Approximate Methods for the Linear and Nonlinear Analysis of Plates and Shallow Shells

Abstract

Berger's equations for the large amplitude deformation of membranes are used to produce a simple approximate expression for the large amplitude deflection of plates. The deformation of shallow shells is also considered and two approximate methods are outlined. Several important problems are discussed, the obtained solution being in good agreement with both experimental data and other approximate results. The main advantage of this technique is its ease of application, as it requires comparatively little computational work. A simple approximate formula for computing the fundamental frequency of a vibrating shallow shell is also presented and is shown to yield very accurate values in the case of a shallow dome and a rectangular panel.     

一般夹层旋转壳在轴对称变形下的大挠度方程

应用最小势能原理建立了具有不同质不等厚薄表层和软夹心的一般夹层旋转壳在轴对称变形下的非线性理论,得到了一组相对简单的夹层壳大挠度方程和边界条件。在分析壳体的变形时,将表层视作薄膜,假设夹心沿厚度方向不可压缩且只能承受横向剪应力,参考面的法线在变形时保持为直线。为便于实际应用,给出几种特殊壳体情况下的大挠度方程。     

Non-symmetrical large deformation of a shallow thin conical shell

I.IntroductionItisimportanttoresearchnon-symmetricalquestionsofshallowcollicalshellsintheoryoronapplication.Asonekindofpressurevessel'sparts,shallowconicalshellsareverycommonlyusedillellgineerillgpractice,becausethedifficultyofmanul\lctul.illgthemis'small.AlthotlghwelookupmanyChineseandtbreignperiodicalswhicharctlblctobefound,wehavenotyeth'ulldarticlesanddocumentsfornon-symmetricalandnolllinearquestionsofshitllowconictllshells.Oval'rccelltyears,ProlbssorWangXinzhiandhiscolleagueshavedonealot…     

Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitude-frequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.     

The singular perturbation method applied to the nonlinear stability problem of a shallow spherical shell (II)

In this paper we consider the nonlinear stability of a thin elastic circular shallow spherical shell under the action of uniform normal pressure with a clamped edge. When the geometrical parameter k is large, the uniformly valid asymptotic solutions are obtained by means of the singular perturbation method. In addition, we give the analytic formula for determining the centre deflection and the critical load, and the stability curve is also derived. This paper is a continuation of the author’s previous paper[11].     

Analysis for the thermoelastic bending of a functionally graded material cylindrical shell

An analytic study for thermoelastic bending of a functionally graded material (FGM) cylindrical shell subjected to a uniform transverse mechanical load and non-uniform thermal loads is presented. Based on the classical linear shell theory, the equations with the radial deflection and horizontal displacement are derived out. An arbitrary material property of the FGM cylindrical shell is assumed to vary through the thickness of the cylindrical shell, and exact solution of the problem is obtained by using an analytic method. For the FGM cylindrical shell with fixed and simply supported boundary conditions, the effects of mechanical load, thermal load and the power law exponent on the deformation of the FGM cylindrical shell are analyzed and discussed.     

Nonlinear vibration analysis of geometrically nonlinear shell structures

The nonlinear vibration analysis of a geometrically nonlinear shell structure is investigated in this study. In general, when the shell structure is subjected to excessive loadings, the large deformation of the shell structures must be considered, and the governing equation of the shell structure becomes nonlinear since the stiffness matrix of the governing equation is related to the deflection. Therefore, the natural frequency of the shell structure is varied with respect to the time which is quite different from that of the linear structures. In order to solve the nonlinearity of the governing equations of the shell structures, the well known Newton-Raphson iteration procedure in conjunction with Newmark scheme is adopted to perform the frequency analysis of the nonlinear-shell structures. Incidentally, the natural frequencies for various curvatures of the shell structures are also investigated from the practical engineering point of view.     

