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Publication Number: FHWA-RD-01-113
Date: October 2002

 

Back-Calculation of Layer Parameters for LTPP Test Sections, Volume II

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FOREWORD

Deflection basin measurements have been made with the Falling Weight Deflectometer (FWD) on all test sections that are included in the Long Term Pavement Performance (LTPP) program. These deflection basin data are analyzed to obtain the structural-response characteristics of the pavement structure and subgrade that are needed to achieve the overall LTPP program objectives.

One of the more common methods for analysis of deflection data is to back-calculate the elastic properties for each layer and provide the elastic layer modulus that are typically used for pavement evaluation and rehabilitation design. This report documents the procedure that was used to back-calculate, in mass, the elastic properties for both flexible and rigid pavements in the LTPP program using layered elastic analyses. Obtaining these elastic layer properties for use in further data analyses and studies regarding pavement performance was the primary objective of this study.

T. Paul Teng, P.E.
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.

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. Standards and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its information. FHWA periodically reviews quality issues and adjusts its programs and processes to ensure continuous quality improvement.

Technical Report Documentation Page

1. Report No.

FHWA-RD-01-113

2. Government Accession No. 3 Recipient's Catalog No.
4. Title and Subtitle

Back-Calculation of Layer Parameters for LTPP Test Sections, Volume II: Layered Elastic Analysis for Flexible and Rigid Pavements

5. Report Date

OCTOBER 2002

6. Performing Organization Code
7. Author(s)

H. L. Von Quintus and A. L. Simpson

8. Performing Organization Report No.

 

9. Performing Organization Name and Address

Fugro-BRE, 8613 Cross Park Drive, Austin, TX 78754

10. Work Unit No. (TRAIS) C6B

11. Contract or Grant No. DTFH61-96-C-00003
12. Sponsoring Agency Name and Address

Office of Engineering R & D Federal Highway Administration, 6300 Georgetown Pike, McLean, Virginia 22101-2296

13. Type of Report and Period Covered

Final Report, May 1997 -- August 2001

14. Sponsoring Agency Code

HCP 30-C

15. Supplementary Notes This report is a project deliverable under the LTPP Data Analysis Technical Support study. ERES Consultants is the prime contractor, and Fugro-BRE is a subcontractor. Fugro-BRE staff members who provided significant contributions to this study include Dr. Peter Jordahl and Mr. Dale Framelli. Contracting Officer's Technical Representative (COTR): Cheryl Allen Richter, HRDI-13
16. Abstract

This report documents the procedure and steps used to back-calculate the layered elastic properties (Young's modulus and the coefficient and exponent of the nonlinear constitutive equation) from deflection basin measurements for all of the LTPP test sections with a level E data status. The back-calculation process was completed with MDOCOMP4 for both flexible and rigid pavement test sections in the LTPP program. The report summarizes the reasons why MODCOMP4 was selected for the computations and analyses of the deflection data, provides a summary of the results using the linear elastic module (Young's modulus) for selected test sections, and identifies those factors that can have a significant effect on the results.

Results from this study do provide elastic layer properties that are consistent with previous experience and laboratory material studies related to the effect of temperature, stress-state, and season on material load-response behavior. In fact, over 75 percent of the deflection basins analyzed with the linear elastic module of MODCOMP4 resulted in solutions with an RMS error less than 2.5 percent. Those pavements exhibiting deflection-softening behavior with Type II deflections basins were the most difficult to analyze and were generally found to have RMS errors greater than 2 percent. In summary, the nonlinear module of MODCOMP did not significantly improve on the number of reasonable solutions, and it is recommended that nonlinear constitutive equations not be used in a batch mode basis.

17. Key Words

MODCOMP, Back-Calculation, Modulus, Young's Modulus, Elastic Properties, Surface Deflection, Stress Sensitivity.

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

144

22. Price
Form DOT F 1700.7 Reproduction of completed page authorized

SI* (Modern Metric) Conversion Factors

TABLE OF CONTENTS

1.0 INTRODUCTION

2.0 SELECTION OF BACK-CALCULATION METHODOLOGY

3.0 BACK-CALCULATION PROCESS

4.0 RESULTS FROM COMPUTATION OF ELASTIC PROPERTIES

5.0 SUMMARY OF FINDINGS AND RECOMMENDATIONS

6.0 REFERENCES

APPENDIX A.0: Back-Calculation of Layer Elastic Properties from LTPP-FWD Deflection Basin

APPENDIX B.0: Test Section Classification and Subgrade Information for the Back-Calculation Process

APPENDIX C: Median Values and Histograms of Young's Modulus Back-Calculated for Different Materials

LIST OF TABLES

  1. LTPP FWD deflection basin testing plan
  2. Factorial used to estimate the theoretical accuracy of the MODCOMP4 solutions
  3. Identification and summary of the data tables used to obtain the input data used for the back-calculation process of the LTPP sites
  4. "Default" material properties for different materials, if not included in the LTPP database
  5. Layer model descriptions
  6. Summary of information included in the computed parameter data tables that are available in the LTPP database
  7. Average Young's modulus calculated for the outer wheelpath and mid-lane for each layer along specific test sections for drop height 4, MPa
  8. Uniformity Classification of LTPP test sections (refer to figures 9-12)
  9. LTPP test sections with an apparent rigid layer that was used in the back-calculated process and the depth of that rigid layer
  10. Listing of LTPP test sections and test dates where a nonstandard sensor placement was used during FWD testing
  11. LTPP test sections with many deflection basins that did not result in a solution with an RMS error of less than 2 percent
  12. Median Young's modulus back-calculated for unbound pavement materials and soils, based on the AASHTO soil classification system, for the LTPP flexible test sections
  13. Median Young's modulus back-calculated within specific temperature ranges for the LTPP flexible test sections

LIST OF FIGURES

  1. Flow chart showing the major steps and decisions used in the linear elastic back-calculation process
  2. Graphical illustration of the definition for the linear response category
  3. Graphical illustration of the definition for the deflection-hardening response category
  4. Graphical illustration of the definition for the deflection-softening response category
  5. Typical normalized deflection basin
  6. Type I normalized deflection basin
  7. Type II normalized deflection basin
  8. Type III normalized deflection basin
  9. Example of a test section with uniform deflections (test section 081053, 15 June 1994) or low variability in the measured deflections
  10. Example of a test section with highly variable deflections (test section 041024, 13 June 1996) or high variability in the measured deflections
  11. Example of a test section with drift (test section 040114, 12 June 1996) or where the deflections consistently change from the approach end to the leave end, defined as drift
  12. Example of a test section with an abrupt change in the measured deflections (test section 040113, 16 August 1995) between the approach and leave ends
  13. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections measured along test section 091803 in August 1994 for different maximum RMS errors
  14. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections measured along test section 131005 in January 1996 for different maximum RMS errors
  15. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections measured along test section 481068 in August 1994 for different maximum RMS errors
  16. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections along test section 081053 in March for different maximum RMS errors
  17. Distribution of the RMS errors for all of the deflection basins measured on the flexible pavement test sections that were analyzed with the linear elastic modulus MODCOMP4
  18. Distribution of the RMS errors for all of the deflection basins measured on the rigid pavement test sections that were analyzed with the linear elastic module of MODCOMP4
  19. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 091803 in August 1994
  20. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 131005 in January 1996
  21. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 481068 in August 1994
  22. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 081053 in March 1995
  23. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 041024
  24. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 131005
  25. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 241634
  26. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 276251
  27. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 040113
  28. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 131005
  29. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 501002
  30. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 561007
  31. Flow chart showing the overall back-calculation process for the linear and nonlinear material characterization modules in MODCOM4
  32. Flow chart for the deflection basin and load-response characterization procedure
  33. BACKCAL MODULUS by A-1-a
  34. BACKCAL MODULUS by A-1-b
  35. BACKCAL MODULUS by A-2-4
  36. BACKCAL MODULUS by A-2-5
  37. BACKCAL MODULUS by A-2-6
  38. BACKCAL MODULUS by A-2-7
  39. BACKCAL MODULUS by A-3
  40. BACKCAL MODULUS by A-4
  41. BACKCAL MODULUS by A-5
  42. BACKCAL MODULUS by A-6
  43. BACKCAL MODULUS by A-7-5
  44. BACKCAL MODULUS by A-7-6
  45. BACKCAL MODULUS of granular unbound AGG layers
  46. BACKCAL MODULUS of granular unbound base layer
  47. BACKCAL MODULUS of granular unbound subbase layer
  48. BACKCAL MODULUS TXT (PAVT TYPE=SEMI RIG LAY TYPE=PCC)
  49. BACKCAL MODULUS TXT (PAVT TYPE=SEMI RIG LAY TYPE=HMA)
  50. BACKCAL MODULUS TXT (PAVT TYPE=SEMI RIG LAY TYPE=CTB)
  51. BACKCAL MODULUS TXT (PAVT TYPE=SEMI RIG LAY TYPE=ATB)
  52. BACKCAL MODULUS TXT (PAVT TYPE=RIGID LAY TYPE=PCC)
  53. BACKCAL MODULUS TXT (PAVT TYPE=RIGID LAY TYPE=HMA)
  54. BACKCAL MODULUS TXT (PAVT TYPE=RIGID LAY TYPE=CTB)
  55. BACKCAL MODULUS TXT (PAVT TYPE=RIGID LAY TYPE=ATB)
  56. BACKCAL MODULUS TXT (PAVT TYPE=FLEX LAY TYPE=HMA)
  57. BACKCAL MODULUS TXT (PAVT TYPE=FLEX LAY TYPE=CTB)
  58. BACKCAL MODULUS TXT (PAVT TYPE=FLEX LAY TYPE=ATB)
  59. BACKCAL MODULUS TXT (PAVT TYPE=SEMI RIG LAY=HMA TEMP=MODER)
  60. BACKCAL MODULUS TXT (PAVT TYPE=SEMI RIG LAY TYPE=HMA TEMP=HIGH)
  61. BACKCAL MODULUS TXT (PAVT TYPE=FLEX LAY TYPE=HMA TEMP=MODERATE)
  62. BACKCAL MODULUS TXT (PAVT TYPE=FLEX LAY TYPE=HMA TEMP=HIGH)
  63. BACKCAL MODULUS TXT (PAVT TYPE=FLEX LAY TYPE=HMA TEMP=COLD)
1. INTRODUCTION

1.1 Background

Deflection basin measurements have been made with the Falling Weight Deflectometer (FWD) on all General Pavement Study (GPS) and Specific Pavement Study (SPS) test sections that are included in the Long-Term Pavement Performance (LTPP) program. This deflection-testing program is being conducted periodically to obtain the load-response characteristics of the pavement structure and subgrade. FWD deflection basin tests are conducted about every 2 years for the SPS project sites and about every 5 years for the GPS sites. There are 64 test sections included in the LTPP Seasonal Monitoring Program (SMP), and these sites are tested about every month over a period of 1 to 2 years. These deflection basin data are intended to provide structural-response characteristics that are needed to achieve the overall LTPP program objectives.

One of the more common methods for analysis of deflection data is to back-calculate the elastic properties for each layer in the pavement structure and foundation. These analysis methods (referred to as back-calculation programs) provide the elastic layer modulus typically used for pavement evaluation and rehabilitation design. At present, interpretation of deflection basin test results usually is performed with static-linear analyses, and there are numerous computer programs that can be used to calculate these elastic modulus values (Young's modulus).

