Limit State Function Research Articles

Mechanically Stabilized Earth Wall Reliability Analysis Using Response Surface Methodology, ANN, ANFIS and Multi-objective Genetic Algorithm

Considering uncertainty in the analysis of geotechnical structures is a necessary condition for optimal and robust design. An alternative method for studying the reliability of a mechanically reinforced earth wall in granular soil is used to account for these uncertainties more rigorously. This allows for the inclusion of various uncertainties in a mathematical risk formulation based on random variables. The deterministic model is a benchmark taken from the literature used in a numerical simulation to determine the maximum horizontal displacement of the wall. In this case, the serviceability limit state is considered, allowing the wall's actual behavior to be described. ANOVA was used to identify the most influential parameters on the system's response. As uncorrelated random variables, only the parameters (E, φ and γ) were considered. The mathematical model serving as the limit state function was numerically predictedby three methods, response surface methodology (RSM), artificial neural network (ANN) and Adaptive Neuro-Fuzzy Inference System (ANFIS), and their predictive capacities were then compared. The results showed that the ANN model outperformed the RSM and ANFIS models regarding prediction. ANN models and multi-objective genetic algorithm (MOGA) are used to optimize the Hasofer-Lind reliability index. The analysis is then carried out by taking into account the various types of functions of parameter distributions, which allowed us to better appreciate the effects of the uncertainties and identify the set of parameters with a high incidence.

Fragility estimation for performance-based structural design of floating offshore wind turbine components

This study proposes a computational and mathematical framework aimed at assessing the reliability of structural components within Floating Offshore Wind Turbines (FOWT) that reflects the various sources of uncertainties coupled between structural analyses, hydrodynamics, and aerodynamics. The limit state functions are represented through structural capacity and environmental demand models for selected structural failure modes that incorporate fully coupled aero-hydro-servo-elastic analysis. The fragility surfaces are developed for a selected benchmark wind turbine for both operating and parking conditions. The fragilities are also estimated under 50-year and 100-year environmental conditions in the selected U.S. coastal regions. It is found that the wind speed variations largely affect the fragility during non-operation, while wave height variations are significant during operation. Increased uncertainties in environmental parameters raised failure probabilities, especially in lower fragility ranges targeted by design codes. Analyses in U.S. coastal environments show both parking and operating conditions can be critical, challenging the previous focus on parking. Sensitivity studies reveal that under mild conditions, structural reliability is influenced by moment of inertia and material strength, but as environmental loads increase, these parameters become equally significant. Increased uncertainties in parameters lead to higher failure risks, especially below 25 m/s wind speeds.

PLIC-FSR-SYS: System reliability analysis based on parallel learning of influential components with filtered sample region

In practical engineering, system reliability analysis is highly concerned since many structures or products have multiple failure modes. Accordingly, this paper develops an innovative method for system reliability analysis by parallel learning of influential component limit-state functions with filtered sample region (PLIC-FSR-SYS) based on Kriging modeling. Different from the traditional adaptive learning methods that train only one component in each iteration when constructing the surrogate of the composite limit-state function, a new strategy is explored to adaptively identify several important components in one iteration so as to train them simultaneously. In the meanwhile, a filtering formula is explored to determine the fatal region so that the unimportant samples can be removed to further accelerate the training process. Based on the join forces of parallel learning of influential components and avoiding the training at unimportant samples, PLIC-FSR-SYS can achieve a fairly efficient system reliability analysis with multiple failure modes. Finally, four different case studies, including an engineering application to the ultra-voltage on-load tap-changer, are conducted to prove the effectiveness of the proposed method. The results indicate that compared to traditional adaptive learning methods, the proposed method makes a significant efficiency improvement for system reliability analysis with multiple failure modes.

