Structural Reliability Analysis Research Articles

An enhanced learning function for bootstrap polynomial chaos expansion-based enhanced active learning algorithm for reliability analysis of structure

Sparse polynomial chaos expansion (PCE) combined with the bootstrap resampling method is a viable alternative to obtain an active learning algorithm for reliability analysis. The existing learning functions in PCE-based active learning algorithms do not consider the joint probability density function (PDF) information. The present study explores a sparse PCE-based active learning algorithm based on a newly proposed learning function that maintains a balance between the misclassification probability and the joint PDF information of sample points. In doing so, the coefficients of the sparse PCE are estimated using a Bayesian compressive sensing regressor, as it is noted to be one of the best-performing regression solvers for PCE, irrespective of sampling schemes. The proposed learning function considers the weight of the joint PDF with the local accuracy measure of bootstrap PCE (bPCE) to add new samples iteratively in the existing training set. The convergence is achieved when the ten consecutive failure estimates are within a negligible discrepancy and also checks the confidence bounds of the bPCE estimates. The effectiveness of the proposed approach is demonstrated using two structural engineering examples and one well-known analytical test function and is found to be quite efficient and accurate in estimating reliability.

Series–Parallel System Analysis and Reliability Research of Grid Structures Considering Adaptive Dynamic Bounding Threshold

There are many bars in grid structures, and their topological relationships are complex, so the workload of calculations involved in researching their reliability is large. In this paper, based on the geometric topological relationship within a grid structure, a series–parallel system analysis of the structure is carried out by means of node analysis, constraint grade division, and hierarchical combination analysis, and the series–parallel analysis results for the structure are used to judge the failure of the structure. In order to avoid the problem of missing important components due to unreasonable selection of the bounding threshold when identifying failed components, an adaptive dynamic bounding threshold method with high accuracy is introduced. The failure probability value of the failure path is calculated using the direct numerical integration method, and the reliability of the whole structure is obtained. Finally, the validity of this method is verified using an example of a square pyramid grid structure. The results show that the method is feasible for series–parallel system analysis, structural failure judgment, and reliability analysis of a grid structure.

Seismic fragility analysis of structures via an Adaptive Gaussian Mixture Model and its application to resilience assessment

Seismic fragility analysis and seismic resilience assessment are pivotal tasks within the performance-based earthquake engineering framework. The former aims to quantitatively assess the probability of engineering structures sustaining various degrees of damage when subjected to earthquakes of varying intensity levels. In contrast to many traditional seismic fragility methods, this paper introduces a precise and versatile approach grounded in reliability analysis, with the goal of eliminating the restrictive lognormal assumption. Initially, the proposed method involves conducting seismic reliability analysis of structures using the Fractional Exponential Moments-based Maximum Entropy Method (FEM-MEM). This approach efficiently yields failure probabilities. Subsequently, an Adaptive Gaussian Mixture Model (AGMM) is presented for fitting a set of failure probability points, producing comprehensive and continuous seismic fragility curves. Notably, AGMM offers remarkable flexibility in modeling fragility curves, ensuring precise estimations for subsequent resilience assessments. Building upon the results of the fragility analysis, the paper proceeds to assess the seismic resilience of structures. The proposed method’s feasibility and effectiveness are validated through two illustrative numerical examples. The results demonstrate that, in comparison to the conventional lognormal assumption, the proposed method offers superior accuracy and broader applicability for conducting seismic fragility analyses of structures.

Performance analysis of electrical signal output of multi-state flexoelectric structures with parameter uncertainties through quasi-Monte Carlo method

Flexoelectric effect is a more universal electromechanical coupling effect than piezoelectric effect. Flexoelectric beams as the main structural component of flexoelectric power signal output have broad application prospects in the next generation of micro–nano electromechanical systems. However, the electrical signal output of flexoelectric structures in macro-scale is far less than the output of the piezoelectric signal. Therefore, it is urgent to explore the influence of the parameter uncertainties on the electrical signal output of the flexoelectric structures, in order to improve the electrical signal output of flexoelectric materials with excellent design performance. Based on the quasi-static theory, the output voltage model and the output charge model of flexoelectric structures as well as the effective piezoelectric coefficient model are constructed. Then the influences of the flexoelectric parameters on the output voltage and the output charge are researched as well as the influence of the effective piezoelectric coefficient. Finally, the influences of uncertain parameters under different electrical states (e.g. the electrical open circuit and short circuit states) on the output performance of flexoelectric signal are studied by the quasi-Monte Carlo method, in order to further provide a reference for the reliability analysis and optimization design of the flexoelectric structures.

