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Efficiently and accurately solving the failure credibility of a structural system plays a significant role in the engineering field under fuzzy environment. A more precise failure credibility can help assess the safety degree of the system, and then reduce the occurrence of security accidents. At present, the extended fuzzy first-order and second-moment method (EFFOSM) is effective to analyze failure credibility for some occasions. However, when it comes to complicated performance functions and fuzzy inputs, the accuracy of EFFOSM is greatly reduced and its efficiency also needs to be improved. To overcome the above shortcomings, this paper proposes two types of novel fuzzy simulation algorithms, namely uniform discretization algorithm (UDA) and bisection simulation algorithm (BSA). For the system involving frequently-encountered continuous and strictly monotone performance functions of regular LR fuzzy interval inputs, these two algorithms are designed to estimate failure credibility with higher efficiency and accuracy. Subsequently, with the aid of the linearization and regularization procedures in EFFOSM, the application of UDA and BSA is extended to non-monotone performance functions of irregular LR fuzzy intervals. To evaluate and verify the performance of the proposed two algorithms, their comparisons with EFFOSM are conducted through some numerical examples and practical problems. The results show that the proposed two algorithms outperform EFFOSM in terms of accuracy and efficiency, and also have wider application range for estimating the failure credibility of strictly monotone performance functions involving regular LR fuzzy interval inputs. Meanwhile, BSA is slightly better than UDA for the less runtime. • Estimation of the failure credibility of a structural system under the fuzzy environment is considered. • Deficiencies of the extended fuzzy first-order and second-moment method (EFFOSM) are analyzed. • Two novel fuzzy simulation algorithms with high accuracy and efficiency are developed for solving the failure credibility. • The application of two suggested algorithms is extended by using the linearization and regularization procedures in EFFOSM. • The accuracy, efficiency and applicability of the proposed algorithms are demonstrated by the comparison with EFFOSM. [ABSTRACT FROM AUTHOR]

A fuzzy first-order and second moment (FFOSM) method is proposed to efficiently estimate time-independent and time-dependent failure credibility in the presence of fuzzy uncertainty. For the time-independent linear performance function with independently, regularly, and identically distributed fuzzy inputs, the analytical relationship is first derived between the failure credibility and fuzzy safety index, and the relation form is established between fuzzy safety index and fuzzy first two moments of the performance function. Second, in the standard space of fuzzy inputs, the minimum infinite norm distance from the coordinate origin to the linear limit state surface is analytically derived, and the point with the smallest infinite norm distance located on the limit state surface is defined as the fuzzy design point. Based on this definition, the geometric and physical meanings are explained for the fuzzy safety index and fuzzy design point, and the FFOSM method for time-independent failure credibility analysis is established for both linear and nonlinear performance function. Third, for time-dependent performance functions, the equivalence is proved between time-dependent fuzzy safety index and the minimum value of transient fuzzy safety indices, on which the iteration steps of the FFOSM method is designed for time-dependent fuzzy safety index and failure credibility estimation. Since the proposed FFOSM method is an approximately analytical approach, it has higher efficiency than the fuzzy simulation method and the cut level optimization method. Finally, the effectiveness and applicability of the proposed method are demonstrated by different examples. [ABSTRACT FROM AUTHOR]

To measure the effects of the fuzzy inputs on structural safety degree, this paper establishes the failure credibility-based global sensitivity by the fuzzy expected value of the absolute difference between the unconditional failure credibility and conditional one. To establish the failure credibility-based global sensitivity, the conditional failure credibility is firstly defined according to the original definition of conditional event and the relationship among the possibility, necessity and credibility, in which no extra assumption is introduced. After that, the equivalent expression of the failure credibility is deduced, on which the Bayesian transformation of the conditional failure credibility is obtained in this paper. Then, a single-loop method based on the sequential quadratic programming is applied to efficiently estimate the defined failure credibility-based global sensitivity. According to the result of the constructed failure credibility-based global sensitivity, designers can pay more attentions to the more important fuzzy inputs to have a better control of the structural safety degree. The presented examples demonstrate the feasibility of the constructed failure credibility-based global sensitivity and the efficiency of the proposed solution. [ABSTRACT FROM AUTHOR]

