Conditional Probability Of Failure Research Articles

Variational Bayesian Monte Carlo simulation for rare event assessment

Estimating rare events in structural engineering demands intensive computational resources, and the complexity of the performance function poses challenges to accurate probability estimation. To efficiently conduct the structural reliability analysis on rare events, we propose a novel variational Bayesian Monte Carlo (VBMC)-based subset simulation (SS) method. The proposed approach can be divided into two aspects: the Monte Carlo simulation (MCS) is first applied to estimate the first-layer unconditional failure probability, then VBMC is employed to estimate subsequent conditional failure probabilities. Finally, the original small failure probability is obtained by the product of the first-layer unconditional failure probability and a series of conditional failure probabilities. The proposed approach inherits the merits of SS as well as VBMC, which converts the tricky rare event estimation into a series of high-frequency event estimations and leverages VBMC instead of the canonical Markov chain to produce the independent and informative next-generation failure-conditional samples. Four case studies, including high-dimensional and discrete state space scenarios, are performed to illustrate the feasibility and generality of the proposed method.

Reference prior for Bayesian estimation of seismic fragility curves

One of the key elements of probabilistic seismic risk assessment studies is the fragility curve, which represents the conditional probability of failure of a mechanical structure for a given scalar measure derived from seismic ground motion. For many structures of interest, estimating these curves is a daunting task because of the limited amount of data available; data which is only binary in our framework, i.e., only describing the structure as being in a failure or non-failure state. A large number of methods described in the literature tackle this challenging framework through parametric log-normal models. Bayesian approaches, on the other hand, allow model parameters to be learned more efficiently. However, the impact of the choice of the prior distribution on the posterior distribution cannot be readily neglected and, consequently, neither can its impact on any resulting estimation. This paper proposes a comprehensive study of this parametric Bayesian estimation problem for limited and binary data. Using the reference prior theory as a cornerstone, this study develops an objective approach to choosing the prior. This approach leads to the Jeffreys prior, which is derived for this problem for the first time. The posterior distribution is proven to be proper (i.e., it integrates to unity) with the Jeffreys prior but improper with some traditional priors found in the literature. With the Jeffreys prior, the posterior distribution is also shown to vanish at the boundaries of the parameters’ domain, which means that sampling the posterior distribution of the parameters does not result in anomalously small or large values. Therefore, the use of the Jeffreys prior does not result in degenerate fragility curves such as unit-step functions, and leads to more robust credibility intervals. The numerical results obtained from two different case studies—including an industrial example—illustrate the theoretical predictions.

Bayesian update of fragility curves for equipment failure probability in seismic probabilistic safety assessment in nuclear power plant

The assessment of equipment failure probability during seismic probabilistic safety assessment (PSA) is a crucial and fundamental component in nuclear power plant seismic PSA. A Monte Carlo program must be formulated to compute the equipment failure probability and uncertainty analysis in seismic PSA. A numerical calculation method is employed for the integration calculation to obtain the conditional failure probability and the uncertainty distribution for each equipment. The fragility parameters of the equipment and the hazard data of the plant site serve as the input parameters for the failure probability of equipment in seismic PSA. In this study, the probability of occurrence of the minimal cut set in equipment failure under earthquake conditions is calculated using the Monte Carlo sampling method, and the uncertainty distribution for each peak ground acceleration interval is presented. Furthermore, we have updated the fragility parameters of the equipment through the utilization of the Bayesian approach. Based on the updated fragility curves, we have recalculated the failure probability of the equipment and compared it with the results before the update.

A Computationally Efficient Method to Determine the Probability of Rainfall-Triggered Cut Slope Failure Accounting for Upslope Hydrological Conditions

We present a new computationally efficient methodology to estimate the probability of rainfall-induced slope failure based on mechanical probabilistic slope stability analyses coupled with a hydrogeological model of the upslope area. The model accounts for: (1) uncertainty of geotechnical and hydrogeological parameters; (2) rainfall precipitation recorded over a period of time; and (3) the effect of upslope topography. The methodology provides two key outputs: (1) time-varying conditional probability of slope failure; and (2) an estimate of the absolute frequency of slope failure over any time period of interest. The methodology consists of the following steps: first, characterising the uncertainty of the slope geomaterial strength parameters; second, performing limit equilibrium method stability analyses for the realisations of the geomaterial strength parameters required to calculate the slope probability of failure by a Monte Carlo Simulation. The stability analyses are performed for various phreatic surface heights. These phreatic surfaces are then matched to a phreatic surface time series obtained from the 1D Hillslope-Storage Boussinesq model run for the upslope area to generate Factor of Safety (FoS) time series. A time-varying conditional probability of failure and an absolute frequency of slope failure can then be estimated from these FoS time series. We demonstrate this methodology on a road slope cutting in Nepal where geotechnical tests are not readily conducted. We believe this methodology improves the reliability of slope safety estimates where site investigation is not possible. Also, the methodology enables practitioners to avoid making unrealistic assumptions on the hydrological input. Finally, we find that the time-varying failure probability shows marked variations over time as a result of the monsoon wet–dry weather.

