Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

User's Guide: Estimation of Key PCC, 

Base, Subbase, and Pavement 

Engineering Properties from Routine

Tests and Physical Characteristics

PUBLICATION NO. FHWA-HRT-12-031 AUGUST 2012
FOREWORD

Material characterization is a basic aspect of pavement engineering and is critical for analysis, 

performance prediction, design, construction, quality control/quality assurance, pavement 

management, and rehabilitation. Advanced tools like the American Association of State 

Highway and Transportation Officials Mechanistic-Empirical Pavement Design Guide, Interim 

Edition: A Manual of Practice, commonly known as the MEPDG, can be used to estimate the 

influence of several fundamental engineering material parameters on the long-term performance 

of a pavement.

(1) Consequently, there is a great need for more information about material 

properties, which are addressed only to a limited extent with currently available resources for 

performing laboratory and field testing. Reliable correlations between material parameters and 

index properties offer a cost-effective alternative, and the derived material property values are

equivalent to the level 2 inputs in the MEPDG. This study initially verified data adequacy in the 

Long-Term Pavement Performance (LTPP) database and also involved retrieving needed data.

(2)

In the next phase of the study, prediction models were developed to help practicing engineers 

estimate proper MEPDG inputs. This report describes the basis for selecting material parameters 

that need predictive models, provides a review of current LTPP program data, and proposes 

several statistically derived models to predict material properties. The models developed under 

this effort have been incorporated into a simple software program compatible with current 

versions of Microsoft Windows® operating system.

Jorge E. Pagán-Ortiz

Director, Office of Infrastructure

Research and Development

Notice

This document is disseminated under the sponsorship of the U.S. Department of Transportation 

in the interest of information exchange. The U.S. Government assumes no liability for the use of 

the information contained in this document. This report does not constitute a standard, 

specification, or regulation.

The U.S. Government does not endorse products or manufacturers. Trademarks or 

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

objective of the document.

Quality Assurance Statement

The Federal Highway Administration (FHWA) provides high-quality information to serve 

Government, industry, and the public in a manner that promotes public understanding. FHWA 

uses standards and policies are used to ensure and maximize the quality, objectivity, utility, and 

integrity of its information. It also periodically reviews quality issues and adjusts its programs 

and processes to ensure continuous quality improvement.
TECHNICAL REPORT DOCUMENTATION PAGE

1. Report No.

FHWA-HRT-12-031

2. Government Accession No. 3. Recipient's Catalog No.

4. Title and Subtitle

User’s Guide: Estimation of Key PCC, Base, Subbase, and 

Pavement Engineering Properties from Routine Tests and Physical 

Characteristics

5. Report Date 

August 2012

6. Performing Organization Code

7. Author(s) 

C. Rao, L. Titus-Glover, B. Bhattacharya, and M.I. Darter

8. Performing Organization Report No.

9. Performing Organization Name and Address

Applied Research Associates, Inc.

100 Trade Centre Drive, Suite 200 

Champaign, IL 61820

10. Work Unit No. (TRAIS)

11. Contract or Grant No.

DTFH61-02-C-00138

12. Sponsoring Agency Name and Address

Office of Infrastructure Research and Development

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101-2296

13. Type of Report and Period Covered

Interim Report

14. Sponsoring Agency Code

15. Supplementary Notes

The Contracting Officer’s Technical Representative (COTR) was Larry Wiser, HRDI LTPP data analysis contract.

16. Abstract

Material characterization is a critical component of modern day pavement analysis, design, construction, quality 

control/quality assurance, management, and rehabilitation. At each stage during the life of a project, the influence of 

several fundamental engineering material parameters on the long-term performance of the pavement can be predicted 

using advanced tools like the American Association of State Highway and Transportation Officials MechanisticEmpirical Pavement Design Guide (MEPDG). Consequently, there is a need for more information about material 

properties, which are addressed only to a limited extent with currently available resources for performing laboratory 

and field testing. Reliable correlations between material parameters and index properties offer a cost-effective 

alternative and are equivalent to the level 2 MEPDG inputs. The Long-Term Pavement Performance (LTPP) database 

provides data suitable for developing predictive models for Portland cement concrete (PCC) materials, stabilized 

materials, and unbound materials, as well as other design-related inputs for the MEPDG. This user’s guide provides a 

summary of the models developed, describes their applications for specific project conditions, and lists their 

limitations. The following models are included:

• PCC materials: Compressive strength, flexural strength, elastic modulus, tensile strength, and coefficient of 

thermal expansion.

• Stabilized materials: Elastic modulus of lean concrete base.

• Unbound materials: Resilient modulus of fine-grained and coarse-grained materials.

• Rigid pavement design features: Pavement curl/wrap effective temperature difference for jointed plain 

concrete pavement and continuously reinforced concrete pavement designs.

17. Key Words

Pavements, LTPP, material properties, MEPDG, 

prediction model, Index properties

18. Distribution Statement

No restrictions. This document is available to the public 

through the National Technical Information Service, 

Springfield, VA 22161.

19. Security Classification (of this report)

Unclassified

20. Security Classification (of this page)

Unclassified

21. No. of Pages

86

22. Price

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized
ii

SI* (MODERN METRIC) CONVERSION FACTORS 

APPROXIMATE CONVERSIONS TO SI UNITS

Symbol When You Know Multiply By To Find Symbol 

LENGTH

in inches 25.4 millimeters mm 

ft feet 0.305 meters m 

yd yards 0.914 meters m 

mi miles 1.61 kilometers km 

AREA 

in2 square inches 645.2 square millimeters mm2

ft2 square feet 0.093 square meters m2

yd2 square yard 0.836 square meters m2

ac acres 0.405 hectares ha

mi2 square miles 2.59 square kilometers km2

VOLUME 

fl oz fluid ounces 29.57 milliliters mL 

gal gallons 3.785 liters L 

ft3 cubic feet 0.028 cubic meters m3 

yd3 cubic yards 0.765 cubic meters m3 

NOTE: volumes greater than 1000 L shall be shown in m3

MASS oz ounces 28.35 grams g

lb pounds 0.454 kilograms kg

T short tons (2000 lb) 0.907 megagrams (or "metric ton") Mg (or "t") 

TEMPERATURE (exact degrees) o

F Fahrenheit 5 (F-32)/9 Celsius o

C 

or (F-32)/1.8 

ILLUMINATION 

fc foot-candles 10.76 lux lx

fl foot-Lamberts 3.426 candela/m2 cd/m2

FORCE and PRESSURE or STRESS 

lbf poundforce 4.45 newtons N 

lbf/in2 poundforce per square inch 6.89 kilopascals kPa 

APPROXIMATE CONVERSIONS FROM SI UNITS 

Symbol When You Know Multiply By To Find Symbol 

LENGTH mm millimeters 0.039 inches in 

m meters 3.28 feet ft 

m meters 1.09 yards yd 

km kilometers 0.621 miles mi 

AREA mm2 square millimeters 0.0016 square inches in2 

m2 square meters 10.764 square feet ft2 

m2 square meters 1.195 square yards yd2 

ha hectares 2.47 acres ac

km2 square kilometers 0.386 square miles mi2 

VOLUME 

mL milliliters 0.034 fluid ounces fl oz 

L liters 0.264 gallons gal 

m3 cubic meters 35.314 cubic feet ft3 

m3 cubic meters 1.307 cubic yards yd3 

MASS 

g grams 0.035 ounces oz

kg kilograms 2.202 pounds lb

Mg (or "t") megagrams (or "metric ton") 1.103 short tons (2000 lb) T 

TEMPERATURE (exact degrees) o

C Celsius 1.8C+32 Fahrenheit oF 

ILLUMINATION 

lx lux 0.0929 foot-candles fc 

cd/m2 candela/m2 0.2919 foot-Lamberts fl

FORCE and PRESSURE or STRESS 

N newtons 0.225 poundforce lbf 

kPa kilopascals 0.145 poundforce per square inch lbf/in2

*SI is the symbol for the International System of Units. Appropriate rounding should be made to comply with Section 4 of ASTM E380. 

(Revised March 2003)
iii

TABLE OF CONTENTS

CHAPTER 1. INTRODUCTION................................................................................................ 1

BACKGROUND ..................................................................................................................... 1

DATA NEEDS......................................................................................................................... 1

Design ................................................................................................................................. 2

Construction........................................................................................................................ 2

Pavement Management....................................................................................................... 4

SCOPE ..................................................................................................................................... 4

CHAPTER 2. MODEL DEVELOPMENT................................................................................. 7

LIST OF MODELS................................................................................................................. 8

PRACTICAL GUIDE AND SOFTWARE PROGRAM................................................... 10

CHAPTER 3. PCC MODELS ................................................................................................... 11

PCC COMPRESSIVE STRENGTH MODELS ................................................................ 11

Compressive Strength Model 1: 28-day Cylinder Strength Model .................................. 12

Compressive Strength Model 2: Short-Term Cylinder Strength Model........................... 14

Compressive Strength Model 3: Short-Term Core Strength Model ................................. 17

Compressive Strength Model 4: All Ages Core Strength Model ..................................... 21

Compressive Strength Model 5: Long-Term Core Strength Model ................................. 24

Relative Comparison of All Compressive Strength Models............................................. 25

PCC FLEXURAL STRENGTH MODELS........................................................................ 29

Validation of Existing Models.......................................................................................... 29

Flexural Strength Model 1: Flexural Strength Based on Compressive Strength.............. 30

Flexural Strength Model 2: Flexural Strength Based on Age, Unit Weight, and 

w/c Ratio ........................................................................................................................... 33

Flexural Strength Model 3: Flexural Strength Based on Age, Unit Weight, and CMC ... 35

PCC ELASTIC MODULUS MODELS.............................................................................. 38

Validation of Existing Models.......................................................................................... 38

Elastic Modulus Model 1: Model Based on Aggregate Type........................................... 40

Elastic Modulus Model 2: Model Based on Age and Compressive Strength................... 42

Elastic Modulus Model 3: Model Based on Age and 28-Day Compressive Strength...... 43

Limitations of Elastic Modulus Models............................................................................ 45

PCC TENSILE STRENGTH MODELS ............................................................................ 45

PCC Tensile Strength Model Based on Compressive Strength ........................................ 45

PCC CTE MODELS............................................................................................................. 47

Current Issue with CTE Overestimation in LTPP Data.................................................... 47

CTE Model 1: CTE Based on Aggregate Type (Level 3 Equation for MEPDG) ............ 48

CTE Model 2: CTE Based on Mix Volumetrics (Level 2 Equation for MEPDG)........... 49

CHAPTER 4. RIGID PAVEMENT DESIGN FEATURES MODELS................................. 53

DELTAT—JPCP DESIGN................................................................................................... 53

Using the JPCP deltaT Model........................................................................................... 59

DELTAT—CRCP DESIGN.................................................................................................. 60

Using the CRCP deltaT Model ......................................................................................... 64
iv

CHAPTER 5. STABILIZED MATERIALS MODELS.......................................................... 65

LCB ELASTIC MODULUS MODEL................................................................................ 65

CHAPTER 6. UNBOUND MATERIALS MODELS .............................................................. 67

RESILIENT MODULUS OF UNBOUND MATERIALS ................................................ 67

Constitutive Model Parameter k1 ...................................................................................... 67

Constitutive Model Parameter k2 ...................................................................................... 67

Constitutive Model Parameter k3 ...................................................................................... 68

REFERENCES............................................................................................................................ 75
v

LIST OF FIGURES

Figure 1. Illustration. MEPDG performance prediction during the design and construction 

stage ................................................................................................................................................ 3

Figure 2. Screenshot. View of Correlations user interface .......................................................... 10

Figure 3. Equation. Prediction model 1 for fc,28d........................................................................... 12

Figure 4. Graph. Predicted versus measured for 28-day cylinder compressive strength model... 13

Figure 5. Graph. Residual error plot for 28-day cylinder compressive strength model ............... 13

Figure 6. Graph. 28-day compressive strength model sensitivity to w/c ratio ............................. 14

Figure 7. Graph. 28-day compressive strength model sensitivity to CMC................................... 14

Figure 8. Equation. Prediction model 2 for fc,t.............................................................................. 14

Figure 9. Graph. Predicted versus measured for short-term cylinder compressive strength 

model............................................................................................................................................. 15

Figure 10. Graph. Residual errors for short-term cylinder compressive strength model ............. 16

Figure 11. Graph. Short-term cylinder compressive strength sensitivity to CMC ....................... 16

Figure 12. Graph. Short-term cylinder compressive strength sensitivity to w/c ratio .................. 16

Figure 13. Graph. Short-term cylinder compressive strength sensitivity to age........................... 17

Figure 14. Equation. Prediction model 3 for fc,t............................................................................ 17

Figure 15. Graph. Predicted versus measured for short-term core compressive strength 

model............................................................................................................................................. 18

Figure 16. Graph. Residual errors for short-term core compressive strength model.................... 18

Figure 17. Graph. Short-term core compressive strength sensitivity to CMC.............................. 19

Figure 18. Graph. Short-term core compressive strength sensitivity to unit weight .................... 19

Figure 19. Graph. Short-term core compressive strength sensitivity to MAS.............................. 19

Figure 20. Graph. Short-term core compressive strength sensitivity to w/c ratio ........................ 20

Figure 21. Graph. Short-term core compressive strength sensitivity to fine aggregate FM......... 20

Figure 22. Graph. Short-term core compressive strength sensitivity to age................................. 20

Figure 23. Equation. Prediction model 4 for fc,t............................................................................ 21

Figure 24. Graph. Predicted versus measured for all ages core compressive strength model...... 22

Figure 25. Graph. Residual errors for all ages core compressive strength model ........................ 22

Figure 26. Graph. All ages core compressive strength sensitivity to w/c ratio ............................ 23

Figure 27. Graph. All ages core compressive strength sensitivity to CMC.................................. 23

Figure 28. Graph. All ages core compressive strength sensitivity to unit weight ........................ 23

Figure 29. Graph. All ages core compressive strength sensitivity to age ..................................... 24

Figure 30. Equation. Prediction model 5 for fc,LT.......................................................................... 24

Figure 31. Graph. Predicted versus measured for long-term core compressive strength model .. 25

Figure 32. Graph. Residual errors for long-term core compressive strength model..................... 25

Figure 33. Graph. Model compressive strength prediction for varying CMC.............................. 26

Figure 34. Graph. Model compressive strength prediction for varying w/c ratio......................... 26

Figure 35. Graph. Model compressive strength prediction for varying unit weights................... 27

Figure 36. Graph. Strength gain in the short-term predicted by three models.............................. 27

Figure 37. Graph. Long-term strength gain predicted by the models........................................... 28

Figure 38. Equation. Mr ................................................................................................................ 29

Figure 39. Graph. Predicted versus measured for validating 0.5 power flexural strength 

model............................................................................................................................................. 30
vi

