Probabilistic Failure Analysis by Importance Sampling Markov Chain Simulation

A probabilistic approach for failure analysis is presented in this paper, which investigates the probable scenarios that occur in case of failure of engineering systems with uncertainties. Failure analysis can be carried out by studying the statistics of system behavior corresponding to the random samples of uncertain parameters that are distributed as the conditional distribution given that the failure event has occurred. This necessitates the efficient generation of conditional samples, which is in general a highly nontrivial task. A simulation method based on Markov Chain Monte Carlo simulation is proposed to efficiently generate the conditional samples. It makes use of the samples generated from importance sampling simulation when the performance reliability is computed. The conditional samples can be used for statistical averaging to yield unbiased and consistent estimate of conditional expectations of interest for failure analysis. Examples are given to illustrate the application of the proposed simulation method to probabilistic failure analysis of static and dynamic structural systems.     

Seismic Reliability Assessment of Bridges with User-Defined System Failure Events

Bridge-level failure event definitions per limit state have evolved from failure of one key bridge component as representative of the whole bridge system to failure of at least one of multiple components. However, an entire set of bridge failure event possibilities exists between these two extremes in the same limit state, such as failure of any two, any three, or any desired subset of bridge components. This paper proposes a closed-form combinatorial method to evaluate all possible ways in which bridge components can fail within and across limit states. It also highlights bridge component importance measures as key by-products of the closed-form solution. Calculations are illustrated with a particular yet illustrative system failure event, called the augmented event, which incorporates failures of at least one component in a given limit state and joint failures of multiple important components in a previous limit state. Bridges in as-built and retrofitted conditions are used to illustrate the augmentation calculation under seismic loads and the application of the proposed system reliability method. The results reveal an increase in median system fragility at the moderate limit states in the range of 4–20% relative to traditional approaches that neglect augmentation. This methodology to connect bridge components to bridge system reliability can readily support infrastructure stakeholder decision making and risk management through an efficient approach that can adapt to evolving system failure event definitions.     

Reliability Analysis of Single-Degree-of-Freedom Elastoplastic Systems. II: Suboptimal Excitations

A method for constructing an approximation of the critical excitation that drives an elastoplastic system from rest to a target threshold at a specified time instant, referred to as the “suboptimal excitation,” is presented in this paper. It is based on the observations gained from study of the critical excitations in the companion paper. Essentially, for the usual case of interest where the failure time is not small compared to the natural period, the duration of the positive and negative pulses of the critical excitation are roughly equal to half of the natural period. This consideration allows for a simple intuitive approximation of the critical excitation. The amplitudes of the positive and negative pulses are obtained in closed forms using energy balance. Numerical investigations show that the critical excitations are well approximated by the suboptimal excitations.     

Aerostatic Reliability Analysis of Long-Span Bridges

The response surface Monte Carlo method (RSMCM) is proposed for reliability analysis of aerostatic response and aerostatic stability for different types of long-span bridges, in which the nonlinear effects due to geometric nonlinearity and deformation-dependent aerostatic loads are taken into consideration. The geometric parameters, the material parameters, and the aerostatic coefficients of the bridge girder are regarded as random variables in the proposed method. RSMCM has higher accuracy in comparison with the traditional response surface method and requires much less computational cost than the conventional Monte Carlo method. The proposed method is applied to reliability analysis of aerostatic response and aerostatic stability of the Hong Kong Ting Kau Bridge, and reasonable results illustrating effectiveness of the method are obtained.     

