A force-based beam element model based on the modified higher-order shear deformation theory for accurate analysis of FG beams

In this paper, a force-based beam finite element model based on a modified higher-order shear deformation theory is proposed for the accurate analysis of functionally graded beams. In the modified higher-order shear deformation theory, the distribution of transverse shear stress across the beam's thickness is obtained from the differential equilibrium equation on stress, and a modified shear stiffness is derived to take the effect of transverse shear stress distribution into consideration. In the proposed beam element model, unlike traditional beam finite elements that regard generalized displacements as unknown fields, the internal forces are considered as the unknown fields, and they are predefined by using the closed-form solutions of the differential equilibrium equations of higher-order shear beam. Then, the generalized displacements are expressed by the internal forces with the introduction of geometric relations and constitutive equations, and the equation system of the beam element is constructed based on the equilibrium conditions at the boundaries and the compatibility condition within the element. Numerical examples underscore the accuracy and efficacy of the proposed higher-order beam element model in the static analysis of functionally graded sandwich beams, particularly in terms of true transverse shear stress distribution.