# A New SORM Method for Structural Reliability with Hybrid Uncertain Variables
## 1 Introduction  
In recent years the influences of uncertainty on dimensions, geometries, material properties, and load have become more and more profound. Accordingly, the analysis of complicated structural reliability has been actively studied [1, 2]. It has been essential in designing and analyzing the structural systems in many applications: manufacturing, civil engineering, and aerospace vehicles [1, 2, 3, 4, 5]. Structural reliability is typically estimated by the probability of failure (POF) and the reliability index based on the probability theory [5, 6, 7, 8]. The probabilistic reliability estimations can be divided into two categories: the first-order reliability method (FORM) [9, 10] and the second-order reliability method (SORM) [9, 11]. Additionally, reliability is evaluated through a considerable amount of stochastic simulations [12, 13]. Due to their well-balanced accuracy, efficiency, and simplicity [14, 15, 16], the FORM and SORM have been widely adopted in many industrial applications [1, 10, 11, 14, 15, 16]. In the FORM, high nonlinearity in the failure surface induces inaccurate and unreasonable POF. The SORM, using the second-order Taylor series (or other polynomials), was proposed to address this problem [17, 11, 18]. Although the POF estimated by the second-order approximation is more accurate than the POF estimated by the first-order approximation, it requires a much more complex computation [11]. Since there is no closed-form expression in a quadratic polynomial function to define the POF, most SORMs adopted a parabolic approximation of the fitted quadratic polynomial surface in order to estimate the POF ofthe limit state [16; 17; 18; 19; 20]. In these methods precise probability distributions of the random variables are inevitably required, which are based on a significant number of experimental samples that are often limited in many applications [1; 7; 14; 21]. Furthermore, it is challenging to know the precise distributions for some variables. Only the variation ranges for parameters can be estimated from the limited data. Thus, their stochastic characteristic are usually described under subjective assumptions [16]. Consequently, severe errors can be induced in the structural reliability analysis [17; 18; 19; 20]. The use of probabilistic analysis for reliability emerged to assess these uncertainties and is now widely used in many industries and societies [6; 7; 8; 9]. The statistical information and experimental data of structural design parameters are often insufficient, and the subjective uncertainty is caused by the lack of knowledge of complex structures. Thus, in the design parameters, loads, and boundary conditions of the system, the uncertainties are random and also of interval hybrid [10; 11; 12; 13; 14]. Reliability analysis with the coexistence of multiple uncertain parameters is defined as a hybrid analysis for uncertain reliability [15; 16; 17; 18; 19; 20]. Random values and interval variables often coexist in practical engineering applications [21; 22]. Hybrid reliability analysis (HRA), defined as a task that quantifies the two types of uncertainties [1], has been considered a core task in structural reliability research [22]. The use of HRA for structures has been extensively researched in recent years [23; 24; 25; 26], including the function approximation [27], probabilistic transformation [28; 29; 30], iterative rescaling [31], the probability bounds approach [32], mixed perturbation Monte Carlo (MC) method [33], optimization algorithm with single-layer nesting [34; 35], two-layer nesting [36; 37], complex nesting [22], and others [38; 39; 40]. [PERSON] et al. proposed an optimization method to solve structural reliability, based on the definition of the shortest distance of the structural reliability index when probability variables and interval variables simultaneously exist [38]. [PERSON] et al.