Distribution Parameters Of Random Variables Research Articles

Sample regeneration algorithm for structural failure probability function estimation

An efficient strategy to approximate the failure probability function in structural reliability problems is proposed. The failure probability function (FPF) is defined as the failure probability of the structure expressed as a function of the design parameters, which in this study are considered to be distribution parameters of random variables representing uncertain model quantities. The task of determining the FPF is commonly numerically demanding since repeated reliability analyses are required. The proposed strategy is based on the concept of augmented reliability analysis, which only requires a single run of a simulation-based reliability method. This paper introduces a new sample regeneration algorithm that allows to generate the required failure samples of design parameters without any additional evaluation of the structural response. In this way, efficiency is further improved while ensuring high accuracy in the estimation of the FPF. To illustrate the efficiency and effectiveness of the method, case studies involving a turbine disk and an aircraft inner flap are included in this study.

Decoupled reliability-based optimization using Markov chain Monte Carlo in augmented space

An efficient framework is proposed for reliability-based design optimization (RBDO) of structural systems. The RBDO problem is expressed in terms of the minimization of the failure probability with respect to design variables which correspond to distribution parameters of random variables, e.g. mean or standard deviation. Generally, this problem is quite demanding from a computational viewpoint, as repeated reliability analyses are involved. Hence, in this contribution, an efficient framework for solving a class of RBDO problems without even a single reliability analysis is proposed. It makes full use of an established functional relationship between the probability of failure and the distribution design parameters, which is termed as the failure probability function (FPF). By introducing an instrumental variability associated with the distribution design parameters, the target FPF is found to be proportional to a posterior distribution of the design parameters conditional on the occurrence of failure in an augmented space. This posterior distribution is derived and expressed as an integral, which can be estimated through simulation. An advanced Markov chain algorithm is adopted to efficiently generate samples that follow the aforementioned posterior distribution. Also, an algorithm that re-uses information is proposed in combination with sequential approximate optimization to improve the efficiency. Numeric examples illustrate the performance of the proposed framework.

New reliability modeling methods for structural systems with hybrid uncertainty

AbstractThe present study investigates the hybrid reliability modeling of structures in which the inputs contain both random variables and interval variables. Hybrid uncertainty is divided into three categories, including random variables mixed with random variables, interval variables mixed with interval variable, and random variables mixed with interval variables. In order to perform the reliability analysis of structural systems, first, the Bayes method is proposed in the present study to obtain distribution parameters of random variables. Moreover, the self‐sample method is introduced to obtain the interval boundaries based on the least available measuring data. Then, the reliability models are established for three situations and the reliability indices are defined and derived accordingly. The abovementioned three types of reliability indices outline the general situation of structural systems. Finally, the specific calculation process is described in details through different examples. Furthermore, the accuracy and efficiency of the proposed method is discussed by comparing the results obtained from the Monte Carlo simulation and those of other methods. The obtained results indicate that the performance of the proposed model in solving reliability modeling problems is better.

Probabilistic Analysis for Structures with Hybrid Uncertain Parameters

In practical engineering problems, the distribution parameters of random variables cannot be determined precisely due to limited experimental data. The hybrid uncertain model of interval and probability can deal with the problem, but it will produce extensive computation and it is difficult to meet the requirement of the complex engineering problem analysis. In this scenario, this paper presents a vertex method for the uncertainty analysis of the hybrid model. By combining the traditional finite element method, it can be applied to the structural uncertainty analysis. The key of this method is to demonstrate the monotonicity between expectation and variance of the function and distribution parameters of random variables. Based on the monotonicity analysis, interval bounds of the expectation and variance are directly calculated by means of vertex of distribution parameter intervals. Two numerical examples are used to evaluate the effectiveness and accuracy of the proposed method. The results show the vertex method is computationally more efficient than the common interval Monte Carlo method under the same accuracy. Two practical engineering examples are to show that the vertex method makes the engineering application of the hybrid uncertain model easy.

