Static Bending Analysis of Small-Sized Structures Using Various Non-Classical Theories

Abstract

In this chapter, the size-dependent static bending response of microbeams is investigated based on modified couple stress and modified strain gradient elasticity theories. In contrast to the classical beam model, the present non-classical beam models contain length-scale parameters and can capture the size effect. If the length-scale parameters are equal to zero, these models transform to the classical one. The equilibrium equations with classical and non-classical corresponding boundary conditions for microbeams are derived by implementing the principle of minimum total potential energy based on the modified couple stress and modified strain gradient theories in conjunction with the Bernoulli–Euler beam theory. The resulting higher-order equation is analytically solved for simply supported (S–S), clamped–free (C–F), clamped–hinged (C–H), and clamped–clamped (C–C) boundary conditions. Finally, some illustrative examples are given to investigate the effects of the length-scale parameters, size dependency, and boundary conditions on the displacements of the small-sized beams. It is observed that the size effect is more prominent for the larger length-scale parameters. In addition, it is found that the divergence between displacements evaluated by the present and classical models becomes more significant for smaller beams.