An Order Statistics Perspective for System Reliability

ABSTRACTStructural reliability integrates the design variables over the safety region characterized by a positive limit state function when the reliability of the entire system is of concern. Calculating the structure function can be challenging for high‐dimensional systems or intricate system architectures. In order to enhance the efficiency of time‐dependent system reliability assessment, we emulate the integral formulation in structural reliability. To elaborate further, we treat each individual unit's lifetime variable as a design variable and subsequently perform calculations involving multiple integrals. Given the ordered nature of unit failure times, we leverage the order statistics distribution to simplify the multiple integrals into a double integral, and then multiply this result by the survival signature to obtain reliability. A two‐terminal nine‐unit network system configuration is illustrated to assess the performance and effectiveness of the proposed method.

Reliability analysis for running safety of vehicle on slab track via an improved second-order fourth-moment approach

The reliability assessment for running safety of vehicle on slab track under temperature action and seismic excitation is challenging due to the extensive computational efforts. This study proposes an improved second-order fourth-moment (ISOFM) approach for the reliability analysis for running safety of vehicle on slab track under temperature action and seismic excitation. The dimensionless limit state function (LSF) for running safety of vehicle on slab track is formulated considering four failure modes. The proposed ISOFM approach is verified by the Monte Carlo simulation (MCS) method. The proposed ISOFM approach greatly reduces the number of computations of nonlinear LSF. The merit of the proposed ISOFM approach lies in significantly enhancing the computation efficiency and precision of reliability considering nonlinear LSF with multi-failure modes. Additionally, the influences of temperature action and seismic excitation on the reliability for running safety of vehicle on slab track are discussed. The reliability for vehicle running safety decreases more significantly with the increment in overall temperature compared to the positive temperature gradient (PTG). The reliability for running safety of vehicle on slab track decreases nonlinearly with the increase of the earthquake intensity.

A hybrid approach combining UD and GA-CV-SVM to evaluate shear performance in high asphalt concrete core

The shear effect on high asphalt concrete core is significant. However, studies on the reliability of 100-meter-scale cores against shear damage remain limited. A key challenge in this research field is establishing the control criteria for the core and improving the computational efficiency of implicit limit state function (LSF). Additionally, the impact of material parameter uncertainty on the shear failure reliability of the core during the dam construction and impoundment stages remains unclear. To address this, a safety evaluation method based on time discretization was proposed, combining uniform design (UD), K-fold cross-validation (K-CV), and genetic algorithm (GA) to optimize the support vector machines (SVM). The core parameters of 52 asphalt concrete-core rockfill dams (ACCRDs) were analyzed, with the statistical values of the basic variables considered in determining the reliability index. The theoretical derivation of the critical shear failure safety index established a stability formula to assess the safety state of the dam core. The significance parameters were identified, and the sample points were generated at each stage using UD, and the Support Vector Regression (SVR) was applied to reconstruct the LSF, and reliability was calculated through the checking point method.

A physics-informed neural network enhanced importance sampling (PINN-IS) for data-free reliability analysis

Reliability analysis of highly sensitive structures is crucial to prevent catastrophic failures and ensure safety. Therefore, these safety-critical systems are to be designed for extremely rare failure events. Accurate statistical quantification of these events associated with a very low probability of occurrence requires millions of evaluations of the limit state function (LSF) involving computationally expensive numerical simulations. Variance reduction techniques like importance sampling (IS) reduce such repetitions to a few thousand. The use of a data-centric metamodel can further cut it down to a few hundred. In data-centric metamodeling approaches, the actual complex numerical analysis is performed at a few points to train the metamodel for approximating the structural response. On the other hand, a physics-informed neural network (PINN) can predict the structural response based on the governing differential equation describing the physics of the problem, without a single evaluation of the complex numerical solver, i.e., data-free. However, the existing PINN models for reliability analysis have been effective only in estimating a large range of failure probabilities (10-1∼10-3). To address this issue, the present study develops a PINN-based data-free reliability analysis for low failure probabilities (<10-5). In doing so, a two-stage PINN integrated with IS (PINN-IS) is proposed. The first stage is employed to approximate the most probable failure point (MPP) appropriately while the second stage enhances the accuracy of LSF predictions at the IS population centred on the approximated MPP. The effectiveness of the proposed approach is numerically illustrated by three structural reliability analysis examples.

