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LIBRARIES
' ‘ _ MICHIGAN STATE UNIVERSITY
. , ‘. ") EAST LANSING, MICH 48824-1048

M 0.2 “:70

This is to certify that the
dissertation entitled

FREQUENCY AND TIME DOMAIN BACKCALCULATION OF
FLEXIBLE PAVEMENT LAYER PARAMETERS

presented by
Yigong Ji

has been accepted towards fulﬁllment
of the requirements for the

PhD. degree in Civil Engineering

AM ,

Major Professor's Signature

5/4/4005"

Date

 

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2/05 c:ElRC/Date&re.indd-p.15

 

' FREQUENCY AND TIME DOMAIN
BACKCALCULATION OF FLEXIBLE PAVEMENT
LAYER PARAMETERS

By

Yigong Ji

A DISSERTATION

Submitted to
Michigan State University
In partial fulﬁllment of the requirements
For the degree of

DOCTOR OF PHILOSOPHY
Department of Civil and Environmental Engineering

April 2005

ABSTRACT

Frequency and Time domain Backcalculation of Flexible Pavement Layer Parameters

By Yigong Ji

In this study, new algorithms method for backcalculating ﬂexible pavement layer
parameters based on dynamic interpretation of FWD deﬂection time histories using
frequency and time-domain solutions have been developed. The backcalculation
procedure is based on the modiﬁed Newton-Raphson method originally adopted in the
MICHBACK program. Singular value decomposition (SVD), in conjunction with scaling
techniques is employed in solving for the inverse problem. The frequency-domain
method uses real and imaginary deﬂection basins as the measured quantities, while the
time-domain method uses either the peak deﬂections and corresponding time lags or
traces of the deﬂection time histories as the measured quantities to be matched by the

backcalculation procedure.

The new associated program called DYNABACK has been written in the
FORTRAN 77 language, and offers two options: (i) frequency-domain analysis, and (ii)
time-domain analysis. The new program has been incorporated into the WindowsTM
based MFPDS program, which allows for user—ﬁ‘iendly features including interactive
input and output screens, and the ability to view and process the deﬂection data before

analyzing it.

The new program was theoretically veriﬁed using synthetic data. Numerical examples

show that the proposed methods are able to backcalculate layer moduli and thicknesses

accurately from synthetically generated FWD data. The applicability of the new program
to interpret ﬁeld tests was evaluated using measured deﬂection time history data from
several FWD tests conducted in Michigan and elsewhere. The analyses included the
comparison of backcalculated layer moduli and damping ratios with MICHBACK results
for various pavement sections and load levels. The backcalculation was done in both
frequency and time domains, where the time-domain solution included backcalculating
layer moduli and thicknesses. The data were obtained from tests involving KUAB and
Dynatest FWD machines. Most pavement sections were analyzed as three- and four-
layer systems with some sections involving a stiff layer at shallow depth. The results
indicate that dynamic backcalculation of layer parameters using ﬁeld data presents some
serious challenges. The frequency-domain method can lead to large errors if the
measured FWD records are truncated before the motions fully decay in time, and the
time-domain method when simultaneously backcalculating layer moduli and thicknesses

produces mixed results.

ACKNOWLEDGMENTS

For his excellent support and advice in both academic and research works, I would like to
express my gratitude to my major advisor Dr. Karim Chatti, whose assistance and

guidance made this work feasible.

The author would like to express his utmost appreciation to Dr. Ronald S. Harichandran

for his constant guidance and encouragement through the course of this study.

I am also indebted to the other members of the committee, Dr. Gilbert Y. Baladi and Dr.

Haydar Radha for their suggestions.

The contributions, suggestions and discussions of several graduate students, especially

Hyung S. Lee and Zhihui Huang are greatly appreciated.

I am thankful to the Michigan Department of Transportation for providing me with the

funding during my graduate studies.

Finally I would like to thank my wife, Liping Feng, for her love and patience, and my

parents and family for their support and understanding. They are always there for me.

iv

TABLE OF CONTENTS

 

 

 

 

CHAPTER 1 - INTRODUCTION - - - - 1
1.1 GENERAL ............................................................................................................. 1
1.2 PROBLEM STATEMENT ......................................................................................... l
1.3 RESEARCH OBJECTIVE ......................................................................................... 2
1.4 REPORT LAYOUT ................................................................................................. 2

CHAPTER 2 - LITERATURE REVIEW 5
2.1 GENERAL ............................................................................................................. 5
2.2 STATIC MATERIAL CHARACTERIZATION .............................................................. 6

2.2.1 Layered elastic model ................................................................................. 6
2.2.2 Nonlinear elastic model .............................................................................. 7

2.3 STATIC BACKCALCULATION METHODS ............................................................... 7

2.4 DYNAMIC PROPERTIES OF PAVING MATERIALS ................................................... 8
2.4.1 Asphalt Concrete ........................................................................................ 8
2.4.2 Granular Materials .................................................................................... 1 1
2.4.3 Fine-grained Soil ...................................................................................... 13

2.5 VISCOELASTIC MATERIAL CHARACTERIZATION ................................................ 15
2.5. 1 Mechanical Models ................................................................................... 16
2.5.2 Creep Compliance Model ......................................................................... 19

2.6 DYNAMIC FORWARD COMPUTATION PROGRAMS .............................................. 20
2.7 DYNAMIC BACKCALCULATION METHODS ......................................................... 21

CHAPTER 3 - FORWARD PROGRAM - 26
3.1 INTRODUCTION .................................................................................................. 26
3.2 ANALYSIS METHODS ......................................................................................... 26

3.2.1 Modeling of Viscoelasticity ...................................................................... 27
3.2.1.1 Hysteretic Damping ............................................................................... 27
3.2.1.2 Viscous Damping ................................................................................... 27

3.2.2 Steady-State Response .............................................................................. 29

3.2.3 Transient Response ................................................................................... 32
3.2.3.1 Frequency-domain Solution ................................................................... 33
3.2.3.2 Interpolation Scheme ............................................................................. 34
3.2.3.3 Summary of Procedure for Calculating Transient Response ................. 36
3.2.3.4 Dynamic Response of a Pavement System due to an FWD Load ......... 37

3.3 ESTIMATING DEPTH-TO-STIFF LAYER (DSL) AND SUBGRADE MODULUS ......... 42
3.3.1 Estimating Depth to Stiff Layer (DSL) .................................................... 42
3.3.2 Estimating Subgrade Modulus .................................................................. 55
3.3.3 Using the Subgrade Modulus and Depth-to-Stiff Layer Estimates in the
Backcalculation Algorithm ....................................................................................... 57

CHAPTER 4 - INVERSE SOLUTION 62
4. I INTRODUCTION .................................................................................................. 62

4.2 FREQUENCY-DOMAIN BACKCALCULATION ........................................................ 63

 

4.2.1 Multi-frequency backcalculation .............................................................. 63
4.2.2 Single-frequency backcalculation with thickness backcalculation ........... 64
4.2.3 Single-frequency backcalculation without thickness backcalculation ..... 65
4.3 TIME-DOMAIN BACKCALCULATION .................................................................... 65
4.3.1 Time-domain backcalculation without thickness backcalculation ........... 65
4.3.2 Time-domain backcalculation with thickness backcalculation ................ 66
4.4 FWD DATA PROCESSING .................................................................................. 67
4.5 INVERSE SOLUTION ............................................................................................ 68
4.5.1 Frequency-domain Backcalculation ......................................................... 69
4.5.2 Time-domain Backcalculation using Peak Deﬂection and Time Lag ...... 77
4.5.3 Time Domain Backcalculation using traces of time histories .................. 79
4.6 SVD METHOD .................................................................................................... 81
4.6.1 Truncating Singular Values ...................................................................... 83
4.6.2 Scaling ...................................................................................................... 83
4.7 MODIFICATIONS TO THE SOLUTION .................................................................... 84
CHAPTER 5 - DYNABACK PROGRAM STRUCTURE AND FEATURES ....... 85
5.1 GENERAL ........................................................................................................... 85
5.2 DATA INPUT ....................................................................................................... 85
5.3 PROCESSING A FWD DEFLECTION DATA FILE .................................................. 86
5.3.1 Reviewing and Processing the Deﬂection Data ........................................ 86
5.3.2 Data Analysis Options .............................................................................. 88
5.4 PRESENTATION OF BACKCALCULATION RESULTS .............................................. 88
5.5 PROGRAM STRUCTURE ...................................................................................... 91
5.6 BACKCALCULATION OF LAYER PROPERTIES ...................................................... 93
5.6.1 Cases A, C, and E ..................................................................................... 93
5.6.2 Cases B, D, and F ..................................................................................... 93
CHAPTER 6 - THEORETICAL VERIFICATION - 98
6.1 INTRODUCTION .................................................................................................. 98
6.2 THEORETICAL FREQUENCY-DOMAIN BACKCALCULATION USING STEADY-STATE
RESPONSE ...................................................................................................................... 98
6.2.1 Effect of Modulus, Thickness and Frequency Combinations ................... 99
6.2.2 Effect of Sub-Layering on Backcalculation Results ............................... 105
6.2.3 Dynamic Backcalculation of Additional Layers ..................................... 107
6.2.4 Backcalculation of the Depth-to-Stiff Layer (DSL) ............................... 108
6.3 THEORETICAL FREQUENCY-DOMAIN BACKCALCULATION USING TRANSIENT
RESPONSE .................................................................................................................... l 10
6.3.1 Comparison of Single and Multiple Frequency Backcalculation
Results. .................................................................................................................. 1 10
6.3.2 Backcalculation of Damping Ratio for Unbound Layers ....................... 113
6.3.3 Uniqueness of Backcalculated Results ................................................... 115
6.3.3.1 Proﬁles with Different AC Layer Moduli ........................................... 116
6.3.3.2 Proﬁles with Different AC Layer Thicknesses .................................... 120
6.3.3.3 Four-Layer Pavement Proﬁle .............................................................. 124

vi

6.3.4 Convergence Characteristics .................................................................. 126

 

6.3.5 Effect of Poisson’s Ratio on Backcalculated Layer Parameters ............. 133
6.3.6 Simulation of Measurement Errors ......................................................... 136
6.3.6.1 Effect of Deﬂection Imprecision on Backcalculated Results .............. 138
6.3.6.2 Effect of Signal Truncation on Backcalculated Results ...................... 144
6.3.6.3 Extrapolation ........................................................................................ 153
6.3.7 Comparison of Dynamic and Static Backcalculation Results ................ 164
6.4 TIME-DOMAIN BACKCALCULATION USING PEAK DEFLECTION AND TIME LAG 166
6.4.1 Sensitivity Analysis ................................................................................ 166
6.4.2 Theoretical Veriﬁcation .......................................................................... 1 7 1
6.4.3 Uniqueness .............................................................................................. l 76
6.4.4 Convergence Characteristics .................................................................. 178
6.5 TIME DOMAIN BACKCALCULATION USING TRACES OF TIME HISTORY ............ 185
6.5. 1 Theoretical Veriﬁcation .......................................................................... l 85
6.5.2 Uniqueness .............................................................................................. 1 86
6.5.3 Convergence Characteristics .................................................................. 187
6.5.4 Effect of Incorrect Damping Ratio Speciﬁcation on Backcalculation
Results. .................................................................................................................. 197
6.6 SUMMARY ....................................................................................................... 200
CHAPTER 7 - FIELD VALIDATION OF DYNABACK - 203
7.1 GENERAL ......................................................................................................... 203
7.2 BACKCALCULATION OF LAYER PARAMETERS FOR SELECTED PAVEMENT
SECTIONS ..................................................................................................................... 203
7.2.1 Michigan Sites ........................................................................................ 204
7.2.1.1 Comparison of Dynamic and Static Backcalculation for Four layer
System.. ............................................................................................................... 206
7.2.1.2 Dynamic Time-domain Backcalculation Using Peak Deﬂections for
Three layer System .............................................................................................. 213
7.2.1.3 Dynamic Time-domain Backcalculation using Traces of Time
History. .............................................................................................................. 217
7.2.2 Texas Site ................................................................................................ 226
7.2.2.1 Comparison of Dynamic and Static Backcalculation for Four-layer
System.. ............................................................................................................... 231
7.2.2.2 Dynamic T ime-domain Backcalculation for Three- layer System ...... 238
7.2.2.3 Dynamic Time-domain Backcalculation using Traces of Time
History... ............................................................................................................. 242
7.2.3 Cornell Site ............................................................................................. 250
7.2.3.1 Comparison of Dynamic and Static Backcalculation for F our-layer
System 251
7.2.3.2 Dynamic Time-domain Backcalculation for Three-layer System ....... 253
7.2.3.3 Dynamic Time-domain Backcalculation using Traces of Time
History... ............................................................................................................. 255
7.2.4 Florence Site ........................................................................................... 263
7.2.4.1 Comparison of Dynamic and Static Backcalculation for Four-layer
System. . . .. ........................................................................................................... 263

vii

7.2.4.2 Dynamic Time-domain Backcalculation for Three-layer System ....... 266
7.2.4.3 Dynamic Time-domain Backcalculation using Traces of Time

 

History... ............................................................................................................. 267
7.2.5 Kansas Site .............................................................................................. 275

7.2.5.1 Dynamic Time-domain Backcalculation for F our-layer System ......... 275

7.2.5.2 Comparison of Dynamic and Static Backcalculation for Three-layer

System 276

7.2.5.3 Dynamic Time-domain Backcalculation using Traces of Time

History. .............................................................................................................. 278
7.3 DISCUSSION ..................................................................................................... 286
CHAPTER 8 - CONCLUSIONS AND RECOMMENDATIONS -- - -- _ 289
8.1 SUMMARY ....................................................................................................... 289
8.2 CONCLUSIONS .................................................................................................. 291
8.3 RECOMMENDATIONS ........................................................................................ 294
BIBLIOGRAPHY ........................................................................... 296

viii

LIST OF TABLES
TABLE 2.1 DYNAMIC BACKCALCULATION PROGRAMS ........................................................ 24

TABLE 2.2 ADVANTAGES AND DISADVANTAGES FOR DYNAMIC BACKCALCULATION

PROGRAMS .................................................................................................................. 25
TABLE 3.1 PROFILE USED FOR COMPARING SAP SI AND GREEN SOLUTIONS .................... 32
TABLE 3.2 PAVEMENT PROFILE CHARACTERISTICS ............................................................. 38

TABLE 3.3 PROFILES USED IN THE ANALYSIS OF SATURATED SUBGRADE WITH BEDROCK...43

TABLE 3.4 PROFILE USED IN THE ANALYSIS OF UNSATURATED SUBGRADE WITH BEDROCK 44

TABLE 3.5 PROFILE USED IN THE ANALYSIS OF UNSATURATED SUBGRADE WITH GWT ...... 44
TABLE 6.1 LIST OF FREQUENCY COMBINATIONS ................................................................. 99
TABLE 6.2 PROFILES USED ................................................................................................ 100
TABLE 6.3 PROFILE WITH COARSE SUB-LAYERING ............................................................ 106
TABLE 6.4 PROFILE WITH FINE SUB-LAYERING ................................................................. 106
TABLE 6.5 BACKCALCULATION RESULTS USING COARSE SUB-LAYERING ......................... 107
TABLE 6.6 THEORETICAL BACKCALCULATION OF A FIVE-LAYER PAVEMENT SYSTEM ...... 107
TABLE 6.7 THEORETICAL BACKCALCULATION OF A SIX-LAYER PAVEMENT SYSTEM ........ 108

TABLE 6.8 COMPARISON OF THEORETICAL AND BACKCALCULATED LAYER PARAMETERS -
DSL=10 Fr ............................................................................................................... 109

TABLE 6.9 COMPARISON OF THEORETICAL AND BACKCALCULATED LAYER PARAMETERS —
DSL=20 Fr ............................................................................................................... 109

TABLE 6.10 COMPARISON OF THEORETICAL AND BACKCALCULATED LAYER PARAMETERS —
DSL=3O FT ............................................................................................................... 110

TABLE 6.1 l PROFILES USED FOR VERIFYING UNIQUENESS OF SOLUTION (VARYING LAYER
MODULI) ................................................................................................................... 1 16

TABLE 6.12 SEED MODULUS VALUES USED FOR VERIFYING UNIQUENESS OF SOLUTION WITH
THREE-LAYER PAVEMENT SYSTEM ............................................................................ I 16

TABLE 6.13 PROFILES USED FOR VERIFYING UNIQUENESS OF SOLUTION
(VARYING AC LAYER THICKNESS) ............................................................................ 120

TABLE 6.14 FOUR-LAYER PROFILE USED FOR VERIFYING UNIQUENESS OF SOLUTION ....... 124

TABLE 6.15 SEED MODULUS VALUES FOR VERIFYING UNIQUENESS OF SOLUTION WITH
FOUR-LAYER PAVEMENT SYSTEM .............................................................................. 124

TABLE 6.16 PAVEMENT STRUCTURE USED TO STUDY THE EFFECTS OF DEFLECTION
IMPRECISION ON BACKCALCULATED RESULTS ........................................................ 138

TABLE 6.17 PAVEMENT STRUCTURE USED TO STUDY THE EFFECTS OF SIGNAL TRUNCATION
ON BACKCALCULATED RESULTS ................................................................................ 144

TABLE 6.18 RMS VALUES FOR DEFLECTION BASINS CORRESPONDING To TRUNCATED
VERSUS UNTRUNCATED SENSOR SIGNALS .................................................................. 150

TABLE 6.19 COMPARISON OF STATIC AND DYNAMIC BACKCALCULATION RESULTS FOR
THREE-LAYER PAVEMENT SYSTEMS .......................................................................... 165

TABLE 6.20 COMPARISON OF STATIC AND DYNAMIC BACKCALCULATION RESULTS FOR

FOUR-LAYER PAVEMENT SYSTEMS ............................................................................ 165
TABLE 6.21 PROFILES USED FOR THE SENSITIVITY ANALYSIS ........................................... 167
TABLE 6.22 PAVEMENT PROFILES FOR SYNTHETIC DATA .................................................. 172

TABLE 6.23 COMPARISON OF DYNAMIC AND STATIC BACKCALCULATION RESULTS USING
SYNTHETIC DATA FOR PROFILE 1 ............................................................................... I 73

TABLE 6.24 DYNAMIC BACKCALCULATION RESULTS (KNOWN THICKNESS) USING
SYNTHETIC DATA FOR PROFILES 2 THROUGH 5 .......................................................... 175

TABLE 6.25 DYNAMIC BACKCALCULATION RESULTS (UNKNOWN THICKNESS) USING
SYNTHETIC DATA FOR PROFILES 2 THROUGH 5 .......................................................... 176

TABLE 6.26 UNIQUENESS OF RESULTS WITHOUT THICKNESS BACKCALCULATION ............ 17 7

TABLE 6.27 UNIQUENESS OF RESULTS WHEN THICKNESS BACKCALCULATION IS ENABLED

................................................................................................................................. 17 7
TABLE 6.28 DYNAMIC BACKCALCULATION RESULTS (KNOWN THICKNESS) USING
SYNTHETIC DATA FOR PROFILES 2 THROUGH 5 .......................................................... 185
TABLE 6.29 DYNAMIC BACKCALCULATION RESULTS (UNKNOWN THICKNEss) USING
SYNTHETIC DATA FOR PROFILES 2 THROUGH 5 .......................................................... 186
TABLE 6.30 UNIQUENESS OF RESULTS WITHOUT THICKNESS BACKCALCULATION USING
TRACES OF TIME HISTORIES ....................................................................................... 187

TABLE 6.31 UNIQUENESS OF RESULTS WHEN THICKNESS BACKCALCULATION IS ENABLED

USING TRACES OF TIME HISTORIES ............................................................................ 187
TABLE 6.32 LIST OF DAMPING RATIO COMBINATION FOR BASE AND SUBGRADE ............... 198
TABLE 7 .1 SENSOR LAYOUT (DISTANCES ARE IN INCHES) — MICHIGAN DATA ................... 204
TABLE 7.2 PROFILE USED FOR US 1 31 SITE (SECTION 50699) ........................................... 205
TABLE 7.3 PROFILE USED FOR US 1 31 SITE (SECTION 67015) ........................................... 205

TABLE 7.4 COMPARISON OF FREQUENCY AND TIME-DOMAIN BACKCALCULATION RESULTS
WITH THOSE FROM MICHBACK — US 131 SITE ....................................................... 207

TABLE 7.5 BACKCALCULATION RESULTS FROM TIME-DOMAIN ANALYSIS - US 131 SITE ..2 1 4
TABLE 7.6 SEED VALUES USED FOR BACKCALCULATION OF MICHIGAN DATA .................. 222
TABLE 7.7 LIST OF THE COMBINATION OF DAMPING RATIO FOR BASE AND SUBGRADE ..... 223
TABLE 7.8 PROFILE USED FOR TEXAS SITE ....................................................................... 231
TABLE 7.9 SENSOR LAYOUT (DISTANCES ARE IN INCHES) ................................................. 231

TABLE 7.10 COMPARISON OF FREQUENCY AND TIME-DOMAIN BACKCALCULATION RESULTS
WITH THOSE FROM MICHBACK - TEXAS SITE ........................................................ 232

TABLE 7.1 I BACKCALCULATION RESULTS FOR TIME-DOMAIN ANALYSIS — TEXAS SITE ...239

TABLE 7.12 SEED VALUES USED FOR TEXAS DATA ........................................................... 246
TABLE 7.13 PROFILE USED FOR CORNELL SITE ................................................................. 250
TABLE 7.14 SENSOR LAYOUT (DISTANCES ARE IN INCHEs) FOR CORNELL SITE ................ 250

TABLE 7.15 COMPARISON OF FREQUENCY AND TIME-DOMAIN BACKCALCULATION RESULTS
WITH THOSE FROM MICHBACK- CORNELL SITE ..................................................... 251

TABLE 7.16 BACKCALCULATION RESULTS FROM TIME-DOMAIN ANALYSIS - CORNELL SITE

................................................................................................................................. 254
TABLE 7.17 DIFFERENT SEED SPECIFICATIONS - CORNELL DATA .................................... 259
TABLE 7.18 PROFILE USED FOR FLORENCE SITE ............................................................... 263
TABLE 7.19 SENSOR LAYOUT (DISTANCES ARE IN INCHES) FOR FLORENCE SITE ............... 264

TABLE 7.20 COMPARISON OF FREQUENCY AND TIME-DOMAIN BACKCALCULATION RESULTS
WITH THOSE FOR MICHBACK - FLORENCE SITE ..................................................... 264

xi

TABLE 7.21 THICKNESS BACKCALCULATION IN TIME-DOMAIN ......................................... 267

TABLE 7.22 SEED VALUE USED FOR FLORENCE DATA ....................................................... 271
TABLE 7.23 PROFILE USED FOR KANSAS SITE ................................................................... 276
TABLE 7.24 PROFILE USED FOR KANSAS SITE WITH COMBINED AC AND ATB LAYER ...... 277
TABLE 7.25 BACKCALCULATION RESULTS FOR KANSAS SIIE ........................................... 27 7
TABLE 7.26 SEED VALUES USED FOR KANSAS DATA ......................................................... 283

xii

LIST OF FIGURES

FIGURE 2.1 A TYPICAL HYSTERESIS LOOP OF ASPHALT CONCRETE AT 25 o C (FROM SOUSA,
1986) ............................................................................................................................ 9

FIGURE 2.2 INFLUENCE OF FREQUENCY AND TEMPERATURE ON THE DYNAMIC MODULI OF
ASPHALT CONCRETE IN COMPRESSION AND SHEAR (FROM SOUSA, 1986) .................... 10

FIGURE 2.3 INFLUENCE OF FREQUENCY AND TEMPERATURE ON THE DAMPING RATIO OF
ASPHALT CONCRETE (FROM SOUSA, 1986) .................................................................. 10

FIGURE 2.4 INFLUENCE OF FREQUENCY AND TEMPERATURE ON THE DAMPING RATIO OF
ASPHALT CONCRETE (FROM SOUSA, 1986) .................................................................. 11

FIGURE 2.5 THE RELATIONSHIP BETWEEN DYNAMIC SHEAR MODULUS AND FREQUENCY FOR
MONTERY SAND #0, 90 PERCENT RELATIVE DENSITY (FROM SOUSA, 1986) ............... 12

FIGURE 2.6 THE RELATIONSHIP BETWEEN INTERNAL DAMPING AND FREQUENCY FOR
MONTERY SAND #0, 90 PERCENT RELATIVE DENSITY (FROM SOUSA, 1986) ............... 13

FIGURE 2.7 THE INFLUENCE OF FREQUENCY OF LOADING ON THE DYNAMIC SHEAR
MODULUS OF VICKSBURG SILTY CLAY (FROM SOUSA, 1986) ...................................... 14

FIGURE 2.8 THE INFLUENCE OF FREQUENCY OF LOADING ON THE DAMPING RATIO OF

VICKSBURG SILTY CLAY (FROM SOUSA, 1986) ........................................................... 15
FIGURE 2.9 MECHANICAL MODELS ..................................................................................... 17
FIGURE 3.1 LINEAR KELVIN’S MODEL ................................................................................ 28

FIGURE 3.2 COMPARISONS OF DYNAMIC DEFLECTION BASINS FROM SAPSI AND GREEN

COMPUTER PROGRAMS ................................................................................................ 32
FIGURE 3.3 REAL PART OF THE DISPLACEMENT TRANSFER FUNCTION ................................ 38
FIGURE 3.4 IMAGINARY PART OF THE DISPLACEMENT TRANSFER FUNCTION ....................... 39
FIGURE 3.5 REAL PART OF THE LOAD .................................................................................. 39
FIGURE 3.6 IMAGINARY PART OF THE LOAD ........................................................................ 40
FIGURE 3.7 REAL PART OF SENSOR DISPLACEMENTS .......................................................... 40
FIGURE 3.8 IMAGINARY PART OF SENSOR DISPLACEMENTS ................................................. 41
FIGURE 3.9 SENSOR DEFLECTION TIME HISTORIES .............................................................. 41
FIGURE 3.10 COMPARISON OF PREDICTED AND ACTUAL DEPTH-TO-BEDROCK .................... 45

xiii

FIGURE 3.1 1 COMPARISON OF PREDICTED AND ACTUAL DEPTH-TO-BEDROCK .................... 46

FIGURE 3.12 COMPARISON OF PREDICTED AND ACTUAL DEPTH-TO-BEDROCK .................... 47
FIGURE 3.13 COMPARISON OF PREDICTED AND ACTUAL DEPTH-TO-BEDROCK .................... 48
FIGURE 3.14 COMPARISON OF PREDICTED AND ACTUAL DEPTH-TO-BEDROCK .................... 49
FIGURE 3.15 COMPARISON OF PREDICTED AND ACTUAL DEPTH-TO-BEDROCK .................... 50
FIGURE 3.16 COMPARISON OF PREDICTED AND ACTUAL DEPTH-To-BEDROCK .................... 51
FIGURE 3.17 COMPARISON OF PREDICTED AND ACTUAL DEPTH-TO-BEDROCK .................... 52
FIGURE 3.18 COMPARISON OF PREDICTED AND ACTUAL DEPTH-TO-WATERTABLE .............. 53
FIGURE 3.19 COMPARISON OF PREDICTED AND ACTUAL DEPTH-TO-WATER TABLE ............. 54

FIGURE 3.20 RESULTS OF ANALYSIS FOR ELASTIC MODULUS OF SUBGRADE CALCULATION 57
FIGURE 3.21 RESULTS OF ANALYSIS FOR ELASTIC MODULUS OF SUBGRADE CALCULATION 58
FIGURE 3.22 RESULTS OF ANALYSIS FOR ELASTIC MODULUS OF SUBGRADE CALCULATION 58
FIGURE 3.23 RESULTS OF ANALYSIS FOR ELASTIC MODULUS OF SUBGRADE CALCULATION 59
FIGURE 3.24 RESULTS OF ANALYSIS FOR ELASTIC MODULUS OF SUBGRADE CALCULATION 59
FIGURE 3.25 RESULTS OF ANALYSIS FOR ELASTIC MODULUS 0F SUBGRADE CALCULATION 60
FIGURE 3.26 RESULTS OF ANALYSIS FOR ELASTIC MODULUS OF SUBGRADE CALCULATION 60

FIGURE 3.27 RESULTS OF ANALYSIS FOR ELASTIC MODULUS OF SUBGRADE CALCULATION 61

FIGURE 4.1 FWD LOAD VERSUS TIME ................................................................................ 70
FIGURE 4.2 DEFLECTIONS VERSUS TIME ............................................................................. 70
FIGURE 4.3 FAST-FOURIER TRANSFORM OF LOAD-TIME HISTORY ...................................... 71
FIGURE 4.4 FAST- FOURIER TRANSFORM OF DEFLECTION .................................................. 72
FIGURE 4.5 FAST-FOURIER TRANSFORM OF TRANSFER FUNCTION ...................................... 73
FIGURE 5.1 TYPICAL PLOTS FOR MEASURED FWD LOAD AND DEFLECTION DATA .............. 87

FIGURE 5.2 TYPICAL OUTPUT PLOTS FROM FREQUENCY-DOMAIN BACKCALCULATION ....... 89

FIGURE 5.3 TYPICAL OUTPUT PLOTS FROM TIME-DOMAIN BACKCALCULATION .................. 90

xiv

FIGURE 5.4 MAIN FLOW CHART FOR DYNABACK ........................................................... 94

FIGURE 5.5 DETAILS OF FREQUENCY-DOMAIN BACKCALCULATION PROCEDURE (CASES A &
B) ............................................................................................................................... 95

FIGURE 5.6 DETAILS OF TIME-DOMAIN BACKCALCULATION PROCEDURE USING PEAK TIME
AND TIME LAG (CASES C & D) ................................................................................... 96

FIGURE 5.7 DETAILS OF TIME-DOMAIN BACKCALCULATION PROCEDURE USING TRACES OF

TIME HISTORY (CASES E & F) ..................................................................................... 97
FIGURE 6.1 PERCENT ERROR IN BACKCALCULATED RESULTS - PROFILE 1 ........................ 101
FIGURE 6.2 PERCENT ERROR IN BACKCALCULATED RESULTS —- PROFILE 2 ........................ 101
FIGURE 6.3 PERCENT ERROR IN BACKCALCULATED RESULTS — PROFILE 3 ........................ 102
FIGURE 6.4 PERCENT ERROR IN BACKCALCULATED RESULTS - PROFILE 4 ........................ 102
FIGURE 6.5 PERCENT ERROR IN BACKCALCULATED RESULTS — PROFILE 5 ........................ 103
FIGURE 6.6 PERCENT ERROR IN BACKCALCULATED RESULTS — PROFILE 6 ........................ 103
FIGURE 6.7 PERCENT ERROR IN BACKCALCULATED RESULTS - PROFILE 7 ........................ 104
FIGURE 6.8 PERCENT ERROR IN BACKCALCULATED RESULTS — PROFILE 8 ........................ 104
FIGURE 6.9 PERCENT ERROR IN BACKCALCULATED RESULTS — PROFILE 9 ........................ 105

FIGURE 6.10 COMPARISON OF AC MODULUS USING SINGLE AND MULTIPLE FREQUENCY
BACKCALCULATION .................................................................................................. 1 1 1

FIGURE 6.1 1 COMPARISON OF AC DAMPING RATIO USING SINGLE AND MULTIPLE
FREQUENCY BACKCALCULATION .............................................................................. 1 1 1

FIGURE 6.12 COMPARISON OF AC THICKNESS USING SINGLE AND MULTIPLE FREQUENCY
BACKCALCULATION .................................................................................................. 1 12

FIGURE 6.13 COMPARISON OF BASE MODULUS USING SINGLE AND MULTIPLE FREQUENCY
BACKCALCULATION .................................................................................................. 1 12

FIGURE 6.14 COMPARISON OF BASE THICKNESS USING SINGLE AND MULTIPLE FREQUENCY
BACKCALCULATION .................................................................................................. l 12

FIGURE 6.15 COMPARISON OF SUBGRADE MODULUS USING SINGLE AND MULTIPLE
FREQUENCY BACKCALCULATION .............................................................................. 1 13

FIGURE 6.16 COMPARISON OF BACKCALCULATED AND ACTUAL AC MODULUS WITHOUT
THICKNESS BACKCALCULATION ................................................................................ I I3

XV

FIGURE 6.17 COMPARISON OF BACKCALCULATED AND ACTUAL AC DAMPING RATIOS
WITHOUT THICKNESS BACKCALCULATION ................................................................ 1 14

FIGURE 6.18 COMPARISON OF BACKCALCULATED AND ACTUAL BASE MODULUS WITHOUT
THICKNESS BACKCALCULATION ................................................................................ 1 14

FIGURE 6.19 COMPARISON OF BACKCALCULATED AND ACTUAL BASE DAMPING RATIOS
WITHOUT THICKNESS BACKCALCULATION ................................................................ 1 14

FIGURE 6.20 COMPARISON OF BACKCALCULATED AND ACTUAL SUBGRADE MODULUS
WI I HOUT THICKNESS BACKCALCULATION ................................................................ I 15

FIGURE 6.21 COMPARISON OF BACKCALCULATED AND ACTUAL SUBGRADE DAMPING
RATIOS WITHOUT THICKNESS BACKCALCULATION .................................................... 1 15

FIGURE 6.22 EFFECT OF SEED MODULI ON BACKCALCULATION RESULTS - LOW AC
MODULUS .................................................................................................................. 1 17

FIGURE 6.23 EFFECT OF SEED MODULI ON BACKCALCULATION RESULTS - MEDIUM AC
MODULUS .................................................................................................................. 1 18

FIGURE 6.24 EFFECT OF SEED MODULI ON BACKCALCULATION RESULTS - HIGH AC
MODULUS .................................................................................................................. 1 19

FIGURE 6.25 EFFECT OF SEED MODULI ON BACKCALCULATION RESULTS -TI-IIN AC LAYER
................................................................................................................................. 12 1

FIGURE 6.26 EFFECT OF SEED MODULI ON BACKCALCULATION RESULTS —MEDIUM THICK
AC LAYER ................................................................................................................ 122

FIGURE 6.27 EFFECT OF SEED MODULI ON BACKCALCULATION RESULTS -THICK AC LAYER
................................................................................................................................. 123

FIGURE 6.28 EFFECT OF SEED MODULI ON BACKCALCULATION RESULTS - FOUR-LAYER
PROFILE WITH MEDIUM-STIFF AC LAYER .................................................................. 125

FIGURE 6.29 EFFECT OF SEED MODULI ON BACKCALCULATION RESULTS - FOUR-LAYER
PROFILE WITH STIFF AC LAYER ................................................................................. 126

FIGURE 6.30 CONVERGENCE OF LAYER PARAMETERS FOR A THREE-LAYER PAVEMENT WITH
THIN AC LAYER — 4.88 Hz ....................................................................................... 127

FIGURE 6.31 CONVERGENCE OF LAYER PARAMETERS FOR A THREE-LAYER PAVEMENT
WITH THIN AC LAYER — 24.4 Hz .............................................................................. 128

FIGURE 6.32 CONVERGENCE OF LAYER PARAMETERS FOR A THREE-LAYER PAVEMENT
WITH THIN AC LAYER — 48.8 Hz .............................................................................. 129

xvi

FIGURE 6.33 CONVERGENCE OF LAYER PARAMETERS FOR A THREE-LAYER PAVEMENT WITH
MEDIUM-THICK AC LAYER — 4.48 Hz ....................................................................... 130

FIGURE 6.34 CONVERGENCE OF LAYER PARAMETERS FOR A THREE-LAYER PAVEMENT WITH
MEDIUM-THICK AC LAYER - 24.4 HZ ....................................................................... 13 1

FIGURE 6.35 CONVERGENCE OF LAYER PARAMETERS FOR A THREE-LAYER PAVEMENT WITH

MEDIUM-THICK AC LAYER — 48.8 Hz ....................................................................... 132
FIGURE 6.36 BACKCALCULATED POISSON’S RATIO AT VARIOUS FREQUENCIES ................ 134
FIGURE 6.37 PERCENT ERROR IN AC MODULI DUE TO CHANGE IN POISSON’S RATIO ........ 135

FIGURE 6.38 PERCENT ERROR IN BASE MODULUS DUE TO CHANGE IN POISSON’S RATIO... 1 35

FIGURE 6.39 PERCENT ERROR IN SUBGRADE MODULUS DUE TO CHANGE IN POISSON’S RATIO
................................................................................................................................. 135

FIGURE 6.40 EFFECT OF DEFLECTION IMPRECISION AND SIGNAL TRUNCATION ON
DEFLECT ION BASIN ERRORS ...................................................................................... 137

FIGURE 6.41 EFFECT OF DEFLECTION PRECISION ON AC THICKNESS BACKCALCULATION
(THICKNESS BACKCALCULATION ENABLED) .............................................................. 139

FIGURE 6.42 EFFECT OF DEFLECTION PRECISION ON BASE THICKNESS BACKCALCULATION
(THICKNESS BACKCALCULATION ENABLED) .............................................................. 140

FIGURE 6.43 EFFECT OF DEFLECTION PRECISION ON DEPTH-TO-STIF F LAYER
BACKCALCULATION (THICKNESS BACKCALCULATION ENABLED) .............................. 140

FIGURE 6.44 EFFECT OF DEFLECTION PRECISION ON AC MODULUS BACKCALCULATION
(THICKNESS BACKCALCULATION ENABLED) .............................................................. 140

FIGURE 6.45 EFFECT OF DEFLECTION PRECISION ON AC DAMPING RATIO
BACKCALCULATION (THICKNESS BACKCALCULATION ENABLED) .............................. 141

FIGURE 6.46 EFFECT OF DEFLECTION PRECISION ON BASE MODULUS BACKCALCULATION
(THICKNESS BACKCALCULATION ENABLED) .............................................................. 141

FIGURE 6.47 EFFECT OF DEFLECTION PRECISION ON SUBGRADE MODULUS
BACKCALCULATION (THICKNESS BACKCALCULATION ENABLED) .............................. 141

FIGURE 6.48 EFFECT OF DEFLECTION PRECISION ON AC MODULUS BACKCALCULATION
(LAYER THICKNESSES ASSUMED) .............................................................................. 142

FIGURE 6.49 EFFECT OF DEFLECTION PRECISION ON BASE MODULUS BACKCALCULATION
(LAYER THICKNESSES ASSUMED) .............................................................................. I42

xvii

FIGURE 6.50 EFFECT OF DEFLECTION PRECISION ON SUBGRADE MODULUS
BACKCALCULATION (LAYER THICKNESSES ASSUMED) .............................................. 142

FIGURE 6.51 EFFECT OF DEFLECTION PRECISION ON AC DAMPING RATIO
BACKCALCULATION (LAYER THICKNESSES ASSUMED) .............................................. 143

FIGURE 6.52 EFFECT OF DEFLECTION PRECISION ON BASE DAMPING RATIO
BACKCALCULATION (LAYER THICKNESSES ASSUMED) .............................................. 143

FIGURE 6.53 EFFECT OF DEFLECTION PRECISION ON SUBGRADE DAMPING RATIO
BACKCALCULATION (LAYER THICKNESSES ASSUMED) .............................................. 143

FIGURE 6.54 EFFECT OF SIGNAL TRUNCATION ON AC MODULUS BACKCALCULATION
(THICKNESSES KNOWN WITH 21:1 MICRON PRECISION) ................................................ 146

FIGURE 6.55 EFFECT OF SIGNAL TRUNCATION ON BASE MODULUS BACKCALCULATION
(THICKNESSES KNOWN WITH i1 MICRON PRECISION) ................................................ 146

FIGURE 6.56 EFFECT OF SIGNAL TRUNCATION ON SUBGRADE MODULUS BACKCALCULATION
(THICKNESSES KNOWN WITH i1 MICRON PRECISION) ................................................ 147

FIGURE 6.57 EFFECT OF SIGNAL TRUNCATION ON STIFF LAYER MODULUS
BACKCALCULATION (THICKNESSES KNOWN WITH i1 MICRON PRECISION) ................ 147

FIGURE 6.58 EFFECT OF SIGNAL TRUNCATION ON AC DAMPING RATIO BACKCALCULATION
(THICKNESSES KNOWN WITH :1 MICRON PRECISION) ................................................ 147

FIGURE 6.59 EFFECT OF SIGNAL TRUNCATION ON BASE DAMPING RATIO BACKCALCULATION
(THICKNESSES KNOWN WITH 21:1 MICRON PRECISION) ................................................ 148

FIGURE 6.60 EFFECT OF SIGNAL TRUNCATION ON SUBGRADE DAMPING RATIO
BACKCALCULATION (THICKNESSES KNOWN WITH :I:1 MICRON PRECISION) ................ 148

FIGURE 6.61 EFFECT OF SIGNAL TRUNCATION ON STIFF LAYER DAMPING RATIO
BACKCALCULATION (THICKNESSES KNOWN WITH i1 MICRON PRECISION) ................ 148

FIGURE 6.62 EFFECT OF SIGNAL TRUNCATION ON DEFLECTION BASINS ............................ 150

FIGURE 6.63 EFFECT OF SIGNAL TRUNCATION ON AC MODULUS BACKCALCULATION
(THICIG‘IESSES KNOWN WITH FULL PRECISION) .......................................................... 150

FIGURE 6.64 EFFECT OF SIGNAL TRUNCATION ON BASE MODULUS BACKCALCULATION
(THICKNESSES KNOWN WITH FULL PRECISION) .......................................................... 151

FIGURE 6.65 EFFECT OF SIGNAL TRUNCATION ON SUBGRADE MODULUS BACKCALCULATION
(THICKNESSES KNOWN WITH FULL PRECISION) .......................................................... 151

xviii

FIGURE 6.66 EFFECT OF SIGNAL TRUNCATION ON STIFF LAYER MODULUS
BACKCALCULATION (THICKNESSES KNOWN WITH FULL PRECISION) .......................... 151

FIGURE 6.67 EFFECT OF SIGNAL TRUNCATION ON AC DAMPING RATIO BACKCALCULATION
(THICKNESSES KNOWN WITH FULL PRECISION) .......................................................... 152

FIGURE 6.68 EFFECT OF SIGNAL TRUNCATION ON BASE DAMPING RATIO BACKCALCULATION
(THICKNESSES KNOWN WITH FULL PRECISION) .......................................................... 152

FIGURE 6.69 EFFECT OF SIGNAL TRUNCATION ON SUBGRADE DAMPING RATIO
BACKCALCULATION (THICKNESSES KNOWN WITH FULL PRECISION) .......................... 152

FIGURE 6.70 EFFECT OF SIGNAL TRUNCATION ON STIFF LAYER DAMPING RATIO

BACKCALCULATION (THICKNESSES KNOWN WITH FULL PRECISION) .......................... 153
FIGURE 6.71 COMPARISON OF DIFFERENT ORDER EXTRAPOLATIONS FOR SENSOR 1 ......... 156
FIGURE 6.72 COMPARISON OF DIFFERENT ORDER EXTRAPOLATIONS FOR SENSOR 2 ......... 157
FIGURE 6.73 COMPARISON OF DIFFERENT ORDER EXTRAPOLATIONS FOR SENSOR 3 ......... 158
FIGURE 6.74 COMPARISON OF DIFFERENT ORDER EXTRAPOLATIONS FOR SENSOR 4 ......... 159
FIGURE 6.75 COMPARISON OF DIFFERENT ORDER EXTRAPOLATIONS FOR SENSOR 5 ......... 160
FIGURE 6.76 COMPARISON OF DIFFERENT ORDER EXTRAPOLATIONS FOR SENSOR 6 ......... 161
FIGURE 6.77 EFFECT OF EXTRAPOLATION ON AC MODULUS BACKCALCULATION ............ 162
FIGURE 6.78 EFFECT OF EXTRAPOLATION ON BASE MODULUS BACKCALCULATION .......... 162

FIGURE 6.79 EFFECT OF EXT RAPOLATION ON SUBGRADE MODULUS BACKCALCULATION .162

FIGURE 6.80 EFFECT OF EXTRAPOLATION ON STIFF LAYER MODULUS BACKCALCULATION
................................................................................................................................. 163

FIGURE 6.81 EFFECT OF EXTRAPOLATION ON AC DAMPING RATIO BACKCALCULATION...163
FIGURE 6.82 EFFECT OF EXTRAPOLATION ON BASE DAMPING RATIO BACKCALCULATION 163

FIGURE 6.83 EFFECT OF EXTRAPOLATION ON SUBGRADE DAMPING RATIO
BACKCALCULATION .................................................................................................. 164

FIGURE 6.84 EFFECT OF EXTRAPOLATION ON STIFF LAYER DAMPING RATIO

BACKCALCULATION .................................................................................................. 164
FIGURE 6.85 EFFECT OF AC MODULUS ON PAVEMENT DEFLECTION AND TIME LAG .......... 168
FIGURE 6.86 EFFECT OF AC DAMPING ON PAVEMENT DEFLECTION AND TIME LAG .......... 168

xix

FIGURE 6.87 EFFECT OF AC THICKNESS ON PAVEMENT DEFLECTION AND TIME LAG ........ 168

FIGURE 6.88 EFFECT OF BASE MODULUS ON PAVEMENT DEFLECTION AND TIME LAG ....... 169
FIGURE 6.89 EFFECT OF BASE DAMPING ON PAVEMENT DEFLECTION AND TIME LAG ........ 169
FIGURE 6.90 EFFECT OF BASE THICKNESS ON PAVEMENT DEFLECTION AND TIME LAG ...... 169

FIGURE 6.91 EFFECT OF SUBBASE MODULUS ON PAVEMENT DEFLECTION AND TIME LAG .170
FIGURE 6.92 EFFECT OF SUBBASE DAMPING ON PAVEMENT DEFLECTION AND TIME LAG .. 170
FIGURE 6.93 EFFECT OF SUBBASE THICKNESS ON PAVEMENT DEFLECTION AND TIME LAG 1 70

FIGURE 6.94 EFFECT OF SUBGRADE MODULUS ON PAVEMENT DEF LECT ION AND TIME LAG

................................................................................................................................. 171
FIGURE 6.95 EFFECT OF SUBGRADE DAMPING ON PAVEMENT DEFLECTION AND TIME LAG
................................................................................................................................. 171
FIGURE 6.96 CONVERGENCE OF LAYER PARAMETERS FOR CASE 1 (NO-THICKNESS
BACKCALCULATION) ................................................................................................. 179
FIGURE 6.97 CONVERGENCE OF LAYER PARAMETERS FOR CASE 2 (NO-THICKNESS
BACKCALCULATION) ................................................................................................. 180
FIGURE 6.98 CONVERGENCE OF LAYER PARAMETERS FOR CASE 3 (NO-THICKNESS
BACKCALCULATION) ................................................................................................. 1 8 1
FIGURE 6.99 CONVERGENCE OF LAYER PARAMETERS FOR CASE 1 (THICKNESS
BACKCALCULATION) ................................................................................................. 182
FIGURE 6.100 CONVERGENCE OF LAYER PARAMETERS FOR CASE 2 (THICKNESS
BACKCALCULATION) ................................................................................................. 183
FIGURE 6.101 CONVERGENCE OF LAYER PARAMETERS FOR CASE 3 (THICKNESS
BACKCALCULATION) ................................................................................................. 184
FIGURE 6.102 CONVERGENCE OF LAYER PARAMETERS FOR CASE 1 (NO-THICKNESS
BACKCALCULATION) ................................................................................................. l 89
FIGURE 6.103 CONVERGENCE OF LAYER PARAMETERS FOR CASE 2 (NO-THICKNESS
BACKCALCULATION) ................................................................................................. 190
FIGURE 6.104 CONVERGENCE OF LAYER PARAMETERS FOR CASE 3 (NO-THICKNESS
BACKCALCULATION) ................................................................................................. 191
FIGURE 6.105 CONVERGENCE OF LAYER PARAMETERS FOR CASE 1 (THICKNESS
BACKCALCULATION) ................................................................................................. 192

FIGURE 6.106 CONVERGENCE OF LAYER THICKNESS FOR CASE 1 (THICKNESS
BACKCALCULATION) ................................................................................................. 193

FIGURE 6.107 CONVERGENCE OF LAYER PARAMETERS FOR CASE 2 (THICKNESS
BACKCALCULATION) ................................................................................................. 194

FIGURE 6.108 CONVERGENCE OF LAYER THICKNESS FOR CASE 2 (THICKNESS
BACKCALCULATION) ................................................................................................. 195

FIGURE 6.109 CONVERGENCE OF LAYER PARAMETERS FOR CASE 3 (THICKNESS
BACKCALCULATION) ................................................................................................. 196

FIGURE 6.1 10 CONVERGENCE OF LAYER THICKNESS FOR CASE 3 (THICKNESS
BACKCALCULATION) ................................................................................................. 197

FIGURE 6.1 1 1 ERROR IN BACKCALCULATED AC MODULUS DUE TO DIFFERENT DAMPING
RATIO COMBINATION FROM BASE AND SUBGRADE .................................................... 199

FIGURE 6.1 12 ERROR IN BACKCALCULATED AC DAMPING RATIO DUE TO DIFFERENT
DAMPING RATIO COMBINATION FROM BASE AND SUBGRADE ..................................... 199

FIGURE 6.1 13 ERROR IN BACKCALCULATED AC THICKNESS DUE To DIFFERENT DAMPING
RATIO COMBINATION FROM BASE AND SUBGRADE .................................................... 199

FIGURE 6.1 14 ERROR IN BACKCALCULATED BASE MODULUS DUE TO DIFFERENT DAMPING
RATIO COMBINATION FROM BASE AND SUBGRADE .................................................... 200

FIGURE 6.1 15 ERROR IN BACKCALCULATED SUBGRADE MODULUS DUE TO DIFFERENT

DAMPING RATIO COMBINATION FROM BASE AND SUBGRADE ..................................... 200
FIGURE 7 .1 TIME HISTORY FROM KUAB FWD ................................................................ 205
FIGURE 7 .2 FILTERED TIME HISTORY FROM KUAB FWD ................................................ 206