Figure 40. Graph. Predicted versus measured for validating 0.667 power flexural strength 

model............................................................................................................................................. 30

Figure 41. Equation. Prediction model 6 for MR.......................................................................... 30

Figure 42. Graph. Predicted versus measured values for flexural strength model based on 

compressive strength..................................................................................................................... 31

Figure 43. Graph. Residual errors for flexural strength model based on compressive strength... 32

Figure 44. Graph. Comparison of flexural strength models based on compressive strength ....... 33

Figure 45. Equation. Prediction model 7 for MRt ......................................................................... 33

Figure 46. Graph. Predicted versus measured values for flexural strength model based on 

age, unit weight, and w/c ratio...................................................................................................... 34

Figure 47. Graph. Residual errors for flexural strength model based on age, unit weight, 

and w/c ratio.................................................................................................................................. 35

Figure 48. Equation. Prediction model 8 for MRt ......................................................................... 35

Figure 49. Graph. Predicted versus measured values for flexural strength model based 

on age, unit weight, and CMC...................................................................................................... 36

Figure 50. Graph. Residual errors for flexural strength model based on age, unit weight, 

and CMC....................................................................................................................................... 36

Figure 51. Graph. Sensitivity of flexural strength predictions to CMC........................................ 37

Figure 52. Graph. Sensitivity of flexural strength predictions to w/c ratio .................................. 37

Figure 53. Graph. Sensitivity of flexural strength predictions to unit weight .............................. 38

Figure 54. Graph. Sensitivity of flexural strength predictions to age........................................... 38

Figure 55. Equation. Ec as a function of square root of compressive strength ............................. 38

Figure 56. Equation. Model form for E as a function of compressive strength with slope 

and intercept.................................................................................................................................. 39

Figure 57. Equation. Ec ................................................................................................................. 39

Figure 58. Equation. E as function of unit weight and compressive strength .............................. 39

Figure 59. Equation. Prediction model 9 for Ec ............................................................................ 40

Figure 60. Graph. Predicted versus measured for elastic modulus model based on 

aggregate type ............................................................................................................................... 41

Figure 61. Graph. Residual errors for elastic modulus model based on aggregate type............... 41

Figure 62. Equation. Prediction model 10 for Ec,t......................................................................... 42

Figure 63. Graph. Predicted versus measured for elastic modulus model based on age 

and compressive strength.............................................................................................................. 43

Figure 64. Graph. Residual errors for elastic modulus model based on age and compressive 

strength.......................................................................................................................................... 43

Figure 65. Equation. Prediction model 11 for Ec,t......................................................................... 43

Figure 66. Graph. Predicted versus measured for elastic modulus model based on age 

and 28-day compressive strength.................................................................................................. 44

Figure 67. Graph. Residual errors for elastic modulus model based on age and 28-day 

compressive strength..................................................................................................................... 45

Figure 68. Equation. Prediction model 12 for ft............................................................................ 45

Figure 69. Graph. Predicted versus measured for tensile strength model .................................... 46

Figure 70. Graph. Residual errors plot for tensile strength model................................................ 46

Figure 71. Graph. Sensitivity of tensile strength prediction model to change compressive 

strength.......................................................................................................................................... 47
vii

Figure 72. Graph. Comparison of average values from other sources and recommended 

CTE values based on aggregate type from LTPP data ................................................................. 49

Figure 73. Equation. Prediction model 14 for CTEPCC ................................................................. 49

Figure 74. Graph. Predicted versus measured for CTE model based on mix volumetrics........... 50

Figure 75. Graph. Residual errors for CTE model based on mix volumetrics............................. 50

Figure 76. Graph. Comparison of CTE model prediction with average values reported in 

literature for each aggregate rock type.......................................................................................... 51

Figure 77. Graph. CTE model prediction versus average values reported in literature for each 

aggregate rock type ....................................................................................................................... 51

Figure 78. Graph. Sensitivity of the CTE model to coarse aggregate content.............................. 52

Figure 79. Equation. Prediction model 15 for deltaT/inch ........................................................... 53

Figure 80. Graph. Predicted versus measured for JPCP deltaT gradient model........................... 54

Figure 81. Graph. Residual errors for JPCP deltaT gradient model ............................................. 55

Figure 82. Graph. Predicted versus measured deltaT based on the JPCP deltaT gradient 

model............................................................................................................................................. 55

Figure 83. Graph. Sensitivity of predicted deltaT to temperature range during month of 

construction................................................................................................................................... 56

Figure 84. Graph. Sensitivity of predicted deltaT to slab width................................................... 56

Figure 85. Graph. Sensitivity of predicted deltaT to slab thickness............................................. 56

Figure 86. Graph. Sensitivity of predicted deltaT to PCC slab unit weight ................................. 57

Figure 87. Graph. Sensitivity of predicted deltaT to PCC w/c ratio............................................. 57

Figure 88. Graph. Sensitivity of predicted deltaT to latitude of the project location ................... 57

Figure 89. Graph. Predicted deltaT for different locations in the United States .......................... 58

Figure 90. Equation. Prediction model 16 for deltaT/inch ........................................................... 60

Figure 91. Graph. Predicted versus measured for CRCP deltaT model ....................................... 61

Figure 92. Graph. Residual errors for CRCP deltaT model.......................................................... 62

Figure 93. Graph. Effect of maximum temperature on CRCP deltaT prediction model.............. 62

Figure 94. Graph. Effect of temperature range on CRCP deltaT prediction model ..................... 63

Figure 95. Graph. Effect of slab thickness on CRCP deltaT prediction model............................ 63

Figure 96. Graph. Effect of geographic location on CRCP deltaT prediction model................... 63

Figure 97. Equation. Prediction model 17 for ELCB ...................................................................... 65

Figure 98. Graph. Predicted versus measured for the LCB elastic modulus model ..................... 65

Figure 99. Graph. Residual errors for the LCB elastic modulus model ....................................... 66

Figure 100. Equation. Mr .............................................................................................................. 67

Figure 101. Equation. Prediction model 18 for k1 ........................................................................ 67

Figure 102. Equation. Prediction model 19 for k2 ........................................................................ 67

Figure 103. Equation. Prediction model 20 for k3 ........................................................................ 68

Figure 104. Graph. Resilient modulus parameter k1 for unbound material types included 

in the model development database .............................................................................................. 68

Figure 105. Graph. Resilient modulus parameter k2 for unbound material types included 

in the model development database .............................................................................................. 69

Figure 106. Graph. Resilient modulus parameter k3 for unbound material types included 

in the model development database .............................................................................................. 69

Figure 107. Graph. Plot of measured versus predicted resilient modulus (using k1, k2, and k3

derived from figure 101 through figure 103)................................................................................ 70
viii

Figure 108. Graph. Plot showing predicted and measured resilient modulus versus bulk 

stress for fine- and coarse-grained soils........................................................................................ 70

Figure 109. Graph. Effect of material type (AASHTO soil class) on predicted resilient 

modulus......................................................................................................................................... 71

Figure 110. Graph. Effect of percent passing 1/2-inch sieve on predicted resilient modulus ....... 72

Figure 111. Graph. Effect of liquid limit on predicted resilient modulus..................................... 72

Figure 112. Graph. Effect of optimum moisture content on predicted resilient modulus............ 73

Figure 113. Graph. Effect of No. 80 sieve on predicted resilient modulus .................................. 73

Figure 114. Graph. Effect of gravel content on predicted resilient modulus................................ 74

Figure 115. Graph. Effect of effective size on predicted resilient modulus................................. 74
ix

LIST OF TABLES

Table 1. Regression statistics for selected prediction model for 28-day PCC cylinder 

strength.......................................................................................................................................... 12

Table 2. Range of data used for 28-day PCC cylinder strength.................................................... 12

Table 3. Regression statistics for short-term cylinder strength model.......................................... 15

Table 4. Range of data used for short-term cylinder strength model............................................ 15

Table 5. Regression statistics for short-term core strength model................................................ 17

Table 6. Range of data used for short-term core strength model.................................................. 18

Table 7. Regression statistics for all ages core strength model .................................................... 21

Table 8. Range of data used for all ages core strength model ...................................................... 21

Table 9. Regression statistics for long-term core strength model................................................. 24

Table 10. Range of data used for long-term core strength model................................................. 24

Table 11. Power models developed for flexural strength prediction using LTPP data for 

validation....................................................................................................................................... 29

Table 12. Regression statistics for flexural strength model based on compressive strength........ 31

Table 13. Range of data used for flexural strength model based on compressive strength.......... 31

Table 14. Regression statistics for flexural strength model based on age, unit weight, 

and w/c ratio.................................................................................................................................. 34

Table 15. Range of data used for flexural strength model based on age, unit weight, 

and w/c ratio.................................................................................................................................. 34

Table 16. Regression statistics for flexural strength model based on age, unit weight, 

and CMC....................................................................................................................................... 35

Table 17. Range of data used for flexural strength model based on age, unit weight, 

and CMC....................................................................................................................................... 36

Table 18. Range of data used for elastic modulus model based on aggregate type...................... 41

Table 19. Regression statistics for elastic modulus model based on age and compressive 

strength.......................................................................................................................................... 42

Table 20. Range of data used for elastic modulus model based on age and compressive 

strength.......................................................................................................................................... 42

Table 21. Regression statistics for elastic modulus model based on age and 28-day 

compressive strength..................................................................................................................... 44

Table 22. Range of data used for elastic modulus model based on age and 28-day 

compressive strength..................................................................................................................... 44

Table 23. Model statistics for tensile strength prediction model.................................................. 46

Table 24. Range of data used for tensile strength prediction model............................................. 46

Table 25. Prediction model 13 for PCC CTE based on aggregate type, x 10-6 inch/inch/°F........ 48

Table 26. Statistical analysis results for CTE model based on mix volumetrics.......................... 49

Table 27. Range of data used for CTE model based on mix volumetrics .................................... 50

Table 28. Regression statistics for JPCP deltaT model ................................................................ 54

Table 29. Range of data used for JPCP deltaT model .................................................................. 54

Table 30. Regression statistics for CRCP deltaT model............................................................... 60

Table 31. Range of data used for CRCP deltaT model................................................................. 61

1

CHAPTER 1. INTRODUCTION

BACKGROUND

Material characterization is vital to pavement analyses and has received increasing focus 

as it forms a critical component in recent improvements to engineering practices. This pertains 

to all aspects of pavement engineering—analysis, design, construction, quality control/quality 

assurance (QC/QA), pavement management, and rehabilitation. At each stage during the life 

of a project, the influence of several fundamental engineering material parameters on the 

long-term performance of the pavement has been recognized. There is a greater emphasis for 

optimizing the performance of concrete pavements, which involves a detailed understanding 

of the variables that affect pavement behavior and the properties of concrete that correspond to 

the desired performance. 

Consequently, there is a need for more information about material properties so that they can be 

characterized accurately for predicting performance or for verifying their quality during the 

construction phase. With limited resources for performing laboratory and field tests to determine 

material properties, the need for a secondary means to obtain these material property values

(i.e., through correlations or predictive models based on data from routine or less expensive 

tests) is obvious. Additionally, the American Association of State Highway and Transportation 

Officials (AASHTO) Mechanistic-Empirical Pavement Design Guide, Interim Edition: A 

Manual of Practice (MEPDG) offers users the option of using inputs obtained through 

correlations.

(1) The MEPDG defines level 2 inputs as those obtained from correlations between 

the primary inputs (level 1 measured) and other parameters that are material-specific or are 

measured through simpler tests. The Long-Term Pavement Performance (LTPP) database 

provides an excellent source of information to develop these correlations using material

properties of field sections.(2)

The current report addresses critical data needs for design, construction, and pavement 

management operations under the LTPP Data Analysis Technical Support contract. This project 

focuses on developing predictive models to estimate Portland cement concrete (PCC) and 

unbound material properties using LTPP data.

DATA NEEDS

Material property data needs in the context of this study are grouped into the following 

three categories:

• Inputs during the design stage.

• QC/QA during construction.

• Scheduling maintenance and rehabilitation in a pavement management program.
2

Design 

In both empirical and mechanistic-empirical design systems, material property inputs are 

essential to characterize pavement behavior and to predict pavement responses, such as the 

magnitudes of stress, strain, and displacement when subjected to applied traffic loads and 

environmental conditions. Furthermore, major pavement distresses are associated directly with 

the material properties of a component (or layer) of the pavement structure. For example, in 

jointed plain concrete pavements (JPCPs), transverse cracking is influenced by PCC flexural 

strength. Faulting can be related to the erodibility of the underlying base/subbase material. 

Punchout development in continuously reinforced concrete pavements (CRCPs) can be related 

to PCC tensile strength.

The MEPDG, developed under National Cooperative Highway Research Program (NCHRP)

Project 1-37A and subsequently improved under NCHRP 1-40D, allows users to model the 

effects of project-specific climate, traffic loads, materials, design features, and construction 

practices mechanistically to predict pavement performance based on distress models calibrated 

with LTPP field sections.

(3,4) The MEPDG is considered a significant improvement over current 

pavement design procedures, and it received the status of an AASHTO interim standard in 

November 2007. The publication User Manual and Local Calibration Guide for the 

Mechanistic-Empirical Pavement Design Guide and Software developed under NCHRP Project

1-40B provides guidance to State highway agencies (SHAs) that are considering implementing

the MEPDG.

(5) It is expected that SHAs will adopt locally calibrated distress models that are 

representative of their specific materials and design conditions. 

The need for a variety of material inputs is being recognized as agencies evaluate the MEPDG 

and streamline efforts for implementation. They continue to face challenges in estimating 

material parameter inputs and understanding their impact on pavement performance. For 

example, agencies do not have measured test data or access to databases and the necessary 

engineering expertise to develop correlations for their needs. Furthermore, due to a lack of 

familiarity with several input categories, they have come to rely on default values to characterize 

their typical materials. These default parameters are often a gross approximation of the true 

value, which may lead to erroneous distress and International Roughness Index (IRI) predictions. 

As another example, the permanent curl/warp gradient in the national calibration was set at 

-10 °F through the slab because it was not possible to obtain an accurate value for this parameter, 

which depends on construction-related conditions. Analysis of selected LTPP data made it 

possible to derive an improved way to estimate this important input for design.

This study provides much needed procedures to obtain several inputs and provide correlations 

to determine the range of material properties based on routine test results and physical 

characteristics. These correlations will supplement the User Manual and Local Calibration 

Guide for the Mechanistic-Empirical Pavement Design Guide and Software to support MEPDG 

implementation efforts.(5)

Construction

Pavement construction practices are being continually enhanced for faster and more efficient 

processes. In addition, new materials and material types are being introduced. For example, 
3

cement compositions and cement types have changed considerably over the years, resulting 

in PCC properties and durability characteristics different from the past. 