Reliability Analysis of Single-Degree-of-Freedom Elastoplastic Systems. I: Critical Excitations

This paper investigates the application of importance sampling method to estimating the first passage probability of single-degree-of-freedom elastoplastic systems subjected to white noise excitations. The importance sampling density is constructed using a conventional choice as a weighted sum of Gaussian distributions centered among design points. It is well known that the design points, or equivalently the critical excitations in the function space, are difficult to obtain for nonlinear hysteretic systems. An efficient method has been developed recently for finding the critical excitations, on which this paper is based. Characteristics of the critical excitation for elastoplastic systems are explored and the efficiency of the resulting importance sampling strategy is critically assessed. It is found that some efficiency is gained by importance sampling over direct Monte Carlo method but to a lesser extent compared to its linear-elastic counterparts. The cause of this drop in efficiency will be investigated. The study calls for revisiting a basic assumption of importance sampling densities constructed using design points, where they are expected to generate samples lying frequently in the failure region, but in reality their capability should not be taken for granted. A companion paper investigates the approximation of the critical excitation that allows its simple determination.     

Stochastic Dynamic Analysis of Inelastic Structures Using Force Analogy Method

The force analogy method which has been proven to be very efficient in dynamic analysis of inelastic structures is here introduced for the first time into the field of stochastic dynamic analysis for inelastic structures. This stochastic force analogy method (SFAM) maintains the advantage of the high efficiency in the numerical computation of the force analogy method in dynamic analysis. According to the SFAM, the variance covariance functions of inelastic dynamic responses, such as displacement, velocity, inelastic displacement of the entire moment-resisting framed structures, and plastic rotation at individual plastic hinge location, can be produced for structures subject to random excitation. Detailed theoretical development of the SFAM is derived, and a simple numerical example using a single degree of freedom system is presented. The reasonability of the proposed method is validated by the good agreement between the results from the proposed SFAM and those obtained from Monte Carlo simulation.     

Bayesian Updating of Structural Models and Reliability using Markov Chain Monte Carlo Simulation

In a full Bayesian probabilistic framework for “robust” system identification, structural response predictions and performance reliability are updated using structural test data D by considering the predictions of a whole set of possible structural models that are weighted by their updated probability. This involves integrating h(θ)p(θ∣D) over the whole parameter space, where θ is a parameter vector defining each model within the set of possible models of the structure, h(θ) is a model prediction of a response quantity of interest, and p(θ∣D) is the updated probability density for θ, which provides a measure of how plausible each model is given the data D. The evaluation of this integral is difficult because the dimension of the parameter space is usually too large for direct numerical integration and p(θ∣D) is concentrated in a small region in the parameter space and only known up to a scaling constant. An adaptive Markov chain Monte Carlo simulation approach is proposed to evaluate the desired integral that is based on the Metropolis-Hastings algorithm and a concept similar to simulated annealing. By carrying out a series of Markov chain simulations with limiting stationary distributions equal to a sequence of intermediate probability densities that converge on p(θ∣D), the region of concentration of p(θ∣D) is gradually portrayed. The Markov chain samples are used to estimate the desired integral by statistical averaging. The method is illustrated using simulated dynamic test data to update the robust response variance and reliability of a moment-resisting frame for two cases: one where the model is only locally identifiable based on the data and the other where it is unidentifiable.     

Efficient Evaluation of Reliability for Slopes with Circular Slip Surfaces Using Importance Sampling

Evaluating the reliability of a slope is a challenging task because the possible slip surface is not known beforehand. Approximate methods via the first-order reliability method provide efficient ways of evaluating failure probability of the “most probable” failure surface. The tradeoff is that the failure probability estimates may be biased towards the unconservative side. The Monte Carlo simulation (MCS) is a viable unbiased way of estimating the failure probability of a slope, but MCS is inefficient for problems with small failure probabilities. This study proposes a novel way based on the importance sampling technique of estimating slope reliability that is unbiased and yet is much more efficient than MCS. In particular, the critical issue of the specification of the importance sampling probability density function will be addressed in detail. Three examples of slope reliability will be used to demonstrate the performance of the new method.     

