Reliability Methods Applicable to Mechanistic–Empirical Pavement Design Method Jennifer Q. Retherford and Mark McDonald While the AASHTO Pavement Design Guide was a major achievement in pavement design, the purely empirical method that it employed has several critical shortcomings. The AASHTO design method is based primarily on regression analysis of performance data obtained from a single road test performed in the late 1950s in Ottawa, Illinois. While the AASHTO design methodology is an extremely important first step toward cost-effective pavement designs and the AASHO Road Test is a major historical milestone in the understanding of the behavior of highway pavements, in its current form, the methodology is insufficient for the needs of pavement design today. The design equations are completely empirical, are based on only one type of subgrade and pavement materials, and do not appropriately account for environmental effects on pavement performance. Small and Winston (2) and Madanat et al. (3) have shown that the equations are affected by censoring bias. Furthermore, these design equations have been extrapolated to design for inputs far beyond those considered in the road test. As a result of these limitations, many pavement sections fail prematurely while other sections far outlive their design lives. The evolution of pavement design in the United States during the past 60 years has progressed from original design methods that relied solely on empirical data to the mechanistic–empirical (ME) methods, including the Asphalt Institute method (4). Most recently, the NCHRP, with sponsorship from AASHTO, published the Mechanistic–Empirical Pavement Design Guide (MEPDG) (5). Released in March 2004, the objective of the MEPDG was “to provide the highway community with a state-of-the-art practice tool for the design of new and rehabilitated pavement structures, based on mechanistic–empirical principles” (5). This is a major step forward in pavement design, as now the MEPDG offers pavement engineers the opportunity to overcome the deficiencies in the empirical design presented in previous design guides. Because of the research implemented in the MEPDG, the highway community can now use applicable principles of engineering mechanics to provide a more economical and technically defensible pavement design. The MEPDG correctly recognizes that the variability in factors such as traffic, climate, subgrade, material properties, and existing pavement conditions influences pavement performance. With this, the MEPDG introduced reliability considerations into its performance prediction equations (i.e., design equations). The approach to reliability design taken in the MEPDG is a residual-based method that compares the results from predicted and measured distress models. The measure of reliability is based on the difference between the predicted and measured results. Pavement design for higher reliability is ultimately based on a more conservative design equation (5). In the end, reliability methods in pavement design provide the greatest benefit when they can systematically account for all variances
Pavement design incorporates a large number of variable design parameters. The combination of the variances associated with input variables contributes to component and system reliability, and this combination of variances can have a significant effect on the predicted performance of the pavement. Reliability analysis provides a solution to this design problem. Numerous reliability methods are available and applicable to pavement design. Previously implemented reliability methods incorporate only a few basic reliability concepts, typically by using the first and second moments of the random variables. Additional reliability methods are available to design engineers, and more advanced methods can be easily used in pavement design. Many advanced reliability methods were viewed as excessively and computationally expensive. Advancements in both the methods and the computational power available to the designer make a number of reliability methods cost-efficient. Ultimately, reliability analysis in pavement design is most useful if it can be efficiently and accurately applied to components of pavement design and the combination of these components in an overall system analysis. This study investigates potential reliability methods and discusses their advantages and disadvantages in the mechanistic–empirical design approach to pavement design. The mechanistic–empirical design procedure is used to establish failure limit states for the reliability analysis. The results from numerous reliability methods are compared with a Monte Carlo simulation. The accuracies of the methods in relation to the simulation method and the computational costs are then compared and discussed.
