The Compressible Elastica on an Elastic Foundation

The nonlinear two-point boundary-value problem describing the compressible elastica on an elastic foundation is formulated exactly within the context of the technical theory of bending as a set of eight first-order differential equations plus appropriate initial-point conditions and terminal-point conditions. The problem is then solved by a shooting method that determines two missing quantities. Graphs of load versus displacements and load versus the missing quantities are presented for various combinations of the system parameters. These results show that the presence of the elastic foundation enables the member to sustain unsymmetric (as opposed to antisymmetric) shapes in its postbuckled state, and that bifurcations from the straight configuration to symmetric buckled modes and bifurcations from symmetric buckled modes to unsymmetric ones depend on two system parameters—a compressibility measure and the foundation modulus. For a given compressibility and foundation stiffness, equilibrium paths are plotted globally, enabling unsymmetric paths to be extended from one bifurcation point to another, with the result that the complete postbuckling process can be traced. Finally, a discussion of path shapes as a function of foundation stiffness is given.