Backcalculation of Layer Parameters for LTPP Test

Sections, Volume I: Slab on Elastic Solid and Slab on

Dense-Liquid Foundation Analysis of Rigid

Pavements

PUBLICATION NO. FHWA-RD-00-086 JANUARY 2001

Research and Development

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, Virginia 22101-2296
FOREWORD

This report documents a study conducted to backcalculate layer material parameters for rigid

pavements included in the Long Term Pavement Performance (LTPP) program. Using the “bestfit” algorithm, layer material properties were backcalculated using both the slab on elastic solid

foundation and the slab on dense-liquid foundation models. The analysis was conducted for all

General Pavement Studies (GPS), Special Pavement Studies (SPS), and Seasonal Monitoring

Program (SMP) test sections. Data tables that include the backcalculation results were developed

for inclusion in the LTPP Information Management System (IMS).

T. Paul Teng, P. E.

Director

Office of Infrastructure

 Research and Development

NOTICE

This document disseminated under the sponsorship of the Department of Transportation in the

interest of information exchange. The United States Government assumes no liability for its

contents or use thereof. This report does not constitute a standard, specification, or regulation.

The United States Government does not endorse products or manufacturers. Trade and

manufacturers’ names appear in this report only because they are considered essential to the

object of the document.
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BACKCALCULATION OF LAYER PARAMETERS FOR LTPP

TEST SECTIONS, Volume I: Slab on Elastic Solid and Slab on

Dense-Liquid Foundation Analysis of Rigid Pavements

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Lev Khazanovich, Shiraz D. Tayabji, and Michael I. Darter

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ERES Consultants, Inc.

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Office of Infrastructure Research and Development

Federal Highway Administration

6300 Georgetown Pike

McLean, Virginia 22101-2296

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Final Report

Aug 1997 to July 1998

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Contracting Officer’s Technical Representative (COTR): Cheryl Allen Richter, HRDI-13

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This report documents the results of backcalculation of layer material properties for rigid pavements included in the Long

Term Pavement Performance (LTPP) program in the United States and Canada using deflection testing data.

This study backcalculated the layer material properties for rigid pavements using the slab on elastic solid foundation and the

slab on dense-liquid foundation procedures. The "best fit" algorithm was used after consideration of alternative methods of

backcalculation. Pre-processing and post-processing utility software were developed to facilitate data handling. The

backcalculation analysis was conducted for all General Pavement Studies (GPS), Special Pavement Studies (SPS), and

Seasonal Monitoring Program (SMP) test sections. Data tables that include backcalculation parameters were developed for

inclusion in the LTPP Information Management System (IMS). Key findings include the following:

1. The "best fit" method was selected as the primary backcalculation method for both dense-liquid (DL) and elastic

solid (ES) subgrade models.

2. Reasonable backcalculation results were obtained for the large majority of GPS, SPS, and SMP sections. Typical

modulus values and ranges are provided for the PCC slab, many types of bases, and the subgrade.

3. Strong correlations were found between backcalculated parameters using DL and ES subgrade models.

4. Temperature curling during the day had a profound effect on the results of backcalculation [making it important to

conduct falling weight deflectometer (FWD) basin testing early in the morning to reduce variability in

backcalculated values].

5. Poor correlation was found between backcalculated and laboratory elastic moduli of the concrete slab.

6. A bonded interface between the slab and base produced the best layer moduli for a large majority of the sections

(center slab backcalculation).

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Backcalculation analysis, concrete pavements,

deflection testing, falling weight deflectometer,

LTPP, pavement testing, rigid pavements.

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No restrictions. This document is available to the

public through the National Technical Information

Service, Springfield, VA 22161.

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iii

TABLE OF CONTENTS

Section Page

1. INTRODUCTION ................................................................ 1

Deflection Testing Details ...................................................... 1

Scope of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Report Organization ........................................................... 2

2. SELECTION OF BACKCALCULATION METHODOLOGY FOR RIGID PAVEMENTS ...... 3

Backcalculation Algorithms for DL Foundation ..................................... 3

Best Fit Procedure for the Elastic Solid (ES) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Effect of a Base Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Effect of Sensor Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Slab Size Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Acceptability of the Results of Backcalculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Backcalculation for LTPP Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3. BACKCALCULATION FOR GPS AND SPS SECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Selection of Pavement Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Backcalculation of FWD Deflection Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Effect of Load Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Variability in Backcalculated Parameters Along Section Length . . . . . . . . . . . . . . . . . . . . . . . 42

Subgrade Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Radii of Relative Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Bonding Condition Between PCC Slab and Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Backcalculated PCC Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Backcalculations for Unbonded PCC Overlay Sections (GPS-9) . . . . . . . . . . . . . . . . . . . . . . . 56

Backcalculated Base Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Variability in Backcalculated Parameters Between FWD Visits . . . . . . . . . . . . . . . . . . . . . . . . 59

Backcalculation Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4. BACKCALCULATION FOR SMP SECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Selection of Pavement Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Backcalculation of FWD Deflection Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Effect of Load Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Variability in Backcalculated Parameters Along Section Length . . . . . . . . . . . . . . . . . . . . . . . 67

Subgrade Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Radii of Relative Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bonding Condition Between PCC Slab and Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Backcalculated PCC Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Effect of Seasonal Variation and Time of Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Backcalculation Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5. LIMITATIONS OF CURRENT BACKCALCULATION PROCEDURES . . . . . . . . . . . . . . . . . . . 91

Comparison of Backcalculated and Laboratory PCC Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Slab Curling Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Dynamic Interpretation of FWD Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Dynamic Effects on Results of Backcalculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
iv

TABLE OF CONTENTS (Continued)

Section Page

6. SUMMARY AND RECOMMENDATIONS FOR CONTINUED RESEARCH . . . . . . . . . . . . . . . 99

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

APPENDIX A. TYPICAL RESULTS OF BACKCALCULATION FOR PCC LTPP GPS

SECTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
v

LIST OF FIGURES

Figure Page

 1 Backcalculated dynamic k-value for LTPP concrete pavement sections, Best Fit 7 versus

Best Fit 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

 2 Backcalculated dynamic k-value for LTPP concrete pavement sections, Best Fit 4 versus

AREA4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

 3 Backcalculated dynamic k-value for LTPP concrete pavement sections, AREA7 versus

Best Fit 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

 4 Backcalculated dynamic k-value from the theoretical deflection basins, Best Fit 7 versus

Best Fit 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

 5 Backcalculated dynamic k-value from the theoretical deflection basins, AREA4 versus

Best Fit 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

 6 Backcalculated dynamic k-value from the theoretical deflection basins, AREA7 versus

Best Fit 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

 7 Coefficient of variation in backcalculated k-value for multiple load drops and load

level, LTPP concrete pavement sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

 8 Effect of modular ratio on backcalculated PCC modulus of elasticity . . . . . . . . . . . . . . . . . . . . . 18

 9 Comparison of backcalculated PCC moduli for two sets of modular ratio, bonded interface

between PCC plate and base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

 10 Comparison of backcalculated PCC moduli for two sets of modular ratio, unbonded interface

between PCC plate and base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

 11 Effect of modular ratio on backcalculated PCC modulus of elasticity . . . . . . . . . . . . . . . . . . . . . 19

 12 Backcalculated dynamic k-value for LTPP SMP sections, Best Fit 7 versus Best Fit 4 . . . . . . . . 22

 13 Backcalculated E for LTPP SMP sections, Best Fit 7 versus Best Fit 4 methods PCC . . . . . . . . . . 23

 14 Distribution of k-values from Best Fit 4 method for the FWD passes for which Best Fit 7

was not successful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

 15 Distribution of E from Best Fit 4 method for the FWD passes for which Best Fit 7 was PCC

not successful . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

 16 Cumulative chart of mean absolute values of relative errors for LTPP GPS and SPS sections

(DL subgrade model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

 17 Cumulative chart of mean absolute values of relative errors for LTPP GPS and SPS sections (ES

subgrade model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

 18 Comparison of mean k-values with 2 and 5 percent cutoff limit for deflection basin acceptance

for LTPP GPS section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

 19 Comparison of mean k-values with 2 and 5 percent cutoff limit for deflection basin acceptance

for LTPP SPS section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

 20 Comparison of mean k-values with 5 and 10 percent cutoff limit for deflection basin acceptance for

LTPP GPS section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

 21 Comparison of mean k-values with 5 and 10 percent cutoff limit for deflection basin acceptance for

LTPP SPS section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

 22 Variation of AREA with radius of relative stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

 23 Percentage of rejected backcalculation basins for FWD visits for GPS sections . . . . . . . . . . . . . 39

 24 Frequency distribution of backcalculated k-values for GPS LTPP sections (screened sections are

those with less than 30 percent deflection basins rejected) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

 25 Frequency distribution of backcalculated k-values for SPS LTPP sections . . . . . . . . . . . . . . . . . 40

 26 Coefficient of variation in backcalculated k-value for multiple load levels (GPS and SPS 

sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
vi

LIST OF FIGURES (Continued)

Figure Page

 27 Coefficient of variation in radius of relative stiffness (DL model) for multiple load levels

(GPS and SPS sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

 28 Coefficient of variation in backcalculated modulus of elasticity of subgrade for multiple

load levels (GPS and SPS sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

 29 Coefficient of variation in radius of relative stiffness (ES model) for multiple load levels

(GPS and SPS sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

 30 Coefficient of variation in backcalculated parameters along project length (DL model) . . . . . . . 44

 31 Coefficient of variation in backcalculated parameters along project length (ES model) . . . . . . . 44

 32 Backcalculated k-values for section 537409 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

 33 Backcalculated modulus of elasticity of subgrade for section 553014 . . . . . . . . . . . . . . . . . . . . . 45

 34 Frequency distribution of backcalculated k-values for GPS and SPS LTPP sections . . . . . . . . . . 47

 35 Frequency distribution of backcalculated subgrade elastic modulus for GPS and SPS LTPP

sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

 36 Comparison of mean k-values of fine- and coarse-grained subgrades (GPS and SPS sections) . . . 48

 37 Comparison of mean moduli of elasticity of fine- and coarse-grained subgrades (GPS and SPS

sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

 38 Subgrade modulus of elasticity versus coefficient of subgrade reaction (GPS and SPS

sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

 39 Frequency distribution of backcalculated k-values for GPS and SPS LTPP sections . . . . . . . . . . 50

 40 Frequency distribution of backcalculated subgrade modulus of elasticity for GPS and

SPS LTPP sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

 41 ES radius of relative stiffness versus DL radius of relative stiffness . . . . . . . . . . . . . . . . . . . . . . 51

 42 Distribution of bond/no bond interface condition for GPS and SPS LTPP sections . . . . . . . . . . . 53

 43 Distribution of bond/no bond interface condition for GPS and SPS LTPP sections with

stabilized bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

 44 Frequency distribution of backcalculated PCC modulus of elasticity for GPS and SPS

LTPP sections (DL model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

 45 Frequency distribution of backcalculated PCC modulus of elasticity for GPS and SPS

LTPP sections (ES model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

 46 Backcalculated PCC modulus of elasticity (ES model) versus backcalculated PCC modulus

of elasticity (DL model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

 47 Coefficient of variation in backcalculated parameters (DL model) for GPS and SPS LTPP

sections between different FWD visits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

 48 Coefficient of variation in backcalculated parameters (ES model) for GPS and SPS LTPP

sections between different FWD visits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

 49 Percentage of accepted backcalculation basins for FWD passes of SMP LTPP sections . . . . . . . . 64

 50 Coefficient of variation in backcalculated k-values for multiple load levels at a particular location

(SMP sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

 51 Coefficient of variation in radius of relative stiffness (DL model) for multiple load levels at 

a particular location (SMP sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

 52 Coefficient of variation in backcalculated modulus of elasticity of subgrade for multiple

load levels at a particular location (SMP sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

 53 Coefficient of variation in radius of relative stiffness (DL model) for multiple load levels at 

a particular location (SMP sections) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

 54 Coefficient of variation in backcalculated parameters along project length for SMP sections 

(DL model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
vii

LIST OF FIGURES (Continued)

Figure Page

 55 Coefficient of variation in backcalculated parameters along project length for SMP sections

(ES model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

 56 Frequency distribution of backcalculated k-values for SMP LTPP sections . . . . . . . . . . . . . . . . . 69

 57 Frequency distribution of backcalculated subgrade elastic modulus for SMP LTPP sections . . . 70

 58 Subgrade modulus of elasticity versus coefficient of subgrade reaction (SMP sections) . . . . . . . 70

 59 Frequency distribution of backcalculated radius of relative stiffness (DL model) for

SMP LTPP sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

 60 Frequency distribution of backcalculated radius of relative stiffness (ES model) for

SMP LTPP sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

 61 ES radius of relative stiffness versus DL radius of relative stiffness (SMP sections) . . . . . . . . . . . 72

 62 Distribution of bond/no bond interface condition for non-stabilized and stabilized 

SMP LTPP sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

 63 Distribution of bond/no bond interface condition for SMP LTPP sections with stabilized

bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

 64 Frequency distribution of backcalculated PCC modulus of elasticity (DL model) for SMP

LTPP sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

 65 Frequency distribution of backcalculated PCC modulus of elasticity (ES model) for SMP

LTPP sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

 66 Backcalculated PCC modulus of elasticity (ES model) versus backcalculated PCC modulus

of elasticity (DL model) for SMP sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

 67 Daily variation in backcalculated k-value, section 133019 (April 1996) . . . . . . . . . . . . . . . . . . . 77

 68 Daily variation in backcalculated subgrade modulus of elasticity, section 133019

(April 1996) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

 69 Daily variation in backcalculated PCC modulus of elasticity (DL model), section 133019

(April 1996) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

 70 Daily variation in backcalculated PCC modulus of elasticity (ES model), section 133019

(April 1996) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

 71 Daily variation in backcalculated k-value, section 493011 (July 1994) . . . . . . . . . . . . . . . . . . . . 80

 72 Daily variation in backcalculated subgrade modulus of elasticity, section 493011

(July 1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

 73 Daily variation in backcalculated PCC modulus of elasticity (DL model), section 493011

(July 1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

 74 Daily variation in backcalculated PCC modulus of elasticity (ES model), section 493011

(July 1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

 75 Daily variation in backcalculated k-value, section 274040 (March 1994) . . . . . . . . . . . . . . . . . . 82

 76 Daily variation in backcalculated subgrade modulus of elasticity, section 274040

(March 1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

 77 Daily variation in backcalculated PCC modulus of elasticity (DL model), section 274040

(March 1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

 78 Daily variation in backcalculated PCC modulus of elasticity (ES model), section 274040

(March 1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

 79 Seasonal variation in backcalculated subgrade moduli, section 133019 . . . . . . . . . . . . . . . . . . . . 85

 80 Seasonal variation in backcalculated PCC modulus of elasticity, section 133019 . . . . . . . . . . . . 85

 81 Seasonal variation in backcalculated subgrade moduli, section 533813 . . . . . . . . . . . . . . . . . . . . 86

 82 Seasonal variation in backcalculated PCC modulus of elasticity, section 533813 . . . . . . . . . . . . 86

 83 Seasonal variation in backcalculated subgrade moduli, section 274040 . . . . . . . . . . . . . . . . . . . . 87
viii

LIST OF FIGURES (Continued)

Figure Page

 84 Seasonal variation in backcalculated PCC modulus of elasticity, section 274040 . . . . . . . . . . . . 87

 85 Comparison of backcalculated and laboratory concrete moduli of elasticity . . . . . . . . . . . . . . . . 92

 86 Quasi-static pavement responses to FWD loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

 87 Measured pavement responses to FWD loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

 88 Effect of pavement inertia on FWD sensor deflections, m*=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

 89 Effect of pavement inertia on FWD sensor deflections, m*=4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

 90 Effect of pavement inertia on FWD sensor deflections, m*=10 . . . . . . . . . . . . . . . . . . . . . . . . . . 96

 91 Effect of pavement inertia on the results of backcalculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

 92 Typical data plot legend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

 93 Backcalculated k-values for section 013998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

 94 Backcalculated radius of relative stiffness (DL model) for section 013998 . . . . . . . . . . . . . . . . 107

 95 Backcalculated PCC elastic modulus (DL model) for section 013998 . . . . . . . . . . . . . . . . . . . . 108

 96 Backcalculated elastic modulus of base (DL model) for section 013998 . . . . . . . . . . . . . . . . . . 108

 97 Backcalculated modulus of elasticity of subgrade for section 013998 . . . . . . . . . . . . . . . . . . . . 109

 98 Backcalculated radius of relative stiffness (ES model) for section 013998 . . . . . . . . . . . . . . . . 109

 99 Backcalculated PCC elastic modulus (ES model) for section 013998 . . . . . . . . . . . . . . . . . . . . 110

 100 Backcalculated elastic modulus of base (ES model) for section 013998 . . . . . . . . . . . . . . . . . . 110

 101 Backcalculated k-values for section 015008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

 102 Backcalculated radius of relative stiffness (DL model) for section 015008 . . . . . . . . . . . . . . . . 112

 103 Backcalculated PCC elastic modulus (DL model) for section 015008 . . . . . . . . . . . . . . . . . . . . 113

 104 Backcalculated elastic modulus of base (DL model) for section 015008 . . . . . . . . . . . . . . . . . . 113

 105 Backcalculated modulus of elasticity of subgrade for section 015008 . . . . . . . . . . . . . . . . . . . . 114

 106 Backcalculated radius of relative stiffness (ES model) for section 015008 . . . . . . . . . . . . . . . . 114

 107 Backcalculated PCC elastic modulus (ES model) for section 015008 . . . . . . . . . . . . . . . . . . . . 115

 108 Backcalculated elastic modulus of base (ES model) for section 015008 . . . . . . . . . . . . . . . . . . 115
ix

LIST OF TABLES

Table Page

 1 Comparison of LTPP and DIPLOMAT results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

 2 Proposed moduli ratios, E /E PCC base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

 3 Bond index assigning rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

 4 Exceptions in assignment of base parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

 5 Effect of base and interface condition on backcalculated moduli of elasticity . . . . . . . . . . . . . 52

 6 Example of change of interface condition from test to test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

 7 Backcalculated PCC moduli of elasticity for unbonded overlay sections . . . . . . . . . . . . . . . . . 57

 8 Mean, maximum, and minimum backcalculated base moduli for GPS and SPS LTPP sections

(DL model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

 9 Mean, maximum, and minimum backcalculated base moduli for GPS and SPS LTPP sections

(ES model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

 10 Coefficients of variation of the mean values of backcalculated parameters from the same

day of testing (DL model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

 11 Coefficients of variation of the mean values of backcalculated parameters from the same

day of testing (ES model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

 12 Coefficients of variation of the mean values of backcalculated parameters from first FWD

passes (DL model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

 13 Coefficients of variation of the mean values of backcalculated parameters from first FWD

passes (ES model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
x
 1

CHAPTER 1. INTRODUCTION 

An extensive data collection effort has been under way since 1989 for the Long Term Pavement 

Performance (LTPP) program. As part of the monitoring data collection, falling weight deflectometers 

(FWDs) are being used to obtain deflection measurements at General Pavement Study (GPS) and Special 

Pavement Study (SPS) test sections in the United States and Canada. The deflection response of the 

pavement to an applied load is an important indicator of the structural capacity, material properties, and 

seasonal variations of the pavement. In the LTPP program, deflection testing is conducted periodically 

(every few years) at GPS and SPS sections. However, at the 64 sections designated for the Seasonal 

Monitoring Program (SMP), FWD testing is conducted more frequently, about 12 to 14 times per year. 

The backcalculation of material response parameters for each layer in a rigid pavement structure using 

the deflection data has been performed using two approaches: 

1. Backcalculation of layer material properties using elastic layer based procedures. This 

approach was used for both flexible and rigid pavements. Under this approach, Program 

MODCOMP was used. This work was published as a separate report.(1)

2. Backcalculation of layer material properties using the slab on elastic solid (ES) or denseliquid (DL) foundation based procedures. This approach is used for rigid pavements 

only. 

This report presents the results of the backcalculation analysis conducted for rigid pavement sections in 

the LTPP program using the slab on elastic solid and dense-liquid foundation based procedures. 

In the past decade, much progress has been made in the development of reliable methods for 

backcalculation of concrete slab and foundation moduli from deflection measurements. Recently, studies 

conducted by Darter et al.(1) and Hall et al.(2) made significant contributions to the improvement of rigid 

pavement backcalculation procedures. Nevertheless, backcalculation for rigid pavements remains a 

challenging problem. To obtain realistic results from backcalculation, a thorough analysis of all factors 

is required. The effects of sensor configuration, base layer, joint spacing, and temperature conditions on 

the backcalculation results should be taken into account. 

Deflection Testing Details 

Backcalculation of rigid pavements utilizes the deflection basins measured at the center of the slab. The 

LTPP deflection basin testing uses seven deflection sensors placed 0, 203, 305, 457, 610, 914, and 1524 

mm from the center of the load plate to define the shape of the deflection basin. 

