Buckling of micromorphic Timoshenko columns

This paper presents exact solutions for the buckling of a micromorphic Timoshenko column under general boundary conditions. The shear effect is introduced through Engesser's shear column theory.This problem can also be formulated as a nonlocal strain gradient Timoshenko column that uses nonlocal kernels for both the strain (curvature and shear strain for the Timoshenko beam) and its derivative. It is shown that both models yield the same governing equations but differ in potential energy definiteness. The micromorphic model predicts hardening effects, while the nonlocal strain gradient model captures both softening and hardening. The buckling problem is formulated as an eighth-order differential eigenvalue problem, associated with eight variationally-consistent boundary conditions. Analytical buckling solutions are obtained for various boundary conditions using Cardano's method. Closed-form solutions of the buckling load may also be obtained for some specific boundary conditions, including the higher-order nonlocal boundary conditions. The role of variationally-consistent higher-order boundary conditions is specifically addressed. The softening or stiffening contributions of the small length-scale terms are discussed for all the considered boundary conditions, including simply-supported, clamped-free, clamped–clamped, and clamped-simply-supported cases. The role of shear contribution is also analyzed for this micromorphic Engesser–Timoshenko column.