Second-order Reliability Method Research Articles

Reliability and sensitivity analysis of delamination growth of composite laminate structures using two efficient sampling methods

In recent years, composite structures have been used in a large number of applications in aerospace, machinery, marine, and civil engineering. However, there are inevitably many uncertainties in the whole life cycle of composite structures, which can easily lead to structural damage and failure. Therefore, it is important to analyze the reliability and sensitivity of composite structures. At present, most of the contributions use the first-order reliability method (FORM) and the second-order reliability method (SORM) to study the reliability of composite structures and compare them with the results of the Monte Carlo simulation. However, both methods have their limitations. FORM cannot guarantee the calculation accuracy for the highly nonlinear limit state equation, and the calculation efficiency of SORM is too low. Therefore, this paper proposes to use importance sampling (IS) and backpropagation neural network-based Monte Carlo (MC-BPNN) to study the reliability, sensitivity, and dispersion of delamination growth of composite laminates. The results show that compared with FORM and SORM, IS and MC-BPNN have higher calculation accuracy and efficiency and can more accurately evaluate the failure degree of composite structures and ensure their safe operation in the field of aerospace equipment. The universality of this method can make it being widely used in the reliability and sensitivity analysis of different composite materials as well as dispersion analysis.

Improved FORM and SORM Based on Improved Modified Symmetric Rank 1 Algorithm and Adaptive Kriging Model

Abstract The first-order reliability method (FORM) is simple and efficient for solving structural reliability problems but may have large errors and converge slowly or even result in divergence when dealing with strongly nonlinear performance functions. For this case, the existing second-order reliability method (SORM) achieves higher computational accuracy but with a consequent decrease in efficiency. To achieve a better balance between accuracy and efficiency, this paper proposes an improved FORM and an improved SORM. First, an improved modified symmetric rank 1 (IMSR1) algorithm, in which the line search strategy for step length is unnecessary, is proposed for iterations of the FORM, and an adaptive Kriging model with a rational update criterion is presented to improve the efficiency of the FORM. Then, an improved FORM with high efficiency and good convergence is proposed. Second, due to the good precision of the adaptive Kriging model at the final design point, the Hessian matrix is available easily without additional computational effort, and an improved SORM with the same efficiency as the improved FORM is presented naturally. Finally, the accuracy, efficiency, and convergence of the proposed methods are verified by numerical and engineering examples.

Dimension reduction for constructing high-dimensional response distributions by accounting for unimportant and important variables

Probability distributions of responses have been widely used in structural analysis and design because of their complete statistical information. In practice, the dimensionality of input variables could easily reach hundreds or thousands, making it computationally expensive to obtain accurate distributions. In this paper, a generalized most probable point (MPP) method is developed to effectively build the response distributions of high-dimensional problems. First, a global index based on one-iteration MPPs is presented for dimension reduction, which is to divide the input variables into important and unimportant variables. After fixing the unimportant variables at their one-iteration MPP components, the MPP components of the important variables are obtained by performing the inverse first-order reliability method (FORM) in the reduced space. Predictive models of the all MPP components are then established to quickly estimate the MPPs of other cumulative distribution function (CDF) values. To accurately calculate CDF points of limit state functions with different shapes, a comprehensive uncertainty analysis method that accommodates the contributions of the important and unimportant variables is proposed by multiple combinations of FORM, second-order reliability method, and second-order saddlepoint approximation. Finally, the response distributions are generated based on Gaussian mixture distribution and all CDF points. The effectiveness of the proposed method is verified by a mathematical example and two engineering cases.

Robust adaptive analysis of dynamic responses of wave energy converters

PurposeThe purpose of this paper is to exploit a new and robust method to forecast the long-term extreme dynamic responses for wave energy converters (WECs).Design/methodology/approachA new adaptive binned kernel density estimation (KDE) methodology is first proposed in this paper.FindingsBy examining the calculation results the authors has found that in the tail region the proposed new adaptive binned KDE distribution curve becomes very smooth and fits quite well with the histogram of the measured ocean wave dataset at the National Data Buoy Center (NDBC) station 46,059. Carefully studying the calculation results also reveals that the 50-year extreme power-take-off heaving force value forecasted based on the environmental contour derived using the new method is 3572600N, which is much larger than the value 2709100N forecasted via the Rosenblatt-inverse second-order reliability method (ISORM) contour method.Research limitations/implicationsThe proposed method overcomes the disadvantages of all the existing nonparametric and parametric methods for predicting the tail region probability density values of the sea state parameters.Originality/valueIt is concluded that the proposed new adaptive binned KDE method is robust and can forecast well the 50-year extreme dynamic responses for WECs.

