Vol. 6, No. 5, 2011

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Consistent loading for thin plates

Isaac Harari, Igor Sokolov and Slava Krylov

Vol. 6 (2011), No. 5, 765–790

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 three-dimensional 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.

Kirchhoff thin-plate theory, structural reduction, Kirchhoff equivalent force, distributed couples, corner forces
Received: 13 July 2010
Revised: 28 December 2010
Accepted: 7 January 2011
Published: 9 September 2011

Proposed: Davide Bigoni
Isaac Harari
School of Mechanical Engineering
Tel Aviv University
69978 Tel Aviv
Igor Sokolov
School of Mechanical Engineering
Tel Aviv University
69978 Tel Aviv
Slava Krylov
School of Mechanical Engineering
Tel Aviv University
69978 Tel Aviv