Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

Simplified Techniques for Evaluation and 

Interpretation of Pavement Deflections 

for Network-Level Analysis: Guide for

Assessment of Pavement Structural

Performance for PMS Applications

PUBLICATION NO. FHWA-HRT-12-025 JUNE 2012
FOREWORD

The objective of this study was to develop an approach for incorporating techniques used to 

interpret and evaluate deflection data for network-level pavement management system (PMS) 

applications. The first part of this research focused on identifying and evaluating existing 

techniques, seeking out those that were simple, reliable, and easy to incorporate into current 

PMS practices, as well as those that produced consistent results. The second part of the research 

detailed the development of guidelines for the application of recommended techniques and

procedures for determining optimum falling weight deflectometer (FWD) test spacing and data 

collection frequency. While there are many viable techniques available for evaluating the 

structural capacity of pavements that use FWD for project-level analysis, many of these 

techniques are time consuming and require an experienced analyst. As a result, using pavement 

deflection testing for network-level analysis has been limited to date. This guide contains 

step-by-step instructions for applying appropriate evaluation techniques for network-level 

(not project-level) measurements and analyses.

Jorge E. Pagán-Ortiz

Director, Office of Infrastructure

Research and Development

Notice

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

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

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

specification, or regulation.

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

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

objective of the document.

Quality Assurance Statement

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

Government, industry, and the public in a manner that promotes public understanding. 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-HRT-12-025

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

4. Title and Subtitle

Simplified Techniques for Evaluation and Interpretation of 

Pavement Deflections for Network-Level Analysis: Guide 

for Assessment of Pavement Structure Performance for PMS 

Applications

5. Report Date 

June 2012

6. Performing Organization Code

7. Author(s) 

R. Stubstad, R. Carvalho, R. Briggs, and O. Selezneva

8. Performing Organization Report No.

9. Performing Organization Name and Address

Applied Research Associates, Inc.

7184 Troy Hill Drive, Suite N

Elkridge, MD 21075-7056

10. Work Unit No. (TRAIS)

11. Contract or Grant No.

DTFH61-02-C-00138

12. Sponsoring Agency Name and Address

Office of Infrastructure Research and Development

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101-2296

13. Type of Report and Period Covered

Guide 

14. Sponsoring Agency Code

15. Supplementary Notes

The Contracting Officer’s Technical Representative (COTR) was Larry Wiser, HRDI-30.

16. Abstract

The objective of this study was to develop an approach for incorporating techniques to interpret and evaluate deflection 

data for network-level pavement management system (PMS) applications. The first part of this research focused on 

identifying and evaluating existing techniques, seeking out those that were simple, reliable, and easy to incorporate into 

current PMS practices, as well as those that produced consistent results. The second part of the research detailed the 

development of guidelines for the application of recommended techniques and procedures for determining optimum 

falling weight deflectometer (FWD) test spacing and data collection frequency. While there are many viable techniques 

available for evaluating the structural capacity of pavements that use FWD for project-level analysis, many of these 

techniques are time consuming and require an experienced analyst. As a result, using pavement deflection testing for 

network-level analysis has been limited to date. This guide provides information for the assessment of pavement structural 

performance for PMS applications. It reflects the general findings presented in the accompanying final report, Simplified 

Techniques for Evaluation and Interpretation of Pavement Deflections for Network-Level Analysis (FHWA-HRT-12-023).

17. Key Words

Pavement performance, Rehabilitation, Maintenance, 

Pavement design, Long-term performance, Flexible 

pavements, Rigid pavements

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

33

22. Price

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

TABLE OF CONTENTS

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

NETWORK-LEVEL MEASUREMENT TECHNIQUES........................................................ 3

REQUIRED INPUT DATA ......................................................................................................... 5

SELECTION OF APPROPRIATE ANALYSIS TECHNIQUES............................................ 7

STOCHASTIC APPROACH TO EVALUATE FWD DEFLECTION DATA....................... 9

ANALYSIS TECHNIQUES FOR FLEXIBLE PAVEMENTS.............................................. 11

STRUCTURALLY BASED ROUGHNESS PERFORMANCE MODELS FOR 

FLEXIBLE PAVEMENTS .................................................................................................. 11

STRUCTURALLY BASED RUTTING PERFORMANCE MODEL FOR 

FLEXIBLE PAVEMENTS .................................................................................................. 12

STRUCTURALLY BASED FATIGUE CRACKING PERFORMANCE MODEL 

FOR FLEXIBLE PAVEMENTS......................................................................................... 13

ANALYSIS TECHNIQUES FOR RIGID PAVEMENTS...................................................... 15

STRUCTURALLY BASED ROUGHNESS PERFORMANCE MODELS FOR 

RIGID PAVEMENTS .......................................................................................................... 15

