An efficient importance sampling approach for reliability analysis of time-variant structures subject to time-dependent stochastic load

Highlights

• The time-variant reliability problem is transformed into a series system.

• An efficient two-step importance sampling function is proposed.

• The method only entails a single straightforward simulation without optimization or linearization with respect to parameters.

• Examples with explicit and implicit limit state functions are presented.

Abstract

Structural performance is affected by deterioration processes and external loads. Both effects may change over time, posing a challenge for conducting reliability analysis. In such context, this contribution aims at assessing the reliability of structures where some of its parameters are modeled as random variables, possibly including deterioration processes, and which are subjected to stochastic load processes. The approach is developed within the framework of importance sampling and it is based on the concept of composite limit states, where the time-dependent reliability problem is transformed into a series system with multiple performance functions. Then, an efficient two-step importance sampling density function is proposed, which splits time-invariant parameters (random variables) from the time-variant ones (stochastic processes). This importance sampling scheme is geared towards a particular class of problems, where the performance of the structural system exhibits a linear dependency with respect to the stochastic load for fixed time. This allows calculating the reliability associated with the series system most efficiently. Practical examples illustrate the performance of the proposed approach.

Introduction

In the past decades, structural reliability theory for time-invariant problems has been widely investigated and developed. Following this framework, it is assumed that the system and its characteristics are static, and random variables are used to characterize their natural variability. Various methods have been developed to carry out static reliability analysis, which can be broadly classified as: asymptotic analytical methods, such as first/s order reliability method (FORM/SORM) [1]; and simulation-based methods, e.g., Monte Carlo simulation (MCS) [2], [3], importance sampling (IS) [4], [5], line sampling [6] and subset simulation [7]. The efficiency of both classes of methods can be improved by applying surrogate methods such as response surface methods [8], Kriging [9], [10] support vector machines [11] and polynomial chaos expansion [12], among others.

Although advances in time-invariant reliability problems have been far reaching, in realistic engineering situations, the model parameters typically change as a function of time, which is termed as time-dependent (or time-variant). This is a result of the fact that engineering structures and systems are often exposed to severe operating or environmental conditions during their service life, which are responsible for the deterioration of structural strength and stiffness with time [13]. Furthermore, the intensity and frequency of loads acting on these systems may also vary with time. A reliability analysis can properly reflect and quantify the effect of time-variant factors by estimating the failure probability of a system/structure over a period of time. Because time is considered, more challenges are faced as compared with traditional, time-invariant reliability analysis. As such, the application of typically applied reliability engineering methods may not be direct in this context.

Reliability analysis considering time-variant properties and loadings has attracted much attention recently and a vast number of methods have been developed. These are roughly classified into three categories: (1) the out-crossing rate based methods; (2) the extreme value methods; and (3) the composite limit state methods. Methods based on out-crossing rate make use of the relationship between the failure probability and the expected mean number of out-crossings of the random process over a prescribed threshold. There are many different approximations to the out-crossing rate available in the literature, see e.g. [14], [15], [16]. However, the main drawback of methods based on out-crossing rate for reliability analysis is that they are based on the assumption of independence and Poisson distribution, which in certain cases may lead to a low accuracy. Methods based on extreme values consider the worst situation of system’s performance over the time interval of interest, and whenever the extreme value of the limit state function exceeds a given threshold, failure occurs. The key challenge in extreme value methods lies in the construction of a proper surrogate model or a probability distribution for the output random process that characterizes the structural performance [17]. In this context, a Gaussian process (GP) model has been used in [18] to represent the extreme system response over time. Later, Hu and Mahadevan [19] proposed a single-loop GP approach where training points of random variables over time are generated at once (instead of tracking time and maximum responses separately). Qian et al. [20] also proposed a single-loop strategy for time-variant system reliability analysis by combining multiple response Gaussian process models. Many surrogate methods are only applicable to cases where no input random process are involved. In [21], [22], surrogate model methods have been proposed to address this issue. However, discrete representation of stochastic processes increases the dimensionality of the problem, posing a challenge for surrogate modeling due to the so-called curse of dimensionality. An alternative strategy to surrogate modelling schemes is to fit a probability distribution for the extreme values by a suitable distribution estimation method. Hu and Du [17] proposed in this context a method for constructing an extreme value distribution, based on the expansion optimal linear estimation method (EOLE) and saddle-point approximation. However, the distribution of extreme values in some cases may be highly non-linear and/or follow a multimodal distribution, posing additional challenges. A third group of methods is based on the concept of a composite limit state function, which serves as an alternative to handle reliability problems with time-variant characteristics. The main idea behind composite limit state methods is that the original time-variant limit state function is discretized into a sequence of instantaneous ones, and then the concept of series system reliability is used to convert the time-variant reliability analysis into a time-invariant one. Jiang et al. [23] used time discretization to convert stochastic processes into random variables, and then the first-order reliability method (FORM) is adopted to compute the probability associated with the linearized limit-state function. Mourelatos et al. [24], based on the concept of composite limit state, used the total probability theorem and FORM to calculate reliability of time-dependent problems. Also, this composite limit state idea allows applying simulation-based methods for static system reliability analysis in time-variant problems. Recently, Li et al. [25] proposed a Generalized subset simulation (GSS) to handle high-dimensional time-variant reliability. Similarly, Du et al. [26] adopted parallel subset simulation to handle time-variant reliability with both deterioration in material properties and dynamic load.

