# An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties
## 1 Introduction  
Structural reliability is regarded as the ability of the device or structure to complete the required functions under the specified conditions within the prescribed design period [1, 2, 3, 4, 5]. Since the advent of this concept in engineering and industrial applications, structural reliability analysis in the probabilistic framework has been developed rapidly. Although a series of effective methods, including the first-order second-moment method [6, 7], second-order second-moment method [8] and system reliability analysis [9], have been proposed. However, there are several time-variant uncertain parameters in practical structures such as external dynamic loads, degradation of material properties and changes of geometric characteristics that affect the time-variant reliability of structures [10]. Accordingly, it is significant importance to conduct to carry out a time-variant reliability analysis for the engineering structures. Conventional time-variant reliability analyzing methods can be mainly divided into five categories [11, 12, 13], i.e. the first-passage method, numerical methods, extreme value density methods, surrogate methods and the quasi-static methods. The time-variant reliability problem mainly originates from the first-passage method proposed by [PERSON] [14] in the 1940s, which lays the foundation for the development of the subsequent crossing rate method. It should be noted that the calculation of the outcrossing rate is essential in [PERSON]'s method. Studies show that, although the [PERSON] method hasremarkable advantages over other methods from the computational point of view, this is accounted for by the hypothesis that the outer crossover is statistically independent. However, this assumption results in the relatively low accuracy of the time-variant reliability approaches. In order to resolve this shortcoming, [PERSON] et al. [15] transformed the calculation of the outcrossing rate into the reliability problem of a static parallel system and established the pHi2 method. Accordingly, they provided a good scheme to determine the span rate efficiently and analyze the time-variant reliability. On this basis, [PERSON] [16] presented the PHi2+ approach to achieve an analytical solution for the crossover rate. The second scheme is the numerical simulation method. [PERSON] and [PERSON] [17] and [PERSON] et al. [18] developed sampling methods and subset simulation methods, respectively [19; 20; 21]. Moreover, studies show that the Monte Carlo simulation (MCS) method can be used to sample a random process or random variable in a limited state function of a structure and bring these random numbers into a function to compute time-variant reliability. Further investigations show that although the Monte Carlo simulation method has high accuracy, it has a relatively low computational efficiency. The third method is the extreme value density method, which uses the probability distribution of the response extremum of the time-variant problem to change the time-variant problem into an invariant problem. In this fashion, [PERSON] and [PERSON] [22] utilized an improved probability density evolution method. Furthermore, [PERSON] and [PERSON] [23] transformed the time-variant reliability analysis into a reliability problem and proposed an innovative sampling method for the response extreme value distribution. It should be pointed out that the proposed method does not require time-variant parameters, thereby improving the calculation efficiency. The fifth method is the surrogate model, which is based on constructing an analytical expression between the input variables and the structural response [24]. Then, an appropriate interpolation algorithm is applied to obtain the analytical expression that meets the accuracy requirements. Accordingly, [PERSON] and [PERSON] [25; 26] embedded the efficient global optimization (EGO) method [27] in an extreme value surrogate model and proposed a nested extreme value response method to identify the extreme values. Moreover, [PERSON] et al.