Reliability analysis of laminated composite structures using finite elements and neural networks

A Critical Study on the Haldar and Mahadevan's Reliability Analysis

Introduction:Reliability analysis is a good tool to deal with the uncertainty and has been widely used in the engineering system. The first order reliability method (FORM) is generally used to calculate the reliability index but FORM is time-consuming and requires derivative computing.Methods:Artificial Bee Colony (ABC) algorithm is a very simple, robust and population-based stochastic optimization algorithm. In this study, an ABC-based reliability analysis was proposed to calculate the reliability index of engineering system through combining Artificial Bee Colony (ABC) algorithm with FORM. FORM was adopted to calculate the reliability index and design point. ABC is used to solve the constrained optimization about FORM. The procedure of ABC-based reliability analysis was presented in detail.Results and Conclusion:The proposed method was verified by two classic examples and then applied to geotechnical engineering. The results show that the ABC algorithm can effectively solve the global optimization problem in FORM. Results demonstrate that ABC-based reliability analysis is a good approach to obtain the reliability index and design point with a good accuracy so that it can be applied to analyze the reliability of a complex engineering system.

The structural integrity of offshore wind turbine (OWT) support structures is affected by one of the most severe damage mechanisms known as pitting corrosion‐fatigue. In this study, the structural reliability of such structures subjected to pitting corrosion‐fatigue is assessed using a damage tolerance modelling approach. A probabilistic model that ascertains the reliability of the structure is presented, incorporating the randomness in cyclic load and corrosive environment as well as uncertainties in shape factor, pit size and aspect ratio. A non‐intrusive formulation is proposed consisting of a sequence of steps. First, a stochastic parametric Finite Element Analysis (FEA) is performed using SMART© crack growth and Design Xplorer© facilities within the software package ANSYS. Secondly, the results obtained from the FEA are processed using an Artificial Neural Network (ANN) response surface modelling technique. Finally, the First Order Reliability Method (FORM) is used to calculate the reliability indices of components. The results reveal that for the inherent stochastic conditions, the structure becomes unsafe after the 18th year, before the attainment of the design life of 20 years. The FEA results are in very good agreement with results obtained from analysis steps outlined in design standard BS 7910 and other references designated as ‘theoretical analysis methods’ in this study. The results predict, for the case study, that the pit growth life is approximately 56% of the total pitting corrosion fatigue life. Sensitivity analysis results show that the aspect ratio of pits at critical size plays a significant role on the reliability of the structure.

For many reinforced concrete structures corrosion of the reinforcement is an important problem since it can result in maintenance and repair actions. Further, a reduction of the load-bearing capacity can occur. In the present paper the Finite Element Reliability Method (FERM) is employed for obtaining the probability of exceeding a critical chloride concentration level at the reinforcement bars, both using Monte Carlo Simulation (MCS) and the First Order Reliability Method (FORM). The chloride ingress is modelled by the Finite Element Method (FEM) and the diffusion coefficient, surface chloride concentration and reinforcement cover depth are modelled by stochastic fields, which are discretized using the Expansion Optimum Linear Estimation (EOLE) approach. The response gradients needed for FORM analysis are derived analytically using the Direct Differentiation Method (DDM). As an example, a bridge pier in a marine environment is considered and the results are given in terms of distributions of time for initiation of corrosion.

<abstract><p>The training of artificial neural networks (ANNs) with rectified linear unit (ReLU) activation via gradient descent (GD) type optimization schemes is nowadays a common industrially relevant procedure. GD type optimization schemes can be regarded as temporal discretization methods for the gradient flow (GF) differential equations associated to the considered optimization problem and, in view of this, it seems to be a natural direction of research to <italic>first aim to develop a mathematical convergence theory for time-continuous GF differential equations</italic> and, thereafter, to aim to extend such a time-continuous convergence theory to implementable time-discrete GD type optimization methods. In this article we establish two basic results for GF differential equations in the training of fully-connected feedforward ANNs with one hidden layer and ReLU activation. In the first main result of this article we establish in the training of such ANNs under the assumption that the probability distribution of the input data of the considered supervised learning problem is absolutely continuous with a bounded density function that every GF differential equation admits for every initial value a solution which is also unique among a suitable class of solutions. In the second main result of this article we prove in the training of such ANNs under the assumption that the target function and the density function of the probability distribution of the input data are piecewise polynomial that every non-divergent GF trajectory converges with an appropriate rate of convergence to a critical point and that the risk of the non-divergent GF trajectory converges with rate 1 to the risk of the critical point. We establish this result by proving that the considered risk function is <italic>semialgebraic</italic> and, consequently, satisfies the <italic>Kurdyka-Łojasiewicz inequality</italic>, which allows us to show convergence of every non-divergent GF trajectory.</p></abstract>

The two first order reliability methods (FORM) for computing the non-probabilistic reliability index, namely the mean-value method and the design-point method, are investigated. A performance comparison is presented between these two methods. The results show that: (1) the value of the reliability index of the mean-value method depends on the specific form of the limit state function, whereas the value of the reliability index of the design-point one does not;(2) the design-point method should be preferentially used in structural reliability assessment. The conclusions are verified by a numerical example.

