Time-Dependent Reliability Analysis of Pavement
Structures under Fatigue Loading
Dilip DEEPTHI, Babu G.L.SIVAKUMAR and Suku LEKSHMI
Civil Engineering, Indian Institute of Science, Bangalore, India
Abstract. The premature failure of well-designed structures in the field is a problem commonly observed worldwide. These early failures and the deterioration in structural resistance of pavements can be attributed to the loading and external environmental factors, damage and improper maintenance during the design life among other factors. The time dependence of performance in the formulation of the reliability problem helps to include the effects of degradation of resistance and the variation of loading with time. In this study, the effect of degradation of the surface layer modulus on the fatigue reliability of pavements is analyzed, where the decrease in the modulus with time is considered as a function of the accumulated damage due to repeated loading. The pavement section considered in the study designed for a period of 15 years at a reliability level of 80%, is seen to have a probability of failure of around 50% after 8 years and 75% after 10 years, when strength degradation is considered. This emphasizes the need for capturing the temporal characteristics of the materials and loading in a time-dependent reliability analysis framework.
Keywords. Time-dependent reliability, strength degradation, premature failure
1. Introduction
As structural systems age in service, their reliability decreases over time or load history. This decrease in reliability may be a result of degradation in strength or from the multiple application effect of the randomly imposed load, particularly in the case of geotechnical structures like pavements; and can be effectively captured using time-dependent or dynamic reliability models. The traffic induced fatigue cracking of flexible pavement structures being a major design criterion (Huang 1993), the prediction of time-dependent fatigue reliability is crucial for the design of pavement structures. Structural fatigue is the result of repeated cycles of stress or strain which causes decay in the stress capacity and ultimately the structural failure (Sanchez-Silva et. al. 2005). A failure may be defined as the event in which the deteriorating resistance (pavement strength) drops below the applied stress (traffic loading). Many factors contribute to the degradation of strength of asphalt pavements. In addition to the resilient modulus degradation of asphalt concrete pavements due to the repeated loading, environmental conditions such as temperature and water contribute immensely to the deterioration of the pavement
structure. Therefore, the flexible pavement deterioration models that can predict the deterioration of the pavement over time and under varying environmental conditions are an integral part of the pavement design process.
The objective of this paper is to examine the effect of degradation of the asphalt layer moduli on the pavement fatigue life through a time-dependent reliability analysis, based on the dynamic reliability model proposed by Gao et al. (2013). As the failure of the pavement is characterized by the number of load repetitions that the pavement can sustain before failure, the reliability analysis is performed with respect to the load application times. Under traffic loading, the damage of the pavement system under consideration gradually accumulates and over a period of time leads to its failure. The damage accumulation is a complex and irreversible phenomenon, and can be treated as a measure of degradation in fatigue resistance of materials (Rathod et al., 2011). The advantage of this method lies in its simplicity, particularly in the absence of inspection data to characterize the degradation process. In the case of pavement layers, the resilient modulus is very sensitive to change in stress conditions and moisture content and is indicative of the pavement strength.
Geotechnical Safety and Risk V T. Schweckendiek et al. (Eds.) © 2015 The authors and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-580-7-358
Therefore, in this chapter, the pavement strength degradation is modelled by considering the resilient modulus as a function of the cumulative damage; and as the failure by fatigue is of interest only the degradation of the asphalt layer modulus with loading is considered. This method is also quite flexible and can be extended to practical examples with the availability of the moduli degradation data from field studies.
2. Time-Dependent Modelling
For reliability analysis, the load-strength interference model is widely used, wherein reliability is defined as the probability that the load does not exceed the strength. The load-strength interference model can be used when the probability distribution function of load and that of the strength, are known, given as:
(1) In the popular Mechanistic-Empirical (M-E) pavement design methodology, this equation can be written as
(2) where is the probability density function (pdf) of the allowable fatigue life in the structure, is the pdf of expected traffic. Fatigue models are traditionally established using S-N curve approach, where the fatigue is usually computed in terms or stress/strain amplitudes or stress/strain ranges of a cycle. The variable N describes the fatigue life (cycles to failure) and S the strain amplitude to reach the failure.
(3) where m and C are empirical constants. Substituting this in Equation 1 and incorporating the uncertainties of the material parameter C, the reliability model adopted for the study is as follows
(4) These models do not reflect the relationship between reliability and the times of load action,
but they can calculate the reliability when a random load is applied once, or for loads applied at specified times. In service life, random loads which act on components and systems are almost always repetitive, and the effect of the times of load action on reliability should be considered. (Wang and Xie, 2008). In the case of pavement design the failure of the pavement is characterized by the number of load repetitions that the pavement can sustain before failure. Furthermore, when only the failure mode of fatigue is taken into consideration, the load process is a discrete process (Gao et. al. 2013) Therefore, the traffic loading can be characterized by load application times and magnitude of load. The effect of times of load action on reliability of components, under the assumption that there is no strength degradation during the service life, is based on the principle that the system will survive all the successive loads less than the highest load it has once resisted. Meanwhile the likelihood that a higher load appears decreases with the increase in the number of applied load cycles. The reliability under repeated random load can be modelled as (O’Connor 2002)
(5) where denotes the component reliability after n times of load action. For pavements, the design life of the pavement is usually considered to be 15 years; hence, the reliability model is developed as a function of the discrete time intervals until the design life. The expected number of load repetitions in each time interval is determined using the annual traffic growth rate function that can be defined as
(6) where N0 is the traffic volume at year t=0, and is the traffic annual growth rate, assumed as 0.06 in this study. The time-dependent reliability of the structure can now be estimated using
(7) where is the probability density function of traffic loading which is considered to be a function of time t.