The load sequence, as stored in the database, for rigid pavement testing is as follows: 

Height No. of Drops Target Load, kN Acceptable Range, kN 

2 

3 

4 

4 

4 

4 

40.0 

53.3 

71.1 

36.0 to 44.0 

48.1 to 58.7 

64.1 to 78.3 

For jointed rigid pavements, deflection basin tests are performed along the mid-lane path at each tested slab, 

and the test locations are designated as J1. The number of panels can vary from as few as 9 or 10 to as 

many as 35 or more on a 152.4-m-long section. Regardless of the total number of panels present, no more 

than 20 panels are tested at a section. 
 2

For the continuously reinforced concrete pavements (CRCP), deflection basin tests are also performed along 

the mid-lane path at a spacing of about 7.6 m, and the test locations are designated as C1. These tests are 

performed at mid-length of an effective panel, which is defined by two adjacent transverse cracks typically 

at a spacing of 0.3 to 2.5 m. For CRCP sections, tests are performed at 20 effective panels. 

It should be noted that slab temperature gradient measurements are conducted at the time of the deflection 

testing. However, no attempt was made to address temperature conditions at the time of testing in the 

backcalculation analysis. For rigid pavements, slab curling does play a critical role in FWD deflection 

measurements. As such, any interpretation of backcalculation results should account for the effect of slab 

curling. 

Scope of Work 

The scope of the backcalculation analysis study for LTPP rigid pavement sections included the following: 

1. Selection of one or more procedures to compute the modulus of elasticity of the concrete 

slab, the base course, and the elastic solid foundation and also to compute the modulus of 

subgrade reaction of the foundation. 

2. Modification, if necessary, of the selected procedures to meet the specific needs of the 

LTPP program. 

3. Development of pre-processing and post-processing utility software to facilitate data 

handling. 

4. Performing backcalculation analysis for SMP, GPS, and SPS test sections. 

5. Development of data tables for uploading appropriate backcalculation analysis results to 

the LTPP Information Management System (IMS). 

6. Performing preliminary assessment of the backcalculated material properties. 

Report Organization 

This report documents the research effort and findings of the LTPP rigid pavement backcalculation analysis 

effort. Chapter 1 discusses background information. Chapter 2 provides details on the selection of the 

backcalculation methodology for rigid pavements. Chapter 3 presents results of the backcalculation analysis 

for GPS and SPS test sections. Chapter 4 presents results of the backcalculation analysis for SMP test 

sections. Chapter 5 discusses limitations of the current backcalculation analysis procedures. A summary 

and recommendations are presented in chapter 6. 

Typical examples of the results from the backcalculation analysis are given in appendix A for GPS test 

sections. 
 3

CHAPTER 2. SELECTION OF BACKCALCULATION METHODOLOGY 

FOR RIGID PAVEMENTS

Several methods for backcalculating the portland cement concrete (PCC) slab, base, and subgrade moduli 

or moduli of subgrade reaction (k) are available. Each method has its strengths and its limitations. The 

following procedures are typically considered for rigid pavements: 

• Backcalculation software and procedures based on elastic layer analysis typically used for 

flexible pavements. 

• Backcalculation procedures specifically developed for rigid pavements that are based on slab 

on elastic solid or slab on dense-liquid models: 

− AREA method-based procedures. 

− Best fit-based procedures. 

The following issues are addressed relative to backcalculation for rigid pavements: 

• AREA method versus Best Fit method: Which should be used? 

• Sensor configurations: Should outer sensors be omitted? 

• Effect of a base layer: How should it be accounted for? 

• Slab size effect: What slab size correction, if any, should be applied? 

• Temperature effects: Should temperature curling correction factors be used? If so, what 

should they be? 

This report does not address the issue of which is the most realistic subgrade characterization method: 

ES model versus DL model. In this study, both models were used. The backcalculated parameters 

obtained using DL and ES models are compared. Several correlations between the models were found, 

and these correlations may be useful for development of subgrade characterization guidelines for the 

2002 Design Guide being developed under National Cooperative Highway Research Program (NCHRP) 

Project 1-37A. 

Backcalculation Algorithms for DL Foundation 

Two backcalculation procedures based on plate theory were evaluated in this study. The Best Fit method 

solves for a combination of the radius of relative stiffness, l, and the coefficient of subgrade reaction, k, 

that produces the best possible agreement between the predicted and measured deflections at each sensor. 

The AREA method is included in the 1993 AASHTO Guide and estimates the radius of relative stiffness 

as a function of the AREA of the deflection basin. This estimation, along with the subsequent 

calculation of subgrade reaction, k, and slab modulus of elasticity, E, is made using simple closed form 

equations. Both methods are based on Westergaard’s solution for the interior loading of a plate 

consisting of a linear elastic, homogeneous, and isotropic material resting on a dense-liquid foundation. 

Under a load distributed uniformly over a circular area of radius, a, the distribution of deflections, w(r), 

may be written as:(3)

( ) f (r, l) k

p w r = (1) 

f r C a ber s C a bei s for r a ( ) =1 − 1( l ) ( ) − 2 ( l ) ( ) 0 < < (2) 
 4

f r C a s C a kei s for r a ( ) = 3 ( l ) ker( ) + 4 ( l ) ( ) > (3) 

where 

al = (a/l) 

 = dimensionless radius of the applied load 

r = radial distance measured from the center of the load 

s = (r/l) 

 = normalized radial distance 

l = (D/k)1/4

 = radius of relative stiffness of plate-subgrade system for the 

dense-liquid foundation 

D = Eh3

/12(1-μ2) 

 = flexural rigidity of the plate 

E = plate elastic modulus 

μ = plate Poisson’s ratio 

h = plate thickness 

k = modulus of subgrade reaction 

p = applied load intensity (pressure) = P/(πa

2

) 

P = total applied load 

Note that ber, bei, ker, and kei are Kelvin Bessel functions that may be evaluated using appropriate 

series expressions available in the literature.(3)

A method for determining the constants C1 through C4 has been proposed by Ioannides.(3) However, that 

method is tedious and is valid only for a relatively small radius of the applied load. A more general and 

simple solution has been proposed by Korenev, who suggests that these constants have the following 

form for any value of the radius of the applied load:(4)

C aλ aλ ker 1 = − ′ (4) 

λ aλ C = a kei′ 2 (5) 

λ aλ C = − a ber′ 3 (6) 

λ aλ C = − a bei′ 4 (7) 

Where ker’, kei’, ber’, and bei’ are the first derivatives of the functions ker, kei, ber, and bei, 

respectively. 

Best Fit Algorithm

The Best Fit backcalculation algorithm finds a combination of concrete elastic modulus and subgrade kvalue for which the calculated deflection profile closely matches the measured profile.(2,5) The problem 

is formulated as the minimization of the error function, F, defined as follows: 

FEk wr W ( ) ii i ( ) ( )

i

n

, = −

=

∑α

2

0

 (8) 
 5

where αi is the weighting factor, w(ri) is the calculated deflection, and Wi is the measured deflection. The 

weighting factor might be set equal to 1, or (1/Wi)

2

, or any other number. The ability to control the 

weights given to the various deflection measurements adds some flexibility to the Best Fit solution 

process. 

Using equation 1, the error function, F, can be presented in the following form: 

( ) () ∑ ( ) =





 



 ≡ = −

n

i

i i Wi f l

k

p F E k F l k

0

2

, , α (9) 

To obtain the minimum of the error function, F, the following conditions should be satisfied: 

= 0

k

F

∂

∂ (10) 

∂

∂

F

l

= 0 (11) 

Substitution of the error function equation into the equation for the first condition yields the following 

equation for the k-value: 

( ) ( )

( )

k p

f l

Wf l

i i k

i

n

i ii k

i

= n

=

=

∑

∑

α

α

2

0

0

 (12) 

Substitution of the error function equation into the equation for the second condition yields the following 

equation for the radius of relative stiffness: 

() ()

( ) ( )

( )

( )

α

α

α

α

ii k i k

i

n

i ik

i

n

i ii

i

n

k

i ii k

i

n

fl f l

f l

Wf l

Wf l

′

=

′

=

=

=

=

∑

∑

∑

∑

0

2

0

0

0

 (13) 

The solution of equation 13 has been facilitated by development of a computer program. The execution 

time per backcalculation on a PC is only a fraction of a second. The primary advantage of the Best Fit 

method is that it can provide the best fit between the calculated and the measured deflections for any 

sensor configuration. 

In this study, the following procedures were used to apply the Best Fit algorithm to backcalculation of 

subgrade k-values: 

1. Assign weighting factors for equation 8. In this study, they were set equal to 0 or 1, depending 

on whether the sensor is being used for backcalculation. 

2. Determine the radius of relative stiffness that satisfies the l equation from equation 13. 
 6

3. Use equation 12 to determine the modulus of subgrade reaction. 

Knowing the calculated values of lk and k, the elastic modulus for the plate, EPL, may be determined from 

the following relationship: 

( )

3

2 4 12 1

PL

PL

PL h

l k E − μ = (14) 

where 

hPL = plate slab thickness 

k = subgrade k-value 

:PL = PCC Poisson’s ratio 

AREA Algorithm

Hoffman and Thompson first proposed the use of a parameter called AREA for interpreting flexible 

pavement deflection basins.(6) This parameter combines the effect of several measured deflections in the 

basin and is defined as follows: 

[ ] ( ) AREA = 1

2W

W r + W r r W r r

0

0 1

i=1

n-1

∑ i ( - ) + ( - ) i+1 i n n n-1 (15) 

where 

Wi = measured deflections (i = 0, n) 

n = number of FWD sensors minus one 

ri = distances between the center of the load plate and sensors 

The AREA algorithm has been used extensively to analyze concrete pavement deflection basins since 

1980. Ioannides et al. identified the unique relationship between AREA and the radius of relative 

stiffness.(7) Hall obtained simple approximations for this relationship for different AASHTO and SHRP 

sensor configurations.(8) The AREA parameter is not truly an area, but rather has dimensions of length, 

since it is normalized with respect to one of the measured deflections in order to remove the effects of 

load magnitude. For a given number and configuration of deflection sensors, AREA may be computed 

using the trapezoidal rule. The AREA method was examined using the following equations for the four 

sensor configurations by Hall et al.:(2)

SHRP sensor configuration (at 0, 203, 305, 457, 610, 914, and 1524 mm) - Method A7: 

A d

d

d

d

d

d

d

d

d

d

d

d

7 4 6 5 6 9 18 12 8

0

12

8

18

0

24

0

36

0

60

0

= + 



 



 + 



 



 + 



 



 + 



 



 + 



 



 + 



 



 (16) 

where A7 = AREA parameter for sensors at 0, 203, 305, 457, 610, 914, and 1524 mm 

SHRP outer sensor configuration (at 305, 457, 610, 914, and 1524 mm) – Method A5: 

A d

d

d

d

d

d

d

d

5 3 6 9 18 12 18

12

24

12

36

12

60

12

= + 



 



 + 



 



 + 



 



 + 



 



 (17) 

where A5 = AREA parameter for sensors at 305, 457, 610, 914, and 1524 mm 

AASHTO sensor configuration (at 0, 305, 610, and 914 mm) – Method A4: 
 7

A d

d

d

d

d

d

4 6 12 12 6 12

0

24

0

36

0

= + 



 



 + 



 



 + 



 



 (18) 

where A4 = AREA parameter for sensors 0, 305, 610, and 914 mm 

AASHTO outer sensor configuration (at 305, 610, and 914 mm) – Method A3: 

A d

d

d

d

3 6 12 6 24

12

36

12

= + 



 



 + 



 



 (19) 

where A3 = AREA parameter for sensors 305, 610, and 914 mm 

Methods A4 and A7 were considered for rigid pavement backcalculation, and methods A3 and A5 were 

recommended for composite pavement backcalculation. 

AREA Method Versus Best Fit Method

The results of backcalculation obtained by the Hall et al. study were used to compare backcalculated kvalues using the Best Fit and AREA procedures and various sensor configurations. The methods and 

sensor configurations are as follows:(2)

Configuration 

Name Procedure 

Sensor Position 

(mm) 

B7 Best Fit 0, 203, 305, 457, 610, 914, and 1524 

B4 Best Fit 0, 305, 610, and 914 

A7 AREA 0, 203, 305, 457, 610, 914, and 1524 

A4 AREA 0, 305, 610, and 914 

It was observed that these procedures produce different results. Figures 1 through 3 show comparisons 

of B7, A4, and A7 versus B4 configuration/procedure, respectively. In this study, the following simple 

relationships were found between the backcalculated k-values using B4 and the remaining methods: 

 kB7 = 0.867kB4 R2

=0.970 (20) 

 kA7 = 0.984kB4 R2

=0.988 (21) 

 kA4 = 1.148kB4 R2

=0.976 (22) 

where kB4, kB7, kA7, and kA4 are k-values obtained using the B4, B7, A7, and A4 methods, respectively. 

The closest relationship was observed between B4 and A7. Good agreement between these two methods 

for a large number of sections was also found in the Federal Highway Administration (FHWA) rigid 

pavement research study.(5) The R2

 values for all but one relationship exceed 0.97, which means that 

these linear relationships explain more than 97 percent of all variability in the results. 
 8

y = 0.8667x

R2

 = 0.9703

N=396

0

50

100

150

200

250

300

0 50 100 150 200 250 300

Backcalculated k-value, Best Fit 4, kPa/mm

Backcalculated k-value, Best Fit 7, kPa/mm

Figure 1. Backcalculated dynamic k-value for LTPP concrete pavement sections, Best Fit 7 versus 

Best Fit 4. 

y = 1.1482x

R2

 = 0.976

N=396

0

50

100

150

200

250

300

0 50 100 150 200 250 300

Backcalculated k-value, Best Fit 4, kPa/mm

Backcalculated k-value, AREA 4, kPa/mm

Figure 2. Backcalculated dynamic k-value for LTPP concrete pavement sections, Best Fit 4 versus 

AREA4. 
 9

y = 0.9839x

R2

 = 0.9879

N=396

0

50

100

150

200

250

300

0 50 100 150 200 250 300

Backcalculated k-value, Best Fit 4, kPa/mm

Backcalculated k-value, AREA 7, kPa/m

m

Figure 3. Backcalculated dynamic k-value for LTPP concrete pavement sections, AREA7 versus 

Best Fit 4. 

The relationships presented were obtained from FWD data collected for a large number of rigid 

pavements having different design and site conditions. Although good correlationships were observed, it 

is necessary to provide a theoretical basis for the discrepancies between the different methods since both 

backcalculation methods are based on plate theory. 

A very important assumption associated with plate theory is that no compression of the upper layer exists 

and all of the deflection is attributed to compression of the subgrade and bending of the plate. As 

observed, the discrepancy between the Best Fit and AREA methods is higher if the maximum deflection 

(deflection under the center of the load plate) is used in backcalculation. At this location, the plate 

theory prediction of deflections deviates the most from the theory of elasticity predictions due to 

compressibility in the concrete layer. Therefore, it is reasonable to suggest that deviation of the PCC 

slab behavior from the plate theory prediction is a significant source of discrepancy between different 

backcalculation methods. 

To investigate this hypothesis, the computer program DIPLOMAT was used.(9,10) Similar to conventional 

layered elastic analysis programs, DIPLOMAT can accommodate multi-layered pavement systems loaded 

by multiple wheel loads. In addition, it allows the user the option to treat one or more of the constructed 

layers as compressible elastic layers or plates and treat the last layer in the pavement system as a Winkler 

foundation. 

For this study, DIPLOMAT was used to model the PCC layer as an elastic layer, rather than as a plate, 

allowing for vertical compression through the thickness of the slab. The subgrade was modeled as a 

Winkler foundation (springs). 
 10

DIPLOMAT runs were conducted using a randomly generated set of inputs for PCC thickness, PCC 

modulus, and modulus of subgrade reaction, k. The established ranges for the PCC thickness, PCC 

modulus, and the modulus of subgrade reaction were 152 to 305 mm, 28 to 56 GPa, and 13.5 to 135 

kPa/mm, respectively. 

Comparisons of results using B7 versus A7, B4 versus A4, and B3 versus A3 procedures showed the 

same trends observed in the Hall et al.(2) study. In every case, the AREA method produces slightly higher 

k-values than the Best Fit method. The A7 and A4 k-values are, on average, 6.8 and 9.8 percent higher 

than the B7 and B4 results, respectively. Figures 4 and 5 show comparison of k-values from B4 versus 

B7 and A4 comparison. This analysis supports the hypothesis that PCC layer compressibility is one of 

the major sources of discrepancy between the Best Fit and AREA methods. Figure 6 shows that good 

agreement was observed between the B4 and A7 methods. This supports the recommendation made by 

Hall et al. to use the A7 method if the Best Fit backcalculation program is not available.(2)

 

As was done for the GPS LTPP data, the relationships between the results of backcalculation for the 

theoretical deflection basins using the B4 method and the remaining methods were obtained. Good 

correlation with R2 greater than 0.988 were observed. Table 1 presents a comparison of the results of 

these studies. Similar trends exist for the LTPP field data and DIPLOMAT theoretical data 

backcalculated k-values. The differences are generally greater for the field data, most likely due to the 

effects of factors not considered in this study [e.g., departure of true pavement behavior from the 

theoretical models and the effects of joints (slab size)]. 

Table 1. Comparison of LTPP and DIPLOMAT results. 

 Mean Difference, percent 

Comparison LTPP DIPLOMAT 

B7 versus B4 procedure 13.3 4.2 

A7 versus B4 procedure 1.6 -2.5 

A4 versus B4 procedure -14.8 -9.6 

Analysis of the results of backcalculation from the Hall et al. study and from the theoretical examination 

indicated that backcalculation method and sensor configuration may significantly affect backcalculated 

moduli.(2) Therefore, two questions need to be answered: 

1. What backcalculation method (AREA or Best Fit) should be used? 

2. What sensor configuration should be selected? 

The first problem was addressed in the Hall et al. study. (2) The coefficients of variation of backcalculated 

k-values for the GPS-3 (jointed plain concrete pavement) sections from multiple drops contained in the 

LTPP database were compared. The results of that analysis are shown in figure 7. For any sensor 

configuration, the Best Fit method yields a lower coefficient of variation in backcalculated k-values from 

multiple drops than the AREA method. Therefore, the Best Fit method was considered the preferred 

backcalculation procedure. The Best Fit method is also less sensitive to the randomness in measurement 

of maximum deflection and it provides better correspondence between measured and calculated 

deflection basins. 
 11

y = 0.959x

R2

 = 0.9971

N = 675

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

k-value using Best Fit 4, kPa/mm

k-value using Best Fit 7, kPa/mm

Figure 4. Backcalculated k-value from the theoretical deflection basins, Best Fit 7 versus Best Fit 4. 

y = 1.0976x

R2

 = 0.9898

N = 675

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

k-value using Best Fit 4, kPa/mm

k-value using AREA 4, kPa/mm

Figure 5. Backcalculated k-value from the theoretical deflection basins, AREA4 versus Best Fit 4. 
 12

y = 1.0251x

R2

 = 0.999

N = 675

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

k-value using Best Fit 4, kPa/mm

k-value using AREA 4, kPa/mm

Figure 6. Backcalculated k-value from the theoretical deflection basins, AREA7 versus Best Fit 4. 

0%

20%

40%

60%

80%

100%

120%

0% 5% 10% 15% 20% 25%

Coefficient of variation at a particular station

Cumulative percent of stations

Best Fit 7

Best Fit 5

Best Fit 4

Best Fit 3

AREA7

AREA4

AREA5

AREA3

Figure 7. Coefficient of variation in backcalculated k-value for multiple load drops and load level, LTPP 

concrete pavement sections. 
 13

Best Fit Procedure for the Elastic Solid (ES) Model 

Since the Best Fit procedure was selected as the primary procedure for DL foundations, the decision was 

made to adopt this algorithm for ES foundations also. The procedure developed in this study is based on 

Losberg’s solution for a plate on an elastic solid.(11) Under a load distributed uniformly over a circular 

area of radius, a, the distribution of deflections, w(r), may be written as: 

( ) f (r) E

p w r

s

= (23) 

where 

μ α α α

α α f(r)= a d elJ r J a

s ∫

∞

+ −

0

(1 )

2 ( ) ( )

3 3

0 1 2 (1 ) (24) 

a = radius of the applied load 

p = applied pressure 

r = radial distance measured from the center of the load 

le = (D/C)1/3

= radius of relative stiffness of plate-subgrade system for the dense-liquid 

foundation 

D = Eh3

/12(1- μPL

2

) 

C = Es/(1- μs

2

) 

ES = modulus of elasticity of subgrade 

μs = Poisson’s ratio for subgrade 

J0 = Bessel function of zero order 

J1 = Bessel function of first order 

Using equation 23, the error function, F, from equation 8 can be presented in the following form: 

( )( ) ∑ ( ) = 













 ≡ = −

n

i

i e i

s

s s i f l W

E

p F E E F E E

0

2

, , α 

To obtain the minimum of the error function, F, the following conditions should be satisfied: 

= 0

Es

F

∂

∂

= 0

el

F

∂

∂

Substitution of the error function equation into the equation for the first condition yields the following 

equation for the subgrade modulus of elasticity: 

( ) ( )

∑ ( )

∑

=

= = n

i

i i i e

n

i

i i e

s

W f l

f l

E p

0

0

2

α

α

(26) 

(25) 

(27) 

(28) 
 14

Substitution of the error function equation into the equation for the second condition yields the following 

equation for the radius of relative stiffness: 

() ()

( ) ( )

( )

∑ ( )

∑

∑

∑

=

=

=

=

′

=

′

n

i

i i i e

e

n

i

i i i

n

i

i i e

n

i

i i e i e

W f l

W f l

f l

f l f l

0

0

0

2

0

α

α

α

α

The solution of equation 29 has been facilitated by development of a computer program. The 

Microsoft® IMSL library was used for numerical evaluation of Bessel functions. The execution time per 

backcalculation on a PC is only a fraction of a second. The primary advantage of the Best Fit method is 

that it is able to provide the best fit between the calculated and the measured deflections for any sensor 

configuration. 

