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O kryteriach oceny elementów skończonych. Od belki Timoshenki do płyty o średniej grubości

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EN
On the criteria for evaluation of finite elements. From Timoshenko beam to Hencky-Boole plate
Języki publikacji
PL
Abstrakty
PL
Celem pracy jest zaproponowanie i weryfikacja względnie prostych kryteriów oceny poprawności sformułowania elementów skończonych - kryteriów niesprzecznych z kryteriami matematycznymi. W pracy zaproponowano cztery kryteria oceny poprawności modelu MES: - analizę spektralną macierzy sztywności elementu, - analizę spektralną macierzy mas elementu, - analizę spektralną macierzy sztywności geometrycznej elementu, - kryterium energetyczne. Kryteria te opisano teoretycznie i zastosowano do analizy wybranych elementów skończonych. Przykłady zostały dobrane tak, aby pokazać możliwość zastosowania wszystkich czterech kryteriów w różnych zadaniach -analizie elementów o różnej liczbie węzłów, różnej geometrii i różnych funkcjach kształtu. Uzyskane wyniki należy uznać za bardzo obiecujące - pozwalają one przeanalizować poprawność dowolnego elementu skończonego w zadaniach statyki, drgań i stateczności początkowej. Analiza spektralna macierzy sztywności elementu pozwala określić, czy element spełnia matematyczny warunek eliptyczności. Energii odkształcenia, której jądrem jest macierz sztywności, nadano interpretację fizyczną w postaci hiper-elipsy energii. Wprowadzono dwa dodatkowe parametry - objętość hiper-elipsy i wskaźnik uwarunkowania macierzy - które uzupełniają warunek eliptyczności o informacje jakościowe o numerycznych własnościach elementu. Analiza spektralna macierzy mas pozwala określić czy element w poprawny sposób odwzorowuje energię kinetyczną dźwigara. Kryterium to jest odpowiednikiem warunku eliptyczności przeniesionym na zadania drgań. Podobnie jak poprzednio objętość hiper-elipsy energii kinetycznej i wskaźnik uwarunkowania pozwalają uzyskać dodatkowe informacje o własnościach numerycznych elementu. Analiza spektralna macierzy sztywności geometrycznej pozwala określić poprawność odwzorowania dodatkowej energii odkształcenia związanej z działaniem dużych sił osiowych. Energetyczne kryterium oceny poprawności sformułowania MES odpowiada matematycznemu warunkowi zgodności elementu, rozszerzonemu na zagadnienia drgań i stateczności początkowej. Kryterium pozwala określić, czy element skończony przy malejących wymiarach odwzorowuje w poprawny sposób gęstość energii sprężystej, energii kinetycznej, dodatkowej energii sprężystej (związanej z działaniem dużych sił osiowych) i pracę sił zewnętrznych. Kryterium energetyczne pozwala określić, czy element jest zbieżny do wyniku dokładnego, rozumianego w sensie energetycznym, oraz dodatkowo ocenić rząd zbieżności elementu. Proponowane kryteria oceny dotyczą w zasadzie modelu przemieszczeniowego MES, w pracy pokazano jednak jak można je uogólnić na modele mieszane. Pracę uzupełnia obszerny przegląd literatury dotyczącej budowy i oceny elementów skończonych belek i płyt o średniej grubości.
EN
The objective of this paper is to propose and to verify relatively simple criteria for evaluation of correctness of the finite element formulations. The criteria should be in agreement with mathematical theorems. Four criteria are proposed for evaluation the finite elements: - spectral analysis of the element stiffness matrix, - spectral analysis of the element mass matrix, - spectral analysis of the element geometric stiffness matrix, - energy criterion. The criteria are described theoretically and implemented for the analysis of selected finite elements. The examples are completed to show the possibility of implementation of the proposed criteria for various situations - the elements with different number of nodes and degrees of freedom, the elements of various geometry, the element with various shape functions, etc. The results are very promising - it is possible to evaluate the correctness of the formulation for any kind of finite element for static, vibration and initial stability problems. Spectral analysis of the element stiffness matrix gives the information if the element satisfy the mathematical condition of ellipticity. The strain energy of the element is described as a hiper-ellipsoid in multi dimensional space od degrees of freedom. Additional parameters - volume of hiper-ellipsoid and matrix condition ratio - gives the additional information on numerical sensitivity of the element. Spectral analysis of the element mass matrix can help to evaluate if the element describes the kinetic energy correctly. This criterion is equivalent to the ellipticity condition for vibration analysis. Volume of hiper-ellipsoid and mass matrix condition ratio are also discussed. Spectral analysis of the element geometric stiffness matrix is to define if the finite element describes correctely the additional strain energy combined with the in-plane forces. Energy criterion of the correctness of the FEM formulation is equivalent to the mathematical condition of consistency, extended from the static analysis to vibrations and initial stability. This criterion can help to evaluate if the densities of strain energy, kinetic energy, geometric strain energy and external load work of the finite element are described properly with the use of certain shape functions, methods of integration, etc. The convergence of the formulation to the correct value of energy can be evaluated, as well as the rate of convergence can be estimated. The proposed set of criterion is extended for the mixed formulation of the FEM. Some examples are given. The paper is completed with the wide analysis of the literature dedicated for the finite element formulation and evaluation in the field of Timoshenko beams and Hencky-Boole plates.