STRUCTURALLY BASED JOINT FAULTING MODELS FOR RIGID 

PAVEMENTS ....................................................................................................................... 16

STRUCTURALLY BASED TRANSVERSE SLAB CRACKING 

PERFORMANCE MODEL FOR RIGID PAVEMENTS ................................................ 16

OPTIMUM SPACING AND TIMING..................................................................................... 19

CONSIDERATION OF TIME OF YEAR AND SEASON .................................................... 21

LOCAL CALIBRATION........................................................................................................... 23

DATA PREPARATION....................................................................................................... 23

LOGISTIC MODEL CALIBRATION............................................................................... 24

VERIFICATION OF ACCURACY.................................................................................... 24

SENSITIVITY ANALYSIS.................................................................................................. 25

Sensitivity Analysis with Reasonable Results.................................................................. 25

Sensitivity Analysis with Unreasonable Results............................................................... 26

REFERENCES............................................................................................................................ 29
iv

LIST OF FIGURES

Figure 1. Graph. Typical logistic model probability function of predictor variable X................. 10

Figure 2. Equation. General formulation of the logistic model .................................................... 10

Figure 3. Equation. General linear formulation for the exponent term of the logistic model ...... 10

Figure 4. Equation. Calculation of I2 ............................................................................................ 11

Figure 5. Equation. Calculation of CI3.......................................................................................... 12

Figure 6. Equation. Calculation of I1 ............................................................................................ 13

Figure 7. Equation. Calculation of CI5.......................................................................................... 15

Figure 8. Equation. Calculation of CI4.......................................................................................... 17

Figure 9. Graph. Example of an ROC curve................................................................................. 24

Figure 10. Graph. Sensitivity of roughness acceptable probability to deflection parameter 

I2 for flexible pavements............................................................................................................... 26

Figure 11. Graph. Sensitivity of roughness acceptable probability to deflection parameter 

Hogg for flexible pavements......................................................................................................... 26

LIST OF TABLES

Table 1. Required FWD or HWD sensor positions to utilize the FWD at the network level......... 5

Table 2. Example of structural decision matrix ............................................................................ 10

Table 3. Variables in the I2 final model for flexible pavements based on roughness 

performance .................................................................................................................................. 11

Table 4. Variables in the CI3 final model for flexible pavements based on rutting 

performance .................................................................................................................................. 12

Table 5. Variables in the I1 final model for flexible pavements based on fatigue cracking 

performance .................................................................................................................................. 13

Table 6. Variables in the CI5 final model for rigid pavements based on roughness 

performance .................................................................................................................................. 15

Table 7. Variables in the D6 final model for rigid pavements based on faulting at joints 

performance .................................................................................................................................. 16

Table 8. Variables in the CI4 final model for rigid pavements based on transverse slab 

cracking performance.................................................................................................................... 17

Table 9. Errors in percentage for 90 percent reliability based on various section lengths, 

sample spacings, and coefficient of variation of 33 percent......................................................... 19
1

INTRODUCTION

This guide can be used to characterize the overall level of a pavement’s structural characteristics 

for network-level analyses using falling weight deflectometers (FWDs) or heavy weight 

deflectometers (HWDs). The deflection data are coupled with other readily available input and 

pavement performance data that can be used to characterize the overall level of the pavement 

structural characteristics for pavement management applications. This guide contains step-bystep instructions for applying appropriate evaluation techniques for network-level (not projectlevel) measurements and analyses. An example of an application of this approach to a pavement 

network sample is provided in the final report, Simplified Techniques for Evaluation and 

Interpretation of Pavement Deflections for Network-Level Analysis.

(1)

3

NETWORK-LEVEL MEASUREMENT TECHNIQUES

Applicable measurement and analysis techniques for assessing the overall network-level 

structural performance of roadways only include the use of load deflection data obtained by the 

commonly used FWD. An HWD can also be used at its lower load levels corresponding to 

typical truck axle loads on highway and roadway pavements. While it is possible that currently 

available or future high-speed load-deflection measurement devices (i.e., the experimental 

rolling wheel deflectometer) may be used as well, the data obtained with these newer devices are 

not considered in these guidelines.

5

REQUIRED INPUT DATA

The input data required for the addition of a structural component to an agency’s pavement 

management system (PMS) include data normally present in an agency’s PMS in addition to

FWD or HWD load deflection data taken at appropriate spatial and temporal intervals systemwide. Preprocessing of these data is not required other than the usual quality control checks for 

FWD field errors, such as misstated sensor positions, faulty deflection sensors or load cells, 

and/or other data entry errors from field testing.