The current focus of QC/QA procedures is on identifying more reliable and faster QC/QA tests 

and determining material properties that are related directly to performance. The MEPDG 

enables performance prediction of the as-built pavement in addition to that of the as-designed 

pavement as long as deviations from design assumptions (i.e., material properties or construction 

practices such as curing or temperature during construction) are identified during the 

construction process (see figure 1). For example, although the density of an unbound material is 

a good indicator of construction quality, the more fundamental resilient modulus is an indicator 

of performance and is a key input to the MEPDG. The ability to predict resilient modulus 

from index properties measured during construction will make the QA process address both 

construction quality and pavement performance issues. Note that in figure 1, material properties 

measured during construction can be used to predict performance in the field and may be 

different from the design/target performance.

Also, performance-related specifications (PRSs) for concrete pavements have been developed in 

recent years. Irick et al., under a Federal Highway Administration (FHWA) study to demonstrate 

a PRS system for rigid pavement construction, considered three key performance indicators: 

PCC strength, PCC slab thickness, and initial serviceability.

(6) Several relationships for the 

prediction of PCC properties were evaluated under this study. PRSs have also been 

implemented on several projects that required many correlations between pavement 

properties and performance.

(7,8)

Figure 1. Illustration. MEPDG performance prediction during the design and construction

stage.
4

This study will bridge the gaps in current knowledge regarding the estimation of more 

fundamental material parameters that influence performance-based index properties or 

other commonly measured properties during construction.

Pavement Management

One of the key needs in managing pavements is an estimation of remaining life. Several SHAs 

use this parameter to program rehabilitation treatments. Various models (including the MEPDG 

models) are useful here in that they can be utilized to predict the remaining life until critical 

levels of each distress and IRI are reached. Also, agencies are now considering the integration of 

construction quality databases with pavement management databases to track the effect of design 

and construction quality on long-term performance. Such efforts lend themselves to more 

accurate performance predictions, whereby the performance of the as-constructed pavement can 

be used for scheduling maintenance and rehabilitation programs (see figure 1). However, many 

model inputs are needed related to the existing pavement, including inputs to characterize 

materials accurately. Index properties from construction QA data can be used to predict 

fundamental material properties that are related to performance.

In summary, the MEPDG provides a tool to specify material characteristics during the design 

and construction processes to achieve desired performance. The same models used in the 

MEPDG for design and construction analyses can be used in the future management of the 

pavement to estimate its remaining structural and functional life. For example, the inputs for a 

10-year-old pavement could be measured from the existing pavement and estimated from the 

MEPDG models to project future slab cracking. The curve can be adjusted to match today’s 

actual performance to improve the prediction. The slab cracking curve can be projected into the 

future to determine when it reaches a critical value to estimate its remaining life. The same could 

be done with joint faulting and IRI.

Therefore, the design, construction, and pavement management stages share a common need for 

determining a variety of material properties based on correlations from index properties and/or 

properties determined from more routine test procedures. This practice has been used in past 

AASHTO pavement design procedures and likely will increase in the future due to the more 

complex fundamental inputs required for the MEPDG procedure. 

SCOPE

This user’s guide provides a summary of the material property relationships that were developed 

through correlations to index properties. Also included are relationships for the prediction of 

the permanent curl/warp equivalent temperature difference, commonly referred to as deltaT, in 

rigid pavement design. A detailed research report prepared under this study is available as a 

separate publication.

(9)

SHAs can use these correlations to characterize material and design parameters as necessary for 

design, QC/QA, and pavement management. Most of the data used in the development of 

prediction models to estimate material properties were obtained from the LTPP Long-Term 

Pavement Performance Standard Data Release 23.0.

(10) Since these correlations are based on 

actual data from LTPP and field calibration sections used in the development of MEPDG distress 
5

prediction models, they are more reliable than default or typical values currently being used. 

This full potential of the MEPDG to predict performance accurately can be realized by providing 

more accurate input values to the procedure. These models can also help improve material 

specifications for use in pavement construction, particularly for PRSs.

Chapter 2 of this user’s guide provides general comments on the analyses performed in 

developing the models so that users understand the validity of the models presented in this guide. 

Each material type is then discussed in a separate chapter. Models for PCC materials, MEPDG 

design features, stabilized materials, and unbound materials are included in chapters 3 through 6,

respectively. For each predictive model, the following are included:

• The mathematical relationship.

• Model statistics.

• Plots showing the quality of prediction (predicted versus measured and residual 

error plots).

• Plots showing the sensitivity of each model parameter.

7

CHAPTER 2. MODEL DEVELOPMENT

The LTPP study database, Long-Term Pavement Performance Standard Data Release Version 

23.0, was used to develop the models.

(10) Material properties and pavement engineering 

properties for which develop predictive models were developed were selected based on

the following:

• Material inputs requirements for the MEPDG design procedure and the sensitivity of the 

specific parameter for performance prediction.

• Typical agency needs for determining material properties during QA. 

• Typical agency needs for determining material properties during routine pavement 

management functions.

• Data availability in the LTPP database.

Predictive models were developed for PCC compressive strength, PCC flexural strength, PCC 

elastic modulus, PCC tensile strength, lean concrete base (LCB) modulus, and unbound materials 

resilient modulus. In addition, rigid pavement design feature inputs properties were developed 

using the MEPDG calibration data. These include the JPCP and CRCP deltaT parameters. For all 

PCC material properties, multiple models were developed for use in different project situations 

and to provide user prediction model alternatives depending on the extent of mix design 

information available.

In developing the models, a uniform set of statistical criteria were used to select independent 

parameters to define a relationship as well as to mathematically formulate prediction functions. 

The analyses examined several statistical parameters in choosing the optimal model and in 

determining the predictive ability of the model. In general, the optimal set of independent 

variables (Mallows coefficient, Cp), the interaction effects (variance inflation factor (VIF)), the 

significance of the variable (p-value), and the goodness of fit (R2

) were verified. Additionally, 

the study validated or refined existing models and developed new relationships. In the analyses, 

the following general observations were made:

• PCC compressive strength could be correlated to several index properties. It was found 

to increase with decreasing water/cementitious (w/c) ratio, increasing cementitious 

materials content (CMC), increasing curing time, increasing unit weight, decreasing 

maximum aggregate size (MAS) for a given level of w/c ratio, and decreasing fineness 

modulus (FM) of the sand. 

• PCC flexural strength could be correlated to the compressive strength using a power

model. These relationships were validated and refined using the LTPP data. It also could 

be correlated to the w/c ratio, unit weight, CMC, and curing time. The correlation was 

improved significantly in the new models with the additional parameters. The flexural 

strength increased proportionally with all parameters listed except w/c ratio, with which it 

had an inverse relationship.
8

• PCC elastic modulus could be correlated to the compressive strength and unit weight 

using a power model, as has been done in past studies. These relationships were validated 

and verified with the data used in this study. Predictions could be made based on 

aggregate type, unit weight, compressive strength, and age with improved correlation. 

The elastic modulus increases with an increase in magnitude of all parameters listed.

• PCC tensile strength was found to correlate well with the compressive strength using a 

power relationship.

• The coefficient of thermal expansion (CTE) of PCC was most sensitive to the coarse 

aggregate type and the volumterics of the mix design.

• JPCP deltaT negative gradient was found to increase with an increase in temperature 

range at the project location for the month of construction and slab width and with a 

decrease in PCC thickness, unit weight, w/c ratio, and latitude of the project location.

• CRCP deltaT negative gradient was found to increase with an increase in maximum 

temperature at the project location for the month of construction and maximum 

temperature range and decrease with the use of chert, granite, limestone, and quartzite.

• The modulus of LCB was found to correlate well with its 28-day compressive strength 

based on a power model.

• The prediction of resilient modulus was possible using parameters k1, k2, and k3 of the 

constitutive model as follows:

o The parameter k1 was found to increase with a decrease in percent passing 

the 1/2-inch sieve, an increase in liquid limit, and a decrease in optimum 

moisture content.

o The parameter k2 was found to increase with a decrease in percent passing the 

No. 80 sieve, liquid limit, and percent gravel and an increase in the maximum 

particle size of the smallest 10 percent of the soil sample.

o The parameter k3 was dependent on the soil classification (coarse-grained versus 

fine-grained materials).

LIST OF MODELS

The following models have been developed under this study.

PCC compressive strength models include the following:

• Compressive strength model 1: 28-day cylinder strength model.

• Compressive strength model 2: Short-term cylinder strength model.

• Compressive strength model 3: Short-term core strength model.
9

• Compressive strength model 4: All ages core strength model.

• Compressive strength model 5: Long-term core strength model.

PCC flexural strength models include the following:

• Flexural strength model 1: Flexural strength based on compressive strength.

• Flexural strength model 2: Flexural strength based on age, unit weight, and w/c ratio.

• Flexural strength model 3: Flexural strength based on age, unit weight, and CMC.

PCC elastic modulus models include the following:

• Elastic modulus model 1: Model based on aggregate type.

• Elastic modulus model 2: Model based on age and compressive strength.

• Elastic modulus model 3: Model based on age and 28-day compressive strength.

The PCC indirect tensile strength model is as follows:

• PCC indirect tensile strength model: Model based on compressive strength.

PCC CTE models include the following:

• CTE model 1: CTE based on aggregate type (level 3 equation for MEPDG).

• CTE model 2: CTE based on mix volumetrics (level 2 equation for MEPDG).

The JPCP design deltaT model is as follows:

• JPCP deltaT model: JPCP deltaT gradient based on temperature range, slab width, slab 

thickness, PCC unit weight, w/c ratio, and latitude.

The CRCP design deltaT model is as follows:

• CRCP deltaT model: CRCP deltaT gradient based on maximum temperature, maximum 

temperature range, and aggregate type.

The LCB elastic modulus model is as follows:

• Elastic modulus model: Elastic modulus based on 28-day compressive strength.

The unbound materials resilient modulus is as follows:

• Resilient modulus model: Resilient modulus using constitutive model based on gradation, 

Atterberg limits, optimum moisture content, and soil classification.
10

PRACTICAL GUIDE AND SOFTWARE PROGRAM

The models described in this user’s guide have been incorporated into a user friendly software 

program, Correlations, which was developed under this study and can be used independently 

from the MEPDG. The software was developed on the Microsoft.NET platform to be compatible 

with the latest versions of the Microsoft Windows® operating systems. It uses a modern user 

interface library to provide a familiar look and feel. It features multiple windows on the user 

interface that are initially docked inside the main window. These windows can be moved 

separately from the main window for better viewing of the inputs or results. 

The program interface features tabs for PCC, design features, stabilized materials, and unbound 

materials. Models that belong to each of these categories are made available through a series of 

radio button selections placed in an accordion control. This placement not only provides the 

ability to make multiple selections, but it also conserves screen space so that the results of the 

calculations can be placed for easy viewing. Once a model is selected, the entry area adds 

controls for the available inputs of the model. Figure 2 shows a screenshot of the program

and displays the various tabs and general layout of the user interface.

Figure 2. Screenshot. View of Correlations user interface.

On all screens of the software, tooltips are provided for feedback on input range. Calculations 

occur after all necessary values have been input. Information about each model is available in 

an information window initially located at the bottom of the screen. This information is contextsensitive to the specific selections that the software user has made. Results of each calculation 

are displayed prominently in the results area window initially placed on the right side of the 

main window.
11

CHAPTER 3. PCC MODELS

Prediction models were developed for PCC compressive strength, PCC flexural strength, 

PCC elastic modulus, PCC tensile strength, and CTE. The following limitations apply to 

all PCC models:

• A fundamental limitation for any model is that the relationship that exists between the 

predicted parameter and the regressors is only valid for the range of data that has been 

included in the dataset. Furthermore, the statistical modeling procedures for the most part 

assume that the variables are normally distributed within the dataset. For example, the 

relationships developed for PCC properties, (e.g., compressive strength prediction model)

are applicable only for mixes with cement types 1 and 2. While one data point with type 3 

cement exists in the database (a JRCP section) compared to 500 data sets with type 1 and 

type 2 cements, the strength gain pattern of a type 3 cement is clearly outcompeted by the 

other 2 cement types in the database. As a result, it might not be evident within this 

dataset that type 3 cements produce higher strengths, especially in the early ages. 

• The model will reflect the intrinsic trends of the dataset used. For example, the data used 

for prediction of the 28-day compressive strength contains target low-strength and highstrength mix designs. If the primary means of achieving higher strengths for the States 

was to increase the cement content, the model will show a high correlation between the 

CMC and strength. However, there are multiple ways to enhance mix compressive 

strength, such as the use of lower w/c ratios, water-reducing agents, higher-strength 

aggregates, curing at higher temperatures and insulation, and use of type 3 cements. This 

is critical when the prediction models are implemented for estimating material properties.

PCC COMPRESSIVE STRENGTH MODELS

Compressive strength is considered a fundamental strength parameter and is used at different 

stages of a project—design, QA, opening time, rehabilitation design, etc. The following models 

are offered for PCC compressive strength, each of which is discussed in subsections to follow:

• Compressive strength model 1: 28-day cylinder strength model (suitable for estimating 

design strength).

• Compressive strength model 2: Short-term cylinder strength model (suitable for 

estimating opening time).

• Compressive strength model 3: Short-term core strength model (suitable for in situ

strength and opening time).

• Compressive strength model 4: All ages core strength model (suitable for estimating in 

situ strength at any age).

• Compressive strength model 5: Long-term core strength model (suitable for estimating 

long-term strength for rehabilitation design).
12

Compressive Strength Model 1: 28-day Cylinder Strength Model

The 28-day compressive strength model developed for cylinder strength is as follows:

f c,28d = 4028.41841− 3486.3501*w/ c + 4.02511*CMC

Figure 3. Equation. Prediction model 1 for fc,28d.

Where:

f’c,28d = 28-day compressive strength, psi.

w/c = Water to cementitious materials ratio.

CMC = Cementitious materials content, lb/yd3.

The model statistics are shown in table 1. The model was developed using 42 data points, and 

the prediction has an R2 value of 54.44 percent and a root mean square error (RMSE) of 871 psi.

Although it was compromised relative to the models discussed above, it provides a more 

meaningful model with a superior predictive ability. Table 2 provides details of the range of 

data used to develop the model. 

Table 1. Regression statistics for selected prediction model for 28-day PCC cylinder 

strength.

Variable

Degrees 

of 

Freedom 

(DF) Estimate

Standard 

Error t-Value Pr > |t| VIF

Intercept 1 4,028.41841 1,681.71576 2.4 0.0215 0

w/c ratio 1 -3486.3501 2,152.99857 -1.62 0.1134 2.40903

CMC 1 4.02511 1.32664 3.03 0.0043 2.40903

Table 2. Range of data used for 28-day PCC cylinder strength.

