Deformation and vibration of a spatial clamped elastica with noncircular cross section

• The spatial buckling and post-buckling behavior of a thin rod with noncircular cross-section are studied.
• Static analysis is employed to determine the bifurcation points of the equilibrium paths.
• The locus of spatial deformation emerges from a bifurcation point on the locus of planar non-self-contact deformation.
• The stability of the equilibrium configurations is determined by means of vibration analysis.

In this paper we study the deformation and vibration of a clamped–clamped spatial rod with noncircular cross section. Focus is placed on the effect of the bending stiffness ratio in the two principal directions of the cross-section on the rod deformation and stability. The equations of motion are formulated within the framework of director theory and solved by shooting method. Static analysis shows that the clamped rod possesses both planar and spatial deformations. The planar deformation may be of self contact or not. The locus of the planar deformation is independent of the bending stiffness ratio. The locus of spatial deformation emerges from a bifurcation point on the locus of planar non-self-contact deformation. The bifurcation point depends on the stiffness ratio. Vibration method is adopted to determine the stability of the deformations. Both force- and displacement-driven procedures are considered. For the planar non-self-contact deformation, the natural frequency of the out-of-plane mode with one nodal point decreases to zero when the end shortening approaches the bifurcation point. The experimental results on a rod with circular cross section agree qualitatively with the theoretical predictions. In particular, the jump phenomena in force-driven experiment during loading and unloading processes are confirmed.