Co-Rotational Formulations for Geometrically Nonlinear Analysis of Beam-Columns Including Warping and Wagner Effects

The warping effects may predominate in geometrically nonlinear analysis of open cross-section members. The formulation of conventional beam-column elements incorporating the warping effects is cumbersome due to the method considering the inconsistency between the shear center and centroid. To develop a concise warping element formulation, this paper presents a transformation matrix to integrate the inconsistent effects into the element stiffness matrix. The co-rotational (CR) method used to establish the element equilibrium conditions in the geometrically nonlinear analysis is adopted to simplify the element formulation and improve the efficiency of nonlinear analysis. A new beam-column element explicitly considering the warping deformation and the Wagner effects is derived based on the CR method and the Euler-Bernoulli beam theory. A detailed kinematic description is provided for considering large deflections and rigid body motions. Based on the mechanical characteristic, the coordinate and the rigid body motion transformation matrices are given. The secant relationship is developed to evaluate the element internal forces accurately and effectively in each iteration. Several verification examples are provided to validate the proposed method's reliability and robustness. The verifications demonstrate that the proposed element leads to considerable computational advantages. The results of this paper are useful for future upgrading of frame analysis software with warping degrees-of-freedom (DOFs).