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Kirchhoff and thomson-tait transformations in the classical theory of plates

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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.

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References

  1. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959; Fizmatgiz, Moscow, 1963.

    Google Scholar 

  2. G. Kirchhoff, “Uber dasGleichgewicht und dieBewegung einer ElastischenScheibe,” J. ReineAngew.Math. 40, 51–58 (1850).

    MATH  Google Scholar 

  3. W. Thomson and P. Tait, Treatise on Natural Philosophy, Part 2 (Cambridge Univ. Press, Cambridge, 1883).

    Google Scholar 

  4. S. D. Poisson, “Memoire sur l’equilibre et le mouvement des corps e’lastiques,” Memoires Acad. Sci. Paris 18, 357–570 (1829).

    Google Scholar 

  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].

    Google Scholar 

  9. E. Reissner, “On the Theory of Bending of Elastic Plates,” J. Math. Phys. 23(4), 184–191 (1944).

    MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  Google Scholar 

  12. A. Kromm, “Uber die Randquerkafter bei gestutzten Platten,” ZAMM 35(6/7), 231–242 (1955).

    MATH  Google Scholar 

  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)].

    Google Scholar 

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Authors and Affiliations

  1. Moscow State Aviation Technological University, Orshanskaya 3, Moscow, 121552, Russia

    V. V. Vasil’ev

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  1. V. V. Vasil’ev
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Correspondence to V. V. Vasil’ev.

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Original Russian Text © V.V. Vasil’ev, 2012, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2012, No. 5, pp. 98–107.

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Vasil’ev, V.V. Kirchhoff and thomson-tait transformations in the classical theory of plates. Mech. Solids 47, 571–579 (2012). https://doi.org/10.3103/S0025654412050111

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