功能梯度截顶圆锥壳的热弹性弯曲精确解

研究了功能梯度材料截顶圆锥壳在横向机械载荷与非均匀热载荷同时作用下的变形问题. 基于经典线性壳体理论推导出了以横向剪力和中面转角为基本未知量的功能梯度薄圆锥壳轴对称变形的混合型控制方程. 假设功能梯度圆锥壳的材料性质为沿厚度方向按照幂函数连续变化的形式. 然后采用解析方法求解,得到了问题的精确解. 分别就两端简支和两端固支边界条件,给出了圆锥壳的变形随其载荷、材料参数等变化的特征关系曲线,重点分析和讨论了载荷参数与材料梯度变化参数对变形的影响.      

基于拟应变法的壳单元大变形分析及在冲击动力问题中的应用

为了简便有效地解决板壳结构的大变形问题，本文针对八节点相对自由度壳单元进行研究。该单元的位移场由壳的中面节点位移和上表面节点的相对位移组成，不带有转动变量。所有的研究都是基于完全的三维位移、应力、应变场。采用拟应变法，对应变场另行假设，能够改善该单元在大变形情况下的计算精度。通过引入Wilson非协调模式，构造了大变形情况下的拟应变场表达式，给出了该单元用于解决非线性动力分析问题的有限元求解方程。通过算例表明，本文针对相对自由度壳单元提出的方法及推导的公式，能够解决冲击动力问题中的大变形问题。     

简支梯形底扁球壳弯曲问题的准格林函数方法

以简支梯形底扁球壳的弯曲问题为例,详细阐明了准格林函数方法的思想.即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将简支扁球壳弯曲问题的控制微分方程化为两个互相耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程...     

轴对称弹性圆锥扁壳非线性自由振动直接摄动解法

本文采用弹性圆锥扁壳中心无量纲振幅和壳体母线的倾角为参数，将挠度、应力函数的导数以及自由振动频率展开为双参量的幂级数形式．用直接摄动法获得各级递推线性偏微分方程．应用变分法求得各级递推方程的近似解答．从而给出弹性圆锥扁壳非线性自由振动频率的基本公式。     

Large deflections of deep orthotropic spherical shells under radial concentrated load: asymptotic solution

Nonlinear behavior of deep orthotropic spherical shells under inward radial concentrated load is studied. The singular perturbation method is developed and applied to Reissner’s equations describing axially symmetric large deflections of thin shells of revolution. A small parameter proportional to the ratio of shell thickness to the sphere radius is used. The simple asymptotic formulas describing load–deflection diagrams, maximum bending and membrane stresses of the structure are derived. The influence of boundary conditions on the behavior of the shell by large deflections is considered. Obtained asymptotic solution is in close agreement with the experimental and numerical results and has the same accuracy (in the asymptotic meaning) as the given equations of nonlinear theory of thin shells.     

简支梁大挠度弹性后屈曲载荷与变形的优化算法

针对简支梁结构大挠度后屈曲载荷与变形的计算问题,本文提出了一种直接求解其后屈曲载荷和变形的优化算法。在简支梁处于大挠度屈曲平衡状态下,将梁结构划分为有限子段,以待求后屈曲载荷为设计变量,根据起点的边界条件和每个子段满足的弯矩变形公式,累积计算出其他各个节点的坐标,以得到的终点坐标满足的边界条件构建目标函数模型。在此基础上,通过MATLAB编制优化程序分析了两个典型算例,并将理论结果与相关软件的计算结果进行对比,从而证明了本文算法的正确性。本文算法求解过程简单、快速,具有一定的实用性,为变截面结构大挠度弹性屈曲稳定性问题的研究提供了参考。     