The development effort for these guidelines utilized data from the over 20-year-old Strategic 

Highway Research Program (SHRP) experimental test program. For this reason, the original 

seven SHRP sensor positions were used to perform the data analyses required (i.e., 0, 8, 12, 18, 

24, 36, and 60 inches (0, 200, 300, 450, 600, 900, and 1,500 mm)).(2) Since only interior or center 

slab tests were utilized to develop the method for reasons described in the following sections, the 

same sensor positions were used for concrete as well as asphalt pavements.

A list of required sensor positions covering four flexible pavement and three rigid concrete 

pavement distress types described in this guide are shown in table 1.

Table 1. Required FWD or HWD sensor positions to utilize the FWD at the network level.

Sensor Position

Flexible 

Pavements Rigid Pavements

0 inches (0 mm) X

8 inches (200 mm) X

12 inches (300 mm) X

18 inches (450 mm) X X

24 inches (600 mm) X

36 inches (900 mm) X

Note: For rigid pavements, only interior or center slabs were tested.

Due to many issues, including the propensity of concrete pavements to warp and curl, joint 

tests are not recommended for network-level testing since they would require a stringent time 

window within which testing would have to occur. Moreover, studies of the Federal Highway 

Administration’s (FHWA) Long-Term Pavement Performance (LTPP) database provided no 

improvement in the derived stochastic relationships when joint tests (e.g., load transfer efficiency 

(LTE)) were included in the analyses.(3) Lastly, the acceptable time window for interior slab 

testing is much longer than for joint testing, making the simple approach of applying FWD load 

deflection data to a network database easier to carry out in the field.

As can be seen in table 1, the most efficient way to proceed for network-level FWD testing is to 

use the original SHRP configuration of sensor positions (0, 8, 12, 18, 24, and 36 inches (0, 200, 

300, 450, 600, and 900 mm)) for the first six sensors. These six sensor positions cover both 

flexible pavement and rigid pavement tests taken at an interior position for jointed concrete 

pavements. Additional FWD/HWD deflection readings should also be taken to cover other 

potential uses of FWD load deflection data, such as LTE for rigid pavement tests at joints with a 
6

trailing sensor and/or additional sensors placed further from the load plate than the requisite 

36 inches (900 mm) for all pavements.

The only required test load for flexible pavements is 9,000 lbf (40 kN) within ±10 percent. 

For rigid pavements, the required test load is either 9,000 or 12,000 lbf (40 or 53 kN) within 

±10 percent depending on an agency’s normal deflection testing practices for rigid pavements. In 

any event, the deflections have to be normalized (scaled up or down) to the target load level in an 

agency’s analysis package of choice.

In this guide, only the 9,000-lbf (40-kN) rigid pavement loading relationships are shown. 

However, in the final report, equally positive relationships also exist for the nominal 12,000-lbf 

(53-kN) load level on concrete pavements if that load level is preferred by the agency.

In order to implement a network-level structural analysis component in a PMS enhanced with 

FWD tests using a stochastic model, the following data elements are needed:

• Pavement age (time elapsed since construction or latest rehabilitation).

• Daily volume of class 9 truck traffic (the average daily volume of 18 wheelers).

• Average annual daily truck traffic (AADTT).

• Average annual precipitation in the geographical area (millimeters).

• Base type for flexible pavements (bound versus unbound pavement type).

• Subgrade type for rigid pavements (fine-grained versus coarse-grained).

The three flexible pavement distress types and the three rigid pavement distress types (with 

overlap) in the suggested stochastic models include the following:

• Roughness (International Roughness Index (IRI)): Flexible versus rigid pavements.

• Rutting: Flexible pavements only.

• Joint faulting: Jointed concrete pavements only.

• Surface cracking: Flexible pavement only.

• Transverse slab cracking: Jointed concrete pavements only.

By implication and based on the above list of pavement distresses included in this guide, 

composite pavements and continuously reinforced concrete pavements were not included due to 

a lack of historical data.
7

SELECTION OF APPROPRIATE ANALYSIS TECHNIQUES

The appropriate analysis technique for use with network-level load-deflection data depends on 

the pavement type and the type of distress the PMS user wishes to use to characterize structural 

condition. At a minimum, it is recommended that the typical distresses observed on the local 

network are used. If this information is not available, the analysis should be conducted using all 

applicable models as described in this section.

For almost all agencies, system-wide load deflection data are not available. Therefore, it is 

recommended that the so-called “global” relationships and threshold values derived through the 

methodology should be used until local calibration of these relationships can take place, which 

could take as long as 5 years. Local calibration will significantly improve the accuracy of these 

models, as reported in the accompanying final report.(1)

It should be noted that the methodology developed was based on data obtained from FWD tests 

conducted along typical State and interstate highways, expressways, and freeways during the 

SHRP LTPP study. Therefore, it is possible that an agency with lighter-duty pavements will find 

that the threshold and cutoff values initially derived through the equations presented in the 

following sections may not be appropriate for every existing PMS, whether in the United States 

or elsewhere. This will eventually be mitigated by the use of traffic levels.