Probability-Based Design of Reinforced Rock Slopes Using Coupled FORM and Monte Carlo Methods

The efficiency of the first-order reliability method (FORM) and the accuracy of Monte Carlo simulations (MCS) are coupled in probability-based designs of reinforced rock slopes, including a Hong Kong slope with exfoliation joints. Load–resistance duality is demonstrated and resolved automatically in a foundation on rock with a discontinuity plane. Other examples include the lengthy Hoek and Bray deterministic vectorial procedure for comprehensive pentahedral blocks with external load and bolt force, which is made efficient and more succinct before extending it to probability-based design via MCS-enhanced FORM. The FORM–MCS–FORM design procedure is proposed for cases with multiple failure modes. For cases with a dominant single failure mode, the time-saving importance sampling (IS) and the fast second-order reliability method (SORM) can be used in lieu of MCS. Two cases of 3D reinforced blocks (pentahedral and tetrahedral, respectively) with the possibility of multiple sliding modes are investigated. In the case of the reinforced pentahedral block, direct MCS shows that there is only one dominant failure mode, for which the efficient method of importance sampling at the FORM design point provides fast verification of the revised design. In the case of the reinforced tetrahedral block, there are multiple failure modes contributing to the total failure probability, for which the proposed MCS-enhanced FORM procedure is demonstrated to be essential. Comparisons are made between Excel MCS and MATLAB MCS.

Modified control variates method based on second-order saddle-point approximation for practical reliability analysis

Abstract. A novel method is presented for efficiently analyzing the reliability of engineering components and systems with highly nonlinear complex limit state functions. The proposed method begins by transforming the integral of the limit state function into an integral of a highly correlated limit state function using the control variates method. The second-order reliability method is then employed within the control variates framework to approximate the highly correlated limit state function as a quadratic polynomial. Subsequently, the probability of failure is obtained through the estimation of the saddle-point approximation and a small number of samples generated by Latin hypercube sampling. To demonstrate the effectiveness of the proposed method, four examples involving mathematical functions and mechanical problems are solved. The results are compared with those obtained using the second-order reliability method (SORM), control variates based on Monte Carlo simulation (CVMCS) with second-order saddle-point approximation (SOSPA), importance sampling (IS) and Monte Carlo simulation (MCS). The findings demonstrate that, while maintaining high-precision reliability results, the proposed method significantly reduces the number of evaluations of the limit state function through a small number of initial samples. The method is capable of efficiently and accurately solving complex practical engineering reliability problems.

An Efficient Reliability Analysis Method Based on the Improved Radial Basis Function Neural Network

Abstract This paper develops an efficient reliability analysis method based on the improved radial basis function neural network (RBFNN) to increase the accuracy and efficiency of structural reliability analysis. To solve the problems of low computational accuracy and efficiency of the RBFNN, an improved RBFNN method is developed by transferring the sampling center of Latin hypercube sampling (LHS) from the mean values of random variables to the most probable point (MPP) in the sampling step. Then, the particle swarm optimization algorithm is adopted to optimize the shape parameters of RBFNN, and the RBFNN model is assessed by the cross-validation method for subsequent reliability analysis using Monte Carlo simulation (MCS). Four numerical examples are investigated to demonstrate the correctness and effectiveness of the proposed method. To compare the computational accuracy and efficiency of the proposed method, the traditional radial basis function method, hybrid radial basis neural network method, first-order reliability method (FORM), second-order reliability method (SORM), and MCS method are applied to solve each example. All the results demonstrate that the proposed method has higher accuracy and efficiency for structural reliability analysis. Importantly, one practical example of an industrial robot is provided here, which demonstrates that the developed method also has good applicability and effectiveness for complex engineering problems.