This report documents the procedure that was used to back-calculate, in mass, the elastic properties for both flexible and rigid pavements in the LTPP program using layered elastic analyses. All data used for this back-calculation study were extracted from the LTPP data release dated October 1997 for the SMP sites and April 1998 for the GPS and SPS sites, and have a level E status (the highest quality data in the LTPP database). All work was completed under the LTPP Data Analysis Technical Support Study (Contract No. DTFH61-96-C-00003).

1.2 LTPP Deflection Testing Program

The LTPP deflection basin testing program uses seven sensors placed at 0, 203, 305, 457, 610, 914, and 1524 mm from the center of the load plate to define the shape of the deflection basin. The loading sequence, as stored in the LTPP database for flexible and rigid pavement testing, is summarized in table 1. The following summarizes the general locations for the deflection tests for each different type of pavement.


Table 1. LTPP FWD deflection basin testing plan.

Pavement Type Drop Height Number of Drops Target Load, kN Acceptable Load Range, kN
Flexible 3 3 Seating ---
Flexible 1 4 26.7 24.0 to 29.4
Flexible 2 4 40.0 36.0 to 44.0
Flexible 3 4 53.4 48.1 to 58.7
Flexible 4 4 71.2 64.1 to 78.3
Rigid, JCP & CRCP 3 3 Seating ---
Rigid, JCP & CRCP 2 4 40.0 36.0 to 44.0
Rigid, JCP & CRCP 3 4 53.4 48.1 to 58.7
Rigid, JCP & CRCP 4 4 71.2 64.1 to 78.3

A more complete description of the testing plan and data storage in the database is provided in LTPP Manual for Falling Weight Deflectometer Measurements -- Operational Field Guidelines, Version 3.1, dated August 2000.

1.3 Objective

The primary objective of this study was to back-calculate the elastic layer properties from deflection basin measurements for use in further data analyses and studies regarding pavement performance. As part of this objective, the elastic layer properties back-calculated from the deflection basin data for the LTPP flexible and rigid pavement test sections were to be included in the LTPP computed parameter database for future use.

A secondary objective of this study was to provide any modifications to the current guidelines that have been prepared for use in back-calculating elastic properties. These guidelines include ASTM D5858 and the procedure written by Von Quintus and Killingsworth.(1) Another secondary objective was to identify those LTPP test sections with unusual load-response characteristics.

1.4 Application of Results to Future Studies

The elastic properties computed for each structural layer in the pavement structure and subgrade strata can be used in future studies of materials-pavement behavior and performance. In fact, these computed parameters will be needed to achieve some of the stated objectives and "outcomes" identified in the LTPP Strategic Plan that was published in 1999.(18) For example, elastic properties can be used directly in developing or validating load-related distress prediction models based on elastic layer theory or used indirectly for selecting test sections with significantly different properties for studying a particular design issue. The following lists some of the LTPP strategic objectives where the results from this study can be used to achieve those objectives.

Some specific applications of these results are listed below.


2.0 SELECTION OF BACK-CALCULATION METHODOLOGY

There are three basic approaches to back-calculating layered elastic moduli of pavement structures: 1) the equivalent thickness method (e.g., ELMOD and BOUSDEF), 2) the optimization method (e.g., MODULUS and WESDEF), and 3) the iterative method (e.g., MODCOMP and EVERCALC). Layer thickness is a critical parameter that must be accurately known for nearly all back-calculation programs, regardless of methodology, although some programs claim to be able to determine a limited set of both Young's modulus and layer thickness (e.g., MICHBACK).(2) Many of the software packages are similar, but the results can be different as a result of the assumptions, iteration technique, back-calculation, or forward-calculation schemes used within the programs.

Within the past couple of decades, there have been extensive efforts devoted to improving back-calculation of elastic-layer modulus by reducing the absolute error or root mean squared (RMS) error (difference between measured and calculated deflection basins) to values as small as possible. The absolute error term is the absolute difference between the measured and computed deflection basins expressed as a percent error or difference per sensor; whereas the RMS error term represents the goodness-of-fit between the measured and computed deflection basins.

The use of these linear elastic layer programs, however, has been only partly successful in analyzing the deflections measured at the LTPP sites. For example, only about 50 percent of the flexible GPS sites were found to have absolute error terms less than the generally considered reasonable value of 2 percent per sensor.(3,4) In addition, results from use of linear elastic models are highly variable, with an undefined reliability over a wide range of conditions.

So the question is: which program should be used to back-calculate Young's modulus for each structural layer in the pavement structure?

The purpose of this section is to document the methodology and software package used for back-calculating pavement layer and subgrade moduli from the deflection basins measured on all LTPP test sections that have a level E data status.

2.1 Back-Calculation Methods

The common analysis method is to back-calculate material response parameters for each layer within the pavement structure from the FWD deflection basin measurements. Many of the back-calculation programs are limited by the number and thickness of the layers used to define the pavement structure but, more importantly, assume that the layers are linear elastic. Most unbound pavement materials and soils are nonlinear. Thus, the calculated layer-modulus represents an "effective" Young's modulus that adjusts for stress-sensitivity and discontinuities or anomalies (such as variations in layer thickness, localized segregation, cracks, and the combinations of similar materials into a single layer).

The back-calculation methods can be grouped into four general categories.

2.1.1 Type of Load Application. At present, interpretation of deflection basin test results is performed with static analyses. There have been many improvements in back-calculation technology within the past 4 to 5 years. These improvements have spawned standardization procedures and guidelines to ensure that there is consistency within the industry and to improve upon the load-response characterization of the pavement structural layers.(5) ASTM D5858 is a procedure for analyzing deflection basin test results to determine layer elastic moduli (i.e., Young's modulus).

There are no similar standardized procedures for back-calculating materials properties of pavement layers using dynamic analysis techniques. In fact, there are only a few programs that have the capability to do dynamic analyses. Thus, the programs using dynamic analyses were not considered for use in calculating elastic-layer modulus from hundreds of deflection basins measured at the same site. Only the static load application analysis methods were considered appropriate for use in a production mode -- mass back-calculation of elastic layer modulus from deflection basins measured along the LTPP test sections.

2.1.2 Type of Material Response Models. Most of the back-calculation procedures that have been used to determine layer moduli are based on elastic layer theory. However, some of the programs based on elastic theory have been modified to account for the viscoelastic (time-dependent) or elastoplastic (inelastic) behavior of materials. Unfortunately, programs that include time-dependent properties or inelastic properties have not been used in a production mode and, more importantly, have not been very successful in producing consistent and reliable solutions.

SHRP, as well as others, studied and evaluated many of these back-calculation procedures to select one method for use in characterizing the subgrade and other pavement layers in order to predict the performance of flexible and rigid pavements.(6,7) The program entitled "MODULUS 4.0" was selected for flexible and composite pavements, whereas a new procedure was developed for rigid pavements as part of the SHRP P-020 Data Analysis Project.(8,9) As stated above, many of these programs are limited by the following:

Thus, it must be understood that the calculated layer modulus represents an "effective" or "equivalent" elastic modulus that accounts for differences in stress states and any discontinuities or anomalies (such as variations in layer thickness, slippage between two adjacent layers, cracks, and the combinations of similar materials into a single layer).

Although there have been extensive efforts devoted to improving back-calculation of layer moduli by reducing the RMS error to values as small as possible, highly variable results from the use of linear elastic models have been found with an undefined reliability over a wide range of conditions. In addition, models that assume a linear response of materials require numerous iterations at varying load levels (different drop heights) to identify the stress sensitivity of unbound pavement materials and soils. From this standpoint, the use of nonlinear elastic layer programs was believed to have merit.

Two programs that have been used with some success and contain a nonlinear structural response capability are MODCOMP and a program developed by the Corps of Engineers at the Waterways Experiment Station.(10,11) The Corps of Engineers program has the capability of a true nonlinear response model but has not been used on a large number of projects and does not have a batch mode processor or data management software to facilitate its use in a production mode. Conversely, MODCOMP can be used on a production mode basis, and its convergence is reasonably fast, but it is a quasi-nonlinear response model. Stated differently, for the same load level, the modulus of a layer does not vary horizontally or vertically within that layer in accordance with changes in stress state. The layer modulus is only varied by stress state between load levels and sensor locations.

Standardized procedures and guidelines are available to assist in this back-calculation process. Some of these include procedures written under the SHRP program ASTM D5858 and the one documented in report FHWA-RD-97-076, Design Pamphlet for the Back-Calculation of Pavement Layer Moduli in Support of the 1993 AASHTO Guide for the Design of Pavement Structures.(1,5,6) All of these programs and guidelines are based on the use of elastic layer response programs. Viscoelastic or elastoplastic response programs have not been used in a sufficient number of projects to substantiate the reliability and adequacy of their use. Therefore, selection of the back-calculation software and procedures were confined to those that are based on the use of elastic layer theory.

2.1.3 Summary. Although many analyses of deflection data can be undertaken, the studies currently underway within the LTPP program, by definition, require highly focused efforts accomplishable within a short period of time. Thus, the study for back-calculating layer moduli looked only at the computation of material properties using existing software that has the capability to analyze massive amounts of deflection data in a reasonable time frame. These requirements basically restrict the back-calculation methodology to a static load analyzed with a linear or nonlinear elastic response model.

2.2 Selection Factors

It should be understood that many different software packages can be used to calculate the elastic modulus of pavement layers and subgrade soils from deflection basins, as demonstrated in report FHWA-RD-97-086, Back-Calculation of Layer Moduli of LTPP GPS Sites.(3) Many of the software packages that have the same type of response model have, in fact, provided reasonably consistent results. The following lists the factors considered for evaluating the different back-calculation programs based on elastic layer theory and were found to be applicable to pavement diagnostic studies and the requirements noted above.

2.3 Selection of Software -- MODCOMP4

The evaluation focused on two areas: (1) the ability and accuracy of the software to determine elastic-layer moduli of the pavement materials and soils and (2) operational characteristics -- ease of use mode, flexibility in analyzing a wide range of pavement structures (flexible and rigid), and material response models. Much of the information used for the software selection process was obtained from previous comparisons and evaluation studies of the software. (See references 3, 7, 12, and 13.)

The primary reasons that MODCOMP4 was selected as the program for calculating the elastic properties of pavement structural layers and subgrade soils from FWD deflection basins measured at the LTPP SMP, GPS, and SPS test sections are listed below.

