Failure Domain Research Articles

An orthotropic multi-surface damage-plasticity FE-formulation for wood: Part I – Constitutive model

Restoration of ancient wooden beams and design of smart wood-based structures are gaining an increasing interest in building industry. In this context, the computational challenge is to develop numerical constitutive models that account for the complex and strongly non-linear behaviour of wood. Wood is a natural composite exhibiting pronounced orthotropic behaviour, and markedly different properties along the parallel and transverse-to-the-fiber directions. It displays a strongly non-linear mechanical behavior, almost elasto-plastic at compressive loadings and elasto-damaging when subjected to tensile and shearing stresses. We devise here a novel constitutive model for wood with a multi-surface failure domain resulting from a set of plastic laws for compressive failure modes and orthotropic damage laws for tensile/shear failure modes. The advantage over existing formulations is that the coexistence of anisotropic damage and plasticity constitutive laws allows to correctly capture brittle failure induced by strain localization as well as the possible occurrence of ductile plastic behavior. Furthermore, the present contribution shows how to numerically treat the simultaneous presence of anisotropic damage and plasticity in a general algorithmic multi-surface framework. It is shown that the obtained numerical results satisfactorily fit to experimental data.

Plastic hinges mechano-reliability analysis in the beams of RC frames structures

Purpose The purpose of this study is to establish a relationship between causes and effects, the respect of materials characteristics values [concrete compressive strength (fc) and steel yield stress (fy)] and the norms of the construction dispositions value (covers). This study is motivated by the post-seismic damages related to the plastification of the reinforced concrete (RC)/beams sections, named plastic hinges. The results are given by fragility curves representing the failure probability (Pf) of the plastic hinges versus covers value. Design/methodology/approach A mechanical-reliability coupling methodology is proposed and performed on three frames (three, six and nine storey). For each frame, seven covers the value of reinforcement steel bars has been taken into account in the beams. After definition of the limit state function G(x), a process of idea to twin-track; deterministic and probabilistic, is considered. Thus, numerical simulations are carried out under ETABS© software, to extract a soliciting moments Ms(x). Then, ultimate moments Mu(x), the result of reliability approach are calculated using Monte Carlo Simulations. In this step, two random variables; concrete compressive strength in 28 days of age (fc) and steel yield stress (fy), have been studied. Findings In the mechanical study, the results show that, the first plastic hinge appears at the beams for all frames. In the reliability study, the (fy) variation shows that all plastic hinges are in failure domain, nevertheless, the (fc) variation leads to have all sections in the safety domain, except A7 and B7 models. The failure probability (Pf) calculation according to (fc) and (fy) shows that an absolute error of 0.5 cm in the steel bars covers can switch the frame from the safety domain to the failure domain. Originality/value The plastic hinges reliability of the RC/ frame structures is independent on the high of the structure. The (fc) random variable according to the used distribution law does not affect the reliability (safety or failure). However, the impact of the steel yield stress variation (fy) is not negligible. The errors in covers affect considerably the strength of the elements.

Reliability analysis of pumping station for sewage network using hybrid neural networks - genetic algorithm and method of moment

The reliability of pumping stations is of great importance for the robust design of wastewater networks. In order to strike a balance between safe design and energy consumption of the pumping system, a complex performance function using multiple-failure mode under various uncertainties is required. In the present research to evaluate the failure probability of wastewater pumping station, a hybrid reliability analysis framework using artificial neural network (ANN) coupled by moment methods is proposed. Drawing upon genetic algorithm (GA), ANN model is trained to approximate the failure domain of the pump. The ANN is approximated input variables captured by the optimal condition of the rotational speed of the pump in turn it is obtained by GA. The main strength of the reliability method is to diminish the computational burden with accurate predictions of safety margin in the pumping systems of Zabol. The machine learning-based backpropagation (BP) for training feedforward neural networks using GA and gradient methods are compared for the regressed process of ANN models. This hybrid reliability framework involves three main levels including i) the pump rotational speed is minimized using GA as optimal hydraulic parameters such as inlet flow, static head, pumping head and outlet flow rate, ii) the limit sate function is approximated using ANN for optimal rotating speed pumps and iii) reliability analysis is disused using MCS and method of the moment for sewage pumping stations located at Zabol (Iran) station. Given the results, the machine learning-based GA for training ANN model provides accurate predictions compared to the ANN-based gradient method. Three moments of reliability method is an efficient and accurate approach to evaluate the reliable conditions of the pumping system.