Failure credibility is popular in measuring safety degree of structure under fuzzy uncertainty, but the heavy computational cost is still a challenge in estimating the failure credibility. To alleviate this issue, an iterative method combining adaptive Kriging and fuzzy simulation (AK-FS) has been developed by Ling et al. However, for the problem with complex performance function, a large candidate sampling pool is needed in the AK-FS, which makes the training process of the Kriging model fairly time consuming. In order to improve the estimation efficiency of failure credibility through reducing the size of candidate sampling pool in AK-FS, an efficient sample reduction strategy based on adaptive Kriging (SR-AK) is proposed in this paper. In the SR-AK, the estimation of failure credibility is transformed into searching two active points in candidate sampling pool. After updating the Kriging model in each circle, current active points can be easily identified. Then, according to the properties of the active points and the prediction characteristics of Kriging model, the samples in current candidate sampling pool can be divided into two sets, i.e., the samples affect the estimation of active points and the samples have no effect on it. Obviously, the samples in the latter set can be deleted from current candidate sampling pool to reduce its size. By using this sample reduction strategy, the process for training Kriging model is accelerated circle by circle, which is very helpful to save the analysis time and improve the computational efficiency in estimating failure credibility. Four examples are employed to demonstrate the performance of the proposed SR-AK in fuzzy safety degree analysis. [ABSTRACT FROM AUTHOR]

Most of the existing methods for estimating time-dependent failure credibility (TDFC) are based on optimization algorithms, which may result in a heavy burden on the computational cost and accuracy issue related to the local optimization. In order to achieve the best compromise between computational accuracy and cost, an efficient method is proposed in this work by embedding a single-loop adaptive Kriging model (S-AK) into the dichotomy searching algorithm (S-AK-DSA). The proposed S-AK-DSA can be regarded as a double-loop procedure. In the inner loop, the Kriging model of the real time-dependent performance function (TD-PF) is updated iteratively to accurately predict the signs of the upper/lower boundary of the TD-PF minimum with respect to the time variable at the given membership level. Based on the inner loop, the outer loop searches TDFC by continuously dichotomizing the searching interval of the TDFC. The advantages of S-AK-DSA are mainly manifested in two aspects. Firstly, S-AK-DSA converts the problem of optimizing the exact value of the upper/lower boundary into the problem of accurately identifying their signs, which can avoid the use of optimization algorithms. Secondly, at different membership levels, the S-AK-DSA method chooses the candidate sample pool and continuously updates the current Kriging model of TD-PF, and the adaptive learning function as well as appropriate stopping criterion can effectively reduce the cost of predicting the signs of the upper/lower boundary and improve the computational accuracy. Four case studies are introduced to demonstrate the feasibility and superiority of the proposed S-AK-DSA approach. [ABSTRACT FROM AUTHOR]

Time-dependent failure credibility (TDFC) in a given service time of interest can reasonably quantify the safety of the structure with fuzzy inputs. However, the estimation of TDFC requires high computational cost because it involves multi-layer optimization, especially for problems with the implicit performance functions in engineering. In order to improve the efficiency of estimating TDFC under satisfied precision, this paper proposes an efficient dichotomy searching algorithm (DSA) combined with double-loop adaptive Kriging (D-AK) model (shorten as D-AK-DSA). In the proposed D-AK-DSA, a double-loop adaptive Kriging model is constructed to provide the directions for the dichotomy searching TDFC. In the inner loop, the AK model is constructed to surrogate the relation of the performance function and time variable at the given fuzzy input realization so as to obtain the minimum performance function with respect to the time for the outer loop, and in the outer loop, the AK model is constructed to identify the sign of the upper/lower boundary of the minimum performance function with the fuzzy inputs at a given membership interval so as to provide the dichotomy searching direction for TDFC. By the proposed dichotomy searching strategy, accurately solving the values of the upper/lower boundaries in evaluating TDFC is innovatively replaced by identifying the signs of them, which greatly reduces the difficulty of constructing the outer AK model. At the same time, the proposed method enhances the convergence of the outer AK surrogate model by establishing a reasonably improved learning function. Three examples verify the accuracy and efficiency of the proposed D-AK-DSA method. [ABSTRACT FROM AUTHOR]

