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STP 1198

Nondestructive Testing

of Pavements and Backcalculation

of Moduli: Second Volume

Harold L. Von Quintus, Albert J. Bush, III, and

Gilbert Y. Baladi, Editors

ASTM Publication Code Number (PCN)

04-011980-08

AsTM

1916 Race Street

Philadelphia, PA 19103

Printed in the U.S.A.

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Library of Congress Cataloging-in-Publication Data

Nondestructive testing of pavements of backcalculation of moduli,

Second volume/Harold L. Von Quintus, Albert J. Bush, III, and

Gilbert Y. Baladi, editors.

p. cm. -- (STP 1198)

Contains papers presented at the symposium held in Atlanta, GA on

23-24 June 1993, sponsored by ASTM Committee D-18 on Soil and Rock and

its Subcommittee D4 on Road Paving Materials.

"ASTM publication code number (PCN) 04-011980-08."

Includes bibliographical references and index.

ISBN 0-8031-1865-1

1. Pavements--Testing~ongresses. 2. Nondestructive testing---Congresses.

I. Von Quintus, H. L. (Harold L.) IL Bush, A. J. (Albert Jasper) IlL

Baladi, Gilbert Y., 1943- IV. ASTM Committee D18 on Soil and Rock.

Subcommittee D-4 on Road and Paving Materials. V. Series:

ASTM special technical publication: 1198

TE250.N572 1994

625.8'028'7~c20 94-24308

CIP

Copyright 9 1994 AMERICAN SOCIETY FOR TESTING AND MATERIALS, Philadelphia, PA. All

rights reserved. This material may not be reproduced or copied, in whole or in part, in any printed,

mechanical, electronic, film, or other distribution and storage media, without the written consent of the

publisher.

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specific clients, is granted by the AMERICAN SOCIETY FOR TESTING AND MATERIALS for users

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the base fee of $2.50 per copy, plus $0.50 per page is paid directly to CCC, 222 ROSEWOOD DR.,

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have been granted a photocopy license by CCC, a separate system of payment has been arranged.

The fee code for users of the Transactional Reporting Service is 0-8031-1865-1--91 $2.50 + .50.

Peer Review Policy

Each paper published in this volume was evaluated by three peer reviewers. The authors

addressed all of the reviewers' comments to the satisfaction of both the technical editor(s) and the

ASTM Committee on Publications.

To make technical information available as quickly as possible, the peer-reviewed papers in this

publication were printed "camera-ready" as submitted by the authors.

The quality of the'papers in this publication reflects not only the obvious efforts of the authors and

the technical editor(s), but also the work of these peer reviewers. The ASTM Committee on

Publications acknowledges with appreciation their dedication and contribution to time and effort on

behalf of ASTM.

Printed in Ann Arbor, MI

December 1994

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Foreword

This publication, Nondestructive Testing of Pavements and Backcalculation of Moduli (Sec￾ond Volume), contains papers presented at the symposium of the same name held in Atlanta,

GA on 23-24 June 1993. The symposium was sponsored by ASTM Committee DI8 on Soil

and Rock and its Subcommittee D4 on Road and Paving Materials. Albert J. Bush, II1, of U.S.

Army Corps of Engineers in Vicksburg, MS, Harold L. Von Quintus of Brent Rauhut Engi￾neering in Austin, TX, and Gilbert Y. Baladi of Michigan State University in East Lansing, MI

presided as symposium chairmen and are the editors of the resulting publication.

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Contents

Overview ix

ANALYTICAL MODELS AND TECHNIQUES

Advanced Backcalculation Techniques---J. UZAN

SHRP's Layer Moduli Backcalculation Procedure---G. RADA, C. RICHTER,

AND P. JORDAHL

Methodology for Identifying Material Properties in Pavements Modeled as Layered

Viscoelastic Half Spaces--N. STUBBS, V. TORPUNURI, R. LYTTON,

AND A. MAGNUSON

Backcalculation of Pavement Layer Moduli, Thicknesses, and Bedrock Depth

Using a Modified Newton Method--R. HARRICHANDRAN, T, MAHWOOD,

R, RAAD, AND G. BALADI

Improved Methods for AC/PCC Pavement Backcalculation and Evaluation--K.