A novel dual-stage adaptive Kriging method for profust reliability analysis

Profust reliability analysis (RA) is established on probability assumption for the model inputs and fuzzy state assumption, and it is a useful tool to measure the safety degree of the structure with fuzzy failure state. In order to overcome the inefficiency of the existing AK-MCS method in profust RA for the structure with small profust failure probability, a novel dual-stage adaptive Kriging (DS-AK) method is proposed. Firstly, to improve the sampling efficiency, the importance sampling (IS) is used to estimate the profust failure probability where the optimal importance sampling density (OISD) is derived in this paper. As the analytical expression of the OISD cannot be acquired, the Kriging model for the true performance function is employed to build an approximate OISD. Then, the profust failure probability can be expressed as the product of the augmented profust failure probability and the correction factor, which are estimated in two different stages of the proposed DS-AK method. In the first stage of the DS-AK method, a Kriging model is constructed and updated to obtain the approximate OISD and generate the IS samples, and the augmented profust failure probability can be estimated as a byproduct. In the second stage of the DS-AK method, the current Kriging model is continuously updated to accurately predict the membership functions of the fuzzy failure domain at the IS samples generated in the first stage, from which the correction factor can be calculated efficiently. Finally, the fuzzy failure probability can be estimated as the product of augmented failure probability estimate obtained in the first stage and the correction factor estimate acquired in the second stage. Results of the validation cases demonstrate the accuracy, efficiency and robustness of the proposed method in estimating profust failure probability.

Extension of AK-MCS for the efficient computation of very small failure probabilities

We consider the problem of estimating a probability of failure pf, defined as the volume of the excursion set of a complex (e.g. output of an expensive-to-run finite element model) scalar performance function J below a given threshold, under a probability measure that can be recast as a multivariate standard gaussian law using an isoprobabilistic transformation. We propose a method able to deal with cases characterized by multiple failure regions, possibly very small failure probability pf (say ∼10−6−10−9), and when the number of evaluations of J is limited. The present work is an extension of the popular Kriging-based active learning algorithm known as AK-MCS, as presented in [1], permitting to deal with very low failure probabilities. The key idea merely consists in replacing the Monte-Carlo sampling, used in the original formulation to propose candidates and evaluate the failure probability, by a centered isotropic Gaussian sampling in the standard space, whose standard deviation is iteratively tuned. This extreme AK-MCS (eAK-MCS) inherits its former multi-point enrichment algorithm allowing to add several points at each iteration step, and provide an estimated failure probability based on the Gaussian nature of the Kriging surrogate.Both the efficiency and the accuracy of the proposed method are showcased through its application to two to eight dimensional analytic examples, characterized by very low failure probabilities: pf∼10−6−10−9. Numerical experiments conducted with unfavorable initial Design of Experiment suggests the ability of the proposed method to detect failure domains.

Implementation of framework for assessment of severe accident management effectiveness in Nordic BWR

Nordic Boiling Water Reactor (BWR) design employs ex-vessel debris coolability in a deep pool of water as a severe accident management (SAM) strategy. Depending on melt release conditions from the vessel and core-melt coolant interactions, containment integrity can be threatened by (i) formation of non-coolable debris bed, or (ii) energetic steam explosion.In order to assess the effectiveness of SAM the Risk Oriented Accident Analysis Methodology framework (ROAAM+) has been developed. The framework is further extension of the approach originally developed and applied by Prof. Theofanous and co-workers.This paper presents the implementation of ROAAM+ probabilistic framework for uncertainty quantification and risk analysis.We further apply ROAAM+ to the analysis of steam explosion risk in Nordic BWR assuming different state-of-knowledge situations and different containment fragilities. We employ an iterative processes of knowledge refinement using risk analysis as a guiding tool in identification of the major sources of uncertainty. We estimate failure domains and discuss ROAAM+ results in terms of possibility vs necessity of containment failure.

