DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

The history and development prospects of one of the methods for solving multidimensional problems of structural mechanics

Vestnik MGSU 12/2015
  • Stankevich Anatoliy Nikolaevich - Kyiv National University of Construction and Architecture (KNUCA) Candidate of Technical Sciences, Associate Professor, chair, Department of Strength of Materials, Kyiv National University of Construction and Architecture (KNUCA), 31 Vozdukhoflotskiy prospekt, Kiev, 03680, Ukraine; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 76-91

The equations describing stress-strain state of the majority of the calculation models in linear statement are considered in frames of the theory of elasticity (statics and dynamics) and thermoelasticity. These equations are defined in three-dimentional space or with account for symmetry in teo-dimentional space and are belong to complicated tasks of mathematical physics. The famous mathematicians and nechanics had already offered solutions for the simplest cases of such tasks. But it is impossible to solve the majority of tasks, especially the dynamic ones, using analythical solutions. In this work the authors deal with the modern methods for solving multidimensional problems of structural mechanics. The attention is paid to the methods of dimension reduction of the initial equations. The development and improvement of the method of lines is considered in detail, the shortcomings of the existing approaches are enumerated and the possible development direction of the method to the new task classes is offered.

DOI: 10.22227/1997-0935.2015.12.76-91

References
  1. Lur’e A.I. Prostranstvennye zadachi teorii uprugosti [Spatial Problems of Elasticity Theory]. Moscow, GITTL Publ., 1955, 491 p. (In Russian)
  2. Lyav A. Matematicheskaya teoriya uprugosti [Mathematical Theory of Elasticity]. Translated from English. Moscow, Leningrad, ONTI Publ., 1935, 674 p.
  3. Novatskiy V. Dinamicheskie zadachi termouprugosti [The Dynamic Thermoelasticity Problems]. Translated from Polish. Moscow, Mir Publ., 1970, 256 p. (In Russian)
  4. Kruz T., Ritstso F. Metod granichnykh integral’nykh uravneniy. Vychislitel’nye aspekty i prilozheniya v mekhanike : sbornik trudov [The Method of Boundary Integral Equations. Computational Aspects and applications on Mechanics : Collection of Works]. Translated from English. Moscow, Mir Publ., 1978, no. 15, 210 p. (Novoe v zarubezhnoy nauke. Mekhanika [New in the Foreign Science. Mechanics]) (In Russian)
  5. Veryuzhskiy Yu.V. Metod potentsiala v staticheskikh zadachakh stroitel’noy mekhaniki : dissertatsiya ... doktora tekhnicheskikh nauk [Potential Methods for Static Problems of Structural Mechanics. Dissertation of the Doctor of Technical Sciences]. Kiev, 1980, 431 p. (In Russian)
  6. Veryuzhskiy Yu.V. Chislennye metody potentsiala v nekotorykh zadachakh prikladnoy mekhaniki [Numerical Methods of Potential in Some Problems of Applied Mechanics]. Kiev, Vishcha shkola Publ.,1978, 183 p. (In Russian)
  7. Benerdzhi P., Batterfild R. Metod granichnykh elementov v prikladnykh naukakh [The Method of Boundary Elements in Applied Sciences]. Translated from English. Moscow, Mir Publ., 1984, 494 p. (In Russian)
  8. Brebbiya K., Uoker S. Primenenie metoda granichnykh elementov v tekhnike [Application of the Method of Boundary Elements in Engineering]. Moscow, Mir Publ., 1982, 248 p. (In Russian)
  9. Vaynberg D.V., Sinyavskiy A.L. Raschet obolochek [Calculation of Shells]. Kiev, Gosstroyizdat USSR Publ., 1961, 119 p. (In Russian)
