Bifurcation of Circular Rings Under Normal Concentrated Loads

1973 ◽  
Vol 40 (1) ◽  
pp. 233-238 ◽  
Author(s):  
P. Seide ◽  
E. D. Albano
Keyword(s):  
Bifurcation Point ◽  
Circular Ring ◽  
Average Pressure ◽  
Uniform Pressure ◽  
Concentrated Loads ◽  
Finite Distortion ◽  
Circular Rings ◽  
Concentrated Forces

The deformation in bending of a circular ring loaded in its plane by concentrated forces is studied. The ring is assumed to be an elastica. The loads are of equal magnitudes and are equally spaced about the ring. Values of loading at which bifurcation of the symmetrical finite distortion shape occurs are determined for forces which remain normal to the ring. It is found that no bifurcation point exists for a ring under three loads. Buckling of a ring under two loads can occur only when the prebuckling configuration is an extremely distorted one. If the number of loads is five or greater, the critical average pressure does not differ greatly from the result for the ring under uniform pressure.

A Fourier Integral Solution for the Plane-Stress Problem of a Circular Ring With Concentrated Radial Loads

1951 ◽  
Vol 18 (2) ◽  
pp. 173-182
Author(s):  
Carl W. Nelson
Keyword(s):  
Plane Stress ◽  
Integral Method ◽  
Fourier Integral ◽  
Circular Ring ◽  
Outer Radius ◽  
Integral Solution ◽  
Concentrated Loads ◽  
Stress Problem ◽  
Plane Stress Problem ◽  
Circular Rings

Abstract A Fourier integral solution for the stresses in a straight bar of uniform cross section loaded by various combinations of loads applied normally to the edges of the bar was published by L. N. G. Filon in 1903 (3). Solutions for the stresses in circular rings, loaded on one or both boundaries by radial loads, have been limited to Fourier-series solutions for closed circular rings (1, 12, 13, 14, 15), except that solutions in closed form have been obtained for the limiting cases which occur either when the inner radius becomes very small or when the outer radius becomes very large. This paper presents a Fourier integral solution for the plane-stress problem of a curved bar bounded by two concentric circles and loaded by radial loads on the circular boundaries. It treats only the particular case of a curved bar in equilibrium under the action of two equal and opposite radial forces, one on each boundary. However, the method can be extended so as to deal with other combinations of loads. Sufficient numerical results are given to show that the Fourier integral method permits the calculation of numerical values of the stresses in the particular case considered. It is the purpose of this paper to show that the Fourier integral method can be used successfully in what is probably the simplest problem of concentrated loads acting on a curved bar and to furnish a background of material for use in less simple problems such as bending of curved bars due to concentrated loads.

Bifurcation of Rings Under Concentrated Centrally Directed Loads

1973 ◽  
Vol 40 (2) ◽  
pp. 553-558 ◽  
Author(s):  
E. D. Albano ◽  
P. Seide
Keyword(s):  
Average Pressure ◽  
Uniform Pressure ◽  
Circular Rings ◽  
The Stability

The stability of the large symmetrical deformations of circular rings under equal and equally spaced centrally directed loads is examined. It is found that for 5 or more loads the critical average pressure (the load divided by the distance between loads) does not differ significantly from the result for uniform pressure. The results for 2, 3, and 4 centrally directed or normal loads are identical.

Design of Laterally Loaded Plating—Concentrated Loads

1983 ◽  
Vol 27 (04) ◽  
pp. 252-264
Author(s):  
Owen Hughes
Keyword(s):  
Experimental Data ◽  
Load Parameter ◽  
Uniform Pressure ◽  
Concentrated Loads ◽  
Rigid Plastic ◽  
Concentrated Load ◽  
Load Effect ◽  
Permanent Set ◽  
Multiple Location ◽  
Single Location

