Secondary bifurcation of a compressible rod with spring-supports

• Postbuckling behavior of a compressible rod near a double eigenvalue is analyzed.
• The existence and type of primary bifurcations depend on compressibility.
• Secondary bifurcations depend on spring stiffness and compressibility.
• There are three groups of bifurcation diagrams depending on the slenderness ratio.
• The asymptotic expansions of postbuckling states are constructed.

We consider the problem of determining the stability boundary and postbuckling behavior of an elastic rod with spring supports at clamped ends. The rod is loaded by a compressive force and the constitutive equations of the rod take into account the compressibility of the rod axis. Using the Liapunov–Schmidt procedure local bifurcation analysis is performed. The spring stiffness is chosen to be in the neighborhood of the critical one corresponding to a double eigenvalue. Due to the splitting of eigenvalues secondary bifurcations occur. The results show that the type of primary bifurcation depends on the slenderness ratio and that there are three groups of bifurcation diagrams. Also, asymptotic expansions of postbuckling states are constructed.