SOME PROBLEMS IN PLATES AND SHELLS

Abstract

The present thesis is an attempt to study some of the problems of static deformation of isotropic plates and shells. It can be divided into three parts. The first part deals with the deformation of clamped circular plates', the second part considers the deformation of spherical shells (both non-shallow and shallow)*, whereas the third part studies the dynamic response of a simply supported rectangular plate on an elastic foundation due to moving loads. The first part of the thesis comprises of the first three chapters. Chapter I gives an introduction in which the development of the subject from the early part of the 17th century upto the present day has been presented in a graded order for beams, plates and shells. Chapter II deals with the deformation of a clamped circular plate when the effect of transverse shear is taken into consideration. The deformation has been considered for three loadings: (i) when the plate is uniformly loaded, (ii) when the plate is loaded along a concentric ring of radius b and (iii) when the plate is uniformly loaded over a concentric disc of radius b. The numerical values of deflection and the bending moments are found in each case. The effect of variation of thickness is studied. The case of a point load situated at the centre has been obtained as a particular case by making b—>o. The effect of transverse shear is obtained by comparing the results with that of classical plate theory. In chapter III the asymmetric deformation of a circular plate, clamped along its boundary has been investi gated. The equations of equilibrium are obtained on lines similar to Mindlin and their solution is obtained in terms of Bessel functions with the aid of three auxiliary variables. The transverse deflection due to point load situated at any point is obtained by considering it as a limiting case of a uniformly distributed load over a small sectorial area. The effect of variation of thickness on the transverse deflection has been discussed with the help of numerical results. The second part of the thesis contains six chapters IV to IX. Chapter IV deals with the deformation of a moderately thick spherical shell under a general loading. Spherical polar coordinate system is employed for the solution. The five equations of equilibrium are uncoupled with the help of auxiliary variables. The solutions of the uncoupled equations are obtained in terms of Associated Legendre functions. Two types of axisymmetric deformations of such shells have been derived. Torsionless deformations have been analysed in chapter V and the torsional deformations have been considered in chapter VI. Series solutions for the following particular loadings have been obtained: (i) ring loading for torsional deformation (ii) a uniform loading over a spherical cap for torsionless deformation. In chapter VII, the analysis of chapter IV has been adapted to shallow shells. Solutions have been obtained in terms of Bessel functions. In chapter VIII, the case of axisymmetric deformations has been derived for shallow shells. Both torsionless and torsional axisymmetric deformations have been considered. Numerical results for loading over a spherical cap are presented. Chapter IX deals with the axisymmetric deformation of a spherical shell on the basis of membrane theory. The expressions for the displacements and stress-resultants have been obtained in terms of Associated legendre functions. The third part of the thesis, consisting of chapter X, deals with the dynamic response of a simply supported rectangular plate on an elastic foundation due to moving load. The plate is also acted upon by axial forces along its sides. The foundation reaction is supposed to be proportional to the transverse deflection. Two types of loads are considered: (i) Drop load and (ii) Static load. The analysis includes the effect of transverse shear as well as rotatory inertia. The numerical work involved in this thesis was done mostly on IBM 1620 Computer at the Computer Centre, Structural Engineering Research Centre, Roorkee. The numerical work for chapter III was done at the Computer CDC 3600 at Tata Institute of Fundamental Research,Bombay.