Large deformations of a heavy cantilever
Author:
Chang Yi Wang
Journal:
Quart. Appl. Math. 39 (1981), 261-273
MSC:
Primary 73H99
DOI:
https://doi.org/10.1090/qam/625473
MathSciNet review:
625473
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Abstract: A cantilever of uniform cross-section and density is held at an angle $\alpha$ at one end. The shape of the cantilever depends heavily on $\alpha$ and a nondimensional parameter $K$ which represents the relative importance of density and length so that of flexural rigidity. Perturbations on the elastica equations for small and large $K$ show good agreement with exact numerical integration. It is found that whenever $K$ reaches a critical value, bifurcations of the solutions occur. This nonuniqueness can be observed by the flipping phenomena as $\alpha$ is increased.
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R. Schmidt and D. A. DaDeppo, Approximate analysis of large deflections of beams, Zeit. angew. Math. Mech. 51, 233–234 (1971)
L. Euler, De curvis elasticis (1744)
A. G. Greenhill, Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportions can grow, Proc. Camb. Phil. Soc. 4, 65–73 (1881)
W. G. Bickley, The heavy elastica, Phil. Mag. Ser. 7 17, 603–622 (1934)
H. Lippmann, O. Mahrenholtz and W. Johnson, Thin heavy elastic strips at large deflexions, Int. J. Mech. Sci. 2, 294–310 (1961)
F. V. Rhode, Large deflections of a cantilever beam with uniformly distributed load, Quart. Appl. Math. 11, 337–338 (1953)
R. Frisch-Fay, The analysis of a vertical and a horizontal cantilever under a uniformly distributed load, J. Franklin Inst. 271, 192–199 (1961)
R. Schmidt and D. A. DaDeppo, Approximate analysis of large deflections of beams, Zeit. angew. Math. Mech. 51, 233–234 (1971)
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Article copyright:
© Copyright 1981
American Mathematical Society