FIGURE 7.3 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS FOR US 1 31
(50699-15) ............................................................................................................... 208

FIGURE 7 .4 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS FOR US 131
(50699-20) ............................................................................................................... 209

FIGURE 7 .5 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS FOR US l 31
(50699-30) ............................................................................................................... 210

FIGURE 7 .6 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS FOR US 131
(50157-13) ............................................................................................................... 211

FIGURE 7 .7 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR US l 31 (50699-15) ................................................................................... 212

FIGURE 7 .8 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR USl3l (50699-20) ................................................................................... 212

FIGURE 7 .9 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR US131 (50699-30) ................................................................................... 212

FIGURE 7.10 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FORUS131 (50157-13) ................................................................................... 213

FIGURE 7 .1 1 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR US 1 31 (50699-15) WITH THICKNESS BACKCALCULATION ........................ 214

FIGURE 7.12 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR US 1 31 (50699-20) WITH THICKNESS BACKCALCULATION ........................ 215

FIGURE 7.13 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR US 1 31 (50699-30) WITH THICKNESS BACKCALCULATION ........................ 215

FIGURE 7.14 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR US 131 (50699-13) WITH THICKNESS BACKCALCULATION ........................ 215

FIGURE 7.15 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR US 131 (50699-15) WITHOUT THICKNESS BACKCALCULATION ................. 216

FIGURE 7.16 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR US 1 31 (50699-20) WITHOUT THICKNESS BACKCALCULATION ................. 216

FIGURE 7.17 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR USl31 (50699-30) WITHOUT THICKNESS BACKCALCULATION ................. 216

FIGURE 7.18 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR US 131 (50699-13) WITHOUT THICKNESS BACKCALCULATION ................. 217

FIGURE 7.19 ITERATION NUMBER TO CONVERGENCE VERSUS 01 FOR USl3l SITE ............. 218

FIGURE 7.20 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 1 .................................................................................................................. 219

FIGURE 7.21 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 2 .................................................................................................................. 219

FIGURE 7 .22 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 3 .................................................................................................................. 219

FIGURE 7.23 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 4 .................................................................................................................. 220

FIGURE 7 .24 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 5 .................................................................................................................. 220

FIGURE 7.25 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 6 ............................................................................................................ 220

FIGURE 7 .26 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR7 ................................................................................................................... 221

FIGURE 7.27 COMPARISON OF BACKCALCULATED LAYER MODULI FROM DIFFERENT MODELS
................................................................................................................................. 221

FIGURE 7.28 COMPARISON OF BACKCALCULATED LAYER MODULI USING DIFFERENT SEED
VALUES ..................................................................................................................... 222

FIGURE 7.29 COMPARISON OF BACKCALCULATED AC MODULUS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 224

FIGURE 7.30 COMPARISON OF BACKCALCULATED AC DAMPING RATIO FOR DIFFERENT
DAMPING RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ......................... 224

FIGURE 7.31 COMPARISON OF BACKCALCULATED AC THICKNESS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 225

FIGURE 7.32 COMPARISON OF BACKCALCULATED BASE MODULUS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 225

FIGURE 7.33 COMPARISON OF BACKCALCULATED SUBGARDE MODULUS FOR DIFFERENT

DAMPING RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ......................... 226
FIGURE 7.34 PAVEMENT PROFILE AND TEST SETUP FOR TEXAS SITE ................................. 227
FIGURE 7.35 NORMALIZED DEFLECTION VERSUS FWD LOAD ......................................... 228

FIGURE 7.36 FWD LOAD AND DEFLECTION TIME HISTORIES (LOAD LEVEL 1 - 6000 LB) -—
TEXAS SITE ............................................................................................................... 229

FIGURE 7.37 FWD LOAD AND DEFLECTION TIME HISTORIES (LOAD LEVEL 2 — 9000 LB) -
TEXAS SITE ............................................................................................................... 229

FIGURE 7.38 FWD LOAD AND DEFLECTION TIME HISTORIES (LOAD LEVEL 3 — 12000 LB) —
TEXAS SITE ............................................................................................................... 230

FIGURE 7.39 FWD LOAD AND DEFLECTION TIME HISTORIES (LOAD LEVEL 4 — 15000 LB) —
TEXAS SITE ............................................................................................................... 230

FIGURE 7.40 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS FOR LOAD
LEVEL 1 - TEXAS SITE .............................................................................................. 233

xxiii

FIGURE 7.41 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS FOR LOAD
LEVEL 2 — TEXAS SITE .............................................................................................. 234

FIGURE 7.42 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS FOR LOAD
LEVEL 3 — TEXAS SITE .............................................................................................. 235

FIGURE 7.43 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS FOR LOAD
LEVEL 4 - TEXAS SITE .............................................................................................. 236

FIGURE 7.44 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS FOR LOAD LEVEL 1 - TEXAS SITE ..................................................................... 237

FIGURE 7.45 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS FOR LOAD LEVEL 2 - TEXAS SITE ..................................................................... 237

FIGURE 7.46 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS FOR LOAD LEVEL 3 - TEXAS SITE ..................................................................... 238

FIGURE 7.47 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS FOR LOAD LEVEL 4 — TEXAS SITE ..................................................................... 238

FIGURE 7.48 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS FOR LOAD LEVEL 1 (WITH THICKNESS) — TEXAS SITE ....................................... 240

FIGURE 7.49 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS FOR LOAD LEVEL 2 (WITH THICKNEss) - TEXAS SITE ....................................... 240

FIGURE 7.50 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS FOR LOAD LEVEL 3 (WITH THICKNEss) —- TEXAS SITE ....................................... 240

FIGURE 7.51 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS FOR LOAD LEVEL 4 (WITH THICKNESS) — TEXAS SITE ....................................... 241

FIGURE 7.52 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS FOR LOAD LEVEL 1 (WITHOUT THICKNESS) — TEXAS SITE ................................ 241

FIGURE 7.53 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME

LAGS FOR LOAD LEVEL 2 (WITHOUT THICKNEss) — TEXAS SITE ................................ 241
FIGURE 7.54 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME

LAGS FOR LOAD LEVEL 3 (WITHOUT THICKNEss) — TEXAS SITE ................................ 242
FIGURE 7.55 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME

LAGS FOR LOAD LEVEL 4 (WITHOUT THICKNESS) — TEXAS SITE ................................ 242
FIGURE 7.56 ITERATION NUMBER TO CONVERGENCE VERSUS (1 FOR TEXAS SITE ............. 243

xxiV

FIGURE 7.57 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 1 .................................................................................................................. 244

FIGURE 7.58 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 2 ............................................................................................................ 244

FIGURE 7.59 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 3 .................................................................................................................. 244

FIGURE 7.60 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORY FOR
SENSOR 4 .................................................................................................................. 245

FIGURE 7.61 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 5 .................................................................................................................. 245

FIGURE 7.62 COMPARISON OF MEASURED AND PREDICT ED DEFLECTION TIME HISTORIES FOR
SENSOR 6 .................................................................................................................. 245

FIGURE 7.63 COMPARISON OF LAYER MODULUS FROM DIFFERENT MODEL ...................... 246

FIGURE 7.64 COMPARISON OF BACKCALCULATED LAYER MODULI USING DIFFERENT SEED
VALUES ..................................................................................................................... 247

FIGURE 7.65 COMPARISON OF BACKCALCULATED AC MODULUS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 248

FIGURE 7.66 COMPARISON OF BACKCALCULATED AC DAMPING FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 248

FIGURE 7.67 COMPARISON OF BACKCALCULATED AC THICKNESS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 249

FIGURE 7.68 COMPARISON OF BACKCALCULATED BASE MODULUS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 249

FIGURE 7.69 COMPARISON OF BACKCALCULATED SUBGRADE MODULUS FOR DIFFERENT
DAMPING RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ......................... 250

FIGURE 7.70 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS — CORNELL
SITE ........................................................................................................................... 252

FIGURE 7.71 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS ......................................................................................................................... 253

FIGURE 7.72 COMPARISON OF MEASURED AND PREDICTED DEFLECTIONS AND TIME LAGS
(WITH THICKNESS BACKCALCULATION) .................................................................... 254

FIGURE 7.73 COMPARISON OF MEASURED AND PREDICTED DEFLECTIONS AND TIME LAGS
(WITHOUT THICKNESS BACKCALCULATION) .............................................................. 254

FIGURE 7.74 ITERATION NUMBER TO CONVERGENCE VERSUS a FOR CORNELL SITE ......... 255

FIGURE 7.75 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 1 .................................................................................................................. 256

FIGURE 7.76 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 2 .................................................................................................................. 256

FIGURE 7.77 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 3 .................................................................................................................. 256

FIGURE 7.78 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 4 .................................................................................................................. 257

FIGURE 7.79 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 5 .................................................................................................................. 257

FIGURE 7.80 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 6 .................................................................................................................. 257

FIGURE 7.81 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 7 .................................................................................................................. 258

FIGURE 7.82 COMPARISON OF MEASURED AND PREDICTED DEFLECT ION TIME HISTORIES
FOR SENSOR 8 ........................................................................................................... 258

FIGURE 7.83 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 9 ........................................................................................................... 258

FIGURE 7.84 COMPARISON OF BACKCALCULATED MODULUS FROM DIFFERENT MODELS ..259

FIGURE 7.85 COMPARISON OF BACKCALCULATED LAYER MODULI USING DIFFERENT SEEDS
................................................................................................................................. 260

FIGURE 7.86 COMPARISON OF BACKCALCULATED AC MODULUS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE ..................................................... 261

FIGURE 7.87 COMPARISON OF BACKCALCULATED AC DAMPING RATIO FOR DIFFERENT
DAMPING RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ......................... 261

1:IGURE 7.88 COMPARISON OF BACKCALCULATED AC THICKNESS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 262

FIGURE 7.89 COMPARISON OF BACKCALCULATED BASE MODULUS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 262

xxvi

FIGURE 7.90 COMPARISON OF BACKCALCULATED SUBGRADE MODULUS FOR DIFFERENT
DAMPING RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ......................... 263

FIGURE 7.91 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS— FLORENCE
SITE ........................................................................................................................... 265

FIGURE 7.92 COMPARISON OF MEASURED AND PREDICTED DEFLECTION BASINS AND TIME
LAGS — FLORENCE SITE ............................................................................................. 266

FIGURE 7.93 COMPARISON OF PEAK DEFLECTIONS AND TIME LAGS (WITH THICKNESS
BACKCALCULATION) - FLORENCE SITE .................................................................... 267

FIGURE 7.94 COMPARISON OF PEAK DEFLECTIONS AND TIME LAGS (WITHOUT THICKNESS
BACKCALCULATION) — FLORENCE SITE .................................................................... 267

FIGURE 7.95 ITERATION NUMBER TO CONVERGENCE VERSUS (1 FOR FLORENCE SITE ....... 268

FIGURE 7.96 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 1 .................................................................................................................. 268

FIGURE 7.97 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 2 .................................................................................................................. 269

FIGURE 7.98 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 3 .................................................................................................................. 269

FIGURE 7.99 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES FOR
SENSOR 4 .................................................................................................................. 269

FIGURE 7.100 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 5 ............................................................................................................ 270

FIGURE 7.101 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 6 ............................................................................................................ 270

FIGURE 7.102 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 7 ............................................................................................................ 270

FIGURE 7.103 COMPARISON OF BACKCALCULATED LAYER MODULI FROM DIFFERENT
MODELS .................................................................................................................... 271

FIGURE 7.104 COMPARISON OF BACKCALCULATED LAYER MODULI USING DIFFERENT SEED
VALUE ....................................................................................................................... 272

FIGURE 7.105 COMPARISON OF BACKCALCULATED AC MODULUS FOR DIFFERENT DAMPING
RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ........................................ 273

xxvii

FIGURE 7.106 COMPARISON OF BACKCALCULATED AC DAMPING RATIO FOR DIFFERENT
DAMPING RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ......................... 273

FIGURE 7.107 COMPARISON OF BACKCALCULATED AC THICKNESS FOR DIFFERENT
DAMPING RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYER .......................... 274

FIGURE 7.108 COMPARISON OF BACKCALCULATED BASE MODULUS FOR DIFFERENT
DAMPING RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ......................... 274

FIGURE 7.109 COMPARISON OF BACKCALCULATED SUBGRADE MODULUS FOR DIFFERENT
DAMPING RATIO COMBINATIONS FOR BASE AND SUBGRADE LAYERS ......................... 275

FIGURE 7.1 10 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS (FOUR LAYER BACKCALCULATION) .................................................................. 276

FIGURE 7 .1 1 1 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTIONS AND TIME
LAGS FOR CASE 1 (THREE LAYER BACKCALCULATION) ............................................. 27 8

FIGURE 7 .1 12 COMPARISON OF MEASURED AND CALCULATED PEAK DEFLECTION AND TIME
LAG FOR CASE 2 (THREE LAYER BACKCALCULATION) ............................................... 27 8

FIGURE 7 .1 13 ITERATION NUMBER TO CONVERGENCE VERSUS a FOR KANSAS SITE ......... 279

FIGURE 7 .1 14 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 1 ............................................................................................................ 279

FIGURE 7 .1 15 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 2 ............................................................................................................ 280

FIGURE 7.1 16 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 3 ............................................................................................................ 280

FIGURE 7 .1 17 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 4 ............................................................................................................ 280

FIGURE 7.1 18 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 5 ............................................................................................................ 281

FIGURE 7.1 19 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 6 ............................................................................................................ 281

FIGURE 7.120 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 7 ............................................................................................................ 281

FIGURE 7.121 COMPARISON OF MEASURED AND PREDICTED DEFLECTION TIME HISTORIES
FOR SENSOR 8 ............................................................................................................ 282

FIGURE 7.122 COMPARISON OF BACKCALCULATED MODULI FROM DIFFERENT MODELS ...282

xxviii

FIGURE 7.123 COMPARISON OF BACKCALCULATED LAYER MODULI USING DIFFERENT SEED
VALUES ..................................................................................................................... 283

FIGURE 7.124 COMPARISON OF BACKCALCULATED AC MODULUS FOR DIFFERENT DAMPING
RATIO COMBINATIONS OF BASE AND SUBGRADE LAYERS .......................................... 284

FIGURE 7.125 COMPARISON OF BACKCALCULATED AC DAMPING RATIO FOR DIFFERENT
DAMPING RATIO COMBINATIONS OF BASE AND SUBGRADE LAYERS ........................... 284

FIGURE 7.126 COMPARISON OF BACKCALCULATED AC DAMPING RATIO FOR DIFFERENT
DAMPING RATIO COMBINATIONS OF BASE AND SUBGRADE LAYERS ........................... 285

FIGURE 7.127 COMPARISON OF BACKCALCULATED AC DAMPING RATIO FOR DIFFERENT
DAMPING RATIO COMBINATIONS OF BASE AND SUBGRADE LAYERS ........................... 285

FIGURE 7.128 COMPARISON OF BACKCALCULATED AC DAMPING RATIO FOR DIFFERENT
DAMPING RATIO COMBINATIONS OF BASE AND SUBGRADE LAYERS ........................... 286

xxix

CHAPTER 1 - INTRODUCTION

1.1 General

The falling weight deﬂectometer (FWD) is a commonly used device for evaluating the
structural condition of pavements. Considerable effort has been expended over the years
to interpret FWD deﬂection basins for determining rehabilitation strategies. This is
. usually done through static backcalculation in which layer moduli are determined by
matching the peak deﬂections measured under a known load with deﬂections generated

through a theoretical model of the pavement.

1.2 Problem Statement

Over the years, many backcalculation procedures for pavement parameters have been
developed. At present, pavement layer moduli can be backcalculated from FWD data
using static and dynamic methods. Static methods use only the peak values of the FWD
response time histories, While dynamic methods use more of the information contained
within the time histories. Since the FWD imparts a dynamic load, Viscoelastic pavement
properties and dynamic effects such as inertia and damping will affect the pavement
response. Static backcalculation neglects these effects and is therefore less accurate than
dynamic backcalculation. Furthermore, dynamic backcalculation uses the richer
information contained Within the FWD response time histories and may therefore have

the potential to backcalculate a greater number of parameters than static backcalculation.

The above considerations indicate a need for dynamic backcalculation of the layer
moduli. The purpose of this study is to develop such a tool. Dynamic backcalculation
Should characterize pavement materials more accurately, and thus lead to a better

prediction of the pavement response using the mechanistic-empirical method of design.

1.3 Research Objective

The objective of this project is to develop a robust dynamic backcalculation computer
program, Whose results are not sensitive to the seed values of layer moduli. In addition,
the algorithm should be able to compute the layer thicknesses and damping ratios

accurately.

The resulting program needs to be user-friendly, providing various options to the user to
view and preprocess the load and deﬂection time histories and deﬂection basins. This
should provide an advanced backcalculation tool to pavement engineers in the context of

a mechanistic based design methodology.

Upon verifying the robustness of the new dynamic backcalculation program, it will be
possible to incorporate it in the Michigan Flexible Pavement Design System (MFPDS)

computer software.

1.4 Report Layout

This report is organized aS follows:

Chapter 2 provides a review of relevant literature on the analysis of asphalt concrete

pavements. Various backcalculation methods of layer moduli and their merits and

limitations are presented. Also some of the difﬁculties related with the backcalculation

process and error sources are discussed.

Chapter 3 describes the forward analysis program. The response of Viscoelastic multi-
layered pavement system due to a FWD loading is presented. The pavement is modeled
as a system of horizontal layers whose material is assumed to be isotropic and linearly
elastic With a hysteretic type damping. The complex response method is introduced, and
the steady state as well as the transient response analyses using frequency-domain

analysis is discussed.

Chapter 4 introduces an efﬁcient iterative method for dynamic backcalculation of
pavement layer properties using the relative difference between measured and computed
deﬂections. A modiﬁed Newton method and its application to the backcalculation of
pavement layer properties are presented for both frequency and time-domain

backcalculation.

Chapter 5 presents the structure and features of the DYNABACK program.

Chapter 6 presents the validation results of the DYNABACK program using theoretical
deﬂection time histories. Important aspects of convergence characteristics and uniqueness
of solutions are examined. Sensitivity analyses for the various layer parameters are
conducted, and the effects of imprecision in deﬂections and duration of deﬂection records

are studied.

Chapter 7 contains the evaluation of the DYNABACK program using measured FWD

test data from pavements across the State of Michigan as well as other sites.

Chapter 8 includes a summary of the ﬁndings and some recommendations for future

research.

CHAPTER 2 - LITERATURE REVIEW

2.1 General

Falling Weight Deﬂectometers (FWD) are Widely used to evaluate the structural
properties of ﬂexible pavements nondestructively. Backcalculation of pavement
properties from FWD data is usually carried out by matching the measured deﬂections
under a known load with theoretical deﬂections generated by an analytical model of the
pavement by varying the elastic moduli. Such procedures usually use error minimization
techniques to minimize either the absolute or the squared error, with or without weighing

factors.

At present, pavement layer moduli can be backcalculated from the FWD deﬂection basin
using the peak values of the deﬂection time histories (static backcalculation) or using the
FWD full time history (dynamic backcalculation). However, the deﬂection basin under a
static load is different from that under dynamic or impulse loads because of Viscoelastic
pavement properties and dynamic effects such as inertia, damping, and resonance.
Dynamic analysis would therefore provide a more accurate estimation of the pavement

modulus.

However, the interpretation of data still remains problematic. This is due to the
limitations associated With the mechanical models incorporated into the backcalculation
procedures and the uniqueness of inverse solutions. The net effect of these limitations is
to increase the uncertainty associated with the values of the estimated in-situ mechanical

properties. Such uncertainties will contribute to reducing an engineer’s conﬁdence in

their ability to properly evaluate the structural integrity of the pavement and estimate its
remaining life. Nevertheless, during the past few decades, there was a signiﬁcant
improvement in the area of pavement modeling and NDT techniques. In the following
sections, the development of pavement models and backcalculation schemes Will be

reviewed and discussed.

2.2 Static Material Characterization

2. 2. I Layered elastic model

The simplest way to characterize the behavior of ﬂexible pavements is based on
Boussinesq’s solution that models a ﬂexible pavement as a homogeneous, isotropic, and
elastic half-space. Later, Bunnister (1943) presented a method for determining stress,
stain and displacement in a two layer system. Based on Bunnister’s method, Acum and
Fox (1951) presented the solution for a three-layered pavement system. Since then, a
large number of computer programs have been developed for calculating the analytical
response of multi-layered ﬂexible pavements to different load and layer interface
conditions, including CHEVRON (Warren and Dieckmann, 1963), BISAR (Dejong et
al,1973), ELSYMS (Kopperman, 1985), and KENLAYER (Huang, 1993). Finite element
analysis is another method that can model a layered elastic system, in Which the layered
pavement is divided into many small “elements”. The stress state in each element is
calculated using the theory of elasticity. Programs such as MICH-PAVE (Yeh, 1989) and
ILLI-PAVE (Raad and Figueroa, 1980) have been developed using the ﬁnite element
method. Other approaches, such as the equivalent thickness method based on the

equivalent layer theory were introduced by Odernark (1949) and Ullidiz (1987).

2. 2.2 Nonlinear elastic model

It is well known that granular materials and subgrade soils are nonlinear with their elastic
modulus varying with the level of stress. Various constitutive equations have been
developed to describe the behavior of nonlinear elastic materials. Computer programs
that can handle non-linear behavior Within the layered elastic theory include
KENLAYER (Huang, 1993) and NELAPAVE (Irwin, 1994). The ﬁnite element
computer programs MICHPAVE and ILLIPAVE can model non-linear material behavior

more accurately.

2.3 Static Backcalculation Methods

Most of the commonly used backcalculation programs are generally based on static
forward models. Exiting static backcalculation methods can be separated into three major

groups depending on the techniques used to reach the solution.

The ﬁrst group is based on iteration techniques, which repeatedly use a forward analysis
method within an iterative process. The layer moduli are repeatedly adjusted until a
suitable match between the calculated and measured deﬂection basins is obtained. A
number of computer programs, such as BISDEF (Bush, 1985), BOUSDEF (Roesset,
1995), CHEVDEF (Bush, 1985), and COMCOMP (Irwin, 1994), have been developed

for back-calculation analysis using this method.

The second group is based on searching a database of deﬂection basins. A forward
calculation scheme is used to generate a database, which is then searched to ﬁnd a best

match for the observed deﬂection basin. The program MODULUS (Uzan, 1994) is one

such example. It uses deﬂection databases generated from the forward program BISAR,
and a Hook-Jeeves pattern search algorithm within a three-point Lagrange interpolation

technique to backcalculate a set of layer moduli.

The third group is based on the use of regression equations ﬁtted to a database of
deﬂection basins generated by a forward calculation scheme. The LOADRATE program
(Chua, 1984) belongs to this category and uses regression equations generated ﬁom a
database obtained by using the ILLIPAVE (Raad, 1980) nonlinear ﬁnite element

program.

A thorough literature review on static backcalculation can be found elsewhere

(Mahmood, 1993).

2.4 Dynamic Properties of Paving Materials

2. 4. I Asphalt Concrete

Laboratory tests indicate that the stress-strain curves for asphalt concrete materials under
harmonic load exhibit a hysteresis loop as Shown in Figure 2.1(Sousa, 1986). The
elliptical shape shows that asphalt concrete properties are linear Viscoelastic materials at
low strains. The Viscoelasicity can be expressed in terms of a modulus and a damping
ratio which can be determined from stress or strain-controlled sinusoidal testing (Sousa,
1986). Experimental results plotted in Figure 2.2 through Figure 2.4 Show that the
dynamic modulus increases with frequency between 0.5 and 20 Hz, while it decreases
with higher temperature. The ﬁgures also Show that dynamic modulus (Slope) is less

affected by frequency at lower temperatures. On the other hand, damping increases With

higher temperature, and it decreases with higher frequency. Poisson’s ratio increases with
increasing of temperature, and decreases with increasing frequency. Also it is well known
that aging makes asphalt concrete lose its Viscoelasticity and become more brittle with

time.

 

82.2443 lbs/in:

 

46.24432 /

 

 

 

 

-5.39274E-04 -5.152742E-04 in/in

Figure 2.1 A typical hysteresis loop of asphalt concrete at 25 ° C (from Sousa, 1986)

Dynamic Moduli E“ G‘(Psl)

 

1.E-01 1.E+00 1.E+01 1.E+02
Frequency (Hz)

Figure 2.2 Inﬂuence of frequency and temperature on the dynamic moduli of asphalt
concrete in compression and Shear (from Sousa, 1986)

0.30 —.

0.20 i

Internal Damping

0.10

 

 

 

0.00 ‘l I , 1 4—4— +4-— L—A—L—vr—L—w 2A.--.
0.00 5.00 10.00 15.00 20.00 25.00
Frequency (Hz)

Figure 2.3 Inﬂuence of frequency and temperature on the damping ratio of asphalt
concrete (from Sousa, 1986)

10

 

3° 1.0 a -2 r 1
E a
.0 . l
c l
o .
3 .
g 0.1 — — m
l i ‘ 1
00.1 __- - - v A. ,__ s. . -. 2 ._l 2__,__ _.L_ .. 2 . __.._;__. 1., _ _ . ,_, -.__ ._ I. .‘_.L_ 1....i
1.E-01 1.E+00 1.E+01 1.E+02

Frequency (Hz)

Figure 2.4 Inﬂuence of frequency and temperature on the damping ratio of asphalt
concrete (from Sousa, 1986)

2. 4.2 Granular Materials

Granular materials are commonly used for the construction of bases and subbases. Due to
the non-linearity of granular materials, the modulus and damping ratio of the base and the
subbase is dependent on three main factors: (1) the strain level; (2) the conﬁning
pressure; and (3) the relative density (Harding and Dmevich, 1972) (Seed, Wong, and et
al, 1986). Figure 2.5 (Sousa, 1986) shows the dynamic shear modulus as a function of
frequency at three strain levels (0.01, 0.1 and 1.0 percent) conducted with three different
levels of effective mean stress (26, 20 and 16. in. Hg). The ﬁgure Shows that the shear
modulus decreases with increasing strain level; i.e., it exhibits no-linear behavior.
However, the modulus is independent of frequency. These results are in excellent

agreement with the results presented by seed and Idriss (1968).

ll

Figure 2.6 (Sousa, 1986) Shows that the internal damping increases with increasing strain

levels, in addition scattering of data in the ﬁgure shows that frequency have a effects on

 

 

 

 

 

 

damping ratio.
25000 —r— h” j-—~—-~ -— ~ -~ A w- w— -.-__, . ~——— _A‘l
:_ Specumen S1 —— Total Stress=16 In.Hg 3
1 ----- Total Stress=20 in.Hg
---------- Total Stress=26 in.Hg
'5'? 20000 T B: .—. ..:: :.~. : :.:.: :9; :11: 5.7:: T.T."_.'.f.’,_’,;‘ @5erng ,.__
9." F ,0— . ° " l
39 l— or ----------- ~0- ______________ Strain( /o) -0.01 l
_ L ‘Y Strain(%) =0.01
-— l 5‘ I
.3 15000 T*"*"’*3.:.: :. ; :* ........ ﬁ .. Tr . " “l
O l ‘3' ----------------------- 9 Strain(%) =0.10 1
E f ‘o_ _________ l
'5 ,. or ” " ‘°‘ _______________ Strain(%) =o.10 1
0 l A . ° Strain(%) =0.1o
c ‘ A,“ u A l
0) 10000 7“" — -_ ~~- -~ w -- —--- W - a — e— _- - ”-1 —-i
.2 L l
E i l
«I .1
C I l
> : .
o i l
5000 j~——- —-—-~ ** ~— -— ~14 —— - -"— ”tr-*4” —m~~~~i
1 Strain(%) =1.0 .;
I" '3' ‘9 ------------------ a ............................. .3 Strain(%) =1.0 E
it"s—3" """""" ‘8‘““"""“";2 Strain(%)=1.0 l
I i

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00
Frequency (Hz)

Figure 2.5 The relationship between dynamic Shear modulus and frequency for Montery
sand #0, 90 percent relative density (from Sousa, 1986)

12

Figure 2.6 (Sousa, 1986) Shows that the internal damping increases with increasing strain

levels, in addition scattering of data in the ﬁgure shows that frequency have a effects on

 

 

 

 

 

 

damping ratio.
25000 T—N ;» u u -* ~ - *1— —* -- .___,__.___ —-——- m M -~
1. Specimen S1 —— Total Stress=16 in.Hg
r- ------ Total Stress=20 in.Hg
.......... Total Stress=26 in.Hg
77: 20000 ”135'(”'94:..L..:‘.’..i.".."..i'jfjg "s‘trgm(o/,);5,a1““'j
n. l A» 8 (°/) 0 0 1‘
v 1 ------- train 0 = . 1
'3 ,— V _________________
<3 ‘L 7°" ‘0 Strain(%) =0.01
— M
.3 15000 F‘ﬁj'é‘,’ ........ ' ............ T3 _____ #- _ '7'“ E;
o 1r ------------------------ a Strain(%) =o.10
E F
a L «4' """""" ‘°‘ _______________ Strain(%) =0.1o
g 1. A . ° Strain(%) =o.10
A’" u A
In 10000 — - —--——~- —— —— —7 - r v - - 1 ————~:
.2 T— .
E ' l
a .
C r 1
>4 1 l
a l
5000 -%~-—-—— -—‘—-~— - - —- LL--- — ,,___._- L___ ~--——-- ..I
r Strain(%) =1.0
i” :13"_“_“_“_'-_- ------------ e ............................. .3 Strain(%) =1.0
r r3; """ ‘3"“‘“""“";_? Strain(%)=1.0

L

l
l
l
j l
0 .+_1_L_L_L_L_.L.&.—L<L.L “ALA WEI—LL“ 4“" I-..;__L.h_l_L.L_4_l_J_z_I_J_J_J_+_l_L_L_L_I_LL_L_L_+_A_LCL_L.L LL...— - .1

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00
Frequency (Hz)

Figure 2.5 The relationship between dynamic Shear modulus and frequency for Montery
sand #0, 90 percent relative density (from Sousa, 1986)

12

Total Stress=16 in.Hg
------ Total Stress=20 in.Hg
.......... Total Stress=26 in.Hg

 

A
O
O

l

1

l

l

l

l

1

l

1

1

1

l

l

l

1

l

1

l

1

«Yr—

f Specimen S1

    
 
 

1

l

l

l

1

i

_——— ......... o . o 1
,,...-—..-—. ----- Strain(/o)=1.0 1
1‘

l

l

l

l

1

1

1

.‘T‘TT‘T—T—TM‘WT’T‘ _.- rr Tl" ‘T “r
l
I
l
l
1
l
l
1
1
1
1
1
1
1
1
1
1
l
1
1
1
l
1
l
l
l
1
l
l

 

8:87.737 -- -° ”" T" H

O) 0.100 — , -

.E ‘ Strain(%) =0.01

a.

E .

N r 1

D L 1

'g 1- 1

3 1 1

.E 1 1

0.010 1; n-____-_._......_1
E Strain(%) =0.10 }
1 1
1 1
1 1
r 1
l l
I 1
0.001 .1144 LR. .-L,.: 1.. .I..- 1.1.T ‘..‘_....'_i_-. .14.. -.-.1.... “1” 1.; 1.1... ...1 7.1.1.1..1...'._l_.‘..'.+;__._...'-._J...1...L.....1.1l
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00

Frequency (Hz)

Figure 2.6 The relationship between internal damping and frequency for Montery sand
#0, 90 percent relative density (from Sousa, 1986)

2. 4.3 Fine-grained Soil

Determination of hysteretic stress-strain loops in soils can be obtained from triaxial
compression tests, Simple shear tests or torsional Shear tests conducted under harmonic
loading conditions. Dynamic properties of a Vicksburg silty clay are shown in Figure 2.7

and Figure 2.8. These ﬁgures illustrate that the internal damping increases With

13

increasing strain amplitude and decreases with increasing frequency; however frequency

of loading does not affect the dynamic shear modulus of clay.

 

 

 

 

 

12000 --_--- - - ----——— --_- —~——
. Specmen CZ —— Total Stress = 29 in.Hg (1st)
——————— Total Stress = 20 in.Hg
........ Total Stress = 11 in.Hg 1
1 CK ..... Total Stress = 29 in.Hg (2nd) 1
10000 -t_-___ _ __.. _ .._ -._ .\_ .-_- ..,_.._....... --_-.._._.,_-1
1 \x ,
A " \\ I
5 l»— Shear Strain \
e; '_ Amplitude (%)=0.0l \ T /
‘U .'.____ ___. .__. ,___ __ _ - - - _ _ . ___- . __ - _ § T,~_—-.- __, .._ ___ ___-____ _ __.-._ .._-)
o 8000 - / ....... a _ ,
= ' T ‘ ~ ~ a
.3 ‘2 t _________________ A~ .‘ . 1
é’ ‘r """ “ 1‘
a 6000 "it! ___ - _ _. _ _ _ .2 -._--." .. R_;_ __.. _ _—‘ ’. x - - c .2 9: 9...: '—‘_‘-‘_Z 1 ' .' '..' '_'__'. FFXLLF'TT.'T:—_1
o I \ '1" ‘3‘
.1: '» Shear Strain ‘ . - "\ L
‘0 Amplitude 1%)=0. I 0 X x j
02 " \\ l
E .-_. _..___-_ _ _ _ __ _ _ _ __°:\§_. .._. _ - ___ .._-.. -1
. 40w N
c i
5‘ .A ----------------- A ------- A
l“ / ° I o - ' ' .*
2000 -._- -—~ ~ -+ ,- ,5- - ~ e; f; 5; .7. e ,-.- ~——
7 Shear Strain 0‘ X """"" }
‘” Amplitude (%)=l.00 fFT' ‘

 

0.1 1.0 10.0
Frequency (Hz)

Figure 2.7 The inﬂuence of frequency of loading on the dynamic Shear modulus of
Vicksburg silty clay (from Sousa, 1986)

14

 

0.25 -- —- -————--—» —_ .. . -2 - __ --_
—— Total Stress = 29 in.Hg (1 st)

 

 

 

0 23 1.- ___ Spa—cin_e_nC2 ﬂ __ .____-__::-.'.':':".- Total Stress = 20 in.Hg .1
' l ........ Total Stress = 11 in.Hg 1
1 ct ..... Total Stress = 29 in.Hg (2nd) ‘
0.20 ;——----—-—--—— —- - ——— \ —-—-- _ --
1 Shear Strain \ .
l Amplitude (%)=l.00 \ .
0.18 .- --_- —--—-— —— —— — - ~— —-\ --_--- —- ----- +1
2: i
a 0.15 _--
E .
“I 1
0 0.13 ., _, ~ -
'5 1 Shear Strain
E 1 Amplitude (%)=0.10
3 0.10 1‘ —~—-—-- - - - -— -
.E
l
0.08 - -
i
0.05 __.. — +- 4 —
1 Shear Strain
I Amplitude (%)=0.0l ' ..
0.03 - _.-_- ._ - +—+ - ~ —— — — —— ~ , <"“
i \
000 -7- ~- -- ~22 - -- _ -_,- 4- --i-- - .7 - 4 - 7 --_- ‘4 ,-- --‘u----_L.i_1
0 1 1.0 10 0

Frequency (Hz)

Figure 2.8 The inﬂuence of frequency of loading on the damping ratio of Vicksburg silty
clay (from Sousa, 1986)

2.5 Viscoelastic Material Characterization
Viscoelastic material response is comprised of elastic and Viscous responses
corresponding to the behavior of a solid and a liquid, respectively. There are two general

methods to characterize Viscoelastic materials (Huang, 1993): (i) by mechanical models,

and (ii) by a creep compliance curve.

15

2.5.1 Mechanical Models

The behavior of an asphalt concrete material can be modeled using a combination of
springs and dashpots (Huang, 1993). The most basic models include the Maxwell and
Kelvin models. The Maxwell model consists of a linear spring and a viscous damper in
series. The Kelvin model consists of a linear spring and a viscous damper in parallel.
More complex Viscoelastic models include the standard solid model and the Burger’s
model. The standard solid model combines Kelvin and spring models in series, while
Burger’s model consists of Kelvin and Maxwell models in series. Figure 2.9 shows the
various mechanical models described above, with o, E, and X denoting stress, elastic

modulus, and viscous fraction, respectively.
Maxwell Model

The response from the Maxwell model is the least realistic; under constant stress loading
(creep), it consists of an instantaneous strain, which is recoverable, followed by a linearly
increasing strain, which is irrecoverable. The equation characterizing the response

behavior of Maxwell model (Huang, 1993) is:
77 . 77 .
0+—0=E—8 11

For viscous damping of Maxwell model, the elastic modulus is replaced by the complex

modulus (Haddad, 1995):

2 2
* . a) . a)
E =E,.a,+.Elm=E ___gj..._g_2 (2.2)
1+0) 5 1+0) 4‘

l6

in which .5 = g , and a) is the circular frequency.

 

 

o
o
o 52
E E L1: )0 El ___1 7w
T I. t___. I?)
l__l 7t
0 o i 2
o
(a) Maxwell (b) Kelvin (c) Burgers

Figure 2.9 Mechanical models

Kelvin Model

 

 

((1) Standard Solid

The response from the Kelvin model is more realistic than that for the Maxwell model;

under constant stress loading (creep), the strain starts at zero, increases non-linearly (with

an exponential term) approaching a maximum value corresponding to the elastic

response. The Kelvin model is usually used to simulate viscous or hysteretic (material)

damping. For viscous damping, the elastic modulus is replaced by the complex modulus:

E* = E(1+iw§d)

(2.3)

in which E is the spring constant, 0) is the circular frequency and id is the damping ratio

(equal to the ratio of the dashpot constant to the spring constant). Note that in this case,

the real part of the modulus is constant and equal to the elastic modulus while the
imaginary part of the modulus (representing damping) varies linearly with increasing

frequency.
Burger Model

Burger’s model, also called the Four-parameter ﬂuid model, consists of both a Maxwell
component and a Kelvin component in series. The complex modulus can be written as
(Al-Khoury, 2001):

E*(a;)= (02[P141 -q2(l-p202)]+iw[p1qzwz -q1(1- 192602)] (2.4)

2
p102 +[l — pzwz)

 

where

,1 ,1 A 2.1
p1=__1_+_;+_2 ”:42.
E1 152 52 151192
1112
:1 =—
41 142 152

The complex shear modulus and the complex Lame’s constant can be determined as (Al-

Khoury, 2001):

 

G'(a) 3K(iwa—a)2q2)

= 2.5
9K(l+iwpl—w2p2)-iwa+w2q2 ( )

18

where K is the bulk modulus.

For hysteretic damping, the complex modulus is constant with frequency, and can (for

small damping) be written as:
5" = £0 + zigd) (2.6)

The best model (among the four models described above) for describing the response of
asphalt concrete is the Burger’s model. While not perfectly suited for real material
behavior, its strain response under constant stress shows many of the characteristics
observed under creep testing in the laboratory. It is important to note, however, that both
the real and imaginary parts of the complex modulus are zero at zero frequency and vary
non-linearly with increasing frequency. The Poisson’s ratio is also complex and

frequency dependent in this case.

Olard and Di Benedeho (2003) proposed a general “ZSZPID” model which simulates

better the Viscoelastic aspect of both asphalt and mixtures.

Note that the SAPSI program (Chen, 1987) allows for the elastic and shear moduli as
well as the Poisson ratio to be complex and frequency dependent, thus allowing for
describing the response according to any Viscoelastic model (i.e., Kelvin, Burger’s or any

other model).
2. 5.2 Creep Compliance Model
The asphalt concrete material may be modeled by using a power law model. Such a

model describes the stress-strain relationship for an asphalt concrete mixture as a creep

l9

compliance function. For the purpose of backcalculation, Magnuson et a1 (1991) used a

three-parameter model:
D(t)= D0 + Dlt’" (2.7)

where, D0 = l/Eo (Elastic response term)
D; = Creep compliance constant (for viscous term)

m = Exponent for nonlinear time dependence

2.6 Dynamic Forward Computation Programs

Most dynamic backcalculation methods use dynamic damped-elastic ﬁnite-layer or ﬁnite
element models for their forward solutions. The ﬁnite layer solutions are based on
Kausel’s formulation (Kausel, 1982) which subdivides the medium into discrete layers
that have a linear displacement function in the vertical direction and satisfy the wave
equation in the horizontal direction. The solution is based on the premise that if the
sublayer thickness is small relative to the wavelength of interest, it is possible to linearize
the transcendental functions and reduce them to algebraic expressions. Examples of
programs containing such solutions include UTFWIBM (Roesset, 1987) and UTFWD
(Chang, 1992), GREEN (Kang, 1998), and SAPSI (Chen, 1987). The computer program
SCALPOT (Magnuson et al, 1991) models the asphalt concrete layer as a Viscoelastic
material using a two-parameter power law model, while the SAPSI program allows the
layer material properties to be complex and frequency-dependent. Al-Khoury et al. (Al-
Khoury, 2001) developed an efﬁcient forward solution for the dynamic analysis of

ﬂexible pavements using the spectral element technique for the simulation of wave

20

propagation in layered systems. The method is able to model each layer as one element

without the need for subdivision into several sublayers.

Finally, Endiran developed a non-linear dynamic model in DYNARK computer program
(Endiran, 1999) which accounts for non-linearities in granular material as well as

subgrade soil.

2.7 Dynamic Backcalculation Methods

Dynamic backcalculation methods are based on either frequency or time domain
solutions. For the former procedure, the applied load and measured deﬂection time
histories are transformed into the frequency domain by using the Fast Fourier Transform
(FFT). Backcalculation of layer parameters is done by matching the calculated steady-
state (complex) deﬂection basin with the frequency component of the measured sensor
deﬂections at one or more frequencies. In time domain backcalculation, the measured
deﬂection time histories are directly compared with the predicted results from the
forward program. One of the advantages of this method is that matching can be achieved
for any time interval desired. Uzan compared both methods and concluded that time
domain backcalculation is preferred over frequency domain backcalculation (Uzon,
1988). A number of computer programs have been developed for dynamic
backcalculation of ﬂexible pavement layer parameters. Each program employs a
particular forward model and a speciﬁc backcalculation scheme. A brief overview of the

programs deve10ped so far is mentioned below.

21

Uzan (Uzan,]994) presented two dynamic linear backcalculation procedures, one in the
time domain and the other in the frequency domain. Both approaches use the program

UTFWIBM as the forward model and Newton’s method as the backcalculation solution.

PAVE-SID (Magnuson, 1991) is a computer program that uses the SCALPOT program
to generate frequency response curves; a system identiﬁcation technique is applied for
matching computed frequency data in order to extract pavement properties. SCALPOT
computes the dynamic response of a horizontally layered Viscoelastic half-space to a time

dependent surface pressure distribution.

BKGREEN (Kang,1998) models the pavement as a layered elastic system in terms of
dynamic Green ﬂexibility inﬂuence functions using Kausel’s formulation of discrete
Green functions for dynamic loads in linear Viscoelastic layered media (1981).
Backcalculation is done at multiple frequencies, and the set of layer moduli is determined
using a non-linear least squares technique. The solution can experience some
computational difﬁculties at certain frequencies due to the numerical complications

associated with implementing inﬁnite integration in computer codes.

Al-Khoury et al. (2001) developed an axisymmetric layered solution as a forward model
using the spectral element technique, and used the modiﬁed Levenberg-Marquardt and

Powell hybrid methods for solving the resulting system of nonlinear equations.

Losa (2002) used the SAPSI program as the forward solution and a nonlinear least
squares optimization technique (Levenberg-Marquardt method) for backcalculating layer

parameters at multiple frequencies. The solution assumes the asphalt concrete and

22

be

S}?

Slim

subgrade moduli to be frequency dependent, while the base modulus is assumed to be

constant with frequency.

FEDPAN (1991) is a ﬁnite element program that can perform both static and dynamic
backcalculation for three-layer pavement systems using the CHEVDEF backcalculation
algorithm (Bush, 1985). This program can simulate the effects of pavement inertia and

damping in the dynamic analysis, and material nonlinearity in the static analysis.

Matsui et a1. (1998) modeled the ﬂexible pavement as an axisymmetric linear elastic
system using the FEM method, in conjunction with Ritz vectors to improve
computational efﬁciency. The backcalculation method was the Gauss-Newton method

coupled with the singular value decomposition (SVD) method with truncation.

Finally, Meier and Rix (1995) developed an artiﬁcial neural network solution that has
been trained to backcalculate pavement layer moduli for 3-layer ﬂexible pavement
systems using synthetic dynamic deﬂection basins. The dynamic pavement response was
calculated using an elastodynamic Green function solution based on Kausel’s formulation

(Kausel and Roesset, 1981).

The various dynamic backcalculation programs are summarized in Table 2.1 along with
their inverse method and the corresponding forward analysis programs. Table 2.2

summarizes the advantages and disadvantages for each method.

23

Table 2.1 Dynamic backcalculation programs

 

Forward

 

 

 

 

Program Domain Inverse Method Program Author, Year
BKGREEN Frequency-domain “niftglli‘z’ﬁgqm GREEN K321399322“) V'

No name Fggliiegzimin Newton’s method UTFWIBM Uzan J. (1994)
PAVE-SID Frequency-domain S ystem identiﬁcation (SlD) SC ALPOT M: £19“ ls )0 n

FEDPAN Time-domain Linear least squares SAP IV Ong (1991)

 

Levenberg-Marquardt

 

 

No name F requency-domam method SAPSI Losa (2002)

No name F requency-domain Newton’s method LAMDA A3333”
Matsui K

No name Time-domain Gauss-Newton method FEM (1998)

 

24

Table 2.2 Advantages and disadvantages for dynamic backcalculation programs

 

 

 

 

 

 

 

 

Author Advantages Disadvantages
1 no thickness backcalculation
2 backcalculation is limited to the narrow
l multi-frequencies (three frequencies) frequency range (O-SHz)
BKGREEN 2 can handle thin layers 3 ﬁxing damping ratio (2%)
3 potentially more layers 4 constant AC modulus
5 blow up in numerical calculation at some
frequencies
1 creep moduli for AC
2 both time domain and frequency
domain . .
Uzan J. 3 full time history match (75 points) in 5 22:11chng backca 1031?]; n
time domain g mpmg ratio( - o)
4 multi-ﬁequency backcalclation
(13 frequencies)
1 creep moduli for AC 1 no thickness backcalculation
Magnuson 2 backcalculates up to 8 parameters 2 ﬁxing damping ratio (2-5%)
includingcreep compliance
1 FEM method 1 no thickness backcalculation
On 2 can handle thin layers 2 constant modulus for every layer
g 3 potentially more layers 3 ﬁxing damping ratio (5%)
1 thickness backcalculation . . .
I blow up in numerical calculation at some
Losa 2 AC frequency-dependent fr .
equencres
Al-Khoury 1 gggidﬁiggelffd‘r/IXC Maxwell, and 1 no thickness backcalculation
2 veriﬁcation only for theoretical data
1 traces of time history are used 1 no thickness backcalculation
Matsui K. 2 Rize vectors are introduced to reduce 2 time consuming

the number of system equations

25

CHAPTER 3 - FORWARD PROGRAM

3.1 Introduction

The forward program SAPSI (Chen, 1987) models the pavement structure as a system of
layers that are inﬁnite in the horizontal direction and underlain by an elastic half-space.
The materials are assumed to be isotropic and linearly elastic with hysteretic damping.
Full interface bonding is assumed at the layer interfaces. The mass densities and elastic
moduli change with depth, from layer to layer, but are assumed to be constant within
each layer. For the present application, the top layer represents the asphalt surface which

is supported by the base, subbase and subgrade.

3.2 Analysis Methods

The forward program uses the complex response method, which uses the notation of
complex algebra to express the response of a pavement system to a harmonic excitation
force. The main assumption of this method is that if the forcing function is expressed in
the complex form then the steady-state response should have the same function of
frequency. This method of dynamic analysis is based on Fourier transformation. To
simplify the analysis, a transient mode of loading can be represented by a series of
harmonic loads with different frequencies and magnitudes using Fourier transformation.
Once the responses to steady-state loadings are obtained in terms of frequency and
magnitude, the response to a transient load in the time-domain can be obtained through

the inverse Fourier transform.

26

3. 2.] Modeling of Viscoelasticity

In Viscoelasticity, the behavior of materials is expressed in terms of a complex modulus,

E ', which couples the elastic and the viscous part of the response. The modulus and
damping ratio of soils are assumed to be independent of frequency, whereas those of

asphalt concrete (AC) are allowed to vary with frequency.
3. 2. 1.1 Hysteretic Damping

According to this model the complex modulus can be written as follows:

15‘=E(1--2,/32 +2iﬂ./1—,32) (3.1)

wherei = J-l , E is Young’s modulus of elasticity, and ,8 is the hysteretic damping

ratio.

For very low damping ( ,8 << 1), the complex modulus can be expressed as:
E’ 5 E(1+i2,8) (3.2)
3.2.1.2 Viscous Damping

Asphalt concrete can be achieved using the Kelvin model, which consists of a spring and
a dashpot in parallel, as shown in Figure 3.1. The spring represent the stress component
(Es ), while the dashpot represents the strain component that is proportional to the stain

rate ( 275' ).

27

«Va

 

 

(T

Figure 3.1 Linear Kelvin’s model

The stress-strain relationship for Kelvin’s model is expressed as

0=E8+ryé

(3.3)

The steady-state response of the system to harmonic forcing function can be expressed as

uteiwt

8=§ﬂ=§geiwt
6x 6x
6x6t 6x

Substituting equation (3.4) in equation (3.3), we obtain

0'=E—+ra)— E—+ia)r7————

au , 62u_( aU aneiwt
6x 6x6! 6x 6):

Therefore

28

(3.4)

= (E + icon): eimt (3.5)

*

E = (E + iam) = ERe + Elm (3.6)

where ERe and E Im are the real part and imaginary part of the modulus in the frequency

domain, respectively.
3. 2.2 Steady-State Response

The steady-state solution of a layered system subjected to a harmonic vertical disk load is
adopted from the SAPSI model (Chen, 1987). For a unifome distributed vertical
pressure q acting over a circular area with radius a, the load vector in the axisymmetric

spatial domain may be expressed as:

{p}={:;}=q{‘f} 0...... (3.7,

In the wave number domain the load can be expressed as:

{5}: (3° — {J1 g”) Jog/MHZ” }drdl9 = %J1(m){?} (3.8)

[’2

where, k is the wave number, r is the radial distance to the center of the load, and Jo and

J 1 are Bessel functions of zeroth and ﬁrst degrees, respectively.

