ANALYSIS OF THIN ELASTIC PLATES BY THE METHOD OF FINITE ELEMENTS

Abstract

The work presented in the thesis deals with the analysis of thin elastic plates in bending by the use of triangular finite elements and a new type of segmental element. The method of finite elements is a powerful numerical technique of finding approximate solutions to the complicated problems. One of the greatest advantage of the method is its versatility. The same general technique applies to any type of elastic continuum with arbitrary boundary and loading conditions. A chapter-wise summery of the work is given below: The first chapter briefly reviews classical thin plate theory on which the finite element analysis is based. Basic plate equations are developed in terras of Cartesian and polar coordinates. These equations are needed in the subsequent chapters. A general description of the finite element method based on the displacement approach is presented in the second chapter. It explains the basic principle and the various steps involved in the method. Basically, the method is a discretization technique. It consists in replacing the actual continuum by a modified structure made up of a number of discrete elements. The elements are interconnected only at a finite number of joints. This modified system is further approximated by a mathematical model. The accuracy and efficiency of the method mainly depends on how closely, a mathematical function applied to the discretized system represents its behaviour as a true continuum. A. function uniquely defined in terms of certain parameters is chosen in such a way that the results converge toward the true solution as the element size is progressively reduced. (v) Various shapes of elements and types of approximations that are possible in the method are discussed. Chapter 3, develops a new shape of segmental element which is ideally suited to a class of problems. Annular and sectorial plates can easily be idealized with these slements. It can solve any arbitrary boundary and loading conditions assigned to the plate. Any number of annular holes can easily be taken care of by these elements. The disadvantage with triangular and quadrilateral elements available for idealizing such problems 13 that these cause an additional geometrical approximation at the curved boundaries. A conforming function in polar coordinates, satisfying all the displacement continuity conditions along the element boundary is developed in this chapter. It is uniquely specified in terms of 20 unknown parameters defined at the nodes of the elements, along the element boundary. The nodal parameters are the deflection and some of its derivatives with respect to r and c . Element stiffness properties are evaluated and presented in a simplified form. In chapter IV, the accuracy and convergence achieved by the segmental element stiffness matrix is demonstrated. The element is applied to a number of plates in bending and the finite element results obtained are compared with exact analytical solutions. The examples illustrate the efficiency and power of the element and the method. Results are shown to be good even with fairly coarse subdivision. Because of its versatility in idealizing plates of all shapes and shells of single and double curvature, the triangular finite element has been chosen for detailed treatment in the chapter V. The simplest, non-conforming function with 9 degrees of freedom is derived in a convenient form so that it offers the basis for a systematic and efficient computer programme. (vi) Finally, in chapter VI, triangular elements are used to idealize a class of rectangular and circular plates with various boundary and loading conditions. Numerical solutions obtained are investigated to find, how the accuracy of the results changes with the geometry, boundary and loading conditions of the plate. Plates with circular boundaries have been intentionally chosen to illustrate the efficiency of triangular elements in idealizing curved boundaries.