Structural reliability reformulation
Introduction
The basic challenge in structural reliability problems is the calculation of the following probability integral:where is the failure probability, is the performance function and is the joint probability density function (PDF) of random variable X. This problem can also be defined in a standard normal space (denoted as u space) as follows:where is the standard normal PDF and is the failure domain in the u space [1].
Although various approaches have been presented to solve Eq. (1), the most accurate one is still the oldest one, i.e. the crude Monte Carlo simulation (MCS), but it suffers from inefficiency in computationally time-intensive problems and those with small failure probability. Attempts made to solve this problem led to the development of efficient approaches such as the first-order second moment (FOSM) [2], first-order and second-order reliability methods (FORM and SORM, respectively) [1], [3], importance sampling [4], directional simulation [5], subset simulation [6], and line sampling [7]. However, they are generally not as accurate as the crude MCS especially in many complex or high-dimensional reliability problems.
The main contribution of the present study is the presentation of a new mathematical framework for the improved structural reliability through reformulating the failure probability integral based on a new reality-oriented insight presented in the following section.
Section snippets
Basic idea: Methodic doubt
The basic idea behind this study is a philosophy known as the methodic doubt [8] according to which, if the general accuracy of a reliability method has not been mathematically proven, the veracity of the result should be considered with doubt. By considering Eq. (1) as the estimation of interest, we can find that only the general accuracy of the crude MCS is mathematically proven. The other methods are derived based on some assumptions, and therefore, when the considered assumptions no longer
Probability expectation: General Monte Carlo interpretation
A solution is presented to solve Eq. (3) by considering the error term as a random variable. The reason is that the error value is unknown, and for different situations, the employed approach can accurately approximate the failure probability ( = 0), overestimate it (0), or underestimate it (0). Here, is also considered as a function of random variable (this issue is clarified in this section by providing more details). As a result, Eq. (3) is expressed as follows using the
Control variates
The CV technique provides a framework to directly compute the probability errors by focusing on the related source and using an efficient reliability method to estimate the imprecise probability . Then, by emphasizing the assumptions behind the employed reliability approach and comparing them with the original problem, this method estimates the error term to refine [9].
This technique can improve the accuracy/efficiency of the existing reliability methods (examples are in [10] and in
Numerical examples
Benchmark reliability examples are solved using the proposed approaches to examine the validity of the proposed propositions (Table 1) with the results listed in Table 2.
As shown, the proposed sequential CV approaches have suitably solved the mentioned examples while the conventional subset simulation has failed to do so. According to these results, the proposed probability expectation approach has desirably increased the accuracy of the existing methods. For problems with less than 20 random
Conclusions
The structural reliability integral has been reformulated based on the methodic doubt. It showed that the assumptions in the existing reliability methods often changes the original failure domain of the problems and causes error in the probability estimation. The failure probability was reformulated by considering this error term in the estimations and two general approaches were proposed to solve the proposed problem, namely, probability expectation and control variates. The probability
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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2021, Applied Mathematical ModellingCitation Excerpt :During the past decades, many efficient advanced MCS methods such as Importance sampling (IS), Line sampling (LS), subset simulation (SS) and sequential space conversion (SESC) have been developed and improved for reducing the error of the crude MCS and efficiently estimating the proposed integrals [6-10]. However, since they are derived based on certain assumptions, their solutions can be erroneous if the assumptions do not hold true [11]. For instance, LS and IS require detailed information about important regions of the integral domain; hence, when this information is not enough, they may fail to suitably approximate the probability integrals.
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2021, Case Studies in Construction MaterialsCitation Excerpt :Khodam Ali et al. [3] presented a new method for performing reliability-based design optimization of structures based on sequential optimization and reliability assessment and illustrated the appropriate efficiency and accuracy of the method by five numerical examples. Rashki Mohsen [4] proposed a reality- oriented concept for improved structural reliability analysis and a general sequential framework for a robust reliability evaluation. In 2015, the International Organization for Standardization (ISO) promulgated the fourth edition of the General Principle on Reliability for Structures (ISO 2394:2015) [5].