Uncertainty Distribution Models for Key Input Parameters for Design and Analysis of Flexible Pavement in Nigeria

JOURNAL OF SCIENCE TECHNOLOGY AND EDUCATION 8(2), JUNE, 2020 ISSN: 2277-0011; Journal homepage: www.atbuftejoste.com Uncertainty Distribution Models for Key Input Parameters for Design and Analysis of Flexible Pavement in Nigeria

Shuaibu, A. A., Kaura, J. M., Murana, A. A., Olowosulu, A. T. Department of Civil Engineering, Ahmadu Bello University, Zaria. ABSTRACT The road transportation sector is cardinal to accessing the ARTICLE INFO growth and socio-economic development of any country. This Article History Received: October, 2019 is because it contributes directly and immensely to support Received in revised form: Dec., 2019 other sectors of the economy to function properly. Flexible Accepted: April, 2020 pavement is a multi-layer system of which its design is mostly Published online: June, 2020 based on empiricism, assuming a deterministic process with final output as pavement thickness. Each layer thickness is KEYWORDS designed and placed in a manner that the wheel loading does Uncertainty, distribution model, layer not become excessive for the subgrade to bear. Various thickness, layer stiffness, variability studies have revealed uncertainties due to variations of thickness and layer modulus which could result in premature failure of the pavement. This variability could be as a result of poor design practice, inadequate quality control, complex traffic loading, complex material properties, lack of fit of design model as well as subgrade characteristic. To address these variabilities many highway agencies were forced into probabilistic design approach to accommodate uncertainties. Of crucial importance to this type of design is the characterization of material properties and establishment of appropriate probability distribution models. This study focused on the development of probability distribution models of key input pavement parameter to the design and analysis of flexible pavement. Data on the geometry and material properties were collected from Pavement Evaluation Unit (PEU), Kaduna and Federal Ministry of Power Works and Housing (FMPWH), Abuja. The probability distribution models of the variation of layer thicknesses, asphalt stiffness, base stiffness and subgrade stiffness were developed using Easy fit 5.5 statistical packages. The results of the studies revealed that the best distribution model for the layer thicknesses as well as stiffness considered in the study is the normal distribution.

INTRODUCTION variability and uncertainty in pavement The large stochastic/random performance and may increase the cost of elements in pavement deterioration caused maintenance and rehabilitation over time. by immeasurable and unpredictable factors Analysis of data obtained by in situ core and (Harvey, 2012) introduces a measure of ground penetration radar indicates spatial

