Limit State Function Research Articles

Reliability analysis of controlled structures based on probabilistic active controller

The reliability of controlled structures has been a subject of considerable concern in recent years. The present study investigated the effects of a probabilistic fuzzy logic controller (PFLC) as an intelligent controller and a linear quadratic regulator (LQR) controller as a classic controller on the reliability of a structure. Monte Carlo reliability analysis has been used to determine the reliability of the controlled structures. A single-degree-of-freedom (SDOF) system and multiple-degree-of-freedom (MDOF) structures with different tendon arrangements were considered. The structures were subjected to random Gaussian white noise as excitation and the variables considered were random Gaussian samples with a dispersion coefficient of 10%. The probability of failure could be estimated by PFLC at up to 10-5 as the threshold level increased in MDOF structures. Moreover, the results indicate that PFLC was more accurate than the LQR controller for a lower probability of failure because the associated limit state function includes stochastic uncertainty.

Evidence-Theory-Based Reliability Analysis From the Perspective of Focal Element Classification Using Deep Learning Approach

Abstract Epistemic uncertainty is widespread in reliability analysis of practical engineering products. Evidence theory is regarded as a powerful model for quantifying and analyzing epistemic uncertainty. However, the heavy computational burden has severely hindered its application in practical engineering problems, which is essentially caused by the repeated extreme analysis of limit-state function (LSF). In order to address the issue, this paper proposes a novel method to solve the evidence-theory-based reliability analysis (ETRA). It transforms the conventional ETRA problem into the classification of three classes of joint focal elements (JFEs) and then solves the classification problem effectively through a deep learning approach. The core of solving an ETRA problem is to determine whether the joint focal element is located in the reliable region, failure region, or intersected with the LSF. A spatial position feature reduction and arrangement method is proposed to classify the JFEs, which can effectively reduce the feature dimension and take into account the integrity and correlation of features. The stacked autoencoders model is then constructed and updated by extracting the spatial position features of the sampled JFEs to achieve high-accuracy classification of the remaining JFEs, and the reliability interval is calculated efficiently according to the classification results. Finally, the effectiveness of the proposed method is demonstrated using several numerical examples.

Influence of Limit State Function’s Form of Geotechnical Structures on Approximate Analytical Reliability Methods

Approximate analytical methods have been frequently used in geotechnical engineering to estimate the reliability of geotechnical structures due to their efficiency and simplicity. The main spirit of these methods is using the moments of the limit state function to estimate the reliability index. However, the moments are strongly dependent on the form of the limit state function, resulting in the fact that these methods are sensitive to the form of limit state functions. This study aims to systematically explore how various equivalent forms of limit state functions affect the performance of several commonly used approximate analytical methods, including the first-order second-moment method, the first-order reliability method, and the point estimation method. The applicable conditions of these methods are illustrated through five typical geotechnical examples. The results indicate that the estimated accuracy for the first-order second-moment method and the point estimation method is affected by the form of the limit state functions. Although the form of the limit state function does not affect the accuracy of the first-order reliability method, it affects computational efficiency. The limit state functions with an equivalent logarithmic form are almost always favorable for the investigated examples and are thus recommended in practice.

An efficient surrogate model for reliability analysis of the marine structure piles

A hybrid random-forest-based subset simulation (RFSS) method for probabilistic assessment of scour around pile groups under waves is proposed. In the RFSS, a random forest is used to replace the true limit state function (LSF); it is updated based on design samples in the first and last levels of the subset simulation method. For modelling, 127 laboratory datasets collected from the literature were used. First, an existing equation for predicting the scour depth around piles was modified using a metaheuristic approach. The performance of the modified equation was compared with existing equations and models. The modified equation was found to be more accurate than the existing formulas and AI-based models. A probabilistic model based on the RFSS was then developed by considering the modified formula as the LSF of scour depth. Solving two numerical problems, one hydraulic engineering problem and one scour problem validated the robustness and accuracy of the structural reliability method. The results showed that the RFSS is a robust and efficient method for solving high-dimensional real-world problems. Furthermore, compared to the Monte Carlo simulation, the RFSS was able to estimate the reliability index with less computational cost and the same accuracy.

