Bagus Hario Setiadji paper for ICPT 2017 (revised)

CLOSED-FORM BACKCALCULATION ALGORITHM
FOR INDONESIA OVERLAY DESIGN PROCEDURE
Bagus Hario SETIADJI a, SUPRIYONO b
Department of Civil Engineering, Faculty of Engineering, Diponegoro University
Jl. Prof. Soedarto, SH., Tembalang, Semarang, Indonesia 50277
a
E-mail: bhsetiadji@ft.undip.ac.id; bE-mail: supriyono.ir@gmail.com

ABSTRACT
So far, condition evaluation of existing pavement by using backcalculation method in 2002 Indonesia
Overlay Design Guide gives inaccurate results. To overcome this problem, another closed-form
backcalculation algorithm was proposed in this study. Closed-form backcalculation algorithm
developed was based on the concept of two-layer flexible pavement backcalculation with area load
(named as 2L-BACK), in which the multi-layer pavement structure was simplified to a two-layer of
pavement structure and the response of the pavement was developed based on uniform loading. In this
study, the proposed algorithm was developed by using Nelder-Mead optimization method. For
evaluating the proposed algorithm, analyses using hypothetical data and also measured data from
LTPP database was conducted. Similar analyses was also conducted on two other bakcalculation
methods, i.e backcalculation method in 2002 Design Guide and a best-fit trial and error
backcalculation method, namely EVERCALC, for comparison. The results of the analysis showed that
2L-BACK and EVERCALC programs were insensitive in determining elastic moduli using deflection

data with + 2% of measurement errors. Using data from LTPP database, it is found that 2L-BACK
program can provide more accurate results than other backcalculation methods evaluated. It is
recommended that 2L-BACK can be used as replacement of backcalculation procedure in 2002
Design Guide.
Keywords: closed-form backcalculation algorithm, overlay, two-layer pavement structure

1.

INTRODUCTION

At this time, pavement preservation program becomes the most popular pavement maintenance and
rehabilitation program as this program is able to maintain the stability of road conditions until the end
of the design life at a relatively low cost (Galehouse, 2003). The components of pavement
preservation program may differ in each country. Federal Highway Administration (2006) stated that
pavement preservation program consisted of preventive maintenance, minor/corrective maintenance
and minor rehabilitation, while in Indonesia as stated in Regulation of the Minister of Public Works
No. 13/PRT/M/2011, pavement preservation program covers routine maintenance, periodic
maintenance, road rehabilitation and reconstruction. The difference of the component of pavement
preservation program could due to factors in the form of funding adequacy and resources availability,
as well as how well the ability of the agency to manage pavement distress and the causes of the

distress.
In Indonesia, the concept of road preservation is not completely understood, so that pavement
rehabilitation (overlay) is generally a type of treatment that most people do, because it is believed to
be able to fix a lot of road damages at one time simultaneously. However, this overlay work requires a
proper design as this treatment is intended to address the structural deficiency of road pavement.
In order to obtain a proper design of overlay, Indonesia Directorate General of Highways provides a
guideline, namely Pt T-01-2002-B Overlay Design Guide (Directorate General of Highway, 2002)
(stated as 2002 Design Guide in the rest of this paper), which adopted the overlay design guide in
1993 AASHTO Guide for Design of Pavement Structure (American Association of State Highway and
Transportation Officials, 1993).

In the guideline, the design of overlay is commenced by predicting the evaluated layer moduli using
closed-form backcalculation equations as follows.

Mr 

0.24 P
dr r








1
d 0  1.5 pa 
D

 M r 1  

a




(1)

3



1

1 
2
D

1  

a


Ep
Ep 

M r 

















(2)

in which: p is non-destructive testing (NDT) load plate pressure (psi), a is NDT load plate radius (in.),
D is total pavement layer thickness (in.), Ep is effective modulus of all pavement layers above the
subgrade (psi), and d0 is deflection measured at the center of the load plate (in.).
However, those equations have a fundamental weakness, i.e. the predicted subgrade resilient modulus
Mr, which is calculated using backcalculation procedure (equations 1 and 2), differs significantly with

