## 1 Introduction  
Structural design optimization is a pivotal discipline in industrial engineering products. Generally, structural design aims at searching the best solution of design variables that minimizes the mass or total costs while satisfying several performance constraints [1]. Various uncertainties are intrinsically involved in the process of engineering structural design and optimization [2]. Traditional design optimization is performed under deterministic conditions that neglect the influence of uncertainties stemming from structure size imprecision, external loads fluctuation, and material property variation [3, 4]. Those ineluctable uncertainties will lead to unreliable results and decrease confidence in deterministic design optimization. Consequently, reliability assessment and reliability-based design optimization (RBDO) have been widely employed in various engineering problems [5], such as vehicle engineering [3], aerospace engineering [6, 7, 8], offshore engineering [9], and civil engineering [10]. As a precondition for reliability assessment and RBDO, an appropriate mathematical framework has to be selected to describe the characterization of uncertainties [2, 11]. Various methods have been used to determine the best designs according to the uncertainty type. The most accepted classification is to divide uncertainty into two categories, i.e., aleatory uncertainty and epistemic uncertainty [12]. Aleatory uncertainty is also referred to as statistical or variability uncertainty, which is inherent in any engineering product. It is supposed that the sample data of aleatory uncertainties is sufficient to characterize their influence on the structural performance in the reliability analysis and optimal design process [13]. Generally speaking, the probability theory-based design frameworks provide reasonable solutions when accurate probability distributions of input factors are available. In this regard, different reliability analyses and RBDO strategies have been proposed to obtain an optimal design [14]. Among these strategies, the first-order reliability method (FORM), second-order reliability method (SORM), and Monte Carlo simulation (MCS) are widely employed due to their capability and simplicity [15; 16; 17]. The reliability index approach (RIA) is developed based on FORM or SORM for the structural design optimization [18; 19]. Generally, real-world structural problems more commonly encounter mixed aleatory and epistemic uncertainties. Nevertheless, the research conducted in this paper focuses on the structural reliability assessment and design under epistemic uncertainty. Epistemic uncertainty describes insufficient information of the input factors due to partial or no sample data available [12]. In this case, the aforementioned probability-theory-based strategies cannot be used for structural reliability assessment and design optimization [20]. Therefore, the non-probabilistic methods show distinctive attraction in dealing with epistemic uncertainty problems under limited sample data [21]. The common non-probabilistic reliability analysis and design theories involve fuzzy set theory [8], interval theory [7], possibility theory [22; 23], convex set theory [24], and evidence theory [25; 26]. However, since the possibility measure and fuzzy set cannot satisfy the self-duality axiom, their estimation results may be counter-intuitive, i.e., the sum of failure possibility and reliability is not equal to one [27]. In addition, interval theory, evidence theory, and convex set theory are based on the probability-interval measure characterized by the interval extension problem and rapid attenuation leading to over-conservative results in the analysis process [28; 29]. Hence, a new axiomatic mathematics called uncertainty theory was developed to deal with epistemic uncertainty, which simultaneously satisfies both subadditivity and self-duality [30]. The uncertainty theory was founded by [PERSON] in 2007 based on uncertain measures [31].