Modeling pavement damage and predicting fatigue cracking of flexible pavements based on a combination of deterministic method with stochastic approach using Miner’s hypothesis
Abstract
Distress in flexible pavements has been a problem in many countries due to the increase in road traffic and vehicle loads. One of the most important distress modes in the design of pavements is fatigue cracking. Despite the fact that there have been considerable efforts in recent years in fatigue performance evaluation and the design process of flexible pavements, there is still a need for further studies to overcome the difficulty in predicting fatigue cracking in terms of damage distribution considering the uncertainty associated with the input parameters of pavement life and traffic repetitions. The purpose of this paper is to develop a methodology for modeling pavement damage and predicting fatigue cracking of flexible pavements based on a combination of deterministic method with stochastic approach using Palmgren–Miner’s hypothesis. Four predictive models are introduced and used in a comprehensive case study to estimate the fatigue life of the pavement surface layer. The solutions are obtained through numerical integration based on Gaussian quadrature method. The results reveal that pavement damage has a broad range of distribution in which the actual traffic load repetitions has a Poisson distribution, while the traffic repetitions to failure follows a lognormal distribution.
Keywords
Damage Fatigue cracking Flexible pavement Deterministic Stochastic Miner Poisson processList of symbols
 \(A\)
Average annual commercial vehicle per day
 \(C\)
Laboratory to field adjustment factor
 \(D\)
Damage index of fatigue cracking
 \(E\)
Stiffness modulus of the asphalt layer
 \(F_{\text{j}} \left( {\text{J}} \right)\)
Cumulative distribution function of \({\text{J}}\)
 \(F_{L}\)
Conversion factor
 \(f_{\text{j}} \left( {\text{J}} \right)\)
Probability density function of \({\text{J}}\)
 h
Pavement thickness
 \(K_{T}\)
Thickness correction factor
 \(k_{i}\)
Laboratory material coefficients
 \(L_{f}\)
Lateral distribution factor
 \(r_{t}\)
Annual traffic growth rate
 \(T\)
Total number of periods
 \(V_{a}\)
Air void content
 \(V_{b}\)
Volumetric asphalt content
 \(X\)
Actual traffic load repetitions
 \(Y\)
Number of load repetitions to cause failure
 \(y_{t}\)
Design period
 \(\beta_{i}\)
Field calibration coefficients
 \(\varepsilon_{t}\)
Tensile strain at the bottom of the asphalt layer
 \(\lambda_{X}\)
Expected number of occurrences
 \(\mu_{j}\)
Mean value of \(j\)
 \(\sigma_{j}\)
Standard deviation of \(j\)
 \(\sigma_{j}^{2}\)
Variance of \(j\)
1 Introduction
Pavement distress, also known as pavement failure, is defined as any signs of break or fracture in the pavement layer, or any indication of poor or undesirable pavement performance [1]. Guo and Prozzi [2] investigated fatigue life of flexible pavements due to repeated loading. There are many types of pavement distress such as fatigue cracking, rutting, potholes, patches, raveling, bleeding, etc. [3]. The previous research revealed that the fatigue cracking, caused by repeated and excessive traffic loading, is the most common distress observed in flexible pavements [4, 5].
Fatigue cracking first appears as a set of microcracks in the wheel paths, and progresses into a network of interconnecting cracks, and eventually leading to potholes [3]. Fatigue cracking, in flexible pavements, consists of two phases: crack initiation and crack propagation which is caused by tensile strains generated in the pavement due to traffic loading and temperature variations [6]. Crack initiation can be measured using different experiments. Despite the various efforts in testing and measuring the crack initiation and propagation, these attempts still suffer in view of their stochastic nature [6]. Further, there are different severity levels to further define the fatigue cracking. According to the pavement distress survey manual [3], low severity fatigue cracking consists of some connecting cracks, and the cracks are not spalled or sealed with no signs of pumping. Moderate severity is reached when the cracks become interconnected, and the cracks may be slightly spalled and may be sealed, and pumping is not evident. A high severity fatigue cracking occurs when pieces may move when subjected to traffic, cracks may be sealed, and pumping is evident. Fatigue cracking can be classified into two main groups: alligator or bottom up fatigue cracking, longitudinal or top down fatigue cracking, and both may look identical on the pavement surface [6]. Alligator fatigue cracking initiates at the bottom of asphalt layer due to mechanical failure caused by the highest tensile stress and strain then propagates randomly upwards to the surface of pavement [7]. Thin pavement layers are most likely to exhibit bottom up fatigue cracking problems, which makes it a problem often aggravated by cold weather [8]. Longitudinal cracking, conceptually similar to alligator cracking, develops at the surface where high localized tensile stress and strain resulting from tirepavement interaction exist and propagates downwards to the bottom of asphalt layer. Thick pavement layers are most likely exposed to top bottom fatigue cracking [9].
In this research, a methodology has been developed for modeling the pavement damage and predicting the fatigue cracking of flexible pavements by combining the deterministic and stochastic techniques (using Palmgren–Miner’s hypothesis) into a general approach. The developed methodology overcomes the shortcomings associated with the detminustic and stochastic models when they are applied individually.
2 Pavement performance models