Structural Reliability Estimation with Vibration-Based Identified Parameters

This paper presents a unique structural reliability estimation method incorporating structural parameter identification results based on the seismic response measurement. In the shaking table test, a three-bent concrete bridge model was shaken to different damage levels by a sequence of earthquake motions with increasing intensities. Structural parameters, stiffness and damping values of the bridge were identified under damaging seismic events based on the seismic response measurement. A methodology was developed to understand the importance of structural parameter identification in the reliability estimation. Along this line, a set of structural parameters were generated based on the Monte Carlo simulation. Each of them was assigned to the base bridge model. Then, every bridge model was analyzed using nonlinear time history analyses to obtain damage level at the specific locations. Last, reliability estimation was performed for bridges modeled with two sets of structural parameters. The first one was obtained by the nonlinear time history analysis with the Monte Carlo simulated parameters which is called nonupdated structural parameters. The second one was obtained by updating the first set in Bayesian sense based on the vibration-based identification results which is called updated structural parameters. In the scope of this paper, it was shown that residual reliability of the system estimated using the updated structural parameters is lower than the one estimated using the nonupdated structural parameters.     

Bounds on System Reliability by Linear Programming

Bounds on system probability in terms of marginal or joint component probabilities are of interest when exact solutions cannot be obtained. Currently, bounding formulas employing unicomponent probabilities are available for series and parallel systems, and formulas employing bi- and higher-order component probabilities are available for series systems. No theoretical formulas exist for general systems. It is shown in this paper that linear programming (LP) can be used to compute bounds for any system for any level of information available on the component probabilities. Unlike the theoretical bicomponent and higher-order bounds, the LP bounds are independent of the ordering of the components and are guaranteed to produce the narrowest possible bounds for the given information. Furthermore, the LP bounds can incorporate any type of information, including an incomplete set of component probabilities or inequality constraints on component probabilities. Numerical examples involving series, parallel and general structural systems are used to demonstrate the methodology.     

Quasi-Monte Carlo Simulation with Low-Discrepancy Sequence for Reinforced Soil Slopes

This paper describes a precise numerical technique to compute the limit state exceedance probability of geosynthetic reinforced soil (GRS) slopes with normally distributed backfill and foundation soils by using the low-discrepancy sequence Monte Carlo (LDSMC) and importance sampling with LDSMC (ISLDSMC) methods. The LDSMC and ISLDSMC methods can effectively compute an accurate limit state exceedance probability of GRS slopes with a limited number of simulations. By using importance sampling, random variables can be generated in an expected failure region, thereby enabling enumeration by the Monte Carlo simulation. The failure region can be searched by the conventional first-order reliability method. To increase the computational efficiency, a low-discrepancy sequence, which is a sequence of quasi-random numbers with uniform distribution, is adopted in this study. The numerical simulation in this study revealed that the LDSMC and ISLDSMC methods can effectively compute an accurate limit state exceedance probability of GRS slopes by performing comparatively fewer simulations than the conventional crude Monte Carlo simulation.     

Finite Element-Based Structural Reliability Assessment Using Efficient Directional Simulation

Reliability analysis of structural systems often requires finite element (FE)-based simulation to estimate failure probabilities. Common simulation methods, even those incorporating variation reduction techniques, usually involve a very large number of FE analyses to achieve acceptable accuracy. A recently developed directional approach significantly improves the efficiency of directional simulation by utilizing deterministic point sets to preserve the underlying joint probability distribution of the random vector describing the structure and by employing neural networks to focus the simulation effort in the significant regions. This paper investigates the application of this method to structural system reliability analysis. The method is illustrated using deformation-based system limit states, proposed for performance-based engineering, for two plane steel frames.     

Consistent Finite-Element Response Sensitivity Analysis

This paper examines the important issue of response sensitivities of dynamic models of structural systems to both material and (discrete) loading parameters. Plasticity-based finite-element models of structural systems subjected to base excitation such as earthquake loading are considered. The two methods for computing the response sensitivities, namely, (1) discretizing in time the time continuous-spatially discrete response equations and differentiating the resulting time discrete-spatially discrete response equations with respect to sensitivity parameters, and (2) differentiating the time continuous-spatially discrete response equations with respect to sensitivity parameters and discretizing in time the resulting time continuous-spatially discrete response sensitivity equations, are clearly distinguished. The discontinuities in time of the response sensitivities arising due to material state transitions in the plasticity models, and their propagation from the quadrature point level to the global structural response level are discussed using a specific one-dimensional plasticity model. The procedure to obtain the exact sensitivities of the numerical nonlinear finite-element response, including proper capture of their discontinuities, is formalized. Application examples illustrating the concepts are presented at the end.     

