Uncertainty propagation in structural reliability with implicit limit state functions under aleatory and epistemic uncertainties

Structural reliability assessment has been widely recognized as vital in engineering product design and development [7]. In the context of structural reliability assessment, uncertainty propagation plays a significant role, which aims to quantify the uncertainties of input factors and calculate the overall uncertainty within the model response in reliability estimation [36]. Before propagating the structure’s uncertainty, a primary issue is to choose a reasonable mathematical theory related to the types of uncertainty, to quantify the uncertainty [12,38]. In practical structural engineering problems, uncertainty can be divided into two categories: aleatory uncertainty derived from inherent randomness of physical behavior, while the epistemic uncertainty arising out of lack of knowledge [10]. Probability theory is regarded as the most effective tools to describe aleatory uncertainty in structural reliability assessment. Over the last decades, numerous reliability assessment methods based on probability theory have been developed, including first-order reliability method (FORM) [23], second-order fourth moment [29] Monte Carlo simulation (MCS) [24], FORM-sampling simulation method [22], envelope function method [28], response surface method (RSM) [6], and Bayesian networks method [26]. Although these probabilistic methods typically make sense in uncertainty quantification and propagation when the structure is mainly affected by aleatory uncertainty, they do not work well in the scenarios involving great epistemic uncertainty [37]. For example, the distribution of input factors may not be precisely obtained due to insufficient sample data. Consequently, several alternative non-probabilistic theories have been developed to describe the epistemic uncertainty in reliability assessment. The general non-probabilistic structural reliability assessment theories consist of fuzzy set theory [9], fuzzy random theory [13], possibility theory [1], interval theory [5, 27], and evidence theory [39]. The fuzzy and possibility measures fail to satisfy the duality property, which will make it difficult for decision-makers to understand the results [33]. Moreover, interval and evidence theories will lead to an over-conservative result due to the interval extension problems [38]. To overcome the shortcomings of the above-mentioned theories, a new mathematical framework called uncertainty theory was introduced to deal with epistemic uncertainty. Uncertainty propagation plays a pivotal role in structural reliability assessment. This paper introduces a novel uncertainty propagation method for structural reliability under different knowledge stages based on probability theory, uncertainty theory and chance theory. Firstly, a surrogate model combining the uniform design and least-squares method is presented to simulate the implicit limit state function with random and uncertain variables. Then, a novel quantification method based on chance theory is derived herein, to calculate the structural reliability under mixed aleatory and epistemic uncertainties. The concepts of chance reliability and chance reliability index (CRI) are defined to show the reliable degree of structure. Besides, the selection principles of uncertainty propagation types and the corresponding reliability estimation methods are given according to the different knowledge stages. The proposed methods are finally applied in a practical structural reliability problem, which illustrates the effectiveness and advantages of the techniques presented in this work. Highlights Abstract

