Singular critical points in the general theory of elastic stability

Problems which deal with the stability of bodies in equilibrium under stress are so distinct from the ordinary applications of the theory of elasticity that it is legitimate to regard them as forming a special branch of the subject. In every other case we are concerned with the integration of certain differential equations, fundamentally the same for all problems, and the satisfaction of certain boundary conditions; and by a theorem due to Kiechiioff we are entitled to assume that any solution which we may discover is unique. In these problems we are confronted with the possibility of two or more configurations of equilibrium , and we have to determine the conditions which must be satisfied in order that the equilibrium of any given configuration may be stable. The development of both branches has proceeded upon similar lines. That is to say, the earliest discussions were concerned with the solution of isolated examples rather than with the formulation of general ideas. In the case of elastic stability, a comprehensive theory was not propounded until the problem of the straight strut had been investigated by Euler, that of the circular ring under radial pressure by M. Lévy and G. H. Halphen, and A. G. Greenhill had discussed the stability of a straight rod in equilibrium under its own weight, under twisting couples, and when rotating.

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Some static and dynamic implications of of the general theory of elastic stability

Archive for Rational Mechanics and Analysis ◽

10.1007/bf00277007 ◽

1965 ◽

Vol 19 (3) ◽

pp. 167-188 ◽

Cited By ~ 44

Author(s):

Millard F. Beatty

Keyword(s):

General Theory ◽

Elastic Stability

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Energy theorems and critical load approximations in the general theory of elastic stability

Quarterly of Applied Mathematics ◽

10.1090/qam/44337 ◽

1952 ◽

Vol 9 (4) ◽

pp. 371-380 ◽

Cited By ~ 1

Author(s):

J. N. Goodier ◽

H. J. Plass

Keyword(s):

General Theory ◽

Critical Load ◽

Elastic Stability

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Imperfection-sensitivity of semi-symmetric branching

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1977.0163 ◽

1977 ◽

Vol 357 (1689) ◽

pp. 193-211 ◽

Cited By ~ 27

Keyword(s):

General Theory ◽

Potential Function ◽

Catastrophe Theory ◽

Bifurcation Theory ◽

Elastic Stability ◽

Structural Systems ◽

Stiffened Plate ◽

Imperfection Sensitivity ◽

Branching Points ◽

Symmetric Points

The general theory of elastic stability is extended to include the imperfection-sensitivity of twofold compound branching points with symmetry of the potential function in one of the critical modes (semi-symmetric points of bifurcation). Three very different forms of imperfection-sensitivity can result, so a subclassification into monoclinal, anticlinal and homeoclinal semi-symmetric branching is introduced. Relating this bifurcation theory to René Thom’s catastrophe theory, it is found that the anticlinal point of bifurcation generates an elliptic umbilic catastrophe, while the monoclinal and homeoclinal points of bifurcation lead to differing forms of the hyperbolic umbilic catastrophe. Practical structural systems which can exhibit this form of branching include an optimum stiffened plate with free edges loaded longitudinally, and an analysis of this problem is presented leading to a complete description of the imperfection-sensitivity. The paper concludes with some general remarks concerning the nature of the optimization process in design as a generator of symmetries, instabilities and possible compound bifurcations.

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