Finite strain theories of extensible and shear-flexible planar beams based on three different hypotheses of member forces

Abstract

This study intends to answer how nonlinear beam theories could be rigorously derived when based on three hypotheses: Haringx/Reissner, Engesser, and Ziegler. For this purpose, Reissner formulation is first summarized for an elastic beam segment undergoing large displacements and strains, and the nonlinear equations obtained are compactly converted into a non-dimensional form using two parameters of shear-flexibility and extensibility. After that, conjugate strain measures corresponding to axial and shear forces based on Engesser and Ziegler's assumptions are consistently derived, and the resulting governing equations are presented in a dimensionless form. Finally, nonlinear problems of extensible and shearable cantilever beams are solved using the 4th order Runge-Kutta method combined with a shooting method. Shear-deformation and extensibility effects are addressed through two examples showing large deflections and post-buckling behaviors of cantilever beams.