  10. Kupradze V.D. O priblizhennom reshenii zadach matematicheskoy fiziki [Approximate Solution of the Problems of Mathematical Physics]. Uspekhi matematicheskikh nauk [Success of Mathematical Sciences]. 1967, vol. 22, no. 2 (134), pp. 59—107. (In Russian)
  11. Parton V.Z., Perlin P.I. Integral’nye uravneniya teorii uprugosti [Integral Equations of Elasticity Theory]. Moscow, Nauka Publ., 1977, 312 p. (In Russian)
  12. Ugodchikov A.G., Khutoryanskiy N.M. Metod granichnykh elementov v mekhanike deformiruemogo tverdogo tela [Boundary Element Method in Solid Mechanics]. Kazan, Izdatel’stvo Kazanskogo universiteta Publ., 1986, 295 p. (In Russian)
  13. Lur’e A.I. K teorii tolstykh plit [On the Theory of Thick Plates]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1942, vol. 6, no. 2—3, pp. 151—168. (In Russian)
  14. Kil’chevskiy N.A. Obobshchenie sovremennoy teorii obolochek [Generalization the Modern Theory of Shells]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1939, vol. 2, no. 4, pp. 427—438. (In Russian)
  15. Vlasov V.Z. Metod nachal’nykh funktsiy v zadachakh teorii uprugosti [The Method of Initial Functions in the Axisymmetric Problem of the Theory of Elasticity]. Izvestiya’ AN SSSR, OTN. [News of the Academy of Sciences of the USSR. Technical Sciences]. 1955, no. 7, pp. 49—69. (In Russian)
  16. Selezov І.T. Doslіdzhennya poperechnikh kolivan’ plastin [Oscillations Research of Transverse Plates]. Prikladna mekhanіka [Applied Mechanics]. 1960, vol. 6, no. 5, pp. 319—326. (In Ukrainian)
  17. Selezov I.T., Kil’chitskaya G.A. Privedenie trekhmernoy dinamicheskoy zadachi termouprugosti dlya sloya postoyannoy tolshchiny [Bringing the Three-Dimensional Dynamic Thermoelasticity Problem for a Layer of Constant Thickness]. Teplovye napryazheniya v elementakh konstruktsiy : doklady nauchnogo soveshchaniya [Thermal Stresses in Structural Elements : Reports of the Scientific Conference]. Kiev, Naukova dumka Publ., 1964, no. 4, pp. 172—179. (In Russian)
  18. Vorovich I.I. Nekotorye matematicheskie voprosy teorii plastin i obolochek [Some Mathematical Problems in the Theory of Plates and Shells]. Trudy Vtorogo Vsesoyuznogo s”ezda po teoreticheskoy i prikladnoy mekhanike [Works of the Second All-Union Meeting on Theoretical and Applied Mechanics]. Moscow, Nauka Publ., 1966, no. 3, pp. 116—136. (In Russian)
  19. Vorovich I.I., Malkina O.S. Napryazhennoe sostoyanie tolstoy plity [Stress State of a Thick Plate]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1967, vol. 31, no. 2, pp. 230—241. (In Russian)
  20. Vorovich I.I., Prokopov V.K. Nekotorye voprosy trekhmernoy teorii uprugosti [Some Questions of the Three-Dimensional Theory of Elasticity]. III Vsesoyuznyy s”ezd po teorii i prikladnoy mekhanike. Annotatsiya dokladov [The 3rd All-Union Meeting on the Theory and Applied Mechanics. Annotation of Reports]. Moscow, Nauka Publ., 1968, p. 81. (In Russian)
  21. Vlasov V.V. Primenenie metoda nachal’nykh funktsiy k raschetu tolstykh plit [Application of the Method of Initial Functions to the Calculation of Thick Plates]. Issledovaniya po teorii sooruzheniy : sbornik [Investigations on Structural Analysis : Collection]. Moscow, Gosstroyizdat Publ., 1961, no. 10, pp. 189—207. (In Russian)
  22. Vlasov V.V. Metod nachal’nykh funktsiy v osesimmetrichnoy zadache teorii uprugosti [The Method of Initial Functions in the Theory of Elasticity]. Raschet plastin i obolochek [Calculation of Plates and Shells]. 1963, no. 34, pp. 31—45. (In Russian)
  23. Vlasov V.Z., Leont’ev N.N. Balki, plity i obolochki na uprugom osnovanii [Beams, Plates and Shells on the Elastic Foundation]. Moscow, Fizmatgiz Publ., 1960, 491 p. (In Russian)
