An equilibrium-based quasi-3D beam model with rational transverse normal stress expression for accurate analysis of functionally graded beams

Although existing quasi-3D beam models can partially account for the influence of transverse stretching deformation, their inability to accurately predict transverse normal stress severely limits their practical value, particularly when applied to functionally graded beams. A key contributing factor to this limitation is the failure of the stress components to satisfy the equilibrium relationships. This paper develops a novel quasi-3D beam models with equilibrium-based transverse normal stress expression for achieving accurate solutions of functionally graded beams. In contrast to the conventional quasi-3D models where stress expressions are derived from constitutive and geometric equations, the expressions of transverse shear stress and transverse normal stress in this study are derived according to the differential equilibrium equations among stresses, ensuring strict adherence of stress solutions to equilibrium conditions. To incorporate the influence of equilibrium-derived stress distributions, the equilibrium-based cross-sectional stiffness matrix is derived, enhancing the theoretical and practical feasibility of the beam model. In the numerical implementation, the mixed element method with generalized internal forces and generalized displacements regarded as two independent fields is employed to construct the beam element, and especially, semi-analytical internal force fields, which partially satisfy the differential equilibrium equations, are introduced to improve the element performance. Numerical results demonstrate that the proposed quasi-3D beam model accurately predicts both displacement and stress fields in functionally graded beams, which exhibits the effectiveness of introducing stress equilibrium relationships.