CLOSED-FORM BACKCALCULATION ALGORITHM FOR INDONESIA OVERLAY DESIGN PROCEDURE
Bagus Hario SETIADJIa, SUPRIYONOb
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 the guideline, the design of overlay is commenced by predicting the evaluated layer moduli using
in which:pis non-destructive testing (NDT) load plate pressure (psi),ais NDT load plate radius (in.), D is total pavement layer thickness (in.), Ep is effective modulus of all pavement layers above the subgrade (psi), andd0is 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 Mrhas 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
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 thicknessD.
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).
in which: dtis true deflection (µm), and r1-r4are random numbers from 0 - 1. The error is generated by equation (3) will be limited in the range of + 2% of true deflectiondtto 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.
100
where Xi is the calculated moduli (ksi) based on true deflection with errors, xi is the initial moduli (ksi).
(c) 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 Lavenberg-Marquardt 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 areEp= 1000 ksi (with a range of modulus of 100 - 5000 ksi), whileEs= 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 valuesd1throughd6were calculated using EVERSTRESS and 2L-DEF (see Table 1).
Table 1. The data set used to determine the validity of the seed modulus No
1 15985 12 200 9 EVERSTRESS 26.134 23.793 20.805 18.107 13.647 8.083
2L-DEF 26.082 23.913 20.877 18.163 13.669 8.081
2 11241 12 200 9 EVERSTRESS 18.378 16.732 14.630 12.733 9.597 5.684
2L-DEF 18.341 16.816 14.681 12.773 9.612 5.682
3 15985 20 200 9 EVERSTRESS 17.351 16.026 14.718 13.581 11.511 8.189
2L-DEF 17.089 16.021 14.742 13.625 11.541 8.197
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
1 200.00 9.00 200.01 9.00 199.97 9.00 0.00 0.00 0.02 -0.01
2 200.00 9.00 200.00 9.00 199.97 9.00 0.00 0.00 0.02 -0.01
3 200.00 9.00 200.23 8.98 199.77 9.01 -0.11 0.22 0.11 -0.11
Remarks: *) Deviation between initial and calculated elastic moduli
Table 3. Calculated elastic moduli from 2002 Design Guide and its deviation against corresponding initial elastic moduli
No of data set
Initial elastic moduli Calculated elastic moduli Deviation *)
Ep, ksi Es, ksi Ep, ksi Es, ksi Ep, % Es, %
1 200.00 9.00 189.20 7.91 5.40 12.11
2 200.00 9.00 189.20 7.91 5.40 12.10
3 200.00 9.00 192.10 7.81 3.95 13.24
Remarks: *) Deviation between initial and calculated elastic moduli
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 moduliEpandEs. 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, 2L-BACK and 2002 Design Guide), as depicted in Tables 5 and 6.
Table 4. Deflection data set with errors No of data set
1a 15985 12 200 9 26.509 24.217 20.955 18.293 13.893 8.016
1b 15985 12 200 9 25.748 23.440 20.493 17.833 13.435 7.948
2a 11241 12 200 9 18.362 16.751 14.693 12.835 9.765 5.694
2b 11241 12 200 9 18.041 16.402 14.790 12.888 9.745 5.657
3a 15985 20 200 9 17.518 16.203 14.904 13.775 11.719 8.307
3b 15985 20 200 9 17.027 16.043 14.742 13.611 11.552 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% forEpandEs, respectively. It means that EVERCALC and 2L-BACK could predict the moduli better than that of 2002 Design Guide.
