Time-Dependent System Reliability Analysis for Bivariate Responses

Introduction:Reliability analysis is a good tool to deal with the uncertainty and has been widely used in the engineering system. The first order reliability method (FORM) is generally used to calculate the reliability index but FORM is time-consuming and requires derivative computing.Methods:Artificial Bee Colony (ABC) algorithm is a very simple, robust and population-based stochastic optimization algorithm. In this study, an ABC-based reliability analysis was proposed to calculate the reliability index of engineering system through combining Artificial Bee Colony (ABC) algorithm with FORM. FORM was adopted to calculate the reliability index and design point. ABC is used to solve the constrained optimization about FORM. The procedure of ABC-based reliability analysis was presented in detail.Results and Conclusion:The proposed method was verified by two classic examples and then applied to geotechnical engineering. The results show that the ABC algorithm can effectively solve the global optimization problem in FORM. Results demonstrate that ABC-based reliability analysis is a good approach to obtain the reliability index and design point with a good accuracy so that it can be applied to analyze the reliability of a complex engineering system.

This paper presents a general mechanical system reliability modeling and analysis method, which consider the correlations of failure modes. Firstly, the performance functions which mathematically represent the failure modes are linearized based on FORM (First Order Reliability Methods), and the correlation coefficient matrix of those functions is calculated. Then, the PFTA (Probabilistic Fault Tree Analysis) method is adopted to establish the system reliability model which can take the correlations into account. Finally, effective algorithms such as G-FOMN (General First Order Multinormal Method) are used to quantify the system reliability, and the sensitivities of the system reliability index with respect to each failure mode, as well as the mean value and standard deviation of each random variable are also calculated. An engineering example demonstrates the applicability of the proposed method.

This study proposes a system reliability analysis of rack storage facilities subjected to forklift colliding events. The proposed system reliability analysis consists of two steps: the first step is to identify dominant failure modes that most contribute to the failure of the whole rack facilities, and the second step is to evaluate the system failure probability. In the first step, dominant failure modes are identified by using a simulation-based selective searching technique where the contribution of a failure mode to the system failure is roughly estimated based on the distance from the origin in the space of the random variables. In the second step, the multi-scale system reliability method is used to compute the system reliability where the first-order reliability method (FORM) is initially used to evaluate the component failure probability (failure probability of one member), and then the probabilities of the identified failure modes and their statistical dependence are evaluated, which is called as the lower-scale reliability analysis. Since the system failure probability is comprised of the probabilities of the failure modes, a higher-scale reliability analysis is performed again based on the results of the lower-scale analyses, and the system failure probability is finally evaluated. The illustrative example demonstrates the results of the system reliability analysis of the rack storage facilities subjected to forklift impact loadings. The numerical efficiency and accuracy of the approach are compared with the Monte Carlo simulations. The results show that the proposed two-step approach is able to provide accurate reliability assessment as well as significant saving of computational time. The results of the identified failure modes additionally let us know the most-critical members and their failure sequence under the complicated configuration of the member connections.

The optimal design of multidisciplinary systems under uncertainty involves three types of iterative analyses- multidisciplinary analyses, reliability analyses, and optimization. If these processes are carried out in a nested manner in reliability based multidisciplinary design optimization (RBMDO), the computational expense can be enormous. Computationally efficient simultaneous analysis and design (SAND) or All-AtOnce methods have been developed earlier for deterministic MDO. Single loop RBDO methods have also been developed to integrate reliability analysis and optimization in a single loop. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints. The formulation allows for solution with a single design optimization loop by using the KKT optimality conditions of the first-order reliability method (FORM) as constraints in the design optimization along with disciplinary consistency constraints. Alternative solution processes which remove the FORM MPP’s from the decision space and many constraints from the problem are developed. The alternative solution methods are more efficient and more robust than using nonlinear programming to solve the single loop formulation. A numerical example is solved with the two proposed solution algorithms. I. Introduction In the design of multidisciplinary systems it is often necessary to make decisions under uncertainty. This requires that multidisciplinary systems modeling be integrated with uncertainty analysis and optimization. A common example of such an engineering design application would be the design of an aircraft wing under uncertain aerodynamic loading conditions, with multiple structural components with uncertain capacities, such that the entire structural system achieves a prescribed level of reliability. The problem has coupled physics, in that the pressure distribution is dependent upon the deflection, and the deflection is dependent on the pressure distribution. Such designs have been performed using multidisciplinary optimization (MDO). Other examples where MDO may be applied include electronics packaging, in which the disciplines of heat transfer and circuit analysis are coupled, the design of new types of aircraft and integrating them into an existing fleet of aircraft, and even the management of interconnected homeland infrastructures. While MDO has matured with the development of different types of algorithms for various types of problems and design organizations, there has been little work to incorporate reliability analysis into MDO. However, system and component reliabilities are often a major concern in the design of multidisciplinary systems. MDO can be computationally expensive due to the need to achieve consistency across multiple disciplinary analysis codes; each disciplinary analysis code may be computationally expensive to run. Further, the optimal design of multidisciplinary systems under uncertainty involves three types of iterative procedures- multidisciplinary analyses, reliability analyses, and optimization. The integration of these three procedures is referred to in this paper as reliability based multidisciplinary design optimization (RBMDO). If these processes are carried out simultaneously, the computational expense can be enormous. Computational savings can often be achieved with simultaneous analysis and design (SAND) or All-At-Once methods, which allow for the solution of the multidisciplinary analyses and the optimization in one single loop. Single loop RBDO methods allow for the integration of reliability analysis and optimization in a single loop. Applying SAND and single-loop RBDO together may allow for large computational savings when applied to problems in RBMDO. This paper develops a novel formulation of RBMDO problems with both system and component level reliability constraints based on the SAND method for MDO and by use of the KKT conditions of the first order reliability method (FORM). The resulting formulation may be solved as a standard nonlinear programming problem. While the single loop formulation presented in this paper is a solution of the RBMDO problem, robustness and efficiency are improved through two improved algorithms which can remove the MPP’s from the decision space, thereby significantly reducing the optimizer’s workload.