The general approach recommended in this guide only provides one additional link in an 

otherwise incomplete pavement management database and should only be used as a planning 

tool to assist network pavement engineers in predicting the likelihood of the development of 

pavement distress levels. As a result, there is a need for future structural improvement to the 

network as a whole, not every maintenance and rehabilitation project on an individual basis. This 

is accomplished through the use of a stochastic approach to evaluate FWD load deflection data 

called the “logistic model,” which is discussed in the following section.

9

STOCHASTIC APPROACH TO EVALUATE FWD DEFLECTION DATA

The objective of a pavement design is to provide a structure that will exhibit adequate structural 

performance throughout its design life. It is expected that the structural condition affects the rate 

the pavement deteriorates over time. By definition, a given pavement structure exhibits 

acceptable performance if distress levels do not exceed their threshold values during a given 

period of time (e.g., fatigue cracking should not exceed 25 percent of the total area within the 

first 15 years of service).

Logistic models describe the probability of an event being acceptable or not acceptable as a 

function of a predictor variable. Therefore, they can be used to establish a link between 

performance and structural condition by modeling the probability of acceptable performance 

using measurements of deflections and parameters (techniques) computed from FWD or HWD 

deflection basins.

The goal of the stochastic approach is to use the probability density function from the logistic 

model to obtain insightful probability information about the pavement structural condition 

required to achieve the desired performance during a desirable (design) period of time. This new 

network-level information obtained from the FWD can then be used to define strategic 

treatments (maintenance or rehabilitation) to be used in the PMS optimization process.

A typical logistic model probability function is shown in figure 1. It describes the likelihood of a

pavement section exhibiting acceptable performance (measured in terms of one particular 

distress) as a function of a given deflection parameter called “predictor variable X.” The limit 

between acceptable and not acceptable performance is given by a parameter called the “cutoff.”

The cutoff value is computed as part of the model calibration process using the receiver 

operating characteristics (ROC) curve analysis. (See chapter 4 in the final report.

(1))

The probability density function is used to determine the thresholds that define structural 

condition. By definition, the cutoff value in a logistic model determines how to convert the 

continuous probability prediction as a dichotomous outcome (i.e., predicted probabilities above 

the threshold are defined as acceptable while others are defined as not acceptable). In addition 

to the cutoff threshold, other values can be defined to further enhance the structural analysis. 

For example, figure 1 describes a second threshold created at a probability of 0.4, and it is 

used to differentiate sections with fair structural condition from poor ones. It is recommended 

that the second threshold be initially defined as one-half of the cutoff value. Note that this 

definition is arbitrary and can be adjusted to better fit an agency’s rehabilitation practices or 

network characteristics.
10

Figure 1. Graph. Typical logistic model probability function of predictor variable X.

Once thresholds and their respective deflection parameters are identified, a structural decision 

matrix (also referred to as a structural rehabilitation matrix) can be generated. Table 2 shows the 

structural decision matrix obtained from the probability density function in figure 1. After the 

structural decision matrix has been established, it can be incorporated into the PMS and used in 

the analysis in a fashion similar to the functional decision matrix to prescribe maintenance and 

rehabilitation strategies.

Table 2. Example of structural decision matrix.

Structural 

Condition

Deflection 

Parameter

Good < 200

Fair 200–350

Poor > 350

The deflection parameter used to stochastically predict pavement performance in terms of typical 

surface distresses is shown in figure 2 and figure 3.

( ) 1

1 b P event

e− = +

Figure 2. Equation. General formulation of the logistic model.

0 11 2 2 n n b a ab ab ab =+ + ++ 

Figure 3. Equation. General linear formulation for the exponent term of the logistic model.
11

ANALYSIS TECHNIQUES FOR FLEXIBLE PAVEMENTS

STRUCTURALLY BASED ROUGHNESS PERFORMANCE MODELS FOR FLEXIBLE 

PAVEMENTS

The input variables needed to utilize the flexible pavement roughness model are as follows:

• The deflection at 9,000-lbf (40-kN) load at 8 inches (200 mm) from the load plate, D2.

• The volume of class 9 trucks regardless of their actual axle loads.

• The expended service life of the pavement section at the time of FWD testing (i.e., the 

elapsed time since construction or latest rehabilitation).

The most accurate roughness (IRI) model identified for flexible pavements is represented by the 

equations in figure 2 and figure 3 utilizing the variable values shown in table 3.

Table 3. Variables in the I2 final model for flexible pavements based on roughness 

performance.