Parameter Minimum Maximum Average

w/c ratio 0.27 0.71 0.42

Cementitious content 376 936 664

Compressive strength 3,034 7,611 5,239

Figure 4 and figure 5 show the predicted versus measured values and the residuals plot for the 

model, respectively. Figure 6 and figure 7 show the sensitivity of this model to w/c ratio and 

CMC. The change in compressive strength appears reasonable for both of the parameters for 

the range of values evaluated. They are also consistent with the data in the database. Within 

practical ranges, a change in CMC from 500 to 650 lb/ft3 increases the 28-day strength from 

approximately 4,700 to 5,300 psi for a w/c ratio of 0.4. Likewise, a decrease in w/c ratio from 

0.5 to 0.35 increases the strength from 4,700 to 5,200 psi.
13

Figure 4. Graph. Predicted versus measured for 28-day cylinder compressive strength 

model.

Figure 5. Graph. Residual error plot for 28-day cylinder compressive strength model.
14

Figure 6. Graph. 28-day compressive strength model sensitivity to w/c ratio.

Figure 7. Graph. 28-day compressive strength model sensitivity to CMC.

Compressive Strength Model 2: Short-Term Cylinder Strength Model

The short-term cylinder compressive strength is expressed as follows:

6358.60655 3.53012* 34.24312* / * 633.3489*ln( ) , f CMC w c uw t c t = + − +

Figure 8. Equation. Prediction model 2 for fc,t.

Where:

fc,t = Compressive strength at age t years, psi.

CMC = Cementitious materials content, lb/yd3.

w/c = Water to cementitious materials ratio.

uw = Unit weight, lb/ft3

.

t = Short-term age up to 1 year.
15

The regression statistics for this model are presented in table 3, and details of the range of data 

used to develop the model are presented in table 4. The model was developed using 79 data 

points, and the prediction has an R2 value of 66.6 percent and an RMSE of 789 psi. The reason 

for an improved R2 compared to the 28-day strength model is not clear from these analyses. 

Table 3. Regression statistics for short-term cylinder strength model.

Variable DF Estimate

Standard 

Error t-Value Pr > |t| VIF

Intercept 1 6,358.60655 1,213.09762 5.24 < 0.0001 0

CMC 1 3.53012 0.90968 3.88 0.0002 2.15941

w/c × unit weight 1 -34.24312 11.00358 -3.11 0.0026 2.152

Ln(age) 1 633.3489 87.49625 7.24 < 0.0001 1.00604

Table 4. Range of data used for short-term cylinder strength model.

Parameter Minimum Maximum Average

w/c ratio 0.27 0.69 0.43

Cementitious content 376 936 660

Unit weight 124 151 143

Pavement age 0.0384 1.0000 0.3081

Compressive strength 2,480 10,032 5,256

Figure 9 and figure 10 show the predicted versus measured plot and the residual plot, 

respectively. Figure 11 through figure 13 show the sensitivity of this model to CMC, w/c ratio, 

and age, respectively. The trends are all reasonable. Figure 11 and figure 12 show the change in 

compressive strength at two ages, 28 days and 1 year, which are almost at the lower and upper 

bounds of ages included in this model. The plot in figure 13 can be considered a strength gain 

curve for typical unit weight and w/c ratios used in mix designs.

Figure 9. Graph. Predicted versus measured for short-term cylinder compressive strength 

model.

0

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 12000

Predicted Compressive Strength, psi

Measured Compressive Strength, psi

N = 79

R2 = 0.666

RMSE = 789 psi
16

Figure 10. Graph. Residual errors for short-term cylinder compressive strength model.

Figure 11. Graph. Short-term cylinder compressive strength sensitivity to CMC.

Figure 12. Graph. Short-term cylinder compressive strength sensitivity to w/c ratio.

3000

5000

7000

9000

11000

300 500 700 900 1100

Compressive Strength (psi)

Cementitious Material Content (lb/yd3)

PCC COMPRESSSIVE STRENGTH

28 Day Strength

1 Year Strength

w/c - 0.4

Unit Weight - 145 pcf

2000

4000

6000

8000

0 0.2 0.4 0.6 0.8

Compressive Strength (psi)

w/c Ratio

PCC COMPRESSSIVE STRENGTH

28 Day 

Strength

1 Year 

Strength

CMC - 600 lb/yd3

Unit Weight - 145 pcf
17

Figure 13. Graph. Short-term cylinder compressive strength sensitivity to age.

Compressive Strength Model 3: Short-Term Core Strength Model

The short-term core compressive strength model is as follows:

-199.84664* FM 611.30879 *ln( )

98.92962 5.70412 * 28.48527 * uw 2570.13151*MAS* w/c ,

t

f c t CMC

+

= + + +

Figure 14. Equation. Prediction model 3 for fc,t.

Where:

fc,t = Compressive strength at age t years, psi.

CMC = Cementitious materials content, lb/yd3

.

uw = Unit weight, lb/ft3.

MAS = Maximum aggregate size, inch.

w/c = Water to cementitious materials ratio.

FM = Fineness modulus of fine aggregate.

t = Short-term age up to 1 year.

The regression statistics for this model are presented in table 5. The model was developed 

using 294 points, and the prediction has an R2 value of 67.61 percent and an RMSE of 

1,122 psi. Table 6 provides details of the range of data used to develop the model. 

Table 5. Regression statistics for short-term core strength model.

Variable DF Estimate

Standard 

Error t-Value Pr > |t| VIF

Intercept 1 98.92962 1,544.34064 0.06 0.949 0

CMC 1 5.70412 0.36589 15.59 < 0.0001 1.23548

Unit weight 1 28.48527 10.59672 2.69 0.0076 1.0182

MAS × w/c ratio 1 2,570.13151 538.267 -4.77 < 0.0001 1.2201

FM 1 -199.84664 120.68288 -1.66 0.0988 1.01426

Ln(age) 1 611.30879 45.08962 13.56 < 0.0001 1.00026

3000

5000

7000

9000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Compressive Strength (psi)

Pavement Age (years)

PCC COMPRESSSIVE STRENGTH

Short 

Term 

Strength
18

Table 6. Range of data used for short-term core strength model.

Parameter Minimum Maximum Average

w/c ratio 0.27 0.69 0.42

Cementitious content 376 999 670

Unit weight 120 163 144

MAS 0.375 1.000 0.683

FM 2.50 4.37 3.05

Pavement age 0.0380 2.2160 0.4230

Compressive strength 1990 11,350 5,596

Figure 15 and figure 16 show the predicted versus measured plot and the residual plot, 

respectively. Figure 17 through figure 22 show the sensitivity of this model to CMC, unit weight, 

MAS, w/c ratio, FM, and age, respectively.

Figure 15. Graph. Predicted versus measured for short-term core compressive strength 

model.

Figure 16. Graph. Residual errors for short-term core compressive strength model.

0

2000

4000

6000

8000

10000

12000

Predicted Compressive Strength, psi

0 2000 4000 6000 8000 10000 12000

Measured Compressive Strength, psi

N = 294

R2 = 0.6761

RMSE = 1122 psi

y = 2E-07x + 0.0005

R2 = 7E-14

-4000

0

4000

0 2000 4000 6000 8000 10000

Residual Compressive Strength, psi

Predicted Compressive Strength, psi
19

Figure 17. Graph. Short-term core compressive strength sensitivity to CMC.

Figure 18. Graph. Short-term core compressive strength sensitivity to unit weight.

Figure 19. Graph. Short-term core compressive strength sensitivity to MAS.

3000

5000

7000

9000

11000

300 500 700 900 1100

Compressive Strength (psi)

Cementitious Material Content (lb/yd3)

PCC COMPRESSSIVE STRENGTH

28 Day 

Strength

1 Year 

Strength

w/c - 0.4

Unit Weight - 145 pcf

MAS - 0.75

FM - 3.0

3000

5000

7000

9000

120 130 140 150 160 170

Compressive Strength (psi)

Unit Weight (pcf)

PCC COMPRESSSIVE STRENGTH

28 Day 

Strength

1 Year 

Strength

CMC - 600 lb/yd3

w/c - 0.4

MAS - 0.75

FM - 3.0

3000

5000

7000

9000

0.2 0.4 0.6 0.8 1 1.2

Compressive Strength (psi)

Maximum Aggregate Size

PCC COMPRESSSIVE STRENGTH

28 Day 

Strength

1 Year 

Strength

CMC - 600 lb/yd3

w/c - 0.4

Unit Weight - 145 pcf

FM - 3.0
20

Figure 20. Graph. Short-term core compressive strength sensitivity to w/c ratio.

Figure 21. Graph. Short-term core compressive strength sensitivity to fine aggregate FM.

Figure 22. Graph. Short-term core compressive strength sensitivity to age.

3000

5000

7000

9000

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Compressive Strength (psi)

Pavement Age (years)

PCC COMPRESSSIVE STRENGTH

Short 

Term 

Strength

CMC - 600 lb/yd3

w/c - 0.4

Unit Weight - 145 pcf

MAS - 0.75

FM - 3.0
21

Compressive Strength Model 4: All Ages Core Strength Model

The compressive strength for cores at all ages is estimated as follows:

-6022.44 -854.46*w/c 4.8656* 68.5337* 533.15*ln( ) , f CMC uw t c t = + + +

Figure 23. Equation. Prediction model 4 for fc,t.

Where:

fc,t = Compressive strength at age t years, psi.

w/c = Water to cementitious materials ratio.

CMC = Cementitious materials content, lb/yd3.

uw = Unit weight, lb/ft3.

t = Short-term age in years.

The regression statistics for this model are presented in table 7. The model was developed using 

580 data points, and the prediction has an R2 value of 55.38 percent and an RMSE of 992 psi. 

Table 8 provides details of the range of data used to develop the model. 

Table 7. Regression statistics for all ages core strength model.

Variable Estimate

Standard 

Error t-Value Pr > |t| VIF

Intercept -6,022.44 2,028.37 -2.97 0.0032 0

w/c ratio -854.46 675.86 -1.26 0.2069 2.15941

CMC 4.8656 0.5737 8.48 < 0.0001 2.152

Unit weight 68.5337 13.4368 5.1 < 0.0001 1.00604

Ln(age) 533.15 22.3343 23.87 < 0.0001 1.00026

Table 8. Range of data used for all ages core strength model.

Parameter Minimum Maximum Average

w/c ratio 0.00 0.72 0.43

Cementitious content 354 999 615

Unit weight 120 163 145

Pavement age 0.0380 45.3840 6.4320

Compressive strength 1,990 11,750 6,430

Figure 24 and figure 25 show the predicted versus measured plot and the residual 

plot, respectively. 
22

Figure 24. Graph. Predicted versus measured for all ages core compressive strength model.

Figure 25. Graph. Residual errors for all ages core compressive strength model.

Figure 26 through figure 29 show the sensitivity of this model to w/c ratio, CMC, unit weight, 

and age, respectively. Again, the sensitivity plots showing the variation in core compressive 

strength with changes in w/c ratio, CMC, and unit weight are presented for 28 days, 1 year, and 

20 years. The rate of strength gain clearly is much higher in the short term (28 days to 1 year) 

than during the next 19 years. Figure 29 can be treated as the strength gain relationship 

representative of a typical mix (w/c of 0.4, CMC of 600 lb/yd3

, and unit weight of 145 lb/ft3). 

0

2000

4000

6000

8000

10000

12000

0 2000 4000 6000 8000 10000 12000

Predicted Compressive Strength, psi

Measured Compressive Strength, psi

N = 580

R2 = 0.5538

RMSE = 992 psi

y = -0.002x + 79.723

R2 = 5E-06

-6000

0

6000

0 2000 4000 6000 8000 10000 12000

Residual Compressive Strength, psi

Predicted Compressive Strength, psi
23

Figure 26. Graph. All ages core compressive strength sensitivity to w/c ratio.

Figure 27. Graph. All ages core compressive strength sensitivity to CMC.

Figure 28. Graph. All ages core compressive strength sensitivity to unit weight.

3000

5000

7000

9000

11000

0.2 0.4 0.6 0.8

Compressive Strength (psi)

w/c Ratio

PCC COMPRESSSIVE STRENGTH

28 Day 

Strength

1 year 

strength

20 Years 

Strength

CMC - 600 lb/yd3

Unit Weight - 145 pcf

3000

5000

7000

9000

11000

300 500 700 900 1100

Compressive Strength (psi)

Cementitious Material Content (lb/yd3)

PCC COMPRESSSIVE STRENGTH

28 Day 

Strength

1 year 

strength

20 year 

strength

w/c - 0.4

Unit Weight - 145 pcf

3000

5000

7000

9000

11000

120 130 140 150 160

Compressive Strength (psi)

Unit Weight (pcf)

PCC COMPRESSSIVE STRENGTH

28 Day 

Strength

1 year 

strength

20 Years 

Strength

CMC - 600 lb/yd3

w/c - 0.4
24

Figure 29. Graph. All ages core compressive strength sensitivity to age.

Compressive Strength Model 5: Long-Term Core Strength Model

The model developed for the long-term strength is expressed as follows:

2 f c,LT = -3467.3508+ 3.63452*CMC + 0.42362*uw

Figure 30. Equation. Prediction model 5 for fc,LT.

Where:

fc,LT = Long-term compressive strength, psi.

CMC = Cementitious materials content, lb/yd3

.

uw = Unit weight, lb/ft3.

The regression statistics for this model are presented in table 9. The model was developed using 

201 data points, and the prediction has an R2 value of 18.03 percent and an RMSE of 1,179 psi. 

Table 10 provides details of the range of data used to develop the model. 

Table 9. Regression statistics for long-term core strength model.

Variable DF Estimate

Standard 

Error t-Value Pr > |t| VIF

Intercept 1 -3,467.3508 1,720.49637 -2.02 0.0452 0

Cementitious 1 3.63452 1.38354 2.63 0.0093 1.024

(Unit weight)2 1 0.42362 0.06634 6.39 < 0.0001 1.024

Table 10. Range of data used for long-term core strength model.

Parameter Minimum Maximum Average

Cementitious content 354 781 550

Unit weight 134 156 147

Compressive strength 4,315 11,750 7,655

3000

5000

7000

9000

11000

0 5 10 15 20

Compressive Strength (psi)

Pavement Age (years)

PCC COMPRESSSIVE STRENGTH

All Ages 

Strength

CMC - 600 lb/yd3

w/c - 0.4

Unit Weight - 145 pcf
25

Figure 31 and figure 32 show the predicted versus measured plot and the residual plot, 

respectively. This model does not have a good predictive ability (see figure 31). While there is 

no significant bias, the error in prediction is fairly high (see figure 32). This model needs to be 

used with caution. Additionally, other means to verify the value would be necessary, such as 

core tests.

Figure 31. Graph. Predicted versus measured for long-term core compressive strength 

model.