the measured one.
To overcome the deviation, Darter et al. (1992) recommends that the calculated resilient modulus
should be multiplied by an adjusment factor C by a maximum of 0.33. The need of this factor then
leads to a question whether all calculated resilient modulus Mr has to multiply with that factor. This is
because the variability of the soil is very large, and there are no guidelines to justify the use of the
adjustment factor for all types of soil. AASHTO (1993) also stated that the adjustment factor is
required for fine-graind soil only, while coarse-grained soil needs further user consideration.
The lack of clarity in the use of adjustment factor in predicting resilient modulus above could be
contributed by the algorithm of closed-form backcalculation procedure in 2002 Design Guide.
Therefore, this research proposed another closed-form backcalculation algorithm developed based on
layer elastic theory. The strong reason why closed-form method selected in this research is that
because the method could offer a unique (single) solution (Livneh, 2015). To evaluate the accurateness
of the proposed algorithm and closed-form backcalculation algorithm in 2002 Design Guide, another
backcalculation method based on best-fit trial and error algorithm was used. This was conducted by
comparing all backcalculation methods using hypothetical deflection data and also measured data
extracted from Long Term Pavement Performance (LTPP) database, which is part of the program of
the Federal Highway Administration (FHWA) (Elkins et al., 2003).

2.

RESEARCH METHODOLOGY

The research methodology consists of several parts as follows.
(a)
Development of proposed backcalculation algorithm
The proposed backcalculation algorithm was a closed-form backcalculation method. It means
that the ultimate goal of this backcalculation algorithm was to derive equations in order to
determine a unique or a single solution. In this research, this task was done by using NelderMead or simplex search optimization method. This method is a multidimensional

unconstrained optimization without derivatives which could produce a good result with only
needs a relative small number of function evaluation (Setiadji, 2010).
Two calculated moduli were determined as a result of backcalculation algorithm in this
research. They are: (i) elastic modulus of subgrade, Es; and (ii) effective elastic modulus of all
layers (composite layer) above the subgrade, Ep. In practice, it is common to find pavement
structure consists of four layers or more. Therefore, to enable the proposed algorithm in
calculating the moduli of pavement layers, a simplification of the pavement structure was
made. This simplification was conducted by considering all layers above the subgrade become
single layer and no change was made for the subgrade (see Figure 1). The thickness of all
layers above the subgrade was summed and equals to the total thickness D.
Pavement structure in practice

Structure pavement in this study
Layer 1

E1, µ1, h1

Layer 2

E2, µ2, h2

Layer 3

E3, µ3, h3

Subgrade

ES, µS

Ep, µp, D

Composite

layer

ES, µS

Subgrade

Figure 1. Simplification of Pavement Structure Used in This Study
(b)

Sensitivity analysis of proposed backcalculation algorithm
The purpose of this sensitivity analysis was to determine how sensitive the proposed
backcalculation algorithm (and also the two comparative backcalculation methods) against
their input parameters, which are generally prone to measurement errors. To do so, the
deflection values, collected in NDT measurement using falling-weight deflectometer (FWD)
device, were added with possibility of errors by using random numbers, as expressed in the
following equation (Pronk, 1998).
d m  d t  0.02d t

 r1  0.5  r2
r1  0.5

2

 r3  0.5  r4
r3  0.5

(3)

in which: dt is true deflection (µm), and r1 - r4 are random numbers from 0 - 1. The error is
generated by equation (3) will be limited in the range of + 2% of true deflection dt to simulate
the magnitude of measurement error commonly produced by FWD device (Irwin et al., 1989).
The deviation of all elastic moduli calculated based on true deflection with errors was
evaluated by means of the following term of error.
 X x
error   i i
 xi

(c)



100


(4)

where Xi is the calculated moduli (ksi) based on true deflection with errors, xi is the initial
moduli (ksi).
Validation of proposed backcalculation
This step was performed by comparing calculated moduli based on measured deflection and
measured moduli itself (using destructive testing method). For validation purpose, all
necessary data, i.e. measured deflection from field measurement and elastic moduli
determined by labotaroy test, were all provided. These data were collected from Long Term
Pavement Performance (LTPP) database, which covers all data (load, environment, and so on)

related to a long-term evaluation of pavement performance in the United States and Canada
(Elkins et al., 2003).
The accuracy of the results of validation process was evaluated using error term (as seen in
equation 4).

3.