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A pavement undergoes deterioration with passing time and traffic. The pavement is usually very stiff after construction, but as a result of traffic and environmental conditions such as temperature and moisture, the state of the pavement changes.To include this effect in the pavement performance model, Collop and Cebon (1995) developed a modulus degradation model using the surface deflection data obtained from two Accelerated Loading Facility trials (Johnson-Clark et. al; Kadar 1991) performed by the Australian Road Research Board (ARRB). A back calculation procedure with appropriate temperature correction (Collop 1994) was used to determine the set of layer moduli that gave the best fit between calculated and measured surface deflection bowls. The moduli were plotted as a function of the cumulative fatigue damage experienced by the test section using Miner's hypothesis of linear damage accumulation, given by
(8)
(9) where E/ Eo is the reduction in elastic modulus of the asphaltic material, D is the cumulative fatigue damage and (E/Eo)c is a constant that determines the level of modulus reduction corresponding to the end of the fatigue life of the asphalt (i.e. D = 1).
The Miner’s law (Miner 1945), which defines damage as the ratio of the number of cycles of operation to the number of cycles to failure at any given stress level, relates the percentage fatigue cracking to damage in a probabilistic manner. The cumulated damage can be interpreted as a measure of degradation in traffic loading. The expected value for damage at time t, when the stress/strain is a continuous function governed by the probability density function , can be expressed as
(10)
where n(t) is the expected traffic at time t and N(s) is the number of loading cycles of stress/strain s required to reach the failure.
According to the Miner damage accumulation rule, strength degrades under multiple applications of random load, which should be modeled as the function of both the load application times and the magnitude of stress/strain. In general, the remaining strength of the mechanical components can be expressed in the following form (Gu et al. 2007)
(11) where n is the load application times and a is the material parameter, D(n) is the cumulative damage caused by load, which is determined by both load application times n and the magnitude of load. Using this principle, the asphalt modulus degradation model is modified in this study as
(12) and K is assumed as 0.5, indicating that the pavement is considered to have failed when the asphalt modulus reduces to half of the initial value. From Miner’s law and the S-N model, the damage after n load repetitions can be derived as
(13) Thus, the asphalt modulus can be estimated using
(14)
3. Numerical Example
The objective of this study is to develop a methodology wherein the effect of the strength degradation with load applications on the pavement relaibility can be incorporated in the mechanistic-empirical design framework. The conventional M-E approach, which is based only on the initial strain conditions of the pavement, does not consider the decrease in pavement strength with load applications. In order to validate the methodology described earlier for use in the pavement analysis in a
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D. Deepthi et al. / Time-Dependent Reliability Analysis of Pavement Structures Under Fatigue Loading
empirical framework, the model is first employed for the reliability estimation of pavement structure without considering the load repetitions and corresponding reduction in strength. This approach is referred to as the conventional reliability analysis in this study. For any given pavement structure, the critical strains imposed can be estimated through a finite-element or linear-elastic pavement analysis model. In a probabilistic approach, the uncertainties of the material and loading parameters that affect the critical strain are considered through probability distribution functions and the probability of exceedence of the allowable fatigue life can be estimated through Monte Carlo simulations, for various levels of traffic.
The pavement structure considered in the study consists of a three layered structure consisting of the asphalt surface, the base layer and the subgrade. The statistical properties of the pavement section are shown in Table 1. The resilient modulus of the asphalt concrete (AC) surface layer was assumed to follow a lognormal distribution, with a coefficient of variation (COV) of 10%. To analyze the fatigue reliability, as assumption is made that only the asphalt layer modulus decrease with traffic load repetitions, while the modulus of the base and subgrade layer are deterministic. The traffic load spectra, P is assumed to follow a lognormal distribution with a mean of 40KN and a COV of 25%.The distress model considered for the estimation of the fatigue life given in the IRC:37-2012, for a mean value of asphalt resilient modulus taken as 3000 MPa, is reduced to the S-N model with parameters, m = 3.89 and C = 2.371E-07. The COV of the material parameter C is assumed to be 20% in the study. The pavement is considered to have failed by fatigue if the expected traffic demand exceeds the fatigue life.