The procedure for backcalculation using the ES subgrade model is similar to that used for DL 

foundations: 

1. Assign weighting factors for equation 8. In this study, they were set equal to 0 or 1, 

depending on whether the sensor is not being used (0) or is being used (1). 

2. Determine the radius of relative stiffness that satisfies the le equation from equation 29. 

3. Use equation 28 to determine the modulus of elasticity of subgrade, Es.

From the calculated values of le and k, the elastic modulus for the concrete layer, EPCC, may be 

determined from the following relationship: 

( )

(1 )

6 1

3 2

2 3

PCC s

PCC e s

PCC h

l E

E

μ

μ

−

− =

where hPCC = PCC slab thickness 

Es = subgrade modulus of elasticity 

:s = subgrade Poisson’s ratio 

:PCC = PCC Poisson’s ratio 

The procedure was verified using the computer program DIPLOMAT. A close agreement (less than 1 

percent difference) between backcalculated and input elastic parameters was observed. 

Effect of a Base Layer 

Concrete pavements are generally analyzed as slab-on-grade structures, with no structural contributions 

attributed to the underlying base or subbase layers. However, it is known that these underlying layers 

can have a significant effect on the structural performance of the pavement, particularly if bonding 

between the slab and base occurs. If such bonding develops, the effective pavement structure is now 

greater, and the manner in which the pavement reacts to loading is altered. Because multi-layered 

concrete pavements are quite common, the ability to evaluate these structures as multi-layered systems is 

quite valuable to both new and rehabilitation design activities. 

(29) 

(30) 
 15

The approach for the backcalculation of two-layered slab-on-grade is discussed in the next sections, 

based on a methodology proposed by Ioannides and Khazanovich.(12) The two constructed layers may be 

bonded or unbonded and are assumed to act as plates. Thus, no through-the-thickness compression is 

assumed. The backcalculation procedure described represents an adaptation of the forward calculation 

approach for such pavement systems, which was presented by Ioannides et al.(13) The resulting scheme 

was combined in a computer program with the Best Fit procedure. 

Unbonded Case

In accordance with the derivations presented by Ioannides et al., two distinct cases may be recognized, 

depending on the interface condition between the two constructed layers.(13) The case of two unbonded 

plates is considered first. Such plates will act independently, although their respective deflected shapes 

will remain identical if there is to be no separation between them. Under these conditions, it has been 

shown that: 

D DD e = +1 2 

where 

D1 = flexural stiffness of the upper plates 

D2 = flexural stiffness of the lower plate 

De = corresponding stiffness of a fictitious “effective,” composite, homogeneous plate, 

which deforms in an identical manner to the actual two-plate system 

In one sense, slab-on-grade backcalculation schemes may be thought of as producing an estimate of De

when applied to a three-layer PCC pavement system. The apparent task that remains, therefore, is to 

subdivide De into its component parts, namely D1 and D2. This cannot be accomplished merely by 

reference to the field measurements of the deflection profile. An additional input parameter is needed. 

This requirement is akin to the need to provide seed moduli for conventional multi-layered AC pavement 

system backcalculation. In this case, it is convenient to introduce the modular ratio, β, of the two plates 

as the additional input parameter. Furthermore, it may be assumed with no loss of generality that the 

thickness of the "effective" plate, he, is equal to the thickness of the upper plate, h1. As a result, the 

backcalculated E-value from a slab-on-grade analysis is Ee, such that: 

( )

E h D e e

e

e

3

2 12 1− = μ 

It is convenient at this point to introduce the additional assumption that: 

μ1 2 = = μ μ e 

Thus, it follows that: 

Eh Eh Eh Eh ee e 3

1

3

1 1

3

2 2

3 = =+ 

where 

E1 = modulus of upper plate 

E2 = modulus of lower plate 

(31) 

(33) 

(34) 

(32) 
 16

h2 = thickness of lower plate 

Therefore, 

E h

h h 1 Ee 1

3

1

3

2

3 = + β 

and 

E h

h h 2 Ee 1

3

1

3

2

3 = +

β

β (36) 

where 

β = E

E

2

1

 (37) 

Given the values for β and for the real plate thicknesses h1 and h2, equations 37 and 36 may be used with 

the Ee value backcalculated from slab-on-grade analysis (assuming he = h1), to yield E1 and E2 for the two 

plates. 

Bonded Case

For the case of two bonded plates, the flexural stiffness of the fictitious "effective," homogeneous, 

composite plate is no longer a linear sum of the two actual plate stiffnesses, but may be derived using the 

parallel axes theorem. Thus: 

Eh Eh Eh x h Eh Eh h x e e h 3

1 1

3

1 1

1

2

2 2

3

22 1

2

2

12 12 2 12 2

=+ − 



 



 + + −+ 



 



 (38) 

where 

x

h h h h

h h =

+ + 



 





+

1

2

2 1

2

1 2

2 2 β

β (39) 

Proceeding as for the unbonded plates, it may be assumed that he = h, which means that the 

backcalculated E-value from slab-on-grade analysis is Ee. Therefore: 

E h

h h hx h hh x h

1 Ee 1

3

1

3

2

3

1

1

2

2 1

2

2

12

2

12

2

=

++ − 



 



 + −+ 



 

 β β 

 (40) 

 (35) 
 17

Equations 39 and 40 for the bonded plates correspond to equations 35 and 36 for the unbonded plates and 

may be used in a manner analogous to the latter in backcalculating E1 and E2 for the two plates. 

Effect of the Moduli Ratio

The backcalculation procedures presented above require the modular ratio as an input parameter. This 

ratio should be assigned based on engineering judgment. It is assumed that, if the ratio is assigned within 

the reasonable limits, the results of backcalculation are insensitive to the ratio. To verify this 

assumption, consider a 225-mm-thick PCC slab placed over a 150-mm-thick base. A bonded interface 

condition is assumed. Let Ee be the backcalculated modulus if the base is ignored. Figure 8 presents the 

relationship between backcalculated PCC modulus, EPCC, normalized to Ee , and PCC to base modular 

ratio. One can observe that if the modular ratio is between 10 and 100, significant change in the ratio 

produces significant change in PCC modulus. If the ratio is greater than 100, then the PCC modulus is 

practically insensitive to the modular ratio. 

This conclusion was further verified in the recent FWHA-sponsored study.(5) Two sets of the ratios 

between the moduli of elasticity of base materials and PCC were assigned. Figures 9 and 10 present 

comparison of backcalculated moduli of PCC slab using two data sets for unbonded and bonded interface 

conditions, respectively. One can observe that the influence of the moduli ratio is not significant in the 

vast majority of the projects. 

The base modulus is more sensitive to change in the modular ratio. Figure 11 shows that this is true even 

for very high values of modular ratios. This indicates that the proposed procedure is not applicable for 

determination of moduli of elasticity of granular bases. Indeed, for granular bases the ratios are high. 

The error in the moduli ratio, which does not affect backcalculated PCC modulus, may lead to erroneous 

backcalculated base modulus. 

Table 2 presents the proposed modular ratios of PCC and base moduli for each type of base layer. It 

should be noted that β from equation 37 is defined as a ratio of base to PCC moduli. That was done to 

make it stable for the case of weak base (β approaches). Therefore, the ratios from table 2 should be 

inverted before using them in the procedure described above. 

Effect of Sensor Configuration 

To develop recommendations regarding the preferred sensor configuration, the results of backcalculation 

for 19 SMP sections (605 FWD passes) using the Best Fit 4 and Best Fit 7 methods were compared. The 

results of backcalculation are summarized in this section. These results were analyzed to develop 

recommendations regarding the preferred sensor configuration. 

The Best Fit 7 method could determine representative values for only 447 FWD passes (74 percent of all 

FWD visits), meaning that the remaining 158 passes did not satisfy convergence tests. The Best Fit 4 

method obtained values for 544 FWD passes (94 percent of the total FWD visits). Figure 12 presents a 

comparison of backcalculated k-values for those FWD visits. Once again, a strong correlation between 

backcalculated k-values from these two methods exists and has the following form: 

7 864 4 0. B B k = k R2 = 0.920 
 18

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 10 100 1000 10000

Epcc/Ebase

Epcc/Ee

Figure 8. Effect of modular ratio on backcalculated PCC modulus of elasticity. 

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Epcc with set of ratios A, Mpsi

Epcc with set of ratios B, Mpsi

1 Mpsi = 6.895 GPa

Figure 9. Comparison of backcalculated PCC moduli for two sets of modular ratio, bonded 

interface between PCC plate and base. 
 19

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20

Epcc with set of ratios A, Mpsi

Epcc with set of ratios B, Mpsi

1 Mpsi = 6.895 GPa

Figure 10. Comparison of backcalculated PCC moduli for two sets of modular ratio, unbonded interface 

between PCC plate and base. 

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000

Epcc/Ebase

Ebase/Ee

Figure 11. Effect of modular ratio on backcalculated PCC modulus of elasticity. 
 20

Table 2. Proposed moduli ratios, EPCC /Ebase. 

LTPP Code Base Type 

Ratio 

β*=1/β

1 Hot-Mixed, Hot-Laid Asphalt Concrete (AC), Dense Graded 10 

2 Hot-Mixed, Hot-Laid AC, Open Graded 15 

3 Sand Asphalt 50 

4 Jointed Plain Concrete Pavement (JPCP) 1 

5 Jointed Reinforced Concrete Pavement (JRCP) 1 

6 Continuously Reinforced Concrete Pavement (CRCP) 1 

7 PCC (Prestressed) 1 

8 PCC (Fiber Reinforced) 1 

9 Plant Mix (Emulsified Asphalt) Material, Cold Laid 20 

10 Plant Mix (Cutback Asphalt) Material, Cold Laid 20 

13 Recycled AC, Hot Laid, Central Plant Mix 10 

14 Recycled AC, Cold-Laid, Central Plant Mix 15 

15 Recycled AC, Cold-Laid, Mixed-in-Place 15 

16 Recycled AC, Heater Scarification/Recompaction 15 

17 Recycled JPCP 100 

18 Recycled JRCP 100 

19 Recycled CRCP 100 

181 Fine-Grained Soils: Lime-Treated Soil 100 

182 Fine-Grained Soils: Cement-Treated Soil 50 

183 Bituminous Treated Subgrade Soil 100 

292 Crushed Rock 150 

302 Gravel, Uncrushed 200 

303 Crushed Stone 150 

304 Crushed Gravel 175 

305 Crushed Slag 175 

306 Sand 250 

307 Soil-Aggregate Mixture (Predominantly Fine-Grained) 400 

308 Soil-Aggregate Mixture (Predominantly Coarse-Grained) 250 

319 Hot-Mixed AC 15 
 21

LTPP Code Base Type 

Ratio 

β*=1/β

320 Sand Asphalt 50 

321 Asphalt-Treated Mixture 50 

322 Dense-Graded, Hot-Laid, Central Plant Mix AC 10 

323 Dense-Graded, Cold-Laid, Central Plant Mix AC 15 

324 Dense-Graded, Cold-Laid, Mixed-in-Place AC 15 

325 Open-Graded, Hot-Laid, Central Plant Mix AC 15 

326 Open-Graded, Cold-Laid, Central Plant Mix AC 15 

327 Open-Graded, Cold-Laid, Mixed-in-Place AC 15 

328 Recycled AC, Plant Mix, Hot Laid 10 

329 Recycled AC, Plant Mix, Cold Laid 15 

330 Recycled AC, Mixed in Place 15 

331 Cement Aggregate Mixture 5 

332 Econocrete 4 

333 Cement-Treated Soil 50 

334 Lean Concrete 2 

335 Recycled Portland Cement Concrete 100 

338 Lime-Treated Soil 100 

339 Soil Cement 10 

340 Pozzolanic-Aggregate Mixture 100 

341 Cracked and Seated PCC Layer 25 

351 Treatment: Lime, All Classes of Quick Lime and Hydrated Lime 100 

352 Treatment: Lime-Flyash 150 

353 Treatment: Lime and Cement Flyash 150 

354 Treated: Portland Cement 50 

355 Treatment: bitumen (Includes All Classes of Bitumen and Asphalt 

Treatments) 

100 

700 AC 15 

730 PCC 1 

999 No Base (Fictitious Base) 10000 
 22

Although these FWD tests were performed at different seasons and times of day, the relationship 

obtained is remarkably close to that obtained from the results of backcalculation for GPS sections under 

the FHWA-RD-96-198 study. This supports the present findings. A similar relationship was found for 

EPCC (see figure 13). The following observations were also made: 

1. The Best Fit 4 method was successful for a substantially higher number of FWD passes than the Best 

Fit 7 method. 

2. For those passes for which the Best Fit 7 method was not successful, more than 90 percent of 

backcalculated k-values and EPCC from the Best Fit 4 method are within reasonable limits (see figures 

14 and 15). 

3. In addition, if the outer sensor is placed near a transverse joint, Westergaard or Losberg’s solutions 

cannot properly describe its deflection, since these solutions are developed for the interior loading 

case. This discrepancy should be larger for the DL model, since the ratio between the edge and 

interior deflections is higher for Westergaard’s model than for Losberg’s model.(14) Because the 

four-sensor configuration was recommended in other studies, the Best Fit 4 method was selected as 

the primary method for rigid pavement backcalculation.(2)

y = 0.8668x

R2

 = 0.8515

N=464

0

20

40

60

80

100

120

0 20 40 60 80 100 120

Backcalculated k-values from 4 sensors, kPa/mm

Backcalculated k-values from 7 sensors, kPa/mm

Figure 12. Backcalculated dynamic k-value for LTPP SMP sections, Best Fit 7 versus Best Fit 4. 
 23

y = 1.1404x

R2

 = 0.9567

N=464

0

50000

100000

150000

200000

250000

300000

350000

0 50000 100000 150000 200000 250000 300000 350000

Epcc, unbonded, DL4, MPa

Epcc, unbonded, DL7, MPa

Figure 13. Backcalculated EPCC for LTPP SMP sections, Best Fit 7 versus Best Fit 4 methods. 

.%

10.%

20.%

30.%

40.%

50.%

60.%

70.%

80.%

90.%

100.%

0 50 100 150 200 250 300

k-value, kPa/mm

Cumulative Percentage

Figure 14. Distribution of k-values from Best Fit 4 method for the FWD passes for which Best Fit 7 was 

not successful. 
 24

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

21 24 28 31 35 38 41 45 48 52 55 59 62 66 69 >79

 Epcc, GPa

Cumulative Percentage

Figure 15. Distribution of EPCC from Best Fit 4 method for the FWD passes for which Best Fit 7 was not 

successful. 

Slab Size Effect 

The backcalculation procedures presented above are based on Westergaard or Losberg’s solutions for 

interior loading of an infinite plate. However, a concrete slab has finite dimensions. Crovetti developed a 

slab size correction for a square slab based on the results of finite element analysis using the computer 

program ILLI-SLAB.(14) For interior loading, he developed the following procedure: 

1. Estimate Rest from the infinite slab size backcalculation procedure. 

2. Calculate L/Rest, where L = square slab size. 

3. Calculate adjustment factors for maximum deflection (d0) and R from the following equation. 1.04831

0.61662

1 0.89434















 −

= − est

est

L

AF e

λ

λ

0.80151

0.71878

1 1.15085 0















 −

= − est

L

d AF e

λ

4. Calculate adjusted d0 = measured d0 * AFd0. 

5. Calculate adjusted R = Rest * AFl. 

6. Backcalculate k-value and concrete E using adjusted d0 and R. 

In a recent LTPP study, Crovetti's procedure was verified using an analytical closed form solution and 

modified for a case of a rectangular slab, eliminating the need to correct k-value based on the maximum 

deflection only.(2) To verify this procedure, an alternative procedure was developed using an analytical 

(41) 
 25

solution for interior loading of a finite size slab obtained by Korenev.(4) The solution generalizes 

Westergaard’s solution for deflection of an infinite slab to the case of a circular slab. To find the deflection 

distribution in a rectangular and not very long slab for points located not too close to the edges, Korenev 

recommended using the solution for a circular slab with a surface area equal to the rectangular slab’s area. 

In this study, Korenev’s recommendation was modified. It is proposed that Crovetti’s correction factors be 

applied using an equivalent square slab, L, that provides the same surface area of the rectangular and square 

slabs, that is, 

where L1 and L2 are slab width and length, respectively. 

This recommendation should be applied only if the slab length is no more than twice the slab width. For 

longer slabs, an equivalent slab size is equal to: 

An alternate correction for k-value was developed. Steps 4 and 6 above are replaced by the following 

equation for k-value: 

This correction factor can be applied with any backcalculation procedures based on Westergaard’s theory, 

including the Best Fit procedures. 

Although the slab size correction procedure is very simple and straightforward, its application requires that 

slab sizes be assigned properly. This might be a significant problem for two reasons: 

• Load transfer to the adjacent slabs may significantly affect effective slab length and width. Even if the 

load transfer efficiency at the transverse joints can be estimated from the tests at J5 and J4 (transverse 

joint leave and approach slab, respectively) locations, the load transfer efficiency at the longitudinal 

joints and at the shoulder are not known. 

• For pavements with a random joint spacing, it is difficult to determine the slab length corresponding to a 

particular station. It is more realistic to apply slab size correction to the representative backcalculated 

moduli based on the average joint spacing, if necessary. 

Based on these observations, it was decided not to apply any slab size correction. 

Acceptability of the Results of Backcalculation

A level of discrepancy between the predicted and measured deflection basins was selected as a criterion 

for acceptance of backcalculation results. A relative error between measured sensor deflections and 

deflections calculated using backcalculated elastic parameters is defined using the following equation: 

L= L1 L2 

L= 2 L1 

AF AF

k k =

d

2

est

λest 0

 

(42) 

(44) 

(43) 
 26

*100%

,

, ,

i m

i c i m

i

w

w − w

ε = (45) 

where εi = relative error for sensor i 

wi,c = computed deflection for sensor i 

wi,m = measured deflection for sensor i 

The mean absolute relative error for a deflection basin is defined using the following equation: 

 4

| | | | | | | | 1 3 5 6 ε ε ε ε

ε

+ + + m = (46) 

where ε1, ε3, ε5, and ε6 are relative errors of the sensors located 0, 305, 610, and 914 mm from the center 

of the FWD plate. 

High mean absolute relative error indicates that the backcalculation results are not reliable. There are 

several reasons why this might happen. Possible error in sensor deflection measurements and deviation of 

the pavement system behavior from the structural model behavior can be mentioned among others (cracked 

slabs). Use of the backcalculated parameters with high mean error for determination of mean elastic 

parameters for the entire pavement sections may make those parameters less reliable. On the other hand, 

rejection of too many basins is also undesirable because it reduces a number of basins used for mean 

parameter determination. In this study, an acceptable level of the mean error was initially selected equal to 

2 percent. To verify this recommendation, the following analysis was performed: 

• Distribution of the mean relative errors for the GPS and SPS LTPP pavement sections were 

analyzed. 

• Mean coefficients of subgrade reaction were calculated for the GPS and SPS LTPP pavement 

sections using different cutoff levels for the mean relative error. 

Figures 16 and 17 present cumulative distribution of the mean relative error for DL and ES foundation 

models, respectively. For both foundation models, the GPS sections exhibited a higher level of 

discrepancy between measured and calculated deflection basins. Several factors may contribute to this 

effect: 

• On average, the GPS sections are older than the SPS sections and exhibit higher levels of 

distress. 

• Pavement structures of the SPS sections are more uniform than those for the GPS sections. 

For the GPS sections, the backcalculation performed using both ES and DL models resulted in about 75 

percent of backcalculated parameters for which εa

 is less than 2 percent. For SPS sections, more than 85 

percent of backcalculated parameters were obtained for which εa

 is less than 2 percent. This leads to the 

conclusion that the tolerance level for εa is a reasonable requirement for basin acceptance in terms of 

numbers of retaining basins. Very few additional backcalculated points are obtained if larger than 2 

percent error are included given the slope of the plots in figures 16 and 17. 
 27

0

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10

Mean relative sensor error, percent

Cumulative percent of basins

GPS

SPS

)LJXUH&XPXODWLYHFKDUWRIPHDQDEVROXWHYDOXHVRIUHODWLYHHUURUVIRU/733*36DQG636

VHFWLRQV'/VXEJUDGHPRGHO

0

10

20

30

40

50

60

70

80

90

100

0 2 4 6 8 10

Mean relative sensor error, percent

Cumulative percent of basins

GPS

SPS

)LJXUH&XPXODWLYHFKDUWRIPHDQDEVROXWHYDOXHVRIUHODWLYHHUURUVIRU/733*36DQG636

VHFWLRQV(6VXEJUDGHPRGHO
 28

Although a large majority of the basins resulted in less than 2 percent error, a significant number of 

basins exhibited a higher level of error. To investigate the effect of inclusion of those basins in the 

determination of the representative parameters for the LTPP sections, the mean values of the coefficients 

of subgrade reactions were determined for the LTPP GPS and SPS pavement sections using 2 percent, 5 

percent, and 10 percent cutoff limits for acceptability of the backcalculated k-values. Figures 18 and 19 

present comparisons of the mean k-values for 2 and 5 percent cutoff limits for the GPS and SPS 

sections, respectively. Figures 20 and 21 present such comparisons of the mean k-values for 5 and 10 

percent cutoff limits for the GPS and SPS sections, respectively. 

