Structural reliability analysis for implicit performance functions using artificial neural network

The Monte-Carlo simulation (MCS), the first-order reliability methods (FORM) and the second-order reliability methods (SORM), are three reliability analysis methods that are commonly used for structural safety evaluation. The MCS requires the calculations of hundreds and thousands of performance function values. The FORM and SORM demand the values and partial derivatives of the performance function with respect to the design random variables. Such calculations could be time-consuming or cumbersome when the performance functions are implicit. Such implicit performance functions are normally encountered when the structural systems are complicated and numerical analysis such as finite element methods has to be adopted for the prediction.To address this issue, this paper presents three artificial neural network (ANN)-based reliability analysis methods, i.e. ANN-based MCS, ANN-based FORM, and ANN-based SORM. These methods employ multi-layer feedforward ANN technique to approximate the implicit performance functions. The ANN technique uses a small set of the actual values of the implicit performance functions. Such a small set of actual data is obtained via normal numerical analysis such as finite element methods for the complicated structural system. They are used to develop a trained ANN generalization algorithm. Then a large number of the values and partial derivatives of the implicit performance functions can be obtained for conventional reliability analysis using MCS, FORM or SORM. Examples are given in the paper to illustrate why and how the proposed ANN-based structural reliability analysis can be carried out. The results have shown the proposed approach is applicable to structural reliability analysis involving implicit performance functions. The present results are compared well with those obtained by the conventional reliability methods such as the direct Monte-Carlo simulation, the response surface method and the FORM method 2.     

Reliability analysis of roof wedges and rockbolt forces in tunnels

The ambiguous nature of the factor of safety is first discussed in the context of a symmetric roof wedge of a circular tunnel, where two different definitions of the factor of safety are shown to be reconcilable when using the reliability index computed with the first-order reliability method (FORM). The probabilities of failure based on the second-order reliability method (SORM) are also obtained for comparison with those of FORM and Monte Carlo simulations. The FORM and SORM analyses are then applied to a circular tunnel supported with elastic rockbolts in a homogeneous and isotropic elasto-plastic ground with the Coulomb failure criterion. The similarities and differences between the ratios of mean values to design-point values, on the one hand, and the partial factors of limit state design, on the other hand, are discussed. Finally, all this is used to show how a reliability-based design can be performed to obtain the length and spacing of rockbolts for a target reliability index.     

Reliability based optimization of laminated composite structures using genetic algorithms and Artificial Neural Networks

The design of anisotropic laminated composite structures is very susceptible to changes in loading, angle of fiber orientation and ply thickness. Thus, optimization of such structures, using a reliability index as a constraint, is an important problem to be dealt. This paper addresses the problem of structural optimization of laminated composite materials with reliability constraint using a genetic algorithm and two types of neural networks. The reliability analysis is performed using one of the following methods: FORM, modified FORM (FORM with multiple checkpoints), the Standard or Direct Monte Carlo and Monte Carlo with Importance Sampling. The optimization process is performed using a genetic algorithm. To overcome high computational cost it is used Multilayer Perceptron or Radial Basis Artificial Neural Networks. It is shown, presenting two examples, that this methodology can be used without loss of accuracy and large computational time savings, even when dealing with non-linear behavior.     

Reliability analysis of tunnel using least square support vector machine

In the reliability analysis of tunnels, the limited state function is implicit and nonlinear, and is difficult to apply based on the traditional reliability method, especially for large-scale projects. Least squares support vector machines (LS-SVM) are capable of approximating the limited state function without the need for additional assumptions regarding the function form, in comparison to traditional polynomial response surfaces. In the present work, the LS-SVM method was adapted to obtain the limited state function. An LS-SVM-based response surface method (RSM), combined with the first-order reliability method (FORM), is proposed for use in tunnel reliability analysis and implementation of the method is described. The reliability index obtained from the proposed method applied to particular tunnel configurations under different conditions shows excellent agreement with Low and Tang’s (2007) method and traditional RSM results, and indicates that the LS-SVM-based RSM is an efficient and effective approach for reliability analysis in tunnel engineering.     

Kriging response surface reliability analysis of a ship-stiffened plate with initial imperfections

The use of structural reliability methods with implicit limit state functions (LSFs) shows the increasing demand for efficient stochastic analysis tools, because the structural behaviour predictions are often obtained by finite element analysis. All stochastic mechanics problems can be solved by Monte Carlo simulation method, nevertheless, in most cases, at a prohibitively high computational cost. Several approximations can be achieved using first-order reliability method (FORM) and second-order reliability method and response surface methods. In this paper, a method that combines the FORM and Kriging interpolation models, as response surface, is proposed. The prediction accuracy of the Kriging response surface obtained from different sampling techniques is assessed, and the failure probability estimates calculated by the FORM using the classical second-order polynomial regression models and the Kriging interpolation models as surrogates of nonlinear LSFs are compared. The usefulness and efficiency of the reliability analysis using the Kriging response surface are demonstrated on the basis of existing results available in the literature and with an application problem of a stiffened plate structure with initial imperfections.     