An efficient method based on AK-MCS for estimating failure probability function

The function of failure probability varying with distribution parameters of random inputs is referred as failure probability function (FPF), it is often required in reliability-based design optimization. However, it is a computational challenging task since large number of expensive model evaluations are needed to estimate the failure probability at every distribution parameter. A method combining adaptive Kriging with Monte Carlo simulation is employed by this paper to efficiently estimate the FPF. Based on the augmented reliability theory and Bayes’ rule, the estimation of FPF is firstly transformed into that of the augmented failure probability and conditional joint probability density function (PDF) of distribution parameters on failure domain. Then, a Kriging model for the actual performance function is iteratively trained by the U-learning function until the stopping criterion is satisfied. The well-trained Kriging model can efficiently and accurately recognize the failure samples in the sample pool, on which the augmented failure probability and conditional joint PDF of distribution parameters on failure domain can be simultaneously estimated without extra model evaluations. The results of test examples illustrate that the method used in this work is more efficient than the existing methods, but its accuracy depends on the PDF approximation algorithms.

An efficient approach for computing analytical non-parametric fragility curves

Fragility curves are used in earthquake engineering for assessing the seismic vulnerability of structures or systems. Direct estimations of fragility curves by means of simulation-based approaches lead generally to relevant computational costs, especially when the failure region is characterized by small probabilities of occurrence. Simplified hypotheses are therefore introduced in the common practice to approximate the dependency between the structural response and the associated seismic intensity level.The study proposes a non-parametric methodology to estimate analytical fragility curves without specific assumptions on their final shape. The approach starts by identifying all the subsets characterized by the same values of the chosen seismic intensity measure parameter. Then, the failure region is mapped by means of a classification algorithm coupled with a polynomial kernel. Finally, the conditional failure probability is computed by associating the samples generated in each subset to the corresponding classification score. A stochastic earthquake model is employed to define the seismic dataset and the uncertainty associated with the ground motion records.Two case studies are analyzed in which the non-parametric methodology is compared against three popular parametric approaches and a reference solution. The proposed approach shows an overall higher accuracy and efficiency, especially in case of rare failure domains.

Subset simulation method including fitness-based seed selection for reliability analysis

Probability estimation of rare events is a challenging task in the reliability theory. Subset simulation (SS) is a robust simulation technique that transforms a rare event into a sequence of multiple intermediate failure events with large probabilities and efficiently approximates the mentioned probability. Proper handling of a reliability problem by this method requires employing a suitable sampling approach to transmit samples toward the failure set. Markov Chain Monte Carlo (MCMC) is a suitable sampling approach that solves the SS transition phase using the failed sample of each simulation level as the seed of next samples. This paper is aimed to study the seed selection effect on the SS accuracy through several seed selection approaches inspired by the genetic algorithm and particle filter and using the main PDF of the variables to assign a mass function probability to each subset sample in the failure domain. Roulette wheel (I, II), tournament and proportional probability techniques are then employed to choose the weighed samples as seeds to be placed in the MCMC to transmit the samples. To examine the capability of each approach, reliabilities of some engineering problems were investigated and results showed that the proposed approaches could find proper failure sets better than the original SS method, especially in problems with several failure domains.

