The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
A Simplified Method for Dynamic Response of Flexible Pavement
And Applications in Time domain Backcalculation
Yigong Ji*, Fuming Wang**, Maotian Luan ***, Zhongyin Guo****
* Michigan State University, East Lansing, MI 48824-1226, USA, jiyigong@msu.edu
** School of Hydraulic & Environmental Engineering, Zhengzhou University, Zhengzhou, Henan, China,
fuming@zzu.edu.cn
*** Department of Civil Engineering, Dalian University, Dalian, Liaoning, China, mtluan@dlut.edu.cn
**** School of Transportation Engineering, Tongji University, Shanghai, China, zhongyin@mail.tongji.edu.cn
Abstract: A new solution and its associated computer program, named SSSM-SIM, for dynamic back-calculation of
multi-layered flexible pavement parameters under FWD tests are presented. A new simplified approach based on
spline semi-analytical method is proposed for dynamic response analysis of a visco-elastic layered flexible pavement. The advantage of the new solution is that it enables a three-dimensional problem to be simplified to a onedimensional numerical problem; therefore the large number of computational cost and storage is greatly reduced.
Both frequency and time domain solutions are developed based on this method. Furthermore, in the inverse process
of identifying parameters, the data of falling weight deflectometer (FWD) and the system identification method are
jointly used in conjunction with the proposed procedure for estimating material properties from responses of a viscoelastic layered flexible pavement. The analysis is based on dynamic time domain fitting in a series of a period of
time. The theoretical results show that (1) system identification method is an efficient back calculation approach that
can converge to a solution quickly in a wide range of initial values of modulus, and (2) good agreement between
measurement and prediction can be achieved. Based on the above work, the deflection data from two in-situ falling
weight deflectometer (FWD) tests are used to determine the flexible pavement material properties. It has been
shown that the backcalculation program can be adapted to these situations. The reasonable results can be obtained
by comparing to other backcalculation method. [The Journal of American science. 2006;2(2):70-81].
FWD response time histories are often truncated in time
(30 or 60 millisecond) and not tend to zero at the end of
the time window. Therefore, the FWD field data in frequency using FFT scheme will be fundamentally
changed by these two reasons (4,7).
Comparing to frequency backcalculation method,
backcalculation analysis in time domain has advantages,
which does not consider the truncation problem and
data drifting in the end of time history (4,7). Therefore,
it will provide a reasonable way to backcalculate the
pavement parameters. However, dynamic time domain
analysis is time-consuming; therefore backcalculation in
time domain requires much more time comparing to
frequency domain backcalculation. In addition, in the
extraction of FWD time history, Chatti et al (8)use the
time lag and peak time to backcalculate the modulus,
damping and thickness; Dong et al (9) use period time
to backcalculate layer modulus and viscous coefficients
in assumption of three dimensional finite element
pavement model; and Uzan (7) use the full time history
to backcalculate AC creep compliance and layer modulus.
This paper presents a dynamic time domain backcalculation program, SSSM-SIM, in which spline semianalytical method is used for calculating dynamic re-
1. Introduction
Pavement evaluation is essential for maintaining
exsisting pavement and rehabilitating the deteriorating
highway infrastructure. Comparing to destructive test,
nondestructive testing (NDT) is fast, economic, accurate
in evaluating the structural capacity. The Falling Weight
Deflectometer (FWD) test is the most popular NDT
equipment. The considerable efforts have been made
over the decades to interpret FWD deflection time history for determining the layer parameters, such as layer
module, damping ratio, thickness and so on.
Over the years, many backcalculation programs have
been developed to interpolate the FWD data, including
both static and dynamic assumptions. However, most
popular backcalculation programs are based on the static assumption, which use only peak values of the FWD
response time histories. Dynamic backcalculation is
more attractive than static backcalculation since it considers dynamic effects of pavement and dynamic characteristics of FWD. At present, based on single frequency or multi-frequency fitting, several dynamic backcalculation computer programs that have been developed
such as PAVE-SID (1), BKGREEN (2), SSSM-BACK
(3), DYNABACK-F (4), and LAMDA (5,6). However,
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The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
sponse of visco-elastic multi-layered pavement system.
By virtue of the data measured from in-situ Falling
Weight Deflectometer (FWD), the proposed procedure
in conjunction with a system identification method and
the singular value decomposition (SVD) method are
applied to backcalculate material properties of pavement. Verification of the proposed methodology is
made through a synthetic data analysis. Finally, a comparison of the backcalculated moduli and damping ratio
from two sets of field data with results from other developed backcalculation programs is also presented. The
accuracy and consistency of backcalculated results from
field data indicate that the proposed method may be
considered as an applicable method to evaluate the material parameters of pavement structure.
R
In which X mn , Ymn , Z mn represents Fourier's series
terms expressed for horizontal and vertical loading by Ji
et al. (11),
X mn
Ymn
2.1 Modeling Viscoelasticity
Z mn
In visco-elastic theory, behavior of materials is
generally expressed in terms of complex modulus
coupling modulus and viscous damping together. The
modulus and damping ratio are independent on frequency
in soils, but are frequency dependent on asphalt concrete
(10). For simplicity reason, it is assumed that all
parameters are frequency independent, which are defined
as below
Ly 

