Kirchhoff plate modelling using finite element method

Abstract

The Kirchhoff plate theory works well for thin plates where the real shear strains are small. In this study, the development of Kirchhoff plate theory using FEM is presented. The equilibrium condition of the problem defined as 2 0 2 2 2 2 2 � � � � � � � � � � � q y M x y M x Mxx xy yy is investigated in providing the appropriate boundary conditions, hence to the establishment of the FE formulation of the problem. The plate elements developed are the two-dimensional triangular element. To meet the convergence criteria, the quadratic interpolation function is adopted and the six nodes triangular element is developed. The deflection w takes the form of � , � 2 . 5 6 2 w x y ��1 �� 2 x �� 3 y �� 4 x �� xy �� y The numerical results of two neighbouring six nodes triangular elements are studied. These elements are considered to be interconnected at specified nodes which lie on the element boundaries where adjacent elements are considered to be connected. In each piece or element, the element shape function Ni , the stiffness matrix K , and the load vector fl are derived. The assemblage of these matrices together with the derivation of boundary vector fb will yield to an approximate solution for the displacement of the problem. The computational scheme is developed by using Matlab programming language on the Windows environment for computing the problem studied.