DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

Account for geometrical nonlinearity in the analysis of reinforced concrete columns of rectangular section by finite element method

Vestnik MGSU 4/2014
  • Agapov Vladimir Pavlovich - Moscow State University of Civil Engineering (MGSU) Doctor of Technical Sciences, Professor, Department of Applied Mechanics and Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoye shosse, Moscow, 129337, Russian Federation; +7 (495) 583-47-52; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Vasil'ev Aleksey Viktorovich - limited liability company "Rodnik" design engineer, limited liability company "Rodnik", 22 Kominterna str., Tver, 170000, Russian Federation; +7 (482) 2-761-004; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 37-43

The superelement of a column of rectangular section made of homogeneous material and intended for linear analysis, developed by authors earlier on the basis of the three-dimensional theory of elasticity, is updated with reference to static analysis of reinforced concrete columns with account for geometrical nonlinearity. In order to get the superelement the column is divided on sections and longwise into eight-node solid finite elements modelling the concrete and two nodes rod elements modelling reinforcement. The elements are connected with one another in the nodes of finite element mesh that provides joint operation of concrete and reinforcement. The internal nodes of the obtained finite element mesh are excluded at the stage of stiffness matrix and load vector of a column calculation. Formulas for calculation of linearized stiffness matrix of a superelement and a vector of the nodal forces statically equivalent to internal stresses are received. The element is adjusted to the computer program PRINS, and can be used for geometrically nonlinear analysis of complex structures containing reinforced concrete columns of rectangular section. Separately standing reinforced concrete column was calculated on longitudinal-transverse bending for the verification of the received superelement. The critical load was determined according to the results of calculation. The determined critical force value corresponds to the theoretical value. Thus, the proposed method of accounting for the geometric nonlinearity in the analysis of reinforced concrete columns can be recommended for practical use.