L 


nπ y 
mπ x  x 

2
2
 sin 

 cos 
(5)
Lx
Ly
Ly 

L 


nπ y 
mπ x  x 
2 
2 


(6)
 sin
cos
Lx
Ly
Ly 

L 


nπ y 
mπ x  x 
2 
2 


(7)
 sin
sin
Lx
Ly
amn  a0 a1  a N T ,
bmn  b0 b1  bN T ,
T
and cmn  c0 c1  c N  are variables to be
determined,     0 1   N  represent simple spline function matrix with its component as following (Ji et al 1999):
(1)
where i   1 , E is the Young’s modulus, and  is
the hysteretic damping ratio.
 z

 k 
 hk

 k  1,k 
layer, N is the total number of sub-layer. Simple
spline function is expressed as
 z  1 , z   1, 0 ;

(9)
1z   1  z , z  0, 1;
 0
, z   1, 1

Equation (2), (3) and (4) can be written in matrix
form
u 
R S
    v     Nmn d mn eit  N d eit (10)
w m 1 n 1
 
The stresses of soil can be given
(11)
   D   DBdeit
a
The square loading area is of dimension a in both x- and
y-directions, respectively. At two vertical sides, the
boundaries are placed sufficiently far away from the loading area so that the reflected radiating stress waves from
the boundaries are damped out before reaching the reign
of interest. The displacement functions u , v , w for
layered half-space with any number of nodal surface are
represented by the product of simple spline functions and
Fourier’s series.
S
u     X mn amn e it  u 0 e it
(2)
v    Ymn bmn e it  v0 e it
(3)
k  1,2  N (8)
In which hk is the thickness of the k-th sub-
2.2 Spline Semi-Analytical Procedure for steady
state solution of Layered pavement System
Refer to Figure 1; a non-homogenous visco-elastic
pavement system on rigid base is subjected to a uniformly
distributed surface load with unity intensity q  12 e it .
R
(4)
m 1 n 1
2. Dynamic Pavement Response
E *  E 1  2i   E1  iE 2
S
w     Z mn cmn e it  w0 e it
In which B is the strain-displacement matrix. D
can be expressed
m 1 n 1
R
S
m 1 n 1
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The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
 d1
d
 2
d
D   3
0
0

