A consistent theory of finite stretches and finite rotations, in space-curved beams of arbitrary cross-section

Attention is focused in this paper on the development of a consistent finite deformation beam theory, and its mixed variational formulation. The shearing deformation, as well as cross-sectional warping displacement, are taken into account in this formulation. Beginning with the equilibrium equations of 3-D continuum body, we obtain the linear momentum balance (LMB), angular momentum balance (AMB) and director momentum balance (DMB) conditions of the beam. The conjugate relationships between the strain and stress measures are obtained through the stress power, in which the AMB condition plays an important role. The use of the strain measures proposed herein, leads to the strain energy function which is invariant under a rigid-body motion. The present formulation is shown to be objective by using a numerical example. On the basis of Atluri's variational principle, we develop a mixed type variational functional for a space-curved beam, undergoing arbitrarily large rotations and arbitrarily large stretches. A choice of a proper finite rotation vector, and unsymmetric curvature strains, makes it possible for constructing a consistent variational principle. The use of the present functional always leads to a symmetric tangent stiffness. The mixed variational functional developed herein leads to a powerful tool for obtaining accurate numerical results of 3-D space-curved beams, undergoing arbitrarily large stretches and rotations.