Homotopy approach for random eigenvalue problem

A novel approach, referred to as the homotopy stochastic finite element method, is proposed to solve the eigenvalue problem of a structure associated with some amount of uncertainty based on the homotopy analysis method. For this approach, an infinite multivariate series of the involved random variables is proposed to express the random eigenvalue or even a random eigenvector. The coefficients of the multivariate series are determined using the homotopy analysis method. The convergence domain of the derived series is greatly expanded compared with the Taylor series due to the use of an approach function of the parameter h. Therefore, the proposed method is not limited to random parameters with small fluctuation. However, in practice, only single‐variable and double‐variable approximations are employed to simplify the calculation. The numerical examples show that with a suitable choice of the auxiliary parameter h, the suggested approximations can produce very accurate results and require reduced or similar computational efforts compared with the existing methods.     

随机结构弹性屈曲的递推求解方法

采用随机收敛的非正交的多项式展式表示未知的随机屈曲特征值和屈曲模态,利用摄动技巧,建立了随机结构弹性屈曲的递推求解方法。算例表明,和基于泰勒展开的摄动随机有限元方法相比,方法的结果能在较宽的随机涨落范围内更好地逼近蒙特卡洛模拟结果,即使只采用前四阶非正交多项式展式,逼近的结果仍然较好。     

Buckling of triangular plates with elastic edge constraints

Summary This paper presents the first-known investigation on the buckling behavior of triangular plates with both translational and rotational elastic edge constraints. The energy functional of a general triangular plate with elastic edge supports is derived, and thep-Ritz method is utilized to derive the governing eigenvalue equation for the buckling problem. Convergence and comparison studies are carried out to verify the validity and accuracy of the solution method. Extensive buckling factors are presented for several selected isosceles and right-angled triangular plates of various edge support conditions and subjected to isotropic inplane compressive load. The influence of elastic edge supports on the buckling factors for triangular plates of various vertex angles (aspect ratios) and boundary conditions is examined.     

Buckling Analysis of Plates Stiffened by Parallel Beams

In this paper a general solution for the elastic buckling analysis of plates stiffened by arbitrarily placed parallel beams of arbitrary doubly symmetric cross section subjected to an arbitrary inplane loading is presented. According to the proposed model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined. The unknown distribution of the aforementioned integrated tractions is established by applying continuity conditions in all directions at the two interface lines, while the analysis of both the plate and the beams is accomplished on their deformed shape. The method of analysis is based on the capability to establish the elastic and the corresponding geometric stiffness matrices of the stiffened plate with respect to a set of nodal points. Thus, the original eigenvalue problem for the differential equation of buckling is converted into a typical linear eigenvalue problem, from which the buckling loads are established numerically. For the calculation of the elastic and geometric stiffness matrices six boundary value problems are formulated and solved using the Analog Equation Method (AEM), a BEM-based method. Numerical examples with practical interest are presented. The accuracy of the results of the proposed model compared with those obtained from a 3-D FEM solution is remarkable.     

A Fourier–Karhunen–Loève discretization scheme for stationary random material properties in SFEM

In order to overcome the computational difficulties in Karhunen–Loève (K–L) expansions of stationary random material properties in stochastic finite element method (SFEM) analysis, a Fourier–Karhunen–Loève (F–K–L) discretization scheme is developed in this paper, by following the harmonic essence of stationary random material properties and solving a series of specific technical challenges encountered in its development. Three numerical examples are employed to investigate the overall performance of the new discretization scheme and to demonstrate its use in practical SFEM simulations. The proposed F–K–L discretization scheme exhibits a number of advantages over the widely used K–L expansion scheme based on FE meshes, including better computational efficiency in terms of memory and CPU time, convenient a priori error‐control mechanism, better approximation accuracy of random material properties, explicit methods for predicting the associated eigenvalue decay speed and geometrical compatibility for random medium bodies of different shapes. Copyright © 2007 John Wiley & Sons, Ltd.     