The displacement vector can be expressed as:

{U} = {:3 = 180 — {J1 gkr) Jo(()kr)i{l7}dk (3.9)

29

The displacements piand force {5}, in the wave number domain are related by

HRH}: {17} (3.10)

where, [K]=[A}l(2 +[B}k+[G]—w2[M], and the respective matrices can be found

elsewhere (Kausel, 198 1).

The closed form expression for the vertical displacement of the surface layer can be

obtained as

w = qa 2 ¢ ¢ 115 (3.11)
S=l Z,S Z,S

l
where ¢ Z S denotes the vertical displacement at the surface load in the sth mode.
9

 

 

1
11s=18° 2 2 10(kr)J1(ka)dk
(k —kS)
= ” J0(ksr)H1(2)(ksr)——%—— for OSrSa
S ksa
7r (2) 1
=——J k H k —— f 2
2m. 1(.r) 0 (.r) [(520 or r .

3O

Consider a pavement with m surface deﬂections. Let the vector { W } represent the
calculated surface deﬂections due to a harmonic vertical load. For each frequency a) , the

steady-state equation (3.10) of motion can be written as:
[K M = {27} (3.12)

,l . ,2 , r 2

representing the elastic and viscous response, respectively.

Although SAPSI had been veriﬁed by Chen (1987) using simple analytical solutions, it
was decided to compare the SAPSI results with those obtained using another dynamic
layered solution: the computer program GREEN by Kang (1998). The validation was
done by comparing SAPSI results with published deﬂection data by Kang (1998). A
three-layer pavement system resting over a subgrade is subjected to a 44.5 kN (10,000 lb)
load on a 300 mm diameter plate. The layer properties of the pavement section are
presented in Table 3.1. The dynamic deﬂections at offset distances of O, 225, 300, 525,
750 and 1350 mm were calculated using SAPSI and GREEN. Figure 3.2 shows the
results for three different ﬁequencies: 0.25, 8 and 25 Hz. Excellent agreement exists at all
frequencies, except for the farthest sensor at 25 Hz. It is believed that for this particular
case, the result from SAPSI is more reasonable than that from GREEN because there is
no reason for the far sensor deﬂection to be higher than the 6th sensor. Given that the
FWD analysis is in the frequency range from 0 to 30 Hz, it can be concluded that the

program SAPSI is suitable as a forward program.

31

Table 3.1 Proﬁle used for comparing SAPSI and GREEN solutions

 

Layer Name Thickness (in) UnitWeight (pct) Poisson Ratio Damping Ratio Modulus (ksi)

 

AC 3 140 0.35 0. O 300

Base 6 125 0.40 0.0 45
Subbase 12 125 0.40 0.0 21
Subgrade oo 1 10 0.45 0.0 7.5

 

 

Deﬂection (mils)

 

 

 

 

0.0 10.0 20.0 30.0 40.0 50.0 60.0

Distance (in)
—e—-GREEN 0.25 Hz —-e—GREEN 8 Hz +GREEN 25 Hz
u... - SAPSI 0.25 Hz +SAPSI 8 Hz —e——SAPSI 25 Hz
Figure 3.2 Comparisons of dynamic deﬂection basins ﬁom SAPSI and GREEN computer

programs

3. 2.3 Transient Response

For an arbitrary transient excitation such as that corresponding to a FWD test, the time
history of the speciﬁed force can be decomposed into different frequency components

using a Fourier transform. Responses corresponding to each frequency are calculated and

combined to obtain the displacement time history.

32

3.2.3.1 Frequency-domain Solution

The ﬁrst step of the analysis is to decompose the excitation function p(t) into its different

frequency components P(w) by means of the discrete Fourier Transform, which is

evaluated numerically by using the Fast Fourier Transform (FFT) algorithm. In order to
use this algorithm, the basic load-time history input needs to be speciﬁed at N points with
a constant time interval over the duration T. In other words, the load-time history can be

expressed as:
P(t) = p(s - At) (3.13)
where At is the time interval T/N. The load-time history can be expressed as:

N/2
P(t)=Re Zp(ws)exr3(iwst) (3.14)
s=0

where the frequencies are deﬁned as (05 =27r /(NAt), and p(ws) are complex load

amplitudes deﬁned as follows:

1 N -1 .
p(m,)-_-_ p(nAt)exp(—za)snAt) .0, F0 and ,J.
N n=0 2
N—l
p(w,)=i Zp(nAt)exp(—ia)snAt) .0, ”1,2,0.-.
N n=0 2

The second step is to obtain the transfer function H ,. (co), which is deﬁned as the response

of the pavement system due to a harmonic excitation with unit amplitude. In this case, the

33

transfer function is the displacement due to a harmonic unit load applied at the surface of
the pavement system. The third step is to obtain the displacement by multiplying the

Fourier Transform of the force with the transfer function:

Ui(ws)=Hi(ws)*p(ws) (3.15)

which is evaluated for the entire range of frequencies. Finally, the displacement time
history can be obtained using an Inverse Fourier Transform that is evaluated numerically.
It should be noted that unlike the continuous Fourier Transform, the Discrete Fourier
Transform (FFT algorithm) assumes that the input function is periodic with a period Tp.
When using the FFT algorithm, the values of the basic parameters involved (e. g. number
of sampled points N, time increment At and total period Tp) have to be selected properly
so that a compromise can be reached between the accuracy of results and the cost of
computation. It should be noted that the transfer function does not need to be computed
for all frequencies, as interpolation techniques can be used effectively to reduce

computation time.
3. 2. 3.2 Interpolation Scheme

In order to calculate the response time history, SAPSI performs the following steps: (1)
The load time history is transformed to the frequency-domain using the Fast Fourier
Transform (FFT). (2) The unit response functions are computed at several particular
frequencies. (3) The unit response functions at other frequencies are estimated using an

interpolation scheme. (4) The load and unit response ﬁrnctions are multiplied in the

34

ﬁequency-domain to obtain the response of the pavement in the ﬁ'equency-domain. (5)

The inverse F FT is applied to yield the pavement response in the time-domain.

The FWD load time history is sampled typically at every 0.1 ms or 0.2 ms. In order to
obtain the time lag between the peak load and computed peak response accurately, a 0.01
ms sampling interval was used in the analysis, with linear interpolation of the load and
measured deﬂections. Since the load has practically no contribution from the harmonics
with frequencies above 75 Hz, the steady-state response is computed in the range from O
to 75 Hz. To reduce the computational effort of the time-domain backcalculation,
responses are computed at a limited number of frequencies (usually less than 10) and an

interpolation scheme is used to obtain the response at other frequencies.

The interpolation technique was developed by Tajirian (1981) to estimate the response at
all frequencies from the calculated response at a limited number of frequencies. The
technique is based on the frequency response function of a two-degree of freedom
system. The response of each degree of freedom subjected to a harmonic load has the

following general form:

_ 61(04 +czw2 +63

U(a2) 3.16
w4 +C4a)2 +c5 ( )

 

where U ((0) is the response at frequency (0, which will be calculated using equation

(3.15), and cl, c2, c3, c4 and 05 are constants. Thus, if the response of the system is
known at ﬁve frequencies, the ﬁve constants may be obtained by solving the following

equation:

35

 

 

 

 

 

 

(014 wlz 1 —a)12U1 —U1 (01‘ frol4U1‘
a); a); 1 ~w§U2 -U2 62 ngz
a);1 mg: l —w§U3 —U3 (:3 >=< w§U3> (3.17)
w4 w4 1 —a)2U4 —U4 4 (04U4
a);1 052 1 —a)%U5 —U5‘ ‘CS’ kwg’UsJ

3. 2. 3.3 Summary of Procedure for Calculating Transient Response

The summary of the transient response computations is as follows:

. The FWD load and deﬂection data are read, and if necessary, interpolated to obtain

values at every 0.01 ms.

. The time history of the load is ﬁlled with zeros beyond the recorded time and

transformed to the frequency-domain using the FFT algorithm.

. The computer program SAPSI is used to compute the unit response functions of the
pavement at several frequencies. Both real and imaginary components of the

responses are obtained.

. The unit response function at 0 Hz is estimated ﬁ'om that at 0.01 Hz, and those at
other ﬁ'equencies are interpolated from values computed at the several frequencies

spanning from 1.52 to 76.29 Hz.

. The load and unit response vectors are ﬁlled with zeros for the frequencies beyond 75

Hz.

36

6. The deﬂections in the frequency-domain are obtained by multiplying the load and

unit response functions at all frequencies.

7. The deﬂection time histories are computed by using the inverse F FT of the deﬂection

vector in the frequency-domain.

3. 2.3.4 Dynamic Response of a Pavement System due to an FWD Load

In order to illustrate the typical behavior of the pavement system subjected to a FWD
load and the type of information that can be extracted from its dynamic response, a
typical ﬂexible pavement with bedrock presented in Table 3.2 was analyzed. Figures 3.2
and 3.3 show the real and imaginary parts of the transfer functions, respectively, for
different sensors. It can be observed that as the frequency increases, the displacement
increases until they reach a peak at the same frequency for all sensors. The low
amplitudes of displacement at high frequencies are the result of inertial effects. The
interpolation scheme mentioned previously is used in this calculation. Figures 3.4 and 3.5
show the real part and the imaginary part of the Fourier Transform of the load,
respectively. The direct Fourier transform of the displacements is obtained by multiplying
the frequency component of the load by the transfer functions for different sensors and
are shown in Figures 3.6 and 3.7. The displacement time histories of the different sensors

are obtained using an Inverse Fourier Transform, and are shown in Figure 3.9.

It is noted that SAPSI program uses real and imaginary parts of modulus as input
parameters. The difference between the hysteretic and the viscous damping models is that
the former model employs the damping ratio to simulate the Viscoelastic property, while

the latter model uses the imaginary part of the modulus. Since the elastic modulus and

37

damping ratio can be expressed in term of the real and imaginary part of the complex
modulus according to equation (3.1), and the Kelvin model has real and imaginary parts

already according to equation (3.6), the SAPSI computer program is suitable for both

models.

Table 3.2 Pavement proﬁle characteristics

 

Layer Name Thickness (inch) Unit Weight (pct) Poisson Ratio Damping Ratio Modulus (ksi)

 

AC 8 145 0.3 0.05 200
Base 12.13 135 0.35 0.03 24
Subgrade 54.69 120 0.40 0.02 18

Stiff layer 00 145 0.15 0.05 500

 

 

 

 

 

 

Real Part (mils/lb)

 

 

 

 

 

Frequency (Hz)

 

sensorl — - - - sensor2 —B—sensor3 —e——sensor4

+ sensorS ——-x—— sensor6 —*— sensor7

Figure 3.3 Real part of the displacement transfer function

38

0.0005
0.0003 «
0.0001
-0.0001 ‘
-0.0003 —————
00005 a
-0.0007 .
0.0009
-0.0011 , ~ , -
-00013 h- __ _
-0.0015 1

 

 

 

 

 

 

 

Imag. Part (mils/lb)

 

 

Frequency (Hz)

 

sensorl ‘ ‘ ‘ ‘ sensor2 —9— sensor3 —°— sensor4
—b— sensor5 —-*—- sensor6 -*-— sensor7

Figure 3.4 Imaginary part of the displacement transfer function

 

2000
1500 ‘
1000 "

500*

Real Part (lb)

 

 

 

-1500

 

0 50 100 150
Frequency (Hz)

Figure 3.5 Real part of the load

39

 

Real Part (mils/lb)

Imag. Part(lb)

 

 

 

 

 

0 50 100 150
Fre que ncy(Hz)

Figure 3.6 Imaginary part of the load

 

 

 

 

 

 

 

0 50 100 150
Frequency (Hz)

sensorl - — - - sensor2 -—B—sensor3 -—o—sensor4
—-A—scnsor5 —-)(——sensor6 +sensor7

 

Figure 3.7 Real part of sensor displacements

40

 

 

Deflection( mils)

 

 

 

 

 

 

 

 

Imag. Part (m ils/lb)

 

 

 

 

Frequency (Hz)

 

sensorl ' ’ r ‘ sensor2 —9— sensor3 -—°— sensor4
—-ﬁ— sensor5 —><— sensor6 -*— sensor7

Figure 3.8 Imaginary part of sensor displacements

 

 

 

 

 

 

 

 

Time (ms)

 

sensorl - — sensor2 —9— sensor3 —°— sensor4
—&— sensor5 -*— sensor6 + sensor7

Figure 3.9 Sensor deﬂection time histories

41

 

 

3.3 Estimating Depth-to-Stiff Layer (DSL) and Subgrade Modulus

3. 3.1 Estimating Depth to StrjfLayer (DSL)

To determine the depth to bedrock and the depth to ground water table, one-dimensional
wave propagation theory was used as suggested by Roesset et al (1995). Two equations
were developed: Equation 3.18 for saturated and Equation 3.19 for unsaturated subgrade.
Equation 3.18 can be used only for bedrock, while Equation 3.19 can be used for both
bedrock and ground water table. Both equations were initially developed by Roesset et a1
(1995), and modiﬁed in this research. The proﬁles used in the veriﬁcation analysis are
shown in Tables 3.3 through 3.5. Two different proﬁles with three different shear wave
velocities were used for the veriﬁcation of Equation 3.18 and one proﬁle with three
different shear wave velocities was used for the veriﬁcation of Equation 3.19. During the
veriﬁcation analysis, the coefﬁcient for Equation 3.18 (saturated subgrade) was modiﬁed
because there was a signiﬁcant difference between the actual and calculated depth to

bedrock using the coefﬁcient proposed by Roesset (1995).

 

VS * Td .
D = _1—35— for saturated subgrade wrth Bedrock (3. 1 8)
Vs * T d .
Db = for unsaturated sub grade wrth Bedrock or Ground Water Table (3. l 9)
(7r - 2.24 * v)

where, VS = S - wave velocity of subgrade material
Td = Natural period of free vibration
v = Poisson ratio of subgrade

42

 

 

Q

A
| v

.1 ~
ﬁth‘ -ot

1... .
- “.th ' '

It»:
‘4'“,
-v~.
_r f
..g _
-5 v
_,-‘
9.-

Table 3.3 Proﬁles used in the analysis of saturated subgrade with bedrock

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. Unit . . S-wave P-wave Elastic ‘3“
11:13:; T111338“ Weight P1322211 Daggg velocity velocity Modulus if:
(DCQ (0)8) (fPS) (ksi) :3
AC 6 145 0.3 0.05 2217 4150 400 j";
Proﬁle Base 6 140 0.35 0.03 700 1460 40 '5'?"
l Subbase 6 130 0.35 0.03 629 1310 30
135 0.495 0.02 500 5020 21.8
Subgrade h* 135 0.495 0.02 600 6000 31.4 _
135 0.495 0.02 765 5000 50.8
Bedrock 00 150 0.2 0.05 3590 5860 1000
. Unit . . S-wave P-wave Elastic .
gag; “11813688 Weight P1313832” 0:23? velocity velocity Modulus ‘1
(p00 (fpS) (fPS) (ksi)
Proﬁle AC 1 145 0.3 0.05 2217 4150 690.4 1
2
Base 12 125 0.35 0.03 700 1460 67.4 ‘
Subgrade h* 1 10 0.495 0.02 500 5020 17.8
1 10 0.489 0.02 750 5110 39.8
Bedrock 00 150 0.2 0.05 3590 5860 1000

 

* Thickness of subgrade layer is varied from 3.5 ft to 31.5 ft

The results of the veriﬁcation analyses are shown in Figures 3.9 through 3.18.
Deﬂection- time histories were calculated using the SAPSI program. The natural period
of the proﬁles and the peak time delay between the 6th and 7th sensors (r = 3 ﬂ and 5 ft,
respectively) were then determined from the deﬂection time records. Since the shear
wave velocity, unit weight and Poisson’s ratio of the subgrade are known, the depth to
stiff layer or the depth to ground water table can be calculated using the equations. The
results indicated that the depth to bedrock and depth to ground water table could be

accurately predicted using these two equations.

43

 

Table 3.4 Proﬁle used in the analysis of unsaturated subgrade with bedrock

 

 

 

 

 

 

 

 

. Unit . . Elastic
Layer Name Image“ Weight P121832“ 13:22:)” veliciva‘ﬁ‘ps) Modulus

112d) " (ksi)

AC 6 145 0.3 0.05 2217 400
Base 6 140 0.35 0.03 700 40
Subbase 6 130 0.35 0.03 629 30
110 0.35 0.02 500 16
Subgrade h* 1 10 0.35 0.02 600 23

110 0.35 0.02 700 31.4

Bedrock 00 150 0.2 0.05 3590 1000

 

"' Thicknesses of subgrade layer is varied from 3.5 ft to 31.5 R

Table 3.5 Proﬁle used in the analysis of unsaturated subgrade with GWT

 

 

 

 

 

 

 

. Unit . . S-wave P-wave Elastic
1‘11: 1:; Tmzllgress Weight P12135331] Dagggng velocity velocity Modulus

(pct) (fpS) (fpS) (ksi)

AC 6 145 0.3 0.05 2217 4150 400
Base 6 140 0.35 0.03 700 1460 40
Subbase 6 130 0.35 0.03 629 1310 30
51.1de h* 1 10 0.35 0.02 300 - 5.8

GWT 00 135 0.495 0.02 500 5000 21.6

135 0.49 0.02 700 5000 42.6

 

* Thicknesses of subgrade layer is varied from 3.5 ft to 31.5 ft

44

 

 

 

 

 

Calculated Depth (it)

   
     

(a)
0‘. .,_. . ,._-
o 5 10 15 20 25 30 35
Measured Depth (ft)
25 fAAAkaivv____AAﬂ
l
A 21
5 1
s 1
t 1.5 1
m l
l
0
(b) E 1 i
O
.3 .
< 0.5 . I
3.5 55 75 95 11,5 16.5 21,5 31.5
Depth to Bedrock (ﬂ)
1‘
S
h
E
l-
til 1
2
(C) 3
0
a:

 

3.5 5.5 7.5 9.5 11.5 16.5 215 31.5
Depth to Bedrock (it)

Figure 3.10 Comparison of predicted and actual depth-to-bedrock

45

Calculated Depth (ft)

 

         

(a)
A 2‘
5
I-
(b) :2 1.5 ..
a:
0
‘5 11.
3 1
M 1
.o
<05. -
,4. 1:] LI, ,
3.5 5.5 75 9.5 11.5 16.5 21.5
Depth to Bedrock (it)
(C)

Relative Error (%)

 

3.5 5.5 7 5 9.5 11.5 16.5 2 .
Depth to Bedrock (ft)

tn
w
Ln

Figure 3.11 Comparison of predicted and actual depth-to-bedrock

46

 

8

 

_1\T11 1 1117 1411 k
6 4 2 o 8 6 4

.11..

A5 ﬁnen— toga—330

)
a
(

M easured Depth (11)

 

 

 

A5 3...;— 333.3.

)
1D
(

Depth to Bedrock (It)

 

 

 

#1 a

4 7. 0 8 6 4

TX; ..ctm 92.29:

)
C
(

Depth to Bedrock (ft)

Figure 3.12 Comparison of predicted and actual depth-to-bedrock

47

0
3

 

7

1‘1 . 14 1‘ a _
5 0 5 0 5
2 2 .l

c: __..—5 3.23.5

)
a
(

30

20
M easured Depth (ﬂ)

 

 

_

v

_

_ m
u a
A m
A w
A m
_ w
_ D
_

_ C

. a “.._

A5 ..otm— 33°3<

)
b
(

0
Ar-

TXL Let”.— «>223—

5

\I
c
(

0

 

Depth to Bedrock (1!)

Figure 3.13 Comparison of predicted and actual depth-to-bedrock

48

5 0 5 0
2 2 .l

c: 539 3.23:5

mm
(

 

5

M easured Depth (ﬂ)

 

45352

c: Sta 8:.

)
b
(

 

Gmn<

0

Depth to Bedrock (ﬂ)

 

0
7-

A

5 O 5

.XL 3...:— 2523—

\l/
C
(

 

30

Depth to Bedrock (It)

Figure 3.14 Comparison of predicted and actual depth-to-bedrock
49

 

‘3

 

 

 

 

_ H
) _
ﬂ .
l\ .
.n .
t _
P _
e .
o a
d .
e
r _
u .
s _
a
e _
M _
_
_
n
.
a
d
a
71 111111111
5 0 5 0 5 0 5 0 4 5. 3 5. 2
3 3 2 2 II. II. 3 2

A5 53: 3.23.5

)
a
(

(b)

S: ..etm 32°3<

9.5 ”.5 16.5 21.5 31.5

7.5
Depth to Bedrock (h)

5.5

3.5

 

3L Sta 3:2»:

Depth to Bedrock (ft)

1cted and actual depth-to-bedrock

Figure 3.15 Comparison of pred

50

 

 

 

 

30
A
S
a 25
a
a
0
a 20
:1
3 15
a
a —
(1 g .0
G
U 5 1
0
o 5 10 15 20 25 30 35
Measured Depth (ﬂ)
3 7. iiiiiiii v'__vv’_
1 1
1
1 __ 1
2.5
A
a:
V
x- 2
Q
l-
a
1.5
(b) 3
E
o 1
(I)
a
< I ‘
0’5 ‘ i I F
0 .77—--_.,._ ,. , D .l i J;
35 5.5 7.5 95 115 I65 21.5 31.5
Depth to Bedrock (ft)
9 7 7 7 7 7 77 7 7 7 7.
8 1
A7 ‘1
§ ‘1
1. 6
O
t 5 1
H
(C) 1’ “ 1
E 3
g 2.

     

35 55 75 9.5 11.5 16.5 21.5 31.5
Depth to Bedrock (it)

Figure 3.16 Comparison of predicted and actual depth-to-bedrock

Calculated Depth (ft)

 

(a)
*3
I-
o
t
(b) m
3
E
o
M
.n
<
. l. .-3
3.5 55 75 9.5 11.5 16.5 21.5 31.5
Depth to Bedrock (It)
10 A*’ v ' ——;
.\°
‘5
c
L
1::
(c) 0
.3
E
0
I

            

35 5.5 7.5 9.5 115 16.5 21.5 315
Depth to Bedrock (ft)

Figure 3.17 Comparison of predicted and actual depth-to-bedrock

52

 

 

 

    

     

  
  

  

 

 

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3 3 2 2 l l 2 l l l l l 7 6 5 4 3 2 l
c: 5%: 333.5 2: Sta 323.2 3.; 2:5 3:23.
) )
a \I C
{\ ﬂw (

Depth to GWT (R)
53

Figure 3.18 Comparison of predicted and actual depth-to-watertable
(Poisson’s Ratio=0.495, Vs=500 ﬁ/s)

 

5 0 5 0 S
2 2 I l

3: 5...: 333.5

)
a
(

35

30

IS

M easured Depth (11)

IO

.111.111ﬁ1

 

.111.1141.111.14.11u111a11q11

‘1
1
4

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II 1

II

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(b)

‘._—._.ﬁ_4m -.VHE 1__EZ 7._ELT4=:L 1.E«..h4 .._

7.5 9.5 11.5 16.5 21.5 31.5
Depth to GWT (n)

5.5

3.5

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

 

 

.4.-121.13- it 'I'

9.5

: 11-11.11...

11_111 1111.1.11._11141l1111 1“
O 0 0 0 0 0 0
6 5 4 3 2 ..l

7%.; not”... 9523—

)
C
(

16.5 21.5 31.5

Depth to GWT (ﬂ)

”.5

7.5

5.5

3.5

Figure 3.19 Comparison of predicted and actual depth-to-water table

0.495, Vs=7OO ft/s)

(Poisson’s Ratio

54

1111
:11

£111

3. 3.2 Estimating Subgrade Modulus

The procedure for estimating the depth-to-stiff layer, described in section 3.3.1 requires
the knowledge of the subgrade modulus values. However, in the analysisof ﬁeld data,
subgrade properties (shear-wave velocity, unit weight and Poisson ratio) are not generally
known, and therefore need to be either measured or assumed. Two different methods

were considered for estimating the elastic modulus of the subgrade:

0 Using the shear wave velocity as estimated from the time difference between two

speciﬁed sensors (r = 3 ﬂ and 5 ft)

0 Using the base damage index (BDI) and shape factor (F2) proposed by Lee et al

(1998).

In the ﬁrst method, using the time difference between two speciﬁed sensors, the shear-
wave velocity can be calculated, and two other properties (Poisson’s ratio and unit
weight) are assumed with typical values; using these values, the elastic modulus of the

subgrade can then be calculated.

hi the second method, EDI and F2 are calculated using Equations 3.16 and 3.17 and then
combined with Equation 3.18, which calculates the surface deﬂection at a distance r from
the applied load for a single layer system. Finally, the elastic modulus of the subgrade is
calculated using Equation 3.19, assuming a typical value for Poisson’s ratio of the

sub grade

Base Damage Index: BDI = 61 — 62 . (3.16)

55

Shape Factor: F2 = ﬂ

5, (3.17)

where: 6' = Deﬂection at a distance of 12 in from the load

(52 = Deﬂection at a distance of 24 in from the load

 

63 = Deﬂection at a distance of 36 in from the load

The results of the analysis showed that the second method was more accurate than the

 

ﬁrst method. Therefore, only the second method is used in the new backcalculation h
program. Figures 3.19 through 3.26 only show the results of the analysis using the ‘
second method.
P(l —v2 J
6r =—————f(r) (3.18)
E sg

where, 6, =surface deﬂection at offset, r, from the applied load

P = Applied load

Esg = Subgrade modulus

v = Subgrade Poisson’s ratio

f (r) = l , with r being the distance ﬁom the applied load
r

_ F2 - P(1 —v2)f(3)
“’3 431)] - 61F2

 

(3.19)

56

3.3.3 Using the Subgrade Modulus and Depth-to-Stiff Layer Estimates in the

Backcalculation Algorithm

In the backcalculation algorithm, the subgrade modulus is ﬁrst estimated using Equation
3.19. An improved estimate of the depth-to-stiff layer or the depth-to- ground-water-table
can then be obtained using the new value for the subgrade modulus. The total duration of
the deﬂection should be several times larger than the actual duration of the load to insure
that all ﬁee vibrations have attenuated. Although the appropriate value depends on the
fundamental natural period of the system and the amount of damping, a duration of 0.15
to 0.2 second is generally sufﬁcient to determine the natural period, Td, of the pavement

system.

Elastic Modulus of Subgrade (psi)

 

Depth to Bedrock (it)

Figure 3.20 Results of analysis for elastic modulus of subgrade calculation

57

 

 

an... guaram... 3.35:. use...

Bedrock

Depth to Bedrock (it)

Figure 3.21 Results of analysis for elastic modulus of subgrade calculation

 

 

 

W. W.

as... 83.35... .._—.252 3.8....

Bedrock

Depth to Bedrock (it)

Figure 3.22 Results of analysis for elastic modulus of subgrade calculation

58

 

 

i

Elmtic Mosulus of Subgrade (psi)

 

 
 

16.5 21.5 31.5 No

Depth to Bedrock (It)

Figure 3.23 Results of analysis for elastic modulus of subgrade calculation

 

i

g
.._.1- .— . » r .-+—. T..—_

i

 

g
4— .—.ﬁ~ .4-- .ﬁ—r—1

Elastic Mosulus of Subgrade (psi)

 

 

O

3.5 5.5 7.5 9.5 11.5 16.5 21.5 31.5 No
Bedrock
Depth to Bedrock (I!)

Figure 3.24 Results of analysis for elastic modulus of subgrade calculation

 

§

§

...umqmwrﬂ—

    

§

3

 

é

 

é

Elastic Mosulus of Subgrade (psi)

Bedrock

Depth to Bedrock (it)

Figure 3.25 Results of analysis for elastic modulus of subgrade calculation

 

‘41‘14r

E

N!
M
8
7 4414

4...; ._.

1..1 V

Elastic Mosulus of Subgrade (psi)

N
‘8"
414r4144rr1r

 

3.5 5.5 7.5 9.5 11.5 16.5 2|.5 31.5 No

Depth to GWI‘ (it)

Figure 3.26 Results of analysis for elastic modulus of subgrade calculation

60

 

16000 ‘
14000
12000 j
10000 .-
8000 j
6000 q

4000:

Elastic Mosulus ofSubgrade (psi)

20001

 

Depth to GWI‘ (it)

Figure 3.27 Results of analysis for elastic modulus of subgrade calculation

61

by 1
111111“
for (
Pffse

6121511

CHAPTER 4 - INVERSE SOLUTION

4.1 Introduction

Backcalculation of pavement layer parameters is an inverse problem, where some of the
layer parameters are estimated by matching the theoretical prediction to the measured
deﬂections such that the measured system response (in the form of the deﬂection basin) is

matched by the theoretical predictions.

In FWD test interpretation, the input is the impulse load applied to the pavement structure
by the Falling Weight Deﬂectometer; the output is the deﬂection time histories at the
different sensors, and the system is the pavement structure. The theoretical formulation
for computing the response of the pavement structure due to the FWD load has been
presented in chapter 3. The dynamic response of the pavement structure depends on the

elastic modulus, damping ratio, thickness, Poisson’s ratio and mass density of each layer.

Current methods of interpretation of FWD test results use the maximum displacement at
each sensor to deﬁne a deﬂection basin, which is interpreted as having resulted from a
statically applied load. This approach neglects the dynamic nature of the test. When the
time histories of the load and displacements are recorded, the additional information
available provides substantial insight into the properties of the system and can improve

the accuracy of the backcalculation results.

The number of deﬂection measurements must exceed (or, theoretically, at least be equal
to) the number of parameters that are to be backcalculated. Most backcalculation schemes

allow for backcalculating 3 to 5 parameters, these being layer moduli. Some schemes

62

allow for backcalculating the depth to stiff layer; however, none of the available

backcalculation solutions allows for backcalculating both layer moduli and thicknesses.

In this study, two new methods based on dynamic interpretation of deﬂection time
histories using frequency and time-domain solutions are developed. The methods allow
for theoretically backcalculating the layer moduli, damping ratios and thicknesses for a
three to ﬁve- layer system. The backcalculation procedure is based on the modiﬁed
Newton-Raphson method originally adopted in the MICHBACK program (1993). The
new program offers two options: (i) frequency-domain and (ii) time-domain

backcalculation.

4.2 Frequency-domain backcalculation

In the frequency-domain solution, the modiﬁed Newton-Raphson method is extended to
include complex valued deﬂection gradients, and the gradient matrix can be expanded to
handle multiple frequencies simultaneously. In addition, methods for estimating the depth
to stiff layer and the seed subgrade modulus, proposed by Roesset (1995) and Lee et al.
(1998), respectively, have been adopted with some modiﬁcations and are implemented in

the new program.

4. 2. I Multi-frequency backcalculation

In this option, the AC modulus is frequency-dependent, while the other layer moduli are
assumed to be constant with frequency. The damping of the AC layer is solved by using
the real and imaginary parts of the backcalculated complex moduli, while damping ratios

of the base and subgrade layers are assumed.

63

The ability of the new solution to analyze complex deﬂection basins at multiple
frequencies simultaneously allows for increasing the number of parameters that can be
backcalculated. Currently, the computer program uses deﬂection basins from three

frequencies. This enables the backcalculation of twelve parameters:

0 Modulus of the asphalt concrete layer at three frequencies; (3)

o Damping ratio of the asphalt concrete layer at three frequencies; (3)
0 Moduli for the base, subbase and subgrade layers; (3)

o Thicknesses for the AC, base and subbase layers. (3)

For the case of a stiff layer at shallow depth, the base and subbase layers can be
combined into one layer, and the program can calculate the depth-to-stiff layer as a third

thickness. In this option, the user may also choose not to backcalculate layer thicknesses.

4. 2.2 Single-frequency backcalculation with thickness backcalculation

In this option, all parameters are allowed to vary with frequency since they are
backcalculated at different (independent) frequencies. However, given the reduced
amount of information (only one complex deﬂection basin), only eight parameters can be

backcalculated (the damping ratios of the base and subgrade are assumed):
o Modulus of the asphalt concrete layer at a given frequency; (1)

o Damping ratio of the asphalt concrete layer at a given frequency; (1)
o Moduli for the base, subbase and subgrade layers; (3)

o Thicknesses for the AC, base and subbase layers. (3)

Similarly to the multi-frequency backcalculation, the base and subbase layers can be
combined into one layer when a stiff layer is suspected to exist at shallow depth, and the
program can calculate the depth-to-stiff layer as a third thickness. Also in this option, the

user may choose not to backcalculate layer thicknesses.

4. 2.3 Single-frequency backcalculation without thickness backcalculation

In this option, layer thicknesses are assumed, and the moduli and damping ratios of all
layers are backcalculated at each frequency. This leads to eight backcalculated

parameters at each frequency;

Moduli of the AC, base, subbase and subgrade layers; (4)

Damping ratio of the AC, base, subbase and subgrade layers. (4)

4.3 Time-domain backcalculation

In the time-domain backcalculation, the gradient matrix is expanded by including
gradients of peak deﬂections and their corresponding times (or traces of time history).

The details of the inverse solution are described in section 4.5.

4. 3. 1 T ime-domain backcalculation without thickness backcalculation

In this option, layer thicknesses are assumed, and the moduli and damping ratios of all
layers are backcalculated using since either peak deﬂections and their corresponding time
lags or traces of time history. This leads to backcalculating as many as eight parameters.

The ﬁrst option is to backcalculate the following:

65

o Moduli of the AC, base, subbase and subgrade layers; (4)

e Damping ratios of the AC, base, subbase and subgrade layers.(4)

The second option is to backcalculate:
- Modulus of the asphalt concrete layer (1)
o Damping ratio of the asphalt concrete layer ( l)

o Moduli for the base, subbase and subgrade layers.(3)

This option is more realistic when using ﬁeld data.

4. 3.2 T ime-domain backcalculation with thickness backcalculation

In this option, all parameters are allowed to vary, and as many as eight parameters can be

backcalculated:
o Moduli of AC, base and subgrade layers; (3)
o Damping ratios of AC, base and subgrade layers; (3)

o Thicknesses of AC and base layers. (2)

The other option backcalculates:

o Modulus of the asphalt concrete layer; (1)

0 Damping ratio of the asphalt concrete layer; (1)

0 Moduli for the base, subbase and subgrade layers; (3)

0 Thicknesses for the AC, base and subbase layers. (3)

66

In all options mentioned above, a least square optimization algorithm 'or singular value

decomposition (SVD) is used to mach the measured and computed deﬂection

4.4 FWD Data Processing

Modeling the dynamic response of the pavement subjected to an FWD pulse requires
calculating the time history of surface deﬂections Ui(t) that would be recorded at
receivers 1' due to a transient uniform disk load P(t) applied to the pavement structure.
The full-time histories of the load and deﬂection are used in the analysis. Because the
FWD load is transient in nature and not harmonic, the Fourier transform is used to
represent the transient load as a series of harmonic loads with different amplitudes at

different frequencies. The same transformation is done for the deﬂection time histories.

As a ﬁrst step the excitation P(t) is decomposed into its different frequency components
P(w) by means of a Fourier transform. This is evaluated numerically using the Fast
Fourier Transform (FFT) algorithm. The second step is to obtain the Fourier transform of

the different sensor displacements, U ,- ((1)).

It should be noted that unlike the continuous Fourier Transform, the Discrete Fourier
Transform (FFT algorithm) assumes that the input function is periodic with a period Tp.
When using the FFT algorithm, the values of the basic parameters involved (e. g. number
of sampled points, N; time increment, At; and total period, T.) have to be selected
properly so that a compromise can be reached between the accuracy of results and the

cost of computation. Finally, it should be noted that the transfer function does not need to

67

be computed for all frequencies, as interpolation techniques can be used effectively to

reduce computation time.

An example load pulse is shown in Figure 4.1. In this example, data are sampled every
0.77 ms and the sampling time is 100 ms. Deﬂection time histories from all seven sensors
are shown in Figure 4.2. The load and deﬂection functions in the ﬁequency-domain are
shown in Figure 4.3 and Figure 4.4. Figure 4.5 shows the real and imaginary parts of the

transfer functions due to a unit harmonic load as a function of frequency.

4.5 Inverse Solution

The objective of any back-calculation solution is to ﬁnd a set of layer parameters such
that the calculated deﬂection basin will match the measured one within a speciﬁed
tolerance. To accomplish this, it repeatedly adjusts the parameter values until a suitable
match is obtained. The discussion below describes the solution in terms of the modulus
being the backcalculated parameter. The same method can be applied to thickness (in lieu

of the modulus); however, it is not included herein for the sake of brevity.

The dynamic backcalculation solution developed in this research is an extension of the
solution used in the MICHBACK program (1993). It uses the modiﬁed Newton method
to obtain a least squares solution of an over determined set of equations. In the
MICHBACK solution, these sets of equations are real-valued and correspond to the peak
deﬂection values, since the backcalculation scheme uses a static solution (CHEVRONX)
to predict the deﬂection basin. In the frequency-domain solution, the equations are

complex-valued and correspond to the steady-state solution at one or multiple

68

frequencies. In the time-domain solution, the real—valued equations are expanded to
correspond to the peak transient deﬂections and their corresponding time lags relative to

the peak load.

4. 5. 1 F requency-domain Backcalculation

Frequency-domain backcalculation uses the harmonic (steady-state) solution in SAPSI to
predict the deﬂection basin at any given frequency. In this case, the equations become
complex-valued, and they can be expanded to include deﬂection basins at multiple

frequencies.

Newton’s method consists of approximating the non-linear curve relating the complex

deﬂections {U(a))} = {Wlm W5" W211 Win" } by a series of straight lines tangent

to the curve at the estimate of the com lex modulus E'. The com lex deﬂection is
P P

2

deﬁned as ﬁlm = w1 + iwi where the real part of the deﬂection corresponds to the elastic

i
response and the imaginary part describes the viscous response. The complex modulus is

deﬁned asE = E1 + iEz. The slope of the straight line is used to obtain the increment,

AEi, which is added to Ei to obtain the improved modulus estimate Ei +1.

69

Load (lb)

Deflection (mils)

 

 

 

 

 

 

 

 

 

 

 

 

 

 
   
 
 

 

 

 

 

 

 

 

-2000
0 20 40 6O 80 100
Time (ms)
Figure 4.1 FWD load versus time
10

8 ._ __ .._ iv W ___.—

6 _ .__ __ -_.
4.___.._ _~-____'v.__..
2 .

o .

O 20 4O 60 80 100
Time (ms)

—9— sensor1 —o— sensor2 + sensor3 + sensor4
—-— sensor5 —1— sensor6 —0— sensor?

Figure 4.2 Deﬂections versus time

70

Magnitude (lb.)

Real Pa rt (lb.)

Imag. Part (lb.)

 

 

 

 

 

 

Freque ncy (Hz)

(a) Magnitude of Fast-Fourier Transform of load

 

 

 

 

 

 

 

Fre que ncy (Hz)

(b) Real part of Fast-Fourier Transform of load

 

 

 

 

 

 

 

 

 

200
O -
-200 -.
-400 4~
-600
0 10 20 3O 40 50
Frequency (Hz)

(0) Imaginary part of Fast-Fourier Transform of load

Figure 4.3 Fast-Fourier Transform of load-time history

71

\
I

Magnitude (mils/lb

Real Part (mils/lb)

Imag. Part (mils/lb:

 

 

 

Freqrency (Hz)
sensorl - - - -sensor2 —9—sensor3 —9—sensor4
—e.—sensor5 -—><-—sensor6 —X—sensor7

 

(a) Magnitude of F ast-Fourier Transform of deﬂection
0.6

 

 

 

 

0 10 20 30 40 50
E'eqrency (Hz)

sensorl - - - -sensor2 -—-B-——sensor3 —e—sensor4
—1&—-sensor5 —x—sensor6 —)1t—sensor7

 

(b) Real part of Fast-Fourier Transform of deﬂection

 

 

 

 

 

Freqrency (Hz)

sensorl - - - -sensor2 -——-E——sensor3 +sensor4
—e1—sensor5 —x—-sensor6 —31(—sensor7

 

(c) Imaginary part of Fast- Fourier Transform of deﬂection

Figure 4.4 Fast- Fourier Transform of deﬂection

72

Real Part (mils/lb)

Imag. Part (mils/lb)

Magnitude (mils/lb)

 

 

 

 

 

 

 

 

0.0015
0.001
0.0005 1
O T
O 10 20 30 40
Frequency (Hz)
sensorl - - - - sensor2 —-9— sensor3 —°— sensor4

-—¢— sensor5 —X—- sensor6 —3K-- sensor7

(a) Magnitude of transfer function
0.0015

 

 

0.001 -4——~ ‘ _ -__ -____.--~ ._.

 
    

  

1 I

II c ‘
. .
A . 5 ' V
- ‘ 1 _
_ . . - - .‘ ' ' .. ' - x -
- - ‘ ' . - ' ' ‘
- ' 7 h \

    

6

 

 

 

0 10 20 30 40
Frequency (Hz)

 

sensorl ' ‘ ‘ ‘ sensor2 + sensor3 —°— sensor4

—-ﬁ—- sensor5 —*—- sensor6 + sensor7

(b) Real part of transfer function
0.0005

 

 

  

-0.0005 41 -

 

 

 

-0.001

 

Fi'equency (Hz)

 

sensorl ' ' ‘ ' sensor2 —9- sensor3 —°— sensor4

—¢— sensor5 —-X— sensor6 + sensor7

(c) Imaginary part of transfer function

Figure 4.5 Fast-Fourier Transform of transfer function

73

Because E is complex, the slope is evaluated for both the real and imaginary parts of the
modulus. A similar approximation is used for the thickness using the increment AH,-. The
expression for the slope of the curve relating deﬂection and thickness is the same as that
for the real part of the modulus. Also, for both moduli and thickness, since the slope is

not known analytically, it is obtained numerically by using the following equations:

W1(51(1+.))_W1[§1) .W2(El(l+r)]-W2(E1)

 

 

 

 

 

 

 

 

6E E=E1 I'E] fill
an? W2(E2(l+r))—W2(E12) W1(E2(l+r))-W1(E2) (41)
__ = -i .
6E E252 rEZ r2352

61? W‘(ﬁ(1+r))—W‘(H) W2(1?(1+r))—W2(’)

— = - +‘ -

5H H=H rH I 71']

 

in which r is sufﬁciently small. This requires additional deﬂections to be computed,

arisin from moduli and thickness values of 173' + l and [:11 +1 , res ectively.
g P

For the described system of n identiﬁed parameters (1 complex moduli and n-1 layer

thicknesses) and m sensors, the slope is represented by the gradient matrix

i_ iv: 92 _
G _[EE E=E1+iE2 aH H=H1 1-{IGIIGZIHB (4'2)

where

74

 

 

 

 

 

 

 

 

 

 

 

 

6W1 aw] _WI 211 an ﬂ
1 1 l 2 2 2
6E1 6E2 6E1 61:31 6E2 6E1
1011 1021- =
6W”, 675m an 6W,” 6W," mm
1 1 l 2 2 2
_6E1 6E2 6E]- _6El 6E2 6E]:
P571 571 5‘71 _
an] 6H2 aHn—l
[H]: : :
aWm Wm 6W,"
_dHl 6H2 6H —1‘
and

 

6W1 _ WiURlEU-WXEUHWi1lR1511-Wi151‘.)

 

 

l l 1
6E k rE k If k

5,7]. : WJZ([R]EIE)—WJ.2(EI%) -1. W}([R]E%)—W}(EI‘:')

 

 

 

2 2 2
615k rE k rE k

 

any zW}(1R1Hk)-W}(Hk)+iWJ-ZqRin)—W}(Hk)

 

de er er

[R] is a diagonal matrix with the k th diagonal element being (1+r) and all other elements

being 1. Thus the partial derivative is estimated numerically by taking the difference in

the jth deﬂection arising from the use of a set of moduli and thicknesses. The increments

to the moduli and thicknesses, {AE, AH}i can then be obtained by solving the equations:

75

{111’ } +10i ]{AE,AH}i = {w} (4.3)

Because equations (4.3) are over determined with m equations and n unknowns, a least

squared solution or SVD method are used to solve for {AE, AH 1i :

The revised moduli and thicknesses are obtained through:

‘ -

{E,H}"+‘ = {E,H}’ +{AE,AH}" (4_4)

The iteration is completed when the changes in layer moduli and thicknesses are smaller

than a set of speciﬁed tolerances:

 

 

 

Ei+l —Ei Ei+l —-Ei 1:114] —Hi
k’lAi ["1 Se , k,2Ai k’2 $82, k 1 k SE] (4.5)
511,1 Ek,2 Ek
wherek= l, 2, ...,n.

In addition, the computed and measured deﬂections must match closely, so that the root-
mean-square error in real and imaginary deﬂections must be smaller than a given

tolerance:

 

 

 

 

76

4. 5.2 T ime-domain Backcalculation using Peak Deﬂection and Time Lag

In the FWD response time history, the peak deﬂection reﬂects the stifﬁiess of the
pavement, and the time lag between the peak of the applied FWD load and a sensor
deﬂection reﬂects the effects of pavement inertia and damping. Only the peak deﬂections
from each sensor and the time lags between the peak load and the peak deﬂections at

each sensor are used in the backcalculation algorithm. The vector of measured responses

T
is therefore {U }={w1 t1 wm tm} , where m is the number of sensors, w, is

the peak deﬂection at sensor i, and t, is the time lag between the peak load and peak

deﬂections.

The unknown properties of pavement layer i are taken to be the real and imaginary parts

of the complex modulus, E11 and E 2, , respectively, and the thickness H ,. . The vector of

unknowns becomes {x}: {[E“ E11] [E21 E21] [H1 HIBT,where

l is the total number of layers in the pavement.

Following the derivation by Harichandran et al. (1994), the increment to the unknown

parameters in iteration i, { Ax} 1, is obtained by solving the linear set of equations
. 1' ~ -
1U} +1Gi’1Ax1’ = {U} (4.7)

~ i . . . . .
where {U } rs the vector of peak deﬂections and time lags computed usrng the estimates of

the pavement layer properties at iteration i, and {01' is the gradient matrix at iteration 1'

given by

77

 

 

 

 

 

 

 

 

 

 

1011,11] .=[1G.11c1211031] ,,. 14.8)
x spay WWW}
where

'am am 21‘ "9:1 9:1. ...ﬂ'

6511 61512 6E1] 5521 61522 6152]

EL at1 3 3L _ai ﬂ.

, 6511 6E12 @151, . W21 6E22 5521

[611' = [621’= =

an aWm @131 9&1 912’. 1’4

6E” 6E12 6E1] 6E2] 6E22 6E2]

2’1 -5"?! in: Pm in. 911

51511 61‘312 (31511 {x}___{£}z _5E21 5522 8E21_{x_£}1

in 3W1 m“

6H1 6H2 6H]

EL EL 91

, 6H1 6H2 6H1

[031' = s

51% 59m 21m

6H1 6H2 6H1

3m. 1321. Elm

_aHl 6H2 5H] {X}={5E}l
The partial derivatives in the gradient matrix must be evaluated numerically using
1.1., 11411111101111)
—— .= ~i ,j=l,2~~,2m ,k=l,2---,3l (4.9)
6x11114121 m.

where U can be the peak deﬂection or the corresponding time lag and x is the layer
parameter (real or imaginary modulus, or thickness). [R] is a diagonal matrix with the k’h

diagonal element being (1 + r) and all other elements being 1. A separate call to the

78

forward calculation program is required to compute the partial derivatives in each column

of the gradient matrix.

Equation (4.7) represents a set of 2m equations in 31 unknowns. Since there are more
equations than unknowns, more robust method for solving the problem is to use the
singular value decomposition (SVD). This algorithm has been implemented in the

program.

After the increments {Ax}i are obtained by solving Equation 4.7, the revised moduli and

thicknesses are obtained ﬁ'om:
{x}" *1 =1x1’ +{Ax1’ (4.10)

The iteration is terminated when the changes in layer moduli and thicknesses are smaller

than a set of speciﬁed tolerances:

 

*i+l *i ‘i+1 “i Ai+1 “i
-E E —-E —

£74ng 2k . 2k 3.91 $43.91 k=1,2,-~,I (4.11)
E’ E' 19'
1k 2k k

4. 5.3 Time Domain Backcalculation using traces of time histories

In this method, the deﬂection time histories are matched within a range of time near the
peak responses. The backcalculation algorithm is similar to that described above, except
that the gradient matrix is expanded to include deﬂection basins at individual time steps

within the speciﬁed range in time. The vector of measured responses is:

{U}=1[W1(rs) 11mm] 111w) wm11f111T (4.12)

79

where m is the number of sensors, w,(t,) is the deﬂection of sensor j at the starting time

and wj(tf) is the deﬂection at the ﬁnal time of the speciﬁed range. The vector of unknowns

1 2 2 T -
E] [E E1] [H1 Hl—ll} .Theincrement

is described as {[151 1

1 l
to the unknown parameters in iteration n, {Ax}n, is obtained by solving Equation (4.10),
~ n. . . . . . . . . .
where {U} 18 the vector of deﬂections at indiwdual time steps, Within the spec1ﬁed time
range, computed using the estimates of the pavement layer properties at iteration n, and

the gradient matrix [0]" at iteration n, in Equation (4.13), is composed of the following

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

submatrices:
”awn Mrs) Mrs)‘ Paws) (W's) M"
6E} 6E5 7E}— 01.912 5?? . 61?"?
awnias) awnias) E awnias) aWrits) Mas) m
6E1] 65; W 51?? as; . 515.121
[GI]: (Mi!!!) (Mitf) S (Mitf) ,[GZ]: awlz’f) Mi’f) awli’f) ’
6E]1 655 615} 61:312 01:3; . as?
an:(’f) awmit f) W an:(tf) awmlt f) an:(tf)
_ 651‘ 65; 651‘ - L 61512 615% 6E12 -

80

pawlas) aW103) aW105)-
6H1 5H2 61‘],

Was) awnias) ,Z, awnias)

 

 

 

6H1 0H2 6H,

[G3]= s s 2 a (4-13)
aWl(tf) 5W1(tf) 5W1(tf)
6H1 6H2 6H1

awm'u f) awm'o f) L. awm'af)

5H1 6H2 6H1

_ d

 

 

 

The partial derivatives in the gradient matrix must be evaluated numerically using
Equation (4.9) and the revised moduli and tolerances are obtained using Equations (4.10)
and (4.11).

