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Normal reactions in a clamped elastic rectangular plate

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Abstract

This paper addresses the classical problem of bending of a clamped thin rectangular elastic plate by a normal uniform load. A brief account of engineering approaches to consider the problem is given. The object of this paper is to provide a full description of the local behaviour of normal reactions and deflection near the corner points of the plate. Among various mathematical and engineering approaches, the method of superposition is effective for solving the problem. Numerical results reveal an analytically predicted local behaviour of normal reactions.

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References

  1. Th. von Kármán, Tooling up mathematics for engineering. Quart. Appl. Math. 1 (1943) 2–6.

    Google Scholar 

  2. A. N. Krylov, A report on Professor Papkovich's treatese “Structural Mechanics of a Ship”. In: Recollections and Essays. Moscow: Acad. Sci. SSSR Press (1956) 475–487 (in Russian).

    Google Scholar 

  3. V. V. Meleshko, Bending of an elastic rectangular clamped plate: Exact versus ‘engineering’ solutions. J. Elasticity 48 (1997) 1–50.

    Article  Google Scholar 

  4. V. V. Meleshko, Biharmonic problem in a rectangle. Appl. Sci. Res. 58 (1997) 217–249.

    Article  Google Scholar 

  5. A. E. H. Love, Biharmonic analysis, especially in a rectangle, and its application to the theory of elasticity. J. London Math. Soc. 3 (1928) 144–156.

    Google Scholar 

  6. A. C. Dixon, The problem of the rectangular plate. J. London Math. Soc. 9 (1934) 61–74.

    Google Scholar 

  7. P. G. Ciarlet, Mathematical Elasticity. Vol.II. Theory of Plates. Amsterdam: Elsevier (1997) 497 pp.

    Google Scholar 

  8. C. B. Biezeno, Graphical and numerical methods for solving stress problems. In: C.B. Biezeno, J. M. Burgers (eds), Proc. First Intern. Congress for Applied Mechanics. Delft: Waltman (1925) 3–17.

    Google Scholar 

  9. S. P. Timoshenko, Bending of rectangular plates with clamped edges. In: Proc. Fifth Intern. Congress for Applied Mechanics. New York: Wiley (1939) 40–43. Reprinted in Collected Papers. New York: McGraw-Hill (1953) pp. 516–522.

    Google Scholar 

  10. F. Grashof, Theorie der Elasticität und Festigkeit. (2nd ed.). Berlin: Gaertner (1878) 408 pp.

    Google Scholar 

  11. H. Lorenz, Technische Elastizitätslehre. München: Oldenbourg (1913) 692 pp.

    Google Scholar 

  12. S. P. Timoshenko, A Course of Theory of Elasticity. Vol. 2. Petrograd: Collins (1916) 416 pp. (in Russian).

    Google Scholar 

  13. A. Föppl and L. Föppl, Drang und Zwang. Vol 1. München: Oldenbourg (1920) 300 pp.

    Google Scholar 

  14. J. Prescott, Applied Elasticity. London: Longmans (1924) 666 pp.

    Google Scholar 

  15. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity. (4th ed.) Cambridge: Cambridge University Press (1927) 643 pp.

    Google Scholar 

  16. A. Nádai, Der elastischen Platten. Berlin: Springer (1925) 326 pp.

    Google Scholar 

  17. S. P. Timoshenko, Theory of Plates and Shells. New York: McGraw-Hill (1940) 492 pp.

    Google Scholar 

  18. K. Girkmann, Flächentragwerke. (6th ed.). Wien: Springer (1974) 632 pp.

    Google Scholar 

  19. B. M. Koialovich, On one partial differential equation of the fourth order (Doctoral dissertation). St Petersburg: St Petersburg University Press (1902) 105 pp. (in Russian). (German abstract in Jahrb. Fortschr. Math. 33 367–368.)

    Google Scholar 

  20. I. G. Bub0nov, Stresses in the bottom plating of ship due to water pressure (Adjunct dissertation). St Petersburg: Venike (1904) 90 pp. (in Russian).

