FINITE DIFFERENCE ANALYSIS OF ROTATIONALLY SYMMETRIC SHELLS USING VARIABLE NODE POINT SPACINGS

Abstract

The system of equations for the analysis of rotationally symmetric shells under time-dependent or static surface loadings has been formulated with the transverse, meridional, and circumferential displacements as the dependent variables. All loading functions must be continuous. The thickness h of the shell may vary along the meridian. Four of the eight natural boundary conditions will be prescribed as time-dependent boundary conditions at each boundary edge of the shell. Surface loadings and inertia forces in the three displacement directions of the shell have been considered. Fourier series are used in the circumferential direction of the shell. Solutions for each Fourier component are found by replacing all derivatives by their finite difference equivalents and solving the resulting system of algebraic equations at successive increments of the time variable. The complete system of equations is solved implicitly for the first time increment, while explicit relations are used to determine the three primary displacements within the boundary edges of the shell for the second and succeeding time increments. The remaining unspecified primary variables are then determined by separate implicit solutions at each boundary for the second and succeeding time increments. Subsequently, all remaining primary and secondary variables are found explicitly. A variable node point spacing may be specified over the full range of the spatial finite difference mesh. A numerical stability criterion which will ensure stable solutions for selected time increments and spatial meshes has been derived and found to agree with results for typical example solutions. Numerical solutions obtained with the computer program which accompanies this development have been found to be stable and in agreement for a wide range of practical values of both spatial and time increments for typical shells and loadings. Static and dynamic solutions for a parabolic shell with fixed boundary conditions at one edge and free boundary conditions at the other edge have been obtained using both constant and variable node point spacings and presented as examples.