Large deflections of rectangular plates

Approximation of large, non-linear deflections of rectangular plates by a Fourier cosine series and evaluation of the resulting energy expressions leads to highly accurate results for maximum plate deflection. Consideration of non-linear terms which arise in the expression for membrane energy leads to a stiffer plate behavior resulting in considerably smaller deflections than can be accounted for by the classical small-deflection theory and by some modern approaches to largedeflection analysis. Results compare favorably to infinite element analysis results. Various boundary conditions are considered.     

Large deflections of heterogeneous anisotropic rectangular plates

An approximate solution is presented for large deflections of clamped, uniformly loaded, unsymmetrically laminated, anisotropic, rectangular plates. Expressing the load and displacements in the form of series, the von Karman-type nonlinear differential equations and immovable boundary conditions are reduced to a series of linear partial differential equations and boundary conditions. The solution obtained by successive approximations can reduce to some existing solutions for large deflections of homogeneous plates. Numerical results based on the first three terms of the truncated series are graphically presented for unsymmetrical cross-ply and angle-ply plates having various values of fiber-reinforced material, number of layers, and aspect ratio. The results in small deflections of coupled laminates are compared with available data.     

A new approach to an analysis of large deflections of thin elastic plates

For thin plates undergoing large deflections, a modified energy expression has been suggested and a new set of differential equations has been obtained in a decoupled form. Accuracy of the equations has been tested for a circular and a square plate with immovable as well as movable edge conditions under a uniform static load. Results obtained are in excellent agreement with other known results. These new equations are more advantageous than Berger's decoupled equations which fail to give meaningful results for movable edge conditions.     

Large deflections of prestressed simply supported rectangular plates under uniform pressure

The von Kármán large deflection equations for laterally loaded rectangular plates are extended to include uniform prestresses parallel to the edges and are solved for uniform load and for edges which are simply supported against movement normal to the plane of the plate and which are either held or free to move as a rigid body in the plane of the plate. Calculated values of center deflection and maximum stress parameters are given as functions of the load parameter for plates of various aspect ratios.     

On the elastostatics of thin periodic plates with large deflections

Thin periodic plates with large (i.e. of the order of plate thickness) deflections are considered. In this note the tolerance and the asymptotic models of these plates are presented. As an example of applications, these models are used to analyse a bending of periodic plates under various loadings.     

Finite-element analysis of rectangular plates with rectangular inserts

The present investigation deals with the stress distribution in the vicinity of rectangular inserts in finite rectangular plates. This problem is more complex due to the singularities at the corners of the inserts. In this paper, the finite-element technique is used to determine the deformations and, subsequently, the stresses. The paper treats the problem in a generalized form in the sense that the size and orientation of the insert are taken as variables. The finite rectangular plate is subjected to a uniform axial tensile load. The material of the plate and that of the insert are considered to be different. Element selections are made which are optimal with regard to accuracy and computational effort. The local element stresses which generate considerable discontinuity at the element nodes are plotted. Averaging process for the local stress calculations is discussed and these are compared with the results available¹ which are obtained by experimental techniques.     

General mathematical theory of large deflections of thin plates with some holes

The theoretical analysis and the numerical computations for the problem of a thin plate with large deflection and some holes become much more difficult due to the multi-valued properties of the stress functionF and the single-valued demands on the displacements. The necessary and sufficient conditions which can assureF to be single-valued are obtained in this paper. At the same time, we prove that the single-valued demands on the displacements are equivalent to 3m functional constraint equationsDC(w,F)=0, wherem is the number of holes. From these conclusions, the single-valued governing-equations of the problem of plates with large deflection and some holes are derived. It is a system of fourth order partial differential equations with 3m unknown constants and constrained equations. A numerical method for solving this problem is presented. The problem of the critical load is considered and an iterative scheme for computing the buckled states is given when a critical load λ is ‘single’.     

Simulation of three-point bending at large deflections by an elastoplastic contact analysis

Three-point bending is simulated by an elaborate numerical procedure based on an elastoplastic, large deflection, contact analysis. A minimization formulation is used, which is equivalent to the incremental form posed as partial differential equations with inequalities. A sequential quadratic programming approach based on the finite-element technique is adopted as a method of solution. To examine the validity of the simulation method, experiments are carried out for specimens that have various widths.Paper was presented at the 1988 SEM Spring Conference on Experimental Mechanics held in Portland, OR on June 5–10.     

作大范围运动矩形板的动力学建模理论研究

研究了初应力法的作大范围运动矩形板的建模理论。根据连续介质理论,考虑应变-位移中的非线性项,用一致质量有限元法对柔性板进行离散,基于Jourdain速度变分原理导出定轴转动下大范围运动为自由的柔性板刚-柔耦合动力学方程。从其刚柔耦合动力学方程出发,考虑在大范围运动已知情况下的结构动力学方程。通过引入准静态概念,把其结构动力学方程转化为准静态方程。对纵向和横向变形节点坐标进行坐标分离,解出与纵向变形相关的准静态方程,得到准静态时的纵向应力表达式,从而获得附加刚度项;并对此非惯性系下作大范围运动柔性板的结构动力学方程进行数值仿真,验证了采用初应力法柔性板的动力学建模方法来计算经历大范围运动的不规则柔性板的动力学响应是可行的,体现了初应力法对柔性板建模的优越性。     

Calculation of shells with large rectangular holes in the presence of finite bending deflections

The stability of shells perforated by large holes has been investigated previously [1–6]. The results published in the cited papers lead to the conclusion that the problem of the stability of shells with large holes is in a stage of vigorous development. The application of standard numerical methods in these problems is impeded by the fact that they are incapable of refining the stress concentration and local stability in the vicinity of corners of the holes and give an approximate smoothed distribution pattern of the forces and torques. Methods of approximation of the solutions in terms of regular functions [2] are complicated by the poor convergence of the corresponding series near the edges of a hole.Structural Engineering Institute, St. Petersburg. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 41–48, February, 1994.     

On the analysis of thick rectangular plates

Summary Thick rectangular plates are investigated using the method of initial functions proposed by Vlasov. The governing equations are derived from the three-dimensional elasticity equations using a MacLaurin series approach. As the governing equations can be obtained in the form of series, approximate theories of any desired order can be constructed easily by proper truncation. An exact solution is obtained for an allround simply supported thick plate using a Navier type solution. A Levy type solution for higher order theories is illustrated for the case of a thick plate with two opposite edges simply supported and other two edges clamped. Numerical results obtained are compared with those of classical, Reissner and Srinivas et al. solutions.

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Übersicht Mit Hilfe der Methode der Initial-Funktionen von Vlasov werden rechteckige Platten untersucht. Die zugehörigen Gleichungen werden aus den Gleichungen für das dreidimensionale Problem durch eine Entwicklung in MacLaurin-Reihen gewonnen. Durch Abbrechen dieser Reihen können Näherungen beliebiger Ordnung erhalten werden. Für den Fall einer allseitig einfach gelagerten dicken Platte wird eine exakte Lösung erhalten, bei der eine Lösung vom Navier-Typ verwendet wird. Eine Lösung vom Levy-Typ höherer Ordnung wird am Beispiel einer dicken Platte abgeleitet, von der zwei gegenüberliegende Ecken einfach gelagert, die anderen fest eingespannt sind. Die numerischen Ergebnisse werden mit den klassischen, von Reissner, Srinivas u. a. erhaltenen Resultaten verglichen.

     

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。