Rocznik
Tom
Strony
3--117
Opis fizyczny
Bibliogr. 619 poz., rys., tab., wykr.
Twórcy
autor
  • Instytut Mechaniki Konstrukcji Inżynierskich, Wydział Inżynierii Lądowej Politechniki Warszawskiej
Bibliografia
  • Część 1 - publikacje ogólne oraz dotyczące oceny elementów skończonych:
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  • [9] Gilewski W., Correctness of plate-bending element with physical shape functions. Finite Element News, 3, 1993, str. 29-34
  • [10] Gilewski W., Evaluation of lumped mass matrices for moderately thick plate. International Conference on "Modern Building Materials, Structures and Techniques", Wilno, Litwa, Vilnius Gediminas Technical University Press "Technika", 2004, str. 747-752
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  • Część 2 - publikacje dotyczące elementów skończonych belek i płyt o średniej grubości:
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  • [50] Chatterjee A., Setlur A.V., A mixed finite element formulation for plate problems. IJNME, 4, 1, 1972, str. 67-84
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  • [52] Davis R., Henshell R.D., Warburton G.B., A Timoshenko beam finite element. J. Sound Vibr., 22, 1972, str. 475-487
  • [53] Khatua T.P., Cheung Y.K., Triangular element for multilayered sandwich plates. Jour. Eng. Mech. Division, Proc. ASCE, EMS, 1972, str. 1225-1238
  • [54] Kikuchi F., Ando Y., Rectangular finite element for plate bending analysis based on Hellinger-Reissner's variational principle. Journ. Nuc. Sci. Technology, 9, 1, 1972, str. 28-35
  • [55] Mau S.T., Tong P., Pian T.H.H., Finite element solutions for laminated thick plates. Journ. Composite Materials, 6, 1972, str. 304-311
  • [56] Nickell R.E., Secor G.A., Convergence of consistently derived Timoshenki beam finite elements. IJNME, 5, 1972, str. 243-253
  • [57] Bergan P.G., Clough R.W., Large deflection analysis of plates and shallow shells using the finite element method. IJNME, 5, 1973, str. 543-556
  • [58] Fried I., Yang S.K., Triangular, nine degrees of freedom, C-0 plate bending element of quadratic accuracy. Quart. Appl. Mathematics, 31, 3, 1973, str. 303-312
  • [59] Mau S.T., Pian T.H.H., Tong P., Vibration analysis of laminated plates and shells by a hybrid stress element. AIAA J., 11, 1973, str. 1450-1452
  • [60] Thomas D.L., Wilson J.M., Wilson R.R., Timoshenko beam finite elements. Journ. Sound and Vibration. 31, 1973, str. 315-330
  • [61] Mawenya A.S., Davies J.D., Finite element bending analysis of multilayer plates. IJNME, 8, 2, 1974, str. 215-225
  • [62] Venkateswara Rao G., Venkataramana J., Prakasa Rao B., Vibrations of thick plates using a high precision triangular element. Nuc. Eng. Design, 31, 1, 1974, str. 102-105
  • [63] Venkateswara Rao G., Venkataramana J., Raju I.S., A high precision triangular plate bending element for the analysis of thick plates. Nuc. Eng. Design, 30, 1974, str. 408-412
  • [64] Hinton E., Owen D.R.J., Shantaram O., Dynamic transient linear and nonlinear behavior of thick and thin plates. Proc. 2nd Brunei Univ. Conf. The Mathematics of Finite Elements and Applications, MAFELAP-1975, J.R. Whiteman, ed., Academic Press, London, 1976, str. 423-438
  • [65] Hinton E., Razzaque A., Zienkiewicz O.C., Davies J.D., A simple finite element solution for plates of homogeneous, sandwich and cellular construction. Proc. Instn. Civ. Engineers, Part 2, 59, 1975, str. 43-65
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  • [67] Venkateswara Rao G., Venkataramana J., Kanaka Raju K., Stability of moderately thick rectangular plates using a high precision triangular finite element. CS, 5, 1975, str. 257-259