Figure 32. Graph. Residual errors for long-term core compressive strength model.

Relative Comparison of All Compressive Strength Models 

The compressive strength models, like any other empirical model, reproduce the trends present 

in the datasets used for each correlation. It is highly recommended that a user estimate the 

strength based on as many models as possible with the information available at the time of 

analysis. This might provide a fair assessment of the ranges of compressive strength likely for 

the project and at different ages. 

4000

6000

8000

10000

12000

4000 6000 8000 10000 12000

Predicted Compressive Strength, psi

Measured Compressive Strength, psi

N = 201

R2 = 0.1803

RMSE = 1179 psi

y = 0.0384x - 291.07

R2 = 0.0003

-4000

0

4000

4000 5000 6000 7000 8000 9000 10000

Residual Compressive Strength, psi

Predicted Compressive Strength, psi
26

Figure 33 through figure 37 show the relationship between compressive strength and CMC, w/c 

ratio, and unit weight, respectively. Figure 36 and figure 37 show the strength gain at short- and 

long-term ages, respectively. Note that relationships have been plotted for typical values for all 

variables, and the raw data used in the models do not necessarily lie on the plots.

Figure 33. Graph. Model compressive strength prediction for varying CMC.

Figure 34. Graph. Model compressive strength prediction for varying w/c ratio.

1000

3000

5000

7000

9000

11000

300 500 700 900 1100

Compressive Strength (psi)

Cementitious Material Content (lb/yd3)

a+b*WC+c*CMC - cyl 28days

a+b*CMC+c*(WC*UW)+d*ln(age) - cyl short term

a+b*WC+c*CMC+d*UW+e*ln(age) - core all ages

a+b*CMC+c*UW+d*(MAS*WC)+e*FM+f*ln(age) - core short term

Raw Data - core

Raw Data - cyl

w/c - 0.4

Unit Weight - 145 pcf

MAS - 0.75

FM - 3.0
27

Figure 35. Graph. Model compressive strength prediction for varying unit weights.

Figure 36. Graph. Strength gain in the short-term predicted by three models.
28

Figure 37. Graph. Long-term strength gain predicted by the models.

The following observations can be made:

• The predictions are within 700 psi of each other for a given CMC (see figure 33). 

However, for more typical ranges of cement contents, the predictions are within 300 to 

400 psi of each other. The short-term core compressive model has the steepest slope for 

this relationship.

• Figure 34 suggests that the predictions have a range as high as 1,500 psi for a given w/c 

ratio, especially at very high w/c ratios. However, within typical ranges (0.3 to 0.5), the 

models predict within a range of 250 to 800 psi. The range slightly increases at lower w/c 

ratios. The short-term cylinder strength has the highest slope in this case.

• Based on the trend presented in figure 35, for a given level of unit weight, the 

compressive strength predictions are within 200 to 300 psi for typical ranges of unit 

weight (140 to 145 lb/ft3

). The prediction can vary by about 800 psi for very high unit 

weight values. Note that the short-term cylinder compressive strength model has not been 

included in this plot, as the variable appears as a transformed variable in the model and 

its effect cannot be isolated.

• Short-term strength predictions by all models that are relevant to short-term strengths 

show predictions within 200 to 400 psi of each other. The predictions are closer in value 

at as the age increases from 14 days to 1 year (see figure 36).

• Figure 37 suggests that the long-term strength predicted by the core all ages model is 

close to the strength predicted by the long-term model. This is essentially because the 

data used in this range are common to both models. 

1000

3000

5000

7000

9000

11000

13000

15000

0 2 4 6 8 10 12 14 16 18 20

Compressive Strength (psi)

Pavement Age (years)

a+b*WC+c*CMC+d*UW+e*ln(age) - core all ages

Raw Data - Core

Series3

CMC - 600 lb/yd3

w/c - 0.4

Unit Weight - 145 pcf

MAS - 0.75

FM - 3.0
29

These observations illustrate the benefit of comparing predictions made by the various models 

available to obtain the range of strength that each project or observation could develop. Any 

other information to substantiate or validate the strength predictions should be utilized whenever 

possible, such as strength values from other projects that have used similar materials and 

mix design.

PCC FLEXURAL STRENGTH MODELS

Validation of Existing Models 

Previous models correlating flexural strength to compressive strength have generally used a 

power model of the following form:

b

r c M = a * f '

Figure 38. Equation. Mr.

Where:

a = 7.5 to 11.7 for b = 0.5.

a = 2 to 2.7 for b = 0.67.

Table 11 shows a summary of the models developed. The regressed constants, a and b, were 

found to be within the range of values reported by the other studies. This validation not only 

provides feasible models, but it also confirms that the data used reasonably represent the broad 

range considered in the various studies. The correlations are presented in figure 39 and figure 40

for the power models with exponents of 0.5 and 0.67, respectively.

Table 11. Power models developed for flexural strength prediction using LTPP data for 

validation.

Model a b R2 N

b

r c M = a * f ' 10.3022 0.50 0.446 185

2.4277 0.67 0.449 185
30

Figure 39. Graph. Predicted versus measured for validating 0.5 power flexural strength 

model.

Figure 40. Graph. Predicted versus measured for validating 0.667 power flexural strength 

model.

Flexural Strength Model 1: Flexural Strength Based on Compressive Strength

This model provides the best correlation between compressive strength and flexural strength with 

the LTPP data. The model form utilizes the power equation. This model will be most useful for 

cases when the compressive strength of the PCC has been determined through a routine cylinder 

break. This model can be expressed as follows:

0.4082 22.7741* '

c MR = f

Figure 41. Equation. Prediction model 6 for MR.

100

300

500

700

900

1100

1300

100 300 500 700 900 1100 1300

Predicted Modulus of Rupture, psi

Measured Modulus of Rupture, psi

N = 185

R2 = 0.4460

RMSE = 84 psi

100

300

500

700

900

1100

1300

100 300 500 700 900 1100 1300

Predicted Modulus of Rupture, psi

Measured Modulus of Rupture, psi

N = 185

R2 = 0.4493

RMSE = 111 psi
31

Where:

MR = Flexural strength, psi.

f'c = Compressive strength determined at the same age, psi.

The regression statistics for this model are presented in table 12. The model was developed 

using 185 data points, and the prediction has an R2 value of 45.2 percent and an RMSE of 69 psi. 

Table 13 provides details of the range of data used to develop the model.

Table 12. Regression statistics for flexural strength model based on compressive strength.

Parameter Estimate

Standard 

Error

Approximate 

95 Percent

Confidence Limits

a 22.7741 6.6362 9.6807 to 35.8674

b 0.4082 0.0338 0.3416 to 0.4748

Table 13. Range of data used for flexural strength model based on compressive strength.

Parameter Minimum Maximum Average

Compressive strength 1,770 10,032 5,431

Flexural strength 467 1,075 754

Figure 42 and figure 43 show the predicted versus measured plot and the residual 

plot, respectively. 

Figure 42. Graph. Predicted versus measured values for flexural strength model based on 

compressive strength.

100

300

500

700

900

1100

1300

100 300 500 700 900 1100 1300

Predicted Modulus of Rupture, psi

Measured Modulus of Rupture, psi

N = 185

R2 = 0.452

RMSE = 69 psi
32

Figure 43. Graph. Residual errors for flexural strength model based on compressive 

strength.

Figure 44 shows a comparison of the power models used to validate the data and develop a new 

correlation. Note that the three power models (the new equation developed for this study as well 

as the validation models) provide close estimates (within 50 psi) in the 4,500- to 5,500-psi 

compressive strength range.

The American Concrete Institute (ACI) and Portland Cement Association (PCA) models are

plotted for comparison. Also plotted are the raw data that were used in the model. Clearly, the 

ACI equation is conservative for this data. It has also been found to give a conservative estimate 

for several large datasets that have been used in flexural strength model prediction. Conversely, 

the PCA model fits the LTPP data more closely. The reasons for this lack of fit of the current 

data with the previous models may be too many to fully explain. The data used in models from 

prior studies often came from mixes batched under controlled laboratory experiments and were 

typical of paving and structural concrete. The mixes used in the current model developed from 

LTPP data relies on only mixes proportioned for typical paving operations. Furthermore, the 

LTPP data used are from many projects widely dispersed around the United States. This in itself

makes the models more robust than any previous data used to make similar correlations. 

The spread in the raw data about the prediction model in figure 44 clearly indicates that there are 

factors other than compressive strength that influence the flexural strength of PCC. Among the 

various factors influencing flexural strength are the mix design parameters and age of the 

concrete. These variables are considered in the other models developed in this study.

y = 0.0131x - 10.037

R2 = 0.0001

-500

0

500

100 300 500 700 900 1100 1300

Residual Modulus of Rupture, psi

Predicted Modulus of Rupture, psi
33

Figure 44. Graph. Comparison of flexural strength models based on compressive strength.

Flexural Strength Model 2: Flexural Strength Based on Age, Unit Weight, and w/c Ratio

Flexural strength model 2 provides a correlation between flexural strength and mix design 

parameters, specifically the unit weight and w/c ratio. Age is also a parameter in this model, 

which helps reduce some of the variability seen in the prediction relative to the predictions 

shown in figure 44. This model will be most useful for cases when the compressive strength of 

PCC is not determined but mix design information is available. Also, the user has the option of 

predicting the 28-day strength value for design or estimating the strength at traffic opening time. 

This model can be expressed as follows:

= 676.0159 -1120.31* w/c + 4.1304 * uw + 35.74627 *ln(t) MRt

Figure 45. Equation. Prediction model 7 for MRt.

Where:

MRt = Flexural strength at age t years, psi.

w/c = w/c ratio.

uw = Unit weight, lb/ft3.

t = Pavement age, years.

The regression statistics for this model are presented in table 14. The model was developed 

using 62 data points, and the prediction has an R2 value of 61.11 percent and an RMSE of 91 psi. 

Table 15 provides details of the range of data used to develop the model. 

400

600

800

1000

1200

Modulus of Rupture (psi)

Compressive Strength (psi)

MR = 10.3022* (f'c ^ 0.5)

MR = 2.42 * (f'c^ 0.67)

MR = 22.7741* (f'c^0.4082)

MR = 7.5*(f'c^0.5) - ACI Model

MR = 9.5*(f'c^0.5) - PCA Model

Raw data
34

Table 14. Regression statistics for flexural strength model based on age, unit weight, and 

w/c ratio.

Variable DF Estimate

Standard 

Error

tValue Pr > |t| VIF

Intercept 1 676.0159 277.7887 2.43 0.0181 0

w/c 1 -1,120.31 141.3573 -7.93 < 0.0001 1.00591

Unit weight 1 4.1304 1.88934 2.19 0.0329 1.00311

Ln(age) 1 35.74627 8.78516 4.07 0.0001 1.00619

Table 15. Range of data used for flexural strength model based on age, unit weight, and 

w/c ratio.

Parameter Minimum Maximum Average

w/c ratio 0.27 0.58 0.40

Unit weight 124 151 142

Pavement age 0.0384 1.0000 0.3169

Flexural strength 467 978 742

Figure 46 and figure 47 show the predicted versus measured plot and the residual 

plot, respectively.

Figure 46. Graph. Predicted versus measured values for flexural strength model based on 

age, unit weight, and w/c ratio.

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Predicted Modulus of Rupture, psi

Measured Modulus of Rupture, psi

N = 62

R2 = 0.6111

RMSE = 91 psi
35

Figure 47. Graph. Residual errors for flexural strength model based on age, unit weight, 

and w/c ratio.

Flexural Strength Model 3: Flexural Strength Based on Age, Unit Weight, and CMC

The model is expressed as follows:

= 24.15063 + 0.55579*CMC + 2.96376*uw + 35.54463*ln(t) MRt

Figure 48. Equation. Prediction model 8 for MRt.

Where:

MRt = Flexural strength at age t years, psi.

CMC = Cementitious materials content, lb/yd3.

uw = Unit weight, lb/ft3

.

t = Pavement age, years.

The regression statistics for this model are presented in table 16. The model was developed using 

62 data points, and the prediction has an R2 value of 70.23 percent and RMSE of 80 psi. Table 17

provides details of the range of data used to develop the model. 

Table 16. Regression statistics for flexural strength model based on age, unit weight, and 

CMC.

Variable DF Estimate

Standard 

Error t-Value Pr > |t| VIF

Intercept 1 24.15063 236.7606 0.1 0.9191 0

CMC 1 0.55579 0.05563 9.99 < 0.0001 1.01522

Unit weight 1 2.96376 1.66087 1.78 0.0796 1.01253

Ln(age) 1 35.54463 7.68504 4.63 < 0.0001 1.00573

y = 1E-05x - 0.0088

R2 = 2E-10

-300

0

300

400 600 800 1000

Residual Modulus of Rupture, psi

Predicted Modulus of Rupture, psi
36

Table 17. Range of data used for flexural strength model based on age, unit weight, and 

CMC.

Parameter Minimum Maximum Average

CMC 388 936 668

Unit weight 124 151 142

Pavement age 0.0384 1.0000 0.3169

Flexural strength 467 978 742

Figure 49 and figure 50 show the predicted versus measured plot and the residual 

plot, respectively.

Figure 49. Graph. Predicted versus measured values for flexural strength model based on 

age, unit weight, and CMC.

Figure 50. Graph. Residual errors for flexural strength model based on age, unit weight, 

and CMC.

Figure 51 through figure 54 present the sensitivity of the mix design-based flexural strength 

models to CMC, w/c ratio, unit weight, and age. Figure 51 and figure 52 show that prediction 

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200 1400

Predicted Modulus of Rupture, psi

Measured Modulus of Rupture, psi

N = 62

R2 = 0.7023

RMSE = 80 psi

y = 2E-06x - 0.0008

R2 = 6E-12

-300

0

300

400 600 800 1000

Residual Modulus of Rupture, psi

Predicted Modulus of Rupture, psi
37

models 7 and 8 do not show any sensitivity to CMC and w/c ratio (see figure 45 and figure 48). 

For typical values of these parameters, the flexural strength prediction from these two models 

could show a difference of about 200 psi for extreme values of w/c ratios. However, within a 

typical range of 0.35 to 0.45, the flexural strength prediction is within 50 psi. Similar trends are 

evident for the w/c ratio parameter. Therefore, if all details about a mix design are available, it is 

highly recommended that both models be used to predict flexural strength so that the user has a 

fair estimate of the flexural strength range. Figure 53 shows that the predictions are close from 

both models. Likewise, figure 54, which is more or less a flexural strength gain model for a 

typical mix design, shows very close predictions from both models.

Figure 51. Graph. Sensitivity of flexural strength predictions to CMC.

Figure 52. Graph. Sensitivity of flexural strength predictions to w/c ratio.