Until recently, the primary design procedure used for pavement design in the United States has been the AASHTO Pavement Design Guide (1), which has incorporated the results of the 1950s era AASHO Road Test into empirical equations that have been the basis of most pavement design implemented in the United States during the past 50 years. The earlier editions of the AASHTO Design Guide marked the beginning of a progression toward pavement design based on scientific research and engineering principles. Remarkably, even as early as the 1950s, the need to account for variability and uncertainty and to include reliability considerations have been recognized and included in the AASHTO Pavement Design Guide. J. Q. Retherford, Vanderbilt University, 5617 Dory Drive, Antioch, TN 37013. M. McDonald, Vanderbilt University, 283 Jacobs Hall, 2201 West End Avenue, Station B 351831, Nashville, TN 37235. Corresponding author: J. Q. Retherford,
[email protected]. Transportation Research Record: Journal of the Transportation Research Board, No. 2154, Transportation Research Board of the National Academies, Washington, D.C., 2010, pp. 130–137. DOI: 10.3141/2154-13
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introduced in the engineering design process. Huang (6) discusses three major sources of uncertainty that contribute to variability in pavement design: inherent variability, statistical uncertainty, and model uncertainty. The first contribution to overall variance is from traffic or design factors. Empirical data are used to determine vehicular loads predicted for a specific design problem, and material properties are obtained from either field or laboratory tests. Errors such as improper measurements or improper measuring techniques are unavoidable to some degree and contribute to a variance in the results measured. The second source of uncertainty, statistical uncertainty, is the result of the uncertainty in the constants derived and used in design equations. Not all constants represent deterministic values, yet most design procedures do not allow for incorporation of these variances in the design procedure. The third source of uncertainty, model uncertainty, arises from inadequacy of the design procedure or lack of fit of a design equation. This variance is due to the use of closed-form design equations that are approximations of a specific distribution. Although they are deterministic, they represent possibly nondeterministic relationships between the unknown variables. Reliability analysis based on probabilistic methods for uncertainty quantification implemented within the MEPDG would provide significant benefits to the highway community. It would aid highway agencies by providing both a basis for quantifying the benefits of quality control and quality assurance and a technically sound basis for computation of pay factors for awards to contractors for meeting certain quality control standards. The combination of these methods also enables the design engineer to account for uncertainty in the design parameters and to design pavements accordingly. The MEPDG has accounted for statistical and model uncertainties by modifying the design equations to fit the empirical data. This modification itself represents a major technical achievement, resulting from many long years of in situ pavement testing, monitoring, and data analysis. This integration does not adequately address inherent variability in the design inputs that use the current practices. However, because of the computational effort required by the MEPDG to perform the analysis of the pavement section and the cost of Monte Carlo simulation (MCS), the MEPDG is currently not able to perform reliability analysis through probabilistic methods of uncertainty propagation. Furthermore, the MEPDG considers only one limit state at a time. It does not consider union or intersection events, and as such, this is a significant limitation in maintenance, repair, and rehabilitation planning. Methods that are capable of handling unions of failure events (also termed systems failure events in structural reliability) are needed. While MCS is by far the most accurate reliability approximation technique available, several other techniques have been employed in the structural reliability community, including those that have achieved accuracy close to that of MCS for many practical engineering design problems but require few model evaluations to implement. Some examples include the mean value first-order second moment (MVFOSM), first-order reliability methods (FORMs), Rosenblueth, and advanced mean value (AMV) (7–11). The authors advance the hypothesis that one or more of these methods are suitable for use in ME pavement design and can likely be implemented in the context of the MEPDG. They further propose that at least one of these methods is capable of handling system reliability. Therefore, the objective of this paper is to evaluate some of these methods for suitability in ME pavement design that incorporates statistical properties of input parameters in relation to computational effort and expense.