Effects of limit state data on constructing accurate surrogate models for structural reliability analyses

Engineering problems are mainly defined in implicit processes; hence, the fully probabilistic analyses, e.g., Monte Carlo simulations (MCS), are expensive to implement. In practice, two approaches to overcome the issues are either reducing the size of simulations or developing surrogate models for actual problems. The latter does not sacrifice the size of MCS and requires less insight into probabilistic calculation; hence, it is preferable to most engineers. This study proposes an efficient framework to develop reliable and accurate surrogate models by considering data at the limit state margins (LS data). Effects of involving LS data in the training process and performances of the proposed metamodels are investigated for most issues relating to reliability analyses, including nonlinear performance functions, multiple failure modes, and implicitly defined problems. Two machine learning algorithms, including artificial neural networks and the Gaussian process, are employed to prove the ability of the proposed method. Investigations reveal that the limit state data plays a vital role in developing accurate surrogate models for reliability analyses, and accumulating them into the training dataset helps quickly construct accurate metamodels. This work contributes a practical framework for reliability analyses because the LS data can be detected easily without insight into probabilistic calculations.

Adaptive directed support vector machine method for the reliability evaluation of aeroengine structure

Machine learning methods have been widely applied to structural reliability analysis, due to the excellent performance in modeling precision and efficiency. In this paper, an adaptive directed support vector machine model (ADSVM) is proposed by integrating the adaptive sampling technology (AST) and the developed directed support vector machine (DSVM), to improve the efficiency and accuracy of turbine blade reliability analysis. In the developed method, the AST is adopted to extract high quality samples, in respect of distributed probability density and the distances between sample values and response values. The DSVM was developed by introducing the penalty coefficients for all the slack variables, to improve the modeling accuracy of SVM approach. One numerical example and the reliability analysis of aeroengine high-pressure turbine blade were performed to validate the developed methods. As illustrated in this numerical investigation, the modeling precision of DSVM is improved by alleviating the lag effect of support vector regression. From the comparison of methods, it is demonstrated that the ADSVM has higher accuracy, efficiency and stability in reliability simulations, which are improved by ∼20 % and ∼46.2 %, respectively. The main efforts of this study are to propose a promising approach, ADSVM, for the reliability analysis of complex structures besides turbine blades, which hold the academical significance in enriching and developing mechanical reliability theory.

Linear Moments-Based Monte Carlo Simulation for Reliability Analysis With Unknown Probability Distributions

Abstract Within the realm of structural reliability analysis, the uncertainties tied to resistance and loads are conventionally embodied as random variables possessing established cumulative distribution functions (CDFs). Nevertheless, real-world scenarios often involve cases where the CDFs of random variables are unknown, necessitating the probabilistic traits of these variables solely through statistical moments. In this study, for the purpose of integrating random variables characterized by an unknown CDF into the framework of Monte Carlo simulation (MCS), a linear moments (L-moments)-based method is proposed. The random variables marked by an unknown CDF are rendered as a straightforward function of a standard normal random variable, and the formulation of this function is determined by utilizing the L-moments, which are typically attainable from the statistical data of the random variables. By employing the proposed approach, the generation of random numbers associated with variables with unknown CDFs becomes a straightforward process, utilizing those derived from a standard normal random variable constructed by using Box-Muller transform. A selection of illustrative examples is presented, in which the efficacy of the technique is scrutinized. This examination reveals that despite its simplicity, the method demonstrates a level of precision that qualifies it for incorporating random variables characterized by unspecified CDFs within the framework of MCS for purposes of structural reliability analysis.

