Variational Principles of Extensible Shearable Elastica—Engesser's Approach

An extensible shearable elastica is a rigorous mathematical model of the Timoshenko beam, of which the cross-sections remain planar, but not necessarily normal to the beam axis after deformation. First, the principle of virtual work for the extensible shearable elastica expressed in terms of the extensional and shear strains of the axis and the rotation of the cross-section in Engesser's approach is derived from the virtual work in three-dimensional solid mechanics. Then, utilizing linear constitutive equations between generalized stresses and strains, we derive the principle of stationary potential energy, also expressed in terms of the extensional and shear strains and rotation. Finally, from the criterion of Trefftz on the second variation of the potential energy, we obtain the buckling equations for the extensible and shearable elastica, which show the effect of the axial and shear stiffness on the buckling load for a cantilever elastica subjected to compressive end load.