Probable Failure Point Research Articles

A Moving Kriging Interpolation Response Surface Method for Structural Reliability Analysis

In order to obtain reliable structural design, it is of extreme importance to evaluate the failure probability, safety levels of structure (reliability analysis) and the effect of a change in a variable parameter on structural safety (sensitivity analysis) when uncertainties are considered. With a computationally cheaper approximation of the limit state function, various response surface methods (RSMs) have emerged as a convenient tool to solve this especially for complex problems. However, the traditional RSMs may produce large errors in some conditions especially for those highly non-linear limit state functions. Instead of the traditional least squares approximation, in the present paper, a new RSM is proposed which employs moving Kriging interpolation, based on the axial experimental points selected from the region where the most probable failure point (MPFP) is likely to exist to construct the substitute response surface. The proposed method is illustrated by comparing with the results obtained from first order reliability method (FORM) and other conventional RSMs with reference to specific structural reliability analysis problems. The results show that the proposed method improves the accuracy of the reliability analysis with a reasonable computational cost, and could more quickly approach the exact solution.

Reliability Based Multidisciplinary Optimization of Aeroelastic Systems with Structural and Aerodynamic Uncertainties

A reliability-based multidisciplinary optimization framework is constructed by coupling high-fidelity commercial solvers for aeroelastic analysis and an in-house code developed for reliability analysis. The finite volume-based flow solver Fluent is used to solve inviscid three-dimensional Euler equations, whereas three-dimensional solid models are updated using Catia parametrically. A mesh-based parallel code coupling interface (MPCCI), is used to exchange the pressure and displacement information between Fluent and Abaqus to perform a loosely coupled aeroelastic analysis. The optimization criteria include both deterministic and probabilistic constraints with both structural and aerodynamic uncertainties, such as in yield strength, Mach number, and angle of attack. To evaluate the probability of failure for the probabilistic constraints a first-order reliability analysis method, the Hasofer–Lind iteration method, is implemented in MATLAB to compute the most probable failure point solution. The integrated framework is validated with structural problems and then extended to more realistic wing configurations with aeroelastic criteria. The presented reliability-based multidisciplinary optimization process is proven to be fully automatic, modular, and practical, which could find potential applications in industrial problems.

A new efficient simulation method to approximate the probability of failure and most probable point

In this paper, a new simulation method for approximating the probability of failure and the most probable point of failure is proposed. The method works by generating uniformly distributed samples in a design space for all random variables and applying the probability density value as a weight index at any sample. The result is a simple approximation of the probability of failure without any limitation becoming feasible. The probability of failure is defined as the ratio of the sum of the weight indices in the failure domain over the sum of the indices in the entire domain. High accuracy in estimating small values of the probability of failure as well as the need for few samples are advantages of this method. Moreover, the possibility of estimating the point and the region with the highest failure probability for different types of limit state functions can be considered as another important advantage of the proposed method. The efficiency and robustness of the method are investigated by solving several examples. The results are compared with the results of common reliability methods, and they demonstrate the efficiency and robustness of the proposed method.

Probabilistic assessment of the structural failure of single-lap bonded joints

Single-lap bonded joints are a common typology, frequently used when designing structural systems. Several authors have studied their behaviour in different situations, involving both failure of the adhesive and also of the adherends. So far, the design of such joints has been based on deterministic postulates. In this work, a non-deterministic approach is followed. The reliability of the joint is assessed with several limit state approximation methods. Furthermore, sampling methods and a global reliability approach are considered. The algorithms are applied to the reliability analysis of a generic single-lap bonded joint, where the specific uncertainties related with this typology are taken into account, including mechanical properties and geometrical parameters of adherends and adhesive. All of them are defined as random variables and different cases are studied. Results show that first- and second-order second-moment and advanced mean value methods give an estimation to calculate the probability of failure at early stages of design, when precise values are not required. However, methods based on approximations of limit state surface around the most probable point of failure present a better response in terms of accuracy. Global reliability methods offer a good alternative to limit state approximations, as they present similar accuracy and usually require fewer iterations. Sampling methods often need extensive use of computational resources and are suitable for those cases with a large number of random variables, where computational cost is comparable with that required by other methods.

