Reliability analysis of nonlinear laminated composite plate structures

A procedure for the reliability analysis of laminated composite plate structures subjected to large deflections under random static loads is presented. The nonlinear analysis of laminated composite plate structures is achieved via a corotational total Lagrangian finite element formulation which is based on the von Karman assumption and first order shear deformation theory. This formulation is applicable for the nonlinear analysis of plate structures with large rotations but moderate deformation and thus accurate enough to predict the behavior of the structures at the point of failure. The reliability assessment of laminated composite plate structures with random strength subjected to random loads is approached by the determination of limit state surfaces in load space. The limit space surfaces are obtained by performing a series of first ply failure analyses following different load paths in load space using the proposed nonlinear structural analysis technique and an appropriate failure criterion. A numerical technique is then proposed to evaluate the reliability of the plate structures. Examples of the reliability analyses of laminated plates with different layer orientations subject to random loads are given for illustration.     

隧道复合衬砌运营期可靠性评价方法

一些隧道的二次衬砌作为重要的结构参与工作,承担荷载。提出了复合衬砌运营期可靠性评价方法,采用不等宽的脉冲随机过程描述钢筋混凝土衬砌的劣化,用施工资料和监测数据+基于连续介质模型的有限元计算+响应面法获得荷载效应的显式,给出了严格意义的可靠性计算公式,并给出算例。     

Finite element response sensitivity analysis using force‐based frame models

This paper presents a method to compute consistent response sensitivities of force‐based finite element models of structural frame systems to both material constitutive and discrete loading parameters. It has been shown that force‐based frame elements are superior to classical displacement‐based elements in the sense that they enable, at no significant additional costs, a drastic reduction in the number of elements required for a given level of accuracy in the computed response of the finite element model. This advantage of force‐based elements is of even more interest in structural reliability analysis, which requires accurate and efficient computation of structural response and structural response sensitivities. This paper focuses on material non‐linearities in the context of both static and dynamic response analysis. The formulation presented herein assumes the use of a general‐purpose non‐linear finite element analysis program based on the direct stiffness method. It is based on the general so‐called direct differentiation method (DDM) for computing response sensitivities. The complete analytical formulation is presented at the element level and details are provided about its implementation in a general‐purpose finite element analysis program. The new formulation and its implementation are validated through some application examples, in which analytical response sensitivities are compared with their counterparts obtained using forward finite difference (FFD) analysis. The force‐based finite element methodology augmented with the developed procedure for analytical response sensitivity computation offers a powerful general tool for structural response sensitivity analysis. Copyright © 2004 John Wiley & Sons, Ltd.     

On the equivalent static load method for flexible multibody systems described with a nonlinear finite element formalism

The equivalent static load (ESL) method is a powerful approach to solve dynamic response structural optimization problems. The method transforms the dynamic response optimization into a static response optimization under multiple load cases. The ESL cases are defined based on the transient analysis response whereupon all the standard techniques of static response optimization can be used. In the last decade, the ESL method has been applied to perform the structural optimization of flexible components of mechanical systems modeled as multibody systems (MBS). The ESL evaluation strongly depends on the adopted formulation to describe the MBS and has been initially derived based on a floating frame of reference formulation. In this paper, we propose a method to derive the ESL adapted to a nonlinear finite element approach based on a Lie group formalism for two main reasons. Firstly, the finite element approach is completely general to analyze complex MBS and is suitable to perform more advanced optimization problems like topology optimization. Secondly, the selected Lie group formalism leads to a formulation of the equations of motion in the local frame, which turns out to be a strong practical advantage for the ESL evaluation. Examples are provided to validate the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Geometrically nonlinear composite beam structures: design sensitivity analysis

The purpose of this paper is to develop a finite element model for optimal design of composite laminated thin-walled beam structures, with geometrically nonlinear behavior, including post-critical behavior. A continuation paper will be presented with design optimization applications of this model. The structural deformation is described by an updated Lagrangean formulation. The structural response is determined by a displacement controlled continuation method. A two-node Hermitean beam element is used. The beams are made from an assembly of flat-layered laminated composite panels. Beam cross-section mass and stiffness property matrices are presented.