  1. First and foremost, the use of a nonlinear constitutive equation to represent the response of unbound pavement materials and soils was believed to provide added value and to result in a higher number of reasonable or adequate solutions. The use of linear elastic response models can be used to estimate the nonlinear properties but require more steps in the back-calculation process. A software package that has the capability to do both is believed to have increased flexibility in its overall usage on this project and in future projects (such as National Cooperative Highway Research Program [NCHRP] Project 1-37A) and should be more adaptable to a wider range of conditions and materials. More importantly, a common opinion of a group of experts was that successful back-calculation using much of the LTPP deflection data would require use of nonlinear characterization.(14) As noted above, MODCOMP was believed to have added value and was selected because it considers different nonlinear constitutive equations in determining the elastic response properties.
  2. The process for back-calculating elastic properties only looked at the computation of material and soil properties using existing software packages that have the capability to analyze large amounts of deflection data in a reasonable time frame. As noted above, this requirement restricted the back-calculation methodology to a static load analyzed with a linear or nonlinear elastic response model. MODCOMP was selected because its convergence is reasonably fast, and it can be used in a production basis to mass-calculate elastic properties from hundreds of deflection basins measured at a site over time.
  3. Elastic layer theory was used to calculate a deflection basin under a specific load for a known set of layer moduli. MODCOMP was used to calculate the layer modulus from these deflection basins. The difference between the calculated and target (or known) modulus was used to estimate the accuracy of the program and to establish a practical error term for the matched deflection basin. The RMS error for each solution and the percent difference between the calculated and the target or known moduli were the parameters considered in the evaluation. The RMS error was found to vary from 0.1 to 1 percent, which is considered very acceptable.
  4. Another very important fact: The revisions and changes made to the MODCOMP program to correct a problem identified in the 1991 SHRP review study significantly improved the stability and success in achieving reasonable solutions.(7) The probability of success for the limited number of basins reviewed in the preliminary study was in excess of 90 percent, which was considered good for the diverse conditions used in the examples.
  5. The majority of the inputs needed to execute MODCOMP in a batch mode process are readily available in the LTPP database.

2.4 Program Accuracy

One of the most difficult questions to be answered is, How accurate are the layer moduli back-calculated from measured deflection basins? In reality, this is an impossible question to answer conclusively for real basins, but it must be addressed to promote confidence in the computed results. This section describes the procedure used to estimate the accuracy of the results.

A small experiment was conducted to check out the capabilities of MODCOMP using LTPP data and the procedure used for classifying deflection data in terms of load response patterns and deflection basin shapes.(1,4) Elastic layer theory (specifically, ELSYM5)1 was used to calculate a deflection basin under a specific load and a known set of layer moduli. The computed deflection basin was considered a measured basin and input into MODCOMP. The program was then used to back-calculate the layer moduli for each layer.

Note: 1ELSYM5 Version 1.1 , corrected in 1993, was the version used for these calculations.

These calculated layer moduli were compared to the known values that were originally used to calculate the deflection basin. The differences between the calculated and target (or known) moduli were used to estimate the accuracy of the program and to establish a practical error term for the matched deflection basins. This is a process that has been used previously by the authors in evaluating the results from other back-calculation programs.

To perform this preliminary analysis, only the linear elastic layer portion of the MODCOMP4 program was used. The nonlinear part was not included in this part of the study. A factorial was developed to cover the expected range of the conditions and pavement structures that are included in the LTPP database. Table 2 shows the factorial that was used for this effort.

Back-calculation was conducted only for those cells where an "X" appears in the factorial (see table 2). These cells were selected because they represented the extreme conditions. The stiffer layer for the "inverted pavements" was established as an asphalt-treated base, 25.4 centimeters in thickness, with an elastic modulus of either 689 MPa or 13,790 MPa. The unbound granular bases for the "conventional pavements" were all 25.4 centimeters in thickness and had an elastic modulus of 2,007 MPa.


Table 2. Factorial used to estimate the theoretical accuracy
of the MODCOMP4 solutions.

Type of
Pavement
Subgrade
Strength
Depth To
A Rigid Layer
Surface Layer:
Moduli, ksi,
6,895 MPa;
Thickness 15.2 cm
Surface Layer:
Moduli, ksi,
6,895 MPa;
Thickness 7.6 cm
Surface Layer:
Moduli, ksi,
2069 MPa;
Thickness 15.2 cm
Surface Layer:
Moduli, ksi,
2069 MPa;
Thickness 7.6 cm
Conventional Strong, 207 MPa Infinite X      
Conventional Strong, 207 MPa Shallow, 1.8 m     X  
Conventional Weak, 69 MPa Infinite X X    
Conventional Weak, 69 MPa Shallow, 1.8 m   X   X
Inverted1 Strong, 207 MPa Infinite X X   X
Inverted1 Strong, 207 MPa Shallow, 1.8 m        
Inverted1 Weak, 69 MPa Infinite     X  
Inverted1 Weak, 69 MPa Shallow 1.8 m     X X

1Inverted, in this case, means a pavement with a stiffer layer (higher modulus material below or supporting a weaker material in the pavement structure).


The RMS error for each solution and the percent difference between the calculated modulus and the target modulus were the study results considered in the evaluation. The RMS error was found to vary from 0.1 to 1 percent, which is considered very acceptable. The following are the average and standard deviations of the RMS error.

Accuracy of the computed results was found to vary with layer depth, which has been observed in other studies. The following summarizes the differences (percent of the target value) as a function of layer within the pavement structures.


Layer Back-Calculated E/Target E Mean Back-Calculated E/Target E Mean Standard Deviation
Surface (7.6 or 15.2 cm in thickness) 1.191 0.5877
Stabilized Base Layers 1.101 0.3427
Unbound Granular Base/Subbase Layers 1.078 0.2322
Subgrade Layers 1.008 0.0420

As shown above, on the average, the back-calculated layer moduli are slightly greater than the target or known values used to calculate the deflection basins. In summary, use of linear elastic back-calculation programs may result in errors on the order of 20 percent for surface layers, 10 percent for stabilized base or subbase layers, 8 percent for unbound granular base/subbase layers, and 1 percent for subgrades.


3.0 BACK-CALCULATION PROCESS

The process that was used to back-calculate the elastic properties of the pavement layers and subgrade soils are summarized in this part of the report. The overall operational process follows the procedure outlined by Von Quintus and Killingsworth in publication FHWA-RD-97-076 and the Instructional Guide for Back-Calculation and the Use of MODCOMP.(1,10) Appendix A of this report is the User's Guide, which defines all of the steps, programs and decision functions included in this process. Figure 1 shows a simplified flow chart of the major steps and decisions used in the back-calculation process. As shown, there are six major steps:

  1. Extraction of data needed for back-calculation.
  2. Classification of test section and deflection basin data.
  3. Determination of inputs for MODCOMP.
  4. Trial computations.
  5. Mass back-calculation of deflection basin data for the site.
  6. Extract and store results in summary tables.

This section of the report discusses and defines three of the six steps in more detail than in appendix A -- Test Section Classification, Determination of Inputs, and Trial Computations.


Figure 1. Flow chart showing the major steps and decisions
used in the linear elastic back-calculation process.

Click on D for alternate textD

3.1 Test Section Classification

The deflection data measured at each site were used to classify the test sections with similar deflection characteristics. Three deflection-site classifications were used: (1) a load-response classification, (2) a deflection basin classification, and (3) a test section uniformity classification. Each is defined in the following paragraphs.

3.1.1 Load-Response Classification. The first step in the test section classification was to determine the load-response characteristics of the pavement structures included in the LTPP database. In general, there are three types of load-response categories:

  1. Linear elastic pavement structures -- Pavement structures having a linear relationship between the load and measured deflections at the center of the loading plate. The slope of the load-response curve can further divide these groups of pavements. Figure 2 shows plots of deflections versus load for some of the GPS sections that display linear elastic load-response behavior.
  2. Deflection-hardening pavement structures -- Pavement structures where the center-load deflection increases with load but at a decreasing rate. The tangent slope of the load response curve can further divide this group of pavements. Figure 3 shows some of the GPS sections with the assumed deflection hardening load-response behavior.
  3. Deflection-softening pavement structures -- Pavement structures where the center-load deflection increases with load but at an increasing rate. The tangent slope of the load-response curve can further divide this group of pavements. Figure 4 shows some of the GPS sections with the assumed deflection-softening load-response behavior.

Figure 2. Graphical illustration of the definition for the linear response category.

Figure 2. Graphical illustration of the definition for the linear response category. The load (kiloPascals) is graphed on the horizontal axis and Center-Load Deflection (microns) on the vertical axis. The graph for Site 41006 is a straight line that goes from the origin to 150 microns at a load of 1000 kiloPascals. Site 472008 is a straight line that goes from the origin to a deflection of about 150 microns at a load of 1150 kiloPascals. Site 481130 is also a straight line, starting near the origin and ending at a deflection of about 575 microns at a load of 1000 kiloPascals.

Figure 3. Graphical illustration of the definition for the deflection-hardening response category.

Figure 3. Graphical illustration of the definition for the deflection-hardening response category. The load (kiloPascals) is graphed on the horizontal axis and Center-Load Deflection (microns) on the vertical axis. The graph for Site 41017 is a straight line that goes from 100 microns (0 kiloPascals) to 575 microns (1000 kiloPascals). For Site 473075 the graph is a straight line that goes from 350 microns (0 kiloPascals) to 950 microns (1000 kiloPascals). Site 481113 is also a straight line goes from 150 microns (0 kiloPascals) to 550 microns (1000 kiloPascals).

Figure 4. Graphical illustration of the definition for the deflection-softening response category.

Figure 4. Graphical illustration of the definition for the deflection-softening response category. The Load (kiloPascals) is graphed on the horizontal axis and Center-Load Deflection (microns) on the vertical axis. The graph for Site 41016 is a straight line that goes from 0 microns at a load of about 150 kiloPascals to 225 microns at a load of 1000 kiloPascals. Site 81057 is a straight line that goes from 0 microns at a load of about 100 kiloPascals to a deflection of about 950 microns at a load of 1000 kiloPascals. Site 531801 is also a straight line, starting at 0 microns (300 kiloPascals) and ending at 200 microns at 1000 kiloPascals.

In determining the load-response behavior of these test sections, a linear model was used (i.e., a linear relationship between load and the deflection measured at the center of the loading plate). The slope and intercept were determined for each set of deflection data, and the multiple correlation coefficient, R2, was determined for these linear relationships for each set of deflection data. The criteria used to characterize the load-response behavior was somewhat judgmental, but was defined by R2 and the deflection intercept plus geometric considerations. The criteria used to establish the load-response behavior are listed below.

Although the intercept values are arbitrary, the plot must go through the origin for zero-load so that the shape of the response curve is determined. If deflection decreases with load, deflection hardening must be occurring in one or more layers. Conversely, deflection increases with increasing loads indicate deflection softening.

It should be noted that the initial characterization for categorizing the GPS test sections was based only on the center-load deflections. A more complete subdivision of the sections would use some of the other deflections measured at specific radial distances from the load plate (for example, 305, 610, and 1,524 mm). This would help in selecting different constitutive equations to be used for different layers and material types. The majority of the LTPP tests sections were found to have linear elastic load-deflection response characteristic; very few of the sections exhibit deflection-softening behavior, as expected.

3.1.2 Deflection Basin Classification. To evaluate the different types of deflection basins, all measured basins were first normalized to the center-load deflection, which is directly under the load. These normalized deflection basin data were divided into four types of basins. This is the same procedure followed by Von Quintus, et al., as documented in report FHWA-RD-97-076.(1) Examples of basins for these different categories are shown in figures 5-8 and defined below:


Figure 5. Typical normalized deflection basin.

Figure 5. Typical normalized deflection basin. The figure shows a graph of 10 sites. The horizontal axis shows Distance (millimeters) and Sensor Location, and the vertical axis shows Normalized Deflection Ratio. For sites 421613, 15008, 87035, and 175020 the graph is nearly a straight line, with all graphs starting at a Normalized Deflection Ratio of -1 (Sensor Location 0, Distance 0 millimeters) and increasing to -0.65, -0.5, -0.4, and -0.4, respectively (Sensor location 1524, Distance 1500 millimeters). For sites 471029, 481077, 11001, 851803, and 121370 the graphs increase steadily until they reach sensor location 305 (300 millimeters distance), where they begin to level off and increase more slowly, ending in ratios between -0.2 to -0.1 (Sensor Location 1524, Distance 1500 millimeters).