One can observe that for the vast majority of the sections a change in cutoff limit does not significantly 

affect mean values for the sections. On the other hand, for those sections where the effect is pronounced, 

an increase in cutoff leads to an increase in mean backcalculated k-values, pushing the latter to 

unrealistically high levels. This leads to the conclusion that the tolerance level of 2 percent for εa is a 

reasonable requirement for basin acceptance to prevent unrealistically high values of the k-value on some 

sections. 

Backcalculation for LTPP Sections 

This section describes the step-by-step procedures that were developed for routine interpretation of FWD 

deflection data and computation of representative elastic moduli for the LTPP test sections: 

 1. Obtain raw FWD deflection data and section information from IMS database. 

 2. Determine backcalculation pavement structure. 

 3. Conduct backcalculation of FWD deflection data. 

 4. Select interface condition between the PCC slab and the base. 

 5. Select backcalculated parameters for PCC layer and base. 

Each step is discussed in detail in the following sections. 

Determine Backcalculation Pavement Structure

Information about LTPP pavement section layers was obtained from IMS table TST_L05B. A table was 

created with information about layer thicknesses, material codes, and other material properties. Based on 

the information obtained from this table, the pavement structures used in backcalculation were 

determined. The following backcalculation system information was assigned: 

• Thickness of the top PCC layer.

• Thickness of the base layer. 

• Ratio between the PCC slab and base moduli. 

The thickness of the PCC layer was assigned as an average thickness of the top PCC layer. If the 

underlying base layer was nonstabilized and all layers beneath this layer were of similar or lower 

stiffness or if the underlying base layer was stabilized and all layers beneath this layer were of lower 

stiffness (based on layer material classification), then the thickness and type of the base layer were 

assigned to be the base layer. The ratio between the PCC slab and base moduli was assigned based on 

base material type. Table 2 presents the proposed modular ratios of PCC and base moduli for each type 

of base layer. 
 29

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

Mean k-value, kPa/mm, 2 percent cutoff

Mean k-value, kPa/mm, 5 percent cutoff

Figure 18. Comparison of mean k-values with 2 and 5 percent cutoff limit for deflection basin acceptance 

for LTPP GPS section. 

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

Mean k-value, kPa/mm, 2 percent cutoff

Mean k-value, kPa/mm, 5 percent cutoff

Figure 19. Comparison of mean k-values with 2 and 5 percent cutoff limit for deflection basin acceptance 

for LTPP SPS section. 
 30

0

150

300

450

600

750

0 150 300 450 600 750

Mean k-value, kPa/mm, 5 percent cutoff

Mean k-value, kPa/mm, 10 percent cutoff

Figure 20. Comparison of mean k-values with 5 and 10 percent cutoff limit for deflection basin acceptance 

for LTPP GPS section. 

0

150

300

450

600

750

0 150 300 450 600 750

Mean k-value, kPa/mm, 5 percent cutoff

Mean k-value, kPa/mm, 10 percent cutoff

Figure 21. Comparison of mean k-values with 5 and 10 percent cutoff limit for deflection basin acceptance 

for LTPP SPS section. 
 31

Engineering judgment was applied if a more complex system was encountered. If necessary, two or more 

layers were combined. In several cases, two layers with thickness equal to h1 and h2 and a PCC modulus 

to base modulus ratio equal to β1 and β2 were replaced by an equivalent layer with a thickness defined as 

1 2 h h h eff = + (47)

and a PCC modulus to base modulus ratio, βeff, equal to 

2 1

2

12 2 1

2

1

1

3

12 2

3

1

3

1

2

12

2

12

β

β β

β





 



  + − +  



 + + −

= h h h x

h h h h x

h

eff (48)

where 

1 12 2

2

12 2 1

2

1

2 2

h h

h h h h

x

β

β

+





 



 + +

= (49)

and 

2

1

12 β

β β = (50)

This ratio was later used for determination of the modulus of elasticity of the equivalent base layer. 

Conduct Backcalculation of FWD Deflection Data

A computer program, ERESBACK 2.0, was modified under this project into a version ERESBACK 2.2 

to preprocess deflection data, perform backcalculation, and calculate a statistical summary for each FWD 

visit.(5) The program includes the following capabilities: 

• Checks for nondecreasing deflections. 

• Averages deflection basin from the same drop height at the same location. 

• Backcalculates the structural properties of the slab and subgrade. Methods utilized include 

two subgrade models for 0, 305, 610, and 914 mm sensor configurations. 

• Reports a summary of the meaningful statistics for each section. 

The program was verified using the following procedure: 

• Random basin results were chosen from the beginning, middle, and end of the 

backcalculation results file. Each basin has Best Fit results for the DL and ES subgrade 

models. 

• The test basins were backcalculated using the AREA backcalculation method for rigid 

pavements. 
 32

• The results of the ERESBACK2.2 and AREA backcalculation were used as input parameters 

for DIPLOMAT to obtain deflections to compare with measured deflections. 

• DIPLOMAT was run for each basin, treating the PCC slab as a rigid plate layer. The 

appropriate subgrade model was chosen (i.e., elastic half-space for ES and independent 

springs for DL). 

• All three sets of deflections (measured from LTPP results and from AREA results) were 

plotted for each point and for each result. 

The three sets of deflection data showed very close agreement in all cases. 

Pre-Process the Raw FWD Deflection Data, Average Deflection Basins for Each Load Level, and 

Determine Basin Type 

In the LTPP program, four individual drops at four load levels are collected at each point. In this study, 

for each load level, the deflection basins and the applied load were averaged using the following 

procedure: 

• If all deflection basins are decreasing (i.e., sensor deflections decrease with an increase of 

distance from the center of the load plate to the sensor) then for each sensor the deflections 

were averaged along with the applied load. 

• If all deflection basins are non-decreasing (i.e., sensor deflection for at least one sensor is 

higher than for a sensor located closer to the load plate) then for each sensor the deflections 

were averaged along with the applied load and an appropriate flag is reported. 

• If at least one deflection basin for a given drop height is decreasing, all non-decreasing 

deflection basins and corresponding applied loads were excluded from averaging. 

Backcalculate Subgrade Moduli and Radii of Relative Stiffness 

The backcalculation program reads averaged deflection basins from the file created in the previous step. 

For each deflection basin with test type J1 (center of slab of JPCP or JRCP) or C1 (center of CRCP), it 

performs backcalculation using the Best Fit procedure using deflections from four sensors located 0, 305, 

610, and 914 mm from the center of the FWD plate. The backcalculated parameters are determined for 

both DL and ES foundation models. Backcalculation for the DL model results in a radius of relative 

stiffness and a coefficient of subgrade reaction. Backcalculation using the ES model results in a radius of 

relative stiffness and a modulus of elasticity of the subgrade. If the resulting radius of relative stiffness is 

not within a reasonable interval (i.e., if it is less than 500 mm or greater than 2250 mm for the DL model 

and less than 500 mm or greater than 2000 mm for the ES model), then the basin is ignored. 

The lower limit was selected based on the consideration that behavior of the pavement systems with too 

low radii of relative stiffness cannot be adequately described using a slab-on-grade model. A layered 

elastic model is a more appropriate analytical tool and a layered elastic backcalculation procedure should 

be used for backcalculation. The upper limit was assigned to recognize that backcalculation cannot also 

be reliable for every rigid system. As was found by Ioannides et al. (1989), an AREA-radius of relative 

stiffness relationship becomes almost flat for high radii of relative stiffness (see figure 22). This means 

that small variability in a measured basin may cause significant variability in the backcalculated radius of 

relative stiffness. Different upper limits for DL and ES models were selected based on the following 

observations: 
 33

• Radii of relative stiffness for the DL and ES models are calculated using different equations. 

Usually, a radius of relative stiffness is higher for the DL than for the ES model. 

• As was found by Ioannides et al. (1989), the AREA-radius of relative stiffness becomes flat 

for lower values of ES radii of relative stiffness than for DL radii of relative stiffness. 

 Note: 1 in = 25.4 mm 

Figure 22. Variation of AREA with radius of relative stiffness.(7)

Calculate Mean Absolute Errors 

For each deflection basin, the mean absolute relative error is calculated using equations 45 and 46. For 

the basins for which backcalculation was not successful (i.e., no set elastic parameters were determined), 

an appropriate flag (relative errors equal to 99.9 percent) are reported. 

For Each Deflection Basin Used in Backcalculation, Determine Backcalculated Moduli of Elasticity of 

the PCC Layer and Base for ES and DL Subgrade Models Separately, Using the Following Procedure: 

1. Determine effective modulus of elasticity assuming effective pavement thickness equal to the 

thickness of the PCC layer. 

2. Determine assumed ratio of moduli of elasticity of PCC and base layer using the base code. 

3. Considering full bond and full slip, determine moduli of elasticity of the PCC layer and the 

 based using the procedure developed by Ioannides and Khazanovich (equations 35, 36, 39, 

 and 40).(12) 
 34

This procedure results in a set of four moduli of elasticity for the PCC layer and four moduli of elasticity 

of base layer (full bond with the base layer, DL model; full slip with the base layer, DL model; full bond 

with the base layer, ES model; full slip with the base layer, ES model). 

Compute Statistical Summary of Backcalculated Parameters for Each FWD Pass

For each FWD pass, the mean and standard deviation parameters are determined for backcalculated radii 

of relative stiffness, subgrade moduli, PCC moduli, and base moduli. Only deflection decreasing basins 

that result in backcalculation with an average absolute error less than 2 percent are used for computing 

these parameters. Note that a statistical summary computation is performed independently for DL and 

ES subgrade models. A basin may be acceptable for the DL model but be rejected for the ES model, or 

vice versa. 

For each FWD pass, the following parameters are determined for backcalculated radii of relative 

stiffness, subgrade moduli, PCC moduli, and base moduli: 

• Mean value. 

• Minimum value. 

• Maximum value. 

• Standard deviation. 

Selection of the mean error equal to 2 percent as a cutoff criterion for acceptability of the backcalculation 

results is discussed below. 

Screen Backcalculated Parameters 

The backcalculated parameters for each deflection basin are compared with the corresponding mean 

value. If the backcalculated value differs more than two corresponding standard deviations from the 

mean value, an appropriate flag is reported for backcalculated values. 

Identify Interface Condition Between the PCC Slab and the Base

In this step, the bonding conditions between the PCC slab and the base are estimated for each FWD pass. 

The following sub-steps are performed. 

1. Import files created in the previous step into a spreadsheet. 

2. If backcalculation is not successful for an FWD pass (no average modulus is reported), then 

the bond index is equal to 3 (interface condition is unknown). 

3. Assign interface condition independently for DL and ES subgrade model backcalculation 

results. If the mean value for PCC modulus of elasticity assuming bonded interface is greater 

than 27 GPa for the DL model or 21 GPa for the ES model, then assign a bonding index 

equal to 1 (full bond interface); otherwise, assign a bonding index equal to 2 (full slip 

interface). The “cut off” limits of 27 and 21 GPa were selected by the group of experts on 

the basis of past experience in rigid pavement backcalculation. 

4. Compare the results of backcalculations for the same sections but from different FWD 

passes. If the results are different, consider changing the interface condition for an FWD 

pass with lower backcalculated PCC modulus to “unbonded” if it will bring the results 
 35

significantly closer to each other. The justification for the interface condition adjustment is 

the observation that the bond interface condition does not necessarily mean presence of a 

physical bond between the PCC and base layers. If a PCC layer and a base are in full 

contact, they usually exhibit strong bond type behavior due to interface friction. However, 

PCC slab curling may cause separation of the PCC slab from the base. In this case, behavior 

of the PCC slab and the base will be more realistically described using unbonded interface 

condition. Since the actual interface condition is unknown, it is reasonable to assume that a 

significant portion of variation in backcalculated PCC moduli comes from the variation in 

the interface condition. Therefore, it is reasonable to adjust interface conditions 

appropriately if it results in less variability in backcalculated moduli. 

5. Compare the results of bonding indices for the same FWD pass determined using the DL and 

the ES models. The following options are possible (engineering judgment should be 

applied): 

Table 3. Bond index assigning rule. 

Bond Index 

DL Model ES Model Select 

3 3 3 

3 2 2 

3 1 1 

2 3 2 

2 2 2 

1 3 1 

1 1 1 

2 1 2 or 1* 

1 2 2 or 1* 

1 - full bond interface; 2 - full slip interface; 3 - unknown interface 

If both methods result in successful backcalculation but different bond indices, select the 

bond index that provides a PCC modulus of elasticity that is more realistic and closer to the 

results from other FWD passes. 

Select PCC Layer and Base Backcalculated Parameters

Based on the bond index determined from the previous step, select appropriate statistical parameters for 

backcalculated moduli of PCC and base layers for inclusion in the LTPP database. 
 36
 

 37

CHAPTER 3. BACKCALCULATION FOR GPS AND SPS SECTIONS 

This section presents the results of backcalculation for GPS and SPS rigid pavement sections. 

The backcalculation analysis was performed for 331 LTPP GPS and SPS rigid pavement test 

sections. Data from a total of 645 FWD visits were analyzed. The deflection data were 

downloaded during the spring of 1998 from IMS table MON_DYNATEST_DROP_DATA. 

Information about LTPP pavement section layers was obtained from IMS table TST_L05B 

(September 1999 release). This section presents specific aspects of implementation of the 

backcalculation procedure described in chapter 2 and discusses the results of backcalculation. It 

will be shown that, for the majority of cases, reasonable results were obtained. 

Selection of Pavement Structure 

As discussed in chapter 2, backcalculation procedures used in this study model rigid pavement 

systems as two-layered plates resting on a DL or ES foundation. To backcalculate subgrade 

elastic parameters, k-value and modulus of elasticity of subgrade, and radii of relative stiffness, 

no additional information is required. However, to determine elastic moduli of the plate layers, 

the thicknesses of these layers must be assigned. The procedure also requires the user to assign a 

ratio between the elastic moduli of these layers. For all LTPP GPS and SPS sections, thickness 

of the upper layer was assigned as the average thickness of the top PCC layer obtained from the 

IMS database. The ratios were assigned based on the material code of the base layer (see table 2 

and the discussion in chapter 2). 

For the majority of the sections, the thickness of the lower layer in the backcalculation analysis 

was assigned as the average thickness of the second from the top layer. Exceptions were made 

for the following sections shown in table 4. 

Backcalculation of FWD Deflection Data 

ERESBACK 2.2(3) was used in this project to pre-process deflection data, perform 

backcalculation, and calculate a statistical summary for each FWD site visit. Since FWD testing 

of LTPP rigid pavement sections is performed for three load levels, up to three deflection basins 

are available for each FWD station (some basins may have been rejected earlier for not passing 

quality control checks). 

The total number of backcalculated basins for the GPS and SPS LTPP sections was 35,502. A 

total of 25,095 and 27,083 basins resulted in successful backcalculation for DL and ES 

foundation models, respectively. This corresponds to 70 and 76 percent of all basins. A basin 

could be rejected for one of three reasons: 

• The deflection basin was nondecreasing. 

• The backcalculation program was not able to determine proper elastic parameters. 

• The mean of absolute values of relative error for all four sensors was higher than the 

tolerable level of 2 percent. 

The main parameter used to evaluate the results of backcalculation was the mean of absolute 

values of relative sensor errors, εa, defined by equation 46. For most sections, only a small 

fraction of the total number of FWD basins was rejected. 
 

 38

Table 4. Exceptions in assignment of base parameters. 

Sections Assigned Base 

 133011 and 133015 The total thickness of the second and the third from the 

top layers was assigned. The ratio was assigned based 

on the code of the third layer 

069048, 069049, 069017, 089019, 089020, 

189020, 209037, 269029, 269030, 276300, 

279075, 289030, 395569, 399006, 399022, 

404155, 429027, 483569, 483845, 489167, 

489355, 899018 

PCC overlays. The thickness of the underlying PCC 

layer was assigned. 

327084 The total thickness of the second, third, and fourth from 

the top layers was assigned. These layers have the same 

code, which was assigned as a code for the base layer. 

284024,285006,313033,385002,403018,4830

03,483010,483699,483719,484146,484152,4

85024,485154,485287,485301,485310,48531

7,485323,485335,485336 

The total thickness of the second and the third from the 

top layers was assigned. The equivalent ratio was 

derived using the procedure described in chapter 2. 

 485283 and 485284 The total thickness of the third and the forth from the 

top layers was assigned. The equivalent ratio was 

derived using the procedure described in chapter 2. 

For some sections, only a few deflection basins along the section obtained acceptable 

backcalculated parameters. Figure 23 shows the frequency distribution of the percentage of 

rejected deflection basins after backcalculation using the DL model. For more than 85 percent of 

the FWD site visits, more than 30 percent of backcalculation basins within the site were 

accepted. On the other hand, many sections with less than 30 percent accepted basins within the 

site exhibited unrealistic mean values of backcalculated parameters. Figures 24 and 25 present 

comparisons of mean values of distributions of backcalculated k-values for all sections and 

sections with greater than 30 percent accepted basins (screened sections) for the GPS and SPS 

tests, respectively. One can observe that screened sections exhibit a smaller percentage of 

unrealistically high k-values. Based on these observations, it was decided to exclude FWD site 

visits for a section with more than 70 percent rejected basins and to not include a statistical 

summary for these sections in the IMS database. 

Effect of Load Level 

It is known that the results of backcalculation for concrete pavements usually do not depend on 

load level if the load level is sufficiently large. The results of this study support this conclusion. 

Figures 26 and 27 show histograms of coefficient of variation in backcalculated k-value at a 

particular location based on backcalculation using the DL model from three load levels. Figures 

28 and 29 show histograms of coefficient of variation in backcalculated modulus of elasticity of 

subgrade and corresponding radii of relative stiffness based on ES backcalculation. The highest 

variability was observed in backcalculated k-values, although the median coefficient of variation 

in k-value is less than 5 percent and, for more than 80 percent of the stations, the coefficient of 

variation is less than 10 percent. Variability in backcalculation results from the ES model was 

smaller than from the DL model, and variability in subgrade parameter is significantly higher 

than in radii of relative stiffness for both models. 
 

 39

GPS

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90 100

Percentage rejected basins within site

Percentage of sections

GPS

Figure 23. Percentage of rejected backcalculation basins for FWD visits for GPS sections. 

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100 120 140 160 180 >200

Backcalculated k-value, kPa/mm

Percentage of sections

all GPS sections

screened GPS sections

Figure 24. Frequency distribution of backcalculated k-values for GPS LTPP sections (screened 

sections are those with less than 30 percent deflection basins rejected). 
 

 40

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Backcalculated k-value, kPa/mm

Percentage of sections

all SPS sections

screened SPS sections

Figure 25. Frequency distribution of backcalculated k-values for SPS LTPP sections. 

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

0 5 10 15 20 25 30

Coefficient of variation at particular location

Cumulative percentage of locations

GPS

SPS

Figure 26. Coefficient of variation in backcalculated k-value for multiple load levels 

(GPS and SPS sections). 
 

 41

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

0 5 10 15 20 25 30

Coefficient of variation at particular location

Cumulative percentage of locations

GPS

SPS

Figure 27. Coefficient of variation in radius of relative stiffness (DL model) for multiple load 

levels (GPS and SPS sections). 

Figure 28. Coefficient of variation in backcalculated modulus of elasticity of subgrade for 

multiple load levels (GPS and SPS sections). 

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

0 5 10 15 20 25 30

Coefficient of variation at particular location

Cumulative percentage of locations

GPS

SPS
 

 42

Figure 29. Coefficient of variation in radius of relative stiffness (ES model) for multiple load 

levels (GPS and SPS sections). 

Variability in Backcalculated Parameters Along Section Length 

In this study, mean values and standard deviations were calculated for all FWD visits to GPS and 

SPS sections. The mean backcalculated values are stored in the LTPP IMS table 

MON_DEFL_RGD_BAKCAL_SECT. However, only parameters from FWD visits that resulted 

in more than 30 percent of acceptable backcalculation basins were recommended for inclusion in 

the IMS database. To further examine how well these mean values represent pavement section 

properties, distributions of coefficients of variation of backcalculated parameters for FWD visits 

were analyzed. Figures 30 and 31 show cumulative frequency distributions of the coefficient of 

variation of backcalculated subgrade moduli, moduli of elasticity of concrete, and radii of 

relative stiffness obtained using the DL and ES models. The coefficient of variation in 

backcalculated parameters is less than 20 percent for about 80 percent of pavement sections. As 

stated by Hall et al.:(2) 

A coefficient of variation in backcalculated k that is less than 20 percent after screening 

of outliers is reasonable. Significantly higher k coefficients of variation suggest 

significant changes in the subgrade soil type, the embankment thickness, or the depth to 

bedrock. 

Other sources of variability, such as variability in layer thickness and layer conditions, may play 

a role. Similar observations apply for moduli of elasticity of subgrade determined using the ES 

model, as well as for concrete moduli of elasticity. 

