Approximate analysis of complex reliability problems

Reliability is approximated for problems involving linear and nonlinear limit states and independent or dependent basic variables from estimators of the distribution of a scalar control variable that depends on the input random parameters and the safety condition. The estimators de developed for the distribution of the control variable can be obtained from moments or simulated values of the basic variables. They have provided most satisfactory approximations for the distribution of sums of independent and correlated, non-Gaussian basic variables and the reliability of a portal frame.     

Fast probability integration by three-parameter normal tail approximation

A new method of fast probability integration is presented, based on a three-parameter normal tail approximation to a non-normal distribution function. Values of the distribution function, the probability density function and its derivative are matched at the approximation point with the approximating function. The approximation point is the known “design point”, used within the framework of second moment reliability theory. The proposed method is as fast as the available two-parameter algorithm, but is apparently more accurate.     

Methods for approximate reliability analysis

Approximate solutions are developed for reliability problems involving constant-in-time and time-dependent random parameters. They are based on estimates of the distribution of scalar control variables obtained from approximations of the safety condition (control variable approach) or on a transformation of the input space of parameters into a space in which the safety condition is nearly linear (linearization approach). The distribution of the control variable or of the random parameters in the transformed space can be estimated from moments of these variables. The control variable approach is most effective when the random parameters are constant in time while the linearization approach is useful for time-dependent reliability studies since it permits application of various rules (e.g., Turkstra's rule) for approximate load combination analysis.     

Investigation of polynomial normal transform

Normal transformation is often used in probabilistic analysis especially when multivariate non-normal random variables are involved. A third-order polynomial normal transformation technique is presented in this paper and its characteristics examined. Four methods based on different statistical information of a random variable are used to determine the polynomial coefficients in this normal transformation technique. The performance of these four methods is investigated by comparing with parametric technique using Rosenblatt transformation that preserves the marginal distribution of a non-normal random variable. From the numerical experiment conducted, this simple technique is found to be quite accurate, and it is less restrictive in its usage for merely requiring the information of the first four statistical moments of a random variable rather than requiring a stronger assumption of the full distribution information in the Rosenblatt transformation. The technique is especially attractive when only samples of the random variables are available.     

Design and sensitivity analysis using the probability-safety-factor method. An application to retaining walls

This paper presents a new method for designing engineering works that makes the classical approach, based on safety factors, and the modern, probability-based, approach compatible, and includes a sensitivity analysis. The method consists of a sequence of classical designs, based on given safety factors, that (a) minimize cost or optimize an alternative objective function, (b) calculate the different failure mode probabilities or their upper bounds, and (c) update the safety factors to satisfy both the safety factors and the failure probability requirements. The process is repeated until convergence. As a result, an automatic design of the engineering work, the safety factors and the corresponding probabilities of failure for all failure modes are obtained. A double safety check is used and the correspondence between safety factors and probabilities of failure for the different modes are easily understood. An advantage of this approach is that the optimization procedure and the reliability calculations are decoupled. In addition, a sensitivity analysis is performed using a method that consists of transforming the data parameters into artificial variables and using the dual associated problem. The method is illustrated by its application to a retaining wall design.     

Structural system reliability assessment using directional simulation

The theory for the estimation of the reliability of a structural system which is subject to one or more load processes requires knowledge of the surface defining the mechanical response of the system. For many systems this surface may be known only implicitly and defined through a method of structural analysis. In the formulation presented herein, the surface is obtained using an established probabilistic structural analysis technique. The reliability estimation is formulated in the load process space. In this space, directional simulation is employed to estimate the outcrossing rate and the initial (zero time) probability of failure and hence to estimate the probability of failure at any subsequent time. Two examples using discrete rectangular pulse (Poisson) processes are described.     

PERMAS-RA/STRUREL system of programs for probabilistic reliability analysis

Programs for reliability analysis of structural, operational and other systems based on first- and second-order reliability concepts were made available as early as 1976 by the Technical University of Munich. In the meantime and since 1987 by RCP Consult GmbH (RCP) the programs have experienced many revisions, improvements and additional developments. The programs now cover the preparatory steps as well as the computational tasks in technical reliability, decision making under uncertainty and in statistical analysis. Important modules of strurel have also been embedded in the finite element program permas developed and maintained by INTES GmbH (INTES). The programs have been used in structural engineering and code making, in hydrology, operations research, financial planning and mathematical statistics and, in particular, in the nuclear power plant, offshore, ship, automotive and aerospace industry.     

