双曲扁壳的非线性稳定

本文采用布勃诺夫——伽辽金方法对在均布荷载作用下矩形底四边简支双曲扁壳的可动边界和不可动边界进行了非线性稳定问题的分析,求得有限变形的屈曲载荷.从P和W_0曲线可以看出双曲扁壳失稳的跳跃现象.另外还导出了矩形底双曲扁壳其它边界条件下的无矩内力,因此可以推广应用到其它边界条件下非线性稳定问题的求解.     

关于弹性扁壳边界补充条件问题

文献[1]曾提出了确定四边简支矩形底扁壳边界应力函数的计算公式,此公式实为文献[2]所称的四边简支情形的补充条件。文献[2]在提出扁壳的广义变分原理的同时,利用此原理解决了许多扁壳边界问题,导出了比较广泛的扁壳边界补充条件,其结果我们曾在实际工作中有效地应用过。现在本文提出一个求扁壳边界补充条件的结构力学方法,此法简明、直观,而且适用于各种边界情形。     

确定矩形底四边简支或滑动固支扁壳在任意法向载荷作用下应力函数边界值φ_Γ的计算公式

本文提出了适用于任意曲面形状的矩形底四边简支或滑动固支扁壳在任意法向载荷作用下确定应力函数边界值φr的一般计算公式,从而明确了根据以φ及w为未知量的基本微分方程来进行计算时对边界条件的提法。此外对基本微分方程在φr不为零时的解法提供了简捷的处理方法。     

考虑横向剪切变形矩形底中厚扁壳的屈曲研究

基于考虑横向剪切变形直角坐标下矩形中厚扁壳的几何方程、本构关系、平衡方程,建立了关于三个中面位移和两个中面转角为独立变量的矩形中厚扁壳小挠度屈曲的基本微分方程。该方程可退化为矩形中厚板屈曲的基本微分方程,从而说明本文推导过程的正确性及一般性。文中矩形中厚扁壳小挠度屈曲的基本微分方程是一组耦合的变系数二阶偏微分方程,对常曲率扁壳使用双重三角级数并将其作为广义坐标对该方程组进行解耦,进一步建立中厚扁壳小挠度屈曲的特征方程,并得到了简支矩形中厚壳屈曲的临界荷载表达式,最后获得了其屈曲的临界荷载曲线及其相应的临界荷载值。该临界荷载曲线及其相应的临界荷载值可以退化为矩形中厚板的临界荷载曲线及临界荷载值。结果表明：本文提出的算法求解过程简便,矩形中厚扁壳临界荷载收敛较快。     

矩形底面球形网格扁壳的动力分析

本文利用连续化方法对矩形底面球形网格扁壳的动力问题进行了研究,推出了一般的动力控制方程。在四边简支的边界条件下,振动方程的解可由两个双重级数的试函数求得。对竖向地震作用下的解也可用类似的方法得到,其计算公式中的相应系数与《建筑抗震设计规范(GBJ11-89)》中的系数相似,并可利用竖向地震反应谱求得。     

四边简支正交各向异性圆柱扁壳的后屈曲和非线性振动

本文在几何非线性三维弹性理论的基础上,通过量级分析导出了考虑横向剪切效应的正交各向异性纤维增强复合材料扁壳的基本方程,并应用伽略金方法求得了四边可动简支正交各向异性圆柱形扁壳后屈曲变形和非线性自由振动问题的数值解。计算结果表明:对于复合材料而言,横向剪切效应是值得注意的。     

中厚夹层圆柱扁壳的屈曲分析

基于一阶剪切变形中厚圆柱壳理论,考虑夹心层的面内刚度,根据中厚夹层圆柱扁壳的几何方程、本构关系、平衡方程,建立了中厚夹层圆柱扁壳小挠度屈曲的微分方程;应用广义傅里叶级数解法,得到了四边简支条件下中厚夹层圆柱扁壳屈曲的临界荷载表达式。该表达式可以退化为硬夹心夹层板的表达式,表明本文推导过程的正确性。本文还讨论了夹心层的弹性模量和泊松比以及中厚夹心圆柱扁壳的长宽比和厚度对扁壳屈曲荷载的影响。结果表明:临界荷载与夹心层的弹性模量和泊松比成正比,与扁壳的长宽比和厚度成反比。     

