Elsevier

Structural Safety

Volume 88, January 2021, 102006
Structural Safety

Structural reliability reformulation

https://doi.org/10.1016/j.strusafe.2020.102006Get rights and content

Highlights

  • A new look at the structural reliability using methodic doubt.

  • A general interpretation of the Monte Carlo simulation is proposed.

  • Control Variate technique is proposed for reducing the reliability errors.

  • A general sequential approach is proposed for robust reliability analysis.

  • Proposed insight can be used for deriving new robust reliability methods.

Abstract

When the general accuracy of a reliability method is not mathematically proven, the correctness of its yielded results may be in doubt. This study emphasizes this observation and proposes a reality-oriented concept for improved structural reliability analysis. The failure probability integral is reformulated based on this insight and two general approaches are presented for probability estimation, namely, probability expectation and control variates. The former is a general interpretation of the Monte Carlo simulation (MCS) based on which the formulation of the existing reliability methods can be used as an indicator function of the MCS, while the latter can remove errors of a reliability method by considering the assumptions employed in it. Using the suggested CV approach and considering the subset simulation as a method of interest, a general sequential framework is proposed for a robust reliability evaluation. Using the presented reality-oriented concept, some popular simulation methods are re-derived and it is shown that the proposed idea can be easily used to derive novel robust reliability methods.

Introduction

The basic challenge in structural reliability problems is the calculation of the following probability integral:Pf=gX0fXxdxwhere Pf is the failure probability, g is the performance function and f 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:Pf=Ω=gU0φuduwhere φ· is the standard normal PDF and Ω 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 (ξ = 0), overestimate it (ξ<0), or underestimate it (ξ>0). Here, Pf^ is also considered as a function of random variable u (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 P̂f. 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 P̂f [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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