, envelope function method [28], response surface method (RSM) [6], and Bayesian networks method [26]. Although these probabilistic methods typically make sense in uncertainty quantification and propagation when the structure is mainly affected by aleatory uncertainty, they do not work well in the scenarios involving great epistemic uncertainty [37]. For example, the distribution of input factors may not be precisely obtained due to insufficient sample data. Consequently, several alternative non-probabilistic theories have been developed to describe the epistemic uncertainty in reliability assessment.
The general non-probabilistic structural reliability assessment theories consist of fuzzy set theory [9], fuzzy random theory [13], possibility theory [1], interval theory [5,27], and evidence theory [39]. The fuzzy and possibility measures fail to satisfy the duality property, which will make it difficult for decision-makers to understand the results [33]. Moreover, interval and evidence theories will lead to an over-conservative result due to the interval extension problems [38]. To overcome the shortcomings of the above-mentioned theories, a new mathematical framework called uncertainty theory was introduced to deal with epistemic uncertainty.
Uncertainty propagation plays a pivotal role in structural reliability assessment. This paper introduces a novel uncertainty propagation method for structural reliability under different knowledge stages based on probability theory, uncertainty theory and chance theory. Firstly, a surrogate model combining the uniform design and least-squares method is presented to simulate the implicit limit state function with random and uncertain variables. Then, a novel quantification method based on chance theory is derived herein, to calculate the structural reliability under mixed aleatory and epistemic uncertainties. The concepts of chance reliability and chance reliability index (CRI) are defined to show the reliable degree of structure. Besides, the selection principles of uncertainty propagation types and the corresponding reliability estimation methods are given according to the different knowledge stages. The proposed methods are finally applied in a practical structural reliability problem, which illustrates the effectiveness and advantages of the techniques presented in this work.
The concepts of chance reliability and chance reli-• ability index (CRI) are proposed.
A new level-2 uncertainty propagation model is • provided for uncertain random structure.
The principles for choosing reasonable uncertain-• ty propagation types are presented.
Uncertainty propagation in structural reliability with implicit limit state functions under aleatory and epistemic uncertainties Uncertainty theory proposed by Liu [18] in 2007 to describe the belief degree of human, has been successfully applied in various areas such as decision making [30], uncertain insurance [19,32], uncertain risk and reliability analysis [34,35]. Uncertainty theory is considered a reasonable and useful tool to express epistemic uncertainty, compared with the theories mentioned above [12]. Since the uncertain measure satisfies the axiom of duality, normality, and subadditivity, the results produced by the uncertainty theory are more in line with real engineering conditions [8]. Hence, in this work, uncertainty theory is chosen to express epistemic uncertainty and describe human thinking processes. In practical structural problems, there are usually two types of input factors that embody different types of uncertainties at the same time. Some input factors may suffer great epistemic uncertainty and are described by uncertainty theory, while some others may be primarily determined by aleatory uncertainty and are modeled based on probability theory. These structures comprising both aleatory and epistemic uncertainties are called uncertain random structures in this paper. It is impossible to analyze the reliability of uncertain random structures only by probability theory or uncertainty theory [38].
To solve this problem, chance theory was established by Liu [20] in 2013 to propagate aleatory and epistemic uncertainties together. Chance theory can be understood as a combination of probability theory and uncertainty theory, which also satisfies normality, duality, and subadditivity theorems. In recent years, chance theory has been successfully used in various fields such as project scheduling [11], uncertain random risk analysis [8], uncertain random programming [25], and systems reliability analysis [31,38]. Especially, a hybrid model of structural reliability analysis based on chance theory was proposed by Zhang [37] in 2019, and a new reliability index was proposed. However, this method has the following disadvantages. Firstly, there is no corresponding reliability analysis method when the implicit limit state function (LSF) contains both random and uncertain variables. Secondly, the defined reliability and reliability index do not involve time dynamic parameters. Thirdly, this reliability analysis method does not consider the problem of uncertainty propagation.
For completeness, this paper uses a uniform design (UD) combined with the least-squares (LS) method to simulate LSF, which adapts to both random and uncertain variables. The UD is a novel kind of experimental design method founded by Fang and Wang [3], defined according to the uniform distribution in number theory [40]. Compared to the orthogonal design (OD), factorial design (FD), and Latin hypercube sampling (LHS) methods, the UD method appears to be more advanced if the number of experimental factors is large and the number of experiments is limited [4].
Besides the above research, this paper's main contribution is to provide a new uncertainty propagation method for structural reliability assessment. Uncertainty propagation aims to estimate structural output responses by propagating the input factors essential for structural reliability assessment and safety design [36]. Normally, uncertainty propagation can be classified into the form of level-1 and level-2 [14]. For level-1 propagation, the values of input factors can be characterized by epistemic or aleatory uncertainties at the same level [2]. For level-2 propagation, the values of input factors are represented by aleatory uncertainties on the first level. Epistemic uncertainties describe the parameters of probability distributions in the second level [34]. These two types of uncertainty propagation methods are commonly used in risk assessment. Comprehensive research about this was reported by Hu et al. [8], who presented a framework for propagation methods corresponding to different knowledge stages in fault tree analysis. However, there are no literature about the level-2 uncertainty propagation modeling and propagation type selection methods for structural reliability assessment. Hence, this paper aims to develop some propagation analysis methods and the principles for the selection of propagation type in structural reliability assessment.
The remainder of this work is organized as follows: Section 2 briefly discusses some important mathematical concepts of uncertainty and chance theory. A new surrogate model combining UD and LS method is proposed for implicit LSF in Section 3. Section 4 provides a novel structural reliability quantification model based on chance theory. Some principles for choosing appropriate uncertainty propagation types are discussed, and corresponding reliability calculation methods are provided in Section 5. In Section 6, a practical engineering case study is carried out to show the proposed method's rationality. Finally, some conclusions are presented in Section 7.