In the design of plating subject to lateral loading, the principal load effect to be considered is the amount of permanent set, that is, the maximum permanent deflection in the center of each panel of plating bounded by the stiffeners and the crossbeams. The present paper is complementary to a previous paper [1]2 which dealt with uniform pressure loads. It first shows that for design purposes there are two types of concentrated loads, depending on the number of different locations in which they can occur; single location or multiple location. The hypothesis is then made that for multiple-location loads the eventual and stationary pattern of plasticity which is developed in the plating is very similar to that for uniform pressure loads, and hence the value of permanent set may be obtained by using the same formula as for uniform pressure loads, with a load parameter Q that is some multiple r of the load parameter for the concentrated load: 0 = rQP. The value of r is a function of the degree of concentration of the load and is almost independent of plate slenderness and aspect ratio. The general mathematical character of this function is established from first principles and from an analysis of the permanent set caused by a multiple-location point load acting on a long plate. The results of this theoretical analysis provide good support for the hypothesis, as do also the relatively limited experimental data which are available. The theory and the experimental data are combined to obtain a simple mathematical expression for r. A more precise expression can be obtained after further experiments have been performed with more highly concentrated loads. Single-location loads produce a different pattern of plasticity and require a different approach. A suitable design formula is developed herein by performing regression analysis on the data from a set of experiments performed with such loads. Both methods presented herein, one for multiple-location loads and the other for single-location loads, are valid for small and moderate values of permanent set and can be used for all static and quasistatic loads. Dynamic loads and applications involving large amounts of permanent set require formulas based on rigid-plastic theory. Such formulas are available for uniform pressure loads and were quoted in reference [1]. A formula for single-location loads has recently been derived by Kling [4] and is quoted herein.

Inplane Vibrations of Circular Rings With a Radially Variable Thickness

1983 ◽  
Vol 105 (1) ◽  
pp. 137-143 ◽  
Author(s):  
H. Lecoanet ◽  
J. Piranda
Keyword(s):  
Experimental Data ◽  
Galerkin Method ◽  
Variable Thickness ◽  
Circular Ring ◽  
Constant Thickness ◽  
Circular Rings ◽  
The Galerkin Method ◽  
Good Agreement

This paper gives some results on inplane vibrations of circular ring with a radially variable thickness. The problem is solved with the Galerkin method [1] making use of the eigenfunctions of a constant thickness ring. Good agreement is obtained between the approximate results and those of the exact calculus or experimental data.

In-Plane Flexural Vibrations of Circular Rings

1969 ◽  
Vol 36 (3) ◽  
pp. 620-625 ◽  
Author(s):  
S. S. Rao ◽  
V. Sundararajan
Keyword(s):  
Natural Frequencies ◽  
Classical Equation ◽  
Circular Ring ◽  
Classical Formula ◽  
Free Ring ◽  
Circular Arcs ◽  
End Conditions ◽  
Circular Rings ◽  
Experimental Values ◽  
Fixed End

An equation of motion governing the free, in-plane vibrations of a circular ring is developed to include the effects of shear deformation and rotatory inertia. This equation is solved to find the natural frequencies of vibration of free rings and stiffened rings and the results compared with those given by a classical formula. The frequencies for a free ring are found to compare well with the experimental values of Kuhl [5]. Natural frequencies of circular arcs are calculated from the classical equation with hinged and fixed end conditions and the results compared with the approximate values given by Den Hartog [8, 9].

Rectangular Plates on Unilateral Edge Supports: Part 2—Implementation; Concentrated and Uniform Loading

1986 ◽  
Vol 53 (1) ◽  
pp. 151-156 ◽  
Author(s):  
J. P. Dempsey ◽  
Hui Li
Keyword(s):  
Unilateral Contact ◽  
Rectangular Plates ◽  
Uniform Pressure ◽  
Computer Implementation ◽  
Concentrated Loads ◽  
Uniform Loading ◽  
Loss Of Contact ◽  
Implementation Aspects ◽  
Laterally Loaded ◽  
Support Reactions

Rectangular plates in unilateral contact with sagged and unsagged supports laterally loaded by centrally concentrated loads and uniform pressure are examined. The loss of contact and the redistribution of deflections, moments, and support reactions are presented. Computer implementation aspects are discussed.

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