Interior formulation of axisymmetric Levinson plate theory

• We show that the axisymmetric Levinson plate theory is an interior theory.
• Interior stress based surface tractions do work on the lateral edges of the plate.
• Consistent variational formulation for an annular Levinson plate is given.
• The obtained plate equations are the same as those derived vectorially.
• Exact Levinson plate finite element is developed.

In this study, we show that the axisymmetric Levinson plate theory is exclusively an interior theory and we provide a consistent variational formulation for it. First, we discuss an annular Levinson plate according to a vectorial formulation. The boundary layer of the plate is not modeled and, thus, the interior stresses acting as surface tractions do work on the lateral edges of the plate. This feature is confirmed energetically by the Clapeyron's theorem. The variational formulation is carried out for the annular Levinson plate by employing the principle of virtual displacements. As a novel contribution, the formulation includes the external virtual work done by the tractions based on the interior stresses on the inner and outer lateral edges of the Levinson plate. The obtained plate equations are consistent with the vectorially derived Levinson equations. Finally, we develop an exact plate finite element both by a force-based method and from the total potential energy of the Levinson plate.

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