DESIGNING AND DETAILING OF BUILDING SYSTEMS. MECHANICS IN CIVIL ENGINEERING

The modeling of the structurefoundation-base system with the use of two-layer beamon an elastic basis with variable coeficcient of subgrade reaction

Vestnik MGSU 10/2013
  • Barmenkova Elena Vjacheslavovna - Moscow State University of Civil Engineering (MGSU) Candidate of Technical Sciences, Associate Professor, Department of Strength of Materials, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Matveeva Alena Vladimirovna - Moscow State University of Civil Engineering (MGSU) postgraduate student, Department of the Strength of materials, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 30-35

In the paper the author presents the results of calculations of the system «structurefoundation-base» in case of using the two-layer and the single-layer beam models on an elastic basis with variable and constant coefficients of subgrade reaction. The analytical solution is obtained using the method of initial parameters. The calculations are carried out in case of building up the structure.The method of calculating two-layer beam with variable flexural rigidity along the length on an elastic foundation was described in the author’s previous articles, while in the present paper variable coefficients of subgrade reaction are taken into account. A two-layer beam is a beam of variable rigidity, the lower layer simulates the foundation, and the upper — the structure, at the same time the weight of each layer is considered.For comparison, the problem is also considered in its traditional statement. That means the problem of single-layer beam bending is solved with cross-section of constant length, which is freely lying on an elastic basis of Winkler’s type.The results of calculations of two-layer and single-layer beams show, that the values of the internal forces and stresses are higher with variable coefficient of subgrade reaction than with the constant one. When comparing the two-layer and the single-layer beam models with the same foundation characteristics, the values of internal forces in two-layer beams are much higher.On the basis of the calculations we can make the following conclusion: in order to obtain more reliable prognosis of the stress-strain state of the system «structure-foundation» on an elastic basis, it is appropriate to carry out calculations with the use of a contact model in the form of a two-layer beam on an elastic basis of Winkler’s type with variable coefficients of subgrade reaction. The model allows us to take account of such factors as rigidity changes in the base and the rigidity of the upper structure.

DOI: 10.22227/1997-0935.2013.10.30-35

References
  1. Garagash B.A. Avarii i povrezhdeniya sistemy «zdanie — osnovanie» i regulirovanie nadezhnosti ee elementov [ Accidents and Damages of the "Base-Structure" System and Reliability Control of its Elements]. Volgograd, VolGU Publ., 2000, 384 p.
  2. Avramidis I.E., Morfidis K. Bending of Beams on Three-parameter Elastic Foundation. International Journal of Solids and Structures. 2006, vol. 43, no. 2, pp. 357—375.
  3. Kerr A.D. Elastic and Viscoelastic Foundation Models. Journal of Applied Mechanics 1964, vol. 31, no. 3, pp. 491—498.
  4. Teodoru I.-B. Beams on Elastic Foundation. The Simplified Continuum Approach. Bulletin of the Polytechnic Institute of Jassy, Constructions, Architechture Section. Vol. LV (LIX), 2009, no. 4, pp. 37—45.
  5. Klepikov S.N. Raschet konstruktsiy na uprugom osnovanii [Calculation of the Structures on Elastic Basis]. Kiev, Budivel'nik Publ., 1967, 184 p.
  6. Barmenkova E.V., Andreev V.I. Izgib dvukhsloynoy balki na uprugom osnovanii s uchetom izmeneniya zhestkosti balki po dline [The Bending of Two-layer Beam on Elastic Basis with Account For the Beam Stiffness Changes along the Length]. International Journal for Computational Civil and Structural Engineering. 2011, vol. 7, no. 3, pp. 50—54.
  7. Andreev V.I., Barmenkova E.V. Izgib dvukhsloynoy balki na uprugom osnovanii s uchetom massovykh sil [The Bending of Two-layer Beam on Elastic Basis with Account For Budy Forces]. XVIII Polish-Russian-Slovak Seminar «Theoretical Foundation of Civil Engineering». Warsaw, 2009, pp. 51—56.
  8. Alekseev S.I., Kamaev V.S. Uchet zhestkostnykh parametrov zdaniy pri rasche-takh osnovaniy i fundamentov [The account of the stiffness parameters of buildings in the calculation of the foundations]. Vestnik TGASU [Proceedings of Tomsk State University foe Architecture and Enfineering]. 2007, no. 3, pp. 165—172.