  24. Gol’denveyzer A.L. Postroenie priblizhennoy teorii izgiba plastinki metodom asimptoticheskogo integrirovaniya uravneniy teorii uprugosti [An Approximate Theory of Bending a Plate Using the Method of Asymptotic Integration of the Equations of Elasticity Theory]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1962, vol. 26, no. 4, pp. 668—686. (In Russian)
  25. Gol’denveyzer A.L. Asimptoticheskiy metod postroeniya teorii obolochek [The Asymptotic Method of Constructing the Theory of Shells]. Materialy I Vsesoyuznoy shkoly po teorii i chislennym metodam rascheta obolochek i plastin [Materials of the 1st All-Russian School on the Theory and Numerical Methods of Calculating Shells and Plated]. Tbilisi, Izdatel’stvo Tbilisskogo universiteta Publ., 1975, pp. 151—213. (In Russian)
  26. Vekua I.N. Teoriya tonkikh pologikh obolochek peremennoy tolshchiny [The Theory of Thin Shallow Shells of Variable Thickness]. Trudy Tbilisskogo matematicheskogo instituta [Works of the Tbilisi Mathematical Institute]. Tbilisi, Metsniereba Publ., 1965. T. 30. 103 p. (In Russian)
  27. Akhiezer V.G. Klassicheskaya problema momentov i nekotorye voprosy analiza, svyazannye s ney [Classical Moment Problem and Some Related Questions in Analysis]. Moscow, Fizmatgiz Publ., 1961, 310 p.
  28. Gobson E.V. Teoriya sfericheskikh i ellipsoidal’nykh funktsiy [Theory of Spherical and Ellipsoidal Functions]. Translated from English. Moscow, Izdatel’stvo inostrannoy literatury Publ., 1952, 476 p. (In Russian)
  29. Vekua I.N. Nekotorye obshchie metody postroeniya razlichnykh variantov teorii obolochek [Some Common Methods of Constructing Different Versions of the Theory of Shells]. Moscow, Nauka Publ., 1982, 288 p. (In Russian)
  30. Isakhanov G.V., Chibiryakov V.K., Smolyar A.M. Chislenno-analiticheskiy metod resheniya zadach statiki tolstykh neodnorodnykh plastin [Numerical-Analytical Method for Solving Statics Problems of Thick Inhomogeneous Plates]. Trudy 13-y Vsesoyuznoy konferentsii po teorii plastin i obolochek [Works of the 13th All-Russian Conference on the Theory of Plates and Shells]. Tallin, 1983, part 2, pp. 130—135. (In Russian)
  31. Isakhanov G.V., Chibiryakov V.K. Razvitie metoda Vekua I.N. v statike i dinamike tolstykh plastin [Development of the Method of I.N. Vekua in Statics and Dynamics of Thick Plates]. Tezisy dokladov 2-y Vsesoyuznoy konferentsii po teorii uprugosti [Abstracts of the 2nd All-Union Conference on Elasticity Theory]. Tbilisi, 1984, 122 p. (In Russian)
  32. Chibiryakov V.K. Obobshchennyy metod konechnykh integral’nykh preobrazovaniy v statike i dinamike netonkikh plastin [Generalized Method of Integral Transformations in Statics and Dynamics of Non-Thin Plates]. Soprotivlenie materialov i teoriya sooruzheniy : respublikanskiy mezhvedomstvennyy nauchnyy sbornik [Strength of Materials and Theory of Structures : Republican Interdepartmental Scientific Collection]. Kiev, Budivel’nik Publ., 1982, no. 40, pp. 90—95. (In Russian)
  33. Chibiryakov V.K., Smolyar A.M. Ob odnom obobshchenii metoda konechnykh integral’nykh preobrazovaniy v teorii tolstykh plastin [On a Generalization of the Method of Integral Transformations in the Theory of Thick Plates]. Soprotivlenie materialov i teoriya sooruzheniy : respublikanskiy mezhvedomstvennyy nauchnyy sbornik [Strength of Materials and Theory of Structures : Republican Interdepartmental Scientific Collection]. Kiev, Budivel’nik Publ., 1983, no. 42, pp. 80—86. (In Russian)
  34. Chibіryakov V.K., Smolyar A.M. Teorіya tovstikh plastin ta obolonok [The Theory of Thick Plates and Shells]. Cherkasi, ChDTU Publ., 2002, 160 p. (In Ukrainian)