四边固支球面扁壳的解析解

本文用解析法四边固支球面扁壳的在均布荷载作用下的内力和挠度进行了计算。     

应用混合变量余能原理求解不同载荷作用下四边简支矩形板的弯曲

由功的互等定理导出了弯曲矩形板混合变量的总余能,应用混合变量余能原理求解四边简支矩形板弯曲问题,给出了四边简支矩形板的混合变量余能表达式,求解了不同载荷作用下四边简支矩形板的弯曲问题,并将计算的不同载荷作用下的挠度和弯矩值与ANSYS有限元的计算值比较,给出了一种求解四边简支矩形板的新方法。     

含面内转动自由度的广义协调曲面矩形扁壳元

扁壳单元中引入结点转角自由度可以在不增加结点的情况下,增加位移场的阶次,提高计算精度,从而显著地提高单元性能。同时在单元中引入泡状位移场,能有效地扩大了单元位移场的解空间,所构造的单元具有计算精度高、对计算网格畸变不敏感的优良特性。本文利用广义协调薄板单元RGC-12的位移函数作为扁壳元的法向位移,利广义协调矩形膜元的位移函数作为扁壳面的切向位移,通过附加面内转动自由度构造了一个具有24个自由度的4结点广义协调曲面矩形扁壳元GRC-S24。在此基础上再增加一个广义泡状位移,又构造了一个具有更高计算精度的曲面矩形扁壳元GRC-S24M。并通过实例分析对这两个单元的收敛性和精度进行了验证。     

FREE VIBRATION OF ANISOTROPIC RECTANGULAR PLATES BY GENERAL ANALYTICAL METHOD

According to the differential equation for transverse displacement function of anisotropic rectangular thin plates in free vibration, a general analytical solution is established. This general solution, composed of the composite solutions of trigonometric function and hyperbolic function, can satisfy the problem of arbitrary boundary conditions along four edges. The algebraic polynomial with double sine series solutions can also satisfy the problem of boundary conditions at four corners. Consequently, this general solution can be used to solve the vibration problem of anisotropic rectangular plates with arbitrary boundaries accurately. The integral constants can be determined by boundary conditions of four edges and four corners. Each natural frequency and vibration mode can be solved by the determinate of coefficient matrix from the homogeneous linear algebraic equations equal to zero. For example, a composite symmetric angle ply laminated plate with four edges clamped has been calculated and discussed.     

Vibrations of shallow shells rectangular in the horizontal projection with two freely supported opposite edges

The exact mode shapes of linear vibrations of a shallow shell rectangular in the horizontal projection with two freely supported opposite edges are obtained. These shapes are used to construct a discretemodel of vibrations of a shallow shell in geometrically nonlinear deformation. The harmonic balance method is used to study the free and forced nonlinear vibrations under internal resonance. The Lyapunov stability of the obtained periodic vibrations is analyzed.     

各向异性矩形板的剪切屈曲分析

根据各向异性矩形薄板剪切屈曲横向位移函数的微分方程建立了一般性的解析解。该一般解包括三角函数和双曲线函数组成的解,它能满足四个边为任意边界条件的问题;该一般解还包括代数多项式解,它能满足四个角的边界条件问题。因此,这一解析解可用于精确地求解任意边界的各向异性矩形板的剪切屈曲问题。其中待定常数可由四边和四角的边界条件来确定,由此得出的齐次线性代数方程系数矩阵行列式等于零可以求得各阶临界载荷及其屈型。结合配点法,利用变形的对称和反对称性,以及对称迭层正方形板均可使计算更简单。以四边平夹的对称角铺设复合材料迭层板为例进行了计算和讨论。     

On the non-classical mathematical models of coupled problems of thermo-elasticity for multi-layer shallow shells with initial imperfections

Mathematical modeling of evolutionary states of non-homogeneous multi-layer shallow shells with orthotropic initial imperfections belongs to one of the most important and necessary steps while constructing numerous technical devices, as well as aviation and ship structural members. In first part of the paper fundamental hypotheses are formulated which allow us to derive Hamilton–Ostrogradsky equations. The latter yield equations governing shell behavior within the applied hypotheses and modified Pelekh–Sheremetev conditions. The aim of second part of the paper is to formulate fundamental hypotheses in order to construct coupled boundary problems of thermo-elasticity which are used in non-classical mathematical models for multi-layer shallow shells with initial imperfections. In addition, a coupled problem for multi-layer shell taking into account a 3D heat transfer equation is formulated. Third part of the paper introduces necessary phase spaces for the second boundary value problem for evolutionary equations, defining the coupled problem of thermo-elasticity for a simply supported shallow shell. The theorem regarding uniqueness of the mentioned boundary value problem is proved. It is also proved that the approximate solution regarding the second boundary value problem defining condition for the thermo-mechanical evolution for rectangular shallow homogeneous and isotropic shells can be found using the Bubnov–Galerkin method.     