Reliability Analysis of Structures by Active Learning Enhanced Sparse Bayesian Regression

Adaptive sampling near a limit state is important for metamodeling-based reliability analysis of structures involving an implicit limit state function. Active learning based on the posterior mean and standard deviation provided by a chosen metamodel is widely used for such adaptive sampling. Most studies on active learning-based reliability estimation methods use the Kriging approach, which provides prediction along with its variance. As with the Kriging approach, sparse Bayesian learning-based regression also provides posterior mean and standard deviation. Due to the sparsity involved in learning, it is expected to be computationally faster than the Kriging approach. Motivated by this, active learning-enhanced adaptive sampling-based sparse Bayesian regression is explored in the present study for reliability analysis. In doing so, polynomial basis functions, which do not involve free parameters, are chosen for the sparse Bayesian regression to avoid computationally expensive parameter tuning. The convergence of the proposed approach is attained based on the stabilization of 10 consecutive failure estimates. The effectiveness of the proposed adaptive sparse Bayesian regression approach is illustrated numerically with five examples.

Efficient model‐correction based reliability analysis of uncertain dynamical systems

AbstractIn this contribution, a model‐correction‐based strategy is applied for efficient reliability analysis of uncertain dynamical systems based on a low‐fidelity (LF) model whose outcomes are corrected in a probabilistic sense to represent the more realistic outcomes of the high‐fidelity (HF) model. In the model‐correction approach utilized, the LF model is calibrated to the HF model close to the so‐called most probable point (MPP) in standard normal space, which allows a more realistic assessment of the considered complex dynamical system. Since only few expensive limit state function evaluations of the HF model are required, an efficient reliability analysis is enabled. In an application example, the LF model describes an existing single span railway bridge modelled as simply supported Euler‐Bernoulli beam subjected to moving single forces representing the axle loads of a moving train. The HF modelling approach accounts for the bridge‐train interaction by modelling the passing train as mass‐spring‐damper (MSD) system, however increasing the computational effort of the limit state function evaluations. The failure probabilities evaluated with the model‐correction approach are contrasted and discussed with the failure probabilities of the sophisticated bridge‐train interaction model. It is shown that the model‐correction‐based approach provides reliable failure probability prediction of the HF model while leading to a significant reduction in computational effort.

Lateral impact performance of wire ropes

Wire ropes are often used in flexible barriers or road engineering to resist lateral impact because of their good reliability, high economy, and construction convenience. However, their application under lateral impact lacks a systematic design approach. In this paper, to ensure the safe use of the structure, the lateral impact test of single wire ropes and wire rope nets is carried out, and the strain, displacement, and acceleration response are analyzed. In addition, the tensile mechanical properties of wire rope strands are determined through static tensile mechanical properties tests, which provide reasonable parameters for the constitutive model used in numerical simulations. Furthermore, considering the influence of wire rope pre-tension ratio, span ratio, and mass ratio, empirical formulas for longitudinal peak tensile force and lateral peak displacement are given. The results show that the wire rope mainly occurs local abrasion at the impact point, fracture, and end connection failure. It is worth noting that when the dimensionless lateral peak displacement (ωm) reaches 0.089, the wire rope breaks. Moreover, according to Lagrange Green’s strain theory, a theoretical calculation formula for the mid-span lateral limit displacement of wire rope under pre-tension is derived and compared with the simulated values of test members. Consequently, according to the damage degree of wire ropes, the damage limit value is determined. Finally, considering the tensile capacity of wire ropes and their displacement responses, a limit state function is proposed to evaluate the damage of existing components or to design new components.

Effect of structural repairs on the load rating and reliability of a prestressed concrete bridge

