On the equations of motion for curved slender beams using tubular coordinates

The equations of motion for a beam are derived from the three-dimensional continuum mechanical point of view. The kinematics involve bending, torsion and shearing as well as cross-sectional displacements. The transplacement of the beam from an arbitrary curved reference placement is considered. For slender beams so called tubular coordinates may be used as global coordinates in the reference placement. An explicit geometrical characterization of slender beams is given. The equations of motion for the slender beam are derived and some consequences of the power theorem are presented. Using the principle of virtual power the classical local beam equations are obtained along with equations representing the local balance of bi-momentum. Constitutive assumptions are introduced where the dependence of stress on the natural deformation measures for the beam is obtained by assuming that the beam consists of a St Venant-Kirchhoff elastic material. Simplified stress-strain relations may be obtained using so called torsion free coordinates.