Doubly weighted moving least squares and its application to structural reliability analysis

In this paper, we proposed a two-stage hybrid reliability analysis framework based on the surrogate model, which combines the first-order reliability method and Monte Carlo simulation with a doubly-weighted moving least squares (DWMLS) method. The first stage consists of constructing a surrogate model based on DWMLS. The weight system of DWMLS considers not only the normal weight factor of moving least squares, but also the distance from the most probable failure point (MPFP), which accounts for reliability problems. An adaptive experimental design scheme is proposed, during which the MPFP is progressively updated. The approximate values and sensitivity information of DWMLS are chosen to determine the number and location of the experimental design points in the next iteration, until a convergence criterion is satisfied. In the second stage, MCS on the surrogate model is then used to calculate the probability of failure. The proposed method is applied to five benchmark examples to validate its accuracy and efficiency. Results show that the proposed surrogate model with DWMLS can estimate the failure probability accurately, while requiring fewer original model simulations.

복합 반응면 기법을 이용한 복합재 적층판의 신뢰성해석

본 논문에서는 복합 반응면 기법을 제안하고 성능을 고찰하였다. 복합 반응면 기법은 MPP의 좌표를 기준으로 하여 근사모델을 반복 계산하는 기법이다. 성능을 검증하기 위해 비선형 함수와 복합재 적층판에 대하여 신뢰성 해석 기법을 적용하여 파괴확률, MPP(Most Probable failure Point), 신뢰도 지수를 계산하고 일반적인 반응면 기법의 결과와 비교하였다. 파괴확률은 비선형 한계상태식을 가정하고 임의의 파괴 기준을 정의하여 계산하였다. 제안한 복합 반응면 기법을 이용하여 파괴확률을 계산한 결과 일반적인 반응면 기법보다 향상된 성능을 나타내었다. In this paper, the hybrid response surface method(HRSM) is proposed and examined. Hybrid response surface method calculate a approximate model repeatedly based on MPP coordinates. To verify the performance, probability of failure, MPP(Most Probable failure Point) and reliability index are calculated for nonlinear function and composite laminated plate by using reliability analysis method and compared with results by using typical response surface method(RSM). Probability of failure is calculated under the assumption of the nonlinear limit state equation and given failure criterion. The results of proposed method shows performance improvement in estimating the probability of failure.

An efficient response surface method using moving least squares approximation for structural reliability analysis

The response surface method (RSM) is widely adopted for structural reliability analysis because of its numerical efficiency. However, the RSM is time consuming for large-scale applications and sometimes shows large errors in the calculation of the sensitivity of the reliability index with respect to random variables. In order to overcome these problems, this study proposes an efficient RSM applying a moving least squares (MLS) approximation instead of the traditional least squares approximation generally used in the RSM. The MLS approximation gives higher weight to the experimental points closer to the most probable failure point (MPFP), which allows the response surface function (RSF) to be closer to the limit state function at the MPFP. In the proposed method, a linear RSF is constructed at first and a quadratic RSF is formed using the axial experimental points selected from the reduced region where the MPFP is likely to exist. The RSF is updated successively by adding one new experimental point to the previous set of experimental points. Numerical examples are presented to demonstrate the improved accuracy and computational efficiency of the proposed method compared to the conventional RSM.

Dynamic adaptability with .NET service components.

In self-adaptive systems components are dynamically modified according to the execution environment requirement, where each component is a probable point of failure. Existing approaches to make such systems more vigorous and safe are both brittle and time intensive. A framework for dynamic adaptation has been designed to automate the component integration process at runtime by accessing the equivalent component from a repository of components. The .NET technology allows developers to adapt run-time components by specifying component behavior using pre- and post-assertions on the component's services. Components can be compared for behavioral equivalence by comparing the assertions. These assertions will help us to compute the utility value for each component in the repository and the component with the highest value is picked for replacement. In this paper, we describe the mechanism for component adaptation using .NET services, by considering a system in a dynamic context with proxy switcher and network switcher components.

Reliability estimation using univariate dimension reduction and extended generalised lambda distribution

This paper presents an analytical approach for structural reliability analysis without requiring the calculation of most probable point of failure. Initially, the primary statistical moments of a multi-dimensional performance function are estimated using the Univariate Dimension-Reduction (UDR) methodology based on additive decomposition of the limit state function. Through moment matching, the UDR-based estimated moments are then used to fit the parameters of Extended Generalised Lambda Distribution (EGLD), and finally the probability of failure is calculated. To evaluate the accuracy and efficiency of the UDR + EGLD approach in comparison to the traditional First-Order Reliability Method (FORM) and direct Monte Carlo Simulation (MCS), five example problems involving nonlinear limit state functions are examined. The results show that UDR + EGLD offers nearly the same level of accuracy as MCS with superior efficiency to FORM. However, UDR + EGLD appears to have tail sensitivity, which limits its application to problems with moderate levels of reliability.