Steel moment frames column loss analysis: The influence of time step size

This paper applies two well-known structural dynamics computational algorithms to the problem of disproportionate collapse of steel moment frames applying the alternate load path method. Any problem of structural dynamics strongly depends on the accuracy and the reliability of the analysis method since the parameters involved in the selection of the appropriate algorithm are affected by the nature of the problem. Disproportionate collapse is herein simulated via a time history analysis used to “turn off” the effectiveness of an element to the structure. For this kind of problem the time step size of the computational algorithm is of major importance for the accuracy of the method and thus, remains a variable throughout the present analyses. Two plane steel moment frames are used for the numerical examples, while all the analyses are performed independently. Firstly the β-Newmark method is applied and secondly the linear Hilbert-Hughes-Taylor a-method is applied and the respective results are compared and discussed in the last part of the paper.     

基于单变量降维模型和坐标旋转的可靠度混合分析方法

经典的一次可靠度方法对于隐式功能函数和强非线性功能函数的可靠度问题存在适用性问题,尽管二次可靠度方法可以一定程度上处理强非线性功能函数的问题,但理论基础和计算过程均颇为复杂,不利于实用。为克服上述问题,将一次可靠度确定验算点的过程与响应面法的思路相结合是一种行之有效的思路。为此,文中首先引入具有普适性的一次可靠度法,其中考虑了相关非正态随机变量的Nataf变换,并引入单边差分法针对性地解决了隐式功能函数求偏导数的问题|其次,根据梯度值引入坐标旋转向量,并对旋转后的功能函数引入单变量函数降维近似模型|再次,结合验算点的函数值、梯度值以及附加点的函数值,确定各分量函数的二次多项式近似,从而获得近似的整体功能函数|然后,采用重要性抽样法计算近似功能函数的失效概率|最后,分别通过数值算例和工程算例对建立方法的精度和效率进行了验证。结果表明建议方法具有高精度、高效率的特点,且无论对于显式和隐式功能函数均具有广泛适用性。     

Reliability analysis of underground excavation in elastic-strain-softening rock mass

This paper deals with the reliability analysis of a circular tunnel in elastic-strain-softening rock mass. Dilatancy angle which varies with softening parameter in different stress conditions is accounted for. Deterministic and probabilistic analyses of the circular tunnel in elastic-strain-softening rock mass are performed. Computational procedures for the first-order and second-order reliability methods (FORM/SORM) are used in the reliability analyses of the elastic-strain-softening model. The results are in good agreement with those from Monte Carlo simulations incorporating importance sampling. Reliability-based design of the required support pressure for the circular tunnel is efficiently conducted. The effect of positive correlation between compressive strength and elastic modulus of the rock mass on the reliability of the tunnel is discussed. The influence of in situ field stress and support pressure as random variables on the probability of failure of the tunnel is investigated.     

Derivation of the exact stiffness matrix for a two-layer Timoshenko beam element with partial interaction

This paper presents the full closed-form solution of the governing equations describing the behaviour of a shear-deformable two-layer beam with partial interaction. Timoshenko’s kinematic assumptions are considered for both layers, and the shear connection is modelled through a continuous relationship between the interface shear flow and the corresponding slip. The limiting cases of perfect bond and no bond are also considered. The effect of possible transversal separation of the two members has been neglected. With the above assumptions, the present work can be considered as a significant development beyond that available from Newmark et al.’s paper [4]. The differential equations derived considering the above key assumptions have been solved in closed form, and the corresponding “exact” stiffness matrix has been derived using the standard procedure basically inspired by the well-known direct stiffness method. This “exact” stiffness matrix has been implemented in a general displacement-based finite element code, and has been used to investigate the behaviour of shear-deformable composite beams. Both a simply supported and a continuous beam are considered in order to validate the proposed model, at least within the linear range. A parametric analysis has been carried out to study the influence of both shear flexibility and partial interaction on the global behaviour of composite beams. It has been found that the effect of shear flexibility on the deflection is generally more important for composite beams characterized by substantial shear interaction.     