Preliminaries
In this section, some fundamental knowledge and results regarding the uncertainty theory and chance theory are introduced.

Uncertainty theory
Uncertainty theory is a fairly new branch of axiomatic mathematics, and has been widely applied in various areas. In the uncertainty theory, the human belief degree of events are quantified by defining uncertain measures. Definition 2.1 (Uncertain measure [15]) Let Γ be a nonempty set, and  be a σ-algebra over Γ. Each element Λ in  is called an event. Then, a set function M is defined as an uncertain measure if it satisfies normality, duality, and subadditivity axioms. Definition 2.2 (Uncertain variable [18]) An uncertain variable is a measurable function τ from an uncertainty space (Γ, , M ) to the set of real numbers, i.e., {τ ∊B} is an event for any Borel set B of real numbers. Definition 2.3 (Uncertainty distribution [15]) The uncertainty distribution Φ(x) of an uncertain variable τ can be defined by Φ(x) =M{τ≤x} for any real number x.
A regular uncertainty distribution Φ(x) is defined as an uncertainty function that is continuous and strictly increasing with respect to x. Example 2.1 An uncertain variable τ is defined as a normal uncertain variable if it has a normal uncertainty distribution: It is denoted by τ τ  (m, σ), where m is the expected value and σ is the standard deviation.
Example 2.2 An uncertain variable τ is defined as linear variable if it has a linear uncertainty distribution: It is denoted by τ τ   ( , ) a b, where a and b are real numbers with Since the uncertainty theory can describe the incomplete information contained in design variables, the epistemic uncertainty (especially human) can be characterized by uncertain variables and uncertainty distribution in the uncertainty space [16,17]. Definition 2.4 (Inverse uncertainty distribution [15]) Let τ be an uncertain variable with regular uncertain distribution Φ(x). The inverse function Φ -1 ( ) u is known as the inverse uncertainty distribution of τ. Theorem 2.1 (Operational law [18]) Let τ τ τ 1 2 , , ,  n be independent uncertain variables with regular uncertainty distributions Φ Φ Φ ,

Chance theory
As a combination of uncertainty and probability theory, the chance theory is applied as a new tool to deal with problems affected by both uncertainty and randomness. The basic concept involves the chance measure of an uncertain random event in a chance space. Definition 2.5 (Chance measure [20]) Let (Γ, , M ) × (Ω, , Pr) be a chance space, and let Θ È ∈  ×  be an event. Then the chance measure of Θ can be defined as:  [20]) An uncertain random variable is a measurable function ξ from a chance space (Γ, , M ) × (Ω, , Pr ) to the set of real numbers such that { } ξ ∈ B is an event in  ×  for any Borel set B of real numbers. Definition 2.7 (Chance distribution [15]) Let ξ be an uncertain random variable. Then its chance distribution is defined as for any real number x.