Fragility Analysis for Water-Retaining Structures Resting on Spatially Random Soil

The maintenance of water-retaining structures involves evaluating their performance against current and future operating water levels. Fragility curves are commonly used for this purpose, as they indicate the conditional probability of failure for various load conditions and accurately characterize a structure’s performance. Monte Carlo simulation (MCS) can be used to determine the fragility curve of water-retaining structures by calculating the probability of failure as the water level changes. However, performing repetitive MCS involves extensive calculations, thus making it inefficient for practical applications. Therefore, it is essential to develop efficient methods that require a minimum number of MCS runs to estimate the fragility curve. This study proposed two methods to estimate the fragility curves of water-retaining structures, thereby allowing for the assessment of failure probabilities related to important quantities such as the steady-state seepage rate, exit gradient, and uplift force, which make them suitable for practical applications. The fragility curves obtained using the proposed methods are valuable for risk assessment, design, and decision-making purposes, as they offer information to evaluate the performance of the water-retaining structure under various water level conditions.

Comparison of seismic analysis methods within analytical approach

Introduction. In the nuclear power industry, ensuring nuclear and radiation safety is of paramount importance, therefore all aspects that may affect safety are considered in a deterministic and probabilistic approaches.
 As part of the safety justification, consideration of earthquakes of various intensity levels is regulated by safety standards. At the same time, the deterministic analysis of the earthquake and the models and calculation methods used are quite well-established, and in the probabilistic way, a relatively large number of different approaches have been developed and used as part of PSA level 1. The results of the evaluation of the conditional probability of failure of system elements, buildings and structures depending on the intensity of seismic impact on the free ground surface of the site depends on the approach used for the calculations.
 Aims. To perform a comparative analysis of various methods for constructing a failure probability curve under seismic impact.
 Materials and methods. A comparative analysis of the use of various approaches to the construction of a failure curve under seismic impact for a relatively simple case is performed. The results of the calculations showed that the scaling method leads to the most conservative results. The use of linearization methods and the method of moments give more realistic estimates, which can have a positive impact on the results of PSA level 1.
 Conclusions. A comparative analysis of some approaches has practical interest and is relevant, since various methods can contribute more than 50% to the core damage frequency, which creates a significant imbalance in the safety of nuclear power plants.

FRAGILITY ANALYSIS OF DUNES REINFORCED WITH GEOSYNTHETIC SAND CONTAINERS

Encroachment of infrastructure on the natural beach system constrains dune volumes and necessitates construction of structural and nonstructural mitigation measures to improve coastal resilience. Nature-based solutions, such as dunes reinforced with geosynthetic sand containers (GSCs), are increasingly being used to stabilize coastlines and protect communities from smaller storm events (e.g. 50-year storms) while at the same time providing flexibility in design considering the uncertainty regarding rates of sea level rise and the increasing destructive power of storm events. Using risk-based hazard assessments, such as fragility curves, it is possible to quantify the resiliency of reinforced coastal systems to address these dynamic conditions. A fragility curve represents the conditional probability of failure of a coastal structure (e.g. natural dune, reinforced dune, seawall, etc.) as a function of a certain stress acting on the structure (typically water level, wave height/period; Gruhn et al. 2012). The main advantage of a fragility curve, compared to a damage function that quantifies a deterministic degree of damage directly to a stress, is that it can account for uncertainties in both the structural resistance (i.e. capacity) and the environmental stress (i.e. demand) of the system. The objective of this research is to present a fragility analysis of a U.S. Federally funded GSC-reinforced dune in Montauk, NY. This dune was constructed in 2016 and experienced significant erosion of the protective berm and sand covering the GSCs during a 1-year storm event that same year.