  35. Chibiryakov V.K. Ob odnom variante uravneniy tsilindricheskogo izgiba netonkikh plastin [On a Variant of the Cylindrical Bending Equations of Non-Thin Plates]. Soprotivlenie materialov i teoriya sooruzheniy : respublikanskiy mezhvedomstvennyy nauchnyy sbornik [Strength of Materials and Theory of Structures : Republican Interdepartmental Scientific Collection]. Kiev, Budivel’nik Publ., 1977, no. 31, pp. 59—67. (In Russian)
  36. Chibiryakov V.K. Chislennoe reshenie zadach statiki i dinamiki tolstykh plastin [Numerical Solution of the Problems of Statics and Dynamics of Thick Plates]. Chislennye metody resheniya zadach stroitel’noy mekhaniki : sbornik nauchnykh statey [Numerical Methods of Solcing the Problems of Structural Mechanics : Collection of Scientific Articles]. Kiev, KISI Publ., 1978, pp. 153—157. (In Russian)
  37. Chibіryakov V.K., Zhupanenko І.V. Vlasnі kolivannya tovstoї tsilіndrichnoї obolonki [Own Fluctuations of Thick Cylindrical Shell]. Opіr materіalіv і teorіya sporud : naukovo-tekhnіchniy zbіrnik [Resistance of Materials and Theory of Structures : Scientific and Technical Collection]. Kiev, KNUBA Publ., 2009, no. 84, pp. 127—133. (In Ukrainian)
  38. Chibіryakov V.K., Zhupanenko І.V. Vlasnі kolivannya tovstostіnnoї obolonki obertannya zmіnnoї tovshchini [Proper Rotation Oscillation of a Thick Shell of Variable Thickness]. Promislove budіvnitstvo ta іnzhenernі sporudi [Industrial Construction and Engineering Structures]. 2010, no. 2, pp. 5—9. (In Ukrainian)
  39. Chibіryakov V.K., Zhupanenko І.V. Metodika rozv’yazannya zadachі pro vlasnі kolivannya plastin obertannya zmіnnoї tovshchini [Methods for Solving the Problem of the Oscillations of Plates of Variable Thickness]. Opіr materіalіv і teorіya sporud : naukovo-tekhnіchniy zbіrnik [Resistance of Materials and Theory of Structures : Scientific and Technical Collection]. Kiev, KNUBA Publ., 2010, no. 86, pp. 30—46. (In Ukrainian)
  40. Chibіryakov V.K., Zhupanenko І.V. Pro odin algoritm rozrakhunku vіsesimetrichnikh kolivan’ krugloї plastini [On One Algorithm of Axially Symmetric Vibrations of Circular Plates]. Opіr materіalіv і teorіya sporud : naukovo-tekhnіchniy zbіrnik [Resistance of Materials and Theory of Structures : Scientific and Technical Collection]. Kiev, KNUBA Publ., 2007, no. 81, pp. 43—50. (In Ukrainian)
  41. Chibiryakov V.K., Boyko K.E. Opredelenie chastot i form sobstvennykh kolebaniy po utochnennym teoriyam plastin [Definition of Frequencies and Forms of Natural Oscillations in Refined Theories of Plates]. Soprotivlenie materialov i teoriya sooruzheniy : respublikanskiy mezhvedomstvennyy nauchnyy sbornik [Strength of Materials and Theory of Structures : Republican Interdepartmental Scientific Collection]. Kiev, Budivel’nik Publ., 1985, no. 47, pp. 74—80. (In Russian)
  42. Chibiryakov V.K., Boyko K.E. Osesimmetrichnye sobstvennye kolebaniya tolstykh plastin peremennoy tolshchiny [Axisymmetric Eigenmodes of Thick Plates of Variable Thickness]. Soprotivlenie materialov i teoriya sooruzheniy : respublikanskiy mezhvedomstvennyy nauchnyy sbornik [Strength of Materials and Theory of Structures : Republican Interdepartmental Scientific Collection]. Kiev, Budivel’nik Publ., 1986, no. 49, pp. 54—58. (In Russian)
  43. Kantorovich L.V. Odin pryamoy metod priblizhennogo resheniya zadachi o minimume dvoynogo integrala [One Direct Method for the Approximate Solution of the Problem of the Minimum of a Double Integral]. Izvestiya AN SSSR. Seriya 7. Otdelenie matematicheskikh i estestvennykh nauk [News of the Academy of Sciences of the USSR. Series 7. Department of Mathematical and Natural Sciences]. 1933, no. 5, pp. 647—652. (In Russian)
  44. Sadskul M.N.O., Obiozor C.N. A Simple Introduction to the Method of Lines. International Journal of Electrical Engineering Education. 2000, vol. 37/3, pp. 282—296.
