# A New SORM Method for Structural Reliability with Hybrid Uncertain Variables
## 1 Introduction
[37] proposed a concept of using the hybrid model for solving structural reliability based on the probability-convex set and the reliability index to measure the structural reliability. Considering the distribution parameters of random variables and the mean and broadband of interval variables, [PERSON] et al. [36] measured the mean value and fluctuation range of structural reliability by defining six sensitivity coefficients and sensitivity analysis. [PERSON] et al. [41] suggested that the limit-state function is affected by interval variables, and there are two closed limit-state surfaces in the normal distribution space. With the limit-state areas, reliability models and algorithms were proposed, respectively. [PERSON] [42] proposed a mathematical definition of the structural reliability index for measuring structural safety. In the model, the parameter and load uncertainties were considered, and two-level nested optimization was adopted with the outer layer used for probabilistic reliability analysis and the inner layer used for interval analysis. In such a way, the optimization problem of the nested double loop was transformed into an approximate single loop minimization problem. Although several works aimed at improving the accuracy and efficiency of the optimization model [35; 43; 44], there is still room to improve the efficiency and stability of the algorithm [43; 44]. In [23], a mixed uncertainty model was proposed with the uncertainty expressed by random distribution and some key parameters in the distribution function expressed by interval values. The structural reliability measurement was represented by the interval values of the reliability index and the performance target interval. However, in the HRAs based on the FORM and SORM, several computation problems need to be solved before practical applied can begin. Accordingly, it is essential to develop an efficient and robust algorithm for the successful practical application of the methods [21; 22]. To this end, this paper proposes the use of the uncertain-polar coordinates SORM (UPSORM) that handles random and interval hybrid variables, significantly reducing computational complexity while achieving comparable precision. The cosine of the angle and distance in polar coordinates are employed in the proposed method [45]. The hybrid problem can be formulated as a standard reliability problem by taking this space, where an n-dimensional limit-state function can be defined using only two random variables. The probability density functions (PDFs) of boundaries of interval variables and random variables can be used to estimate the two variables. In such a way, the integral method can efficiently compute the interval of failure probability for the new bidimensional function. A unified reliability framework is proposed to address the reliability of a complex system in a scenario where stochastic and interval variables are simultaneously available. The key idea behind the proposed method is to build a generalized bivariate limit-state function in the polar space, instead of the traditional multivariate limit-state function, by aggregating both probabilistic and non-probabilistic information. To this end, the UPSORM approach is developed based on an advanced polar transformation technique. As the dimension of the original reliability problem is significantly reduced, quantities of interest such as the upper and lower bounds of reliability are expected to be estimated with higher precision and improved efficiency. The rest of this paper is organized as follows: the traditional SORM for structural reliability analysis is first reviewed in Section 2. The proposed framework of UPSORM is described in Section 3; Section 3.1 discusses the polar transformation with random and interval uncertainty, deriving the distribution of the new variable in the polar space. The proposed computation model is described in Section 3.2, and the procedure for the range of failure probability is given in Section 3.3.