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NATURAL TRANSVERSE VIBRATIONS OF A PRESTRESSED ORTHOTROPIC PLATE-STRIPE

Vestnik MGSU 2/2012
  • Egorychev Oleg Aleksandrovich - Moscow State University of Civil Engineering (MSUCE) Doctor of Technical Sciences, Professor 8 (495) 320-43-02, Moscow State University of Civil Engineering (MSUCE), 26 Jaroslavskoe shosse, Moscow, 129337, Russia.
  • Egorychev Oleg Olegovich - Moscow State University of Civil Engineering (MSUCE) Doctor of Technical Sciences, Professor 8 (495) 287-49-14, Moscow State University of Civil Engineering (MSUCE), 26 Jaroslavskoe shosse, Moscow, 129337, Russia; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Brendje Vladimir Vladislavovich - Moscow State University of Civil Engineering (MSUCE) Senior Lecturer 8 (499) 161-21-57, Moscow State University of Civil Engineering (MSUCE), 26 Jaroslavskoe shosse, Moscow, 129337, Russia; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 11 - 14

The article represents a new outlook at the boundary-value problem of natural vibrations of a homogeneous pre-stressed orthotropic plate-stripe. In the paper, the motion equation represents a new approximate hyperbolic equation (rather than a parabolic equation used in the majority of papers covering the same problem) describing the vibration of a homogeneous orthotropic plate-stripe. The proposed research is based on newly derived boundary conditions describing the pin-edge, rigid, and elastic (vertical) types of fixing, as well as the boundary conditions applicable to the unfixed edge of the plate. The paper contemplates the application of the Laplace transformation and a non-standard representation of a homogeneous differential equation with fixed factors. The article proposes a detailed representation of the problem of natural vibrations of a homogeneous orthotropic plate-stripe if rigidly fixed at opposite sides; besides, the article also provides frequency equations (no conclusions) describing the plate characterized by the following boundary conditions: rigid fixing at one side and pin-edge fixing at the opposite side; pin-edge fixing at one side and free (unfixed) other side; rigid fixing at one side and elastic fixing at the other side. The results described in the article may be helpful if applied in the construction sector whenever flat structural elements are considered. Moreover, specialists in solid mechanics and theory of elasticity may benefit from the ideas proposed in the article.

DOI: 10.22227/1997-0935.2012.2.11 - 14

References
  1. Egorychev O.O. Kolebanija ploskih elementov konstrukcij [Vibrations of Two-Dimensional Structural Elements]. Moscow, ASV, 2005, pp. 45—49.
  2. Arun K Gupta, Neeri Agarwal, Sanjay Kumar. Free transverse vibrations of orthotropic viscoelastic rectangular plate with continuously varying thickness and density// Institute of Thermomechanics AS CR, Prague, Czech Rep, 2010, Issue # 2.
  3. Filippov I.G., Cheban V.G. Matematicheskaja teorija kolebanij uprugih i vjazkouprugih plastin i sterzhnej [Mathematical Theory of Vibrations of Elastic and Viscoelastic Plates and Rods]. Kishinev, Shtinica, 1988, pp. 27—30.

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MODELING OF THE REAL SYSTEM «STRUCTURE-FOUNDATION-BEDDING» THROUGH THE EMPLOYMENT OF A MODEL OF A TWO-LAYER BEAM OF VARIABLE RIGIDITY RESTING ON THE ELASTIC BEDDING

Vestnik MGSU 6/2012
  • Andreev Vladimir Igorevich - Moscow State University of Civil Engineering (National Research University) (MGSU) Doctor of Technical Sciences, Professor, corresponding member of Russian Academy of Architecture and Construction Sciences, chair, Department of Strength of Materials, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Barmenkova Elena Vyacheslavovna - Moscow State University of Civil Engineering (MGSU) Candidate of Technical Science, Associate Professor, Department of the Strength of materials, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 37 - 41