DOI: 10.22227/1997-0935.2014.4.37-43

References
  1. Geniev G.A., Kissyuk V.N., Tyupin G.A. Teoriya plastichnosti betona i zhelezobetona [Plasticity Theory of Concrete and Reinforced Concrete]. Moscow, Stroyizdat Publ., 1974, 316 p.
  2. Yashin A.V. Kriterii prochnosti i deformirovaniya betona pri prostom nagruzhenii dlya razlichnykh vidov napryazhennogo sostoyaniya [Strength and Strain Criteria of Concrete at Simple Loading for Various Kinds of the Stress State]. Raschet i proektirovanie zhelezobetonnykh konstruktsiy [Analysis and Design of Reinforced Concrete Structures]. Moscow, 1977, pp. 48—57.
  3. Karpenko N.I. Obshchie modeli mekhaniki zhelezobetona [General Models of Reinforced Concrete Mechanics]. Moscow, Stroyizdat Publ., 1996, 396 p.
  4. Chen W.F. Plasticity in Reinforced Concrete. J. Ross Publishing, 2007. 463 p.
  5. Gedolin L., Deipoli S. Finite Element Studies of Shear-critical R/C Beams. ASCE Journal of the Engineering Mechanics Division. 1977, vol. 103, no. 3, pp. 395—410.
  6. Ngo D., Scordelis A.C. Finite Element Analysis of Reinforced Concrete. J. Am. Conc. Inst., 1967, vol. 64, pp. 152—163.
  7. Kotsovos M.D. Effect of Stress Path on the Behaviour of Concrete under Triaxial Stress States. J. Am. Conc. Inst., vol. 76, no. 2, pp. 213—223.
  8. Nam C.H., Salmon C.G. Finite Element Analysis of Concrete Beams. ASCE J. Struct. Engng. Div. Vol. 100, no. ST12, pp. 2419—2432.
  9. Willam, K.J., Warnke E.P. (1975). Constitutive Models for the Triaxial Behavior of Concrete. Proceedings of the International Assoc. for Bridge and Structural Engineering. Vol. 19, pp. 1—30.
  10. Hinton E., Owen D.R.J. Finite Element Software for Plates and Shells. Pineridge Press, Swansea, U.K., 1984, 403 pp.
  11. Beglov A.D., Sanzharovskiy R.S. Teoriya rascheta zhelezobetonnykh konstruktsiy na prochnost' i ustoychivost'. Sovremennye normy i Evrostandarty [The Theory of Strength and Buckling Analysis of the Reinforced Concrete Structures. Modern Norms and Eurostandards]. Saint Petersburg, Moscow, ASV Publ., 2006, 221 p.
  12. Mailyan D.R., Muradyan V.A. K metodike rascheta zhelezobetonnykh vnetsentrenno szhatykh kolonn [The Method of Calculating Eccentrically Compressed Reinforced Concrete Columns]. Inzhenernyy vestnik Dona [The Engineering Bulletin of Don]. 2012, no. 4 (part 2). Available at: http://www.ivdon.ru/magazine/archive/n4p2y2012/1333.
  13. Agapov V.P., Vasil'ev A.V. Modelirovanie kolonn pryamougol'nogo secheniya ob"emnymi elementami s ispol'zovaniem superelementnoy tekhnologii [Modeling Columns of Rectangular Cross-section with Superelement Technology]. Stroitel'naya mekhanika inzhenernykh konstruktsiy i sooruzheniy [Structural Mechanics of Engineering Buildings and Structures]. 2012, no. 4, pp. 48—53.
  14. Agapov V.P. Issledovanie prochnosti prostranstvennykh konstruktsiy v lineynoy i nelineynoy postanovkakh s ispol'zovaniem vychislitel'nogo kompleksa «PRINS» [Strength Analysis of Three-dimensional Structures with Computer Program PRINS]. Prostranstvennye konstruktsii zdaniy i sooruzheniy (issledovanie, raschet, proektirovanie, primenenie): sbornik statey [Three-dimensional Structures of Buildings (Investigation, Calculation, Design, Application): Collection of Articles]. Moscow, 2008, no. 11, pp. 57—67.
  15. Agapov V.P., Vasil'ev A.V. Superelement kolonny pryamougol'nogo secheniya s geometricheskoy nelineynost'yu [Superelement of the Rectangular Cross Section Column Having Physical Nonlinearity]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2013, no. 6, pp. 50—56.

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SUPERELEMENT OF THE RECTANGULARCROSS SECTION COLUMN HAVING PHYSICAL NONLINEARITY

Vestnik MGSU 6/2013
  • Agapov Vladimir Pavlovich - Moscow State University of Civil Engineering (MGSU) Doctor of Technical Sciences, Professor, Department of Applied Mechanics and Mathematics, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoye shosse, Moscow, 129337, Russian Federation; +7 (495) 583-47-52; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Vasil’ev Aleksey Viktorovich - Rodnik Limited Liability Company design engineer, Rodnik Limited Liability Company, 22 Kominterna St., Tver, 170000, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 50-56

The superelement of the rectangular cross section column designed by the authors earlier for the linear analysis purposes is now applied to analyze the same column with account for the geometric nonlinearity. The superelement is composed of eight solid finite elements. The stiffness matrix technique, the initial stress matrix and the analysis of the vector of unbalanced nodal forces are described.The procedure for excluding internal degrees of freedom of a superelement, using the layer-by-layer reduction method, is described in detail. All calculation formulas are provided in the article. The element, developed by the authors, was adapted to PRINS finite element software; therefore, it can be used to perform the nonlinear analysis of building structures. The console beam, having a rectangular cross section, was analyzed in transverse longitudinal bending to verify the developed element. The comparison of the theory and calculations using PRINS software proved the accuracy of the proposed technique.