RESULTS AND ANALYSIS

3.1

Development of Proposed Backcalculation Algorithm

The proposed closed-form two-layer backcalculation algorithm (2L-BACK) in this research was
derived from the closed-form three-layer backcalculation algorithm 3L-BACK (Setiadji and Fwa,
2010) by simplifying its forward calculation, namely as 3L-DEF. The simplification was performed by
eliminating one layer of 3L-DEF program to make it as 2L-DEF. To develop the proposed 2L-BACK
backcalculation algorithm, Nelder-Mead optimization method was used, as similar with the
development of 3L-BACK (Setiadji and Fwa, 2010).
In order to compare the accurateness of 2L-BACK (and also 2L-DEF), a best-fit trial and error
backcalculation algorithm, that is EVERCALC, was used in this study. EVERCALC (with
EVERSTRESS as its forward calculation) is a backcalculation algorithm that used LavenbergMarquardt algorithm as its general minimization method, so that this backcalculation algorithm can
converge quickly with only a small number of calls (Setiadji, 2010). EVERCALC program requires
inputs such as seed moduli (E), number of iterations and error tolerance limits (ε). The number of
iterations, error tolerance limit, as well as seed moduli must be analyzed by trial and error to ensure
that the backcalculation metode can run as what expected. The values of seed moduli used in this
study are Ep = 1000 ksi (with a range of modulus of 100 - 5000 ksi), while Es = 45 ksi (with a range of
modulus of 1 - 100 ksi). Maximum iteration was set equals to 1000 times and the error tolerance limit
was set equals to 0.001%.
The main input data, includes deflection data, layer thicknesses (h1 – h3), the magnitude of the load
(P), sensor offets and corresponding deflections (d1 – d6), was similar for all backcalculation methods
and collected from LTPP database.
In this study, to evaluate the reliability of all backcalculation algorithms, then three sets of artificial
data were generated, as depicted in Table 1. The initial data set was developed by varied the load and
layer thickness. With the combination of the pavement layer properties, i.e. load, thickness and layer
moduli, then the the deflection values d1 through d6 were calculated using EVERSTRESS and 2L-DEF
(see Table 1).
Table 1. The data set used to determine the validity of the seed modulus
No
of
data
set
1

Load
(P),
lbs.

Layer
thickness
(h), in.

Ep,
ksi

Es,
ksi

15985

12

200

9

2

11241

12

200

9

3

15985

20

200

9

Name of
forwardcalculation
programs

d 1,
mils

d 2,
mils

d 3,
mils

d 4,
mils

d 5,
mils

d 6,
mils

EVERSTRESS
2L-DEF
EVERSTRESS
2L-DEF
EVERSTRESS
2L-DEF

26.134
26.082
18.378
18.341
17.351
17.089

23.793
23.913
16.732
16.816
16.026
16.021

20.805
20.877
14.630
14.681
14.718
14.742

18.107
18.163
12.733
12.773
13.581
13.625

13.647
13.669
9.597
9.612
11.511
11.541

8.083
8.081
5.684
5.682
8.189
8.197

As seen in Table 1, EVERSTRESS and 2L-DEF could produce similar deflections with small
discrepancy, i.e. less than 2%. From the table, it is also indicated clearly that the deflection values are
lower at lower load P or higher thickness D, and vice versa. Based on deflection values and other
pavement properties in Table 1, layered elastic moduli Ep and Es can be determined using

backcalculation programs EVERCALC and 2L-BACK (Table 2), and backcalculation procedure in
2002 Design Guide (Table 3).
Table 2. Calculated elastic moduli from EVERCALC and 2L-BACK and the deviation between them
and corresponding initial elastic moduli
No of
Initial elastic
Calculated elastic moduli
data
moduli
EVERCALC
2L-BACK
set
E1, ksi
E2, ksi
E1, ksi
E2, ksi
E1, ksi
E2, ksi
1
200.00
9.00
200.01
9.00
199.97
9.00
2
200.00
9.00
200.00
9.00
199.97
9.00
3
200.00
9.00
200.23
8.98
199.77
9.01
Remarks: *) Deviation between initial and calculated elastic moduli

Deviation *)
EVERCALC
2L-BACK
E1, %
E2, %
E 1, %
E2, %
0.00
0.00
0.02
-0.01
0.00
0.00
0.02
-0.01
-0.11
0.22
0.11
-0.11