Variables (bi) ai p-value

I2 239.849 0.000005706

Current life -0.189 0.01174

Class 9 volume -0.0006781 0.0001037

Constant 0.8375 0.01071

Cutoff 0.812

bi = Load transfer efficiency.

ai = Variables presented in figure 3.

Ii = Reciprocal indicator for the ith deflection value.

The variable I2 in table 3 is calculated using the equation in figure 4.

Figure 4. Equation. Calculation of I2.

Where D2 is the deflection 8 inches (200 mm) from the center of the FWD load plate.

The accuracy in logistic models is computed based on the ROC curve. The area under the ROC 

curve provides the accuracy of the model, which can be considered the equivalent of the R2 value 

used for linear models. In addition to accuracy, the ROC curve also provides the threshold that 

defines how the results from the probability density function are interpreted in a binary system 

(i.e., the threshold used to separate probabilities that are referred to acceptable and not 

acceptable). This threshold is called the cutoff value. For the model described in table 3 and 

figure 4, the accuracy is 0.72, and the cutoff value is 0.812.

I2 = 1

D2
12

STRUCTURALLY BASED RUTTING PERFORMANCE MODEL FOR FLEXIBLE 

PAVEMENTS

The input variables needed to utilize the flexible pavement rutting model are as follows:

• The deflection at 9,000-lbf (40-kN) load 12 inches (300 mm) from the load plate, D3.

• The deflection at 9,000-lbf (40-kN) load 18 inches (450 mm) from the load plate, D4.

• The volume of class 9 trucks regardless of their actual axle load levels.

• The average annual precipitation in the area (millimeters).

The most accurate rutting model identified for flexible pavements utilizes the variable values in 

table 4 and the equation in figure 5.

Table 4. Variables in the CI3 final model for flexible pavements based on rutting 

performance.

Variables (bi) ai p-value

CI3 -0.01146 0.00001594

Precipitation (mm) -0.0005259 0.04018

Class 9 volume -0.0007688 0.00005421

Constant 2.6586 1.0136×10-13

Cutoff 0.792

bi = Load transfer efficiency.

ai = Variables presented in figure 3.

CIi = Curvature index for the ith deflection value.

1 inch = 25.4 mm

The variable CI3 in table 4 is calculated using the equation in figure 5.

Figure 5. Equation. Calculation of CI3.

Where D3 is the deflection 12 inches (300 mm) from the center of the FWD load plate and D4 is

the deflection 18 inches (450 mm) from the center of the FWD load plate. The accuracy as 

measured by the area under the ROC curve is 0.66, and the cutoff value is 0.792.

CI3 = D3 - D4
13

STRUCTURALLY BASED FATIGUE CRACKING PERFORMANCE MODEL FOR 

FLEXIBLE PAVEMENTS

The input variables needed to utilize the flexible pavement fatigue cracking model are 

as follows:

• The deflection at 9,000-lbf (40-kN) load measured at the center of the load plate, D1.

• AADTT.

• The pavement type of the base course is a binary variable with a bound (value of 1) or 

unbound (value of 0) base layer.

The most accurate fatigue cracking model utilizes the equations in figure 2 and figure 3 with the 

variable values in table 5.

Table 5. Variables in the I1 final model for flexible pavements based on fatigue cracking 

performance.

Variables (bi) ai p-value

I1 154.764 0.0006015

AADTT -0.0005073 0.0007847

Pavement type 0.3774 0.07598

Constant -0.2202 0.4422

Cutoff 0.605

bi = Load transfer efficiency.

ai = Variables presented in figure 3.

Ii = Reciprocal indicator for the ith deflection value.

The variable I1 in table 5 is calculated using the equation in figure 6.

Figure 6. Equation. Calculation of I1.

Where D1 is the deflection at the center of the FWD load plate. The accuracy as measured by the 

area under the ROC curve is 0.72, and the cutoff value is 0.605.

I1 = 

1

D1

15

ANALYSIS TECHNIQUES FOR RIGID PAVEMENTS

Two sets of models have been developed for rigid pavements—one for a target FWD load of 

9,000 lbf (40 kN) and one for a target load of 12,000 lbf (53 kN). The overall accuracy of the 

9,000-lbf (40-kN) load relationships are at least as good as the relationships for the 12,000-lbf 

(53-kN) FWD load level. Accordingly, only the 9,000-lbf (40-kN) equations are shown in this 

section. Agencies that generally use a 12,000-lbf (53-kN) FWD load level for rigid pavements 

should refer to the final report for additional information.(1)

STRUCTURALLY BASED ROUGHNESS PERFORMANCE MODELS FOR RIGID 

PAVEMENTS

The input variables needed to utilize the rigid pavement roughness model are as follows:

• The deflection at 9,000-lbf (40-kN) load at 24 inches (600 mm) from the load plate, D5.

