The bending,stability and vibrations of rectangular plates with free edges on elastic foundations

This paper discusses the problems of the bending, stability and vibrations of the rectangular plates with free boundaries on elastic foundations. In the present paper we select a flexural function, which satisfies not only all the boundary conditions of free edges but also the conditions at free corner points, and consequently we obtain a better approximate solution. The energy method is used in this paper.     

A free rectangular plate on elastic foundation

This article will discuss the bending problems of the rectangular plates with free boundaries on elastic foundations. We talk over the two cases, that is, the plate acted on its center by a concentrated force and the plate subjected to by a concentrated force equally at four corner points respectively. We select a fiexural function which satisfies not only all the geometric boundary conditions on free edges wholly but also the boundary conditions of the total internal forces. We apply the variational method meanwhile and then obtain better approximate solutions.     

矩形夹层板稳定性分析

本文在Reissner 理论基础上，应用功的互等定理推导了夹层矩形板稳定问题的基本解，在已推导出的夹层板基本解的基础上，利用功的互等法求解了两对边固定一边简支一边自由、两邻边简支另两邻边自由且角点支承、两邻边简支另两邻边自由且角点悬空三种不同边界条件下夹层板的稳定问题，给出了挠曲面方程及其对应的执行方程；进行了数值计算，并与有限元结果进行对照分析．结果表明：本文方法求解过程更简单，提供了一种求解夹层板稳定问题的新方法，计算结果对解决工程实际问题具有一定的参考价值．     

扁球面网壳的混沌运动研究

在圆形三向网架非线性动力学基本方程的基础上,用拟壳法给出了圆底扁球面三向网壳的非线性动力学基本方程．在固定边界条件下,引入了异于等厚度壳的无量纲量,对基本方程和边界条件进行无量纲化,通过Galerkin作用得到了一个含二次、三次的非线性动力学方程．为求Melnikov函数,对一类非线性动力系统的自由振动方程进行了求解,得到了此类问题的准确解．在无激励情况下,讨论了稳定性问题．在外激励情况下,通过求Melnikov函数,给出了可能发生混沌运动的条件．通过数字仿真绘出了平面相图,证实了混沌运动的存在．     

两邻边自由另两边任意支撑的矩形厚板静力问题的一般解

本文采用胡海昌教授提出的厚板方程,并用作者所提出的滑支边和广义滑支边的概念,再加上广义简支边的概念,用叠加法求解两邻边自由另两边任意支撑的矩形厚板静力问题一般解。     

On rate-type viscoelasticity and the second law of thermodynamics

We deduce an energy identity which must be satisfied by the smooth solutions of the system of equations governing the dynamics of body with quasilinear rate-type constitutive equation. We give conditions when a unique energy function exists for rate-type viscoelasticity. In the semilinear case we give the conditions when a unique, positive and convex energy function exists and we obtain estimates in energy for the smooth solutions of initial-boundary value problems. A viscoelastic approach to nonlinear elasticity is discussed. Finally, an example shows that the second law of thermodynamics does not imply stability.     

具有自由边的复合材料层合板的解析解

从三维弹性力学基本方程出发,通过假设自由边的边界位移函数,建立了正交异性层合板的状态方程,给出了对边自由,对边简支矩形板的解析解.此解满足层合板的基本方程和层间连续条件.用本文的方法比较容易处理层合板的自由边.算例表明,数值结果具有较高的精度.     

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.     

Interaction of plane harmonic waves with a thin elastic inclusion of zero flexural rigidity

We solve the problem on the interaction of plane elastic harmonic waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) that is under plane strain conditions. The normal forces applied by the medium to the inclusion side edges are taken into account. Because of the small thickness of the inclusion, we assume that its flexural rigidity is zero and that the shear displacements at any of its points coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself can be found from the corresponding equation of the theory of plates. The solution method consists in representing the displacements as discontinuous solutions of the Lamé equations and then determining the unknown jump from a singular integral equation. This equation is solved numerically by the collocation method, and formulas for the approximate calculation of the stress intensity factors near the inclusion ends are obtained.     

Mode jumping in the buckling of struts and plates: a comparative study

A detailed investigation of the phenomenon of mode jumping in compressed struts on stiffening foundations and elastic plates of varying lengths is performed, with emphasis on the effects of altering boundary conditions. The variety of possible modal interactions is presented in a concise form using the parameter space of Arnol'd tongues, borrowed from non-linear dynamical systems theory. For the strut system, a full range of end conditions from simply supported to clamped is examined. For the plate, simply supported and clamped flexural conditions along both long (unloaded) and short (loaded) edges are considered, together with in-plane conditions ranging from free to pull in, to fully restrained. For each system, simply supported end conditions are found to provide protection against early mode jumping in a so-called “safety envelope”, but this is eroded as the end conditions are systematically altered from simply supported to clamped. For the plate system, mode jumping is induced at an earlier stage in the loading process by restricting the long (unloaded) edges against in-plane movement, but is delayed by clamping the same edges against rotation.     

