Bi-directional higher-order shear deformable mixed finite element formulation including couple effects for stresses of functionally graded curved 3d beams

This paper presents a formulation based on higher-order shear deformation functions in both normal and binormal directions on the cross section to include the influence of in-plane shear stresses in the case of curved beams subjected to out-of-plane forces. Besides, another distinctive feature is the introduction of accuracy elements at the stress concentration zones to widen the range of St. Venant principle. Within this scope, a two-noded curved mixed finite element formulation with twenty-eight degree of freedoms (DOFs) in total is derived. The curved axial geometry is defined over the exact functions of the gradient of arch length and curvature. The volume fraction of functionally graded (FG) material constituents is based on the power-law distribution and the rule of mixture. The functional including the coupling effects is derived via the Hellinger Reissner formulation. The normal/shear stresses are determined over the curvatures on the section and stress equilibrium condition. Quite satisfactory converged results with respect to the solid finite elements are obtained via advantageous DOFs for the bi-directional higher-order shear deformable mixed finite elements which reduces the computational time and space. Besides, in order to increase the precision at the stress concentration zones, the length ratios of the mixed finite elements at these zones are related to the geometric features and material parameters. Finally, the influences of geometric features, material constituents and the power-law index on the stress distribution of FG spatial/planar curved beams are presented both via the bi-directional higher-order mixed finite elements and the solid finite elements.