Bending of Circular and Ring-Shaped Plates on an Elastic Foundation

1954 ◽  
Vol 21 (1) ◽  
pp. 45-51
Author(s):  
Herbert Reismann
Keyword(s):  
Circular Plate ◽  
Elastic Foundation ◽  
Series Solution ◽  
Circular Disk ◽  
Symmetric Problem ◽  
Concentrated Load ◽  
Transverse Loads ◽  
Infinite Extent ◽  
Theory Of Plates ◽  
Pure Moment

Abstract This paper develops a method for the evaluation of deflections, moments, shears, and stresses of a circular or ring-shaped plate on an elastic foundation under transverse loads. A series solution is derived for plates subjected to edge and/or concentrated loads and is given in terms of tabulated functions. It is exact within the assumptions underlying the classical theory of plates and includes, as a particular case, the known solution of the corresponding radially symmetric problem. Two examples displaying radial asymmetry are worked. A solution is given for (a) a circular plate resting on an elastic foundation, clamped at the boundary and subjected to an arbitrarily placed concentrated load, and (b) a plate of infinite extent, resting on an elastic foundation and clamped to the boundary of a rigid circular disk to which a pure moment is applied.

Bending of an Infinite Beam on an Elastic Foundation

1937 ◽  
Vol 4 (1) ◽  
pp. A1-A7 ◽  
Author(s):  
M. A. Biot
Keyword(s):  
Elastic Foundation ◽  
Poisson Ratio ◽  
Elementary Theory ◽  
Three Dimensional ◽  
Bending Moment ◽  
Two Dimensional ◽  
Concentrated Load ◽  
Elastic Continuum ◽  
Infinite Beam ◽  
A Value

Abstract The elementary theory of the bending of a beam on an elastic foundation is based on the assumption that the beam is resting on a continuously distributed set of springs the stiffness of which is defined by a “modulus of the foundation” k. Very seldom, however, does it happen that the foundation is actually constituted this way. Generally, the foundation is an elastic continuum characterized by two elastic constants, a modulus of elasticity E, and a Poisson ratio ν. The problem of the bending of a beam resting on such a foundation has been approached already by various authors. The author attempts to give in this paper a more exact solution of one aspect of this problem, i.e., the case of an infinite beam under a concentrated load. A notable difference exists between the results obtained from the assumptions of a two-dimensional foundation and of a three-dimensional foundation. Bending-moment and deflection curves for the two-dimensional case are shown in Figs. 4 and 5. A value of the modulus k is given for both cases by which the elementary theory can be used and leads to results which are fairly acceptable. These values depend on the stiffness of the beam and on the elasticity of the foundation.

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.

The Method of Finite Differences in Solutions Concerning Circular Plate

2011 ◽  
Vol 490 ◽  
pp. 305-311
Author(s):  
Henryk G. Sabiniak
Keyword(s):  
Circular Plate ◽  
Finite Differences ◽  
Machine Building ◽  
Polar Coordinates ◽  
Circular Plates ◽  
Technical Problems ◽  
Theory Of Plates ◽  
Worm Gearing ◽  
Mixed Derivatives ◽  
Plate Deflection

Finite difference method in solving classic problems in theory of plates is considered a standard one [1], [2], [3], [4]. The above refers mainly to solutions in right-angle coordinates. For circular plates, for which the use of polar coordinates is the best option, the question of classic plate deflection gets complicated. In accordance with mathematical rules the passage from partial differentials to final differences seems firm. Still final formulas both for the equation (1), as well as for border conditions of circular plate obtained in this study and in the study [3] differ considerably. The paper describes in detail necessary mathematical calculations. The final results are presented in identical form as in the study [3]. Difference of results as well as the length of arm in passage from partial differentials to finite differences for mixed derivatives are discussed. Generalizations resulting from these discussions are presented. This preliminary proceeding has the purpose of searching for solutions to technical problems in machine building and construction, in particular finding a solution to the question of distribution of load along contact line in worm gearing.

Flexure of a Circular Plate With a Ring of Holes

1962 ◽  
Vol 29 (3) ◽  
pp. 489-496 ◽  
Author(s):  
H. Kraus
Keyword(s):  
Circular Plate ◽  
Shear Force ◽  
General Boundary Condition ◽  
Uniform Pressure ◽  
Simply Supported ◽  
Moment Distribution ◽  
Theory Of Plates ◽  
Circular Holes ◽  
The Moment ◽  
Kirchhoff Theory

The problem of the moment distribution resulting from a uniform pressure load acting over the surface of a circular plate containing a ring of equally spaced circular holes with, and without, a central circular hole is solved within the framework of the Poisson-Kirchhoff theory of plates. A general boundary condition is applied at the outer rim of the plate to make the solution valid for a range of conditions from the simply supported case to the clamped case. The edges of the perforations are allowed to be either free or to have a net shear force acting. Numerical results in the form of curves are given for typical cases, and the results of a photoelastic test are also presented.

Bending of a Cylindrically Aeolotropic Circular Plate With Eccentric Load

1952 ◽  
Vol 19 (1) ◽  
pp. 9-12
Author(s):  
A. M. Sen Gupta
Keyword(s):  
Boundary Conditions ◽  
Circular Plate ◽  
Uniform Thickness ◽  
Eccentric Load ◽  
Concentrated Load ◽  
Different Boundary Conditions ◽  
Small Deflection ◽  
Deflection Theory ◽  
Clamped Edge

Abstract The problem of small-deflection theory applicable to plates of cylindrically aeolotropic material has been developed, and expressions for moments and deflections produced have been found by Carrier in some symmetrical cases under uniform lateral loadings and with different boundary conditions. The author has also found the moments and deflection in the case of an unsymmetrical bending of a plate loaded by a distribution of pressure of the form p = p0r cos θ, with clamped edge. The object of the present paper is to investigate the problem of the bending of a cylindrically aeolotropic circular plate of uniform thickness under a concentrated load P applied at a point A at a distance b from the center, the edge being clamped.

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