Linear Limit State Function Research Articles

Interval reliability sensitivity analysis using Monte Carlo simulation and mouth brooding fish algorithm (MBF)

In reliability analysis and reliability-based design approach, sensitivity analysis establishes a relation between the variation in random variable parameters and that in reliability. The analysis of sensitivity is also employed to determine key random variables with the greatest contribution to reliability. The sensitivity analysis of a non-linear limit state function is relatively complicated and time-consuming. Therefore, a novel approach is proposed in this study to calculate the reliability sensitivity parameter in terms of an interval. In the proposed algorithm, the non-linear limit state function is initially converted to a linear limit state function by the Monte Carlo simulation method. Then, the interval random variables are introduced to include both epistemic and aleatory uncertainties so as to obtain a realistic result. In the second step of the proposed methodology, the mouth-brooding fish algorithm (MBF) is applied to capture the upper and lower bounds of the reliability sensitivity parameters. Several illustrative examples were then presented, showing the capability of the proposed approach in obtaining results accurately and efficiently. According to the results, the percentage of improvement in the sensitivity analysis of random variables in the proposed method, compared to the classical method, by using the MBF optimization algorithm to the parameters of random variables due to the variation Coefficient is more than 40 percents. This means that the length of the response interval has increased.

First order control variates algorithm for reliability analysis of engineering structures

• Indicator of linearized limit state function served as control variates of original indicator. • First order failure probability regulated by control variates technique using few samples. • 50 benchmark reliability problems gathered and solved by proposed method. • Accuracy close to simulations and efficiency close to first order reliability method . • Method is suitable for nonlinear/high-dimensional problems with small probability. This study introduces an efficient method for safety evaluation of the structures with small failure probability. The proposed approach reformulates basic failure probability formula based on control variates method and suggests to firstly substitute the nonlinear limit state function with a linear limit state function for taking the advantages of linear problems during reliability process. Reliability analysis of linear problems would be fully accurate and dimension insensitive. Subsequently, the control variates method imposes the effect of limit state function nonlinearity on the obtained first-order failure probability by using a very small sampling size simulation. The result is a new reliability formulation that is highly suitable for solving nonlinear and high dimension problems. The capabilities of the method examined by solving several benchmarks numerical/engineering problems involving non-normal random variables, complex/noisy limit state functions, and nonlinear high dimension problems. Compared with mainstream reliability methods, for all solved problems, it is demonstrated that the proposed approach presents accuracy close to Monte Carlo simulation while the required number of performance function valuation is close to first-order reliability method.

Equilibrium reliability measure for structural design under twofold uncertainty

In structural reliability engineering, designer may encounter twofold uncertainty that arises from the existence of the insufficient statistical data and the incorporated experts’ information. For this case, the randomness and fuzziness coexist in each uncertain element. This paper aims to develop a new approach to evaluating structural reliability and obtain the optimal structural design under twofold uncertainty. Specifically, we first introduce a metric termed equilibrium chance to formulate equilibrium reliability index (ERI) for modeling structural reliability under linear limit-state function and nonlinear limit-state function. Second, a new class of equilibrium reliability-based structural design (ERSD) optimization problem with equilibrium reliability constraint is developed. Under mild conditions, we provide the equivalent reformulation of the equilibrium reliability constraint and demonstrate its convexity. Finally, we introduce the fuzzy simulation (FS) scheme to obtain the feasible solution generated by equilibrium reliability constraint, and a hybrid heuristic algorithm by integrating the FS, genetic algorithm (GA) and local search (LS) is designed to solve the proposed ERSD optimization problem. Several numerical examples are carried out to illustrate the applicability and effectiveness of our proposed reliability modeling method.

