Bending and Buckling of an Elastically Restrained Circular Plate

1952 ◽  
Vol 19 (2) ◽  
pp. 167-172
Author(s):  
H. Reismann
Keyword(s):  
Circular Plate ◽  
Classical Theory ◽  
Physical Significance ◽  
Buckling Loads ◽  
Simply Supported ◽  
Elastic Boundary ◽  
Theory Of Plates ◽  
Degree Of Restraint ◽  
Elastically Restrained ◽  
Clamped Edge

Abstract This paper considers the effect of an elastic boundary restraint upon the deflection, moments, and critical (buckling) loads of a circular plate. The solutions given are based on the classical theory of plates and are exact within the assumptions underlying this theory. They include, as particular limiting cases, the known solutions for the simply supported and rigidly clamped edge. The physical significance of the results obtained is discussed in detail with particular emphasis upon the degree of restraint along the clamped edge.

Buckling of Circular Plate on Elastic Foundation

1965 ◽  
Vol 87 (3) ◽  
pp. 323-324 ◽  
Author(s):  
L. V. Kline ◽  
J. O. Hancock
Keyword(s):  
Circular Plate ◽  
Middle Surface ◽  
Elastic Plates ◽  
Buckling Loads ◽  
Simply Supported ◽  
Small Deflection ◽  
Edge Conditions ◽  
Deflection Theory ◽  
Plate On Elastic Foundation ◽  
Clamped Edge

The buckling loads are found for the simply supported and clamped-edge conditions for a circular plate on a springy foundation under the action of edge loading in the middle surface of the plate. The small deflection theory of bending of thin elastic plates has been used.

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 an Elastically Restrained Circular Plate Under Normal Loading Over a Sector

1958 ◽  
Vol 25 (1) ◽  
pp. 37-46
Author(s):  
W. A. Bassali ◽  
R. H. Dawoud
Keyword(s):  
Boundary Condition ◽  
Circular Plate ◽  
Complex Variable ◽  
Variable Method ◽  
General Boundary Condition ◽  
Single Treatment ◽  
Normal Loading ◽  
Simply Supported ◽  
Thin Circular Plate ◽  
Elastically Restrained

Abstract The complex variable method is used to find the deflection, bending and twisting moments, and shearing forces at any point of a thin circular plate normally loaded over a sector and supported at its edge under a general boundary condition including the usual clamped and simply supported boundaries. In this way separate treatments for these two cases are avoided and a single treatment is available.

The deflection of a thin circular plate with an eccentric circular hole

1976 ◽  
Vol 11 (2) ◽  
pp. 107-124 ◽  
Author(s):  
E Ollerton
Keyword(s):  
Circular Plate ◽  
Circular Hole ◽  
Theoretical Investigation ◽  
Outer Edge ◽  
Plate Surface ◽  
Concentrated Load ◽  
Simply Supported ◽  
Thin Circular Plate ◽  
Clamped Edge

A theoretical investigation of the small deflections of a thin circular plate is reported. The plate has a flat circular clamp at the outer edge and a similar clamp at the inner edge, which is placed eccentrically. These supports can be arranged to prescribe either a clamped edge or a simply supported edge, and all combinations of the two types are investigated. The plate can be subjected to a concentrated load at the centre of the inner clamp, moments about two perpendicular axes of the inner clamp, or pressure on the plate surface between the clamps. Deflections and slopes of the inner clamp have been determined, and in all cases the new values tend towards established values for the case of a central inner clamp, as the eccentricity of the inner clamp is reduced.

Bending of Elastically Restrained Rectangular Plates Under Uniform Pressure

1958 ◽  
Vol 62 (575) ◽  
pp. 834-836 ◽  
Author(s):  
C. Lakshmi Kantham
Keyword(s):  
Natural Frequencies ◽  
Unit Length ◽  
Rectangular Plates ◽  
Uniform Pressure ◽  
Simply Supported ◽  
Edge Conditions ◽  
Definition Of ◽  
Elastically Restrained ◽  
Clamped Edges ◽  
Clamped Edge

In the bending and vibration of plates it is found that the values of maximum deflection and natural frequencies, respectively, vary considerably from the simply-supported to clamped edge conditions. For an estimation of these characteristics in the intermediate range a generalised boundary condition may be assumed, of which the simply-supported and clamped edges become limiting cases. While Bassali considers the ratio of edge moment to the cross-wise moment as a constant, Newmark, Lurie and Klein and other investigators, in their analyses of various structures, consider that moment and slope at an end are proportional.Here the definition of elastic restraint as given by Timoshenko, α=βM, is followed, where α is the slope at any edge, M the corresponding edge moment per unit length while β is the elastic restraint factor. β→0 and β→∞ represent the two limiting cases of simply-supported and clamped edge conditions.