• The deflection at 9,000-lbf (40-kN) load at 36 inches (900 mm) from the load plate, D6.

• The volume of class 9 trucks regardless of their actual axle loads.

The most accurate roughness (IRI) model identified for rigid pavements is represented by the 

equation in figure 3 using the variable values in table 6.

Table 6. Variables in the CI5 final model for rigid pavements based on roughness 

performance.

Variables (bi) ai p-value

CI5 -0.078 0.00406

Class 9 volume -0.0003877 0.115

Constant 2.057 0.00002168

Cutoff 0.665

bi = Load transfer efficiency.

ai = Variables presented in figure 3.

CIi = Curvature index for the ith deflection value.

The variable CI5 in table 6 is calculated using the equation in figure 7.

Figure 7. Equation. Calculation of CI5.

Where D5 is the deflection 24 inches (600 mm) from the center of the FWD load plate and D6 is 

the deflection 36 inches (900 mm) from the center of the FWD load plate. The accuracy as 

measured by the area under the ROC curve is 0.65, and the cutoff value is 0.665.

CI5 = D5 - D6
16

STRUCTURALLY BASED JOINT FAULTING MODELS FOR RIGID PAVEMENTS

The input variables needed to utilize the rigid pavement joint faulting model are as follows:

• The deflection at 9,000-lbf (40-kN) load 36 inches (900 mm) from the load plate, D6.

• The subgrade type (fine-grained (cohesive) versus coarse-grained (non-cohesive)).

The most accurate joint faulting (i.e., step faulting) model identified for rigid pavements is 

represented by the equations in figure 2 and figure 3 using the variable values in table 7.

Table 7. Variables in the D6 final model for rigid pavements based on faulting at joints 

performance.

Variables (bi) ai p-value

D6 -0.009225 0.138

Subgrade type -0.495 0.2003

Constant 1.499 0.005818

Cutoff 0.635

bi = Load transfer efficiency.

ai = Variables presented in figure 3.

Where D6 is the deflection 36 inches (900 mm) from the center of the FWD load plate. The 

accuracy as measured by the area under the ROC curve is 0.64, and the cutoff value is 0.635.

STRUCTURALLY BASED TRANSVERSE SLAB CRACKING PERFORMANCE 

MODEL FOR RIGID PAVEMENTS

The input variables needed to utilize the rigid pavement transverse slab cracking model are 

as follows:

• The deflection at 9,000-lbf (40-kN) load 18 inches (457 mm) from the load plate, D4.

• The deflection at 9,000-lbf (40-kN) load 24 inches (610 mm) from the load plate, D5.

• The volume of class 9 trucks regardless of their actual axle loads.

The most accurate transverse slab cracking model identified for rigid pavements is represented 

by the equations in figure 2 and figure 3 using the variable values in table 8.
17

Table 8. Variables in the CI4 final model for rigid pavements based on transverse slab

cracking performance.

Variables (bi) ai p-value

CI4 -0.254 0.0003759

Class 9 volume -0.001012 0.001084

Constant 3.630 0.0000003134

Cutoff 0.7

bi = Load transfer efficiency.

ai = Variables presented in figure 3.

CIi = Curvature index for the ith deflection value.

The value for CI4 in table 8 is calculated using the equation in figure 8.

Figure 8. Equation. Calculation of CI4.

The accuracy as measured by the area under the ROC curve is 0.77, and the cutoff value is 0.70.

CI4 = D4 - D5

19

OPTIMUM SPACING AND TIMING

Optimum test spacings for FWD measurements are a function of the criteria used to determine 

what “optimal” means. One approach is to use statistical methods to determine the minimum 

number of deflection measurements that accurately (i.e., within an acceptable level of error) 

reflect the true structural condition of the entire highway segment in question. Another approach 

is to collect as much information as the budget, equipment, human resources, and allotted time 

will allow while minimizing risk to the travelling public and equipment operators.

Table 9 can be used to calculate the expected error in the average deflection as a result of a 

selected sampling strategy (spacing) for a given section length at a 90 percent reliability level.

Table 9. Errors in percentage for 90 percent reliability based on various section lengths,

sample spacings, and coefficient of variation of 33 percent.

