Explicit high-order finite-difference analysis of rotationally symmetric shells

Abstract

The system of equations for the analysis of rotationally symmetric shells subjected to time-dependent loadings and boundary conditions has been formulated with the transverse, meridional, and circumferential displacements as the dependent variables in the field equations. Inertia forces are considered in each of the three coordinate directions of the shell. The solution for each Fourier harmonic is obtained by employing ordinary finite-difference representations of the derivatives and high-order finite-difference representations for the meridional coordinate derivatives. The complete system of equations is solved implicitly for the first time increment, while an explicit solution for variables within the boundary edges of the shell, together with separate implicit solutions at each boundary, is utilized for the second and succeeding increments. To aid in the choice of a increment, the equations for the three lower frequencies of vibration of the shell for each Fourier component of response have been derived by the Rayleigh-Ritz method. The developed equations have been programmed in FORTRAN IV language to provide solutions for general shell geometries and loadings by electronic computer. Solutions obtained with the program for typical shells and loadings have been found to be stable and in agreement for a wide range of practical values of both spatial and increments. Solutions for a typical shell and loading together with comparison of the stability requirements with the stability requirements for other formulations and finite-difference representations have been included.