Table 3. Calculated elastic moduli from 2002 Design Guide and its deviation against corresponding
initial elastic moduli
No of
Initial elastic moduli
Calculated elastic moduli
data set
Ep, ksi
Es, ksi
Ep, ksi
Es, ksi
1
200.00
9.00
189.20
7.91
2
200.00
9.00
189.20
7.91
3
200.00
9.00
192.10
7.81
Remarks: *) Deviation between initial and calculated elastic moduli

Deviation *)
E p, %
Es, %
5.40
12.11
5.40
12.10
3.95
13.24

From Table 2, it seems that the deviation between the initial and calculated elastic moduli is very low
(less than 0.5%). It showed the two programs are very accurate in determining the layered elastic
moduli Ep and Es. On the other hand (as seen in Table 3), the average absolute errors of elastic moduli
produced by 2002 Design Guide were almost 10% or twenty times worse than the errors produced by
EVERCALC and 2L-BACK..
3.2

Sensitivity Analysis of Proposed Backcalculation Algorithm

For the purpose of sensitivity analysis, the deflections in Table 1 were recalculated by involving
random numbers in the deflection values using equation (3) (Pronk, 1998). The errors added to each
deflection value were set maximum + 2% of true deflections (see Table 4). Based on these deflections,
calculated elastic moduli (Ep and Es) were determined using the three methods (EVERCALC, 2LBACK and 2002 Design Guide), as depicted in Tables 5 and 6.
Table 4. Deflection data set with errors
No of data set
with different
random numbers
1a
1b
2a
2b
3a
3b

Load
(P),
lbs.
15985
15985
11241
11241
15985
15985

Layer
thickness
(D), inch
12
12
12
12
20
20

E p,
ksi

Es,
ksi

d1,
mils

d2,
mils

d3,
mils

d4,
mils

d 5,
mils

d 6,
mils

200
200
200
200
200
200

9
9
9
9
9
9

26.509
25.748
18.362
18.041
17.518
17.027

24.217
23.440
16.751
16.402
16.203
16.043

20.955
20.493
14.693
14.790
14.904
14.742

18.293
17.833
12.835
12.888
13.775
13.611

13.893
13.435
9.765
9.745
11.719
11.552

8.016
7.948
5.694
5.657
8.307
8.248

Tables 5 and 6 show that the deviations between the initial and calculated elastic moduli produced by
EVERCALC and 2L-BACK can achieve 4.91% and 4.27% for Ep and Es, respectively, while the
deviations between initial and calculated elastic moduli produced by 2002 Design Guide can be up to
9.36% and 14.77% for Ep and Es, respectively. It means that EVERCALC and 2L-BACK could predict
the moduli better than that of 2002 Design Guide.
All the backcalculation methods initially determined Es, and use the value of Es to determine Ep. From
Tables 3 and 6, it can be seen that the use of equation (1) in 2002 Design Guide cannot give sufficient
accurateness for predicting subgrade resilient modulus Mr or Es. This could be contributed by the fact

that resilient modulus Mr or Es in equations (1) is independent from the properties of another layer.
The calculation of Ep in equation (2) has accommodated properties of structural layer includes Es,
however, its accurateness in predicting Ep are still lower than that of proposed backcalculation method.
Table 5. Calculated elastic moduli from EVERCALC and 2L-BACK using deflection data with errors
No of
Initial elastic
Calculated elastic moduli
data
moduli
EVERCALC
2L-BACK
set
Ep, ksi Es, ksi
Ep, ksi
Es, ksi
Ep, ksi
Es, ksi
1a
200.00
9.00
193.71
8.96
196.15
8.96
1b
200.00
9.00
202.40
9.15
205.77
9.16
2a
200.00
9.00
197.61
8.95
195.53
8.94
2b
200.00
9.00
210.32
8.94
208.53
8.93
3a
200.00
9.00
200.15
8.85
201.49
8.86
3b
200.00
9.00
208.78
8.89
205.81
8.95
Remarks: *) Deviation between initial and calculated elastic moduli

Deviation *)
EVERCALC
2L-BACK
E p, %
Es, %
E p, %
Es, %
3.15
0.44
1.93
0.44
-1.20
-1.67
-2.89
-1.78
1.19
0.56
2.24
0.67
-5.16
0.67
-4.27
0.78
-0.08
1.67
-0.75
1.56
-4.39
1.22
-2.91
0.56