DOI: 10.22227/1997-0935.2013.6.50-56

References
  1. Belokonev E.N., Abukhanov A.Z., Belokoneva T.M., Chistyakov A.A. Osnovy arkhitektury zdaniy i sooruzheniy [Fundamentals of Architecture of Buildings and Structures]. Rostov-on-Don, Feniks Publ., 2009, 324 p.
  2. NASTRAN Theoretical Manual. NASA, Washington, 1972.
  3. Basov K.A. ANSYS. Spravochnik pol’zovatelya [ANSYS. User’s Manual]. Moscow, DMK-Press Publ., 2005, 637 p.
  4. Bathe K.J., Wiener P.M. On Elastic-plastic Analysis of I-Beams in Bending and Torsion. Computers and Structures. 1983, vol. 17, pp. 711—718.
  5. Klinkel S., Govindjee S. Anisotrophic Bending-torsion Coupling for Warping in Non-linear Beam. Computational Mechanics. 2003, no. 31, pp. 78—87.
  6. Ayoub A., Filippou F.C. Mixed Formulation of Nonlinear Steel-concrete Composite Beam. J. Structural Engineering. 2000, ASCE, no. 126, pp. 371—381.
  7. Hjelmstad K.D., Tacirouglu E. Mixed Variational Methods for Finite Element Analysis of Geometrically Non-linear, Inelastic Bernoulli-Euler Beams. Communications in Numerical Methods of Engineering. 2003, no. 19, pp. 809—832.
  8. Zienkiewicz O.C., Taylor R.L. The Finite Element Method for Solid and Structural Mechanics. McGraw-Hill, 2005, 631 p.
  9. Bathe K.J. Finite Element Procedures. Prentice Hall, Inc., 1996, 1037 p.
  10. Agapov V.P., Vasil’ev A.V. Modelirovanie kolonn pryamougol’nogo secheniya ob”emnymi elementami s ispol’zovaniem superelementnoy tekhnologii [Modeling Rectangular Section Columns Using 3D Elements and the Superelement Technology]. Stroitel’naya mekhanika inzhenernykh konstruktsiy i sooruzheniy [Structural Mechanics of Engineering Constructions and Structures]. 2012, no. 4, Moscow, RUDN Publ., pp. 48—53.
  11. Agapov V.P. Shugaev V.V. Issledovanie prochnosti prostranstvennykh konstruktsiy v lineynoy i nelineynoy postanovkakh s ispol’zovaniem vychislitel’nogo kompleksa «PRINS» [Research into Strength of Spatial Structures Based on Linear and Non-linear Problem Definitions Using PRINS Software]. Prostranstvennye konstruktsii zdaniy i sooruzheniy (issledovanie, raschet, proektirovanie, primenenie). [Spatial Constructions of Buildings and Structures (Research, Analysis, Design and Application). Collection of works, no. 11, Moscow, MOO «Prostranstvennye konstruktsii» Publ., 2008, pp. 57—67.

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Possibility of using finite element method in the form of classical mixed method for geometrical nonlinear analysis of hinged-rod systems

Vestnik MGSU 12/2015
  • Ignat’ev Aleksandr Vladimirovich - Volgograd State University of Architecture and Civil Engineering (VSUACE) Candidate of Technical Sciences, Associate Professor, Department of Structural Mechanics, Volgograd State University of Architecture and Civil Engineering (VSUACE), 1 Akademicheskaya str., Volgograd, 400074, Russian Federation.
  • Ignat’ev Vladimir Aleksandrovich - Volgograd State University of Architecture and Civil Engineering (VSUACE) Doctor of Technical Sciences, head, Department of Structural Mechanics, Volgograd State University of Architecture and Civil Engineering (VSUACE), 1 Akademicheskaya str., Volgograd, 400074, Russian Federation.
  • Onishchenko Ekaterina Valer’evna - Volgograd State University of Architecture and Civil Engineering (VSUACE) external student, Department of Structural Mechanics, Volgograd State University of Architecture and Civil Engineering (VSUACE), 1 Akademicheskaya str., Volgograd, 400074, Russian Federation.