Distribution reconstruction and reliability assessment of complex LSFs via an adaptive Non-parametric Density Estimation Method

Complex limit state functions (LSFs), often characterized by strong nonlinearity, non-smoothness, or discontinuity, pose challenges for structural reliability analysis in engineering practices. Conventional methods for uncertainty propagation and reliability assessment may struggle to handle these issues effectively. This paper introduces a novel approach to adaptively reconstruct the unknown distributions of complex LSFs. The Non-parametric Density Estimation Method based on Harmonic Transform (NDEM-HT) is employed as the tool for this purpose. An adaptive strategy is then proposed to determine the number of harmonic moments required in NDEM-HT for achieving high accuracy. Specifically, the Adaptive Kernel Density Estimation (AKDE) method is also adopted to provide an initial estimation of the rough distribution. Subsequently, the optimal number of harmonic moments is determined by minimizing the relative entropy between the distributions obtained by AKDE and NDEM-HT. The efficacy of the proposed method is demonstrated through five numerical examples, considering various types of complex LSFs. Comparative results are also provided employing MCS along with both conventional and state-of-the-art methods.

Quantitative method for the probability of structural damage based on moment theory

In the context of uncertainty in the inverse problem of damage identification, this study proposes a moment theory based probabilistic damage identification (MbPDI) method. Noise errors in the identification process are treated as uncertain parameters, while structural stiffness parameters are treated as functions of noise errors. A limit state function for damage identification is defined, and its probability is computed. Utilizing the concept of moment theory, the expansion series is performed for the limit state function around the most probable point of failure, with gradient information derived using sensitivity matrices. Based on the gradient information and uncertainty inputs, damage probabilities for various elements can be obtained. Additionally, this paper addresses the calculation of structural damage probabilities under the non-normal distribution of random variables by transforming non-normal variables into equivalent normal variables. The proposed probability index and expected index of structural damage in the probabilistic damage identification method has a clear physical meaning, and the calculation has high accuracy. The proposed method is validated through a mathematical example and a numerical simulation. Meanwhile, the influence of damage extent, noise level, and modal order on identification results are analyzed. Experimental validation of the method is also presented.

Probabilistic assessments of running ductile fractures in dense-phase and supercritical CO2 pipelines

Running ductile fracture (RDF) is a severe failure mode of high-pressure pipelines. Dense-phase and supercritical CO2 pipelines are particularly susceptible to RDF due to the unique characteristics of the depressurization process resulting from a fracture of the pipeline. The present study carries out probabilistic analyses of RDF in dense-phase and supercritical CO2 pipelines. The well-known Battelle two-curve method is employed to establish the limit state function for RDF by comparing the arrest pressure with the saturation pressure. The arrest pressure is computed using the Battelle through-wall crack model with the adjustment recently proposed in the literature. The saturation pressure is computed based on the one-dimensional isentropic decompression assumption and a rigorous equation of state. The first-order reliability method is employed to evaluate the probabilities of RDF in hypothetical CO2 pipelines designed per DVN-RP-F104 with representative pipe attributes and initial operating conditions by considering uncertainties in the relevant pipe geometric and material properties. The analysis results indicate that the probability of RDF can vary by several orders of magnitude depending on the pipe attributes and initial operating conditions. Furthermore, the results shed light on the key uncertainties associated with the pipe geometric and material properties that influence the probability of RDF.

Probabilistic investigation of piezoelectric application for reliability enhancement of composite laminates under edge delamination

In this research a novel probabilistic numerical approach is developed to investigate the effectiveness of piezoelectric (PZT) materials to enhance the reliability of composite laminates under edge delamination onset (EDO). To achieve this, finite element (FE) modeling is employed as an implicit limit state function (LSF) based on a quadratic failure criterion and the concept of critical length. An interactive interface that integrates FE analysis and reliability analysis is then established to execute both processes simultaneously until the convergence condition of the reliability index is satisfied. To validate current FE analysis, the interlaminar stress peaks, and resulting edge delamination probability obtained in this study are compared with available data. Subsequently, the probability of EDO in angle-ply composite laminates under applied longitudinal strain and an applied electric field on the PZT layer is assessed using both the first-order reliability method (FORM) and the second-order reliability method (SORM). The Monte Carlo simulation (MCS) is employed to verify the numerical results. Findings reveal that employment of PZT reduces EDO probability. Moreover, as the applied voltage increases, the critical strain required for definitive EDO also increases, and the positive influence of PZT on enhancing reliability becomes more pronounced, particularly with increased thickness under higher applied strain.