Prestressed girders often deteriorate over time due to environmental and man-made stressors, lowering the strength and serviceability of bridge structures. Although structural repairs are implemented to improve the load carrying capacity of the structure, the presence of numerous unknowns leads to high uncertainty in estimating the adequacy of repairs. For instance, the cross-section of the remaining strands, material properties, applied external loads, and workmanship assumptions made throughout the repair process introduce ambiguities when estimating the adequacy of the repairs. This study evaluates the efficiency of re-tensioning repairs of prestressed concrete bridge span girders. The repairs include field splicing, re-tensioning, of deteriorated or damaged strands by torquing a splicing coupler. The evaluation in this study considers component, system reliability, and load ratings while accounting for several uncertainties, such as structural repair, material properties, and external loads. This paper introduces an approach to account for prestressing strands damage and repair uncertainties while also accounting for other uncertainties. In this regard, five cases are studied: as-built, repaired, and three varying degrees of damage cases. First, the distribution for structural demand and capacity accounting for uncertainty in loads, material properties, and repair process is defined for each girder in the prestressed concrete bridge span. In doing so, Monte-Carlo simulation is employed to determine the distributions. Accordingly, the limit state function of the girders is defined from the obtained distributions. Then, the component reliability of each AASHTO (American Association of State Highway and Transportation Officials) Type II girder is calculated from the obtained reliability indices based on the determined limit state functions. Finally, a system reliability model of the span is developed from the component reliability of each girder. Some advantages and disadvantages of using component and system reliability index versus load rating in damaged and repaired prestressed concrete bridge girders are also discussed. Several critical conclusions are made regarding the uncertainties in structural repair, material properties and external loads, and their impact on the load rating and the component and system reliability of the prestressed concrete bridge structure girders.

Interval reliability sensitivity analysis using Monte Carlo simulation and mouth brooding fish algorithm (MBF)

In reliability analysis and reliability-based design approach, sensitivity analysis establishes a relation between the variation in random variable parameters and that in reliability. The analysis of sensitivity is also employed to determine key random variables with the greatest contribution to reliability. The sensitivity analysis of a non-linear limit state function is relatively complicated and time-consuming. Therefore, a novel approach is proposed in this study to calculate the reliability sensitivity parameter in terms of an interval. In the proposed algorithm, the non-linear limit state function is initially converted to a linear limit state function by the Monte Carlo simulation method. Then, the interval random variables are introduced to include both epistemic and aleatory uncertainties so as to obtain a realistic result. In the second step of the proposed methodology, the mouth-brooding fish algorithm (MBF) is applied to capture the upper and lower bounds of the reliability sensitivity parameters. Several illustrative examples were then presented, showing the capability of the proposed approach in obtaining results accurately and efficiently. According to the results, the percentage of improvement in the sensitivity analysis of random variables in the proposed method, compared to the classical method, by using the MBF optimization algorithm to the parameters of random variables due to the variation Coefficient is more than 40 percents. This means that the length of the response interval has increased.

An efficient method for computing slope reliability calculation based on rigorous limit equilibrium

Abstract Traditional rigorous limit equilibrium methods satisfy all equilibrium conditions and usually have high accuracy, however, which are less efficient for slope reliability analysis. The main reason is that the limit state functions are highly nonlinear implicit functions of safety factor. Complex numerical iterations are required, which may sometimes lead to computational convergence problems. A new method for computing slope reliability calculation with high efficiency and accuracy was proposed. This method was based on the rigorous limit equilibrium method by modifying normal stresses over the slip surface. The critical horizontal acceleration factor K c {K}_{c} , which can be expressed explicitly, was used to replace the implicit safety factor as a representation of slope stability. The difference between K c {K}_{c} and the known value K c 0 {K}_{c0} was used as the limit state function. Two slope examples were analyzed. The results showed that the calculation results of this method were in good agreement with those of the traditional Morgenstern–Price limit equilibrium method, but the computational efficiency was significantly improved. When this method was combined with the subset simulation method, the calculation time was only a few seconds. Therefore, this method can be used for rapid calculation of slope reliability.

Robust reliability‐based design approach by inverse FORM with adaptive conjugate search algorithm

AbstractThe concept of reliability‐based design (RBD) of geotechnical works has seen great success in recent years, nevertheless, its application in real‐world engineering practice is still limited. This phenomenon could be mainly attributed to two aspects: the engineer's unfamiliarity of reliability algorithms even in their simplest forms, and the lack of direct connection between reliability algorithms and deterministic design tools. To promote the geotechnical RBD application, this study proposes an inverse first‐order reliability method (FORM) integrating adaptive conjugate direction search technique (RBD via conjugate FORM) to solve classical RBD problems where the number of (multiple) limit state functions is usually equal to the number of unknown design parameters. Notably, the RBD via conjugate FORM can readily back‐calculate those unknown design parameters subject to prescribed reliability index or probability of failure, by means of iterative evaluation of the parametric sensitivities. The proposed algorithm is first verified by several well‐documented examples. Comparative studies show that when the reliability problem to solve is highly nonlinear, the RBD results based on steepest descent search direction may fall in oscillation and non‐convergence, while the proposed RBD via conjugate FORM algorithm performs robustly and efficiently. It is then illustrated in combination with standalone numerical codes to conduct risk‐based optimization of unknown design parameters for two soil slope stability models. The first slope RBD model achieves designing the slope angle under specified reliability index, and the second slope RBD model is to optimize multiple design parameters of the stabilization pile, which can meet different slope reliability requirements under static and seismic loading conditions simultaneously.

