG and from the Monte Carlo
simulation. There seems to be fairly good agreement between LDR predictions in the Monte
Carlo and the MEPDG simulations (Lansing: at 29 years, DMEPDG = 0.78 and DMonte Carlo =
0.73, and Miami: at 40 years, DMEPDG = 0.61 and DMonte Carlo = 0.62). Note that the reason why
damage is less than one in both cases is because the simulations were stopped when KI = KIC in
the fracture fatigue equation.
Damage was assessed in the fracture model by the crack extension, a. These values
were normalized by the crack extension at failure (Lansing, aeq,f = 19.2 mm and Miami, aeq,f
= 24.1 mm) so the damage could be compared to the LDR. As shown in Figures 52a and 52c, the
fatigue fracture model yields greater damage than its counterpart linear model. The LDR’s seem
to under-predict damage by approximately 30%, (1-DMEPDG), on average. This is happening
because the linear model does not predict damage well after a significant overload (SR

0.7-

0.9). This is consistent with the findings of Oh (1991) and the Lemaitre (1992). In addition, it
seems that the R ratio does not counteract the load history effect under these specific random
stress histories.

170

(a)

(b)
Figure 51: (a) Histogram of modulus of subgrade reaction, k for Miami and Lansing, (b)
Histogram of Equivalent Linear Temperature Gradients for Miami and Lansing
Simulations
Calibration of Percent Slabs Cracked
One way to account for the under-prediction of damage when using the linear damage equation
is to re-calibrate the % slabs crack algorithm, shown in equation (93) by inserting a calibration
factor, C1. The variable FD represents the fatigue damage, which is defined from 0 to 1. The
MEPDG defines FD = 1 when the % slabs cracked is equal to 50%.

171

𝑟

(

)

(93)

In MEPDG, C1 is equal to one. For the % slabs cracked to be 50% when FD from the

non-linear fatigue model is 1, C1 must be equal to 1.25 for the Lansing case and 1.6 for the
Miami simulation. Note that, for different types of loading history, the discrepancy between nonlinear damage and linear damage changes and C1 will change accordingly. Therefore, a
sensitivity analysis would need to be conducted to explain the effect of the fatigue parameters n,
p, q, log C, and stress distribution on the prediction error between non-linear and linear damage.
In addition, the effect of size and boundary conditions should also be introduced within the
fatigue cracking equation.

172

(a)

(b)
Figure 52: (a) Lansing: linear and non-linear damage, (b) Lansing: % slabs cracked, linear
and non-linear damage (c) Miami: linear and non-linear damage, (d) Miami: % slabs
cracked, linear and non-linear damage

173

Figure 52 (cont’d)

(c)

(d)

174

CHAPTER VIII

CONCLUSION

Summary of findings
Concrete is a quasi-brittle material that exhibits a large residual stress zone along the wake of a
dominant crack tip. This residual zone is governed by an assumed exponential bridging stress
distribution that acts to resist further crack opening. The bridging stresses influence both the
capacity and crack growth in concrete structures.
A new method to determine the cyclic bridging parameters: f’t, wc, c1, and c2,

under

quasi-static loading for a TPBSEN specimen was proposed in chapter IV. The cyclic bridging
stresses were determined through inverse analysis using an internal moment and CMOD
equilibrium condition to match smoothened Papp-aeq data. The inelastic deformations were
neglected after each cycle; therefore, the bridging parameters are said to represent a cyclic event
and not a monotonic one. This was confirmed by the fact that the bridging parameters c1, and c2
are greater than published results for monotonic loading (approximately 5 and 25 compared to 3
and 7); the larger values show that the bridging distribution decays more rapidly, which is
consistent with the degrading nature of a cyclic bridging stress distribution.
The bridging stresses were then used to determine a corrected (equivalent) crack
resistance curve. The corrected resistance was obtained by integrating the weight function from
zero to the corrected crack length, acorr, using successive approximation. Next, the equivalent
175

resistance was obtained by integrating the weight function from zero to the equivalent crack
length, aeq, using an equivalent tensile strength f’teq without using successive approximation.
Converting the corrected resistance to an equivalent one is useful so it can be used in a fatigue
loading scenario; where it is easiest to record equivalent crack lengths with compliance
measurements.
From chapter IV, the following conclusions could be made:


A unique set of bridging parameters can be determined for cyclic, quasi-static TPBSEN
specimens using a moment and CMOD equilibrium condition.