An adaptive trivariate dimension-reduction method for statistical moments assessment and reliability analysis

An adaptive trivariate dimension-reduction method is proposed for statistical moments evaluation and reliability analysis in this paper. First, the raw moments of the performance function can be estimated by means of the trivariate dimension-reduction method, where the trivariate, bivariate and univariate Gaussian-weighted integrals are involved. Since the trivariate and bivariate integrals control the efficiency and accuracy, delineating the existence of bivariate and trivariate cross terms is performed, which could significantly reduce the numbers of trivariate and bivariate integrals to be evaluated. When the cross terms exist, the trivariate and bivariate integrals are numerically evaluated directly by the high-order unscented transformation, where the involved free parameters are provided. When the cross terms don’t exist, the trivariate and bivariate integrals can be further decomposed to be the lower-dimensional integrals, where the high-order unscented transformation is again adopted for numerical integrations. In that regard, the first-four central moments can be computed accordingly and the performance function’s probability density function can be reconstructed by fitting the shifted generalized lognormal distribution model based on the first-four central moments. Then, the failure probability can be computed by a one-dimensional integral over the performance function’s probability density function in the failure domain. Three numerical examples, including both the explicit and implicit performance functions, are investigated, to demonstrate the efficacy of the proposed method for both the statistical moments assessment and reliability analysis.

New approach for face stability assessment of tunnels driven in nonuniform soils

Abstract A novel approach to assess the face stability of a plane strain tunnel driven in nonuniform soils is proposed herein. The failure mechanism of a tunnel face is established by a discretization-based approach within the framework of the limit analysis method. It is generated point by point and the vertical length of each element is fixed, which is beneficial for addressing stability problems involving depth-dependent soil parameters. The cohesion, internal friction angle, and unit weight of soils are modelled as varying with depth, which is closer to the true characteristics of the soils. First, the proposed approach is degraded to a conventional upper bound analysis by considering a uniform soil case to validate the proposed approach. A practical problem concerning longitudinal weak strata positioned ahead of the tunnel face is investigated using the presented approach. To highlight the effects on the critical support pressure and failure domain of the tunnel face, the thicknesses, positions, and soil parameters of weak strata are further discussed. Finally, numerical simulations are performed to demonstrate the accuracy of the proposed approach in the face stability assessment of tunnels affected by weak strata.

Application of directional importance sampling for estimation of first excursion probabilities of linear structural systems subject to stochastic Gaussian loading

This contribution addresses the estimation of first excursion probabilities of linear structural systems subject to stochastic Gaussian loading by means of simulation. This probability is estimated by combining existing knowledge on the geometry of the associated failure domain with Directional Importance Sampling. In this way, the space associated with the stochastic loading is explored by generating some random directions according to a prescribed importance sampling distribution; then, each random direction is analyzed taking advantage of the linearity of the response with respect to the stochastic loading. Such an approach allows estimating small failure probabilities with high accuracy and precision while requiring a reduced number of samples. The application of Directional Importance Sampling is illustrated by means of a series of examples, indicating that failure probabilities in the order of 10-3 or less can be estimated reliably with a reduced number of samples, even in problems comprising involved structural models.

Probabilistic safety model and its efficient solution for structure with random and interval mixed uncertainties

Random and interval mixed uncertainties (RIMU) universally exist in engineering, thus it is necessary to construct a reasonable model for analyzing the probabilistic safety degree in this case. Aiming at analyzing the probabilistic safety degree efficiently, the existing double-loop nested optimization (DLNO) algorithm for estimating the upper and lower bounds of failure probability (FP) is firstly investigated under RIMU. It is proved that the upper and lower bounds of FP obtained by DLNO actually equal to the maximum and minimum of all approximate FPs with respect to the possible discrete points of interval variables. Secondly, it is found that the failure domain corresponding to the upper bound of FP is similar to that of time-dependent reliability under random uncertainty. Based on this similarity, the inequality relations are established for the real upper and lower bounds of FP and those obtained by DLNO, and the Kriging surrogate model methods are proposed to solve probabilistic safety model under RIMU efficiently. Additionally, this efficient solution strategy is further extended to the time-dependent structure under RIMU. Finally, numerical and engineering examples are used to demonstrate the reasonability and efficiency of proposed methods.

