Implicit Limit State Function Research Articles

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.

An efficient approach for evaluating the reliability of engineering structures using support vector machine with clustering algorithm

ABSTRACT An implicit limit state function is predicted with a support vector machine (SVM) for reliability analysis of engineering structures to reduce the number of finite element analyses. The accuracy and predictability of the SVM method are reduced considerably by noise in data. In this paper, density-based spatial clustering of applications with noise (DBSCAN) is applied to reduce the noise in training samples for SVM regression. Then, the SVM model is linked with Monte Carlo simulation (MCS) to find out the reliability of the engineering structures. Four different examples of static and dynamic problems are solved to show acceptability and efficiency of the proposed method. It is observed that the proposed method is suitable for a smaller number of performance function calls. Direct MCS, artificial neural network-based MCS and response surface methods have been used to examine the effectiveness of the algorithm.

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.

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.

Uncertainty analysis and reliability improvement of planetary roller screw mechanism using active learning Kriging model

The uncertainties in the geometry, material and operation conditions may cause structural failure of the planetary roller screw mechanism (PRSM). The uncertainty analysis model is the key to the reliability assessment of the PRSM, however, the relevant studies have been rarely reported in the past. This paper focuses on establishing a preliminary mathematical model of the PRSM considering uncertain factors. The quasi-Monte Carlo (QMC) method is introduced to improve the solving efficiency of the multidimensional and nonlinear implicit limit state function (LSF). Then, the parameter sensitivities of the uncertain factors to the load distribution and contact characteristics are comprehensively ranked by the design of experiment (DoE). The computational cost for constructing the active learning Kriging (ALK) model of PRSM is decreased by only selecting the most sensitive variables. Moreover, the ALK model and QMC method (ALK-QMC) are combined to explore how the main factors affect the structural reliability of PRSM, which further guides the implementation of multi-objective optimization to improve the reliability by the developed NSGA-II-Downhill algorithm. Finally, the theoretical model and optimization results are verified by the finite element method.

Cohesive zone model based reliability analysis for a sandwich pipe

A cohesive zone model based reliability analysis for a sandwich pipe (SP) is presented in this paper. The collapse or burst strength of SP is predicted by a cohesive zone model based nonlinear finite element (FE) analysis. The interlayer adhesion is simulated by cohesive zone model in accordance with the maximum shear strength. The bilinear traction separation law is employed in the cohesive zone model. A quadratic stress criterion and an energy-based model are used for simulating the damage initiation and evolution. The validity of the FE analysis is then validated by the comparison between model prediction and experimental results. A first-order reliability method coupled with a response-surface method is then used for reliability estimate dealing with implicit limit state functions corresponding to failure modes of collapse and burst. A sensitivity analysis to investigate the influence of design variables on the reliability of SP is further conducted and the results show that working pressure, yield strength of material, and modelling uncertainties are relatively important design variables in predicting the collapse and burst strength of SP, while initial ovalities of internal and external pipes could be regarded as constants in the reliability analysis for improving the computational efficiency.

Active Kriging-Based Adaptive Importance Sampling for Reliability and燬ensitivity Analyses of Stator Blade Regulator

The reliability and sensitivity analyses of stator blade regulator usually involve complex characteristics like high-nonlinearity, multi-failure regions, and small failure probability, which brings in unacceptable computing efficiency and accuracy of the current analysis methods. In this case, by fitting the implicit limit state function (LSF) with active Kriging (AK) model and reducing candidate sample pool with adaptive importance sampling (AIS), a novel AK-AIS method is proposed. Herein, the AK model and Markov chain Monte Carlo (MCMC) are first established to identify the most probable failure region(s) (MPFRs), and the adaptive kernel density estimation (AKDE) importance sampling function is constructed to select the candidate samples. With the best samples sequentially attained in the reduced candidate samples and employed to update the Kriging-fitted LSF, the failure probability and sensitivity indices are acquired at a lower cost. The proposed method is verified by two multi-failure numerical examples, and then applied to the reliability and sensitivity analyses of a typical stator blade regulator. With methods comparison, the proposed AK-AIS is proven to hold the computing advantages on accuracy and efficiency in complex reliability and sensitivity analysis problems.

