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V.V. Vasil'ev, "Kirchhoff and Thomson-Tait Transformations in the Classical Theory of Plates," Mech. Solids. 47 (5), 571-579 (2012)
Year 2012 Volume 47 Number 5 Pages 571-579
DOI 10.3103/S0025654412050111
Title Kirchhoff and Thomson-Tait Transformations in the Classical Theory of Plates
Author(s) V.V. Vasil'ev (Moscow State Aviation Technological University, Orshanskaya 3, Moscow, 121552 Russia, vvvas@dol.ru)
Abstract The transformation of the torque into the transverse force is considered; this transformation is traditional in the educational literature [1] and was proposed by Kirchhoff [2] and Thomson and Tait [3] to match the order of the differential equation of the classical theory of plates with the number of boundary conditions. It is shown that this transformation is not universal and its mathematical and physical justification depends on the conditions of the plate fixation and loading. It is shown that this justification is absent for the most widely used problems of bending of a rectangular plate freely supported and fixed on the contour.
Keywords theory of plates, boundary conditions
References
1.  S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959; Fizmatgiz, Moscow, 1963.
2.  G. Kirchhoff, "Uber das Gleichgewicht und die Bewegung einer Elastischen Scheibe," J. Reine Angew. Math. 40, 51-58 (1850).
3.  W. Thomson and P. Tait, Treatise on Natural Philosophy, Part 2 (Cambridge Univ. Press, Cambridge, 1883).
4.  S. D. Poisson, "Memoire sur l'equilibre et le mouvement des corps e'lastiques," Memoires Acad. Sci. Paris 18, 357-570 (1829).
5.  P. A. Zhilin, "On the Poisson and Kirchhoff Theories of Plates from the Standpoint of the Contemporary Theory of Plates ," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 48-64 (1992) [Mech. Solids (Engl. Transl.)].
6.  P. A. Zhilin, "On the Classical Plate Theory and the Kelvin-Tait Transformation," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 4, 134-140 (1995) [Mech. Solids (Engl. Transl.) 30 (4), 124-129 (1995)].
7.  N. A. Alfutov, "On Some Paradoxes of the Theory of Elastic Plates," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 65-72 (1992) [Mech. Solids (Engl. Transl.)].
8.  V. G. Rekach, Solution of Elasticity Problems. Handbook (Vysshaya Shkola, Moscow, 1966) [in Russian].
9.  E. Reissner, "On the Theory of Bending of Elastic Plates," J. Math. Phys. 23 (4), 184-191 (1944).
10.  V. V. Vasil'ev, "Classical Theory of Plates: Historical Perspective and State-of-the-Art," Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 46-58 (1998) [Mech. Solids (Engl. Transl.) 33 (3), 35-45 (1998)].
11.  M. Schafer, "Uber eine Verfeinerung der Klassischen Theorie dünner schwach gebegener Platten," ZAMM 32 (6), 161-171 (1952).
12.  A. Kromm, "Uber die Randquerkafter bei gestutzten Platten," ZAMM 35 (6/7), 231-242 (1955).
13.  M. P. Sheremet'ev, B. L. Pelekh, and O. P. Dyachina, "A Study of the Effect of Shear Strains on the Bending of a Square Slab by a Concentrated Force," Prikl. Mekh. 4 (4), 1-7 (1968) [Int. Appl. Mech. (Engl. Transl.) 4 (4), 1-4 (1968)].
Received 25 May 2012
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