Collapse of Heavy Cantilevered Elastica With Frictional Internal Support

Abstract

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 elástica. Experiments are conducted on strips of transparency fllm. 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 stiffiiess 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.

Slip Morphology of Elastic Strips on Frictional Rigid Substrates

“…Equation (1) suggests that the instantaneous shape of an elastica determines its own boundary condition. This feature is a particular characteristic of our system and differs from the behavior in a standard setup employed in previous elastica problems [41][42][43]. First, we numerically investigate the planar deformations of a strip in the absence of gravity by changing the values of the coefficient of static friction, µ, and the height of the strip, y 0 , and classify the deformations into three distinct states explained below.…”

“…A fundamental process common to a variety of the problems listed above is the postbuckling behavior of an elastic strip [38][39][40][41][42][43], that is subject to a vertical compressive stress on a rigid substrate [Fig. 1(a)].…”

“…Both tangential and vertical external forces must be applied at the clamped end, i.e., F (0) = (F x , F y ), where F x , F y , are yet to be determined. Solving the force-balance equation with this condition and substituting it into the momentum balance equation leads us to the shape equation for θ(s) [42,43], which may be written as EIθ ′′ (s) = −F x cos θ(s) − F y sin θ(s). Although the equation for θ(s) can be analytically solved using elliptic integrals [49][50][51], the result for small strain, ǫ y ≪ 1, is useful for our aim here.…”

Variable-Length Link-Spring Model of Wire-Bonding Looping Process

“…Some researchers have studied the arc length of a beam, either with one end subjected to a moment at a hinge and able to slide freely over a support [22]; with one support elevated above the other [23]; or as a strip that has a specified length and deformation due to its own weight [24]. A more closely related problem is the slender beam that is clamped at one side but can slide along the axial direction at the opposite side [25], [26].…”

Snap-buckling in asymmetrically constrained elastic strips

“…A thin geometric motif such as a plate or strip is a building block for more complex solid structures in nature [11][12][13], industry [14][15][16][17][18][19], and everyday life [20][21][22][23][24], and is currently a target of active research in var- ious scientific fields [25][26][27][28][29][30][31][32][33][34][35]. In classical Euler buckling, one of two possible bending directions is selected by spontaneous symmetry breaking.…”