Pages 47-58

At the present time a great number of works have been published, in which the problems of numerical solution of geometrical nonlinear tasks of calculating different types of structures are considered. Nevertheless the problem of the certainty of the numerical solution of geometrical nonlinear tasks of rod structures deformation (large displacements) still provokes great interest. The quality of the solution for a certain task is proved only by the coincidence of the results obtained before using two different methods or with the experiment. The authors consider the numerical solution algorithm of geometrical nonlinear tasks of the deformation of hinged-rod systems (large displacements and turns) both in case of high and gentle loading basing on the finite element method in the form of classical mixed method being developed by the authors. Solving the problem of static deformation of a flat mechanical hinged-rod system consisting of two linear-elastic rods the authors show the simplicity and efficiency of the algorithm when finding all the range equilibrium system states. The quality of the solution is proved by the coincidence of the results in case of gentle and heavy loading of the system and with the results of other investigations.

DOI: 10.22227/1997-0935.2015.12.47-58

References
  1. Belytschko T., Liu W., Moran B. Nonlinear Finite Elements for Continua and Structures. J.Wiley & Sons, 2000, 300 p.
  2. Bonet J., Wood R. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, 1997, 248 p.
  3. Crisfield M.A. Non-Linear Finite Element Analysis of Solids and Structures. J. Wiley & Sons, 1996, vol. 1, 362 p.
  4. Kyther P., Wie D. An Introduction to Linear and Nonlinear Finite Element Analysis. Birkhauer Verlag, 2004, 445 p. DOI: http://dx.doi.org/10.1007/978-0-8176-8160-9.
  5. Reddy J.N. An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, 2004, 488 p.
  6. Danilin A.N., Zuev N.N., Snegovskiy D.V., Shalashilin V.I. Ob ispol’zovanii metoda konechnykh elementov pri reshenii geometricheski nelineynykh zadach [On the Use of Finite Element Method when Solving Geometry Nonlinear Tasks]. SAPR i grafika [CAD and Graphics]. 2000, no. 4, pp. 26—31. (In Russian)
  7. Perel’muter A.V., Slivker V.I. Ustoychivost’ ravnovesiya konstruktsiy i rodstvennye problemy [Equilibrium Stability of Structures and Related Problems]. Moscow, SKAD SOFT Publ., 2007, 653 p. (In Russian)
  8. Kheydari A., Galishnikova V.V. Pryamoy uprugoplasticheskiy raschet stal’nykh ferm s bol’shimi peremeshcheniyami na predel’noe ravnovesie i prisposoblyaemost’ [Straight Elastic-Plastic Calculation of the Limit Equilibrium and Adaptability of Steel Trusses with Large Displacements]. Stroitel’naya mekhanika inzhenernykh konstruktsiy i sooruzheniy [Structural Mechanics of Engineering Constructions and Buildings]. 2014, no. 3, pp. 51—64. (In Russian)
  9. Gorodetskiy A.S., Evzerov I.D. Komp’yuternye modeli konstruktsiy [Computer Models and Structures]. Kiev, «Fakt» Publ., 2007, 394 p. (In Russian)
  10. Pokrovskiy A.A., Khechumov R.A. Smeshannaya forma MKE v raschetakh sterzhnevykh sistem s uchetom fizicheskoy i geometricheskoy nelineynostey [Mixed Form of FEM in Calculation of Truss Systems with Account for Physical and Geometric Nonlinearity]. Stroitel’naya mekhanika i raschet sooruzheniy [Structural Mechanica and Calculation of Structures]. 1991, no. 2, pp. 5—11. (In Russian)
  11. Pokrovskiy A.A., Khechumov R.A. Predel’noe i zapredel’noe sostoyanie sterzhnevykh sistem [Limit and Beyond Limit State of Truss Systems]. Stroitel’naya mekhanika i raschet sooruzheniy [Structural Mechanics and Calculation of Structures]. 1991, no. 4, pp. 18—21. (In Russian)
  12. Nazarov D.I. Geometricheski nelineynyy analiz v metode konechnykh elementov, real’nosti i mify [Geometric Nonlinear Analysis in Finite Element Method, Reality and Myths]. Problemy dinamiki, prochnosti i iznosostoykosti mashin [Problems of Dynamics, Stability and Durability of Machines]. 2000, no. 6. (In Russian)
  13. Nazarov D.I. Obzor sovremennykh programm konechno-elementnogo analiza [Review of the Modern Programs of Finite Element Analysis]. SAPR i grafika [CAD and Graphics]. 2000, no. 2, pp. 52—55. (In Russian)
  14. Kurguzov V.D. O chislennom reshenii geometricheski nelineynykh zadach stroitel’noy mekhaniki [On Numerical Solution of Geometric Nonlinear Tasks of Structural Mechanics]. Izvestiya vuzov. Stroitel’stvo [News of Higher Educational Institutions. Construction]. 2009, no. 3—4, pp. 14—22. (In Russian)
  15. Evzerov I.D., Geraymovich Yu.D., Laznyuk M.V., Marchenko D.V. Chislennoe reshenie zadach sil’nogo izgiba [Numerical Solution of Strong Bend Tasks]. Sayt podderzhki pol’zovateley SAPR [Site of CAD User Support]. Available at: http://www.cad.dp.ua/obzors/lira.php/. Date of access: 30.10.2015. (In Russian)
  16. Levyakov S.V. O chislennom reshenii geometricheski nelineynykh zadach statiki uprugikh konstruktsiy [On Numerical Solution of Geometric Nonlinear Tasks of Elastic Structures’ Statics]. Sayt podderzhki pol’zovateley SAPR [Site of CAD User Support]. Available at: http://www.cad.dp.ua/obzors/fem3.php/. Date of access: 30.10.2015. (In Russian)
  17. Ignat’ev V.A., Ignat’ev A.V., Zhidelev A.V. Smeshannaya forma metoda konechnykh elementov v zadachakh stroitel’noy mekhaniki [Mixed Form of Finite Element Method in Problems of Structural Mechanics]. Volgograd, VolgGASU Publ., 2006, 172 p. (In Russian)
  18. Ignat’ev V.A., Ignat’ev A.V., Galishnikova V.V., Onishchenko E.V. Nelineynaya stroitel’naya mekhanika sterzhnevykh sistem. Osnovy teorii. Primery rascheta [Nonlinear Structural Mechanics of Truss Systems. Foundation of the Theory. Calculation Examples]. Volgograd, VolgGASU Publ., 2014, 84 p. (In Russian)
  19. Petrov V.V. Nelineynaya inkremental’naya stroitel’naya mekhanika [Nonlinear Incremental Structural Mechanics]. Moscow, Infra — Inzheneriya Publ., 2014, 480 p. (In Russian)
  20. Petrov V.V. Metod posledovatel’nykh nagruzheniy v nelineynoy teorii plastinok i obolochek [Method of Continuous Loadings in Nonlinear Theory of Plates and Shells]. Saratov, Izdatel’stvo Saratovskogo gosudarstvennogo universiteta Publ., 1975, 120 p. (In Russian)