HALL AND M. DARTER

Concrete Pavement Backcalculation Using ILLI-BACK--A. IOANNIDES

Dynamic Analysis of FWD Loading and Pavement Response Using a Three￾Dimensional Dynamic Finite--Element Program--s. ZAGHLOUL, T. WHITE,

V. DRNEVICH, AND B. COREE

38

53

68

83

103

125

MEASUREMENT AND CALCULATION TECHNIQUES IN THE FIELD AND LABORATORY

Verification of Pavement Response Models--a. ULLIDTZ, J. KRARUP,

AND T. WAHLMAN

Field Validation of a Methodology to Identify Material Properties in Pavements

Modeled as Layered Viscoelastic Halfspaces---v. s. TORPUNUR, N. STUBBS,

R. L. LYTTON, AND A. H. MAGNUSON

Comparing Laboratory and Backcalculated Layer Moduli on Instrumented

Pavement Sections---T. AKRAM, T. SCULLION, AND R. E. SMITH

V

143

159

170

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In-Situ and Laboratory Characterization of Nonlinear Pavement Layer Moduli--

K. P. GEORGE AND W. UDDIN

A Pavement Evaluation Procedure Incorporating Material Non-Linearity--

J. R. DE ALMEIDA, S. F. BROWN, AND N. H. THOM

Effect of Material Stress Sensitivity on Backcalculated Moduli and Pavement

Evaluation--R. N. STUBSTAD, J. P. MAHONEY, AND N. F. COETZEE

203

218

233

PROBLEMS/ERRORS ASSOCIATED WITH BACKCALCULATION METHODS AND DESIGN PARAMETERS

Misleading Results from Nondestructive Testing--A CASE STUDY--J. W. HALL, JR.,

AND P. S. MCCAFFREY, JR. 251

Stochastic Analysis of Errors in Remaining Life Due to Misestimation of Pavement

Parameters in NDT--K. M. VENNALAGANTI, C. FERREGUT, AND S. NAZARIAN 261

Interpretation of Dynamic Survey Measurement on Pavement with Treated

Roadbase--P. LEPERT, AND P. CAPRIOLI 277

Phase LAG Effects on Analysis of FWD Data--P. E. SEBAALY, AND S. HOLIKAq'TI 291

The Effect of Annular Load Distributions on the Backcalculated of Moduli of

Asphalt Pavement Layers---J. A. CROVETTI, AND M. Y. SHAHIN 309

Effects of Pavement-Falling Weight Deflectomer Interaction on Measured

Pavement Response---K. M. BODDAPATI, AND S. NAZARIAN 326

NDT FOR OTHER PAVEMENT USES

Radar for Pavement Thickness Evaluation--K. R. MASER, T. SCULLION,

W. M. K. RODDIS, AND E. FERNANDO 343

A Comparison of Laboratory and Field Subgrade Moduli at the Minnesota Road

Research Project--D. A. VAN DEUSEN, C. A. LENNGREN, AND D. E. NEWCOMB 361

Detection of Multi-Course Pavement Layers by the SASW Method--N. GUCUNSrd 380

Strategies for Application of the Falling Weight Deflectomer to Evaluate Load

Transfer Efficiency at Joints in Jointed Concrete Pavements--o. J. JACKSON,

M. R. MURPHY, AND A. WIMSATT 395

Field Testing and Structural Evaluation of Selected Concrete Pavement Sections in

Florida---c. L. WU, AND M. TIA 404

Backcalculation of System Parameters for Jointed Rigid Pavements---

D. R. HILTUNEN AND R. ROQUE 440

vi

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Evaluation of Support Conditions Under Jointed Concrete Pavement Slabs--

J. A. CROVETTI AND M. R. T. CROVETTI

Determination of Voids Under Rigid Pavements Using Impulse Method￾s. NAZARIAN, S. REDDY AND M, BAKER

Evaluation of NDT Equipment for Measuring Voids Under Concrete Pavements---

w. UDDIN AND W. R. HUDSON

455

473

488

PROPOSED STANDARD GUIDE

The Quest for a Standard Guide to NDT Backealculation--R. w. MAY AND

H. L. VON QUINTUS

Author Index

Subject Index

505

521

523

vii

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Overview

In June 1988, the first International Symposium on Nondestructive Testing (NDT) of Pave￾ments and Backcalculation of layer moduli was held. Since then, another symposium on NDT

and backcalculation of layer moduli was held in August of 1991 and was sponsored by the

Transportation Research Board. Both of these symposia were well attended, and showed that

there was a strong interest within the transportation community in the area of NDT and the use

of deflection data for evaluating and designing pavement structures. Unfortunately, these two

symposia also showed that the industry was divided regarding the adequacy and use of state￾of-the-art evaluation procedures for determining structural capacity of pavement structures.