Artificial Neural Network Technique For Seismic Fragility Analysis Of A Storage Tank Supported By Multi-Storey Frame

Abstract Fragility function, which defines the conditional probability of exceeding a limit state given an intensity measure (IM) of the earthquake, is an essential ingredient of modern approaches like the performance-based earthquake engineering methodology. However, the generation of such curves generally entails a high computational effort to account for epistemic and aleatory uncertainties associated with structural analysis and seismic load. Moreover, a certain probability function, such as the log-normal distribution, is usually assumed in order to carry out the conditional probability of failure of a structure, without any prior information on the correct probability distribution. In this paper, an artificial neural network (ANN) model is proposed to carry out fragility curves in order to avoid the aforementioned problems. In this respect, this paper investigates the following aspects: (i) implementation of an efficient algorithm to select proper seismic intensity measures as inputs for ANN, (ii) derivation of surrogate models by using the ANN techniques, (iii) computation of fragility curves by means Monte Carlo Simulations and (iv) validation phase.

Nonparametric Representation for Seismic Fragility Assessment of Earth Dams with Spatially Variable Soil Properties

Seismic fragility is the conditional failure probability of reaching certain performance states under various intensities of seismic load, which is a significant part of the seismic risk and resilience assessment of structures and infrastructure systems. However, conventional seismic fragility curves are constructed based on the assumption of specific distribution models (i.e., a lognormal distribution), which might lead to a lack of fit and cannot reflect the true structures of data. Therefore, the validity of these assumptions remains questionable. In this study, a nonparametric approach that was based on dynamic reliability theory was employed to calculate the seismic failure probabilities under various intensities of seismic loads and seismic fragility curves were constructed for an earth dam, in which the spatial variability in the soil properties of the earth dam was properly considered by random field (RF) models. In addition, the results from the earth dams with homogeneous, random variables (RV), and RF models were compared to illustrate the effect of spatially variable soil properties on the seismic fragility of earth dams. The numerical results indicated that more comprehensive, reliable, and safe results could be acquired during the seismic fragility analysis of earth dams when the spatially variable dam properties were considered.

Game-Theoretic Strategies for Cyber-Physical Infrastructures Under Component Disruptions

Networked infrastructures of recursively defined systems composed of discrete cyber and physical components are considered. The components of basic systems at the finest levels can be disrupted by cyber or physical means, and can be reinforced to survive at certain costs. A problem of ensuring the infrastructure performance is formulated as a game between a provider and an attacker, who probabilistically choose components to reinforce and attack, respectively. The disruptions of this infrastructure are characterized using the aggregate failure correlation function that specifies the conditional failure probability of the infrastructure given the failure of an individual system at that level. The survival probabilities of basic systems satisfy simple product-form, first-order differential equations expressed in terms of the multiplier functions. The utility functions of the provider and attacker are composed of the reward and cost terms, both expressed in terms of the component reinforcement and attack probabilities. The Nash equilibrium of this game is characterized, along with the sensitivity functions of the survival probabilities of basic systems that highlight their dependence individually on the cost-benefit terms, the correlation functions, and the multiplier functions. These results are illustrated using simplified models of a distributed cloud servers infrastructure, a 5G data network infrastructure, a high performance computing federation, and a smart energy grid infrastructure.

Bayesian Network–Based Fault Diagnostic System for Nuclear Power Plant Assets

Future advances in nuclear power technologies call for enhanced operator advice and autonomous control capabilities that can leverage simpler designs and increased safety features to reduce reliance on human labor. One of the first tasks in the development of such capabilities is the formulation of symptom-based conditional failure probabilities for the plant structures, systems, and components (SSCs) of interest. The primary goal is to aid plant personnel in (1) deducing the probabilistic performance status of the monitored SSCs and (2) detecting impending faults/failures. The task of estimating conditional failure probability is a bidirectional inference problem, and a logical approach is to use the Bayesian network (BN) method. As a knowledge-based explainable artificial intelligence tool and a probabilistic graphical model, BN offers reasoning capability under uncertainty, graphical representation emulating physical behavior of the target SSC, and interpretability of the model structure and results. This paper provides a systematic overview of the BN technique and the software tools for implementing BN models, along with the associated knowledge representation and reasoning paradigm. Both operational data and expert judgment can be readily incorporated into the knowledge base of a BN model. The challenges with data availability are highlighted, and the general approach to target SSC identification is presented. The focus is on failure-prone and risk-important balance of plant assets, especially for cases with strong operator involvement. Two example case studies on the failure of (1) a centrifugal pump and (2) an electric motor are conducted to demonstrate the usefulness and technical feasibility of the proposed BN-based fault diagnostic system using an expert system shell.