Impact of the correlation between random variables on the reliability assessment of the corroded pipelines

In the present work, an Algerian pipeline made of API X70 steel, used for transporting gas and subjected to corrosion attack for three decades, was investigated. Using an intelligent inspection tool, the sizes and orientations of the defects were determined to analyze and improve the accuracy of reliability and/or probability of failure estimates for the damaged pipelines. A probabilistic methodology has been developed to assess the reliability index and the sensitivity of each random variable, within the limit state function, of a gas-transporting steel pipeline that has been corroded for three decades. The reliability analysis utilizes the Second Order Reliability Method, which is based on the BFGS algorithm. In addition, the corrosion defect depth, pipe wall thickness, and corrosion defect length are the most dominant variables within the limit state function, followed by ultimate stress and operating pressure, respectively. Meanwhile, for a failure scenario, thus, it is essential to include a potential parameter such as the correlation coefficient between random variables within the objective function during reliability assessment. For instance, not taking into account the correlation between random variables can lead to an overestimation of the reliability index by 8.89% for a defect depth to wall thickness ratio d/t of 20%. This overestimation increases to 19.16% when the d/t ratio increases to 70%. If the correlation coefficient is significant, it can cause an increase of 8.86% in the error of reliability index assessment for a defect with a depth to wall thickness ratio of 20%, and an increase of 14.58% for a defect with a ratio of 50%. Besides, these avoid the conservative practice resulting from the negligence of the existing correlation between random variables within the limit state function. Hence, considering the correlation coefficients during reliability assessment contributes to setting optimal maintenance schedules for pipelines made of steel under localized corrosion attack.

Reliability analysis of flap deflection angle based on collaborative intelligent moving Kriging model

To effectively monitor the reliability of civil aircraft flap deflection angle, combined with the quick access recorder(QAR), the collaborative intelligent moving Kriging(CIMK) method is proposed by absorbing the Kriging model, decomposition and co-ordination strategy, equilibrium optimizer(EO), and moving least square(MLS). Among them, the decomposition coordination strategy is used to deal with the relationship between the flaps left and right deflection angles. MLS is employed to select effective modeling samples and solve the undetermined coefficients of Kriging model. EO method is applied to determine optimizing the local compact support region radius of MLS. Firstly, the fault reason for flap left-right asymmetry is analyzed to clarify the main characteristic parameters in QAR data. Secondly, combined with the QAR data of relevant influencing parameters, the civil aircraft flap deflection model(limit state function) is constructed by using CIMK. Then, the reliability and influence of civil aircraft flap deflection angle are analyzed by Monte Carlo(MC) sampling method. The results show that when the flap deflection angle is 3°, the reliability is 0.450 2, and the important factors affecting the flap deflection angle are Mach number, left angle of attack, right angle of attack, etc. Compared with the response surface method(RSM), Kriging, support vector machine(SVM), and back-propagation-artificial neural network(BP-ANN), the average absolute error accuracy of the proposed method is relative improved by 53.02%, 51.43%, 49.03%, and 44.04%, the average relative error accuracy is relative improved by 68.36%, 66.76%, 64.41%, and 62.64%, and the modeling efficiency is relative improved by 50.62%, 26.35%, and 43.01% respectively compared with Kriging, SVM and BP-ANN. When the number of simulations is 103, the analysis accuracy is relative improved by 8.82%, 7.25%, 6.22%, and 3.98% respectively.