  • [68] Cook R.D., Further development of an effective plate bending element. CS, 6. 2, 1976, ·str. 93-97
  • [69] Mawenya A.S., Shear in isoparametric beam and plate bending elements. Proc. Instn. Civ. Engineers, Part 2, 61, 1976, str. 197-204
  • [70] Rock T.A., Hinton E., A finite element method for the free vibration of plates allowing for transverse shear deformations. CS, 6, 1, 1976, str. 37-44
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  • [74] Hughes T.J.R., Taylor R.L., Kanoknukulchai W., A simple and efficient finite element for plate bending. IJNME, 11, 10, 1977, str. 1529-1543
  • [75] Noor A.K., Mathers M.D., Finite element analysis of anisotropic plates. IJNME, 11, 2, 1977, str. 289-307
  • [76] Ramm E., A plate/shell element for large deflections and rotations. In: Formulations and Computational Algorithms in Finite Element Analysis, K.J. Bathe, ed., MIT, Cambridge, MA, 1977, Chap. 10, str. 265-293
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  • [79] Hughes T.J.R., Cohen M., Haroun M., Reduced and selective integration techniques in the finite element analysis of plates. Nuc. Eng. Design, 46, 1978, str. 203-222
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  • [81] MacNeal R.H., A simple quadrilateral shell element. CS, 8, 1978, str. 175-183
  • [82] Parish H., Geometrical nonlinear analysis of shells. CMAME, 14, 1978, str. 159-178
  • [83] Pugh E.D.L., Hinton E., Zienkiewicz O.C., A study of quadrilateral plate bending elements with "reduced" integration. IJNME, 12, 1978, str. 1059-1079
  • [84] Bathe K.J., Bolourchi S., Large displacement analysis of three-dimensional beam structures. IJNME, 26, 1979, str. 961-986
  • [85] Cheung Y.K., Chan H.C., A family of rectangular bending elements. CS, 10, 4, 1979, str. 613-619
  • [86] Hinton E., Bicanic N., A comparison of lagrangian and serendipity Mindlin plate elements for free vibration analysis. CS, 10, 1979, str. 483-493
  • [87] Panda S.C., Natarajan R., Finite element analysis of laminated composite plates. IJNME, 14, 1979, str. 69-79
  • [88] Parish H., A critical survey of the 9-node degenerated shell element with special emphasis of thin shell application and reduced integration. CMAME, 20, 1979, str. 323-350
  • [89] Thangham P.V., Reddy D.V., Sodhi D.S., Frequency analysis of thick orthotropic plates on elastic foundation using a high precision triangular plate bending element. IJNME, 14, 4, 1979, str. 531-544
  • [90] Akay H.U., Dynamic large deflection analysis of plates using mixed finite elements, CS, 11, 1, 1980, str. 1-11
  • [91] Batoz J.L., Bathe KJ., Ho L.W., A study of three-node triangular plate bending elements. IJNME, 15, 1980, str. 1771-1812
  • [92] Bathe K.J.,Bolourchi S., A geometric and material nonlinear plate and shell element. CS, 11, 1980, str. 23-48
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  • [95] Kanaka Raju K., Hinton E., Nonlinear vibrations of thick plates using Mindlin plate elements. IJNME, 15, 2, 1980, str. 249-257
  • [96] Kączkowski Z., Płyty - obliczenia statyczne. Arkady, Warszawa, 1980
  • [97] Pica A., Hinton E., Transient and pseudo-transient analysis of Mindlin plates. IJNME, 15, 2, 1980, str. 189-208
  • [98] Pica A., Wood R.D., Postbuckling behavior of plates and shells using a Mindlin shallow shell formulation. CS, 12, 1980, str. 759-768
  • [99] Pica A., Wood R.D., Hinton E., Finite element analysis of geometrically nonlinear plate behavior using a Mindlin formulation. CS, 11, 3, 1980, str. 203-215
  • [100] Reddy J.N., A penalty plate-bending element for the analysis of laminated anisotropic composite plates. IJNME, 15, 1980, str. 1187-1206