In the paper, the authors provide the results of analysis of a real construction facility performed with the help of a model of a two-layer beam of variable rigidity resting on the elastic bedding. The bottom layer of a two-layer beam simulates the foundation, the upper payer stands for the structure, and the weight of each layer is taken into consideration. The characteristics of the upper layer change alongside its length. Analytical and numerical methods of calculation were applied to solve this problem.
The analytical solution is based on the method of initial parameters and backed by the practical data extracted from "Frame and Towerlike Buildings: Mattress Foundation Design Manual". According to the above manual, whenever the length-to-width ratio of a building exceeds 1.5, one-dimensional pattern composed of a composite beam resting on the elastic bedding may be used. The beam is divided into several sections, and deflection of each section is identified. It is equal to the settlements of the bedding surface. The rigidity change alongside the length of each section is assumed to be permanent, i.e. the beam is considered as the one that demonstrates its piecewise-constant rigidity.
The following conclusion can be made on the basis of the calculations performed by the authors: the calculation of the «structure-foundation-bedding» system may require a simplified model representing composite beams and plates resting on the elastic bedding. More accurate models, such as sets of finite elements, are recommend for use in conjunction with simplified ones.

DOI: 10.22227/1997-0935.2012.6.37 - 41

References
  1. Barmenkova E.V., Andreev V.I. Izgib dvukhsloynoy balki na uprugom osnovanii s uchetom izmeneniya zhestkosti balki po dline [Deflection of the Two-layer Beam Resting on the Elastic Bedding with Consideration for the Beam Rigidity Change Alongside Its Length]. International Journal for Computational Civil and Structural Engineering, vol. 7, no. 3, 2011, pp. 50—54.
  2. Klepikov S.N. Raschet konstrukcij na uprugom osnovanii [Analysis of Structures Resting on Elastic Bedding]. Kiev, Budivel’nik Publ., 1967, 184 p.
  3. Rukovodstvo po proektirovaniyu plitnykh fundamentov karkasnykh zdaniy i sooruzheniy bashennogo tipa. Design of Mattress Foundations of Frame Buildings and Towerlike Structures. The Manual. Scientific and Research Institute of Beddings and Foundations named after N.M. Gersevanov. Moscow, Stroyizdat Publ., 1984, 263 p.
  4. SP 50-101—2004 [Construction Rules 50-101—2004]. Proektirovanie i ustroystvo osnovaniy I fundamentov zdaniy i sooruzheniy [Design and Construction of Beddings and Foundations of Buildings and Structures]. Moscow, FGUP TsPP Publ., 2005.
  5. Andreev V.I., Barmenkova E.V. Izgib dvukhsloynoy balki na uprugom osnovanii s uchetom massovykh sil [Deflection of the Two-layer Beam Resting on the Elastic Bedding with Consideration for the Bulk Forces]. Proceedings of the XVIII Polish-Russian-Slovak Seminar «Theoretical Foundation of Civil Engineering». Warsaw, 2009, pp. 51—56.

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NATURAL TRANSVERSE VIBRATIONS OF AN ORTHOTROPIC PLATE-STRIP WITH FREE EDGES

Vestnik MGSU 7/2012
  • Egorychev Oleg Aleksandrovich - Moscow State University of Civil Engineering (MSUCE) Doctor of Technical Sciences, Professor, Moscow State University of Civil Engineering (MSUCE), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Egorychev Oleg Olegovich - Moscow State University of Civil Engineering (MSUCE) Doctor of Technical Sciences, Professor 8 (495) 287-49-14, Moscow State University of Civil Engineering (MSUCE), 26 Jaroslavskoe shosse, Moscow, 129337, Russia; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Brende Vladimir Vladislavovich - Moscow State University of Civil Engineering (MSUCE) Senior Lecturer, +7 (499) 161-21-57, Moscow State University of Civil Engineering (MSUCE), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 26 - 30

In the article, the authors present their new formulation of the problem of the boundary value of natural vibrations of a homogeneous pre-stressed orthotropic plate-strip in different boundary conditions. A new approximate hyperbolic (in contrast to most authors) equation of oscillations of a homogeneous orthotropic plate-strip is used in the paper in the capacity of an equation of motion. Besides, the authors propose their newly derived boundary conditions for a free edge of the plate. The authors employ the Laplace transformation and a non-standard representation of the general solution of homogeneous differential equations with constant coefficients. The authors also provide a detailed description of the problem of free vibrations of a homogeneous orthotropic plate-strip, if rigidly attached in the opposite sides. The results presented in this article may be applied in the areas of construction and machine building, wherever flat structural elements are used. In addition, professionals in mechanics of solid deformable body and elasticity theory may benefit from the findings presented in the article.

