Uncertainty propagation in structural reliability with implicit limit state functions under aleatory and epistemic uncertainties

Structural reliability assessment has been widely recognized as vital in engineering product design and development [7]. In the context of structural reliability assessment, uncertainty propagation plays a significant role, which aims to quantify the uncertainties of input factors and calculate the overall uncertainty within the model response in reliability estimation [36]. Before propagating the structure’s uncertainty, a primary issue is to choose a reasonable mathematical theory related to the types of uncertainty, to quantify the uncertainty [12,38]. In practical structural engineering problems, uncertainty can be divided into two categories: aleatory uncertainty derived from inherent randomness of physical behavior, while the epistemic uncertainty arising out of lack of knowledge [10]. Probability theory is regarded as the most effective tools to describe aleatory uncertainty in structural reliability assessment. Over the last decades, numerous reliability assessment methods based on probability theory have been developed, including first-order reliability method (FORM) [23], second-order fourth moment [29] Monte Carlo simulation (MCS) [24], FORM-sampling simulation method [22], envelope function method [28], response surface method (RSM) [6], and Bayesian networks method [26]. Although these probabilistic methods typically make sense in uncertainty quantification and propagation when the structure is mainly affected by aleatory uncertainty, they do not work well in the scenarios involving great epistemic uncertainty [37]. For example, the distribution of input factors may not be precisely obtained due to insufficient sample data. Consequently, several alternative non-probabilistic theories have been developed to describe the epistemic uncertainty in reliability assessment. The general non-probabilistic structural reliability assessment theories consist of fuzzy set theory [9], fuzzy random theory [13], possibility theory [1], interval theory [5, 27], and evidence theory [39]. The fuzzy and possibility measures fail to satisfy the duality property, which will make it difficult for decision-makers to understand the results [33]. Moreover, interval and evidence theories will lead to an over-conservative result due to the interval extension problems [38]. To overcome the shortcomings of the above-mentioned theories, a new mathematical framework called uncertainty theory was introduced to deal with epistemic uncertainty. Uncertainty propagation plays a pivotal role in structural reliability assessment. This paper introduces a novel uncertainty propagation method for structural reliability under different knowledge stages based on probability theory, uncertainty theory and chance theory. Firstly, a surrogate model combining the uniform design and least-squares method is presented to simulate the implicit limit state function with random and uncertain variables. Then, a novel quantification method based on chance theory is derived herein, to calculate the structural reliability under mixed aleatory and epistemic uncertainties. The concepts of chance reliability and chance reliability index (CRI) are defined to show the reliable degree of structure. Besides, the selection principles of uncertainty propagation types and the corresponding reliability estimation methods are given according to the different knowledge stages. The proposed methods are finally applied in a practical structural reliability problem, which illustrates the effectiveness and advantages of the techniques presented in this work. Highlights Abstract