  45. Vinokurov L.P. Reshenie prostranstvennoy zadachi teorii uprugosti v peremeshcheniyakh [Solution of the Spatial Problem of Elasticity in Displacements]. Byulleten’ Khar’kovskogo inzhenerno-stroitel’nogo instituta [Bulletin of Kharkov Engineering and Construction Institute]. 1940, no. 18. (In Russian)
  46. Slobodyanskiy M.G. Sposob priblizhennogo integrirovaniya uravneniy s chastnymi proizvodnymi i ego primenenie k zadacham teorii uprugosti [The Method of Approximate Integration of Partial Differential Equations and Its Application to the Problems of the Theory of Elasticity]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1939, vol. 3, no. 1, pp. 75—82. (In Russian)
  47. Faddeeva V.N. Metod pryamykh v primenenii k nekotorym kraevym zadacham [Method of Lines Applied to Some Boundary Problems]. Trudy matematicheskogo instituta imeni V.A. Steklova [Works of the Mathematical Institute Named after V.A. Steklov]. 1949, vol. XXVIII, pp. 73—103. (In Russian)
  48. Kantorovich L.V., Frumkin P.V. Primenenie odnogo metoda priblizhennogo resheniya uravneniy v chastnykh proizvodnykh k resheniyu zadachi o kruchenii prizmaticheskikh sterzhney [On the Application of a Method for the Approximate Solution of Partial Differential Equations and the Problem of Torsion of Prismatic Bars]. Trudy Leninigradskogo instituta inzhenerov promyshlennogo stroitel’stva [Works of the Leningrad Institute of Industrial Construction Engineers]. 1937, no. 4, pp. 111—112. (In Russian)
  49. Slobodyanskiy M.G. Prostranstvennye zadachi teorii uprugosti dlya prizmaticheskikh tel [Spatial Problem of Elasticity Theory for Prismatic Bodies]. Uchenye zapiski Moskovskogo gosudarstvennogo universiteta. Mekhanika [Scientific Notes of the Moscow State University. Mechanics]. 1940, no. 39. (In Russian)
  50. Slobodyanskiy M.G. Otsenka pogreshnosti iskomoy velichiny pri reshenii lineynykh zadach variatsionnym metodom [Errors Estimate for the Unknown Quantity in the Solution of Linear Problems Using the Variational Method]. DAN SSSR [Reports of the Academy of Sciences of the USSR]. 1952, vol. 86, no. 2, pp. 243—246. (In Russian)
  51. Slobodyanskiy M.G. Otsenki pogreshnosti priblizhennogo resheniya v lineynykh zadachakh, svodyashchayasya k variatsionnym, i ikh primenenie k opredeleniyu dvustoronnikh priblizheniy v staticheskikh zadachakh teorii uprugosti [Errors Estimation of Approximate Solutions to Linear Problems, which are Reduced to the Variationals and their Application to the Determination of Two-sided Approximations for Static Problems of Elasticity]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1952, vol. 16, no. 4, pp. 449—464. (In Russian)
  52. Slobodyanskiy M.G. Otsenka pogreshnostey priblizhennykh resheniy lineynykh zadach [Errors Estimation of Approximate Solutions of Linear Problems]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1953, vol. 17, no. 2, pp. 229—244. (In Russian)
  53. Slobodyanskiy M.G. O priblizhennom reshenii lineynykh zadach, svodyashchikhsya k variatsionnym [On the Approximate Solution of Linear Problems Reduced to Variationals]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1953, vol. 17, no. 5, pp. 623—626. (In Russian)
  54. Slobodyanskiy M.G. Priblizhennoe reshenie nekotorykh kraevykh zadach dlya ellipticheskogo deferentsial’nogo uravneniya i otsenka pogreshnosti [Approximate Solution of Some Boundary Value Problems for Elliptic Deferential Equation and Error Estimate]. DAN SSSR [Reports of the Academy of Sciences of the USSR]. 1953, vol. 89, no. 2. (In Russian)