均匀受压矩形管的屈曲后板组效应研究

本文根据势能驻值原理,建立了均匀受压矩形管的大挠度弹性屈曲理论,用于分析屈曲后相邻板件间的相互约束作用,即屈曲后板组效应。该方法通过引入拉格朗日乘子,使系统总势能取驻值的同时,形函数能满足一系列基本的,但非完备的边界条件;在此基础上采用增量法将非线性方程组线性化,从而确定给定荷载下矩形管的受力状态及变形。由于系统总势能是以解析式的形式给出,且仅需取有限的级数项就可达到满意的精度,故该半解析法具有计算效率高的特点。目前许多国家的钢结构设计规范中,在计算受压板件的有效宽度时,常采用板组约束系数来考虑板组效应。由于有效宽度法本身考虑了板的屈曲后强度,而约束系数多数却是建立在小挠度弹性屈曲理论上的,故有必要分析板件的屈曲后板组效应。本文详细讨论了矩形管的屈曲后板组效应,并与小挠度理论给出的计算结果进行比较,得出了均匀受压矩形管屈曲后板组效应减弱的结论。     

THE SCHRÖDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of [50] and [51]. In this paper: (A) The Love-Kirchhoff equation of small deflection problem for elastic thin shell with constant curvature are classified as the same several solutions of Schrodinger equation, and we show clearly that its form in axisymmetric problem;(B) For example for the small deflection problem, we extract me general solution of the vibration problem of thin spherical shell with equal thickness by the force in central surface and axisymmetric external field, that this is distinct from ref. [50] in variable. Today the variable is a space-place, and is not time;(C) The von Kármán-Vlasov equation of large deflection problem for shallow shell are classified as the solutions of AKNS equations and in it the one-dimensional problem is classified as the solution of simple Schrodinger equation for eigenvalues problem, and we transform the large deflection of shallow shell from nonlinear problem into soluble linear problem.     

球形扁壳在冲击载荷作用下的超临界变形

本文利用Pogorelov提出的薄亮稳定性几何学理论,研究了球形扁壳在冲击作用下的超临界变形行为。这种方法是建立在实验观察中,壳结构的大变形是以近似于一种等距变换的方式产生的。首先,给出了壳体变形能的近似表达式,在此基础上,考虑了两种不同的在扁球壳顶部的冲击方式,利用能量原理,得到了描述运动的控制方程。从而给出扁球壳中心最大凹陷半径随冲击载荷变化的近似表达式,并将此结果与实验进行了比较,二者吻合的还是比较好的。     

夹层扁球壳的非线性稳定性

基于Reissner假设和变分原理,给出夹层扁球壳在均布压力作用下的大挠度方程,采用修正迭代法求得了夹层扁球壳非线性稳定问题的解析解,得到两类边界条件下临界屈曲载荷的表达式,讨论了几何参数和物理参数对临界屈曲载荷的影响     

等截面抛物线无铰拱挠度影响线实用解析解

为解决弧长微分表达式复杂导致抛物线拱挠度解析计算繁琐的难题,采取有简单弧长微分表达式的近似曲线拟合抛物线的方法,通过以沿曲线的曲率和为量化指标评价3种曲线(直线)的拟合度,最终得出悬索线拟合度最优的结论。在此基础上,利用基本力学原理,推导出抛物线无铰拱挠度影响线解析计算公式,最后应用于实例,并与有限元分析结果进行对比。算例结果表明,公式计算与有限元分析结果最大相差不超过1%,具有非常高的工程精度,可用于工程实践。     

Nonlinear stability of double-deck reticulated circular shallow spherical shell

Based on the variational equation of the nonlinear bending theory of doubledeck reticulated shallow shells, equations of large deflection and boundary conditions for a double-deck reticulated circular shallow spherical shell under a uniformly distributed pressure are derived by using coordinate transformation means and the principle of stationary complementary energy. The characteristic relationship and critical buckling pressure for the shell with two types of boundary conditions are obtained by taking the modified iteration method. Effects of geometrical parameters on the buckling behavior are also discussed.     

用高阶剪切变形理论研究双模量梁的弯曲

利用高阶剪切变形理论研究了双模量梁的弯曲变形问题,推导出了双模量梁的挠曲线方程及弯曲正应力公式．讨论分析了翘曲函数的指数n对挠度、正应力的影响．研究结果表明：拉压弹性模量的差异对梁的弯曲应力有较大影响．把高阶剪切变形理论的计算结果与弹性理论计算结果进行比较,可知该方法计算精度非常高．