Structural Reliability Analysis for Implicit Performance with Legendre Orthogonal Neural Network Method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be estimated by a numerical analysis such as finite element method. The response surface method could be used to reduce the computational effort required for reliability analysis when the performance functions are implicit. However the conventional response surface method is time-consuming or cumbersome if the number of random variables is large. This paper presents a Legendre orthogonal neural network (LONN)-based response surface method to predict the reliability of a structure. In this method, the relationship between the random variables and structural responses is established by a LONN model. Then the LONN model is connected to a reliability method, i.e. first-order reliability methods (FORM) to predict the failure probability of the structure. Numerical example has shown that the proposed approach is applicable to structural reliability analysis involving implicit performance functions.

Fuzzy reliability analysis can be implemented using two discrete optimization maps in the processes of reliability and fuzzy analysis. Actually, the efficiency and robustness of the iterative reliability methods are two main factors in the fuzzy-based reliability analysis due to the huge computational burdens and unstable results. In the structural fuzzy reliability analysis, the first-order reliability method (FORM) using discrete nonlinear map can provide a C membership function. In this paper, a discrete nonlinear conjugate map is proposed using a relaxed-finite step size method for fuzzy structural reliability analysis, namely Fuzzy conjugate relaxed-finite step size method fuzzy CRS. A discrete conjugate map is stabilized using two adaptive factors to compute the relaxed factor and step size in FORM. The framework of the proposed fuzzy structural reliability method is established using two linked iterative discrete maps as an outer loop, which constructs the membership function of the response using alpha level set optimization based on genetic operator, and the inner loop, implemented for reliability analysis using proposed conjugate relaxed-finite step size method. The fuzzy CRS and fuzzy HL-RF methods are compared to evaluate the membership functions of five structural problems with highly nonlinear limit state functions. Results demonstrated that the fuzzy CRS method is computationally more efficient and is strongly more robust than the HL-RF for fuzzy-based reliability analysis of the nonlinear structural reliability problems.

The seismic reliability analysis of complex structure is carried out based on the response surface method and finite element method. Firstly, the appropriate design points are selected based on the mean values and standard deviations of the basic random variables. Secondly, the finite element method is employed to obtain the values of the limit state function of the complex structure. Thirdly, with selected design points and the obtained values of the limit state function of the complex structure, a polynomial function is constructed to approximate the original implicit limit state function. Then, with the established explicit polynomial limit state function and available methods of structural reliability analysis, the seismic reliability of the complex structure is estimated. Numerical analyses show that the established method is simple to use for the evaluation of the reliability analysis of complex structure.

To improve the efficiency and accuracy of traditional least squares method–based polynomial response surface method (RSM) for reliability analysis of structure, the application of various adaptive metamodeling approaches is notable. The moving least squares method (MLSM)–based RSM is the simplest one and found to be effective in this regard. But, its performance in reliability analysis of structure largely depends on the proper choice of the parameter of weight function involved. In the present study, a generalized scheme to appropriately obtain the hyper-parameter of the MLSM-based RSM to approximate implicit responses of structure for reliability analysis is proposed. The algorithm is hinged on the fact that for reliability analysis, one is interested in the sign of the approximated limit state function (LSF) rather than its magnitude. Thereby, it is sufficient to obtain the hyper-parameter for which the first derivative of the probability of failure as obtained from the approximated LSF with respect to the hyper-parameter is zero. The effectiveness of the proposed algorithm is elucidated through three numerical examples. The improvement achieved by the proposed MLSM-based RSM has been compared with the reliability results obtained by the MLSM-based RSM considering the commonly recommended value of the hyper-parameter and also by the approach where the parameters are obtained by leave one out cross-validation procedure.