4.6 SVD method

SVD is a very powerful set of techniques for dealing with sets of equations or a matrix
that are either singular or else numerically very close to singular. SVD methods are based

on the following theorem of linear algebra, which is in the reference (Press et al, 1989):

Any M><N matrix [Alwhose number of rows M is greater than or equal to its number of
columns N, can be written as the product of an M X N column-orthogonal matrix [U], an
N X N diagonal matrix [(0] with positive or zero elements (the singular values), and the

transpose of an N X N orthogonal matrix [V]

The matrix [A] can be decomposed as three matrices as follows:

81

“’2

 

 

 

 

 

 

 

_ “N. ‘
Where,U -UT = l ,V -V" =1, and [w]is diagonal matrix.

For the following ill conditioned system of equations

— _ F—

 

 

 

 

 

 

 

 

 

 

 

 

 

where,U -UT =1, V - VT =1, and [-1—] is adiagonal matrix.
a)

82

 

 

 

 

(4.14)

(4.16)

(4.17)

4. 6.1 T runcating Singular Values

Reference (Press, 1989) deﬁnes the condition number of a matrix as a ratio of the largest

of 01,- to the smallesta) j .

Condition number = 3mg— (4.18)
‘0 min

A matrix [A] is singular or ill conditioned if its condition number is too large. Reference

(Press, 1989) suggested this number should be adjusted according to the experiment with

the speciﬁc problem.
Deﬁne the threshold of truncation as:

a) j < wmax .10 ‘Th’eshO’d (4.19)

where j=1,2, ------ ,N

After selecting the threshold of a condition number, the SVD algorithm will simply

replace J— with zero.
0 J

4. 6.2 Scaling

The objective of scaling is to reduce the possibility of ill conditioning in the inverse
problem. In this process, the nonlinear equation (4.3) including parameters of modulus,
damping ratio, and thickness will be simultaneously identiﬁed. Since the absolute value

among modulus, thickness, and damping ratio are not compatible, the elements of the

83

gradient matrix vary from a very small value to a large value. Therefore during SVD

process, the value of a) j varies ﬁ'om a very small number to large number

correspondingly. The condition number can be determined using equation (4.18). A
matrix is singular if its condition number is inﬁnite, or if it is too large (Press, 1989).
Take the example of Michigan USl3l site. If no scaling is performed, the value of the
condition number is within the order of 106 in all iterations, potentially causing ill-
conditioning. Scaling will make all the elements in the gradient matrix compatible
through dividing by a number compatible to the values of the modulus, damping ratio,
and thickness, respectively. When scaling is performed in the Michigan USl31 site, the
condition numbers decreases to the order of 103 in all iterations, thus diminishing the
possibilities of an ill-conditioned problem and increasing the likehood of getting a

reasonable solution during the inversion process.

4.7 Modiﬁcations to the Solution

The MICHBACK backcalculation solution allows for modifying the Newton method
algorithm such that the total number of calls to the forward calculation program can be
reduced (Mahmood, 1993). In the modiﬁed method several iterations are performed with
a gradient matrix before it is revised. Although, the convergence in the modiﬁed
approach is slower than the normal method, the n forward calculations required for
calculating the gradient matrix during each iteration can be reduced. This method was

adopted in the current algorithm.

84

CHAPTER 5 - DYNABACK PROGRAM STRUCTURE AND FEATURES

5.1 General

The DYNABACK program has been written in Fortran 77 computer language and
includes several source ﬁles. The source ﬁles are written using the Microsoft Fortran
complier version 4.0. User-ﬁ'iendly features have been designed to facilitate the use of
the program by any pavement engineers. The program can read the output ﬁles of KUAB
Falling Weight Deﬂectometer (FWD) used by MDOT, and other FWDs such as Dynatest.
The program provides a range of options to the user to view and process the deﬂection
data before it is analyzed. In this chapter, the general structure and the features of the
program are described. The function of each of the program subroutines is brieﬂy

introduced in Appendix A.

5.2 Data Input

Deﬂection data can be entered using the deﬂection output ﬁle format of KUAB used by
MDOT; the ﬁle can be read and processed by the program automatically. The cross-
sectional data, Poisson’s ratios, type of the layers being analyzed, the desired
convergence criteria, expected ranges of the layer parameters, the number of sampled
points, the number of points in the FFT algorithm must be enter using the keyboard.
Users are allowed to specify the layout of deﬂection sensors using the keyboard input.
The default weighting factor (for matching deﬂections) allocated to each sensor is 1.0,
but can be changed by the user if desired. The input information is stored in a data ﬁle,

which can be edited on the interactive screen at any stage.

85

When processing the deﬂection data from a ﬁle, comprehensive keyboard input is
required only at the start of the analysis. All the essential information required to operate
the program is stored in easily accessible data ﬁles. The various options provided to view,

process, and analyze the deﬂection data from FWD ﬁles are discussed in the next section.

5.3 Processing 3 FWD Deﬂection Data File

The DYNABACK program can read output ﬁles containing deﬂection data generated by
the KUAB software. The system is also ﬂexible for Dynatest software and only minor
changes would be required. The programs provide features that make the pre-processing
of the deﬂection data easy and efﬁcient. The program allows for plotting the measured
data so that the user may be able to identify and if necessary, remove any outlier
indicative of unwanted sensor measurements. The program also plots the frequency
content of the load and deﬂections in order for the user to decide on the range of
frequencies to be used in the analysis. The highlights of the various features in the

program are covered in the following sections.

5.3.1 Reviewing and Processing the Deﬂection Data

This feature is useful in checking for any anomalies in the measured load or sensor time
histories. The user can view the peak deﬂection basin as well as the time lags
corresponding to the peak response for each sensor. Figure 5.1 shows an example of these
plots. The program can also plot the deﬂection and load time histories simultaneously so

that the user can easily select the duration (length) of the record to be used in the analysis.

86

For example, according to Figure 5.1, the user may select 100 milliseconds as the

duration for the analysis.

 

1W1) lime History

“mud The “day Odo

 

 

 
 

 

 

.._ 4 [0‘0
g 4 5-1
9 " 1 3-2
E :C I 9-3
Tmr , 9-4
9 z: ‘ 5-5
m»- 5‘5
1E ‘ 5-7
6 A
0 50 100 150
Turn. (mg)

Peak Deﬂection d micron! Sensors

 

  

 

 

 

 

 

 

 

A
C 0 ' ' ' j f T ' r ' .
8 1» .
4h 1
bm- 1
q
.E 1 a
5200 -
<5 I. I
II 1
.. 4
3 106—- ~
0 q
x I:
E 0 A A . l A 4 ‘ l ‘ L d
T I I T T V V Y
0 4 a)
Water- (fret-u}
MLagofOﬂen-d Sensors
12:. 1 - - I ~ . - . . - 4
A I
db
E '01? 1
a a. ,
.1 , .(
. 53. 4
E 4' 1
F t; -j
4' I: ;
2 2:7 1
1h 1
h A 1 : l a a V = _3_ i 5 ‘
U 1 SD

 

 

 

 

 

Figure 5.1 Typical plots for measured FWD load and deﬂection data

87

5.3.2 Data Analysis Options

DYNABACK allows for backcalculating layer parameters in either the frequency or time
domain. The features highlighted in the previous section are designed to help the user
decide on whether to use time-domain or ﬁequency—domain solution. For example, if the
deﬂection time histories are cutoff prematurely so that the ﬁ'ee vibration response is not
allowed to decay to zeros, the user may consider using the time-domain solution to avoid
the problems associated with truncation for the frequency-domain solution. Conversely, if
the curve of time lag versus sensor location is very irregular, then the user may want to
try the frequency-domain solution. Furthermore, in the ﬁequency-domain solution, the
user can choose the frequencies at which the backcalculation is to be done. The results for
the various frequencies are averaged by the program automatically. Naturally, the user
may want to try both frequency and time domain solutions and compare the results from

both analyses.

5.4 Presentation of Backcalculation Results

The backcalculated results are saved in a ﬁle, which can be printed, or can be viewed on
the screen. In single frequency backcalculation, the results can be seen graphically to
observe the variation of the backcalculation parameters with frequency. For cases where
the backcalculated results have reached either the upper or lower bounds, as speciﬁed by
the user, a warning message is posted in the output for the user to consider. The program

also provides the measured and simulated deﬂections for a visual inspection of the match.

88

Typical results from frequency and time-domain backcalculation are shown in Figures

5.2 and 5.3, respectively.

1" l eque ntry Backcalc “lat ion Results

 

9:01 ”(called vs Calculoied

 

 

 

 

 

- 1‘9. I I l ' r T l I l I I l T ‘
i- 035 i
E :
E- 0.5: ~ _ .55. j
§-ML \ H-“ €
-05... N
""1 5 1'0 1'5 2'0 2'5 5'0. 3'5 4‘0 4'5 5'0 5'5 50
unseen CALCULATED °"'°"°' (“h”)
Imaginary Measured vs. Calculated
1‘9: " rT l' ‘ Y ' f I ' T ‘ 1 V r t V I l U I

 

 

Deﬂedlm (mis)

 

 

 

 

 

 

 

nn' . r . A . A 1 1 r l 1 1 A L 1 . 1
“'0 5 1'0 1'5 2'0 2'5 50 3'5 4'0 4'5 5'0 5'5 an
orgasm CALCULAYEO °'"°"°' (”Ml
Magnitude Measured vs. Calculoied
ecu : : : 1 viwv: “iv : . . i .
:5 02-22- *4
f we:- «z
E Me:- -1
0.55- _:
1'3“ I I . I I l . 1 . 1 . I . n A I 1 4
o 5 to 15 20 25 so 35 no 45 50 55 co
MUSLIPEO cucuwm °“°"°' 0W3

Frequency: 14.848000

l<<Freqml 0K l Frequency >>§

 

 

 

Figure 5.2 Typical output plots from frequency-domain backcalculation

89

1W1) lime History

 

Otﬂeclloﬂ (ml!)
9“ ‘g‘ - cg: : :§: : g

Poolr Ooﬂocuon (mull)

Pooh 11in. Log {ma}

 

 

“courted Us. Calculdod Time Hilary Odo

 

 

 

 

8
8

“courted w. Calculated Peck Median

‘ f "f l ' i
-i

1111. .l
A A 4A_. 1 A A l A A A

cil"'?o"'4’oﬁ"eb

ammo WWO Wm OW)

 

 

Wed vs. Cdculuted "me Log

 

 

 

 

 

 

 

 

Figure 5.3 Typical output plots from time-domain backcalculation

90

 

5.5 Program Structure

The main ﬂow chart of the program is presented in Figure 5.4. The program ﬁrst reads
the inventory data (layer thicknesses, assumed material properties, sensor conﬁguration,
etc.) and the load and deﬂection time histories. Seed values for the layer parameters that
are to be backcalculated are also input at this point. The program also allows for the
options of estimating the subgrade modulus and the depth-to-stiff layer (or ground water
table). These values are obtained using the regression equations presented in Chapter 3.
Once the input data are entered, the program allows the user to select the main
backcalculation method; i.e., either frequency- or time-domain analysis. For each
method, the user has the option to backcalculate layer thicknesses or to assume them to
be ﬁxed. If the layer thicknesses are ﬁxed, the program will backcalculate the damping
ratios for all layers; if layer thicknesses are backcalculated the program ﬁxes the damping
ratios of the unbound materials (which generally do not vary signiﬁcantly) and allows for
backcalculating only the damping ratios of the asphalt concrete layer. Once
backcalculation is performed, the results can be viewed graphically and the results are
saved in both summary and detailed formats in separate ﬁles which can be viewed or
printed, as desired. The details of frequency- and time-domain backcalculation

procedures are shown in Figure 5.5, Figure 5.6, and Figure 5.7, respectively.

In frequency-domain backcalculation (Figure 5.5) the load and deﬂection time histories
are ﬁrst corrected to the frequency domain using the FFT algorithm. The user selects the
frequencies at which backcalculation is to be performed, and the program then calculates

the steady-state (harmonic) response at these prescribed frequencies. For each frequency,

91

the program computes the gradient matrix according to the option chosen by the user
(e.g., the matrix will be different depending on whether layer thicknesses are
backcalculated or not). The program then revises the real moduli (Cases A and B),
imaginary moduli of speciﬁed layers (all layers for Case A; AC layer only for Case B)
and thickness (Case B). Then the program calculates the real and imaginary deﬂections as
well as the corresponding RMS errors. The procedure is repeated until the convergence
criteria are met or the maximum number of iterations is reached. Once the analysis is

completed, the results can be plotted and printed in either a summary or detailed format.

In time-domain backcalculation (Figure 5.6 or Figure 5.7) the program ﬁrst determines
the peak deﬂections and corresponding time lags from the measured sensor records or
traces of time histories. The user can view the frequency contents of the measured load
and deﬂection, and select the frequencies at which the steady-state response is to be
calculated for determining the transient response (to save on computational time, the
forward solution allows for calculating the response at a limited number of frequencies
and interpolating the response at the remaining frequencies). The program then calculated
the transient deﬂections and determines the peak values and the corresponding time lags
(or traces of time histories) from the calculated responses. Next, the program computes
the gradient matrix according to the option chosen by the user (either with or without
thickness backcalculation). The program then revises the real moduli for all layers (Cases
C, D, E, and F), imaginary moduli of the speciﬁed layers (Case C and E) and layer
thicknesses (Case D and F), and calculates the transient deﬂections as well as the
corresponding peak values and time lags, together with their respective RMS errors. The

procedure is repeated until the convergence criteria are met or the maximum number of

92

iterations is reached. Once the analysis is completed, the results can be plotted and

printed in either a summary or detailed format.

5.6 Backcalculation of Layer Properties

For each method of analysis (i.e., frequency- and time-domain backcalculation) the

backcalculation tasks performed by the program can be divided in to two major groups:

1. Backcalculation of all layer moduli and damping ratios without thickness
backcalculation (Cases A, C, and D for frequency— and time-domain method,

respectively)

2. Backcalculation of all layer moduli, AC damping ratio and layer thicknesses (Cases

B, D , and F for frequency- and time-domain methods, respectively)
5. 6.] Cases A, C, and E

For these cases, layer thicknesses are fixed and the program backcalculates the moduli
and damping ratios of all layers. The user may choose to use the estimated depth-to-stiff
layer if the program detects the presence of a stiff layer from the ﬁ'ee vibration response.

In that case, the program will use the estimated depth-to-stiff layer.
5.6.2 Cases B, D, and F

For these cases, in addition to layer moduli, the layer thicknesses (including the depth—to-
stiff layer) are allowed to be backcalculated. The program also allows for backcalculating

the AC damping ratio; however the damping ratios of the remaining layer are ﬁxed.

93

 

START

/ Read inventory data /

 

/ Read FWD deﬂec tion and load data /

 

[ Input seed values for layer parameters I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
   
    
 

  
  

 

 

Yes Estimate modulus
of subgrade?
Use equation (3.5) to get?
initial moduli of subgrade
Incorporate stiff Yes
layer or GWT? 1
Use equation (3.14) to get
initial depth to stiff layer
(GWT)
Frequency-domain SCICCI Time-domain
backcalculation
method Perform traces
, of time history
No Perform Yes Perform peak deﬂection
thickness and time lag or traces of
backcalculation? Perform pea time history?
deﬂection and *
time lags

    

 

Perform thickness
backcalculation?

 
   
    
  

 

Perform thickness
backcalculation?

 

 

 

 

 

 

Perform backcalculation

l l

 

 

 

 

 

Graphics: view results Print results

 

 

 

 

 

Figure 5.4 Main Flow chart for DYNABACK

94

lCasesAtch

 

Convert load and deﬂection time histories into
frequency-domain using FFT

Select frequencies at which
backcalculation is to be performed

Calculate steady-state deﬂections at
prescribed frequencies

i

For each frequency, compute gradient matrix and revise:
Real moduli for all layers (Cases A & B), imaginary moduli of
speciﬁed layers (all layers for Case A; AC layer only for Case B)
and thickness (Case B)

l

[ Calculate real and imaginary deﬂections and RMS errors ]

 

 

 

 

 

 

 

 

 

 

 

 

     
 

 

 

1'

[ WARNING: No convergence J [ Graph and print output ]

 

 

 

Figure 5.5 Details of frequency-domain backcalculation procedure (Cases A & B)

95

ICasesc&D

 

Determine peak deﬂections and time lags from
measured FWD data

Select frequencies at which steady-
state responses will be calculated

Calculate transient deﬂections and
determine peak values and time lags

l

Compute gradient matrix and revise:
Real moduli for all layers (Cases C & D), imaginary moduli of
specified layers (all layers for Case C; AC layer only for Case D)
and thickness (Case D)

l

Calculate transient deﬂections, corresponding peak
values & time lags, and RMS errors

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     
 

 

 

 

 

I WARNING: No convergence I I Graph and print output I

 

 

Figure 5.6 Details of time-domain backcalculation procedure using peak time and time
lag (Cases C & D)

96

ICasesEacF

 

Determine traces of time history from measured FWD
data

Select frequencies at which steady-
state responses will be calculated

Calculate transient deﬂections and
determine traces of time history

1

Compute gradient matrix and revise:
Real moduli for all layers (Cases E & F), imaginary moduli of
speciﬁed layers (all layers for Case E; AC layer only for Case F)
and thickness (Case F)

 

 

 

 

 

 

 

 

 

 

 

 

l

Calculate transient deﬂections, corresponding traces of
time history, and RMS errors

 

 

 

 

 

 

 

Y
I WARNING: No convergence I I Graph and print output I

 

 

 

Figure 5.7 Details of time-domain backcalculation procedure using traces of time history
(Cases E & F)

97

CHAPTER 6 - THEORETICAL VERIFICATION

6.1 Introduction

In this chapter, the theoretical aspects of the backcalculation program are validated using
theoretical deﬂection basins generated by SAPSI. Numerical examples have been
included to highlight various aspects of the program using both frequency and time-
domain backcalculations, including the ability to backcalculate layer moduli and
damping, layer thicknesses, the depth-to-stiﬂ‘ layer, as well as the possibility of
backcalculating these parameters for proﬁles with different stiffness and thickness
characteristics with a larger number of layers. In addition, the effects of deﬂection
measurement accuracy and signal truncation in time on backcalculation results are
investigated. Finally, convergence characteristics and the uniqueness of backcalculation
results are investigated. Sensitivity analysis of the backcalculated results to the various

layer parameters is conducted.

6.2 Theoretical F requency-Domain Backcalculation using Steady-State Response

The usefulness and robustness of the new backcalculation solution was veriﬁed through a
large number of backcalculation examples with theoretical deﬂection basins. It should be
noted that this option is not currently available in the latest version of DYNABACK
because of implementation problems with ﬁeld data. The effects of layer thickness (i.e.,
thin/thick layers), layer stiffness (stiff/soﬁ layer combinations), and multiple frequencies
(low, medium, and high frequency combinations) were investigated. For each pavement

section, layer thicknesses and properties were input into SAPSI and the theoretical

98

complex deﬂections at lateral distances of 0, 8, 12, 18, 24, 36, and 60 inches from the
center of the loaded area were generated. The load magnitude was 10,000 lb, and a

circular contact area was used with a radius of 5.91 inches.

6. 2. I Effect of Modulus, Thickness and Frequency Combinations

The purpose of this exercise is to insure that the backcalculation algorithm works for a
variety of proﬁles and frequency combinations. Nine different proﬁles and twenty-seven
frequency combinations were used, for a total of 243 runs. The list of frequency
combinations is presented in Table 6.1. The proﬁles used and the backcalculation results
are shown in Table 6.2. The errors in backcalculated results are summarized in Figure 6.1
through Figure 6.9. The results indicate excellent agreement between backcalculated and

actual parameters for more than 90 percent of the cases.

Table 6.1 List of frequency combinations

 

 

(22:13:25: n Low Medium High CZﬁCbISStCiZn Low Medium High
1 0.63 6.34 15.85 15 2.53 10.14 25.36
2 0.63 6.34 20.29 16 2.53 13.95 15.85
3 0.63 6.34 25.36 17 2.53 13.95 20.29
4 0.63 10.14 15.85 18 2.53 13.95 25.36
5 0.63 10.14 20.29 19 3.8 6.34 15.85
6 0.63 10.14 25.36 20 3.8 6.34 20.29
7 0.63 13.95 15.85 21 3.8 6.34 25.36
8 0.63 13 .95 20.29 22 3.8 10.14 15.85
9 0.63 13.95 25.36 23 3.8 10.14 20.29
10 2.53 6.34 15.85 24 3.8 10.14 25.36
11 2.53 6.34 20.29 25 3.8 13.95 15.85
12 2.53 6.34 25.36 26 3.8 13.95 20.29
13 2.53 10.14 15.85 27 3.8 13.95 25.36
14 2.53 10.14 20.29

 

 

 

 

99

Table 6.2 Proﬁles used

 

 

 

 

 

 

 

 

 

 

Proﬁle No. Layer Thickness (in) Damping Ratio Modulus (ksi)

AC 3 0.08 250
1 Base 12 0.03 30
Subbase 12 0.03 15
Subgrade oo 0.02 10

AC 3 0.08 250
2 Base 6 0.03 30
Subbase 24 0.03 15
Subgrade oo 0.02 5

AC 6 0.08 500
3 Base 6 0.03 30
Subbase 24 0.03 15
Subgrade oo 0.02 5

AC 6 0.08 500
4 Base 6 0.03 50
Subbase 12 0.03 15
Subgrade oo 0.02 10

AC 6 0.08 500
5 Base 12 0.03 30
Subbase 12 0.03 15
Subgrade oo 0.02 5

AC 6 0.08 500
6 Base 12 0.03 30
Subbase 12 0.03 15
Subgrade oo 0.02 10

AC 6 0.08 500
7 Base 12 0.03 30
Subbase 12 0.03 15
Subgrade co 0.02 1 5

AC 9 0.08 750
8 Base 6 0.03 30
Subbase 12 0.03 15
Subgrade oo 0.02 5

AC 9 0.08 1000
9 Base 6 0.03 30
Subbase 12 0.03 15
Subgrade oo 0.02 5

 

100

     

Freq. Combination

 

Parameters

A Maw semi...
mo<E

Figure 6.1 Percent error in backcalculated results - proﬁle 1

1.2

-—..

 

? \ 0.2 BMW!)

 

        

Freq. Conﬁrmation

£9453me
£94335

6$~m
65:.

 

Figure 6.2 Percent error in backcalculated results — proﬁle 2

101

Enor(%)

Error(%)

 

Parameters

Figure 6.4 Percent error in backcalculated results — proﬁle 4

102

   
   
 

0 EITOl(%)

" -s

Freq. Corrbination

..
‘5:‘::..
'$—°:::*“
:—:5:::
- _-
2:553 mm)
....o
R‘s-$.15?
. \\
Pararmters

 

Figure 6.6 Percent error in backcalculated results — proﬁle 6

103

 

0 Enor(%)

 

   

Freq. Corrbination

 

Figure 6.8 Percent error in backcalculated results — proﬁle 8

Error(%)

  
     

r?
BAS ),

 

mtg/1C

BAS
F S BAS
BGRAD
1
BAS
TH(SU
'15
no
El
3
O
a

Figure 6.9 Percent error in backcalculated results - proﬁle 9

6. 2.2 Effect of Sub-Layering on Backcalculation Results

SAPSI uses the ﬁnite layer method to calculate pavement response. It models the
pavement system using the ﬁnite element method with each layer representing an element
of ﬁnite thickness in the vertical direction and inﬁnite extent in the horizontal direction.
Each pavement layer must be subdivided into several sublayers to insure accuracy of the
results. Therefore its accuracy depends on the sublayer thicknesses. The effect of the
ﬁneness in the sub-layering was investigated by running the backcalculation program
using coarse and ﬁne sublayers. Table 6.3 and Table 6.4 show the coarse and ﬁne layered
proﬁles, respectively. The analysis was performed on proﬁle 5, with the frequency

combination of 3.8, 6.3, and 25.4 Hz.

105

Table 6.5 shows the results from the coarse and ﬁne layers, respectively. It can be seen

that the relative error is high when the coarse layers are used. Using the ﬁner layers

improved the solution signiﬁcantly.

Table 6.3 Proﬁle with coarse sub-layering

 

 

 

 

 

 

 

 

 

 

 

Pavement Layer Sublayer Number Thickness (in)

1 0.25

2 0.25

3 0.5

AC 4 1 .0

5 2.0

6 2.0

7 4.0

Base 8 8.0
Subbase 9 12.0
Subgrade 10 12.0

 

 

 

Table 6.4 Proﬁle with ﬁne sub-layering

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Pavement Layer Sublayer Number Thickness (in)
1 0.25
2 0.25
3 0.3 12
4 0.391
5 0.488
AC 6 0.610
7 0.763
8 0.789
9 0.954
10 1.19
1 l 1.49
12 1.86
Base 13 2.33
14 2.91
15 3.41
16 4.26
Subbase 17 7.74
Subgrade 18 12.0

 

 

 

106

Table 6.5 Backcalculation results using coarse sub-layering

 

 

 

 

 

 

Backcalculated Parameters
True value
Layer Coarse Sublayering Fine Sublayering
Name M d 1
o uus Ratio . Modulus Damping Thickness Modulus Damping Thickness
(ksr) Thickness (1n) (psi) Ratio (in) ( si) Ratio (in)
Damping p
AC (2H2) 500 0.08 500261 0.08 500023 0.08
AC( 12Hz) 500 0.08 6 500246 0.08 6.00 500045 0.08 6.00
AC(24Hz) 500 0.08 5003 16 0.08 500036 0.08
Base 30 0.03 12 29983 --- 13 .92 29994 --- 12.04
Subbase 15 0.02 12 15125 --- 10.03 14980 --- 11.93
Subgrade 5 0.02 00 4999 --- -- 5001 -- --

 

 

6. 2.3 Dynamic Backcalculation of A dditional Layers

The program has been expanded to backcalculate the complex moduli and thicknesses for

pavement systems with up to six layers and four frequencies. For 5- and 6-layer systems,

the backcalculation has to be done at four frequencies. The results for 5-1ayer and 6-layer

systems are shown in Table 6.6 and Table 6.7, respectively. The results indicate that the

program is able to backcalculate layer parameters of pavements with up to six layers.

Table 6.6 Theoretical backcalculation of a ﬁve-layer pavement system

 

 

 

True value Backcalculated Parameters
Layer
Name Modulus Damping Thickness Modulus Damping Thickness
(psi) Ratio (in) (psi) Ratio (in)

AC(2Hz) 500000 0.05 500000 0.05
AC(lZl-Iz) 500000 0.05 6 500000 0.05 6
AC(22Hz) 500000 0.05 500000 0.05
AC(28Hz) 500000 0.05 500000 0.05

Base 1 25000 0.03 10 25000 --- 10

Base 2 20000 0.03 10 20000 --- 9.999
Subbase 1 17000 12 17000 12.001
Subgrade l 5000 0.02 1 5000 «-

 

107

Table 6.7 Theoretical backcalculation of a six-layer pavement system

 

 

 

True value Backcalculated Parameters

Layer

Name Modulus (psi) Damping Ratio “1181:; ess Modulus (psi) Damping Ratio T111813; ess
AC(2Hz) 500000 0.05 500000 0.05
AC( 12Hz) 500000 0.05 500000 0.05
AC(22Hz) 500000 0.05 6 500000 0.05 6
AC(28Hz) 500000 0.05 500000 0.05

Base 1 25000 0.03 10 25000 --- 10

Base 2 20000 0.03 10 20000 --- 10
Subbase 1 17000 12 17000 12.037
Subbase 2 16000 0.03 12 16000 --- 11.963
Subgrade 15000 0.02 15000 --- ---

 

6. 2. 4 Backcalculation of the Depth-to-Sti/f Layer (DSL)

In a layered pavement system, the stiff layer can be incorporated by assigning a high
modulus to the bottom layer. However, the depth to this layer may or may not be known.
Previous work by Roesset (1995) presented a set of regression equations to estimate the
depth-to-stiff layer for different conditions (saturated subgrade, unsaturated subgrade,
etc.). However, the estimation of depth-to-stiff layer based solely on regression equations
can give rise to signiﬁcant errors in the backcalculated moduli. Therefore, it is necessary
to develop an iterative process to improve the estimation. The iterative process for
estimating the depth-to-stiff layer in the new program is similar to that of backcalculating
the layer thickness. Also, when the modulus of the bedrock is greater than 500,000 psi, it
becomes very difﬁcult for the backcalculation routine to identify the modulus. This
difficulty arises because the deﬂection basin is not sensitive to the modulus of the stiff
layer beyond this value. Consequently, for the case of a pavement proﬁle with stiff layer,
the stiff layer modulus is ﬁxed at 1,000,000 psi. An example illustrating the ability of the
backcalculation program to backcalculate the depth-to-stiff layer using the main routine

for modulus and thickness backcalculation is presented herein.

108

Table 6.8 through Table 6.10 show the results of backcalculation with three different

DSL values (10, 20 and 30 ft). The asphalt properties were assumed as frequency

independent, as shown in the table. The results show that the program backcalculates the

DSL very accurately. Note that these results are based on the steady-state solution in

SAPSI.

Table 6.8 Comparison of theoretical and backcalculated layer parameters — DSL=10 ft

 

 

 

True value Backcalculated Results
Layer Name Modulus Damping Thickness Modulus Damping Thickness
(psi) Ratio (in) (psi) Ratio (in)
AC(3.8 Hz) 500000 0.05 50000l 0.05
AC(7.6 Hz) 500000 0.05 6 500001 0.05 6
AC(12.0 Hz) 500000 0.05 500001 0.05
Base 35000 0.03 10 35000 --- 10
Subgrade 15000 0.02 120 15000 --- 120
Bedrock 100000 0.05 00 -- -- --

 

 

Table 6.9 Comparison of theoretical and backcalculated layer parameters — DSL=20 ﬁ

 

 

 

True value Backcalculated Results
Layer Name Modulus Damping Thickness Modulus Damping Thickness
(psi) Ratio (in) (psi) Ratio (in)
AC(3.8 Hz) 500000 0.05 500007 0.05
AC(7.6 Hz) 500000 0.05 6 500007 0.05 6
AC(12.0 Hz) 500000 0.05 500007 0.05
Base 35000 0.03 10 35000 ~—- 10
Subgrade 15000 0.02 240 15000 --- 240
Bedrock 100000 0.05 00 --- --- ---

 

 

109

Table 6.10 Comparison of theoretical and backcalculated layer parameters — DSL=30 ﬂ

 

 

 

 

True value Backcalculated Results
Layer Name Modulus Damping Thickness Modulus (psi) Damping Thickness
(psi) Ratio (in) Ratio (in)
AC(3.8 Hz) 500000 0.05 500000 0.05
AC(7.6 Hz) 500000 0.05 6 500000 0.05 6
AC(12.0 Hz) 500000 0.05 500000 0.05
Base 35000 0.03 10 35000 --- 10
Subgrade 15000 0.02 360 15000 --- 360
Bedrock 100000 0.05 00 --- --- ---

 

6.3 Theoretical Frequency-Domain Backcalculation using Transient Response

The preceding sections dealt with backcalculation results using steady-state (harmonic)
deﬂection basins at different frequencies. In this section, transient deﬂection time
histories were generated artiﬁcially using SAPSI. The backcalculation program converts
these time histories into the frequency-domain, then backcalculates the layer parameters
at different frequencies. The results presented herein are based on backcalculation at
either a single frequency or multiple frequencies. However, note that the backcalculation

at multiple frequencies is not implemented in the software.

6. 3.1 Comparison of Single and Multiple Frequency Backcalculation Results

The results using single and multiple frequency backcalculation are shown in Figure 6.10
through Figure 6.15 for the AC modulus, damping ratio and thickness, base modulus and

thickness, and subgrade modulus, respectively.

Examination of the results indicates that both single frequency and multiple frequency
backcalculation algorithms produce very good results, including backcalculation of layer

thicknesses. One exception is when the base thickness is backcalculated using the

110

 

 

combination of frequencies (19.5, 22 and 24.4 Hz). This implies that single frequency

backcalculation can produce similar results to those from multiple frequency

backcalculation when the number of pavement layers is limited to three.

 

 

 

 

 

 

A 1600000 -—
.._ ,,,/I
a i."
;’ 1400000 a” W * -__- T;:'B"' __,__-__.2___-__-
g
_ u
.g 1200000r*”'-‘—u”‘——‘ -———A~—~e
O I ,- X
2 1000000 ~~-~~-—-~——e—— —————~
U u
< 800000 -
O 5 10 15 20 25 30
Frequency (Hz)
"El—- Actual AC ---<+— Single ~A-- multl
“X"— mult2 WX— mult3 '+— mult4

Figure 6.10 Comparison of AC modulus using single and multiple ﬁequency
backcalculation

 

 

 

 

3 0.20
’3 9
a:
0.151? ~— ~- -2 -- ~— —* ——-—~ _, a ————~-
E” K
=- 0.10 -~ - «1 ‘f—h —— ------ _—
anh__n~~bsummmmw_
1* a
< 0.00 .
0 5 10 15 20 25 30
Frequency (Hz)
-*9-- Actual AC ~0— Single —-A- multl
-- *— mu1t2 X— mult3 ~+- mult4

Figure 6.1] Comparison of AC damping ratio using single and multiple ﬁ'equency
backcalculation

lll

 

 

 

 

2 8.014
g 8.012 -- -—-—————»-- -—+~+~+ — -
f; 8.010 ~~— - - — —ﬁ——~--————— —~-~—kﬁ— — — .
3 8.008 ..r-_-, , ~—-W~-* — —- ——~———7—«—~—
5 8.006 1--” - —~~*———~ - m—————k—#
,2 8.004 -L__ - - -——~--'—~7—— ——~'—~——————-—~
(f: 8.002 --_- — ————-——*~——————~——— — —

8.000 -___. I- I—auu wH—O— euro-— 5 -
S; 7.998 4 ,

0 5 10 15 20 25 30
Frequency (Hz)
'3 ActualAC 0“Single 4 multl
'X -mult2 ﬂ!” mult3 -+— mult4

Figure 6.12 Comparison of AC thickness using single and multiple frequency

 

 

 

 

 

backcalculation

.17 21200

@- 21000 — + , A» x «Aux —*A—- ---—x -._..._..___-__.__

3 20800 ~—— — — —~ —— A ~— —-— -—— —— — — ~————— —— ——~—

8 3040,0600 T g. a a .1313: :1: ‘j

E L . --°_ -- ._ -_ _V __ ._:-*-‘____;:____

3:0200 ix x—-x _ 0'

0 5 10 15 20 25 30
Frequency (Hz)
-0 — Actual AC —«>— Single —4—— multl
X" mult2 ”X” mult3 ’4'“ mult4
Figure 6.13 Comparison of base modulus using single and multiple frequency
backcalculation
12.9
m 12.8 - —— — — —— ,________,ﬁ,___+_+__ —i
E“ 13% ” V” ‘V “ "“‘ ‘ ""‘”—" — “
.2 51225 -.- __ —— e —— “A -_ #ﬁ.__r —— — —
5512.4+——— ———————-———~—-——~-—~
:2. “12.3 --_“ —— — — — ___-“ ——— ————————
g 3% I..-H i 3-4-£415 B «D :p— n: H: i
12

 

 

 

0 5 10 15 20 25 30
Frequency (Hz)
~ 0 - Actual AC 0’ Single “A” multl
”X -_ mult2 4- mult3 —+*- mult4

Figure 6.14 Comparison of base thickness using single and multiple frequency
backcalculation

112

 

 

 

“_f 15010
'g 15005 0 ++' +
2915000 404440484'9494 ° °4 44
0 8. ° '
gv14995 4 44444 4 .
go 14990 ’8 >84 :8. a76582515.:aiia’ ’ ’ 1
5 14985 ’
0 5 10 15 20 25 30
Frequency(Hz)
4‘3 ActualAC 044Single A -mult1
4*4 mult2 4 *4 mult3 4+4 mult4

Figure 6.15 Comparison of subgrade modulus using single and multiple frequency
backcalculation

6.3.2 Backcalculation of Damping Ratio for Unbound Layers

In this section, the program capability to backcalculate the damping ratios of the unbound
pavement layers (in addition to the AC layer) is investigated. Figure 6.16 through Figure
6.21 show backcalculated results for the modulus and damping ratio of the AC, base and
subgrade layers, respectively, using single frequency backcalculation with known layer

thicknesses. The backcalculated values show very good agreement with the actual values.

 

 

 

 

 

10000000 77:4 .;_74V:7L4 .: thi r *1; 4’ ’15":".:"...7
E .
m
=
3 1000000 ,7
O
E
3 " 1’

100000 .
01 1 10 100

Frequency (Hz)
-'9— Backcalculated -6- Actual

Figure 6.16 Comparison of backcalculated and actual AC modulus without thickness
backcalculation

113

 

Damping Ratio
O

 

 

 

.o
O
.._-

l

1

l

Freque my
-9— Backcalculated —9- Actual

Figure 6.17 Comparison of backcalculated and actual AC damping ratios without
thickness backcalculation

 

 

 

 

30000 4 44 22.22
g525000 44 44 7444447777777
3- - .. - - .. - .. 5,
§ 20000 v v v’ Yr4"4 4" Y ~V v.22v ,,,,,,
2
315000 44444444444444444444
N
1::

10000

0 5 10 15 20 25 30
Frequency(Hz)

+ Backcalculated -°— Actual

Figure 6.18 Comparison of backcalculated and actual base modulus without thickness

 

 

 

 

backcalculation
1 2 7 : 2 2 :7 22 : 22
O T ’2 T 2 22 T '2. '2’
'5 T2: :2‘2’m;: 2:2::’;’22
s2 2- 22 2222
E) 0 l 7 ,7: z; ,7 :~ '5 ~: ’ :
an 2 2' L7 2 2 T T '2’2'2’L" . T 2
E 2ﬂfit’ {’22;2:2:;:‘2”2’ :122
S 1:321? ~249—44-L‘ﬁ ::
0.01 — ‘
0 5 10 15 20 25 30
Frequency(Hz)

~9— Backcalculated -°- Actual

Figure 6.19 Comparison of backcalculated and actual base damping ratios without
thickness backcalculation

ll4

 

25000

zomo -12A2222__-2222_2.__2 _ __.-..- 22E2 ‘2 -2_##2 _2 2k__ 2

15000 ~ — ——e—a—G—e——a—a—-a—a—a—e~~~~

 

 

 

Subgrade Modulus (psi)

0 5 10 15 20 25 30
Frequency (Hz)
+ Backcalcuhted -°— Actual

Figure 6.20 Comparison of backcalculated and actual subgrade modulus without
thickness backcalculation

 

 

 

 

 

0.03
.2
2; 0.025 -~~~~ 2222--_ 2.222.
a:
.3 0.02 444W-.__22
a.
5. 0.015 222222—22222222222222222
:3

0.01 .

0 5 10 15 20 25 30

Fre que ncy (Hz)
—9— Backcalculated —0— Actual

Figure 6.21 Comparison of backcalculated and actual subgrade damping ratios without
thickness backcalculation

6. 3.3 Uniqueness of Backcalculated Results

Many backcalculation programs suffer from the disadvantage that the backcalculated
results are highly dependent on the seed modulus values provided by the user. The farther
the guess is from the true values, the higher are the chances of converging to a wrong
solution. The convergence of Newton’s method is, in general, problem dependent.
However, backcalculation of layer properties from FWD deﬂection data appears to be a

well behaved problem (Mahmood, 1993). For static backcalculation of ﬂexible

115

 

 

pavements, the results obtained using Newton’s method seem to be independent of the
starting value (Mahmood, 1993). In the following subsections, the sensitivity of the
backcalculated results obtained by the frequency-domain solution to the seed values is
investigated. The effect of layer thickness and moduli as well as the number of pavement

layers on the uniqueness of backcalculation results are also considered.

6. 3. 3.] Proﬁles with Different A C Layer Moduli

The uniqueness of backcalculation results are considered for pavement proﬁles with
different AC layer moduli. The properties of the three layer ﬂexible pavements used in
the analysis are listed in Table 6.11. The seed moduli values are listed in Table 6.12. The
results are shown in Figure 6.22 through Figure. The Figures show that the results are
generally good for all three cases, although they tend to be slightly better at lower
frequencies. Also, the results from the frequency-domain solution are not affected by the
seed moduli. The only difference is in the number of iterations required to meet the given

convergence criteria.

Table 6.1 1 Proﬁles used for verifying uniqueness of solution (varying layer moduli)

 

 

 

 

Layer Low Actuaidhggiiius 1k“) Hi h Thickness

Name E._. (inch)
E1 E2 E1 E2 E 1 E2

AC 300 30 500 50 800 80 9

Base 45 2.70 75 4.5 45 2.7 8

Subgrade 7.5 0.30 15 0.6 7.5 0.3 00

 

 

 

 

 

 

 

Table 6.12 Seed modulus values used for verifying uniqueness of solution with three-
layer pavement system

 

 

 

Case number AC base Subgrade
E1(ksi) E2(ksi) E 1(ksi) E2(ksi) E 1(ksi) E2(ksi)

Case 1 1000 100 1000 100 1000 100

Case 2 l 0.1 1 0.06 l 0.04

 

ll6

 

 

 

 

1

 

 

Modulus (psr)
. 8 8 8 8
‘

 

0 5 10 15 20 25 30 35

Frequency(Hz)
+Actual —°— Seed casel +Seed case2

(a) AC

 

 

1

 

 

 

 

 

Modulus (psr)
. 8 8 8 8
1
I
1
1
1
1
1
1

0 5 10 15 20 25 30 35

Frequency(Hz)
+Actual -°- Seed easel -£r— Seed case2

(b) Base

 

 

 

 

 

 

 

Modulus (psi)

 

 

 

 

Fre que ncy(Hz)
+Actual + Seed casel -¢— Seed case2

(c) Subgrade

Figure 6.22 Effect of seed moduli on backcalculation results - low AC modulus

ll7

 

 

 

 

 

 

 

8 8" 8 8 8
1+,
1.
la

Frequency(Hz)
+Actual + Seed casel + Seed case2

(a) AC

 

100000

 

 

 

 

Modulus (psi)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Frequency(Hz)
+Actual -°- Seed casel + Seed case2
(b) Base
10000
.4.: 9000 44444 —— 4— ___—"—-4-—4——44444-4~
E»
E 8000 +4444— 4~444~4— 3 I} ———-—-:
.§ 7000 44 4- *4
2 6000 —«— —— 4 — 4 — ,_ —
5000 .
0 5 10 15 20 25 30 35
Fre que ncy(Hz)
+Actual + Seed casel —6— Seed case2
(c) Subgrade

Figure 6.23 Effect of seed moduli on backcalculation results - medium AC modulus

ll8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1000000
E- 900000 —444 4 444 4444444 4444 “4444444444 4‘44 4 _—
m
5
'5
'8 800000 44"""E l w ’1
E
700000 . .
0 5 10 15 20 25 30 35
P re que ncy(Hz)
494- Actual -°— Seed casel -&— Seed case2
(a) AC
70000
E- 60000 4 4444* ~44 — 44444444*444444444444444—4
m
E.
=
'8 50000 4-———
2 3
40000 . . 4 . 4.
0 5 10 15 20 25 30 35
Fre que ncy(Hz)
-9— Actual —°— Seed casel -A— Seed case2
(b) Base
3.
m
3
'5
15
e
E

 

 

 

 

Fre que ncy(Hz)
4-9— Actual 449— Seed casel —&- Seed case2

(c) Subgrade
Figure 6.24 Effect of seed moduli on backcalculation results - high AC modulus

ll9

6. 3. 3.2 Profiles with Different A C Layer T hicknesses

The uniqueness of the solution was investigated for pavement proﬁles with different AC
layer thicknesses. The properties of the three layer ﬂexible pavements used in the
analysis are listed in Table 6.13. The same seed moduli values that were listed in Table
6.12 above were used. The results are shown in Figure 6.25 through Figure 6.27. The
Figures show that the results are generally good for all three cases, although they tend to
be slightly better at the lower frequencies. Again, the results from the frequency-domain
solution are not affected by the seed moduli. The only difference is in the number of

iterations required to meet the given convergence criteria.

Table 6.13 Proﬁles used for verifying uniqueness of solution
(varying AC layer thickness)

Modulus Thickness in
E 1 E2 Thin Medium

600 30 5 9
45 2.7 8
7.5 0.3 00

 

120

 

 

 

 

 

 

 

8 888' 8
1..

0 5 10 15 20 25 30 35

 

 

 

 

 

 

 

 

 

F re que ncy(Hz)
4944Actual 4+4 Seed easel 4&4 Seed case2
(a) AC
60000
8- 50000 »-~-~~~— — . 25.
8 W; :1
'3'
'8 40000 44 44 44 44——44 44 44—4—4 44
2
30000 i 1 1 r
0 5 10 15 20 25 30 35
Frequency(Hz)
49— Actual 404 Seed casel 416- Seed case2
(b) Base

 

 

Modulus (psi)

 

 

 

 

Fre que ncy(Hz)
4944Actual + Seed casel 46— Seed case2

(c) Subgrade
Figure 6.25 Effect of seed moduli on backcalculation results —thin AC layer

121

 

 

 

 

Modulus (psr)
1.
:1
ﬂ

 

0 5 10 15 20 25 30 35

F re que ncy(Hz)
49—Actual 4°44 Seed casel 49:4 Seed case2

(a) AC

 

 

8

 

 

Modulus (psr)

8

1

1

1
_ 1:1
1:

Q “/8

J 4

 

 

 

 

40000 4
30000
0 5 10 15 20 25 30 35
Frequency(Hz)
449— Actual 440- Seed casel 4&— Seed case2
(b) Base
10000
9000 4444 44 44 44 ~4444444— 444444444‘44

 

 

 

 

 

Modulus(psr)
88
11
I1
1
$1»
M,
f 11 .
I |1
1
1
. 121
i m

 

 

 

6000 -22 _2 _2 __222_ ___2
5000 . 4
0 5 10 15 20 25 30 35
Frequency(Hz)
49—Actual + Seed case] —A— Seed case2
(c) Subgrade

Figure 6.26 Effect of seed moduli on backcalculation results -—medium thick AC layer

122

 

 

 

 

 

 

 

 

 

608000
9 606000 4 4
353
2 604000 *4
E 602000 -1——- ~—
2 600000 » A
598000 I r 4
0 5 10 15 20 25 30 35
F re que ncy(Hz)
49—Actual 449-4Seed casel +Seed case2
(a) AC
60000

 

Modulus (psn)
w
1
I! 1
El 1
1:
l3 ‘
1 a

 

 

 

 

40000 “___ __ _ _
30000 4 r
0 5 10 15 20 25 30 35
Fre que ncy(Hz)
49-4 Actual 40— Seed casel 4&— Seed case2
(b) Base

 

 

 

 

 

Modulus (psi)

 

 

 

 

 

0 5 10 15 20 25 30 35

Frequency(Hz)
49—Actua1 404- Seed casel 4&— Seed case2

(c) Subgrade
Figure 6.27 Effect of seed moduli on backcalculation results —thick AC layer

123

6.3.3.3 F our-Layer Pavement Proﬁle

A four layer pavement system was analyzed, for the cases of medium stiff and stiff AC
moduli. The properties of the four layer ﬂexible pavements used in the analysis are listed
in Table 6.14. The seed moduli values are listed in Table 6.15. The results are shown in
Figure 6.28 and Figure 6.29. The Figures show that, for both cases, the results are
generally good and are not affected by seed moduli at frequencies below 25 Hz. The

solution diverges at higher frequencies.

Table 6.14 Four-layer proﬁle used for verifying uniqueness of solution

 

 

 

 

 

 

 

 

 

Modulus (ksi)
Layer name Medium-high High “”336”
E 1 E2 E 1 E2
AC 500 50 800 80 9
Base 45 2.7 75 4.5 8
Subbase 15 0.9 25 1.5 8
Subgrade 7.5 0.3 15 0.6 00

 

 

 

 

Table 6.15 Seed modulus values for verifying uniqueness of solution with four-layer
pavement system

 

 

 

Case number AC Base Subbase Subgrade
E1(ksi) E2(ksi) E 1(ksi) E2(ksi) E 1(ksi) E2(ksi) E 1(ksi) EZLksi)

Case 1 1000 100 1000 100 1000 100 1000 100

Case 2 l 0.1 l 0.06 l 0.06 l 0.04

 

124

 

 

g Mguluépﬂé

o .

