Instability of Flexible Strip Hanging Over Edge of Flat Frictional Surface

Abstract

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.