Center manifold reduction of geometrically nonlinear beams: From 3D to 1D models

Abstract

This paper presents a novel center manifold-based approach for reducing the geometrically nonlinear three-dimensional continuum description of beam structures to efficient one-dimensional models. The beam considered is composed of hyperelastic material and features uniform cross-sectional geometry and material properties along its axial line. While Mielke previously proved that Saint-Venant’s solution resides within a twelve-dimensional center manifold, its construction for nonlinear dimensional-reduction of beams remains unexplored. This study presents the first explicit construction of the center manifold by approximating the warping field, sectional strains, and one-dimensional equilibrium equations as polynomials of the six stress resultant components, enabling dimensional reduction for beams undergoing geometrically nonlinear deformations. The method begins by decomposing beam kinematics into rigid-section motion and a warping field. A finite element semi-discretization of the cross-section is employed, and Hamilton’s variational principle yields equilibrium equations as nonlinear ordinary differential equations along the beam’s axial coordinate, enabling center manifold reduction. A critical challenge arises from the kinematic decomposition, requiring two interdependent mappings: sectional strains and warping field as nonlinear functions of stress resultants. Unlike conventional center manifold methods, which use a single set of invariance equations, this approach demands two distinct, coupled invariance equations. Additionally, the non-uniqueness of the kinematic decomposition leads to inherently singular cohomological equations, differing from traditional center manifold reduction. To address these challenges, four key advancements are proposed: (1) Two coupled set of invariance equations are established: one ensures sectional strains and warping fields satisfy three-dimensional equilibrium, and the other ensures the composed mapping for stress resultants reduces to the identity mapping. (2) The left- and right-null spaces of the cohomological equations, arising from non-unique kinematic decomposition, are rigorously identified, ensuring solution existence. (3) Uniqueness of compliance and warping matrices is established via energetic analysis of the system’s Hamiltonian, guaranteeing symmetry in higher-order compliance matrices. (4) Distributed functions of loads are reformulated as differential equations, transforming inhomogeneous systems into augmented homogeneous ones for dimensional reduction. Finally, the reduced-order one-dimensional beam model is solved using a mixed finite element formulation to enforce complementary-energy-based nonlinear constitutive laws derived from the center manifold reduction. Numerical examples validate the accuracy and computational efficiency of the proposed method, demonstrating its capability to capture geometric nonlinearities in beams with complex cross-sections, heterogeneous materials, initial curvatures, and distributed loading conditions.