Failure credibility is popular in measuring safety degree of structure under fuzzy uncertainty due to its excellent property of self-duality. Existing methods for estimating failure credibility can be mainly divided into two categories, i.e., the simulation-based methods and the optimization-based methods. The simulation-based methods are universal and robust, but time-consuming in dealing with high-dimensional problems. The optimization-based methods are efficient and flexible, but the precision of the failure credibility estimate will be impacted by the local optimum solution resulted from some optimization techniques. Thus, in order to overcome the disadvantages of existing methods, an efficient method by combining genetic algorithm and active learning Kriging (AK-GA) is proposed to estimate failure credibility in this paper. Firstly, by use of the fuzzy simulation, the estimation of failure credibility is transformed into finding for the failure/safety sample that processes the maximum joint membership function (MF) from the fuzzy simulation pool, which makes the credibility estimation less difficult to be computed. Secondly, the genetic algorithm is used to find the failure/safety sample with maximum joint MF. Thirdly, to drastically improve the computational efficiency of the proposed method, active learning Kriging is embedded to accurately and efficiently distinguish the failure and safety sample in computing the fitness function of genetic algorithm. Four mathematical and engineering test examples are used to demonstrate the accuracy and efficiency of the proposed AK-GA for estimating failure credibility. [ABSTRACT FROM AUTHOR]

The hinged slab bridge is widely used in medium- and small-span bridges because of its simple structure and convenient construction. However, hinge joints damage is the main defect of this kind of bridge, and it is difficult to express the deterministic damage degree of hinge joints in the detection process. A system reliability evaluation method considering fuzzy detection information of hinge joints damage and member failure credibility is proposed in this paper. Firstly, the membership function is used to quantitatively express the fuzzy detection information of hinge joints, and the fuzzy variable is transformed to an equivalent random variable. Secondly, the functional relationship between the transverse distribution coefficient and hinge joints damage is constructed by the modified hinge-jointed plate method and response surface method, and the reliability of the member considering the fuzzy detection information of hinge joints damage is calculated by the first-order second-moment method (FOSM). Then, the failure credibility is introduced to represent the different possibilities of system failure caused by member failure, and a system reliability assessment method of different failure criteria considering member failure credibility is established based on copula theory. Finally, the applicability of the proposed method is verified by taking the reinforced concrete hinged slab bridge as a numerical example. [ABSTRACT FROM AUTHOR]

The failure credibility can be used to measure the safety level of the structure under the fuzzy inputs, but the computational efficiency for estimating the failure credibility is still a challenge. A novel method by combining the adaptive Kriging with fuzzy simulation (AK-FS) is proposed to efficiently estimate the failure credibility. The proposed method firstly employs the FS to transform the estimation of failure credibility into a classification problem, which can be viewed as a bi-level strategy. In the inner loop, a Kriging model for the actual complicated performance function is actively trained by U-learning function in the sample pool generated by FS until the convergent condition is satisfied, on which the samples are divided into failure group and safety group by the well-trained Kriging model instead of the actual performance function in the outer loop. Finally, the failure credibility is obtained by respectively searching the maximum joint membership degrees of the samples in these two groups. The proposed AK-FS method only evaluates the actual performance function in the process for constructing the Kriging model. Since the U-learning function can iteratively construct a sufficiently accurate Kriging model with model evaluations as little as possible, thus the failure credibility can be efficiently and accurately estimated by the proposed AK-FS method. The advantages of the proposed AK-FS method are demonstrated by several examples. [ABSTRACT FROM AUTHOR]