Algorithms for optimal replica placement under correlated failure in hierarchical failure domains

In data centers, data replication is the primary method used to ensure availability of customer data. To avoid correlated failure, cloud storage infrastructure providers model hierarchical failure domains using a tree, and avoid placing a large number of data replicas within the same failure domain (i.e. on the same branch of the tree). Typical best practices ensure that replicas are distributed across failure domains, but relatively little is known concerning optimization algorithms for distributing data replicas. Using a hierarchical model, we answer how to distribute replicas across failure domains optimally. We formulate a novel optimization problem for replica placement in data centers. As part of our problem, we formalize and present a new criterion for optimizing a replica placement. Our overall goal is to choose placements in which correlated failures disable as few replicas as possible.In this work, we provide two optimization algorithms for dependency models represented by trees. We first present an O(n+ρlog⁡ρ) time dynamic programming algorithm for optimally placing ρ replicas of a single block on the leaves (representing servers) of a tree with n vertices. We next consider the problem of optimally placing replicas of multiple blocks of data, where every block may have a different replication factor. For this problem, we give a dynamic programming algorithm that runs in O(nρmax3δ2mpoly(δ)), where m denotes the number of blocks, ρmax denotes the maximum replication factor of a block, and δ denotes the maximum difference in the replication factors of any two blocks. The running time of the algorithm is polynomial when the δ, which we refer to as the skew, is a constant.

Reliability sensitivity analysis with subset simulation: application to a carbon dioxide storage problem

Reliability sensitivity analysis (RSA) is a sensitivity analysis to measure the effect of modelling parameters on the predicted reliability of a system. It can be used for reliability-based design, safety management, etc. The output-classification-based version of RSA compares the failure-conditional probability density function (PDF) of model parameters with their unconditional PDF to measure sensitivity. The main challenge is to estimate failure-conditional PDFs. Usually, these PDFs can be estimated through the failure samples obtained by Monte Carlo simulation. However, practical systems usually have a small failure probability. For such cases, the brute-force Monte Carlo simulation requires a larger number of samples to obtain enough failure samples. Therefore, the computational cost is very high. In this paper, we propose to use subset simulation to estimate the output-classification-based reliability sensitivity index. Subset simulation introduces a series of intermediate failure events which are easier to sample from, and then iteratively samples in each constrained failure region until the target failure event is reached. Compared to brute-force Monte Carlo simulation, subset simulation samples in a direction towards the target failure domain. Therefore, the failure samples can be obtained more efficiently. We apply subset simulation to perform RSA for a carbon dioxide storage benchmark problem. We show that subset simulation can estimate the output-classification-based reliability sensitivity index more efficiently compared to brute-force Monte Carlo simulation.

An efficient method for estimating failure probability of the structure with multiple implicit failure domains by combining Meta-IS with IS-AK

For efficiently estimating the failure probability of the structure with multiple implicit failure domains, a method abbreviated as Meta-IS-AK is proposed by combining the adaptive Kriging Meta model Importance Sampling (Meta-IS) and Importance Sampling based Adaptive Kriging (IS-AK). In the proposed method, the failure probability is equivalently expressed as a product of the augmented failure probability and the correction factor, then two steps are respectively established to solve two terms. In the first step, Meta-IS algorithm is executed to generate IS samples. The augmented failure probability can be estimated as a byproduct in the first step. In the second step, all these IS samples compose a sample pool, in which the AK model is subsequently reconstructed for accurately predicting failure domain indicators instead of the actual implicit limit state function. Then the failure domain indicator at each IS sample and further the correction factor can be efficiently estimated. From the strategy of the proposed method, it can be seen that the proposed Meta-IS-AK possesses both the advantages of the Meta-IS method suitable for multiple failure domains and efficiency of the AK model for accurately predicting the failure domain indicators at all IS samples, which is demonstrated by the numerical and engineering examples.