Design sensitivities are imbedded into the finite element modeling and assembled in order to perform the structural design sensitivity analysis. The adjoint structure method is used. The lamina orientation and the laminate thickness are selected as the design variables. Displacement, failure index, critical load and natural frequency are considered as performance measures. The critical load constraint calculated as the limit point of the nonlinear response is also considered, but a new method is proposed, replacing it by a displacement constraint.     

Finite element response sensitivity analysis using three‐field mixed formulation: general theory and application to frame structures

This paper presents a method to compute response sensitivities of finite element models of structures based on a three‐field mixed formulation. The methodology is based on the direct differentiation method (DDM), and produces the response sensitivities consistent with the numerical finite element response. The general formulation is specialized to frame finite elements and details related to a newly developed steel–concrete composite frame element are provided. DDM sensitivity results are validated through the forward finite difference method (FDM) using a finite element model of a realistic steel–concrete composite frame subjected to quasi‐static and dynamic loading. The finite element model of the structure considered is constructed using both monolithic frame elements and composite frame elements with deformable shear connection based on the three‐field mixed formulation. The addition of the analytical sensitivity computation algorithm presented in this paper extends the use of finite elements based on a three‐field mixed formulation to applications that require finite element response sensitivities. Such applications include structural reliability analysis, structural optimization, structural identification, and finite element model updating. Copyright © 2006 John Wiley & Sons, Ltd.     

有限元法和退火进化算法相结合分析结构模糊可靠性

结构的失效除了具有随机性，还应具有模糊性。本文在介绍一种修正的联合概率密度函数的基础上，采用有限元法和退火进化算法相结合来研究结构的模糊可靠度。在每一模糊失效水平下，有限元法用来计算荷载效应项，并将荷载效应项代入原联合概率密度函数形成修正的联合概率密度函数。为了解决进化算法的早熟收敛问题，采用模拟退火算法与进化算法相结合，以保证更有效地搜索到最可能失效点(设计点)。解决不存在显式极限状态方程的大部分实际结构的可靠度研究的困难。数例结果表明该法可直接应用现有的确定性的有限元程序，并且具有很好的效率和精度。     

Stochastic finite element analysis of a cable-stayed bridge system with varying material properties

Stochastic seismic finite element analysis of a cable-stayed bridge whose material properties are described by random fields is presented in this paper. The stochastic perturbation technique and Monte Carlo simulation (MCS) method are used in the analyses. A summary of MCS and perturbation based stochastic finite element dynamic analysis formulation of structural system is given. The Jindo Bridge, constructed in South Korea, is chosen as a numerical example. The Kocaeli earthquake in 1999 is considered as a ground motion. During the stochastic analysis, displacements and internal forces of the considered bridge are obtained from perturbation based stochastic finite element method (SFEM) and MCS method by changing elastic modulus and mass density as random variable. The efficiency and accuracy of the proposed SFEM algorithm are evaluated by comparison with results of MCS method. The results imply that perturbation based SFEM method gives close results to MCS method and it can be used instead of MCS method, especially, if computational cost is taken into consideration.     

结构可靠性分析的模拟重要抽样方法

提出了一种基于Kriging模拟的重要抽样方法用以结构可靠度计算。对于含有隐式极限状态方程的问题,重要抽样方法将大部分的计算时间用于结构有限元分析,这使得重要抽样的计算效率被大大降低了。而Kriging方法能够较好地模拟高度非线性的极限状态方程,并以此模拟计算代替真实的结构分析,其主要目的是为了减少蒙特卡罗方法的计算量。将此方法用于两个框架结构可靠度分析的实例,表明了该方法的有效性和较高的计算效率。     

Thermally-induced bending-torsion coupling vibration of large scale space structures

In this paper, a finite element scheme is developed to solve the problem of thermally-induced bending-torsion coupling vibration of large scale space structures, which are usually composed of thin-walled beams with open and closed cross-section. A two-noded finite element is proposed to analyze the transient temperature field over the longitudinal and circumferential direction of a beam. Since this temperature element can share the same mesh with the two-noded beam element of Euler–Bernoulli type, a unified finite element scheme is easily formulated to solve the thermal-structural coupling problem. This scheme is characterized with very strong nonlinear formulation, due to the consideration of the thermal radiation and the coupling effect between structural deformations and the incident normal heat flux. Moreover, because the warping is taken into account, not only the thermal axial force and thermal bending moments but also the thermal bi-moment are presented in the formulation. Consequently, the thermally-induced bending-torsion coupling vibration can be simulated. The performance of the proposed computational scheme is illustrated by the analysis of the well-known failure of Hubble space telescope solar arrays. The results reveal that the thermally-induced bending-torsion coupling vibration is obviously presented in that case and could be regarded as a cause of failure.     