Failure Analysis of Transmission Line Towers

Transmission line towers, though designed per code provisions, may fail during mandatory testing required in many countries. Different types of premature failures that were observed during full-scale testing of transmission line towers at Tower Testing and Research Station, Structural Engineering Research Centre, Chennai (CSIR-SERC) are studied, and the results are discussed in detail. The failures are modeled using finite-element software, and the analytical results and the test results are compared with various code provisions. The nonlinear finite-element analysis program NE-Nastran was used to model the elastoplastic behavior of towers. Bracing members with slenderness ratios above 170 become ineffective, even though they have to carry insignificant forces. Importance of design assumptions and connection detailing in overall performance of towers were studied. Nonlinear finite-element analysis is useful in understanding the system behavior and for prediction of the failure pattern and ultimate load. Based on the test results, the importance of studying these failures is highlighted and significant conclusions were drawn.     

Bayesian Network Enhanced with Structural Reliability Methods: Application

The enhanced Bayesian network (eBN) methodology described in the companion paper facilitates the assessment of reliability and risk of engineering systems when information about the system evolves in time. We present the application of the eBN: (1) to the assessment of the life-cycle reliability of a structural system; (2) to the optimization of a decision on performing measurements in that structural system; and (3) to the risk assessment of an infrastructure system subject to natural hazards and deterioration of constituent structures. In all applications, observations of system performances or the hazards are made at various points in time and the eBN efficiently includes these observations in the analysis to provide an updated probabilistic model of the system at all times.     

Reliability Analysis of Laterally Loaded Piles Involving Nonlinear Soil and Pile Behavior

An alternative approach of analyzing laterally loaded piles in the ubiquitous spreadsheet platform is presented. The numerical procedure couples nonlinear pile flexural rigidity (EpIp) with nonlinear p-y analysis. The deterministic study is then extended to carry out reliability analysis, which reflects the uncertainties and correlation structure of the underlying parameters. The reliability index is evaluated based on the alternative intuitive perspective of an expanding equivalent ellipsoid in the original space of the random variables. This paper investigates two modes of failure: deflection and bending moment, and considers non-normal random variables. Spatial variability of the soil medium is accounted for by incorporating an autocorrelation model. The spreadsheet-based reliability approach can also be coupled with stand-alone programs via the response surface method. The probabilities of failure inferred from reliability indices agree well with Monte Carlo simulations. Simple reliability-based design is demonstrated, in which the appropriate pile section or length that satisfies target reliability in one or more limit states is sought.     

Bayesian Model Updating Using Hybrid Monte Carlo Simulation with Application to Structural Dynamic Models with Many Uncertain Parameters

In recent years, Bayesian model updating techniques based on measured data have been applied to system identification of structures and to structural health monitoring. A fully probabilistic Bayesian model updating approach provides a robust and rigorous framework for these applications due to its ability to characterize modeling uncertainties associated with the underlying structural system and to its exclusive foundation on the probability axioms. The plausibility of each structural model within a set of possible models, given the measured data, is quantified by the joint posterior probability density function of the model parameters. This Bayesian approach requires the evaluation of multidimensional integrals, and this usually cannot be done analytically. Recently, some Markov chain Monte Carlo simulation methods have been developed to solve the Bayesian model updating problem. However, in general, the efficiency of these proposed approaches is adversely affected by the dimension of the model parameter space. In this paper, the Hybrid Monte Carlo method is investigated (also known as Hamiltonian Markov chain method), and we show how it can be used to solve higher-dimensional Bayesian model updating problems. Practical issues for the feasibility of the Hybrid Monte Carlo method to such problems are addressed, and improvements are proposed to make it more effective and efficient for solving such model updating problems. New formulae for Markov chain convergence assessment are derived. The effectiveness of the proposed approach for Bayesian model updating of structural dynamic models with many uncertain parameters is illustrated with a simulated data example involving a ten-story building that has 31 model parameters to be updated.     