Advanced UD-LS surrogate model for implicit limit state functions
In the practical structural reliability problems, the analytical expression of LSF is generally unknown. The traditional RSM of structural reliability analysis is iteratively obtained based on the probabilistic reliability index (PRI). Furthermore, the traditional RSM is only suitable for random variables and requires a large number of test sample data. Thus, a new surrogate model is established by combining UD with the LS method considering both of aleatory and epistemic information.
The structure's response is obtained by experiment or finite element analysis, and the sample points used to fit the surrogate model are determined by the design of experiments (DOE) methods. Compared with traditional DOE methods, the UD method is more stable and efficient [4]. UD can maintain the results with high stability and accuracy even with a small sample data. Similar to the OD approach, the UD method can be used to generate experiment points by a series of designed UD tables. The representation of a specific UD table is where U denotes the UD table, m represents the number of levels and the number of experiments required, n is the number of input factors, and * represents the UD table with a smaller deviation and better uniformity. This work presents only a brief introduction of the UD method, and interested readers can refer to relevant research literature [3,4,40]. The quadratic polynomial surrogate model without the cross-terms is chosen as the response surface function of the structure.
undetermined coefficients vector in the surrogate model [4]. According to the LS approach, b can be estimated based on b a a a y = − ( ) where a is the regression coefficients vector with Some indexes are used for validation to verify the surrogate model's fitting performance and check the accuracy. Among them, the coefficient of determination R 2 is the most crucial measurement index: where f ( ) x is the expected value of all the real responses f i ( ) x , and f i  ( ) x are the simulation values of the responses. The closer the value of R 2 to 1, the higher is the accuracy of UD-LS surrogate model fitting.
According to the stress-strength interference model and the UD-LS surrogate model, the LSF G( ) x,α of a structural system under mixed aleatory and epistemic uncertainties can be expressed as: is the input factor that affects the structural functioning, threshold S is the allowable threshold of structural response, and α is a dynamic input parameter associated with time.

Structural reliability assessment method under mixed aleatory and epistemic uncertainties
In a complex structural system, some design variables may have enough samples for estimating their probability distribution, which can be described by random variables. Nonetheless, some other design variables may lack sufficient data, which can be estimated by domain experts and regarded as uncertain variables. The structure cannot be simply considered to be a random or uncertain structure model under mixed aleatory and epistemic uncertainties [37]. This section put forward an advanced structural reliability assessment method for this issue depending on chance measure and belief reliability theory.

Uncertainty quantification for structural reliability based on chance measure
Let (Γ, , M ) × (Ω, , Pr ) be a chance space, and the LSF of structure contains uncertain random input factors 12 . In the present work, the input factors are uniformly described by uncertain random variables ξ ξ = ( , , , ) ξ ξ ξ 1 2  n , then the chance reliability of structure based on the chance measure can be defined as follows.
Definition 4.1 Assuming that G( ) ξ ξ ,α is the LSF of a structure, in which ξ ξ is the vector of uncertain random variables, the chance reliability is defined as the chance measure of the reliability event { ( ) } G ξ ξ ,α > 0 at α: Because of the duality of chance measure, the chance measure of a failure event { ( ) } G ξ ξ ,α ≤ 0 at α can be derived as: Consequently, the uncertainty of a safety event at α in structure can be quantified by Ch reliability ( ) α with a numerical value of [ 0 , 1] . Ch failure ( ) α describes the confidence how a failure even will be happened at α. Obviously, the higher the Ch failure ( ) α , more is the possibility that the failure event will occur at α. The theorem to be defined below provides computational methods for practical engineering applications.