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Solving geometrically nonlinear tasks of the statics of hinged-rod systems basing on finite element method in the form of classical mixed method

Vestnik MGSU 2/2016
  • Ignat’ev Aleksandr Vladimirovich - Volgograd State University of Architecture and Civil Engineering (VSUACE) Candidate of Technical Sciences, Associate Professor, Department of Structural Mechanics, Volgograd State University of Architecture and Civil Engineering (VSUACE), 1 Akademicheskaya str., Volgograd, 400074, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Ignat’ev Vladimir Aleksandrovich - Volgograd State University of Architecture and Civil Engineering (VSUACE) Doctor of Technical Sciences, head, Department of Structural Mechanics, Volgograd State University of Architecture and Civil Engineering (VSUACE), 1 Akademicheskaya str., Volgograd, 400074, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Onishchenko Ekaterina Valer’evna - Volgograd State University of Architecture and Civil Engineering (VSUACE) external student, Department of Structural Mechanics, Volgograd State University of Architecture and Civil Engineering (VSUACE), 1 Akademicheskaya str., Volgograd, 400074, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 20-33

The most widely used numerical method used in linear calculation of building structures is finite element method in traditional form of displacements. Different software is developed on its basis. Though it is only possible to check the certainty of these numerical solutions, especially of non-linear tasks of engineering structures’ deformation by the coincidence of the results obtained by two different methods. The authors solved geometrically nonlinear task of the static deformation of a flat hinged-rod system consisting of five linear elastic rods undergoing great tension-compression strains. The solution was obtained basing on the finite element method in the form of classical mixed method developed by the authors. The set of all equilibrium states of the system, both stable and unstable, and all the limit points were found. The certainty of the solution was approved by the coincidence of the results obtained by other authors basing on traditional finite element method in displacements.