An efficient reliability evaluation method involving correlated random variables under distribution parameter uncertainty

To fully describe the safety of the structure involving correlated random variables under the distribution parameter uncertainty, it is necessary to obtain not only the mean value of the conditional failure probability, but also to grasp its complete probability characteristics. Due to the large variability of the conditional failure probability, it is difficult to directly use an explicit function to fit its probability distribution. Therefore, the focus of this paper is to first approximate the distribution of the conditional reliability index involving correlated random variables, and then obtain the quantile of the conditional failure probability in form of an explicit formula. For this purpose, the limit state function involving the correlated random variables should be first transformed into the standard normal space based on the three-parameter lognormal transformation, and then the first-three central moments of the conditional reliability index can be calculated by combining the bivariate dimensionreduction integration method combined with the point-estimate method, so that the distribution of the conditional reliability index can be obtained. Finally, an explicit expression of the quantile values of the conditional failure probability can be derived from the distribution of the corresponding conditional reliability index. Two illustrative examples show that the results obtained from the proposed method are almost similar to those of the Monte Carlo simulations, and the proposed method has a wide range of applications.

A novel framework for seismic fragility analysis with the combination of Box-Cox transformation and Bayesian inference

Fragility curves describe the conditional failure probability that the structural demand reaches or exceeds a limit state under a given intensity measure, which is extensively used in performance-based earthquake engineering. To improve the performance of current methodologies, a novel framework for seismic fragility analysis with the combination of Box-Cox transformation and Bayesian inference is proposed in the present study. A long-span cable-stayed bridge is taken as a case study, and the numerical model of the bridge is established within the OpenSees platform. The probabilistic seismic demand models are established with the Bayesian inference for the Box-Cox transformed data and developing the fragility models with binary Bayesian regression analysis. The numerical results reveal that the proposed framework can establish the nonlinear probabilistic seismic demand models and improve the performance of the classical methods. In addition, the binary Bayesian logistic regression-based fragility model eliminates the assumptions of the classical analytical approaches, and robust results can be obtained. Based on the derived fragility curves, the classical cloud method usually underestimates the failure probability of the components in severe damage states. In contrast, the proposed framework can accurately predict seismic demands at a large intensity level.

Influence of hydrodynamic pressure on fragility of high-pier continuous rigid frame bridge subjected to ground motion

To investigate the seismic fragility of deep-water and high-pier continuous rigid frame bridge subjected to the combined action of hydrodynamic pressure and ground motion, a representative deep-water and high-pier continuous rigid frame bridge is examined to study the influence of hydrodynamic pressure and ground motion on structural fragility. The corresponding damage indicators and conditional exceedance probabilities are defined and quantified. The probabilistic seismic demand model (e.g., pier and bearing) is analyzed using incremental dynamic analysis (IDA) in the nonlinear finite element software OpenSEES. A comparative analysis is conducted to identify the effect of hydrodynamic pressure on the dynamic response of the bridge. Hydrodynamic pressure plays an important role in influencing the conditional failure probability of piers and bearings. Damage probability increases with increasing water depth, particularly for the pier top in the longitudinal direction.

Estimation of failure probability-based-global-sensitivity using the theorem of Bayes and subset simulation

The failure probability-based-global-sensitivity can measure the effect of the uncertainty of model input on the failure probability of the structure or system. In this paper, a novel simulation method based on the theorem of Bayes and subset simulation is proposed to efficiently estimate the failure probability-based-global-sensitivity. In the proposed method, the unconditional failure probability is estimated by the efficient subset simulation through the estimation of the probability of a series of conditional failure domain. Then, based on the Bayes formula in the conditional failure domain involved in subset simulation, an equivalent transformation formula is derived for the conditional failure probability on fixed model inputs. Next, according to the deduced equivalent transformation formula and the failure samples obtained by the subset simulation, the conditional failure probability on arbitrary inputs realization can be directly computed without any extra model evaluations. Finally, the failure probability-based-global-sensitivity can be estimated by the unconditional failure probability acquired by the subset simulation and those conditional failure probabilities obtained by the proposed equivalent transformation formula. Additionally, the preconditioned Crank–Nicolson algorithm is used in the subset simulation and the interval approximation approach is employed in estimating the conditional failure probability so as to further improve the robustness of the proposed method.

Efficient method for fully quantifying the uncertainty of failure probability

It has been recognized that the uncertainty of distribution parameters has a significant effect on the outcome of structural reliability analysis. In this study, an efficient and accurate method is proposed to fully quantify failure probability, considering the uncertainty of the distribution parameters. This method obtains predictive failure probability by integrating the probability space of the conditional reliability index instead of directly utilizing the point-estimate method on the conditional failure probability as in previous studies. In the proposed method, a three-parameter lognormal distribution is suggested to approach the distribution of the conditional reliability index. The first three central moments of the conditional reliability index are estimated utilizing the point-estimate method combined with the bivariate dimension-reduction method. Based on this, the analytical solutions for the quantiles and probability distribution of the conditional failure probability can be readily derived without redundant calculations. Four illustrative examples were investigated to verify the efficiency and accuracy of the proposed method. The results show that the proposed method produces sufficiently accurate results in an efficient and simple way. It also presents a complete picture of the structural reliability evaluation results considering distribution parameter uncertainty in a wide range of applications.