  • [101] Roufaeil O.L., Dawe D.J., Vibration analysis of rectangular Mindlin plates by the finite strip method. CS, 12, 6, 1980, str. 833-842
  • [102] Spilker R.L., A hybrid stress finite-element formulation for thick multilayer laminates. CS, 11, 6, 1980, str. 507-514
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  • [104] Spilker R.L., Munir N.I., A serendipity cubic-displacement hybrid-stress element for thin and moderately thick plates. IJNME, 15, 8, 1980, str. 1261-1278
  • [105] Spilker R.L., Munir N.I., Comparison of hybrid-stress element through-thickness distribution corresponding to a high-order plate theory. CS, 11, 6, 1980, str. 579-586
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  • [107] Belytschko T., Tsay C.S., Liu W.K., A stabilization matrix for the bilinear Mindlin plate bending plate element. CMAME, 29, 1981, str. 313-327
  • [108] Cheung M.S., Chan M.Y.T., Static and dynamic analysis of thin and thick sectorial plates by the finite strip method. CS, 14, 1-2, 1981, str. 79-88
  • [109] Hughes T.J.R., Liu W.K., Nonlinear finite element analysis of shells. Part I. Three-dimensional shells. Part II. Two-dimensional shells. CMAME, 26, 1981, str. 331-362, CMAME, 27, 1981, str. 167-181
  • [110] Hughes T.J.R., Taylor R.L., The linear triangular bending element. Conf. The Mathematics of Finite Elements and Applications IV, Brunel Univ., England, 28.04-01.05.1981, Academic Press 1982, str. 127-142
  • [111] Hughes T.J.R., Tezduyar T.E., Finite elements based upon Mindlin plate theory with particular reference to the four-node bilinear isoparametric element. J Appl. Mechanics, 48, 3, 1981, str. 587-596
  • [112] Mohr G.A., Application of penalty factors toa doubly curved quadratic shall element. CS, 14, 1981, str. 15-19
  • [113] Noor A.K., Peters J.M., Mixed model and reduced/selective integration displacement models for nonlinear analysis of curved beams. IJNME, 17, 1981, str. 615-631
  • [114] Parish H., Large displacements of shells including material nonlinearities. CMAME, 27, 1981, str. 183-214
  • [115] Pica A., Hinton E., Efficient transient dynamic plate bending analysis with Mindlin elements. Earthq. Eng. Struct. Dyn., 9, 1, 1981, str. 23-31
  • [116] Reddy J.N., Finite element modelling of layered, anisotropic plates and shells. Shock Vib. Dig., 13, 1981, pp. 3-12
  • [117] Reddy J.N., Chao W.C., A comparison of closed-form and finite element solutions of thick laminated anisotropic rectangular plates. Nuc. Eng. Des., 64, 2, 1981, str. 153-167
  • [118] Reddy J.N., Chao W.C., Large-deflection and large amplitude free vibrations of laminated composite material plates. CS, 13, 1/3, 1981, str. 341-347
  • [119] Reddy J.N., Chao W.C., Non-linear bending of thick rectangular, laminated composite plates. IJ Non-Linear Mechanics, 16, 3/4, 1981, str. 291-301
  • [120] Reddy J.N., Huang C.L., Singh I.R., Large deflections and large amplitude vibrations of axisymmetric circular plates. IJNME, 17, 1981, str. 527-541
  • [121] Spilker R.L., High order three-dimensional hybrid-stress elements for thick-plate analysis. IJNME, 17, 1, 1981, str. 53-69
  • [122] Tessler A., On a conforming, Mindlin-type plate element. Conf. The Mathematics of Finite Elements and Applications IV, Brunel Univ., England, 28.04-01.05.1981, Academic Press 1982, str. 119-125
  • [123] Tessler A., Dong S., On a hierarchy of conforming Timoshenko beam elements. CS, 14, 1981, str. 335-344
  • [124] Tsach U., Locking of thin plate/shell elements. IJNME, 17, 1981, str. 633-643
  • [125] Batoz J.L., An explicit formulation for an efficient triangular plate-bending element. IJNME, 18, 1982, str. 1077-1089
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