Fuzzy Reliability Analysis With Fuzzy Random Variables Based on Perturbation Principle

In structural reliability calculation, there are fuzzy uncertainties in the distribution parameters of random variables, which bring the problem of large computation and poor precision. In order to improve accuracy and efficiency of structural reliability, a novel structural reliability calculation method with fuzzy random variables is proposed from the perspective of perturbation principle. First, the fuzzy characteristic parameters of the structural fuzzy random variables are expressed as the sums of their true values and fuzzy perturbations. At the same time, the perturbations are equivalently selected. Second, the probability density function and the performance function are decomposed into a series of intervals under different level cut sets, and the Taylor series is expanded on the upper and lower limits of the intervals. Finally, according to the definition of structural reliability, based on the direct integration method, the sigmoid function is introduced into the reliability function to approximate the step function, and the fuzzy reliability is obtained. The simulation results show the effectiveness and accuracy of the proposed method, and it provides a feasible way for structural reliability analysis with fuzzy variables or fuzzy random variables.

Structural reliability analysis with fuzzy random variables using error principle

In structural reliability calculation, there are fuzzy uncertainties in the distribution parameters of random variables, which bring the problem of large computation and poor precision. In order to improve the accuracy and efficiency of structural reliability, a novel structural reliability calculation method with fuzzy random variables is proposed from the perspective of error propagation. Firstly, fuzzy variables are transformed into uncertain interval variables according to the fuzzy decomposition theorem. Secondly, by using the error transfer principle, the sigk. function is introduced into the reliability function to approximate the step function, and a structural reliability error analysis model based on the direct integration method is established. On this basis, the equivalent error of the fuzzy variable is determined by traversing the interval value of the membership function at [0, 1] level cut set, and then the structural reliability interval values corresponding to each cut set are obtained. The examples are investigated to demonstrate the efficiency and accuracy of the proposed method, which provides a feasible way to analyze and calculate the structural reliability with uncertain variables such as fuzzy random variables, random variables and fuzzy variables.

Hybrid Uncertain Analysis for Exterior Acoustic Field Prediction with Interval Random Parameters

For exterior acoustic field problems that lack sufficient information to construct precise probability distributions, an interval random model is introduced to deal with the uncertain parameters. In the interval random model, the probability variables are employed to treat the uncertain parameters, whereas some distribution parameters of random variables are modeled as interval variables instead of precise values. Based on the interval random model, the interval random finite element equation for exterior acoustic fields is established and a hybrid uncertain analysis method is presented to solve the exterior acoustic field problem with interval random variables. In the presented method, by temporarily neglecting the uncertainties of interval variables, a first-order stochastic perturbation method is adopted to calculate the expectation and the variance of the response vector. According to the monotonicity of the expectation and variance of the response vector with respect to the interval variables, the lower and upper bounds of the expectation and variance of the response vector can be calculated by the vertex method. In addition, in order to ensure accuracy of the proposed method, the subinterval technique is introduced and investigated. The numerical example of a square flexible shell model is presented to demonstrate the effectiveness of the proposed method.

Topology optimization of structure for dynamic properties considering hybrid uncertain parameters

In the design and manufacturing of mechanical components, the dynamic properties of continuum structure are one of the most significant performances. At the same time, the uncertainty is widespread in these dynamic problems. This paper presents a robust topology optimization methodology of structure for dynamic properties with consideration of hybrid uncertain parameters. The imprecise probability uncertainties including materials, geometry and boundary condition are treated as an interval random model, in which the probability distribution parameters of random variables are modeled as the interval variables instead of given precise values. Two dynamic properties, including dynamic-compliance and eigenvalue, are chosen as the objective function. In addition, different excitation frequency or eigenvalue is discussed. In this work, the bi-directional evolutionary structural optimization (BESO) method is adopted to find the optimal robust layout of the structure. A series of numerical examples is presented to illustrate the optimization procedure, and the effectiveness of the proposed method is demonstrated clearly.

Development of a stochastic model of failure of structural materials in creep at hardening stage

The results of research of patterns of long-term failure of metals and alloys in isothermal creep considering all its stages are presented. The approach to prediction of long-term strength characteristics of structural materials in isothermal creep under uniaxial steady load is proposed. This approach is original in the probabilistic interpretation of kinetic equations describing the creep, as well as a variety of failure criteria and their relative characteristics. Using the experimental creep curves, the adequate stochastic model was constructed and the method of identification of unknown constants of the model was developed. The calculations of basic probability characteristics of the time to failure on the example of experimental data for PA6 aluminum alloy specimens at various stress and temperatures levels were carried out. According to the research, it is enough to use the quadratic dependencies as dependencies of distribution parameters of random variables a and b, as well as the parameters c and g on the stress and temperature. The choice of the type of dependencies of higher degrees has little impact on calculation results. The proposed model can be used for prediction of the time to failure of various structural materials, as well as the development of methods, standards and guidelines in the field of reliability and durability assessment of structural materials.