 

 

 

10 20 30
Frequency(Hz)
-—B—Actual +Seed case] +Seed case2

(a) AC

 

 

 

 

 

 

Frequency(Hz)
—B—-Actual -e—Seed casel +Seed case2

(c) Subbase

§§°§§§§§

 

O

10 20 3O
Frequency(Hz)

—B—Actual —e—Seed easel +Seed case2

 

 

 

 

 

 

 

 

 

(b) Base
A 1200000
3 1000000 ~1— ,. a
5, 800000 A? /L 1
'5 600000 J
‘3 400000 /
2 200000 1 1C —«

0 - .
0 10 20 30
Fremencymz)

—8—Actual —o—Seed easel +Seed case2

(d) Subgrade

Figure 6.28 Effect of seed moduli on backcalculation results — four-layer proﬁle with
medium-stiff AC layer

125

 

 

 

   

 

 

 

 

 

 

 

1000000 140000
E 900000 '44 4444444444 444 44 4 4 'g120000 *__..____ —— -7b-\—A
s 300000 ‘ ” 1' E 100000 -, _ _ _
'5 00°00 ” 3 80000
'2 600000 . '8 60000 _~ ._ *
2500000444 4444-4444444 2

0 10 20 30 O 10 20 30
F remencvf Hz) E'eqnencyﬂlz)
-B—Actual —e—Seed easel -¢—Seed case2 —8—Actual +Seed casel +Seed case2
(a) AC (b) Base

 

 

 

 

 

Modulus (psi)

 

 

 

 

   

Frequency(Hz) Frequency(Hz)
—B—Actua| —e—Seed casel —A—-Seed case2 —9—Actual —9—Seed casel —A—Seed case2

(c) Subbase (d) Subgrade

Figure 6.29 Effect of seed moduli on backcalculation results — four-layer proﬁle with stiff
AC layer

6. 3.4 Convergence Characteristics

Newton’s method is, in general, a rapidly converging and accurate optimization
technique. The convergence characteristics have been tested in this section using the
deﬂection data generated by SAPSI. The results for a three layer pavement with thin and
medium-thick AC layer (see Table 6.13 for the proﬁles) are shown in Figure 6.30 through
Figure 6.32, and Figure 6.33 through Figure 6.35, respectively. The results show that the
solution converged within 10 iterations irrespective of the seed values. For the proﬁle

with medium-thick AC layer, the solution converges aﬁer 12 iterations. These results

126

indicate that the frequency-domain solution has very good convergence characteristics

when using synthetic data, suggesting the theoretical algorithm for backcalculation in the

frequency-domain is satisfactory.

 

 

 

p/p __actuai

 

 

 

 

 

 

Iteration Number Iteration Number

 

AC Modulus + AC Danping

 

 

 

EIE_actuaI

B/p _actual

 

 

 

 

 

 

 

   

Iteration Number

Iteration Number

+ Base Modulus —I-— Base Danping

1.6
1.4
1.2 ~ ~

 

 

 

 

 

 

0.8 - «#-
0.6 w—_ .,_-_.
0.41“ _—
0.2 A» —

1
1
1
1
1
1

EIE actual
p/p _actuai

 

 

 

 

 

 

 

 

Iteration Number Iteration Number

+ Subgrade Nbdulus + Subgrade Darroing

Figure 6.30 Convergence of layer parameters for a three-layer pavement with thin
AC layer — 4.88 Hz

127

 

1.2 2.5

2 {W— ——>——~—4
1.5 4 -

0.5 ~—

 

 

B/B_ actual
1

l

1

1

1

(

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ite rtaion Number Iteration Number
AC Modulus +AC Darrping
2.5
5. ~
3. a t._ __ __ --_. _. _s
u ' l
m 9
0.5 ~—
0 V

 

 

 

Iteration Number iteration Number

-—3lt— Base Modulus —I— Base Danping

 

 

 

 

 

 

 

 

 

 

 

 

6 1
5 A
a 0.8 .._—___ .._- ___” _ -._ a. 3
I.I.II o 6 on.
m - “‘4 ‘g “‘— "— "—4 44 " \ 2
0.4 1v *1 *4 ___; ———--—— a
02 1‘ ,___-__ ___ __ ___ 1
0 0 . .
0 2 4 6 8 10 0 2 4 6 8 10
Iteration Number Iteration Number
—o— Subgrade Modulus + Subgrade Dan'oing

Figure 6.31 Convergence of layer parameters for a three-layer pavement
with thin AC layer — 24.4 Hz

128

EIE actual

EIE_actual

 

 

_L
N

.N’

0'1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1
0.8 4» §
0.6 3|
G.
0.4 -< a
0.2 0.5-—————————————~4——
0 0
0 2 4 6 8 10
Iteration Number Iteration Number
AC Modulus -+— AC Dancing
5 1
5
'2’ 4 «1
g. 3
CD.
a 2
1
O
Iteration Number Iteration Number
-—)lt-— Base Modulus —l— Base Gaming
2 . 8
7 ,
1.5 ~ _ 6 1
:3: 5 u
1 , 8| 4
e 3 ~
0.5 ~~~h —-— ._ — —---- .-_,- “'2 A
1 _,
0 0
0 2 4 6 8 10 0 2 4 6 8 10
Iteration Number Iteration Number
—e—— Subgrade Nbdulus + Subgrade Darrping

Figure 6.32 Convergence of layer parameters for a three-layer pavement
with thin AC layer — 48.8 Hz

129

 

 

 

 

 

 

 

 

 

 

' ‘ “WK—(__T
.3. °'8 _“” ‘4 4“ 44-4-4 g ,5
8 0.6 ~»-——— ___- _- —.—__.~-__.2 8
uu' I 1 __ __ _______\’ >
m 0.4 1—————_ _- _ -7 _ _g g
0.2 —1——————- .2 ___ *._.___. 0.5 —
° 0
0 2 4 6 8 10 0 2 4 6 8 1o
Iteration Number Iteration Number
AC Nde'US . AC mﬂping

 

 

 

 

EIE actual

B/ﬁIactual
1
1
1
1 1
I 1
1 1
1 1

 

 

 

 

.0“ Y 2

Iteration Number Iteration Number

 

 

 

-)u(— Base Modulus + Base Darrping

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.6 1
1.4 1 — - ——- w— _____..__E
ii
_ 1.2 «E _
g 1 MI {5;
8' 0.0 .. __.Wi_._~#_ 111'
E 0.6 .___ _— :3
0.4 ~———-——-——~ a
0.2 «m—— 7— mm
0 0
0 2 4 6 8 10 0 2 4 6 8 10
Iteration Number Iteration Number
—0— Subgrade Modulus + Subgrade Danping

Figure 6.33 Convergence of layer parameters for a three-layer pavement with medium-
thick AC layer - 4.48 Hz

130

 

 

 

 

 

 

 

 

 

 

2.5
1.___ _ _2_,...- u
g 0.8-4 _,72,W,* -__-m g
3.0.6 ———— _ _ _ -__-_ _ 3|
a.
$1.4 _ ”___.” a
02 __ __ _ ,_,,_--,_,_ 05 ——— ——m -2.-_#___
0 0 1
0 2 4 6 8 10 0 2 4 6 8 10
Iteration Number Iteration Number
AC Modulus + AC Dancing

 

 

 

EIE actual
BIB _actual

 

 

 

 

 

 

 

 

Iteration Number Iteration Number

-—ul(— Base Modulus + Base Danping

 

 

BIB _actuai

 

 

 

 

 

 

 

 

Iteration Number Iteration Number

—e— Subgrade Modulus + Subgrade Dan'ping

Figure 6.34 Convergence of layer parameters for a three-layer pavement with medium-
thick AC layer — 24.4 Hz

131

 

 

 

 

 

O 2 4 6 8 10 12 14

Iteration Number

 

AC Modulus

 

 

 

 

4 ___
3.5
g 2.5
uu' 2
g 1.5 ,
1 .
0.5 1
0’ 1
O 2 4 6 8 10 12 14
Iteration Number
+BaseModulus

 

 

EIE actual

 

 

 

0 2 4 6 8 10 12 14
IterationNumber

—e— Subgrade Modulus

 

BIB _actuai

B/ﬂ _actual

0/0 _actual

...

O-hNOD-bUIOINQCDO

A

oarowacnaawoecoo

 

 

 

 

 

2 4 6 8101214

Iteration Number

+ AC Darrping

 

4 6 8 10 12 14
IterationNumber

+ Base Dancing

 

1‘1
1

 

 

 

 

T

4 6 8 10 12 14
Iteration Iteration

+ Subgrade Darrping

Figure 6.35 Convergence of layer parameters for a three-layer pavement with medium-
thick AC layer — 48.8 Hz

132

 

 

6. 3.5 Effect of Poisson ’s Ratio on Backcalculated Layer Parameters

Several studies were conducted to assess the effect of Poisson’s ratios of the various
layers on calculated deﬂections. Using static analysis Pichmani (1972) concluded that
only Poisson’s ratio of the roadbed soil has some appreciable effect on the surface
deﬂections. Variations in the Poisson’s ratios of the other layers were found to have little
effect on the surface deﬂections. This and other similar ﬁndings have led to a general
consensus that since Poisson’s ratios of the pavement layers have little inﬂuence on the
surface deﬂections, their effect on the backcalculated layer moduli may be neglected. No
study appears to have investigated the direct effect of Poisson’s ratios on the dynamic
backcalculation layer moduli. In this study, this issue was investigated and results are

presented in this section.

First an attempt was made to backcalculate the modulus, thickness, and Poisson’s ratio of
the AC layer for a simple proﬁle. The proﬁle with a 9 inch AC layer in Table 6.13 was
used for this example. The results shown in Figure 6.36 indicate that the Poisson’s ratio
of the AC layer cannot be backcalculated since it reaches either the upper or the lower

boundary.

133

 

 

.0
cn

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O
'5 04 .1— ——— --_—__--h— ___—__-. ",-,-2
N
m /
y 03 4 4 44 4 4 51—9—54 J“ 3 5 V44\4 Upper limit
I:
302 ———— _“w,
.2 01 V\
o a __ __ __ _ __ ___—.._. .__ _. _ - ___ ...- .2_ , - __
(L Lower limit
0 1 14 1 ,
0 5 10 15 20 25 30

Frequency (Hz)
—9— Actual —o— Backcalculated

Figure 6.36 Backcalculated Poisson’s ratio at various frequencies

In light of these results, it was decided to look at the effect of Poisson’s ratio on the
backcalculated results. The deﬂection basins used in the previous sections were generated
by using constant Poisson’s ratios of 0.35, 0.40 and 0.45 for the AC, base and the
roadbed soil, respectively. To assess the effect of Poisson’s ratio on the backcalculated
layer moduli, the value of Poisson’s ratio was varied by 0.05 from the true value for one
layer at a time. The results of this analysis for the same 9 inch AC pavement are shown in
Figure 6.37 through Figure 6.39. The results indicate that for frequencies below 20 Hz,
the effects of variations in Poisson’s ratios on the backcalculated moduli are negligible,
with the error being within 2%, 5% and 6% for the AC, base and subgrade modulus,
respectively. The error is higher at higher frequencies, reaching 4%, 20% and 27% for the
AC, base and subgrade modulus, respectively, at 44 Hz. Based on these results, it appears
that it would be prudent to limit backcalculation to frequencies lower than about 20 Hz in

order to minimize the errors caused by the variation in Poisson’s ratio.

l34

 

 

 

 

5
E
E1
5 _, ,
-5
0 5 1015 20 25 30 354045
Fre que ncy (Hz)
49— AC-0.05 49- AC 410.05 44— Base-0.05

4"— Base+0.05 4*4 Subgrade-0.05 44*— Subgrade+0.05

Figure 6.37 Percent error in AC moduli due to change in Poisson’s ratio

 

 

 

 

 

Frequency (Hz)
49— AC-0.05 40— AC +0.05 4&4 Base-0.05
4*4 Base+0.05 + Subgrade-0.05 4+4 Subgrade+0.05

Figure 6.38 Percent error in base modulus due to change in Poisson’s ratio

 

 

 

 

 

25
20 4
A 15
s lg:
h ,
E -5 _,
a -10-
-15 4 4
-20
-25
0 5 10 15 20 25 30 35 40 45
Frequency (Hz)
494 AC-0.05 4+4 AC +0.05 44:4 Base-0.05

4*4 Base+0.05 4*4 Subgrade-0.05 41— Subgrade+0.05

Figure 6.39 Percent error in subgrade modulus due to change in Poisson’s ratio

135

6. 3.6 Simulation of Measurement Errors

To investigate the possible reasons for the erratic behavior of the backcalculated layer
parameters with frequency observed using measured FWD data from the ﬁeld, synthetic
deﬂection time histories were generated using SAPSI with different precision levels and

durations.

One source of error is the precision of the deﬂection measurements. The precision of the
sensor readings in Dynatest and KUAB FWD machines is about :1: 0.1 and a: 1
micrometer, respectively. It is generally believed that since the deﬂections at the outer
sensors are comparatively smaller, imprecision at these sensors have a large contribution
towards the overall error especially for the lower layer. In this section, the effects of
imprecision in deﬂections at different sensor locations on the backcalculated layer

parameters are examined. For simplicity the maximum error of 3:1 micrometer is used.

Another source of error for frequency-domain analysis is the truncation in the duration of
the load and deﬂection time histories. Note that the ﬂuctuation of the backcalculated
parameters along frequency is basically due to the truncated FWD sensor records. The
Fourier spectrum of the truncated signal is not the same as the original signal, leading to a
different deﬂection basin at a given frequency, and hence resulting in poor
backcalculation results. Taking an average value across the frequency-domain, while
technically incorrect, may lead to more reasonable estimates of the backcalculated
parameters. To investigate the effect of signal truncation, dynamic backcalculation was

conducted using both truncated (60 ms) and longer (200 ms) load and deﬂection time

136

histories. An alternative solution to this problem is to perform the time-domain

backcalculation.

The combination of these two sources of error can lead to very large errors in sensor
deﬂections, as shown in Figure 6.40. Such errors will inevitably lead to erroneous

backcalculation results.

30
20 ~
10 ~1

0
-10 —
-20 ~
-30 - —
-40 ~~
-50 ~
-60

 

 

 

 

 

 

Realtive Error (%)

 

 

 

0 5 10 15 20 25 30 35
Frequency (Hz)

494 60 ms full precision 494 200ms i lmicron 44:4 60ms i lmicron

(a) Sensor 1

 

 

 

 

 

 

 

Realtive Error (%)

 

 

 

 

 

Frequency (Hz)

494 60 ms full precision 404 200ms :t lmicron 4:44 60ms :t lmicron

(b) Sensor 6
Figure 6.40 Effect of deﬂection imprecision and signal truncation on
deﬂection basin errors

137

6. 3. 6.1 Effect of Deﬂection Imprecision on Backcalculated Results

The effect of deﬂection imprecision on backcalculation results was investigated for two
cases: (i) when moduli and layer thicknesses are backcalculated, and (ii) when layer
moduli and damping ratios are backcalculated. The pavement structure used in this

analysis is shown in Table 6.16.

Table 6.16 Pavement structure used to study the effects of deﬂection imprecision on
backcalculated results

 

 

Thickness . . . Unit Weight . .
Layer Name (inch) Modulus (ksr) Dampmg Ratio (pct) Porsson Ratio
400 to 700 0.135 to 0.02
AC 8 (from 5 to 25 Hz) (from 5 to 25 Hz) 145 0'3
Base 12.13 20.4 0.03 135 0.35
Subgrade 54.69 15 0.02 125 0.4
Stiff layer 00 100 0.05 145 0.15

 

Case (I) —— Backcalculation of Layer Moduli and Thicknesses

In this case, layer moduli and thicknesses were backcalculated while the damping ratios
of the unbound layers were assumed. Figure 6.4] through Figure 6.47 show the
backcalculated parameters using surface deﬂections with full precision and i 1 micron
precision. The error in backcalculated layer thicknesses varies with the layer type and the
frequency at which the backcalculation was performed. The maximum error in the
backcalculated AC thickness was 2.5%, which is very reasonable. For the base layer, the
errors in thickness varied from 7.5% to 16%, which is relatively large. For the subgrade
depth (or depth-to-stiff layer,) the maximum error was 5.5%. Therefore, it appears that
base thickness is the most affected by deﬂection imprecision. In terms of modulus
backcalculation (for the case when layer thicknesses are also backcalculated,) the error in

AC modulus (and damping ratio) was negligible. For the base modulus, the error is

138

within 5% except for one case (15 Hz) where the error is 20%. For the subgrade modulus,

the maximum error was close to 7%.
Case (11) — Backcalculation of Layer Moduli and Damping Ratios

In this case, layer thicknesses are assumed while the moduli and damping ratios of all
layers are backcalculated. Figure 6.48 through Figure 6.53 show the backcalculated
parameters using surface deﬂections with full precision and i: 1 micron precision. The
errors in backcalculated layer moduli (Figure 6.48 through Figure 6.50) are signiﬁcantly
lower than when layer thicknesses were backcalculated, with the error being within 1%
for the AC and subgrade layers, and the maximum error being short of 4% for the base
layer. The errors in backcalculated damping ratios (Figure 6.51 through Figure 6.53) are
insigniﬁcant except for one case (15 Hz) where the backcalculated base damping ratio

was 2.3% as compared to the actual value of 3%.

8 7 h_#.m___. ___ __2_ ___—.._ ___.”,...___._..___._.____._2

1°
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

Thickness (in)
1
1
1
1
1
1
1
1
1
1
1
1

\1
oo
1
1
in).
<11
:h
1
1

 

 

 

 

.41
m

Fre que ncy(Hz)
49— Actual thickness 404 Full precision —A— i1 micron

Figure 6.41 Effect of deﬂection precision on AC thickness backcalculation (thickness
backcalculation enabled)

139

 

15 ‘4
l4 4

l3

Thickness (in)

 

 

 

 

 

 

 

Freque ncy(Hz)
49- Actual thickness 404 Full precision 4t:— 2t1 micron

Figure 6.42 Effect of deﬂection precision on base thickness backcalculation (thickness

Thickness (In)
‘5 8. ‘6 & 8 a 5‘ d 8

backcalculation enabled)

 

1

1

1

l

 

1
1
1
1

 

 

 

 

 

 

10 15 20 25 30 35
F re que ncy(Hz)

494 Actual thickness 494 Full prec'sion —e— :tl micron

Figure 6.43 Effect of deﬂection precision on depth-to-stiff layer backcalculation
(thickness backcalculation enabled)

Modulus (psi)

1 000000

1 00000

 

 

 

 

 

:: i+ ”4 4‘ 4:"— 4 :Tl'._;li":i 1
1 414 -4 4:41 1
__ 1”" 4.. $1.1...»
~—— 1 —— + *7» 1~T++~
1 1y~444444 14444
»— ~4—41—4—1.—1—-.1—.1—1L...__—_L——i —+—.. $14—11-
1 1 1 1 1 1 1 1 1 1 ‘ 1 g 1 1 1 1
4444 144 44 444141 144‘ 44444 444‘. ”1"41‘47—1 T
1 1 1 1 ' 1 | 1 | 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1
‘ . .
10 100

Freque ncy (Hz)

49— Actual AC Modulus 40— full decimal 413'— 1 micron

Figure 6.44 Effect of deﬂection precision on ac modulus backcalculation

(thickness backcalculation enabled)

140

 

0.15
0.12 r

10

 

.09
8353

 

 

Damping Rat

0.03 " *-

 

 

Frequency (Hz)
+ Actual damping ratio + Full precision + 2%] micron

Figure 6.45 Effect of deﬂection precision on AC damping ratio backcalculation
(thickness backcalculation enabled)

 

25000
23000 no“ . .0
21000 4 _ —~ W:

19000 1— ___
17000 -1_ ._1_ . ____ _ A _ M
15000

 

Modulus (psi)

 

 

 

Frequency (Hz)
-9— Actual modulus -°— Full precision —ﬁ- i1 micron

Figure 6.46 Effect of deﬂection precision on base modulus backcalculation (thickness
backcalculation enabled)

 

18000
17000 w— -— » — —--- -—-———...___ _ -_

16000 ~~ ~-
-. mxgxﬂ “__..”

 

 

15000 W ——
14000 WW—
13000 -1~—-————— -—-— WWW—W
12000

Modulus (psi)

 

 

 

Frequency (Hz)
—9— Actual modulus + Full precision —A— :hl micron

Figure 6.47 Effect of deﬂection precision on subgrade modulus backcalculation
(thickness backcalculation enabled)

141

 

1000000 1 . 1 - . 1 1 ' 1 ,
,, HHH._;_._‘H_ ,1 f H 7!. 1 T 1-1;.“ _ ' HH_H H HH1H7 if. 1H.
. o~_- i ‘1 » = I -1 ,, _ a. ._

Modulus (psi)

 

 

 

 

Frequency (Hz)
-9— Actual AC Modulus + full decimal +=tl micron

Figure 6.48 Effect of deﬂection precision on AC modulus backcalculation (layer

 

 

 

 

 

 

 

 

thicknesses assumed)

25000
5: 23000 —
é
... 21000 W
E.
.E 19000 c
2 17000 WW WWWW v—WW -— _— ___—___..-

15000 1 1 1 1 .

O 5 10 15 20 25 30
Freque ncv (Hz)

—9— Actual modulus -°— Full precision —¢— :tl micron

Figure 6.49 Effect of deﬂection precision on base modulus backcalculation (layer

 

 

 

 

 

 

 

 

 

thicknesses assumed)

18000
a 17000 WW W—u #_________ WW WW WWW—WW
E 16000 W W— W W
E 15000 1 ___ r 9; a===o=—-¢ «— W—
E 14000 W —W ___. — W W

13000 WW WW WWW __ W— W

12000 1 1 1

0 5 10 15 20 25 30
Frequency (Hz)

-9- Actual modulus -0— Full precision -t- i=1 micron

Figure 6.50 Effect of deﬂection precision on subgrade modulus backcalculation (layer
thicknesses assumed)

142

 

0.15

 

 

 

 

 

 

g 0.12 W W W \ W W WW WWWW

d?

a 0.09 - ______________ \ ___._.______.__.___.___.____4____._

.5

Q. 4W ——— W W W _,_ WW— __._ W—WW—WW—W ,1

E 0.06

a:

a 0.03 -1WWW W WW WWW WWW
O L 1 1 : l

0 5 10 15 20 25 30

Fre que ncy (Hz)
+ Actual damping ratio + Full precision + 3:] micron

Figure 6.51 Effect of deﬂection precision on AC damping ratio backcalculation (layer
thicknesses assumed)

 

 

 

 

 

 

 

0.05
.2
g 0.04 ,n.__.-_._______,z.._~...w__._____,____
CD 0.03 "”’ "“— "_‘ WW -W———WW
.E
=- .02 ~*— —- WW-WW—W~WW-W “___.
g 0
a 0.01 rW WW W WW W WWW WW—W— _— —«—_..
0 1 T
0 5 10 15 20 25 30
Frequency (Hz)

+ Actual damping ratio -9- Full precision -&— i1 micron

Figure 6.52 Effect of deﬂection precision on base damping ratio backcalculation (layer
thicknesses assumed)

 

—— -.——— .-Ws

 

 

 

Damping Ratio
O
O
N
1
1
1 1
1

 

 

 

 

O 5 10 15 20 25 30
Freque ncy (Hz)

—°— Actual damping ratio —9— Full precision + £1 micron

Figure 6.53 Effect of deﬂection precision on subgrade damping ratio backcalculation
(layer thicknesses assumed)

143

6. 3. 6.2 Effect of Signal Truncation on Backcalculated Results

The effect of signal truncation on backcalculation results was investigated for two cases:
(i) deﬂections matched within i1 micron and (ii) deﬂections matched with full precision.
The program was not able to backcalculate layer thicknesses when the deﬂection records
were truncated, so only the results without thickness backcalculation are shown. The

pavement structure used in this analysis is shown in Table 6.17.

Table 6.17 Pavement structure used to study the effects of signal truncation on
backcalculated results

 

 

1E1: fr: “($121588 Modulus (ksi) Damping Ratio Umzpilsight Poisson Ratio
AC 8 210 0.30 145 0.3
Base 12.13 18 0.15 135 0.35
Subgrade 54.69 24.1 0.06 125 0.4
Stiff layer 00 100 0.07 145 O. 15

 

Case (I) — Backcalculation with :tl Micron Precision

Figure 6.54 through Figure 6.61 show the backcalculated parameters using 200 ms and
60 ms (truncated) records. The results clearly show that there are errors associated with
the truncation of the load and deﬂection time histories. Using the longer (200 ms)
records, DYNABACK was able to backcalculate layer moduli and damping ratios
correctly. However, when the truncated (60 ms) records were used, backcalculated
parameters showed an erratic behavior with frequency. Basically, the frequency content
of the motion is modiﬁed when the response is truncated before it fully decays. This will
result in deﬂection basins that are different enough to change the backcalculation results.
Figure 6.62 shows examples of deﬂection basins using the truncated and full time

histories, while Table 6.18 shows the corresponding RMS values at different frequencies.

144

The table shows that these values can be very high at certain frequencies. The lowest
RMS values occur at the frequencies where the response is maximal. For the real part of
the deﬂection basin, this occurs at 0 Hz; while for the imaginary part of the deﬂection
basin, it occurs at about 10 Hz for this proﬁle. This suggests that if the truncation
problem cannot be avoided in FWD measurements, deﬂection matching should be done

at these frequencies, for the real and imaginary parts separately.

The error in the AC modulus varies from -8% to + 17% (Figure 6.54). For the base layer,
the error varies from -10% to +12% (Figure 7.50), and for the subgrade layer, the error
varies from -4% to +22% (Figure 6.56). The error for the stiff layer modulus varies from
-l9% to + 32% (Figure 6.57). More importantly, the erratic behavior with frequency that
was observed in the backcalculated parameters from ﬁeld FWD records is similar to that
shown in the above ﬁgures (see Chapter 7). Therefore, it can be safely stated that this

erratic behavior is indeed caused by the truncation in time.

The percent errors in backcalculated damping ratios are larger than those for moduli. The
error in the AC damping ratio varies from -50% to + 30% (Figure 6.58). For the base
layer, the error varies ﬁ'om -47% to +73% (Figure 6.59), and for the subgrade layer, the
error varies from -48% to +60% (Figure 6.60). The error for the stiff layer modulus varies

from -71% to + 128% (Figure 6.61).

Case (11) — Backcalculation with Full Precision

Figure 6.63 through Figure 6.70 show the backcalculated parameters using 200 ms and
60 ms (truncated) records with full precision deﬂection matching. Comparison of these
results with those from case (i) shows practically no difference. This means that the

145

errors caused by truncation outweigh those that may be caused by sensor deﬂection

imprecision.

 

 

 

 

 

0 10 20 30 4O 50

Frequency (Hz)
+200ms +60ms +Actualmodulus

Figure 6.54 Effect of signal truncation on AC modulus backcalculation (thicknesses
known with 3:] micron precision)

 

21000
20000 +
19000 4
18000 4 “' -
17000 ~
16000

 

Modulus (psi)

 

 

 

 

Frequency (Hz)

+200ms +60ms +Actualmodulus

Figure 6.55 Effect of signal truncation on base modulus backcalculation (thicknesses
known with i1 micron precision)

146

 

 

 

 

 

 

Modulus (psr)
B B S: '4‘:
§ § § ‘8”

Freque ncy (Hz)

—9— 200 ms -°— 60 ms + Actual value

Figure 6.56 Effect of signal truncation on subgrade modulus backcalculation (thicknesses
known with +1 micron precision)

 

      
 
 

 

 

 

140000
E 120000 1 1" "% 7“‘7”T;“"‘*7 '# “6* A; '~ ‘ W‘ _
E 100000 q»_ _ _A :
é
a
2 80000 4”“ *" " *‘ --_ —‘_— W
60000 -
0 10 20 30 40 50

Frequencv (Hz)
+' 200 ms +60ms +Actualmodulus

Figure 6.57 Effect of signal truncation on stiff layer modulus backcalculation
(thicknesses known with $1 micron precision)

 

 

 

 

 

 

 

 

 

0.5
.2
23
an
.E
n.
E
a
G
0
0 10 20 30 40 50

Frequency (Hz)

+ 200 ms -°- 60 ms + Actual damping ratio

Figure 6.58 Effect of signal truncation on AC damping ratio backcalculation (thicknesses
known with :L-l micron precision)

147

 

 

 

Dam ping Ra no
0 O
—- N
1
1 '
1
1:
m
1 ID
1
1 u
1
1

 

 

 

Frequency (Hz)
—9- 200 ms -°— 60 ms -9- Actual damping ratio

Figure 6.59 Effect of signal truncation on base damping ratio backcalculation
(thicknesses known with $1 micron precision)

 

 

 

 

 

 

 

 

 

 

 

0.1
‘3 008 -
a 006 -‘
w o
.E
=- 0.04 a
E
Q 0.02 ~- ___.__
0 1 1 . 1 .
O 10 20 30 4O 50
Frequency (Hz)

-9- 200 ms -0— 60 ms + Actual damping ratio

Figure 6.60 Effect of signal truncation on subgrade damping ratio backcalculation
(thicknesses known with :tl micron precision)

 

 

 

 

 

 

 

 

 

0.18

.2 0.16
1;; 0.14
a: 0012 '
50 .l
i 0.08
5 8'3
‘3 0:02

0

0 10 20 30 4O 50
Frequency (Hz)

—9— 200 ms —°— 60 ms + Actual damping ratio

Figure 6.61 Effect of signal truncation on stiff layer damping ratio backcalculation
(thicknesses known with $1 micron precision)

148

Reel Deflection (mils)

 

_._.200 ns 11 m'cron

Real Deﬂection (mile)

 

Reel Deflection (mile)

 

—&—200rrs t1m'cron

 

-1

 

 

0.5

 

 

 

 

 

 

 

 

32‘
.5. 1
_g- ‘14L‘:_” — ,_ r“ "r— r— __ r— W'— ——
\
\\
g..m._msw"_-----
\
é’ “
_ o T T
0 20 40 60
Distance (in) Distance (In)
200 ns ful precision - - - - 60 ms ful precision 200 ms ful precision - - - - 60 rm ful precision

 

....-..60rrst1nicron —o—200rrst1m‘cron ---a---60rrst1m'cron

 

 

 

 

 

 

 

 

200 ms full precision
__.— 200 rm :1: 1 rn'cron

(a) 4.88 Hz
-1.5
E
5 1
g -1 1W -— —— WW W WWWW—W
.. \\
\
g-os WWW , WWWW WWWWW
E
" O 1 .
0 20 40 60 0 20 40 60
ﬁnance (in) nuance iinl

 

--_-60 ms ful precision
...a---60rnst1m'cron

200 ms ful precision

_ _ _ - 60 ms full precision
' —a—200 ms 1 1 m'cron

....---60n'st1mcron

(b) 19.53 Hz

 

 

 

 

 

 

 

 

 

A 0.8
g i
E. 0.6 .,
r:
.2
g 0.4 -_
0.2 <
§
- O
0 20 40 60 0 20 40 60
Distance (in) Distance (in)
200 rrs full precision — — — - 60 rrs full precision 200 rrs full precision - — - - 60 n3 ful precision

 

---‘---60n's:t1m'cron —o—200rrst1rricron ---A---60rmt1nicron

(c) 39.06 Hz

149

Figure 6.62 Effect of signal truncation on deﬂection basins

Table 6.18 RMS values for deﬂection basins corresponding to truncated versus

 

 

 

 

 

 

 

 

 

 

untruncated sensor signals
RMS(%)
Frequency (Hz) Real Part Imag. Part
0 30/0 N/A
4.88 34% ~ 18%
9.77 280/0 3°/o
14.65 N/A 11%
19.53 12% 10%
24.41 9% 51%
29.3 9% 297%
34.18 56% 12%
39.06 46% 7%
43.95 751% 39%
48.83 40% 204%
250000 ~
5: 230000 ~
8.
‘5 210000 «
2
3
'8 190000 4
5 170000 W W — WWWW —— WWWWWWW
150000

0

Frequency (Hz)

—9— 200 ms -°— 60 ms +Actual modulus

Figure 6.63 Effect of signal truncation on AC modulus backcalculation (thicknesses

21000

2’

19000
18000

Modulus (psi)

16000.

17000 *

known with full precision)

 

 

 

 

 

10 20 3O 4O 50
Freque ncy (Hz)

—9— 200 ms —¢— 60 ms —°— Actual modulus

150

Figure 6.64 Effect of signal truncation on base modulus backcalculation (thicknesses
known with full precision)

 

25000

I)

24500 4*

 

24000 '

Modulus (ps

23500 « “_n

 

 

 

 

23000

Freque ncy (Hz)

—9— 200 ms —°— 60 ms -9— Actual value

Figure 6.65 Effect of signal truncation on subgrade modulus backcalculation (thicknesses
known with full precision)

 

140000
120000 "
100000
80000 +

 

 

 

Modulus (psi)

 

 

Frequency (Hz)
+200ms +60ms +Actualmodulus

Figure 6.66 Effect of signal truncation on stiff layer modulus backcalculation
(thicknesses known with full precision)

 

 

 

 

 

 

 

0.5
.2
SE (L4 .
,§”(13 w
o.
S (12
a

0. l

O 10 20 30 40 50
Frequency (Hz)

—9— 200 ms —°— 60 ms —¢— Actual damping ratio

151

Figure 6.67 Effect of signal truncation on AC damping ratio backcalculation (thicknesses

 

 

 

 

 

 

known with full precision)
0.3
.2
52 0.2 —
N)
G
'a.
E 0.1
a
G
0 _ 1
0 10 20 30 40 50

Frequency (Hz)
+200ms +60ms +Actualdampingratio

Figure 6.68 Effect of signal truncation on base damping ratio backcalculation
(thicknesses known with full precision)

 

0.1
0.08 r
0.06 r
0.04
0.02 ‘

0 _

 

 

 

 

Dam ping Ratio

 

 

 

Frequency (Hz)
+200ms —°—60ms +Actualdampingratio

Figure 6.69 Effect of signal truncation on subgrade damping ratio backcalculation
(thicknesses known with full precision)

 

 

1

Damping Ratio
9999 9999
ON oo—Nhaoo

 

 

1 .

 

 

Frequency (Hz)

-9— 200 ms —°— 60 ms + Actual damping ratio

152

Figure 6.70 Effect of signal truncation on stiff layer damping ratio backcalculation
(thicknesses known with full precision)

6. 3. 6.3 Extrapolation

In frequency-domain backcalculation, the FWD load and deﬂection time histories are
transformed to the frequency domain using the Fast Fourier Transform (FFT) algorithm.
The FFT works on a digitized signal which is a series of discrete values sampled at ﬁxed
intervals of time. The FFT sample size must be a power of two. Therefore, the process
which is called zero-packing is used to obtain this sampling size. Since the sampling time
is limited, sensor deﬂection time histories are truncated before they die out. Because the
discrete Fourier Transform assumes periodicity, the truncated signal is converted to
periodic signal with a discontinuity at the point of truncation. It was shown that the pulse
discontinuity produces an undesirable effect on the FFT of the pulse (Chatti et al, 2003,

Uzan, 1994 and Magnuson, 1988).

Extrapolation is used to predict the future of a time history from a record of its past. The

extrapolation equation using linear prediction is expressed as (Press, 1989):

N
yn = Z djyn—j +xn (1)
i=1

where, x” is the discrepancy of the prediction at time step n, and of,- are the linear

prediction (LP) coefﬁcients. These coefﬁcients characterize the known signal in terms of

a ﬁnite number of poles that best represent its spectrum in the complex z-plane.

153

The equation can predict the next value y" of a time series from the previous N
values yw , j = 1~-N . N should be chosen as a small number (Press, 1989). In this

section, the extrapolation is conducted for different sensors using different N-values for
the different sensors. Figure 6.71 through Figure 6.76 show extrapolations of FWD time
histories with different N-values for the different sensors. The following conclusions can

be made ﬁ‘om the ﬁgures:

1. The value of N will affect the extrapolated portion of the record. The smallest
value of N will lead to a line with no decay. Increasing N will cause the

extrapolated portion to become non-linear, with ﬂuctuations that decay with time.

2. For the different FWD sensors, different N-values may need to be selected to
match the different time histories. In this analysis, N-values of 2, 3, 4, 5, 6, and 7

were used.

Based on the above results, two cases of tail extrapolation corrections were used: Case 1
with N1: 5, N2: 4, N3: 4, N4: 4, N5: 4, and N6: 3 corresponding to sensorl through
sensor 6, respectively; and case 2 with N1: 7, N2: 4, N3: 4, N4= 3, N5: 2, and N6: 2.
The ﬁ'equency-based backcalculation is then performed using the corrected time
histories. Figure 6.77 through Figure 6.84 show the backcalculation results for the

various parameters. According to these results, the following conclusions can be made:

1. Backcalculation results obtained using the extrapolated (“corrected”) time

histories are still different from the true values. Therefore extrapolation can not

154

solve the truncation problem, and it is necessary for frequency-based

backcalculation to use the full time history if it does not decay to zero.

The choice of N is key in the extrapolation correction; a different choice of

extrapolation order (N) will cause different backcalculation results.

155

 

 

 

20 .-_7 —\-.— _,_ W WWW » W - W— ~ WWW -—-- ,___ WW
15 -2, -1“ _ __ _ ___! --- Figure (b) _ __---

 

 

 

 

Deﬂection (mile)

 

 

 

 

 

 

 

 

0 50 100 150 200
Time (ms)
--—- =1 ------- N-2 --—--N=3 -----N=
...... N=5 ___— N'6 — N=7
(a)
2
E 1
E, .
1:: _1 ..
8
-2
60 80 100 120 140 160 180 200
Time (ms)
___-N:1 ....... N:2 _._.-N=3 ....-. :4
W N=5 W N=6 W N=7
(b)

Figure 6.71 Comparison of different order extrapolations for sensor 1

156

 

 

 

 

Deﬂection (mile)

 

 

 

-5
0 50 100 150 200
Time (ms)
--—- =1 ------- N=2 -—----N=3 —----N=4
_— N=5 __ N=6 WW- N=7
(a)
1.5

 

 

 

 

Deflection (mils)

 

 

 

  

120 140 160 180 200

Time(ms)
———- =1 ------- N=2 --—--N=3 —----N=
WN=5 WN= WN=7

(b)

Figure 6.72 Comparison of different order extrapolations for sensor 2

157

 

 

 

Deﬂection (mile)

 

 

 

 

 

150 200
Time(ms)
—---N=1 ------- N=2 -----N=3 -----N=
WN=5 WN=6 WN=?
(a)

 

 

 

 

Deﬂection (mile)

 

 

 

 

60 80 100 120 140 160 180 200

Time(ms)
—---N=1 ------- N=2 —----N=3 —----N=
WN=5 WN=6 WN=?

(b)

Figure 6.73 Comparison of different order extrapolations for sensor 3

158

 

k *_7 —‘ 1
W WWW W 4
‘——1)

 

Figure (b)

 

 

 

 

 

 

 

Azzscoﬁxex_

100 150 200

Time (ms)

50

(0

 

 

 

 

 

WV... ”

1. V1 _ W W. .—

r I. V.

. . 2 3.13;: ..
..., ..

_ .. Dig/H...»—

3. 1. 1. 3. 5

0 0 0 0 D

Agzscoa8£x_

120 140 160 180 200

00

1

80

60

Time (ms)

....N:4

-._.-N=3
WN

2

_ N=6

--N

1

W N=5

—---N

=7

00

Figure 6.74 Comparison of different order extrapolations for sensor 4

159

 

Figure (b)

 

 

 

 

 

 

 

 

 

E -_ ____-_-_2
E. /
5
§ W“ 1
I:
8 __ -.-..-
-2
0 50 100 150 200
Time (ms)
—---N=1 ------- N=2 --—--N=3 ----- =4

_— N=5 __ N=6 a- _ .. N=7

 

 

 

 

(a)
0.5
_15 0.3 W = ‘ ___- W WWW W W W W- WW—W W --WW-——--
g 01 ‘
5 - 1,9“ . _ nix—”R "’
§ .01. W 1.5-$.57
. x i v
c 3'” v_____,_...::: -----------
8 -0.3 ~ '_:_:._._________/.._‘g_=_—_-_—>_:____ WW— W— WW-WWW
-O.5
60 80 100 120 140 160 180 200
Time (ms)
———-N=1 ------- N=2 -----N=3 ---—-N=

W N=5 W N=6 W N=7

(b)

Figure 6.75 Comparison of different order extrapolations for sensor 5

160

 

 

 

1.5 .\ Figure (b)

 

 

 

 

 

 

 

 

 

A , X
g 1 ~L———l— -rW———— WW WWrW—WWW-WW—W WWW-W WWW-WWW W-W
g 054 -i_/ *1
5 com f ,i _ a .
g -0.5 WWWI!
C I
8 -1 _ _ AL..- __ ___~_-________A___-__
-1.5
O 50 100 150 200
Time (ms)
-—--N=1 ------- N=2 --—--N=3 —--—--N=4

___—- N=5 —— N=6 __._. — N=7

(a)

 

 

Deﬂection (mils)

 

 

 

 

60 80 100 120 140 160 180 200

Time (ms)

__..- :1 ....... N=2 _._.-N=3 _.._.N:
___—N=5 ——N=6 WN=7

(b)

Figure 6.76 Comparison of different order extrapolations for sensor 6

161

 

 

 

Eéé

;

 

100000 r

Modulus (psi)

 

 

 

O 10 20 30 40 50
Frequency (Hz)

—o—Actual value —a—-Case 1 Wei—Case 2

Figure 6.77 Effect of extrapolation on AC modulus backcalculation

 

60000
50000 W
40000 W .,
30000 W
20000 W -
10000 --

0

 

 

Modulus (psi)

 

 

 

 

 

0 1O 20 30 40 50
Frequency (Hz)

+Actual value —e—Case 1 +Case 2

Figure 6.78 Effect of extrapolation on base modulus backcalculation

 

60000
50000<rv
40000,,_,
300004rm --
20000 +—“
10000 0- -— WWW —- u,_u__aaﬂw_ﬂﬂ_cﬂ._~s-

0

 

 

Modulus (psi)

 

 

 

 

0 10 20 30 40 50

Frequency (Hz)
—e—Actual value +Case 1 +Case 2

Figure 6.79 Effect of extrapolation on subgrade modulus backcalculation

162

 

250000 ~--»
200000
150000 ~
100000 ~
50000 e 7
0

A

 

Modulus (psi

 

 

0 10 20 30 40 50
Frequency (Hz)

—e—Actual ialue —a— Case 1 + Case 2

Figure 6.80 Effect of extrapolation on stiff layer modulus backcalculation

 

 

 

 

Frequency (Hz)

+Actual value ~e—Case 1 Wag—Case 2

Figure 6.81 Effect of extrapolation on AC damping ratio backcalculation

 

0.5
0.4 ,,
u,0.3 W
a. 0.2
0.1 ,, ,,
0

ratio

Dam

 

 

 

 

Frequency (Hz)

—o—Actuai wlue —e——Case 1 +Case 2

Figure 6.82 Effect of extrapolation on base damping ratio backcalculation

163

 

 

 

 

 

 

 

Frequency (Hz)

—e—Actual value —a—Case 1 We—Case 2

Figure 6.83 Effect of extrapolation on subgrade damping ratio backcalculation

 

 

 

 

 

 

 

 

 

 

Frequency (Hz)

—o—Actual value —a—Case 1 +Case 2

Figure 6.84 Effect of extrapolation on stiff layer damping ratio backcalculation

6. 3. 7 Comparison of Dynamic and Static Backcalculation Results

Although dynamic backcalculation of layer moduli from deﬂection time histories should
be more realistic in identifying pavement moduli, deﬂection data collected from the FWD
have mostly been analyzed by using the static layered analysis methods. In this section,
the backcalculation results from static and dynamic backcalculation are compared. Time
histories of surface deﬂections were generated using SAPSI. The peak load and
deﬂection values were input in the MICHBACK program for static backcalculation. The

pavement proﬁles used are those described above (see Table 6.11 and Table 6.13). The

164

results are shown in Table 6.19 and Table 6.20 for three-layer and four-layer pavement
systems, respectively. The results indicate large differences in the static and dynamic
backcalculated results. For the three-layer pavement system, MICHBACK overestimates
the AC modulus by 4 to 29% and the subgrade modulus by 24% to 89%. The percent
error for the base modulus varies from -42% to +33%. For the four-layer pavement
system, the results from MICHBACK are not reliable, especially for the base and subbase
layer moduli, where the base modulus is underestimated by a factor of 4 to 6, and the
subbase modulus is overestimated by a factor of 2 to 9. It should be noted, however, that
the forward solution in MICHBACK is the CHEVRONX static layered elastic program,

which may produce very different results from those from SAPSI.

Table 6.19 Comparison of static and dynamic backcalculation results for three-layer
pavement systems

 

 

 

DYNABACK MICHBACK
Case AC Base Subgrade AC Base Subgrade

(1‘50 (1611 (1‘50 0‘30 (1‘31) (1‘51)

Soﬁ AC modulus 300 45 7.5 342.8 59.7 10.6
Medium-stiff AC modulus 500 75 15 519.3 85.3 18.6
Stiff AC modulus 800 45 7.5 1034.9 35.2 11.8
Thin AC layer 600 45 7.5 563.9 59.4 9.6
Medium-thick AC layer 600 45 7.5 747.3 48.4 11.4
Thick AC layer 600 45 7.5 747.3 26.1 14.2

 

Table 6.20 Comparison of static and dynamic backcalculation results for four-layer
pavement systems

 

 

 

DYNABACK MICHBACK
Case AC Base Subbase Subgrade AC Base Subbase Subgrade
(‘60 (1‘80 (160 (160 (160 (kSi) (kSi) (kSi)
Med-Stiff AC modulus 500 45 115 7.5 375.0 7.5 100.0 9.9
StiffAC modulus 800 75 25 15 613.8 16.6 54.1 11.3

 

165

6.4 Time-domain Backcalculation using Peak Deﬂection and Time Lag

As mentioned above, FWD response time histories are oﬁen truncated in time and do not
tend to zero at the end of the time window due to drifts in the measurement system. These
inaccuracies can yield signiﬁcant errors when transforming the measured data to the
frequency-domain, and subsequently difﬁculties are encountered in the backcalculation.
Time-domain backcalculation is more attractive than frequency-domain backcalculation

because the inaccurate regions of the FWD response time histories can be ignored.

In order to minimize computational effort, only two pieces of information from the
deﬂection time histories at each sensor are used: (a) the peak deﬂection, and (b) the time

delay between the peak of the load and the peak of the deﬂection.

6. 4.1 Sensitivity Analysis

The peak deﬂections are inﬂuenced by the layer stifﬁiesses and the time lags are
inﬂuenced by the pavement inertia and damping characteristics. The proﬁle used for this
study is shown in Table 6.21. In order to investigate the relationship between the
response characteristics (deﬂections and time lag) and layer properties (moduli, damping
and thickness), a sensitivity analysis was conducted. The layer moduli of a four-layer
ﬂexible pavement section of medium AC thickness were varied and their effects on the
calculated surface deﬂections and time lags were studied by plotting them against the
layer moduli, damping ratio, and thickness (see Figure 6.85 through Figure 6.95). Six
time histories were computed at the locations r = 0, 12, 24, 36, 48 and 60 inches from the

load.

166

 

Table 6.21 Proﬁles used for the sensitivity analysis

 

Layer Thickness (in) Damping Ratio Modulus (ksi)

 

AC 8 0.05 400
Base 12 0.03 40
Subgrade 55 0.02 30
Bedrock oo 0.02 400

 

The following observations can be made:

1. For the AC layer, thickness has greater effect on deﬂection and time lag than modulus,
and damping ratio. As thickness increased, the deﬂection decreased; however, the time
lag may or may not increase for different sensors. As modulus increased, the deﬂection
decreased, and the time lag may or may not increase for different sensors. As the

damping ratio increased, the deﬂection increased.

2. For the base and subbase layers, the results are similar to those for the AC layer.
Thickness has greater effect on deﬂection and time lag than modulus and damping ratio.
Increasing the thickness or modulus resulted in decrease of the deﬂection, and the time
lag either increased or decreased depending on the sensor. Increasing damping ratio

resulted in an increase of the deﬂections as well as the time lags.

3. For subgrade layer, as modulus increased, the deﬂection and the time lag decreased.

As damping ratio increased, the deﬂection and the time lag increased.

167

U!
0
O

   

§ 400 7;
2 E
E. 300 g,
g 200 a
5 e
w 100 p.
n
0
200000 400000 600000 800000 200000 400000 600000 800000
Modulus (Psi) Modulus (Psl)
Wo—Sensor1 +Sensor2 +Sensor3 +Sensor1 +Sensor2 +Sensor3
Sensor4 —-—— Sensors —o—Sensor6 Senscr4 Sensors +Sensor6

 

 

 

Figure 6.85 Effect of AC modulus on pavement deﬂection and time lag

   

400 16
‘5‘ 350 A 14
5 300 g 12
g 250 3 10
5 200 3 8
g 150 «E» 6
g 100 F 4
a 50 2
0 0

0.05 0.10 0.15 0.20 0.05 0.10 0.15 0.20

DampingRatio Damping Ratio

‘ 332:8” I 32:33:?) ': 332:3:3 —o—Sensor1 —I—Sensor2 +Sensor3

Sensor4 Sensor 5 —e—Sensor6

 

 

 

 

Figure 6.86 Effect of AC damping on pavement deﬂection and time lag

\l
O
O

 

 

Old"
00
OO

 

§
0
O

 

N
O
0
Time Lag (ms)

Deﬂection (Micron)
U
0
O

100

 

 

 

 

 

 

 

 

2 4 6 8 10 12 2 4 6 8 10 12

Thickness (In.) Thickness (ln.)