Figure 6. Type I normalized deflection basin.

Figure 6. Type I normalized deflection basin. The figure shows a graph of 10 sites. The horizontal axis shows Distance (millimeters) and Sensor Location, and the vertical axis shows Normalized Deflection Ratio. Sites 185518, 53011, and 404166 have graphs that are nearly straight lines that start at Deflections of -1 (Sensor Location 0, Distance 0 millimeters) and increases steadily to Deflections between -0.7 and -0.4 (Sensor Location 1524, Distance 1500 millimeters). Site 537409 has Deflections starting at -1 (Distance of 0 millimeters), increasing sharply to -0.6 (200 millimeters), dropping to -1.3 (300 millimeters) and -1.6 (600 millimeters), increasing sharply to -0.1 (900 millimeters), and staying flat at -0.1 (1500 millimeters). Site 404160 has Deflections starting at -1 (Distance of 0 millimeters), decreasing steadily to -1.2 (450 millimeters), increasing sharply to -0.2 (600 millimeters), and staying flat at -0.2 (1500 millimeters). Sites 313018 and 63013 both start at Deflections of -1 (0 millimeters) and drop to -1.2 and -1.6, respectively (900 millimeters) and then increase sharply to -0.1 and -0.3 (1500 millimeters). Sites 289030 and 67456 stay in the deflection ranges of -1.5 to -0.9 and -1.1 to -0.9, respectively. Site 463013 has deflections that remain at -1 (0 to 600 millimeters) and then increases to -0.1 (900 millimeters), holding steady at -0.1 (1500 millimeters).

Figure 7. Type II normalized deflection basin.

Figure 7. Type II normalized deflection basin. The figure shows a graph of 9 sites. The horizontal axis shows Distance (millimeters) and Sensor Location, and the vertical axis shows Normalized Deflection Ratio. Sites 41002, 307088, 41017, 566029, 41036, and 567775 show a similar trend of starting at -1 (0 millimeters) and increasing relatively steadily to about -0.5 to -0.1 (450 millimeters) and then flattening out to -0.2 to 0 (1500 millimeters). Sites 54021, 63005, and 204067 have deflections that increase in nearly a straight line from -1 (0 millimeters) to -0.3, -0.2, -0.4, respectively (1500 millimeters).

Figure 8. Type III normalized deflection basin.

Figure 8. Type III normalized deflection basin. The figure shows a graph of 6 sites. The horizontal axis shows Distance (millimeters) and Sensor Location, and the vertical axis shows Normalized Deflection Ratio. Sites 13998, 63024, 189020, 295473, 415021, and 429027 have deflections that increase relatively steadily from -1 (0 millimeters) to around -0.35 to -0.65 (1500 millimeters).

In general, Type I and III deflection basins occur most frequently for PCC surfaced pavements. It is believed that these deflection basins may be characteristic of those areas with voids, a loss of support, a severe thermal gradient causing curling of the PCC slab, and/or a combination of these conditions. Conversely, Type II deflection basins occur most frequently for dense-graded asphalt concrete surfaced pavements. As stated above, all Type I and III basins were excluded from the actual back-calculation process.

3.1.3 Test Section Uniformity Classification. The final step included in the classification of each LTPP test section was to determine the uniformity or variability of the deflections measured along the test section. The uniformity of each LTPP test section was classified into one of four different categories, listed below.

  1. Uniform load response measured throughout the test section -- the coefficient of variation (COV) for the test section, using the center-load deflections, is less than 15 percent (figure 9).
  2. Extensive variability measured along the test section -- the COV for a test section or subsection, using the center-load deflections, is greater than 25 percent (figure 10).
  3. Drift exists within the test section -- the deflections are consistently increasing or decreasing from one end of the site to the other (figure 11).
  4. There is significant difference in the load response between each end of the test section, suggesting that the test section may need to be subdivided into two segments (figure 12).

The site-uniformity characterization defines whether the test section can be considered initially as one uniform section or whether the test section should be treated as multiple sections. Appendix B of this report lists those LTPP test sections that were subdivided into two subsections.

Those test sections with significant differences in the load response between the ends of the test sections (figure 12) were flagged and subdivided into two subsections. The pavement layer thickness was then checked to determine whether different layer thicknesses or material types exist between the two ends of the section. If significant differences were found to exist, that section was subdivided based on the differences in the deflection data.


Figure 9. Example of a test section with uniform deflections (test section 081053, 15 June 1994) or low variability in the measured deflections.

Figure 9. Example of a test section with uniform deflections (test section 081053, 15 June 1994) or low variability in the measured deflections. The figure shows a graph of 7 sensors. The horizontal axis shows Stations (meters) and the vertical axis shows Deflections (microns). The Deflections of all seven sensors appear in nearly horizontal lines, each varying by less than 50 microns as they go from Stations at -10 meters to 60 meters.

Figure 10. Example of a test section with highly variable deflections (test section 041024, 13 June 1996) or high variability in the measured deflections.

Figure 10. Example of a test section with highly variable deflections (test section 041024, 13 June 1996) or high variability in the measured deflections. The figure shows a graph of 7 sensors. The horizontal axis shows Stations (meters) and the vertical axis shows Deflections (microns). The Deflections of sensor 7 is nearly horizontal, varying by only 50 microns going from Stations at 90 m to about 160 m. The graphs of sensors 1-6 show much more variability, with sensor 1 showing the most variability (about 150 microns) and sensor 6 showing the least (about 150 microns) as they move from Stations at 90 meters to about 160 meters.

Figure 11. Example of a test section with drift (test section 040114, 12 June 1996) or where the deflections consistently change from the approach end to the leave end, defined as drift.

Figure 11. Example of a test section with drift (test section 040114, 12 June 1996) or where the deflections consistently change from the approach end to the leave end, defined as drift. The figure shows a graph of 7 sensors. The horizontal axis shows Stations (meters) and the vertical axis shows Deflections (microns). In this graph, the Deflection recorded by all seven sensors shows a consistent, gradual increase as it moves from 90 meters to around 160 meters.

Figure 12. Example of a test section with an abrupt change in the measured deflections (test section 040113, 16 August 1995) between the approach and leave ends.

Figure 12. Example of a test section with an abrupt change in the measured deflections (test section 040113, 16 August 1995) between the approach and leave ends. The figure shows a graph of 7 sensors. The horizontal axis shows Stations (m) and the vertical axis shows Deflections (microns). In this graph, the Deflection recorded by all seven sensors shows a relatively horizontal graph going from 90 to 130 m, then jumps by 50 to 250 microns when it reaches around 130 m. After that deflections level off and begin to slowly decrease as it moves to 160 m.

3.2 Determination of Inputs

Appendix A provides a detailed discussion on determination of the inputs to MODCOMP4 for each test section. Table 3 summarizes the sources for the input data needed for the back-calculation of layer modulus from deflection basin data for the LTPP sites. The following paragraphs overview some of the more important input issues.


Table 3. Identification and summary of the data tables used to obtain the input data used for the back-calculation process of the LTPP sites.

Input Parameter Data Source or Table Remarks
Deflection Basins MON_DEFL_DROP_
DATA
Average deflection basin for each drop height and test point.
Pavement Temperature MON_DEFL_TEMP_
VALUES_DEPTH
Mid-depth temperatures used to calculate the seed modulus of HMA layers using the dynamic modulus equation.(15)
Pavement Cross-Section, Material Type and Layer Thickness TST_L05B The pavement cross-section used to back-calculate layer modulus.
Pavement Cross-Section, Material Type and Layer Thickness TST_L05A The pavement cross-section used to back-calculate layer modulus.
Asphalt Viscosity INV_PMA_ASPHALT Data elements needed in the dynamic modulus equation.
HMA Aggregate Gradation TST_AG04 Data elements needed in the dynamic modulus equation.
HMA Aggregate Gradation INV_GRADATION Data elements needed in the dynamic modulus equation.
Air Voids TST_AC02 Data elements needed in the dynamic modulus equation.
Air Voids TST_AC03 Data elements needed in the dynamic modulus equation.
Air Voids INV_PMA_ORIG_MIX Data elements needed in the dynamic modulus equation.
Density, HMA and ATB TST_AC02 Data elements needed to estimate overburden pressure -- nonlinear solutions.
Density, Unbound Layers and Soils TST_ISD_MOIST Data elements needed to estimate overburden pressure -- nonlinear solutions.
Number of Subgrade Layers TST_SAMPLE_LOG Data element used to subdivide subgrade.
Depth to Water Table TST_SAMPLE_LOG Data element needed to subdivide subgrade.
Depth to Apparent Rigid Layer TST_SAMPLE_LOG  
Poisson's Ratio Table 4 Default values.
Lateral Earth Pressure Coefficient Table 4 Default values -- nonlinear solutions.

3.2.1 Deflection Basins. The deflection data were extracted from data table MON_DEFL_DROP_DATA of the LTPP database. MODCOMP limits the number of possible basins for one pavement structure to eight. In the LTPP program, four individual drops at four load levels are collected at each point (refer to table 1). The average deflection basin and applied load for each drop height were used for the computations.

A few pavements were studied using two measured basins at each load level versus using the average of the measured basins at each load level. Based on the limited comparisons completed, use of the average deflection basin at each drop height provided back-calculated moduli comparable to those averaged from separate back-calculations for each basin. The differences were generally within the simulation error previously discussed.

3.2.2 Pavement Cross-Section. All layer thickness and material types were extracted from data tables TST_L05B or TST_L05A in the Information Management System (IMS) database. The initial pavement cross-section (combination of pavement layers and subgrade soil stratas) used in the back-calculation process was obtained from previous work completed by Von Quintus and Killingsworth.(3) When the material types and/or layer thicknesses for the approach and leave ends of a test section were significantly different, as defined in Appendix A, that test section was flagged and put aside for further review. The longitudinal variation in the deflection basin data was reviewed carefully to subdivide the test section into two segments with different cross-sections, as noted above.

3.2.3 Material Properties and Temperatures. Various material properties and the temperature during FWD testing were extracted from different data tables in the database. These data were used to calculate the starting or "seed" modulus of the hot mix asphalt (HMA) layers for the linear solutions using the Witczak dynamic modulus regression equation.(15) The mid-depth temperatures during FWD testing were extracted from data table MON_DEFL_TEMP_VALUES_DEPTHS. The material properties were extracted from data tables INV_PMA_ASPHALT (asphalt viscosity), TST_AG04 or INV_GRADATION (gradation), and TST_AC02 and TST_AC03 (bulk and Rice specific gravities to calculate air voids) or INV_PMA_ORIG_MIX (air voids).

3.2.4 Layer-Sensor Assignment. MODCOMP assigns specific sensors to certain layers. Initially, the default layer-sensor assignment was used. In many cases (10 to 25 percent of the test sections), however, the initial layer-sensor assignments were changed manually to reduce the RMS error. One of the primary reasons for changing the automatic layer-sensor assignments was that the computed deflection basin was insensitive to a specific layer. For those cases, Young's modulus was simply assumed for the insensitive layer, and the sensor was reassigned to an adjacent layer. The assumed value or modulus was based on the initial computations and previous laboratory test results for similar materials.