DOI: 10.22227/1997-0935.2016.2.20-33

References
  1. Belytschko T., Liu W., Moran B. Nonlinear Finite Elements for Continua and Structures. J Wiley & Sons, 2000, 300 p.
  2. Bonet J., Wood R. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, 1997, 248 p.
  3. Crisfield M.A. Non-Linear Finite Element Analysis of Solids and Structures. J. Wiley & Sons, 1996, vol. 1, 362 p.
  4. Kyther P., Wie D. An Introduction to Linear and Nonlinear Finite Element Analysis. Birkhauer Verlag, 2004, 445 p. DOI: http://dx.doi.org/10.1007/978-0-8176-8160-9.
  5. Reddy J.N. An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, 2004, 488 p.
  6. Danilin A.N., Zuev N.N., Snegovskiy D.V., Shalashilin V.I. Ob ispol'zovanii metoda konechnykh elementov pri reshenii geometricheski nelineynykh zadach [On the Use of Finite Element Method when Solving Geometry Nonlinear Tasks]. SAPR i grafika [CAD and Graphics]. 2000, no. 4, pp. 26—31. (In Russian)
  7. Perel’muter A.V., Slivker V.I. Ustoychivost’ ravnovesiya konstruktsiy i rodstvennye problemy [Equilibrium Stability of Structures and Related Problems]. Moscow, SKAD SOFT Publ., 2007, vol. 1, 653 p. (In Russian).
  8. Galishnikova V.V. Stability Analysis of Space Trusses. International Journal for Computational Civil and Structural Engineering. 2009, vol. 5, no. 1—2, pp. 35—44.
  9. Galishnikova V.V. Chislennyy analiz ustoychivosti ravnovesiya prostranstvennykh ferm v geometricheski nelineynoy postanovke [Numerical Analysis of the Stability of Space Trusses in Geometrical Nonlinear Statement]. Stroitel’naya mekhanika inzhenernykh konstruktsiy i sooruzheniy [Structural Mechanics of Engineering Structures and Constructions]. 2010, no. 1, pp. 42a—50. (In Russian)
  10. Gorodetskiy A.S., Evzerov I.D. Komp’yuternye modeli konstruktsiy [Computer Models and Structures]. Kiev, «Fakt» Publ., 2007, 394 p. (In Russian)
  11. Kurguzov V.D. O chislennom reshenii geometricheski nelineynykh zadach stroitel'noy mekhaniki [On Numerical Solution of Geometric Nonlinear Tasks of Structural Mechanics]. Izvestiya vuzov. Stroitel’stvo [News of Higher Educational Institutions. Construction]. 2009, no. 3—4, pp. 14—22. (In Russian)
  12. Evzerov I.D., Geraymovich Yu.D., Laznyuk M.V., Marchenko D.V. Chislennoe reshenie zadach sil’nogo izgiba [Numerical Solution of Strong Bend Tasks]. Sayt podderzhki pol’zovateley SAPR [Site of CAD User Support]. Available at: http://www.cad.dp.ua/obzors/lira.php/. Date of access: 30.10.2015. (In Russian)
  13. Poceski A. Mixed Finite Element Method. Springer-Verlag Berlin Heidelberg, 1992, 356 p. DOI: http://dx.doi.org/10.1007/978-3-642-84676-2.
  14. Pokrovskiy A.A., Khechumov R.A. Smeshannaya forma MKE v raschetakh sterzhnevykh sistem s uchetom fizicheskoy i geometricheskoy nelineynostey [Mixed Form of FEM in Calculation of Truss Systems with Account for Physical and Geometric Nonlinearity]. Stroitel’naya mekhanika i raschet sooruzheniy [Structural Mechanic and Calculation of Structures]. 1991, no. 2, pp. 5—11. (In Russian)