The bridging parameters, f’t, c1, and c2 were shown to be size-independent. However, the
parameter wc was not.



T

The total fracture toughness, KIC , was size dependent because of the size dependency of
wc.



s

The specific fracture toughness, KIC , and the equivalent critical crack extensions, aeq,c,
were size dependent.



The bridging stresses were used to determine a corrected and equivalent crack resistance
curve.
o The equivalent crack resistance was determined by integrating the weight function
from zero to the equivalent crack length, aeq, without using successive
approximation. This was done to ensure the COD field was consistent with that of
an equivalent elastic crack.
176

o The equivalent crack resistance curve has a smaller plateau region than the
corrected crack resistance
o The equivalent crack resistance varies as a function crack extension and applied
loading.


Five parameters govern the shape of the load dependent equivalent crack
f

resistance curve: KRC , aeq,u, m, 1,and 2. The last two parameters
govern the shape of the post peak slope and describe the relationship
between load and crack resistance.


f

The parameters KRC , aeq,u are related to the total fracture toughness,
T

T2

KIC , and the non-dimensional length (wcE/KIC ) respectively.

In chapter V, a new method was developed to determine the cyclic crack resistance curve
using load and crack rate data obtained from a typical fatigue fracture test. This method provides
a way of quantifying the bridging stresses developed during fatigue loading. To verify the
method, a series of tests, both under quasi-static and fatigue loading, were conducted. The
following conclusions could be made:


The equivalent critical crack extension, aeq,c, can be calculated directly by determining
the crack length at which the rate of stress intensity is equal to the rate of resistance.



The quasi-static resistance curve can be related to the fatigue crack resistance curve as
follows:

177

o Determine the critical crack extension under quasi-static loading, which is
statistically equivalent to the critical crack extension, aeq,c, under fatigue
loading.
o Determine the optimal shift factor  and power m to describe the shape of the
f

measured KRc curve using a least squares technique.
o Determine the ultimate crack extension in fatigue by adding the shift factor  to
aeq,c.


The ultimate fatigue crack resistance is approximately 44% of the quasi-static fracture
resistance at peak load.



The power m under quasi-static loading is approximately 3.5 times smaller than that
under fatigue loading.



For finite size PCC specimens, the fracture toughness under fatigue loading is greater
than that under quasi-static loading because of the rising nature of the quasi-static
resistance curve beyond peak load.



Fatigue can be predicted continuously with the use of a fatigue crack resistance curve.

In chapter VI, a new method to determine an equivalent fatigue crack resistance curve
was proposed. It consisted of a two-tier non-linear optimization scheme that used cumulative
crack data to calibrate the Paris coefficients and the fatigue resistance parameters. The initial
seed values given to the crack resistance parameters were the average values obtained from
quasi-static loading. The parameter was assumed to be constant for all specimens of the same
178

size so as to satisfy the condition described in chapter V where the post-peak slope must be predefined. Under variable and random amplitude loading, the post-peak slope was allowed to vary
(which was governed by however, the initial slope corresponding to a load of zero, remained
constant.
A unique set of fatigue parameters was considered to be found when the model error, 12

R , was equal to 0.009 0.001. Using this method, the following conclusions could be made:


Using a crack resistance curve that accounts for a varying post peak slope allows one to
characterize the longer transition zone (between the deceleration and acceleration stage)
under random loading.
o The Paris parameters and crack resistance curves are insensitive to the type of
loading regime if the crack resistance curve is allowed to vary as a function of
applied load. If the post-peak slope is held constant under random loading, the
equivalent crack extension must be larger to accommodate the larger transition
zone.