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000 1200

Predicted MR, psi

Cementitious Material Content, lb/yd3

MR model as f(w/c)

MR model as f(CMC)

Raw data

w/c = 0.4

unit weight = 145 pcf

age = 28 days

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8

Predicted MR, psi

w/c ratio

MR model as f(w/c)

MR model as f(CMC)

Raw data

CMC = 600 lb/yd3

unit weight = 145 pcf

age = 28 days
38

Figure 53. Graph. Sensitivity of flexural strength predictions to unit weight.

Figure 54. Graph. Sensitivity of flexural strength predictions to age.

PCC ELASTIC MODULUS MODELS

Validation of Existing Models 

Existing models correlate elastic modulus to compressive strength and unit weight. The 

following represent the regressed models using LTPP data for existing model forms:

c c E = a * f '

Figure 55. Equation. Ec as a function of square root of compressive strength.

Regressed coefficients for figure 55 are as follows:

• a = 55,294.

Regression statistics for figure 55 are as follows:

• N = 514.

• R2 = 11.8 percent.

0

200

400

600

800

1000

1200

1400

100 110 120 130 140 150 160

Predicted MR, psi

Unit Weight, pcf

MR model as f(w/c)

MR model as f(CMC)

Raw data

w/c = 0.4

CMC = 600 lb/yd3

age = 28 days

0

200

400

600

800

1000

1200

1400

0 0.2 0.4 0.6 0.8 1

Predicted MR, psi

Age, years

MR model as f(w/c)

MR model as f(CMC)

w/c = 0.4

CMC = 600 lb/yd3

unit weight = 145 pcf
39

E = a* f'c + b

Figure 56. Equation. Model form for E as a function of compressive strength with slope 

and intercept.

Regressed coefficients for figure 56 are as follows: 

• a = 31,624.6.

• b = 2,013,192.

Regression statistics for figure 56 are as follows:

• N = 514.

• R2

=11.8 percent.

b

c c E = a * f '

Figure 57. Equation. Ec.

Regressed coefficients for figure 57 are as follows: 

• a = 388,082.

• b = 0.2809.

Regression statistics for figure 57 are as follows:

• N = 514.

• R2 = 12.3 percent.

b c E = a *(UW ) *( f'c)

Figure 58. Equation. E as function of unit weight and compressive strength.

Regressed coefficients for figure 58 are as follows: 

• a = 80,849.3.

• b = 0.3648.

• c = 0.2527.

Regression statistics for figure 58 are as follows:

• N = 514.

• R2 = 10.8 percent.
40

The quality of prediction in the validated models is poor, as indicated by the R2 values reported 

for figure 55 through figure 58. This trend is common with elastic modulus models, especially 

considering that the data used in this study were not generated from controlled laboratory 

experiments. Also, while compressive strength is the most commonly used strength parameter 

and correlations with the compressive strength can be implemented most easily, there is an 

inherent drawback in correlating modulus to compressive strength. Modulus does not test the 

material to its limits. Instead, it is more indicative of the elastic deformational characteristics of 

the material. The data contain modulus measured at a wide range of ages. Therefore, the new 

models developed utilized other mix parameters that impact modulus, including age.

Elastic Modulus Model 1: Model Based on Aggregate Type

The PCC elastic modulus model can be expressed as follows:

c Dagg E (4.499*(UW ) *( f'c) ) * 2.3481 0.2429 =

Figure 59. Equation. Prediction model 9 for Ec.

Where:

Ec = PCC elastic modulus, psi.

UW = Unit weight, lb/ft3.

f'c = Compressive strength.

Dagg = Regressed constant depending on aggregate type as follows:

• = 1.0 for andesite, limestone, and sandstone.

• = 0.9286 for basalt.

• = 1.0079 for chert.

• = 0.9215 for diabase.

• = 1.0254 for dolomite.

• = 0.8333 for granite.

• = 0.9511 for quartzite.

The development of the model required the use of a model form that accommodates aggregate 

type as categorical variables (assigned values of 1, 0). The values for Dagg were initialized to 

1.0 at the start of the analyses and allowed to iteratively determine individual values for each 

aggregate type. The model had 71 observations, an R2 value of 35.8 percent, and an RMSE of 

approximately 500,000 psi. 

The model indicates that the factor that accounts for the aggregate type, Dagg, has a value of 1.0 

for andesite, limestone, and sandstone. Basalt, diabase, granite, and quartzite have lower Dagg

values and therefore lower modulus values than mixes using andesite, limestone, and sandstone 
41

aggregates. Likewise, chert and dolomite have higher values. Table 18 provides details of the 

range of data used to develop the model. 

Table 18. Range of data used for elastic modulus model based on aggregate type.

Parameter Minimum Maximum Average

Compressive strength 1,990 11,310 7,550

Unit weight 137 156 146

Elastic modulus 1,450,000 6,800,000 4,629,646

Figure 60 and figure 61 show the predicted versus measured plot and the residual plot, 

respectively. The R2 value is reasonable and therefore presented as a feasible model. 

Figure 60. Graph. Predicted versus measured for elastic modulus model based on 

aggregate type.

Figure 61. Graph. Residual errors for elastic modulus model based on aggregate type.

y = 0.0178x - 84187

R2 = 0.0002

-2000000

0

2000000

0 2000000 4000000 6000000 8000000

Residual Elastic Modulus, psi

Predicted Elastic Modulus, psi
42

Elastic Modulus Model 2: Model Based on Age and Compressive Strength

The model can be expressed as follows:

1.3 0.2118

, )) 0.03

59.0287 *( ) *(ln( − = t E f'c c t t

Figure 62. Equation. Prediction model 10 for Ec,t.

Where:

Ec,t = Elastic modulus at age t, years.

f'ct = Compressive strength at age t, years.

t = Age at which modulus is determined, years.

The model uses 371 data points, has an R2 value of 26.14 percent, and an RMSE of about 

900,000 psi. Table 19 shows the results of the nonlinear analysis, and table 20 provides details of 

the range of data used to develop the model. 

Table 19. Regression statistics for elastic modulus model based on age and compressive 

strength.

Parameter 

Constants Estimate

Standard 

Error

Approximate 

95 Percent 

Confidence Limits

a 59.0287 2.8881 53.3495 to 64.7079

b -0.2118 0.0284 -0.2677 to -0.1559

Table 20. Range of data used for elastic modulus model based on age and compressive 

strength.

Parameter Minimum Maximum Average

Compressive strength 1,990 12,360 7,361

Pavement age 0.0384 45.3836 14.0900

Elastic modulus 1,450,000 6,800,000 4,586,545

The measured versus predicted plot and the residuals plot for this model are shown in figure 63

and figure 64, respectively.
43

Figure 63. Graph. Predicted versus measured for elastic modulus model based on age and 

compressive strength.

Figure 64. Graph. Residual errors for elastic modulus model based on age and compressive 

strength.

Elastic Modulus Model 3: Model Based on Age and 28-Day Compressive Strength

Since the 28-day compressive strength is usually available for PCC materials, a predictive 

model based on age and the 28-day compressive strength can be useful in many situations. 

The relationship developed for these variables can be expressed as follows:

))0.00524

0.03

375.6*( ) *(ln( 1..1

, 28

t E f'c c t = −day

Figure 65. Equation. Prediction model 11 for Ec,t.

1000000

3000000

5000000

7000000

9000000

1000000 3000000 5000000 7000000 9000000

Predicted Elastic Modulus, psi

Measured Elastic Modulus, psi

N = 371

R2 = 0.2614

RMSE = 949404 

y = 0.5885x - 3E+06

R2 = 0.4198

-4000000

0

4000000

1000000 3000000 5000000 7000000 9000000

Residual Elastic Modulus, psi

Predicted Elastic Modulus, psi
44

Where:

Ec,t = Elastic modulus at age t, years.

f'c28-day = 28-day compressive strength.

t = Age at which modulus is determined, years.

The model used 46 data points, had an R2 value of 16.32 percent, and an RMSE of about 

1,183,400 psi. Table 21 shows the results of the nonlinear analysis, and table 22 provides details 

of the range of data used to develop the model. 

Table 21. Regression statistics for elastic modulus model based on age and 28-day 

compressive strength.

Parameter 

Constants Estimate

Standard 

Error

Approximate 95 Percent 

Confidence Limits

a 375.6 31.4592 312.5 439.3

b 0.00524 0.0714 -0.1388 -0.1492

Table 22. Range of data used for elastic modulus model based on age and 28-day

compressive strength.

Parameter Minimum Maximum Average

28-day compressive strength 3,034 7,912 5,022

Pavement age 0.0384 4.5288 0.9153

Elastic modulus 1,450,000 6,221,000 4,732,101

The measured versus predicted plot and the residuals plot for this model are shown in figure 66

and figure 67, respectively. This model uses data up to an age of 1 year. It is more appropriate 

for estimating the short-term modulus of a project and for supplementing strength estimates used 

to determine opening time for traffic.

Figure 66. Graph. Predicted versus measured for elastic modulus model based on age and 

28-day compressive strength.

1000000

3000000

5000000

7000000

9000000

1000000 3000000 5000000 7000000 9000000

Predicted Elastic Modulus, psi

Measured Elastic Modulus, psi

N = 46

R2 = 0.1632

RMSE = 1,183,400 psi
45

Figure 67. Graph. Residual errors for elastic modulus model based on age and 28-day 

compressive strength.

Limitations of Elastic Modulus Models

An examination of the statistics proposed for determining elastic modulus suggests that they do 

not possess the predictive ability of the other material parameters presented in this study. The 

models are considered fair but not excellent. They provide users with an option of moderate 

estimates when no information about the elastic modulus is available. Therefore, it is

recommended that users exercise caution when using the predicted elastic modulus values 

for analyses.

PCC TENSILE STRENGTH MODELS

PCC Tensile Strength Model Based on Compressive Strength

This model development served as both a validation and development of a new correlation using 

the LTPP database. The model form used was a power equation and can be expressed as follows:

0.4785 ft = 8.9068*( f'c)

Figure 68. Equation. Prediction model 12 for ft.

Where:

ft = Indirect tensile strength of the PCC material.

f'c = Compressive strength of the mix determined at the same age.

The model statistics are presented in table 23. The model was developed using 541 data points 

with an R2 value of 42.09 percent and an RMSE of 61 psi. Table 24 provides details of the range 

of data used to develop the model. 

y = 0.7443x - 4E+06

R2 = 0.623

-3000000

0

3000000

1000000 3000000 5000000 7000000 9000000

Residual Elastic Modulus, psi

Predicted Elastic Modulus, psi
46

Table 23. Model statistics for tensile strength prediction model.

Parameter Estimate

Standard 

Error

95 Percent

Confidence Limits

Coefficient 8.9068 2.0204 4.9381 to 12.8756

Power 0.4785 0.0256 0.4282 to 0.5288

Table 24. Range of data used for tensile strength prediction model.

Parameter Minimum Maximum Average

Compressive strength 1,990 12,360 6,763

Tensile strength 316 1,012 600

Figure 69 and figure 70 show the predicted versus measured plot and the residual errors plot, 

respectively. Figure 71 shows the sensitivity of the model to compressive strength. The 

relationship developed shows that for typical ranges of compressive strength (i.e., 3,000 to 

6,000 psi), the PCC tensile strength varies from about 400 to 570 psi, which is a reasonable range 

for this strength parameter.

Figure 69. Graph. Predicted versus measured for tensile strength model.

Figure 70. Graph. Residual errors plot for tensile strength model.

y = 0.4231x + 346.07

R2 = 0.4209

100

300

500

700

900

1100

1300

100 300 500 700 900 1100 1300

Predicted tensile strength, psi

Measured tensile strength, psi

N = 541

R2 = 0.4209

RMSE = 61 psi

y = 0.0054x - 3.2756

R2 = 2E-05

-500

0

500

100 300 500 700 900 1100 1300

Tensile strength, (Pred - Meas), psi

Predicted tensile strength, psi
47

Figure 71. Graph. Sensitivity of tensile strength prediction model to change compressive 

strength.

PCC CTE MODELS

Current Issue with CTE Overestimation in LTPP Data

CTE tests of the PCC specimens from LTPP sections were performed by FHWA’s TurnerFairbank Highway Research Center (TFHRC) using the AASHTO TP 60 protocol.

(11) TFHRC 

initiated an inter-laboratory study during which an error was discovered with the protocol and 

procedure used to measure concrete CTE. The source of the error was in the assumption of a 

single CTE value for the calibration specimen. Testing performed at independent laboratories 

revealed that a CTE value must be determined for each calibration specimen, and the calibration 

specimen should be tested over the same range of temperature over which the concrete CTE is 

determined—50 to 122 °F. Not meeting these two conditions caused an overestimation of the 

reported CTE by approximately 0.83 inch/inch/°F. Since all of the initial LTPP testing for CTE 

had been done in one laboratory with one calibration specimen, the calibration offset can be 

corrected in the database, and it has been corrected in Long-Term Pavement Performance

Standard Data Release 24.0.

(12)

This overestimation of the CTE has significant ramifications, especially in light of the fact that 

the TFHRC has tested over 2,100 specimens for the LTPP program and the fact that the LTPP 

database was the primary source for the national calibration of the AASHTO MEPDG rigid 

pavement performance models. The national calibration coefficients for all JPCP and CRCP 

performance models may be invalid, and the models may need to be recalibrated. As a result, 

local implementation efforts may be delayed.

The impact of this error in the CTE values on the current study was described in an internal

status report submitted to LTPP. LTPP Standard Data Release Version 23.0 contained the 

uncorrected CTE values; therefore, the CTE models developed in this study are not applicable 

for the corrected data.

(10) However, the models demonstrate the ability to develop correlations,

and the procedures herein may be repeated for the corrected data.
48

CTE Model 1: CTE Based on Aggregate Type (Level 3 Equation for MEPDG)

CTE test data were averaged for each aggregate type, which constitutes level 3 inputs for 

MEPDG. Table 25 lists the average PCC CTE for each aggregate type as found in the literature. 

The data are in general agreement, providing a degree of confidence in the level 3 MEPDG input 

recommendations. The average CTE values determined from the data subset are recommended 

by this study.

Table 25. Prediction model 13 for PCC CTE based on aggregate type, x 10-6 inch/inch/°F.

Aggregate 

Type

Average 

From 

Literature

Average 

From All 

LTPP Data

Average From Data 

Used in Model 

(Recommended)

Basalt 4.85 5.11 4.86

Chert 6.55 6.24 6.90

Diabase 4.85 5.33 5.13

Dolomite 5.75 5.79 5.79

Gabbro 4.85 5.28 5.28*

Granite 4.55 5.62 5.71

Limestone 4.25 5.35 5.25

Quartzite 6.85 6.07 6.18

Andesite 4.85 4.99 5.33

Sandstone 6.05 5.98 6.33

N 228 91

*There were no samples with a Gabbro aggregate type in the data used in the model. 