DOI: 10.22227/1997-0935.2012.7.26 - 30

References
  1. Uflyand Ya.S. Rasprostranenie voln pri poperechnykh kolebaniyakh sterzhney i plastin [Wave Propagation in the Event of Transverse Vibrations of Rods and Plates]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1948, vol. 12, no. 33, pp. 287—300.
  2. Lyav A. Matematicheskaya teoriya uprugosti [Mathematical Theory of Elasticity]. Moscow-Leningrad, ONTI Publ., 1935, 674 p.
  3. Egorychev O.O. Kolebaniya ploskikh elementov konstruktsiy [Vibrations of Flat Elements of Structures]. Moscow, ASV Publ., 2005, pp. 45—49.
  4. Egorychev O.A., Egorychev O.O., Brende V.V. Vyvod chastotnogo uravneniya sobstvennykh poperechnykh kolebaniy predvaritel’no napryazhennoy plastiny uprugo zakreplennoy po odnomu krayu i zhestko zakreplennoy po-drugomu [Derivation of a Frequency Equation of Natural Transverse Vibrations of a Pre-stressed Elastic Plate, If One Edge Is Fixed Rigidly and the Other One is Fixed Elastically]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 246—251.
  5. Filippov I.G., Cheban V.G. Matematicheskaya teoriya kolebaniy uprugikh i vyazkouprugikh plastin i sterzhney [Mathematical Theory of Vibrations of Elastic and Viscoelastic Plates and Rods]. Kishinev, Shtiintsa Publ., 1988, pp. 27—30.
  6. Gupta A.K., Aragval N., Kumar S. Svobodnye kolebaniya ortotropnoy vyazkouprugoy plastiny s postoyanno menyayushcheysya tolshchinoy i plotnost’yu [Free Transverse Vibrations of an Orthotropic Visco-Elastic Plate with Continuously Varying Thickness and Density]. Institute of Thermal Dynamics, Prague, Czech Republic, 2010, no. 2.
  7. Egorychev O.A., Egorychev O.O., Brende V.V. Sobstvennye poperechnye kolebaniya predvaritel’no napryazhennoy ortotropnoy plastinki-polosy uprugo zakreplennoy po odnomu krayu i svobodnoy po drugomu [Natural Transverse Vibrations of a Pre-stressed Orthotropic Plate, If One Edge Is Fixed Elastically and the Other One Is Free]. Vestnik MGSU [Proceedings of Moscow State University of Civil Engineering]. 2010, no. 4, vol. 3, pp. 252—258.
  8. Lol R. Poperechnye kolebaniya ortotropnykh neodnorodnykh pryamougol’nykh plastin s nepreryvno menyayushcheysya plotnost’yu [Transverse Vibrations of Orthotropic Non-homogeneous Rectangular Plates with Continuously Varying Density]. Indian University of Technology, 2002, no. 5.
  9. Egorychev O.A., Brende V.V. Sobstvennye kolebaniya odnorodnoy ortotropnoy plastiny [Natural Vibrations of a Homogeneous Orthotropic Plate]. Department of Industrial and Civil Engineering, 2010, no. 6, pp.
  10. Lekhnitskiy S.G. Teoriya uprugosti anizotropnogo tela [Theory of Elasticity of an Anisotropic Body]. Moscow, Nauka. Fizmatlit Publ., 1977.

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OPTIMIZATION OF INHOMOGENEOUS THICK-WALLED SPHERICAL SHELL IN THE TEMPERATURE FIELD

Vestnik MGSU 12/2012
  • Andreev Vladimir Igorevich - Moscow State University of Civil Engineering (National Research University) (MGSU) Doctor of Technical Sciences, Professor, corresponding member of Russian Academy of Architecture and Construction Sciences, chair, Department of Strength of Materials, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Bulushev Sergey Valer'evich - Moscow State University of Civil Engineering (MGSU) master student, Moscow State University of Civil Engineering (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 40 - 46