Corresponding author: Shuaibu, A. A. [email protected], Department of Civil Engineering, Ahmadu Bello University, Zaria © 2020, Faculty of Technology Education, ATBU Bauchi. All rights reserved 320 JOURNAL OF SCIENCE TECHNOLOGY AND EDUCATION 8(2), JUNE, 2020 ISSN: 2277-0011; Journal homepage: www.atbuftejoste.com variation in pavement thickness layer. design inputs are, therefore fundamentals Likewise, the variation of layer modulus in developing a probabilistic-based design (Maji et al., 2016). that evaluates reliability (Dalla Valle and Uncertainty in engineering analysis Thom, 2016). An important aspect to this and design is commonly defined as approach is the appropriateness or fitness knowledge incompleteness due to inherent of distribution models to the substantive deficiencies in acquired knowledge. situation and/or the data sample of interest It can be used to characterize the state of a and this is the solution the test of goodness- system as being unsettled or in doubt, such of-fit (GOF) seek to proffer. as the uncertainty of the outcome (Ayyub The test is very relevant in risk and Klir, 2006). Because uncertainty is an analysis and is used to check how close the important dimension in the analysis of risks, data set distribution is to the hypothesized the nature and ways of dealing with these theoretical distribution (Babashamsi et al., uncertainties have been of interest to the 2016). The simplest case of a GOF test is a highway community for a long time. statistical null hypothesis of a completely Probability is the mechanism specified distribution against all possible employed to manage the uncertainty that alternative distributions (Anderson, 1995). underlies all real-world data and Thus, the outcome of the GOF test is used phenomena. It assists in gauging the belief to statistically verify an assumed theoretical and to quantify the lack of certitude that is distribution used in design and analysis or inherent in the process that generates the to disprove it. Fitting each random variable data been analyzed (Martinez and Martinez, into the best probability distribution model 2002). For example, to adequately simulate is key to performance prediction and a real system, one needs to understand the ultimately management decisions on any probability distributions that correctly road pavement under considerations. model the underlying processes. According to Kaura, (2015), in the Uncertainty of probabilistic input assignment of distribution models to a consists of specifying the types of random variable, two or more distribution probability distributions of the variables may appear to be plausible probability under consideration as well as the distribution models. In that case, the test of parameters of the distributions goodness-of-fit (GOF) can be used to (Oberguggenberger and Fellin, 2004). An delineate the relative degree of validity of adequate understanding of the influence of the different distributions. input parameter variability on the output of Goodness-of-Fit tests is a engineering computations requires that the procedure to determine whether a sample uncertainty itself be captured in n observation, x1, … xn of certain distribution mathematical terms (Oberguggengerger, is considered as a sample from a given 2004). In the specific area of pavement specified distribution (Anderson, 2011). engineering, techniques such as Calculating statistics of Goodness-of-Fit probabilistic analyses, general stochastic helps to rank the fitted distributions analysis and random fields have been used according to the quality of fit over the raw to explore the impact of material variability data (Mehrannia, and Pakgohar, 2014). The and uncertainty quantification in pavement ultimate ends for performing this GOF design and performance (Castillo and Caro, analysis is to obtain information which 2014). helps take best-informed decisions under Quantification and analyzing uncertainty conditions using best-fitted variability of pavement materials and distribution. This distribution is a model of Corresponding author: Shuaibu, A. A. [email protected], Department of Civil Engineering, Ahmadu Bello University, Zaria © 2020, Faculty of Technology Education, ATBU Bauchi. All rights reserved 321 JOURNAL OF SCIENCE TECHNOLOGY AND EDUCATION 8(2), JUNE, 2020 ISSN: 2277-0011; Journal homepage: www.atbuftejoste.com the stochastic process, which is faced in the parameters with great influence on the real world. variability of predicted fatigue performance Material variability can be without considering variable loads are; described in statistical terms such as mean, thickness and stiffness of asphalt concrete standard deviation together with its layer. Also, for rutting, the parameters of probability density distribution (Dalla Valle, great influence are; base thickness, asphalt 2005). A useful dimensionless way of concrete thickness, and stiffness of expressing the variability of a material’s subgrade. However, when the effect of property is to use the ratio of the standard variable load was considered, the output deviation over the mean, known as the variability for fatigue and rutting was coefficient of variation (COV). Knowledge of doubled. This is an indication that traffic is a the coefficient of variation of each design single most important parameter in input is extremely important to more predicting fatigue and rutting life of the accurately estimate their influence on the pavement. predicted pavement life (Dalla Valle and Dalla Valle and Thom, (2016), Thom, 2016). researched the reliability of pavement As earlier stated, the sources of design. The research revealed that the variability in pavement engineering have parameters with the greatest influence on been of interest to researchers for a very the variability of predicted fatigue long time. Researchers have carried out performance are the asphalt stiffness studies in the field of reliability analysis on modulus and thickness. The parameters pavement and have concluded on with the greatest influence on the parameters that are crucial to pavement variability of predicted deformation performance. Since these parameters are performance are the granular sub-base key, they must be treated as random thickness, the asphalt thickness and the variables in a probabilistic non-arbitrary subgrade stiffness. manner to account for variabilities that are Maji et al., (2016) developed a probabilistic associated with them to avoid the approach for asphaltic overlay design by premature failures and to maintain the considering variability in input parameter. A pavements at a satisfactory level service. sensitivity analysis of pavement layer Chou, (1990) carried out a study on moduli performed in this study revealed reliability design procedure for airport that the asphaltic overlay thickness is flexible pavements. He concluded that the sensitive to the resilient modulus of base performance of convectional flexible layer for fatigue and subgrade layer for pavement is sensitive, in descending order rutting. to variable of gear load, the thickness of Prozzi et al., (2005), carried out a granular base, subgrade modulus and study on the simulation-based reliability of thickness and modulus of bituminous pavement structures using empirical- concrete surface course respectively for mechanistic models. The design of the subgrade strain-failure criterion, and to structure in question is based on the variation of gear load, thickness and AASHTO design approach. A detailed modulus of asphalt concrete, thickness and examination of the failed pavements modulus of granular base and subgrade showed that parameters of great influence modulus for the bituminous strain-failure on the pavement performance are the criteria. surface asphalt thickness and model error. Timm et al., (2000) and Dalla Valle From the foregoing, subgrade, and Thom, (2016), submitted that the base and asphalt concrete surfacing moduli Corresponding author: Shuaibu, A. A. [email protected], Department of Civil Engineering, Ahmadu Bello University, Zaria © 2020, Faculty of Technology Education, ATBU Bauchi. All rights reserved 322 JOURNAL OF SCIENCE TECHNOLOGY AND EDUCATION 8(2), JUNE, 2020 ISSN: 2277-0011; Journal homepage: www.atbuftejoste.com as well as their respective thicknesses Ministry of Power Works and Housing (excluding that of subgrade) have been (FMPWH), Abuja. Pavement input identified as key input parameters that parameters used for the study are extracted have great influence on fatigue and from Region 1 of the trunk road study of deformation/rutting lives of pavement. Nigeria (Claros et al.,1986). Specifically, Therefore, they are considered as random master test section MTS, 1 of Claros et al., variables in this study and fitted to obtain (1986) based on the information presented the probably best distribution models. in Nigeria overlay design procedure was Though many reliability studies used. The MTS is located on road A236. The have been carried out in Nigeria, most of A236 highway runs from the A2 highway at these studies rely solely on distribution Zaria, Kaduna State to the A3 highway at Jos models specified by authors abroad based the capital of Plateau State. (Zaria - on their peculiarities. This might not Pambegua - Saminaka – Plateau). necessarily fit into the Nigerian situation. Road A236 is a trunk route with Thus, this study attempt to determine two lanes in opposite directions. The road uncertainty, appropriate distribution has a total length of 170 km, and width of models based on Nigerian conditions, for 7.3m (FMPWH, 2016). The road still has pavement design and analysis. similar features as described from the previous study of Claros et al., (1986) and METHODOLOGY there are no current studies to advance the The data on the geometry and existing data. material properties of flexible pavement The input parameters of interest were collected from the pavement are as shown in Figure 1 below. evaluation unit (PEU), Kaduna and Federal