Bayesian active learning line sampling with log-normal process for rare-event probability estimation

Line sampling (LS) stands as a powerful stochastic simulation method for structural reliability analysis, especially for assessing small failure probabilities. To further improve the performance of traditional LS, a Bayesian active learning idea has recently been pursued. This work presents another Bayesian active learning alternative, called ‘Bayesian active learning line sampling with log-normal process’ (BAL-LS-LP), to traditional LS. In this method, we assign an LP prior instead of a Gaussian process prior over the distance function so as to account for its non-negativity constraint. Besides, the approximation error between the logarithmic approximate distance function and the logarithmic true distance function is assumed to follow a zero-mean normal distribution. The approximate posterior mean and variance of the failure probability are derived accordingly. Based on the posterior statistics of the failure probability, a learning function and a stopping criterion are developed to enable Bayesian active learning. In the numerical implementation of the proposed BAL-LS-LP method, the important direction can be updated on the fly without re-evaluating the distance function. Four numerical examples are studied to demonstrate the proposed method. Numerical results show that the proposed method can estimate extremely small failure probabilities with desired efficiency and accuracy.

Metamodel-Based Directional Importance Sampling for Structural Reliability Analysis

Metamodel-Based Directional Importance Sampling for Structural Reliability Analysis

Active Kriging-based conjugate first-order reliability method for highly efficient structural reliability analysis using resample strategy

Efficient structural reliability analysis method is crucial to solving reliability analysis of complex structural problems. High-computational cost and low-failure probability problems greatly limit the efficiency in structural reliability analysis problems, causing the safety and reliability of the structure to be questioned. In this work, a highly efficient structural reliability analysis method coupling active Kriging algorithm with conjugate first order reliability method (AK-CFORM) is proposed. Specifically, the resample strategy is considered to reduce the number of samples evaluated in each active learning process; the uniform sampling is used to better balance global and local optimal problems; the conjugate map is used to improve the robustness of analytical first order reliability method; and the approximate numerical differential formula is proposed to solve the problems of non-convergence when solving the gradient of the Kriging surrogate model. Finally, three numerical cases and four engineering cases are used to illustrate the effectiveness and robustness of the proposed method. The results show that the proposed AK-CFORM has greater advantages in the number of calling system response and surrogate model with robust and accurate performance.

Stochastic analysis of structural reliability of complex engineering systems

Purpose: The assessment of risks and prediction of reliability of complex engineering systems, in particular onshore gas pipelines subjected to external corrosion. Two methods are proposed using the structural reliability analysis.Methodology: Two strategies are considered for inspection and maintenance service of hazardous production facilities.Research findings: The model of structural reliability is proposed to estimate the metal rupture from external corrosion of the pipe section. Models are presented for its mechanical failure with respect to stochastic processes of loads and resistance.Value: The inhomogeneous Poisson point process is used to simulate the formation of new defects, and the Poisson distribution is used to simulate their growth. The first method focuses on the analysis of external corrosion of gas pipelines with metal loss and predicts rupture at a reference pipe segment, constructed with respect to average pipe rupture characteristics from the PHMSA database. The second model predicts reliability for untreated sections subject to external corrosion with metal loss.

Importance sampling enhanced by adaptive two-stage Kriging model and active subspace for analyzing rare probability with high dimensional input vector

For the high dimensional and small failure probability events widely existing in engineering, structural reliability analysis method still faces challenges at present. Since active subspace (AS) method can effectively reduce the dimension of input vector space, and metamodel-based importance sampling (IS) can efficiently estimate small failure probability, this paper combines AS with IS to propose an adaptive two-stage Kriging model method (AS-IS-AK2) for rare failure event with high dimensional problem. The proposed AS-IS-AK2 includes two stages. In the first stage, a new learning function is proposed to construct the Kriging model for roughly estimating the performance function values of all IS samples, and the gradient information is provided as a by-product for AS. In the second stage, an adaptive Kriging method is constructed in the reduced dimension space to accurately judge the states of all IS samples. Since AS reduces the dimension of input vector space, and IS reduces the required size of candidate sample pool for obtaining the convergent failure probability estimation, the computational complexity is reduced in reconstructing the Kriging model to judge the states of IS samples. The results of three examples verify the advantage of the proposed method for high dimensional problem with small failure probability.