Reliability-based design optimization is much more computationally expensive than deterministic design optimization. To alleviate the computational demand, the First Order Reliability Method (FORM) is usually used in reliability-based design. Since FORM requires a nonlinear transformation from non-normal random variables to normal random variables, the nonlinearity of a constraint function may increase. As a result, the transformation may lead to a large error in reliability calculation. In order to improve accuracy, a new reliability-based design method with Saddlepoint Approximation is proposed in this work. The strategy of sequential optimization and reliability assessment is employed where the reliability analysis is decoupled from deterministic optimization. The accurate First Order Saddlepoint method is used for reliability analysis in the original random space without any transformation, and the chance of increasing nonlinearity of a constraint function is therefore eliminated. The overall reliability-based design is conducted in a sequence of cycles of deterministic optimization and reliability analysis. In each cycle, the percentile value of the constraint function corresponding to the required reliability is calculated with the Saddlepoint Approximation at the optimal point of the deterministic optimization. Then the reliability analysis results are used to formulate a new deterministic optimization model for the next cycle. The solution process converges within a few cycles. The demonstrative examples show that the proposed method is more accurate and efficient than the reliability-based design with FORM.

Efficient system reliability analysis illustrated for a retaining wall and a soil slope

Fuzzy reliability analysis can be implemented using two discrete optimization maps in the processes of reliability and fuzzy analysis. Actually, the efficiency and robustness of the iterative reliability methods are two main factors in the fuzzy-based reliability analysis due to the huge computational burdens and unstable results. In the structural fuzzy reliability analysis, the first-order reliability method (FORM) using discrete nonlinear map can provide a C membership function. In this paper, a discrete nonlinear conjugate map is proposed using a relaxed-finite step size method for fuzzy structural reliability analysis, namely Fuzzy conjugate relaxed-finite step size method fuzzy CRS. A discrete conjugate map is stabilized using two adaptive factors to compute the relaxed factor and step size in FORM. The framework of the proposed fuzzy structural reliability method is established using two linked iterative discrete maps as an outer loop, which constructs the membership function of the response using alpha level set optimization based on genetic operator, and the inner loop, implemented for reliability analysis using proposed conjugate relaxed-finite step size method. The fuzzy CRS and fuzzy HL-RF methods are compared to evaluate the membership functions of five structural problems with highly nonlinear limit state functions. Results demonstrated that the fuzzy CRS method is computationally more efficient and is strongly more robust than the HL-RF for fuzzy-based reliability analysis of the nonlinear structural reliability problems.

Time-dependent structural systems (TDSSs) in engineering involve high dimensionality, nonlinearity, and complex uncertainties, complicating the reliability analysis compared to time-independent assessments. To address these challenges, this paper proposes an extremum-based back propagation neural network (BPNN) method for TDSS reliability analysis. The method adopts a double-loop structure. Specifically, the inner loop finds the minimum of the time-dependent performance function for a given realization of the random variables. This transformation converts the time-dependent problem into an equivalent time-invariant one. Then, the outer loop constructs a BPNN surrogate model to map the relationship between the random variables and the performance function minima. To improve computational efficiency, an adaptive sample selection strategy is integrated into the training process. This technique selects samples near the failure boundary to iteratively update the BPNN, ensuring high accuracy with a small training set. Once the stopping criterion is satisfied, the failure probability is estimated using Monte Carlo simulation (MCS). The trained BPNN model is used to rapidly predict the extremum for the large-scale sample pool. The proposed method is verified through three practical engineering cases: a four-bar mechanism, an aero-engine turbine disc, and a cantilever tube. Results show that the method remains accurate and efficient. The successful applications confirm the rationality and engineering applicability of the proposed model.