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

0 5 10 15 20 25 30

Coefficient of variation at particular location

Cumulative percentage of locations 

GPS

SPS
 

 43

For radii of relative stiffness, a lower level of a coefficient of variation is required because other 

results of backcalculation are very sensitive to this parameter. As a rule of thumb, a coefficient 

of variation in backcalculated radius of relative stiffness less than 10 percent after screening of 

outliers is considered reasonable. For both DL and ES subgrade models, more than 80 percent of 

pavement sections have a coefficient of variation less than 10 percent. 

Although the majority of the pavement sections exhibit acceptable coefficients of variation, for 

some sections variability of backcalculated parameters along section length was found to be 

significant. In this case, mean values may not be representative parameters for the sections. 

One of the options explored in this study was to divide the LTPP sections with high variability in 

backcalculated values into smaller subsections with more uniform characteristics. This option 

was rejected for two reasons: 

• Several sections with high variability did not show noticeable trends in 

backcalculated parameters from one end to another. Therefore, no uniform 

subsections could be identified. For example, the coefficient of variation of 

backcalculated k-value for section 537409 from the FWD visit on April 15, 1997, 

was 41 percent. However, as shown in figure 32, the section cannot be divided into 

two uniform subsections. 

• Several sections showed uniform subsections for one FWD visit but showed 

different trends for another visit. Figure 33 shows backcalculated subgrade modulus 

of elasticity for section 553014 for different test dates. It can be seen from the 

results from April 15, 1993, that the subgrade is becoming stiffer near the end of the 

sections. The FWD testing conducted on August 18, 1993, shows the reverse trend, 

with subgrade stiffness decreasing near the end of the section. 

In addition, variation in backcalculated values may be caused by changes in curling conditions 

during FWD testing throughout a day rather than variation in the material properties. Therefore, 

it was recommended that the mean values be reported in the NIMS along with the corresponding 

standard deviations, and minimum and maximum values because high variability in 

backcalculation results may be useful information for future research studies. 

Subgrade Moduli 

One purpose of backcalculation is to evaluate the level of support provided by the lower layer in 

the pavement system, including natural subgrade. Backcalculation using DL and ES subgrade 

models results in characterization of subgrade using two different parameters: k-value and 

modulus of elasticity of subgrade. GPS and SPS LTPP rigid pavement section support 

conditions vary from soft, fine-graded material to rock subgrade. Therefore, it is not surprising 

that backcalculated parameters vary widely. 
 

 44

Figure 30. Coefficient of variation in backcalculated parameters along project length 

(DL model). 

Figure 31. Coefficient of variation in backcalculated parameters along project length 

(ES model). 

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Coefficient of variation at particular section

Cumulative percentage of sections

k-value

lk

Epcc

0

10

20

30

40

50

60

70

80

90

100

0 20 40 60 80 100

Coefficient of variation at particular section

Cumulative percentage of sections

Esubgr

le

Epcc
 

 45

0

40

80

120

160

200

0 30 60 90 120 150

Station

k-value, kPa/mm

537409

23-Apr-92 rejected

23-Apr-92 accepted

15-Apr-97 rejected

15-Apr-97 accepted

Figure 32. Backcalculated k-values for section 537409. 

Figure 33. Backcalculated modulus of elasticity of subgrade for section 553014. 

0

100

200

300

400

500

0 30 60 90 120 150

Station

Esubg , MPa

553014

02-Aug-90 rejected

02-Aug-90 accepted

15-Apr-93 rejected

15-Apr-93 accepted

18-Aug-93 rejected

18-Aug-93 accepted
 

 46

Figures 34 and 35 present the frequency distribution of mean backcalculated k-values and moduli 

of elasticity for GPS and SPS LTPP rigid pavement sections. As expected, the majority of the 

sections exhibited k-values between 40 and 110 kPa/mm and subgrade moduli of elasticity 

between 100 and 275 MPa. Only a few sections exhibited much higher subgrade moduli. The 

most likely explanation of this effect is the presence of stiff subbase layers and coarse-grained 

subgrade (sections 323013, 465020, 455017, and 455034) or shallow bedrock (sections 485301 

and 42C430). 

As in the recent FHWA-sponsored study (Hall et al.(2)), poor correlation between subgrade type 

and subgrade parameters was found. This is thought to happen because the subgrade types 

identified in the LTPP database may only describe the top 1 to 2 m of material beneath the 

pavement layers. However, on average, coarse-grained subgrade exhibited higher values of 

subgrade reaction and modulus of elasticity than fine-grained subgrades, as shown in figures 36 

and 37. The results of the statistical t-test confirmed the significance of the difference (p-value is 

less than 0.0001 for k-values and moduli of elasticity). 

The results of backcalculation for the rigid LTPP GPS and SPS sections were used to compare 

backcalculated mean coefficient of subgrade reactions for each section with the corresponding 

backcalculated subgrade moduli of elasticity. Figure 38 shows that, as expected, higher moduli 

of elasticity correspond to higher coefficients of subgrade reaction, and a very good correlation 

was observed. A simple linear regression resulted in the following model: 

k = 0.296 Esubgr 

R2

 = 87.2% 

N = 596 

SEE = 9.37 kPa/mm 

where 

k = coefficient of subgrade reaction, kPa/mm 

Esubgr = subgrade modulus of elasticity, MPa 

Even though this equation shows a very good correlation between the k-value and modulus of 

elasticity of the subgrade, it should be used with caution. Specifically, the equation provides a 

relationship between (1) the backcalculated subgrade k-value (based on plate-theory) with (2) the 

backcalculated modulus of elasticity of an elastic half-space (based on plate-theory) for the same 

LTPP section pavement structure. This subgrade modulus of elasticity (based on plate-theory) 

may be different from the modulus of elasticity based on layered elastic theory using a 

backcalculation program such as MODCOM because different "structures" are used for 

describing the constructed layers (i.e., plate-theory versus elastic layered-theory 

characterization). Moreover, accounting for the presence of a rigid layer in MODCOM will also 

significantly alter the backcalculated subgrade modulus of elasticity results. Therefore, 

considerable caution should be exercised when using this equation to estimate a subgrade k-value 

given a backcalculated subgrade modulus of elasticity obtained from elastic layered theory. 
 

 47

Figure 34. Frequency distribution of backcalculated k-values for GPS and SPS LTPP sections. 

0

20

40

60

80

100

120

25

50

75

100

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

>500

Backcalculated Esubg, MPa

Number of observations

Figure 35. Frequency distribution of backcalculated subgrade elastic modulus for GPS and SPS 

LTPP sections. 

0

20

40

60

80

100

120

140

10

30

50

70

90

110

130

150

170

190

Backcalculated k-value, kPa/mm

Number of observations
 

 48

Figure 36. Comparison of mean k-values of fine- and coarse-grained subgrades (GPS and SPS 

sections). 

Figure 37. Comparison of mean moduli of elasticity of fine- and coarse-grained subgrades 

(GPS and SPS sections).

201.9

232.8

0

50

100

150

200

250

Fine-grained Coarse-grained

k-value, kPa/mm

p-level=0.0000011

57.8

70.0

0

10

20

30

40

50

60

70

80

Fine-grained Coarse-grained

k-value, kPa/mm

p-level=0.000000026
 

 49

Figure 38. Subgrade modulus of elasticity versus coefficient of subgrade reaction 

(GPS and SPS sections). 

Radii of Relative Stiffness 

Radius of relative stiffness is an important characteristic of a rigid pavement structure that 

combines concrete modulus of elasticity of stiff pavement layers, their thickness, and elastic 

moduli of subgrade (k-value for DL model and Esubgr for ES model). Stiffer pavement systems 

have a higher radius of relative stiffness for both DL and ES models. 

Figures 39 and 40 present the frequency distribution of mean backcalculated radius of relative 

stiffness for DL and ES subgrade models for GPS and SPS LTPP rigid pavement sections. As 

expected, the majority of the sections exhibited radii of relative stiffness between 800 and 

1200 mm for the DL model and between 600 and 900 mm for the ES model. 

y = 0.2961x

R2

 = 0.8722

0

20

40

60

80

100

120

140

160

180

0 100 200 300 400 500 600

Backcalculated subgrade modulus of elasticity, MPa

Backcalculated subgrade k-value, kPa/mm
 

 50

Figure 39. Frequency distribution of backcalculated k-values for GPS and SPS LTPP sections. 

Figure 40. Frequency distribution of backcalculated subgrade modulus of elasticity for GPS and 

SPS LTPP sections. 

0

10

20

30

40

50

60

70

80

90

100

600

700

800

900

1000

1100

1200

1300

1400

1500

Radius of relative stiffness (DL model), mm

Number of observations

0

20

40

60

80

100

120

500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150

Radius of relative stiffness (ES model), mm

Number of observations
 

 51

The results of backcalculation for the rigid LTPP GPS and SPS sections were used to compare 

backcalculated mean radii of relative stiffness for the DL and ES models. Figure 41 shows that, 

as expected, higher radii of relative stiffness for the DL model correspond to higher radii of 

relative stiffness for the ES model. An excellent correlation between these two parameters was 

observed. A simple linear regression resulted in the following model: 

lk = 1.280 les + 102.7 

R2

 = 98.7% 

N = 596 

SEE = 17.8 mm 

where 

lk = radius of relative stiffness (DL model), mm 

les = radius of relative stiffness (ES model), mm 

Figure 41. ES radius of relative stiffness versus DL radius of relative stiffness.

Bonding Condition Between PCC Slab and Base 

The presence of a stabilized base may significantly alter PCC pavement responses (stresses and 

deflections) to applied loading. In turn, an interface condition between the PCC slab and the 

stabilized base dramatically affects the significance of base contribution to the overall pavement 

stiffness. The importance of accounting for the presence of a stabilized base and interface 

condition can be demonstrated by analyzing backcalculation results for two LTPP GPS and SPS 

rigid pavement sections, 105004 and 204052. Section 105004 is a 225-mm-thick CRCP placed 

over a 100-mm-thick cement-aggregate mixture. Section 204052 is 225-mm-thick JPCP placed 

y = 1.2798x + 102.7

R2

 = 0.9874

0

500

1000

1500

2000

2500

0 200 400 600 800 1000 1200 1400 1600

Backcalculated ES radius of elasticity, mm

Backcalculated DL radius of elasticity, mm
 

 52

on top of a 100-mm-thick econocrete base. Table 5 presents backcalculated PCC moduli of 

elasticity for these sections based on three different scenarios: 

• The base layer is ignored. 

• The base layer is in full friction with the PCC layer (full bond interface condition). 

• The base layer is in full slip with the PCC layer (no bond interface condition). 

Table 5. Effect of base and interface condition on backcalculated moduli of elasticity. 

Section No Base No Bond Full Bond 

 EPCC, MPa Ebase, MPa EPCC, MPa Ebase, MPa EPCC, MPa Ebase, MPa 

105004 31034  30501 6100 20283 4056 

204052 8203  79569 19892 49871 12468 

Neglecting the presence of the base layer overestimates modulus of elasticity of the concrete 

layers. This is especially clear for section 204052. For this section, even the assumption of 

unbonded interface between the PCC and base layer produces unrealistically high modulus of 

elasticity for both PCC and base layers. An assumption that the interface is bonded brings 

backcalculated moduli within the reasonable range. On the other hand, the assumption of full 

bond produces unrealistically lower elastic modulus for section 105004. Therefore, an unbonded 

interface condition should be assigned. 

Full bond behavior of the pavement system exhibited during FWD testing does not necessarily 

indicate presence of a physical bond between the PCC slab and the base. The PCC slab may be 

in full contact with the base at the center of the slab and have substantial friction at that location 

while being lifted off the slab because of temperature curling and moisture warping at the slab 

edges. Moreover, at the same location, the slab may experience different friction conditions with 

the base layer. For example, the results for section 285025 shown in table 6 indicate that the 

PCC slab was in full friction with the base during the tests conducted in 1989 but showed full 

slip in the test conducted in 1994. 

Table 6. Example of change of interface condition from test to test. 

Section Test Date No Bond Full Bond 

 EPCC, MPa Ebase, MPa EPCC, MPa Ebase, MPa 

285025 6-NOV-1989 60101 4007 49901 3327 

285025 31-OCT-1994 47701 3180 39600 2640 

Among 597 PCC and base section layer moduli obtained in this study using the DL model, 103 

were assigned unbonded interface between the layers. In the remaining 494 cases, the layers 

were assigned a bonded interface condition, as shown in figure 42. For the ES foundation model, 

in 603 of 603 cases, a bonded interface was assigned. It should be noted, however, that if the 

bond/no bond condition did not dramatically affect backcalculated moduli, as was the case for 

the majority of nonstabilized bases, the interface was assumed to be bonded. However, even for 

stabilized bases, the bonded interface was usually observed, as shown in figure 43. 
 

 53

0

100

200

300

400

500

600

DL ES

Number of observations

BOND

NO BOND

Figure 42. Distribution of bond/no bond interface condition for GPS and SPS LTPP sections. 

0

50

100

150

200

250

300

DL ES

Number of observations

BOND

NO BOND

Figure 43. Distribution of bond/no bond interface condition for GPS and SPS LTPP sections 

with stabilized bases. 
 

 54

Backcalculated PCC Moduli

Backcalculated modulus of the PCC layer is an important parameter for mechanistic-empirical 

evaluation and design procedures. In this study, PCC layer properties were determined for 597 

and 603 FWD visits of GPS and SPS LTPP sections using DL and ES subgrade models, 

respectively. Figures 44 and 45 show frequency histogram distributions for PCC moduli 

backcalculated using the DL and ES subgrade models. The majority of the backcalculated 

moduli are in the reasonable range from 25,000 to 55,000 MPa, supporting the conclusion of the 

robustness of the backcalculation approach. 

Analysis of these plots shows that backcalculation using the ES subgrade model results in 

consistently lower PCC elastic moduli than backcalculation using the DL subgrade model. This 

is to be expected, because backcalculated moduli are not actual material properties but rather 

parameters of the corresponding structural systems used in backcalculation. Since an ES 

foundation provides significant shear redistribution and DL provides no shear load redistribution, 

it is reasonable to expect that higher stiffness of the upper layers resting on a DL foundation is 

required to provide the same deflections produced by a corresponding slab on an ES foundation. 

Figure 46 shows backcalculated PCC moduli using the ES subgrade model versus backcalculated 

PCC moduli using the DL subgrade model. A very good correlation between these two sets of 

moduli is observed. A linear regression analysis resulted in the following relationships: 

EPCC,DL = 1.312 EPCC,ES

R2

 = 94% 

N = 591 

SEE = 1.8 GPa 

where 

EPCC,ES = PCC modulus of elasticity backcalculated using ES model, MPa 

EPCC,DL = PCC modulus of elasticity backcalculated using DL model, MPa 

This relationship indicates that backcalculated modulus of the PCC layer depends on the 

foundation model used in backcalculation. Therefore, if a transition from the ES to the DL 

foundation model needs to be made, moduli of elasticity of the upper layers also need to be 

adjusted. It is important to remember, however, that these relationships for the foundation 

parameters, radii of relative stiffness, and PCC moduli of elasticity were developed based on 

equivalency of the deflection basins under an interior loading condition. Development of the 

corresponding relationships based on equivalency of the maximum stresses or on equivalency of 

the pavement responses under the edge or corner loading conditions may require adjustments. 
 

 55

0

5

10

15

20

25

30

15000

20000

25000

30000

35000

40000

45000

50000

55000

60000

65000

70000

75000

>80000

Backcalculated PCC moduli of elasticity (DL model), MPa

Percentage of sections

Figure 44. Frequency distribution of backcalculated PCC modulus of elasticity for GPS and SPS 

LTPP sections (DL model). 

0

5

10

15

20

25

30

15000

20000

25000

30000

35000

40000

45000

50000

55000

60000

65000

70000

75000

>80000

Backcalculated PCC moduli of elasticity (ES model), MPa

Percentage of sections

Figure 45. Frequency distribution of backcalculated PCC modulus of elasticity for GPS and SPS 

LTPP sections (ES model). 
 

 56

EPCC,DL = 1.3121 EPCC,ES

R2 = 0.9404

0

20000

40000

60000

80000

100000

120000

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Backcalculated PCC modulus of elasticity (ES model), MPa

Backcalculated PCC modulus of elasticity 

(DL model), MPa

Figure 46. Backcalculated PCC modulus of elasticity (ES model) versus backcalculated PCC 

modulus of elasticity (DL model). 

Backcalculation for Unbonded PCC Overlay Sections (GPS-9) 

The backcalculated PCC moduli presented in this report do not include moduli for sections with 

unbonded PCC overlays. It is recommended that backcalculated PCC moduli for these sections 

not be uploaded into the IMS database, on the basis of the following observations: 

• The backcalculated moduli are extremely sensitive to the assigned ratio between elastic 

moduli of the PCC overlay and the underlying PCC slab. Assigning this ratio is not a trivial 

task. On the one hand, old PCC in the underlying slab can be much stiffer than younger PCC 

of the unbonded overlay. On the other hand, cracks in the underlying slab may significantly 

reduce effective modulus of elasticity of this layer. In this study, the ratio between the 

moduli of unbonded PCC overlays and underlying slabs was assigned to be equal to 1 for all 

sections. However, analysis of figures 8 and 11 (chapter 2) shows that even a small change 

in this ratio could dramatically affect backcalculated values for both PCC layers. Table 7 

shows the results for the top layer PCC slabs (unbonded overlay). These values are very 

erratic and some are unrealistic. 

• For the many sections for which multiple FWD visits were available (for example, sections 

069048, 069049, 089020, 279075, and 899018), low repeatability of the results was 

observed. 
 

 57

Table 7. Backcalculated PCC moduli of elasticity for unbonded overlay sections. 

 EPCC, GPa 

Section Date DL Model ES Model 

 Unbonded Bonded Unbonded Bonded 

069048 11-Jan-90 50491.5 13137.1 38221.1 9944.5 

069048 29-Jan-97 34358.1 8939.5 26055.2 6779.2 

069049 29-Nov-89 85092.1 21283.7 66463.9 16624.3 

069049 31-Mar-95 34439.4 8614.2 26730 6685.8 

069107 16-Nov-89 62922.6 15991.5 49579.1 12600.3 

089019 28-Jun-89 32395.6 8201.9 24099.5 6101.5 

089019 8-Apr-92 39641.8 10036.5 29945.6 7581.6 

089020 27-Jun-89 20017.3 5008.9 21407.3 5356.8 

089020 9-Apr-92 55683.7 13933.8 42468.7 10627 

134118 20-Aug-90 33315.5 8362.2 24613.2 6177.9 

134118 10-Sep-92 31374.9 7875.1 23123.7 5804 

189020 18-Jul-90 66778.6 16694.6 49062.7 12265.7 

189020 25-Jul-94 51093.2 12773.3 39707.3 9926.8 

209037 7-Sep-89 34240.4 9666.3 25709.8 7258.1 

209037 13-May-94 32928.4 9295.9 24937.2 7039.9 

269029 11-Jul-90 140774.1 35420.7 104394.1 26267 

269029 4-Nov-93 NA NA 71415.8 17969.2 

269030 19-Jun-90 119291.3 31525 95157.7 25147.2 

269030 1-Nov-93 74622 19720.3 54330.6 14357.9 

276300 17-Jul-89 70621 17658.1 53838.2 13461.7 

279075 5-Sep-90 49435 13098.1 37932.2 10050.3 

279075 8-Sep-94 26050.1 6902.1 19478.2 5160.9 

279075 1-Jun-95 57239.5 15165.9 44986.7 11919.5 

289030 31-Oct-89 253363.9 65322.9 183010.1 47184.1 

289030 3-Dec-92 159949.1 41238.5 138265.1 35647.9 

316701 4-Aug-89 59704.4 14967.6 45010.9 11284 

395569 20-Jul-94 26332.5 6772.6 19957.6 5133 

399006 21-Jul-94 88284.7 22316.5 68184.5 17235.6 

399022 18-Oct-90 72454.5 18635.1 65976.6 16969 

399022 19-Jul-94 55228.9 14204.7 43020.8 11064.8 

404155 5-Jun-90 72099.4 19829.7 54528.1 14997 

404155 20-May-93 50587.6 13913.2 37716.9 10373.4 

429027 5-Jun-89 47438.9 13407.6 35212.7 9952.1 

429027 26-Jul-95 28735.6 8121.5 23375.5 6606.6 

483569 1-May-90 81509.8 20415.5 64988.1 16277.4 

483569 4-Jan-96 68708 17209.1 53917.6 13504.6 

483845 8-May-90 62454.7 15683.5 47084.6 11823.8 

483845 16-Aug-91 78141.6 19622.8 57496 14438.3 

483845 22-Aug-96 NA NA 44472.8 11167.9 

489167 6-Apr-90 136670.7 35141.2 110215.5 28339 

489167 3-Jan-96 93737.3 24102.1 79789.6 20515.8 

489355 8-Aug-90 NA NA 49625.7 12420.5 

489355 16-Aug-93 NA NA NA NA 

899018 5-Oct-94 79600.5 21466 62554 16869 

899018 12-Jun-95 51070.5 13772.3 38978 10511.3 
 

 58

Backcalculated Base Moduli 

The main purpose for introduction of a base layer in the backcalculation analysis is to account 

for a structural contribution of the base layer to the overall pavement stiffness. This allows 

engineers to obtain much more realistic values for PCC moduli than when the base is ignored. 