A general analytical solution for elastic vibration of rectangular thin plates

A general solution of differential equation for lateral displacement lunction of rectangular elastic thin plates in free vibration is established in this paper. It can be used to solve the vibration problem of rectangular plate with arbitrary boundaries. As an example, the frequency and its vibration mode of a rectangular plate with four edges free have been solved.     

Stress state of nonthin orthotropic shells with varying thickness and rectangular planform

The stress-strain state of a shallow orthotropic shell with rectangular planform and thickness varying in two coordinate directions is studied. A refined problem formulation is used. Different boundary conditions are considered. A numerical analytic approach based on the spline approximation and discrete orthogonalization is developed. The stress-strain state of shallow orthotropic shells whose thickness is varied keeping its mass constant is studied Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 91–102, August 2008.     

Natural vibrations of a cantilever thin elastic orthotropic cylindrical shell

The natural vibrations of a cantilever thin elastic orthotropic circular cylindrical shell are studied. Dispersion equations for the determination of possible natural frequencies of cantilever closed shells and open shells with Navier hinged boundary conditions at the longitudinal edges are derived from the classical dynamic theory of orthotropic cylindrical shells. It is proved that there are asymptotic relationships between these problems and the problems for a cantilever orthotropic strip plate and for a cantilever rectangular plate and the eigenvalue problem for a semi-infinite closed orthotropic cylindrical shell with free end and for the same but open shell with Navier hinged boundary conditions at the longitudinal edges. A procedure to identify types of vibrations is presented. Orthotropic cylindrical shells with different radii and lengths are used as an example to find approximate values of the dimensionless natural frequency and damping factor for vibration modes __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 68–91, May 2008.     

四角点支承四边自由矩形薄板屈曲问题的新解析解

角点支承矩形薄板的屈曲问题是板壳力学的一类重要课题，控制方程和边界条件的复杂性导致寻求该类问题的解析解十分困难。虽然各类近似/数值方法可用于解决此类难题，但作为基准的精确解析解在公开文献中鲜有报道。本文基于近年来提出的辛叠加方法，解析求解了四角点支承四边自由矩形薄板的屈曲问题。首先将问题拆分为两个子问题，接着利用分离变量与辛本征展开推导出子问题的解析解，最后通过叠加获得原问题的解。由于求解过程从基本控制方程出发，逐步严格推导，无需假定解的形式，因此本文解法是一种理性的解析方法。数值算例给出了不同长宽比和不同面内载荷比情况下，四角点支承四边自由矩形薄板的屈曲载荷和典型屈曲模态，并经有限元方法验证，确认了解析解的正确性。     

Numerical and experimental dynamic characteristics of thin-film membranes

Presented is a total-Lagrangian displacement-based non-linear finite-element model of thin-film membranes for static and dynamic large-displacement analyses. The membrane theory fully accounts for geometric non-linearities. Fully non-linear static analysis followed by linear modal analysis is performed for an inflated circular cylindrical Kapton membrane tube under different pressures, and for a rectangular membrane under different tension loads at four corners. Finite-element results show that shell modes dominate the dynamics of the inflated tube when the inflation pressure is low, and that vibration modes localized along four edges dominate the dynamics of the rectangular membrane. Numerical dynamic characteristics of the two membrane structures were experimentally verified using a Polytec PI PSV-200 scanning laser vibrometer and an EAGLE-500 8-camera motion analysis system.     

Vibrations of ribbed shallow rectangular shells on an elastic foundation

A procedure to determine the natural frequencies and modes of ribbed shallow shells with rectangular planform on an elastic foundation is developed. The effect of the radius of curvature and the Winkler and Pasternak coefficients of subgrade reaction on the natural frequencies and modes of a shallow spherical shell with a square planform is analyzed. It is revealed that not only the natural frequencies but also the modes of the shell depend on the coefficients of subgrade reaction