Hence, the average from the entire dataset is recommended.

Figure 72 shows a plot of recommended CTE values versus average CTE values obtained 

from other sources. While they are in fairly good agreement, the values recommended from this 

study are slightly higher for most cases. This can be explained by the overestimation of CTE 

during testing.
49

Figure 72. Graph. Comparison of average values from other sources and recommended 

CTE values based on aggregate type from LTPP data.

CTE Model 2: CTE Based on Mix Volumetrics (Level 2 Equation for MEPDG)

The PCC CTE model based on mix volumetrics was established as follows:

* 6.4514*(1 ) CA CTEPCC = CTECA VCA + −V

Figure 73. Equation. Prediction model 14 for CTEPCC.

Where:

CTECA = Constant determined for each aggregate type as shown in table 26.

The model statistics are presented in table 26, and details of the range of data used to develop the 

model are presented in table 27. The model has 89 data points, an R2 value of 44.15 percent, and 

an RMSE of 0.35006 psi. 

Table 26. Statistical analysis results for CTE model based on mix volumetrics.

Parameter Comment Estimate

Standard 

Error

95 Percent

Confidence Limits

c N/A 6.4514 0.1889 6.0758 6.827

d CTECA for basalt 3 0 3 3

e CTECA for chert 6.4 0 6.4 6.4

f CTECA for diabase 3.4835 1.2824 0.9337 6.0333

g CTECA for dolomite 5.1184 0.408 4.3071 5.9297

h CTECA for gabbro 3.75 N/A N/A N/A

i CTECA for granite 4.7423 0.4188 3.9096 5.5749

j CTECA for limestone 3.2886 0.3579 2.5771 4.0001

k CTECA for quartzite 6.1 0 6.1 6.1

l CTECA for andesite 3.6243 1.4539 0.7336 6.515

m CTECA for sandstone 4.5 0 4.5 4.5

N/A = Not applicable.
50

Table 27. Range of data used for CTE model based on mix volumetrics.

Parameter Minimum Maximum Average

Coarse aggregate content 582 2,730 1,811

Coarse aggregate specific gravity 2.42 2.86 2.65

w/c ratio 0 0.71 0.45

Coarse aggregate volume fraction 0.13 0.62 0.41

Mortar volume 0.38 0.87 0.59

The predicted versus measured plot and the residual error plots are presented in figure 74 and 

figure 75, respectively. 

Figure 74. Graph. Predicted versus measured for CTE model based on mix volumetrics.

Figure 75. Graph. Residual errors for CTE model based on mix volumetrics.

The constant, C, determined as 6.4514, is equivalent to the CTE of the mortar. (At TFHRC, 

using the AASHTO TP 60 uncorrected values, a CTE value of 6.2 for mortar containing silica 

sand was determined, validating this equation.

(12)

) Since the mortar (all components of the mix 

design except the coarse aggregate) occupies a large volume of the matrix, it was necessary 

for the model to predict higher CTE for increased mortar proportions (or decreasing coarse 

aggregate proportions). In optimizing the model and selecting the representative CTE for each 

aggregate type, it was ensured that the CTE of the aggregate was not above 6.4514.

y = -0.0443x + 0.233

R2 = 0.0014

-2

0

2

0 1 2 3 4 5 6 7

Residual error, x10E-6

Predicted CTE, x10E-6
51

Figure 76 and figure 77 show a comparison of the predicted CTE values with average values 

reported in literature for each aggregate type. Figure 78 shows the sensitivity of the model to 

coarse aggregate content. As expected, CTE decreases as the coarse aggregate content increases 

(or mortar volume decreases). While this is true for most cases, it was also observed that 

for aggregates with high CTE values, such as chert and quartzite, the CTE of the aggregate 

approaches the CTE of the mortar, thereby showing little or no sensitivity to coarse aggregate 

content. As with all other models, the user is advised to verify model predictions with other 

sources of information. If possible, both CTE models should be evaluated simultaneously to 

obtain a range.

Figure 76. Graph. Comparison of CTE model prediction with average values reported in 

literature for each aggregate rock type.

Figure 77. Graph. CTE model prediction versus average values reported in literature for

each aggregate rock type.
52

Figure 78. Graph. Sensitivity of the CTE model to coarse aggregate content.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

500 1000 1500 2000 2500 3000

PCC CTE, in/in/deg F x 10E-6

Coarse aggregate content in mix lb/yd3

Basalt

Chert

Quartzite

Andesite

Peridotite
53

CHAPTER 4. RIGID PAVEMENT DESIGN FEATURES MODELS

The models developed for the prediction of MEPDG-specific inputs fall under the design 

features category. In developing these models, the dependent variable (e.g., deltaT for JPCP 

design) was determined through performing several trial and error runs of the MEPDG and 

establishing the optimum value that minimizes the error prediction. The independent variables 

were obtained from the LTPP database or MEPDG calibration files.

The MEPDG design files used to generate the dependent variable data were obtained from the 

model calibration performed under NCHRP 1-40D, which produced the MEPDG software 

program version 1.0 in 2007.

(4) However, minor changes and software bug fixes have been 

performed since then, and the official version available at the time of this study was the MEPDG 

software version 1.1. Therefore, these models presented under this section are valid only for use 

with the distress calibration model of version 1.1 of the MEPDG software. The prediction 

models presented here for the estimation of design feature inputs therefore may not be valid 

once the products of future MEPDG updates and revisions are released.

deltaT—JPCP DESIGN

The equation developed to estimate the deltaT gradient variable can be expressed as follows:

uw w c latitude

deltaT inch TR PCCTHK

+ 0.01677* + 1.14008* / + 0.01784*

/ = −5.27805 - 0.00794* - 0.0826*SW + 0.18632*

Figure 79. Equation. Prediction model 15 for deltaT/inch.

Where:

deltaT/inch = Predicted average gradient through JPCP slab, °F/inch.

TR = Difference between maximum and minimum temperature for the month of construction, °F.

SW = Slab width, ft.

PCCTHK = JPCP slab thickness, inch.

uw = Unit weight of PCC used in JPCP slab, lb/ft3

.

w/c = Water to cement ratio.

latitude = Latitude of the project location, degrees.

The model considers climate (TR, latitude), design (SW, PCCTHK), and material (uw, w/c) 

parameters. The model statistics are presented in table 28. The model was developed with 

147 data points, has an R2 value of 49.67 percent, and an RMSE of 0.3199 psi. Table 29 provides 

details of the range of data used to develop the model. 
54

Table 28. Regression statistics for JPCP deltaT model.

Variable DF Estimate

Standard 

Error t-value Pr > |t| VIF

Intercept 1 -5.27805 1.06943 -4.94 < 0.0001 0

TR 1 -0.00794 0.00396 -2 0.047 1.86047

SW 1 -0.0826 0.03432 -2.41 0.0174 1.07141

PCCTHK 1 0.18632 0.0195 9.55 < 0.0001 1.0642

uw 1 0.01677 0.00669 2.51 0.0133 1.22792

w/c 1 1.14008 0.2914 3.91 0.0001 1.14857

latitude 1 0.01784 0.0072 2.48 0.0144 1.85265

Table 29. Range of data used for JPCP deltaT model.

Parameter Minimum Maximum Average

Temperature range 21.2 64.5 47.4

Slab width 12.0 14.0 12.5

PCC thickness 6.4 14.3 9.6

Unit weight 134 156 147

w/c ratio 0.27 0.72 0.46

Latitude 27.93 49.60 39.58

Figure 80 shows the predicted versus measured for the proposed JPCP deltaT gradient model, 

while figure 81 shows the residual errors. Note that the measured data here refers to the deltaT

gradient determined by matching MEPDG prediction to field performance. Figure 82 shows the 

predicted versus measured deltaT for the model.

Figure 80. Graph. Predicted versus measured for JPCP deltaT gradient model.

-2.5

-2

-1.5

-1

-0.5

0

-2.5 -2 -1.5 -1 -0.5 0

Predicted DeltaT gradient, degF/inch

DeltaT gradient estimated by matching MEPDG and field performance, deg 

F/inch 

N=147

R2 = 49.67

RMSE = 0.31992
55

Figure 81. Graph. Residual errors for JPCP deltaT gradient model.

Figure 82. Graph. Predicted versus measured deltaT based on the JPCP deltaT gradient 

model.

Figure 83 through figure 89 present the sensitivity analysis performed to examine the impact of 

varying the model parameters on its prediction. The parameters included are temperature range, 

slab width, slab thickness, unit weight, w/c ratio, and latitude. For each sensitivity analysis, the 

variable of interest was varied while holding all other variables constant at their typical values. 

Typical values used in this analysis were 24 °F temperature range, 12-ft slab width, 10-inch slab 

thickness, 145 lb/ft3 unit weight, 0.40 w/c ratio, and 40 degrees latitude.

y = 3E-06x + 4E-06

R2 = 8E-12

-2.500

-2.000

-1.500

-1.000

-0.500

0.000

0.500

1.000

-2.5 -2 -1.5 -1 -0.5 0

Error in prediction, degF/inch

Predicted deltaT, deg F/inch 

-16

-14

-12

-10

-8

-6

-4

-2

0

-16 -14 -12 -10 -8 -6 -4 -2 0

Predicted DeltaT, degF

DeltaT estimated by matching MEPDG and field performance, deg F

Delta-T Model - Temperature Differential Plot
56

Figure 83. Graph. Sensitivity of predicted deltaT to temperature range during month of 

construction.

Figure 84. Graph. Sensitivity of predicted deltaT to slab width.

Figure 85. Graph. Sensitivity of predicted deltaT to slab thickness.
57

Figure 86. Graph. Sensitivity of predicted deltaT to PCC slab unit weight.

Figure 87. Graph. Sensitivity of predicted deltaT to PCC w/c ratio.

Figure 88. Graph. Sensitivity of predicted deltaT to latitude of the project location.

-15

-10

-5

0

130 135 140 145 150 155 160

Delta T in 10in slab, deg F

PCC unit weight, lb/ft3
58

Figure 89. Graph. Predicted deltaT for different locations in the United States.

The following are brief observations from these sensitivity analyses:

• For the typical values used for each of these variables, the deltaT gradients estimated are

in a reasonable range.

• An increase in local climate temperature range increases the temperature gradient (see 

figure 83). The local climate temperature range is indicative of the level of temperature 

drop the project location can experience. The larger the difference in the temperature 

between day and night (assuming paving is performed in the daytime), the larger the 

negative temperature gradient locked into the slab as the slab hardens within a 

24-h period. 

• Wider slabs produce a larger built-in gradient (see figure 84), as has been validated 

in several field studies. The total thermal expansion is larger for a longer/wider slab;

therefore, the resulting curvature of the slab induces a greater lift-up at the slab corners. 

The data did not show a significant effect of the slab length or joint spacing parameter.

• Thicker slabs reduce the deltaT gradient, as shown in figure 85. This is the expected 

trend, as thicker slabs (due to a greater weight) tend to restrain the corners from curling 

up as the concrete hardens. This figure also shows that for very thin slabs (< 8 inches), 

the effect is reversed. The physical significance of this cannot be fully explained or 

supported with data. Therefore, it is necessary to evaluate the sensitivity to each 

parameter while selecting deltaT for each project.

• The larger unit weight of the PCC material used in the JPCP slab also reduces the 

magnitude of built-in gradient (see figure 86) primarily because of the restraint provided 

by the heavier slab during hardening.

• Lower w/c ratios have a higher rate of hydration; therefore, the PCC slab remains plastic 

for a shorter duration of time. Strength gain offers the slab the rigidity necessary to bear 

against the base and does not allow the slab corners to curl up. Therefore, lower w/c 

ratios tend to have higher built-in gradients, as seen in figure 87. Furthermore, at very 

low w/c ratios, the PCC mix undergoes autogeneous shrinkage, which increases the 

potential for higher gradients in the slab.
59

• Figure 88 and figure 89 show the effect of latitude on predicted deltaT gradients. The 

United States lies between 30 and 50 degrees latitude in the Northern Hemisphere. The 

full range of latitudes is covered in figure 88. While this plot might appear to show 

deltaT’s high degree of sensitivity to the latitude parameter, for routine predictions using 

this model, the temperature range is a critical input. In other words, a given maximum 

temperature in the southern United States could have a much different temperature range 

relative to a location in the northern United States with the same maximum temperature. 

Therefore, the latitude parameter has to be evaluated combined with the temperature 

range parameter as shown in figure 89. The predicted deltaT for several locations in the 

United States are presented.

Using the JPCP deltaT Model

This section provides an example for the use of the JPCP deltaT model developed under this 

study. The section used to describe the process is the LTPP Specific Pavement Studies 2 section 

04_0213 located in Maricopa County, AZ, and constructed in July 1993. The following latitude, 

design, and material inputs required for the deltaT prediction model can be obtained from the 

MEPDG inputs:

• Latitude: 33.45 degrees north.

• PCC thickness: 8.3 inches.

• Slab width: 14 ft.

• PCC unit weight: 145.3 lb/ft3

.

• PCC w/c ratio: 0.365.

The temperature range input to this model is the difference between the mean monthly maximum 

and minimum temperatures for the month of July from historical climate data records (as climate 

data included in the MEPDG). If the user does not have this information readily available, the 

data to compute the temperature range can be determined from the output file of the MEPDG 

analysis of this section. The output file (i.e., titled “04_0213.xls”) contains a worksheet titled 

“Climate” with key climate data for the specific location (or the virtual climate station created). 

This worksheet includes the monthly climate summary with minimum and maximum 

temperature by month for all years of data used under the headings “Min. Temp. (°F)” and “Max. 

Temp. (°F),” respectively. (Note that this summary also includes “Average Temp. (°F),” “Max. 

Range (°F),” “Precip. (in.),” “Average Wind (mph),” “Average Sun (%),” “Number Wet Days,”

and “Max. Frost (in.).” However, these data are not of relevance to the deltaT model.

For the month of July, the average minimum and maximum temperatures are 73 and 111.7 °F,

respectively. The difference between these temperatures is 38.7 °F.

Using these inputs, the deltaT gradient can be calculated as -1.7457138 °F/inch. For the slab 

thickness of 8.3 inches, this is equivalent to a deltaT of -14.5 °F. This value is significantly 
60

higher than the default -10 °F/inch. This input can be revised in an MEPDG file and reanalyzed 

to evaluate the predicted transverse cracking performance.

deltaT—CRCP DESIGN

The equation developed to estimate the CRCP deltaT gradient variable is as follows:

+ 0.11299 * PCCTHK

+1.55013*Granite +1.40009 *Limestone + 2.01838*Quartzite

deltaT / inch = 12.93007 - 0.15101* MaxTemp - 0.10241* MaxTempRange + 3.279 *Chert 

Figure 90. Equation. Prediction model 16 for deltaT/inch.

