Rapidly convergent mixed fem for geometrically nonlinear analysis of laminated composite beams integrating 3d strain effects

Abstract This study presents a mixed finite element (MFE) formulation designed to efficiently determine the geometrically nonlinear behavior of laminated composite beams, ensuring rapid convergence and reduced computational cost. This is achieved by incorporating all 3D strain components into the constitutive equations while satisfying the beam theory stress-free surface conditions. Von Kármán nonlinear strains are derived from a displacement field including three displacements and three rotations per node. The governing equations, obtained from the first variation of the Hellinger–Reissner functional, are linearized via an incremental formulation and solved iteratively using the Newton–Raphson algorithm. The proposed MFE is based on Timoshenko beam theory and enhanced by the integration of the cross-sectional warping deformations over a displacement-based FE formulation. The two-noded MFE involves twelve degrees of freedom per node and achieves rapid convergence with substantially reduced computational cost. Its performance is assessed through comparison with advanced beam formulations featuring refined kinematics, as well as 3D solid element simulations for asymmetric [0°/90°] cross-ply laminated beams. It provides satisfactory convergent results via very few degrees of freedom compared to the 20-node brick finite elements and 4-node shell finite elements of ANSYS. Parametric studies discuss the influence of cross-ply lamination, material anisotropy, and geometric design on the ratio of geometrically nonlinear to linear displacements, highlighting the significance of design-induced nonlinearities in high-performance structural applications.