Novel Method Based on Sequence Shepard Interpolation for Structual Reliability Analysis

A novel method based on sequence Shepard interpolation is presented for reliability analysis of structure with implicit limit state equation.Using experiment points of uniform distribution generated by hammersly point set,the response values of samples from importance sampling probability density function are predicted by Shepard interpolation method,and the failure probability of structure is estimated simultaneously.In the process of importance sampling,the most probable failure point searched in current iteration is used as the center of the experiment point set and the density center of the importance sampling probability density function in the next iteration.Comparing with Monte Carlo method,three numerical examples show that the sequence Shepard interpolation is more efficient.The reliability analysis of flap mechanism verifies the engineering applicability of this method.

순차적 샘플링과 크리깅 메타모델을 이용한 신뢰도 기반 최적설계

RBDO approach based on a sampling method with the Kriging metamodel and Constraint Boundary Sampling (CBS), which is sequential sampling method to generate metamodels is proposed. The major advantage of the proposed RBDO approach is that it does not require Most Probable failure Point (MPP) which is essential for First-Order Reliability Method (FORM)-based RBDO approach. The Monte Carlo Sampling (MCS), most well-known method of the sampling methods for the reliability analysis is used to assess the reliability of constraints. In addition, a Cumulative Distribution Function (CDF) of the constraints is approximated using Moving Least Square (MLS) method from empirical distribution function. It is possible to acquire a probability of failure and its analytic sensitivities by using an approximate function of the CDF for the constraints. Moreover, a concept of inactive design is adapted to improve a numerical efficiency of the proposed approach. Computational accuracy and efficiency of the proposed RBDO approach are demonstrated by numerical and engineering problems.

Structural reliability assessment based on probability and convex set mixed model

This paper investigates the reliability assessment of structures exhibiting both stochastic and bounded uncertainties by using a probability and convex set mixed model. The safety measure of a structure is quantified by a reliability index defined by a nested minimization problem. An iterative procedure is developed for seeking the worst-case point and the most probable failure point in the standard uncertainty space. Numerical examples are given to demonstrate the applicability of the probability and convex set mixed model representation in the structural reliability assessment, as well as to illustrate the validity and effectiveness of the proposed numerical method.

Structural Failure Probability Estimation Using HDMR and FFT

This paper presents a new and alternative method based on High Dimensional Model Representation (HDMR) and fast Fourier transform (FFT) to estimate the structural failure probability of structural systems subject to random loads, material properties and geometry. The proposed methodology is based on the limit state/performance function approximation and the convolution theorem to estimate the structural failure probability. The limit-state function is obtained by linear approximation of the first-order HDMR component functions at the most probable failure point, and the convolution integral is solved efficiently using the FFT technique. The proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The accuracy and efficiency of the proposed method is demonstrated through numerical examples involving implicit performance functions.

Projection Method for Validating Reliability Analysis of Soil Slopes

The mean-value first-order second-moment (MFOSM) method is the main method for estimating the reliability of slopes in practice. It is well recognized that the method occasionally underestimates the probability of failure for inhomogeneous soil slopes, but there is as of yet no commonly accepted procedure to identify this problem. This study presents a solution by projecting MFOSM results to obtain an equivalent most probable failure point in the material space as defined by Hasofer and Lind. A stability analysis using this material point then provides a check for the validity of the MFOSM solution. If the resulting critical factor of safety is one, the MFOSM results are valid; whereas, if a factor of safety of less than one is obtained, the reliability index is overestimated and the failure probability underestimated.

A Single-Loop Approach for System Reliability-Based Design Optimization

Abstract An efficient approach for series system reliability-based design optimization (RBDO) is presented. The key idea is to apportion optimally the system reliability among the failure modes by considering the target values of the failure probabilities of the modes as design variables. Critical failure modes that contribute the most to the overall system reliability are identified. This paper proposes a computationally efficient, system RBDO approach using a single-loop method where the searches for the optimum design and for the most probable failure points proceed simultaneously. Specifically, in each iteration, the optimizer uses approximated most probable failure points from the previous iteration to search for the optimum. A second-order Ditlevsen upper bound is used for estimating the system failure probability considering the joint failure probabilities of failure modes. Also, an easy to implement active strategy set is employed to improve algorithmic stability. The approach is demonstrated on two design examples involving a beam and an internal combustion engine.