Data-driven reduced order model and simplicial homology global optimization for reliability analysis and application

A novel framework of reliability analysis was developed in this study to consider the uncertainty of geomaterials and geological conditions by combining the reduced-order model (ROM), reliability analysis, and numerical model. The reliability method was used to determine the reliability index using the simplicial homology global optimization (SHGO) based on the ROM. The developed method was verified and illustrated using three numerical examples and a simple slope. The limit state curve in all three numerical examples was in excellent agreement with the actual curve. The reliability index and failure probability were also in excellent agreement with those of the actual limit state function using the first-order reliability method (FORM) and Monte Carlo simulation, respectively, indicating that the ROM method can present the limit state function well. The results showed that the developed method is feasible and effective for reliability analysis of geotechnical and geological engineering problems with a complex, nonlinear, and implicit limit state function. Furthermore, the developed method is effective, efficient, and accurate for reliability analysis. It provides an excellent way to approximate the limit state function to avoid the time-consuming numerical model in a practical engineering system.

Dimensional Reduction-Based Moment Model for Probabilistic Slope Stability Analysis

Uncertainty is an inevitable factor that influences the function analysis, design, and safe operation in engineering systems. Due to the complexity property and unclear failure mechanism, uncertainty is an intrinsic property of slope engineering. Hence, stability analysis and design cannot meet the demands of slope engineering based on the traditional deterministic method, which cannot deal with uncertainty. In this study, a practical reliability approach was developed to consider the uncertainty factor in slope stability analysis by combining the multiplicative dimensional reduction method (MDRM) and first-order second moment (FOSM). MDRM was used to approximate the complex, nonlinear, high-dimensional, and implicit limit state function. The statistical moment of safety factor was estimated based on the moment method using MDRM. FOSM is adopted to compute the reliability index based on the statistical moment of the safety factor. The proposed method was illustrated and verified by an infinite slope with an analytical solution. The reliability index and failure probability were compared with Monte Carlo simulations (MCS) in various cases. Then, it was applied to a slope based on numerical solutions. The results show that the proposed method is feasible and effective for probabilistic slope stability analysis. The reliability index obtained from the proposed method shows high consensus with the traditional response surface method (RSM). It shows that the proposed method is effective, efficient, and accurate. MDRM provides a practical, simple, and efficient probabilistic slope stability analysis approach.

Three Methods in Reliability Assessment of Engineering Structure

The work attempts to choose the handy methods for analyzing structural reliability. Comparative analysis of the methods was performed on an example of dome truss susceptible to stability loss from the condition of node snapping. In the reliability analysis of structure the load magnitudes (P), the axial stiffness of bars (EA), coordinate nodes (Z) are represented by random variables. The criterion of structural failure is expressed by the condition of non-exceeding the admissible load multiplier. The Hasofer-Lind reliability index was determined. In analysis were used three approaches differing way of defining the limit state function: Approach 1 – using of implicit limit state function, Approach 2 – using of explicit neural state functions, Approach 3 – using of the Hybrid Monte Carlo method.

A New Approach to Symmetry Reliability: Combination of Forward and Inverse Reliability Principle and Its Application to Frame Structures and Bamboo Bridges

Reliability theory is the core basis of engineering design, mainly including forward reliability theory and inverse reliability theory. Forward reliability theory is used to obtain the reliability index using the known design parameters, that is, it is a mapping function that translates the design parameters to the reliability index. Inverse reliability theory is used to obtain the design parameters using the known reliability index, that is, it is a mapping function that translates the reliability index to the design parameters. In other words, forward reliability theory and inverse reliability theory together constitute a method of dual mapping, which is the specific application of symmetry theory in the reliability field. In this paper, a new inverse reliability analysis method is proposed, which can satisfy the requirements of the target reliability index while obtaining the design parameters, without additional calculation and verification of reliability. The method simplifies the reliability inverse problem to the problem of the nonlinear equation, which is solved by identifying the design parameters, and finally obtains the design parameters by iterating the reliability index for each design parameter to gradually approach the target reliability index. For high-dimension and complex problems, the Levenberg–Marquardt method is introduced to avoid the problem of sensitivity to initial values and iterative divergence when identifying the design parameters. The implicit limit state function problem is solved by the interactive operation between ANSYS software and MATLAB software using finite element theory. The accuracy of the proposed method in this paper is verified by several numerical examples, the applicability of the implicit limit state function is verified by a single-story frame structure, and the engineering applicability of the proposed method is demonstrated with a bamboo bridge.

