International Applied Mechanics

, Volume 41, Issue 9, pp 1043–1058 | Cite as

Superposition Method in Thermal-Stress Problems for Rectangular Plates

  • V. V. Meleshko


The classical two-dimensional biharmonic problem for a rectangular domain is considered. Some historical aspects of the problem are elucidated. The superposition method turns out to be efficient in solving thermoelastic-equilibrium problems in a rectangle. The relationship between the mathematical and engineering approaches to these problems is studied. A few typical examples are given


biharmonic problem thermoelastic stresses in rectangle infinite systems of linear algebraic equations 


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  1. 1.
    B. A. Boley and J. H. Weiner, Theory of Thermal Stresses, Wiley, New York (1960).Google Scholar
  2. 2.
    P. S. Bondarenko, “Uniqueness problem for infinite systems of linear equations,” Mat. Sb., 29, No.2, 403–418 (1951).MATHMathSciNetGoogle Scholar
  3. 3.
    I. G. Bubnov, “Stresses in ship bottom plating due to submergence pressure,” Morsk. Sb., 312, No.10, 119–138 (1902).Google Scholar
  4. 4.
    I. G. Bubnov, Stresses in Ship Bottom Plating due to Submergence Pressure [in Russian], Izd. A. E. Vineke, St-Petersburg (1904).Google Scholar
  5. 5.
    I. G. Bubnov, Naval Structural Mechanics [in Russian], Izd. Mor. Min., St-Petersburg (1914).Google Scholar
  6. 6.
    B. E. Geitvud, Thermal Stresses in Aircraft, Missiles, Turbines, and Nuclear Reactors [in Russian], IL, Moscow (1959).Google Scholar
  7. 7.
    V. T. Grinchenko, Equilibrium and Steady-State Vibrations of Finite Elastic Bodies [in Russian], Naukova Dumka, Kiev (1978).Google Scholar
  8. 8.
    V. T. Grinchenko and A. F. Ulitko, “Boundary-value problem of thermoelasticity for a rectangular plate,” Tepl. Napryazh. Elem. Konstr., 5, 138–146 (1965).Google Scholar
  9. 9.
    V. T. Grinchenko and A. F. Ulitko, Equilibrium of Canonical Elastic Bodies, Vol. 3 of the five-volume series Three-Dimensional Problems of Elasticity and Plasticity [in Russian], Naukova Dumka, Kiev (1985).Google Scholar
  10. 10.
    L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis [in Russian], Fizmatgiz, Moscow (1962).Google Scholar
  11. 11.
    V. G. Karnaukhov, I. K. Senchenkov, and B. P. Gumenyuk, Thermomechanical Behavior of Viscoelastic Bodies under Harmonic Loading [in Russian], Naukova Dumka, Kiev (1985).Google Scholar
  12. 12.
    A. D. Kovalenko, Plates and Shells in Turbomachine Rotors [in Russian], Izd. AN USSR, Kiev (1955).Google Scholar
  13. 13.
    A. D. Kovalenko, Circular Plates of Varying Thickness [in Russian], Fizmatgiz, Moscow (1959).Google Scholar
  14. 14.
    A. D. Kovalenko, An Introduction to Thermoelasticity [in Russian], Naukova Dumka, Kiev (1965).Google Scholar
  15. 15.
    A. D. Kovalenko, Fundamentals of Thermoelasticity [in Russian], Naukova Dumka, Kiev (1970).Google Scholar
  16. 16.
    A. D. Kovalenko, Thermoelasticity of Plates and Shells [in Russian], Izd. KGU, Kiev (1971).Google Scholar
  17. 17.
    A. D. Kovalenko, Thermoelasticity [in Russian], Vyshcha Shkola, Kiev (1975).Google Scholar
  18. 18.
    A. D. Kovalenko, Selected Works [in Russian], Naukova Dumka, Kiev (1976).Google Scholar
  19. 19.
    B. M. Koyalovich, On a Partial Differential Equation of the Fourth Order [in Russian], Izd. Imp. Akad. Nauk, St-Petersburg (1902).Google Scholar
  20. 20.
    B. M. Koyalovich, “Analysis of infinite systems of linear algebraic equations,” Izv. Fiz.-Mat. Inst. im. V. A. Steklova, 3, 41–167 (1930).Google Scholar
  21. 21.
    N. N. Lebedev, Thermal Stresses in Elasticity Theory [in Russian], ONTI, Leningrad-Moscow (1937).Google Scholar
  22. 22.
    A. I. Lur'e, Three-Dimensional Problems in Elasticity Theory [in Russian], GITTL, Moscow (1955).Google Scholar
  23. 23.
    A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, Cambridge Univ. Press (1959).Google Scholar