Pavement performance or damage prediction models are one of the major challenges facing researchers and design engineers. In the sixties and seventies, several studies of pavement response due to repreated loads were carried out based on many laboratory fatigue tests on asphalt pavement. As a result, the fatigue life of the asphalt pavement was established based on a relationship between material coefficients and stress or strain levels induced by repeatedly applied loading [2]. Extensive research has been conducted to model and predict fatigue cracking in the last few years. Based on the prediction results of performance models, these models may be classified as either deterministic or probabilistic. Deterministic models can be divided into mechanistic, empirical and mechanistic–empirical (M–E) models [10, 11]. Mechanistic models are based on the theories of mechanics, in which stresses and strains of a pavement layer can be obtained using simple assumptions and simplifications, such as isotropic, linearelastic and homogeneous material, small strain and static loading. While these models are simple and can provide the general response behavior of the pavement, they are not practically effective to describe pavement deterioration due to high nonlinearity in the behavior of pavement materials which are not only anisotropic but also depend on time, temperature and other parameters [12]. Empirical models are based on the results of experiments and statistical techniques [13]. They are employed to overcome the limitations of simplified theoretical models used in the mechanistic approach. The empirical models link the pavement distress with the traffic loadings and pavement deflection and provide the number of load repetitions required to cause pavement failure [13]. One of the major disadvantages of the empirical models is that they are experiment based and developed for particular pavemment area and thus cannot be directly applied for different pavement sections. In other words, they can be used only to a particular section area. M–E models combine the mechanistic and the empirical approaches into one general model to take advantage of the merits of each model and also to overcome some of the shortcomings associated with those models once applied individually [13]. In M–E models, the strains generated at the critical locations due to single wheel load of the vehicle are, first, identified, and then the empirical fatigue model can be used to determine the pavement life. Probabilistic models, which predict distribution of events and occurrences, are represented by transition probability process models and reliability analysis to estimate deterioration with age for different combination of variables [10, 11 and 13].
One of the first empirical models was reported in 1929 [12]. In the fifties, pavement performance received some attention from several researchers utilizing empirical models. Since then, several empirical models were developed to link pavement performance data to design input parameters. The most common empirical model used worldwide is American Association of State Highway and Transportation Officials (AASHTO) which has been improved many times over the years for different pavements [12].
In 2007, Schwartz and Carvalho [12] published a design guide for the NCHRP 137A project including most of the M–E method to predict pavement distresses due to traffic load incorporating the environmental conditions. Moreover, the NCHRP 137A project replaced the equivalent single axle load (ESAL) by distribution of vehicular loads. The M–E PDG model used to predict fatigue cracking using the Asphalt Institute method in 1991 was calibrated using 82 LTPP (Long Term Pavement Performance) section data in 24 states across the USA. First, the pavement damage is determined and then the damage is converted into cracked area. Several revised M–E fatigue models have been suggested by other researchers such as ElBasyouny and Witczack, in 2005 [14]. The number of repetitions to cause failure for a given load can be expressed as follows [12]:
\(\beta_{1}\),\(\beta_{2} ,\beta_{3}\) represent the field calibration coefficients.
 (a)
Alligator fatigue cracking (bottom–up fatigue cracking)
 (b)
Longitudinal fatigue cracking (top–down fatigue cracking)
While deterministic models have received appropriate attention in improving methods to determine pavement failure, they may not be accurate enough to predict the pavement distress due to the uncertainty and variability of some pavement parameters. Thus, many probabilistic models have been proposed to cover the stochastic nature of the pavement performance. Golabi et al. [19] in 1982 developed a pavement management system to capture the dynamic and probabilistic aspects of pavement maintenance using the Markov chain method. Madanat et al. [20] in 1995 used a structured econometric approach using a joint discrete–continuous model to predict pavement cracking initiation and progression. Further studies have been suggested by several researchers to investigate the pavement deterioration based on Markov chain processes such as that of Li et al. [21] who in 1996 developed a Markov probabilistic method to determine a pavement deterioration rates in which the transition probability matrices are determined based on a reliability analysis using Monte Carlo simulation technique. An improved Markov chain model based on the probability distributions using timebased models was proposed by Mishalani and Madanat [22] in 2002. In 2010, Retherford and McDonald [23] discussed the advantages and disadvantages of reliability methods used in M–E approach for the pavement design.
Other efforts are made for the pavement performance prediction considering the uncertainties and random factors in the pavement deterioration process using techniques such as neural networks, fuzzy logic and hybrid systems [11]. However, these attempts are still in the research and development stage. Despite the various efforts in improving the pavement performance prediction effectively based on either deterministic or probabilistic methods, these attempts suffer from the restrictions associated with the difficulty of considering accurately the dynamic and stochastic nature of pavement distress modes.
3 Pavement damage and predicting fatigue cracking
In order to evaluate the fatigue cracking as given by Eq. (24), the PDF of D is required which are generally unknown, although the majority of the previous research works [27, 28] assumed that the pavement damage is either normally or lognormally distributed. In this study, two pavement damage models (1 and 2) based on Poisson distribution to characterize the actual traffic load repetitions have been proposed and compared with previously developed models (3 and 4) [25, 26, 27, 28].
3.1 Damage model 1
Assuming that the mean is a large value, the Poisson distribution maybe approximated by a normal distribution [29] with mean and variance as independent parameters defined as:
Taking logarithm on both sides of Eq. (13) yields:
3.2 Damage model 2
According to Fieller [32] and Hinkley [33] the approximate form for such distribution (ratio of two normally distributed random variables) is given by the following expression:
3.3 Damage Model 3
3.4 Damage Model 4
4 Numerical results and discussion
The integrals in the damage models are evaluated numerically using the Gaussian quadrature method implemented using a computer program code developed in the Matlab environment.
Calculated fatigue cracking (FC) for different models
Model 1  Model 2  Model 3  Model 4  