The hinged slab bridge is widely used in medium- and small-span bridges because of its simple structure and convenient construction. However, hinge joints damage is the main defect of this kind of bridge, and it is difficult to express the deterministic damage degree of hinge joints in the detection process. A system reliability evaluation method considering fuzzy detection information of hinge joints damage and member failure credibility is proposed in this paper. Firstly, the membership function is used to quantitatively express the fuzzy detection information of hinge joints, and the fuzzy variable is transformed to an equivalent random variable. Secondly, the functional relationship between the transverse distribution coefficient and hinge joints damage is constructed by the modified hinge-jointed plate method and response surface method, and the reliability of the member considering the fuzzy detection information of hinge joints damage is calculated by the first-order second-moment method (FOSM). Then, the failure credibility is introduced to represent the different possibilities of system failure caused by member failure, and a system reliability assessment method of different failure criteria considering member failure credibility is established based on copula theory. Finally, the applicability of the proposed method is verified by taking the reinforced concrete hinged slab bridge as a numerical example.

Highlights • The concept of time-dependent failure credibility (TDFC) is proposed. • The double-loop optimization based (DLOB) method is developed to estimate TDFC. • The single-loop optimization based method (SLOB) is developed to estimate TDFC. • Results demonstrate that SLOB is more efficient than DLOB method. Abstract To measure the safety degree of the time-dependent structure under the fuzzy uncertainty, a time-dependent failure credibility (TDFC) is defined. The credibility measure has the excellent property of self-duality, thus the sum of the TDFC and the credibility of the time-dependent safety event, i.e., the complementary of the time-dependent failure event, equals to one. To solve the TDFC, two methods, including direct double-loop optimization (DLOB) method and transformed single-loop optimization (SLOB) method, are proposed. In two methods, the TDFC is firstly expressed as a bi-level problem by its definition, where the outer is the one-dimensional rooting and the inner is the min-min or max-min optimization for different cases. The outer one-dimensional rooting is solved by the dichotomy in DLOB method and SLOB method. The inner min-min or max-min problem is solved by different strategies in DLOB method and SLOB method. In the DLOB method, the min-min or max-min problem is directly solved by double-loop nested optimization strategy. In the SLOB method for improving the computational efficiency of the DLOB method, the min-min and max-min problems are respectively transformed to the single-loop optimizations by corresponding simple transition and trajectory-following algorithm, and in the trajectory-following algorithm, the complex-step method is employed for estimating the first order derivative for high precision. Several examples are used to demonstrate the rationality of the TDFC and the significance of two solutions. The results of the examples show that the proposed TDFC can reasonably describe the safety degree, and the SLOB method can drastically reduce the computational cost compared with the DLOB method under the acceptable precision. [ABSTRACT FROM AUTHOR]

In order to measure structural safety with fuzzy uncertainty, a new root finding method of failure credibility is presented by taking advantage of the simple relation on input and output credibility inverse distributions. A brief introduction is given of the fuzzy credibility measure and the credibility inverse distribution at first. As the theoretical basis of root finding method, principles of transformation from failure credibility solving into a root finding process through credibility inverse distribution functions, are fully explained from the standpoint of root screening conditions as well as computing features. After organizing the fuzzy root finding algorithm for both explicit and implicit performance functions, arithmetical differences of fuzzy safety are meanwhile compared to random reliability to reveal the essence of their different type of uncertainty. Finally the applicability and effectiveness of fuzzy root finding method are testified by several engineering cases. [ABSTRACT FROM AUTHOR]