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RELIABILITY METHODS The early concepts for reliability analysis in the context of pavement design have been summarized in Huang (6) and Hudson (12). Before 1965, the safety factor method was applied in the design of portland cement concrete pavements. The safety factor method, however, failed to account properly for different magnitudes of uncertainties associated with the design and load parameters, which can significantly affect the reliability of the pavement. Later, Lemer and Moavenzadeh (13) employed the MCS technique to compute this reliability. In the MCS technique, the uncertainties in the random variables are described by appropriate probability distributions. However, a large number of iterations needing extremely large amounts of computer time were required, rendering the technique infeasible for all but the simplest problems to obtain a result with a small variance, so the approach never gained widespread application until recently and now only for a simplified approach to pavement design. As an alternative to MCS, Darter and Hudson (14) characterized the problem by two random variables: NF, the number of allowable axle load applications to failure, and NA, the number of actual load applications. The condition of the pavement can then be described by the limit state function g = log(NF) − log(NA). The condition of the pavement is considered to have deteriorated below acceptable limits when NA exceeds NF, or equivalently, when g ≤ 0. By assuming lognormal distributions for NF and NA, the probability of failure (PF) is obtainable as PF = Φ ( −β c )
(1)
where Φ(.) equals the cumulative distribution function of standard normal random variable and βc =
E [ g] σ [ g]
where βc = reliability index, E[g] = g[M], and σ [ g ] = ∇g T [ C ] ∇ g X = M
(2)
The variables E[g] and σ[g] are the mean and standard deviation of g, respectively. These moments are calculated by finding the moments of the first-order Taylor expansion of the limit state equation, g. The Taylor series expansion is truncated at the linear terms, providing the first-order approximation of the mean and variance of the limit state equation. The result is that the mean is calculated by evaluating the limit state function at the mean values of the random, dependent variables. Similarly, the variance involves the covariance and mean values of the variables, as represented in Equation 2. This method of using the mean and covariance of the random, dependent variables to determine the reliability of the limit state function is known as the MVFOSM method. Given the computational simplicity of this method, subsequent work in probabilistic pavement design adopted a similar approach in the procedure used in the determination of the moments of NF (15–18). All these papers apply second-moment reliability methods, particularly the MVFOSM method (7, 19). The Rosenblueth method (10) is similar to the FOSM method in that it calculates reliability from mean and variance of g(x). However,
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these moments are calculated by evaluating g(.)g(.) at all 2n combinations of the n random inputs, each taken at one standard deviation above and below the mean. The mean of the performance function is given by
E [ g] =
1 ⴱ∑g 2n
(3)
where n is the number of variables. The variance is calculated as V [ g ] = E ⎡⎣ g 2 ⎤⎦ − ( E [ g ])
2
(4)
The probability of failure is determined in the same manner as in the FOSM method, by calculating the reliability index and evaluating the cumulative distribution function. While the MVFOSM and Rosenblueth methods have enjoyed popularity in pavement engineering, they have several important limitations. First, more information beyond the first and second moments is typically available to the design engineer. In practical problems, the researcher will likely have data from which higher-order moments and full probability distributions can be determined. This condition renders second moment methods biased, as a reliability analyst must consider all information available. Furthermore, the assumption of a normal distribution for the distribution of the limit state function evaluated in the space of the original random variables is not necessarily valid. But the most important limitation of the MVFOSM and Rosenblueth methods is the lack of invariance with respect to equivalent formulations of the limit state equation, first explained by Ditlevsen (20). The reliability estimates resulting from different but equivalent expressions of the limit state function can be different when these methods are used. Although limit states can be expressed in mechanically equivalent terms, such as stress or strength, the statistical results for these methods will not be mathematically equivalent. Not only is variance a problem with these methods, but also quantification of accurate correlations of failure modes is not possible. For that reason, MVFOSM is not used in system reliability calculations. The FORMs are an improvement to the FOSM method and require additional computation. The FOSM method has a number of deficiencies, one of which is the absence of the probabilistic distribution properties of the random variables. A FORM uses the variable properties, transforms all the variables into equivalent normal variates, and ultimately determines the reliability index by solving for the limit state defined by the performance function. There are multiple methods of solving FORM. The FORM I method (8) used here requires an iterative approach, and the FORM II method (9) incorporates an algorithm to solve for the reliability index. A third method, FORM III, uses a generalized reduced-gradient search algorithm and is implemented with the solver function in Microsoft Excel. In FORM, a limit state function g(x) is used to characterize the state of the system as failed or safe, and failure and safety domains are characterized as follows:
{ F } = { x: g ( x ) ≤ 0 } { S } = { x: g ( x ) > 0 }
(5)
where {F} and {S} are the failure and safety sets, respectively, and g(x) is the limit state or failure surface Lx = {x : g(x) = 0} that
divides the entire x-space into the above distinct sets. The limit state functions are derived from the individual distresses. The probability of “failure” of the pavement section is defined as follows: PF = P { g ( x ) ≤ 0} =
∫
g( x ) ≤ 0
f x ( x ) dx
(6)
where fx(x) is the joint probability density of variables x1, x2, . . . , xn. The reliability is then the probability that the design criterion is not exceeded, or 1 − PF. An analytical evaluation of the integral in Equation 6 is possible in only a few special cases, and hence numerical integration is necessary. However, the limits of integration become intractable whenever the number of random variables exceeds two or three. In FORM, there are four important steps in the calculation of the probability of failure for an individual component distress mode: 1. Definition of a transformation from the original x-space to the standard uncorrelated normal u-space. In the case of uncorrelated variables the transform is given by u = Φ −1 ( F ( x ))
(7)
Convenient transformations are defined in Liu and Der Kiureghian (21) for the general case of correlated variables with prescribed marginal distributions. 2. Calculation of the most probable point of failure (MPP), the solution to the constrained optimization problem: u* = arg min (冷冷 u 冷冷 g ( x ) = 0 )
(8)
3. Calculation of the reliability index β. β is in general equal to ␣u*, in which ␣ is the negative normalized gradient row vector of the limit state surface in the u-space and is pointing toward the failure domain. For most practical problems, β is greater than zero, in which case β is also equal to 储u*储. The magnitude of the elements of the ␣-vector gives information about the sensitivity (relative contribution to the variance of the limit state function) (22). 4. Calculation of the probability of failure. In FORM, the limit state surface is approximated by the hyperplane β − ␣u = 0 to simplify the integration boundary. The probability of failure is approximated as PF1 = Φ (−β). The results of the individual component reliabilities in FORM can also be used to estimate system reliability. A pavement is best represented as a serial system of components defined by individual limit state functions, for the pavement is considered failed if any one of the individual component distresses is exceeded. For pavements in general, the serial system failure probability is ⎫ ⎧ PF , SYS = P ⎨∪ gi ( x ) ≤ 0 ⎬ ⎭ ⎩ i
(9)
Let B be the vector of reliability indices for each of the limit states and the elements of the matrix R be the dot products of the corresponding ␣-vectors for each distress mode. Then for a series system, the system failure probability is given by 1 − Φ(B, R), where Φ(B, R) is the standard normal multivariate cumulative distribution
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function (CDF) with correlation matrix R. For the bivariate case, it can be shown that Φ (β1 , β 2 , ρ1, 2 ) = Φ (β1 ) Φ (β 2 ) ρ1, 2
+
∫ 0
⎡ β 2 + β 22 − 2ρβ1β 2 ⎤ exp ⎢ − 1 ⎥ dρ 2 2 (1 − ρ2 ) 2π 1 − ρ ⎣ ⎦ 1
(10)
If more than two limit states are considered, then one may elect to use bounding formulae such as those in Ditlevsen (23) or to evaluate the multinormal CDF by using a numerical scheme. The FORM I method presented here performs the previously outlined procedure by using an algorithm introduced by Rackwitz (8). Specifically, the algorithm begins by defining the limit state equation, assuming both an initial value for the reliability index and initial values for the random variables. The mean and standard deviations of the equivalent normal distribution for all the random variables are calculated and used to evaluate the partial derivatives of the performance function at each random variable. The evaluated partial derivatives and standard deviations of the normal equivalents are used to determine the direction cosines. Once these calculations have been performed, a new design point can be evaluated for each random variable by the following equation: x ∗i = μ NXi − ␣ iβσ XiN
(11)
where x i* = new design point; N N µ Xi and σ Xi = mean and standard deviation in the equivalent normal space, respectively; ␣i = direction cosine; and β = reliability index. These steps are repeated in this method until the direction cosines converge to a predetermined tolerance. Once the direction cosines converge, a new value of β can be calculated by forcing the performance function to zero by treating β as the unknown variable and solving for β. This last step is repeated until the reliability index converges. Once the reliability index is determined, the final step is to determine the probability of failure by evaluating the cumulative distribution function at the reliability index. The FORM II method used in this study is a modification to the FORM I method, which can be cumbersome or impossible if the reliability index cannot be obtained by evaluating the performance function equal to zero. This method implements an algorithm that linearizes the performance function and performs iterations on the basis of partial derivatives of the performance function. The initial procedure is the same as that for the FORM I method. The partial derivatives are calculated, and then the partial derivatives in the equivalent normal space are evaluated. These partial derivatives represent the components of the gradient vector of the performance function in the equivalent standard normal space (22) and are calculated by Equation 12: ⎛ ∂g ⎞ * ⎛ ∂g ⎞ N ⎜⎝ ∂x ′ ⎟⎠ = ⎜⎝ ∂x ⎟⎠ σ Xi i i