 0
d2
d3
d1
d3
d3
0
1
0
0
0
0
0
0 0 0
0 0 0
0 0 0
G
1 0 0
0 1 0

0 0 1
In general, it can be considered that the load
frequency above 75 Hz will not contribute the transient
response in time domain solution (8). The details of the
time-domain response computations are as follows:
(12)
1.
2.
d1  21   (1  2 ) ,
d 2  d 3   (1   ) ,  is soil Poisson's ratio, G is
In which
3.
shear modulus for every layer.
According to complex damping principle, horizontal
and vertical dynamic responses can be obtained by solving Lagrangian equations of the system. Because of the
4.
,
orthogonality of Fourier s series X mn , Ymn , Z mn , all
terms will be uncoupled and each term can be analyzed
separately. Therefore, dynamic equations can be expressed in the following form while the time factor eit
is removed
  M 
2
mnmn
3. System Identification Method for Pavement Parameter Backcalculation
In this part, the system identification technique (14),
in conjunction with the mechanical model of pavement
structure and FWD test data, is employed to estimate
material parameters of different layers. It is implemented by minimizing a relative error function that quantifies the difference between the computed deflection by
the proposed procedure for dynamic response of layered
system and the experimentally derived deflection from
FWD testing data. For pavement system, two types of
material parameters, i.e., modulus and damping ratio,
are to be identified. The flowchart of this procedure is
given in Figure 2.
Assuming that the model of pavement can be defined
by elastic modulus and damping ratio of all layers under
consideration with the total number 2N of parameters, in
the time of t i , the deformation vector of the system

 1  2i K mnmn d mn   f mn (13)
In which
Ly
N
Lx
M mnmn     k   N mn N mn dxdydz
k 1
2
Ly
Lx
2
hk
Ly
N
T
2
2
Lx
K mnmn      Bmn DBmn dxdydz
k 1
2
Ly
Ly
N
Lx
2
hk
T
2
2
Lx
 f mn     N mn a1 dxdydz
k 1
2
Ly
hk
2
2
Lx
T
2
2
In which  k is the density of the k-th layer.
circular frequency

is
2.3 Time Domain solution of Layered Pavement System
W c will be determined by
W c  f ( E1 ,  1 , E 2 ,  2 , , E N ,  N , t i ) (14)
The above spline semi-analytical method is based on the
frequency domain analysis method. In order to calculate
the time response of FWD, the following procedure will
be complemented. First, the load time history is transformed to the complex load in frequency-domain using
the Fast Fourier Transform (FFT). Second, unit steady
state response is calculated in each frequency from
equation (4). Third, the complex load and unit steady
state response are multiplied to obtain the steady state
response of the pavement due to the complex load in the
frequency domain. Last, the response in the frequency
domain is re-transformed to time domain using the inverse FFT. In order to reduce the computational effort
of time history, an interpolation scheme proposed by
Tajirian (12) can be used for saving computing time.
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The computer program SSSM is used to compute
the unit steady state response of the pavement at
sixteen frequencies (from 0., 4.88 to 73.24 Hz).
The time history of the FWD load is transformed to
complex load in the frequency domain using the
FFT.
The steady state deflections of pavement in the frequency domain are obtained by multiplying the
complex load and unit steady state response in the
frequency domain.
The complex steady state response of pavement is
transformed to obtain the deflection time histories
in the time domain.
Then the data of the j-th sensor of the FWD system is
expressed as
W jc  f j ( E1 , 1 , E2 ,  2 ,, E N ,  N , t i )
If any function f j ( E1 ,  1 , E 2 ,  2 , , E N ,  N , t i ) is
expanded using a Taylor’s series in the vicinity of E and
β in a given time of t i , the error function will be given
as following while neglecting higher order terms based on
modified Newton's iteration procedure
e j  f j E  E, β  β   f j E, β   f E E  f  β
(15)
f j
f j
f j
f j

E1 
 1   
E N 
 N
E1
 1
E N
 N
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The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
Ei 1  Ei
Ei
For the system with s sensors, governing equations can
be established
F ti Xti  eti
(16)
 
In which
F is called as the sensitivity matrix at t i ,
difference vector between measured deformations and
computed deformations of the system at t i ,
Et  E1 1 E N  N T
et  e1 e2 es T
i
RMS 
i
Ft
i
W1
 1
W 2
 1
W1 
 N 