3.2.5 Poisson's Ratio. Poisson's ratio is not available in the LTPP database. This material property was assumed for each type of material and based on previous guidelines. The values of Poisson's ratio, used for different materials and soils, are given in table 4.

3.2.6 Multiple Subgrade Layers. A 6-m boring was drilled adjacent to the pavement at all LTPP sites. This boring is defined as a shoulder probe and used to identify the different soil strata and determine the depth to a rigid layer and water table. The soil profile obtained from the 6-m shoulder probe was used to identify whether the subgrade should be divided into two or more layers, and if so, at what depth. The shoulder probes were obtained from data table TST_SAMPLE_LOG in the IMS. In general, the subgrade was divided into multiple layers when the following conditions were found: (1) significantly different soils were encountered, (2) water or very wet soils were encountered, and (3) extremely stiff or hard soils were noted on the boring log.

3.2.7 Depth to an Apparent Rigid Layer. The soil profile or shoulder probe boring also was used to determine the depth to an apparent rigid layer, and this depth was compared to the calculated depth using the latest revision of MODULUS.(16) In many cases, lower RMS errors were calculated when no apparent rigid layer was used in the back-calculation process. The depth to an apparent rigid layer was kept constant when that depth was determined from the boring log. Appendix B identifies when a rigid layer was used and at what depth for the test section.

3.2.8 Constitutive Equation. MODCOMP4 has the capability to consider different constitutive equations to model the response of the pavement materials and subgrade soils. These constitutive equations are summarized in table 5. Each has been used with some degree of success, and the results from a previous study indicate that the constitutive model used has a definite influence on the results. Obviously, it is desirable to use the same model for all unbound pavement materials and subgrade soils so that the results can be compared across the board. Unfortunately, none of the models used consistently converged to an acceptable solution for all pavements included in this study.

The bulk stress and minor principal stress models (refer to table 5) appear to be appropriate for use for both fine-grained and coarse-grained materials converged with fewer iterations and resulted in lower RMS errors. The second stress invariant, vertical stress, and major principal stress models (refer to table 5) were abandoned because of the number of solutions that did not converge when these equations were used. Where convergence was not achieved, different constitutive equations were used in follow-on computations to achieve more desirable results. The octahedral shear stress and Cornell constitutive models were not used.

The constitutive equation recommended for use by Von Quintus from a previous study (defined as the so-called Universal Model) uses three parameters, or k-values.(4) That model is unavailable in MODCOMP, and there is an insufficient number of sensors in the LTPP database to support its use to calculate nonlinear elastic properties. Thus, the two equations referred to in the AASHTO Design Guide (the bulk stress and deviator stress models) were used for the first set of runs.(17) The bulk stress model was used for coarse-grained soils, and the deviator stress model was used for fine-grained soils. The minor principal stress model was used in the back-calculation process when the first two did not result in any solution. These three equations are listed below and in table 5.


Table 4. "Default" material properties for different materials, if not included in the LTPP database.

Material Description Material Code Median Wet Density, pcf Median Dry Density, pcf Percent Median Moisture Content Poisson's Ratio At-Rest Earth Pressure Coefficient
Cohesive Soil: High Plasticity Clay Soils 101-137 129 109 18.3 Soft 0.45, Stiff 0.40, Hard 0.30, (0.30 - 0.48) Soft 0.80, Stiff 0.70, Hard 1.00
Cohesive Soil: Low Plasticity Clay Soils 101-137 129 109 18.3 Soft 0.45, Stiff 0.40, Hard 0.40, (0.30 - 0.48) Soft 0.80, Stiff 0.70, Hard 1.00
Silty Soils 141-148 130 113 15.4 Above Water Table 0.35, Below Water Table 0.40, (0.30 - 0.45) Above Water Table 0.70, Below Water Table 0.70
Sandy Soils 201-217 129 116 10.7 Above Water Table 0.35, Below Water Table 0.40, (0.20 - 0.40) Above Water Table 0.60, Below Water Table 0.80
Gravelly Soils 251 - 267 137 123 9.6 0.35 0.50
Lime Treated Soils 338 124 100 24.3 Percent Lime >4 0.30, Percent Lime < 40.35, (0.20 - 0.40) *
Sand Base 306-309 132 124 5.8 0.35 (0.20 - 0.40) 0.70
Gravel Base and Crushed Stone Base 302-305, 337 136 129 5.2 0.35 (0.20 - 0.40), 0.30 (0.20 - 0.40) 0.60, 0.50
Asphalt Treated Base 321-330 118 --- --- Temp: <40°F 0.25, Temp: 40-100°F 0.30, Temp: >100°F 0.35, (0.20 - 0.40) *
Cement Treated Base 331-335 125 --- --- 0.20 (0.15 - 0.25) *
Asphalt Concrete Mixtures (Surface, Bases)   --- --- --- Temp: <40°F 0.25, Temp: 40-100°F 0.30, Temp: >100°F 0.40, (0.15 - 0.45) *
* These layers or materials should be treated initially as linear-elastic material.

Table 5. Layer model descriptions.(10)

Model Description
0 Linear model of the form E = constant. In general, most asphalt concrete, Portland cement concrete, and treated (i.e., stabilized) soil materials conform to this constitutive relationship.
1 Bulk stress model of the semi-logarithmic form E = k1 exp (S*k2), where

S = Theta = Sigmaz + Sigmat + Sigmar

(Sigmaz = vertical stress, Sigmat = tangential stress, and Sigmar = radial stress at the mid-depth of the layer). The parameter Theta is known as the bulk stress or as the first stress invariant. It was originally developed as a log-log model, but since Theta can be either positive or negative, it works better as a semi-log model. It is used for untreated, coarse-grained base and subbase gravels. In terms of overall usage, the bulk stress model is one of the most popular and most commonly used models.

2 Deviator stress model of the semi-logarithmic form E = k1 exp (S*k2), where

S = Theta = Sigma1 - Sigma2

the difference between the major principal stress and the minor principal stress at the mid-depth of the layer. It can be used for fine-grained untreated materials. Although this model differs from the traditional linear deviator stress model (E = k1S + k2), it seems to work very well.

3 Minor principal stress model of the semi-logarithmic form E = k1 exp (S*k2), where

S = Sigma3

The parameter Sigma3 is the minor principal stress, which can either be tensile (negative) or compressive (positive). Also known as the "confining stress model," the Sigma3 model was commonly used before the bulk stress model was conceived. It was originally developed as a log-log model, but since Sigma3 can be either positive or negative, it works better as a semi-log model. It can be used for coarse-grained, untreated materials.

4 Second stress invariant model of the semi-logarithmic form E = k1 exp (S*k2), where

S = J2/(1 + Tauoct)

The parameter J2 is the second stress invariant.

J2 = Sigmaz Sigmat + Sigmat Sigmar + Sigmat Sigmaz + Taurz2

Where Sigmaz, Sigmat and Sigmar are as defined in Model 1, and Taurz is the shear stress in the rz plane, Tauoct is the octahedral shear stress, which is similar to a root-mean-square deviator stress.

Tauoct = Alpha[(Sigmaz - Sigmat)3 + (Sigmat - Sigmar)2 + (Sigmar - Sigmaz)2 + 6 Taurz2]½

This constitutive relationship has been found to be useful for both base course and subgrade materials, and the coefficient k1 appears to be primarily a function of material density and soil moisture tension. It was originally developed as a log-log model, but since J2 can be negative it works better as a semi-log model.

5 Octahedral shear stress model of the semi-logarithmic form E = k1 exp (S*k2), where

S = (1 + Tauoct)

This relationship has been found to be useful for fine-grained materials and for frozen soils, and the coefficient k1 appears to be a function of soil temperature and moisture content. It was originally developed as a log-log model but since toct can be negative, it works better as a semi-log model.

6 Vertical stress model of the log-log form E = k1 S k2, where

S = Sigmaz

The parameter Sigmaz is the vertical stress, which is always compressive (i.e., positive). This model is very similar in some respects to Model 7. The difference is that the direction of Sigmaz is always vertical, while the direction of Sigma1 in Model 7 rotates with distance from the load.

7 Major principal stress model of the log-log form E = k1 S k2 (k0)k3, where

S = Sigma1

The parameter Sigma1 is the major principal stress, which is generally always compressive. Dynatest uses this model in their ELMOD back-calculation program. It is included here for completeness. As used in ELMOD, the overburden stress is not included in the major principal stress, so the overburden stress is not included in MODCOMP3 either.

8 Cornell constitutive model of the log-log form E = k1 S k2 (k0)k3, where

S = (Thetao2 + Thetap2)0.22(1-P200) (1 + Tauoct)-0.34P200

The parameter Thetao is the initial bulk stress, due only to overburden stress. The parameter Thetap is the bulk stress at peak load, including both the overburden stress and the load stress. The parameter Tauoct is the octahedral shear stress at peak load, which also includes the overburden stress.

The parameter P200 is the percentage of the materials gradation that passes the number 200 sieve, and it is a required input for both known and unknown models. It is entered on data file line 5 as the K@(I) parameter (see Table 3 of the MODCOMP User's Manual). The parameter k0 is the lateral earth pressure coefficient, which represents the anisotropy of the overburden stress. The coefficient k1 has been found to be a function of material density and soil moisture content, along with other material properties. The exponent k3 = -0.69, is treated as a constant in the program.

The Cornell model has been developed for use with both coarse-grained and fine-grained untreated materials. It has seen limited use, mainly with the glacial materials of the Northeastern U.S.


3.2.9 Material Densities. Wet and dry densities for the pavement layers and subgrade are included in the LTPP database for many of the test sections and were extracted from data table TST_ISD_MOIST. When the densities were available, they were used in the back-calculation process. However, when the densities were unavailable for a specific layer and site, the median density for that type of material, calculated from all other data, was used. These default values are listed in table 4.

3.2.10 Lateral Earth Pressure. MODCOMP has the capability to consider the effect of overburden pressures in back-calculating nonlinear elastic properties. The lateral earth pressure coefficient (Ko) is used to compute the contribution to the lateral stress at different depths from the overburden pressure. Lateral earth pressure coefficients, however, are not included in the LTPP database. A range of basins with varying conditions was used to evaluate the sensitivity of Ko to obtain reasonable solutions (low RMS error). For reasonable variations (plus or minus 25 percent) around the default values included in MODCOMP, it was found that Ko does not have a significant effect on the calculated results. Because there is no reliable method to estimate the actual Ko value to be used for each section, the values given in table 4 were used and are materials dependent. For a linear elastic solution, Ko has no effect on the back-calculated moduli.

3.3 Trial Computations -- Execution of Back-Calculation Process

The procedure used to back-calculate the elastic properties of each layer is not fully automated but is an iterative process and requires engineering judgment. To begin the computation process, a limited number of points are randomly selected along the test section and the elastic properties calculated for those basins. The reason for starting with a limited number of basins is to make any necessary revisions to the inputs for reducing the RMS error to an acceptable level in a short period of time, prior to initiating the mass back-calculation of all deflection basins measured at a test section.

The number of random test points selected were generally in the range from four to eight and depended on the amount of variation of the measured deflection basins within the subsection. MODSHELL was used to analyze the basins measured at those random test points. MODSHELL is a menu-driven program that enables the creation of new data files, editing of existing data files, processing of files with MODCOMP4, and viewing the output files on the screen.

