Nonlinear Limit State Function Research Articles

A new method for parameter sensitivity estimation in structural reliability analysis

For the parameter sensitivity estimation with implicit limit state functions in the time-invariant reliability analysis, the common Monte Carlo simulation based approach involves multiple trials for each parameter being varied, which will increase associated computational cost and the cost may become inevitably high especially when many random variables are involved. Another effective approach for this problem is featured as constructing the equivalent limit state function (usually called response surface) and performing the estimation in FORM/SORM. However, as the equivalent limit state function is polynomial in the traditional response surface method, it is not a good approximation especially for some highly non-linear limit state functions. To solve the above two problems, a new method, support vector regression based response surface method, is therefore presented in this paper. The support vector regression algorithm is employed to construct the equivalent limit state function and FORM/SORM is used in the parameter sensitivity estimation, and then two illustrative examples are given. It is shown that the computational cost of the sensitivity estimation can be greatly reduced and the accuracy can be retained, and results of the sensitivity estimation obtained by the proposed method are in satisfactory agreement with those computed by the conventional Monte Carlo methods.

Artificial Bee Colony Algorithm for Reliability Analysis of Engineering Structures

Artificial bee colony algorithm is a noval optimization method which is inspired by bee colony foraging behavior. Its use in the structure reliability field presents not only the advantage of its facility of implementation, but also the capability to obtain the design point and the failure probability with good accuracy. And by this method, the reliability index of nonlinear and complex limit state function which iteration scheme may fail to converge could be obtained with efficiency. It is demonstrated by four examples that the present method is reliable and accurate in reliability analysis of engineering structures.

A Second-Order Reliability Method With First-Order Efficiency

The first-order reliability method (FORM) is efficient but may not be accurate for nonlinear limit-state functions. The second-order reliability method (SORM) is more accurate but less efficient. To maintain both high accuracy and efficiency, we propose a new second-order reliability analysis method with first-order efficiency. The method first performs the FORM to identify the most probable point (MPP). Then, the associated limit-state function is decomposed into additive univariate functions at the MPP. Each univariate function is further approximated by a quadratic function. The cumulant generating function of the approximated limit-state function is then available so that saddlepoint approximation can be easily applied in computing the probability of failure. The accuracy of the new method is comparable to that of the SORM, and its efficiency is in the same order of magnitude as the FORM.

On multinormal integrals by Importance Sampling for parallel system reliability

The ability to compute multinormal integrals to any required accuracy is a key issue for an efficient computation of failure probabilities, particular important in context with system reliability analysis. Hence in this paper, an accurate Importance Sampling procedure to compute multinormal integrals in high dimensions is presented. The novel method allows to sample exclusively in the failure domain which substantially increases the efficiency of the Importance Sampling procedure. The proposed approach is extended for slightly non-linear limit state functions which typically result from non-Gaussian distributed input variables and linear limit state functions of the response in linear structural analysis. The suggested procedure is easy to implement, accurate and convenient for practical applications.

Conditional probability Markov chain simulation based reliability analysis method for nonnormal variables

Based on fast Markov chain simulation for generating the samples distributed in failure region and saddlepoint approximation (SA) technique, an efficient reliability analysis method is presented to evaluate the small failure probability of non-linear limit state function (LSF) with non-normal variables. In the presented method, the failure probability of the non-linear LSF is transformed into a product of the failure probability of the introduced linear LSF and a feature ratio factor. The introduced linear LSF which approximately has the same maximum likelihood points as the non-linear LSF is constructed and its failure probability can be calculated by SA technique. The feature ratio factor, which can be evaluated on the basis of multiplicative rule of probability, exhibits the relation between the failure probability of the non-linear LSF and that of the linear LSF, and it can be fast computed by utilizing the Markov chain algorithm to directly simulate the samples distributed in the failure regions of the non-linear LSF and those of the linear LSF. Moreover, the expectation and variance of the failure probability estimate are derived. The results of several examples demonstrate that the presented method has wide applicability, can be easily implemented, and possesses high precision and high efficiency.

System reliability analysis with saddlepoint approximation

System reliability is usually estimated through component reliability, which is commonly computed by the First Order Reliability Method (FORM). The FORM is computationally efficient, but may not be accurate for nonlinear limit-state functions. An alternative system reliability analysis method is proposed based on saddlepoint approximation. Unlike the FORM that linearizes limit-state functions in a transformed random space, the proposed method linearizes the limited-state functions without any transformation. After the linearization, the joint probability density of limit-state functions is estimated by the multivariate saddlepoint approximation. Without the nonnormal-to-normal transformation, the present method is more accurate than the FORM when the transformation increases the nonlinearity of limit-state functions. As demonstrated in the two examples, the new method is also as efficient as the FORM.