If a size dependent resistance curve is inserted into the Paris fatigue equation, logC and n
become unique.
f

o The equivalent fatigue crack resistance curve parameters, KRC , aeq,u, m, 1,
and 2, found with the optimization technique, are statistically equivalent to those
obtained in chapter IV, under quasi-static loading.

179



When the crack resistance curve was not inserted into the fatigue equation and only the
steady-state (or acceleration) region was considered, similar results were obtained to
previously published results; logC is larger for the larger size specimens.



f

The specific fracture toughness at failure in fatigue, KIC , was statistically equal to the
s

specific crack resistance, KR , under quasi-static loading at the same equivalent crack
extension.
o This result implies that the cyclic bridging distribution, on average, does not
change as a function of load history.


The cyclic bridging law can be used to predict the capacity of a beam in fatigue and the
equivalent fatigue crack resistance.

In chapter VII, a Monte Carlo truck traffic simulation was conducted for two identical
pavement cross-sections in two different climatic regions, Lansing, MI and Miami, FL. Fatigue
damage was assessed using a modified Paris law in the spirit of the Slowik model; however
here, fatigue resistance curve was incorporated into the model. The parameters were determined
by a two-tiered calibration procedure based on eight laboratory specimens subjected to both
uniform and normal distribution random loading sequences. The fatigue model was used to
construct an S-N curve and was calibrated to match a field distress calibrated S-N curve. Three
f

parameters (logC, KRc , and n) were strategically chosen to be calibrated so that out of plane
fracture, size effect, and frictional effects, can be accounted for. A Monte Carlo simulation was
then conducted to assess the significance of R ratio and load history.

180

The damage predicted in the Monte Carlo simulation using the calibrated S-N curve
compared well with the bottom up damage prediction in MEPDG. However, the results showed
that damage will accumulate more rapidly with the fatigue fracture model. On average, the LDR
seems to under-predict damage by approximately 30%, although the error varies depending on
stress distribution. The results also show that the dominant R ratio frequency is between 0 and
0.1 and does not counteract the load history effect.

Recommendations for future research
The results shown in this work are very promising. Namely, the ability to determine the cyclic
bridging parameters using an inverse analysis under quasi-static loading, and using this
information to predict the capacity and the fatigue resistance in a TPBSEN can be useful for
design.
The limitations of this research, however, are that the fatigue bridging stresses were not
physically measured; rather, they were found with an optimization scheme. In addition, using the
cyclic bridging parameters as a means to predict fatigue crack resistance is a conservative
idealization because there may be some initial fracture toughness that was not accounted.
Furthermore, the methods developed here need to be verified with different geometries,
e.g. CT specimens. This method should be extended to three dimensions, so that it could directly
applicable to a slab fatigue fracture scenario. In order to achieve this, weight functions specific
for fully supported notched beams and slabs need to be developed. Once the weight functions
are determined, the capacity of the systems can be determined by either using principal stress
equilibrium, moment equilibrium, or an energy based equilibrium condition.
181

From a material standpoint, it would be useful to use this methodology to determine the
fatigue resistance for various concrete mixes; especially with the current demand for re-claimed
material. Moreover, it would be useful to understand how the fatigue resistance changes in these
materials as a function of moisture condition, and as a function of freeze thaw cycles.
Finally, it would be useful to determine the crack bridging stresses at loads near the
endurance limit. At this load level, the crack does not grow rapidly, which would allow one to
investigate the bridging stress erosion process more closely. Furthermore, the ‘low bound’
fatigue threshold resistance curve can be used as a means to establish a fatigue reliability index
in the pavement design process.