-¢—Sensor1 —I—Sensor2 +Sensor3
. +Sensor1 +Sensor2 +Sensor3
Sensor4 Sensors Sensors —-—Sensor4 Sensor5 —o—Sensor6

 

 

 

Figure 6.87 Effect of AC thickness on pavement deﬂection and time lag

168

20

 

 

 

 

 

 

 

 

 

 

g i? 15
.2
a E:
g 3 10
O 0
e E 5
8 100 T; j Wet—_t—W
o I - i 7 0
10000 30000 50000 70000 90000 10000 30000 50000 70000 90000
Modulus (Psi) Modulus (Psi)

+ Sensor 1 + Sensor 2 + Sensor 3 + Sensor 1 —I-— Sensor 2 + Sensor 3
Sensor 4 Sensor 5 + Sensor 6 Sensor 4 Sensor 5 —o— Sensor 6

 

 

 

 

Figure 6.88 Effect of base modulus on pavement deﬂection and time lag

400
350
300
250
200
1 50
1 00

 

Time Lag (ms)

   

Deﬂection (Micron)

0|
OO

16
14
12
10
8
6
4
2
0
0

0.00 0.05 0.10 0.15 0.20 .00 0.05 0.10 0.15 0.20

Damping Ratio Damping Ratio

+ Sensor 1 + Sensor 2 + Sensor 3 + Sensor 1 + Sensor 2 + Sensor 3
Sensor 4 Sensor 5 —e— Sensor 6 Sensor 4 Sensor 5 -e-— Sensor 6

 

 

 

 

Figure 6.89 Effect of base damping on pavement deﬂection and time lag

 

 

 

 

 

 

 

 

 

 

 

 

 

500 20
S -
.§ 400 4 —. . - - g 15
E. 300 ‘5
S 3 10
g 200 o
5
g 100 i W. j E
0 . 0
4 6 8 10 4 6 8 10
Thickness (in.) Thickness (in.)

—o— Sensor 1 + Sensor 2 + Sensor 3 —0— Sensor 1 + Sensor 2 + Sensor 3
Sensor 4 Sensor 5 —0— Sensor 6 Sensor 4 Sensor 5 —e— Sensor 6

 

 

 

 

Figure 6.90 Effect of base thickness on pavement deﬂection and time lag

169

 

500
400 4 _-___ _— W
300

 

 

 

 
  

 

 

 

 

Time Lag (ms)

 

 

Deﬂection (Micron)

 

 

200 J
100 - _. a “'13
o ' I a l
1 0000 20000 30000 40000 1 0000 20000 30000 40000

Modulus (P81) Modulus (Psi)

—e— Sensor 1 + Sensor 2 + Sensor 3 + Sensor 1 + Sensor 2 + Sensor 3
Sensor 4 Sensor 5 —0— Sensor 6 Sensor 4 Sensor 5 —0— Sensor 6

 

 

 

 

Figure 6.91 Effect of subbase modulus on pavement deﬂection and time lag

   

400 20

‘5‘ 350 A
E 300 g 15
g 250 '6
g 200 5 10
o 150 o
a 100 E 5
3 50

0 0

0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20

Damping Ratio Damping Ratio

—o—- Sensor 1 + Sensor 2 + Sensor 3 + Sensor 1 + Sensor 2 + Sensor 3
Sensor4 Sensor 5 —e—Sensor6 Sensor4 Sensors —e—Sensor6

 

 

 

 

Figure 6.92 Effect of subbase damping on pavement deﬂection and time lag

   

500 20 A
5 400 —~
.2 g 15
E 300 7;,
5 3 10
33 200 a
6% 100 E 5
o 0 0

4 6 8 10 4 6 8 10
Thickness (in.) Thickness (in.)

—9— Sensor 1 + Sensor 2 + Sensor 3 + Sensor 1 + Sensor 2 + Sensor 3
Sensor 4 Sensor 5 —0— Sensor 6 Sensor 4 Sensor 5 —e-— Sensor 6

 

 

 

 

Figure 6.93 Effect of subbase thickness on pavement deﬂection and time lag

170

Deﬂection (Micron)

+ Sensor 1
Sensor 4

 

Deﬂection (Micron)

+ Sensor 1
Sensor4

 

300
200

_L
O
o

o .
5000

 

 

I F
\-

 

1 0000 1 5000
Modulus (Psi)

 

+ Sensor 2 + Sensor 3
Sensor 5 —o— Sensor 6

20000

Time Lag (ms)

—0— Sensor 1

 

20

12%
5 ‘VI

0 l
5000

 

 

 

 

1 0000 1 5000 20000

Modulus (Psi)

+ Sensor 2 + Sensor 3

Sensor 4 Sensor 5 —o-— Sensor 6

 

Figure 6.94 Effect of subgrade modulus on pavement deﬂection and time lag

400
350

300 ~

250

200 4

150

50
O

0.00

 

‘1’" ’

 

4i

100-4W

 

 

 

 

 

 

 

0.05 0.10 0.15
Damping Ratio

 

—e— Sensor 2 + Sensor 3
Sensor 5 —e— Sensor 6

Time Lag (ms)

+ Sensor 1

 

20

.5
0'1

10

 

0.05 0.15 0.20

Damping Ratio

+ Sensor 2 + Sensor 3
Sensor 5 —o— Sensor 6

0.10

Sensor4

 

Figure 6.95 Effect of subgrade damping on pavement deﬂection and time lag

6. 4.2 Theoretical Veriﬁcation

Synthetic FWD data were generated for several pavement structures consisting of asphalt,

base and subgrade layers using the SAPSI computer program. The pavement proﬁles are

given in Table 6.22. The computed vertical displacement time histories obtained from the

SAPSI program were used as input for the backcalculation. Seven time histories were

computed at the locations r = O, 8, 12, 18, 24, 36, and 60 inches from the load.

171

Table 6.22 Pavement proﬁles for synthetic data

 

 

 

 

 

 

Proﬁle Layer Name Thickness Modulus Damping Unit Poisson’s
(in.) (ksi) Ratio Weight Ratio
(pct)
AC 7.2 250 .10 145 0.3
Proﬁle 1 Base 10 30 .03 135 0.35
Subgrade oo 10 .02 125 0.45
AC 4 250 .10 145 0.35
Proﬁle 2 Base 36 20 .03 135 0.35
Subgrade 00 10 .03 125 0.4
AC 8 400 .075 145 0.35
Proﬁle 3 Base 12 3O .02 135 0.35
Subgﬂle oo 5 .05 125 0.45
AC 12 600 .05 145 0.35
Proﬁle 4 Subgrade 6O 10 .03 135 0.4
Stiff layer 00 500 .01 125 0.2
AC 12 500 .05 145 0.35
Proﬁle 5 Base 30 10 .03 135 0.4
Sat. subgrade 00 44.1 .01 125 0.48

 

Two types of dynamic backcalculation were performed: (a) Estimation of the layer
moduli and damping ratios using the correct thicknesses, and (b) estimation of the layer
moduli, damping ratios and thicknesses. Static backcalculation using MICHBACK (10)
was also per-formed using the peak deﬂections from the synthetic time histories of
Proﬁle 1. Table 6.23 shows the static and dynamic backcalculation results for Proﬁle 1.
The percentage errors for each back-calculated parameter are shown within parentheses.
Using dynamic analysis, the backcalculated layer moduli and damping ratios are
essentially exact if the thicknesses are assumed to be known. Errors occur if the
thicknesses are also backcalculated, but the impact of these errors is small. For the latter
case, the errors in the backcalculated moduli and thicknesses are within 2%. The errors in
the backcalculated damping ratios are as high as 33%, but the impact of this on the time
histories is negligible since the damping ratios are small to begin with. Also, since the
damping ratios are not used at present in pavement rehabilitation decisions, these errors

are not signiﬁcant in practice.

172

Table 6.23 Comparison of dynamic and static backcalculation results using synthetic data
for proﬁle 1

 

Dynamic Backcalculation

 

 

 

 

 

 

 

 

 

5 rue Seed Known Layer Unkown Layer Backfatlldiflation
alue Value Thicknesses Thicknesses (MICHBACK)
Case (a) Case (b)

AC modulus (ksi) 250 450 250.0 (0%) 250.8 (0.3%) 207.7 (-16.9%)
AC damping ratio 0.1 0.3 0.099 (0%) 0.10 (0.0%) W*
AC thickness (in.) 7.2 12 W 7.23 (0.4%) —

Base modulus (ksi) 30 70 29.99 (0%) 29.43 (-1.9%) 39.50 (31.7%)
Base damping ratio 0.03 0.1 0.03 (0%) 0.02 (-33%) W
Base thickness (in.) 10 15 W 10.10 (-1.1%) W

Subgrade modulus 10 70 10.0 (0%) 10.0 (0%) 10.15 (1.5%)

Aksi)

Subg. damping ratio 0.02 0.1 0.02 (0%) 0.022 (10.0%) W

 

* Results are not applicable

A sensitivity analysis (not shown here for brevity) revealed that peak deﬂections and time
lags were least sensitive to damping ratios. Deﬂections were more sensitive to modulus
than to thickness, and time lags were mostly sensitive to AC thickness and subgrade

moduli.

The static backcalculation (from MICHBACK) underestimates the AC modulus by
16.9% and compensates for this by overestimating the base modulus by 31.7%. These
errors are partly due to the fact that the forward analysis routine in MICHBACK is the

CHEVRONX pro-gram, which produces slightly different results than those from SAPSI.

Table 6.24 shows dynamically backcalculated pavement layer parameters for proﬁles 2
through 5, when layer thicknesses are known. The percentage errors for each
backcalculated parameter are shown within parentheses. The backcalculated layer moduli
and damping ratios are essentially exact for three of the four proﬁles. For proﬁle 4, where
there is a stiff layer with a modulus of 500 ksi, there are large errors in the backcalculated

damping ratios as well as in the modulus of the stiff layer. The errors are not of practical

173

signiﬁcance, though, since damping ratios are very small and the stiff layer modulus is
high. More importantly, the errors in backcalculated AC, base and subgrade moduli are
within 2%. The RMS errors for all cases are very low, and the convergence rate was good
except for proﬁle 4. Table 6.25 shows dynamically backcalculated pavement layer
parameters for proﬁles 2 through 5, when layer thicknesses are unknown. The
backcalculated layer moduli, damping ratios and thicknesses are essentially exact for
proﬁles 2 and 3. However, large errors occur for proﬁles 4 and 5. The highest errors
observed are for proﬁle 4, which has a stiff layer condition with unknown modulus value.
Note that if this value is given, then the solution converges with very accurate results. For
proﬁle 5, which has a saturated subgrade (simulating a ground water table condition), the
results are not as good as those for proﬁles 2 and 3, but much better than those for proﬁle
5. This is because while a ground water table does present a stiff layer condition, the stiff
layer modulus is not nearly as high as that of proﬁle 5. Finally, as expected, the RMS
errors are somewhat higher when layer thicknesses are backcalculated; however, all cases

except for proﬁle 4 converged.

174

Table 6.24 Dynamic backcalculation results (known thickness) using synthetic data for

 

 

 

 

 

 

 

 

 

 

proﬁles 2 through 5
Proﬁle 2 Proﬁle 3 Proﬁle 4 Proﬁle 5
True Results True Results True Results True Results
values values values values
AC
250 399 608 500
“$1211” 250 (0.0%) 400 (0.2%) 600 (1.4%) 500 (0.0%)
dagcin 0 1 0'10 0 075 0‘07 0 05 0'03 0 05 0'05
mgo g ‘ (0.0%) ' (-6.7%) ' (40%) ' (0.0%)
Base
20.0 30.0 9.8 10.0
”83;“ 20 (0.0%) 30 (0.0%) 10 (-2%) 10 (0.0%)
Base
. 0.03 0.02 0.05 0.03
darnaigéng 0.03 (0.0%) 0.02 (0.0%) 0.03 (67%) 0.03 (0.0%)
Subgrade
10.0 5.0 661.7 44.0
“11,1111” ‘0 (0.0%) 5 (0.0%) 500 (32%) 44" (0.2%)
Subgrade
. 0.03 0.05 0.04 0.01
dagggng 0.03 (0.0%) 0.05 (0.0%) 0.01 (300%) 0.01 (0.0%)
No. of
iterations 10 3O
RMS 0.02% 0.17% 0.8% 0.09%

 

175

 

 

Table 6.25 Dynamic backcalculation results (unknown thickness) using synthetic data for

 

 

 

 

 

 

 

 

 

 

 

 

proﬁles 2 through 5
Proﬁle 2 Proﬁle 3 Proﬁle 4 Proﬁle 5
True Results True Results True Results True Results
values values values values
AC
243 400 568 501
”111$“ 250 (-2.8%) 400 (0.0%) 600 (5%) 500 (0.2%)
AC 0.1 0.072 0.01 0.06
damping 0.1 (0.0%) 0.075 (4%) 0.05 (_80%) 0.05 (20%)
AC
. 4.0 8.0 12.8 12.0
min“ 4 (0.0%) 8 (0.0%) 12 (6.7%) 12 (0.0%)
Base
20.0 30.0 6.0 10.7
”1&3“ 20 (0.0%) 30 (0.0%) 10 (40%) 1° (7%)
Base 0.03 0.02 0.11 0.01
damping 0'03 QO%) 0'02 (0.0%) 0'03 (266%) 0'03 (-66%L
Base
. 35.9 12.0 42.4 32.0
Th1csknes 36 (0.3%) 12 (0.0%) 60 (29%) 3° (6%)
Subgrade
10.0 5.0 2015 44.0
riff-:11“ 10 (0.0%) 5 (0.0%) 500 (303%) 44" (0.2%)
Subgrade 0.03 0.05 0.10 0.02
damping 0'03 (0.0%) 0‘05 (0.0%) 0'01 (900%) 0'01 (100%)
. N02“ 13 19 13
Iteratlons
RMS 0.77% 0.02% 1.9% 0.79%

 

6. 4.3 Uniqueness

Synthetic FWD data were generated for a pavement structure consisting of asphalt, base

and subgrade layers using SAPSI. The proﬁle 1 provided in Table 6.22 was used. The

computed vertical displacement time histories obtained from SAPSI were used as input

for the backcalculation process. Seven time histories were computed at the locations r =

0, 8, 12, 18, 24, 36, and 60 inches from the load. Table 6.26 and Table 6.27 summarize

the results with and without thickness backcalculation using different seed values. The

results show that the backcalculated moduli and thicknesses are independent of seed

values, proving the uniqueness of the solution. For damping, the differences covered by

176

 

varying the seed values are higher, although they are not of practical signiﬁcance since

the damping ratios vary between 1% and 3% and their effect in the response is negligible.

Table 6.26 Uniqueness of results without thickness backcalculation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Case 1 Case 2 Case 3
Seed value Results Seed value Results Seed value Results
AC 250.0 247.2 248.0
modulus 45° (0%) 80 (1.12%) 350 (-0.8%)
AC . 0.099 0.101 0.099
damping 0'3 (1.0%) 0'01 (1.0%) 0'2 (0.1%)
Base 29.99 29.97 30.05
modulus 70 (0.03%) 10 (0.1%) 20 (0.17%)
Base 0.030 0.028 0.031
dampin 0'1 (0.0%) 0'01 (6.67%) 0'01 (3.33%)
Subgrade 70 10.00 4 9.97 20 9.97
modulus (0.0%) (0.395 (0.3%)
Subgrade 0 1 0.020 0 01 0.020 0 01 0.019
damping ' (0.0%) ' (0.0%) ° (5.0%)
Table 6.27 Uniqueness of results when thickness backcalculation is enabled
Case 1 Case 2 Case 3
Seed value Results Seed value Results Seed value Results
AC 250.5 250.4 250.2
modulus 450 (0.2%) 80 (0.2%) 350 (0.1%)
AC 0.1 l 0.15 0.1 l
damping 0'3 (10%) 0'01 (50%) 0'2 (10%)
AC 12 7.23 4 7.34 8 0 7.2
Thickness (0.4%) (l .9%) ' (0.0°@
Base 29.66 28.9 30.0
modulus 70 (1.1%) 10 (3.7%) 20 (0.0%)
Base 0.02 0.01 0.02
damping 0'1 (33.3%) 0'01 466.70g 0'01 (33.3%)
Base 15 9.82 5 10.0 7 0 9.9
Thickness (0.8%) (0.0%) ° (1.0%)
Subgrade 70 10.00 4 10.0 20 10.0
modulus (0%) (0.0%) (0.0%)
Subgrade 0 1 0.027 0 01 0.01 0 01 0.017
damp' g ' (35%) ' (50%) ' (05.0%)

 

177

 

 

6.4.4 Convergence Characteristics

The convergence characteristics were studied using synthetic data. Figure 6.96 through
Figure 6.101 show convergence plots for layer moduli, damping ratios and thicknesses
with and without thickness backcalculation when using different seed values. When the
layer thicknesses are ﬁxed, the solution converges within 10 iterations. When including
thickness backcalculation, the solution converges within 10 to 20 iterations. These results
show excellent convergence characteristics even when layer thicknesses are

backcalculated, with the convergence rate being slightly faster for the cases when layer

thicknesses are not backcalculated.

178

 

 

 

EIE_actual

EIE actual

 

 

 

 

 

 

6 8
Number of iteration

—-— AC Modulus

 

 

 

 

 

 

 

 

 

 

 

 

 

0
0 2 4 6 8 10
Number of iteration
+3886 Modulus

8

7 g __i.____. -9 E
5 -- __.i __ ___-v ___ __..
5 d I _ _ _
4 _ “___. 22
3 - --_ v.
2 , _—
1 .._—sz
o . d

O 2 4 6 8 10

Number of Iteration

—o— Subgrade Modulus

179

3/3 _actuai

B/B_actuai
0 -¥ N w & 01 0'1 \l

O‘quhalmﬂm

B/p _actuai

 

A

 

 

 

 

Number of iteration

—0— AC Darrping

 

 

 

 

 

 

 

Number of iteration

—I— Base Danping

 

 

 

 

 

 

 

 

Number of iteration

-t— Subgrade Danping

Figure 6.96 Convergence of layer parameters for case 1 (no-thickness backcalculation)

 

 

 

 

 

 

 

 

 

 

 

 

g 0.8 4
all 0.6 4
S 0.4

0.2 4 WW W— - _ _ _ _ -._ ___

0 .
O 2 4 6 0 2 4 6
lumber of iteration Mm ber of Iteration
—-— AC Nbdulus + AC Damping
1.2

 

 

EIE_actuai

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

0
0 2 4 6
Number of iteration Number of iteration
+ Base Modulus —I— Base Dan’ping
1 2 1.2
1 .
g T; 0.8
8' 8' 0.6 l
S a 0.4 <-
0.2 -. W W W W W —— W— 0.2 .. —— WW —
0 0
0 2 4 6 0 2 4 6
number of iteration number of Iteration
—e— Subgrade Modulus + Subgrade Danping

Figure 6.97 Convergence of layer parameters for case 2 (no-thickness backcalculation)

180

 

 

 

o _L_L_L
mdN#O)

EIE_actual
BIB _actuai

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.6
0.4
0.2
0 , 0 W . , .
0 2 4 6 8 10 O 2 4 6 8 10
Number of iteration Number of Iteration
AC Modulus + AC Darrping
1.2
1 ._ __... .._- _._ __fA_—_1_.
g g 0.8 WWWWWW .
a 3'06 WWW-WW WWW
a.
S a. 0.4 1 W ___—.4
0.2 W ___..__ -WWW— W
0 0
O 2 4 6 8 10 0 2 4 6 8 10
Number of iteration Number of iteration
+ Base Modulus + Base Darrping
2.5 3

 

 

 

 

 

EIE_actuai

B/ﬂ _actual

 

 

 

 

 

 

 

 

 

0 2 4 6 8 10
Number of iteration Number of iteration

—e— Subgrade Modulus + Subgrade Danping

Figure 6.98 Convergence of layer parameters for case 3 (no-thickness backcalculation)

181

 

N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

315 W-WWWWWW—WW—WWW—WW g
8'1WW.W- W -—___ 8|
1” a:
was .._ ___-_._,_..__..- a
0 , 0 W .
0 5 10 15 0 5 1° 15
Number of iteration Number Of ""3“”
—-— AC Modulus .._ AC Dan'ping ‘
3 |
E 6
3
8| 4
a
a 2
0 0 .
0 5 10 15 0 5 1° ‘5
Mmber of its ration Mmber Of iteration
+ Base Nbdulus —I— Base Dancing
12
_ 10
'5 a
:1 a 8
:4, :4, s
S e 4
a 2
. 0
0 5 10 15 O 5 10 15
Number of iteration Number of iteration
—e— Subgrade Modulus + Subgrade Darrping
2 2
a 1.5 i/JK WW WW -
3
g 1 1“ - _ - _W—W—W
' \[f
5 0,5 .-_ _- __...-. ___ ______-
_ O .
0 5 10 15 0 5 10 15
Mm ber of iteration Number of iteration
—l—AC Thickness Base Thickness

Figure 6.99 Convergence of layer parameters for case 1 (thickness backcalculation)

182

 

 

1.2

 

 

 

 

 

 

 

_ 1 W
3 0.8 W i
3 :
«I 0.6 l
E 0.4 WW W W W 4
0.2 W W W W W W Wg‘
0 i
0 5 10 15 20
Number of Iteration
——ACModuius
2.5 1
:- 2‘ i
g 1.5 r l
05$ W W;
0
0 5 10 15 20
Number of iteration

+ Base Modulus

 

 

 

 

0 5 10 15 20

 

   

Number of iteration

—o— Subgrade Modulus
2.5 g
_ 2
g 1
III 1
1 3
E l
i
l

 

 

0.5 W
0
0 5 10 15 20
Number of iteration
—+—-AC Thickness

 

B/B _aetual
5 Z;

.0
01

 

 

+AC Darrping

B/B _actual

 

 

Number of iteration

+ Base Danping

 

BIB _actual

 

 

 

 

Mm ber of iteration

+ Subgrade Danping

 

(JIM

_k

 

_diIH actuah

 

 

GUI

 

0 5 10 15 20
lumber of iteration

_— Base Thickness

Figure 6.100 Convergence of layer parameters for case 2 (thickness backcalculation)

183

_L
(’1

 

 

 

 

 

8 1 q _ - - .
‘6
'1
iii .._ ___~-* 2222* ___22__.__
ii 0.5

0

0 5 10
Number of iteration
AC Modulus

 

 

 

 

 

Number of iteration

+ Base Modulus

N
01

 

 

 

ElE_actual
_- 3 yo
i i

' i
' l l
(D | I
i I

l ‘ l
.. l (
11 l l
1 ‘ i
i . )
I J

O
01
i
i
ii
i
i

i
i
i
ii
i

 

 

O .

 

Number of iteration

+ Subgrade Nbdulus

 

 

 

 

Number of iteration
—4— AC Thickness

 

 

 

 

 

 

 

 

 

 

 

5
_ 4 ___________________
‘6'
333 WWW—W WWWWWWWW
"I
92 1 W WWWWWW
n1 . ___.____ _ __ _. y
0 .
0 5 10
Number of Iteration
+AC Dan'ping
10 E
g 3 .._- _- __-_..__-____2__ ..l
t; 6 4s ___—”__..— .__. ___— .__.___._.__._ ....-.
“I 4 A. ._ A ___—__..
9 I 8.4. ‘
ea. 2 .
0 QWJ—a—f . ,
O 5 10
Number of ﬂotation
—I— Base Danping
3

BIB _actual

 

 

 

 

Number of its ration

+ Subgrade Danping

 

 

 

 

 

 

2
'5 1.5 —
8
all 1
E
10.5 ..
0
0 5 10
Mmber of iteration
—-— Base Thickness

Figure 6. 101 Convergence of layer parameters for case 3 (thickness backcalculation)

184

 

 

 

6.5 Time Domain Backcalculation using Traces of Time History

6.5. 1 Theoretical Verification

Synthetic FWD data from pavement proﬁles in Table 6.22 were used in this section for
theoretical veriﬁcation. Table 6.28 shows the pavement layer parameters for proﬁles 2
through 5, when layer thicknesses are known. The percentage errors for each back-
calculated parameter are shown within parentheses. The backcalculated layer moduli and
damping ratios are excellent for all cases. It is noted that proﬁle 4, which has a stiff layer
condition, has the largest error (5%), and the corresponding RMS is 0.06%.T able 6.29
shows dynamically backcalculated pavement layer parameters for proﬁles 2 through 5,
when layer thicknesses are unknown. It shows that all cases except proﬁle 4 are excellent
in identifying the parameters. Case 4 has a stiff layer condition, which makes it harder to

backcalculate the parameters. However, all cases converged.

Table 6.28 Dynamic backcalculation results (known thickness) using synthetic data for

 

 

 

 

 

 

 

 

 

 

proﬁles 2 through 5
Proﬁle 2 Proﬁle 3 Proﬁle 4 Proﬁle 5
True Results True Results True Results True Results
values values values values
AC modulus 250 400 600.2 500
(ksi) 250 (0.0%) 400 (0.0%) 600 (g0%) 500 (0.0%)
AC damping 0.10 0.075 0.05 0.05
ratio 0'1 (0.0%) 0'075 (0.0%) 0'05 (0.0%) 0'05 (0.0%)
Base modulus 20.0 30.0 10 10.0
(ksi) 20 (0.0%) 30 (0.0%) 10 (0.0%) 10 (0.0%)
Base damping 0.03 0.02 0.032 0.03
ratio 0'03 (0.0%) 0'02 (0.0%) 0'03 (6.7%) 0'03 (0.0%)
Subgrade 10.0 5.0 523 44.1
modulus (ksi) ‘0 (0.0%) 5 (0.0%) 500 (5%) 44" (0.0%)
Subgrade 0.03 0.05 0.01 0.01
damping ratio 0'03 (0.0%) 0'05 (0.0%) 0'01 (0.0%) 0'01 (0.0%)
No. of
iterations 9 15 10
RMS 0.00% 0.00% 0.06% 0.00%

 

185

Table 6.29 Dynamic backcalculation results (unknown thickness) using synthetic data for

 

 

 

 

 

 

 

 

 

 

 

 

proﬁles 2 through 5
Proﬁle 2 Proﬁle 3 Proﬁle 4 Proﬁle 5
True Results True Results True Results True Results
values values values values
AC modulus 250 400 500 500
(ksi) 250 (0.0%) 400 (0.0%) 600 (0.0%) 500 (0.0%)
AC 0.1 0.072 0.005 0.05
damping 0'1 (0.0%) 0075 (4%) 0'05 (0.0%) 0‘05 (0.0%)
AC 4 4.0 8 8.0 12 12.8 12 12.0
Thickness (0.0%) Q.O%L (0.0%) (0.0%)
Base modulus 20.0 30.0 6.0 10.0
(ksi) 20 (0.0%) 30 (0.0%) 10 (0.0%) 10 (0.0%)
. 0.03 0.02 0.03 0.03
Base damping 0.03 (0.0%) 0.02 (0.0%) 0.03 (0.0%) 0.03 (0.0%)
Base 36 12.0 60 30.0
Thickness 36 (0.0%) ‘2 (0.0%) 60 (0.0%) 30 (0%)
Subgrade 10.0 5.0 466.1 44.1
modulus (ksi) ‘0 (0.0%) 5 (0.0%) 500 (6.8%) 44" (0.0%)
Subgrade 0.03 0.05 0.01 0.01
damping 0'03 (0.0%) 0'05 (0.0%) 0'01 (0.0%) 0'01 (0.0%)
. N°z°f 12 11 19 16
11613110113
RMS 0.00% 0.00% 0.07% 0.00%

 

6. 5.2 Uniqueness

In this subsection, proﬁle 1 provided in Table 6.22 was used for checking the uniqueness
of the backcalculated parameters, Table 6.28 and Table 6.29 show that the method using
traces of time histories can coverage to the true value successfully, and has more accurate
results than that using peak deﬂection and time lag. Therefore, it is believed that

backcalculation using traces of time histories is superior to other methods.

186

Table 6.30 Uniqueness of results without thickness backcalculation using traces of time

 

 

 

 

 

 

 

 

 

 

histories

Seed value Results Seed value Results Seed value Results
AC 250.0 250.0 250.0
modulus 450 (0.0%) 80 (0.0%) 350 (0.0%)

AC 0. l 0. l 0. l
dampiug 0'3 @.0%) 0'01 (0.0%) 0'2 (0.0%)
Base 70 30.00 10 300.0 20 300.0
modulus (0.0%) (0.0%) (0.0%
Base 0.030 0.030 0.030
damping 0'1 Lo.o%) 0"“ 10.0%) 0'01 (0.0%)
Subgrade 70 10.00 4 10.20 20 10.00
modulus (0.0 /o) (0.0 /o) (0.0 /o)
Subgrade 0.1 0.020 0.01 0.030 0.01 0.020
dampmg (0.0 /o) (0.0 /o) (0.0 /o)

 

Table 6.31 Uniqueness of results when thickness backcalculation is enabled using traces
of time histories

 

 

 

 

 

 

 

 

 

 

 

 

Seed value Results Seed value Results Seed value Results
AC 250.0 250.0 250.0
modulus 45° (0.00/g 80 (0.0%) 350 9.0%)
AC 0.1 0.1 0.1
dampimg 0'3 (0.0%) 0'01 (0.0%) 0'2 (0.0%)
AC 12 7.20 4 7.20 8 o 7.20
Thickness (0.0%) (0.0%) ' (0.0%)
Base 30.00 30.00 30.00
modulus 70 (0.0%) ‘0 (0.0%) 20 (0.0%)
Base 0.030 0.030 0.030
damping 0'1 (0.0%) 0'01 (0.0%) 0'01 (0.0%)
Base 15 10.00 5 10.00 7 0 10.00
Thickness (0.0%) (0.0%) ' 40.0%)
Subgrade 70 10.00 4 10.00 20 10.00
modulus (0.0%) (0.0%) (0.0%)
Subgrade 0.02 0.02 0.02
damping 0'1 (0.0%) 0'01 «10%) 0'01 (0.0%)

 

6. 5.3 Convergence Characteristics

The convergence characteristics of time domain solution are studied in this section using
the pavement proﬁle listed in Table 6.22. Figure 6.102 through Figure 6.110 show the

convergence plots for layer moduli, damping ratios and thickness with and without

187

thickness backcalculation when using different seed values. It shows that the
backcalculated parameters can converge to the true value within a few iterations, and that
the number of iterations when layer thicknesses are known is less than cases for unknown
layer. This conclusion is similar to that made for the method using peak deﬂection and

time lag.

188

 

 

EIE actual

 

 

 

 

O 3 6 9 12
Iteration Number

15

 

AC Modulus

 

 

 

 

 

 

 

 

 

 

 

 

 

'3
5
‘6
<
0.5 —— — — _— -_-_ _2 J____* _
0 .
0 3 6 9 12 15
Iteration Mmber
—JI(— Base Modulus
1O
9
8 .
7:. 7 I
s 6 ‘
<1 5 —
S 4 ‘
3 ..
2 .
1
0 . . . .
0 3 6 9 12 15
Iteration Number
—0— Subgrade Modulus

a/p _actual

p/p _actual

B/p _actual

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 .
O 3 6 9 12 15
Iteration Number
—+—AC Dancing
9 Q
8 ———-———1
7 —1—1— ___ [L _ _. __1
6 _ _ _ l __
5 t.— \ .
«~~+u
3 ”l
2 l
1 u 6V9
0
012 15
Iteration Number
+Base Danping
o l \
9 1 _ _ -- -____ _. -
7 _.
6 _ \ ___-
5 |———— -————— _w_ ——-—
4 P ————»—
3 2 ___..__ _ _.
2 r ___\______2
1 .- ._ ..s
0 1 T
0 3 6 9 12 15

Ite ration Number

+ Subgrade Darrping

Figure 6.102 Convergence of layer parameters for case 1 (no-thickness backcalculation)

189

EIE_Actual

0.09.0

EIEactual
o 'N :3 a) on

 

 

 

 

 

Iteration Number

 

AC Modulus

 

)1 J

 

 

 

 

3 6 9 12 15

Iteration Mun ber
+ Base Modulus

 

 

 

 

 

3 6 9 12 15

Ite ration Number
+ Subgrade Nbduius

BIB _actual

actual

/B

blb actual

 

1.2+

 

 

 

 

3 6 9 12 15
iteration Mam ber

+ AC Dan'ping

 

 

 

 

 

 

 

 

 

 

 

Iteration Number
+Base Darrping
5
4. ___m _2 __ ___2____
3. __2__4.m___2
2. ____ -22 -22“-
1 ___.-L“.._ ...___ _w____
I
0

 

3 6 9 12 15

Iteration Num ber
+ Subgrade Dan'ping

Figure 6.103 Convergence of layer parameters for case 2 (no-thickness backcalculation)

190

 

EIE actual

 

 

 

 

 

 

 

 

 

 

 

 

0
0 3 6 9 12
Iteration Number
AC Modulus
'5 1:
3 .
8
1 ' .
g - a
O
0 3 6 9 12 15

Iteration Number

—)1t— Base Modulus
3 1
* i
a . '
3 1
° 1
“I 2
“E11 7 7 1
l
0 l
0 3 6 9 12 15

Iteration Num ber

—e— Subgrade Nbdulus

Figure 6.104 Convergence of layer parameters for case 3 (no-thickness backcalculation)

BIB _actual

ﬂlﬂ _actual

.N
01

 

.3
01 M
1 A

 

 

 

 

Iteration Number

+ AC Danping

 

 

 

 

+ Base Danping

_L

 

 

all} _actual

 

 

L

 

O—‘NQAUIONQ‘DO

Iteration Mm ber

+ Subgrade Danping

191

_ EIE actual

EIE actual

EIE actual

 

.5
(II

A
.1
T

.0
01

 

 

 

 

3 6 9

Iteration Number
AC Modulus

 

12

15

 

 

 

 

Iteration Number

—)lt—— Base Modulus

 

 

 

 

 

 

Ite ration Num be r

—e— Subgrade Modulus

 

BIB _actual
N

M) _actual

c—l

5/p_actual
o-snw-bcnoasloocoo

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.3.5.5.;
OmemON-bouoo

 

 

i, A444

0 3 6 9 12 15
Number of Iteration
+ACDanping

' / \11 :2

ll _. ll 2__

*‘Ezé

0 3 6 9 12 15
Iteration Number
+BaseDanping

 

3 6 9 12 15

iteration Number
+ Subgrade Dan'ping

Figure 6.105 Convergence of layer parameters for case 1 (thickness backcalculation)

192

2.5 1.8

1.6
1.4
1.2

 

| 0.8
0.6
0.4
0.2

HIH actual
HII-I actual

 

 

 

 

 

Iteration Number Iteration Number

—-+— AC Thickness

 

Base Thickness

Figure 6.106 Convergence of layer thickness for case 1 (thickness backcalculation)

193

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 -——‘ ——~—+-~1
0 , . .
0 3 6 9 12 15 0 3 6 9 12 15
Iteration Number iteration Number
AC Modulus + AC Darrping
8 5
71 ________,,_2~2
6 ._ 2 ____ __2 -._ -2 -_ 4* -1 4 **—~ ___——*’A
S 5« — —~— a ~ ,_o_____ _ S 3 ._..__-___ “___ ___WC.
‘6 ‘6
«'44 _-_,_,-__w___. «I
g 3“ _———— _— __-___.s 92 ———— 1 ———*—‘
Q.
2 {Z R '_‘ _“ 1 g; ___.
1 l- ___—..- .-_»- .—.——_a l
0 0 2. . .
0 3 6 9 12 15 0 3 6 9 12 15
Iteration Mmber Iteration Number
+ Base Modulus + Base Darrping
1.2 i 7
l
1 «r—Wi .1. 6
.. 2' .... 5
g 0.8 - .1 g 4
§| 0.6 «1 — —— ~—- —* --—— -~~ 3'
11.1 a 3
m 0.4 ‘ 77— V77 7 Vi vni ._______ "—i a 2
0.2 +—- —-—— -~-— #—- __ .— ~ ~ 1
0 . 0
0 3 6 9 12 15 0 3 6 9 12 15
Iteration Number Iteration Number
—e— Subgrade Modulus + Subgrade Dan'ping

Figure 6.107 Convergence of layer parameters for case 2 (thickness backcalculation)

194

 

 

HII-I_actual

HIH actual

 

 

 

 

 

 

 

 

 

0 3 6 9 12 15 0 3 6 9 12 15
Iteration lumber Iteration Number

 

-—+— AC Thickness Base Thickness

Figure 6.108 Convergence of layer thickness for case 2 (thickness backcalculation)

195

1.6

 

 

 

 

 

 

 

 

 

 

 

1.4 «
_ 1.2 .
«II 0.8 .
0.6
0.2 .2--- _,__. -7— , .._ _.,_-_,_,_-
0 .
0 3 6 9 12 15
Iteration Mmber
ACModulus
12
1o . . _ _ _ , . __-_ .___-_--_____,
'3 8 -7 J- _
3
3' 6 , 2 _,__ _
5 4. - - _ __
2 . _ __ __ __
0
O 3 6 9 12 15
Iteration Number

+ Base Nbdulus

 

EIE_actual

 

 

 

 

Iteration Mmber

+ Subgrade Modulus

 

B/ﬂ _actual

ﬁle _actual

 

 

 

 

 

 

0 s .
0 3 6 9 12 15
Iteration Number
—e—AC Darrping
4.5
4 .- _ ._ ._ _-___ o
3 5 1 k ______

 

 

 

 

 

 

 

 

 

 

 

 

0 3 6 9 12 15
Iteration Mimber
+ Base Damping
10
9
8
7
6
5
4
3
2
1
o .
Iteration Number
+ Subgrade Danping

Figure 6.109 Convergence of layer parameters for case 3 (thickness backcalculation)

196

 

 

 

 

 

 

 

 

 

 

 

 

 

2.5
2 +—--—-— A ~ —— —» —7 —~ —— -2
To '3
31.5 . 3
0 O
“I 4 “I
1 < I
5 a
0.5 l
0
0 3 6 9 12 15
Iteration Number Iteration Number
—1— AC Thickness Base Thickness

 

Figure 6.1 10 Convergence of layer thickness for case 3 (thickness backcalculation)

6. 5.4 Effect of In correct Damping Ratio Specification on Backcalculation Results

The effect of incorrectly specifying the damping ratio of the base and subgrade on
backcalculated moduli and layer thicknesses was investigated for the three layer
pavement having the properties shown in proﬁle 1 of Table 6.22. The actual base and
subgrade damping ratio are 3% and 2%, respectively. Either the base damping or
subgrade damping was speciﬁed with the error ranging from -50% to 50%. Table 6.32
shows all nine combination of damping ratio and case 5 corresponds to when the true
value for the base and subgrade layers are used; i.e., reference case. The ratios of
backcalculated to actual values are shown in Figure 6.111 through Figure 6.115. The
results show that the effect of incorrectly specifying the subgrade damping ratio can lead
to relatively large errors in larger moduli and thicknesses, except for the subgrade layer.
The most predictions are for case 9, 6 and 3, respectively. These correspond to an
assumed subgrade damping ratio that is higher (by 50%) than the actual value. Case 2 and
8 correspond to using the correct subgrade damping ratio and an incorrect base damping

ratio. The errors in backcalculated parameters are much less signiﬁcant in these two

197

cases, suggesting that the effect of base damping ratio is not as important. Case 1, 4, and
7 correspond to an assigned subgrade damping ratio that is lower (by 50%) than the
actual value. The errors in backcalculated parameters are not as severe as those in those
in case 3, 6, and 9, suggesting that underestimating the damping ratio of the subgrade is
not as risky as overestimating it. For casel, while the error in backcalculated AC
damping ratio is high (about 70%), its practical signiﬁcant is low. Finally, the
backcalculated subgrade modulus is not affected by any of the incorrectly speciﬁed

damping ratio values.

Table 6.32 List of damping ratio combination for base and subgrade

 

 

18:25:11: “22:10 Base Subgrade
1 0.015 0.01
2 0.015 0.02
3 0.015 0.03
4 0.03 0.01
5 0.03 0.02
6 0.03 0.03
7 0.045 0.01
8 0.045 0.02
9 0.045 0.03

 

 

198

 

 

Ratio of Estimated/Exact

I 2 3 4 5 6 7 8
Combination of Damping Ratio
El AC Modulus

Figure 6.11 1 Error in backcalculated AC modulus due to different damping ratio
combination from base and subgrade

 

 

Ratio of Estimated/Exact

l 2 3 4 5 6 7 8 9
Combination of Damping Ratio
E3 AC Damping ratio

Figure 6.112 Error in backcalculated AC damping ratio due to different damping ratio
combination from base and subgrade

 

 

‘6

I

I

E

.3;

a

.E

‘I'a'

{:1

B-

O . ,

.2 ‘ ‘

g l 2 3 4 5 6 7 8 9
Combination of Damping Ratio

El AC Thickness

Figure 6.1 13 Error in backcalculated AC thickness due to different damping ratio
combination from base and subgrade

199

 

 

 

l 2 3 4 5 6 7 8 9

Combination of Damping Ratio

Ratio of Estimated/Exact

D Base Modulus

Figure 6.1 14 Error in backcalculated base modulus due to different damping ratio
combination from base and subgrade

 

....
{It

...
O

9
U1

 

Ratio of Estimated/Exact
O
'o

l 2 3 4 5 6 7 8 9
Combination of Damping Ratio

E3 Subgrade Modulus

Figure 6.1 15 Error in backcalculated subgrade modulus due to different damping ratio
combination from base and subgrade

6.6 Summary

In this chapter, time histories of FWD surface deﬂections generated theoretically were
used to verify the capabilities of the newly developed dynamic backcalculation program.
In general, it was found that the frequency response-based backcalculation method can
lead to large errors in deﬂection basins if the FWD records are truncated before the
motions fully decay in time. The errors due to sensor imprecision were found to be less
signiﬁcant.

200

Based on the analysis, the following conclusions were drawn for ﬁ'equency-domain

backcalculation:

The backcalculation results are all in excellent agreement with the true values. Both the
average root mean square error (RMS) on the calculated and actual deﬂection basins and
the relative errors on layer moduli and thicknesses are practically zero, indicating that the

program has the ability of backcalculate the moduli and thicknesses accurately.

Theoretical backcalculation shows that among the modulus, damping ratio, thickness and
Poisson’s ratio, the modulus is the easiest to backcalculate followed by damping ratio,

thickness and Poisson’s ratio.

Theoretical backcalculation shows that the frequency backcalculation program gives
satisfactory convergence of layer moduli and thicknesses when using untruncated
deﬂection time histories. However backcalculation results at higher frequencies are less

accurate than those obtained at low frequencies.

Although Poisson’s ratio of the AC layer is frequency-dependent, assuming a constant
value for it will not affect the results signiﬁcantly because the backcalculated results are

not sensitive to reasonable variations in this parameter.

The following conclusions were drawn for time-domain backcalculation:

Backcalculation based on synthetic time histories generated by SAPSI shows excellent
stability and accuracy for both methods (peak deﬂection and traces). However, since the
traces of time history uses more information than the peak deﬂection, it is recommended

that the backcalculation using traces of time history be used for ﬁeld application.

201

The time-domain approach can match selected features of the measured time histories
directly, while ignoring the inaccurate measurement regions in time. Therefore, from this
point of view, the time—domain backcalculation is better than the frequency-domain

backcalculation.

Numerical examples illustrate that the method is able to backcalculate layer moduli and
thicknesses accurately from synthetically generated FWD data for a three layer pavement
system. Backcalculation of layer damping ratios is less accurate, but the inﬂuence of this

error on the pavement response is insigniﬁcant.

202

 

CHAPTER 7 - FIELD VALIDATION OF DYNABACK

7.1 General

Measured deﬂection time history data from several FWD tests conducted in Michigan
and elsewhere were analyzed to evaluate the applicability of the DYNABACK to
interpret ﬁeld tests. The analyses include the comparison of backcalculated layer moduli
and damping ratios with MICHBACK results for various pavement sections and load
levels. The backcalculation was done in both frequency and time-domains, where the
time-domain backcalculation included backcalculation of thickness and modulus. The
data were obtained from tests involving KUAB and Dynatest FWDs. Most pavement
sections were analyzed as four-layer systems with some sections involving a stiff layer at

shallow depth.

7.2 Backcalculation of Layer Parameters for Selected Pavement Sections

The selected pavement test sections include sites in Texas, Cornell University, Florence
(Italy), Michigan and Kansas (LTPP study). The data were analyzed using DYNABACK
and MICHBACK. For the frequency-domain solution, the results shown in this chapter
are from single frequency backcalculation with the average values at the different
frequencies reported. For the Texas site, different load levels were considered. The
following describes the various test sites analyzed and the results of the backcalculation

analyses.

203

7.2.1 Michigan Sites

The selection of the pavement test sections was accomplished in consultation with
technical advisory group from the Michigan Department of Transportation (MDOT). The
main criterion used in the selection is that the pavement sections be representative of the
spectrum of pavement cross-sections, paving materials used in the state of Michigan and
that the cross-section information be available. While dynamic FWD test data were
available for three different projects, only the USl3l project had complete coring data

available. Therefore, only the results for this project are presented and discussed herein.

Layer thicknesses were obtained by coring, which was done at several locations. The
cores were obtained by using a power auger equipped with a 6-inch coring bit. FWD tests

were conducted at several locations along the road.

The data were sampled every 0.1 ms. and sampling time was 100 ms. The sensor

spacings are shown in Table 7 . 1.

Table 7.1 Sensor layout (distances are in inches) — Michigan data

 

D1 D2 D3 D4 D5 D6 D7
0 8 12 I8 24 36 60

 

A signiﬁcant feature in the FWD records is the truncation of the pulses at the end of the
100 ms sample times. Ideally, the pulses should die out or go to zero at the end of the
sample period. Instead, the deﬂection pulses usually cross the time axis and become
negative before they are truncated. An effort was made to get longer deﬂection records;
however, it appears that the accuracy of the sensor measurements after unloading is not

acceptable, as the free vibrations show unrealistic trends in time. Figure 7.1 shows an

204

 

example of an “acceptable” time history from the KUAB FWD system, and Figure

shows the ﬁltered time history. Tables 7.15 and 7.16 show the pavement proﬁles for two

control sections along US l 31

Table 7.2 Proﬁle used for USl3l site (section 50699)

 

 

Bore hole Subbase Thickness (in)
Number AC Thickness (in) Base Thickness (up
15 7.2 6 46.8
20 7.2 4.8 48
30 7.0 4.8 42

 

Table 7.3 Proﬁle used for USl3l site (section 67015)

 

 

 

 

 

 

 

 

 

 

 

Bore hole Subbase Thickness (in)
Number AC Thickness (in) Base Thickness (in)
13 6 9.6 44.4
200

.... 160

r:

2

o

E 120

c:

:9- so

8

E .0

0 .'
Time (ms)
Sensor1 — - — - Sensor2 ------- Sensor3 — - - - - Sensor4
— - - — - Sensors Sensor6 Sensor?

Figure 7.1 Time history from KUAB FWD

205

 

_—

200

 

 

 

 

 

 

 

E 150 -__.__ ___ _..__.I 1:: _-_..__ _- _ln.,-,u_-’_._.
. - x
g f .' _ -. x
5 120 - _"._2___- ’7'?” ~\—~ , __2 ~ww~v~v “2--.-
E . '9'! , I- 5 \}....
g 30 , -2 2%-- __ ’21:- 2“: 1 ___ﬁ * mm
3 , I r as. "—
It: . -,' A“
O f [I .\\.\.
D 40 um ' " _,_____ - 5‘33. ___—__--”..w,_~.__l
,.v q ,_ , {’41 \T‘E:\
'--...---.v:?’. ‘H‘dﬁg,e .
0 ,y Lord-5:5?» . 2M..- -- - _, -_
0 20 40 60 80 100
Time (ms)
Sensor1 — — — - Sensor2 ------- Sensor3 — - — - - Sensor4
— - - —- - Sensors Sensor6 Sensor?

 

 

Figure 7.2 Filtered time history from KUAB FWD

7.2. I. 1 Comparison of Dynamic and Static Backcalculation for Four layer System

Table 7.4 summarizes the backcalculation results when the proﬁles are characterized by a
4—layer system and layer thicknesses are known. The backcalculated moduli are generally
high for all stations; however the time-domain backcalculation results seem to give
unreasonable results. Note that the backcalculated damping ratios for both dynamic
solutions are not reasonable. The static and dynamic backcalculation solutions give
similar values for the AC, subbase and subgrade moduli, but different values for the base

layer modulus.

206

Table 7.4 Comparison of frequency and time-domain backcalculation results with those
from MICHBACK — USl3l site

 

 

 

 

 

 

 

 

F requency-domain Time-domain Static
Test Site Layer Dynamic Dynamic Backcalculation B k lculation
Backcalculation ac ca
Modulus Damping Modulus Damping Modulus

iksi) (ksii iksi)

AC 1953 0.08 2231 .04 2257
Base 272 0.12 42 .16 99
15 Subbase 34 0.18 32 .26 36
Subgrade 39 0.07 60 .05 38

AC 1146 0.01 981 .08 1065
Base 150 0.12 42 .05 75
20 Subbase 46 0.15 31 .28 43
Subgrade 18 0.15 27 .02 28

AC 697 0.13 838 .12 834
Base 228 0.02 57 .ll 89
30 Subbase 27 0.16 27 .31 27
Subgrade 39 0.08 48 .01 45

AC 1334 0.13 1444 0.14 1443
13 Base 60 0.11 18 0.21 22
Subbase 53 0. 12 69 0.08 63
Subgrade 29 0.18 28 0.02 32

 

Figure 7.3 through Figure 7.6 show the measured and predicted deﬂection basins at low,

intermediate and high frequencies for the different sites. The comparisons are fair to

poor, with the measured deﬂection basins showing irregular patterns for some test

sensors and at certain ﬁequencies. Figure 7.7 through Figure 7.10 show the measured and

predicted peak deﬂections and time lags. The agreement is fair for peak deﬂections and

poor for time lags, with the measured time lag curves showing irregular patterns. These

irregularities could be caused by errors in sensor locations and/or synchronization

problems in the time readings from different sensors.

207

 

 

 

 

 

 

 

 

 

   

E .15

E E.