A new random interval method for response analysis of structural–acoustic system with interval random variables

For the response analysis of the structural–acoustic system without sufficient information, an interval random model is introduced. In the interval random model, the uncertain parameters are assumed as random variables, while some probability distribution parameters of random variables are modeled as interval parameters instead of precise values. Based on the interval random model, the interval random dynamic equation is constructed and a new random interval method named as the random moment – change of variable interval stochastic perturbation method (RM-CVISPM) is proposed. In RM-CVISPM, the response of the structural–acoustic system is approximated as a linear function of interval random variables based on the first-order stochastic perturbation analysis. Then, the expectation and variance of the obtained response are calculated by the random moment method. The probability density of the obtained response is yielded by the change of variable technique. Finally, the intervals of expectation, variance and probability density of response are estimated by an interval perturbation method. Numerical results on a shell structural–acoustic model and an automobile passenger compartment with flexible front panel verify the effectiveness and efficiency of RM-CVISPM.

The interval PHI2 analysis method for time-dependent reliability

The uncertainty modeling and time-dependent reliability analysis for time-dependent structures with a large number of inherently uncertain parameters is very important in engineering. When evaluating the reliability of structures under stochastic loadings through traditional methods, the distribution parameters of random variables, such as strength, stiffness and damping, are usually determined, but the accurate value of distribution parameters can hardly be known in practical engineering. In this paper an interval PHI2 (I-PHI2) method is proposed to solve the time-dependent reliability for random problems with the interval distribution parameters. I-PHI2 method can provide the reliability interval for assessing design reliability over the whole lifecycle of uncertain structures. Firstly, the time-dependent reliability analysis model is built for uncertain problem of interval distribution parameters. Secondly, the interval distribution parameters of random variables and stochastic processes are determined by use of the interval analysis method and then transferred into the standard coordinate space. Thirdly, the analytical expression of out-crossing rate in which there are many interval parameters are derived and the corresponding upper and lower bounds of out-crossing rate can be obtained. Finally, based on the traditional time-independent reliability tools such as FORM, an efficient algorithm for time-dependent structure is developed to obtain the upper and lower bounds of time-dependent reliability index and failure probability. Two numerical examples are presented for verification of the validity of I-PHI2 method.

An interval random perturbation method for structural‐acoustic system with hybrid uncertain parameters

SUMMARYAn interval random model is introduced for the response analysis of structural‐acoustic systems that lack sufficient information to construct the precise probability distributions of uncertain parameters. In the interval random model, the uncertain parameters are treated as random variables, whereas some distribution parameters of random variables with limited information are expressed as interval variables instead of precise values. On the basis of the interval random model, the interval random structural‐acoustic finite element equation is constructed, and an interval random perturbation method for solving this interval random equation is proposed. In the proposed method, the interval random matrix and vector are expanded by the first‐order Taylor series, and the response vector of the structural‐acoustic system is calculated by the matrix perturbation method. According to the linear monotonicity of the response vector, the lower and upper bounds of the response vector are calculated by the vertex method. On the basis of the lower and upper bounds, the intervals of expectation and standard variance of the response vector are obtained by the random interval moment method. The numerical results on a shell structural‐acoustic model and an automobile passenger compartment with flexible front panel demonstrate the effectiveness and efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

Interval design point method for calculating the reliability of structural systems

The evaluation of reliability for structural system is important in engineering practices. In this paper, by combining the design point method, JC method, interval analysis theory, and increment load method, we propose a new interval design point method for the reliability of structural systems in which the distribution parameters of random variables are described as interval variables. The proposed method may provide exact probabilistic interval reliability of structures whose random variables can have either a normal or abnormal distribution form. At last, we show the feasibility of the proposed approach through a typical example.