Time-dependent reliability analysis model under fuzzy state and its safety lifetime model

In view of the lack of time-dependent reliability analysis model (TDRA) under fuzzy state, which is ubiquitous in engineering, a TDRA model under fuzzy state is proposed in this paper, followed by the corresponding safety lifetime model. To establish the TDRA model under fuzzy state, this paper firstly transforms the TDRA model under binary state into a form expressed by the time-dependent failure domain indicator function, and then the TDRA model under fuzzy state is derived based on the basic principle of the time-independent reliability analysis (TIRA) model under fuzzy state. By introducing an auxiliary variable and establishing the time-dependent generalized performance function, the TDRA model under fuzzy state is transformed into a generalized one under binary state, where the failure domain and safety domain are clearly defined. Then, the single-loop Kriging (SLK) surrogate model approach is used to efficiently estimate the time-dependent failure probability (TDFP) in the special service time interval under fuzzy state. Based on the generalized TDRA model and its efficient estimation, a safety lifetime model constrained by the target TDFP under fuzzy state and a corresponding efficient solving method are presented. Finally, examples are used to verify the rationality of the TDRA model and the safety lifetime model under fuzzy state established in this paper, and the efficiency of the algorithm is also validated.

Improved cross entropy-based importance sampling with a flexible mixture model

The probability of a rare event or failure event is defined through a potentially high-dimensional integral, whose integration domain is often only known point-wise in terms of the outcome of a numerical model. The probability of failure can be estimated efficiently through importance sampling (IS), provided that an effective IS density is chosen. The cross entropy (CE) method is an adaptive sampling approach that determines the IS density through minimizing the Kullback–Leibler divergence between the theoretically optimal IS density and a chosen parametric family of distributions. We propose an improved version of the classical CE method that introduces a smooth transition to make better use of the samples from intermediate sampling levels for fitting the sought IS density. The improved CE method is combined with a novel flexible parametric distribution model that is able to handle low- and high-dimensional problems as well as problems with multimodal failure domains. A set of numerical examples demonstrate that the proposed approach performs consistently better than the classical CE method in various problem settings.

Global reliability sensitivity estimation based on failure samples

Global reliability sensitivity analysis (RSA) can help to assess the effects of input random variables X on the probability of failure Pr(F) of an engineering system. Conventionally, this requires repeated evaluations of the conditional failure probability Pr(F|Xi = xi) for multiple values of the input random variable Xi and for all Xi of interest. Such a solution is straightforward but computationally expensive. In this paper, we propose a new method to perform global RSA, which requires as an input only samples of X that fall in the failure domain. Such samples are a by-product of many sampling-based reliability analysis methods. The proposed method constructs the Pr(F|Xi = xi) by application of Bayes’ rule, based on the probability density function (PDF) of X conditioned on system failure F. This conditional PDF is approximated with a kernel density estimation from the failure samples. In this way, the reliability sensitivities of all the input random variables can be computed following a sampling-based reliability analysis with no additional computation cost. The approach is investigated on numerical examples in conjunction with crude Monte Carlo simulation, importance sampling and subset simulation. The results demonstrate the computational advantages over existing single-loop sampling methods for global RSA.

Full probabilistic design of earth retaining structures using generalized subset simulation

Reliability-based design (RBD) of earth retaining structures has attracted considerable attentions due to several distinctive elements, such as correlated load and resistance, multiple correlated failure modes, etc. These elements can be rationally considered using full probabilistic design approaches based on Monte Carlo simulation (MCS). However, MCS-based full probabilistic design approaches are often criticized because of a lack of computational efficiency and resolution at small probability levels. This paper develops an efficient MCS-based full probabilistic design approach, which directly makes use of Generalized Subset Simulation (GSS) to perform RBD of earth retaining structures. The proposed method allows simultaneously approaching failure domains of different possible designs in a prescribed design space by a single GSS run and generating their corresponding failure samples in an efficient manner. As a result, a large number of failure samples can be generated for numerical representation of failure domains of possible designs. Then, the failure probabilities of possible designs are calculated for identifying feasible designs and the design point of each design can be approximately determined as the most probable failure sample of the design according to its probabilistic interpretation. Finally, the proposed approach is illustrated through two design examples of gravity retaining wall.