(12)
where (∂g/∂xi) and (∂g/∂x′i )* are the partial derivatives in the origN inal and equivalent normal spaces, respectively, and σ Xi is the
standard deviation in the standard normal space. New design points are determined in the equivalent standard normal space by using the Rackwitz–Fiessler formula: x k′∗+1 =
1
( )
∇g x k′∗
2
( )
( )
( )
t ⎡ ⎤ ⴱ ⎢∇g x k′∗ ⴱ x k′∗ − g x k′∗ ⎥ ⴱ ∇g x k′∗ ⎣ ⎦
(13)
where ∇g( xk′*) represents the gradient vector of the performance function. The reliability index can then be calculated as the root sum of the squares of the design variables. The new values of the design points should be used to repeat the process until the reliability index converges. The probability of failure is determined in a way similar to the method described for the FORM I method. The third FORM method, FORM III, is also used to determine the probability of failure by calculating the cumulative distribution function at the reliability index. The reliability index is evaluated by using the Excel solver, minimizing β by modifying all random variables, subject to the performance function being equal to zero. The reliability index is calculated as the square root of the sum product of the equivalent normal values of the design variables, and the probability is calculated by evaluating the cumulative distribution function at β. The final reliability to be studied in the context of ME pavement design is the AMV method. This method is similar to the FORM method but makes one simplifying assumption. The AMV method assumes that, when the limit state function approaches zero, that point represents the most probable point. Therefore, the limiting function can be forced to zero by changing the β value. This method has an advantage over the second moment method in accuracy because it, like FORM, uses computation in the rotationally symmetric standard uncorrelated normal space. However, while AMV in general is not as accurate as FORM due to its imprecise calculation of the MPP, it needs to evaluate the gradients of the limit state function only once. Because the u-space gradients, evaluated at the origin in u-space, are used to approximate the ␣-vector, system reliability analysis can be performed.
COMPUTATIONAL EXPERIMENTATION The ultimate purpose of this experiment is to determine the most efficient reliability method that incorporates input parameter statistics and provides the most accurate probability of failure. Once the probabilities of failure are evaluated, the accuracy of each reliability method is determined by considering the MCS technique as a baseline index best representing the probability of failure. A simulation process was chosen as a baseline because these processes ultimately produce the statistical properties of a performance function essentially by brute force. The performance function is evaluated at many randomly generated simulation points, the results of which are used to represent the performance function’s distribution. A large number of simulation points must be evaluated to obtain a true representation of the performance function. This requirement necessitates the use of an analytical problem to evaluate the accuracy of the proposed reliability methods. Two failure modes—subgrade rutting and fatigue cracking—for a conventional pavement structure are investigated. Four reliability methods (MVFOSM, Rosenblueth, FORM, and AMV) are applied to these two distress models to determine the probability of failure of these components. Then, these components are considered as a system, and reliability analyses for the series system are performed.
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where
Distress Models Generally, distress models, or transfer functions, are used to calculate the expected number of load repetitions that will fail the pavement section. Many distress models have been introduced by various entities. The models all follow a general formula, but the difference is introduced in the constants. Transfer functions for fatigue cracking generally take the form N f = f1 ⴱ ( ⑀ t ) 2 ( E1 ) 3 f
f
(14)
where Nf = number of load repetitions until failure by fatigue cracking, ⑀t = tensile strain at bottom of the hot-mix asphalt, E1 = asphalt concrete modulus of elasticity, and f1, f2, f3 = empirically determined constants. Because the magnitude of f2 is generally much larger than that of f3, the effect of the modulus of elasticity is negligible and the expression becomes Nf = f1 ⴱ ( ⑀ t ) 2 f
(15)
For rutting, transfer functions typically take the form Nr = f4 ⴱ ( ⑀ v )
− f5
(16)
where Nr = number of load repetitions until failure by rutting, ⑀v = vertical compressive strain on the top of subgrade, and f4, f5 = empirically determined constants. The equations used for this study incorporated the constants derived by the Illinois Department of Transportation (24) for the fatigue model and the Asphalt Institute (4) for the rutting model. Nf = ( 5 ⴱ 10 −6 ) ⴱ ( ⑀ t )
−3
N r = (1.365 ⴱ 10 −9 ) ⴱ ( ⑀ v )
(17) −4.4477
(18)
The asphalt tensile strain and the subgrade compressive strain equations used are calculated in accordance with algorithms developed by Thompson and Elliott (25). The equations were determined through the use of ILLI-PAVE, a computer program developed in 1980. The computer program was used to run 168 pavement configurations, and the resulting algorithms are as follows:
h1 = h2 = E1 = K1 =
HMA thickness, base thickness, HMA modulus, and breakpoint resilient modulus of the subgrade.