As a result of the first symposium in 1988, ASTM Subcommittees D18.10 and D04.39 have

been extensively involved in the preparation of standardized procedures for NDT and the

evaluation of deflection data. Standardized procedures have been prepared and approved for

collecting deflection data with different devices. These are listed below for reference purposes:

D 4602 Standard Guide for Nondestructive Testing of Pavements Using Cyclic Loading

Dynamic Deflection Equipment

D 4694 Standard Test Method for Deflections with a Falling-Weight-Type Impulse Load

Device

D 4695 Standard Guide for General Pavement Deflection Measurements

The task of standardizing backcalculation procedures, however, has been more difficult,

because of the diversity of opinions and procedures currently in use by the transportation

industry. The first draft of a standard guide for backcalculation of layer moduli from deflection

measurements was balloted in 1986. The latest draft balloted in 1992 received numerous neg￾ative ballots that were found to be persuasive. More recently, there have been numerous re￾search projects completed by individual transportation agencies and as part of the Strategic

Highway Research Program (SHRP).

With these recent advancements and the need to develop concurrence within the transpor￾tation industry to develop a standardized evaluation procedure, Subcommittees D18.10 and

D04.39 suggested to the Executive Committees that ASTM sponsor the second International

Symposium on Nondestructive Testing of Pavements and Backcalculation of Moduli. This

Second International Symposium was held in Atlanta, Georgia in June, 1993. The attendance

at this symposium exceeded 80, representing 12 different countries and 25 states in the United

States. An attendance list is included at the end of this publication.

The symposium was divided into four sessions (two sessions per day) and one panel work￾shop or discussion on issues related to standardization of backcalculation procedures. The

papers presented at this Second International Symposium focused in the area of backcalculation

of layer moduli techniques and comparisons of material moduli as measured in the laboratory

to values calculated from field deflection measurements. Information from these papers and

discussion were used to establish whether a backcalculation procedure could be standardized

based upon the current state-of-the-art technology. The format of the presentations was divided

into four sessions followed by a panel discussion. Each of the sessions were subdivided into

two parts as follows:

ix

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SESSION 1DAnalytical Models and Techniques for Backcalculation of Layer Moduli (5

Papers).

Chairman--Dr. Albert J. Bush III, U.S. Army Corps of Engineers, Waterways Experiment

Station, Vicksburg, MS.

Part 1 of Session 1: Recent Developments and Tools to be Used in the Future for

Evaluating Pavements Based on Backcalculation Techniques (2 Papers).

Keynote Speaker--Dr. Jacob Uzan, Professor, Israel Institute of Technology (Technion),

Israel, "Advanced Backcalculation Techniques."

Part 2 of Session 1: Methods and Procedures Used for Backcalculation of Material and

Pavement Properties (4 Papers).

SESSION 2DMeasurement and Calculation Techniques in the Field and Laborator~

Chairman--Mr. Harold L. Von Quintus, President, Brent Rauhut Engineering Inc., Austin,

TX.

Part 1 of Session 2: Verification of backcalculation techniques and comparisons of

laboratory measured values with those calculated from field measurements or deflections

(4 papers).

Part 2 of Session 2: Characterization of Pavement Materials and the Effects of Non

Linearity on Backcalculation of Layer Moduli (4 papers).

SESSION 3--NDT for Pavement Structural Evaluation, Design and Rehabilitation.

Chairman--Dr. Albert J. Bush III, U.S. Army Corps of Engineers, Waterways Experiment

Station, Vicksburg, MS.

Part 1 of Session 3: Problems/errors associated with backcalculation methods in terms

of pavement evaluation, and backcalculation of design parameters for concrete pave￾ments (4 papers).

Part 2 of Session 3: Analysis of deflection measurements and effects of load distributions

on pavement response (4 papers).

SESSION 4---NDT for Other Pavement Uses: Use of the Results From NDT to Determine

Layer Thickness, Joint Efficiency, and Void Detection (5 Papers).

Chairman--Dr. Gilbert Y. Baladi, Professor, Michigan State University, East Lansing,

MI.

