Lyapunov–Schmidt bifurcation analysis of a supported compressible elastic beam

The archetypal instability of a structure is associated with the eponymous Euler beam, modelled as an inextensible curve which exhibits a supercritical bifurcation at a critical compressive load. In contrast, a soft compressible beam is capable of a subcritical instability, a problem that is far less studied, even though it is increasingly relevant in the context of soft materials and structures. Here, we study the stability of a soft extensible elastic beam on an elastic foundation under the action of a compressive axial force, using the Lyapunov–Schmidt reduction method which we corroborate with numerical calculations. Our calculated bifurcation diagram differs from those associated with the classical Euler–Bernoulli beam, and shows two critical loads, pcr±(n), for each buckling mode n. The beam undergoes a supercritical pitchfork bifurcation at pcr+(n) for all n and slenderness. Due to the elastic foundation, the lower order modes at pcr−(n) exhibit subcritical pitchfork bifurcations, and perhaps surprisingly, the first supercritical pitchfork bifurcation point occurs at a higher critical load. The presence of the foundation makes it harder to buckle the elastic beam. Overall, our study has uncovered the subtle interplay between elasticity and foundation effects in a minimal setting for an extensible beam with experimentally testable predictions.