The authors consider the central symmetric problem of the theory of elasticity of inhomogeneous bodies for thick-walled spheres exposed to the external pressure in a stationary temperature field. The essence of the inverse problem lies in the identification of such dependence of the elastic modulus on the radius whereby the stress state of the sphere is the same as the pre-set one.
Maximal stresses in thick-walled shells exposed to internal or external pressures occur in the proximity to the internal contour. Thus, destruction in this area is initiated upon the achievement of the limit state, while the rest of the shell is underused. The essence of the problem solved in the paper is the following. The problems are solved using the simultaneous exposure to forces and temperature loads.The two theories of strength are considered at once: a maximum normal stress theory and a maximum shear stress theory. It is proven that according to the first theory maximum stresses in an inhomogeneous shell are 1.35 times smaller than those in the homogeneous shell. The stress reduction rate equals to 2.5, if the maximum shear stress theory is employed. Thus, the introduction of artificial inhomogeneity leads to the optimization of shells by reducing their thickness or increasing loads.

DOI: 10.22227/1997-0935.2012.12.40 - 46

References
  1. Andreev V.I., Potekhin I.A. Optimizatsiya po prochnosti tolstostennykh obolochek [Optimization of Strength of Thick-walled Shells]. Moscow, MGSU Publ., 2011, 86 p.
  2. Andreev V.I. The Method of Optimization of Thick-walled Shells Based on Solving Inverse Problems of the Theory of Elasticity of Inhomogeneous Bodies. Computer Aided Optimum Design in Engineering XII. WITpress Publ., 2012, pp. 189—201.
  3. Lekhnitskiy S.G. Radial’noe raspredelenie napryazheniy v kline i poluploskosti s peremennym modulem uprugosti [Radial Distribution of Stresses in the Wedge and in the Half-plane with a Variable Modulus of Elasticity]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1962, vol. XXVI, no. 1, pp. 146—151.
  4. Lomakin V.A. Teoriya uprugosti neodnorodnykh tel [Theory of Elasticity of Inhomogeneous Bodies]. Moscow, MGU Publ., 1976, 368 p.
  5. Andreev V.I. Nekotorye zadachi i metody mekhaniki neodnorodnykh tel [Some Problems and Methods of Mechanics of Heterogeneous Bodies]. Moscow, ASV Publ., 2002, 208 p.
  6. Andreev V.I., Minaeva A.S. Postroenie na osnove pervoy teorii prochnosti modeli ravnonapryazhennogo tsilindra, podverzhennogo silovym i temperaturnym nagruzkam [The Inverse Problem for an Inhomogeneous Thick-walled Cylinder Exposed to Power and Thermal Loads]. Privolzhskiy nauchnyy zhurnal [Volga Scientific Journal]. 2011, no. 4, pp. 34—39.
  7. Andreev V.I., Minaeva A.S. Modelirovanie ravnonapryazhennogo tsilindra, podverzhennogo silovym i temperaturnym nagruzkam [Simulation of a Stress-ration Cylinder Exposed to Forces and Thermal Loads]. International Journal for Computational Civil and Structural Engineering. Vol. 7, no. 1, 2011, ðð. 71—75.
  8. Kamke E. Spravochnik po obyknovennym differentsial’nym uravneniyam [Handbook of Ordinary Differential Equations]. Moscow, Nauka Publ., 1976, 576 p.
  9. Bronshteyn I.N., Semendyaev K.A. Spravochnik po matematike dlya inzhenerov i uchashchikhsya vtuzov [Handbook of Mathematics for Engineers and Students of Technical Colleges]. Moscow, Nauka Publ., 1986, 544 p.
  10. Andreev V.I., Potekhin I.A. Ravnoprochnye i ravnonapryazhennye konstruktsii. Modeli i real’nost’ [Full Constant Stress Structures. Models and Reality]. XVIII Russian-Slovak-Polish Seminar “Theoretical Foundation of Civil Engineering”. Arkhangel’sk 01.07 – 05.07.2009. Warszawa, 2009. Proceedings, pp. 57—62.