A reliability analysis method based on the intersection area division of hypersphere and paraboloid

In terms of the approximate failure probability of the limit state function (LSF), First Order Reliability Method (FORM) directly solves the failure probability through the most probable point (MPP), but it can only achieve high accuracy in the LSF with low linearity and non-linearity. Second Order Reliability Method (SORM) performs second-order expansion on the basis of MPP to improve reliability accuracy. However, the approximation may be wrong, and even has a large error in high-dimensional nonlinear problems. In order to get more precise approximation of SORM, this paper proposes an area partition method based on the Intersection Area Division Method (IADM) of Hypersphere and Paraboloid. IADM constructs a hypersphere and an approximate parabolic function of the LSF at MPP. The intersection area of the hypersphere and the approximate paraboloid is defined as the failure region from the geometric of the approximate paraboloid function, and the area of the failure region is calculated by the spherical integral. Finally, the approximation of the failure probability is determined by integrating the ratio of the area to the area of the hypersphere. Several examples proved that IADM is more stable compared with SORM which avoids the generation of wrong solutions and improves the accuracy.

Subset simulation based simplified approach for pipeline with multiple irregular corrosion defects in time-dependent reliability analysis

Pipeline system plays an important role in the natural gas and petroleum transportation, and it is widely employed in the engineering application. However, the pipeline system is subjected to corrosion given the industry environment condition, or soil condition when it is buried. In this case, it is important to assess the remaining life of corroded pipeline. Consequently, it is important to predict the failure probability considering the corrosion growth over time and operating pressure. Nevertheless, the prediction of failure probability in corroded pipeline in not an easy task, due to the fact that a realistic corrosion usually has an irregular geometry, especially, when multiple irregular corrosion is involved in the analysis. To simplify the problem, this work presents a simplified procedure for time-dependent reliability analysis to predict the failure probability in pipeline with multiple irregular corrosion defects, considering two failure modes, the burst and leak mode. The approach is based on Subset Simulation and Weighted Depth Difference method, where the multiple irregular corrosion is treated by a discretization procedure and a weighting coefficient is evaluated in every discretization points. Then, this weighting coefficient is introduced into burst pressure assessment, which is employed by burst failure mode limit state function. In this work, the corrosion growth is modelled by power function corrosion model, and initial corrosion depth is generated randomly. The Subset Simulation is employed to evaluate the failure probability, where the Markov Chain Monte Carlo is adopted to evaluate the conditional probability and Metropolis-Hasting algorithm is employed to solve the problem. Finally, several scenarios with single and multiple irregular corrosion defects are analyzed to demonstrate the effectiveness of presented procedure.

A novel null-hypothesis approximation method of limit state function in multi-failure mode reliability analysis of structural system

A novel null-hypothesis approximation method of limit state function in multi-failure mode reliability analysis of structural system

Utilization of stochastic ground motion simulations for scenario-based performance assessment of geo-structures

