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DAPS™ Software (
Deflection Analysis of Pavement Structure)

Back-calculation of Layer Stiffness from Deflection Data using the Singular Value Decomposition Technique.

Theoretical background
Program Installation
Program Use

Construction Data
View FWD Bowls
Process FWD Data


Pavement structures provide a vital part of our transportation system. Efficient maintenance is necessary in order to achieve maximum cost benefit from the huge investment made by transportation agencies over the years. As part of maintenance planning and analysis, agencies have been making use of deflection testing equipment such as the Falling Weight Deflectometer. This type of equipment yields information regarding the structural performance in terms of deflections that are used to calculate layer stiffness moduli. DAPS – Deflection Analysis of Pavement Structures – is a rapid, accurate and reliable method for performing back-calculation of deflection results.

The software is written using modern windows programming languages – with this version being developed for the Windows 98 operating system.

Theoretical background

As deflection data is increasingly used to accurately define pavement response to loading the need to develop robust back-calculation procedures has become greater. With the speed of computers increasing more mathematically intensive procedures can be used for developing solutions of large data sets. Recently, a Deflection Analysis of Pavement Structures (DAPS) software has been developed, which enables the rapid calculation of layer stiffness moduli using a singular decomposition technique.

The back analysis algorithm solves for both a two layer elastic system and the thickness of subgrade, or a three layer elastic system and the thickness of subgrade. A least square solution process is applied, employing all the measured deflections as parameters characterizing the bowl.

A rigid base beneath the subgrade is assumed (‘bedrock’). This is an accepted method, to some extent, to allow for known effects of non-linearity within the subgrade soil. The rigid base depth is used as an unknown to be solved for, along with the layer modulii.

Seed values for the AC and subgrade stiffnesses are obtained from equations published by Thompson (1989), using deflections d0 to d3. These are used to generate trail values of the parameters characterizing the bowl (i.e. the deflections). If a granular base is assumed to be present, the granular base resilient modulus seed value is estimated by empirical relations (Thompson, 1982). If a three-layer system without granular base is to be solved, the subgrade E estimate can be used for the base layer also. An arbitrary fixed initial trail value of subgrade thickness is employed, viz. 7m.

As described so far, it is evident that there are 7 known parameters, and either 3 or 4 unknowns, viz. 2 E’s and 1T, or 3E’s and 1T. Since there are more parameters than unknowns, an overdetermined set of simultaneous equations can be set up relating changes in the unknowns to changes in the deflections by means of a matrix of partial derivatives, dpi/dUj,where p are the deflections and U are the unknowns (either E values or thickness value).

A least squares solution to these simultaneous equations is obtained by an iterative process using, at each iteration, a solution of the overdetermined equation set by the Singular Value Decomposition technique (Press et al., 1986).

The difference between the computed deflections based on the initial unknown’s estimates (seed values), and the measured deflections are hence minimized by the following procedure for updating the unknowns:

Pkak = rk


Pk = the kth iteration of the matrix of partial derivatives dpi/dUj of the parameters p1, I=1 to 7, with respect to the ‘unknown’ layer modulii and thickness Uj, j=1 to 3 or 1 to 4.

ak = the kth difference vector, which is the differences Uj,k+l –Uj,k between the modulii/thickness used in the Pk matrix and the new modulii/thickness Uj,k+l to be used in the (k+1)th iteration.

rk = the residual vector of differences between the most recently computed parameters and the parameters represented by the measured deflections.

In the above equations, the partial derivatives comprising the P matrix are estimated numerically, by Elastic Layer analysis. At present, no limits are applied to E values generated by the minimization procedure.

This back-calculation procedure is considered to be suitable for:

  • Full depth asphalt concrete (AC)
  • Conventional (AC surface plus granular base)
  • AC + high-strength stability base (HSSB)
  • CC over granular base.

Program Installation

The program is supplied on a single CD-ROM. The user runs the setup.exe program using normal windows execution (e.g. from run menu a:\setup.exe) and then follows the instructions given by the software.

The setup creates a default directory of c:\daps and c:\daps\data1. The data directory is the default data directory for storing deflection data for analysis. The user can change this if necessary along with the program directory during the installation process.

Following installation it is recommended that the user restarts the computer to ensure that all setting changes have taken place.

Program use

The user runs the program from the Start-Program menu or using other standard window methods and then starts by selecting the file menu.