Influence of cross correlation between nominal load and resistance on reliability-based design for simple linear soil-structure limit states

Cross correlations between nominal load and resistance terms in limit state functions for geotechnical soil–structure interaction problems can be expected. A closed-form solution for the reliability index for a simple linear limit state function is used to examine the influence of nominal load and resistance correlations on computed margins of safety. The formulation also includes the contribution of the underlying accuracy of the load and resistance equations (method bias) and bias dependencies with the magnitude of nominal load and resistance values assumed in the limit state design function. Sensitivity analyses and example problems for the external sliding limit state for a cantilever wall and the pullout limit state for internal stability of reinforced soil walls with different soil reinforcement types are presented. Ignoring nominal correlations where they exist is shown to underestimate the reliability index in some cases and to overestimate the reliability index in other cases. In the example problems, these differences are shown to exceed one order of magnitude in terms of probability of failure, but in the sensitivity analyses using a wider range of input parameter values, the differences can be several orders of magnitude.

Reliability assessment of vertically loaded masonry walls

For the safety assessment of vertically loaded masonry walls, the Eurocode 6 considers a linear limit state function with only one basic parameter for the compressive strength of masonry. This assumption disregards the contribution of the uncertainty in other material parameters, like the elastic modulus, which is more important than the compressive strength in slender masonry walls. Furthermore, Eurocode 0 gives no clear advice on how to deal with the partial safety factors for non-linear problems in general.A reliability assessment has been carried out for the vertically loaded masonry walls, in order to clearly understand the contribution of the uncertainty of different material parameters on the evaluation of safety. The limit state function of the problem has been calculated using the transfer-matrix method. The full probabilistic approach has been applied using Monte Carlo simulations which helped to propose a new approach for the use of the partial safety factors in the non-linear analysis.

Assessment of fracture reliability analysis of crack growth in spring clip type Vossloh SKL14

Spring clips type Vossloh SKL14 (fastening system of track) are vulnerable to fatigue damage in lifetime due to excitations caused by traffic loads. This article has tried to develop a method for reliability analysis of spring clips type SKL14, based on fracture mechanics approach. First of all, a linear dynamic analysis of track has been done for dynamic response calculations under various traffic loads. The displacement time histories are applied on finite element method of type Vossloh SKL14 to achieve cyclic stresses. The equivalent stress range is described by lognormal distribution. Then, fracture reliability analysis has been done for crack propagation assessment in spring clip based on Paris’s law. This fatigue crack growth is dominated by a Mode I mechanism. A linear limit state function based on fracture mechanics is derived in terms of random variables. First-order reliability method has been employed for reliability estimation. At the end, the influence of various random variables on overall probability of failure has been studied through sensitivity analysis.

The importance measure on the non-probabilistic reliability index of uncertain structures

In the non-probabilistic uncertainty structural analysis, the input uncertain variables of it, such as loads and material properties, will be propagated to the output responses, which include the displacement, stress and compliance, etc. To measure the effect of these non-probabilistic input variables on the output response, two new uncertainty importance measures on the non-probabilistic reliability index are discussed. For the linear limit state function, the analytical solutions of the importance measures are derived. To reduce computational effort, the discretization method and the surrogate model method are presented to calculate the two importance measures in case of the non-linear limit state. Finally, four examples demonstrate that the proposed importance measures can effectively describe the effect of the input variables on the reliability of the structure system, and the established methods can effectively obtain the two importance measures.

Reliability Sensitivity Analysis Based on Mean Value First Order Esscher's Approximation

A simple and effective computational procedure is presented for computing probability of failure of mechanical and structural systems with linear limit state function by using convolution theorem,fast probability integration and Esscher's approximation.An efficient and accurate mean-value first order Esscher's approximation method(MVFOEA) is proposed by approximating a performance function with the first order Taylor expansion at the mean-value point of random input variables.By combining MVFOEA with sensitivity analysis technique,a new reliability sensitivity analysis method is proposed.Because of the use of complete distribution information,MVFOEA is generally more accurate than mean value first order second moment(MVFOSM).MVFOEA requires the basic random variables should be independent for each other and have moment generating functions.The features of the proposed method are demonstrated with three numerical examples.

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.