Modifications to the ‘directional simulation in the load space’ approach to structural reliability analysis

‘Directional simulation in the load space (DS-LS)’ is a simulation-based technique used to perform reliability analysis of structures subjected to time-invariant or time-variant random loads. To perform DS-LS a location must first be chosen for an ‘origin of simulation’. The origin may be positioned in either the safe or failure region of the load space, and its precise location (with respect to these regions) influences the DS-LS formulation needed to evaluate reliability correctly. The current formulation requires the origin to be positioned in the safe region. However, even for simple structures, the ‘exact’ location of the safe and failure region is not always known explicitly ‘a priori’. Modifications to allow for the possibility of positioning the origin not only in the ‘safe’ region but in the ‘failure’ region are proposed in this paper. Some numerical examples involving one or more stationary continuous Gaussian loads and the simulation of directions by ‘Monte Carlo’ and ‘the hyperspace division method’ are presented to demonstrate the validity of the proposed formulations. Some comments on convergence are made.     

A stochastic finite element method for fatigue reliability analysis of gear teeth subjected to bending

A stochastic finite element method (SFEM) is developed for accurate structural reliability analysis. Using the second-order three-moment reliability analytical model, this method takes into account such random factors as load, material parameters and especially geometry randomness. The calculation of the bending fatigue strength reliability of a cantilever beam is carried out as a numerical example to verify the present method. Monte-Carlo FEM and SFEM based on the first-order second-moment model are used in the example to compare with the proposed method. By incorporating the fatigue theory of gears, the present method is then used to analyze the bending fatigue strength reliability of a spur gear. The effects of random variables' coefficient of variation and skewness and the gear's correction factor (not random variable) on the gear's reliability are also investigated.     

On the Reliable Solution of Contact Problems in Engineering Design

In this paper we examine briefly the reliability of solution needed for the accurate and effective analysis of engineering design problems involving contact conditions. A general finite element formulation for treating the frictional contact problem using constraint functions is first summarized. Then we address general reliability issues and those related to the selection of appropriate elements that provide optimal performance. These elements of course do not lock and would provide the best solution an analyst can expect when simulating a design problem. Reliability issues specific to the contact formulation are also presented. A promising procedure to increase the reliability of an analysis is the method of finite spheres. The method does not require a mesh and in particular can be used with a finite element discretization as described in the paper. Finally, the results of several illustrative analysis problems are given.     

Simulation of two dimensional unilateral contact using a coupled FE/EFG method

In this paper, simulation of two dimensional unilateral contact problems using a coupled finite element/element free Galerkin method is proposed. For the analysis, the element free Galerkin method and Galerkin formulation for two dimensional elasticity problems are considered. Then, the penalty method for imposition of contact constraint is proposed. The finite element shape functions are used in the penalty term of contact constraint. Finally, the accuracy of the presented method is verified through some examples. The numerical results have demonstrated that the presented approach is simple and accurate for frictionless contact analysis of 2D solids.     

A spline wavelet finite‐element method in structural mechanics

The wavelet‐based methods are powerful to analyse the field problems with changes in gradients and singularities due to the excellent multi‐resolution properties of wavelet functions. Wavelet‐based finite elements are often constructed in the wavelet space where field displacements are expressed as a product of wavelet functions and wavelet coefficients. When a complex structural problem is analysed, the interface between different elements and boundary conditions cannot be easily treated as in the case of conventional finite‐element methods (FEMs). A new wavelet‐based FEM in structural mechanics is proposed in the paper by using the spline wavelets, in which the formulation is developed in a similar way of conventional displacement‐based FEM. The spline wavelet functions are used as the element displacement interpolation functions and the shape functions are expressed by wavelets. The detailed formulations of typical spline wavelet elements such as plane beam element, in‐plane triangular element, in‐plane rectangular element, tetrahedral solid element, and hexahedral solid element are derived. The numerical examples have illustrated that the proposed spline wavelet finite‐element formulation achieves a high numerical accuracy and fast convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