Reliability analysis of structures by a three-stage sequential sampling based adaptive support vector regression model

A three-stage adaptive support vector regression (SVR) based metamodel is built by sampling training data sequentially close to a limit state function (LSF). The approach alleviates the difficulty of scarcity of samples in the reduced space for reliability evaluation of a structure involving implicit LSF. Specifically, importance sampling is proposed to ensure a sufficient number of simulation points near the approximated failure plane. A design of experiment is initially constructed by a space-filling design over the entire domain. The optimum choices of the hyper-parameters of the SVR model are then determined by minimizing the generalized root mean square error (GRMSE). A subset of Monte Carlo simulation samples with magnitude of approximated LSF less than the noted GRMSE values are selected. Subsequently, the data points are added sequentially from the subset, based on the maximin criterion. Finally, the SVR model is iteratively updated to improve the reliability estimation by adding more data from the latest subset until convergence. An improved stopping condition is proposed to avoid false convergence. The effectiveness of the proposed approach along with estimation of very small probability of failure is elucidated through three numerical examples.

An Active Learning Polynomial Chaos Kriging metamodel for reliability assessment of marine structures

Metamodel combined with simulation type reliability method is an effective way to determine the probability of failure (Pf) of complex structural systems and reduce the burden of computational models. However, some existing challenges in structural reliability analysis are minimizing the number of calls to the numerical model and reducing the computational time. Most research work considers adaptive methods based on ordinary Kriging with a single point enrichment of the experimental design (ED). This paper presents an active learning reliability method using a hybrid metamodel with multiple point enrichment of ED for structural reliability analysis. The hybrid method (termed as APCKKm-MCS) takes advantage of the global prediction and local interpolation capability of Polynomial Chaos Expansion (PCE) and Kriging, respectively. The U learning function drives active learning in this approach, while K-means clustering is proposed for multiple point enrichment purposes. Two benchmark functions and two practical marine structural cases validate the performance and efficiency of the method. The results confirm that the APCKKm-MCS approach is efficient and reduces the computational time for reliability analysis of complex structures with nonlinearity, high dimension input random variables, or implicit limit state function.

Reliability-based prediction of landslide using efficient Monte Carlo Simulation

Several studies have attempted to develop thresholds for prediction of landslide initiation due to antecedent rainfall by using the physical-based approach. The approach offers a better understanding of the underlying process of the landslide mechanism. In the previous studies, however, determinate values of hydro-mechanical soil parameters are commonly considered, neglecting the uncertainties of the variables. Thus, this study aims to propose a relatively computationally, inexpensive reliability approach to address the uncertainties of the multivariate. The application of the proposed model is demonstrated by taking a case study from Sabah, Malaysia. The Latin Hypercube Sampling was adopted for efficient sampling, while the dependencies of the multivariate were modelled using a higher-order copula. The infiltration process and slope stability were assessed based on a fully-coupled seepage-deformation and shear strength reduction technique, respectively in a finite element programme. The probabilities of failure for various antecedent rainfall condition were computed by using Monte Carlo Simulation. In order to reduce the computational cost of the simulation, the Multilayer Perceptron regressor was applied to form the non-linear surrogate model for an implicit limit state function.

Accurate Structural Reliability Analysis Using an Improved Line-Sampling-Method-Based Slime Mold Algorithm

Line sampling (LS) is a robust and powerful simulation technique to reduce the computational burden provided by Monte Carlo simulation (MCS) for the reliability analysis of engineering structures. However, when dealing with highly nonlinear and implicit limit-state functions, LS yields instable results as nonconvergence or divergence. In this study, a novel framework that integrates the LS method with the slime mold algorithm (LS-SMA) is proposed to solve complex structural reliability problems. SMA is a new metaheuristic population-based algorithm inspired by the behavior and morphological changes in slime molds that can well solve multivariable optimization problems. In the proposed method, the determination of the important direction of LS is formulated as an unconstrained optimization problem according to the LS theory. Then SMA is employed to solve this optimization problem to decrease the computational cost. Thus, the LS-SMA is able to overcome the drawbacks of LS such as the local convergence and divergence. Seven numerical problems were utilized to investigate the LS-SMA applicability, where its performance was compared with MCS, subset simulation (SS), importance sampling (IS), LS, first-order reliability method (FORM), and first-order control variate method (FOCM). The results demonstrate that the proposed LS-SMA can be applied with high efficiency for solving the reliability problems that involve highly nonlinear or dimensional and complex implicit limit-state functions.