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.

A new look at the response surface method for reliability analysis using chaos theory

To overcome excessive computation errors and convergence failures encountered in an iterative calculation of the reliability index using the response surface method (RSM) for some nonlinear limit state functions, this study investigates an essential factor based on chaotic dynamics theory. The bifurcation diagrams of the reliability index are presented for some typical nonlinear limit state functions, and the computation results from the mapping functions due to the RSM iterations show the complicated dynamic phenomena such as the periodic oscillation, as well as bifurcation and chaos. From the numerical examples, it is concluded that the parameter of selection range f plays an important role in the convergence of the RSM iteration, and an improved RSM iterative algorithm is proposed with regard to the incorporation of the iterative sequential function of selection range f. The proposed method is shown to be efficient and to yield accurate results.

An Indicator Response Surface Method for Simulation-Based Reliability Analysis

An accurate and efficient Monte Carlo simulation method is presented for limit-state-based reliability analysis at both component and system levels, using a response surface approximation of the failure indicator function. The cross-validated moving least squares method is used to construct the response surface of the indicator function, based on an optimum symmetric Latin hypercube sampling technique. The proposed method can handle problems with complicated limit state(s). Also, it can easily handle implicit, highly nonlinear limit-state functions, with variables of any statistical distributions and correlations. The method appears to be particularly efficient for multiple limit state and multiple design point problems. Three structural reliability examples are used to highlight its superior accuracy and efficiency over traditional reliability methods.

Safety of a deep water marine riser under random loads

The safety of a deep water marine riser subjected to high axial and bending stresses was addressed with respect to potential fatigue failure. A three-dimensional non-linear dynamic analysis of a marine riser was carried out for response calculations. The response histories so obtained were employed for the study of the fatigue reliability analysis of the riser under random wave and random wave plus top vessel motion. A fatigue damage estimation of the riser was made using Miner–Palmgren's rule (S–N curve approach). A non-linear limit state function based on the above approach was derived in terms of random variables. The advanced first-order, second-moment method (Afosm) and Monte Carlo simulation technique were employed for reliability estimation. The design point important for probabilistic design is located on the failure surface. The influence of various random variables on overall probability of failure was studied through sensitivity analysis. The effect on riser fatigue reliability of uncertainties in various random variables is highlighted.

Reliability of articulated tower joint against random base shear

An Articulated tower is one of the compliant offshore structures connected to the sea-bed through a universal joint which is the most vulnerable location of the tower that sustains the randomly fluctuating shear stresses. The time history response of the bottom hinge shear is obtained and presented in the spectral form. The fatigue and fracture reliability assessment of the tower joint against randomly varying shear stresses have been carried out. Non-linear limit state functions are derived in terms of important random variables using S-N curve and fracture mechanics approaches. Advanced First Order Reliability Method is used for reliability assessment. Sensitivity analysis shows the influence of various variables on the hinge safety. Fatigue life estimation has been made using probabilistic approach.

Determination of point of maximum likelihood in failure domain using genetic algorithms

The point of maximum likelihood in a failure domain yields the highest value of the probability density function in the failure domain. The maximum-likelihood point thus represents the worst combination of random variables that contribute in the failure event. In this work Genetic Algorithms (GAs) with an adaptive penalty scheme have been proposed as a tool for the determination of the maximum likelihood point. The utilization of only numerical values in the GAs operation makes the algorithms applicable to cases of non-linear and implicit single and multiple limit state function(s). The algorithmic simplicity readily extends its application to higher dimensional problems. When combined with Monte Carlo Simulation, the proposed methodology will reduce the computational complexity and at the same time will enhance the possibility in rare-event analysis under limited computational resources. Since, there is no approximation done in the procedure, the solution obtained is considered accurate. Consequently, GAs can be used as a tool for increasing the computational efficiency in the element and system reliability analyses.

The PHI2 method: a way to compute time-variant reliability

Time-variant reliability problems appear in the engineering practice when (a) the material properties of the structure deteriorate in time or (b) random loading modelled as random processes is involved. This paper presents a method called PHI2 which is based on the outcrossing approach and allows to solve such problems using classical time-invariant reliability tools such as FORM/SORM methods. The PHI2 method is first presented. Then it is benchmarked with the well-established ‘asymptotic methods’ [Stochast. Process. Appl. 13 (1988) 195; J. Offshore Mech. Arctic Engng 113 (1991) 241; Probab. Engng Mech. 10 (1995) 53; J. Struct. Engng 25 (1998) 1] on three examples dealing with scalar or vector processes and linear or non-linear limit state functions. The PHI2 method appears more accurate in all cases. As an application example, the method is finally applied on a case representing a mechanical system (a beam) placed in an environment that can have exceptional configuration.