A novel method for analysis of offshore sustainable energy systems

In the present research, we have proposed a new adaptive kernel density estimation method formulated on the theory of linear diffusion processes. By examining the calculation results we have found that in the tail region, our proposed new adaptive kernel density estimation distribution curve becomes very smooth and fits quite well with the histogram of the measured ocean wave dataset at the US National Data Buoy Center station 46026. Carefully studying the calculation results also reveals that the 50-year extreme Power-Take-Off heaving force value forecasted based on the environmental contour derived using the new method is 3021700N, which is much larger than the value 2458700N forecasted via the Rosenblatt-inverse second-order reliability method contour method. Consequently, our proposed new adaptive kernel density estimation method formulated on the theory of linear diffusion processes can forecast well and efficiently forecast the 50-year extreme design force values for offshore sustainable energy systems.

Optimum Design of Unsaturated Finite Clayey Slopes Using Second-Order Reliability Method

This paper discusses the second-order reliability-based design optimization (SORBDO) of unsaturated finite slopes. An efficient search algorithm is proposed to locate the most probable slip surfaces using the Morgenstern–Price method of slices. The variability of fitting parameters of the soil water characteristic curve (SWCC) and shear strength parameters have been taken into account for evaluating the reliability of unsaturated finite slopes. Six unsaturated shear strength (USS) models are considered for the analysis of unsaturated soils. The most conservative model for the optimum design of finite slopes is recommended. Moreover, the results indicate that the mean and standard deviation of the fitting parameters of SWCC have a significant effect on the critical centers, critical slip surfaces, and reliability index of the unsaturated finite slopes. The results demonstrate a significant change in the critical centers associated with critical slip surfaces due to the change in the mean and coefficient of variation of the fitting parameters of SWCC. Statistical analysis of the experimental data shows that a positive correlation exists between fitting parameters of SWCC related to air entry value (af) and residual water content (mf). Negative correlations are found between the fitting parameters of SWCC related to air entry value (af) and slope (nf). Similarly, the SWCC fitting parameters related to residual water content (mf) and slope (nf) have negative correlations for clayey soils. It is observed that the correlated value of the reliability index is overestimated with reference to the uncorrelated value for clayey soil slopes. The proposed design charts are beneficial for engineers and practitioners for the sustainable design of unsaturated finite slopes. Overall, the developed rigorous SORBDO framework offers a more rational approach for assessing critical failure surfaces and critical centers, considering the interdependency of random variables.

Reliability analysis of compressed timber studs on the buckling criterion

Introduction. Ensuring safety is a priority goal in the design, construction and operation of building and structures. A quantitative assessment of safety can be the failure probability of a structural element. The article presents algorithms for reliability analysis of timber studs by the buckling criterion under central compression force.
 
 Materials and methods. The FOSM (First Order Second Moment) method is a classic approach for solving many reliability analyses tasks in the engineering sector. This method uses the second statistical moments of random variables (mathematical expectation and variance) and the method of statistical linearization with the decomposition of the function into a Taylor series of the first order. However, it is often necessary to deal with nonlinear limit state functions, where the statistical linearization method can lead to incorrect results. In this case, it is necessary to use other algorithms for reliability analysis: the Hasofer – Lind method, SORM (Second Order Reliability Method), etc.
 
 Results. The numerical example of reliability analysis for compressed timber studs on the buckling criterion is considered. To illustrate the problem of the research, the dimensions of the cross-section of the studs and the strength of the wood are taken as random variables, and the load is considered a deterministic value. In the following, an algorithm for reliability analysis is described for a timber studs under random load. The classical FOSM approach showed an overestimated reliability index by 35 % compared to the results of the Monte Carlo numerical experiment and the analytical solution using the Hasofer – Lind algorithm.
 
 Conclusions. The use of the traditional FOSM method for reliability analysis of wooden studs according to the buckling criterion can lead to overestimated reliability index estimates. It is unacceptable from the point of view of the structural safety. The calculation and analysis of the reliability of timber studs should be carried out on the basis of the Hasofer – Lind algorithm, the SORM method or other optimization methods.

