Based on classical probability theory, traditional probabilistic reliability analysis has been more and more perfect. The main purpose of probabilistic reliability analysis is to assess reliability or failure probability. Many practical methods, such as Monte Carlo simulation, the importance sampling method [
1], the response surface method [
2], the first-order reliability method (FORM) [
3], the second-order reliability method (SORM) [
4], the subset simulation [
5], the directional method [
6], the line sampling method [
7] and the asymptotic method for SORM [
8], have been proposed to achieve this aim and apply it to practical engineering problems.
However, the traditional probabilistic reliability model requires precise probability density functions of the random variables, which are difficult to obtain in many practical applications because the samples available in practical engineering problems are limited. Although the principle of maximum entropy has been employed as an efficient technique to model the concerned uncertainty with a probabilistic distribution [
9], it has been pointed out that classical probability reliability may be extremely sensitive to the statistical distribution of the data and even small errors in the inputs may yield misleading results in some cases [
10,
11]. This implies that the traditional probabilistic reliability model may be unable to deal with some problems with incomplete information (or limited samples of the inputs). Fortunately, many novel strategies such as the non-probabilistic model [
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20], fuzzy variables and the fuzzy uncertainty propagation model [
21,
22,
23,
24] as well as the fuzzy randomness model [
25,
26] have been provided to deal with such cases.
Compared with the precise probability density functions of the uncertain variables, the bounds of the variables for many engineering problems, however, may be difficult to obtain from the information available. Recently, non-probabilistic reliability models such as the convex set, the interval set and the fuzzy set have been presented as attractive supplements to the traditional probabilistic reliability model [
16]. Since the convex set models, including ellipsoid and hyper-box models, were first proposed by Ben-Haim and Elishakoff to describe the uncertain-but-bounded variables [
10,
11,
12,
13,
14,
15], non-probabilistic techniques have become popular in non-deterministic dynamic finite element analysis [
17] and the capabilities of the approaches have been discussed in detail [
16]. Recently, the multi-ellipsoid convex set model was also proposed to deal with the case when the uncertain-but-bounded variables can be classified into many uncorrelated groups and each group can be defined by a single ellipsoid convex set [
18,
19,
20]. As revealed in several studies [
18,
19,
20], the multi-ellipsoid convex model can be regarded as the extension of the single ellipsoid and hyper-box model—in other words, the single ellipsoid and hyper-box model are two specific instances of the multi-ellipsoid convex set model. According to the non-probabilistic reliability concept stated by Ben-Hain and Elishakoff [
10,
11,
12,
13,
14,
15], the non-probabilistic reliability index of the multi-ellipsoid convex set can be expressed as the maximum allowable variability of the systems, which can be determined by the infinity norm of the vector consisting of Euclidean norms of the uncorrelated group uncertain-but-bounded vectors [
18,
19,
20]. As for the fuzzy variables described by membership functions, the uncertainty propagation in mechanical systems has been investigated widely [
21,
22] and the membership levels method [
23,
24] has been employed as an efficient means to evaluate problems with fuzzy uncertainties. The basic principle of the membership levels method, which is used in this paper, is that at each membership level, each fuzzy variable reduces into an interval with a lower and an upper bound, and then the bounds of the output responses can be obtained by optimization or any other technique [
23,
24]. In other words, if the fuzzy variables are depicted by membership functions, the membership functions of the responses can be approximated by the membership levels method. This is termed uncertain propagation with fuzzy variables. In addition, recently, the previous models have been also employed to deal with various degrees of uncertainty in practical engineering problems [
27,
28,
29,
30,
31,
32,
33,
34]. An improved dimensional approach to multidisciplinary interval uncertainty analysis is developed in which the extreme values of each interval variable used to determine the system response boundaries are solved using Chebyshev polynomial approximation and an iterative criterion [
27]. Zhou developed a fault-tree-based system reliability method to predict the failure probability of system components by non-probabilistic interval models [
28]. The unknown but constrained parameters were used as interval variables and the eigenvalues of the elastic stiffness matrix, geometric stiffness matrix and uncertain parameters were divided into deterministic and perturbative parts using perturbation theory [
29]. Heng et al. proposed a novel dynamic model updating procedure to efficiently update the interval and nonstationary correlation coefficient matrix (NPCCM) of the modal parameters and to establish their accurate and reliable uncertainty bounds [
30]. Xu et al. proposed a two-layer dimensional analysis procedure for the fuzzy finite element method to determine the minimum and maximum (min/max) points at zero cut for each slice of the response surface [
31]. The fuzzy finite element method was also employed to solve the issues in eigenfrequency and deflection analysis [
32], nonlinear free vibration analysis [
33] and high dimensional model representation [
34].
In many engineering applications, the uncertainty of the systems may result from many different sources. In such cases, the information of some uncertain variables may be abundant and the precise probability distribution functions (PDFs) can be accurately estimated by the data available; whereas in other cases with limited samples, which are not sufficient to ensure the accurate PDFs, may be defined by uncertain-but-bounded variables. In addition, some uncertain factors may be relevant to human knowledge and expert experience, which is commonly considered as epistemic uncertainty and fuzzy variables are fit to this situation. Based on the realistic problems, the models dealing with mixed uncertain variables such as the model with random variables and fuzzy uncertainties [
35,
36,
37,
38], the combination of random variables and uncertain-but-bounded variables [
39,
40,
41] and the mixture of random variables and intervals [
42,
43] have been proposed to overcome the difficulty. As the studies mentioned above reveal, a number of attempts have been made for mixed uncertainties analysis. However, most of the existing papers focus on either the mixed model of the random variables and fuzzy inputs or the combination of the random variables and uncertain-but-bounded variables. For engineering applications with limited available samples and epistemic uncertainty, the combination of uncertain-but-bounded variables and fuzzy variables has an advantage in dealing with such a situation. It is necessary to perform an investigation for uncertain propagation for this case. Based on non-probabilistic reliability theory and the membership levels method, the main goal of uncertain propagation is to estimate the membership function of the non-probabilistic reliability index. According to the basic idea of the membership levels method, since each fuzzy variable defined by a membership function degenerates into an interval with a lower and an upper bound at each membership level, the output response of the structure with fuzzy variables and uncertain-but-bounded variables will be bounded within an interval, where the output response here is the so-called non-probabilistic reliability index. When the membership level varies within the bound [0,1], the same procedure can be performed, and then the membership function of the non-probabilistic reliability index can be estimated.