基于同伦分析方法的随机结构静力响应求解

该文提出了一种基于同伦分析方法的求解含随机参数结构的静力响应的新方法。该方法将随机静力平衡方程重新进行同伦构造,利用含随机变量和趋近函数的同伦级数展式来表示结构的随机静力位移响应,该同伦级数的各阶确定性系数和趋近函数可通过对一系列的变形方程求解得到。由于趋近函数的引入,该同伦级数解相较于传统的摄动法有更大的收敛范围,对于含较大变异性随机参数的结构也能获得不错的求解精度。同时,该文提出了一种降维策略来提高该方法的计算效率。数值算例表明,与目前广泛应用的广义正交多项式展开法（GPC）相比,从计算精度上看,该文方法的3阶展开与GPC 2阶展开相当,该文方法的6阶展开与GPC 4阶展开相当,而计算时间上前者均明显少于后者。此外,该文方法也可以方便地应用到随机结构的几何非线性分析当中,并具有较好的计算精度和计算效率。     

A boundary element method for elastic buckling analysis of assembled plate structures

In this paper, we first discuss the integral equation formulation for the buckling problem of a single plate, using the biharmonic fundamental solution for the plate bending problems. The so called boundary-element method previously proposed by the senior author is applied to the numerical solution of the resulting set of integral equations. The total set of simultaneous equations are derived for nodal unknowns taken out of the whole domain, and reduced to an algebraic set of eigenvalue equations. The proposed method is method to the solution of elastic buckling of assembled plate structures. A few examples are computed and results obtained are compared with other solutions to demonstrate the potential usefulness of the proposed method.     

Buckling analysis of delaminated laminates with consideration of contact in buckling mode

This paper concerns the buckling load analysis of laminates with a pre‐existing delamination, using the finite element method based on the Mindlin plate theory. To deal with the contact problem in the buckling mode, an effective algorithm is presented. In this method, an iterative updating process based on the first‐order sensitivity analysis and the quadratic programming technique is proposed to compute the fictitious forces in contacting areas at first. These fictitious forces are then transferred into the stiffness parameters of some artificial springs. The original stiffness matrix of system can be modified, using these artificial springs. Finally, the penetration between two delaminated layers in the buckling mode can be prevented effectively. Numerical examples show that this method is very efficient to solve the contact problem in eigenvalue analysis from the viewpoint of its accuracy, stability and convergence speed. The effects of contact and delamination size on the buckling load analysis are also investigated. Copyright © 1999 John Wiley & Sons, Ltd.     

Q460高强钢焊接工字形截面单伸臂梁整体稳定性能与设计方法研究

为研究Q460高强钢焊接工字形截面单伸臂梁的整体稳定性能以及完善规范中关于此类构件的设计方法,进行了6根集中荷载作用下Q460高强钢焊接工字形截面单伸臂梁的整体稳定性能试验,测量了试件的初始几何缺陷和截面残余应力。试验结果表明：当单伸臂梁整体失稳的控制梁段位于简支跨时,单伸臂梁的整体稳定承载力随伸臂长度比的增大而减小,截面高宽比较伸臂长度比对单伸臂梁整体稳定承载力的影响更大。在试验基础上,运用有限元程序创建考虑初始几何缺陷和残余应力的有限元模型对试验进行模拟,模拟结果与试验结果吻合良好;基于试验验证的有限元模型,分析荷载形式、荷载比例、简支跨长度、伸臂跨长度和截面高宽比等因素对Q460高强钢焊接工字形截面单伸臂梁整体稳定承载力的影响规律。基于特征值屈曲分析结果,回归出单伸臂梁的弹性屈曲临界弯矩计算公式,并通过对试验结果和有限元参数分析结果的回归提出了Q460高强钢焊接工字形截面单伸臂梁的极限弯矩计算公式。     

Computational p-element method on the effects of thickness and length on self-weight buckling of thin cylindrical shells via various shell theories