SESSION 5--Panel Discussion on Backcalculation of Layer Moduli

Chairman--Dr. Gilbert Y. Baladi, Professor, Michigan State University, East Lansing,

MI.

Discussion paper presented by Richard May, Asphalt Institute, Lexington, KY and

Harold L. Von Quintus, Brent Rauhut Engineering, Austin, TS entitled "The Quest for

a Standard Guide to NDT Backcalculation".

Panel participants: Dr. Albert J. Bush III., U.S. Army of Engineers, Waterways Exper￾iment Station, Vicksburg, MS. Dr. Jacob Uzan, Israel Institute of Technology (Tech￾nion), Israel; Richter, Federal Highway Administration, Turner Fairbanks, Washington,

DC; Dr. Ullditz, Technical University of Denmark, Denmark, and Luckanen, Braun

Intertec, Minneapolis, MN.

Papers m this STP are presented on those topics in the four sessions listed previously. These

papers include examples of different backcalculation of layer moduli procedures, comparisons

X

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between laboratory measured and field calculated values, as well as, the more common ex￾amples on the use of deflection testing to evaluate pavement structures. The papers published

represent eight different countries, eleven different states, and thirteen different educational

agencies. It is the hope of the organizers of this symposium that the papers presented will

provide the readers with much of the latest information in the areas of pavement evaluation

using NDT techniques, and application of that data for use in pavement design.

One of the goals and objectives of this symposium was to determine if the industry could

find a common ground to standardize a backcalculation procedure. In specific, this was the

focus of the panel discussion at the end of the symposium. This panel discussion was preceded

by a paper entitled "The Quest for a Standard Guide to NDT Backcalculation" (presented by

Mr. Richard May) and a presentation by Dr. Albert Bush (Symposium Cochairman and D4.39

Subcommittee Chairman) entitled "Where We Go From Here."

From the question and answers during the panel discussion, it was the general consensus

that backcalculation of layer moduli from deflection measurements will definitely be used in

the future for the rehabilitation design and evaluation of pavement structures. The question

however, is still: what is the reliability of these values? Specifically, it was the general consensus

of the panel and attendees that the accuracy of backcalculated moduli is model dependent and

unknown, as well as those values measured in the laboratory because there is a diversity of

opinion on the simulation of field conditions in the laboratory. For example, there is controversy

within the industry on whether backcalculation procedures should be based on a dynamic or

static analysis, and what values actually represent the "truth," both in the laboratory or from

field measurements.

In summary, most participants, concurred that there needs to be a standard "baseline" of

values from which to compare on a project, material, or pavement bases, and that one should

not become paralyzed by the imperfection of the procedures. More importantly, research must

be merged into practice on a consistent basis and one way to accomplish this is through the

standardization process. As such, a procedure needs to be standardized and that procedure

should concentrate on user oriented issues. Thus, the editors, panel, as well as most symposium

participants involved in these discussions, believe that some standardized procedure should be

pursued to ensure that a common set of values can be compared.

The editors wish to thank all those who participated in this symposium and who contributed

to this STP. Special thanks are given to the authors, the reviewers of the papers, ASTM Com￾mittees D18 and D4 for sponsoring the symposium, and to the members of Subcommittees

D18.10 and D04.39 for their valuable input and efforts. Last but not least, the editors would

like to express their deep appreciation to the ASTM staff for their assistance in preparing for

this symposium and in its preparation. The high professional quality of ASTM publications

would not be possible without their dedicated and professional efforts.

Dr. Albert J. Bush III

U.S. Army Corps of Engineers, Waterways

Experiment Station, Vicksburg, MS; symposium

cochairman and coeditor.

Mr. Harold L. Von Quintus

President, Brent Rauhut Engineering, Austin, Texas,

symposium cochairman and editor

Dr. Gilbert Y. Baladi

Professor of Civil Engineering, Michigan State

University, East Lansing, Michigan, symposium

cochairman and coeditor

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Analytical Models and Techniques

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Jacob Uzan ~

ADVANCED BACKCALCULATION TECHNIQUES

REFERENCE: Uzan, J., "Advanced Backcalculation Techniques,"

Nondestructive Testin~ of Pavements and Backcalculation of ~odu~i

(Second Volume~. ASTM STP 1198, Harold L. Von Quintus, Albert J.