Probability

Length 

(mile)

Spacing (mile)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

90 percent

1 14.18 17.94 21.24 24.22 26.98 29.57 32.02 34.35 36.59

2 10.10 12.71 14.98 17.03 18.93 20.70 22.37 23.96 25.48

3 8.28 10.38 12.21 13.86 15.38 16.80 18.13 19.41 20.62

4 7.19 9.00 10.56 11.97 13.27 14.48 15.63 16.71 17.75

5 6.44 8.05 9.44 10.69 11.84 12.91 13.92 14.88 15.80

6 5.89 7.35 8.61 9.74 10.78 11.75 12.67 13.54 14.36

7 5.46 6.80 7.97 9.01 9.97 10.86 11.70 12.49 13.25

8 5.11 6.36 7.45 8.42 9.31 10.14 10.92 11.66 12.36

9 4.82 6.00 7.02 7.93 8.76 9.54 10.27 10.96 11.62

10 4.58 5.69 6.65 7.51 8.30 9.03 9.72 10.38 11.00

1 mi = 1.61 km

Table 9 shows that if 10 10-mi (16.1-km) highway segments are tested at 0.2-mi (0.322-km) 

intervals, the average of the deflections measured in each section will be within ±4.58 percent of 

their true value for 9 of the 10 sections.

Practical considerations, such as access to the facilities (due to traffic or other factors), might 

force an agency to adopt wider test point spacings. If so, table 9 can be used to quantify the 

additional error in measurement accuracy. The main point to take away from this is that an 

agency should do what is reasonably possible under their own specific circumstances; if a less 

than desirable test spacing interval is all that is possible, it is better than nothing at all and will 

generally provide reasonable results for network-level applications.

21

CONSIDERATION OF TIME OF YEAR AND SEASON

There are two seasonal pavement conditions during which deflection testing should be avoided: 

frozen pavements and spring thaw. Neither of these conditions produces pavement deflections 

that are representative of the pavement response at most other times of the year. Agencies should 

conduct seasonal testing to determine when these occur and how long they last and then avoid 

collecting measurements during these times.

Analysis of LTPP deflection data collected over many years indicates that flexible pavements 

can be tested at the network level at 5-year cycles. Rigid pavements may be tested at up to 

10-year intervals.

23

LOCAL CALIBRATION

The process for selecting the best fitting technique for local calibration is relatively simple. A 

complete example of calibration is described in chapter 4 of the final report.

(1) A summary of the 

steps is provided in this section. 

The process for selecting the best model consists of the following steps:

1. Data preparation.

2. Logistic model calibration.

3. Verification of accuracy.

4. Sensitivity analysis.

DATA PREPARATION

Data availability is important. Logistic regression models are obtained from the statistical 

analysis of several independent variables that potentially have an impact on pavement 

performance. Initially, no variable should be eliminated without being properly assessed. 

Typical groups of variables include deflection, structure, traffic, climate, and performance. 

It is important that any available variable in these categories is used; however, dependent 

variables that are directly related to each other (interdependent) should be avoided.

Deflections obtained from FWD testing are used to compute deflection parameters to be used as 

independent variables in the logistic model. A detailed literature review of the most used and 

more successful applications was carried out as part of this research and is described in chapter 3 

in the final report.(1)

It is recommended that thickness and material type be used as structure variables. In addition, 

two dichotomous variables based on the structure characteristics are recommended: one related 

to the subgrade type (coarse or fine) and the other related to the base type (bound or unbound).

It is recommended that traffic data be limited to truck volume using AADTT and the volume of 

the two most predominant classes, classes 5 and 9 in the FHWA truck classification system.

For climate conditions, two variables are recommended: precipitation and temperature. For 

simplification, annual averages can be used for both. It is expected that there will not be 

significant variations in climatic conditions for local calibration. If that is the case, climatic 

variables may be excluded from the analysis.

The main characteristic of a logistic model is the prediction of the probability of an event 

occurring given a set of independent variables. The assumption is that the rate of deterioration 

increases as the structural condition worsens, and the probability of a pavement failing 

prematurely can be calculated based on a combination of deflection parameters and site 

characteristics. Therefore, pavement performance must be transformed in a dichotomous 
24

variable representing the occurrence of this event (acceptable performance after a number of 

years of service). Service life can be arbitrarily chosen or estimated to reflect the expected 

performance in the network as a whole.

LOGISTIC MODEL CALIBRATION

The calibration and evaluation of selected deflection techniques are based on the logistic model 

approach and the notion that the best deflection analysis techniques yield the most accurate 

models. Each deflection parameter available can be used to develop a logistic model for each 

performance measure in the dataset. A statistical package should be used to expedite the process, 

although a spreadsheet can be used to program the optimization algorithm. The general 

formulation of the logistic model is shown in figure 2 and figure 3.

The model’s exponent, b, is calibrated to minimize the false positives and false negatives in the 

probability analysis. The significance of each variable is computed, and the least significant 

variable is removed from the calculation. This process continues until no improvement in 

accuracy can be made or the maximum number of iterations is achieved.

VERIFICATION OF ACCURACY

Logistic model accuracy is best evaluated through an analysis of an ROC curve, which allows 

researchers to select possibly optimal models and discard suboptimal ones independently from 

the class distribution of the predictor. An example of an ROC curve is provided in figure 9.