The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching

1961 ◽  
Vol 28 (2) ◽  
pp. 288-291 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Flexural Vibration ◽  
Frequency Equation ◽  
Numerical Results ◽  
Special Technique ◽  
Point Matching ◽  
Buckling Loads ◽  
Simply Supported ◽  
Flexural Vibrations

The bending by uniform lateral loading, buckling by two-dimensional hydrostatic pressure, and the flexural vibrations of simply supported polygonal plates are investigated. The method of meeting the boundary conditions at discrete points, together with the Marcus membrane analog [1], is found to be very advantageous. Numerical examples include the calculation of the deflections and moments, and buckling loads of triangular square, and hexagonal plates. A special technique is then given, whereby the boundary conditions are exactly satisfied along one edge, and an example of the buckling of an isosceles, right-angled triangle plate is analyzed. Finally, the frequency equation for the flexural vibrations of simply supported polygonal plates is shown to be the same as that for buckling under hydrostatic pressure, and numerical results can be written by analogy. All numerical results agree well with the exact solutions, where the latter are known.

A new classical theory of electrons

1951 ◽  
Vol 209 (1098) ◽  
pp. 291-296 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Quantum Theory ◽  
Classical Theory ◽  
Physical Significance ◽  
Gauge Transformations

In the theory of the electromagnetic field without charges, the potentials are not fixed by the field, but are subject to gauge transformations. The theory thus involves more dynamical variables than are physically needed. It is possible by destroying the gauge transformations to make the superfluous variables acquire a physical significance and describe electric charges. One gets in this way a simplified classical theory of electrons, which appears to be more suitable than the usual one as a basis for a passage to the quantum theory.

scholarly journals Green's function of the clamped punctured disk

1977 ◽  
Vol 20 (2) ◽  
pp. 175-181 ◽  
Author(s):  
Mitsuru Nakai ◽  
Leo Sario
Keyword(s):  
Green’S Function ◽  
Circular Plate ◽  
Green's Function ◽  
American Mathematical Society ◽  
Point Load ◽  
Thin Elastic Plate ◽  
Constant Sign ◽  
Boundary Circle ◽  
Simply Supported ◽  
Image Position

If a thin elastic circular plate B: ∣z∣ < 1 is clamped (simply supported, respectively) along its edge ∣z∣ = 1, its deflection at z ∈ B under a point load at ζ ∈ B, measured positively in the direction of the gravitational pull, is the biharmonic Green's function β(z, ζ) of the clamped plate (γ(z, ζ) of the simply supported plate, respectively). We ask: how do β(z, ζ) and γ(z, ζ) compare with the corresponding deflections β0(z, ζ) and γ0(z, ζ) of the punctured circular plate B0: 0 < ∣ z ∣ < 1 that is “clamped” or “simply supported”, respectively, also at the origin? We shall show that γ(z, ζ) is not affected by the puncturing, that is, γ(·, ζ) = γ0(·, ζ), whereas β(·, ζ) is:on B0 × B0. Moreover, while β((·, ζ) is of constant sign, β0(·, ζ) is not. This gives a simple counterexmple to the conjecture of Hadamard [6] that the deflection of a clampled thin elastic plate be always of constant sign:The biharmonic Gree's function of a clampled concentric circular annulus is not of constant sign if the radius of the inner boundary circle is sufficiently small.Earlier counterexamples to Hadamard's conjecture were given by Duffin [2], Garabedian [4], Loewner [7], and Szegö [9]. Interest in the problem was recently revived by the invited address of Duffin [3] before the Annual Meeting of the American Mathematical Society in 1974. The drawback of the counterexample we will present is that, whereas the classical examples are all simply connected, ours is not. In the simplicity of the proof, however, there is no comparison.

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.

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