A novel mixed uncertainty support vector machine method for structural reliability analysis

For structures with both random and fuzzy uncertainty parameters, a novel method for obtaining the membership function of reliability in the fuzziness failure criterion is presented using a Random Fuzzy Support Vector Machine based on the Particle Swarm Optimization method (PSO-RFSVM). The proposed method is used to solve the structural reliability problem with implicit limit state function in the presence of mixed variables. Under each cut level, a new distance measure between the mixed variables is presented, named the advanced Yang distance; then, the RFSVM model can be constructed using the kernel function built by the advanced Yang distance and random fuzzy mixed sampling points. Furthermore, to obtain satisfactory fitting, a PSO algorithm is used to optimize the super-parameters in RFSVM. The limit state function is subsequently approximated at the given cut level, and then, the reliability bound under the given cut level is readily obtained. The PSO-RFSVM method provides a new efficient analysis framework in the case of implicit approximation under mixed uncertainty when the classical method has excessive consumption in virtual prototype simulation. Two examples are used to demonstrate the validity and advantage of PSO-RFSVM compared to the Extreme Response Surface based on Simulated Annealing (SAERS) method and Monte Carlo simulation (MCS).

Uncertainty propagation in structural reliability with implicit limit state functions under aleatory and epistemic uncertainties

Uncertainty propagation plays a pivotal role in structural reliability assessment. This paper introduces a novel uncertainty propagation method for structural reliability under different knowledge stages based on probability theory, uncertainty theory and chance theory. Firstly, a surrogate model combining the uniform design and least-squares method is presented to simulate the implicit limit state function with random and uncertain variables. Then, a novel quantification method based on chance theory is derived herein, to calculate the structural reliability under mixed aleatory and epistemic uncertainties. The concepts of chance reliability and chance reliability index (CRI) are defined to show the reliable degree of structure. Besides, the selection principles of uncertainty propagation types and the corresponding reliability estimation methods are given according to the different knowledge stages. The proposed methods are finally applied in a practical structural reliability problem, which illustrates the effectiveness and advantages of the techniques presented in this work.

A Systematic Review of Structural Reliability Methods for Deformation and Fatigue Analysis of Offshore Jacket Structures

This paper presents the state of the art in Structural Reliability Analysis (SRA) methods with a view of identifying key applications of each method and its proposed variations, qualifying characteristics, advantages, and limitations. Due to the increasing complexity and scale of modern offshore jacket structures, it becomes increasingly necessary to propose an accurate and efficient approach for the assessment of uncertainties in their material properties, geometric dimensions, and operating environments. SRA, as a form of uncertainty analysis, has been demonstrated to be a useful tool in the design of structures because it can directly quantify how uncertainty about input parameters can affect structural performance. Herein, attention was focused specifically on the probabilistic fracture mechanics approach because this accounts accurately for fatigue reliability mostly encountered as being dominant in the design of such structures. The well-established analytical/approximate methods such as the First- and Second-Order Reliability Methods (FORM/SORM) are widely used as they offer a good balance between accuracy and efficiency for realistic problems. They are, however, inaccurate in cases of highly non-linear systems. As a result, they have been modified using methods such as conjugate search direction approach, saddle point approximation, subset simulation, evidence theory, etc. in order to improve accuracy. Initially, direct simulations methods such as the Monte Carlo Simulation Method (MCS) with its various variance reduction techniques such as the Importance Sampling (IS), Latin Hypercube Sampling (LHS), etc. are ideal for structures having non-linear limit states but perform poorly for problems that calculate very low probabilities of failure. Overall, each method has its own merits and limitation, with FORM/SORM being the most commonly used, but recently, simulation methods have increasingly been used due to continuous advances in computation powers. Other relevant methods include the Response Surface Methods (RSM) and the Surrogate Models/Meta-models (SM/MM), which are advanced approximation methods and are ideal for structures with implicit limit state functions and high-reliability indices. Combinations of advanced approximation methods and reliability analysis methods are also found in literature as they can be suitable for complex, highly non-linear problems.

Structural Reliability Assessment of Steel Four-Bolt Unstiffened Extended End-Plate Connections Using Monte Carlo Simulation and Artificial Neural Networks

Connections account for a pivotal function in the safety of a structure. The structural design should be based on minimal probability of failure in its lifetime, which is addressed by the probabilistic methods for the evaluation of structural reliability. However, conventional reliability techniques, including Monte Carlo simulation (MCS), require plenty of time for cases with implicit limit state function (LSF). Accordingly, the present study aims to analyze the structural reliability of the steel four-bolt unstiffened extended end-plate connections using integrated artificial neural network (ANN) and MCS approaches. The ANN-based MCS exhibits a higher speed compared to the conventional application as the implicit LSF is estimated by the ANN model. The finite element modeling of the connections provides the data required for ANN training. Following the achievement of the LSF, the MCS method is used to assess the connection reliability. Considering a target reliability index of 3.5, a resistance reduction factor was obtained to be 0.82 which is smaller than the current system resistance factor used in AISC 358 connections.