The new chance reliability index based on uncertain random variables
PRI in probability space is a vital indicator for quality of structure, and it can be used to describe the structural reliability under aleatory uncertainty. However, the PRI cannot accurately measure the reliability under mixed aleatory and epistemic uncertainties. For completeness, a novel chance reliability index (CRI) is defined using the expected value and variance of the uncertain random variable, showing the reliable degree of a structure in chance space.
of a structure contains independent random variables η η η 1 2 , , ,  p with probability distributions 1 2 , , , p Ψ Ψ Ψ  , and independent uncertain variables τ τ τ 1 2 , , ,  q with regular uncertainty distributions ϒ ϒ ϒ 1 2 , , ,  q , respectively. Then the CRI of the structural system at α can be given as follows: where E G [ ( , )] ξ ξ α is the expected value, V G [ ( , )] ξ ξ α is the variance of LSF, and ξ ξ = ξ ξ ξ 1 2 ,  p q + is an uncertain random vector. If the LSF is continuous and strictly increasing with respect to τ τ τ 1 2 , , ,  k , and strictly decreasing with respect to τ τ τ k k q +1 +2 , , ,  , then according to the theorems 2.4 and 2.5, the expected value and variance of LSF at α can be calculated as: The chance-measure-based CRI is offered as a tool to measure the confidence that a reliability event will occur in the structural system affected by both aleatory and epistemic uncertainties. A larger β α chance ( ) indicates a better possibility that the reliability event will occur.

Joint uncertainty propagation method for structural reliability assessment
Uncertainty propagation plays a significant role in reliability problem, which aims to estimate structural output responses by propagating the input factors essential for reliability assessment and safety design. To make it possible for decision-makers to find an appropriate uncertainty propagation types under different knowledge stages, a new joint uncertainty propagation technique is presented in this section. Therefore, the selection principles of uncertainty propagation types are developed in section 5.1. Section 5.2 briefly introduces the level-1 uncertainty propagation method, especially the propagation of uncertain random structure. The novel level-2 joint uncertainty propagation method for structural reliability assessment is proposed in Section 5.3.

The principles of uncertainty propagation types selection
In general, uncertainty propagation can be classified into the form of level-1 and level-2. The quantification and propagation of uncertainty runs through the whole analysis process. To explain the uncertainty propagation of level-1 and level-2 types, the probability theory is utilized to express aleatory uncertainties, while the uncertainty theory is used to describe the epistemic uncertainties. G( ) x is assumed to be the LSF established in Section 3, where x = ( , , , ) x x x n 1 2  is the input factors vector, and G is the output. To analyze the uncertainty of output G, the uncertainty expressions of the input factors needs to be studied, in addition to their propagations through LSF.
According to the knowledge stage of reliability analyst, the reliability evaluation types can divided into different stages. For example, uncertainty propagation types can be divided into five different stages, shown in Fig.1. At stage 1, the reliability analyst has no sample data on 12 , , , n xx x  . So uncertain variables are used to describe all the input factors. In this situation, the uncertainty propagations are in level-1 type. At stage 2, the reliability analyst collects more sample data and improves his knowledge. The distribution function type of 1 x is known, which is the probability distribution type. Nonetheless, the shaping parameters of this probability distribution are still lack-ing and can be described by uncertain variables. 23 , , n xx x  are still described by uncertain variables. In this case, the uncertainty propagations will be in level-2 type. At stage 3, the knowledge of reliability analyst improves further. The probability distribution of 1 x is determined completely, while 23 , , n xx x  are still described by uncertain variables, and the uncertainty propagations turns into level-1 type. At stage 4, the knowledge of reliability analyst improves by obtaining the probability distributions type of 2 x and 3 x , but their shaping parameters are still lacking and can be described by uncertain variables. 1 x and 4 , n x x  are perfectly described by probability distributions. In this situation, the uncertainty propagations turn into level-2 type. At stage 5, the probability distributions of all input factors are determined completely due to elimination of epistemic uncertainties, and random variables are used to describe all input factors. Meanwhile, the uncertainty propagations turn into level-1 type.
In summary, the selection of uncertainty propagation types depends on the knowledge stage and sample data owned by the reliability analyst. For more general situations, the decision-makers can match any circumstances in practical engineering by increasing the number of stages.

