UNIFIED NONLINEAR-ANALYSIS FOR NONHOMOGENEOUS ANISOTROPIC BEAMS WITH CLOSED CROSS-SECTIONS

A unified methodology is presented for analysis of nonhomogeneous, anisotropic beams. Based on geometrically nonlinear, three-dimensional elasticity, and subject only to the restrictions that strain and local rotation are small compared to unity and that warping displacements (the deformation in and out of the cross-sectional plane) are small relative to cross-sectional dimensions, a two-dimensional analysis is derived that enables the determination of sectional elastic constants for the beam. Effects typical of open-section beams, such as restrained warping, are assumed to be negligible; thus, the analysis is well suited for beams with closed cross sections. From application of the finite element method, it is observed that the equations containing warping degrees of freedom are identical in form to those of an existing, purely linear analysis in the literature. There are two differences in the analysis as a whole, however: 1) the linear strain measures of the published analysis are replaced by certain generalized strain measures that are nonlinear functions of the displacement of the beam reference axis and rotation of the cross-sectional frame; and 2) the linear global equilibrium equations are replaced by exact, nonlinear, intrinsic ones. The structure of the governing equations tells us that the warping solutions can be affected by large deformation and that this could alter the incremental stiffness of the section. For a certain range of deformation, however, the elastic constants based on the reference state are adequate. As a result, it is shown that sectional constants derived from the published, linear analysis can be used in the present nonlinear, one-dimensional analysis governing the global deformation of the beam, which is based on the intrinsic equations for nonlinear beam behavior. The global behavior is determined by means of a mixed finite element analysis based on the weak form of all equilibrium, constitutive, and kinematical equations, including boundary conditions and continuity requirements. In spite of the simplicity of the approach, excellent correlation is obtained with published experimental results for both isotropic and anisotropic beams undergoing large deflections.