  55. Slobodyanskiy M.G. O preobrazovanii problemy minimuma funktsionala k probleme maksimuma [On the Transformation of the Problem of Functional Minimum to the Problem of Maximum]. DAN SSSR [Reports of the Academy of Sciences of the USSR]. 1953, vol. 91,no. 4, pp. 733—736. (In Russian)
  56. Slobodyanskiy M.G. O postroenii priblizhennogo resheniya v lineynykh zadachakh [On the Construction of Approximate Solutions to Linear Problems]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1955, vol. 19, no. 5. (In Russian)
  57. Vinokurov L.P. Pryamye metody resheniya prostranstvennykh i kontaktnykh zadach dlya massivov i fundamentov [Direct Methods for Solving Spatial and Contact Problems for Soils and Foundations]. Kharkov, KhGU Publ., 1956, 280 p. (In Russian)
  58. Vinokurov L.P. Raschet plit na uprugom poluprostranstve s primeneniem inzhenerno-diskretnogo metoda [Calculation of Plates on Elastic Half-Space Using Integro-Discrete Method]. Vestnik inzhenerov i tekhnikov [Proceedings of Engineers and Technicians]. 1951, no. 4, pp. 166—171. (In Russian)
  59. Vinokurov L.P. Priblizhennye metody resheniya differentsial’nykh uravneniy stroitel’noy mekhaniki [Approximate Methods for Solving Differential Equations of Structural Mechanics]. Trudy KhISI [Works of Kharkov Institute of Construction and Engineering]. 1951, no. 3. (In Russian)
  60. Vinokurov L.P. Raschet kolodtsa na uprugom osnovanii, nagruzhennogo silami, ne lezhashchimi v ploskosti krivizny kolodtsa [Calculation of a Well on an Elastic Foundation Loaded with the Forces That Do Not Lie in the Plane of the Well Curvature]. Vestnik inzhenerov i tekhnikov [Proceedings of Engineers and Technicians]. 1938, no. 1. (In Russian)
  61. Vinokurov L.P. Priblizhennyy metod resheniya ploskikh zadach teorii uprugosti [An Approximate Method of Solving Plane Elasticity Problems]. Trudy KhISI [Works of Kharkov Institute of Construction and Engineering]. 1949, vol. II. (In Russian)
  62. Petrov Yu.P. Raschet na izgib uprugikh pryamougol’nykh plastin diskretnym metodom [Discrete Method Calculation of Bending of Orthotropic Elastic Plates]. Trudy Khar’kovskogo aviatsionnogo instituta [Works of Kharkov Aviation Institute]. 1961, no. 18, pp. 86—99. (In Russian)
  63. Petrov Yu.P. Osnovy rascheta na izgib plastin diskretnym metodom [Bases of Bending Calculation of Plates Using Discrete Method]. Trudy Khar’kovskogo aviatsionnogo instituta [Works of Kharkov Aviation Institute]. 1961, no. 18, pp. 67—83. (In Russian)
  64. Petrov Yu.P. Raschet na izgib kosozashchemlennoy konsol’noy plastiny peremennoy tolshchiny [Bending Calculation of Sidely Restrained Console Plate of Variable Thickness]. Trudy Khar’kovskogo aviatsionnogo instituta [Works of Kharkov Aviation Institute]. 1963, no. 22, pp. 62—78. (In Russian)
  65. Petrov Yu.P. Raschet na izgib diskretnym metodom ortotropnykh uprugikh plastin [Bending Calculation of Orthotropic Elastic Using Discrete Method]. Trudy Khar’kovskogo aviatsionnogo instituta [Works of Kharkov Aviation Institute]. 1963, no. 22, pp. 79—86. (In Russian)
  66. Petrov Yu.P. Raschet na izgib plastin s lineynym izmeneniem tolshchiny diskretnym metodom [Bending Calculation of Plates with Linear Variation of Thickness Using Discrete Method]. Trudy Khar’kovskogo aviatsionnogo instituta [Works of Kharkov Aviation Institute]. 1961, no. 18, pp. 79—86. (In Russian)
