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Geometrical interpretation of the Timoshenko–Ehrenfest theorem

In this paper, geometrical interpretation of the Timoshenko–Ehrenfest theorem is given. It is based on a vector-valued version of continuum mechanics, which considers the deformed body as a surface in extended coordinate space. Scalar parameters of the metrics of this hypersurface determine the strain. In the bending beam case, this surface is formed by a twisted cylinder rolled into a cone. Accordingly, three kinds of metric parameters, the stretch ratio, the twist and cone angles responsible for stretching, shear, and bending determine the shape of the elastic curve. The balance equations for generalized forces taking into account shear deformation and rotational bending effects are derived. A new formula for the shear coefficient comes out of the derivation. It is shown that its magnitude 5/6 for quasistatic Timoshenko beam remains fair for the exact form of the cross-section at the free end.