    Google Scholar 

  21. I. G. Boobnoff, On the stresses in a ship's bottom plating due to water pressure. Trans. Inst. Naval Arch. 44 (1902) 15–52. Reprinted in Engineering 73 (1902) 384–385, 390–392.

    Google Scholar 

  22. I. G. Bubnov, Stresses in the bottom plating of ship due to water pressure. Morsk. Sb. 312 (1902) No 10 119–138 (in Russian).

    Google Scholar 

  23. W. Ritz, Ñber eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. J. reine angew. Math. 135 (1908) 1–61.

    Google Scholar 

  24. H. Hencky, Der Spannungszustand in rechteckigen Platten (Dissertation). München: Oldenbourg (1913) 96 pp.

    Google Scholar 

  25. S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. (2nd ed.). New York: McGraw-Hill (1959) 580 pp.

    Google Scholar 

  26. F. Pietzker, Festigkeit der Schiffe. Berlin: Reichs-Marine-Arm (1911) 176 pp. English translation: The Strength of Ships. New York: SNAME (1959) 184 pp.

    Google Scholar 

  27. I. G. Bubnov, Structural Mechanics of a Ship. Part II. St Petersburg: Naval Academy Press (1914) 310 pp. (in Russian).

    Google Scholar 

  28. Yu. A. Shimanski, Bending of Plates. Leningrad: ONTI (1934) 223 pp. (in Russian).

    Google Scholar 

  29. P. F. Papkovich, Structural Mechanics of a Ship. Part 2. Leningrad: Sudpromgiz (1941) 960 pp. (in Russian).

    Google Scholar 

  30. C. E. Inglis, Sresses in rectangular plates clamped at their edges and loaded with a uniformly distributed pressure. Trans. Inst. Naval Arch. 67 (1925) 147–165.

    Google Scholar 

  31. B. G. Galerkin, Rods and plates. Series in some problems of elastic equilibrium of rods and plates. Vest. Inzh. 1 (1915) 897–908. Reprinted in Collected Papers, Vol. 1. Moscow: Acad. Sci. SSSR Press (1952) pp. 168–195 (in Russian).

    Google Scholar 

  32. Rayleigh, Lord, Hydrodynamical notes. Phil. Mag. (6)21 (1911) 177–195.

    Google Scholar 

  33. W. R. Dean and P. E. Montagnon, On the steady motion of viscous liquid in a corner. Proc. Cambridge Phil. Soc. 45 (1949) 389–395.

    Google Scholar 

  34. M. L. Williams, Surface stress singularities resulting from various boundary conditions in angular corners of plates under bending. In: Proc. First U. S. Congress of Applied Mechanics. Chicago: Illinois Institute of Technology Press (1951) pp. 325–336.

    Google Scholar 

  35. K. Wieghardt, Ñber das Spalten und Zerrüben elastischen Körper. Z. Math. Phys. 55 (1907) 60–103. English translation: On splitting and cracking of elastic bodies. Fatigue Fracture Engng Mater. Struct. 18 (1995) 1371–1405.

    Google Scholar 

  36. S. Woinowsky-Krieger, The bending of a wedge-shaped plate. Trans. ASME. J. Appl.Mech. 20 (1953) 77–81.

    Google Scholar 

  37. V. V. Meleshko and A. M. Gomilko, Infinite systems for a biharmonic problem in a rectangle. Proc. R. Soc. Lond. A 453 (1997) 2139–2160.

    Google Scholar 

  38. N. Koshliakov, On a general summation formula and its application. C. R. (Doklady) Acad. Sci. URSS 4 (1934) 187–190 (in Russian, with English summary).

    Google Scholar 

  39. W. L. Ferrar, Summation formulae and their relation to Dirichlet's series, II. Compositio Math. 4 (1937) 394–405.

    Google Scholar 

  40. H. Buchholz, Eine einfache Reihentransformation bei einer sehr allgemeien Fourierschen Reihe. Z. angew. Math. Mech. 22 (1942) 277–286.

