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Publication Number: FHWA-HRT-05-152
Date: February 2006

Guidelines for Review and Evaluation of Backcalculation Results

Chapter 1. Introduction


Backcalculation has long created exciting opportunities along with puzzling obstacles to the analyst assigned to the task of deriving layered elastic parameters based on falling weight deflectometer (FWD) load-deflection data. Each available program has its own set strengths and weaknesses, and no two programs give exactly the same set of results. With luck, the results of two different Backcalculation programs may be close to one another.

An approach called forward calculation has been developed through the Federal Highway Administration (FHWA) Long-Term Pavement Performance (LTPP) program in a report titled, Review of the Long-Term Pavement Performance Backcalculation Results-Final Report.(1) This spreadsheet-based procedure may be used to screen other methods of determining layered elastic properties or as a stand-alone method of determining layered elastic properties for routine pavement rehabilitation design.

This document presents a screening approach for review and evaluation of backcalculated moduli. This approach, which allows users to choose any Backcalculation program they wish, offers forward calculated values that may be used to compare the results of the two or more methods of evaluation. Ideally, if these approaches give similar (but still not identical) results, one can be reasonably confident that the results obtained through either method will be reasonable and tenable for further use in pavement evaluation and rehabilitation design.

Backcalculation in a Nutshell

Most Backcalculation programs, including those used to generate the back calculated modulus data in the LTPP-computed parameter tables, involve the use of numerical integration subroutines that are capable of calculating FWD pavement deflections and other parameters, given the stiffnesses (or moduli) of the various pavement layers and their thicknesses, etc. If all assumptions are correct, meaning each layer is an elastic layer, is isotropic and homogeneous, and all other boundary conditions are correct, then it is possible to iterate various combinations of moduli until there is a virtually perfect match between measured and theoretical FWD deflections. In this manner, a solution to the problem of deriving moduli from deflections, instead of the other way around, is obtained.

A serious drawback to this approach is the fact that one or more of the many input assumptions mentioned above may be incorrect and therefore do not apply to the actual in situ pavement system where FWD was used to measure deflections. In spite of this potential drawback, many of the moduli derived through Backcalculation will appear both reasonable and rational, based on common engineering sense and a working knowledge of pavement materials. This conclusion appears to be especially true when relatively intact, well-defined, and undistressed pavement sections are tested with FWD.

For any pavement system, the engineer using a Backcalculation program of choice should be very well versed in its proper use and inherent limitations. Such an expert is able to fine-tune the input data to better model the layered elastic system in a manner that is both rational and suits the particular Backcalculation program and the structure of the input data in an advantageous manner. Accordingly, Backcalculation is arguably more of an art than a science.

Through forward calculation, presented below, it is now possible to screen back calculated results to see if these results are in the ballpark. For routine testing that is not research related, forward calculation may also be used as a stand-alone method of pavement analysis and rehabilitation design.

Introduction to Forwardcalculation

Through the forward calculation method presented below, it was possible to screen the entire LTPP Backcalculation database, which consists of a series of back calculated computed parameter tables based on the pre-1998 FWD data in the database. Forwardcalculation was used to generate a set of layered elastic moduli that is independent of the back calculated values, for comparison purposes, to screen the back calculated moduli in the database. This approach is based on the premise that two substantially different approaches to calculated layered elastic parameters from the same deflection data should produce at least somewhat similar moduli if one is to believe that either approach is credible.

In its current form, forward calculation only involves the use of certain portions of the FWD deflection basin to derive an apparent or effective modulus (stiffness) of the subgrade and/or the bound surface course, using closed-form as opposed to iterative solutions. In other words, there is only one directly calculated solution for each of these values, given the deflection data and the layer thicknesses.

The forward calculation formulae used to deduce the subgrade modulus mainly use deflections measured at larger distances from the load as well as the center deflection, while the surface course modulus is mainly a function of the near-load deflections and/or the radius of curvature of the deflection basin.

The advantages of forward calculation are as follows:

  1. Since the subgrade and bound surface course stiffnesses obtained are not dependent on the other moduli within the pavement system, as is the case with Backcalculation, there is a unique solution to each problem.
  2. Forwardcalculation is easy to understand and use, whereas Backcalculation is more of an art than a science. Forwardcalculation can be performed by anyone, while Backcalculation requires expert engineering judgment along with the art of running the iterative program of choice. The art is in the evaluation of the reasonableness of the results and proper selection of the model and other input parameters used for Backcalculation.
  3. Using an elastic layered system and the MODCOMP program, forward calculation techniques developed for the LTPP database produce considerably less scatter in the flexible pavement system results (for the same layer and test section) than do Backcalculation programs run in batch mode.

Nothing in the way of pavement analysis comes without its own unique drawbacks. As such, these drawbacks are not limited to Backcalculation alone, as for example:

  1. Since the subgrade and surface course stiffnesses are calculated independently of one another through different forward calculation formulae, in combination the values obtained may or may not be reasonable with respect to the total center deflection.
  2. The forward calculated bound surface course modulus has to be a single value, with all bound layers combined into a single, effective surface course layer.
  3. To obtain a third, intermediate layer stiffness (if present), such as a granular layer or crushed rock base, one can assume that the surface and subgrade stiffnesses are correct and then fit the center deflection to the remaining unknown stiffness of, perhaps, a base course layer. This base layer determination approach suffers from the same drawback as Backcalculation-one layer's modulus is dependent on the other layers' analysis results.
  4. It is also possible to use a ratio between the subgrade modulus calculated through forward calculation and apply the modular ratio relationship for unbound base materials developed by Dorman and Metcalf (2) or else apply any other suitable ratio between an intermediate layer and the subgrade or surface course, as deemed appropriate. However, there is no assurance that such a solution is correct, so ultimately one must apply the test of reasonableness to the results.
  5. Since forward calculation produces approximate values (particularly for the base or intermediate layer/s), these should only be used as modulus estimates, as for example for screening back calculated moduli, quality assurance/quality control (QA/QC) applications, or routine pavement evaluation purposes.

Organization of Report

This report is organized into three chapters. This chapter presents a background of the development of the guidelines, followed by an introduction to Backcalculation and an introduction to forward calculation. In chapter 2, development of the forward calculation technique is described and discussed. Chapter 3 documents the use of the accompanying forward calculation spreadsheets, as well as of the forward calculation results in reviewing and screening Backcalculation or any other modulus determination results.

The Microsoft® Excel spreadsheets containing all formulae used in phase I of this study have been provided to FHWA, so all forward calculation input quantities are totally transparent to those who wish to use the methodology, whether for screening or in rehabilitation design. To this end, four spreadsheets are available-two for asphalt-bound surfaces (using SI and U.S. Customary units) and two for cement-bound surfaces (SI and U.S. Customary). These spreadsheets can be obtained by contacting LTPP Customer Support Services by phone at 202-493-3035 or by e-mail at


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