Dynamic response and reliability analysis of tall buildings subject to wind loading

This paper explores the wind stochastic field from a new viewpoint of stochastic Fourier spectrum (SFS). The basic random parameters of the wind stochastic field, the roughness length z₀ and the mean wind velocity at 10 m height U₁₀, as well as their probability density functions (PDF), are obtained. It provides opportunities to use probability density evolution method (PDEM), which had been proved to be of high accuracy and efficiency, in computing the dynamic response and reliability of tall buildings subject to the wind loading. Principals and corresponding numerical solving algorithm of the PDEM are first presented. Then, the adopted model of the wind stochastic field is described briefly. The simulation method of the fluctuating wind velocity based on the SFS is introduced. Finally, as an example of the application of the PDEM, a 20-storey frame subject to wind loading is investigated in detail. The responses, including the mean value and the standard deviation, and the reliabilities of the frame are evaluated by the PDEM. The results demonstrate that the PDEM is applicable and efficient in the dynamic response and reliability analysis of wind-excited tall building.     

Directional methods for structural reliability analysis

Directional simulation reduces the dimension of the limit state probability integral by identifying a set of directions for integration, integrating either in closed-form or by approximation in those directions, and estimating the probability as a weighted average of the directional integrals. Most existing methods identify these directions by a set of points distributed on the unit hypersphere. The accuracy of the directional simulation depends on how the points are identified. When the limit state is highly nonlinear, or the inherent failure probability is small, a very large number of points may be required, and the method can become inefficient. This paper introduces several new approaches for identifying directions for evaluating the probability integral — Spherical t-design, Spiral Points, and Fekete Points — and compares the failure probabilities with those determined in a number of examples in previously published work. Once these points have been identified for a probability integral of given dimension, they can be used repeatedly for other probability integrals of the same dimension in a fashion analogous to Gauss Quadrature.     

利用Birkhoff多项式求解结构动力响应的新方向

结合解析方法和数值方法优点,提出了一种求解动力响应的新方法。在求解动力响应的Duhamel积分中,利用分段Birkhoff插值多项式逼近任意动力荷载,并推导了相关公式。由于分段Birkhoff多项式的Duhamel积分有精确解,因而和现在常用的逐步积分法相比,本文方法不但具有高得多的计算精度,避免了收敛性和稳定性的问题,而且大大减少了计算工作量。     

Sample-specific linearization in reliability analysis of off-shore structures

The reliability analysis of off-shore structures under dynamic non-Gaussian wave, wind and water current loading is considered using an outcrossing approach and directional simulation in the space of the load processes. To circumvent the non-Gaussianity of the load processes for which results for mean outcrossing rate are not generally available, the “sample-specific linearization method” is proposed. This method is based on well-known results for Gaussian processes and uses the functional relationships between the non-Gaussian wave, wind and water current loads and the corresponding velocities. The proposed method is applied to a simple model off-shore structure. The structure is analyzed also to obtain probability of failure by means of Monte Carlo simulation and using time domain simulation of the load processes. This was found to be extremely expensive of computer time. The results were found to compare very favourably with those obtained by the proposed method, both in terms of accuracy and computation required.     

Distribution of structural collapses and optimum reliability for infrequent environmental loads

Structures located in a particular zone of interest may be subjected to the same environmental events or the same natural phenomena that occur infrequently. That is, the structures are subjected to the same environmental loading parameters such as the peak ground acceleration, the reference wind velocity pressure, and the total ice accumulation. Therefore, the structural collapses are dependent or correlated. The correlation may affect the probability distribution of the number of structural collapses and the optimum reliability level for upgrading the exiting structures and infrastructures. This is investigated in this paper. It is shown that the coefficient of variation of the number of structural collapses is an increasing function of the correlation coefficient. This increase can be important to the selection of optimum reliability level for retrofitting of existing structures and infrastructures when the risk aversion factor is considered. An approach by taking the correlation among structural collapses into account in selecting the optimum upgrading level of a group of existing structures is presented and illustrated by a numerical example.     

Role of non-destructive evaluation in time-dependent reliability analysis

Uncertainties exist in the ability of nondestructive evaluation (NDE) methods to detect and measure flaws. Probabilistic methods can be used for characterizing these uncertainties and for updating flaw sizes predicted from stochastic fatigue crack growth analysis. The impact of NDE on condition assessment and reliability of structures, in which fatigue crack growth is occurring as a result of cyclic random service loads, is examined. Examples of probability-based condition assessment using magnetic particle or ultrasonic inspection and its potential impact on time-dependent fatigue reliability analysis are illustrated for a steel miter gate at a lock operated by the US Army Corps of Engineers.     