This paper is concerned with the development of a global p-element method for the analysis of self-weight buckling of thin cylindrical shells. Such buckling problems occur when cylindrical shells are subject to high-g acceleration, for instance the launching of rockets and missiles under high propulsive power. The cylindrical shells may have any combination of free, simply supported and clamped ends. A p-element computational method has been developed based on various thin shell theories including Donnell, Sanders and Goldenveizer-Novozhilov models. The strain energy for the global element during buckling is formulated and an eigenvalue equation is derived. Unlike the conventional buckling problem where the eigenvalue is directly solved, a pre-determined buckling parameter is fixed at the outset for a geometric-dependent stiffness and a recursive numerical procedure is developed to compute the effect of critical buckling length. The critical buckling length is found to be proportional to thickness to a power of approximately 0.9. The effects of shell thickness and length on buckling parameter are also investigated. Comparison of results from various shell theories indicates solutions of the Sanders and Goldenveizer-Novozhilov shell theories are in excellent agreement while the Donnel shell theory is good for buckling of short cylindrical shells.The work described in this paper was fully supported by grants from City University of Hong Kong [Project Nos. 7001186 (BC) and 7100256 (BC)].     

含裂纹损伤圆弧曲梁弹性屈曲的有限元网格自适应分析

该文建立圆弧形曲梁裂纹的截面损伤缺陷比拟方案,实现裂纹大小(深度)、位置、数目的模拟。引入变截面Euler-Bernoulli梁的h型有限元网格自适应分析方法,求解含裂纹损伤圆弧曲梁弹性屈曲问题,得到优化的网格和满足预设误差限的高精度屈曲荷载和屈曲模态解答。数值算例表明该算法中网格非均匀加密可适应裂纹损伤引起的屈曲模态变化,应用于各类曲梁夹角和裂纹损伤分布工况下的弹性屈曲研究,定量分析了裂纹损伤程度对圆弧曲梁的屈曲荷载和屈曲模态的影响,检验了该文算法的精确性和可靠性。     

Matrix analysis of local instability in plates,stiffened panels and columns

A displacement method of matrix analysis for local instability of plates, stiffened panels and thin-walled columns is presented. The analysis is applicable to stiffened panel and columns for which the cross-section is made up of thin flat plates. For these cases it may be assumed that during buckling deformation no flat component of the cross-section is translated in its own plane and the edge lines at the junctions between flats remain fixed in space. The analysis leads to the standard eigenvalue equation from which the buckling stress can be determined. The elastic and geometrical stiffness matrices derived for this analysis depend on the wavelength of the buckled pattern and this dependence is of a simple form since all coefficients in the resulting stiffness matrices contain the buckling wavelength only as a common factor allowing for considerable simplification in any numerical computations. With this new formulation of local instability analysis very few elements are required to obtain high accuracy for the buckling stress. Several examples illustrating typical applications of this new method have been included.     

基于完全概率的随机结构动力特性分析

基于加权残值法,提出了一种求解随机结构特征值概率密度函数的方法。首先利用加权残值法得到特征值关于随机结构参数构成的随机向量的函数表达式,应用随机向量函数的概率分布函数表达式,通过确定积分区间、变量替换、积分顺序变换等一系列数学上的处理,获得结构特征值的概率密度函数的完整信息。通过两算例的结果与Monte-Carlo模拟法结果相比较,表明该方法具有较高的精度和计算效率。     

Classes of exact solutions for buckling of multi-step non-uniform columns with an arbitrary number of cracks subjected to concentrated and distributed axial loads

The governing differential equation for buckling of a multi-step non-uniform column subjected to combined concentrated and distributed axial loads, each step of which has an arbitrary number of cracks with or without spring supports, is expressed in the form of bending moment. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the column. In this paper, the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of functional transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for six important cases. The exact buckling solutions of one-step non-uniform columns are thus found. Then a new approach that combines the exact buckling solution of a one-step column and the transfer matrix method is presented to establish the eigenvalue equation for buckling of a multi-step non-uniform column with spring supports. The main advantage of the proposed method is that the eigenvalue equation for buckling of a non-uniform column with an arbitrary number of cracks, any kind of two end supports and various spring supports at intermediate points can be conveniently determined from a second order determinant. Due to the decrease in the determinant’s order as compared with previously developed procedures the computational time required by the present method can be reduced significantly. A numerical example is given to examine the accuracy of the proposed method and to investigate the effect of cracks on buckling of a multi-step non-uniform column.     

