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Research Papers

Instability of Flexible Strip Hanging Over Edge of Flat Frictional Surface

[+] 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

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

1

Corresponding author.

J. Appl. Mech 78(3), 031011 (Feb 15, 2011) (6 pages) doi:10.1115/1.4003359 History: Received November 11, 2009; Revised October 06, 2010; Posted January 05, 2011; Published February 15, 2011; Online February 15, 2011

The conditions for an overhanging flexible strip to slide off a flat surface are investigated. This problem may be applicable to pieces of paper, fabric, leather, and other flexible materials, including plastic and metallic strips used herein for experimental comparisons. The critical overhang length depends on (a) the length, weight per unit length, and bending stiffness of the strip, (b) the coefficients of friction (CoFs) between the strip and both the surface and its edge, and (c) the inclination of the surface. The strip is modeled as an inextensible elastica. A shooting method is applied to solve the nonlinear equations that are based on equilibrium, geometry, and Coulomb friction. Three types of equilibrium shape are obtained. In the most common type, one end of the strip overhangs the edge and the other end contains a segment that is in contact with the surface. In another type, contact only occurs at the nonoverhanging end and at the edge. The third type involves the strip balancing on the edge of the surface. The ratio of the critical overhang length to the total strip length is plotted as a function of the surface CoF, edge CoF, and weight parameter for a horizontal surface. In most cases, this ratio increases as the CoFs and the strip’s bending stiffness increase, and decreases as the strip’s weight per unit length increases. The rotation of the strip at the edge tends to increase as the strip’s weight per unit length, the strip’s length, and the surface CoF increase, and to decrease as the strip’s bending stiffness increases. Inclined surfaces are also considered, and the critical overhang length decreases as the surface slopes more downward toward the edge. The theoretical results are compared with experimental data, and the agreement is good.

Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of strip on flat surface: contact length C, liftoff at K, surface edge at D, overhang length B, and surface inclination β

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Figure 2

Coordinate systems: origin at edge, arc lengths S1 and S2, rotations θ1 and θ2 relative to surface with inclination β, and coordinates (X1,Y1) and (X2,Y2)

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Figure 3

Forces acting on strip: weight W per unit length, normal force RK at liftoff, normal force RD at edge, frictional force μeRD at edge, and frictional force μs(WC cos β+RK) along contact length C; normal forces along contact length C are not shown

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Figure 4

Shape I for horizontal surface (β=0), nondimensional weight per unit length w=100, CoFs μs=μe=0.4, nondimensional overhang length bcr=0.449, and nondimensional contact length c=0.067

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Figure 5

Shape II for β=0, w=100, μs=μe=0.6, and bcr=0.555, point contact at left end

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Figure 6

Shape III for β=0, w=100, μe=0.8, and bcr=0.632, contact only at edge

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Figure 7

Element of strip above surface

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Figure 8

Element of strip on overhanging segment

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Figure 9

Critical overhang length as function of CoF μ=μs=μe for β=0 and w=50, 100, and 1000, transition between shapes I and II at dot (shape I left of dot) and transition between shapes II and III at kink

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Figure 10

Critical overhang length as function of edge CoF for β=0, μs=0.4, and w=50, 100, and 1000; dots denote the transition between shapes I and II (shape I left of dot)

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Figure 11

Critical overhang length as function of surface CoF for β=0, μe=0, and w=50, 100, and 1000, shape I

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Figure 12

Critical overhang length as function of w for β=0, μe=0, and μs=0.2, 0.4, and 0.6, shape II for small values of w, otherwise, shape I

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Figure 13

Rotation θD at edge as function of w for β=0, μe=0, and μs=0.2, 0.4, and 0.6, shape II for small values of w, otherwise, shape I

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Figure 14

Critical overhang length as function of surface inclination for μs=0.4, μe=0, and w=50, 100, and 1000, shape I

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Figure 15

Uplift overhang length as function of w for β=−0.2, 0, and 0.2; bu is an approximate upper bound for bcr

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Figure 16

Critical overhang length as function of w for β=0 and μ=μe=μs=0.404 and 0.466, including experimental data points ● for feeler gauge (μ=0.466) and ◆ for transparency film (μ=0.404)

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