The linear elastic module of MODCOMP was used to calculate the elastic modulus, and those results were used as the starting point for the nonlinear analysis. Results from the initial solutions were reviewed to determine whether the production runs should be executed or changes should be made to the inputs before proceeding to the production runs. The decision on whether to proceed was based on, in order of importance:

  1. Magnitude of the RMS errors -- revisions were made until the RMS error was reduced to 2 percent or less, or to the lowest possible value.
  2. Computed elastic modulus (Young's modulus) -- revisions were made if high and low moduli were calculated for the type of materials identified. High and low values for specific materials were based on judgment using the seed values designated in ASTM D5858 and the procedure recommended by Von Quintus and Killingsworth, as guidance.(1,3) If the change made to the input resulted in higher RMS errors or a larger difference between the calculated and measured deflection basin, the revision was not implemented in the mass back-calculations for that test section.
  3. Changes in the elastic modulus from one iteration to the other -- revisions were made when the solutions (elastic modulus for each layer) did not begin to converge within the specified number of iterations, generally 10.

If the RMS errors were large (greater than 2 percent) or the calculated elastic moduli were questionable for the type of material, the inputs were checked and adjustments were made to the layer combinations, layer-sensor assignments, and/or the use or omission of an apparent rigid layer. MODSHELL was used to recalculate the elastic moduli with those changes. This iterative process was continued until a "reasonable or acceptable" solution was achieved. As stated above, a reasonable or acceptable solution was one with an RMS error less than or equal to 2 percent with elastic moduli that were considered typical for the material type. These revisions to the input parameters were then used for the mass back-calculation of elastic layer modulus along that test section.

For some test sections, extremely high or low moduli were computed for one or more layers in the pavement structure with good RMS error values. When this occurred, changes were made to the inputs, and MODSHELL was used to recalculate Young's modulus, as noted above. If the final RMS error was less than 2 percent for the trial run that resulted in the extremely high or low moduli and much greater than 2 percent for the other trial runs, the trial run resulting in the high or low moduli was used for the production runs.


4.0 RESULTS FROM COMPUTATION OF ELASTIC PROPERTIES

This section of the report provides an overview and brief summary of some of the pertinent details from the calculation of elastic properties using MODCOMP4. A detailed analysis and comparison of the results was beyond the scope of work for this study. Thus, the elastic properties calculated from the deflection basin were reviewed to determine whether the results seemed reasonable. Appendix C of this report tabulates the median Young's modulus calculated for different materials and includes histograms of the calculated Young's modulus for different materials and soils.

4.1 Some Basic Facts from the Back-Calculation

The magnitude of the effort to complete the back-calculation of elastic properties for all deflection basin data in the LTPP database for both the flexible and rigid pavements included:

The deflection data were analyzed using more than six PC's, ranging from a 400-MHz PentiumII® with a 2-Gb partition down to a 166-MHz Pentium II with a 1-Gb hard drive. The execution time per solution ranged from approximately:

The total computational time to complete the back-calculation of elastic properties from the deflection basin data for all LTPP test sections was approximately:

4.2 Computed Parameter Database

The elastic layer properties for the LTPP flexible and rigid pavement test sections have become a part of the LTPP database. The computed elastic layer properties can be found in six LTPP database tables, which are listed below and summarized in table 6.

  1. The first table (MON_DEFL_FLX_BAKCAL_BASIN) includes the average deflection basins that were used in the back-calculation process.
  2. The second table (MON_DEFL_FLX_BAKCAL_LAYER) includes the material code, thickness, density, and other properties used for each layer in the pavement structure.
  3. The third table (MON_DEFL_FLX_BAKCAL_POINT) contains the calculated Young's modulus using MODCOMP for each of the pavement layers, including the subgrade. The computed moduli are provided for each station or deflection basin measured with the FWD that is included in the LTPP database.
  4. The fourth table (MON_DEFL_FLX_BAKCAL_SUMMARY) includes summary statistics of Young's moduli calculated for each load level and structural layer for each deflection survey or site visit.
  5. The fifth table (MON_DEFL_FLX_NMODEL_POINT) includes the nonlinear elastic properties for the constitutive equations used in the MODCOMP back-calculation program, as well as Young's modulus determined for the highest load level used in the deflection-testing program at each test section.
  6. The sixth table (MON_DEFL_FLX_NMODEL_SUMMARY) is a summary of the nonlinear solutions. This table identifies the model or constitutive equation used to represent each material type and a summary of Young's modulus calculated at the highest load level used during the deflection-testing program.

The summary tables (the fourth and sixth tables) also include the average, standard deviation, and minimum and maximum values for those parameters where the mean values were determined and reported.

4.3 Allowable Maximum RMS Error

Most back-calculation programs use some sort of iterative or optimization technique to minimize the difference between the calculated (for a specific set of elastic layer properties) and measured deflection basins. Obviously, the absolute error (percent error per sensor) and RMS error (goodness-of-fit) vary from station to station and depend on the pavement's physical features that have an effect on the deflection basin measured with the FWD. For example, thickness variations, material density variations, surface distortion, and cracks, which may or may not be visible at the surface, can cause small irregularities within the measured deflection basin, which are not consistent with elastic layer theory.

These irregularities result in differences between the measured and calculated deflection basins. In fact, some of the differences between the calculated and measured basins are so large that the solution is considered highly questionable or that no elastic layered solution exists for that measured deflection basin for the simulated pavement structure (layer type and thickness).

There has been serious discussion and debate over the maximum absolute or RMS error to help decide when an adequate solution or set of layer moduli is considered reasonable and can be used in other analyses. Maximum error values that have been used to judge whether the solution is acceptable vary from 1 to 3 percent for the RMS error and 1 to 2.5 percent for the absolute error or difference.


Table 6. Summary of information included in the computed parameter data tables that are available in the LTPP database.

Name of LTPP
Database Table
for Computed
Parameters
Data Element or Parameter Description of Table
MON_DEFL_FLX_
BAKCAL_BASIN
  • Lane number
  • FWD -- Pass number
  • Point location number
  • Time of measurement
  • Drop height
  • Construction number
  • Deflection at each sensor
  • Temperature for each test point
Average deflection basin for each drop height at each test point that was used in the back-calculation process
MON_DEFL_FLX_
BAKCAL_LAYER
  • Construction number
  • Material type
  • Layer thickness
  • Density
  • Poisson's ratio
  • Depth to apparent rigid layer
Pavement cross-section used in the back-calculation process
MON_DEFL_FLX_
BAKCAL_POINT
  • Lane number
  • FWD -- Pass number
  • Point location number
  • Time of measurement
  • Drop height
  • Calculated pavement temperature
  • Layer modulus
  • RMS error
Young's modulus calculated for each layer and drop height at each test point or location
MON_DEFL_FLX_
BAKCAL_SUMMARY
  • Mean pav't. temp.
  • Min. pav't. temp.
  • Max. pav't. temp.
  • Standard deviation of temperature
  • Mean Young's modulus
  • Min. Young's modulus
  • Max. Young's modulus
  • Standard deviation of Young's modulus
  • Average RMS error
  • Total number of basins
  • Section characterization
Section averages and statistics for the calculated Young's modulus for each layer and test date
MON_DEFL_FLX_
NMODEL_POINT
  • Lane number
  • FWD -- Pass number
  • Point location number
  • Time of measurement
  • Calculated pavement temperature
  • Layer modulus for highest drop height
  • RMS error
  • Constitutive equation number
  • Coefficient and exponent of equation
Young's modulus calculated for each layer for the maximum drop height and the coefficient and exponent of the constitutive equation
MON_DEFL_FLX_
NMODEL_SUMMARY
  • Mean pav't. temp.
  • Min. pav't. temp.
  • Max. pav't. temp.
  • Standard deviation of temperature
  • Mean Young's modulus for highest drop height
  • Min. Young's modulus for highest drop height
  • Max. Young's modulus for highest drop height
  • Standard deviation of Young's modulus
  • Model number
  • Section characterization
  • Mean RMS error
Section averages and statistics for Young's modulus calculated for each layer and test date for the maximum drop height

For the calculation of section average elastic properties, different maximum RMS error values were used to determine the average Young's modulus of a layer for a site and test date. Figures 13-16 graphically compare the average Young's modulus for different maximum RMS errors. The COV for the values shown in figures 13-16 varies from 5 to 15 percent along a test section. In general, a maximum RMS error of 2 to 3.5 percent had little to no effect on many of the test sections. In other cases, however, there were consistent and systematic changes in the average layer modulus with the maximum RMS error used to calculate the section average layer modulus.


Figure 13. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections measured along test section 091803 in August 1994 for different maximum RMS errors.

Figure 13. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections measured along test section 091803 in August 1994 for different maximum RMS errors. The figure contains 4 graphs, with each graph showing 4 different FWD load levels. The horizontal axis of each graph shows Percent Error at 2, 2.5, 3, 3.5, 4, and all; and the vertical axis shows Modulus. In the first graph, E(1), the Modulus for each of the 4 FWD load levels varies inconsistently from each percent error point. For Load Level 1, the Modulus varies from 5700 to about 6050. For Load 2, Modulus varies from 6050 to nearly 6300. Load 3 varies only from 6000 to 6150 and Load 4 varies from 5900 to 6000. In the second graph, E(2), all FWD graphs are nearly straight lines that tend to decrease slightly as they go from 2 percent error to all percent error. Level 1 and Level 4 are nearly identical, going from about 200 modulus (2 Percent Error) to about 175 (All Percent Error). Levels 2 and Level 3 are even more identical, going from about 175 Modulus (2 Percent Error) to nearly 150 Modulus (2.5 and 3 Percent Error) and dropping to 125 Modulus (All Percent Error). In the graph of E(3), Modulus for all 4 loads stays below 15000, except at All Percent Error, where Modulus is over 50,000 and nearly 20,000 for Load 1 and Load 2, respectively. In the graph of E(4), Modulus for all 4 Loads is very consistent, staying between 440 and 520.

Figure 14. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections measured along test section 131005 in January 1996 for different maximum RMS errors.

Figure 14. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections measured along test section 131005 in January 1996 for different maximum RMS errors. The figure contains 4 graphs, with each graph showing 4 different FWD load levels. The horizontal axis of each graph shows Percent Error at 2, 2.5, 3, 3.5, 4, and all; and the vertical axis shows Modulus. In the graph of E(1), Modulus is almost flat across all percent error for all 4 load levels. Load 1 is around 141500, load 4 around 14100, load 2 14000, and load 3 around 13800. In E(2), loads are also consistent across all percent errors. Load 4 is at 250, load 3 is at 240, load 2 is at 225, and load 4 is between 215 and 225. In E(3), graphs of load 2-3 are straight lines around 150 Modulus. Load 1 is nearly straight at around 450. In E(4), graphs of all 4 loads show no change across the different Percent Errors, lying between 350 and 450.

Figure 15. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections measured along test section 481068 in August 1994 for different maximum RMS errors.