‘5 c

g 8

0 0

g 0

= 5

.3 u

.5 a

a E

0 .
0 20 40 60
Distance from center of load(In) Distance from center of load(In)
+ Measured & Calculated + Measured 6 Calculated
(a) Real and imaginary deﬂection basins at 2.44 Hz

g Q 0.14 -

E. E 0.12

= S

3 =1 0.1

g g 0.08

5 o 0.06

U '8

a 5', 0.04

a? g 0.02

.. 0 .
Distance from center of load(In) Distance from center of load(in)
+ Measured & Calculated + Measured er Calculated

(b) Real and imaginary deﬂection basins at 14.65 Hz

 

 

 

   

E .11"
s s—
c

g 8
° 8
3 1:
O O
15 ".
i o:
a E

0 20 4O 60
Distance from center of load(in) Distance from center of load(in)
-o- Measured .3 Calculated 4* Measured «8- Calculated

(c) Real and imaginary deﬂection basins at 24.41 Hz

Figure 7.3 Comparison of measured and predicted deﬂection basins for USl31 (50699-
15)

208

 

   

 

 

 

 

 

 

 

 

 

 

 

 

.. 0.7 ~ A 0
g 0.6 g .0.1
F: o 5 5
g - 3 -o.2 -
g 0.4 E -0 3 ‘
8 0.3 115:: '
a -0.5
g 0.1 E
0 ﬂ r a -O.6 .
O 20 40 60 O 20 40 60
Distance from center of load(in) Diemnce from center of load(in)
4* Measured 4% Calculated + Measured «B Calculated
(a) Real and imaginary deﬂection basins at 2.44 Hz
3 *~ .2
“g -0.05 1 ‘5'
= -0.1
g 8
g -o.15 - g
E e
.5 —0.2 °_
7' -o.25 — °
6? . , g
'0.3 o i ,
O 0 20 40 60
Distance from center of load(In) Distance from center of load(In)
+ Measured +3~ Calculated + Measured 4} Calculated

(b) Real and imaginary deﬂection basins at 14.65 Hz

 

 

 

 

 

 

 

 

 

 

... A 005 _m-_i-....,w~......._...,._..._.2..-..2..-.........._._--_.2.........- ,.

g a 0.04 - *

E. g. 0.03 z

.5 g 0.02 «

o 8 0.01 )

° 6:

s . 0 ~ -

'0 '0

g 6 -0.01 “

m g -o.o2 « -» .
-0.03 . a a

0 20 40 60
Distance from center of load(in) Distance from center of load(In)
+Measured eCalculated +Measured ECalculated

(c) Real and imaginary deﬂection basins at 24.41 Hz

Figure 7.4 Comparison of measured and predicted deﬂection basins for US131 (50699-
20)

209

 

 

Real deflection(mils) Real deflection(mlls)

Real deflection(mlle)

 

 

O

   
 

 

 

E
g -0.1 ; 7
g -o.2 (7 77 7 7
8 ~03 L7 7 7 7 —
‘5 l
'u -O.4 ‘ rrrrrrrrr , 7 7 7
€05 77 77 7 7
‘ -O.6 l

0 20 40 60

Distance from center of load(in) Distance from center of load(In)
+Measured ‘73- Calculated +Measured sea-Calculated

(a) Real and imaginary deﬂection basins at 2.44 Hz

 

 

 

-o.05 3 .

-o.1 .2,
-O.15 '9

-o.2 8 .
-o.25 E

-0.3 5',
-o.35 g -

_0-4 “ A *7 v "—1——‘. _

O 20 40 60
Distance from center of load(in) Distance from center of load(in)
+ Measured -9 Calculated -o- Measured G Calculated

(b) Real and imaginary deﬂection basins at 14.65 Hz

 

 

 

A 008 ‘,_.2. ... 222.2. . .2..22. .2222“ 2..- 2.. .222.....
g .
15.
E
o
a
0
0
‘5
. ‘5
' ‘ 5"
0 _
O 20 40 60
Distance from center of load(in) Distance from center of load(in)
-o- Measured 8* Calculated + Measured fr Calculated

(c) Real and imaginary deﬂection basins at 24.41 Hz

Figure 7.5 Comparison of measured and predicted deﬂection basins for US131
(50699-30)

210

 

 

Real deflection(mlls) Real deﬂection(mlle)

Real deﬂection(mile)

 

99.09.0999
O-‘wawmﬂm

 

.6
-lO

-0.2

.b
on

.6 .
a:

' A
I v
\

   

.6
m

 

Imag. deflection(mile)
b
A

.b
V

 

 

Distance from center of load(in) Distance from center of load(in)

4» Measured 8» Calculated + Measured & Calculated

(a) Real and imaginary deﬂection basins at 2.44 Hz

 

 

 

 

E
E.
C
o
a
3
o
1:
E"
_ 0 .
O 20 40 60
Distance from center of load(in) Distance from center of load(in)
+ Measured «9 Calculated + Measured 1% Calculated

(b) Real and imaginary deﬂection basins at 14.65 Hz

 

 

 

Imag. deﬂection(mlls)

 

 

 

 

 

Distance from center of load(ln) Distance from center of load(In)

+ Measured *3 Calculated 4* Measured «9 Calculated

(c) Real and imaginary deﬂection basins at 24.41 Hz

Figure 7.6 Comparison of measured and predicted deﬂection basins for US 131
(50157-13)

211

 

 

Time IagIms)

Deflection(mlls)
O -b N (a) ab 01 O) \I oo (.0

 

 

 

 

 

 

O 20 40 60
Distance from center of load(in) Distance from center of load(ln)
+ Measured ~3~ Calculated + Measured & Calculated

Figure 7.7 Comparison of measured and calculated peak deﬂections and time lags for
USl3l (50699-15)

 

 

 

   

3 ° 2' .2
g 8
c ‘5
:3 a
e i
8
Distance from center of load(in) Distance from center of load(In)
+ Measured 6* Calculated 4» Measured & Calculated

Figure 7.8 Comparison of measured and calculated peak deﬂections and time lags for
USIBl (50699-20)

 

 

 

 

 

   

12 1. . ,, 22
g .2
.6. E.
.5 a
a 6
8
Distance from center of load(in) Distance from center of load(in)
+ Measured 7:7 Calculated + Measured 45» Calculated

Figure 7.9 Comparison of measured and calculated peak deﬂections and time lags for
US131 (50699-30)

212

 

 

—L_&

 

 

 

 

 

 

2
2* °‘ 7
.6. 8 .. E;
I: a
a 4 g
s 2 ..

0 T 2

0 20 40 60
Distance from center of load(in) Distance from center of load(In)
+ Measured a Calculated + Measured -& Calculated

Figure 7.10 Comparison of measured and calculated peak deﬂections and time lags for
US131 (50157-13)

7.2.1.2 Dynamic T ime-domain Backcalculation Using Peak Deﬂections for Three

layer System

In this analysis, the base and subbase layers were combined, and the program was
allowed to backcalculate layer thicknesses. Table 7.5 shows the backcalculation results
from time-domain analysis. The error in the backcalculated AC thickness varies from -
17% to 39%. For the combined base and subbase layers, the errors in the backcalculated
thickness are very large, with the value reaching the boundary in 3 out of 4 cases. The
effect of thickness backcalculation on layer moduli is more pronounced for the AC layer,

and the backcalculated damping ratios are erratic.

Figure 7.11 through Figure 7.18 show the surface deﬂection and time lag in different
sites. The ﬁgures show fair to poor agreement between measured and predicted response,
with the measured time lags showing irregular patterns. These irregularities could be
caused by errors in sensor locations and/or synchronization problems in the time readings

from different sensors.

213

 

l-m-Il-m_lj I]?

 

Table 7.5 Backcalculation results from time-domain analysis — USl31 site

 

 

 

 

 

 

 

   

 

 

. Without thickness With Thickness Backcalculation
Test Slte Layer .
Backcalculatlon
ﬁdulus Damping 33331111“ Damping Thickness (in.)
AC 2350 0.05 1473 0.01 6.7
15 Base 42 0.27 74 0.30 25.0
Subgrade 37 0.01 41 0.05 ---
AC 1038 0.07 1424 0.01 4.3
20 Base 43 0.24 73 0.30 20.5
Subgrade 28 0.01 27 0.12 ---
AC 905 0.08 958 0.03 6.3
30 Base 31 0.30 61 0.37 10.0
Subgrade 47 0.01 36 0.13 «-
AC 726 0.03 742 0.33 10.0
13 Base 47 0.18 40 0.07 35.8
Subgrade 31 0.01 31 0.12 ---
8
E 7
:2 2%.
5 4 3
§a g
E 2 i-'-'
1
0 , 0
0 20 40 60 20 40 60
Distance from center of load(in) Distance from center of load(in)
4» Measured «7? Calculated + Measured 45* Calculated

Figure 7.1 1 Comparison of measured and calculated peak deﬂections and time lags for
US l 31 (50699-15) with thickness backcalculation

214

 

 

_L
O

 

Deflection(mils)
o N sh O) a:

 

 

 

 

 

Distance from center of load(in) Distance from center of load(in)
+ Measured *8 Calculated 4» Measured 9 Calculated

Figure 7.12 Comparison of measured and calculated peak deﬂections and time lags for
US l 31 (50699-20) with thickness backcalculation

 

 

 

 

 

 

 

 

 

 

g ‘10 ‘ A

g 87 g

C

g 6 1 3

a 4 E

a 2 .-

o , .
0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+Measured & Calculated +Measured emlwlatw

Figure 7.13 Comparison of measured and calculated peak deﬂections and time lags for
US 1 31 (50699-30) with thickness backcalculation

 

 

 

 

 

 

 

 

 

 

 

 

 

14 .
:1
3 12 .
:5. 10 - g
o 6 1 E
r» =
2 -. 2 ~~ — ~—— A
0 . 0
0 20 4O 60 0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+ Measured er Calculated + Measured a Calculated

Figure 7.14 Comparison of measured and calculated peak deﬂections and time lags for
US 131 (50699-13) with thickness backcalculation

215

 

 

w: .

 

 

 

 

   

 

 

 

 

 

8
A 7
I
= 6 "‘
.E. 5. E
s 1 ‘5'
a; g
0 .
§2~ :

1 -

0 1 . 0 .

0 20 40 60 0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+ Measured 8* Calculated 4» Measured *3 Calculated

Figure 7.15 Comparison of measured and calculated peak deﬂections and time lags for
US 131 (50699—15) without thickness backcalculation

 

 

 

 

DeﬂectionImlIs)

 

 

 

  

 

 

40 60

Distance from center of load(in) Distance from center of load(In)
4» Measured +3 Calculated + Measured 5% Calculated

Figure 7.16 Comparison of measured and calculated peak deﬂections and time lags for
US131 (50699-20) without thickness backcalculation

 

 

 

 

 

 

 

 

 

 

 

12 2. — 2
E 10 1* A
.5. 8 , g
r: 1:»
=3 6 . e
.2. 4 E
a 2 _

0 z . . . .

0 20 40 60 0 20 40 60
Distance from center of load(in) Distance from center of load(in)
4» Measured a Calculated + Measured a Calculated

Figure 7.17 Comparison of measured and calculated peak deﬂections and time lags for
US 131 (50699-30) without thickness backcalculation

216

Time Iag(me)

 

Deflection(mils)

 

 

 

 

 

0 20 40 60

Distance from center of load(in) Distance from center of load(ln)
4» Measured 6— Calculated + Measured *3 Calculated

Figure 7.18 Comparison of measured and calculated peak deﬂections and time lags for
US l 31 (50699-13) without thickness backcalculation

7.2.1.3 Dynamic T ime-domain Backcalculation using Traces of Time History

In chapter 4, the need for truncating the singular value that are smaller than the critical
allowable error in order to insure convergence was discussed in the context of the SVD

method. Recall the equation (4. 16)
m}- < wmax 40‘“ (4.16)

If a singular value is smaller than them max . 10 ’a , the rank of the gradient matrix is
reduced, then reducing the ill-conditioning problem and insure converge. An appropriate
threshold value for or needs to be selected carefully. In this section, backcalculation
analysis is conducted for various sites using different or-value to investigate the

COUVCTgCI'ICC process.

Figure 7.19 shows that the iteration number until convergence is a function of or in
US131 ﬁeld data. The results indicate that the threshold value for or of 3 or 4 is

appropriate for dynamic back calculation. Using an or value greater than 4 will lead to no

217

“I

 

 

convergence, while an or value less than 3 lead to inaccuracy in the solution to the inverse

problem.

50

 

 

Iteration Number

 

 

 

Threshold number

+ Hysteretic damping +Viscous damping

Figure 7.19 Iteration number to convergence versus or for USl3l site

In this part, the same proﬁle listed in Table 7.2 is used for the four-layer system in US131
site (section 50699_30). The backcalculation result from three different models including
DYNABACK with hysteretic and viscous damping models and an exiting FEM method,
which was developed by Matsui (1998) are shown in Figure 7.20 through Figure 7.26. It
can be seen that the match from DYNBACK is slightly between that from the FEM
model. Figure 7.27 shows the comparison of backcalculated layer moduli from the three
models. The relative difference in layer moduli between hysteretic damping model and
the viscous damping model is 16% for AC, 1% for base, -15% for subbase, and 13% for
subgrade. The difference between the results from the FEM model and DYNABACK are
large, especially for the base layer modulus (92%). This may be because of the difference

of assumptions in the forward models.

218 ’

11.0

 

 

 

 

 

 

 

 

’2‘
E 10.0
.§ 9.0
3:3 8.01
8
7.0 1 7 . . . .
30.8 32.8 34.8 36.8 38.8 40.8 42.8 44.8
Trme (ms)
— D1-measured —-— D1 ﬁsteretic damping
—+—D1—viscous damping +D1- M

Figure 7.20 Comparison of measured and predicted deﬂection time histories for sensor 1

9.0

 

 

8.0 2.

7.0 ~

 

Deflection (mils)

 

 

6.0 . T . .
30.8 32.8 34.8 36.8 38.8 40.8 42.8 44.8

Time (ms)

—— DZ-measured —-— DZ—Igésteretic damping
——.— D2-viscous damping -e-— 02- M

 

Figure 7.21 Comparison of measured and predicted deﬂection time histories for sensor 2

 

 

 

 

 

 

 

A 8.0 ~
g
E 7.0
C
.9
A: 6.0
8 5.0 . . . , . .
30.8 32.8 34.8 36.8 38.8 40.8 42.8 44.8
Time (ms)
— D3-measured —-— D3-hysteretic damping

—-o— D3—vlscous damping + D3-FEM

Figure 7.22 Comparison of measured and predicted deﬂection time histories for sensor 3

219

 

Deflection (mils)

 

 

 

 

30.8 32.8 34.8 36.8 38.8 40.8 42.8 44.8

Time (ms)
— D4-measured —-— D4-hysteretic damping
—.— D4-viscous damping + D4-FEM

Figure 7.23 Comparison of measured and predicted deﬂection time histories for sensor 4

 

 

 

 

6.0
E
E 5.0
_S 4.0
+82; 3.0
8
2.0
30.8 32.8 34.8 36.8 38.8 40.8 42.8 44.8
Trme (ms)
— D5-measured 77- DS—hysteretic damping

+ DS-viscous damping + DS-FEM

Figure 7.24 Comparison of measured and predicted deﬂection time histories for sensor 5

3.5
3.0
2.57
2.0
1.5 777777 7
1.0
30.8 32.8 34.8 36.8 38.8 40.8 42.8 44.8
Trme (ms)

— DG-measured —-— DG-hysteretic damping
—e— DG-viscous damping + D6-FEM

 

 

 

Deflection (mils)

 

 

Figure 7.25 Comparison of measured and predicted deﬂection time histories for sensor 6

220

N
o

 

 

 

 

 

 

iv?

E15

:9: 1.07

.9 0.5

8

0.0 . . . ‘
30.8 32.8 34.8 36.8 38.8 40.8 42.8 44.8
Time (ms)
—- D7-measured —-— 07-hysteretic damping

—.— D7-viscous damping + D7-FEM

Figure 7.26 Comparison of measured and predicted deﬂection time histories for sensor7

Us 1 31__50699_30

 

1.00E+06

1.00805 ——

Modulus (psD

1.00E+04 i

 

1.005103

AC Base Subbase Stiff layer
Layer

DYNAMIC-hysteretic damping I DYNABACK-viscous damping I FEM I MICHBACK

Figure 7.27 Comparison of backcalculated layer moduli from different models

The viscous damping model did not necessarily improve the backcalculation results in the
backcalculation of the ﬁeld FWD data; therefore the hysteretic damping is employed in
the uniqueness analysis. The three different values of the seed moduli used in the

program are provided in Table 7.6. The backcalculated results are shown in Figure 7.28.

221

The results show that the backcalculation program can yield excellent uniqueness results
for different layers.

Table 7.6 Seed values used for backcalculation of Michigan data

 

 

 

 

 

 

 

 

 

 

Layer Case 1 Case 2 Case 3
Seed Seed S d Seed S d
Layer Modulus Dam in Modulus Darfrein Modulus D :16.
(ksi) p g (ksi) P g (ksi) a ““8
AC 450 0.1 300 0.2 680 0.2
Base 70 0.08 45 0.05 105 0.05
Subbase 70 0.05 40 0.04 85 0.05
Subgrade 70 0.03 30 0.03 75 0.05
US131_50669_30 (hysteretic damping)
1.00E+06
T7: 1.00E+O5
EL-
(I)
2
8
2 1.00E+04 .
1.00E+03 2 . I
Base Subbase Subgrade

 

llCasei lCase2ClCase31

Figure 7.28 Comparison of backcalculated layer moduli using different seed values

Thickness backcalculation is considered next using the hysteretic damping model, since
the viscous damping model did not necessarily improve the backcalculation results, and
the interpretation of damping is easier. In general, simultaneous backcalculation of layer
moduli and thicknesses is diﬁicult. Therefore only the AC layer thickness is

backcalculated along with AC, base, and subgrade layer moduli and AC damping ratio.

222

Damping ratio for base and subgrade are ﬁxed in the backcalculation analysis. The
various combinations of damping ratios for base and subgrade layer are listed in Table
7.7. The backcalculated results as shown in Figure 7.29 through Figure 7.33. The results
indicate that the combination of damping ratio only signiﬁcantly affect the AC damping
ratio, which decreases with the combination number, (increasing damping ratios of base
and subgrade layers). This can be expected because of compensation in the damping ratio
of different layers. The AC thickness can also be affected varying by 13% to 21% from
the true value. The effect of assuming different base and subgrade damping ratio on the
backcalculated layer moduli is negligible. The implication of this ﬁnding is that, the
backcalculated results deduced using the wrong damping ratios for base and subgrade

layer should be reasonable even for ﬁeld evaluation.

Table 7.7 List of the combination of damping ratio for base and subgrade

 

 

Damping

Ratio Base Subgrade
Combination

l 0.05 0.03
2 0.05 0.07
3 0.05 0.10
4 0.10 0.03
5 0.10 0.07
6 0.10 0.10
7 0.15 0.03
8 0.15 0.07
9 0.15 0.10

 

 

223

 

USI31_50699__30 El AC_Modqus

 

 

 

1.00E+07
5;" MICHBACK result
3
V)
g 1.00E+06 . * , 2, *\ ,, 2 222 72, r * _,2 -—l‘
U t on
o . .
2
1.00E+05

 

 

Combination of Danping Ratio

Figure 7.29 Comparison of backcalculated AC modulus for different damping ratio
combinations for base and subgrade layers

 

USI31_50699_30 IAC_dan'ping

 

 

 

Damping Ratic

 

Corrbination ofDarrping Ratio

Figure 7.30 Comparison of backcalculated AC damping ratio for different damping ratio
combinations for base and subgrade layers

224

US l 3 1_50699_30 E AC_Thickness

 

Thickness (inch)

 

l 2 3 4 5 6 7 8 9
Combination of Damping Ratio

Figure 7.31 Comparison of backcalculated AC thickness for different damping ratio
combinations for base and subgrade layers

 

0513 1_50699_30 Base_Modu1us
1.00E+05 r MlCHBACKresult
51‘ 1:3.-
234?
a 82:1
_; 1.00E+04 ,
1.00E+03 *

 

 

l 2 3 4 5 6 7 8 9
Combination of Darrping Ratio

Figure 7.32 Comparison of backcalculated base modulus for different damping ratio
combinations for base and subgrade layers

225

US131_50699_30 El Subgrade_Modulus

1.00E+OS MICHBACK result

1.00E+04

Modulus (psi)

1
l
-1
1
l
l

 

 

1.00E+03 7 ' '

Combination of Danping Ratio

Figure 7 .33 Comparison of backcalculated subgarde modulus for different damping ratio
combinations for base and subgrade layers

7.2.2 Texas Site

This site is located in Texas near J acksboro on State Highway 281. The pavement section
was tested using a Dynatest FWD. The data for this site were provided by the
Transportation Research Board (TRB) A2B05 committee. Surface deﬂections for various

load levels (6000, 9000, 12000 and 16000 pounds) were measured and vertical
displacements were recorded simultaneously with a Multi-Depth Deﬂectometer (MDD)
at three depths (3.7, 12.4 and 23.4 in). The pavement’s surface deﬂections were recorded
at six points, 12 inches apart, starting at the center of the FWD load plate. The applied
loads and deﬂections were recorded at 0.2 ms interval for 60 ms. The MDDs have
anChOI‘S deep in the subgrade or bedrock, and the movement of these anchors during
typical FWD-MDD tests has been recorded. The bedrock depth is approximately 6.2 11
below the surface. Figure 7.34 shows the test setup and pavement proﬁle. The pavement
is made up of an 8 in. asphalt concrete surface and a 12 in. ﬂexible base layer on top of

the sub grade.
226

FWD

203
\ 1' I? W7- W3 W4 W5 W6

1770—. F1 Fl ['1 Fl 1'1

 

 

top I ‘ Thickness=203 mm
middle —> £
BASE

Thickness=3 80 mm

 

bottom

SUBGRADE

 

 

Figure 7.34 Pavement proﬁle and test setup for Texas site

Figure shows the peak deﬂections versus peak load, normalized to the lowest load level
values for the six sensors. The curves show higher than 1:1 ratios, indicating that the
pavement system exhibits some nonlinear behavior. The nonlinearity is lowest for the

ﬁrst sensor, and generally increases for the farther sensors.

227

 

2.8 2 2- 2222222.

 

 

 

 

   

 

1—°——sensor1' ‘1 \

2.6 -. 21---o~--sensor2 2 1 “##1#wa __
l--a---sensor3 1 1/ I'fﬁl’o

2.4 -~i--x--sensor42_2l2 2.2. ’ 3,37“;
:-—+——sensor5 .1 1 ’

 

2.2 *1 — "— -sensor 6 f—"J m"
221

 

 

 

 

 

 

 

 

 

 

 

s -—~-——1—4 ‘
...l 1 1
82‘—“T‘— 1
a 1 1
g 1.8 “*~- -—~P-~- t—A~-—— —
E T f 1
1'6 1 I’ / V _1‘" 1 '_T
O. I |
I [ 1 1
1 4 L I; .2222 2222 2222_2_.22__222
.l/ ‘ 1
I ' . 1
l 2 42“ I 1 1 “
1 1
1 . T 7. 1
1 1.5 2 2.5 3 3.5
Normalized Deﬂection

Figure 7.35 Normalized deﬂection versus FWD load

Figure 7.36 through Figure 7.39 show the time histories of the FWD load and measured

sensor deﬂections for the four different load levels.

228

 

7000
6000
5000
4000 5
3000 u
2000

1 000

0

-1 000

Deflection (mile)
0 A N on A 01 a: N on (D

 

“.40.-“ 20 w... .._..,30_._40__

I
A

Time (ms)

Sensor 1 — -- - Sensor 2 ------- Sensor 3 — - - — . Sensor4
Sensor 5 Sensor 6 ——Load

 

 

 

Figure 7.36 FWD load and deﬂection time histories (load level 1 —- 6000 lb) -— Texas site

 

_I_l
WU"

 

 

A
A

 

 

Deﬂection (mils)

 

(DOING)

 

 

—l
I

 

 

 

I
A

Time (ms)
Sensor 1 — —-— - Sensor 2 ------- Sensor 3 — - - — - Sensor4

 

Sensor 5 Sensor 6 -—Load

 

 

Figure 7.37 FWD load and deﬂection time histories (load level 2 — 9000 lb) — Texas site

229

 

 

 

 

 

 

 

24 14000
«» 12000
19 r
... «- 10000
g
E 14 ‘* 300° 3
C a
8 6000 1::
o 9 J g
a r- 4000
Q
a 4 2000
4 4
r 0
-1 ‘ -2000
Time (ms)
Sensor 1 - .. - Sensor 2 ------- Sensor 3 — - - - - Sensor4

 

Sensor 5 Sensor 6 —Load

 

 

Figure 7.38 FWD load and deﬂection time histories (load level 3 — 12000 lb) — Texas site

 

 

 

 

 

 

 

 

 

29 16000
«~ 14000
24 _.W a
<~ 12000
75
= 19~ -—» 10000
‘E' 3
S «L 8000 :5
14 7 1:
a: 9 ~ «~ 4000
o
-- 2000
4 .
L 0
-1 ”i“ -2000
60
T1me(ms)
Sensor1 —-a—-Sensor2 ------- Sensor3 —--—-Sensor4

 

Sensor 5 Sensor 6 —Load

 

 

Figure 7.39 FWD load and deﬂection time histories (load level 4 — 15000 lb) — Texas site

The ﬂuctuations in the free vibration response confirm the presence of a stiff layer at
shallow depth, which traps the energy from the FWD load within the pavement system,
thuS causing the propagating waves to reﬂect back and forth. The fact that the response of

the ﬁrst and second sensors exhibit less vibrations, with the ﬁrst sensor deﬂection

230

remaining positive even after the load reaches zero indicates high damping in the
pavement system. This can be attributed to nonlinear material behavior in some of the
pavement layers. The combination of stiff layer and material nonlinearity makes this site

particularly challenging for backcalculation.

Table 7.8 and Table 7.9 show the proﬁle used in the backcalculation exercise and the

sensor layout, respectively.

Table 7.8 Proﬁle used for Texas site

 

 

Layer
Name Thickness (in) Unit Weight (pct) Poisson Ratio
AC 8 145 0.35
Base 12 135 0.40
Subgrade 55 120 0.45
Stiff layer 00 145 0.25

 

Table 7.9 Sensor layout (distances are in inches)

 

D1 D2 D3 D4 D5 D6
0 12 24 36 48 60

 

72 2. 2. I Comparison of Dynamic and Static Backcalculation for F our-layer System

Normal practice for static analysis is to take the peaks of each of the deﬂection pulses
from different sensors and form a deﬂection basin. The static force is taken as the peak of
the corresponding force pulse. MICHBACK was used for the static backcalculation in

order to investigate the difference between dynamic and static backcalculation result

Table 7.10 shows the results from frequency-domain, time-domain and static
baCkcalculations. The backcalculated moduli from time and frequency-domain analyses
agree with those from static analysis except for the stiff layer. However, the

backcal culated damping from both dynamic solution are not consistent.

231

Table 7.10 Comparison of frequency and time-domain backcalculation results with those
from MICHBACK — Texas site

 

 

 

 

 

Load Frequency—domain . . . Static
Level Layer Backcalculation Time-domain Backcalculation Backcalc
ulatron
, Modulus
Modulus (ksi) Damping Modulus (ksi) Damping (ksi)
AC 208 0.23 195 0.12 191
Base 23 O. 15 22 0.06 29
6000 lb Subbase 29 0.08 26 0.1 l 22
Stiff Layer 119 0.04 97 0.01 66
AC 216 0.20 197 0.02 214
Base 20 0.16 20 0.337 21
9000 lb Subbase 28 0.06 26 0.03 23
Stiff Layer 95 0.02 87 0.01 51
AC 212 0.22 163 0.522 203
Base 19 0.17 27 0.01 22
12000
1b Subbase 26 0.07 20 0. 10 20
Stiff Layer 98 0.01 160 0.01 52
AC 228 0.25 167 0.55 214
Base 18 0.17 27 0.04 20
16000
lb Subbase 26 0.07 19 0.01 19
Stiff Layer 125 0.01 202 0.01 48

 

Figure 7.40 through Figure 7.43 show the match between measured and predicted
deﬂection basins from frequency backcalculation. The match between measured and
predicted deﬂection basins is better at low and high frequencies for real deﬂections. For

intermediate frequencies, the match is better for imaginary deﬂections.

232

Real deﬂection(mlle)

Real deﬂection(mlls)

Real deflection(mlls)

 

 

 

a

 

 

Imag. deflection(mlls)

 

 

 

 

20 40
Distance from center of load(in)
4* Measured fr Calculated

60
Distance from center of load(in)

+ Measured 8 Calculated

(a) Real and imaginary deﬂection basins at 2.44 Hz

 

 

 

 

 

 

 

 

 

 

 

 

 

A o _..

g -0.05

E -0.1

a .015

g -0.2

o -0.25

‘6

5, .03 ~~---/

g -035 ,/

" -0.4 . . .
0 20 40 60

 

Distance from center of load(ln)
+ Measured 6 Calculated

Distance from center of load(in)
+ Measured 4} Calculated

(b) Real and imaginary deﬂection basins at 9.77 Hz

'2

 

 

 

-0.06 Je— ~

 

 

 

 

 

 

 

 

Imag. deﬂection(mlls)

 

 

 

 

-0.16 . r
20 40
Distance from center of load(in)

+ Measured 4+ Calculated

 

20
Distance from center of load(ln)
+ Measured & Calculated

60

(c) Real and imaginary deﬂection basins at 26.86 Hz

Figure 7.40 Comparison of measured and predicted deﬂection basins for load level 1 —

Texas site

233

.0
N
o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

z E
g 0'6 t a 0.05 «»
g 0.5 ‘5'
§ 0.4 g -0.1
" a
3 0.2 a. .02
n . e...
E 0.1 1» ~ — E 4
0 -0.25 .
0 20 40 60 0 20 40 60
Dletance from center of load(in) Distance from center of load(in)
+ Measured 43» Calculated + Measured 9- Calculated
(a) Real and imaginary deﬂection basins at 2.44 Hz
A 0.08 A
.2 0.07 g
a e 4’
‘5 0'06 a
3 0.05 g 4’
é? 0.04 3 .0
. 0.03 ‘5
.3 1: -0
T. 0.02 a,
g 0.01 - g '0-5
0 ' '06 T r 1
0 20 40 60 0 20 40 60
Distance from center of load(in) Distance from center of load(ln)
+ Measured «3 Calculated + Measured «8 Calculated

(b) Real and imaginary deﬂection basins at 9.77 Hz

 

 

 

 

 

 

Real deflection(mlls)
Imag. deflection(mlls)

 

 

 

   

Distance from center of load(ln) Distance from center of load(ln)
+ Measured 6 Calculated + Measured e Calculated

(c) Real and imaginary deﬂection basins at 26.86 Hz

Figure 7.41 Comparison of measured and predicted deﬂection basins for load level 2 —
Texas site

234

 

 

 

 

 

   

 

 

 

 

.. 0.9 A 0 -.
:3 08 t g 005
g 0.7 .5. 431
g 0.6 . 5
z: 05 '3 -O.15
% 0‘4 . -0.2 i
3 3'3 * ° 0.25
g 0:1 . g 03%
0 T “a " 035 r a
0 20 40 60 0 20 40 60
Distance from center of load(ln) Distance from center of load(ln)
+Measured 5» Calculated +Measured 9 Calculated

(a) Real and imaginary deﬂection basins at 2.44 Hz

.0
N

.o
_L
U!

 

 

0.05

Real deﬂecﬂon(mlls)
O

 

 

 

 

  

40 60

Dlstance from center of load(ln) Distance from center of load(ln)
+ Measured 6* Calculated + Measured 6 Calculated

(b) Real and imaginary deﬂection basins at 9.77 Hz

 

 

 

 

 

 

 

 

 

 

g a 0.01 - -
.2. g. 0 - ..
g g -0.01 4
g :2? 0.02 «
'5 .3 —0.03 w
2 a, —0.04 _i ‘-
0 -0 05 1 ,
n: g -
-0.06 . ‘
0 20 40 60
Distance from center of load(ln) Distance from center of load(ln)
+Measured eCalculated +Measured eCalculated

(0) Real and imaginary deﬂection basins at 26.86 Hz

Figure 7 .42 Comparison of measured and predicted deﬂection basins for load level 3 —
Texas site

235

 

 

 

 

 

 

 

 

 

 

g E 0.05
755 g. 0.1 1L ——
g g 015 -
g 5 -0.2
'5 ,3 0.25
'0
.5 a, .03
£5 E —0.35 L
-0.4 . . '
0 20 40 60
Distance from center of load(in) Distance from center of load(ln)
+ Measured 9 Calculated 4» Measured 9 Calculated

(a) Real and imaginary deﬂection basins at 2.44 Hz

 

 

 

 

 

 

 

Real deﬂection(mlls)

 

 

 

Distance from center of load(in) Distance from center of load(ln)
+ Measured 8» Calculated 4» Measured 6 Calculated

(b) Real and imaginary deﬂection basins at 9.77 Hz

 

 

 

 

 

 

E .l.’
s s 0 ~
= 5
c :3 -0.05
1'; s
c ‘3 -0.1
.3 'u
T; E” -0.15 'l
m -

-0.2

0 20 40 60
Distance from center of load(ln) Distance from center of load(ln)
+ Measured «3 Calculated 4» Measured «9 Calculated

(c) Real and imaginary deﬂection basins at 26.86 Hz

Figure 7.43 Comparison of measured and predicted deﬂection basins for load level 4 —
Texas site

236

 

Figure 7.44 through Figure 7.47 show matched deﬂection from time-domain
backcalculation. The match for peak deﬂections is better than that for time lags. This
could be due to errors in sensor locations or in time synchronization of the data

acquisition system.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 . 9 1 ---------- ~
... 8 8 «
2 7 .. 7 ~~ ,
e6 E64 i
g 5 l g 5 ‘
g 4 "' 4 r
g 3 E 3
52~ ‘2 —a

1 « 1 +—

0 v 0 T a

0 20 40 60 0 20 40 60
Distance from center of load(in) Distance from center of load(ln)
+Measured eCalculated +Measured eCalculated

Figure 7.44 Comparison of measured and predicted deﬂection basins and time lags for
load level 1 — Texas site

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14 l ”WV, I if 7 H N A 7 if H] 9 _H' TAT
ti
A 12 ... __mr,,,_lﬁ ,_____,A__ M 8 4”
u ,,. .
E 1° ‘L\xr " __ __.. 3 g
s a ~— \:~ ~ e— ++— —~-— Ma 3» s~
g 6 .2... _. ___ - ___ ___—.7 .. 4 —
a. ”_‘J;\K,._ g3:
. 2 _‘
8 2 r ,_____, —'——— \$\E 1 -
0 7 l l 0 l l
0 20 40 60 0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+Measured eCalculated +Measured eCalculated

Figure 7.45 Comparison of measured and predicted deﬂection basins and time lags for
load level 2 — Texas site

237

 

 

 

 

 

 

 

 

 

 

 

 

 

20 1'" - ~ ~ - .. 8 -
9 7
’6
= 15 i A 6 a
.5. E 5
g a
a 10 2 4
as
8 5 « F 2 S p.
1 «~ -:
0 0
0 20 40 60 0 20 40 60
Distance from center of load(ln) Dismnce from center of load(ln)
+ Measured 6 Calculated + Measured e Calculated

Figure 7.46 Comparison of measured and predicted deﬂection basins and time lags for
load level 3 — Texas site

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3O 9 1“ —~ a ,_ L
8
g 25 A 7 - ~
g 20 g 6
C a 5
o ..
E 15 .9 4 .
g 10 . g 3 E
a 5 _ 2 s
1 _.
0 r 0
0 20 40 60 0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+ Measured «9 Calculated 4» Measured & Calculated

Figure 7.47 Comparison of measured and predicted deﬂection basins and time lags for
load level 4 — Texas site

7.2.2.2 Dynamic Time-domain Backcalculation for T hree- layer System

Due to the limited number of sensors, only 8 parameters including layer moduli, damping
ratios and thicknesses can be backcalculated. The results are listed in Table 7.11. The
results indicate that the error in backcalculated AC layer thickness varies between -25%

and 4% while that in depth-to-stiff layer varies between 23% and 41%.

The effect of thickness backcalculation on backcalculated layer moduli is signiﬁcant for

the stiff layer, in unreasonablely high modulus value. On the other hand, thickness

238

 

 

backcalculation has not affected while it is not signiﬁcant for the base layer. The effect
on backcalculated AC layer modulus is variable. For both options, the backcalculated

damping ratios are not reasonable.

Table 7.11 Backcalculation results for time-domain analysis - Texas site

 

 

 

 

 

 

 

 

Load La Dynamic Backcalculation Dynamic Backcalculation
Level yer Without Thickness With Thickness
Modulus Damping Modulus Damping Thickness
(ksi) (ksi) (in-)
AC 192 0.1 l 300 0.73 6.0
6000 lb Base 23 0.12 27.3 0.09 94.8
Stiff layer 125 0.58 3996 0.01 «-
AC 154 0.49 172 0.29 8.3
9000 lb Base 25 0.04 25 0.10 82.3
Stiff layer 67 0.45 3382 0.02 «-
AC 151 0.53 230 0.35 6.8
12000 lb Base 25 0.08 24 0.12 84.0
Stiff layer 201 0.02 4000 0.02 ---
AC 182 0.42 366 0.49 7.5
15000 lb Base 21 0.14 21 0.10 92.8
Stiff layer 104 0.46 306 0.27 ---

 

 

 

 

Comparisons of measured and simulated deﬂection time histOries are shown in Figure
7.48 through Figure 7.55. The match for peak deﬂections is signiﬁcantly better than that
for time lags. Again, this could be due to errors in sensor location or in time
synchronization of the data acquisition system. Also, the effect of thickness

backcalculation on matching the peak deﬂection and time lags is not visible.

239

Deﬂection(mlls)
O -I» N (.0 -5 0| 0) V on

 

 

Distance from center of load(in)
+ Measured e Calculated

 

 

 

 

 

 

Distance from center of load(in)
+ Measured & Calculated

Figure 7.48 Comparison of measured and predicted deﬂection basins and time lags for
load level 1 (with thickness) — Texas site

DeﬂectloMmlls)

 

 

 

 

Distance from center of load(in)
4» Measured e Calculated

 

 

   
  
  

 

 

 

 

 

 

 

 

a 7 j
7 s- a.
a 5 “" "'*‘*l
s 4 — ,
E 3 3”" —1
F 2 _.
1:
0 4‘? T l '
0 20 4o 60

Distance from center of load(ln)
+ Measured 9- Calculated

Figure 7.49 Comparison of measured and predicted deﬂection basins and time lags for
load level 2 (with thickness) — Texas site

Deflectlon(mlls)

 

 

 

 

Distance from center of load(in)
+ Measured 6 Calculated

 

 

 

WW

 

   

 

 

 

 

Tlmelag(ms)
oamoemowoe

l
1"

 

20 40
Distance from center of load(ln)
4» Measured 4} Calculated

0

Figure 7.50 Comparison of measured and predicted deﬂection basins and time lags for
load level 3 (with thickness) — Texas site

240

 

 

 

 

 

 

 

 

30«- — 4 91“ ~47 ~ «~4— —

8a

3251K—‘rr—*r*—’——~—rr -7-

§20¢>~——~~444—--~‘—~ £6

5 35

= 4

8 E3:

‘5 :2

a 1
o r

 

 

 

 

 

0 20 40 60
Distance from center of load(ln) Distance from center of load(in)
+ Measured 9 Calculated + Measured 1} Calculated

Figure 7.51 Comparison of measured and predicted deﬂection basins and time lags for
load level 4 (with thickness) - Texas site

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9 9-— —~ , ~ , , ~ ~ - ;
A 8 ‘K._ 8 ’l‘
§7+_1_______,_____L__L ....7
s 6 _ E; 6 -
C 5 4 a 5 i
O :-
g 4 . 8 4 1 i
g 3 E 3 3 -r
3 2 ‘ l: 2 2

1 . 1 L— 4

0 0 l 1

0 20 40 60 0 20 4O 60
Distance from center of load(in) Distance from center of load(in)
+Measured eCalculated +Measured eCalculated

Figure 7.52 Comparison of measured and predicted deﬂection basins and time lags for
load level 1 (without thickness) - Texas site

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14 - ~ ~ _ ~~ 9 1- _ - A 1;:
a ‘2 g" 3 i "
E 10 l a 6 ..

c 8 . 3 5 «

'3 6 - 3 4 "

a 4 . 3 3

8 2 - p 2
1 _l

0 . 1 0 . . '
0 20 40 60 0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+ Measured 18» Calculated +Measured eCalculated

Figure 7.53 Comparison of measured and predicted deﬂection basins and time lags for
load level 2 (without thickness) — Texas site

241

 

 

 

 

 

 

 

20 — — ~ — — 8 —
5. 7 1

3" A _j ‘_ i
= 6 5
E. E, 5 l -
5 an i
8 g g ”
g i: 2 S

1

0 . i

0 20 40 60
Distance from center of load(in) Distance from center of load(in)
4» Measured 4:? Calculated + Measured & Calculated F

Figure 7.54 Comparison of measured and predicted deﬂection basins and time lags for

 

 

 

 

 

 

 

 

 

 

 

 

 

load level 3 (without thickness) — Texas site i
t , if
E A 10 1
s .3; 8 1-
c
:3 3 6
8 g 4 1
8 2 1
0 1 .
0 20 4O 60
Distance from center of load(in) Distance from center of load(in)
+ Measured 6 Calculated + Measured 1% Calculated

Figure 7.55 Comparison of measured and predicted deﬂection basins and time lags for
load level 4 (without thickness) — Texas site

7.2.2.3 Dynamic T ime-domain Backcalculation using Traces of Time History

In the ﬁrst part, the relation between convergence and threshold (or ) is investigated.
Figure 7.56 shows the number of iterations until convergence as a function of or, The

appropriate value for on is 3 or 4 in this site.

242

 

 

 

 

 

 

Iteration Number

 

 

 

 

 

1 2 3 4 5 6 7
Threshold number
+ Hysteretic damping -~— Viscous damping

Figure 7.56 Iteration number to convergence versus on for Texas site

In the second part of the analysis, a three-layer pavement system with known thickness is
used for backcalculation. The pavement proﬁle is listed in Table 7.8. Since there is a
present in this site a modulus of 500,000 psi was assumed for it, The FWD test result for
a load level of 9000 lb was used for backcalculation. Figure 7.57 though Figure 7.62
show the traces of time history from both measurement and calculation. It can be seen
that the program can provide a good match between the calculated and measured
response. It is also noted that the matches in the last two sensors have large differences.
This may be caused by the assumption of an arbitrary value for the stiff layer modulus,

which will affect the propagation of the wave trapped above the stiff layer.

Figure 7.63 summarizes the backcalculation results in terms of modulus values. The
backcalculation results indicate that the relative difference in layer moduli using
hysteretic and viscous damping models can be relatively large (18% for AC, —48% for
the base, and 43% for the subgrade layer) The difference between backcalculated moduli
from the FEM model (Massui, 1998) and DYNABACK with hysteretic damping are 2%,

7% and 13 for the AC, base and subgrade layer, respectively.

243

 

15.0

 

 

 

 

 

 

 

 

 

 

.117 13.0 : ~ .22..
g -
'3 90 ‘
§ .
2'3 7.0 l-—— ___ —— .._.w ~~— -—~— — a .4
5.0 .
15.0 17.0 19.0 21.0 23.0 25.0
Time (ms)
——D1 measured _._ D1-hysteretic damping
.—.—- D1-viscous damping - 1- - D1-FEM

Figure 7.57 Comparison of measured and predicted deﬂection time histories for sensor 1

8.0
7.0 ~L—-
6.0 -

 

 
  

 

\

r7

 

Deflection (mils)

 

 

 

5.0 i
4.0 a, T
15.0 17.0 19.0 21.0 23.0 25.0
Time (ms)
-— DZ-measured ---—- DZ-hysteretic damping
+ DZ-viscous damping + DZ-FEM

Figure 7.58 Comparison of measured and predicted deﬂection time histories for sensor 2

 

 

 

 

 

 

 

_’_§ 4.0
E
E 3.0 «A
.9. 4
g 2.0 +
8 1.0 . , T
15.0 17.0 19.0 21.0 23.0 25.0
Time (ms)
—D3-measured —-— DB—hFésteretic damping
.._—DB-viscous damping +03- M

Figure 7.59 Comparison of measured and predicted deﬂection time histories for sensor 3

244

 

 

3.0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.0? 2.5 we" ——~— ___.
E 2.0 7+ “1
c 1.5 -. ~ _— i *_
.g 1.0 1
g 0.5
8 0.0 . .
15.0 17.0 19.0 21.0 23.0 25.0
Time (ms)
—D4-rneasured . —-— m-léésteretic damping
—+— D4-vuscous damping + 04- M F
Figure 7.60 Comparison of measured and predicted deﬂection time history for sensor 4
2.0
i
1.3 1.5
E 1.0 r
8
'3 0.5
Q)
“E, 0.0
D 15.0 17.0 19.0 21.0 23.0 25.0
Time (ms)
-—-DS-measured -—-— D5-hysteretic damping
—.— DS-viscous damping + 05-FEM

Figure 7.61 Comparison of measured and predicted deﬂection time histories for sensor 5

2.0

1.0

Deﬂection (mils)
O
O

 

1 . . 19.0 21.0
-1.0
Time (ms)
— DG-measured —-— DG—hysteretic damping

_.— DG-u'scous damping + DG-FEM

Figure 7.62 Comparison of measured and predicted deﬂection time histories for sensor 6

245

Texas_load level 9000 lb

 

1.00E+06

 

3 1.00905
w
2
3
g 1.00904»
1.00E+03 ~ " “ 1 " " *
AC Base Subgrade
Layer

DYNABACK-hysteretic damping I DYNABACK-uscous damping FEM I MICHBACK

Figure 7.63 Comparison of layer modulus from different model

In the third part of the analysis, the uniqueness was analyzed. The different seed

parameters that were used in the program are shown in Table 7.12. The backcalculated

results are shown in Figure 7.64.

Table 7.12 Seed values used for Texas data

 

 

 

Layer Case 1 Case 2 Case 3
Seed Seed Seed
Layer Modulus Dasrfigiing Modulus Dizzig Modulus 3513118
(ksi) LkaL (ksi)
AC 350 0.10 850 0.15 250 0.20
Base 70 0.10 90 0.05 10 0.15
Subgrade 70 0.02 90 0.05 8 0.10

 

246

Texas_Load Level 9000 lb (hysteretic damping)
1 .00E+06 w-

 

1.00E+05 , 7

Modulus (psi)

 

 

1.00E+04

 

Base Subgrade

 

‘ECase1lCase2EICase3l
L#_ —, , _ 4)

Figure 7.64 Comparison of backcalculated layer moduli using different seed values

In the fouth part of the analysis, only the hysteretic damping model is used. The
thicknesses of base and subgrade layer are combined, and the modulus of AC, base, and
stiff layers were backcalculated. Various combinations of damping ratios for the base and
subgrade layer were used, as listed in Table 7.7. Figure 7.65 through Figure 7.69 show
the backcalculation results. It can be seen that the backcalculated AC damping ratio
deceases with the increasing damping ratios of base and subgrade layers because of
compensation. The thickness of the AC layer varies from -12% to 4% from the true
value (a good result). Also layer moduli are not signiﬁcantly affected by the different

assumptions of damping ratios for base and subgrade layer.

247

Site3_9000 lb. E AC_Modulus

 

1.00E+06 _

MICHBACK result

 

1.00305 2 "

Modulus (psi)

 

 

1.00E+04 - * . .
l 2 3 4 5 6 7 8 9
Combination of Damping Ratio

Figure 7.65 Comparison of backcalculated AC modulus for different damping ratio
combinations for base and subgrade layers

 

 

 

 

Site3_9000 lb. AC_danping

0.5
'0 0.4
32
w 0.3
.E
g 0.2

0.1

0.0 ‘ 1 '

l 2 3 4 5 6 7 8 9
Contination of Danping Ratio

Figure 7.66 Comparison of backcalculated AC damping for different damping ratio
combinations for base and subgrade layers

248

Site3_9000 lb. E! AC_Thickness

 

_
>190 :0 9
0° C O

Thickness (inch)

9‘
o

 

 

S"
c

Combination ofDanping Ratio

Figure 7.67 Comparison of backcalculated AC thickness for different damping ratio
combinations for base and subgrade layers

 

Site3_9000 lb. E Base_Modulus
1.00E+05
MICHBACK result

(5
3
V}
g 1.00E+04 *
'U
o
2

1.00E+03

 

l 2 3 4 5 6 7 8 9
Combination of Dan'ping Ratio

Figure 7.68 Comparison of backcalculated base modulus for different damping ratio
combinations for base and subgrade layers

249

Site3_9000 lb El Stiff layer_Modulus

1 .00E+06

1.00E+05 .

Modulus (psi)

 

 

1.00E+04

Combination ofDanping Ratio

Figure 7.69 Comparison of backcalculated subgrade modulus for different damping ratio
combinations for base and subgrade layers

7.2.3 Cornell Site

Table 7.13 shows the pavement cross-section of the Cornell test site. Table 7.14 shows

the FWD sensor layouts, which is unique in the sense that it includes nine sensors with

the farthest sensor at almost 6 11 from the load.

Table 7.13 Proﬁle used for Cornell site

Layer Name Thickness (in) Unit Weight (pcf) Poisson Ratio
AC 4.5 145 0 3

 

Base 15 135 0.55
Subbase 110 135 0.40
Subgrade oo 125 0.45

 

Table 7.14 Sensor layout (distances are in inches) for Cornell site

D1 D2 D3 D4 D5 D6 D7 D8 D9
0 8 l2 18 24 36 47 59 71

250

7.2. 3. 1 Comparison of Dynamic and Static Backcalculation for F our-layer System

The analysis was ﬁrst conducted on a 4-layer pavement system. However, the results for
dynamic analysis compare reasonably well with those from MICHBACK as shown in
Table 7.15, with the exception of the subgrade modulus. Both analyses predict a very low
modulus for the base layer and a high subgrade modulus which is not reasonable from an
engineering point of view. Also, the backcalculated damping ratios for the unbound
materials are unreasonably high. This may be indicative of non-linear behavior for these

materials.