Moment methods for structural reliability

First-order reliability method (FORM) is considered to be one of the most reliable computational methods. In the last decades, researchers have examined the shortcomings of FORM, primarily accuracy and the difficulties involved in searching for the design point by iteration using the derivatives of the performance function. In order to improve upon FORM, several structural reliability methods have been developed based on FORM, such as second-order reliability method (SORM), importance sampling Monte-Carlo simulation, first-order third-moment reliability method (FOTM), and response surface approach (RSA). In the present paper, moment methods for structural reliability are investigated. Five moment method formulas are presented and investigated, and the accuracy and efficiency of these methods are demonstrated using numerical examples. The moment methods, being very simple, have no shortcomings with respect to design points, and requires neither iteration nor the computation of derivatives, and thus are convenient to be applied to structural reliability analysis.     

A general procedure for first/second-order reliabilitymethod (FORM/SORM)

First/second-order reliability method (FORM/SORM) is considered to be one of the most reliable computational methods for structural reliability. Its accuracy is generally dependent on three parameters, i.e. the curvature radius at the design point, the number of random variables and the first-order reliability index. In the present paper, the ranges of the three parameters for which FORM/SORM is accurate enough are investigated. The results can help us to judge when FORM is accurate enough, when SORM is required and when an accurate method such as the inverse fast Fourier transformation (IFFT) method is required. A general procedure for FORM/SORM is proposed which includes three steps: i.e. point fitting limit state surface, computation of the sum of the principal curvatures K_s and failure probability computation according to the range of K_s. The procedure is demonstrated by several examples.     

Assessing small failure probabilities by combined subset simulation and Support Vector Machines

Estimating small probabilities of failure remains quite a challenging task in structural reliability when models are computationally demanding. FORM/SORM are very suitable solutions when applicable but, due to their inherent assumptions, they sometimes lead to incorrect results for problems involving for instance multiple design points and/or nonsmooth failure domains. Recourse to simulation methods could therefore be the only viable solution for these kinds of problems. However, a major shortcoming of simulation methods is that they require a large number of calls to the structural model, which may be prohibitive for industrial applications. This paper presents a new approach for estimating small failure probabilities by considering subset simulation proposed by S.-K. Au and J. Beck from the point of view of Support Vector Machine (SVM) classification. This approach referred as ²SMART (“Two SMART”) is detailed and its efficiency, accuracy and robustness are assessed on three representative examples. A specific attention is paid to series system reliability and problems involving moderately large numbers of random variables.     

Turbulence generated by a wind tunnel of multi-fan type in uniformly active and quasi-grid modes

In an attempt to simulate atmospheric turbulence at high-Reynolds numbers, an innovative wind tunnel of “multi-fan type” is developed. The airflow is driven by an array of fans (9 columns × 11 rows), each of which is independently controlled by a computer. A driving mode (“uniformly active mode”) is applied, which has been typically used for turbulence modeling using a smaller wind tunnel. Another mode with grid-like distributed fans that are driven steadily (“quasi-grid mode”) is introduced and found to produce turbulence characteristics similar to the conventional grid turbulence. Comparison of turbulence characteristics of the two modes reveals the turbulence structure of the uniformly active mode.     

Comparison of response surface and neural network with other methods for structural reliability analysis

The evaluation of the failure probability and safety levels of structural systems is of extreme importance in structural design, mainly when the variables are eminently random. Some examples of random variables on real structures are material properties, loads and member dimensions. It is necessary to quantify and compare the importance of each one of these variables in the structural safety. Many researchers studied structural reliability problems and nowadays there are several approaches for these problems. Two recent approaches, the Response Surface (RS) and the Artificial Neural Network (ANN) techniques, have emerged attempting to solve complex and more elaborated problems. In this work, these two techniques are presented, and comparison are carried out using the well known First Order Reliability Method (FORM), Direct Monte Carlo Simulation and Monte Carlo Simulation with Adaptive Importance Sampling technique with approximated and exact limit state functions. Problems with simple limit state functions (LSF) and closed form solutions of the failure probability are solved in order to highlight the advantages and shortcomings using these techniques. Some remarks are outlined regarding the fact that RS and ANN techniques have presented equivalent precision levels. It is observed that in problems where the computational cost of structural evaluations (looking for the failure probability and safety levels) is high, these two techniques may turn feasible the evaluation of the structural reliability through simulation techniques.     

Observations on FORM in a simple geomechanics example

This technical note studies the First Order Reliability Method (FORM) applied to a plane strain Mohr-Coulomb drained (c′, tan ?′) element test. The influences of distribution types and linear correlation between random variables are studied. The approximation of assuming a “first order” limit state function is assessed by comparison with direct integration of the probability distribution function in the failure region. The results indicate that FORM overestimates p_f when random variables are lognormally distributed and underestimate p_f when random variables are normally distributed.     