箱梁的剪力滞效应分析

在变分法薄壁箱梁剪力滞基本微分方程的基础上,提出了一种与现有普通梁单元配合使用的分析箱梁剪力滞效应的有限元方法,导出了剪力滞单元系数矩阵和广义荷载列阵计算公式。针对变分法分析剪力滞问题的特点,给出了按照剪力滞广义平衡与变形协调条件进行结构系统分析、组集结构总剪力滞系数矩阵的方法。分析了连续梁和悬臂梁这两种不同结构类型、不同边界支承条件的箱梁在不同荷载条件下的剪力滞效应,并与变分法解析结果作了对比,验证了该文方法的广泛适用性和可靠性。     

Reliability analysis of plane elasticity problems by stochastic spline fictitious boundary element method

In this paper a stochastic spline fictitious boundary element method (SFBEM) is proposed for reliability analysis of plane elasticity problems in conjunction with the advanced first-order-second-moment (AFOSM) method. The AFOSM method has been demonstrated to be a reliable and practical approach to the structural reliability analysis, yielding results of reasonable accuracy for the engineering applications. And as a modified method for the conventional indirect boundary element method, SFBEM can provide accurate numerical solutions at high efficiency in deterministic analyses. For the purpose of structural reliability analysis, SFBEM is introduced during the iteration process of the AFOSM method, to obtain the required values of structural responses and their derivations with respect to the random variables considered. The use of SFBEM in the formulation of the AFOSM method makes it unnecessary to construct an explicit expression to the implicit limit state function of the problem, leading to a higher efficiency and better accuracy. The present approach is validated by comparing calculated solutions with those of Monte Carlo simulation for a number of example problems and a good agreement of the results is achieved.     

Karhunen–Loève decomposition of random fields based on a hierarchical matrix approach

The simulation of the behavior of structures with uncertain properties is a challenging issue, because it requires suitable probabilistic models and adequate numerical tools. Nowadays, it is possible to perform probabilistic investigations of the structural performance, which take into account a space‐variant uncertainty characterization of the structures. Given a structural solver and the probabilistic models, the reliability analysis of the structural response depends on the continuous random fields approximation, which is carried out by means of a finite set of random variables. The paper analyzes the main aspects of discretization in the case of 2D problems. The combination of the well‐known Karhunen–Loève series expansion, the finite element method and the hierarchical matrices approach is proposed in the paper. Copyright © 2013 John Wiley & Sons, Ltd.     

Cohesive surface model for fracture based on a two-scale formulation: computational implementation aspects

The paper describes the computational aspects and numerical implementation of a two-scale cohesive surface methodology developed for analyzing fracture in heterogeneous materials with complex micro-structures. This approach can be categorized as a semi-concurrent model using the representative volume element concept. A variational multi-scale formulation of the methodology has been previously presented by the authors. Subsequently, the formulation has been generalized and improved in two aspects: (i) cohesive surfaces have been introduced at both scales of analysis, they are modeled with a strong discontinuity kinematics (new equations describing the insertion of the macro-scale strains, into the micro-scale and the posterior homogenization procedure have been considered); (ii) the computational procedure and numerical implementation have been adapted for this formulation. The first point has been presented elsewhere, and it is summarized here. Instead, the main objective of this paper is to address a rather detailed presentation of the second point. Finite element techniques for modeling cohesive surfaces at both scales of analysis (FE\(^2\) approach) are described: (i) finite elements with embedded strong discontinuities are used for the macro-scale simulation, and (ii) continuum-type finite elements with high aspect ratios, mimicking cohesive surfaces, are adopted for simulating the failure mechanisms at the micro-scale. The methodology is validated through numerical simulation of a quasi-brittle concrete fracture problem. The proposed multi-scale model is capable of unveiling the mechanisms that lead from the material degradation phenomenon at the meso-structural level to the activation and propagation of cohesive surfaces at the structural scale.