Bayesian Network Enhanced with Structural Reliability Methods: Methodology

We combine Bayesian networks (BNs) and structural reliability methods (SRMs) to create a new computational framework, termed enhanced BN (eBN), for reliability and risk analysis of engineering structures and infrastructure. BNs are efficient in representing and evaluating complex probabilistic dependence structures, as present in infrastructure and structural systems, and they facilitate Bayesian updating of the model when new information becomes available. On the other hand, SRMs enable accurate assessment of probabilities of rare events represented by computationally demanding physically based models. By combining the two methods, the eBN framework provides a unified and powerful tool for efficiently computing probabilities of rare events in complex structural and infrastructure systems in which information evolves in time. Strategies for modeling and efficiently analyzing the eBN are described by way of several conceptual examples. The companion paper applies the eBN methodology to example structural and infrastructure systems.     

New Multidimensional Visualization Technique for Limit-State Surfaces in Nonlinear Finite-Element Reliability Analysis

Structural reliability problems involving the use of advanced finite-element models of real-world structures are usually defined by limit-states expressed as functions (referred to as limit-state functions) of basic random variables used to characterize the pertinent sources of uncertainty. These limit-state functions define hyper-surfaces (referred to as limit-state surfaces) in the high-dimensional spaces of the basic random variables. The hyper-surface topology is of paramount interest, particularly in the failure domain regions with highest probability density. In fact, classical asymptotic reliability methods, such as the first- and second-order reliability method (FORM and SORM), are based on geometric approximations of the limit-state surfaces near the so-called design point(s) (DP). This paper presents a new efficient tool, the multidimensional visualization in the principal planes (MVPP) method, to study the topology of high-dimensional nonlinear limit-state surfaces (LSSs) near their DPs. The MVPP method allows the visualization, in particularly meaningful two-dimensional subspaces denoted as principal planes, of actual high-dimensional nonlinear limit-state surfaces that arise in both time-invariant and time-variant (mean out-crossing rate computation) structural reliability problems. The MVPP method provides, at a computational cost comparable with SORM, valuable insight into the suitability of FORM/SORM approximations of the failure probability for various reliability problems. Several application examples are presented to illustrate the developed MVPP methodology and the value of the information provided by visualization of the LSS.     

Analysis of Regulatory Guidance Values for Residential Surface Soil Arsenic Exposure

Residential surface soil regulatory guidance values (RGVs) specify the threshold at which soil contamination requires action. Usually, RGVs are risk-based values based on child ingestion, inhalation, and dermal exposure. The U.S. Environmental Protection Agency, 45 U.S. states, and 27 other nations have developed arsenic surface soil RGVs. Regulating arsenic poses unusual problems because it presents both cancer and noncancer risks, and its background concentration often exceeds health-based risk levels. Statistical analyses are presented to characterize 119 arsenic surface soil RGVs. State values vary between 0.039 and 200 mg/kg. Worldwide values vary between 1.7 and 687 mg/kg. The U.S. and worldwide values resemble lognormal probability distributions but the data cannot be mingled since worldwide values are significantly higher. An analysis of 40 arsenic background studies yielding averages between 1.3 and 45.1 mg/kg is also presented. Monte Carlo simulations of screening model calculations are used to explore the impact of coefficient uncertainty. Results indicate that 95% of cancer-based results should fall between 0.004 and 2.7 mg/kg and 95% of noncancer results should fall between 1.0 and 40 mg/kg. Although U.S. state arsenic RGVs vary by 3.7 orders of magnitude, most values appear to fall within the bounds of plausible risk- and background-based values.