Figure 15. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections measured along test section 481068 in August 1994 for different maximum RMS errors. The figure contains 4 graphs of Maximum Modulus by Percent Error, with each graph showing 4 different FWD load levels. The horizontal axis of each graph shows Percent Error at 2, 2.5, 3, 3.5, 4, and all; and the vertical axis shows Modulus. In the graph of E(1), Load 1 Modulus ranges from about 1650 to 1750, Load 2 Modulus ranges from about 1700 to 1850, and Loads 3 and 4 range from 1800 to 1850. In E(2), the Modulus declines as the Percent Error increases. Loads 1-3 the Modulus ranges from 130 to 80. Load 4 the Modulus goes from about 160 down to about 130. In E(3), the Modulus increases as the Percent Error increases. For Loads 1-3, the Modulus ranges from about 200 to about 250. For Load 4, the Modulus ranges from 175 to 200. In E(4), Modulus is relatively flat. Loads 2-4 have Modulus around 186 and 188, but at All Percent Error, Modulus changes to 191, 189, and 183 for Loads 2, 3, and 4, respectively. For Load 1, Modulus starts at 103 and drops to 101 (4.0) and then increases to 106 (All Percent Error).

Figure 16. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections along test section 081053 in March 1995 for different maximum RMS errors.

Figure 16. Graphical illustration of the average Young's modulus for the pavement layers and subgrade computed from deflections along test section 081053 in March 1995 for different maximum RMS errors. The figure contains five graphs, with each graph showing four different FWD load levels. The horizontal axis of each graph shows Percent Error at 2, 2.5, 3, 3.5, 4, and All; and the vertical axis shows Modulus. For the graph of E(1), Loads 2, 3, and 4 have a Modulus of about 5500, 5800, and 5200, respectively. For Load 1, Modulus ranges from about 5500 to 5100. In graph E(2), Modulus shows a gradual increase for all 4 load levels, with ranges of 300-600, 450-600, 550-700, and 800-900 for Loads 1, 2, 3, and 4, respectively. In E(3), modulus is about 100-200 for all 4 Loads, except for a high of about 1500 for Load 1 at All Percent Error. In E(4), Modulus for all 4 loads shows large increases, starting at about 1000-2000 (2.0) to a maximum of about 6000-9000 (All). In E(5), Modulus is mostly the same across the graph, with all 4 loads staying in the 60-75 range.

As expected, the maximum RMS error has a greater effect on the lower drop heights (smaller deflections) than on the higher drop heights (larger deflections). In general, the average elastic moduli calculated from all solutions with RMS error values of 3 percent or less are no different than those average values calculated with lower RMS error values.

In the third and fifth computed parameter tables, identified above (see table 6), all results were stored if a solution was obtained, even though some of the RMS errors for particular test points were greater than 4 percent. In the computed parameter summary tables (the fourth and sixth tables), only those solutions with RMS errors equal to or less than 2 percent were used in determining the average layer modulus and standard deviation along each test section on a specific test date. Selection of this maximum RMS error value of 2 percent is arbitrary and based on previous studies.

A total of 1,467,000+ deflection basins measured on the flexible pavement test sections were analyzed with MODCOMP4, and a total of 186,000+ basins measured on the rigid pavement test sections were analyzed. Figures 17 and 18 graphically illustrate the distribution of the RMS errors for the deflection basins measured on the flexible and rigid test sections that were analyzed with the linear elastic module of MODCOMP4, respectively. As shown, over 75 percent of the solutions have RMS errors less than 3 percent and are considered acceptable.


Figure 17. Distribution of the RMS errors for all of the deflection basins measured on the flexible pavement test sections that were analyzed with the linear elastic module of MODCOMP4

Figure 17. Distribution of the RMS errors for all of the deflection basins measured on the flexible pavement test sections that were analyzed with the linear elastic module of MODCOMP4. The figure contains 3 bar graphs: Flexible GPS (654,191 Deflection Basins), Flexible SMP (747,810 Deflection Basins), and Flexible SPS (65,803 Deflection Basins). RMS Error of 0-0.5, 0.51-1, 1.01-1.5, 1.51-2, 2.01-2.5, 2.51-3, 3.01-3.5, 3.51-4, 4.01-4.5, 4.51-5, and >5.01 is graphed on the horizontal axis and Frequency (%) 0-30 on the vertical axis. All three graphs show a similar trend, with the highest Frequency (23-25%) at 0.51-1 RMS Error, which gradually decreases to a minimum Frequency (1-2%) at 4.51-5 RMS Error. At >5.01 RMS Error, the Frequency ranges from about 8-16%.

Figure 18. Distribution of the RMS errors for all of the deflection basins measured on the rigid pavement test sections that were analyzed with the linear elastic module of MODCOMP4.

Figure 18. Distribution of the RMS errors for all of the deflection basins measured on the rigid pavement test sections that were analyzed with the linear elastic module of MODCOMP4. The figure contains 3 bar graphs: Rigid GPS (127,102 Deflection Basins), Rigid SMP (35,009 Deflection Basins), and Rigid SPS (24,812 Deflection Basins). RMS Error of 0-0.5, 0.51-1, 1.01-1.5, 1.51-2, 2.01-2.5, 2.51-3, 3.01-3.5, 3.51-4, 4.01-4.5, 4.51-5, and >5.01 is graphed on the horizontal axis and Frequency (%) on the vertical axis. For Rigid GPS, the maximum Frequency is 28% at an RMS Error of 0.51-1, which decreases to as low as 2 as the RMS Error increases. The graph of Rigid SMP has Frequencies below 15% across the graph, with a minimum Frequency of 2 at a 4.51-5 RMS Error. For Rigid SPS, Frequencies are below 10% across the graph, except at RMS Error >5.01 where the Frequency is about 40%.

4.4 Brief Evaluation of Reasonableness of Solutions

A brief evaluation of the results was conducted to determine the reasonableness of the solutions. This review was focused mainly on the variation of the calculated moduli along the test section length, the change in the calculated moduli with season or month and with mid-depth temperature, and the effect of test load on the resulting moduli. This section of the report also presents examples of how these results can be used in pavement performance and/or material behavior studies.

4.4.1 Longitudinal Variation of Elastic Moduli. Figures 19-22 graphically illustrate examples of the longitudinal variation of the computed elastic moduli (Young's modulus) for selected test sections and different time periods. These results are typical of many other test sections. Some have extensive variability with both distance and time, whereas others are relatively uniform along the test section and with time.

These graphical comparisons are useful in judging whether the physical conditions of the pavement and subgrade soils may be significantly changing along the test section. In general, as the variability of the measured deflections increased, the variability of the calculated elastic layer moduli also increased and/or the number of reasonable solutions within a test section decreased.

For many test sections, a very stiff layer was computed for the subbase or stabilized subgrade layer of the pavement structure. These computed moduli are unrealistic and are not representative of the material, as shown in table 7 for test section 081053. These unrealistic moduli generally occurred for the more extreme Type II deflection basins (see figure 7). This condition was identified previously by Von Quintus and Killingsworth in a back-calculation study using the MODULUS software package.(3) Appendix C lists those test sections where the results for many of the test points are questionable.


Figure 19. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 091803 in August 1994.

Figure 19. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 091803 in August 1994. The figure consist of 4 graphs, with Station on the horizontal axis, Modulus on the vertical axis, and 4 drops heights graphed. For E(1) the graph of all 4 drops are nearly identical, with Modulus increasing and decreasing as it goes from 300 to 525, with ranges from about 4000 to 8000. In E(2), the graphs of the 4 drops are similar, with Modulus gradually increasing from station 300 to station 400, dropping sharply around 450, increasing around 500 sharply and then dropping again around 525. Modulus ranges from 50 to just over 250 for all 4 drops. In E(3) Modulus increases and decreases regularly for Drop 1, ranging from near zero-10,000. Drop 2 had Modulus of near zero throughout, except at Station 475 (Modulus of 175,000). Drop 3 and 4 had Modulus of near zero throughout. (Note, Modulus was graphed on a scale of 50,000, therefore Modulus near zero could actually be several 1000). In E(4) the graphs of the 4 drop heights show considerable amount of variability going from Station 300 to Station 525. Overall, the Modulus ranges from 200-800.

Figure 20. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 131005 in January 1996.

Figure 20. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 131005 in January 1996. The figure consist of 4 graphs, with Station on the horizontal axis, Modulus on the vertical axis, and 4 drops heights graphed. For E(1), the graph of all 4 drops are nearly identical, with Modulus ranging from about 10000 to 175000. In E(2), the graphs of the 4 drops are also nearly the same, with Modulus ranging from 50-475. In E(3), Modulus is under 200, with the exception of two points, for all graphed points for Drops 2-4. Drop 1 has 5 points (out of 11) that are above 200, ranging from 500-1400. In E(4) the graphs of the 4 drops are similar, ranging from 300-500.

Figure 21. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 481068 in August 1994.

Figure 21. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 481068 in August 1994. The figure consist of 4 graphs, with Station on the horizontal axis, Modulus on the vertical axis, and 4 drops heights graphed. For E(1), the graph of all 4 drops are nearly identical, with Modulus ranging from about 1000 to 2500. In E(2), the graphs of all 4 drops stay between 20-40, except outliers at Station 0 and 200. At Station 0, Modulus ranges from 4-160 and at Station 200 it ranges from 20-140. In E(3), Modulus varies erratically for all 4 drops and shows no consistency among them. Overall, Modulus ranges from 100-500. In E(4), the graphs of the 4 drops are nearly identical, ranging from 150-250.

Figure 22. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 081053 in March 1995.

Figure 22. Longitudinal variation of Young's modulus in the wheelpath for each pavement layer and subgrade along test section 081053 in March 1995. The figure consists of 5 graphs, with Station on the horizontal axis, Modulus on the vertical axis, and 4 drops heights graphed. For E(1), the graph of all 4 drops are nearly identical, with Modulus ranging from about 2000 to 8000. In E(2), the graphs of Modulus for all 4 drops show similar patterns and range from 50-1400. In E(3), Modulus is zero for all 4 drops, except at Station 100. There Modulus is 2500 for Drop 1 and 23000 for Drop 1. In E(4), the graphs of the 4 drops show similar trends, with Modulus ranging from 0-120000. In E(5), the graphs of the 4 drops are identical (Modulus ranges from 25-100), except at Station 100, where modulus is 250 (Drop 1) and 150 (Drop 2).

4.4.2 Wheelpath Versus Non-Wheelpath Measurements. At all of the LTPP test sections, deflections are measured both in the "outer wheelpath" (within the wheelpath) and in the "mid-lane" (between wheelpaths). The outer wheelpath measurements are identified as F3 in the database, whereas the mid-lane measurements are identified as F1. The back-calculation of layer moduli was completed for both sets separately.

Table 7 summarizes the average layer moduli computed for four test sections. For many of the test sections, there is no significant difference in the computed Young's modulus for the between-wheelpath and within-wheelpath measurement locations (e.g., test section 091803). However, significant differences were found for other test sections (e.g., test sections 081053 and 481068).


Table 7. Average Young's modulus calculated for the outer wheelpath and mid-lane for each layer along specific test sections for drop height 4, MPa.