W 2 

 N 


 t
4.1 Frequency Domain Solution
In order to verify the frequency domain solution, this
paper demonstrates the numerical accuracy through
comparing SSSM results with published deflection data
by the GREEN program (2). A three-layer pavement
system resting over a subgrade is subjected to a 44.5 kN
(10,000 lb) load on a 0.3 m diameter plate. The layer
properties of the pavement layer are presented in Table
1. According to the exited result (11), the dimensions of
Lx  12 m , Lx  12 m , H  9 m , Fourier’s series
domain solution. The partial derivatives in the gradient
matrix can be calculated numerically by using 2N+1
runs of SSSM computer program.
At the same time, the governing equations at any other
time point t i 1 can be established, therfore the total gov-
R  50 , S  50 and 15 sub-layer element are selected
for the following analysis. The dynamic deflections at
offset distances of 0, 0.225, 0.300, 0.525, 0.750 and
1.350 m were calculated using SSSM and GREEN. Figure 3 shows the results for two different frequencies:
0.25 and 8 Hz. Excellent agreement exists at both frequencies. It can be concluded that the program SSSM is
suitable as a forward program.
erning equations can be expressed as
F total E  etotal
(17)
 
In which
e  e  
F  F F  F 
etotal 
e 
T
ti
T
T
t i 1
T
tn
T
ti
ti 1
tn
n is the number of selected time point that be used for
4.2 Theoretical Bakcalculation Analysis Synthetic
Data
minimization.
Equation (17) can be rewritten as
T
T
etotal
Ftotal
Ftotal E  Ftotal
Synthetic FWD data were generated for hypothetical
pavement structures consisting of asphalt, base and subgrade layers using the SSSM computer program, and it
is showed in Figure 3. The pavement profiles are given
in Table 2. The computed vertical displacement time
histories obtained from the SSSM program were used as
input for the backcalculation. Seven time histories were
computed at the locations r = 0, 0.305, 0.457,0.610,
0.914, 1.219 and 1.524 m from the load. The FWD
loading is applied on a square area with a side length of
0.266 m, according to equivalent circular FWD load,
and time step is 0.0002 second.
(18)
Solving this equation (18), the revised moduli and
damping ratio are obtained from
Ei 1  Ei  Ei
(19)
Equation (17) may be of ill condition from a
mathematical viewpoint; therefore the singular value
decomposition (SVD) technique (14) is used to solve this
equation. The coverage criteria can be defined as
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w mj is the measured deflection, and wcj is
4. Numerical Examples
is created using a time
i
2

 100 % (21)


the calculated deflection.
W s

 N
s is sensor number.
The sensitivity matrix F
m
c
1 s n  w j  w j

n  s i 1 j 1 w m
j

In which
 




W s
 1
(20)
In which,  1  0.01 . The combined root mean
square (RMS) is calculated using the following equation:
E represents parameter vector at t i , and e is the
 W1
 E
 1
 W 2
  E1
 
 W
 s
 E1
 1
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Life Science Journal, 2(1), 2005, Ji, et al, Method for Dynamic Response of Flexible Pavement
Layer Name
AC
Base
Subbase
Subgrade
Table 1. Profile used for comparing SSSM and GREEN solutions
Thickness (mm) Mass density ( kg/m 3 ) Poisson Ratio Damping Ratio
0.762
2242
0.35
0.0
0.152
2002
0.40
0.0
0.305
2002
0.40
0.0
1762
0.45
0.0

Modulus (MPa)
2069
310.3
144.8
51.71
Table 2. Pavement profiles for synthetic data
Layer Name
AC
Base
Subgrade
Thickness (m)
0.203
0.254

Mass density ( kg/m 3 )
Poisson Ratio
0.3
0.35
0.45
2322
2162
2001
Damping Ratio
0.05
0.03
0.02
Modulus (MPa)
2068
310
52
Table 3. Pavement profiles for field data
Test Site
Layer Name
Thickness (m)
Unit Weight
( kg/m 3 )
Poisson’s Ratio
Kansas
AC
Base
Subgrade
0.287
0.152