薄壁杆件侧向稳定的近似闭合解

到目前为止，除了简文工字型梁在纯弯情况下有临界荷载的闭合解外在其它荷载条件下，欲求薄壁杆件的临界荷载一般都借助于数值方法．本文利用Ｇａｌｅｒｋｉｎ加权余量法提出一个简文工字型梁在横向荷载作用下临界荷载的计算公式．利用这个式子算出的值与试验结果、Ｔｉｍｏｓｈｅｎｋｏ的结果、用最优化技术得到的结果，以及用有限单元法等得到的结果吻合得很好，说明本文提出的公式能迅速、有效地计算薄壁杆件的横向临界荷载．     

随机结构复合随机振动分析的概率密度演化方法

提出了随机结构复合随机振动分析的概率密度演化方法。通过引入扩展状态向量,构造具有随机初始条件的状态方程,导出了复合随机振动反应的概率密度演化方程。结合精细时程积分方法和Lax-Wendroff差分格式对概率密度演化方程提出了数值求解方法。进行了八层层间剪切框架结构复合随机振动反应的概率密度演化分析,证明提出的方法具有计算高效、收敛性稳定与精度高的特点。研究表明随着时间的增长,复合随机振动反应概率密度曲线趋于复杂,基于正态分布假定的二阶矩分析方法可能造成可靠度分析结果的显著偏差。与仅考虑结构参数随机性和仅考虑输入随机性时的结构反应相比,复合随机振动反应概率密度曲线峰值降低、分布变宽,且随机涨落显著增强。     

Delamination buckling of laminated plates

The subject of this paper is the buckling of laminated plates, with a pre-existing delamination, subjected to in-plane loading. Each laminate is modelled as an orthotropic Mindlin plate. The analysis is carried out by a combination of the finite element and asymptotic expansion methods. By applying the finite element method, plates with general delamination regions can be studied. The asymptotic expansion method reduces the number of unknown variables of the eigenvalue equation to that of the equation for a single Kirchhoff plate. Numerical results for the critical buckling load are presented for several examples. The effects of the shape, size and position of the delamination on the buckling load are studied through these examples.     

Wittrick–Williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters

The well‐established Wittrick–Williams algorithm is used to derive novel and general proofs that show that the eigenvalues of systems with constraints can be bracketed by replacing the constraints by positive and negative pairs of either ordinary or inertial penalty parameters. It is also shown that convergence occurs from both above and below when the numerical values of these parameters are increased towards infinity. The proofs are applicable in many contexts but are derived in that of structural systems, for which the eigenvalues are either buckling load factors or the squares of natural frequencies of vibration; ordinary penalty parameters are stiffnesses of translational and rotational springs; and inertial penalty parameters are either masses or rotary inertias. The penalty parameters can be used to constrain a system or to impose constraints between systems. It is shown that the use of inertial penalty parameters has several advantages compared with using ordinary ones. Then the pth eigenvalue of a system with n constraints is bounded closely from above by the (p+n)th eigenvalue of the system with very large positive inertial penalty parameters and from below by the pth eigenvalue, when large negative values are used instead. This work is expected to enhance the versatility of numerical eigenproblem methods, e.g. the Rayleigh–Ritz method. Copyright © 2007 John Wiley & Sons, Ltd.     

Localized wrinkling instabilities in radially stretched annular thin films

Summary We investigate a pre-stressed annular thin film subjected to a uniform displacement field along its inner boundary. This loading scenario leads to a variable stress distribution characterized by an orthoradial component that may change sign along a concentric circle within the annular domain. When the intensity of the applied field is strong enough, elastic buckling occurs circumferentially, leading to a localized wrinkling pattern near the inner edge. Using a linear non-homogeneous pre-bifurcation state, the eigenvalue problem describing this instability is cast as a singularly-perturbed fourth-order linear differential equation with variable coefficients. The dependence of the lowest eigenvalue on various non-dimensional quantities is numerically investigated using the compound matrix method. These results are complemented by a WKB analysis which suggests that the qualitative and quantitative features of the full model can be described by a simplified second-order eigenvalue problem which takes into account the finite stiffness of the system.