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.

In this work, a novel framework is proposed for the risk based design optimization of engineering systems by minimizing the demand on the system components’ accuracy (which directly relates to their cost). The fundamental development of this work is an analytical upper bound for calculating the probability of failure. This is in contrast with First Order Reliability Method (FORM), where a lower bound is used in calculating the probability of failure. FORM is one of the most popular methods for reliability analysis of engineering systems. In this paper, we show that FORM results in an optimistic measure of risk, hence potentially catastrophic in engineering design. A more accurate measure of failure is proposed by utilizing an analytical upper bound for the distribution of reliability index (the length of the most probable point vector to origin). This distribution is a function of the eigenvalues of the linearized limit state function in the normal space which results in a better understanding of failure phenomenon. The proposed formulation is computationally efficient and straightforward to solve, since it only involves finding eigenvalues in each iteration. This algorithm is applicable to any linearizable continuous limit state function with any type of distribution for the design variables. The method is applied to two examples and its accuracy is compared with the Monte Carlo simulation and FORM, demonstrating its effectiveness and value.

SummaryIn this study, a post optimization technique for a correction of inaccurate optimum obtained using first‐order reliability method (FORM) is proposed for accurate reliability‐based design optimization (RBDO). In the proposed method, RBDO using FORM is first performed, and then the proposed second‐order reliability method (SORM) is performed at the optimum obtained using FORM for more accurate reliability assessment and its sensitivity analysis. In the proposed SORM, the Hessian of a performance function is approximated by reusing derivatives information accumulated during previous RBDO iterations using FORM, indicating that additional functional evaluations are not required in the proposed SORM. The proposed SORM calculates a probability of failure and its first‐order and second‐order stochastic sensitivity by applying the importance sampling to a complete second‐order Taylor series of the performance function. The proposed post optimization constructs a second‐order Taylor expansion of the probability of failure using results of the proposed SORM. Because the constructed Taylor expansion is based on the reliability method more accurate than FORM, the corrected optimum using this Taylor expansion can satisfy the target reliability more accurately. In this way, the proposed method simultaneously achieves both efficiency of FORM and accuracy of SORM. Copyright © 2015 John Wiley & Sons, Ltd.

An investigation was performed to study the effect of uncertainty in applied loads, helix angles, and failure stresses on the probability of failure of composite cylinders. Also, the effect of the factor of safety on the probability of failure of the cylinder was studied. The probability of failure for a composite cylinder subject to axisymmetric loading was obtained using the First Order Reliability method (FORM). The failure of the cylinder was studied based on first ply failure. Ply failure was predicted using the non-smooth Hashin criterion. Hashin criterion is based on writing a different interaction equation for separate failure modes. Four failure modes are considered associated with matrix or fiber failure, in either tension or compression. The accuracy of FORM as applied to this problem was verified by comparing the FORM results to those obtained by Monte-Carlo Simulation method with importance sampling. i x µ = mean values of the design parameters i x

Overcoming the drawbacks of the FORM using a full characterization method

In this paper tunnel probabilistic structural analysis (TuPSA) was performed using the first order reliability method (FORM). In TuPSA, a tunnel performance function is defined according to the boundary between the structural stability and instability. Then the performance function is transformed from original space into the standard normal variable space to obtain the design point, reliability index, and also the probability of tunnel failure. In this method, it is possible to consider the design factors as the dependent or independent random parameters with arbitrary probability distributions. A software code is developed to perform the tunnel probabilistic structural analysis (TuPSA) using the FORM. For validation and verification of TuPSA, a typical tunnel example with random joints orientations as well as mechanical properties has been studied. The results of TuPSA were compared with those obtained from Monte-Carlo simulation. The results show, in spite of deterministic analysis which indicates that the rock blocks are stable, that TuPSA resulted in key-blocks failure with certain probabilities. Comparison between probabilistic and deterministic analyses results indicates that probabilistic results, including the design point and probability of failure, are more rational than deterministic factor of safety.

Advantages of employing a full characterization method over FORM in the reliability analysis of laminated composite plates

Risk analysis using stochastic reliability methods applied to two cases of deterministic water quality models

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