    Google Scholar 

  41. G. G. Macfarlane, The application of Mellin transforms to the summation of slowly convergent series. Phil. Mag. (ser. 7) 40 (1949) 188–197.

    Google Scholar 

  42. C. B. Biezeno, Over een vereenvoudiging en over een uitbreiding van de methode van Ritz. Christiaan Huygens 3 (1924) 69–75.

    Google Scholar 

  43. W. Ritz, Oeuvres. Paris: Gauthier-Villars (1911) 541 pp.

    Google Scholar 

  44. D. Pistriakoff, Bending of a thin plate. Izv. Kiev. Politekh. Inst. 10 (1910) 311–373 (in Russian, with French summary).

    Google Scholar 

  45. S. P. Timoshenko, On application the normal coordinates method for calculation of bending of rods and plates. Izv. Kiev. Politekh. Inst. 10 (1910) 1–49 (in Russian, with French summary). (German abstract in Jahrb. Fortschr. Math. 41 903–905.)

    Google Scholar 

  46. H. Lorenz, Angenäherte Berechnung rechteckiger Platten. Z. Ver. deut. Ing. 57 (1913) 623–625.

    Google Scholar 

  47. L. S. Leibenzon, Variational Methods of Solution of Problems in Theory of Elasticity.Moscow: GTTI (1943) 286 pp. Reprinted in: Collected Papers, Vol. 1. Moscow: Acad. Sci. SSSR Press (1952) 177–463 (in Russian).

    Google Scholar 

  48. C. B. Biezeno and J. J. Koch, Over een nieuwe methode ter berekening van vlakke platen, met toepassing op eenige voor de techniek belangrijke belastingsgevallen. Ingenieur 38 (1923) 25–36.

    Google Scholar 

  49. C. B. Biezeno and R. Grammel, Technische Dynamik. Berlin, Springer (1939) 1056 pp.

    Google Scholar 

  50. K. Sezawa, The stress on rectangular plates. Engineering 116 (1923) 188–191.

    Google Scholar 

  51. H. W. March, The deflection of a rectangular plate fixed at the edges. Trans. Amer. Math. Soc. 27 (1925) 307–317.

    Google Scholar 

  52. L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis. Groningen: Noordhoff (1958) 681 pp.

    Google Scholar 

  53. F. Oberhettinger, Tables of Mellin Transforms. Berlin: Springer (1974) 275 pp.

    Google Scholar 

  54. V. M. Abramov, The problem of contact of an elastic infinite half-plane with an absolutely rigid rough foundation. Dokl. Akad. Nauk SSSR 17 (1937) 173–178. (German abstract in Jahrb Fortschr. Math. 63 745.)

    Google Scholar 

  55. O. C. Zienkewicz and R. L. Taylor, The Finite Element Method, Vol. 2: Solid Mechanics. (5th ed.). Oxford: Butterworth – Heinemann (2000) 459 pp.

    Google Scholar 

  56. Y. Sawaki and N. Kamiya, Adaptive boundary element formulations for plate bending analysis. In: M.H. Aliabadi (ed.), Plate Bending Analysis with Boundary Elements. Southampton: Computational Mechanics Publications (1998) 225–248.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Hydromechanics, National Academy of Sciences, 03057, Kiev, Ukraine

    V.V. Meleshko

  2. Institute of Hydromechanics, National Academy of Sciences, 03057, Kiev, Ukraine

    A.M. Gomilko & A.A. Gourjii

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  1. V.V. Meleshko
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  2. A.M. Gomilko
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  3. A.A. Gourjii
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Meleshko, V., Gomilko, A. & Gourjii, A. Normal reactions in a clamped elastic rectangular plate. Journal of Engineering Mathematics 40, 377–398 (2001). https://doi.org/10.1023/A:1017501111057

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