Bush, III, and Gilbert Y. Baladi, Eds., American Society for

Testing and Materials, Philadelphia, 1994.

ABSTRACT: The backcalculation procedures are separated into five

categories; (a) static linear elastic, (b) static nonlinear elastic, (c)

dynamic linear using frequency domain fitting, (d) dynamic linear using

time domain fitting and (e) dynamic nonlinear analysis. In this paper

each category is described and case studies are presented comparing

their results. Advanced techniques require more complete material

characterization models. In the nonlinear elastic procedure a universal

k I - ~ model (an extension of the bulk modulus model) is used. In the

dynamic analysis technique a generalized power law relationship is used

for the asphaltic layer. Two case studies are presented. In the first

the nonlinear elastic scheme was found to give excellent results at

matching deflection bowls at four different load levels for each of two

test sites analyzed. In these analyses only the k I of the k I - ~ model

was backcalculated. It was found that the backcalculated k I in all

layers including the asphalt concrete are larger than those measured in

the laboratory. In the second case study dynamic analysis techniques are

applied to full wave shape data obtained from the SHRP data base. Both

the frequency and time domain procedures are shown to yield reasonable

results.

KEYWORDS: linear, nonlinear, dynamic backcalculation, elastic and

viscoelastic material, flexible pavements

INTRODUCTION

Backcalculation of moduli of pavement material, is nowadays widely

used for structural evaluation and rehabilitation. The number of

existing procedures and computer programs for this purpose are

relatively large (Rada et al. 1992). Moreover different modulus values

may be obtained from these different programs (Lytton 1989; Chou and

Lytton 1991). This indicates that backcalculation is very sensitive to

I Associate Professor of Civil Engineering, TECHNION, Israel Institute

of Technology, Haifa, Israel 32000

3

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4 NONDESTRUCTIVE TESTING: SECOND VOLUME

the kind of analysis and assumptions underlying the analysis. It is the

intent of the paper to present a synthesis of the backcalculation

procedures and a discussion of their limitations.

All backcalculation procedures use error minimization techniques

to minimize either the absolute or the squared error, with or without

weighing factors. The most common backcalculation procedure is based on

static loading type of analysis and linear elastic material response.

However, most of the loading devices apply either a vibratory load or an

impact load. Also, the pavement materials are in most cases far from

being linear elastic. In recognition of these conditions, the need for

applying more advanced techniques using nonlinear and dynamic analysis

is becoming stronger everyday. Researchers and engineers are more aware

of the badness (not goodness) of fit, and of assumptions leading to

large errors in the fit. This paper presents the author's evaluation of

these advanced techniques and recommendations for future work. However,

it should be kept in mind that these analysis procedures are not

currently ready to be implemented due to the lack of knowledge on the

behavior of the pavement materials and to the lack of confidence in the

accuracy of the time history deflection bowls. A section of the paper

will be devoted to discuss the material characterization most

appropriate for backcalculation.

Examples are presented to illustrate the procedures developed.

These are based on data collected using Falling Weight Deflectometer

(FWD) equipment. The procedures work with relatively low frequency

components (such as those induced under both FWD and truck loading).

They do not include high frequency loading procedures such as spectral

analysis of surface waves (SASW).

BACKCALCULATION CATEGORIES

The backcalculation procedures can be separated into several

categories, depending on the type of load representation - static versus

dynamic and on the type of material characterization - linear versus

nonlinear for elastic, viscoelastic and/or plastic materials. A

discussion of each is presented below. All backcalculation procedures

use error minimization techniques to minimize either the absolute or the

squared error, with or without weighing factors.

Static Linear Backcaleulation

In the simplest case which is widely used today, the load is

assumed to be static and the material is assumed to be linear elastic.

In this case only the peak load and peak surface response deflections

are used in the backcalculations (Figure la). The problem reduces to

finding the unit response of the pavement that will correspond to the

measured response. The unit response of the pavement is computed using

appropriate computer programs for linear elastic multi-layer systems.

The unit response is defined by the set of moduli of the pavement

layers. Therefore, the problem is reduced again to finding the set of

moduli that produces a unit pavement response close to the measured one.

When several load levels are applied in the test, each load level

is analyzed separately and separate sets of moduli are obtained. The

procedures for backcalculation using the above scheme are numerous.