PurposeBecause of the high computational efficiency, response surface method (RSM) has been widely used in structural reliability analysis. However, for a highly nonlinear limit state function (LSF), the approximate accuracy of the failure probability mainly depends on the design point, and the result is that the response surface function composed of initial experimental points rarely fits the LSF exactly. The inaccurate design points usually cause some errors in the traditional RSM. The purpose of this paper is to present a hybrid method combining adaptive moving experimental points strategy and RSM, describing a new response surface using downhill simplex algorithm (DSA-RSM).Design/methodology/approachIn DSA-RSM, the operation mechanism principle of the basic DSA, in which local descending vectors are automatically generated, was studied. Then, the search strategy of the basic DSA was changed and the RSM approximate model was reconstructed by combining the direct search advantage of DSA with the reliability mechanism of response surface analysis.FindingsThe computational power of the proposed method is demonstrated by solving four structural reliability problems, including the actual engineering problem of a car collision. Compared to specific structural reliability analysis methods, the approach of modified DSA interpolation response surface for structural reliability has a good convergent capability and computational accuracy.Originality/valueThis paper proposes a new RSM technology based on proxy model to complete the reliability analysis. The originality of this paper is to present an improved RSM that adjusts the position of the experimental points judiciously by using the DSA principle to make the fitted response surface closer to the actual limit state surface.

A hybrid self-adaptive conjugate first order reliability method for robust structural reliability analysis

In the reliability analysis of safety critical complex engineering structures, a very large number of the system parameters can be considered as random variables. The difficulty in computing the failure probability using the classical first- and second-order reliability methods (FORM and SORM) increases rapidly with the number of variables or ‘dimension’. There are mainly two reasons behind this. The first is the increase in computational time with the increase in the number of random variables. In principle, this problem can be handled with superior computational tools. The second reason, which is perhaps more fundamental, is that there are some conceptual difficulties typically associated with high dimensions. This means that even when one manages to carry out the necessary computations, the application of existing FORM and SORM may still lead to incorrect results in high dimensions. This paper is aimed at addressing this issue. Based on the asymptotic distribution of quadratic form in Gaussian random variables, two formulations for the case when the number of random variables n →∞ is provided. The first is called ‘strict asymptotic formulation’ and the second is called ‘weak asymptotic formulation’. Both approximations result in simple closed-form expressions for the probability of failure of an engineering structure. The proposed asymptotic approximations are compared with existing approximations and Monte Carlo simulations using numerical examples.

A new bivariate dimension reduction method for efficient structural reliability analysis

Efficient response surface method for high-dimensional structural reliability analysis

A method is presented to efficiently approximate the failure probability of structures subjected to time-variant loads, where the system of loads and structure may have uncertain parameters. The method uses response surface methodology in conjunction with the fast integration technique suggested by Wen and Chen, to provide a limit-state formulation that is computationally simple to solve based on a small number of response time histories. The system reliability may then be quickly computed by first-order reliability method (FORM)/ second-order reliability method (SORM) or Monte Carlo simulation. Sensitivity analysis is performed to determine the effect on the failure probability of changes to the system parameters, which can be important when determining whether uncertainty in a given system parameter is significant. An empirical measure of the accuracy of the response surface approximation is presented.

Reliability-based design optimization is much more computationally expensive than deterministic design optimization. To alleviate the computational demand, the First Order Reliability Method (FORM) is usually used in reliability-based design. Since FORM requires a nonlinear transformation from non-normal random variables to normal random variables, the nonlinearity of a constraint function may increase. As a result, the transformation may lead to a large error in reliability calculation. In order to improve accuracy, a new reliability-based design method with Saddlepoint Approximation is proposed in this work. The strategy of sequential optimization and reliability assessment is employed where the reliability analysis is decoupled from deterministic optimization. The accurate First Order Saddlepoint method is used for reliability analysis in the original random space without any transformation, and the chance of increasing nonlinearity of a constraint function is therefore eliminated. The overall reliability-based design is conducted in a sequence of cycles of deterministic optimization and reliability analysis. In each cycle, the percentile value of the constraint function corresponding to the required reliability is calculated with the Saddlepoint Approximation at the optimal point of the deterministic optimization. Then the reliability analysis results are used to formulate a new deterministic optimization model for the next cycle. The solution process converges within a few cycles. The demonstrative examples show that the proposed method is more accurate and efficient than the reliability-based design with FORM.