Probabilistic seismic performance assessments of engineered structures can be highly sensitive to the seismic input excitation and its variability. In the present study, the scenario-based performance assessment recommended by Federal Emergency Management Agency (FEMA) P-58 guidelines is adopted to estimate seismic fragility of concrete dams for various seismic hazard scenarios. Due to the scarcity of recorded ground motions and thereby their poor representation of uncertainties, stochastic ground motion simulation methods are utilized to obtain the required input excitations. Moreover, to understand the uncertainty in ground motion simulation models, two broadband stochastic simulation models are used to generate input excitations representing six seismic hazard scenarios defined by earthquake magnitude, source-to-site distance, and soil conditions.Optimal intensity measure parameters for each scenario are identified using a systematic procedure that considers criteria such as efficiency, practicality, proficiency, sufficiency, and hazard compatibility. Fragility curves and surfaces are derived using the cloud analysis technique, taking into account various damage measures and limit state functions. The study finds that the derived fragility curves are particularly sensitive to the selection of earthquake scenarios, the choice of records, and the methods used to calculate fragility curves, with less sensitivity observed to different engineering demand parameters. Given this sensitivity, particularly to ground motion selection, the study highlights the necessity of incorporating both model-to-model variability (epistemic uncertainty) and record-to-record variability (aleatory uncertainty), alongside the established material and modeling uncertainties, in the probabilistic seismic assessment.

Extension of K-nearest neighbors and introduction of an applicable prediction criterion for a novel Monte Carlo simulation-based method in structural reliability

Efficiency is an important issue in structural reliability analysis. In this paper, a novel method is proposed to increase the efficiency of Monte Carlo Simulation by means of a machine learning algorithm. Using K-nearest neighbors method, the safe or failure region of many generated samples can be predicted without evaluating the limit state function. The proposed method represents three novel parts as its main core. In the first part of the proposed method, the classical K-nearest neighbors algorithm has been extended to a new fuzzy level such that the proposed method could predict even if all K neighbors are not in the same region. In previous similar methods, there is a prerequisite that K neighbors must be all in the failure region or all in the safe region. This extension to the fuzzy level is carried out by assigning a specific failure weight to each neighbor. In the second part of the proposed method, an innovative diagnostic criterion has been defined to check whether the region of a sample could be predicted or the sample has to be evaluated in the limit state function. The criterion describes the relative position of the newly generated sample with respect to the neighbors. This newly defined criterion is an effective alternative for the laborious action of drawing the convex hull around K neighbors, which is done as a time-consuming part in previous similar methods. Promotion of the accuracy or efficiency in the proposed method could be set using the adjusting parameter of the represented criterion. Since the implementation of the algorithm for each individual sample needs much time, in the third part, the proposed method introduces a technique to perform function evaluations or predictions in clusters. This part effectively reduces the computational burden, i.e. the number of function evaluations and run time. The performance of the proposed method, applying three novel and newly represented parts, has been verified using a number of mathematical and structural examples.

Second-order reliability analysis of an energy pile with CPT data

As pile foundations and ground source heat producers, energy piles are becoming more and more popular. Even after much research, engineers find their design to be challenging. An effective way for using the Second-Order Reliability method (SORM) in foundation engineering scenarios including explicit limit state functions is shown in this article. The suggested SORM process is simple to implement in a spreadsheet and is based on an approximate paraboloid fitted to the limit state surface close to the design point. Using well-established closed-form SORM formulae, the failure probability is automatically determined using the primary curvatures of the limit state surface and the reliability index. An energy pile foundation engineering design examples are analyzed using the proposed method and discussed. Comparison with FORM, SORM and Genetic Programming (GP)-based FORM and SORM are made. In the case of determination of group capacity of pile and thermal load the input random variables are qc0, qc1, qc2, fs and E. The reliability index and probability of failure value are found to be 9.34 for FORM, 9.55 for SORM, 9.24 for GP-based FORM, and 9.45 for GP-based SORM. The probability of failure value for FORM 4.60E-21, SORM 3.24E-21, GP-based FORM 1.20E-20, and GP-based SORM is 8.50E-21. The relative percentage error for the average second-order approximation corresponding to FORM is also calculated, and found that Pf2 is 29.48 % in the case of simple SORM and 29.15 % in the case of GP-based SORM. The Breitung provides the most accurate estimation, and the Cai_Elishakoff provides the least accurate estimate.