Reliability analysis of smart laminated composite plates under static loads using artificial neural networks

The applications of smart structures with integrated piezoelectric elements have been expanding in the last few decades due to the abilities of such structures to withstand mechanical loads and operate as sensors or actuators using their electromechanical coupling. The available manufacturing techniques can result in uncertainties in the structure's geometric parameters, which, coupled with uncertainties in material properties, can lead to unexpected failures or unreliable performance. This paper presents a reliability analysis of a smart laminated composite plate made of a graphite/epoxy cross-ply substrate with a piezoelectric fiber-reinforced composite (PFRC) actuator layer under static electrical and mechanical loads. A coupled finite element (FE) model was developed in COMSOL Multiphysics, from which nondimensional stresses and displacements were calculated. To investigate the effects of randomness in the material and geometric properties, an artificial neural network (ANN) model was developed and trained using generated FE data. Monte Carlo Simulation (MCS) and First- and Second-Order Reliability Methods (FORM/SORM) were then used to shed light on the significance of considering randomness in the various material and geometric parameters and the effect of such uncertainty on the resulting nondimensional stresses and displacements. A coefficient of variation (CV) study identified the piezoelectric stress coefficient as the most significant contributing factor to the variation of all nondimensional parameters. Variation in the nondimensional parameters also increases under the application of an electric load. ANN-based FORM, SORM, and MCS all indicate a pattern of low probability of failure until a threshold value of about 3% of input parameter variation is reached, beyond which there is a rapid nonlinear increase in failure probability with increasing input parameter variation.

Stochastic Reliability-Based Design Optimization Framework for the Steel Plate Girder with Corrugated Web Subjected to Corrosion.

This paper proposes the framework for reliability-based design optimization (RBDO) of structural elements with an example based on the corrugated web I-girder. It tackles the problem of topological optimization of corroding structures with uncertainties. Engineering restrictions follow a concept of the limit states (LS) and extend it for stability and eigenfrequency assessment. The reliability constraints include all the LS; they are computed according to first- and second-order reliability methods. The RBDO example minimizes the bridge girder cross-section while satisfying the structural reliability level for the ultimate and the serviceability limit states, stability, and eigenfrequency. It takes into consideration two uncorrelated random effects, i.e., manufacturing imperfection and corrosion. They are both Gaussian; the first of them is applied at assembly time, while the second is applied according to the time series. The example confronts three independent FEM models with an increasing level of detailing, and compares RBDO results for three concurrent probabilistic methods, i.e., the iterative stochastic perturbation technique (ISPT), the semi-analytical method, and the Monte Carlo simulation. This study proves that the RBDO analysis is feasible even for computationally demanding structures, can support automation of structural design, and that the level of detailing in the FEM models influences its results. Finally, it exemplifies that reliability restrictions for LS are much more rigorous than for their deterministic counterparts, and that the fastest ISPT method is sufficiently accurate for probabilistic calculations in this RBDO.

Representative slip surface identification and reliability analysis of slope systems in spatially variable soils

ABSTRACT A slope system is a series system with numerous potential slip surfaces (PSSs), and its failure probability is commonly evaluated by several significant failure surfaces, or representative slip surfaces (RSSs). Previous efforts have mainly identified the RSSs in spatially variable soils from the perspective of the correlations between different PSSs, the effects of the failure probabilities of the PSSs were rarely considered. With the goal of identifying RSSs from the perspective of the system failure probability, a method adopting the second-order reliability method (SORM) and the multimodal optimisation is proposed. In this method, the spatial variability of soil properties along the slip surface is characterised by local averaging to reduce the number of variables in SORM. Equations for calculating the correlation coefficient between different PSSs with correlated variables are derived. The task of RSS identification is transformed as a multimodal optimisation problem, and the PSSs that make great contributions to the system failure probability are determined as RSSs. The proposed method and the derived equations are demonstrated using two slope examples. The results show that the proposed method is capable of identifying RSSs with significant contributions, and it provides a proper estimate of the system failure probability.

