Discussion of “Elastic Instability of Unbraced Space Frames”

1. When a straight cylindrical rod is bent into a circle by couples applied at its ends, the resulting state of stress is given, with sufficient accuracy for practical purposes, by the well-known theory of St. Venant. In that theory qunatities of the second and higher orders in therms of strain are neglected, and the resulting solution asserts that the stress is purely longitudinal, so that the rod may be thought of as an assembly of cylindrical fibres, each of which behaves independently of its neighbours. It is evident that this description cannot be exact; for a fibre bent into a circle cannot be kept in tension unless radial forces operate to maintain equilibrium, and in the case considered such forces can come only from actions between adjacent fibres. The apparent paradox is explained by the consideration that those action are of the second order in terms of the curvature, and accordingly are neglected in St. Venant's theory. In connection with a certain problem of elastic instability it was thought desirable to attempt a more accurate description for a particular case, namely, a rod of deep and thin rectangular section. It was found that the equations of equilibrium can be integrated independently of any simplifying assumption, and the stress-distribution determined for curvature of any magnitude. The results have no great practical importance, sice they show that St. Venant's theory gives a close approximation to the facts within that range of strains which actual materials can sustain elastically; but they have some theoretical interest, and accordingly are presented in this paper.

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Closure to “Dynamic Analysis of Inelastic Space Frames”

Journal of the Engineering Mechanics Division ◽

10.1061/jmcea3.0001451 ◽

1971 ◽

Vol 97 (4) ◽

pp. 1296-1296

Author(s):

Robert K. Wen ◽

Fereydoon Farhoomand

Keyword(s):

Dynamic Analysis ◽

Space Frames

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Buckling of Multiple-Bay Ring-Reinforced Cylindrical Shells Subject to Hydrostatic Pressure

Journal of Applied Mechanics ◽

10.1115/1.4010749 ◽

1953 ◽

Vol 20 (4) ◽

pp. 469-474

Author(s):

W. A. Nash

Keyword(s):

Cylindrical Shell ◽

Hydrostatic Pressure ◽

Strain Energy ◽

Elastic Strain Energy ◽

Middle Surface ◽

Elastic Buckling ◽

Elastic Instability ◽

Total Potential ◽

External Forces ◽

Work Done

Abstract An analytical solution is presented for the problem of the elastic instability of a multiple-bay ring-reinforced cylindrical shell subject to hydrostatic pressure applied in both the radial and axial directions. The method used is that of minimization of the total potential. Expressions for the elastic strain energy in the shell and also in the rings are written in terms of displacement components of a point in the middle surface of the shell. Expressions for the work done by the external forces acting on the cylinder likewise are written in terms of these displacement components. A displacement configuration for the buckled shell is introduced which is in agreement with experimental evidence, in contrast to the arbitrary patterns assumed by previous investigators. The total potential is expressed in terms of these displacement components and is then minimized. As a result of this minimization a set of linear homogeneous equations is obtained. In order that a nontrivial solution to this system of equations exists, it is necessary that the determinant of the coefficients vanish. This condition determines the critical pressure at which elastic buckling of the cylindrical shell will occur.

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