Distributions of Random Variables The asphalt tensile strain and subgrade compressive strain design equations incorporate four design variables, and the statistical properties of these variables are required for the reliability analysis. The variables are represented by statistical means and standard deviations from various sources. Table 1 summarizes the values chosen for this investigation. The HMA thickness and subgrade properties are derived from results presented by Darter and Hudson (14). The resilient modulus for the subgrade is obtained from work by Rada and Witczak (26) and represents properties of a crushed-stone granular material. The HMA modulus mean and standard deviation are both obtained from Shell nomographs and equations from the Asphalt Institute as summarized by Huang (6). The value used for experiment here is applicable for a temperature of 70°F and a load frequency of 4 Hz. The performance functions for fatigue cracking and rutting compare the calculated number of load repetitions to an assumed number of load repetitions. This assumed number of load repetitions is treated as a constant; however, further analysis could be performed to incorporate the variance of this term as well. The assumed number of load repetitions per year is calculated according to Equation 18, which represents the number of equivalent single-axle loads per year (6). N 18 = ( ADT )0 ⴱ T ⴱ T f ⴱ G ⴱ D ⴱ L ⴱ ( 365) Y where ADT0 = average daily traffic, T = percentage of trucks in the average daily traffic, Tf = number of 18-kip single-axle load applications per truck, G = growth factor, D = directional distribution factor, L = lane distribution, and Y = design period. The value for N18 used for analysis here assumed an average daily traffic count of 2,000 vehicles, 15% of the traffic classified as truck traffic, 0.2 load applications per truck, a growth factor of 2, distribution factor of 0.5, and a lane distribution equal to 1. The resulting number of equivalent single-axle loads per year is 21,900.
TABLE 1
log ( ⑀ t ) = 2.9496 + 0.1289 ⴱ h1 −
0.1595 ⴱ log ( h2 ) − 0.0807 h1
ⴱ h1 ⴱ log ( E1 ) − 0.0408 ⴱ log ( K1 )
(19)
log ( ⑀ c ) = 4.5040 − 0.0738 ⴱ h1 − 0.0334 ⴱ h2 − 0.3267 ⴱ log ( E1 ) − 0.0231 ⴱ K1
(20)
(21)
Probability Distributions of Random Variables
Random Variable
Probability Distribution
Mean
Standard Deviation
h1 (in.) h2 (in.) E1 (ksi) K1 (ksi)
Normal random variable Normal random variable Normal random variable Normal random variable
3.1 12.5 1,600 7.21
0.48 1.25 100 1
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TABLE 2
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Probability of Failure of Distress Modes and Systems According to Various Reliability Methods Probability of Failure
Error
Method
Fatigue
Rutting
Fatigue (%)
Rutting (%)
FOSM FORM I FORM II FORM III Rosenblueth AMV Monte Carlo
.2678034 .2303859 .2303864 .2303864 .2036254 .2294162 .2232
.13066336 .02738818 .02738787 .02738786 .11578284 .02738744 .02804
19.98 3.22 3.22 3.22 8.77 2.79
366 2.32 2.33 2.33 312.9 2.33
NUMERICAL RESULTS The results of the reliability analyses for the fatigue cracking distress and rutting distress components, and the system reliability results, are presented in Table 2. The probability of failure of the components indicates relatively consistent results regardless of the reliability method applied. The error, calculated by the percentage error method, compares the results of each reliability methods to the MCS. The simulation is treated as the standard because of the efficiency of the method to produce a distribution that most closely represents the actual distribution for the distress