Analysis of Pavement Structure)
Back-calculation of Layer Stiffness from Deflection Data using the
Singular Value Decomposition Technique.
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.
software is written using modern windows programming languages –
with this version being developed for the Windows 98 operating
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.
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.
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.
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
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).
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
= 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.
= 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
= the residual vector of differences between the most recently
computed parameters and the parameters represented by the measured
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
back-calculation procedure is considered to be suitable for:
depth asphalt concrete (AC)
(AC surface plus granular base)
+ high-strength stability base (HSSB)
over granular base.
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
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.
installation it is recommended that the user restarts the computer
to ensure that all setting changes have taken place.
The user runs the program from the Start-Program menu
or using other standard window methods and then starts by selecting
the file menu.
file menu enables the user to choose an area to work in:
of these areas will be considered in detail in the following text.
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
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
information stored will be as follows:
– Asphalt Concrete
– Portland Cement
– Granular Base
– Granular Sub-base
ratio must be less that 0.5.Thickness and density inputs are
in metric units.
the user has completed the data entry in this area he uses
the Save or Save As file commands to store the file.
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.
ratio must be less that 0.5.Thickness and density inputs
are in metric units.
the user has completed the data entry in this area he uses the Save or
Save As file commands to store the file.
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.
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
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.
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.
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
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
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
M.R. "ILLI-PAVE Based NDT Analysis Procedures’, Nondestructive
testing of pavements and Backcalculation of modulii, ASTM STP 1026,
M.R. Discussion, 5th Int.Conf.Struct. Design of Asphalt Pavements,
W.H., Flannery, B.P., Teukolsky, S.A. & Vetterling, W.T.
‘Numerical Recipes", Cambridge Univ. Press, Cambridge, 1986