Consistent loading for thin plates

Structural models are well-established for the governing operators in solid mechanics, yet the reduction of loads (data) is often performed in an ad hoc manner, which may be inadequate for the complex load distributions that often arise in modern applications. In the present work we consistently convert three-dimensional data to the form required by Kirchhoff thin-plate theory, in a variational framework. We provide formulas for all types of resultant structural loads and boundary conditions in terms of the original three-dimensional data, including proper specification of corner forces, in forms that are readily incorporated into computational tools. In particular, we find that in-plane components of threedimensional loads engender distributed couples, contributing to an effective distributed transverse force and boundary shear force, the latter generalizing the notion of the celebrated Kirchhoff equivalent force. However, in virtual work we advocate a representation of the twisting moment in a form that involves neither the Kirchhoff equivalent force nor corner forces. An interpretation of the structural deflections as through-the-thickness averages of the continuum displacements, rather than their values on the midplane, yields explicit formulas for the thin-plate essential boundary data. The formulation facilitates the solution of problems that would otherwise pose formidable challenges. Numerical results confirm that appropriate use of the thin-plate model economizes computation and provides insight into the mechanical behavior, while preserving a level of accuracy comparable with the full three-dimensional solution.