Time-domain seismic reliability of nonlinear structures

A novel reliability analysis technique is presented to estimate the reliability of real structural systems. Its unique feature is that the dynamic loadings can be applied in time domain. It is a nonlinear stochastic finite element logarithm combined with the response surface method (RSM). It generates the response surface around the most probable failure point and incorporates information of the distribution of the random variables in the RSM formulation. It is verified using the Monte Carlo simulation technique, and is found to be very efficient and accurate. Most sources of nonlinearlity and uncertainty can be explicitly incorporated in the formulation. The flexibility of connections, represented by moment-relative rotation(M-θ) curves, is addressed. After the Northridge earthquake of 1994, several improved steel connections were proposed. Structural Sesimic Design Associates (SSDA) tested several full-scale proprietory slotted web beam-column connections. The authors suggested(M-θ) curves for this connection using actual test data. Behaviours of steel frames, assuming the connections are fully restrained, partially restrained, consisting of pre and post-Northridge connections are evaluated and compared. Desirable features of the post-Northridge connections observed during testing are analytically confirmed. Laterally weak steel frame is then strengthened with concrete shear walls. Capabilities and the advanced nature of the method are demonstrated with the help of realistic examples.

New Decoupled Framework For Reliability-Based Design Optimization

Traditionally, reliability-based design optimization (RBDO) has been formulated as a nested optimization problem. The inner loop, generally, involves the solution to optimization problems for computing the probabilities of failure of the critical failure modes, and the outer loop performs optimization by varying the decision variables. Such formulations are by nature computationally intensive, requiring numerous function and constraint evaluations. To alleviate this problem, researchers have developed iterative decoupled RBDO approaches. These methods perform deterministic optimization and reliability assessment in a sequential manner until a consistent reliability-based design is obtained. The sequential methods are attractive because a consistent reliable design can be obtained at considerably lower computational cost. However, the designs obtained by using these decoupled approaches cannot guarantee production of the true solution. A new decoupled method for RBDO is developed in this investigation. Postoptimal sensitivities of the most probable point (MPP) of failure with respect to the decision variables are introduced to update the MPPs during the deterministic optimization phase of the proposed approach. A damped Broyden‐Fletcher‐Goldfarb‐Shanno method is used to significantly reduce the cost of obtaining these sensitivities. It is the use of postoptimal sensitivities that differentiates this new decoupled RBDO approach from previous efforts.

A sequential approximate programming strategy for reliability-based structural optimization

Although reliability-based structural optimization (RBSO) is recognized as a rational structural design philosophy that is more advantageous to deterministic optimization, most common RBSO is based on straightforward two-level approach connecting algorithms of reliability calculation and that of design optimization. This is achieved usually with an outer loop for optimization of design variables and an inner loop for reliability analysis. A number of algorithms have been proposed to reduce the computational cost of such optimizations, such as performance measure approach, semi-infinite programming, and mono-level approach. Herein the sequential approximate programming approach, which is well known in structural optimization, is extended as an efficient methodology to solve RBSO problems. In this approach, the optimum design is obtained by solving a sequence of sub-programming problems that usually consist of an approximate objective function subjected to a set of approximate constraint functions. In each sub-programming, rather than direct Taylor expansion of reliability constraints, a new formulation is introduced for approximate reliability constraints at the current design point and its linearization. The approximate reliability index and its sensitivity are obtained from a recurrence formula based on the optimality conditions for the most probable failure point (MPP). It is shown that the approximate MPP, a key component of RBSO problems, is concurrently improved during each sub-programming solution step. Through analytical models and comparative studies over complex examples, it is illustrated that our approach is efficient and that a linearized reliability index is a good approximation of the accurate reliability index. These unique features and the concurrent convergence of design optimization and reliability calculation are demonstrated with several numerical examples.

ReliabIlity analysis of containment building subjected to earthquake load using response surface method

The seismic safety of reinforced concrete containment building can be evaluated by probabilistic analysis considering randomness of earthquake, which is more rational than deterministic analysis. In the safety assessment of earthquake-resistant structures by the deterministic theory, it is not easy to consider the effects of random variables but the reliability theory and random vibration theory are useful to assess the seismic safety with considering random effects. The reliability assessment of reinforced concrete containment building subjected to earthquake load includes the structural analysis considering random variables such as load, resistance and analysis method, the definition of limit states and the reliability analysis. The reliability analysis procedure requires much time and labor and also needs to get the high confidence in results. In this study, random vibration analysis of containment building is performed with random variables as earthquake load, concrete compressive strength, modal damping ratio. The seismic responses of critical elements of structure are approximated at the most probable failure point by the response surface method. The response surface method helps to figure out the quantitative characteristics of structural response variability. And the limit state is defined as the failure surface of concrete under multi-axial stress, finally the limit state probability of failure can be obtained simply by first-order second moment method. The reliability analysis for the multiaxial strength limit state and the uniaxial strength limit state is performed and the results are compared with each other. This study concludes that the multiaxial failure criterion is a likely limit state to predict concrete failure strength under combined state of stresses and the reliability analysis results are compatible with the fact that the maximum compressive strength of concrete under biaxial compression state increases.

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.