  24. 24.
    V. M. Maizel', Thermal Problem in Elasticity Theory [in Russian], Izd. AN USSR, Kiev (1951).Google Scholar
  25. 25.
    G. N. Maslov, “Thermoelastic-equilibrium problem,” Izv. NII Gidrotekhn., 23, 130–219 (1938).Google Scholar
  26. 26.
    Von E. Melan and H. Parkus, Warmespannungen infolge Stationarer Temperaturfelder, Springer Verlag, Vienna (1953).Google Scholar
  27. 27.
    V. V. Meleshko, “Normal reactions in a clamped rectangular plate,” in: Problems of Solid Mechanics [in Russian], Izd. SPb. Univ., St-Petersburg (2002), pp. 220–233.Google Scholar
  28. 28.
    V. V. Meleshko, “Biharmonic problem in a rectangle: From bending of elastic plates to mixing of viscous fluids,” Nauk. Zap. Kyiv. Nats. Univ. im. T. Shevchenka, 8, 45–62 (2003).MathSciNetGoogle Scholar
  29. 29.
    V. V. Meleshko, “Biharmonic problem in a rectangle: History and current status,” Mat. Met. Fiz.-Mekh. Polya, 47, No.3, 45–68 (2004).MathSciNetGoogle Scholar
  30. 30.
    V. V. Meleshko, “Biharmonic problem in a rectangle,” Izv. VUZov, Severo-Kavkaz. Region, Estestv. Nauki, Special Issue, 60–71 (2004).Google Scholar
  31. 31.
    N. I. Muskhelishvili, Some Basic Problems in the Mathematical Theory of Elasticity [in Russian], Izd. AN SSSR, Moscow (1954).Google Scholar
  32. 32.
    W. Nowacki, Issues of Thermoelasticity [Russian translation], Izd. AN SSSR, Moscow (1962).Google Scholar
  33. 33.
    P. F. Papkovich, Theory of Elasticity [in Russian], Oborongiz, Leningrad-Moscow (1939).Google Scholar
  34. 34.
    H. Parkus, Instationare Warmespannungen, Springer, Wien (1959).Google Scholar
  35. 35.
    S. P. Timoshenko, A Course on the Theory of Elasticity [in Russian], pt. 1, Izd. Inzh. Putei Soobshch., St-Petersburg (1914).Google Scholar
  36. 36.
    S. P. Timoshenko, Theory of Elasticity [in Russian], ONTI, Leningrad-Moscow (1934).Google Scholar
  37. 37.
    S. P. Timoshenko, Plates and Shells [in Russian], GITTL, Moscow-Leningrad (1948).Google Scholar
  38. 38.
    S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York (1970).Google Scholar
  39. 39.
    A. F. Ulitko, Method of Vector Eigenfunctions in Three-Dimensional Problems of Elasticity [in Russian], Naukova Dumka, Kiev (1979).Google Scholar
  40. 40.
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York (1965).Google Scholar
  41. 41.
    G. B. Airy, “On the strains in the interior of beams,” Phil. Trans. Roy. Soc. London, A153, 49–79 (1863).Google Scholar
  42. 42.
    N. M. Borodachev, “Solution of the three-dimensional thermoelastic problem in stresses,” Int. Appl. Mech., 39, No.4, 438–444 (2003).CrossRefMATHGoogle Scholar
  43. 43.
    N. M. Borodachev, “Thermoelastic solution for stresses,” Int. Appl. Mech., 41, No.3, 264–271 (2005).CrossRefGoogle Scholar
  44. 44.
    P. S. Bruckman, “On the evaluation of certain infinite series by elliptic functions,” Fibonacci Quart., 15, 293–310 (1977).MATHMathSciNetGoogle Scholar
  45. 45.
    A. F. Bulat, “Rock deformation problems,” Int. Appl. Mech., 40, No.12, 1311–1322 (2004).CrossRefGoogle Scholar
  46. 46.
    A. M. J. Davis, “Infinite systems for a biharmonic problem in a rectangle: discussion of non-uniqueness,” Proc. Roy. Soc. London, A459, 409–412 (2003).ADSGoogle Scholar
  47. 47.
    J. M. C. Duhamel, “Memoire sur le calcul des actions moleculaires developpees par les changements de la temperature dans les corps solides,” Mem. Acad. Sci. Savants Etrang., 5, 440–498 (1838).Google Scholar
  48. 48.
    V. T. Grinchenko, “The biharmonic problem and progress in the development of analytical methods for the solution of boundary-value problems,” J. Eng. Math., 46, 281–297 (2003).CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    E. Grosswald, “Comments on some formulae of Ramanujan,” Acta Arithmetica, 21, 25–34 (1972).MATHMathSciNetGoogle Scholar
  50. 50.
    A. N. Guz, “Design models in linearized solid mechanics,” Int. Appl. Mech., 40, No.5, 505–516 (2004).CrossRefMathSciNetGoogle Scholar