Steam turbine maintenance planning based on forecasting of life consumption processes and risk analysis

Flexible operation of coal-fired power plants contributes to the intensification of the life consumption processes, which is a serious problem especially in the case of units with a long in-service time. In steam turbine rotors, the crack propagation rate and material wear caused by low-cycle fatigue increase. The aim of the research is an attempt to forecast the development of these processes and to estimate the probability of critical elements damage, such as the high-pressure and intermediate-pressure rotors. In the stress state analyses, the finite element method (FEM) is used, the Monte Carlo method and the second order reliability method (SORM) is apply to calculate the probability of failure. It is proposed to use risk analysis to plan preventive maintenance of the turbine. The optimal intervals for carrying out diagnostic tests and prophylactic repairs is determined for various operating scenarios and various failure scenarios. This enables a reduction of the costs while ensuring the safety of the turbine's operation.

Punching Shear Strength of FRP-Reinforced Concrete Slabs without Shear Reinforcements: A Reliability Assessment.

The recent failure of buildings because of punching shear has alerted researchers to assess the reliability of the punching shear design models. However, most of the current research studies focus on model uncertainty compared to experimentally measured strength, while very limited studies consider the variability of the basic variables included in the model and the experimental measurements. This paper discusses the reliability of FRP-reinforced concrete slabs’ existing punching shear models. First, more than 180 specimens were gathered. Second, available design codes and simplified models were selected and used in the calculation. Third, several reliability methods were conducted; therefore, three methods were implemented, including the mean-value first-order second moment (MVFOSM) method, the first-order second moment (FOSM) method, and the second-order reliability method (SORM). A comparison between the three methods showed that the reliability index calculated using the FOSM is quite similar to that using SORM. However, FOSM is simpler than SORM. Finally, the reliability and sensitivity of the existing strength models were assessed. At the same design point, the reliability index varied significantly. For example, the most reliable was the JSCE, with a reliability index value of 4.78, while the Elgendy-a was the least reliable, with a reliability index of 1.03. The model accuracy is the most significant parameter compared to other parameters, where the sensitivity factor varied between 67% and 80%. On the other hand, the column dimension and flexure reinforcement are the least significant parameters compared to other parameters where the sensitivity factor was 0.4% and 0.3%, respectively.

Finite element reliability analysis of edge delamination onset due to interlaminar stresses in composite laminates

A novel approach is employed to develop an efficient algorithm by which Finite Element Reliability Analysis (FERA) is conducted to predict the Edge Delamination Onset (EDO) caused by interlaminar stresses. The Finite Element Method (FEM) has made it possible to calculate the EDO probability process through different interfaces of composite laminates. The FEM and reliability analyzes are performed by ANSYS and MATLAB, respectively. Then, an interactive interface is created to link ANSYS and MATLAB. Simultaneous integration of these two analyses results in stochastic algorithm development to predict the probability of EDO. The probability of EDO is calculated with the aid of First-Order Reliability Method (FORM) and Second-Order Reliability Method (SORM), and the results deduced from these methods are verified by Monte Carlo Simulation (MCS). Three different states are considered for the probabilistic prediction of the edge delamination onset including pure mode I, pure mode III and mixed mode I/III. The effects of fibre angle, laminate’s thickness, different interfaces and scattering of random variables on EDO probability are investigated. The results are qualitatively validated with published experimental data, showing a good agreement between the probabilities achieved by the FERA algorithm and the experimental observations.

A probabilistic approach to estimate the remaining life and reliability of corroded pipelines

Considering corrosion rate during the remaining life assessment of aging pipelines is fundamental to calculate the interval between two consecutive inspections. A total of 798 internal and external corrosion defects have been detected, using the Magnetic Flux Leakage intelligent inspection tool, over 48 km of a pipeline length located in the west region of Algeria. The statistical analysis has shown that there is a strong correlation between the corrosion defect length and the corrosion defect circumferential width, with a significant correlation coefficient equal to 82.87%. A probabilistic methodology is presented for the assessment of the remaining life of a corroded pipeline transporting gas, and a finite element method (FEM) was used to assess the pipeline failure pressure. The numerical FEM modeling results were compared with the commonly used codes-models for calculating limit pressure to establish a more realistic and accurate engineering model. The reliability analysis of an API 5L X60 steel made Algerian natural gas pipeline, in service for thirty years, and exposed to active corrosion attack, is presented. The sensitivity analysis of the basic random variables within the nonlinear limit state function was carried out to bring out the relative contribution of each variable affecting the remaining life of corroded pipelines. The reliability analysis is carried out by using Breitung's formula, based on the second-order reliability method (SORM). The reliability assessment of the corroded pipeline is based on the usage of the notched failure assessment diagram (NFAD), different codes for the calculation of the failure pressure, and the numerical results using the finite element analyses (FEA) software ANSYS.