Seismic fragility analysis of the transfer and purging chamber in a nuclear power plant based on incremental dynamic analysis

The seismic fragility of the transfer and purging chamber in a nuclear power plant is assessed in this paper based on the incremental dynamic analysis method. Select the peak ground acceleration as the ground motion intensity measure, and select the compressive strain of concrete, steel plate stress, and peak displacement of the structure as the damage measures. This paper calculates the dynamic response of the transfer and purging chamber under 160 seismic conditions and analyzes the conditional failure probabilities of no damage, slight damage, recoverable damage, and serious damage. The result shows that the conditional failure probabilities of concrete and steel plate have remarkable differences under the design basis ground motions, and both are lower than the criterion restriction. The conditional failure probability of peak displacement of the cantilever in the transfer and purging chamber under the design basis ground motions exceeds the criterion restriction. It indicates that peak displacement of the cantilever is overlarge in one direction caused by insufficient constraints and too small local stiffness. The failure probability of the transfer and purging chamber that does not meet operational basis performance demand is 6.32 × 10−2‱, and the failure probability of the transfer and purging chamber that does not meet safety shutdown performance demand is 1.76 × 10−3‱.

Cyclonic damage assessment of rural houses for the east coastal region of India

Tropical cyclones are one of the most frequently occurring natural hazards in the coastal regions of India that cause extensive damage to life as well as property. Undertaking damage assessment of houses in a cyclone-prone region is important for cyclone disaster mitigation. In this work, a systematic procedure has been outlined for conducting damage assessment of rural houses caused due to cyclonic wind using a component approach. By combining the damage states and using the damage ratio of the component, a damage probability matrix (DPM) is determined. By Integrating the probability of cyclonic wind speed with the conditional probability of failure of rural houses and conditional mean damage, the overall damage estimate for a specified region is obtained. Three types of analysis are carried out showing three types of fragility curves, which include, annual damage vs. wind velocity, probability of complete failure of houses vs. wind velocity, and probability of failure of some important components vs. wind velocity. The effects of several types of wind speed distributions, component distribution parameters, and wind speed are investigated.

Conditional capital surplus and shortfall across renewable and non-renewable resource firms

This study examines the conditional capital surplus and shortfall dynamics of renewable and non-renewable resource firms. To this end, this study uses the systemic risk index by Brownlees and Engle (2017) and considers two conditional systemic events, namely, a stock market crash and a commodity price crash. The results indicate that generally, companies in the resource sector tend to have conditional capital shortfall before 2000 and conditional capital surplus after 2000 owing to the boom of the commodity sector stocks and moderate capital structure management adopted by these companies. This finding is especially valid for resource firms in developed countries, whose observations dominate the dataset used in this study. Furthermore, the analysis using the panel vector autoregressive model indicates a positive influence of commodity price and geopolitical uncertainties on the conditional capital shortfall. These uncertainties have also been proven to increase the conditional failure probability of resource firms in the sample. Lastly, the performance analysis shows that potential capital shortfall is positively related to market returns, reflecting a high-risk high-return trade-off for the resource sector.

More Causes Less Effect: Destructive Interference in Decision Making.

We present a new experiment demonstrating destructive interference in customers’ estimates of conditional probabilities of product failure. We take the perspective of a manufacturer of consumer products and consider two situations of cause and effect. Whereas, individually, the effect of the causes is similar, it is observed that when combined, the two causes produce the opposite effect. Such negative interference of two or more product features may be exploited for better modeling of the cognitive processes taking place in customers’ minds. Doing so can enhance the likelihood that a manufacturer will be able to design a better product, or a feature within it. Quantum probability has been used to explain some commonly observed “non-classical” effects, such as the disjunction effect, question order effect, violation of the sure-thing principle, and the Machina and Ellsberg paradoxes. In this work, we present results from a survey on the impact of multiple observed symptoms on the drivability of a vehicle. The symptoms are assumed to be conditionally independent. We demonstrate that the response statistics cannot be directly explained using classical probability, but quantum formulation easily models it, as it allows for both positive and negative “interference” between events. Since quantum formalism also accounts for classical probability’s predictions, it serves as a richer paradigm for modeling decision making behavior in engineering design and behavioral economics.