Asymmetric snap-buckling of a column restrained by a stiff wire

Abstract

A relatively simple example of asymmetric snap-through buckling in a continuous structure is the nonlinear problem of a cantilevered column restrained at its tip by a stiff wire, which is inclined at an acute angle to the column centerline, and loaded at its tip by a force perpendicular to the centerline. A parameter called μ, which is the nondimensional ratio of the flexural rigidity of the column to the combined extensional stiffness of the wire and the column centerline, determines the essential features of the buckling. If μ is zero, or is small compared to unity, the bending of the column is small enough to justify the use of linear bending theory for the column. Hence, even though the constraint is nonlinear, the solution to this problem is obtained in closed form. The critical point for the structure is found to be an asymmetric branching point for μ=0, while for μ positive, the critical point is a snap-through type. The effect of μ is similar to that induced by initial imperfections in more complex structures. For very small μ, the critical load is markedly decreased from the value for μ=0. Moreover, the graph of the load vs. tip deflection has the appearance of having an acute discontinuity in slope at the critical point for μ very small, although it is actually found that the graph has a horizontal tangent there.