The seismic safety of reinforced concrete containment building can be evaluated by probabilistic analysis considering randomness of earthquake, which is more rational than deterministic analysis. In the safety assessment of earthquake-resistant structures by the deterministic theory, it is not easy to consider the effects of random variables but the reliability theory and random vibration theory are useful to assess the seismic safety with considering random effects. The reliability assessment of reinforced concrete containment building subjected to earthquake load includes the structural analysis considering random variables such as load, resistance and analysis method, the definition of limit states and the reliability analysis. The reliability analysis procedure requires much time and labor and also needs to get the high confidence in results. In this study, random vibration analysis of containment building is performed with random variables as earthquake load, concrete compressive strength, modal damping ratio. The seismic responses of critical elements of structure are approximated at the most probable failure point by the response surface method. The response surface method helps to figure out the quantitative characteristics of structural response variability. And the limit state is defined as the failure surface of concrete under multi-axial stress, finally the limit state probability of failure can be obtained simply by first-order second moment method. The reliability analysis for the multiaxial strength limit state and the uniaxial strength limit state is performed and the results are compared with each other. This study concludes that the multiaxial failure criterion is a likely limit state to predict concrete failure strength under combined state of stresses and the reliability analysis results are compatible with the fact that the maximum compressive strength of concrete under biaxial compression state increases.

Objectives: Quantification of uncertainties and its implications on stability assessment of geotechnical problems can very well be captured through probabilistic analysis in which parameters are considered to be random variables. Thus, randomness of the response of the system is also captured and safety assessment is reported in terms of a safety index, commonly called as “reliability index”. In this study, probabilistic approach in conjunction with conventional deterministic approach is used for the analysis of a deep excavation problem and advantages of probabilistic approach are highlighted. Methods/Statistical Analysis: Probabilistic analysis requires establishment of connection between input parameters and output responses, for which response surface method (RSM) concepts can effectively be utilized. RSM helps developing regression model between input and output which can be used for the probabilistic analysis through First Order Reliability Method (FORM). Advanced finite element numerical tool, i.e., PLAXIS 3D is utilized for the purpose of the analysis. Findings: Considering this typical case of deep excavation problem, it is highlighted that probabilistic analysis when used in conjunction with the deterministic approach, it brings rationality in decision making while dealing with uncertainty and provides mathematical basis for treating uncertainty. Application/Improvements: Conventionally, stability analyses of a deep excavation system are carried out through limit equilibrium based approaches and the parameter used to define extent of stability is factor of safety. With advent of computers and finite element or finite difference based computing facilities, strength reduction technique is adopted to estimate the factor of safety. These numerical tools also help in predicting the deformation pattern and judging the serviceability requirements. In either case, the soil parameters used as input in the numerical tools are taken a constant value uniformly. Choice of appropriate value of design input parameter is based on past experience, good engineering judgment and confidence in the adopted analysis methodology. Due to various uncertain sources involved in the estimation of the input properties of a natural material like soil, such as, natural variability, testing errors, model transformation, it is impractical to assign a single value to design parameter. Deterministic approach, which involves single value of design parameters for stability assessment is considered to be simple and straight forward. It is demonstrated that quantification of uncertainties and its influence on stability assessment of deep excavation problem can be captured in probabilistic framework.

This research investigates the collapse responses of a concrete moment frame considering modeling uncertainties. These modeling uncertainties are considered for evaluating a collapse response related to the modified Ibarra-Krawinkler moment-rotation parameters for beam and column elements of a given structure. To analyze these uncertainties, the correlations between the model parameters in one component and between two structural components were considered. Latin Hypercube Sampling (LHS) method was employed to produce independent random variables. Moreover, Cholesky decomposition was adopted to produce correlated random variables. Performing 281 simulations for the uncertainties involved considering their inter-correlations, incremental dynamic analysis (IDA) was done using 44 far-field accelerograms to determine structural collapse responses. Collapse responses of each simulation, including mean collapse capacity, mean collapse drift, and mean annual frequency, were obtained. Then, the collapse responses were predicted using the response surface method and artificial neural network. The results show that the Correlation coefficients (R) between the target data resulted from incremental dynamic analysis (IDA), output data resulted from response surface method (RSM), and artificial neural network (ANN) were obtained for the collapse responses above 0.98. The maximum prediction errors for mean collapse capacity and mean collapse drift are less than 5% and for mean annual frequency less than 10% under response surface method (RSM), and artificial neural network (ANN).

Probabilistic analysis of laterally loaded piles using response surface and neural network approaches