## 1 Introduction
Uncertain measure quantifies the human belief degree that an event may happen that satisfies duality axiom, normality axiom, and subadditivity axiom, which compensates for the defects of the aforementioned measures [29]. Therefore, the uncertainty theory is regarded as a suitable method when describing epistemic uncertainty. The uncertainty theory has been studied by many researchers and successfully applied to various fields such as financial analysis [32], uncertain risk analysis [33], and decision-making [34]. In reliability analysis, [PERSON] et al. [20] presented a reliability quantification method known as Belief Reliability based on the uncertainty theory. [PERSON] [35] defined a Cornell uncertainty index for structural reliability with uncertainty theory. [PERSON] [36] developed an uncertainty propagation methodology for structural reliability analysis with different knowledge stages. Nevertheless, these aforementioned models do not involve the challenging issue of structural design optimization. Moreover, the premise of these methods is that the limit state function (LSF) is strictly monotonic with respect to the input factors. [PERSON] [37] formulated a mathematical framework for uncertain programming where he defined some basic concepts about system reliability design problems. [PERSON] [38] employed uncertain programming framework to solve uncertain optimal assignment issue in which profit is uncertain. [PERSON] et al. [39] proved that some uncertain programming, including uncertain variables with continuous distributions, can be equivalently converted to a deterministic model. Additional optimization research based on uncertainty theory can be found in [40]. To the best of our knowledge, however, there is no existing method based on uncertainty theory for structural reliability analysis and design optimization under epistemic uncertainty. Thus, a new framework to evaluate structural reliability and guiding the design optimization based on uncertainty theory is explored in this paper to supplement the existing non-probabilistic methods. In the proposed framework, an uncertainty reliability indicator (URI) formulation is defined for structural reliability analysis based on the uncertain measure. This formulation can demonstrate the belief degree and how epistemic uncertainty may affect the structural reliability. Subsequently, the crisp equivalent analytical and uncertain simulation (US) approaches are proposed to solve the URI. Based on the URI, a class of URI-based design optimization (URBDO) model is established to meet a specified level of uncertainty reliability. The URBDO model can provide an alternative methodology for the early design of structural problems with limited experimental data. Furthermore, crisp equivalent programming (CEP) and US combined with genetic algorithm (USGA) methods are constructed to solve the URBDO model. The remainder of this research is structured as follows. The commonly used mathematical concepts of uncertainty and uncertain programming are reviewed in Section 2. A new URI and corresponding two solving methods for structural reliability analysis under epistemic uncertainty are explained in Section 3. The presented general URBDO model is introduced in Section 4. Then, CEP and USGA strategies are provided to acquire the optimal results of the URBDO model. Several engineering applications are investigated to demonstrate the significance of the proposed approaches in Section 5. The application summary and future research directions of the proposed method are discussed in Section 6. Finally, the concluding remarks are summarized in Section 7.