ON SOME VIBRATION PROBLEMS OF BEAMS AND PLATES

Abstract

This thesis is an attempt to study some of the problems of flexural vibrations of beams and plates and is divided into two parts. The first part deals with problems relating to the flexural vibrations of uniform beams and naturally curved rods which include th effects of rotatory inertia and shear. The equations for flexural vibrations of a uniform cantilever beam according to Timoshenko theory have been numerically solved and it is shown that results similar to those of Anderson for a beam supported at both ends can be obtained. The equations of flexural vibrations of a deep beam obtained by a method of successive approximations are studied and solved for the case of a cantilever beam for various ratios of cepth to length. These are compared with the solution of Timoshenko's equation for the same beam. Here we include the effects of shear and rotatory inertia, but do not assume that the cross-sections of the beam remains plane as assumed by Timoshenko. For naturally curved elastic rods, the equations f0r flexural vibrations given by Morley, where the neutral axis forms a plane curve of constant radius, is worked out. The effects of rotatory inertia and shear are included in the same way as in Timoshenko theory for straight rods and the effect of the extension of the neutral axis is also included. Numerical solutions for a rod fixed at both ends are obtained for rods of different curvatures and compared with the results for a straight uniform beam. The equations of vibrations of a short thin arch in the shape of catenary are also derived, the displacements being confined to the plane of the catenary. Numerical solutions of ii the equations for the symmetrical case of fundamental mode of vibration are obtained. These equations and their solutions would be of significance in the study of vibrations of the 325 feet-span concrete shell arch designed and constructed at the University of Roorkee. • In the second part a method of successive approximations is presented to obtain the vibrations equations of an Isotropic elastic plate under conditions which are more general than those of Timoshenko theory. Here, we include the secondary effects of rotatory inertia and shear, but do not assume that the displace ments follow a linear law as was done by Timoshenko or Mindlin. The governing equations for the flexural vibrations of plates are solved•separately for square and circular plates for various ratios of thickness to the sice or diameter of the respective plates. Solutions are also given of the equations for flexural vibrations of isotropic elastic plates according to Minclin's theory for the case of simply supported uniformly loaded thin square and circular plates and the comparison is ^a6e between the results of two equations. The former method of analysis yields results more accurate than those by the existing methods. The method therefore offers a means of solving vibrational problems to a high degree of accuracy. Lastly an attempt has been made to make a comparative study of the equation derived by the method of successive approximations with the exact theory, Mindlin1s theory and Lagrange's classical plate theory through the results for the case of straight crested waves. The contents of each chapter are outlined below. iii FART I. Chapter I. The equations for flexural vibrations of a uniform beam according to Timoshenko theory are studied and numerically solved for the case of a cantilever beam for various slenderness ratios. Chapter II. A method of successive approximations to derive the equation for flexural vibrations of a deep beam is studied. The equation is then solved numerically for the case of a cantilever beau for various values of depth to lengtn ratios. Charter III. Morley's equations for flexural vibrations of a naturally curved elastic rod are obtained and solutions are given for the case of a rod fixed at both ends for various curvatures. Chapter IV. The equations of vibrations of a short thin arch in the shape of catenary are derived and solved for the symaetrical case of the fundamental mode of vibration. PART II. Chapter V. A equation for flexural vibrations of an isotropic elastic plate is derived by the method of successive approximations. Chapter VI. The equation of the above chapter has been solved for the case of a simply suppored thin square plate. Chapter VII. The equation of the chapter V has been solved for another case of a simply supported thin circular plate. Chapter VIII. Minclin's equation for flexural vibrations of isotropic elastic plates is obtained and a numerical solution is given for the case of a simply supported thin square plate. Chapter IX. The equation of the above chapter has been solved for another case of a simply supported thin circular plate. iv Chapter X. In this concluding chapter, an attempt has been made to make a comparative study of the equation derived by the method of successive approximations in Chapter V with the exact theory, Mindlin's theory and Lagrange's Classical plate theory through the resiilts for tke case of strajgit crested waves. The work, presented here, is original research by the author except sections 1.1, 2.1, 3,2 and 8.2 which have been put in to present a connected account of the whole. The material of Chapters I,VI ana IX is being published in the Bulletins of Calcutta Mathematical Society. The Chapters V and VIII are published in the proceedings of National Institute of Sciences of India and Chapter VII in the University of Roorkee Research Journal. The material of Chapter IV was presented in the second symposium of Earthquake Engineering held at Roorkee in December, 1962.