The general solution of bending of a spherical thin shallow shell with a circular hole at the center under arbitrary transverse loads

A general method is presented for solving shallow shell problems with finite boundaries and with an arbitrarily placed load that is uniformly distributed over a circular area of radius r0. A known solution for the distributed load on an unbounded shell is used to describe the load effects, and this particular solution is combined with Reissner’s general solution of the shallow shell equations in such a manner that all the boundary conditions are satisfied. Numerical results have been obtained for a shallow shell, clamped at the outer boundary and having a circular polar aperture free of tractions and support.

Download Full-text

Note on General Bi-Harmonic Analysis for a Plate Containing Circular Holes

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100021502 ◽

1941 ◽

Vol 37 (1) ◽

pp. 29-33 ◽

Cited By ~ 3

Author(s):

A. E. Green

Keyword(s):

General Solution ◽

Harmonic Analysis ◽

Circular Hole ◽

Elastic Constants ◽

Infinite Plate ◽

Stress Distributions ◽

Zero Force ◽

Circular Holes

In a previous paper a general solution was given for problems of stress distributions in a plate containing circular holes of varying sizes arranged in any manner. This work was a generalization of special methods used by various writers for particular arrangements of holes. The types of stress distributions were, however, confined to those which produce zero force resultants on each hole and the solutions were therefore independent of the elastic constants. Bickley has studied distributions of stress round one circular hole in an infinite plate when the force resultants on the hole are no longer zero, and a few other problems of this type have been dealt with by other writers.

Download Full-text

Torsional Vibrations of Pretwisted Cantilever Plates

Journal of Mechanical Design ◽

10.1115/1.3453962 ◽

1978 ◽

Vol 100 (3) ◽

pp. 528-534 ◽

Cited By ~ 10

Author(s):

K. Gupta ◽

J. S. Rao

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Fourth Order ◽

Shallow Shell ◽

Torsional Vibrations ◽

Cantilever Plate ◽

Order Ordinary Differential Equation ◽

Aspect Ratios ◽

Constant Coefficients ◽

Thin Shallow Shell

A pretwisted cantilever plate is treated as a thin shallow shell. Its potential and kinetic energies in torsional vibration are determined by assuming an appropriate displacement field. Applying Hamilton’s principle, the problem is reduced to a fourth-order ordinary differential equation with constant coefficients, which is solved to obtain the first four torsional frequencies of vibration. Plates of aspect ratios varying from 1.0 to 8.0 are analyzed with pretwist angles varying from 0 to 90 deg. Results of the present analysis are compared with existing theoretical and experimental results.

Download Full-text

Point Load on a Shallow Elliptic Paraboloid

Journal of Applied Mechanics ◽

10.1115/1.3625124 ◽

1966 ◽

Vol 33 (3) ◽

pp. 575-585 ◽

Cited By ~ 11

Author(s):

Kevin Forsberg ◽

Wilhelm Flu¨gge

Keyword(s):

Power Series ◽

Axial Symmetry ◽

Shallow Shell ◽

Point Load ◽

Singular Solutions ◽

Elliptic Paraboloid ◽

Concentrated Loading ◽

Shell Equations ◽

Shell Geometry ◽

Thin Shallow Shell

The present work is a study of a thin shallow shell having a specific type of deviation from axial symmetry, i.e., the portion of an elliptic paraboloid near its vertex. The singular solutions to the homogeneous shallow-shell equations are expressed as power series in terms of a parameter γ, which is a measure of the deviation of the shell geometry from axial symmetry. These singular solutions can be directly related to concentrated loading at the vertex of the shell. The solution converges in the range γ = 0 (sphere) to γ = 1/2 (cylinder). Detailed graphical results are presented for the stress resultants and radial deflection of a shell subjected to a point load at its vertex.

Download Full-text