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

Abstract

Attention is focused in this paper on the development of a consistent ®nite 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 spacecurved beam, undergoing arbitrarily large rotations and arbitrarily large stretches. A choice of a proper ®nite 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. IntroductionThis paper deals with the development of a consistent theory of ®nite stretches and ®nite rotations, in spacecurved beams of an arbitrary cross-section. The ®nite rotation vector plays an important role for describing the kinematics of the beam. The shearing deformation, as well as cross-sectional warping displacement, are taken into account in this formulation. Using the kinematic and kinetic variables for the present theory of a curved beam in Atluri's variational principle (Atluri, 1979(Atluri, , 1980(Atluri, , 1983Atluri and Cazzani, 1995), a consistent multi-®eld variational theorem for a space-curved beam undergoing arbitrarily large stretches and rotations, under bending and torsion, is developed.Beginning with the equilibrium equations of a 3-D continuum body, we obtain the linear momentum balance (LMB), angular momentum balance (AMB) and director momentum balance (DMB) equations of the beam. The balance equation for the bimoment is also obtained. With the use of the AMB and DMB equations, the conventional moment equilibrium equation is derived. The AMB equation plays an important role in the expression of stress power of the beam, in which the deformation gradient tensor and the ®rst Piola±Kirchhoff stress tensor are used. In addition to the conventional strain measures, we introduce new strain measures for constructing a consistent beam theory. The conjugate relationships between the strain and stress measures are obtained through the stress power.The invariance condition of strain energy function is discussed, and the important role of the AMB equation is emphasized. The A...

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