Partially Bayesian active learning cubature for structural reliability analysis with extremely small failure probabilities

The Bayesian failure probability inference (BFPI) framework provides a well-established Bayesian approach to quantifying our epistemic uncertainty about the failure probability resulting from a limited number of performance function evaluations. However, it is still challenging to perform Bayesian active learning of the failure probability by taking advantage of the BFPI framework. In this work, three Bayesian active learning methods are proposed under the name ‘partially Bayesian active learning cubature’ (PBALC), based on a cleaver use of the BFPI framework for structural reliability analysis, especially when small failure probabilities are involved. Since the posterior variance of the failure probability is computationally expensive to evaluate, the underlying idea is to exploit only the posterior mean of the failure probability to design two critical components for Bayesian active learning, i.e., the stopping criterion and the learning function. On this basis, three sets of stopping criteria and learning functions are proposed, resulting in the three proposed methods PBALC1, PBALC2 and PBALC3. Furthermore, the analytically intractable integrals involved in the stopping criteria are properly addressed from a numerical point of view. Five numerical examples are studied to demonstrate the performance of the three proposed methods. It is found empirically that the proposed methods can assess very small failure probabilities and significantly outperform several existing methods in terms of accuracy and efficiency.

Model updating of highway slope under seismic intensity conditions considering spatially varying soils

Abstract Understanding the mechanisms underlying earthquake-induced landslides and assessing seismic responses are crucial for effective mitigation strategies. Earthquakes typically involve a mainshock followed by aftershocks, posing challenges to structures weakened by the mainshock. Highway slope structures, especially those in unsaturated soft-soil slopes, are vulnerable to aftershocks, amplifying the damage caused by the mainshock-aftershock (MSAS) sequence. While existing research primarily focuses on the effects of mainshocks on certain structures, there is a notable gap regarding the damage sustained by unsaturated slope structures under MSAS conditions. Addressing this gap is vital for comprehensive risk assessment and mitigation. To address these challenges, we propose a stochastic model updating approach for seismic reliability analysis. This approach integrates subset simulation with adaptive Bayesian updating and dimensionality reduction using the Karhunen–Lòeve expansion. Shaking table tests on a slope structure with unsaturated red clay soil are conducted to investigate the effects of matrix suction on performance degradation and failure mechanisms. The results reveal spatial variability in soil property parameters, underscoring the need to incorporate this variability into inverse analyses. Traditional deterministic methods or probability-based approaches may overlook such variability. Also, the results indicated our proposed approach enables effective prediction of seismic responses for unsaturated slopes subjected to MSAS sequences. By considering spatial variability and the effects of matrix suction, our method offers a comprehensive framework for seismic reliability analysis of unsaturated slope structures.

Structural Reliability and Optimization Using Differential Geometric Approaches

This study investigates how differential geometry ideas can be used to effectively carry out structural optimization and reliability analysis. Strong mathematical representations and methods for examining intricate surfaces and forms are provided by differential geometry. The basic ideas of differential geometry, such as tensors, manifolds, and curvature, are initially introduced in the work. Then, to account for ambiguities in the geometry, the probability theory in tangent spaces is developed. As a result, structural reliability can be determined using propagating uncertainty. To enable reliability-based design optimization, differential geometry representations are linked with optimization techniques. The proposed differential geometry-based approach is applied in a number of case studies to trusses, airplane wings, car bodies, and ship hulls. The outcomes show a significant increase in productivity and scalability compared to conventional finite element methods. The article offers new tools for dealing with uncertainty.

AOK‐ES: Adaptive optimized Kriging combining efficient sampling for structural reliability analysis

AbstractThe pivotal problem in reliability analysis is how to use as few actual assessments as possible to obtain an accurate failure probability. Although adaptive Kriging provides a viable method to address this problem, unsatisfied Kriging surrogate accuracy and reliability modeling samples often lead to an unacceptable computing burden. In this paper, an adaptive optimized Kriging combining efficient sampling (AOK‐ES) is proposed: first, to enhance the Kriging approximation ability, a high‐fidelity optimized Kriging model (OKM) is established; further, to ensure the samples quality of OKM modeling and reliability calculation, an improved Latin hypercube sampling method and an optimized importance sampling approach are developed correspondingly. Six different types of case studies demonstrate the superiority of the proposed AOK‐ES. The analysis results demonstrate that the proposed AOK‐ES holds the potential to reduce computing cost while ensuring accuracy.

