Numerical analysis of rotationally symmetric shells under transient loadings

Abstract

This report involves the development of procedures for determining the dynamic response of rotationally symmetric open ended thin shells of revolution under time dependent distributed impulsive and thermal loadings. The solution is restricted to thermal loadings which are small in the circumferential direction but which may vary in any manner in the meridional direction of the shell. Inertia forces are considered in a direction normal to the middle surface and in a direction along the meridians of the shell. Time dependent boundary conditions may be prescribed at each of the two edges of the shell. The field equations are derived in the form of eight first-order partial differential equations with respect to the meridional direction of the shell. The solution for each Fourier harmonic is obtained by employing finite difference representations for all time and spatial derivatives. The complete system of equations is solved implicitly for the first time increment, whereas explicit relations are used for the meridional and transverse displacements for the second and succeeding time increments. The developed equations have been programmed in FORTRAN IV language for solution by computer. Solutions obtained with this program for typical shells and loadings are found to be stable and in agreement for a range of values of the space and time increments. It is concluded that the finite difference methods employed here constitute a most expeditious procedure for obtaining reliable and accurate values of the response histories of rotationally symmetric shells of revolution under transient loadings.