On Some Problems in Bending of Thick Circular Plates on an Elastic Foundation

1956 ◽  
Vol 23 (2) ◽  
pp. 195-200
Author(s):  
Daniel Frederick
Keyword(s):  
Circular Plate ◽  
Elastic Foundation ◽  
Classical Theory ◽  
Infinite Plate ◽  
Circular Inclusion ◽  
Circular Plates ◽  
Axially Symmetric ◽  
Governing Equations ◽  
Uniform Loading ◽  
Rigid Circular Inclusion

Abstract The governing equations and solutions for the nonsymmetrical bending of circular plates resting on an elastic foundation are presented using the theory developed by E. Reissner. Also included are two examples in which numerical comparisons have been made with the predictions of the classical theory. These are (a) the axially symmetric bending of a finite circular plate on an elastic foundation under a partial uniform loading, and (b) the nonsymmetric bending of an infinite plate on an elastic foundation with a rigid circular inclusion.

Symmetrically Loaded Circular Plates under the Combined Action of Lateral and End Loading

1959 ◽  
Vol 10 (3) ◽  
pp. 266-282 ◽  
Author(s):  
Raymond Hicks
Keyword(s):  
Circular Plate ◽  
Infinite Plate ◽  
Lateral Load ◽  
Simultaneous Action ◽  
Bending Moments ◽  
Circular Plates ◽  
Simply Supported ◽  
Combined Action

Expressions are obtained for the radial and tangential bending moments in a circular plate under the combined action of (a) a lateral load concentrated on the circumference of a circle and an end tension or compression, and (b) a uniformly distributed lateral load, having a diameter less than the diameter of the plate, and an end tension or compression. For both types of loading, solutions are obtained for plates which are simply-supported and for plates with an arbitrary end rotation.In addition, the following limiting cases are considered: (i) concentrated lateral load with end tension or compression, and (ii) an infinite plate under the simultaneous action of an end tension and a lateral load concentrated on the circumference of a circle of finite diameter.

Axisymmetric Vibrations of Circular Plates on Tensionless Elastic Foundations

1990 ◽  
Vol 57 (3) ◽  
pp. 677-681 ◽  
Author(s):  
Zekai Celep ◽  
Dogˇan Turhan
Keyword(s):  
Circular Plate ◽  
Elastic Foundation ◽  
Dynamic Load ◽  
Static Solution ◽  
Forced Vibrations ◽  
Initial Configuration ◽  
Contact Radius ◽  
Circular Plates ◽  
Time Dependency ◽  
Stepwise Change

The present study deals with the axisymmetric vibrations of a circular plate subjected to a concentrated dynamic load at its center. The plate is assumed to be supported on an elastic foundation that reacts in compression only. At first the static solution is obtained and contact radius is determined. Later, this is used as an initial configuration of the forced vibrations. The forced vibrations are assumed to be due to the time dependency of the load. The solution is carried out by Galerkin’s method and by using modal functions of the completely free plate. Numerical results are illustrated in figures for stepwise change of the loading.

scholarly journals Stress Concentration in a Rubber Sheet Under Axially Symmetric Stretching

1967 ◽  
Vol 34 (4) ◽  
pp. 942-946 ◽  
Author(s):  
Wei Hsuin Yang
Keyword(s):  
Differential Equation ◽  
Phase Plane ◽  
Order Differential Equation ◽  
Outer Boundary ◽  
Circular Inclusion ◽  
Axially Symmetric ◽  
Rubber Sheet ◽  
First Order ◽  
Radial Loading ◽  
Rigid Circular Inclusion

A class of axially symmetric problems, concerning a highly elastic, circular rubber sheet with (a) a centered circular hole, (b) a rigid circular inclusion under outward radial loading at outer boundary, and (c) a rigid outer boundary and a concentric hole under inward radial loading around the hole, is solved. The solution of (a) has been obtained by Rivlin and Thomas [1] by solving simultaneously a set of differential equations numerically. In this paper, their equations are reduced to a single second-order differential equation governing the deformation function ρ(r). This is further reduced to two decoupled first-order equations after introducing the phase plane (λ1 – λ2 plane). The solutions are obtained conveniently in the phase plane by Picard’s method and by straightforward numerical integration.

scholarly journals Bending of an incomplete annular plate and related problems

1979 ◽  
Vol 14 (3) ◽  
pp. 103-109 ◽  
Author(s):  
J R Barber
Keyword(s):  
Stress Concentration ◽  
Closed Form ◽  
Circular Plate ◽  
Annular Plate ◽  
Infinite Plate ◽  
Concentration Data ◽  
Closed Form Solutions ◽  
Concentrated Forces

Closed-form solutions and stress-concentration data are obtained for the problem of a sector of an annular plate subjected to moments and transverse forces on its radial edges. Closed-form solutions are also given for a semi-infinite plate or a circular plate subjected to a system of concentrated forces and/or moments at the edge.

Large Amplitude Vibrations of Circular Plate on a Uniform Elastic Foundation

1972 ◽  
Vol 94 (1) ◽  
pp. 43-49 ◽  
Author(s):  
R. Bolton
Keyword(s):  
Circular Plate ◽  
Elastic Foundation ◽  
Normal Modes ◽  
Single Mode ◽  
Linear Mode ◽  
Mode Amplitude ◽  
Plate Vibrations ◽  
Edge Conditions ◽  
Large Amplitude Vibrations ◽  
Mode Response

Herrmann’s equations, the dynamic analogues of the von Karman equations, are solved for a circular plate on a linear elastic foundation by assuming a series solution of the separable form involving unknown time functions. The spatial functions include both regular and modified Bessel functions and are chosen to satisfy the linear mode shape distributions of the plate as well as the usual edge conditions. Total differential equations governing the symmetric plate motions are derived using the Galerkin averaging techniques for a spatially uniform load. By extending the concept of normal modes to nonlinear plate vibrations, comparisons between normal mode response and single mode response, as functions of the first mode amplitude, are shown for different values of the elastic foundation parameter. Results are obtained for plates with simply supported and clamped edges and with both radially moveable and immoveable edges. These results are used to discuss the limitations of single-mode response of circular plates, both with and without an elastic foundation.

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