Table 6. Calculated elastic moduli from 2002 Design Guide using deflection data with errors
No of
Initial elastic moduli
Calculated elastic moduli
data set
Ep, ksi
Es, ksi
Ep, ksi
Es, ksi
1a
200.00
9.00
183.02
7.98
1b
200.00
9.00
193.35
7.87
2a
200.00
9.00
181.29
7.90
2b
200.00
9.00
198.74
7.81
3a
200.00
9.00
193.13
7.67
3b
200.00
9.00
192.53
7.70
Remarks: *) Deviation between initial and calculated elastic moduli

3.3

Deviation
E p, %
8.49
3.33
9.36
0.63
3.44
3.74

*)
Es, %
11.33
12.56
12.22
13.22
14.78
14.44

Backcalculation Analysis Using LTPP Data

For the purpose of real comparison among the backcalculation methods used in this study, some
required measured data from roads in the State of Utah were extracted from LTPP database. The soil
at those roads is categorized as non-cohesive soil (or category A3 according to AASHTO soil
classification). Deflection data at designated sensor offsets generated at loads in the range of 15,373
lbf – 15,494 lbf was also collected, together with total layer thickness above the subgrade.
Those data then was used as input for backcalculation process using EVERCALC, 2L-BACK and
2002 Design Guide. Especially for EVERCALC program, it was assumed that the initial seed moduli,
range of initial seed moduli, and Poisson's ratio were 1000 ksi, 30 - 5000 ksi and 0.35, respectively,
fore layer 1 (or composite layer), and 45 ksi, 1 - 100 ksi, and 0.5, respectively, for the subgrade. The
results of the backcalculation analysis are presented in Table 7.
Table 7. Result of backcalculation analysis using deflections form LTPP database
Measured
Es
10.00
8.30
8.80
9.10

Calculated Es from different method
2002 Design
EVERCALC
2L-BACK
Guide
11.64
11.54
8.40
11.94
11.90
8.90
11.83
11.88
8.90
11.83
11.87
8.90

Deviation between measured & calculated Es (%)
2002 Design
EVERCALC
2L-BACK
Guide
-16.40
-15.40
13.90
-43.90
-43.40
-5.00
-34.40
-35.00
-0.80
-30.00
-30.40
1.70

From Table 7, it was found that the deviation varies (in absolute value) by 16.4% - 43.9% (2002
Design Guide), 15.5% - 43.4% (EVERCALC), and 0.8% - 13.9% (2L-BACK). It is very surprising
that EVERCALC produced high errors in this analysis. This might be contributed by errors involved
in the deflection used that could be more than 2%.

From these results, it appears that the proposed backcalculation algorithm, i.e. 2L-BACK, can provide
more accurate prediction of the modulus of subgrade compared to the others. In addition, it is also
very obvious that the calculated modulus produced by 2002 Design Guide was not reflected the
adjustment factor C stated previously since the ratio between measured and calculated modulus exceed
the criterion of C, i.e. maximum 0.33.

4.

CONCLUSIONS

Several conclusions can be drawn from the results as follows:
(a)
Load - deflection model of two layers was made by assuming that entire layers above the
subgrade were considered as one composite layer. By simplifying the pavement structure into
two layers, the main focus of this model is actually the subgrade itself. This is because the
composite layer becomes more difficult to be predicted as a result of various materials that
compose the layer.
(b)
A sensitivity analysis on EVERCALC and the proposed backcalculation algorithm (2LBACK) indicates that both programs were not sensitive to the deflections with errors less than
+ 2%. On the contrary, 2002 Design Guide showed the drawback of its backcalculation
procedure by consistantly producing errors in predicting resilient modulus of subgrade that
was more than 10%.
(c)
A real backcalculation analysis was also conducting by using data from Long Term Pavement
deflection performance (LTPP) database, and the results showed that 2L-BACK could show
superior results compared with other backcalculation methods for the type of non-cohesive
soil.
(d)
The ratio of the calculated resilient modulus obtained using 2002 Design Guide to the
measured one showed that the adjustment factor C in 2002 Design Guide was not relevant
anymore to be used since the ratio exceed the maximum criterion. Therefore, it seems that the
proposed backcalculation algorithm 2L-BACK could be used as a replacement for closed-form
backcalculation algorithm in 2002 Design Guide, although further validation using more field
data is still required to give more evidence on the robustness of the proposed algorithm.

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