To compare the results from the reliability model with the conventional reliability analysis estimates, the model is used to estimate the reliability without considering the load repetitions and the corresponding degradation of the strength. The reliability analysis is carried out using Equation 4, at different levels of expected traffic and the obtained reliability estimates are validated by a carrying out a Monte Carlo simulation with 106 simulations. From Figure 1, it is observed that the reliability estimates
obtained from the reliability model perfectly match the estimates from MCS.
Table 1. Statistical properties of the pavement section
Layer Parameter Mean Std. dev
AC Surface E1 (MPa) 3000 300 C 31,250 3125 m 5.0 - Base E2 (MPa) 300 - Subgrade E3 (MPa) 100 -
3.1. Effect of Incorporating Load Applications on Pavement Reliability
To demonstrate the use and validity of the methodology in pavement design reliability, the reliability model is first carried out without considering the degradation of the asphalt modulus. The reliability analysis is carried out with respect to time, using Equation 7. Although describing the traffic loading as a random process would be the most accurate way, for the sake of simplicity, the increase in the mean value is estimated according to Equation 6, keeping the COV at a constant level of 10%. Based on this analysis, the reliability of the structure at the end of 15 years (design life) is presented in Figure 1 at different levels of traffic.
From the results it can be seen that the estimates from the reliability model are comparable with those obtained through MCS, particularly at higher levels of reliability. At higher traffic levels, the effect of incorporating the load repetitions is seen to result in a small decrease of reliability. Thus it can be seen that a validation of the methodology adopted in pavement analysis is obtained. The analysis also implies that in the absence of strength degradation, the use of the conventional reliability analysis is sufficient.
Figure 1. Validation of Reliability model with MCS
The advantage of the time-variant reliability model is that for any given expected traffic level, the decrease of reliability can be easily obtained. In Figure 2 the variation of the reliability with time is shown for three traffic levels of 40, 50 and 60msa.
Figure 2. Time-variant reliability for different traffic levels
From this figure, the expected reliability after any given period of time can be estimated. For instance, if the expected traffic loading for the pavement structure is 50msa, the reliability of the structure is estimated as 80%. However, the reliability after a period of 10 years in the absence of strength degradation is around 97%.
3.2. Time-Dependent Reliability of Pavements with Degradation of Asphalt Modulus
Having validated the reliability model, the analysis is extended to the time dependent
reliability of the pavement structure, with the consideration of the modulus degradation with load repetitions. The first step to establish the strength degradation as a function of the damage in the pavements is to estimate the value of E[Sm]. In its general form, E[Sm] can be evaluated as
(15) where is the strain density function. In this study in an attempt to incorporate the loading and material uncertainties in the evaluation of the stochastic damage function, a stochastic response surface methodology was adopted. The probability density function of the tensile strain values is estimated using the Collocation-Based SRSM. From this strain density function the value of E[Sm], and thereby the damage after n load repetitions can be estimated using Equation 13. Once the damage has been estimated, the reduced value of the resilient modulus has been estimated using the degradation model developed by Collop. The reliability of the structure with the passage of time is estimated using
(16) where St is the critical strain value at time t, the value of which increases with the decrease in modulus and can be determined from the pavement model. Thus, the reliability of the pavement structure over time, incorporating the effect of strength (resilient modulus) degradation is estimated, as seen in Figure 3.
Figure 3. Time-variant reliability with asphalt modulus degradation dc de e f c f t R t i cS N C m t e ¿ ¾ ½ ¯ ® »¼ º «¬ ª
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For a traffic level of 50msa (design reliability of 80%), a comparison of the reliability over time, with and without the consideration of strength degradation has been presented in Figure 4. To validate the results obtained from the time-variant reliability model (Equation 16) used in the study, a MCS is carried out for 104 simulations and is seen in Figure 4 to been in close agreement with reliability model estimates. It can be easily observed from that for a pavement designed to last for a design period of 15 years with a reliability of 80%, reaches this level of reliability after a period of about 6 years due to the reduction in the asphalt modulus. Also, after this point the reliability decreases rapidly, with a probability of failure of around 50% after 8 years and 75% after 10years. This implies that the probability of the pavement failing prematurely before its design period is extremely high, and emphasizes the importance of periodic repair and maintenance on the reliability of pavements.
Figure 4. Effect of asphalt modulus degradation on time-variant reliability
4. Conclusions
In the design of pavement structures, it is important to know how resistance degradation affects the reliability of a system over a period of time. A time-dependent reliability model for the estimation of the pavement reliability considering the strength degradation with time has been proposed. The following conclusions can be drawn from the study:
1) The time-variant reliability model can be efficiently solved by evaluating the multi-integral which has been validated through a MCS.
2) The reliability estimates obtained from the time-variant reliability model agree closely with those obtained from a conventional reliability analysis, in the absence of the degradation of pavement strength with time. The degradation in the pavement strength which can be modelled as a function of the cumulative damage will result in an increase in the critical strains with time. This results in a drastic reduction in the reliability from the desired level, much earlier than the design life of the pavement structure. Incorporating this degradation of strength in a time-dependent reliability analysis provides a way in which the problem of premature failures can be understood and tackled.
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