Mixed finite element formulation for laminated composite beams incorporating transverse stretching effect

This study presents a two-node C⁰-continuous mixed finite element formulation for the static flexural analysis of laminated composite beams within a quasi- three dimensional (3D) higher-order beam theory incorporating transverse shear and transverse normal deformation effects. The formulation is derived from the Hellinger-Reissner variational principle, in which displacement variables and generalised stress resultants are treated as independent nodal unknowns. The novelty of the proposed approach lies in combining transverse-stretching higher-order beam kinematics with a compact mixed finite element implementation that enables the direct nodal recovery of axial force, bending moment, higher-order bending moment, transverse shear resultant and transverse normal resultant. Unlike purely displacement-based formulations, the present approach evaluates strain measures through the sectional compliance relation and reduces the need for derivative-based stress-resultant recovery in post-processing. Transverse shear and normal deformations are represented through distinct shear functions, and the kinematic description uses two deflection-related terms with one axial displacement and one sectional rotation. The governing equations are obtained from the principle of virtual work in conjunction with the Hellinger-Reissner framework, so that displacement and stress-resultant fields are treated as independent nodal unknowns. This treatment enables direct recovery of strain components and through-thickness axial stresses without ad hoc post-processing or shear-correction factors. The convergence results show that the formulation preserves its accuracy for both thick and slender laminated beams, without the artificial stiffness increase typically associated with shear-locking-prone displacement-based elements. Parametric studies illustrate the influence of lamination angles, modulus ratios, and foundation stiffness on deflection and stress distributions. In general, the proposed method provides an easy-to-implement tool for the precise analysis of static bending of laminated composite beams.