Saddlepoint approximation based structural reliability analysis with non-normal random variables

The saddlepoint approximation (SA) can directly estimate the probability distribution of linear performance function in non-normal variables space. Based on the property of SA, three SA based methods are developed for the structural system reliability analysis. The first method is SA based reliability bounds theory (RBT), in which SA is employed to estimate failure probability and equivalent normal reliability index for each failure mode firstly, and then RBT is employed to obtain the upper and the lower bounds of system failure probability. The second method is SA based Nataf approximation, in which SA is used to estimate the probability density function (PDF) and cumulative distribution function (CDF) for the approximately linearized performance function of each failure mode. After the PDF of each failure mode and the correlation coefficients among approximately linearized performance functions are estimated, Nataf distribution is employed to approximate the joint PDF of multiple structural system performance functions, and then the system failure probability can be estimated directly by numerical simulation using the joint PDF. The third method is SA based line sampling (LS). The standardization transformation is needed to eliminate the dimensions of variables firstly in this case. Then LS method can express the system failure probability as an arithmetic average of a set of failure probabilities of the linear performance functions, and the probabilities of the linear performance functions can be estimated by the SA in the non-normal variables space. By comparing basic concepts, implementations and results of illustrations, the following conclusions can be drawn: (1) The first method can only obtain the bounds of system failure probability and it is only acceptable for the linear limit state function; (2) the second method can give the estimation of system failure probability, and its error mostly results from the approximation of Nataf distribution for the joint PDF of the structural system performance functions and the linearization of the performance functions; (3) the SA based LS method can obtain the estimator of system failure probability, which converges to the actual value along with the increase of sample size. The SA based LS method considers the influence of nonlinearity of limit state function on the failure probability, and it is acceptable for the structural system both with a single failure mode and with multiple failure modes, therefore it has the widest applicability.

Domain Decomposition Method for Calculating the Failure Probability of Linear Dynamic Systems Subjected to Gaussian Stochastic Loads

In this paper the problem of calculating the probability of failure of linear dynamic systems subjected to random vibrations is considered. This is a very important and challenging problem in structural reliability. The failure domain in this case can be described as a union of linear failure domains whose boundaries are hyperplanes. Each linear limit state function can be completely described by its own design point, which can be analytically determined, allowing for an exact analytical calculation of the corresponding failure probability. The difficulty in calculating the overall failure probability arises from the overlapping of the different linear failure domains, the degree of which is unknown and needs to be determined. A novel robust reliability methodology, referred to as the domain decomposition method (DDM), is proposed to calculate the probability that the response of a linear system exceeds specified target thresholds. It exploits the special structure of the failure domain, given by the union of a large number of linear failure regions, to obtain an extremely efficient and highly accurate estimate of the failure probability. The number of dynamic analyses to be performed in order to determine the failure probability is as low as the number of independent random excitations driving the system. Furthermore, calculating the reliability of the same structure under different performance objectives does not require any additional dynamic analyses. Two numerical examples are given demonstrating the proposed method, both of which show that the method offers dramatic improvement over standard Monte Carlo simulations, while a comparison with the ISEE algorithm shows that the DDM is at least as efficient as the ISEE.

Wedge simulation method for calculating the reliability of linear dynamical systems

In this paper the problem of calculating the probability of failure of linear dynamical systems subjected to random excitations is considered. The failure probability can be described as a union of failure events each of which is described by a linear limit state function. While the failure probability due to a union of non-interacting limit state functions can be evaluated without difficulty, the interaction among the limit state functions makes the calculation of the failure probability a difficult and challenging task. A novel robust reliability methodology, referred to as Wedge-Simulation-Method, is proposed to calculate the probability that the response of a linear system subjected to Gaussian random excitation exceeds specified target thresholds. A numerical example is given to demonstrate the efficiency of the proposed method which is found to be enormously more efficient than Monte Carlo Simulations.

Multinormal integrals by importance sampling for series system reliability

A structural system with multi-failure modes can be modeled as a series system if it fails whenever any of the failure mode occurs. Applying FORM, failure probability of a series system can be expressed using a complementary standard multinormal integral. However, the integral is increasingly more difficult as the dimension increases. Importance sampling method can be used to deal with such multi-fold integration. Considering the fact that the optimal importance sampling function can be determined for a linear limit state function in a uncorrelated standard normal space, this paper proposes an importance sampling function for multinormal integral as a linear combination of such optimal sampling functions. The accuracy and applicability of the method are investigated using numerical examples.