Level-1 uncertainty propagation in structural reliability assessment
As the example mentioned in Section 5.1, there are three different knowledge stages in level-1 uncertainty propagation type, namely, stage 1, stage 3 and stage 5. For stage 1, the uncertain variables describe all the input factors of LSF, and uncertainty propagations analysis is handled through a pure uncertainty model. The uncertain reliability and index can be obtained based on the uncertainty theory and the operational laws, and the specific calculation methods of uncertain structure can be referred to the literature [35]. In stage 3, some input factors have sufficient data, so their probability distributions can be obtained, while some input factors lack data and can only be described by uncertain variables. By analyzing the joint propagation of uncertainty and probability, the methods proposed in the Sections 4.1 and 4.2 are used to estimate the reliability and index of uncertain random structure. The structure that corresponds to stage 5 is called random structure, and the uncertainty propagations analysis for random Fig. 1. Uncertainty propagation types of structure corresponding to different stages structure can be implemented by traditional pure probability model. The probabilistic reliability and index of random structure can be estimated by the classical methods such as FORM and MCS.

Level-2 joint uncertainty propagation in uncertain random structure reliability assessment
As the example mentioned in Section 5.1, there are two different knowledge stages in the level-2 uncertainty propagation type, namely, stage 2 and stage 4. For stage 2, some input factors in the LSF are expressed by probability distributions, of which the shaping parameters are described by uncertainty distributions, while other input factors can be expressed by uncertainty distributions. The next two subsections will introduce the detailed calculation methods in these two stages.
Let a LSF of structural system contain p q + input factors, of which p input factors are expressed by random variables, and the shaping parameters of the probability distributions are described by uncertain variables, while q input factors are expressed by uncertain variables. The level-2 joint propagation in stage 2 can be considered as a more general situation of propagations in stage 3. Consequently, a new level-2 uncertainty analysis method and the corresponding reliability calculation model are provided for stage 2 in this work.
Assuming that p input variables are represented as η η η 1 2 , , ,  p , and each probability distribution of η i is represented as Ψ i i i ( ) η θ , in which θ i represents the shaping parameters of probability distribution. The shaping parameters are described by uncertainty distributions Φ i i ( ) θ . Let τ τ τ where F y y y p ( ; , , , ) ; 1 2 0  α is the root u of the Equation (15) for any real numbers 12 , , , p y y y  .
Moreover, the CRI of the structural system at α can be computed from Equation (17), where the expected value and variance of LSF ξ ξ α at α can be computed as follows: (1 ), 1 0 Therefore, the chance reliability and CRI of the structural system is no longer a point value, but varies between the lower and upper bounds of shaping parameters with the uncertainty distribution Φ i i ( ) θ . The level-2 joint propagation in stage 4 represents more general circumstances of random structure in stage 5. Thus, the reliability and index of random structure can also calculated by the traditional pure probability methods such as FORM and MCS. Consider a LSF G n ( , , ) η η α 1  ; of structure contains n input factors, m input factors are expressed by the probability distributions Ψ i i i ( ) η θ , and the shaping parameters θ i of Ψ i i i ( ) η θ are described by the uncertainty distributions Φ i i ( ) θ , while n m − input factors are expressed by random variables with no epistemic uncertainties. The variation range of reliability and index can be calculated by replacing the original probability distributions with Ψ i i i ( ) η θ in classical FORM. Thus, the probabilistic reliability and index of the structure are also no longer a point value, and the variation range can be obtained based on the uncertainty distribution of the shaping parameters.

An illustrated example
In this section, the propagation analysis methods developed herein, are applied to a practical engineering application of turbine disk reliability assessment. The description of turbine disk and the implementation of UD-LS surrogate model are introduced in Section 6.1. Section 6.2 shows the specific application process of the uncertainty propagation method proposed in this work. Some results and discussions on the advantages of the proposed method are given in Section 6.3.