Figure 7.70 shows the measured and calculated deﬂections in frequency. The match is
better for real deﬂections at low frequencies while it is better for imaginary deﬂection at
intermediate frequencies. Figure 7.71 shows the comparison of measured and predicted
peak deﬂections and time lags. The match is fairly good for peak deﬂections and poor for

the time lags.

Table 7.15 Comparison of frequency and time-domain backcalculation results with those
from MICHBACK- Cornell site

 

 

 

Frequency-domain Time-domain Static
Bckcalculation Bckcalculation Backcalculation
Modulus (ksi) Damping Modulus (ksi) Damping Modulus (ksi)
AC 1752 0.10 1972 0.10 2013
Base 7 0.11 8 0.42 7
Subbase 29 0.08 20 0.13 21
Subgrade 216 0.24 58 0.14 34

 

 

 

251

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

0.8 A o - -
A ﬂ
=3 0-7 g -0.05 -
5 0.6 .6
8 0.5 g ’0-1
§ 0.4 g 0.15 -1
§ 0.3 8 -02
" 02 - '
"7' ' °’ -0 25 <
a? 0-1 E ' 1
0 , 1 " -0.3 , r f . a
0 20 40 60 80 0 20 40 60 80
Distance from center of load(ln) Distance from center of load(in)
+Measured ..3. Calculated +Measured E-Calculated
(a) Real and imaginary deﬂection basins at 2.44 Hz
.3 a -0.1 _3 <2.- — —
g ‘2' 0.21_— —— —/",7/ E
E g 3.3 / I
"a .4 —_-———4-— -- --~—— ——~
g g .05 .___ “Z;
:9 5’ ‘0'6 1 / l ' __
g 3 ~07 ~ j
m - -0.8 . r si
0 20 40 60 q 80
Distance from center of load(in) Distance from center of load(in)
+Measured GCalculated +Measured ~:+Calculated
(b) Real and imaginary deﬂection basins at 9.77 Hz
.. o -,
g i -0.1
g g -O.2 ~ -
g g-o.3~————— _— ——mw w"-
3 § -0.4 — ————“ #* M -
E :3, -0.5 « —— — s
g g -0.6
a - '0] , a f T
0 20 40 60 80
Distance from center of load(ln) Distance from center of load(in)
+Measured eCalculated +Measured eCalculated

(0) Real and imaginary deﬂection basins at 14.65 Hz

Figure 7.70 Comparison of measured and predicted deﬂection basins — Cornell site

252

WL

 

 

 

 

Deﬂection(mlls)

 

 

 

 

Distance from center of load(in) Distance from center of load(in)
4» Measured 9 Calculated + Measured 6* Calculated

Figure 7.7] Comparison of measured and predicted deﬂection basins and time lags

7.2.3.2 Dynamic T ime-domain Backcalculation for T hree-layer System

In this section, the base and subbase layers were combined and the program was allowed
to backcalculate the layer thicknesses. The time-domain backcalculation results are listed
in Table 7.16. The error in the backcalculated thicknesses was about 33% for the AC
layer and about -40% for the combined base and subbase layer. The effect of thickness
backcalculation on layer moduli was signiﬁcant for the AC layer as well as the subgrade.
Also, the backcalculated damping ratio values are unreasonably high for the AC and base
layers. The backcalculated subgrade modulus and damping ratio for the case when
thickness backcalculation was allowed are unacceptable. Comparisons of measured and
simulated deﬂections and time lags are shown in Figure 7.72 and Figure 7.73, for the
cases with and without thickness backcalculation. The matching is not good, in both

C3868.

253

 

 

Table 7.16 Backcalculation results from time-domain analysis — Cornell site

 

 

 

Dynamilcl: Bailicalltclulation Dynamic Backcalculation (with thickness
(Em out C . ess backcalculation)
ackcalculation)
Modulus (ksi) J Damping Modulus (ksi) Damping Thickness(in.)
AC 1903 0.11 687 0.21 6
Base 17 0.38 14 0.14 74
Subgrade 23 0.05 400 0.001 «-

 

 

 

 

 

 

 

 

 

 

 

325a.‘ ******* ii—1 A
szo‘ E;
c
.3151- 3
3 g
=10 F
8 5

0

Dlstance from center of load(in)
+ Measured & Calculated

80 0 20 40 60 80
Distance from center of load(in)
+ Measured 9 Calculated

Figure 7.72 Comparison of measured and predicted deﬂections and time lags (with
thickness backcalculation)

 

 

,. 1 14-———-——— ——— —- —_ — n. a
£20“ * * r r - ‘ E12 ._.-__ — .-_—w a”, _-
E 15 S E . _ . _ __
g 2
u .0
a 10 g
8 5 q
0

 

 

0 20 40 60
Distance from center of load(in)
+ Measured 8 Calculated

 

 

80 0 20 40 60 80

Distance from center of load(ln)
+ Measured *3 Calculated

Figure 7.73 Comparison of measured and predicted deﬂections and time lags (without
thickness backcalculation)

254

7.2.3.3 Dynamic T ime-domain Backcalculation using Traces of Time History

In the ﬁrst part, Figure 7.74 shows the relationship of or versus iteration number, and a

value for or of 3 or 4 can be considered to be appropriate for this site.

 

50-

 

Iteration Number

 

 

 

Threshold number
+ Hysteretic damping —.—Viscous damping

Figure 7.74 Iteration number to convergence versus or for Cornell site

In the second part of the analysis, a four-layer system is employed for backcalculation
purposes. Traces of time history are used in the analysis. Figure 7.75 through Figure 7.82
show measured and predicted deﬂections for the various sensors. The backcalculated
layer moduli from various models are compared to see the similarities and differences in
Figure 7.84. The comparisons show that the relative difference for layer modulus
between the hysteretic and viscous damping is —2% for AC layer, 20% for base, -7% for
subgrade and —69% for stiff layer. The difference between the FEM method and
DYNABACK with hysteretic damping is 1% for AC layer, -32% for base, 47% for
subgrade, and —48% for stiff layer. The value seems to show a very good consistency for

modulus in this site.

255

 

 

 

 

Deflection (mils)

 

 

 

0 ‘ ' 7 I
14.6 16.6 18.6 20.6 22.6 24.6 26.6
Time (ms)
—D1-measured —-— D1-hésteretic damping
+ D1-viscous damping + D1-F M

Figure 7.75 Comparison of measured and predicted deﬂection time histories for sensor 1

 

 

 

.117 20 1
§ l
g 15
i
8 10 1 . 1
14.6 16.6 18.6 20.6 22.6 24.6 26.6
Time (ms)
— DZ-measured —-— DZ-hysteretic damping

—.— DZ-viscous damping + DZ-FEM

Figure 7.76 Comparison of measured and predicted deﬂection time histories for sensor 2

 

N
O

 

 

 

 

 

 

01

Deflection( ils)
8 61‘
l
l l

14.6 16.6 18.6 20.6 22.6 24.6 26.6
Time (ms)

— DB—measured —-— DS-hysteretic damping

—o— D3-viscous damping + DB-FEM

Figure 7.77 Comparison of measured and predicted deﬂection time histories for sensor 3

256

 

 

 

 

 

 

33 15
E
.9
g 5 sn_m_____.__ .._____. . __..—v ___. .
3 0 ‘ ,
14.6 16.6 18.6 20.6 22.6 24.6 26.6

—- D4-measured —+— D4-hysteretic damping
——.— D4-viscous damping + D4-FEM

Figure 7.78 Comparison of measured and predicted deﬂection time histories for sensor 4

 

15

10 M
5 -_ -. 1
0 .
14.6 16.6 18.6 20.6 22.6 24.6 26.6

Time (ms)

— D5-measured —-— DS-hysteretic damping
—-.— DS—viscous damping + DS-FEM

 

 

 

 

 

 

Deflection (mils)

Figure 7.79 Comparison of measured and predicted deﬂection time histories for sensor 5

 

 

 

 

 

 

 

g 6
E,
C 4 H w
.9
is; 2 —-— -
8 0 . .
14.6 16.6 18.6 20.6 22.6 24.6 26.6
Time (ms)
—DG-measured —-— DG-hysteretic damping

—.— DG-viscous damping + DG-FEM

Figure 7.80 Comparison of measured and predicted deﬂection time histories for sensor 6

257

 

 

 

 

 

 

 

A 4
g
s31——-~—-~‘———— - e E
.5 2 -, _
8
a: 1 ~—~ — —— ——- ———~—
8

0 . . . ,

14.6 16.6 18.6 20.6 22.6 24.6 26.6

Time (ms)
— D7-measured —-— D7-hysteretic damping

—.— D7-viscous damping + D7-FEM

Figure 7.81 Comparison of measured and predicted deﬂection time histories for Sensor 7

 

 

 

 

 

 

 

 

 

 

 

2
E,
C
.9
8
z
8 -2 - —
-3
14.6 16.6 18.6 20.6 22.6 24.6 26.6
Time (ms) _
— DB-measured _._ DB-constant model
—.— DB-viscous damping + DS-F EM

. Figure 7.82 Comparison of measured and predicted deﬂection time histories for Sensor 8

 

 

 

 

 

 

 

 

 

 

 

 

A 2
.313 t
g 1 _-
C
.3 01» 2 : ,,..—V 7 s
E -1 __..
"8

-2

14.6 16.6 18.6 20.6 22.6 24.6 26.6

Time (ms)

— DQ-measured —o— DQ-constant model —0— DQ—v‘scous damping

Figure 7.83 Comparison of measured and predicted deﬂection time histories for Sensor 9

258

Cornell

 

1.00E+07

1.00E+06

1.00E+05

Modulus (psi)

1.00604

      

1.00E+03 "5
Stiff layer

AC Base Subbase
Layer

:5 DYNAMIC-hysteretic damping I DYNABACK-viscous damping I FEM I MICHBACK

Figure 7.84 Comparison of backcalculated modulus from different models

In the third part of the analysis, the uniqueness was analyzed for the four layer pavement

system. The different seed parameters that were used in the program are shown in Table

7.17. The backcalculated results using different seeds that are listed show the excellent

agreement in Figure 7.85.

Table 7.17 Different seed Speciﬁcations - Cornell data

 

 

 

Layer Case 1 Case 2 Case 3
Seed Seed Seed
Layer Modulus Digging Modulus ijgiing Modulus 31:2?“
(ksi) (ksi) (ksi)
AC 450 0.1 250 0.2 850 0.3
Base 80 0.08 10 0.05 100 0.08
Subgrade 60 0.05 10 0.05 90 0.05
Stiff layer 50 0.03 30 0.03 10 0.03

 

259

 

 

 

 

Comell (hysteretic damping)
1.00E+07 ;

 

1.00E+06

1.00E+05 ,

Modulus (psi)

1.00E+04

 

 

1.00E +03

AC Base Subgrade Stiff layer

 

{a Casen1 I Case 2 El Case 3

 

Figure 7.85 Comparison of backcalculated layer moduli using different seeds

In the fourth part of the analysis, a three-layer pavement system with unknown AC
thickness and hysteretic damping was used. Various damping ratio combinations of base
and subgrade layer as shown in Table 7.7. Six parameters are backcalculated, including
modulus of AC. base, and subgrade and damping ratio and thickness for AC. Figure 7.86
through Figure 7.90 show the backcalculation results. Figures show the AC damping ratio
deceases with the increasing of damping ratio for base and subgrade. The AC thickness
varies from 25% to 46% from the true value. The AC, base and subgrade moduli were not

affected.

260

Cornell AC_Modulus

 

1.00E+07
MICHBACK result

r yww—

 

 

1.0013106 7,

Modulus (psi)

1.00505 ~

 

 

 

1.00E+04 .
l 2 3 4 5 6 7 8 9

Combination of Dan‘ping Ratio

Figure 7.86 Comparison of backcalculated AC modulus for different damping ratio
combinations for base and subgrade

Cornell AC_danping

 

 

 

Damping Ratic

 

 

Combination of Damping Ratio

Figure 7 .87 Comparison of backcalculated AC damping ratio for different damping ratio
combinations for base and subgrade layers

261

Cornell AC_Thickness

 

Boring average thickness

 

Thickness (inch)

 

 

Conbination of Damping Ratio

Figure 7.88 Comparison of backcalculated AC thickness for different damping ratio
combinations for base and subgrade layers

MI Cl-IB A CK result ‘Comell I Base_Modulus

 

Modulus (psi)

 

Corrbination of Damping Ratio

Figure 7 .89 Comparison of backcalculated base modulus for different damping ratio
combinations for base and subgrade layers

262

Comell Cl Subgrade_Modulus

 

1.00E+05

MICHBACK result \

1.00E+04

1.00E+03

Modulus (psi)

 

 

1.00E+02 ' " “ ‘ ‘ " .
l 2 3 4 5 6 7 8 9
Corrbination of Danping Ratio

Figure 7.90 Comparison of backcalculated subgrade modulus for different damping ratio
combinations for base and subgrade layers

7.2.4 Florence Site

The site in Florence, Italy, consists of an asphalt concrete surface layer overlying a
cement-treated base. Table 7.18 shows the pavement cross-section for the test site, and

Table 7.19 shows the FWD sensor layouts.

Table 7.18 Proﬁle used for Florence site

Layer Name Thicknesstin) Unit Weightgpsf) Poisson Ratio
AC 4 0.35

 

138
CTB 5.5 150 0.20
Subgrade 90 1 16 0.45
Stiff layer 00 120 0.15

 

7. 2. 4. I Comparison of Dynamic and Static Backcalculation for F our-layer System

Table 7.20 shows the backcalculation results. The results from time-domain analysis are
somewhat more reasonable than those for frequency-domain analysis, while the results
for static backcalculation are not reasonable, showing low values for the cement-treated
base and bedrock moduli and a very high value for the subgrade modulus. The damping

263

ratios (for dynamic analysis) for the AC layer and subgrade are also unreasonably high.
Figure 7.91 shows the measured and predicted deﬂection basins at low, intermediate and
high frequencies. There is generally poor agreement in both shape and magnitude. Figure
7.92 shows the measured and predicted peak deﬂections and time lags. The agreement is

fair but not acceptable for backcalculation purposes.

Table 7.19 Sensor layout (distances are in inches) for Florence site

 

D 1 D2 D3 D4 D5 D6 D7
0 12 18 24 35 47 59

 

Table 7.20 Comparison of frequency and time-domain backcalculation results with those
for MICHBACK — Florence site

 

 

Fre uenc -domain . . . Static
BailckcaIZulation Time-domain Backcalculation Backcalculation
Modulus (ksi) Damping Modulus (ksi) Damping Modulus (ksi)
AC 300 0.32 562 0.52 440
CTB 495 0.01 624 0.01 200
Subgrade l l 0.13 9 0.27 124
Bedrock 124 0.05 1989 0.03 27

 

 

 

 

264

_C)
w
l

 

 

 

 

 

 

 

 

 

 

 

3 ; E
--E- 0.25 g
g 0.2 l 5
=1 1 g
g 0.15 e
= ‘5
3 0.1 1 1:
7' 0.05 ~ °
2 E
0 . .
0 20 40 60 0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+ Measured -3 Calculated + Measured +3 Calculated

(a) Real and imaginary deﬂection basins at 3.66 Hz

 

 

 

   

 

 

 

 

 

 

 

 

 

- 0— ,.-
.'—é’ €0.02 ~ ——m~ «m— —/£~-1/ 9
TE ‘5’ -0.04 . . _
g g -0.06
8 g -0.08 +—
3 ° 01 —
7. 5':
g g -0.12 -
-0.14 r , a
0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+Measured ~8Calculated +Measured G Calculated

(b) Real and imaginary deﬂection basins at 15.89 Hz

 

 

 

   

 

 

 

 

 

 

 

g * 9’ 0.14 l 1 ; .
E 0 ‘E’ 0.12 —
g .005 ° 0.1 1
=3 ' a
g g 0.08 1
c .
'3 -0.1 < , r I, : '8 333 .
_ / 3 ° . 1 '- ' .-
8 .015 :9"ng #A 4“” ——‘N 7*- I-” I g 0 02 .1___ _ a
a: j _ - 5
-0.2 r 1 1 0 . ‘
0 20 40 60 0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+ Measured 1} Calculated + Measured 6- Calculated

(c) Real and imaginary deﬂection basins at 23.19 Hz

Figure 7.91 Comparison of measured and predicted deﬂection basins— Florence site

265

...;
.5

Deflection(mils)
Time lag(ms)

 

 

 

 

   

Distance from center of load(in) Distance from center of load(in)
4» Measured ~63» Calculated 4» Measured «9 Calculated

Figure 7.92 Comparison of measured and predicted deﬂection basins and time lags —
Florence site

7.2. 4.2 Dynamic T ime-domain Backcalculation for T hree-layer System

In this analysis, the AC and CTB layers were combined again and the program was
allowed to backcalculate layer thicknesses. The time-domain backcalculation results are
listed in Table 7.21. The error in the backcalculated thicknesses was about 17% for the
combined AC and CTB layer and about 31% for the subgrade layer above the bedrock.
The effect of thickness backcalculation on layer moduli was signiﬁcant for all layers with
the difference ranging from -55% to 34%. The backcalculated damping ratios are
unreasonably high. Comparisons of measured and simulated deﬂections and time lags are
listed in Figure 7.93 and Figure 7.94, for the cases with and without thickness
backcalculation. Matching of peak deﬂections is better than that for time lags, and the

results are slightly better when layer thicknesses are known.

266

Table 7.21 Thickness backcalculation in time-domain

 

 

 

 

 

 

 

Dynamic Backcalculation
Without Thwkness With Thickness Backcalculation
Backcalculation
Modulus (ksi) Damping Modulus (ksi) Damping Thickness(in.)
AC + CTB 862 0.11 547 0.23 11.1
Subgrade 9 0.18 12 0.12 118.2
Bedrock l 18 0.47 52 0.50 ---

101K

\\

 

 

 

 

 

 

 

 

 

   

 

E ‘ T ‘ _ _ A
s a l g
c O
§ 6 S
e4 +e~~—- g
a 2 I
o ' T 1
0 20 40 60
Dishnce from center of load(in) Distance from center of load(in)
+Measured ECalaulated +Measured 1S+Calaulated
Figure 7.93 Comparison of peak deﬂections and time lags (with thickness
backcalculation) — Florence site
12 14 -
4 12 l—
g A
E, g 10 f » - ——
c "' 8 7 -
a 3 6 e.
0 E ;
G .
E 1: 4 “
2 "1L __ _ _. ___ __ _ _ _. _.4
0 l a
0

 

Distance from center of load(in)
+ Measured & Calculated

20 40
Distance from center of load(in)
+ Measured 9 Calculated

Figure 7 .94 Comparison of peak deﬂections and time lags (without thickness
backcalculation) - Florence site

7.2.4.3 Dynamic T ime-domain Backcalculation using Traces of Time History

In the ﬁrst part, the relation of or versus iteration number is shown in Figure 7.95. An

a—Value of 3 or 4 is appropriate for this site.

267

 

 

 

 

Iteration Number

 

 

 

 

 

Threshold number
-+— Hysteretic damping —.—Viscous damping

Figure 7.95 Iteration number to convergence versus 01 for Florence site

In the second part of the analysis, a three-layer pavement system consists of the AC layer,
a cement treated base (CTB) and a subgrade layer is used. Figure 7.96 to Figure 7.102
show the comparisons of predicted and measurement time histories for different sensors.
Figure 7.103 shows the backcalculated values for different layers using different models.
The difference in backcalculated moduli using hysteretic and viscous damping is -21%
for AC, 75% for base, and -21% for subgrade. The difference between the
backcalculated moduli from the FEM method and DYNABACK with hysteretic damping

is —33% for the AC and 50% for the base, and -5% for the subgrade.

 

12.00
11.00 - -—
10.00 ~
9.00 1.- -
8.00 1‘ -

7.00 . . . .
20.6 22.6 24.6 26.6 28.6 30.6 32.6 34.6
Time (ms)

— D1 measured —-— D1-2ésteretic damping
—+— D1-viscous damping + 01- M

 

 

 

 

Deflection (mils)

 

 

 

 

Figure 7.96 Comparison of measured and predicted deﬂection time histories for sensor 1

268

 

 

 

 

 

9.00
.11?
E, 8.00 ~ W -——-——— —-
C
.9
g 7.00 + -- . ____-_ —_— —- u— ~ ~—
8 1 ,

6.00 ,

20.6 22.6 24.6 26.6 28.6 30.6 32.6 34.6
Time (ms)

—D2-measured —o—DZ-t'¥steretic damping
+DZ-u'scous damping +02— M

Figure 7.97 Comparison of measured and predicted deﬂection time histories for sensor 2

 

 
  

 

 

 

A 8.00

é’

g 7.00 -~—— ___...

C

.9

§ 6.00 .

8 5.00 . . .

20.6 22.6 24.6 26.6 28.6 30.6 32.6 34.6
Time (ms)

— DB—measured —-— D3—hysteretic damping
—+— D3-viscous damping + D3-FEM

Figure 7.98 Comparison of measured and predicted deﬂection time histories for sensor 3

 

8.00
7.00
6.00
5.00
4.00 ~
3.00
2.00 . . ,
20.6 22.6 24.6 26.6 28.6 30.6 32.6 34.6

Time (ms)
— D4-measured —-— D4-hysteretic damping

—.— D4-viscous damping + D4-FEM

 

 

 

 

 

 

 

Deflection (mils)

 

 

Figure 7.99 Comparison of measured and predicted deﬂection time histories for sensor 4

269

 

6.00

 

 

 

 

 

 

 

’2‘
E 5.00
.§ 4.00
§ 3.00
8

2.00 . . . . -

20.6 22.6 24.6 26.6 28.6 30.6 32.6 34.6
Time (ms)

— DS-measured —-— DS-hysteretic damping
—-.— D5-u’scous damping —+- D5—FEM

Figure 7.100 Comparison of measured and predicted deﬂection time histories for sensor 5

4.00 ,
3.50
3.00
2.50
2.00
1.50
1.00 -

20.6 22.6 24.6 26.6 28.6 30.6 32.6 34.6

Time (ms)

 

 

 

 

 

 

 

Deflection (mils)

 

 

 

-— DG-measured —-— DG-hysteretic damping
—-o— D6-viscous damping —.— DG-FEM

Figure 7.101 Comparison of measured and predicted deﬂection time histories for sensor 6

 

 

 

 

 

 

 

 

 

7—2‘
.5,
C
.9
‘6
.13
8 .
0.00 . . . . _
20.6 22.6 24.6 26.6 28.6 30.6 32.6 34.6
Time (ms)
— D7-measured —-— D7-hysteretic damping
+ D7-viscous damping + D7-FEM

Figure 7.102 Comparison of measured and predicted deﬂection time histories for sensor 7

270

Florence

 

 

1.00E+07
r: 1.00E+06
U)
E
3 1.00905
3
8
5 1.00904
1.005+03 . . .. .
AC Base Subgrade
Layer

El DYNABACK—hysteretic damping I DYNABACK-viscous damping I FEM I MICHBACK

Figure 7.103 Comparison of backcalculated layer moduli from different models

In the third part of the analysis, the uniqueness of the backcalculated results was analyzed
for three layers including AC, base and subgrade. The random seed parameters that were
used in the program are shown in Table 7.22. Figure 7.104 shows that the backcalculation

results are not affected by random seed values.

Table 7.22 Seed value used for Florence data

 

 

 

Layer Case l Case 2 Case 3
Seed Seed Seed
Layer Modulus D3512?” Modulus Dasrggiing Modulus Dasneiggng
(ksi) (ksi) (ksi)
AC 300 0.4 530 0.2 230 0.3
Base 140 0.08 180 0.05 400 0.03
Subgrade 13 0.08 15 0.05 15 0.08

 

271

Florence (hysteretic damping)

 

1.00E+07

1.00E+06 ,

1.00E+05 - ,

Modulus (psi)

1.00E+04 -

 

 

1.00E+03 ,

AC Base Subgrade

Case 1 lCaseIZ UCase 1:

l__f

 

Figure 7.104 Comparison of backcalculated layer moduli using different seed value

In the fourth part of the analysis, a three-layer pavement system was used. The AC
thickness is leﬁ unknown, and various damping ratio combinations for base and subgrade
layers were used (see Table 7.7). The aggregate base and subgrade were combined as one
layer base. Figure 7.105 through Figure 7.109 show the backcalculation results using
hysteretic damping. The ﬁgure shows that the layer moduli are not signiﬁcantly affected
by the chain of damping ratio for the base and subgrade. The AC damping ratio is
somewhat more affected, however its variation is not of practical signiﬁcance. The error
in AC thickness backcalculation varies ﬁ'om 23% to 49%; this is may not be acceptable

for ﬁeld application.

272

Florence El AC_Modulus

 

l .00E+06

MICHBACK result

1.00E+05

Modulus (psi)

 

1.00E+04 . . . .
1 2 3 4 5 6 7 8 9
Corrbination of Drawing Ratio

Figure 7.105 Comparison of backcalculated AC modulus for different damping ratio
combinations for base and subgrade layers

Florence I AC_danping

 

 

Damping Ratir

 

 

 

Combination of Damping Ratio

Figure 7.106 Comparison of backcalculated AC damping ratio for diﬁermt damping ratio
combinations for base and subgrade layers

273

 

 

 

 

Florence E AC_Thickness

 

Boring average thickness

Thickness (inch)

 

Combination of Damping Ratio

Figure 7.107 Comparison of backcalculated AC thickness for different damping ratio
combinations for base and subgrade layer

 

 

Florence Base_Modulus
1.00E+07 E
MICHBACK result

g 1.00E+06 a»
m .
:1 .
“:5
‘3
2 1.00E+05

1.00E+04 ,

 

 

 

l 2 3 4 5 6 7 8 9
Conbination ofDarrping Ratio

Figure 7.108 Comparison of backcalculated base modulus for different damping ratio
combinations for base and subgrade layers

274

Florence El Subgrade_Modulus

 

1.00E+05 r"—'
MICHBACK result

 

1.00E+04 ‘ *

 

Modulus (psi)

 

 

1.00E+03

l 2 3 4 5 6 7 8 9
Combination ofDamping Ratio

Figure 7.109 Comparison of backcalculated subgrade modulus for different damping
ratio combinations for base and subgrade layers

7.2.5 Kansas Site

Backcalculation was also performed using FWD data collected in the ﬁeld as a part of
LTPP study in Kansas (Section ID No. 20-0103-1). Two proﬁles were used: One using
four layers with thicknesses as determined from cores; the other using a 3-layer system
with the combined AC and ATB layers. For the three-layer system, backcalculation was
done with and without assuming layer thicknesses. Again, the MICHBACK program was
used to perform static backcalculation for comparison purposes. The FWD data contained
eight deﬂection time histories for sensors located at r = 0, 8, 12, 18, 24, 36, 48 and 60

inches from the load. The accuracy of each sensor was about :t 0.1 pm.

7.2.5.1 Dynamic Time-domain Backcalculation for Four-layer System

The four-layer pavement proﬁle and backcalculation results are shown in Table 7.23. The
results appear to be reasonable, although the subgrade modulus is higher than the base
modulus. This is typical of backcalculation results, but is not necessarily realistic. The

275

damping ratio values are also unrealistic. The measured and calculated peak deﬂections

and time lags are shown in Figure 7.110. The match is poor, especially for the time lags.

Table 7.23 Proﬁle used for Kansas site

 

Layer Thickness (in) Unit Weight (pct) Poisson’s Ratio Modulus (ksi) Damping ratio

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Name
AC 3.6 145 0.3 640 .33
Base 7.7 135 0.35 436 .54
Subbase 6 135 0.35 18 .09
Subgra_de 00 125 0.45 25 .29
6 16 --....m-.- H---
A 5
.2 A
E. 4 E
g 3 3
a 2 g
8 1
0 .
0 20 40 60
Distance from center of load(in) Distance from center of load(in)
+ Measured +3- Calculated + Measured & Calculated

Figure 7.110 Comparison of measured and calculated peak deﬂections and time lags
(four layer backcalculation)

7.2. 5.2 Comparison of Dynamic and Static Backcalculation for T hree-layer System

For the combined proﬁle in Table 7.24, the results of the dynamic and static
backcalculation are given in Table 7.25. The errors in the backcalculated AC and base
thicknesses in Case 2 compared to the thicknesses reported from cores are shown within
parentheses. In the dynamic backcalculation, the AC modulus decreases by 14% between
Cases 1 and 2 mainly because the backcalculated AC thickness for Case 2 is 22% larger
than the AC thickness used in Case 1. The backcalculated base thickness in Case 2 is

9.5% larger than the reported thickness from cores.

276

The measured and predicted surface peak deﬂections and time lags are shown in Figure

7.111 and Figure 7.112. The following observations are made from these ﬁgures.

The magnitude of the peak displacement and the time of its occurrence are very well
matched by the simulation whether the layer thicknesses are assumed to be known

(Figure 7.111), or when the layer thickness are assumed to be unknown (Figure 7.112).

Table 7.24 Proﬁle used for Kansas site with combined AC and ATB layer

 

Layer Name Thickness (in) Unit Weight (pct) Poisson’s Ratio

 

AC 1 1.3 145 0.3
Base 6 135 0.35
Subgrade 00 125 0.45

 

Table 7.25 Backcalculation results for Kansas site

 

 

 

Dynamic Backcalculation Static
True Value Seed Value Encluding All Parameters Backcalculatro
Thicknesses (Case 2) n usmg
(Case 1) MICHBACK
AC 3123““ Unknown 350 446.6 383.1 479.4
AC da‘i‘pmg Unknown 0.2 0.15 0.21 —
ratio
AC thickness 11 3 8 _ 13.80 _
(in.) ' (Case 2 only) (22.1%)
Base “19‘1““ Unknown 20 5.43 5.50 4.25
(km)
Base “.mpmg Unknown 0.1 0.22 0.18 —
ratio
. 12.3 6.57
Base thickness 6 (Case 2 only) — (9.5%) —
subgrade Unknown 10 42 1 41 4 53 21
modulus (ksi) ' ' '
Subgrade
damping ratio Unknown 0.1 0.19 0.21 —

 

277

in

 

 

 

 

 

 

 

 

6 "' ' '?
La 5 —~
5 4 n é
a a
g 3 E;
8 1 . 5

0 ﬂ '1

0 20 40 60 0 20 40 60
Distance from center of load(ln) Distance from center of load(ln)
+ Measured 9 Calculated + Measured 8r Calculated

Figure 7.1 11 Comparison of measured and calculated peak deﬂections and time lags for
case 1 (three layer backcalculation)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14 -, , a —~ ~ «
5 l
4 12 -
7 5 l ' “ 10 4
= 8
E 4 .. .
E a 8 4
o 3 J 2
§ 2 4 - g 6 no.
s =
1 2 ~~—— ~———— — ~
0 i , . 0 . .
0 20 40 60 0 20 40 60
Distance from center of load(ln) Distance from center of load(ln)
+Measured €~Calculated +Measured a Calculated

Figure 7.1 12 Comparison of measured and calculated peak deﬂection and time lag for
case 2 (three layer backcalculation)

7.2.5.3 Dynamic T ime-domain Backcalculation using Traces of Time History

First, the relation between convergence and a is investigated; Figure 7.113 shows that an

or—value of 3 or 4 is appropriate.

278

 

 

 

 

Iteration Number
8
l
l
l
l

 

 

 

1hreshold number
—+— Hysteretic damping +Viscous damping

Figure 7.113 Iteration number to convergence versus or for Kansas site

In this following section, the combined proﬁle listed in Table 7.24 will be used in the
backcalculation analysis. Figure 7.114 to Figure 7.121 show the traces of time histories
for different sensors. Figure 7.122 demonstrates that the three models give close
prediction for this site. The difference between the DYNABACK with hysteretic and
viscous damping is -3% for AC, -3% for base, and 4% for subgrade. The difference
between the backcalculated moduli from FEM and those from DYNABACK with

hysteretic damping is —2% for AC, and 45% for base, and —6% for subgrade.

 

 

 

 

 

 

 

 

Deﬂection (mils)
N 0) #- 0| 0)
l l

 

 

 

21.4 23.4 25.4 27.4 29.4 31.4 33.4 35.4

Time (ms)
— D1 —measured —-— Dl hysteretic damping

+ D1 -viscous damping —.n— D1 -FEM

Figure 7.114 Comparison of measured and predicted deﬂection time histories for sensor 1

279

 

 

 

 

 

 

 

 

l

Deflection (mils)
O —|~ N 00 b GI 03

 

 

 

21.4 23.4 25.4 27.4 29.4 31.4 33.4 35.4
Time (ms)
-— DZ-measured _._. DZ-hysteretic damping
+ DZ-viscous damping + DZ-FEM

Figure 7.1 15 Comparison of measured and predicted deﬂection time histories for sensor 2

 

 

6
E 5
E 4
.8 3
§ 2
5 1
o
O
21.4 23.4 25.4 27.4 29.4 31.4 33.4 35.4
T1me(ms)
—D3-measured ——-— DS—hysteretic damping

-+— DB-viscous damping + D3-FEM

Figure 7.116 Comparison of measured and predicted deﬂection time histories for sensor 3

 

 

 

 

 

Deflection (mils)
N w
>

 

 

 

0 . .
21.4 23.4 25.4 27.4 29.4 31.4 33.4 35.4
Time (ms)
— D4»measured —-— D4-hysteretic damping

+ D4-viscous damping + D4-FEM

Figure 7.117 Comparison of measured and predicted deﬂection time histories for sensor 4

280

 

 

 

 

 

 

 

 

4
’u?
E 3 . _
.5 2 »—-——~ __.
25‘, _ _ _ __ ___-’
8
0 . . .
21.4 23.4 25.4 27.4 29.4 31.4 33.4 35.4
Time (ms)
— DS-measured —-— DS—hysteretic damping
+ DS-viscous damping —o— DS—F EM

Figure 7.118 Comparison of measured and predicted deﬂection time histories for sensor 5

 

 

 

 

 

 

 

 

 

4
In:
E 3 4— _ ﬂ
.5 2
4* 1- ...ﬁ .
8
0 . . .
21.4 23.4 25.4 27.4 29.4 31.4 33.4 35.4
Time (ms)
— DG-measured —-— DS-hysteretic damping

+ DG-viscous damping + DS-F EM

Figure 7.1 19 Comparison of measured and predicted deﬂection time histories for sensor 6

 

 

 

 

 

 

 

 

 

2
E15 7477* ﬁ—eieehk
.5 1 - _— —— — -
:8; v v
‘8' 0.5 —
0 r r . . .
21.4 23.4 25.4 27.4 29.4 31.4 33.4 35.4
Time (ms)
— D7-measured —-— hysteretic damping
+ D7-u'scous damping + D7-FEM

Figure 7.120 Comparison of measured and predicted deﬂection time histories for sensor 7

281

 

 

 

 

 

2
E
E,
C
.9
.4?
8
21.4 23.4 25.4 27.4 29.4 31.4 33.4 35.4
Time (ms)
— D8-measured —-—— D8-hysteretic damping
+ D8-viscous damping + D8-FEM

Figure 7.121 Comparison of measured and predicted deﬂection time histories for sensor 8

Modulus (psr)

1.00E+06
1.00E+05
1.00E+04 ,,

1.00E+03

Kansas

 

 

 

AC Base Subgrade

Layer

DYNABACK-hysteretic damping I DYNABACK-viscous damping I FEM I MICHBACK

Figure 7.122 Comparison of backcalculated moduli from different models

In the second part of the analysis, different seed parameters were used for the uniqueness

analysis (see Table 7.26). The backcalculated results are shown in Figure 7 . 123.

282

Table 7.26 Seed values used for Kansas data

 

 

 

Layer Case 1 Case 2 Case 3
Seed Seed Seed
Layer Modulus Digging Modulus Damping Modulus Damfatilng
(ksi) (ksi) (ksi)
AC 250 0.1 850 O. 15 100 0.2
Base 70 0.05 10 0.08 15 0.1
Subgrade 70 0.03 10 0.05 15 0.05

 

Kansas (hysteretic damping)

 

1.00E+06

1.00E+05 -

Modulus (psi)

1.00E+04 - ,

 

 

 

   

Base Subgrade

lFCase 1 I Case 271:] Case :1)

Figure 7.123 Comparison of backcalculated layer moduli using different seed values

In the third part of the analysis, the same proﬁle is used except that the AC layer
thickness is unknown. Various combinations of damping for base and subgrade, as listed
in Table 7.7, are used for the backcalculation. The results are shown in Figure 7.124
through Figure 7.128. Backcalculated moduli for all combinations are in agreement with
the reference case (Figure 7.122) meaning that for this site the choice of base and
subgrade damping ratio did not affect the backcalculated layer moduli. Also, the AC
damping ratio did not vary much (16% to 24%). Finally, the backcalculated thickness AC

varies from 4% to 8% from the true value, which is an excellent result.

283

Kansas 5 AC_Modulus

 

 

 

”0306 : MICHBACKresult
'3‘: T341
3. . , p.
3 1.00E+05 7: .7 . .53
‘3 ...é
3:32
1.00304 — ...:

 

 

l 2 3 4 5 6 7 8 9
Corrbination of Danping Ratio

Figure 7.124 Comparison of backcalculated AC modulus for different damping ratio
combinations of base and subgrade layers

 

 

 

 

 

 

 

Kansas ﬂ AC_darrping

0.5
.._, 0.4
3
w 0.3 *7 i - —
.E
g 0.2 .

0.1 -

0.0 -

1 2 3 4 5 6 7 8 9
Contination of Damping Ratio

Figure 7.125 Comparison of backcalculated AC damping ratio for different damping ratio
combinations of base and subgrade layers

284

Kansas AC_Thickness

 

Boring average thickness
13.0 a 7.7 if \

 

15"» B

”*3

in?
5..

142‘

Thickness (inch)

.7":

 

 

Combination of Danping Ratio

Figure 7.126 Comparison of backcalculated AC damping ratio for different damping ratio
combinations of base and subgrade layers

Kansas 5 Base_Modulus

 

Modulus (psi)
é.
é

 

1 .OOE+02

Contination of Damping Ratio

Figure 7.127 Comparison of backcalculated AC damping ratio for different damping ratio
combinations of base and subgrade layers

285

Kansas El Subgrade_Modulus

 

 

 

 

 

MICHBACK result

1.00E+05 '— \
r;
E:
W
g 1.00E+04
'0
O
E

1.00E+03

l 2 3 4 5 6 7 8 9
Combination ofDarrping Ratio

Figure 7.128 Comparison of backcalculated AC damping ratio for different damping ratio
combinations of base and subgrade layers

7.3 Discussion

The discrepancies between measured and calculated deﬂection basins can be attributed to
either measurement errors (both in deﬂection amplitude and time or arrival) or the

inability of the theory to produce realistic responses for backcalculation purposes.

Measurement errors could be random or systematic. No matter what the nature of the
error is, the consequence is a variation in the deﬂection. Moreover, truncated time records
cause systematic errors in the frequency-based backcalculation solution. This will result

in deﬂection basins that are different enough to change the backcalculation results.

While the program can theoretically backcalculate more parameters than typically
allowed, the use of ﬁeld data causes the program not to coverage. The program will then
select the parameters corresponding to the lowest RMS automatically, which will

potentially cause errors in the backcalculation results.

286

When using ﬁeld data, time-domain backcalculation is preferred over frequency-domain
backcalculation because the inaccurate regions of the FWD response time histories can be
ignored and because of the truncations typically imposed on sensor time records.
However, time-domain backcalculation is computationally much more intensive than
frequency-domain backcalculation. Finally the use of an interpolation scheme and a cut
off frequency in the forward calculation may potentially cause some errors in time-

domain backcalculation results.

111 conditioning is a serious issue in dynamic backcalculation when using ﬁeld data. How
to deal with it will determine not only the convergence in the backcalcultion process but
also the quality of matches between measured and calculated deﬂection time histories.
The use of the singular value decomposition (SVD) method with the appropriate
truncation of the smaller singular value does improve the convergence of dynamic
backcalculation solution and achieve better matches between measured and calculated
deﬂections. The recommended threshold for the relative allowable error is about 10’3 to
104 when using ﬁeld data. It should be noted that there is no need for truncation using
synthetic FWD data since the same forward program is used for backcalculation. The
ﬁeld backcalction results show that viscous damping does not necessary lead to more
accurate result than hysteretic damping. It may mean that viscous damping does not
describe ﬂexible pavement response better than hysteretic damping. While other
rheologic method may be better suited for describing the real response of pavement
materials, then use in the backcalculation problem is not possible at this time because of

the increased number of parameters to be backcalculated.

287

In summary, dynamic backcalculation of layer parameters using ﬁeld data presents some
serious challenges. The frequency-domain method can lead to large errors if the
measured FWD records are truncated before the motions ﬁilly decay in time, and the
time-domain methods produce mixed results. At this point, it is recommended that time-
domain solutions should be further explored when analyzing ﬁeld data, mainly because

of the truncation problem associated with the ﬁ'equency-domain solution.

Simultaneous backcalculation of layer moduli and thicknesses is a difﬁcult problem to
solve when using ﬁeld data. The results presented in this chapter showed that it is
possible to backcalculate the AC thickness (with some error) when using three-layer
pavement systems with assumed damping values for the base and subgrade. However,

this problem needs to be fully studied.

288

 

CHAPTER 8 - CONCLUSIONS AND RECOMMENDATIONS

8.1 Summary

In this study, a new method for backcalculating ﬂexible pavement layer parameters based
on dynamic interpretation of FWD deﬂection time histories using frequency and time-
domain solutions have been developed. The method allows for theoretically
backcalculating the layer moduli, damping ratios and thicknesses for a three to ﬁve- layer
system. The new associated program called DYNABACK has been written in the
FORTRAN 77 language, and offers two options: (i) frequency-domain analysis, and (ii)

time-domain analysis.

The new program uses the SAPSI program (Chen, 1987) as its forward routine. SAPSI
models the pavement structure as a system of layers that are inﬁnite in the horizontal
direction and underlain by an elastic half-space. The materials are assumed to be isotropic
and linearly elastic with hysteretic damping. Full interface bonding is assumed at the
layer interfaces. The mass densities and elastic moduli are assumed to be constant within
each layer. The steady-state solution in SAPSI is used for the frequency-domain

backcalculation, while the transient solution is used for the time-domain backcalculation.

The dynamic backcalculation procedure is based on the modiﬁed Newton-Raphson
method originally adopted in the MICHBACK program (Mahmood, 1993). Either the
least squares method or singular value decomposition (SVD), with scaling and truncation
can be used to solve the over determined set of equations. In the MICHBACK solution,

this set of equations is real-valued and correspond to the peak deﬂection values, since the

289

backcalculation scheme uses a static solution (CHEVRONX) to predict the deﬂection
basin. In the frequency-domain solution, the equations are complex-valued and
correspond to the steady-state solution at one or multiple frequencies. In the time-domain
solution, the real-valued equations are expanded to correspond to the peak transient
deﬂections and their corresponding time lags relative to the peak load or to include traces

of time history near the peaks.

In addition, methods for estimating the depth to stiff layer and the seed subgrade
modulus, proposed by Roesset (1995) and Lee et.al. (1998), respectively, have been

adopted with some modiﬁcations and are implemented in the new program.

The new program has been incorporated into the WindowsTM based MFPDS program,
which allows for user—friendly features including interactive input and output screens,

and the ability to view and process the deﬂection data before analyzing it.

The new program was theoretically veriﬁed using synthetic data, and its application to
mechanistically-based pavement design and rehabilitation was evaluated using ﬁeld

FWD data.

For the theoretical veriﬁcations, time histories of FWD surface deﬂections generated
from SAPSI were used to verify the capabilities of the newly developed dynamic
backcalculation program. The backcalculation was done using both frequency and time-
domain solutions. Various pavement proﬁles of different combinations of layer
thicknesses and moduli with up to ﬁve layers were analyzed. Some proﬁles included

cases where there was a shallow bedrock or ground water table. In addition to conducting

290

 

a sensitivity analysis, the effects of signal truncations in time and imprecision of the

measured sensor deﬂections were also investigated theoretically.

To evaluate the applicability of the DYNABACK to interpret ﬁeld tests, measured
deﬂection time history data from several FWD tests conducted in Michigan and
elsewhere were analyzed. The selected pavement test sections included sites in Texas,
Cornell University, Florence (Italy), Michigan and a SPS-l site in Kansas. For the Texas
site, different load levels were considered. The analyses included the comparison of
backcalculated layer moduli and damping ratios with MICHBACK results for various
pavement sections and load levels. The backcalculation was done in both ﬁ'equency and
time domains, where the time-domain solution included backcalculating layer moduli and
thicknesses. The data were obtained from tests involving KUAB and Dynatest FWD
machines. Most pavement sections were analyzed as three- and four- layer systems with

some sections involving a stiff layer at shallow depth.

8.2 Conclusions

Based on the theoretical veriﬁcation analysis, the following conclusions were drawn for

frequency-domain backcalculation:

l. The backcalculation results are all in excellent agreement with the true values.
Both the average root mean square error (RMS) on the calculated and actual
deﬂection basins, and the relative errors on layer moduli and thicknesses are
practically zero, indicating that the program has the ability of backcalculate the

moduli and thicknesses accurately.

291

2. Theoretical backcalculation shows that among the modulus, damping ratio,
thickness and Poisson’s ratio, the modulus is the easiest to backcalculate followed

by damping ratio, thickness and Poisson’s ratio.

3. Theoretical backcalculation shows that the frequency backcalculation program
gives satisfactory convergence of layer moduli and thicknesses when using
untruncated deﬂection time histories. However backcalculation results at higher

frequencies are less accurate than those obtained at low frequencies.

4. Although Poisson’s ratio of the AC layer is frequency-dependent, assuming a
constant value for it will not affect the results signiﬁcantly because the

backcalculated results are not sensitive to reasonable variations in this parameter.

5. The frequency response-based backcalculation method can lead to large errors in
deﬂection basins if the FWD records are truncated before the motions fully decay

in time. The errors due to sensor imprecision were found to be less signiﬁcant.

The following conclusions were drawn from the theoretical veriﬁcation analysis for time-

domain backcalculation:

6. Backcalculation based on synthetic time histories generated by SAPSI shows
excellent stability and accuracy, therefore Newton-Raphson method could be used

with the time-domain backcalculation.

7. The time-domain approach can match selected features of the measured time

histories directly, and ignore the inaccurate measurement regions in time.

292

 

Therefore, from this point of view, the time-domain backcalculation is better than

the frequency-domain backcalculation.

8. Numerical examples have illustrated that the method is able to backcalculate layer
moduli and thicknesses accurately from synthetically generated FWD data for a
three layer pavement system. Backcalculation of layer damping ratios are less

accurate, but the inﬂuence of this error on the pavement response is insigniﬁcant.

In terms of ﬁeld evaluation of the new backcalculation solutions, the results were not
satisfactory. The discrepancies between measured and calculated deﬂection basins can be
attributed to several factors including:
— Sensor measurement errors;
— Time synchronization errors in the data acquisition systems for sensor
measurements;
— Truncated time records, which cause systematic errors in the frequency-based
backcalculation solution;

— Improper characterization of damping effects.

While the program can theoretically backcalculate more parameters than typically
allowed, using ﬁeld data causes the program not to converge. The program will instead
select the parameters corresponding to the lowest RMS automatically, which will

potentially cause errors in the backcalculation results.

The following conclusions were reached from the analysis involving ﬁeld FWD data:

293

1. When using ﬁeld data, time-domain backcalculation is preferred over
frequency-domain backcalculation because the inaccurate regions of the FWD
response time histories can be ignored and because of the truncations typically

imposed on sensor time records.

2. Ill conditioning is a serious issue in dynamic backcalculation when using ﬁeld
data. The use of singular value decomposition (SVD) method with scaling and
appropriate truncation of the smallest singular values does improve the
convergence of the solution and achieve better matches between measured and
calculated time histories. The recommended threshold for the relative allowable

error is about 10'3 to 10".

3. Simultaneous backcalculation of layer moduli and thicknesses is a difﬁcult
problem to solve when using ﬁeld data. However, the results presented in this
research show that it is possible to backcalculate the AC layer thickness (with
some error) when using three-layer pavement system with assumed damping

value for the base and subgrade.

8.3 Recommendations

Based on the results from this research, the following recommendations are made:

1. Dynamic backcalculation of layer parameters using ﬁeld data presents some
serious challenges. The frequency-domain method can lead to large errors if the
measured FWD records are truncated before the motions fully decay in time, and

the time-domain methods produce mixed results. At this point, it is recommended

294

 

that time-domain solutions are used when analyzing ﬁeld data, mainly because of

the truncation problem associated with the frequency-domain solution.

. Simultaneous backcalculation of layer moduli and thicknesses is a difﬁcult

problem to solve when using ﬁeld data. This problem needs to be studied further.

. Determining the depth to bedrock and the depth to ground water table requires
the recording of free vibrations from FWD tests. The recorded time histories
from existing FWD system usually not long enough for this purpose. Therefore

their duration need to be increased to allow for at least two free vibration cycles.

. Temperature is a very important factor affecting the behavior of asphalt concrete
layer. Incorperating the variation of AC layer modulus as a function of

temperature with depth should be considered in the future research.

. SAPSI computer program is based on the assumption of linear Viscoelastic
behavior. In reality asphalt concrete is a nonlinear Viscoelastic material, and its
response depends on load level and duration as well as temperature. Also
nonlinear characteristics for unbound granular materials and ﬁne-grained soil
materials should be considered. This non-linear effect should be considered in

future research.

. The effect of Poisson’s ratio is another factor for backcalculation. Poisson’s ratio

is a function of temperature and frequency of loading. The current

backcalculation solution assumes a constant real value for poisson’s ratio. Other

295

rhelogic model that account for frequency-dependent complex moduli and

poisson’s ratio should be considered.

7. There is a need for investigating the relationship between damping obtained ﬁ'om

backcalculation and pavement distress.

296

 

10.

ll.

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