FC  0.3187  0.3172  0.1035  0.0681 
The results reveal that the damage distribution is neither normal nor lognormal as the previous research works assumed. According to the derived model (model 1) the damage follows a broad range distribution which represents the distribution of a ratio of two random variables X and Y, where X represents the actual traffic load repetitions which follows the Poison distribution, and Y stands for the traffic repetitions to cause fatigue cracking which is lognormally distributed.
5 Conclusion

The damage distribution is neither normal nor lognormal.

Damage model 1 has a broad range distribution of pavement damage.

Model 1 shows the highest expectation of fatigue cracking.

For a long design period, the expected fatigue cracking using model 1 is higher than those of the other models in most periods.
Thus, the damage distribution is neither normal nor lognormal as the previous research works assumed, and a better estimation of fatigue cracking in asphalt pavements based on damage model can be carried out using model 1, in which the equivalent actual traffic load repetitions has the Poisson distribution, while the traffic repetitions to cause failure follows the lognormal distribution.

Laboratory and field tests of fatigue cracking should be carried out to validate the accuracy of the proposed damage distribution model.

Some parameters in this research are considered constant such as the stiffness modulus of the asphalt layer and the related tensile strain. However, they can be stochastic values and follow a known distribution due to variability of certain factors such as environmental conditions.

Further investigation is needed by considering different traffic distributions.
Notes
Acknowledgements
Support from Natural Science and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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