182

APPENDICES

183

APPENDIX A: Quasi-static Loading Test Results

184

(b)

(a)

(c)

(d)
S

Figure 53: Specimen L1: (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths

185

(a)

(b)

(c)

(d)

S

Figure 54: Specimen L2: (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
186

(a)
(b)

(d)

(c)
S

Figure 55: Specimen L3 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
187

(a)

(b)

(c)

(d)

S

Figure 56: Specimen L4 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
188

(a)

(b)

(c)

(d)

S

Figure 57: Specimen L5 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
189

(b)

(a)

(d)

(c)
S

Figure 58: Specimen L6 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
190

(a)

(b)

(c)

(d)

S

Figure 59: Specimen S1 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
191

(a)

(b)

(c)

(d)

S

Figure 60: Specimen S2 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
192

(a)

(b)

(c)

(d)

S

Figure 61: Specimen S3 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
193

(b)

(a)

(d)

(c)
S

Figure 62: Specimen S4 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
194

(a)

(b)

(c)

(d)

S

Figure 63: Specimen S5 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
195

(a)

(b)

(d)

(c)
S

Figure 64: Specimen S6 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
196

(a)

(b)

(c)

(d)

S

Figure 65: Specimen S7 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
197

(a)

(b)

(c)

(d)

S

Figure 66: Specimen S8 (a) P-CMOD, (b) KR curve, (c) global equilibrium, (d) equivalent and corrected crack lengths
198

Specimen
L1
L2
L3
L4
L5
L6
S1
S2
S3
S4
S5
S6
S7
S8

Table 18: Quasi-static specimen geometry (mm)
a0
b
15
104
15
104
15
104
35
104
35
104
35
104
9.7
49.4
9.41
49.4
8.85
49.34
8.66
47
8.08
49.9
7.96
51.48
8.04
50.38
9.26
49.67

199

D
100
100
100
100
100
100
48.08
48.7
50.1
49.84
49.94
49.5
50.75
51.68

S
400
400
400
400
400
400
200
200
200
200
200
200
200
200

APPENDIX B: Fatigue Loading Test Results

200

(a)

(b)

(c)

Figure 67: Specimen P07 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq. crack extension, (c)
Experimental and model eq. crack extension v. N
201

(a)

(b)

(c)

Figure 68: Specimen P09 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq. crack extension, (c)
Experimental and model eq. crack extension v. N
202

(a)

(b)

(c)

Figure 69: Specimen P11 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq. crack extension, (c)
Experimental and model eq. crack extension v. N
203

(a)

(b)

(c)

Figure 70: Specimen B02 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
204

(a)

(b)

(c)

Figure 71: Specimen B04 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
205

(a)

(b)

(c)

Figure 72: Specimen B05 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
206

(a)

(b)

(c)

Figure 73: Specimen B06 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
207

(a)

(b)

(c)

Figure 74: Specimen B10 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
208

(a)

(b)

(c)

Figure 75: Specimen P02 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
209

(a)

(b)

(c)

Figure 76: Specimen P08 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
210

(a)

(b)

(c)

Figure 77: Specimen P15 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
211

(a)

(c)

(b)

Figure 78: Specimen P17 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
212

(a)

(b)

(c)

Figure 79: Specimen B01 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
213

(a)

(c)

(b)

Figure 80: Specimen B08 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
214

(a)

(c)

(b)

Figure 81: Specimen P10 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
215

(a)

(c)

(b)

Figure 82: Specimen P19 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
216

(a)

(c)

(b)

Figure 83: Specimen P22 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
217

(a)

(c)

(b)

Figure 84: Specimen S21 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq. crack extension, (c)
Experimental and model eq. crack extension v. N
218

(a)

(c)

(b)

Figure 85: Specimen S24 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq. crack extension, (c)
Experimental and model eq. crack extension v. N
219

(a)

(c)

(b)

Figure 86: Specimen S25 (a) Load Histogram, (b) Experimental and model eq. crack rate v. eq. crack extension, (c)
Experimental and model eq. crack extension v. N
220

(a)

(c)

(b)

Figure 87: Specimen S04 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
221

(a)

(b)

(c)

Figure 88: Specimen S10 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
222

(a)

(c)

(b)

Figure 89: Specimen S11 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
223

(a)

(b)

(c)