Robustness of Structural Health Monitoring Systems for Offshore Energy Structures based on Damage Characteristics

Offshore energy structures such as fixed/floating wind turbines, tidal turbines and wave energy converters are exposed to loads coming from different sources. Recent advances in SHM offer many unique opportunities to assess the structural integrity of offshore energy infrastructure. The process of SHM generally involves the use of a system of sensors mounted on a structure; the processing of signals received from the sensors to attain damage sensitive features; and the detection of damage in the structure. Over the past decades, most of the research in SHM has been focused on improving and developing new sensor technology, signal processing, and damage detection algorithms. However, there are very few studies assessing the performance of SHM systems with respect to damage detection, localization, and quantification. Moreover, most of the performance analysis models for SHM systems rely only on 'probability of detection (POD)', that is an assessment index used in NDT to evaluate the probability of detecting a damage as a function of its size. For SHM systems, the replicability of POD is influenced by the environment, measurement noise, and the deterioration of sensors and instruments. Therefore, the distribution of POD is often determined by performing an extensive number of test sets with existing damage of varying magnitude. This paper aims to reduce the volume of experimental data required, as well as the uncertainties associated with POD prediction by using a model-assisted probability of detection (MAPOD) approach. We develop a Bayesian network (BN) method to evaluate the performance of SHM systems in terms of damage detection, localization, and assessment. In addition to POD, the probability of accurate localization (POAL) and probability of accurate assessment (POAA) are also incorporated into the analysis. The BN and limit state functions are applied to a fixed offshore energy structure equipped with a vibration based SHM system.

Performance-based target reliability analysis of offshore wind turbine mooring lines subjected to the wind and wave

Reliability assessment is a crucial aspect of the design and operation of structures, particularly in balancing safety and cost considerations. This paper introduces a novel method for evaluating the performance-based target reliability of floating wind turbine platforms in offshore environments. The method focuses on the platform's motion modes and wave frequencies, which significantly influence the system's structural integrity and performance. An improved limit state function is proposed to enhance the accuracy of reliability calculations, specifically for steady-state conditions. The platform's six degrees of freedom motions are carefully analyzed to investigate their dependence on wave frequencies. By considering the time response of these motions and accounting for uncertainties in wave characteristics, wave impact directions, and wind effects, a comprehensive reliability analysis is conducted to assess the stability modes of the platform. This paper introduces the term 'Reliability Performance-Based' (RPB) analysis as a new concept to evaluate the system's reliability at a given performance level. Furthermore, an optimal target reliability index is defined to address the economic aspect of the design process. The proposed methodology's PEB analysis focuses on capturing uncertainties in wave characteristics and wind effects on floating wind turbine platforms. This includes a detailed examination of wave and wind-induced loads and their propagation through the system concerning its performance level. Statistical models were integrated to quantify these uncertainties, applying Monte Carlo simulations to assess their effects on the platform's reliability. This approach allows for a nuanced understanding of the interactions between environmental factors and structural responses, enhancing the precision of our reliability assessments. It enables the consideration of economic efficiency alongside safety, ensuring a balanced approach to the design and operation of the floating wind turbine platform. By providing a comprehensive reliability assessment framework, it aids in the optimization of design and decision-making processes for floating wind turbine platforms.

An improved approximate integral method for nonlinear reliability analysis

In order to evaluate the failure probability corresponding to the Limit State Function (LSF) in structural reliability, the First Order Reliability Method (FORM) linearizes the LSF and directly calculates the failure probability based on the Most Probable Point (MPP). But this method is unable to effectively handle nonlinear problems. The Second Order Reliability Method (SORM), on the other hand, utilizes the Hessian matrix to obtain curvature information at the MPP and computes the failure probability with Breitung's second-order approximation formula. However, SORM is unable to maintain satisfactory computational accuracy in highly nonlinear problems due to its lack of detailed analysis on the shape of the limit state surface. To address this issue, this paper proposes an Improved Approximation Integration Method (IAIM). The initial approximation of the limit state surface is based on the intersection region between the hypersphere and the n-dimensional paraboloid. Subsequently, a statistical analysis of the deviation between the aforementioned region and its projected area is conducted to construct a corresponding fitting function, which is then utilized for dimensionality reduction in the multidimensional integration of the approximation area. Finally, several examples are presented to demonstrate the accuracy and feasibility of the proposed IAIM.