Abstract For lack of safety measure under the widely existing fuzzy inputs and fuzzy state in the aircraft icing process, a safety analysis model is established to quantify the safety degree of aircraft icing under the fuzzy inputs and fuzzy state. Three-fold contribution is included in the established model. Firstly, by analyzing the recognition capability and properties, the failure credibility model possessing wide recognition capability and self-duality property is selected to measure the safety degree under the fuzzy inputs and binary state. Secondly, by a strictly mathematical derivation, an easily extended definition of the failure probability under random inputs and fuzzy state is proved, on which the failure credibility model under the fuzzy inputs and fuzzy state is established finally. After a numerical example is used to illustrate the feasibility of the established failure credibility model under the fuzzy inputs and fuzzy state in view of different perspectives, a real application of the established model is completed for the safety analysis of an aircraft icing problem. [ABSTRACT FROM AUTHOR]

Failure credibility can quantify safety degree of structure under fuzzy uncertainty, and fuzzy first-order and second-moment (FFOSM) method is an efficiently and accurately analytical method for solving failure credibility. However, classical FFOSM is only suitable for problem with re gularly and identically distributed fu zzy i nput (Re-FuI). To extend FFOSM to problem with ir regularly and diversely distributed fu zzy i nput (Ir-FuI), an eq uivalent re gularization (EqRe) for Ir-FuI is proposed by use of property that failure credibility only depends on fuzzy design point (FDP). By ensuring that credibility distribution and membership functions of equivalent Re-FuI are separately equal to those of Ir-FuI at FDP, the proposed EqRe transforms Ir-FuI into equivalent Re-FuI. Then the extended FFOSM (E-FFOSM) is established to efficiently estimate failure credibility of problem with Ir-FuI, in which equivalent Re-FuI with respect to Ir-FuI is obtained by EqRe at each iterative FDP. The proposed E-FFOSM inherits the high efficiency and high accuracy of classical FFOSM, and avoids the limitation inapplicable to problem with Ir-FuI. Four examples are used to verify the feasibility and accuracy of the E-FFOSM. [ABSTRACT FROM AUTHOR]

How to accurately and efficiently estimate the failure credibility is widely concerned in safety analysis under fuzzy uncertainty. To solve this issue, an iterative method combining the adaptive Kriging with fuzzy simulation (AK-FS) was put forward by Ling et al. But for the problem with multidimensional fuzzy inputs and/or high safety degree, AK-FS is inefficient. In order to improve the computational efficiency of AK-FS, a novel hypercube-based fuzzy simulation (HBFS) combined with adaptive Kriging (shorten as AK-HBFS) is proposed in this paper. The first key technique of the proposed method is to develop the HBFS for constructing the candidate sampling pool. In the HBFS, the active point is defined at first. Then, the estimation of the failure credibility is transformed into seeking for the active point. By using an adaptive hypercube shrinkage strategy, the active point can be quickly searched from the samples whose joint membership functions (JMFs) are smaller than the JMF of the active point. Thus, the size of the candidate sampling pool in the AK-HBFS is smaller than that of the original AK-FS, and it helps save the training time of the Kriging model in the AK-HBFS compared to the AK-FS. The second key technique of the proposed method is that the process for finding the active point is efficiently realized by the U-learning function based AK model, rather than calling the true performance function directly. The merits of the proposed AK-HBFS are demonstrated by four test cases. [ABSTRACT FROM AUTHOR]

Since failure credibility has some perfect properties such as the self-duality, it is developed to be a popular failure measure indicator under the fuzzy uncertainty. In order to globally measure the contribution of fuzzy input to the failure credibility, the failure-credibility-based importance measure (FC-IM) is proposed in this paper. It is defined by the fuzzy average difference between unconditional failure credibility and conditional one on fixing the fuzzy input over its entire fuzzy input distribution range, and the mathematical properties of the proposed FC-IM are also discussed. Additionally, an efficient method by combining adaptive Kriging and double-loop fuzzy simulation is developed to estimate FC-IM. Finally, three examples are presented to illustrate the rationality of the proposed FC-IM and the feasibility of the proposed estimation method for FC-IM. [ABSTRACT FROM AUTHOR]