Cross-checking asymptotics and numerics in the hardening/softening behaviour of Timoshenko beams with axial end spring and variable slenderness

Two approximate solutions for the nonlinear free oscillations of a planar Timoshenko beam are compared to each other. The beam has an axial spring that permits to consider different boundary conditions, from axially free (which has a softening nonlinear behaviour) to perfectly axially restrained (which has a hardening nonlinear behaviour). The first approximation is analytical and is obtained by the asymptotic development method, while the second is numerical and is obtained by the finite element method. The comparison is made in terms of backbone curves describing the dependence of the (nonlinear) frequency on the oscillation amplitude. Very good agreement is found, for both slender and thick beams, and for varying stiffness of the end spring. This is a cross-check verification of the reliability of both approximate solutions