mode. The results indicate that the error for most methods is less than 10%, but more impressive, the FORM methods all produce an error less than 4%. Although the FOSM method is one of the simplest processes to implement, the deficiencies seem to be indicated by the decrease in accuracy. The importance of the minimum error produced by a single method should not be overshadowed by the consistency of that error. The Rosenblueth method produced an error of only 8.77% for the fatigue-cracking component but an error of 312.9% for the cracking component. Consistency of results, along with a reasonably accurate result, indicates that the FORM or AMV methods are well suited for pavement design related to fatigue cracking and subgrade rutting. Component reliability is important, but the typical pavement design procedure calls for an understanding of the performance of a complete system. System reliability analysis was performed and is represented here as the combination of the failure and rutting components. Again, the FOSM method is inaccurate in quantifying correlations among failure modes due to the method’s invariance and is therefore not included in the system-based calculations. The analysis of the system produced results that also favor FORM and AMV as suitable reliability methods for pavement design. The Rosenblueth method produces a large error for the system analysis, but further, the method again seems to be inconsistent in comparison with the
TABLE 3
Probability of Failure System
Error System (%)
N/A .026409 .026409 .026409 .499201417 .025373 .02666
N/A 0.94 0.94 0.94 1,772 4.83
component Rosenblueth analysis, rendering it less appealing as a reliable method. The original disregard of the MCS method as a method of performing reliability analysis was due, in strong part, to the computational effort required to perform the simulation. It is therefore of interest here to investigate the effort required to perform these alternate probability methods. As previously discussed, the FORM I method is an iterative process, but the remainder of the methods used are closed form. Therefore, computational effort is minor. Table 3 outlines a comparison in computation in relation to computational time and function count. Gradients were evaluated by using finite differencing with n + 1 function evaluations required to obtain the final solution. Table 4 verifies the relatively cheap computational cost of all reliability methods, in comparison with simulation methods. As anticipated, the FOSM and Rosenblueth methods provide the cheapest computational effort. The FORM methods range in required power depending on the method implemented. AMV required very little computational effort. The FORM methods are more expensive due to the number of iterations required to perform the analysis. Although the function counts seem relatively reasonable from this experiment, an increase in the number of variables will significantly increase the computational effort required because gradient evaluations are more expensive and more iterations are required to achieve convergence, though this is likely to be small for most problems in comparison to the effort required for MCS.
DESIGN OFFSETS FOR ROUTINE USE Though application of reliability methods is straightforward for engineers with advanced training in probability and statistical methods, relatively few in the highway community have the advanced training required to implement these methods in design. In addition,
Computational Values for Design Point at Desired Reliability Index
Random Variable
Mean
h1 h2 E1 K1
3.1 in. 12.5 in. 1,600 ksi 7.21 ksi
NOTE: β = 2, PF = 97.7%.