Reliability analysis of nuclear containment without metallic liners against jet aircraft crash

The present study presents a methodology for detailed reliability analysis of nuclear containment without metallic liners against aircraft crash. For this purpose, a nonlinear limit state function has been derived using violation of tolerable crack width as failure criterion. This criterion has been considered as failure criterion because radioactive radiations may come out if size of crack becomes more than the tolerable crack width. The derived limit state uses the response of containment that has been obtained from a detailed dynamic analysis of nuclear containment under an impact of a large size Boeing jet aircraft. Using this response in conjunction with limit state function, the reliabilities and probabilities of failures are obtained at a number of vulnerable locations employing an efficient first-order reliability method (FORM). These values of reliability and probability of failure at various vulnerable locations are then used for the estimation of conditional and annual reliabilities of nuclear containment as a function of its location from the airport. To study the influence of the various random variables on containment reliability the sensitivity analysis has been performed. Some parametric studies have also been included to obtain the results of field and academic interest.

Reliability analysis of automotive body-door subsystem

Abstract This paper proposes an efficient method for the reliability analysis of a vehicle body-door subsystem with respect to one of the important quality issues—the door closing energy. The developed method combines optimization-based and simulation-based approaches and is particularly applicable for problems with highly non-linear and implicit limit state functions. The proposed approach consists of two major parts. In the first part, an optimization-based method is used to search for the most probable point (MPP) on the limit state. This is achieved by using an adaptive response surface constructed through an optimal symmetric Latin hypercube design of experiments. In the second part, a multi-modal adaptive importance sampling method is proposed using the MPP information from the first part as the starting point. It is demonstrated through numerical examples that the proposed method is superior to existing methods in terms of efficiency and accuracy. The proposed method is illustrated for application to the reliability estimation with respect to the door closing energy problem. A generalized framework for reliability estimation is then established for problems with large numbers of random variables and complicated limit states.

Fatigue and fracture reliability of TLP tethers under random loading

Tension leg platform (TLP), a deep water oil exploration offshore compliant system, is vulnerable to considerable fatigue damage over its design life period due to the dynamic excitations caused by the oscillating waves and wind. In the presence of random loading like, that produced by the wind and waves, reliability against fatigue and fracture failure becomes important. The reliability assessment against fatigue failure considers the uncertainties associated with the parameters and procedures employed for the fatigue damage estimation. This fatigue estimate requires a dynamic response analysis under various environmental loadings. In the present study a non-linear dynamic analysis of the platform has been carried out for response calculations. The response histories so obtained have been employed for the study of fatigue reliability analysis of TLP tethers under long crested random sea and associated wind. Fatigue damage estimation of tether joints is made using Palmgren–Miner's rule ( S– N curve approach) and fracture mechanics approaches. The stress ranges employed are described by Rayleigh distribution. Non-linear limit state functions based on above two approaches are derived in terms of random variables. First order reliability method (FORM) and Monte Carlo simulation technique have been employed for reliability estimation. The influence of various random variables on overall probability of failure has been studied through sensitivity analysis.

Estimation of failure probabilities for intersections of non-linear limit states

This paper is concerned with estimating the failure probability associated with the intersection of two or more non-linear limit state functions. Previous work has relied on the use of general non-linear minimization routines to establish the intersection point of the limit states and then to use tangents to bound the region for estimating the failure probability. The present paper outlines two approaches based on successive approximation both to find the intersection point and to estimate the intersection probability. It is shown that the results for selected examples converge to the probabilities estimated using Monte Carlo analysis. Comments about extending the results to using second order or asymptotic probability estimates close the paper.

Higher-order failure probability calculation using nonlinear approximations

Use of approximations for calculating the structural failure probability under uncertainties is a necessary aspect. But efficient use of suitable approximations at different stages of reliability prediction and design makes this valuable tool practical for many large scale structures. The objective of this paper is to address the modern approximation concepts and their utility in structural reliability estimation. Emphasis is placed on building the higher order approximations using intervening variables to closely represent the nonlinear limit state functions. The most probable failure point (MPP) search can be carried either in X-space or U-space for a wide variety of random variable distributions. Several test problems with highly nonlinear performance functions are used to demonstrate the accuracy of the proposed method.