Variational formulation of the static Levinson beam theory

• Clapeyron's theorem is applied to Levinson's beam equations.
• It is found that the interior beam end stresses are an integral part of the theory.
• Consistent variational formulation for the Levinson beam theory is given.
• The obtained beam equations are the same as those derived vectorially.
• Exact Levinson beam finite element is developed.

In this communication, we provide a consistent variational formulation for the static Levinson beam theory. First, the beam equations according to the vectorial formulation by Levinson are reviewed briefly. By applying the Clapeyron's theorem, it is found that the stresses on the lateral end surfaces of the beam are an integral part of the theory. The variational formulation is carried out by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the stresses on the end surfaces of the beam. This external virtual work contributes to the boundary conditions in such a way that artificial end effects do not appear in the theory. The obtained beam equations are the same as the vectorially derived Levinson equations. Finally, the exact Levinson beam finite element is developed.

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