Several of the methods for reliability analysis which have been developed so far consider load as stochastic excitation which is dependent on time, while parameters related with structural behavior are represented either as (static) random variables or even deterministic. For example, Au and Beck [27] proposed an efficient importance sampling method for linear systems; later, Misraji et al. [28] applied a directional importance sampling scheme for reliability analysis of structural systems subject to stochastic Gaussian loading. When uncertainties on different types of parameters are simultaneously considered, i.e., random structural variables, time-variant structural parameters (due to deterioration, etc.) and stochastic load processes, the reliability problem comprises a time-variant structure (system), which is subjected to time-variant loads. In this context, most of the current methods for reliability are based on approximate analytical methods, i.e., FORM [23], [24], or resort to surrogate models, i.e., through building extreme value surrogate models, as in [29], [18], [19], [20], [21], [22], which have their own potential shortcomings. In the context of simulation-based methods, MCS and subset simulation can be used to solve reliability problems involving time-variant structures subject to stochastic load. However, their practical application may become unfeasible due to the prohibitively high computational associated with uncertainty propagation. Hence, there is still a large space for developing simulation-based methods for solving this kind of reliability problems in an efficient manner.

In view of the aforementioned difficulty in estimating the reliability of time-variant structures subject to time-dependent loads, this contribution focuses on a specific type of problems, namely conditional linear time-variant systems. By time-variant, it is meant that the time-variant structure (with time-variant structural properties) is subjected to time-variant loads (stochastic process); and by conditional linear, it is meant that at a particular instant of time and for a nominal structural parameter setting, the output response has a linear relationship with the load. The proposed approach is developed within the context of the composite limit state concept and transforms the time-variant problem into a series systems reliability problem. As linear time-variant systems are considered, an extremely efficient importance sampling density (ISD) function is proposed to compute the reliability of the transformed series system. The importance sampling scheme splits and explores the stochastic space spanned by the static random variables and the time-variant space spanned by the stochastic processes in two steps. First, samples are generated in the space associated with static parameters, after which conditional samples are generated according to a specially designed ISD in the time-variant space. This allows to compute the reliability associated with the transformed series system efficiently. The innovative aspects of this study with respect to the state-of-the-art are as follows:

• A new tool for structural time-variant reliability analysis is presented, in which input random variables, structural degradation processes as well as stochastic excitation processes are included.

• A simulation-based method which can produce satisfactory, accurate estimations of the failure probability is formulated.

• The most salient feature of the proposed approach is that it entails a single straightforward simulation scheme, where neither optimization is applied to find the extreme response values over time nor approximate linearization with respect to parameters is required.

This contribution is organized as follows. In Section 2, the definition and the transformation of time-dependent reliability problem is discussed. Then, the mathematical formulation of the proposed framework is developed in Section 3. Section 4 illustrates the performance of the proposed approach through a number of application examples. Section 5 closes the paper with discussions and an outlook for future work.