  67. Shkelev L.T. Ispol’zovanie metoda pryamykh dlya resheniya bigarmonicheskogo uravneniya [Using the Method of Lines for Solving Biharmonic Equation]. Referativnaya informatsiya o zakonchenykh nauchno-issledovatel’skikh rabotakh v VUZakh USSR. Stroitel’naya mekhanika, raschet sooruzheniy : sbornik [Reference Information on the Concluded Scientific and Research Works in the Higher Institutions of the USSR. Structural Mechanics, Calculation of Structures : Collection]. Kiev, Vyshcha shkola Publ., 1971, no. 2. (In Russian)
  68. Shkelev L.T. Raschet plastin proizvol’noy formy v polyarnykh koordinatakh. Ploskoe izgibnoe napryazhennoe sostoyanie [Calculation of Plates of Arbitrary Forms in Polar Coordinates. Flat Bending Stress State]. Referativnaya informatsiya : sbornik [Reference Information : Collection]. 1971, no. 2. (In Russian)
  69. Shkelev L.T. Reshenie kraevoy zadachi dlya bigarmonicheskogo uravneniya metodom pryamykh v polyarnykh koordinatakh [Solution of the Boundary Problem for Biharmonic Equation by the Method of Lines in Polar Coordinates]. Referativnaya informatsiya : sbornik [Reference Information: Collection]. 1972, no. 3. (In Russian)
  70. Korbakov A.F. Razvitie i primenenie metoda pryamykh k issledovaniyu slozhnogo napryazhennogo i deformirovannogo sostoyaniya plastin i plastinchatykh sistem : dissertatsiya ... doktora tekhnicheskikh nauk [Development and Application of the Method of Lines to Study Complex Stress and Strain State of Plates and Plate Systems. Dissertation of the Doctor of Technical Sciences]. Kiev, 1985. (In Russian)
  71. Morskov Yu.A. Raschet izgibaemykh plastin proizvol’noy formy [Calculation of Bent Plates of Arbitrary Shape]. Referativnaya informatsiya o zakonchenykh nauchno-issledovatel’skikh rabotakh v VUZakh USSR. Stroitel’naya mekhanika, raschet sooruzheniy : sbornik [Reference Information on the Concluded Scientific and Research Works in the Higher Institutions of the USSR. Structural Mechanics, Calculation of Structures : Collection]. Kiev, 1977, no. 9. (In Russian)
  72. Morskov Yu.A. Reshenie nekotorykh zadach izgiba dvukhsvyaznykh plastin proizvol’nogo ochertaniya [Solution of Some Bending Problems of Doubly Connected Plates of Arbitrary Shape]. Soprotivlenie materialov i teoriya sooruzheniy : sbornik [Strength of Materials and Theory of Structures : Collection]. Kiev, Budіvel’nik Publ., 1977, no. 31. (In Russian)
  73. Morskov Yu.A. Primenenie metoda pryamykh v polyarnykh koordinatakh k resheniyu zadach izgiba plastin proizvol’noy formy [Application of the Method of Lines in Polar Coordinates for the Problems of Bending Plates of Arbitrary Shape]. Soprotivlenie materialov i teoriya sooruzheniy : sbornik [Strength of Materials and Theory of Structures : Collection]. Kiev, Budіvel’nik Publ., 1979, no. 34. (In Russian)
  74. Morskov Yu.A. Priblizhennyy metod rascheta na prochnost’ plastin i plastinchatykh sistem (na osnove metoda pryamykh) : dissertatsiya ... kandidata tekhnichaskikh nauk [Approximate Method for Calculating the Strength of Plates and Plate Systems (Basing on the Method of Lines) : Dissertation of the Candidate of Technical Sciences]. Kiev, 1979. (In Russian)
  75. Odinets E.A. Opredelenie napryazhennogo i deformirovannogo sostoyaniya mnogosloynykh plastin metodom pryamykh : dissertatsiya ... kandidata tekhnichaskikh nauk [Determination of Stress and Strain State of Multi-Plates Using the Method of Lines : Dissertation of the Candidate of Technical Sciences]. Kiev, 1988. (In Russian)
  76. Stankevich A.N. Razvitie metoda pryamykh k raschetu sostavnykh tsilindricheskikh obolochek : dissertatsiya ... kandidata tekhnichaskikh nauk [Development of the Method of Lines to the Calculation of the Components of Cylindrical Shells : Dissertation of the Candidate of Technical Sciences]. Kiev, 1996. (In Russian)