Table1: Input Parameters for Road A236 (FMPWH, 2016). Layer Materials Layer Layer Modulus (N/mm2) Poison Ratio Thickness (mm) Asphalt concrete AC 63.50 6205.28 0.35 Granular base GB 139.70 620.53 0.20 Granular sub-base GSB 71.12 310.26 0.35 Granular Subgrade GS 7620 179.26 0.40

The statistics of the design input parameters used in this study are presented in Table 2.0 below

Table 2: Statistics of the design input parameters Property Mean Coefficient Reference COV Standard Assumed (μ) of Used Deviation Distribution Variation (%) Asphalt concrete 63.50 0.03 – 0.12 Timm,et 7 4.45 Normal thickness TAC al., (2000), Granular base 139.70 0.10 – 0.15 Noureldin 12 16.76 Normal thickness TGB et al., (1994)

Granular 7620 - - - Deterministic Subgrade thickness TGS

Asphalt concrete 6205.28 0.10 – 0.40 Timm,et 25 1551.32 Normal modulus EAC al., (2000)

Granular base 620.53 0.5 – 0.60 Timm,et 30 186.16 Normal modulus EGB al., (1999)

Granular sub- 310.26 0.10 – 0.30 Noureldin 15 46.54 Normal base modulus et al., EGSB (1994)

Granular 179.26 0.20 – 0.45 Noureldin 35 62.74 Normal Subgrade et al., modulus EGS (1994)

Generation of the Random Variables variables (Babashamsi et al., 2016), a study Generation of random numbers for by Li, (1997) as cited by Dalla Valle,(2005) modelling purposes is widely used and well revealed that difference between the known in the world of science (Dalla Valle, results from assumed normal and 2005). The random number generation tool lognormal distributions in simulation in Easyfit 5.5 statistical computer package exercises is negligible. It is important to (2011) developed by Mathwave was used to note that no two random number generate 100,000 random numbers for each generation is the same, but they must key input parameter in line with previous statistically agree at all times. Thus, best fit studies by (Dalla Valle and Thom, 2016). models are obtained devoid of the assumed This simulation is only possible if the mean theoretical model used. and coefficient of variation are known. Another factor of consideration is 2.2 Distributions Fitting and Tests of probability distribution; the normal Goodness-of-Fit (GOF) distribution was used in the random In assigning of distribution models number generation as have been used by to each random variable, the Kolmogorov- several authors because of its relative Smirnov (K-S) test of goodness-of-fit was simplicity. used. It was chosen because of its relative Though the lognormal distribution advantage. It is a very relevant test in risk is said to be best describe construction analysis used to check how close the data

