Buckling of centrosymmetric anisotropic beam structures within strain gradient elasticity

Buckling of centrosymmetric anisotropic beams is studied within strain gradient theory. First, the three dimensional anisotropic gradient elasticity theory is outlined. Then the dimension of the three dimensional theory is reduced, resulting in Timoshenko beam as well as Euler–Bernoulli beam theories. The governing differential equations together with the consistent (classical and non-classical) boundary conditions are derived for centrosymmetric anisotropic beams through a variational approach. By considering von Kármán nonlinear strains, the geometric nonlinearity is taken into account. The obtained nonlinear formulation can be used to study the postbuckling configuration. The analysis of size effect on anisotropic beam structures is missing in the literature so far, while the present model allows one to characterize the size effect on the buckling of the centrosymmetric anisotropic micro- and nano-scale beam structures such as micropillars. As a specific case, the governing buckling equation is obtained for the more practical case of orthotropic beams. Finally, the buckling loads for orthotropic simply supported Timoshenko and Euler–Bernoulli beams as well as a clamped Euler–Bernoulli beam are obtained analytically and the effect of the internal length scale parameters on the buckling load is depicted.