The fact that the majority of the sections exhibited a bonded interface condition with the base 

layer indicates that the structural contribution of the latter is significant. Unfortunately, the 

presence of the stiff concrete layer on top of the base layer does not allow reliable 

backcalculation of the base moduli. Use of plate theory as a structural model for the pavement 

system makes it theoretically impossible. In this study, the base moduli were estimated as 

fractions of the PCC moduli using the ratios presented in table 2. Therefore, it is of interest to 

evaluate the reasonableness of the obtained moduli. 

Tables 8 and 9 present mean, minimum, and maximum values of the backcalculated base moduli 

of GPS and SPS LTPP sections using DL and ES foundation models, respectively, for each base 

type. The backcalculated moduli are in reasonable ranges for all stabilized bases. For 

nonstabilized bases, the moduli obtained using the ES model are reasonable but often lower than 

corresponding backcalculated subgrade moduli. This is reasonable, since modeling of a granular 

layer as a plate layer (i.e., assuming an infinite shear modulus) significantly overestimates its 

stiffness in forward calculation. In backcalculation, it results in lower elastic modulus. 

Fortunately, a significant change in moduli ratio for a nonstabilized base has a small effect on 

backcalculated PCC moduli, although it significantly affects base moduli. 

Table 8. Mean, maximum, and minimum backcalculated base moduli for GPS and SPS LTPP 

sections (DL model). 

Base Material Backcalculated Base Modulus, MPa 

Mean Minimum Maximum 

Asphalt-Treated Mixture 5746 3365 9178 

Gravel, Uncrushed 178 115 418 

Crushed Stone 225 177 280 

Crushed Gravel 200 119 367 

Sand 133 116 156 

Soil-Aggregate Mixture (Predominantly Fine-Grained) 81 76 84 

Soil-Aggregate Mixture (Predominantly Coarse-Grained) 160 101 267 

Hot-Mixed AC 2227 1647 3565 

Sand Asphalt 583 575 590 

Asphalt-Treated Mixture 787 458 1298 

Dense-Graded, Cold-Laid, Central Plant Mix AC 2048 1932 2191 

Open-Graded, Hot-Laid, Central Plant Mix AC 2089 1953 2358 

Cement Aggregate Mixture 6209 3409 15144 

Econocrete 10339 8598 11957 

Lean Concrete 14483 1898 21015 

Soil Cement 2936 1989 4559 
 

 59

Table 9. Mean, maximum, and minimum backcalculated base moduli for GPS and SPS LTPP 

sections (ES model). 

Base Material Backcalculated Base Modulus, MPa 

Mean Minimum Maximum 

Asphalt-Treated Mixture 5746 3365 9178 

Gravel, Uncrushed 178.2565 115.1285 417.792 

Crushed Stone 225.4349 177.4473 279.6053 

Crushed Gravel 200.4254 119.4006 367.0246 

Sand 132.8438 116.2008 156.2364 

Soil-Aggregate Mixture (Predominantly Fine-Grained) 80.96969 75.95625 84.2965 

Soil-Aggregate Mixture (Predominantly CoarseGrained) 159.6977 101.0056 266.6072 

Hot-Mixed AC 2937 2154 4670 

Sand Asphalt 774 762 785 

Asphalt-Treated Mixture 988 558 1687 

Dense-Graded, Cold-Laid, Central Plant Mix AC 2686 2535 2801 

Open-Graded, Hot-Laid, Central Plant Mix AC 2850 2613 3372 

Cement Aggregate Mixture 8145 4096 17167 

Econocrete 13357 11736 14847 

Lean Concrete 19656 3709 27519 

Soil Cement 3929 2583 5867 

Variability in Backcalculated Parameters Between FWD Visits 

 

Several GPS and SPS LTPP sections were tested more than once. The backcalculation results 

obtained from different FWD visits were compared and the coefficients of variation were 

calculated. Figures 47 and 48 present cumulative frequency distributions of the coefficients of 

variation of backcalculated parameters for the ES and DL foundation models, respectively. The 

coefficient of variations of backcalculated subgrade moduli and concrete elastic moduli are less 

than 20 percent for about 80 percent of pavement sections (variability between visits). At the 

same time, more than 90 percent of the GPS and SPS LTPP sections exhibited a coefficient of 

variation in backcalculated radius of relative stiffness of less than 10 percent. 

Although the majority of the pavement sections exhibit low coefficients of variation between 

visits, for some sections it was found to be much larger. Several factors may contribute to high 

variability, including seasonal variation in subgrade properties and variation in temperature 

gradients through the PCC slab thickness that result in slab curling variation. 
 

 60

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coefficient of Variation at Particular Section

Cumulative Percentage of Sections

radius of relative 

stiffness

Epcc

k-value

Figure 47. Coefficient of variation in backcalculated parameters (DL model) for GPS and SPS 

LTPP sections between different FWD visits. 

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coefficient of Variation at Particular Section

Cumulative Percentage of Sections

radius of relative 

stiffness

Epcc

Esubgr

Figure 48. Coefficient of variation in backcalculated parameters (ES model) for GPS and SPS 

LTPP sections between different FWD visits. 
 

 61

Backcalculation Analysis Results 

The results of point-by-point backcalculation for the GPS and SPS sections are given in the 

LTPP IMS table MON_DEFL_RGD_BAKCAL_POINT. A summary of the backcalculation 

analysis results for the GPS and SPS sections is given in the LTPP IMS table 

MON_DEFL_RGD_BAKCAL_SECT. The results are presented in terms of the mean values, 

minimum values, maximum values, and standard deviations of the following parameters: 

Dense-Liquid Model

• k-value (modulus of subgrade reaction). 

• EPCC (modulus of elasticity of the PCC slab). 

• Ebase (modulus of elasticity of the base layer). 

 Elastic Solid Model

• Esubgr (modulus of elasticity of the subgrade). 

• EPCC (modulus of elasticity of the PCC slab). 

• Ebase (modulus of elasticity of the base layer). 

Also, the bond condition between the concrete slab and the base layer is reported. 

Typical plots of the above parameters are given in appendix A. 
 

 62
 63

CHAPTER 4. BACKCALCULATION FOR SMP SECTIONS 

The FWD data collected for the SMP LTPP sections allow for analysis of the effect of season 

and time of day on backcalculated values. Backcalculation analysis was performed for 19 SMP 

sections, with a total of 571 FWD passes. The number of passes is much higher than the number 

of sections because the sections could be tested several times a day and several times per year. 

The deflection data were downloaded during the fall of 1997 from IMS table 

MON_DYNATEST_DROP_DATA. Information about LTPP pavement section layers was 

obtained from IMS table TST_L05B. 

Selection of Pavement Structure 

The SMP LTPP sections were modeled in this study as two-layered plates resting on DL or ES 

foundations. For all LTPP SMP sections, thickness of the upper layer was assigned as an 

average thickness of the top PCC layer of the LTPP section obtained from the IMS database. For 

the majority of the sections, the thickness of the lower layer in the backcalculation model was 

assigned as an average thickness of the second from the top layer, and the ratios between PCC 

and base moduli were assigned based on the material code of the base layer (see table 2). 

Backcalculation of FWD Deflection Data 

ERESBACK 2.2 was used to process the raw FWD deflection data, average deflection basins for 

each load level, determine basin type, backcalculation, and post-process backcalculation results. 

The total number of backcalculated basins for the SMP LTPP sections was 10,626. The number 

of basins resulting in successful backcalculation were 8,188 and 8,771 for DL and ES foundation 

models, respectively. This corresponds to 77 and 82 percent of all basins. 

For most sections, only a small fraction of the total number of FWD basins was rejected. 

However, for some sections, only a few deflection basins were available to obtain acceptable 

backcalculated parameters. Figure 49 shows a distribution of the percentage of accepted 

deflection basins per section after backcalculation using DL and ES models. For the DL and ES 

foundation models, 37 and 48 percent of FWD passes resulted in 100 percent accepted deflection 

basins. Only a small fraction of the FWD passes that exhibited more than 30 percent of 

backcalculation basins were accepted (5 and 3 percent for DL and ES foundation models, 

respectively). Based on this observation, it was decided to exclude FWD passes with more than 

30 percent rejected basins and to exclude mean values for these basins in the IMS database. 
 64

Figure 49. Percentage of accepted backcalculation basins for FWD passes of SMP LTPP 

sections. 

The backcalculated mean values for 19 SMP LTPP PCC rigid pavement sections are presented in 

the LTPP IMS table MON_DEFL_RGD_BAKCAL_SECT. The backcalculation results are 

summarized in the following sections. 

Effect of Load Level

As was observed for GPS and SPS rigid pavement sections, the results of backcalculation for 

SMP concrete pavement sections did not depend on load level. Figures 50 and 51 show 

histograms of coefficient of variation in backcalculated k-value at a particular location based on 

backcalculation using the DL model from three load levels. Figures 52 and 53 show histograms 

of coefficient of variation in backcalculated modulus of elasticity of subgrade and corresponding 

radii of relative stiffness based on ES backcalculation. The highest variability was observed in 

backcalculated k-values, although the median coefficient of variation in k-value is less than 4 

percent (for almost 95 percent of the stations, the coefficients of variation are less than 10 

percent). Variability in backcalculation results from the ES model was smaller than from the DL 

model, and variability in subgrade parameter is higher than in radii of relative stiffness for both 

models. 

0

50

100

150

200

250

300

0 10 20 30 40 50 60 70 80 90 100

Percentage of accepted basins for an FWD pass

Number of FWD passes

DL Model ES Model
 65

0%

20%

40%

60%

80%

100%

0% 5% 10% 15% 20% 25%

Coefficient of variation at particular location

Cumulative percent of locations

Figure 50. Coefficient of variation in backcalculated k-values for multiple load levels at a 

particular location (SMP sections). 

0%

20%

40%

60%

80%

100%

0% 5% 10% 15% 20% 25%

Coefficient of variation at particular location

Cumulative percentage of locations

Figure 51. Coefficient of variation in radius of relative stiffness (DL model) for multiple load 

levels at a particular location (SMP sections). 
 66

Figure 52. Coefficient of variation in backcalculated modulus of elasticity of subgrade for 

multiple load levels at a particular location (SMP sections). 

Figure 53. Coefficient of variation in radius of relative stiffness (DL model) for multiple load 

levels at a particular location (SMP sections). 

0%

20%

40%

60%

80%

100%

0% 5% 10% 15% 20% 25%

Coefficient of variation at particular location

Cumulative percentage of locations

0%

20%

40%

60%

80%

100%

0% 5% 10% 15% 20% 25%

Coefficient of variation at particular location

Cumulative percentage of locations
 67

Variability in Backcalculated Parameters Along Section Length 

In this study, mean values and standard deviations along the project length were calculated for all 

FWD passes to SMP sections. However, only parameters from FWD passes that resulted in more 

than 70 percent of acceptable backcalculation basins were recommended for inclusion in the IMS 

database. To further examine how well these mean values represent pavement section properties, 

distributions of coefficients of variation of backcalculated parameters for FWD passes were 

analyzed. 

Figures 54 and 55 show cumulative frequency distributions of the coefficient of variation of 

backcalculated subgrade moduli, moduli of elasticity of concrete, and radii of relative stiffness 

obtained using the DL and ES models, respectively. The coefficient of variation in 

backcalculated parameters is less than 20 percent for more than 80 percent of SMP sections. 

Sections 063042 and 893015 exhibited the highest variability in PCC modulus of elasticity. 

Variability in PCC thickness (both sections) or base thickness (section 063042) are the most 

likely reason for variability in backcalculated PCC modulus of these sections. Variability in 

backcalculated moduli does not remain constant for the same section—it substantially changes 

from one FWD pass to another. For example, for section 893015 it varies from 10 percent on 

October 6, 1994, to 46 percent on April 18, 1995. 

Figure 54. Coefficient of variation in backcalculated parameters along project length for SMP 

sections (DL model). 

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Coefficient of variation at particular section

Cumulative percentage of FWD passes

Epcc

k-value

lk
 68

Figure 55. Coefficient of variation in backcalculated parameters along project length for SMP 

sections (ES model). 

Subgrade Moduli 

Backcalculation using DL and ES subgrade models results in characterization of subgrade using 

two different parameters: k-value and modulus of elasticity of subgrade. Figures 56 and 57 

present frequency distributions of mean backcalculated k-values and subgrade moduli of 

elasticity for SMP LTPP rigid pavement sections. As expected, the majority of the sections 

exhibited k-values between 40 and 110 kPa/mm and moduli of elasticity between 100 and 

275 MPa. As expected, several sections exhibited higher subgrade moduli from FWD testing 

during wintertime. 

The results of backcalculation for the rigid LTPP SMP sections were used to compare 

backcalculated mean coefficient of subgrade reactions, k, for each section with the corresponding 

backcalculated subgrade moduli of elasticity, Esubgr. Figure 58 shows that, as expected, higher 

moduli of elasticity correspond to higher coefficients of subgrade reaction. A linear regression 

resulted in the following model: 

k = 0.282 Esubgr 

R2

 = 86.3% 

N = 556 

SEE = 9.12 kPa/mm 

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0% 10% 20% 30% 40% 50% 60% 70%

Coefficient of variation at particular section

Cumulative percentage of FWD passes

Epcc

Esubgr

le
 69

where 

k = coefficient of subgrade reaction, kPa/mm 

Esubgr = subgrade modulus of elasticity, MPa 

This equation is remarkably similar to the corresponding relationship obtained for the GPS and 

SPS sections. This verifies the stability of the backcalculation procedure. Figure 58 shows, 

however, that this relationship is not extremely accurate for high values of k-value and modulus 

of elasticity. The majority of those values are obtained from FWD testing of the sections located 

in a very cold climate in wintertime; however, the results of backcalculation from those FWD 

passes are not very reliable. Indeed, the measured deflections were relatively small, therefore, 

small measurement errors could significantly alter results. Moreover, it appears the DL model 

increases greatly for a frozen subgrade. 

Radii of Relative Stiffness 

To further examine results of backcalculation for the SMP LTPP sections, the distribution of 

backcalculated radii of relative stiffness was analyzed. Figures 59 and 60 present the frequency 

distribution of mean backcalculated radius of relative stiffness for DL and ES subgrade models 

for SMP LTPP rigid pavement sections. As expected, the majority of the sections exhibited radii 

of relative stiffness between 800 and 1200 mm for the DL model and between 600 and 900 mm 

for the ES model. 

Figure 56. Frequency distribution of backcalculated k-values for SMP LTPP sections. 

0

20

40

60

80

100

120

140

0

20

40

60

80

100

120

140

160

180

>200

k-value, MPa/m

Number of observations
 70

Figure 57. Frequency distribution of backcalculated subgrade elastic modulus for SMP LTPP 

sections. 

k-value = 0.283 Esubgr

R2 = 0.8646

0

50

100

150

200

250

0 100 200 300 400 500 600

Esubgr, MPa

k-value, kPa/mm

Figure 58. Subgrade modulus of elasticity versus coefficient of subgrade reaction (SMP 

sections). 

0

20

40

60

80

100

120

0

50

100

150

200

250

300

350

400

450

>500

Subgrade modulus of elasticity, MPa

Number of observations
 71

Figure 59. Frequency distribution of backcalculated radius of relative stiffness (DL model) for 

SMP LTPP sections. 

0

20

40

60

80

100

120

140

160

500

550

600

650

700

750

800

850

900

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

Radius of relative stiffness (ES model), mm 

Number of observations

Figure 60. Frequency distribution of backcalculated radius of relative stiffness (ES model) for 

SMP LTPP sections. 

0

20

40

60

80

100

120

140

160

500

600

700

800

900

1000

1100

1200

1300

1400

1500

Radius of relative stiffness (DL model), mm 

Number of observations
 72

The results of backcalculation for the rigid LTPP SMP sections were used to compare 

backcalculated mean radii of relative stiffness for the DL and ES models. Figure 61 shows that, 

as expected, higher radii of relative stiffness for the DL model correspond to higher radii of 

relative stiffness for the ES model. An excellent correlation between these two parameters was 

observed. A simple linear regression resulted in the following model: 

 lk = 1.2444 les + 130.73 

R2

 = 99.2% 

N = 556 

SEE = 13.77 mm 

where 

lk = radius of relative stiffness (DL model), mm 

les = radius of relative stiffness (ES model), mm 

It should be noted that although this relationship looks slightly different from the corresponding 

relationships obtained for SPS and GPS sections, they result in similar predicted radii of relative 

stiffness (DL model) if the corresponding radius of relative stiffness (ES model) is within 

reasonable limits for highway pavements. 

Figure 61. ES radius of relative stiffness versus DL radius of relative stiffness (SMP sections).

lk = 1.2444 le+ 130.73

R2

 = 0.9921

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 200 400 600 800 1000 1200 1400 1600

Modulus of relative stiffness (ES model), mm

Modulus of relative stiffness (DL model), mm
 73

Bonding Condition Between PCC Slab and Base 

Among 491 PCC and base section layer moduli obtained in this study using the DL model, only 

19 were assigned an unbonded interface between the layers. In the remaining 472 cases, the 

layers were assigned a bonded interface condition, as shown in figure 62. For the ES foundation 

model, the same proportion between bonded and unbonded interface was observed. If the 

bond/no bond condition did not dramatically affect backcalculated results, the interface condition 

was assumed to be bonded. In the majority of cases, the bonded interface was observed even for 

stabilized bases, as shown in figure 63. It is important to note that the backcalculation is done at 

the slab center. Bonding may be different at the slab edge. 

Backcalculated PCC Moduli

In this study, PCC layer properties were determined for 524 FWD passes of 19 SMP LTPP 

sections using DL and ES subgrade models. Figures 64 and 65 show frequency histogram 

distributions for PCC moduli backcalculated using the DL and ES subgrade models, respectively. 

The majority of the backcalculated moduli are in the reasonable range (from 25 to 55 GPa), 

supporting the conclusion of the robustness of the backcalculation approach. 

0

100

200

300

400

500

600

DL ES

Number of observations

BOND

NO BOND

Figure 62. Distribution of bond/no bond interface condition for non-stabilized and stabilized 

SMP LTPP sections. 
 74

0

20

40

60

80

100

120

140

160

180

DL ES

Number of observations

BOND

NO BOND

Figure 63. Distribution of bond/no bond interface condition for SMP LTPP sections with 

stabilized bases. 

Figure 64. Frequency distribution of backcalculated PCC modulus of elasticity (DL model) for 

SMP LTPP sections. 

0

50

100

150

200

250

20000 30000 40000 50000 60000 70000 80000 90000 100000

Backcalculated PCC modulus of elasticity (DL model), MPa

Number of observations
 75

Figure 65. Frequency distribution of backcalculated PCC modulus of elasticity (ES model) for 

SMP LTPP sections. 

Figure 66. Backcalculated PCC modulus of elasticity (ES model) versus backcalculated PCC 

modulus of elasticity (DL model) for SMP sections. 

0

50

100

150

200

250

300

20000 30000 40000 50000 60000 70000 80000 90000 100000

Backcalculated PCC modulus of elasticity (ES model), MPa

Number of observations

y = 1.3023x

R2

 = 0.9692

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 10000 20000 30000 40000 50000 60000 70000

Backcalculated Epcc (ES model), MPa

Backcalculated Epcc (DL model), MPa
 76

However, several backcalculated PCC moduli are much higher than acceptable (greater than 

80 GPa). These moduli were backcalculated from FWD passes of the sections located in a very 

cold climate (Manitoba, Minnesota, and Quebec) during wintertime, and there are indications 

that the subgrade was frozen during these tests. Although, theoretically, it should not 

significantly affect backcalculated PCC moduli, analysis of FWD deflection data SMP sections 

clearly shows that currently available backcalculation procedures may produce misleading results 

if a subgrade is frozen. Moreover, the majority of FWD passes of Manitoba, Minnesota, and 

Quebec sections during winter resulted in unsuccessful backcalculation. Therefore, it is not 

recommended to upload these high moduli to the IMS database. 

Figure 66 shows backcalculated PCC moduli using the ES subgrade model versus backcalculated 

PCC moduli using the DL subgrade model. A very good correlation between these two sets of 

moduli is observed. A linear regression analysis resulted in the following relationship: 

EPCC,DL= 1.3023 EPCC,ES

R2

 = 96.92% 

N = 531 

SEE = 1925 MPa 

where 

EPCC,ES = PCC modulus of elasticity backcalculated using ES model, GPa 

EPCC,ES = PCC modulus of elasticity backcalculated using DL model, GPa 

Although this relationship is based on the results of backcalculation from the FWD passes of the 

pavement sections made during different times of the year and time of the day, it is very close to 

the corresponding model developed for the GPS and SPS sections. This supports the conclusion 

that when a transition from the ES to the DL foundation model needs to be made, moduli of 

elasticity of the upper layers also needs to be adjusted.It should be noted, however, that 

unrealistically high PCC moduli obtained from FWD testing of section 833802 (Manitoba) in 

November and December 1993 and December 1994 were excluded from the analysis. 