Table 3.0: Statistical parameters of asphalt concrete thickness Distribution Distribution S/No Name Parameters 1 Frechet =17.425 =61.294 2 Gumbel Max =3.4657 =61.515 3 Lognormal =0.07041 =4.1488 5 Normal =4.4449 =63.515 5 Weibull =17.852 =65.44

Table 4.0: Result of Kolmogorov Smirnov Test of Goodness of Fit for Asphalt Concrete Thickness S/No Distribution α = 0.2 α = 0.1 α = 0.05 α = α = 0.01 KS Rank Model 0.02 Statistics 0.00339 0.00387 0.00429 0.0048 0.00515 1 Normal No No No No No 0.00204 1 2 Lognormal Yes Yes Yes Yes Yes 0.01552 2 3 Weibull Yes Yes Yes Yes Yes 0.0557 3 4 Gumbel Max Yes Yes Yes Yes Yes 0.07243 4 5 Frechet Yes Yes Yes Yes Yes 0.0857 5

The test was performed at 0.2, 0.1, asphalt concrete thickness. Also considering 0.05, 0.02 and 0.01 level of significance the KS statistic for all the distribution (that is α = 0.2, 0.1, 0.05, 0.02 and 0.01). A models, the normal distribution has the hypothesis was put forward that each of the lowest (ranked 1) which implies it is the five distribution models; Normal, best fit for asphalt concrete. Lognormal, Gumbel (extreme type I), From the foregoing, it can be Weibull and Frechet can be used to model concluded that the probability distribution asphalt concrete thickness. model for asphalt concrete thickness for From Table 4.0, the critical values Nigerian site considered is normal. This is in for the Kolmogorov Smirnov test statistics line with the findings of Timm et al., (1999); for normal distribution at all the confidence Timm et al., (2000); Dalla Valle and Thom, levels considered were all greater than KS (2016); Retherford, and McDonald, (2010). statistics obtained from the test. Therefore, the hypothesis that normal distribution can Granular Base Thickness TGB be used to model asphalt concrete As explained in the previous thickness is accepted. Also, it is clear from section with the asphalt concrete, the same the table that the critical values of KS at all procedure was followed to arrive at similar confidence level is less KS statistics for the results in the granular base layer. The lognormal, Weibull, Gumbel max and statistical parameter (mean, standard Frechet. Therefore, these distributions deviation and distribution model) of the models are not considered acceptable for granular base is presented in Table 5.0.

The Kolmogorov-Smirnov test of base and the result is as presented in Table goodness of fit was used to determine the 6.0. suitable distribution model for granular

Table 6.0: Result of Kolmogorov Smirnov Test of Goodness of Fit for Granular Base Thickness S/No Distribution α = 0.2 α = 0.1 α = 0.05 α = α = 0.01 KS Rank Model 0.02 Statistics 0.00339 0.00387 0.00429 0.0048 0.00515 1 Normal No No No No No 0.00247 1 2 Lognormal Yes Yes Yes Yes Yes 0.002489 2 3 Weibull Yes Yes Yes Yes Yes 0.0464 3 4 Gumbel Max Yes Yes Yes Yes Yes 0.07098 4 5 Frechet Yes Yes Yes Yes Yes 0.0955 5

Table 6.0 above shows the result of level is less KS statistics for the Weibull, the Kolmogorov-Smirnov test of goodness Gumbel max and Frechet. Therefore, these of fit for granular base thickness. The test distributions models are not considered was performed at 0.2, 0.1, 0.05, 0.02 and acceptable for asphalt concrete thickness. 0.01 level of significance (that is α = 0.2, 0.1, Also considering the KS statistic for 0.05, 0.02 and 0.01). A hypothesis was put all the distribution models, the normal forward that each of the five distribution distribution has the lowest, followed by models; normal, lognormal, Weibull, lognormal. This implies that the normal Gumbel max and Frechet can be used to distribution model is the best fit for model granular base thickness. granular base thickness then followed by From the table, for normal and lognormal. lognormal distributions, the critical values From the foregoing, it can be concluded for the Kolmogorov Smirnov test statistics that the probability distribution model for at all the confidence levels considered were granular base thickness for Nigerian site all greater than KS statistics obtained from considered is normal (ranked 1), then the test. Therefore, the hypothesis that lognormal (ranked 2). This is in line with the normal and lognormal distribution can be findings of Timm et al., (1999); Timm et al., used to model granular base thickness is (2000); Dalla Valle and Thom, (2016); accepted. Also, it is clear from the table that Retherford, and McDonald, (2010). the critical values of KS at all confidence

