Finite element static and stability analysis of gradient elastic beam structures

In this paper, the static and stability stiffness matrices of a gradient elastic flexural Bernoulli-Euler beam finite element are analytically constructed with the aid of the basic and governing equations of equilibrium for that element. The flexural element has one node at every end with three degrees of freedom per node, i.e., the displacement, the slope and the curvature. The stability stiffness matrix incorporates the effect of axial compressive force on bending. Use of these stiffness matrices for a plane system of beams enables one through a finite element analysis to determine its response to static loading and its buckling load. Because the exact solution of the governing equation of the problem is used as the displacement function, the resulting stiffness matrices and the obtained structural responses are also exact. Examples are presented to illustrate the method and demonstrate its merits.