Figure 90: Specimen S12 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
224

(a)

(b)

(c)

Figure 91: Specimen S13 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
225

(a)

(b)

(c)

Figure 92: Specimen S14 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
226

(a)

(b)

(c)

Figure 93: Specimen S15 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
227

(a)

(b)

(c)

Figure 94: Specimen S16 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
228

(a)

(b)

(c)

Figure 95: Specimen S17 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
229

(a)

(b)

(c)

Figure 96: Specimen S18 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
230

(a)

(b)

(c)

Figure 97: Specimen S01 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
231

(a)

(b)

(c)

Figure 98: Specimen S03 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
232

(a)

(b)

(c)

Figure 99: Specimen S05 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c) Experimental
and model eq. crack extension v. N
233

(a)

(b)

(c)

Figure 100: Specimen S08 (a) Load History, (b) Experimental and model eq. crack rate v. eq. crack extension, (c)
Experimental and model eq. crack extension v. N
234

Table 19: Beam geometry/properties and number of cycles to failure
Specimen
P07
P09
P11
B02
B04
B05
B06
B10
P02
P08
P15
P17
B01
B08
P10
P19
P22
S21
S24
S25
S04
S10
S11
S12
S13
S14
S15
S16
S17
S18
S01
S03
S05
S08

a0

b

D

S

E

15
15
15
50.94
51.77
51.47
50.77
50
15
15.97
35
35
50
50
15
35
35
9.42
8
8.075
9.3
6.41
7.3
6.04
5.4
7.7
10.04
7.53
7.22
8.17
9.3
11.29
6.45
7.25

104
104
104
107.1
107.9
109.4
107.24
104
104
108.2
104
104
104
104
104
104
104
49.945
51.08
52.35
50.1
51.3
53.05
49.4
48.4
50.45
50.48
51.25
49.84
52.55
52.4
51.35
53.25
50.7

100
100
100
102
104
101.8
100.63
100
100
102.01
100
100
100
100
100
100
100
48.55
49.2
49.22
49.95
50.22
51.1
51.15
49.95
47.85
51.24
48.8
48.56
50.85
48.95
52.15
47.2
51.25

400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200
200

29711
29711
29688
25046
27540
28112
25648
24367
26989
29068
26497
29359
25513
24367
28767
33969
30823
25816
25630
19450
25373
19476
20975
22209
17620
15337
21842
22658
22991
21305
25585
21646
20011
23878

235

Nf
1482
397
40520
3561
1290
6681
5109
4827
943
27594
144668
11244
133
58
14504
82
18974
10135
10038
2535
45024
34855
117276
166620
125
3019
1124
22464
16823
75685
534
249
108
6225

APPENDIX C: Pavement Edge Stress Results

236

(a)

(c)

(b)

(d)

Figure 101: Stress Interpolation for (a) single axle, positive temperature gradient, (b) single axle, negative thermal gradient,
(c) tandem axle (front axle at mid-slab), positive thermal gradient (d) tandem axle, (front and back axle equidistant from midslab), positive thermal gradient
237

(a)

(b)

(d)

(c)

Figure 102: Stress Interpolation for (a) tandem axle (front axle at mid-slab), negative thermal gradient, (b) tandem axle, (front
and back axle equidistant from mid-slab), negative thermal gradient (c) tridem axle, (middle axle at mid-slab), positive
thermal gradient, (d) tridem axle, (front and second axle equidistant from mid-slab), positive thermal gradient

238

(a)
Figure 103: Pavement Edge Stress Results for Lansing, MI (a) peak stress and stress range
histogram

239

(b)
Figure 103 (cont’d): (b) R-ratio histogram

240

(c)
Figure 103 (cont’d): (c) Stress ratio histogram

241

Figure 104: Pavement Edge Stress Results for Miami, FL (a) peak stress and stress range
histogram

242

(b)
Figure 104 (cont’d): (b) R-ratio histogram

243

(c)
Figure 104 (cont’d): (c) Stress ratio histogram

244

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