Reliability analysis of ground–support interaction in circular tunnels using the response surface method

The first-order and the second-order reliability method (FORM/SORM) are used to evaluate the failure probability of three performance functions of the ground–support interaction in circular tunnels subjected to hydrostatic stresses. The response surface method (RSM) is used to enable reliability analysis of the implicit convergence-confinement method. The friction angle, cohesion and elastic modulus of the rock mass are considered as basic random variables and are first assumed to obey normal distributions. The quadratic polynomial with cross terms is employed as response surface function to approximate the limit state surface (LSS) at the design point. The strategies for the RSM are presented. The failure probability with respect to different criteria are obtained from FORM/SORM and compared to those generated from Monte Carlo simulations. The results show that the support installation position has great influence on the probability of the three failure modes under consideration. Comparison between analysis using correlated and uncorrelated friction angle and cohesion indicates that the influence of the correlation on the reliability analysis depends on the support installation position and the orientation of the LSS. The reliability analysis involving correlated non-normal distributions and the reliability-based design of the support are also investigated.     

Asymptotic analysis of asymmetric reliability margin

By a first-order-reliability-method (FORM)-based asymptotic analysis the influence of the coefficient of skewness of resistance upon the probability of failure of a reliability problem is studied. The fundamental case of the reliability margin with three-parameter log-normal resistance and normal action effects is considered. First-order-term formulas for the reliability index and weight factors corresponding to the design point and some other points of failure surface are obtained. The study of the points of failure surface is aimed at finding a convenient separated form of a deterministic condition for a reliability verification at the target level with an explicit occurrence of the coefficient of skewness. Following a suggested procedure, some error estimates of several simplified reliability verifications are presented.     

Estimation of torsional-flutter probability in flexible bridges considering randomness in flutter derivatives

Torsional-flutter instability is an aeroelastic phenomenon of interest to the bridge engineer, corresponding to a torsionally unstable vibration regime of the deck driven by wind excitation and appearing beyond a certain critical wind velocity. In this study a method for the derivation of the flutter probability for long-span bridges with bluff decks is proposed.In the first part of this study the deterministic problem is addressed. In contrast with the classical solution method in the frequency domain based on a numerical procedure for assessing the critical wind velocity, a single-mode “closed-form” algorithm for the derivation of the critical velocity was investigated. A polynomial representation of the aeroelastic-loading coefficients (flutter derivatives), necessary for torsional-flutter analysis, was utilized.In the second part an algorithm for estimating the torsional-flutter probability was developed, considering randomness in bridge properties, and flutter derivatives in particular due to their preeminent role in torsional-flutter velocity estimation.Experimental errors in the extraction of flutter derivatives from wind tunnel tests were analyzed. The “closed-form” algorithm, developed in the first part, allowed for a direct numerical solution of the flutter probability in a simple way.The torsional-flutter probability for three simulated bridge models with rectangular closed-box and truss-type girder deck was numerically determined. A set of experimental data, available from the literature, was employed. The simulations enabled the validation of the proposed algorithm.     

组合梁构件正常使用极限状态可靠度分析

基于结构可靠性原理,讨论了组合梁正常使用极限状态的可靠度分析方法。根据《钢结构设计规范》(GB 50017-2003)中的挠度表达式进行了可靠度评估。把组合梁的设计参数看作随机变量,建立两种不同的计算模式,采用考虑基本变量概率分布类型的一次二阶矩方法得出在不同荷载效应比值的可靠指标。通过分析知,在通常范围之内,挠度可靠指标fβ随荷载效应比值ρ的增大而增大。由计算结果得出,按“规范”(GB50017-2003)设计的钢-混凝土构件在正常使用极限状态控制设计的条件下,可靠指标在0.0~1.5之间。     

Feasibility of Face Surcharging during Deep Settlement-Free Tunneling in Dense Urban Settings

A procedure is proposed for the computational modeling of the driving of a deep tunnel by a tunnel-boring machine (TBM) with an active surcharge on the face, which is realized in the GEO-MIGG program. Basic factors affecting the stress-strain state (SSS) of the soil bed-TBM- buildings and structures of the urban setting system are demonstrated in a trial example. The feasibility of using the proposed procedure is confirmed by comparing results of computational modeling of a practical example of tunneling for the third transportation beltway around Moscow in the area of Lefortovo using a TBM manufactured by the German firm Herrenknecht with data derived from field observations.