Hybrid uncertain analysis of acoustic field with interval random parameters

For the acoustic field prediction without sufficient information, an interval random model is introduced to treat with the uncertain parameters in which the probability distribution parameters of random variables are modeled as interval variables instead of given precise values. Based on the interval random model, the hybrid uncertain acoustic dynamic equilibrium equation is constructed and a hybrid uncertain analysis method is proposed. In the proposed method, the interval random matrix and vector are expanded by the first-order Taylor series and the sound pressure response vector is obtained by the matrix perturbation technique. A monotonic analysis is conducted for the sound pressure response vector and based on it the lower and upper bounds of the sound pressure response vector are obtained by the vertex method. According to the response bounds, the intervals of expectation and variance of the sound pressure response vector are calculated by the random interval moment method. Two numerical examples which include a simple problem with explicit expression and a complex engineering application are presented to demonstrate the effectiveness of the present method.

System Reliability Study Based on Bearing Capacity Update and Deformation Iteration

For the sensitive of distribution parameters of random variables, system reliability research should be focus on identifying the significant failure modes, while failure probability as a quantitative reference index, only combining both of them can better estimate system reliability. On the basis of load incremental step theory, a method for identifying the significant failure modes which could more accurately simulate the real failure process for considering the change of bearing capacity and spatial position is proposed in this paper. Then, the corresponding program is prepared with APDL on ANSYS platform. On the one hand, the efficiency and accuracy of identification is improved, on the other hand, the usefulness of this method is enhanced. Finally, the reliability of a space grid engineering project is analyzed and discussed by this method; moreover, the system reliability index β is calculated as well. These analysis results are helpful to provide scientific basis for structural optimization, structural testing, and structural reinforcement and so on.

Stochastic sensitivity analysis using HDMR and score function

Probabilistic sensitivities provide an important insight in reliability analysis and often crucial towards understanding the physical behaviour underlying failure and modifying the design to mitigate and manage risk. This article presents a new computational approach for calculating stochastic sensitivities of mechanical systems with respect to distribution parameters of random variables. The method involves high dimensional model representation and score functions associated with probability distribution of a random input. The proposed approach facilitates first-and second-order approximation of stochastic sensitivity measures and statistical simulation. The formulation is general such that any simulation method can be used for the computation such as Monte Carlo, importance sampling, Latin hypercube, etc. Both the probabilistic response and its sensitivities can be estimated from a single probabilistic analysis, without requiring gradients of performance function. Numerical results indicate that the proposed method provides accurate and computationally efficient estimates of sensitivities of statistical moments or reliability of structural system.

Reliability sensitivity analysis with random and interval variables

AbstractIn reliability analysis and reliability‐based design, sensitivity analysis identifies the relationship between the change in reliability and the change in the characteristics of uncertain variables. Sensitivity analysis is also used to identify the most significant uncertain variables that have the highest contributions to reliability. Most of the current sensitivity analysis methods are applicable for only random variables. In many engineering applications, however, some of uncertain variables are intervals. In this work, a sensitivity analysis method is proposed for the mixture of random and interval variables. Six sensitivity indices are defined for the sensitivity of the average reliability and reliability bounds with respect to the averages and widths of intervals, as well as with respect to the distribution parameters of random variables. The equations of these sensitivity indices are derived based on the first‐order reliability method (FORM). The proposed reliability sensitivity analysis is a byproduct of FORM without any extra function calls after reliability is found. Once FORM is performed, the sensitivity information is obtained automatically. Two examples are used for demonstration. Copyright © 2009 John Wiley & Sons, Ltd.

Reliability design method for steam turbine blades

Based on theories of probability and statistics, and taking static stresses, dynamic stresses, endurance strength, safety ratios, vibration frequencies and exciting force frequencies of blades as random variables, a reliability design method for steam turbine blades is presented. The purport and calculation method for blade reliability are expounded. The distribution parameters of random variables are determined after analysis and numerical calculation of test data. The fatigue strength and the vibration design reliability of turbine blades are determined with the aid of a probabilistic design method and by interference models for stress distribution and strength distribution. Some blade reliability design calculation formulas for a dynamic stress design method, a safety ratio design method for fatigue strength, and a vibration reliability design method for the first and second types of tuned blades and a packet of blades on a disk connected closely, are given together with some practical examples. With these methods, the design reliability of steam turbine blades can be guaranteed in the design stage. This research may provide some scientific basis for reliability design of steam turbine blades.