Table 7.0: Statistical Parameters of Granular Sub-base Thickness Distribution Distribution S/No Name Parameters 1 Frechet =7.757 =65.249 2 Gumbel Max =8.3275 =66.31 3 Lognormal =0.15493 =4.2526 4 Normal =10.68 =71.117 5 Weibull =8.2038 =75.412

The Kolmogorov Smirnov test of suitable distribution model for granular goodness of fit was used to determine the base as presented in Table 8.0 below.

Table 8.0: Results of Kolmogorov-Smirnov Test of Goodness of Fit for Granular sub-base Thickness S/No Distribution α = 0.2 α = 0.1 α = 0.05 α = α = 0.01 KS Rank Model 0.02 Statistics 0.00339 0.00387 0.00429 0.0048 0.00515 1 Normal No No No No No 0.00238 1 2 Lognormal Yes Yes Yes Yes Yes 0.03348 2 3 Weibull Yes Yes Yes Yes Yes 0.03814 3 4 Gumbel Max Yes Yes Yes Yes Yes 0.07272 4 5 Frechet Yes Yes Yes Yes Yes 0.10431 5

The test was performed at 0.2, 0.1, thickness is accepted. At the confidence 0.05, 0.02 and 0.01 level of significance level of 0.2, the Kolmogorov Smirnov test (that is α = 0.2, 0.1, 0.05, 0.02 and 0.01). A statistics for lognormal distribution is hypothesis was put forward that each of the greater than that of KS statistic. Thus, the five distribution models; normal, lognormal, lognormal distribution is acceptable for Weibull, Gumbel max and Frechet can be granular sub-base, but only at a confidence used to model granular sub-base thickness. level of 0.2. From the table, for normal Also, it is clear from the table that distribution, the critical values for the the critical values of KS at all confidence Kolmogorov Smirnov test statistics at all the level is less KS statistics for the Weibull, confidence levels considered were all Gumbel max and Frechet. Therefore, these greater than KS statistics obtained from the distributions models are not considered test. Therefore, the hypothesis that normal acceptable for granular sub-base thickness. distribution can be used to model sub-base Also considering the KS statistic for all the

Table 9.0: Statistical parameters of Asphalt Concrete Modulus Distribution Distribution S/No Name Parameters 1 Frechet =20895.0 =3.2341E+7 =-3.2335E+7 2 Gumbel Max =1204.8 =5513.2 3 Lognormal =0.04227 =10.51 =-30508.0 4 Normal =1545.2 =6208.7 5 Weibull =4.8526 =7402.9 =-586.46

The Kolmogorov-Smirnov test of goodness of fit was used to determine the suitable distribution model for asphalt concrete modulus as presented in Table 10.0.

Table 10.0: Result of Kolmogorov Smirnov Test of Goodness of Fit for Asphalt concrete modulus S/No Distribution α = 0.2 α = 0.1 α = 0.05 α = α = 0.01 KS Rank Model 0.02 Statistics 0.00339 0.00387 0.00429 0.0048 0.00515 1 Normal No No No No No 0.00149 1 2 Lognormal Yes Yes Yes Yes Yes 0.0097 2 3 Weibull Yes Yes Yes Yes Yes 0.02182 3 4 Gumbel Max Yes Yes Yes Yes Yes 0.07154 5 5 Frechet Yes Yes Yes Yes Yes 0.06143 4

Table 11.0: Statistical parameters of Granular base Modulus Distribution Distribution S/No Name Parameters 1 Frechet =3.5311 =709.31 =-211.48 2 Gumbel Max =145.73 =536.03 3 Lognormal =0.03949 =8.4652 =-4129.4 4 Normal =186.91 =620.15 5 Weibull =4.5057 =835.78 =-143.61

The Kolmogorov Smirnov test of suitable distribution model for granular goodness of fit was used to determine the base modulus as presented in Table 12.0.