Figure 9. Graph. Example of an ROC curve.

The ROC curve is drawn using the true positive rate (TPR) and false positive rate (FPR). TPR is 

determined by the number of true positive predictions normalized by the total number of positive 

observed values. Conversely, FPR defines how many incorrect positive results occur among all 

negative samples available during the test.
25

The ROC space is defined by FPR and TPR as the x- and y-axes, respectively, which depict the 

relative trade-off between TPR and FPR. TPR is often described as the sensitivity of the model, 

while (1 − FPR) is often seen as the true negative rate (TNR) or specificity. Therefore, the ROC 

curve is also referred to as the sensitivity versus (1 − specificity) curve. Each case consisting of 

prediction and observed values represents one point in the ROC space. Details about the ROC 

curve can be found in the chapter 4 of the final report.(1)

The area under the ROC curve defines the goodness of fit of the predictions. In an analogy with 

measures of linear optimization techniques, the area under the ROC curve can be viewed as a 

stand-in for the R2 value. In addition to the goodness of fit, the ROC curve is used to define the 

cutoff value of the model. The cutoff is an equally important parameter that defines how the 

results from the probability density function can be interpreted in a binary system (i.e., the 

threshold used to separate probabilities are referred to as acceptable and not acceptable). The 

cutoff is defined as the probability value that corresponds to the point in the ROC curve closest 

to the perfect classification point (0,1). When the cutoff is used to convert the predicted 

probabilities into binary outcomes, the model yields the highest level of accuracy (i.e., highest 

TPR and TNR).

SENSITIVITY ANALYSIS

The last step in the analysis process is the sensitivity of the logistic models to its variables. 

It is important that the model produces as accurate results as possible and that the model is 

accompanied by reasonable probabilities with reasonable values of the deflection parameter 

under consideration. Plots of predicted probability versus the deflection parameter can 

provide valuable information about the reasonableness of the model for a given set of site 

characteristic variables (e.g., a high truck volume, unbound base, cold climate, etc.). If

results from the sensitivity plots are unreasonable, a new model must be calibrated to 

eliminate unrealistic outcomes.

The sensitivity analysis is executed by evaluating the predicted probability for a variety of values 

of the deflection parameter governing the logistic model under consideration. The objective is to 

verify if reasonable deflection parameter values produce expected and not counterintuitive 

outcomes. Two practical examples using the logistic model are provided below.

Sensitivity Analysis with Reasonable Results

Figure 10 describes the sensitivity curve of an I2 model based on roughness performance. 

In this case, the cutoff value is 0.812, which corresponds to I2 = 0.1092 1/mil (0.0043 1/ )

(or D2 equals 9.05 mil (232 )). This is a reasonable value for a well-designed, high traffic 

volume pavement structure. Therefore, figure 10 indicates that flexible pavements with D2

deflections below 9.05 mil (232 ) have greater than 80 percent chance of exhibiting 

acceptable roughness performance for the next 12 years and can be classified as good in 

terms of their structural condition.

μm

μm

μm
26

Figure 10. Graph. Sensitivity of roughness acceptable probability to deflection parameter 

I2 for flexible pavements.

Sensitivity Analysis with Unreasonable Results

In contrast, figure 11 shows the sensitivity curve of the Hogg subgrade modulus model based on 

roughness. The cutoff value of 0.802 yields a Hogg value of 65,000 psi (447,850 kPa), which is 

very high and unreasonable for subgrade moduli. The Hogg model in this case is biased toward 

low probabilities of acceptable performance.

Figure 11. Graph. Sensitivity of roughness acceptable probability to deflection parameter 

Hogg for flexible pavements.
27

This type of sensitivity analysis was performed for all models developed and proved to be 

essential in order to rule out models that at first glance provided very good statistics but were 

unreasonably biased.

An example using the logistic model based on Maryland State Highway Administration data is 

presented in the accompanying final report.

(1)

29

REFERENCES

1. Carvalho, R., Stubstad, R., Briggs, R., Selezneva, O., Mustafa, E., and Ramachandran, A. 

(2012). Simplified Techniques for Evaluation and Interpretation of Pavement Deflections for 

Network-Level Analysis, Report No. FHWA-HRT-12-023, Federal Highway Administration, 

Washington, DC.

2. Strategic Highway Research Program. (1993). Manual for FWD Testing in the Long-Term 

Pavement Performance Program, 9, Report No. SHRP-P-661, Version 2.0, National 

Research Council, Washington, DC.

3. Turner-Fairbank Highway Research Center. (2011). How to Get LTPP Data, Federal 

Highway Administration, Washington, DC. Obtained from: http://www.fhwa.dot.gov/

research/tfhrc/programs/infrastructure/pavements/ltpp/getdata.cfm.


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