In the past, residual factor R in strain-softening soil slopes has been included, either directly or indirectly, as a deterministic variable in both deterministic and probabilistic studies. This paper discusses the uncertainties associated with R and outlines a systematic approach for the reliability analysis of a natural slope in which shear strength parameters and pore pressure ratio are random variables, each assumed with a lognormal probability distribution. For the residual factor R, seven probability distribution options under the generalized beta-distribution system are considered. Slope reliability is computed based on the first order reliability method (FORM) and validated against Monte-Carlo simulation (MCS). Results obtained from two illustrative examples indicate that the probability of failure, with R as one of six random variables, can be orders of magnitude higher than that based on five random variables with R considered as a deterministic parameter. The magnitude of influence of R as a random variable is, however, highly dependent on its probability distribution, the left-skewed triangular distribution having the most significant influence in both the examples. Results of sensitivity analyses reveal that, for almost all of its assumed probability distributions, R is the most dominant among the six random variables. Effects of variation of some of the statistical and correlation properties of the other random variables, viz. the shear strength parameters and the pore pressure ratio, on the results of reliability analyses are also studied.

In order to evaluate the failure probability of a complicated structure, the structural responses usually need to be estimated by some numerical analysis methods such as finite element method (FEM). The response surface method (RSM) can be used to reduce the computational effort required for reliability analysis when the performance functions are implicit. However, the conventional RSM is time-consuming or cumbersome if the number of random variables is large. This paper proposes a Legendre orthogonal neural network (LONN)-based RSM to estimate the structural reliability. In this method, the relationship between the random variables and structural responses is established by a LONN model. Then the LONN model is connected to a reliability analysis method, i.e. first-order reliability methods (FORM) to calculate the failure probability of the structure. Numerical examples show that the proposed approach is applicable to structural reliability analysis, as well as the structure with implicit performance functions.

In this work, a novel framework is proposed for the risk based design optimization of engineering systems by minimizing the demand on the system components’ accuracy (which directly relates to their cost). The fundamental development of this work is an analytical upper bound for calculating the probability of failure. This is in contrast with First Order Reliability Method (FORM), where a lower bound is used in calculating the probability of failure. FORM is one of the most popular methods for reliability analysis of engineering systems. In this paper, we show that FORM results in an optimistic measure of risk, hence potentially catastrophic in engineering design. A more accurate measure of failure is proposed by utilizing an analytical upper bound for the distribution of reliability index (the length of the most probable point vector to origin). This distribution is a function of the eigenvalues of the linearized limit state function in the normal space which results in a better understanding of failure phenomenon. The proposed formulation is computationally efficient and straightforward to solve, since it only involves finding eigenvalues in each iteration. This algorithm is applicable to any linearizable continuous limit state function with any type of distribution for the design variables. The method is applied to two examples and its accuracy is compared with the Monte Carlo simulation and FORM, demonstrating its effectiveness and value.

This paper presents the system reliability analysis of a three-dimensional steel frame designed per AISC LRFD with respect to the collapse limit state under the dead and live loads.The system reliability is evaluated using the first-order reliability method (FORM)-based adaptive response surface approach.The uncertainties in the material properties, geometric properties of frame members, initial geometric imperfection of the structure, dead load and live load as well as the spatial variability of the live load are accounted for in the reliability analysis.The analysis results suggest that the system reliability of the example frame is similar to that of the planar steel frames designed per AISC LRFD and evaluated in a previous study.It is observed that the spatial variability of the live load leads to a decreased system reliability.Results of the sensitivity analysis indicate that the failure probability of the example frame increases by almost one order of magnitude if the coefficient of variation of the steel yield strength increases from 0.06 to 0.1.Furthermore, the system reliability decreases drastically if the upper bound of the magnitude of initial geometric imperfection is greater than 0.4% of the overall height of the example frame.

A hybrid self-adaptive conjugate first order reliability method for robust structural reliability analysis

System reliability analysis of rock wedge stability considering correlated failure modes using sequential compounding method

abstract: The Monte Carlo simulation (MCS) and First-Order Reliability Method (FORM) provide a reliability analysis in axisymmetric deep tunnels driven in elastoplastic rocks. The Convergence-Confinement method (CV-CF) and Mohr-Coulomb (M-C) criterion are used to model the mechanical interaction between the shotcrete lining and ground through deterministic parameters and random variables. Numerical models synchronize tunnel analytical models and reliability methods, whereas the limit state functions control the failure probability in both ground plastic zone and shotcrete lining. The results showed that a low dispersion of random variables affects the plastic zone's reliability analysis in unsupported tunnels. Moreover, the support pressure generates a significant reduction in the plastic zone's failure, whereas the increase of shotcrete thickness results in great reduction of the lining collapse probability.