2322
2162
2002
0.3
0.35
0.45
Texas
AC
Subgrade
Stiff layer
0.203
1.691

2322
2162
2322
0.3
0.35
0.25
Table 4. Comparison of different backcalculation results Using field data from site in Kansas
True
Value
Seed
Value
SSSM-SIM
DYNABACK-T
Static
Backcalculation
(MICHBACK)*
Dynamic Backcalculation
AC modulus
(MPa)
Unknown
2413.8
3240.7
3080.0
3306.2
AC damping
ratio
Unknown
0.2
0.25
0.15
N/A
Base modulus
(MPa)
Unknown
137.9
43.9
37.5
29.3
Base damping
ratio
Unknown
0.1
0.32
0.22
N/A
Subgrade modulus (MPa)
Unknown
69.0
489.7
290.3
367.0
Subgrade
damping ratio
Unknown
0.1
0.20
0.19
N/A
RMS
—
—
10.73%
5.85%
N/A
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The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
Table 5. Comparison of different backcalculation results using field data from site in Texas
SSSM-SIM
DYNABACK-T
Static
Backcalculation
(MICHBACK)
400
237.4
262.8
201.7
Unknown
0.05
0.21
0.14
N/A
Base modulus
(MPa)
Unknown
40
29.6
24.7
21.9
Base damping
ratio
Unknown
0.025
0.17
0.12
N/A
Subgrade modulus (MPa)
Unknown
400
49.0
4000
58.5
Subgrade
damping ratio
Unknown
0.02
0.05
0.02
N/A
RMS
—
—
8.0%
5.8%
N/A
True
Value
Seed
Value
AC modulus
(MPa)
Unknown
AC damping
ratio
Dynamic Backcalculation
h1
Load=eiwt /a2
a
x
y
a
hk
H
z
Rigid layer
Figure 1. Multi-layered pavement system
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The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
FWD Load
Deflection
Unknown
Pavement System
(Measured)
Error
Deflection
Visco-Elastic
Pavement Model
(Calculated)
E and 
Parameter Adjustment
Algorithm
0.001
0.001
0.0008
0.0008
Deflection (mm)
Deflection (mm)
Figure 2. System identification method
0.0006
0.0004
0.0002
0.0006
0.0004
0.0002
0
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.2
0.4
Distance (m )
GREEN in 0.25 Hz
0.6
0.8
1
1.2
1.4
Distance (m )
SSSM in 0.25 Hz
GREEN in 8 Hz
SSSM in 8 Hz
400
350
300
250
200
150
100
50
0
-50 0
Sensor 1
Sensor 5
700
600
500
Load (kPa)
Deflection (Micron)
Figure 3. Comparisons of steady state response between SSSM and GREEN computer programs
400
300
200
100
20
40
Tim e (m s.)
Sensor 2
Sensor 3
Sensor 6
Sensor 7
60
0
-100 0
Sensor 4
20
40
Time (ms.)
Figure 4. FWD load and time history using theoretical bakcalculation
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The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
Ratio of
Estimated/Exact
Ratio of
Estimated/Exact
1.2
1
0.8
0.6
0.4
0.2
1.2
1
0.8
0.6
0.4
0.2
0
0
0
2
4
6
0
8
2
Iteration Num ber
AC Modulus
base Modulus
4
6
8
Iteration Num ber
subgrade Modulus
AC Damping
base Damping
subgrade Damping
(a) Seed value = 10% of true value
20
Ratio of
Estimated/Exact
Ratio of
Estimated/Exact
3
2.5
2
1.5
1
0.5
0
15
10
5
0
0
2
4
6
8
10
0
2
Iteration Num ber
AC Modulus
base Modulus
4
6
8
10
Iteration Num ber
subgrade Modulus
AC Damping
base Damping
subgrade Damping
5
4
Ratio of Estimated/Exact
Ratio of Estimated/Exact
(b) Seed value = 200% of true value
AC Modulus
base Modulus
subgrade Modulus
3
2
1
0
0
2
4
6
8
10
9
8
7
6
5
4
3
2
1
0
AC Damping
base Damping
subgrade Damping
0
10
2
4
6
8
Iteration Num ber
Iteration Num ber
(c) Seed value = 500% of true value
Figure 5. Convergence procedures of modulus and damping ratio using different seed
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The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