Effect of Seasonal Variation and Time of Testing

The collected FWD data allow for analysis of the effect of season and time of day on 

backcalculated values. Typically, several passes are conducted each day on SMP sections to 

study the variations that may occur over a single day. Almost all sections showed dependence of 

backcalculated parameters on the time of the testing. Figures 67 and 68 show the subgrade 

support k-value and ES, respectively, for section 133919 (Georgia) obtained from 

backcalculation of the four FWD passes conducted in April 1996 at 7:30 a.m., 11 a.m., 2 p.m., 

and 5 p.m. The results show great variation over the course of the day. The lowest subgrade 

moduli are determined from the 2 p.m. testing when backcalculated subgrade moduli are about 

three times lower than from 7 a.m. testing. Backcalculated PCC moduli of elasticity followed the 

same trend for both DL and ES models, as shown in figures 69 and 70, although the difference 

among backcalculated values from different FWD passes was not as striking as for subgrade 

moduli. 
 77

Section 133019; 24-Apr-96

0

10

20

30

40

50

60

70

80

90

0 5 10 15 20 25

Station

k-value, kPa/mm

7:30 a.m., Ta=37, Ts=34 11:10 a.m., Ta=61, Ts=73

2:10 p.m., Ta=72, Ts=97 5 p.m., Ta=75, Ts=90

Figure 67. Daily variation in backcalculated k-value, section 133019 (April 1996). 

Section 133019; 24-Apr-96

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90

Station

Backcalculated E subgr , MPa

7:30 a.m., Ta=37, Ts=34 11:10 a.m., Ta=61, Ts=73

2:10 p.m., Ta=72, Ts=97 5 p.m., Ta=75, Ts=90

Figure 68. Daily variation in backcalculated subgrade modulus of elasticity, section 133019 

(April 1996). 
 78

Section 133019; 24-Apr-96

0

5

10

15

20

25

30

35

40

45

50

0 10 20 30 40 50 60 70 80 90

Station

Epcc (DL model), GPa

7:30 a.m., Ta=37, Ts=34 11:10 a.m., Ta=61, Ts=73

2:10 p.m., Ta=72, Ts=97 5 p.m., Ta=75, Ts=90

Figure 69. Daily variation in backcalculated PCC modulus of elasticity (DL model), section 

133019 (April 1996). 

Section 133019; 24-Apr-96

0

5

10

15

20

25

30

35

40

0 10 20 30 40 50 60 70 80 90

Station

Epcc (ES model), GPa

7:30 a.m., Ta=37, Ts=34 11:10 a.m., Ta=61, Ts=73

2:10 p.m., Ta=72, Ts=97 5 p.m., Ta=75, Ts=90

Figure 70. Daily variation in backcalculated PCC modulus of elasticity (ES model), section 

133019 (April 1996). 
 79

Similar effects were observed for many sections located in different climatic regions. Figures 71 

through 78 show the effect of time testing on the backcalculated parameters obtained for section 

493011 (Utah) and 274040 (Minnesota). Once again, a significant change in backcalculated 

subgrade moduli from time of day was observed. For section 493011, change in backcalculated 

PCC moduli was also significant. 

Tables 10 and 11 present mean and maximum coefficient of variation of the mean section 

backcalculated parameters for the SMP LTPP sections from different FWD passes made on the 

same day of testing. For 12 of the 16 SMP sections for which multiple FWD passes were 

available, a coefficient of variation in backcalculated k-value greater than 20 percent was 

observed on at least 1 day of testing. Although variability in radius of relative stiffness and PCC 

modulus of elasticity is lower, six sections (040215, 063042, 313018, 48 4142, 493011, and 

893015) also exhibited significant (greater than 20 percent) variability in backcalculated PCC 

modulus. Sections 063042 and 133019 exhibited coefficient of variation in backcalculated 

radius of relative stiffness greater than 10 percent in at least one day of testing. 

It is reasonable to conclude, therefore, that the time of day of FWD testing may significantly 

affect backcalculated moduli. This effect is most likely due to temperature differences and the 

resulting slab curling. Therefore, accounting for the effect of PCC slab curling is very important 

for reliable interpretation of FWD deflection data. However, development of a procedure for 

slab curling correction was outside the scope of this project. 

As expected, the season of testing was found to affect backcalculated subgrade moduli. This 

effect is highly confounded, however, with the effect of the time of testing. To reduce the latter 

effect, only first FWD passes for each day of testing were considered in the analysis of seasonal 

variation in backcalculated values. 

Figures 79 through 84 show mean backcalculated subgrade moduli and PCC moduli for sections 

133019 (Georgia), 274040 (Minnesota), and 533813 (Washington) obtained from different days 

of testing for the first FWD pass on each day. Section 133019 shows much lower subgrade 

moduli in March 1996 than for any other days of testing. This could be explained by the effect 

of heavy early spring rain that softens the subgrade. Section 533813 shows lower subgrade 

moduli in June than in other months of the testing. For other testing times, these two sections 

show quite consistent backcalculated values for both subgrade and concrete layers. 

Minnesota Section 274040 exhibited lower backcalculated values in the end of spring and early 

summer (June 1994, April–May 1996) and much higher backcalculated values for winter and 

early spring months. Freezing subgrades resulted not only in higher subgrade moduli, but also in 

increased backcalculated PCC moduli. This increase is much higher than can be expected by an 

increase in bending stiffness of the constructed layers due to frozen base and upper subgrade. 

However, the dynamic nature of FWD testing may explain this discrepancy. Therefore, 

accounting for dynamic effects in backcalculation may significantly increase reliability. 

Tables 12 and 13 present coefficients of variation in mean backcalculated parameters from the 

first FWD passes for the SMP LTPP sections for DL and ES models, respectively. It can be 

observed that variability is higher than that observed for GPS sections. This may be due to more 

frequent testing and, therefore, better ability to catch the spring thaw with SMP testing. It is also 

clear that, for the majority of the sections, variability in backcalculated results from different 

FWD passes in one day may be significantly higher than seasonal variability. This suggests that 
 80

Section 493011; 11-Jul-94

0

20

40

60

80

100

120

0 2 4 6 8 10 12 14 16 18

Station

k-value, kPa/mm

9:50 a.m., Ta=84, Ts=81 1 p.m., Ta=95, Ts=99 15:40 p.m., Ta=102, Ts=104

Figure 71. Daily variation in backcalculated k-value, section 493011 (July 1994). 

Section 493011; 11-Jul-94

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12 14 16 18

Station

Backcalculated E subgr , MPa

9:50 a.m., Ta=84, Ts=81 1 p.m., Ta=95, Ts=99 15:40 p.m., Ta=102, Ts=104

Figure 72. Daily variation in backcalculated subgrade modulus of elasticity, section 493011 

(July 1994). 
 81

Section 493011; 11-Jul-94

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0 2 4 6 8 10 12 14 16 18

Station

Epcc (DL model), GPa

9:50 a.m., Ta=84, Ts=81 1 p.m., Ta=95, Ts=99 15:40 p.m., Ta=102, Ts=104

Figure 73. Daily variation in backcalculated PCC modulus of elasticity (DL model), section 

493011 (July 1994). 

Section 493011; 11-Jul-94

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12 14 16 18

Station

Epcc (ES, unbonded), 10 6 MPa

9:50 a.m., Ta=84, Ts=81 1 p.m., Ta=95, Ts=99 15:40 p.m., Ta=102, Ts=104

Figure 74. Daily variation in backcalculated PCC modulus of elasticity (ES model), section 

493011 (July 1994). 
 82

Section 274040; 31-Mar-94

0

10

20

30

40

50

60

70

80

120 125 130 135 140 145 150 155 160

Station

k-value, kPa/mm

8:30 a.m., Ta=41, Ts=28 11:40 a.m., Ta=54, Ts=55

Figure 75. Daily variation in backcalculated k-value, section 274040 (March 1994). 

Section 274040; 31-Mar-94

0

50

100

150

200

250

120 125 130 135 140 145 150 155 160

Station

Backcalculated E subgr , MPa

8:30 a.m., Ta=41, Ts=28 11:40 a.m., Ta=54, Ts=55

Figure 76. Daily variation in backcalculated subgrade modulus of elasticity, section 274040 

(March 1994). 
 83

Section 274040; 31-Mar-94

0

5

10

15

20

25

30

35

40

45

120 125 130 135 140 145 150 155 160

Station

Epcc (DL modeld), GPa

8:30 a.m., Ta=41, Ts=28 11:40 a.m., Ta=54, Ts=55

Figure 77. Daily variation in backcalculated PCC modulus of elasticity (DL model), 

section 274040 (March 1994). 

Section 274040; 31-Mar-94

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

120 125 130 135 140 145 150 155 160

Station

Epcc (ES model), GPa

8:30 a.m., Ta=41, Ts=28 11:40 a.m., Ta=54, Ts=55

Figure 78. Daily variation in backcalculated PCC modulus of elasticity (ES model), 

section 274040 (March 1994). 
 84

Table 10. Coefficients of variation of the mean values of backcalculated parameters from the 

same day of testing (DL model). 

k-Value Radius of Relative 

Stiffness 

PCC Modulus of 

Elasticity 

State Section 

Mean Max Mean Max Mean Max 

Arizona 040215 0.200 0.485 0.045 0.071 0.101 0.232 

California 063042 0.084 0.221 0.045 0.136 0.120 0.293 

Georgia 133019 0.412 0.694 0.076 0.135 0.140 0.239 

Indiana 183002 0.063 0.063 0.005 0.009 0.040 0.053 

Kansas 204054 0.036 0.085 0.008 0.019 0.008 0.017 

Minnesota 274040 0.081 0.374 0.020 0.096 0.016 0.045 

Nebraska 313018 0.140 0.207 0.023 0.043 0.157 0.235 

New York 364018 0.111 0.228 0.022 0.058 0.038 0.095 

North Carolina 370201 0.061 0.183 0.024 0.037 0.082 0.152 

Pennsylvania 421606 0.102 0.215 0.022 0.055 0.043 0.074 

Texas 484142 0.107 0.193 0.021 0.046 0.076 0.231 

Texas 484143 0.099 1.503 0.072 1.188 0.099 1.437 

Utah 493011 0.183 0.560 0.027 0.086 0.219 0.614 

Washington 533813 0.135 0.537 0.024 0.098 0.055 0.169 

Manitoba 833802 0.052 0.213 0.014 0.060 0.047 0.146 

Quebec 893015 0.076 0.200 0.030 0.080 0.090 0.249 

Table 11. Coefficients of variation of the mean values of backcalculated parameters from the 

same day of testing (ES model). 

Subgrade 

Modulus of 

Elasticity 

Radius of Relative 

Stiffness 

PCC Modulus of 

Elasticity 

State Section 

Mean Max Mean Max Mean Max 

Arizona 040215 0.169 0.417 0.052 0.080 0.103 0.213 

California 063042 0.061 0.122 0.050 0.150 0.122 0.322 

Georgia 133019 0.359 0.588 0.092 0.155 0.114 0.171 

Indiana 183002 0.058 0.061 0.006 0.010 0.038 0.051 

Kansas 204054 0.033 0.073 0.010 0.023 0.005 0.010 

Minnesota 274040 0.062 0.293 0.021 0.112 0.020 0.047 

Nebraska 313018 0.139 0.215 0.025 0.044 0.158 0.234 

New York 364018 0.095 0.188 0.026 0.065 0.035 0.065 

North Carolina 370201 0.059 0.195 0.025 0.049 0.073 0.153 

Pennsylvania 421606 0.075 0.120 0.020 0.040 0.048 0.077 

Texas 484142 0.091 0.231 0.020 0.052 0.065 0.189 

Texas 484143 0.017 0.064 0.014 0.033 0.032 0.068 

Utah 493011 0.193 0.517 0.029 0.077 0.210 0.583 

Washington 533813 0.115 0.444 0.026 0.111 0.055 0.155 

Manitoba 833802 0.043 0.200 0.015 0.070 0.047 0.139 

Quebec 893015 0.058 0.208 0.029 0.047 0.087 0.249 
 85

0

50

100

150

200

250

300

350

6-Mar-89

14-Jan-93

31-Mar-95

1-Aug-95

18-Sep-95

16-Oct-95

28-Nov-95

22-Jan-96

22-Feb-96

26-Mar-96

24-Apr-96

21-May-96

20-Aug-96

25-Sep-96

16-Oct-96

Date

Esubgr , MPa, and k-value, 0.25 kPa/mm

k ES

Figure 79. Seasonal variation in backcalculated subgrade moduli, section 133019. 

 0

 5

 10

 15

 20

 25

 30

 35

 40

 45

6-Mar-89

14-Jan-93

31-Mar-95

1-Aug-95

18-Sep-95

16-Oct-95

28-Nov-95

22-Jan-96

22-Feb-96

26-Mar-96

24-Apr-96

21-May-96

20-Aug-96

25-Sep-96

16-Oct-96

Date

Backcalculated E pcc, GPa

DL ES

Figure 80. Seasonal variation in backcalculated PCC modulus of elasticity, section 133019. 
 86

0

50

100

150

200

250

300

27-Apr-92

29-Jun-95

18-Jul-95

25-Sep-95

30-Oct-95

20-Nov-95

18-Dec-95

21-Feb-96

17-Apr-96

22-May-96

25-Jun-96

29-Jul-96

28-Aug-96

Date

Esubgr , MPa, k-value, 0.25 kPa/mm

k Es

Figure 81. Seasonal variation in backcalculated subgrade moduli, section 533813. 

 0

 10

 20

 30

 40

 50

 60

 70

27-Apr-92

29-Jun-95

18-Jul-95

25-Sep-95

30-Oct-95

20-Nov-95

18-Dec-95

21-Feb-96

17-Apr-96

22-May-96

25-Jun-96

29-Jul-96

28-Aug-96

Date

Backcalculated E pcc, GPa

DL ES

Figure 82. Seasonal variation in backcalculated PCC modulus of elasticity, section 533813. 
 87

0

50

100

150

200

250

28-Jun-89

28-Jul-93

21-Sep-9322-Sep-93

18-Oct-93

16-Nov-93

16-Dec-93

14-Apr-94

4-May-94

23-Jun-94

25-Aug-94

30-Sep-94

13-Oct-94

10-Nov-94

31-Mar-95

14-Apr-9528-Apr-95

11-May-95

16-Jun-95

9-Oct-96

22-Apr-97

6-May-97

28-May-97

Date

Esubgr , Mpa, k-value, 0.25 kPa/mm

k Es

Figure 83. Seasonal variation in backcalculated subgrade moduli, section 274040. 

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

100.00

28-Jun-89

28-Jul-93

21-Sep-9322-Sep-93

18-Oct-93

16-Nov-93

16-Dec-93

14-Apr-94

4-May-94

23-Jun-94

25-Aug-94

30-Sep-94

13-Oct-94

10-Nov-94

31-Mar-95

14-Apr-9528-Apr-95

11-May-95

16-Jun-95

9-Oct-96

22-Apr-97

6-May-97

28-May-97

Date

Backcalculated E pcc, GPa

DL4 ES4

Figure 84. Seasonal variation in backcalculated PCC modulus of elasticity, section 274040. 
 88

Table 12. Coefficients of variation of the mean values of backcalculated parameters from first 

FWD passes (DL model). 

State Section k-Value PCC Modulus of 

Elasticity 

Radius of Relative 

Stiffness 

Arizona 040215 0.125 0.057 0.138 

California 063042 0.125 0.077 0.204 

Georgia 133019 0.199 0.040 0.091 

Indiana 183002 0.113 0.039 0.072 

Kansas 204054 0.085 0.031 0.093 

Minnesota 274040 0.386 0.089 0.531 

Nebraska 313018 0.214 0.071 0.122 

New York 364018 0.177 0.027 0.124 

North Carolina 370201 0.210 0.061 0.080 

North Carolina 370205 0.204 0.051 0.120 

North Carolina 370208 0.110 0.024 0.128 

North Carolina 370212 0.045 0.045 0.132 

Pennsylvania 421606 0.191 0.051 0.087 

Texas 484142 0.099 0.051 0.118 

Texas 484143 0.079 0.032 0.056 

Utah 493011 0.156 0.044 0.161 

Washington 533813 0.236 0.054 0.167 

Manitoba 833802 0.652 0.151 0.801 

Quebec 893015 0.357 0.095 0.546 

Table 13. Coefficients of variation of the mean values of backcalculated parameters from first 

FWD passes (ES model). 

State Section Subgrade 

Modulus of 

Elasticity 

PCC Modulus of 

Elasticity 

Radius of Relative 

Stiffness 

Arizona 040215 0.093 0.064 0.150 

California 063042 0.087 0.087 0.230 

Georgia 133019 0.173 0.052 0.088 

Indiana 183002 0.088 0.045 0.079 

Kansas 204054 0.080 0.041 0.104 

Minnesota 274040 0.287 0.099 0.588 

Nebraska 313018 0.125 0.075 0.128 

New York 364018 0.163 0.028 0.112 

North Carolina 370201 0.172 0.072 0.091 

North Carolina 370212 0.181 0.063 0.068 

Pennsylvania 421606 0.125 0.063 0.154 

Texas 484142 0.054 0.049 0.127 

Texas 484143 0.096 0.042 0.080 

Utah 493011 0.066 0.059 0.128 

Washington 533813 0.055 0.034 0.061 

Manitoba 833802 0.140 0.046 0.149 

Quebec 893015 0.208 0.063 0.175 
 89

it is important to conduct FWD basin testing early in the morning to reduce variability in 

backcalculated values.

Backcalculation Analysis Results 

A summary of the backcalculation analysis results for the SMP sections is given in the LTPP 

IMS table MON_DEFL_RGD_BAKCAL_SECT. The results are presented in terms of the mean 

values, minimum values, maximum values, and standard deviations of the following parameters: 

 DL Model

• k-value (modulus of subgrade reaction). 

• EPCC (modulus of elasticity of the PCC slab). 

• Ebase (modulus of elasticity of the base layer). 

 ES Model

• Esubgr (modulus of elasticity of the subgrade). 

• EPCC (modulus of elasticity of the PCC slab). 

• Ebase (modulus of elasticity of the base layer). 
 90
 91

CHAPTER 5. LIMITATIONS OF CURRENT BACKCALCULATION PROCEDURES 

Although backcalculated parameters determined in this study were found to be realistic for the 

majority of the LTPP rigid pavement sections, this study also identified some limitations of the 

current backcalculation procedures for rigid pavements. It was noted that backcalculated values 

may vary significantly due to factors such as temperature at the time of testing, slab curling 

conditions, time of day of the testing, bonding conditions, and time of year of the testing. 

Current rigid pavement backcalculation technology is inappropriate to adequately address all 

these aspects. The LTPP database, however, provides an excellent opportunity to conduct an indepth study of these factors. 

This chapter summarizes observed problems with rigid pavement backcalculation and discusses 

research needed to address the problems. 

Comparison of Backcalculated and Laboratory PCC Moduli

It is commonly believed that the backcalculated EPCC is approximately equal to the static EPCC. 

However, only limited information on comparison of these parameters is available. Darter et al. 

reported that the mean value of backcalculated elastic modulus for three American Association 

of State Highway and Transportation Officials (AASHTO) Road Test Loop1 sections was 

approximately equal to the elastic modulus obtained from dynamic tests on beam samples.(1) 

However, the scatter was too wide to make any definite conclusions. 

Analysis of backcalculated concrete moduli was beyond the scope of the previous LTPP study.(2)

 In this study, a comparison of backcalculated moduli of elasticity of concrete versus elastic 

moduli obtained from the cores was performed for GPS-3 sections. Only a weak correlation 

between these moduli is observed. Figure 85 presents backcalculated moduli versus laboratory 

moduli for LTPP sections with an aggregate base. These sections were selected to eliminate the 

effect of a stiff stabilized base on the results of backcalculation. However, even for these 

sections, the correlation between backcalculated and laboratory moduli is not strong. 

Many factors may contribute to this discrepancy. The choice of subgrade model, the effect of 

temperature curling, and dynamic effects significantly affect the results of backcalculation. In 

addition, the laboratory testing was performed on only a few samples taken beyond the ends of 

the section where deflections were measured. Proper accounting for these factors should close 

the gap between backcalculated and laboratory moduli. 

Slab Curling Correction 

Discussion of the results of backcalculation for SMP sections identified the need for 

development of slab curling correction factors. Several SMP sections exhibited significant 

reduction in backcalculated subgrade moduli from morning to midday testing. Also, 

backcalculated PCC moduli appeared to depend on the time of day of the testing. Development 

of appropriate correction factors should be of high priority for the following reasons: 

• It is not always possible to conduct FWD testing in the early morning when thermal 

gradients are usually very low. 

• FWD testing of GPS sections may be conducted under different temperature 
 92

conditions. In this case, backcalculated parameters in the beginning of the section 

are not comparable with the backcalculated parameters at the end of the section. 

This is possibly why in several cases, reduction in backcalculated subgrade moduli at 

the end of the section is observed. 

y = 0.1923x + 25.202

R2

 = 0.1859

0

10

20

30

40

50

60

70

80

0 10 20 30 40 50 60 70 80 90 100

Backcalculated Epcc, GPa

Laboratory Epcc, GPa

Figure 85. Comparison of backcalculated and laboratory concrete moduli of elasticity. 

Dynamic Interpretation of FWD Test Results 

Analysis of backcalculated PCC moduli shows significant variations in this parameter from 

different test dates for some sections. One of the possible explanations of this effect is that the 

current backcalculation procedures for rigid pavements assume quasi-static pavement behavior 

during FWD testing (see figure 86). Field test data indicate, however, that this assumption may 

not be valid. If the pavement would behave quasi-statically, the peak of the applied load would 

occur at exactly the same time as the peaks of the sensor deflections. Figure 87 presents a typical 

load and deflection time history during FWD testing. This figure shows a significant lag 

between the peak of the applied load and among the peaks of the sensor deflections. 
 93

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time

Deflection, mm

0

5

10

15

20

25

30

35

40

45

50

Total FWD Load, kN

D0

D200

D300

D450

D600

D900

D1500

load

Figure 86. Quasi-static pavement responses to FWD loading. 