A single-loop time-variant reliability evaluation via a decoupling strategy and probability distribution reconstruction

In this paper, a single-loop approach for time-variant reliability evaluation is proposed based on a decoupling strategy and probability distribution reconstruction. The most attractive feature of the proposed method is that the reliability at a specified time instant can be captured by performing time-invariant reliability analysis only once. In this method, the expansion optimal linear estimation is first employed to discretize the loading stochastic process. Then, a decoupling strategy that decouples the loading stochastic process and degradation processes is developed to formulate a single-loop method for time-variant reliability analysis, where an equivalent extreme value limit state function (EEV-LSF) is obtained. To improve the accuracy and robustness, the Box–Cox transformation is applied to get a transformed EEV-LSF. The maximum entropy method with fractional exponential moments is employed to robustly derive the probability distribution of transformed EEV-LSF. Once the probability distribution is captured, the time-variant failure probability can be readily computed. To handle a large number of random variables, a weighted sampling method is applied for moment assessment to ensure an efficient solution. Numerical examples including a complex real-world case are studied to validate the proposed method, where pertinent Monte Carlo simulations and PHI2 method are conducted for comparisons.

Performance-based system reliability analysis for steel moment frames

This study aims to bridge the gap between two major design philosophies in structural design: load and resistance factor design (LRFD) and performance-based design (PBD). While LRFD prioritizes the design of structural members, PBD starts by controlling the system's target parameter. The main area for improvement of LRFD is paying attention to the structure's performance. At the same time, PBD needs to pay attention to the sufficiency of internal members in the presence of local damages. To bring these two attitudes closer together, the study proposes a system-reliability-performance-based design analysis to simultaneously measure the entire structure's safety level associated with the reliability of elements. As a state-of-the-art, new performance-based limit state functions are introduced to measure structural reliability indices corresponding to desired performance levels. Multi-story frames were designed and evaluated based on incremental dynamic analysis. The entire structure was designed and controlled using the linear static analysis method, and static and dynamic nonlinear analyzes were performed to evaluate its behavior. Dynamic analysis of acceleration mapping was also performed to investigate the performance of the structural components. The proposed combined design algorithm aims to develop a new set of performance-based load combinations and provide a strong interlock between LRFD and PBD. The study's results show that the IO level has the lowest performance-based system reliability index, while the CP level has the highest. It was observed that as the number of story levels increases, the performance-based system reliability index decreases due to an upsurge in the possibility of failure patterns. Therefore, the current LRFD practices are unlikely to provide the IO's perfect reliable level for the entire system. The study concludes that the new approach can enhance the safety of structures and prevent catastrophic failure.

Modified replica exchange-based MCMC algorithm for estimation of structural reliability based on particle splitting method

We consider the problem of evaluation of the reliability integral when the limit state functions could have complicating features such as (a) discontinuity/rapid changes in values of GU in the space spanned by U, (b) existence of multiple, possibly unimportant, regions of failure with rapid changes in performance function near one or more of the important regions, and (c) multiple regions of failure with substantial contributions to failure probability. We tackle this problem by using the particle splitting framework and introduce an improved Markov chain Monte Carlo sampler based on replica exchange strategy. This is shown to enhance the capacity of samples to detect and explore important regions of failure. The application of the bootstrap technique to deduce the sampling variance of the estimator of the probability of failure is developed. Also given are a few examples of problems in structural mechanics where limit state functions with the aforementioned difficulties arise. The performance of the proposed method in dealing with these difficulties is compared with those of existing methods.

Enriched global–local multi‑objective optimization scheme for fuzzy logic controller-driven magnetorheological damper-based structural system

To construct an optimal and reliable fuzzy logic controller (FLC) for magnetorheological (MR) damper-based structural systems, a reliability-based design optimization (RBDO) problem is formulated considering two competing objectives, namely, minimizing structural responses and control cost. This work incorporates a two-stage enrichment scheme with the non-dominated sorting genetic algorithm (NSGA-II)-based multi-objective optimization method to solve the RBDO problem. In the global stage of the proposed method (denoted as TS-MO method), an initial Kriging surrogate model is constructed in the augmented space and updated by sequentially adding enrichment points within the whole design space to obtain a coarse Kriging surrogate. In the local stage, within the multi-objective optimization iterations, the surrogate model is further refined using enrichment points selected in the vicinity of the current set of design points near the limit state function based on a distance-refined U learning function. Two convergence criteria with dynamic thresholds are employed to ensure the local precision of the trained surrogate model. The efficacy of the TS-MO method is demonstrated by conducting comparison works with two previous methods, i.e., the deterministic design optimization and global Kriging-based multi-objective optimization method (DDO-GK-MO) and the unified method.

