Second-order Reliability 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.

Machine learning and traditional approaches in shear reliability of steel fiber reinforced concrete beams

In the field of structural engineering, the exploration of steel fibre reinforced concrete (SFRC) beams has recently intensified, particularly due to their improved tension and shear performance of structure. This study pioneers a novel reliability analysis of shear capacity predictions for SFRC beams, distinctively classifying the datasets into high-strength (HSFRC) and normal-strength (NSFRC) categories. A comprehensive database of 142 HSFRC and 265 NSFRC beams serves as the foundation for this analysis, which critically examines the standard Gaussian distribution in shear design models and proposes the Lognormal and Weibull distributions as more precise alternatives. Employing advanced First-order (FORM) and Second-order Reliability Methods (SORM), the study covers a broad spectrum of load conditions, including dead, live, snow, wind, and seismic loads, to evaluate various empirical, semi-empirical and machine learning proposed shear capacity prediction formulas. One of the key innovations of this research is the development of differentiated resistance coefficients for various risk levels in the reliability analysis, allowing future structural designers to tailor their designs according to specific risk profiles. This approach significantly enhances the balance between economic efficiency and structural safety based on the evolution of different target reliability indexes. The study reveals that existing design equations for SFRC beams generally lean towards conservatism. This study found that formulas derived from machine learning exhibited superior prediction ability compared to traditional theoretically derived regression formulas. When compared the proposed machine learning formulas, the Tarawneh's formula demonstrated better prediction ability than the Kara's formula when applied to larger datasets. However, high predictive power does not necessarily equate to high reliability. Machine learning formulas prioritise predictive accuracy, often at the expense of insufficient redundancy for ensuring safety. Overall, it introduces Kara's formula for NSFRC beams, which stands out with its superior predictive performance, offering an optimal balance of safety and cost-efficiency. Ashour's formula is also identified as a more effective and safer option for HSFRC beams. Complementing these findings, the extensive sensitivity analysis of the collected data not only confirms the robustness of the conclusions but also prepares for the integration of broader datasets in the future.

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.

A Curved Surface Integral Method for Reliability Analysis of Multiple Failure Modes System with Non-Overlapping Failure Domains

Abstract In the study of reliability of systems with multiple failure modes, approximations can be obtained by calculating the probability of failure for each state function. The first-order reliability method and the second-order reliability method are effective, but they may introduce significant errors when dealing with certain nonlinear situations. Simulation methods such as line sampling method and response surface method can solve implicit function problems, but the large amount of calculation results in low efficiency. The curved surface integral method (CSI) has good accuracy in dealing with nonlinear problems. Therefore, a system reliability analysis method (CSIMMS) is proposed on the basis of CSI for solving multiple failure modes system reliability problems with non-overlapping failure domains. The order of magnitude of the failure probability is evaluated based on the reliability index and the degree of nonlinearity, ignoring the influence of low order of magnitude failure modes, and reducing the calculation of the system failure probability. Finally, CSIMMS and other methods are compared by three numerical examples, and the results show the stability and accuracy of the proposed method.

Response Extremes of Floating Offshore Wind Turbine Based on Inverse Reliability and Environmental Contour Method

Floating structures are subject to complex marine conditions. To ensure their safety, reliability analysis needs to be conducted during the design phase. However, because of the complexity of traditional full long-term analysis, the environmental contour method (ECM) based on the inverse reliability method, which can combine accuracy and efficiency, is extensively used. Due to the unique environment in the South China Sea, the probabilistic characteristics of three-dimensional (3D) environmental parameters of wind, wave and current are investigated. The ECs of the target sea are established via the ECM based on both the inverse first-order reliability method (IFORM) and inverse second-order reliability method (ISORM). It is found that the sea state forecasted by ISORM is more extreme and may lead to a more conservative design than IFORM. Furthermore, the wind–wave–current combination coefficient matrixes developed using the 3D ECs are proposed for the design of FOWTs in the South China Sea. The validity and practicality of the contours and matrixes are tested by using a floating offshore wind turbine (FOWT) as a numerical example. Then, the short-term response of the structure under the combined wind, wave and current conditions is calculated, providing a theoretical reference for the design of FOWTs.

Investigating the Reliability of an Existing Top Angle and Seat Pad Semi-Rigid Connection System Using Advanced Modelling Techniques

The investigation introduces an advanced model procedure for evaluating the structural reliability of semi-rigid frame connections in existing aged buildings. Specifically, the focus is on the top angle and seat pad (TA–SP) semi-rigid connection, which was not initially considered in the current design standards. The approach employs a plastic hinge model to predict the ultimate strength of the connection and its beam-to-column behaviour. In order to increase computational efficiency, the investigation leverages the nonlinear behaviour of the finite element (FE) model to validate critical parameters. The statistical properties of the existing connection were obtained based on past experimental data, highlighting the weakest elements in the system. The first-order and second-order reliability methods and Monte Carlo simulations were employed to estimate the reliability index. Percentile errors were assessed to understand their impact on higher-order interactions. This new technique of identification quantifies the probability of the system failure interactions. Notably, a 45% lesser error aligns with the target reliability index, while a 114.5% larger error indicates a significant deviation from actual failure probability values. Each methodology introduced adheres to the current standard, and the system reliability analysis provides a vigorous conclusions scheme framework for assessing the existing TA–SP semi-rigid connection.

Reliability analysis of aerospace planetary roller screw mechanism considering multiple failure modes

Abstract Faults in planetary roller screw mechanisms under typical aerospace operating conditions are usually caused by the combined effects of multiple coupled failure modes. Reliability analysis methods that ignore correlation often lead to significant errors. In this paper, the failure models of the planetary roller screw mechanism under fatigue elastic failure, wear failure, and fracture failure are analyzed based on the stress-strength interference model, and failure functional functions are provided. The improved O. Ditlevsen second-order reliability bound theory is used to characterize the relationships between multiple failure modes. Using this method, reliability analysis and calculation of planetary roller screw pairs considering multiple failure modes have been performed, verifying the rationality and effectiveness of this method.

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.

First- and Second-Order Reliability Analysis of Rainfall-Induced Kotropi Slope Failure

First- and Second-Order Reliability Analysis of Rainfall-Induced Kotropi Slope Failure

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.