Double Curvature Bending of Variable-Arc-Length Elasticas

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This paper deals with the double curvature bending of variable arc-length elasticas under two applied moments at fixed support locations. One end of the elastica is held while the other end portion of the elastica may slide freely on a frictionless support at a prescribed distance from the held end. Thus, the variable deformed length of the elastica between the end support and the frictionless support depends on the relative magnitude of the applied moments. To solve this difficult and highly nonlinear problem, two approaches have been used. In the first approach, the elliptic integrals are formulated based on the governing nonlinear equation of the problem. The pertinent equations obtained from applying the boundary conditions are then solved iteratively for solution. In the second approach, the shooting-optimization method is employed in which the set of governing differential equations is numerically integrated using the Runge-Kutta algorithm and the error norm of the terminal boundary conditions is minimized using a direct optimization technique. Both methods furnish almost the same stable and unstable equilibrium solutions. An interesting feature of this kind of bending problem is that the elastica can form a single loop or snap-back bending for some cases of the unstable equilibrium configuration.