Reliability analysis of linear dynamical systems using approximate representations of performance functions

A method to carry out reliability analysis of linear systems with random structural parameters and random excitation is presented. Probability that design conditions are satisfied within a given time period is used as a measure of system reliability. An efficient importance sampling technique is used to estimate the failure probabilities. This technique is combined with high quality approximations of the performance functions in terms of uncertain structural parameters. The number of dynamic analyses required during the estimation process is minimized since the simulations are carried out by evaluating explicit expressions. The explicit quantities correspond to approximate representations of the performance functions in terms of the uncertain structural parameters. A numerical example corresponding to problem 2, sub-case 2 of the benchmark study is presented to illustrate the accuracy and computational efficiency of the proposed computational procedure.     

Asymptotic analysis of asymmetric reliability margin

By a first-order-reliability-method (FORM)-based asymptotic analysis the influence of the coefficient of skewness of resistance upon the probability of failure of a reliability problem is studied. The fundamental case of the reliability margin with three-parameter log-normal resistance and normal action effects is considered. First-order-term formulas for the reliability index and weight factors corresponding to the design point and some other points of failure surface are obtained. The study of the points of failure surface is aimed at finding a convenient separated form of a deterministic condition for a reliability verification at the target level with an explicit occurrence of the coefficient of skewness. Following a suggested procedure, some error estimates of several simplified reliability verifications are presented.     

Improved functional perturbation method for reliability analysis of randomly heterogeneous beams

The strength reliability of randomly heterogeneous beams is studied. The beams are considered as brittle, and failure by the weakest link criterion is assumed. The structure is statically indeterminate, thus the stress field is a function of the random morphology. The probabilistic beam strength is a coupled functional of morphology and stresses. Correlation between local strength and local modulus is also considered, and its effect on reliability is investigated. Heterogeneity is confined to the longitudinal direction only. An improved analytical solution is found by a new, optimized functional perturbation method (FPM). The improvement is achieved by two operations: generalizing the previously used FPM to account for multifunctional dependency, and choosing the perturbation to be around the proper homogeneous case for each type of morphology. It is shown that the improvement of the optimized method for reliability analysis is significant, and depends on the type of local strength-modulus statistical correlation. In addition, analytical results for very large and very small correlation lengths are obtained, and their validity range is examined by numerical solutions.     

Finite element reliability analysis of chloride ingress into reinforced concrete structures

For many reinforced concrete structures corrosion of the reinforcement is an important problem since it can result in maintenance and repair actions. Further, a reduction of the load-bearing capacity can occur. In the present paper the Finite Element Reliability Method (FERM) is employed for obtaining the probability of exceeding a critical chloride concentration level at the reinforcement bars, both using Monte Carlo Simulation (MCS) and the First Order Reliability Method (FORM). The chloride ingress is modelled by the Finite Element Method (FEM) and the diffusion coefficient, surface chloride concentration and reinforcement cover depth are modelled by stochastic fields, which are discretized using the Expansion Optimum Linear Estimation (EOLE) approach. The response gradients needed for FORM analysis are derived analytically using the Direct Differentiation Method (DDM). As an example, a bridge pier in a marine environment is considered and the results are given in terms of distributions of time for initiation of corrosion.     

Progressive failure for seismic reliability analysis

Progressive failure of structural systems subject to seismic excitations is investigated. The system survives the first rupture of an element section (brittle failure) and failure occurs when a predetermined state of excessive displacement is achieved. This study is developed in probabilistic terms and may be seen as part of a simplified approach to seismic reliability analysis whose formulation is in progress. In particular, previous research is systematized and improved on from a mechanical point of view.     

Ductile structural system reliability analysis using adaptive importance sampling

This paper presents a hybrid technique for efficient system reliability estimation of large ductile framed structures. The proposed procedure starts with a simple enumeration scheme, but quickly changes to an adaptive importance sampling scheme to make the process more efficient and easier to implement. The method solves the problem of including the effect of multiple failure sequences in an importance sampling scheme, for the system reliability estimation of large structures. The enumeration method is used to identify the first complete failure sequence. This failure sequence defines the initial failure domain for starting the adaptive sampling process. A weighted multi-modal sampling density is used to account for the contribution of different regions in the sampling domain to the system failure probability. As the simulation progresses, the failure domain is gradually modified to include the effect of other significant failure sequences and arrive at an accurate estimate of the system failure probability.