Harmonic transform-based non-parametric density estimation method for forward uncertainty propagation and reliability analysis

Reconstructing the probability density function (PDF) of the limit state function (LSF) is regarded as a non-intrusive method for forward uncertainty propagation and reliability analysis. In this paper, a novel non-parametric density estimation method is proposed to derive the unknown distribution of the LSF for this purpose, where the efficiency and accuracy are ensured. The main innovation of the paper is to derive the PDF of the LSF via a novel concept, i.e., the harmonic transform, regardless of the complexity of the problem. First, the non-parametric method for PDF estimation based on complex fractional moments/Mellin transform is briefly revisited, where the limitations are also pointed out. Inspired from Mellin transform, the concept of harmonic transform is put forward, which overcomes the disadvantages of Mellin transform. Then, the unknown PDF is recovered via the inverse harmonic transform, where the analytical formula is derived accordingly. Numerical estimation method is also given for practical implementations, where some critical issues are also pointed out. Detailed step-by-step procedures of the proposed method are also outlined. The proposed method is a non-parametric one since it does not require a distribution model in advance. Some commonly-used distributions are first applied to validate the efficacy of the proposed method. Then, four numerical examples involving different types of problems are extensively investigated to validate the proposed method for forward uncertainty propagation and reliability analysis. The results demonstrate that the proposed method is effective for deriving the PDFs for both static and dynamic reliability analyses with nonlinear explicit or implicit LSFs.

A non-gradient-based reliability method using a new six-item instruction for updating design point

Reliability analysis of structures without spending too much computational cost is still a challenge. In this paper, an efficient non-gradient-based reliability method is proposed for analyzing the structures in which limit state functions are not explicitly known. Starting from an initial guess of design point, the proposed method updates this estimation of design point to approach the real one. A new comprehensive six-item instruction, represented for updating the estimations of the design point, is the main component of the proposed method in finding the solution. Due to including various conditions in the proposed instruction, the improvement of MPP candidates does not stop easily. To investigate the performance of this reliability method, at first, it is implemented in a number of mathematical problems of the literature. Then, structural problems, including a five-story steel building with implicit limit state functions and uncertain data are analyzed to show the applicability of the proposed method in real structures. The results of the mathematical and structural problems clearly show the efficiency and applicability of the proposed method.

Time-Variant System Reliability Analysis of Concrete Sewer Pipes under Corrosion Considering Multiple Failure Modes

This paper presents a framework for time-variant system reliability analysis of concrete sewer pipes considering multiple modes of failure. Sewer pipelines are among critical infrastructure systems, designed to collect and transport wastewater away from communities. Such pipes are typically subjected to corrosion and subsequent damage because of continuous contact with corrosive materials. Corrosion is here modeled in terms of the reduction in the pipe wall thickness. In the proposed framework, prevailing uncertainties in the characteristics of wastewater fluid, geometric properties of fluid cross section, and corrosion-related parameters of concrete are quantified as random variables. Next, various modes of failure under bending, shear, excessive cracking, and cover loss are formulated as multiple limit-state functions. This leads to a general system reliability problem defined as a series system of parallel subsystems, i.e., cut sets. Further, by introducing the corrosion model into the limit-state functions, a time-variant system reliability problem is established in which the integration domain is conditioned to the corrosion level at each point in time. By integrating over the conditional probabilities through a rejection sampling method, the failure probability is computed given the corrosion level over the desired time span. The results show that the failure probability increases sharply with time and corrosion level. For instance, the failure probability exceeds 80% for pipes over 50 years and at corrosion levels of over 40%. Moreover, the sensitivity analysis shows that the acid-consumption capacity of the pipe material, the pH-related factor, and the proportion of S2− dissolved highly influence the failure probability. The maximum effects of these parameters occur within 0%–10% corrosion. The results of this study can be used to evaluate the failure probability of sewer pipes given their age and current corrosion level.