The file menu enables the user to choose an area to work in:

  • New Construction Data
  • Open Construction Data
  • View FWD Bowls
  • Run Back Analysis
  • Exit

Each of these areas will be considered in detail in the following text.

Construction Data

Pavement construction information must be defined to enable back calculation. This is done via the creation of a construction information file. The file should have the same root name as the FWD data file and would be normally located in the default directory c:\daps\data unless otherwise specified by the user – see working directory notes below.

The pavement construction information can be defined for various lengths of the pavement by reference to the station numbers. In the example given, three construction types have been defined for the FWD data file. The user must define a length of pavement associated with each construction type.

The information stored will be as follows:

Material Types:
  • AC – Asphalt Concrete
  • PC – Portland Cement
  • GB – Granular Base
  • SB – Granular Sub-base
  • SG – Subgrade

Poisson’s ratio must be less that 0.5.Thickness and density inputs are in metric units.

When the user has completed the data entry in this area he uses the Save or Save As file commands to store the file.

An alternative to starting a new construction information file is to use the data in an existing file by opening and then use the Save As command to a new file name after completion /modification of data.


Poisson’s ratio must be less that 0.5.Thickness and density inputs are in metric units.


When the user has completed the data entry in this area he uses the Save or Save As file commands to store the file.

An alternative to starting a new construction information file is to use the data in an existing file by opening and then use the Save As command to a new file name after completion modification of data.

View FWD Bowls

When selecting the menu option View FWD Bowls the user is presented with the form as illustrated.

Data files in the working directory are listed under the data item in the grid. The program recognizes most common FWD file formats, including one EXCEL spreadsheet format. Initially, all other information are blank. Hitting the Summarize button fills in this information. This summary can be saved to the hard disk. This process determines the number of deflection bowls in the data file and allows the user to review the distance /station information to be reviewed with ease. Often the user will want to summarize data before setting construction information for a file

Summarized Data

The plus symbol next to the data filename indicates that a CDT (construction data) file has been completed, which is necessary to perform the back-calculation. A box towards the upper right corner of the screen contains general information in the file. Title, test date, distance information, number of stations and drops per station are displayed in the grid.

Single data points can be analyzed by interactively clicking on a deflection bowl (providing a CDT file has been made). Information displayed on the deflection bowl graph includes the layer stiffness and the calculated stiffness for each pavement layer. The thickness of the soil layer is the calculated thickness to an assumed rock foundation. The statistic RMSerr/Max (Root Mean Square error divided by Maximum Deflection) is also given. A RMSerr/Max of less than 4% is considered as a satisfactory deflection bowl match.

In locations where tests have been carried out over joints/cracks a note is given on the screen stating that "No back-analysis" has been conducted. Often deflection tests are conducted on jointed pavements at several stress levels to determine the performance of the joint. By clicking on the "Load Defln" (Load versus Deflection) radio button it is possible to view a graph of the pavement deflection versus the stress level.

Process FWD Data

A file which can be processed is indicated by a (+) next to the data file name. The file is selected and processed by selecting the "run" button. Information is updated as the analysis takes place in the display grid. The moduli of the layers are displayed (E1 to E3) along with the thickness of the soil layer to bedrock (Sthk), the calculated horizontal strain (tensile) at the underside of the bound layer (epsH) and the vertical strain at the top of the soil layer (epsV).

In addition the %rms (Root Mean Square error divided by Maximum Deflection) is given in the last column. As discussed earlier a rms error of less than 4% is considered as a satisfactory deflection bowl match. The summary window indicates general statistical information on the analysis such as number of bowls successfully analyzed. The user can save the results by clicking on the Save Results button. In addition, the user is able to review the data and make any changes to the construction information, if required,and rerun the analysis. The file, when saved, is stored in a BLS file that is formatted to enable easy import into a spreadsheet for further manipulation and/or analysis.


Thompson, M.R. "ILLI-PAVE Based NDT Analysis Procedures’, Nondestructive testing of pavements and Backcalculation of modulii, ASTM STP 1026, 1989.

Thompson, M.R. Discussion, 5th Int.Conf.Struct. Design of Asphalt Pavements, Vol.II pp.293-294,1982.

Press, W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T. ‘Numerical Recipes", Cambridge Univ. Press, Cambridge, 1986