However, most have two essential differences. These differences are: (I)

the forward computation of the unit response which are based on

numerical integration (such as Peutz et al. 1968; BISAR User's Manual

1972; WESLEA - Van Cauwelaert et al. 1989) or some approximation (such

as the Method cf Equivalent Thickness MET - Ullidtz 1977; Lytton 1989),

and (2) the error minimization scheme. These differences can lead to

different backcalculated moduli. However, when both the forward

computation programs and the minimization schemes are correct, the

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Linear Ela~%!ic - Static

Given:

Peak applied load I = max P

Peak surface deflections R = max d k

(k= number of sensors )

Find:

Hk(E) such as

max d k • Hk(E ) * max P

N n~J[l~ar_E/astic - Static

Given:

Peak applied loads I = max Pj

Peak surface deflections R = max dkj

( k= number of sensors )

( j= number of load levels)

Find:

Hk(E) such as

max d~ = Hk(E ) * max Pj

Linear - Dynam~

~torv load

_Given:

Load functions I = P exp(iwt)

Deflection functions R = d k exp(iwt+&k)

( k= number o/sensors )

Find:

Hk(E*) such as

R ~- Hk(E')* I

(a) {b) {cJ

FIG. 1 a-c-- Backcalculation schemes (a) linear elastic (b) nonlinear elastic and

(c) dynamic for steady state vibratory load

c

N

>

Z

0

Z

>

z

0

m

>

g

t￾O

E

-I

2:

(71

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6 NONDESTRUCTIVE TESTING: SECOND VOLUME

backcalculated moduli are in general similar (Lytton 1989). It is worth

mentioning that the emphasis is on the correctness of the computation,

not on the ease or speed of computation. For example, the use of MET

may in some cases (for example, for varying, decreasing and increasing

moduli with depth) may lead to an unacceptable error in the forward

computation of the response of the pavement, and thus in the

backcalculation.

Static, Nonlinear Backcalculation

In the static, non-linear elastic backcalculation, only the peaks

of the loads and surface deflections are used (Figure ib). In contrast

with the linear scheme, all load level~ are used simultaneously. In

other words, the problem is to find the response function of the

pavement that will correspond to the measured response at all load

levels. The unit response is defined by the set of material parameters

and is usually computed using Finite Element (FE) computer programs

(Uzan and Scullion 1990). The material characterization is of prime

importance in this category of backcalculation and will be discussed in

a separate section. There is only one such computer program for non￾linear elastic backcalculation developed by the writer. Results of

analysis with this program will be presented later. However, other

computer programs using linear elastic multi-layer systems and different

kinds of approximations exist and are rather widely used (ELMOD -

Ullidtz 1977; PADAL - Brown et al. 1987; MODCOMP - Irwin and Szebenyi

1991; FWDCHECK - PCS/Law Engineering 1990). These programs incorporate

an inter-relationship between surface deflection at a particular radial

location on a deflection bowl and the elastic stiffness of a particular

pavement layer. It is difficult to evaluate the correctness of these

programs, because they are not truly non-linear analyses. Moreover,

they do not represent true pavement materials because they do not

account for dilation and lack of tensile strength effects whenever these

conditions prevail.

The non-linear elastic backcalculation still assumes that the

permanent deformation is small compared to the resilient one. In other

words, the state of stress within the pavement structure is low relative

to the ultimate strength. This situation applies to pavements with a

moderate to thick AC layer, not with thin surfacing of less than 50 mm

(2 inches). In these cases of thin pavements and/or relatively heavy

applied loads, the non-linear elastic theory should be replaced by the

non-linear elasto-plastic theory. This will account for any permanent

deformation and stress redistribution caused during loading as compared

to the elastic behavior. It must be noted that the response function of

the pavement depends on both the non-linear elastic and the plastic

parameters. The number of parameters may become too large to be

resolved by the backcalculation procedure alone, and additional

information concerning the material properties must be supplied before

the analysis is initiated.

Dynamic Linear Backcaloulation

The dynamic backcalculation applies to the NDT equipment that

apply either a steady state vibratory load or an impact load. In the

case of steady state vibratory load with a finite number of frequencies,

the problem is reduced to finding the unit response of the pavement that

will correspond to the measured response (Figure lc). The unit response

is defined by the set of complex moduli of the pavement layers.

Therefore, the problem is reduced again to finding the set of complex

moduli (or viscoelastic parameters) that produces a unit response close

to the measured one.

When several responses at different frequencies are measured, each

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