AK-SESC: a novel reliability procedure based on the integration of active learning kriging and sequential space conversion method

To deal with evaluating small failure probabilities, AK–SESC: a novel approach integrating an active learning Kriging meta-model (AK-MCS) and the SESC, a sequential space conversion method, is suggested. The efficiency of the proposed approach relies on the advantages of the AK-MCS and its updating feature to evaluate the actual performance function and the superiority of SESC in estimating small failure probabilities. Although there are effective methods for small probabilities, the beauty of this approach is that it is derived from the probability integral with no simplifications while providing results of high accuracy.Different problems were solved to study the AK–SESC applicability. The main effort of this method is reducing the function call numbers of the original SESC while reaching the same accuracy as Monte Carlo Simulation (MCS). The reliability analysis results were compared with the main reliability methods of the Importance Sampling (IS), Subset Simulation (SubSim), Line Sampling (LS), First and also Second-Order Reliability Method (FORM and SORM). The solved problems indicate that the proposed approach provides accurate answers with much fewer function calls than SESC. So, it can be a promising method for reliability analyses involving nonlinear or high-dimensional performance functions with small failure probabilities.

Structural system reliability analysis based on multi-modal optimization and saddlepoint approximation

This paper studies reliability analysis of structures with multiple failure modes. The analysis consists of three parts. First, the memetic animal migration optimization algorithm, applied for multiple optimum points detection in a multimodal optimization problem, is used to search for the most probable points of the limit state functions of main failure modes. Next, the multi-point first-order reliability method is used to estimate the failure probability of structures with multiple failure modes. Last, the saddlepoint approximation based second-order reliability method is further extended to system reliability analysis, aiming to improve the accuracy of system reliability prediction.

Probability Methods for Stability Design of Open Pit Rock Slopes: An Overview

The rock slope stability analysis can be performed using deterministic and probabilistic approaches. The deterministic analysis based on the safety concept factor uses fixed representative values for each input parameter involved without considering the variability and uncertainty of the rock mass properties. Probabilistic analysis with the calculation of probability of failure instead of the factor of safety against failure is emerging in practice. Such analyses offer a more rational approach to quantify risk by incorporating uncertainty in the input variables and evaluating the probability of the failure of a system. In rock slope engineering, uncertainty and variability involve a large scatter of geo-structural data and varied geomechanical test results. There has been extensive reliability analysis of rock slope stability in the literature, and different methods of reliability are being employed for assessment of the probability of failure and the reliability of a slope. Probabilistic approaches include Monte Carlo simulation (MCS), the point estimate method (PEM), the response surface method (RSM), first- and second-order reliability methods (FORMs and SORMs), and the first-order second-moment method (FOSM). Although these methods may be complicated, they provide a more complete definition of risk. Probabilistic slope stability analysis is an option in most commercial software; however, the use of this method is not common in practice. This paper provides an overview of the literature on some of the main probabilistic reliability-based methods available for the design of the rock slope in open pit mining. To demonstrate its applicability, the paper investigates the stability of a rock slope in an open pit mine in the Goldfields region, Western Australia. Two different approaches were adopted: deterministic stability analysis using two-dimensional limit equilibrium and finite element shear strength reduction methods using SLIDE and RS2 software, respectively, and probabilistic analysis by applying the MCS and RSM methods in the limit equilibrium method. In this example, the slope stability analysis was performed using the Spencer method with Cuckoo search optimization to locate the critical slip surface. The results obtained were compared and commented on.