THE METHOD OF THE RECIPROCAL THEOREM OF FORCED VIBRATION FOR THE ELASTIC THIN RECTANGULAR PLATES(Ⅱ)—RECTANGULAR PLATES WITH TWO ADJACENT CLAMPED EDGES

In this paper,applying the method of reciprocal theorem,we give the distributions ofthe amplitude of bending moments along clamped edges and the amplitude of deflectionsalong free edges of rectangular plates with two adjacent clamped edges under harmonicdistributed and concentrated loads.     

On problems of optimal design of shallow shell with double curvature on elastic foundation

The present paper discusses a method of optimal design of the shallow shell with double curvature on the elastic foundation Substantially we take the initial flexural function as the control function or design variable which will be found and the potential energy of the external loads as the criterion of quality of the optimal design of the shallow shell with double curvature, therefore the functional of the potential energy will be aim function. The optimal conditions and the isoperimetric conditions belong to the constrained conditions. thus we obtain the necessary conditions of the optimal design for the given problems, at the same time the conjugate function is introduced, then the problems are reduced to the solutions of two boundary value problems for the differential equation of conjugate function and the initial flexual function.     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的位移解建立的板与地基变形协调方程相结合,先按对称性分解,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

A free rectangular plate on the two-parameter elastic foundation

This paper provides a rigorous solution of a free rectangular plate on the V.Z. Vlazov two-parameter elastic foundation by the method of superposition[1]. In this paper we derive basic solutions under the various boundary conditions. To superpose these basic solutions the most generally rigorous solution of a free rectangular plate on the two-parameter elastic foundation can be obtained. The solution strictly satisfies the differential equation of a plate on the two-parameter elastic model foundation, the boundary conditions of the free edges and the free corner conditions. Some numerical examples are presented The calculated results show that when the plane dimension of plate is given and the ratio between the laver depth and the plate thick is equal to 15, the two-parameter elastic model is near the Winkler’s. It shows that the Winkler model can be applied to the thinner layer.     

Transverse vibration of rectangular plates elastically supported at points on edges

This paper studies transverse vibration of rectangular plates with two opposite edges simply supported, other two edges arbitrarily supported and free edges elastically supported at points. A highly accurate solution is presented for calculating inherent frequencies and mode shape of rectangular plates elastically supported at points. The number and location of these points on free edges may be completely arbitrary. This paper uses impulse function to represent reaction and moment at points. Fourier series is used to expand the impulse function along the edges. Characteristic equations satisfying all boundary conditions are given. Inherent frequencies and mode shape with any accuracy can be gained.     

The method of the reciprocal theorem of forced vibration for the elastic thin rectangular plates (II)—Rectangular plates with two adjacent clamped edges

In this paper, applying the method of reciprocal theorem, we give the distributions of the amplitude of bending moments along clamped edges and the amplitude of deflections along free edges of rectangular plates with two adjacent clamped edges under harmonic distributed and concentrated loads.     

广义边界梁的动力学建模与动态特征

对于广义边界条件Euler-Bernoulli梁,采用相对描述方式建立了可描述梁整体运动和相对变形的几何非线性及其线性化动力学模型,应用线性变换得到了该类梁的线性经典动力学方程,得到了广义边界条件下梁的横向振动代数特征方程、特征函数及特征值的退化表达式.算例分析了边界小扰动对固支-固支梁横向振动特征的影响规律.     

不同支承条件矩形中厚板的精确解

本文在Ｒｅｉｓｓｎｅｒ板理论中和学应力变量方法得到挠度函数ｗ（ｘ，ｙ）和应力函数ψ（ｘ，ｙ）解。在这些解中，选择一些三角级数和多项式作为问题的挠度作为函烽和应力函数，从而得到一些矩形厚板问题的解，例如矩形悬臂厚板，三边固定一边自由矩形厚板等。这里不需要繁锁也叠加。     

角点悬空弯曲厚矩形板的边界积分法

在边界积分法中引用了拟基本系统矩形板,在该拟基本系统与实际系统之间应用功的互等定理,得到一挠曲面方程的积分表达式,只要对此表达式进行极简单的积分便可得到该挠曲面方程,这比直接求解Reissner挠度控制方程要简单,边界积分法的求解过程概念清晰,计算伊始便给出了挠曲面方程的总体表达式.以Reissner厚板理论为基础,应用边界积分法研究了角点悬空厚矩形板的弯曲问题,给出了在集中荷载作用下两邻边固定另两邻边自由且角点悬空弯曲厚矩形板的封闭解析解,并给出了相应的数据和图表以供工程上的应用和参考.     

A series boundary integration method for the bending analysis of anisotropic plates

A series boundary integration method is given which exactly satisfies the fundamental equation of bending analysis of general anisotropic plates or laminated plates in Kirchhoffs sense. With a unified deflection series, the method may be applied to the plates having different planforms and support conditions. Several groups of representative examples are calculated. The examples include circular, square and triangular plates, and their boundaries include clamped edges, simply supported edges, free edges and free corner. Numerical results indicate rapid convergency for both deflection and stress resultants and demonstrate wide applicability of the method.This subject was supported by the Natural Science Foundation of Guangdong Province.