120
100
80
60
40
20
0
-20
-40
Deflection (Micron)
Deflection (Micron)
150
100
50
0
-50
0
20
Sensor 1 Measurment
Sensor 1 SSSM-SIM
40
0
60
T ime (ms)
Sensor 1 DYNABACK-T
20
40
Sensor 2 Measurment
Sensor 2 SSSM-SIM
(b)
100
80
80
60
Deflection (Micron)
Deflection (Micron)
(a)
60
40
20
0
-20
40
20
0
-20
-40
-40
0
20
40
0
60
T ime (ms)
Sensor 3 DYNABACK-T
Sensor 3 Measurment
20
40
60
T ime (ms)
Sensor 4 DYNABACK-T
Sensor 4 Measurment
Sensor 4 SSSM-SIM
Sensor 3 SSSM-SIM
(c)
(d)
80
50
60
40
Deflection (Micron)
Deflection (Micron)
60
T ime (ms)
Sensor 2 DYNABACK-T
40
20
0
-20
30
20
10
0
-10
-20
-40
0
20
40
0
60
T ime (ms)
Sensor 5 DYNABACK-T
Sensor 5 Measurment
20
40
60
T ime (ms)
Sensor 6 DYNABACK-T
Sensor 6 Measurment
Sensor 6 SSSM-SIM
Sensor 5 SSSM-SIM
(e)
(f)
25
20
15
10
5
0
-5
-10
-15
Deflection (Micron)
Deflection (Micron)
20
15
10
5
0
-5
-10
0
20
Sensor 7 Measurment
40
0
60
T ime (ms)
Sensor 7 DYNABACK-T
20
Sensor 8 Measurment
Sensor 7 SSSM-SIM
40
T ime (ms)
Sensor 8 DYNABACK-T
Sensor 8 SSSM-SIM
(g)
(h)
Figure 6. Comparisons of measured and simulated deflection time histories for site in Kansas: (a) sensor
1 (b) sensor 2 (c) sensor 3 (d) sensor 4 (e) sensor 5 (f) sensor 6 (g) sensor 7 (h) sensor 8.
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The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
200
340
320
Deflection (Micron)
Deflection (Micron)
360
300
280
260
240
220
200
180
160
140
120
100
16
18
20
Sensor 1 Measurment
Sensor 1 SSSM-SIM
22
24
16
T ime (ms)
Sensor 1 DYNABACK-T
18
Sensor 2 Measurment
20
Sensor 2 SSSM-SIM
(b)
100
55
90
50
Deflection (Micron)
Deflection (Micron)
24
T ime (ms)
Sensor 2 DYNABACK-T
(a)
80
70
60
50
40
45
40
35
30
25
20
16
18
20
Sensor 3 Measurment
22
24
16
T ime (ms)
Sensor 3 DYNABACK-T
18
Sensor 4 Measurment
Sensor 3 SSSM-SIM
20
22
24
T ime (ms)
Sensor 4 DYNABACK-T
Sensor 4 SSSM-SIM
(c)
(d)
25
Deflection (Micron)
35
Deflection (Micron)
22
30
25
20
15
10
5
0
20
15
10
5
0
16
18
20
Sensor 5 Measurment
22
24
16
T ime (ms)
Sensor 5 DYNABACK-T
18
Sensor 6 Measurment
Sensor 5 SSSM-SIM
20
22
T ime (ms)
Sensor 6 DYNABACK-T
Sensor 6 SSSM-SIM
(e)
(f)
Figure 7. Comparisons of measured and simulated deflection time histories for site in Texas: (a) sensor
1 (b) sensor 2 (c) sensor 3 (d) sensor 4 (e) sensor 5 (f) sensor 6.
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The Journal of American Science, 2(2), 2006, Ji, et al, Method for Dynamic Response of Flexible Pavement
The SSSM-SIM program is performed to identify layer
moduli and damping ratio, that are the total six parameters.
The seed values are assumed to be 10%, 200% and 500%
of the true value, respectively. Figure 5 shows the dynamic
backcalculation iteration vs. ration between identified parameter and true values. The following observations are
made from these figures:
 Curve of damping ratio is more fluctuate than
that of moduli in backcalculation procedure,
therefore damping ratio is more sensitive than
moduli, and therefore it is more difficult to
identify than moduli.
 The SSSM-SIM program is not essentially influenced by seed value, and it can provide coverage of true value in limited iterations steadily, therefore it is a robustness of backclaculation solution.
 Seed value will affect the iteration times, and
reasonable estimated seed value will decrease
the iteration times.