Structure description and LSF simulation
Turbine disk is the key rotating component of modern aircraft engines, driven based on high-temperature gas in the engine combustion chamber. Because the turbine disk converts the thermal energy in the gas into mechanical energy to drive the engine, its reliability level will directly affect the performance of the entire engine.
As shown in Fig. 2(a), the three-stage turbine disk of a low-pressure compressor in a turbofan engine was selected as the research object. The pins on the roulette wheel are evenly and symmetrically distributed along the circumference. According to engineering analysis, the chief input factors affecting the reliability of roulette wheel include material characteristics, load and speed. In this work, the material of hollow pin was 3Cr13, while the material of roulette wheel was TC11. The blade load was applied perpendicularly to the hollow pin, and the average value of the load on each hollow pin was 24925N. The relevant parameters of input factors are shown in Table 1. Since the main failure mode of the turbine disk requires that the maximum stress value is greater than the allowable strength threshold S , the maximum stress value can be obtained by finite element analysis.
Because the shape and load of the turbine disk are completely symmetrical, 1/37 part of the turbine disk is considered to describe the entire structure. The average value of each input factor was chosen as the variable value, and the turbine disk was simulated using AN-SYS 18.2 at a speed of 1000 rad/s. According to the simulation results  Fig. 3, the stress-strain level at the junction between the roulette wheel and hollow pin is the highest, which is the dangerous failure point of the structure.
Since with the increase in rotating speed ω(rad/ s) , the maximum stress at the dangerous point will increase, the reliability of turbine disk will continue to degrade. According to the UD-LS surrogate model introduced in Section 3, a UD table  Table 2.
According to the stress-strength interference model, the LSF of the three-stage turbine disk is established as:   the LSF simulated in this experiment has a high degree of fit, which lays a good foundation for the next step of uncertainty propagation analysis.

Joint propagation of uncertainty and probability
Based on the different knowledge and sample data stages possessed by the reliability analyst on input factors, the reliability assessment can be implemented based on the uncertainty propagation model proposed in Section 5. According to the selection principles proposed in Section 5.1, the uncertainty propagation of the turbine disk can be obtained as shown in Table 3. In stage 1, the reliability analyst does not have detailed sample data on all input variables. So a domain expert is invited to estimate the values of input factors. Seven normal uncertainty distributions are used to represent the expert's beliefs corresponding to the input factors. In stage 2, the knowledge stage of reliability analyst is improved, and the distribution of roulette wheel density ρ 1 is confirmed as a normal probability distribution N ( , ) µ σ ρ ρ still uncertain, and a domain expert is invited to estimate the values of µ ρ 1 . Therefore, a linear uncertainty distribution  (3.78,5.18) is used to represent the expert's beliefs on the expected value µ ρ 1 , but the other six input factors are still expressed as normal uncertainty distributions. In stage 3, the knowledge stage of reliability analyst is improved further by obtaining sufficient data about roulette wheel density. So the normal probability distribution of ρ 1 is determined completely. Also, the normal uncertainty distributions of the other six input factors remain unchanged. In stage 4, the knowledge stage of reliability analyst is improved, and the distributions of seven input variables is determined as a normal probability distribution. However, the expected values of N E E ( , ) µ σ  T is the vector of coefficients, which is estimated by the method introduced in Section 3. The coefficient of determination is calculated as 2 0.99784 R = by Equation (10), and is very close to 1. Hence,    In other words, all input factors are perfectly described by a normal probability distribution. The above-mentioned specific reliability assessment processes under different knowledge and sample data stages are based on the joint uncertainty propagation method proposed in Section 5.