that resilient modulus Mr orEs 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 predictingEpare 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
data set
Initial elastic moduli
Calculated elastic moduli Deviation *)
EVERCALC 2L-BACK EVERCALC 2L-BACK
Ep, ksi Es, ksi Ep, ksi Es, ksi Ep, ksi Es, ksi Ep, % Es, % Ep, % Es, %
1a 200.00 9.00 193.71 8.96 196.15 8.96 3.15 0.44 1.93 0.44
1b 200.00 9.00 202.40 9.15 205.77 9.16 -1.20 -1.67 -2.89 -1.78
2a 200.00 9.00 197.61 8.95 195.53 8.94 1.19 0.56 2.24 0.67
2b 200.00 9.00 210.32 8.94 208.53 8.93 -5.16 0.67 -4.27 0.78
3a 200.00 9.00 200.15 8.85 201.49 8.86 -0.08 1.67 -0.75 1.56
3b 200.00 9.00 208.78 8.89 205.81 8.95 -4.39 1.22 -2.91 0.56
Remarks: *) Deviation between initial and calculated elastic moduli
Table 6. Calculated elastic moduli from 2002 Design Guide using deflection data with errors No of
data set
Initial elastic moduli Calculated elastic moduli Deviation *)
Ep, ksi Es, ksi Ep, ksi Es, ksi Ep, % Es, %
1a 200.00 9.00 183.02 7.98 8.49 11.33
1b 200.00 9.00 193.35 7.87 3.33 12.56
2a 200.00 9.00 181.29 7.90 9.36 12.22
2b 200.00 9.00 198.74 7.81 0.63 13.22
3a 200.00 9.00 193.13 7.67 3.44 14.78
3b 200.00 9.00 192.53 7.70 3.74 14.44
Remarks: *) Deviation between initial and calculated elastic moduli
3.3 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
CalculatedEsfrom different method Deviation between measured & calculated Es(%)
2002 Design
Guide EVERCALC 2L-BACK
2002 Design
Guide EVERCALC 2L-BACK
10.00 11.64 11.54 8.40 -16.40 -15.40 13.90
8.30 11.94 11.90 8.90 -43.90 -43.40 -5.00
8.80 11.83 11.88 8.90 -34.40 -35.00 -0.80
9.10 11.83 11.87 8.90 -30.00 -30.40 1.70
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 factorCstated previously since the ratio between measured and calculated modulus exceed the criterion ofC, 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 (2L-BACK) 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.
REFERENCES
American Association of State Highway and Transportation Officials (AASHTO) (1993) AASHTO Guide for Design of Pavement Structure.
Darter, M.I, Elliot, R., and Hall, K.T. (1992) Revision of AASHTO Pavement Overlay Design Procedures, Appendix Documentation of Design Procedures, NCHRP Project 20-7/Task 39. Directorate General of Highway (2002) Pedoman Perencanaan Tebal Perkerasan Lentur. Pd
T-1-2002-B.
Elkins, G.E., Schmalzer, P., Thompson, T. and Simpson, A. (2003) Long-Term Pavement Performance Information Management System – Pavement Performance Database User Guide, FHWA Report No. FHWA-RD-03-088, Washington, DC, USA.
Federal Highway Administration (2006) FHWA Memorandum on Pavement Preservation Definitions, Pavement Preservation Compendium II, US Department of Transportation.
Galehouse, L. (2003) Strategic Planning for Pavement Preventive Maintenance: Michigan Department of Transportation’s Mix of Fixes Program, Pavement Preservation Compendium, Federal Highwa of Administration (FHWA).
Irwin, L.H., Yang, W.S., and Stubstand, R.N. (1989) Deflection Reading Accuracy and Layer Thickness Accuracy in Backcalculation of Pavement Layer Moduli, Non-destructive Testing of Pavement and Backcalculatoon of Moduli, ASTM STP 1026, American Society for Testing and Materials, West Conshohocken, PA., pp. 229-244.
Minister of Public Works (2011) Road Inspection and Maintenance, Regulation of the Minister of Public Works Republic of Indonesia No. 13/PRT/M/2011.
Pronk, A.C. (1998) Implementation Problems and Reliability of Falling Weight Deflectometer (FWD) Measurements on Three Layer Systems, Proceedings of Association of Asphalt Paving Technologist, Williamsburg, VA, Vol. 57.
Setiadji, B.H. (2010) Closed-form Backcalculation Algorithms for Pavement Analysis, Doctoral Thesis, National University of Singapore.