A modified interval perturbation statistical energy analysis method for structural reliability analysis of vibro-acoustic systems with interval uncertain parameters

Structural reliability analysis has become a critical aspect of engineering design for vibro-acoustic systems. With the increasing complexity and uncertainty of design parameters, it is essential to develop methods and tools that can accurately account for all the sources of uncertainty. This paper presents a novel approach for structural reliability analysis of vibro-acoustic systems with interval parameters within the framework of Statistical Energy Analysis (SEA). By introducing the Sherman-Morrison-Woodbury formula and the sub-interval technique, a modified interval perturbation statistical energy analysis is proposed to predict the interval bounds of energy and stress response. Then, the non-probabilistic structural reliability is assessed by using the volume ratio theory. Case studies of a plate-cavity system and a launch fairing model, in which the interval uncertainties in external loads and material properties are considered, are presented to demonstrate the effectiveness of the proposed approach. The results show that the proposed method can effectively handle uncertainties and provide accurate estimates of the probability of failure.

Active Learning-Based Kriging Model with Noise Responses and Its Application to Reliability Analysis of Structures

This study introduces a reliability analysis methodology employing Kriging modeling enriched by a hybrid active learning process. Emphasizing noise integration into structural response predictions, this research presents a framework that combines Kriging modeling with regression to handle noisy data. The framework accommodates either constant variance of noise for all observed responses or varying, uncorrelated noise variances. Hyperparameters and the variance of the Kriging model with noisy data are determined through maximum likelihood estimation to address inherent uncertainties in structural predictions. An adaptive hybrid learning function guides design of experiment (DoE) point identification through an iterative enrichment process. This function strategically targets points near the limit-state approximation, farthest from existing training points, and explores candidate points to maximize the probability of misclassification. The framework’s application is demonstrated through metamodel-based reliability analysis for continuum and discrete structures with relatively large degrees of freedom, employing subset simulations. Numerical examples validate the framework’s effectiveness, highlighting its potential for accurate and efficient reliability assessments in complex structural systems.

Stochastic modelling of in-structure concrete strength based on stochastic damage model and Burr distribution

The probability measure of low-quality concrete is essential for the reliability analysis of concrete structures. However, this problem is usually neglected, and the normal distribution or lognormal distribution is often selected as the probability distribution of concrete strength. In this study, a better solution for this problem is given by theoretically deriving of the Burr distribution based on the stochastic damage model. A large amount of in-situ test data in engineering structures is applied to perform a K-S test of different distribution types and to fit the distribution parameters. As a result, the advantage of Burr distribution in representing the tail probability is explained by both theoretical derivation and fitting results. And the Burr distribution is accepted by the K-S test in every strength grade while the other distribution types are all partly rejected.

Time-dependent reliability analysis of structural systems based on parallel active learning Kriging model

Time-dependent reliability analysis quantifies the failures of structural systems due to time-dependent uncertainties, such as material degradation and dynamic loads. The active learning Kriging model methods are widely used in structural reliability analysis to replace extensive time-consuming finite element simulations. However, they can only update one training sample and one failure mode per iteration, which limits their application to time-dependent, parallel computing, and multiple failure modes problems. In this study, we propose a new parallel active learning Kriging model for time-dependent reliability analysis, which can update multiple training samples and multiple failure modes per iteration. It includes the following strategies: (1) a novel parallel learning function is proposed, which combines the correlation function and U learning function to allow for the selection of multiple training samples per iteration; (2) an adaptive adjustment strategy for the number of parallel samples is proposed, which takes into account the prediction probability of parallel samples; (3) the proposed parallel learning function is integrated into time-dependent reliability analysis with multiple failure modes, enabling simultaneous updates of multiple training samples and failure modes, thus greatly reducing the number of iterations and computational time; and (4) a new stopping criterion is proposed to improve the efficiency of the estimation of failure probability. The proposed method can be applied to series or parallel time-dependent structural systems with multiple failure modes. We demonstrate the effectiveness of the proposed method through three examples, and the proposed method can achieve a balance between the computational time and function calls while maintaining a high level of accuracy in the estimation of time-dependent failure probability.