Research Papers

Collapse of Heavy Cantilevered Elastica With Frictional Internal Support

[+] Author and Article Information
R. H. Plaut1

Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061rplaut@vt.edu

D. A. Dillard, A. D. Borum

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061


Corresponding author.

J. Appl. Mech 78(4), 041011 (Apr 13, 2011) (5 pages) doi:10.1115/1.4003755 History: Received September 16, 2010; Revised March 05, 2011; Posted March 07, 2011; Published April 13, 2011; Online April 13, 2011

Equilibrium states are investigated for a heavy flexible strip that is fixed at one end and rests on an internal frictional support. Large vertical deflections are admitted. In the analytical portion of the study, the strip is modeled as an inextensible elastica. Experiments are conducted on strips of transparency film. For a sufficiently large gap between the fixed end and the internal support, the strip slips through the gap and collapses downward. For moderate gaps, two continuous ranges of equilibrium shapes exist, one with relatively small deflections within the gap and one with large deflections within the gap. The results for a given gap size depend on the length, weight per unit length, and bending stiffness of the strip, and the coefficient of friction between the strip and the internal support. The case of a frictionless support is also analyzed. The experimental results agree well with those from the analysis for cases with small deflections within the gap, but exhibit a larger range of stable large-deflection equilibrium states before collapse occurs.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

Schematic of strip with length A+B=L

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Figure 2

Forces acting on strip: weight W per unit length, normal force N, frictional force F, and reactions at fixed end

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Figure 3

Rotation θC at support as function of gap d for w=50, μ=−0.5, −0.3, −0.1, 0, 0.1, 0.3, and 0.5

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Figure 4

Equilibrium shapes for w=50, d=0.6: (a) μ=−0.5, θC=−0.295; (b) μ=0.5, θC=−0.243; (c) μ=−0.5, θC=0.567; and (d) μ=0.5, θC=1.369

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Figure 5

Rotation θC as function of gap d for w=20, μ=−0.5, −0.4, −0.35, −0.3, 0, and 0.5

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Figure 6

Magnification of part of Fig. 5 plus curves for μ=−0.2 and 0.2 and curve for slip-off (a=1)

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Figure 7

Maximum value dmax as function of w for μ=0, 0.1, 0.3, 0.5, and 0.9; points left of × are associated with slip-off; equilibrium shapes do not exist if d>dmax

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Figure 8

Load parameter w as function of rotation θC for d=0.75, μ=0, 0.1, 0.3, and 0.5; curves end when slip-off occurs

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Figure 9

Arc length a in gap as function of gap d for w=51.4, μ=−0.207 and 0.207; dots represent experimental results




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