LTPP Test
Section
Layer
Designation
Mid-Lane
Mean, MPa
Mid-Lane
Standard
Deviation, MPa
Mid-Lane
Coefficient of
Variation,
Percent
Outer
Wheelpath
Mean, MPa
Outer
Wheelpath
Standard
Deviation, MPa
Outer
Wheelpath
Coefficient of
Variation,
Percent
081053 (March 1995) 1 5,955 1,639 27.5 5,081 1,191 23.4
081053 (March 1995) 2 1,355 528 38.9 641 370 57.8
081053 (March 1995) 3 95 12 12.6 129 44.8 34.7
081053 (March 1995) 4 127,086 107,095 84.3 32,559 38,370 117.5
091803 (August 1994) 1 6,076 959 15.8 5,864 996 17.0
091803 (August 1994) 2 190 60 31.6 189 54 28.6
091803 (August 1994) 3 379 173 45.6 282 85 30.1
091803 (August 1994) 4 450 110 24.4 421 128 30.4
131005 (January 1996) 1 15,097 1,463 9.6 13,258 2,034 15.3
131005 (January 1996) 2 291 130 44.7 216 98 45.4
131005 (January 1996) 3 145 33 22.8 153 57 37.3
131005 (January 1996) 4 368 26 7.1 342 27 7.9
481068 (August 1994) 1 1,695 200 11.8 1,980 311 15.7
481068 (August 1994) 2 216 104 48.1 57 51 89.5
481068 (August 1994) 3 79 49 62.0 318 116 36.5
481068 (August 1994) 4 193 5 2.6 179 8 4.5

Based on a brief review of the results, there is no consistent difference between the within-wheelpath and non-wheelpath measurements. For some of the test sections, however, low moduli were calculated for layers directly under the surface for within-wheelpath measurement locations. In most such cases, a higher modulus was calculated for the lowest layer, as shown in table 7 for test section 481068. These computations are believed to be unrealistic, even though the computations for both within- and between-wheelpaths were completed in the same within- and between-wheelpaths production run using the same pavement structure. Slight thickness variations, caused by plastic flow or lateral movement of the underlying materials close to the surface, may be a cause of this condition.

4.4.3 Seasonal or Monthly Effects. Figures 23-26 graphically illustrate examples of the monthly variation of the computed elastic moduli for the different layers of selected test sections. As shown, the moduli of the asphalt concrete layers increase for the winter months and decrease for the summer months. In addition, the modulus of the unbound aggregate layers and subgrade soils becomes extremely high during periods of possible freezing temperatures below the surface, as shown by the LTPP test section in Minnesota (276251).

4.4.4 Temperature Effects. Figures 27-30 graphically illustrate examples of the computed elastic moduli for the asphalt concrete surface layer as a function of mid-depth temperature. As shown, the modulus of the asphalt concrete layer increases with decreasing temperatures. However, there are some cases where there are inconsistent changes in modulus with temperature. Some of these test sections were identified as having potential stripping in the HMA layer or were found to have extreme variations in the underlying support layers. Appendix C includes a summary of the overall average Young's modulus calculated at different temperature intervals for all HMA surface layers. As tabulated and expected, the average layer modulus decreases with increasing pavement temperature.

4.4.5 Time Effects. The SPS projects were used to look at systematic increases in the elastic moduli of the HMA and PCC surface layers as a result of hardening and curing. However, there are insufficient data to identify statistically any changes (increases) in the HMA and PCC moduli with time.

4.4.6 Stress Sensitivity from Linear Elastic Solutions. The linear elastic solutions were used to determine the change in computed elastic moduli (Young's moduli) with increasing load level. For fine-grained soils, the modulus decreased with increasing load level, as expected. For the coarse-grained soils, the opposite was the case, again an expected result, based on laboratory tests. These unbound materials or soils are stress-sensitive, and consistent changes were calculated with increasing load levels, as shown for some of the layers in figures 19-21.


Figure 23. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 041024.

Figure 23. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 041024. The figure consists of 4 graphs, with Time (July 95 to July 96) on the horizontal axis and Average Modulus on the vertical axis. For the graph of E(1), Average Modulus starts around 2000 (July to 93) and then increases from 2000 to 8000 (August 95 to November 95), drops to about 2000 (December 95 to May 96) and then stays around 2000 (May 96 to July 96). In E(2), Modulus increases from 50-200 (July 95 to November 95), drops to 100 (December 95), peaks at 800 (February 96), decreases to 200 (February 96 to June 96), and increases slightly to 250 (July 96). In E(3), Modulus varies between 1500 and 2500 (July 95 to December 95), peaks at 3250 (December 95), and then drops to around 250 and holds steady (February 96 to July 96). In E(4), Modulus is around 75-100 (July 95 to December 95), then increases to around 200 (January 96 to July 96).

Figure 24. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 131005.

Figure 24. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 131005. The figure consists of 4 graphs, with Time (August 95 to August 96) on the horizontal axis and Average Modulus on the vertical axis. In E(1), Average Modulus starts around 5000 (August 95 to Sep 85) and then increases to around 14000 (Sep 85 to December 85), stays at 14000 (December 85 to February 96), then drops to around 4000 (February 96 to August 96). In E(2), between August 95 and January 96, Modulus varies between 150 and 250, then decreases to nearly 100 (January 96 to April 96), increases to 150 (April 96 to June 96), and drops back to 100 (June 96 to August 96). In E(3), Modulus increases from 400 to 550 (August 95 to November 95), drops to 300 (November 95 to January 96), increases to 500 (January 96 to March 96), drops to 400 (April 96), stays at 400 (April 96 to June 96), and then increases to 900 (June 96 to August 96). In E(4) Modulus stays relatively constant, ranging from 60-80.

Figure 25. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 241634.

Figure 25. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 241634. The figure consists of 4 graphs, with Time (May 95 to August 96) on the horizontal axis and Average Modulus on the vertical axis. For the graph of E(1), the Average Modulus Decreases from 7000 to 2000 (May 95 to Sep 95), peaks at 8000 (December 95), drops to 6000 (March 96), increases to 7000 (May 96), and drops to about 3000 (July 96). In E(2) Modulus is 1750 from May 95 to Sep 95, increases to 7000 (December 95) and decreases to around 2000 from December 95 to July 96. In E(3), Modulus is less than 250 from May 95 to August 96, except for a peak of 4500 (May 96). In E(4) Modulus is fairly constant, ranging from about 150 to nearly.

Figure 26. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 276251.

Figure 26. Monthly variation of Young's modulus for each pavement layer and subgrade at test section 276251. The figure consists of 4 graphs, with Time (May 94 to May 97) on the horizontal axis and Average Modulus on the vertical axis. For the graph E(1), Average Modulus drops from 7500 to 5000 (May 94 to July 94), increases to 45000 (July 94 to January 95), drops to just under 5000 (January 95 to May 95), and increases to about 10000 (January 95 to Sep 96). In E(2) Modulus is less than 50 (May 94 to May 97), except for peaks of 19000 (January 95) and 11000 (February 97). In E(3), Modulus is about 1000 (May 94 to May 97), except for peaks of 20000 (March 95) and 30000 (March 97). In E(4), Modulus stays between 60 and 70 (May 94 to May 97), except for a peak of 110 (May 95).

Figure 27. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 040113.

Figure 27. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 040113. The figure shows temperature (degree Celcius) on the horizontal axis, and AC Modulus (megaPascals) on the vertical axis; and Drops 1-4, averages, and exponential average (y=22223e to the (-0.0429x)) are graphed. The figure illustrates the inverse relationship between backclaculated AC layer modulus and temperature for section 040113. The data fit an exponential curve very well, with an R square of 0.9347.

Figure 28. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 131005.

Figure 28. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 131005. The figure shows temperature (degree Celcius) on the horizontal axis, and AC Modulus (megaPascals) on the vertical axis; and Drops 1-4, averages, and exponential average (y=27533e to the (-0.055x)) are graphed. The figure illustrates the inverse relationship between backclaculated AC layer modulus and temperature for section 131005. The data fit an exponential curve very well, with an R square of 0.984.

Figure 29. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 501002.

Figure 29. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 501002. The figure shows temperature (degree Celsius) on the horizontal axis, and AC Modulus (megaPascals) on the vertical axis; and Drops 1-4, averages, and exponential average (y=21222e to the (-0.0813x)) are graphed. The figure illustrates the inverse relationship between backclaculated AC layer modulus and temperature for section 501002. The data fit an exponential curve very well, with an R square of 0.935.

Figure 30. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 561007.

Figure 30. Graphical comparison of the computed Young's modulus and mid-depth pavement temperature measured along SMP test section 561007. The figure shows temperature (degree Celsius) on the horizontal axis, and AC Modulus (megaPascals) on the vertical axis; and Drops 1-4, averages, and exponential average (y=9740.1e to the (-0.013x)) are graphed. The figure illustrates the inverse relationship between backclaculated AC layer modulus and temperature for section 561007. The data do not fit an exponential curve very well, with an R square of 0.1915.

Changes in the stress sensitivity of some layers also were observed along some of the LTPP test sections. For example, layer 2 in test section 091803 (figure 19) and layers 2 and 4 at test section 481068 (figure 21) illustrate consistent changes in the calculated moduli for different drop heights. The question becomes: Is this a real condition or simply an anomaly from the back-calculation process? It is believed that the stress sensitivity can change but is dependent on the physical conditions of the soils or material (moisture content, gradation, permeability, etc.).

The HMA layer is assumed to be a linear elastic material -- the modulus should not change with changes in load level. However, the computed elastic moduli resulting from the linear solutions were found to increase and decrease consistently with increasing load levels for many test sections (i.e., layer 1 in figure 22). Although these changes in moduli are not consistent with laboratory test results, they have been observed from other back-calculation studies using different programs. This observation could be just an artifact of the back-calculation process, the result of compensating errors or opposite changes in elastic modulus between two different but adjacent layers with increasing drop heights, and/or a stiffening or weakening effect with increased loads.

Again, the question becomes, "Is this a real condition or simply an anomaly from the back-calculation process?" It is believed that this condition may be more related to the interface condition between layers near the surface or damage that has occurred (accumulated) in the bound surface layers, rather than the true stress sensitivity of the layer.

4.5 Linear Versus Nonlinear Solutions

One of the primary purposes of this study was to compute the nonlinear elastic properties of the pavement materials and subgrade soils. In fact, the largest effort was devoted to calculating these nonlinear elastic properties. One of the questions to be answered at the beginning of the data study was whether the nonlinear solutions were worth the effort. Unfortunately, this question cannot be answered until a detailed comparison is completed between laboratory-derived properties and those computed from deflection basins.

Overall, fewer test points had acceptable RMS errors for the nonlinear solutions than for the linear elastic ones. Three possible explanations for this observation are listed below.

  1. The first and most likely explanation is that there are more unknowns when using the nonlinear constitutive equations. The more unknowns there are, the more difficult it is to find a solution (the elastic properties for each layer), all other things being equal.
  2. A quasi-nonlinear elastic structural response model simply does not adequately simulate the real-world, nonlinear response of the pavement and soil interaction from vertical loads.
  3. Young's modulus, calculated for the surface layers, varies with drop height at some of the test sections, as noted above, but for the nonlinear solutions it is assumed to be independent of drop height. This assumption increases the RMS error between the measured and computed deflection basins because elastic layer response models do not simulate real-world conditions.

Another observation was that the HMA elastic moduli (bound layers) for the linear solutions were less than those for the nonlinear solutions but greater than the nonlinear solutions for the unbound layers. There are probably compensating differences between the bound or surface layers and unbound, subsurface layers. The cause for these differences is unknown but may be related to the interface condition of the surface layers/lifts, surface distortion, and/or microcracking of the HMA layers.

To begin to answer some of these questions would require a detailed analysis and comparison of the results with the laboratory tests and observations of the cores taken from the test sections, which was beyond the scope of work for this study.

 

 


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The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
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