  77. Shkelev L.T., Morskov Yu.A., Romanova T.A., Stankevich A.N. Metod pryamykh i ego ispol’zovanie pri opredelenii napryazhennogo i deformirovannogo sostoyaniy plastin i obolochek [Method of Lines and Its Use in Determining the Stress and Strain State of Plates and Shells]. Kiev, 2002, 177 p. (In Russian)
  78. Shkelev L.T., Stankevich A.N., Poshivach D.V., Morskov Yu.A., Karbakov A.F. Primenenie metoda pryamykh dlya opredeleniya napryazhennogo i deformirovannogo sostoyaniy prostranstvennykh i plastinchatykh konstruktivnykh elementov [Application of the Method of Lines to Determine the Stress and Strain States of Spatial and Plate Structural Elements]. Kiev, KNUSA Publ., 2004, 136 p. (In Russian)
  79. Godunov S.K. O chislennom reshenii kraevykh zadach dlya sistem lineynykh obyknovennykh differenial’nykh uravneniy [On Numerical Solution of Boundary Value Problems for Systems of Linear Ordinary Differential Equations]. Uspekhi matematicheskikh nauk [Success of Mathematical Sciences]. 1961, vol. 16, no. 3 (99), pp. 171—174. (In Russian)
  80. Vlaykov G.G., Grigorenko A.Ya., Shevchenko S.N. Nekotorye zadachi teorii uprugosti dlya anizotropnykh tsilindrov s nekrugovym poperechnym secheniem [Some Problems in the Theory of Elasticity for Anisotropic Cylinders with Non-Circular Cross-Section]. Kiev, 2001, 143 p. (In Russian)
  81. Vlaykov G.G., Grigorenko A.Ya. Nekotorye osesimmetrichnye zadachi statiki i dinamiki anizotropnykh tel tsilindricheskoy formy [Some Axisymmetric Problems of Statics and Dynamics of Anisotropic Cylindrical Bodies]. Kiev, 1998, 58 p. (In Russian)
  82. Korbach V.G. Algoritm chislennogo resheniya mnogotochechnykh kraevykh zadach mekhaniki deformirovannogo tverdogo tela [Algorithm for the Numerical Solution of Multipoint Boundary Value Problems in the Mechanics of Deformable Solid]. Prochnost’ konstruktsiy letatel’nykh apparatov : sbornik nauchnykh trudov [Stability of the Structures of Flying Machines : Collection of Scientific Papers]. Kharkov, Khar’kovskiy aviatsionnyy institut Publ., 1990, pp. 88—95. (In Russian)

Download

MODIFIED METHOD OF LINES

Vestnik MGSU 8/2016
  • Stankevich Anatoliy Nikolaevich - Kyiv National University of Construction and Architecture (KNUCA) Candidate of Technical Sciences, Associate Professor, chair, Department of Strength of Materials, Kyiv National University of Construction and Architecture (KNUCA), 31 Vozdukhoflotskiy prospekt, Kiev, 03680, Ukraine; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 34-47

The main idea of the method of solving multidimensional boundary problems is reduction of initial differential equations in partial derivatives to the system of ordinary differential equations. One-dimensional resolving equations allow extending the potential of the method of lines. Although efficient and highly-precise numerical methods have been developed for solution of one-dimensional and initial-boundary value problems their use is impossible in the method of lines. The author considers a new method of lines which is used in order to reduce the dimensionality of multidimensional problems of structural mechanics. The method is used for calculation of thick plates, plates of variable thickness, heterogeneous and multilayered plates. It is proposed to replace finite-difference relations by projection relations which will extend the potential of the method of lines and will allow using the method in the dynamic problems.

DOI: 10.22227/1997-0935.2016.8.34-47

Download

Results 1 - 2 of 2