The Effect of the Axial and Shear Stiffnesses On Elastic Rod’s Stability

This article is about the nonlinear problems of the theory of elastic Cosserat – Timoshenko’s rods in the material (Lagrangian) description. The variational definition for the problem as finding the stationary point of the Lagrangian functional and differential formulation of static problems were given. The exact stability functional and stability equations of the plane problem for physically linear elastic rods taking into account the axial, shear and bending stiffnesses were received. The exact value of the critical load was obtained taking into account the axial, shear and bending deformations in the problem of the stability of a rod compressed by an axial force. In the present paper the stability of classical simplified rod’s models such as the Timoshenko beam and the Euler–Bernoulli beam was investigated. Also, the stability of third simplified rod’s model, based on beam’s axial and bending stiffnesses, was explored. The stability functionals, the stability equations and critical loads formulations for this three types of simplified models were derived as a particular case of the general theory. There were made the comparisons of described solutions which regards all the rod’s stiffnesses and solutions, based on simplified models. The effect of the axial and shear stiffnesses on rod’s stability was analyzed.