because the backcalculation itself is a nonlinear
solution, therefore more reasonable and more
accurate dynamic model is encouraged to use
to ensure that reliable results can be made.
There are small deviations between the measured and simulated time histories, so the point
to point backcalculation method should be better than the peak deflection and time lags, since
it considers more point’s information.
The simulated deflection time histories of both
SSSM-SIM and DYNABACK-T program are
negative at the beginning. This may be due to
the hysteretic damping assumption and calculation errors at the beginning of IFFT procedure,
which is similar to errors of time domain analysis in the beginning. Therefore, time history
around the peak deflections, not the entire time
history, is recommended to use in backcalculation.
5. Conclusions
4.3 Field Data Aplication
This paper presents a simplified approach based on
spline semi-analytical method for analysis of dynamic
response of pavement. In the proposed method, the
pavement is divided into a number of sub-layers; the
displacement functions are expressed as the product of
basic spline function and Fourier's series, which enables
to simplify a three-dimensional problem to a onedimensional numerical problem. This paper also discussed the backcalculation in time domain fitting base
on the system identification method (SIM). The numerical regularization technique, named SVD is explored for
ensuring better stability of the inversion procedure in
iterations. The numerical results show that the system
identification method is an efficient backcalculation
approach that can provide coverage to a solution quickly in a wide range of initial values of module. The reasonable and acceptable results can be obtained when
using field data from FWD. Therefore a practical and
versatile analysis access is established for the study on
evaluation of capacity of flexible pavement.
The field deflection data from Kansas and Texas
were select from Chatti et al (8), in which the
DYNABACK-T was developed using peak deflection
and time lags to identify the pavement parameter and
MICHBACK program was used to verify. For the purpose of comparison, results of SSSM-SIM,
DYNABACK-T and MICHBACK are listed in Table 5
for Kansas and Table 6 for Texas. The FWD data consists of eight deflection time histories for sensors located at r = 0, 0.203, 0.304, 0.457, 0.609, 0.914, 1.219 and
1.524 m from the load, deflection time histories are used
for minimization from 23 to 35 millisecond. The data
from the site in Texas contain six deflection time histories for sensors located at r = 0, 0.304, 0.609, 0.914,
1.219 and 1.524 m from the load, the time history is
used from 16 to 24 milliseconds.
In order to compare the difference between SSSMSIM and DYNABACK-T, the measured and predicted
full deflection time histories are shown in Figures 6 for
the Kansas data, while the measured and predicted deflection time histories in specific input time period are
shown Figure 7 for the Texas data. The following observations are made from these figures:
 For both the Kansas and Texas site, both
SSSM-SIM and DYNABACK-T program can
generate the similar response curve. It maybe
due to the reason that both are based on the
frequency solution method, although the threedimenstional assumption and axial symmetry
assumption are made separately.
 The different forward model assumption will
lead to different backcalculation results. It is
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