Where:

deltaT/inch = Predicted gradient in CRCP slab, °F/inch.

MaxTemp = Maximum temperature for the month of construction, °F.

MaxTempRange = Maximum temperature range for the month of construction, °F.

PCCTHK = JPCP slab thickness, inch.

Chert = 1 if PCC mix coarse aggregate is chert, or 0 if otherwise.

Granite = 1 if PCC mix coarse aggregate is granite, or 0 if otherwise.

Limestone = 1 if PCC mix coarse aggregate is limestone, or 0 if otherwise.

Quartzite = 1 if PCC mix coarse aggregate is quartzite, or 0 if otherwise.

The model considers climate (MaxTemp and MaxTempRange), design parameters (PCCTHK), 

and material (Aggregate type) parameters. The model statistics are presented in table 30. The 

model was developed with 35 data points, has an R2 value of 82.5 percent, and an RMSE of 

0.27932 psi. Table 31 provides details of the range of data used to develop the model. 

Table 30. Regression statistics for CRCP deltaT model.

Variable DF Estimate

Standard 

Error t-Value Pr > |t| VIF

Intercept 1 12.93007 1.98459 6.52 < 0.0001 0

MaxTemp 1 -0.15101 0.01793 -8.42 < 0.0001 3.46347

MaxTempRange 1 -0.10241 0.01869 -5.48 < 0.0001 2.00933

Chert 1 3.279 0.30508 10.75 < 0.0001 2.24965

Granite 1 1.55013 0.22656 6.84 < 0.0001 4.96262

Limestone 1 1.40009 0.18956 7.39 < 0.0001 4.00053

Quartzite 1 2.01838 0.39449 5.12 < 0.0001 1.93773

PCCTHK 1 0.11299 0.0705 1.6 0.1207 1.68624
61

Table 31. Range of data used for CRCP deltaT model.

Parameter Minimum Maximum Average

Maximum temperature 78.4 99.2 90.3

Temperature range 24.8 40.4 30.4

Chert 0 1 0.06

Granite 0 1 0.31

Limestone 0 1 0.46

Quartzite 0 1 0.03

PCC thickness 5.6 9.5 8.4

Figure 91 shows the predicted versus measured for the proposed CRCP deltaT gradient model, 

while figure 92 shows the residual errors. Note that the measured data here refers to the deltaT

gradient determined by matching MEPDG prediction to field performance. 

Figure 91. Graph. Predicted versus measured for CRCP deltaT model.
62

Figure 92. Graph. Residual errors for CRCP deltaT model.

Figure 93 through figure 96 show the sensitivity of the deltaT differential calculation to the 

parameters maximum temperature of the project location, maximum temperature range, CRCP 

slab thickness, and geographic location, respectively. The trends observed in the model—CRCP 

deltaT increasing with increasing maximum temperature and increasing temperature range—

are reasonable. While the effect of slab thickness shows a linear relationship with the deltaT

gradient, the magnitude of the coefficient for this variable results causes the deltaT differential 

(CRCP deltaT gradient × thickness) to assume a nonlinear relationship with the deltaT

differential, peaking at about 10 inches. Figure 96 shows the deltaT predictions for projects 

selected from LTPP sites in Texas, Illinois, Virginia, Mississippi, Oregon, and Georgia.

Figure 93. Graph. Effect of maximum temperature on CRCP deltaT prediction model.
63

Figure 94. Graph. Effect of temperature range on CRCP deltaT prediction model.

Figure 95. Graph. Effect of slab thickness on CRCP deltaT prediction model.

Figure 96. Graph. Effect of geographic location on CRCP deltaT prediction model.

-25

-20

-15

-10

-5

0

20 30 40 50

Delta T in 10in slab, deg F

Temperature range, deg F

-12.1

-12

-11.9

-11.8

-11.7

-11.6

-11.5

-11.4

6 8 10 12 14

Delta T, deg F

Slab thickness, in

-1.3

-31.0

-11.4 -1.5

-25.1

-14.5 -12.4

-40.0

-30.0

-20.0

-10.0

Delta T in 10in slab, deg F

0.0

Location

TX, Cont.=Feb1982, MaxTemp=78.1F,Range=37.3F 

TX, Cont.=July1987, MaxTemp=102.7F,Range=27.4F 

IL, Cont.=May1986, MaxTemp=89.6F,Range=30.2F 

MS, Cont.=Aug1970, MaxTemp=96F,Range=27.3F 

OR, Cont.=Jun1972, MaxTemp=83.2F,Range=42.8F

GA, Cont.=Aug1972, MaxTemp=95.7
64

The sensitivity analyses show reasonable trends but do not demonstrate that the model is 

robust. It is not clear, from an engineering standpoint, if the range of predicted values and their 

magnitudes are practical and realistic. The wide range of deltaT has a significant effect on design 

thickness. The data used to develop the model show very strong correlations, and it is likely that 

the predictions are valid, at least within a certain range of inputs. The current analyses and the 

data available are not adequate to determine these ranges. It is therefore recommended that this 

model be used with extreme caution.

Using the CRCP deltaT Model

The CRCP deltaT model shares similarities with the JPCP deltaT model. The section used to 

describe the process is the LTPP General Pavement Studies section in Illinois, 17_5020, which

was constructed in May 1986. The CRCP thickness is 8.6 inches, and the PCC mix used a 

limestone aggregate. The following inputs can be directly obtained from the MEPDG input file:

• PCCTHK: 8.6 inches.

• Chert: 0.

• Granite: 0.

• Limestone: 1.

• Quartzite: 0.

The maximum temperature and maximum temperature range can be obtained by running the 

design file and deriving this input from the worksheet titled “Climate.” For the month of May, 

the maximum temperature and maximum temperature range for this location were 89.6 and 

39.2 °F, respectively. Using these inputs, the CRCP deltaT gradient can be calculated as 

-1.3214 °F/inch. For the slab thickness of 8.6 inches, this is equivalent to a deltaT of -11.36 °F. 

This value is comparable to the -10-°F default. This input can be revised in an MEPDG file and 

reanalyzed to predict punchout development over time.
65

CHAPTER 5. STABILIZED MATERIALS MODELS

As the LTPP database contains limited data on modulus values and index properties of stabilized 

materials, the only prediction model that can be developed for modulus prediction is that for 

LCB materials, which is included in this section.

LCB ELASTIC MODULUS MODEL

The model developed can be expressed as follows:

58156 ,28 716886 ' ELCB = f c d +

Figure 97. Equation. Prediction model 17 for ELCB.

Where:

ELCB = Elastic modulus of the LCB layer.

f'c, 28d = 28-day compressive strength of the LCB material.

The predicted versus measured and the residual errors plots for this relationship are presented in 

figure 98 and figure 99, respectively. The model has an R2 value of 41.24 percent, an RMSE of 

541,600 psi, and uses 11 data points. The compressive strength values range from 770 to 

2,800 psi with an average value of 1,400 psi.

Figure 98. Graph. Predicted versus measured for the LCB elastic modulus model.

y = 0.4124x + 2E+06

R2 = 0.4124

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

3.5E+06

4.0E+06

0.0E+00 1.0E+06 2.0E+06 3.0E+06 4.0E+06 5.0E+06

Predicted elastic modulus, psi

Measured elastic modulus, psi

N=11

R2 = 0.4124

RMSE = 541619 
66

Figure 99. Graph. Residual errors for the LCB elastic modulus model.

y = -8E-06x + 5.9484

R2 = 4E-11

-1.00E+06

-5.00E+05

0.00E+00

5.00E+05

1.00E+06

1.50E+06

0.0E+00 1.0E+06 2.0E+06 3.0E+06 4.0E+06

Error in prediction

Predicted elastic modulus, psi
67

CHAPTER 6. UNBOUND MATERIALS MODELS

RESILIENT MODULUS OF UNBOUND MATERIALS

The following model recommended for predicting the resilient modulus of unbound materials is 

based on the constitutive equation for modeling resilient modulus behavior when subjected to 

various stress states:

2 3

1

k

a

oct

k

a

r a P P

M k P 





























 = θ τ

Figure 100. Equation. Mr.

Where:

 = Bulk stress = .

= Principal stress.

= Confining pressure.

Pa = Atmospheric pressure.

= Octahedral normal stress =1/3 ( ).

k1, k2, k3 = regression constants that are a function of soil properties, as defined in figure 101

through figure 103 of this report.

This model can be used for various soil types, and the model attributes (k1, k2, and k3) for a 

given soil type remain the same regardless of stress state. Furthermore, models used to predict 

constitutive model attributes for a given set of soil properties are recommended to characterize 

resilient modulus behavior rather than developing models individually for each possible 

combination of expected stress states. 

Constitutive Model Parameter k1

Figure 101. Equation. Prediction model 18 for k1.

Model statistics for k1 are as follows:

• R2 = 0.16 percent.

• Standard error of estimate (SEE) = 237.4.

• N = 1,029.

Constitutive Model Parameter k2

Figure 102. Equation. Prediction model 19 for k2.

θ σ1 + σ2 + σ3

σ1

σ2, σ3

τoct σ1 + 2 σ3

k1 = 1446.2 - 4.56764*PCTHALF + 4.92*LL - 27.73*OPTMOIST

k2 = 0.45679 - 0.00073376*PCTNO80 - 0.00269*LL + 0.00060555*PCTGRVL + 12.97*D10
68

Model statistics for k2 are as follows:

• R2 = 0.67 percent.

• SEE = 0.0934.

• N = 1,032.

Constitutive Model Parameter k3

Figure 103. Equation. Prediction model 20 for k3.

Where:

PCTHALF = Percent passing 1

/2-inch sieve.

LL = Liquid limit, percent.

OPTMOIST = Optimum moisture content, percent.

PCTNO80 = Percent passing No. 80 sieve.

PCTGRVL = Percent gravel fraction (0.078- to 2.36-inch size).

D10 = Maximum particle size of the smallest 10 percent of soil sample.

In the development of these models, a wide range of k1, k2, and k3 parameters were used, which 

varied by soil class. Histograms showing the distribution of k1, k2, and k3 values by soil class are 

shown in figure 104 through figure 106, respectively.

Figure 104. Graph. Resilient modulus parameter k1 for unbound material types included in 

the model development database.

k3 = -0.188 (for fine grained soils)

k3 = -0.153 (for coarse grained materials)

862 858 834

994

1132

792 841 877 897

677

845

0

200

400

600

800

1000

1200

1400

A-1-a

A-1-bA-2-4A-2-6A-2-7

A-3A-4A-5A-6

A-7-5A-7-6

Resilient Modulus Parameter, k1

Unbound Material Type (AASHTO Soil Class)
69

Figure 105. Graph. Resilient modulus parameter k2 for unbound material types included in 

the model development database.

Figure 106. Graph. Resilient modulus parameter k3 for unbound material types included in 

the model development database.

Model prediction accuracy and reasonableness were evaluated by reviewing the plot of predicted 

and measured resilient modulus for all individual resilient modulus test values used in model 

development as presented in figure 107. Figure 108 presents a plot of measured and predicted 

resilient modulus versus bulk stress for all fine- and coarse-grained materials included in the 

model development database. 

0.7

0.6

0.5

0.3

0.4

0.6

0.3

0.2 0.2

0.1

0.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

A-1-a

A-1-b

A-2-4A-2-6A-2-7

A-3A-4A-5

A-6

A-7-5A-7-6

Resilient Modulus Parameter, k2

Unbound Material Type (AASHTO Soil Class)

-0.1

-0.3

-0.6

-0.5

-0.3

-0.4

-0.8

-0.5

-0.8

-0.6

-0.8 -0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

A-1-a

A-1-bA-2-4A-2-6A-2-7

A-3A-4A-5A-6

A-7-5A-7-6

Resilient Modulus Parameter, k3

Unbound Material Type (AASHTO Soil Class)
70

Figure 107. Graph. Plot of measured versus predicted resilient modulus (using k1, k2, and 

k3 derived from figure 101 through figure 103).

Figure 108. Graph. Plot showing predicted and measured resilient modulus versus bulk 

stress for fine- and coarse-grained soils.

y = 0.9388x

R2 = 0.5661

N = 23056

0

10000

20000

30000

40000

50000

60000

0 10000 20000 30000 40000 50000 60000

Predicted Mr, psi

Measured Mr, psi

0

5000

10000

15000

20000

25000

30000

35000

0 20 40 60 80 100

Mr, psi

Bulk Stress, psi

Predicted Mr (Fine-Grained Soils) Predicted Mr (Coarse-Grained Soils)

Measured Mr (Fine-Grained Soils) Measured Mr (Coarse-Grained Soils)
71

Sensitivity analysis results are presented in figure 109 through figure 115. The results of the 

sensitivity analysis are summarized as follows:

• Increasing bulk stress results in significantly higher level of resilient modulus for coarsegrained materials. Soil type has a significant impact on predicted resilient modulus. 

• Increasing the amount of finer materials results in a decrease in resilient modulus.

• Increasing the amount of gravel results in an increase in resilient modulus.

• Increasing effective size results in an increase in resilient modulus.

• Increasing optimum moisture content results in a decreased in resilient modulus.

• Increasing liquid limit results in an increase in resilient modulus.

Figure 109. Graph. Effect of material type (AASHTO soil class) on predicted resilient 

modulus.

0

5000

10000

15000

20000

25000

30000

35000

40000

0 20 40 60 80 100

Mr, psi

Bulk Stress, psi

A-1-b A-2-4 A-6 A-7-5
72

Figure 110. Graph. Effect of percent passing 1/2-inch sieve on predicted resilient modulus.

Figure 111. Graph. Effect of liquid limit on predicted resilient modulus.

20000

22500

25000

27500

30000

32500

35000

0 20 40 60 80 100

Predicted Mr, psi

Percent Passing 1/2-in Sieve

20000

22500

25000

27500

30000

32500

35000

0 10 20 30 40 50

Predicted Mr, psi

Liquid limit, percent
73

Figure 112. Graph. Effect of optimum moisture content on predicted resilient modulus.

Figure 113. Graph. Effect of No. 80 sieve on predicted resilient modulus.

5000

10000

15000

20000

25000

30000

35000

0 10 20 30 40

Predicted Mr, psi

Optimum moisture content, percent

20000

22500

25000

27500

30000

32500

35000

0 20 40 60 80 100

Predicted Mr, psi

Percent Passing No. 80 Sieve
74

Figure 114. Graph. Effect of gravel content on predicted resilient modulus.

Figure 115. Graph. Effect of effective size on predicted resilient modulus.

20000

22500

25000

27500

30000

32500

35000

0 20 40 60 80

Predicted Mr, psi

Percent Gravel Content

20000

22500

25000

27500

30000

32500

35000

0 0.005 0.01 0.015

Predicted Mr, psi

Effective Size (D10), in
75

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