  51. 51.
    H. Hencky, Der Spannungszustand in rechteckigen Platten, Oldenbourg, Munchen, (1913).Google Scholar
  52. 52.
    C. E. Inglis, “Two dimensional stresses in rectangular plates,” Engineering, 112, 523–524 (1921).Google Scholar
  53. 53.
    C. E. Inglis, “Stresses in rectangular plates clamped at their edges and loaded with a uniformly distributed pressure,” Trans. Inst. Naval Arch., 67, 147–165 (1925).Google Scholar
  54. 54.
    A. D. Kovalenko, Thermoelasticity. Basic Theory and Applications, Wolters-Noordhoff, Groningen (1969).Google Scholar
  55. 55.
    G. Lame, Lecons sur la Theorie Mathematique de l'elasticite des Corps Solides, Mallet-Bachelier, Paris (1852).Google Scholar
  56. 56.
    E. Mathieu, “Memoire sur l'equation aux differences partielles du quatrieme ordre ΔΔu=0 et sur l'equilibre d'elasticite d'un corps solide,” J. Math. Pures Appl. (Ser. 2), 14, 378–421 (1869).MATHGoogle Scholar
  57. 57.
    E. Mathieu, “Memoire sur l'equilibre d'elasticite d'un prisme rectangle,” J. Ecole Polytech., 30, No.49, 173–196 (1881).Google Scholar
  58. 58.
    E. Mathieu, Theorie de l'elasticite des Corps Solides, Gauthier-Villars, Paris (1890).Google Scholar
  59. 59.
    J. C. Maxwell, “On reciprocal diagrams in space, and their relation to Airy's function of stress,” Proc. London Math. Soc., 2, 102–105 (1868).Google Scholar
  60. 60.
    V. V. Meleshko, “Equilibrium of elastic rectangle: Mathieu-Inglis-Pickett solution revisited,” J. Elasticity, 40, 207–238 (1995).CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    V. V. Meleshko, “Steady Stokes flow in a rectangular cavity,” Proc. Roy. Soc. London, A452, 1999–2022 (1996).ADSGoogle Scholar
  62. 62.
    V. V. Meleshko, “Bending of an elastic rectangular clamped plate: Exact versus “engineering” solutions,” J. Elasticity, 48, 1–50 (1997).CrossRefMATHMathSciNetGoogle Scholar
  63. 63.
    V. V. Meleshko, “Biharmonic problem in a rectangle,” Appl. Sci. Res., 58, 217–249 (1998).MATHMathSciNetGoogle Scholar
  64. 64.
    V. V. Meleshko, “Selected topics in the history of the two-dimensional biharmonic problem,” Appl. Mech. Revs., 56, 33–85 (2003).Google Scholar
  65. 65.
    V. V. Meleshko and A. M. Gomilko, “Infinite systems for a biharmonic problem in a rectangle,” Proc. Roy. Soc. London, A453, 2139–2160 (1997).ADSMathSciNetGoogle Scholar
  66. 66.
    V. V. Meleshko and A. M. Gomilko, “Infinite systems for a biharmonic problem in a rectangle: further discussion,” Proc. Roy. Soc. London, A460, 807–819 (2004).ADSMathSciNetGoogle Scholar
  67. 67.
    V. V. Meleshko, A. M. Gomilko, and A. A. Gourjii, “Normal reactions in a clamped elastic rectangular plate,” J. Eng. Math., 40, 377–398 (2001).CrossRefMathSciNetGoogle Scholar
  68. 68.
    F. E. Neumann, “Die Gesetze der Doppelbrechung des Lichtes in comprimirten oder ungleichformig erwarmten uncrystallinischen Korpern,” Abhandl. Konigl. Akad. Wissen, Berlin, 2, 1–254 (1841).Google Scholar
  69. 69.
    G. Pickett, “Application of the Fourier method to the solution of certain boundary problems in the theory of elasticity,” Trans. ASME, J. Appl Mech., 11, 176–182 (1944).MathSciNetGoogle Scholar
  70. 70.
    K. Schroder, “Das Problem der eingespannten rechteckigen elastischen Platte. I. Die biharmonische Randwertaufgabe fur das Rechteck,” Math. Ann., 21, 247–326 (1949).MathSciNetGoogle Scholar
  71. 71.
    P. N. Shankar, V. V. Meleshko and E. I. Nikiforovich, “Slow mixed convection in rectangular containers,” J. Fluid Mech., 471, 203–217 (2002).CrossRefADSMathSciNetGoogle Scholar
  72. 72.
    S. P. Timoshenko, “The approximate solution of two-dimensional problems in elasticity,” Phil. Mag. (Ser. 6), 47, 1095–1104 (1924).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. V. Meleshko
    • 1
  1. 1.T. Shevchenko Kiev National UniversityKievUkraine

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