Figure 87. Measured pavement responses to FWD loading.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.001 0.002 0.003 0.004 0.005 0.006

Time, s

FWD deflections, mm

0

10

20

30

40

50

60

Total FWD load, kN

D0

D200

D300

D450

D600

D900

D1500

FWD Load
 94

In this study, Westergaards’s solution for the interior loading of a slab-on-grade was generalized for 

a case of dynamic loading. The following nondimentional parameter, m*, affecting dynamic 

response of a PCC slab was identified: 





 





T

2

k

m

m =

2

* π (51) 

where 

 m = mass of the unit area of the plate and the movable portion of subgrade 

 T = duration of the applied FWD load 

 k = coefficient of subgrade reaction 

A closed form analytical solution for determining deflection time history from FWD-type loading 

was developed. A numerical evaluation of that solution has been facilitated by development of a 

computer program. The execution time per backcalculation on a PC is only a fraction of a 

second. In the future, this solution can serve as a basis for development of an efficient dynamic 

backcalculation procedure for rigid pavements.

To investigate the importance of pavement inertia, the dynamic behavior of a slab-on-grade was 

simulated. The following slab parameters were assumed: E = 34.5 GPa; μ = 0.15; h = 225 mm; k = 

27.1 kPa/mm. The nondimensional mass, m*

, was varied from 0 (quasi-static behavior) to 10. The 

FWD load was modeled as a load uniformly distributed over a circle with a radius equal to 150 mm. 

The following time history was assumed: 

g(t )=1 - (t ) * * cos (52) 

where t* is the nondimensional time defined as: 

T

t

t* = 2π (53)

Figures 88 to 90 present solutions for m* equal to 1, 4, and 10, respectively. If the nondimensional 

deflection m* is equal to 0 (quasi-static loading; see figure 86), then the peak of the applied load 

coincides in time with the peaks of all sensor deflections. With an increase in m*, the lag between 

the load peak and the deflection peaks increases, and this increase is higher for the outer sensors. 

Dynamic Effects on Results of Backcalculation

The maximum applied load deflections were used as input parameters to the conventional static 

backcalculation procedure. Figure 91 presents the ratio between backcalculated E and k and actual 

E and k with respect to m*

. One can see that backcalculated moduli (from a conventional 

backcalculation procedure) are higher than actual plate elastic moduli, and an increase in m*

increases discrepancy between backcalculated and actual moduli of elasticity. Although 

backcalculated k-values are much lower than those used in forward calculation, they are less 

sensitive to m*

. Since the "actual" k-value of subgrade is not known anyway, one may assume that 

this discrepancy is accounted for by converting kdyn into kstatic. 
 95

0

0.05

0.1

0.15

0.2

0.25

01234567

Nondimensional time

Deflection, mm

0

5

10

15

20

25

30

35

40

45

50

Total FWD load, kN

D0

D200

D300

D450

D600

D900

D1500

load

Figure 88. Effect of pavement inertia on FWD sensor deflections, m*=1. 

0

0.05

0.1

0.15

0.2

0.25

0123456

Nondimensional time

Deflection, mm

0

5

10

15

20

25

30

35

40

45

50

Total FWD load, kN

D0

D200

D300

D450

D600

D900

D1500

load

Figure 89. Effect of pavement inertia on FWD sensor deflections, m*=4. 
 96

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0123456

Nondimensional time

Deflection, mm

0

5

10

15

20

25

30

35

40

45

50

Total FWD load, kN

D0

D200

D300

D450

D600

D900

D1500

load

Figure 90. Effect of pavement inertia on FWD sensor deflections, m*=10. 

Figure 91. Effect of pavement inertia on the results of backcalculation. 

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12

m*

Eback / E

0

0.5

1

1.5

2

2.5

Kbaxk / k

E k-value
 97

To further investigate the importance of the inertia effect on the results of backcalculation, it is 

necessary to estimate the nondimensional mass, m*, for a typical rigid pavement. The inertia of the 

pavement system, m, comes from the inertia of the concrete slab and subgrade, and these inertia are 

additive, therefore: 

where 

msl = mass of a unit area of concrete slab 

msubgr = mass of a unit area of subgrade moved by FWD load 

Therefore, 

The parameter msubgr is not known and should be determined from backcalculation. It is of interest, 

however, to estimate m*

sl to check if it has a reasonable value to explain dynamic effects. Indeed, 

where 

γ = density of concrete 

Assuming that the slab is 225 mm thick with a density equal to 0.027 N/cm3

, the k-value is equal to 

27.1 kPa/mm, and the duration of impulse is equal to 0.025 s, m*

 is equal to 1.28. Therefore, for 

this set of parameters, the influence of the inertial properties of the concrete layer significantly 

affects the results of backcalculation. Addition of the subgrade inertia makes the dynamic effects 

even more pronounced. 

Variation in the pavement inertia due to effects of temperature and moisture on the base and 

subgrade layers may explain seasonal variation in backcalculated elastic parameters. If a PCC 

pavement is tested at the same location at different times of the year, it is reasonable to suggest that 

the backcalculated subgrade elastic properties will show significant changes, but the backcalculated 

elastic properties of the concrete will be similar. However, use of the conventional static 

backcalculation procedures may lead to surprising results. It was observed in this study that several 

SMP rigid pavement sections located in a cold climate exhibited much higher PCC moduli of 

elasticity in the winter than in the rest of the year. The fact that the frozen subgrade increases the 

flexural stiffness of the pavement (reflected in the higher backcalculated PCC modulus) explains 

only a part of the discrepancy. Therefore, it is reasonable to suggest that the remaining part of the 

difference came from the increase in the mass of the plate moved by the FWD load. 

This discussion leads to the following conclusions: 

1. Dynamic effects can significantly affects the measured deflections. 

2. The mass of the subgrade moved by the impulse affects backcalculated moduli. 

m= msl+ msubgr (54) 

m* = m*sl + m*subgr (55) 





 





T

2

k

h

m =

2

sl

sl

* π

γ (56) 
 98

3. The variability in backcalculated moduli from one test to another may be explained by 

dynamic behavior of rigid pavements under FWD loading. 

4. An efficient backcalculation procedure may be developed. This solution may use the 

results of static backcalculation and correct it using a correction factor obtained from 

the analysis of lags between deflection peaks. 
99

CHAPTER 6. SUMMARY AND RECOMMENDATIONS FOR CONTINUED RESEARCH 

The analyses reported in this document were intended to calculate elastic layer properties (the coefficients 

and exponents of the constitutive equation) from deflection measurements and recommend representative 

elastic parameters for each section for inclusion in the LTPP IMS database. This report presented a 

discussion on the selection of the backcalculation methodology and results of the backcalculation for 

GPS, SPS, and SMP rigid pavement sections. The following are highlights of the study: 

• The Best Fit method was selected as the primary backcalculation method for the LTPP rigid 

pavement sections for both DL and ES subgrade models. 

• Four sensors (0, 305, 610, and 914 mm) were used for rigid pavement backcalculation. 

• A computer program that includes pre-processor, backcalculation, and post-processor 

modules was developed and utilized for backcalculation of GPS, SPS, and SMP rigid 

pavement sections. 

• In the majority of the cases, reasonable backcalculated values were obtained for the PCC 

slab, a variety of base types, and many different subgrades. 

• Strong correlation was found between backcalculated parameters using DL and ES subgrade 

models. 

• It was found that temperature curling during a day has a profound effect on the results of 

backcalculation. Therefore, it is important to conduct FWD basin testing early in the morning 

when temperature gradients are low to reduce variability in backcalculated values. 

• Poor correlation was found between backcalculated and laboratory elastic moduli of concrete. 

• A large proportion of unstabilized or stabilized base courses were modeled as bonded to the 

PCC slabs to produce reasonable backcalculated material moduli (center slab 

backcalculation). 

The next logical step should be to improve the backcalculation procedure for rigid pavements to address 

the problems identified in this study. Following are the most urgent needs: 

• Develop PCC slab temperature curling correction factors for the results of backcalculation. 

• Develop correction factors accounting for dynamic behavior of rigid pavements under FWD 

loading and verify them using LTPP deflection history data. 

• Adjust backcalculated parameters obtained in this study using the correction factors for 

temperature curling and dynamic behavior. 

These additional studies will significantly improve the reliability of backcalculated parameters, narrow 

the gap between backcalculated and laboratory determined PCC moduli, and significantly improve our 

understanding of rigid pavement behavior. 
100
101

REFERENCES 

1. Darter, M.I., K.T. Hall, and C. Kuo. (1995). Support Under Portland Cement Concrete Pavements. 

NCHRP Report 372. Washington, DC: National Cooperative Highway Research Program. 

2. Hall, K.T., M.I. Darter, T.E. Hoerner, and L. Khazanovich. (1997). LTPP Data Analysis—Phase I: 

Validation of Guidelines for k-Value Selection and Concrete Pavement Performance Prediction. 

Technical Report FHWA-RD-96-198. 

3. Ioannides, A.M. (1990). “Dimensional Analysis in NDT Rigid Pavement Evaluation.” Journal of 

Transportation Engineering. ASCE. Vol. 116, No. 1. New York, NY. 

4. Korenev, B.G. (1954). “Problems of Analysis of Beams and Plates on Elastic Foundation.” 

Gosudarstvennoe Izdatel’stvo Literatury po Stroitel’stvu i Arkhitekture. Moscow, USSR (in Russian). 

5. Smith, K.D. et al. (1995). Performance of Concrete Pavements Volume III—Improving Concrete 

Pavement Performance. Final Report. Contract DTFH61-91-C-00053. McLean, VA: Federal 

Highway Administration. 

6. Hoffman, M.S., and M.R. Thompson. (1981). “Mechanistic Interpretation of Nondestructive Testing 

Deflections.” Civil Engineering Studies. Transportation Engineering Series No. 32. Illinois 

Cooperative Highway and Transportation Research Program Series No. 190. University of Illinois, 

Urbana, IL. 

7. Ioannides et al. (1989). “Interpretation of Falling Weight Deflectometer Test Results Using 

Principles of Dimensional,” Proceedings, Fourth International Conference on Concrete Pavement 

Design and Rehabilitation, Purdue University, W. Lafayette, IN, April 23-25, pp. 231-247. 

8. Hall, K.T. (1991). Performance, Evaluation, and Rehabilitation of Asphalt-Overlaid Concrete 

Pavements. Thesis presented to the University of Illinois, at Urbana, IL, in partial fulfillment of the 

requirements for the degree of Doctor of Philosophy. 

9. Khazanovich, L. (1994). Structural Analysis of Multi-Layered Concrete Pavement Systems. Ph.D. 

Thesis, University of Illinois, Urbana, IL. 

10. Khazanovich, L. and A.M. Ioannides. (1995). “DIPLOMAT: An Analysis Program for Both 

Bituminous and Concrete Pavements.” Transportation Research Record 1482. Washington, DC: 

Transportation Research Board. 

11. Losberg, A. (1960). “Structurally Reinforced Concrete Pavements.” Doktorsavhandlingar Vid 

Chalmers Tekniska Högskola, Göteborg, Sweden. 

12. Ioannides, A.M., and L. Khazanovich. (1994). “Backcalculation Procedure For Three-Layered 

Concrete Pavements.” Proceedings, Fourth International Conference on Bearing Capacity of Roads 

and Airfields. Minneapolis, MN. 

13. Ioannides, A.M., L. Khazanovich, and J.L. Becque. (1992). “Structural Evaluation of Base Layers in 

Concrete Pavement Systems.” Transportation Research Record 1370. Washington, DC: 

Transportation Research Board. 

14. Ioannides, A.M., M.R. Thompson, and E.J. Barenberg. (1985). “A Finite Element Analysis of Slabson-Grade Using a Variety of Support Models.” Proceedings, Third International Conference on 

Concrete Pavement Design and Rehabilitation, Purdue University, W. Lafayette, IN, April 23-25, pp. 

309-324. 

15. Von Quintus, H.L. and A. Simpson (1999). Documentation of the Back-Calculation of Layer 

Parameters for LTPP Test Sections, Volume II: Layered Elastic Analysis for Flexible and Rigid 

Pavements, Report submitted to FHWA, contract number DTFH61-96-C-00003, Work Order 9, Task 

2. 
102
103

APPENDIX A. TYPICAL RESULTS OF BACKCALCULATION FOR 

PCC LTPP GPS SECTIONS 

This appendix consists of two sections: a brief description of the presentation of the results followed by 

typical backcalculation analysis. 

The first item of the section analysis report for each section is an information table. The first set of 

information is drawn from IMS table TST_L05B. It contains the following data items: 

Section Number: This is a 6-digit number, the first two digits of which refer to the SHRP LTPPassigned state ID. The last four digits correspond to the SHRP section ID. 

State: This field contains the name of the state to which the section belongs. 

Layer Information: The thicknesses of each layer along with a description of layer type is presented 

under headings Layer 1, Layer 2, etc. As in the IMS database, the layers are 

numbered starting from the subgrade layer to the topmost layer in the section. 

For example, consider a pavement section comprising PCC, base, and subgrade 

layers. Here, Layer 1 would correspond to the subgrade, Layer 2 is the base, and 

Layer 3 is the PCC. 

The second set of information in the table, presented under the heading Backcalculation System, is drawn 

from the table MON_DEFL_RGD_BAKCAL_LAYER. It contains the following data items: 

Layer Information: The final pavement profile selected to perform the backcalculation analysis is 

presented under headings Layer 1 and Layer 2. Subgrade layer information is not 

included. 

Ratio: This refers to the modulus ratio of the PCC and the base layers (EPCC/Ebase), a 

numerical value included in the MON_DEFL_RGD_BAKCAL_LAYER table. 

The relevant PCC backcalculation data are presented in plots for each parameter as discussed later. 

Because the subgrade is modeled both as dense liquid and elastic solid, two analyses are performed for 

each pavement section. Four graphs are plotted for each case showing the variation of the PCC modulus, 

base modulus, the radius of relative stiffness, and the subgrade parameter (either the k-value or the elastic 

modulus) for each point of the section. 

The paragraphs below contain brief information regarding the section plots. 

Section Number 

The section number for each plot is located above all other entries in the legend. A sample plot legend is 

included in figure 92 for clarity. 

Filled Versus Hollow Legends 

The plots present data of interest obtained from FWD tests over several visits to the site. Each visit is 

indicated by a unique legend entry. For example in figure 92, the data for the test performed on 31-Aug89 is identified with a square point, whereas the data for the test performed on 16-Apr-93 is identified 

using a triangular point, etc. Further, any symbol of a given shape could either be filled or empty, as 

shown in figure 92. A filled legend indicates that the data point was used in computing the mean value of 
104

the quantity it represents (PCC modulus, base modulus, k-value, etc.). An empty legend indicates that the 

data point has been excluded from the computation of the mean. This may occur due to test type, the 

error predicted for that quantity exceeds the tolerance, or it does not fall within twice the standard 

deviation. 

Representation of the Mean Value 

The recommended mean value for a data set is represented by a line through the data under consideration, 

as well as by a numerical value in the legend box located at the right top corner of each plot (see figure 

92). The numerical value shown will have the same units as the individual points. The mean value for 

any given test date is placed immediately below the symbol designation for that particular test date, as 

shown in figure 92. 

 

It is possible, in some cases, that the mean value is not plotted on the figures for certain data sets. This is 

because no mean value recommendation was made. Figure 92 illustrates one such instance, the 16-Apr90 test date. 

Indication of Bonded and Unbonded PCC-Base Interface

The suffix “b” or “u” is attached to the legend date text to represent either a bonded PCC and base layer 

interface or an unbonded interface, respectively. 

High Cut-off Limit 

The data range shown for the graphs are those that have been determined to be the “allowable” range for 

that particular example. The dense-liquid modulus of subgrade reaction, k, may range from 0 to 200 

kPa/mm, for example. Data points lying along the y = “max” line of any plot might carry a value that is 

either y = “max” or greater. For example, any k greater than 200 kPa/mm will plot as 200 kPa/mm. 

Blank Output Charts 

It may occur that the analysis procedure prints out a blank figure with a message that a quantity has not 

been backcalculated. This message indicates that the backcalculation procedure does not calculate moduli 

of elasticity either because there was no thickness information available or there was an unbonded PCC 

overlay present in the system. 
105

Figure 92. Typical data plot legend. 
106

Example 1 

Section: 013998 

State: Alabama 

Layer 5: Portland Cement Concrete (CRCP) 

 Thickness: 208 mm 

Layer 4: Soil Cement 

 Thickness: 145 mm 

Layer 3: Fine-grained Soils 

 Thickness: 152 mm 

Layer 2: Lime-Treated Soil 

 Thickness: 152 mm 

Layer 1: Coarse-grained soil: silty sand 

 Thickness: NA 

Backcalculation system: 

Layer 1: Thickness 208 mm 

Layer 2: Thickness 145 mm; Ratio = 10 
107

0

40

80

120

160

200

0 30 60 90 120 150

Station

k-value, kPa/mm

13998

13-Sep-90

mean=93.1

Figure 93. Backcalculated k-values for section 013998. 

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 30 60 90 120 150

Station

Radius of relative stiffness (DL model), mm 

13998

13-Sep-90-b

mean=938.4

Figure 94. Backcalculated radius of relative stiffness (DL model) for section 013998. 
108

0

20000

40000

60000

80000

100000

0 30 60 90 120 150

Station

EPCC (DL model), MPa

13998

13-Sep-90-b

mean=57723.6

Figure 95. Backcalculated PCC elastic modulus (DL model) for section 013998. 

0

20000

40000

60000

80000

100000

0 30 60 90 120 150

Station

Ebase (DL model), MPa

13998

13-Sep-90-b

mean=5772.4

Figure 96. Backcalculated elastic modulus of base (DL model) for section 013998. 
109

0

100

200

300

400

500

0 30 60 90 120 150

Station

Esubgr , MPa

13998

13-Sep-90-b

mean=312.4

Figure 97. Backcalculated modulus of elasticity of subgrade for section 013998. 

0

200

400

600

800

1000

0 30 60 90 120 150

Station

Radius of relative stiffness (ES model), mm

13998

13-Sep-90-b

mean=637.6

Figure 98. Backcalculated radius of relative stiffness (ES model) for section 013998. 
110

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

0 30 60 90 120 150

Station

EPCC (ES model), MPa

13998

13-Sep-90-b

mean=41216.6

Figure 99. Backcalculated PCC elastic modulus (ES model) for section 013998. 

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

0 30 60 90 120 150

Station

Ebase (ES model), MPa

13998

13-Sep-90-b

mean=4121.7

Figure 100. Backcalculated elastic modulus of base (ES model) for section 013998. 
111

Example 2 

Section: 015008 

State: Alabama 

Layer 4: Portland Cement Concrete (CRCP) 

 Thickness: 234 mm 

Layer 3: HMAC 

 Thickness: 155 mm 

Layer 2: Crushed Stone 

 Thickness: 198 mm 

Layer 1: Coarse-grained soil: silty sand 

 Thickness: NA 

Backcalculation system: 

Layer 1: Thickness 234 mm 

Layer 2: Thickness 155 mm; Ratio = 15 
112

0

40

80

120

160

200

0 30 60 90 120 150

Station

k-value, kPa/mm

15008

17-Sep-90 

mean=50.7

16-Sep-94 

mean=46.6

Figure 101. Backcalculated k-values for section 015008. 

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 30 60 90 120 150

Station

Radius of relative stiffness (DL model), mm 

13998

13-Sep-90-b

mean=938.4

Figure 102. Backcalculated radius of relative stiffness (DL model) for section 015008. 
113

0

20000

40000

60000

80000

100000

0 30 60 90 120 150

Station

EPCC (DL model), MPa

13998

13-Sep-90-b

mean=57723.6

Figure 103. Backcalculated PCC elastic modulus (DL model) for section 015008. 

0

20000

40000

60000

80000

100000

0 30 60 90 120 150

Station

Ebase (DL model), MPa

13998

13-Sep-90-b

mean=5772.4

Figure 104. Backcalculated elastic modulus of base (DL model) for section 015008. 
114

0

100

200

300

400

500

0 30 60 90 120 150

Station

Esubgr , MPa

13998

13-Sep-90-b

mean=312.4

Figure 105. Backcalculated modulus of elasticity of subgrade for section 015008. 

0

200

400

600

800

1000

0 30 60 90 120 150

Station

Radius of relative stiffness (ES model), mm

13998

13-Sep-90-b

mean=637.6

Figure 106. Backcalculated radius of relative stiffness (ES model) for section 015008. 
115

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

0 30 60 90 120 150

Station

EPCC (ES model), MPa

13998

13-Sep-90-b

mean=41216.6

Figure 107. Backcalculated PCC elastic modulus (ES model) for section 015008. 

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

100000

0 30 60 90 120 150

Station

Ebase (ES model), MPa

13998

13-Sep-90-b

mean=4121.7

Figure 108. Backcalculated elastic modulus of base (ES model) for section 015008. 
116