Direction Cosine at β = 2
Design Checking Point (u*)
Design Value (x*)
0.956896109 0.206162992 0.192354387 0.06961643
−1.913792218 −0.412325984 −0.384708774 −0.13923286
2.181379735 11.98459252 1,561.529123 7.07076714
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TABLE 4 Computation Effort of Reliability Methods Comparison Function Count Method
Fatigue
Rutting
System
FOSM FORM I FORM II FORM III Rosenblueth AMV Monte Carlo
5 50 20 35 5 10 100,000
5 30 25 45 5 15 100,000
N/A 1 1 1 5 1 100,000
it is not always desirable to incur the expense of using these methods every time a routine pavement section is designed. One important benefit of the use of analytical reliability methods in the context of the MEPDG is that these methods can provide a rigorous and justifiable basis for finding design values for the random variables that can allow design by using the MEPDG without the end user having to be experienced in reliability methods. The objective of using analytical reliability methods is to offset the random variables from their means so that only one point is required for design of the pavement. This concept is similar to that used in the Steel Construction Manual (27) and in the American Concrete Institute concrete design manual (ACI-318-05) and commentary (ACI381R-05), in which the loads and resistances are factored and the structural element is designed to be safe given the factored loads and resistances. Design values for the variables can be calculated from the relationship u* = β␣. Once the ␣-vector is known, assuming that changes in ␣ are small with respect to changes in the design, the design checking point can be found by multiplying the ␣-vector by the desired reliability index to find u*. The desired reliability index can be predetermined by the designer as the reliability index that corresponds to the preferred probability of failure. Essentially, this method solves the reliability problem backwards by first determining the probability of failure and then using that information to determine the extreme values for the design parameters that achieve this probability. Once the u*-point is determined, it is transformed to the x-space to find the design values of the random variables. Table 3 provides the results of this process for finding a design point in the case of the above example with a desired reliability index of 2. Table 3 includes both the mean values of the design parameters and the design values required to obtain the desired reliability index. By using MCS with the probability distributions in Table 3, the probability of the sampled section having an allowable number of repetitions greater than the section with the checking point properties was 97%. One can observe that the calculated design values are all less than the original mean values. In a manner similar to the methods for steel or concrete design, the values have been reduced by a resistance factor. With further work, standard resistance and load factors can be included in the MEPDG for the convenient use of design engineers. From this illustration, it is clear that not only can reliability-based design that uses probabilistic methods be made accessible to the general highway community without requiring extensive training in probabilistic methods; intelligent use of these methods can practically prevent the need for a computationally intensive reliability process.
CONCLUSION The application of reliability methods that are based on probabilistic uncertainty propagation is underutilized in pavement design, but it can be of great benefit. Previous codes and design guides have depended heavily on empirical data over established reliability methods in an attempt to avoid perceived computational costs. However, in exchange for cheap computation, pavement designs have been historically overdesigned, but more importantly, inconsistently designed. The application of mechanistic design procedures has increased efficiency of design, but even the most current pavement design procedures have forgone the use of probabilistic methods due to their perceived high computational expense. This is not necessary, as advances have been made in both the reliability methods and the computational power of design engineers. Reliability methods such as those presented here provide reliabilitybased design that is capable of incorporating the variability of the parameters of pavement design on the basis of each design case. These reliability methods have been shown to be efficient methods of design that require a minimal amount of computational time or cost. The FOSM and Rosenblueth methods both prove to be efficient ones that significantly sacrifice accuracy of results. The FOSM and Rosenblueth methods are also limited to normal and lognormal distributions for the random variables. These methods are also less accurate than FORM and AMV. FORM is a reasonably accurate method for evaluating the component and system reliability for ME pavement design. However, the FORM method will tend to increase in computational effort as the number of variables increases. Further, convergence issues may arise. In particular, the FORM I method requests that the designer perform numerous iterations to verify convergence of the direction cosines and the reliability index. Under certain circumstances, the original input parameters, such as the initial reliability index, can cause oscillations and the algorithm will not converge. AMV appears to be the best of all methods studied with regard to accuracy and efficiency. Ultimately, the reliability methods presented here, in particular the AMV method, can be used efficiently to perform component and system reliability analysis. The computational effort required for all these methods is reasonable and obtainable by a majority of design engineers. The reliability methods all provide reasonably accurate solutions and avoid the intensive computation time required to perform simulation techniques, such as MCS. Because these methods are based on the use of distributional information about the random variables, they can help (a) quantify the benefits of quality control and management in the field and (b) provide a rigorous justification for pay factors for contractors who meet quality control targets. Use of the most-accurate probabilistic data as input for the design calculations will tend to produce solutions that accurately represent construction conditions. Finally, it is not necessary for all highway engineers to be versed in probabilistic methods to benefit from their application. The probabilistic methods described here offer the ability to pursue a future edition of the MEPDG that assures pavement reliability in a manner similar to that in load and resistance factor design. Inclusion of these methods in the MEPDG would not only make the benefits of probabilistic methods accessible to practicing engineers without backgrounds in probabilistic methods but would allow the MEPDG to assure reliability while still requiring only deterministic inputs to the analysis.
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The Flexible Pavement Design Committee peer-reviewed this paper.