Table 12.0: Result of Kolmogorov Smirnov Test of Goodness of Fit for Granular base Modulus S/No Distribution α = 0.2 α = 0.1 α = 0.05 α = α = 0.01 KS Rank Model 0.02 Statistics 0.00339 0.00387 0.00429 0.0048 0.00515 1 Normal No No No No No 0.00176 1 2 Lognormal Yes Yes Yes Yes Yes 0.00632 2 3 Weibull Yes Yes Yes Yes Yes 0.01869 3 4 Gumbel Max Yes Yes Yes Yes Yes 0.07116 4 5 Frechet Yes Yes Yes Yes Yes 0.12273 5

Table 13.0: Statistical parameters of Granular Sub-base Modulus Distribution Distribution S/No Name Parameters 1 Frechet =7.6928 =284.32 2 Gumbel Max =36.489 =289.06 3 Lognormal =0.15597 =5.7251 4 Normal =46.799 =310.12 5 Weibull =8.1551 =328.94

The Kolmogorov Smirnov test of subbase modulus and the result is goodness of fit was used to determine the presented in Table 14.0 below. suitable distribution model for granular

Table 14.0: Result of Kolmogorov-Smirnov Test of Goodness of Fit for Granular Sub-base Modulus S/No Distribution α = 0.2 α = 0.1 α = 0.05 α = α = 0.01 KS Rank Model 0.02 Statistics 0.00339 0.00387 0.00429 0.0048 0.00515 1 Normal No No No No No 0.0233 1 2 Lognormal Yes Yes Yes Yes Yes 0.0335 2 3 Weibull Yes Yes Yes Yes Yes 0.03778 3 4 Gumbel Max Yes Yes Yes Yes Yes 0.07246 4 5 Frechet Yes Yes Yes Yes Yes 0.10439 5

Table 15.0: Statistical parameters of Granular Sub-base Modulus Distribution Distribution S/No Name Parameters 1 Frechet =3.4152 =235.4 =-97.839 2 Gumbel Max =48.908 =150.66 3 Lognormal =0.03724 =7.4266 =-1502.2 4 Normal =62.727 =178.89 5 Weibull =4.5814 =284.58 =-81.421

The Kolmogorov Smirnov test of subgrade and the result is presented in goodness of fit was used to determine the Table 16.0. suitable distribution model for granular

Table 16: Result of Kolmogorov-Smirnov Test of Goodness of Fit for Granular Subgrade Modulus S/No Distribution α = 0.2 α = 0.1 α = 0.05 α = α = 0.01 KS Rank Model 0.02 Statistics 0.00339 0.00387 0.00429 0.0048 0.00515 1 Normal No No No No No 0.00224 1 2 Lognormal Yes Yes Yes Yes Yes 0.00646 2 3 Weibull Yes Yes Yes Yes Yes 0.01963 3 4 Gumbel Max Yes Yes Yes Yes Yes 0.07047 4 5 Frechet Yes Yes Yes Yes Yes 0.12818 5

Corresponding author: Shuaibu, A. A. [email protected], Department of Civil Engineering, Ahmadu Bello University, Zaria © 2020, Faculty of Technology Education, ATBU Bauchi. All rights reserved 332 JOURNAL OF SCIENCE TECHNOLOGY AND EDUCATION 8(2), JUNE, 2020 ISSN: 2277-0011; Journal homepage: www.atbuftejoste.com The test was performed at 0.2, ii. The distribution models that best 0.1, 0.05, 0.02 and 0.01 level of fit the layer thicknesses, as well significance (that is α = 0.2, 0.1, 0.05, 0.02 as their modulus (stiffness) for and 0.01). A hypothesis was put forward probabilistic assessment of that each of the five distribution models; pavement performance, are the Normal, Lognormal, Weibull, Gumbel max normal distribution followed by and Frechet can be used to model the lognormal. granular subgrade modulus. From the table, for normal The findings of this study are distribution, the critical values for the valuable in providing designers and Kolmogorov Smirnov test statistics at all researchers with the best distribution the confidence levels considered were all models for flexible pavement greater than KS statistics obtained from components thickness and stiffness for the test. Therefore, the hypothesis that purpose of probabilistic design and normal distribution can be used to model assessment of flexible pavement in granular subgrade modulus is accepted. Nigerian situations. For the Lognormal, Weibull, Gumbel max and Frechet distributions, the REFERENCES Kolmogorov-Simirnov statistics at all Anderson, T. W, (2011.), Anderson– confidence level was greater than that of Darling Tests of Goodness-of-Fit, the KS statistics leading to the rejection of InternationalEncyclopedia of these distributions models as the best fit. 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