  15. Pokrovskiy A.A., Khechumov R.A. Predel’noe i zapredel’noe sostoyanie sterzhnevykh sistem [Limit and Beyond Limit State of Truss Systems]. Stroitel’naya mekhanika i raschet sooruzheniy [Structural Mechanics and Calculation of Structures]. 1991, no. 4, pp. 18—21. (In Russian)
  16. Ignat’ev V.A., Ignat’ev A.V., Zhidelev A.V. Smeshannaya forma metoda konechnykh elementov v zadachakh stroitel’noy mekhaniki [Mixed Form of Finite Element Method in Problems of Structural Mechanics]. Volgograd, VolgGASU Publ., 2006, 172 p. (In Russian)
  17. Ignat’ev V.A., Ignat’ev A.V., Galishnikova V.V., Onishchenko E.V. Nelineynaya stroitel’naya mekhanika sterzhnevykh sistem. Osnovy teorii. Primery rascheta [Nonlinear Structural Mechanics of Truss Systems. Foundation of the Theory. Calculation Examples]. Volgograd, VolgGASU Publ., 2014, 84 p. (In Russian)
  18. Nazarov D.I. Geometricheski nelineynyy analiz v metode konechnykh elementov, real’nosti i mify [Geometric Nonlinear Analysis in Finite Element Method, Reality and Myths]. Problemy dinamiki, prochnosti i iznosostoykosti mashin [Problems of Dynamics, Stability and Durability of Machines]. 2000, no. 6. (In Russian)
  19. Nazarov D.I. Obzor sovremennykh programm konechno-elementnogo analiza [Review of the Modern Programs of Finite Element Analysis]. SAPR i grafika [CAD and Graphics]. 2000, no. 2, pp. 52—55. (In Russian)
  20. Levyakov S.V. O chislennom reshenii geometricheski nelineynykh zadach statiki uprugikh konstruktsiy [On Numerical Solution of Geometric Nonlinear Tasks of Elastic Structures]. Statics Sayt podderzhki pol’zovateley SAPR [Site of CAD User Support]. Available at: http://www.cad.dp.ua/obzors/fem3.php/. Date of access: 30.10.2015. (In Russian)
  21. Toroptsev A.V. Reshenie chetyrekh testovykh zadach dlya Nazarova D.I. [Solution of Four Test Tasks for Nazarov D.I.]. Sayt podderzhki pol’zovateley SAPR [Site of CAD User Support]. Available at: http:// www.cad.dp.ua/obzors/paper1.php/. Date of access: 30.10.2015. (In Russian)
  22. Ignat’ev A.V., Ignat’ev V.A., Onishchenko E.V. Vozmozhnost’ ispol’zovaniya metoda konechnykh elementov v forme klassicheskogo smeshannogo metoda dlya geometricheski nelineynogo analiza sharnirno-sterzhnevykh sistem [Possibility of Using Finite Element Method in the Form of Classical Mixed Method for Geometrical Nonlinear Analysis of Hinged-Rod Systems]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2015, no. 12, pp. 47—58. (In Russian)
  23. Petrov V.V. Nelineynaya inkremental’naya stroitel’naya mekhanika [Nonlinear Incremental Structural Mechanics]. Moscow, Infra — Inzheneriya Publ., 2014, 480 p. (In Russian)
  24. Petrov V.V. Metod posledovatel’nykh nagruzheniy v nelineynoy teorii plastinok i obolochek [Method of Continuous Loadings in Nonlinear Theory of Plates and Shells]. Saratov, SGU im. N.G. Chernyshevskogo Publ., 1975, 119 p. (In Russian)

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