Results and discussion
Let the range of turbine disk speed ω be [0,2175] . Then the reliability and index depending on ω in different stages can be calculated based on the methods developed in this paper. It is clear that with the increase in speed, the reliability and indexes of the turbine disk will degenerate because of the increase in stress.
As shown in Fig. 4, the reliability and indexes under three different stages in level-1, namely pure uncertainty in stage 1, uncertain random in stage 3 and pure probability in stage 5 are compared. The results estimated from level-2 in stage 2 are shown in Fig. 5, where the reliability and index of turbine disk fluctuate with the unknown parameter µ ρ 1 . In particular, when the rotation speed ω 1200rad / s ω = , the reliability and index takes the values ) 0.9648, 0.9895 ( and ) 1.8533, 2.3081 ( , respectively. Fig. 6 shows the variation of reliability and index with unknown parameters µ E 1 and µ v 1 at speed ω 1200rad / s ω = , where the reliability and index take values in ) 0.9445, 0.9975 ( and ) 1.5935, 2.8018 ( , respectively. The practical engineering example illustrates the specific implementation process of the presented method in detail, and the reliability of turbine disk are obtained in different knowledge and sample data stages.
Besides, the simulation results of the reliability and index for different knowledge stages at a specific speed ω 1200rad / s ω = are presented in Table 4. It is worth noting that the specific speed is selected arbitrarily and the same comparisons can be implemented at any speed value. As presented in Table 4, the reliability is transformed from the interval ) 0.9648, 0.9895 ( in stage 2 to the determined value 0.9748 in stage 3 due to the increase in sample data. Moreover, the reliability is transformed from the interval ) 0.9445, 0.9975 ( in stage 4 to the determined value 0.9871 in stage 5, which is a good explanation for the process of eliminating epistemic uncertainties. Similar conclusions can be obtained from the reliability indexes in different stages. Besides, as shown in Fig. 4 and Fig.5, when the reliability values are close to 1 at some speed values, the reliability index can be employed to distinguish the reliability differences at these speed values. In the context of structural reliability assessment, the description of epistemic uncertainty is an inevitably common problem. Classical probability theory cannot be employed to express epistemic uncertainty since the real frequency cannot be obtained due to lack of data. Fuzzy measure and possibility measure do not satisfy duality property, and hence the description of epistemic uncertainty is not reasonable enough. Evidence and interval theory leads to the problem of interval expansion in practical applications. Uncertainty theory is a newly proposed mathematical framework that firmly conforms to the normality, duality and subadditivity theorems. This paper uses the uncertainty theory to describe epistemic uncertainty because it is more suitable for describing the human thinking processes. Also, the probability theory is chosen to represent aleatory uncertainty, and the chance theory is selected to deal with the situation when aleatory and epistemic uncertainties exist simultaneously. The results of case study shows that the level-1 and level-2 joint propagation can be explained very well by combining the above three theories. Consequently, the practical engineering application shows that the various knowledge stages outcome the different reliability levels, and the results highlight that the presented methods are effective and could deliver clear messages to decision-makers.  The principles for choosing reasonable uncertainty propaga- (4) tion types are presented for structural reliability assessment.
Decision-makers can evaluate the structural reliability corresponding to the different knowledge and sample data stages based on the uncertainty propagation method is proposed in this paper. As the model presented in this work is based on monotonic conditions, further research is required to focus on non-monotonic situations. Another interesting and important issue is to determine the distribution type of input factors based on small sample data. Acknowledgement This work was supported by the National Natural Science Foundation of China (No. 51675026 and No. 71671009), and Aeronautical Science Foundation of China under Grant (2018ZC74001).

Conclusions
In this paper, a novel uncertainty propagation method is proposed for the structural reliability assessment under mixed aleatory and epistemic uncertainties. To enable the analyst to calculate the structural reliability according to the different knowledge stages, the principles of selecting the uncertainty propagation types and the corresponding reliability estimation methods are presented. In summary, the main contributions of this paper are as follows: A new UD-LS surrogate model is proposed to solve the im- (1) plicit LSF problem involving random and uncertain variables. The concepts of chance reliability and CRI are defined to de- (2) scribe structural reliability under mixed aleatory and epistemic uncertainties.
A novel level-2 uncertainty analysis method and the corre- (3) sponding reliability calculation model are provided for uncertain random structures.