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Analytical solution of physically nonlinear problem for an inhomogeneous thick-walled cylindrical shell

Vestnik MGSU 11/2015
  • Andreev Vladimir Igorevich - Moscow State University of Civil Engineering (National Research University) (MGSU) Doctor of Technical Sciences, Professor, corresponding member of Russian Academy of Architecture and Construction Sciences, chair, Department of Strength of Materials, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .
  • Polyakova Lyudmila Sergeevna - Moscow State University of Civil Engineering (National Research University) (MGSU) Master student, Department of Strength of Materials, Moscow State University of Civil Engineering (National Research University) (MGSU), 26 Yaroslavskoe shosse, Moscow, 129337, Russian Federation; This e-mail address is being protected from spambots. You need JavaScript enabled to view it .

Pages 38-45

Among the classical works devoted to Solid Mechanics a significant place is occupied by the studies taking into account the physical and geometric nonlinearity. Also there is enough of works, which concern linear problems taking into account the inhomogeneity of the material. At the same time there are very few publications, which take into account both effects (non-linearity and inhomogeneity). This is due to the lack of experimental data on the influence of various factors on the parameters defining the non-linear behavior of the materials. Thus it is of great importance to study the influence of inhomogeneity when solving the problems of structures made of physically nonlinear materials. This article provides a solution to one of the problems of the nonlinear theory of elasticity taking into account the inhomogeneity. The problem is solved in an axisymmetric formulation, i.e. all the parameters of the nonlinear relationship between the intensities of stresses and strains are functions of the radius. The article considers an example - the stress distribution in the inhomogeneous soil massif with a cylindrical cavity.

DOI: 10.22227/1997-0935.2015.11.38-45

References
  1. Andreev V.I., Malashkin Yu.N. Raschet tolstostennoy truby iz nelineyno-uprugogo materiala [Calculation of Thick-Walled Pipe of a Nonlinear-Elastic Material]. Stroitel’naya mekhanika i raschet sooruzheniy [Structural Mechanics and Calculation of Structures]. 1983, no. 6, pp. 70—72. (In Russian)
  2. Birger I.A. Nekotorye obshchie metody resheniya zadach teorii plastichnosti [Some Common Methods for Solving the Problems of the Theory of Plasticity]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1951, vol. 15, no. 6, pp. 765—770. (In Russian)
  3. Novozhilov I.V. Ob utochnenii predel’nykh modeley mekhaniki [On a Refinement of Limit Models of Mechanics]. Nelineynaya mekhanika [Nonlinear Mechanics]. Moscow, Fizmatlit Publ., 2001, 432 p. (In Russian)
  4. Stupishin L.U., Nikitin K.E. Numerical Research Methodology of Free Oscillations of Geometrically Nonlinear Shell Using the Mixed Finite Element Method. Advanced Materials Research. 2014, vol. 988, pp. 338—341. DOI: http://dx.doi.org/10.4028/www.scientific.net/AMR.988.338.
  5. Stupishin L.U., Nikitin K.E. Determining the Frequency of Free Oscillations Geometrically Nonlinear Shell Using the Mixed Finite Element Method. Applied Mechanics and Materials. 2014, vols. 580—583, pp. 3017—3020. DOI: http://dx.doi.org/10.4028/www.scientific.net/AMM.580-583.3017.
  6. Grigorenko Ya.M., Vasilenko A.T., Pankratova N.D. Nesimmetrichnaya deformatsiya tolstostennykh neodnorodnykh sfericheskikh obolochek [Asymmetrical Non-Uniform Deformation of the Thick-Walled Spherical Shells]. Doklady AN USSR [Reports of the Ukrainian Academy of Sciences ]. Series A, 1981, no. 6, pp. 42—45. (In Russian)
  7. Kolchin G.B. Raschet elementov konstruktsiy iz uprugikh neodnorodnykh materialov [Calculation of Structural Elements Made of Inhomogeneous Elastic Materials]. Kishinev, Kartya Moldovenyaske Publ., 1971, 172 p. (In Russian)
  8. Kolchin G.B. Ploskie zadachi teorii uprugosti neodnorodnykh tel [Plane Problems of Elasticity Theory of Inhomogeneous Bodies]. Kishinev, Shtiintsa Publ., 1977, 119 p. (In Russian)
  9. Ol’shak V., Rykhlevsky Ya., Urbanovskiy V. Teoriya plastichnosti neodnorodnykh tel [Theory of Plasticity of Heterogeneous Bodies]. Translated from English. Moscow Mir, 1964. 156 s. (In Russian)
  10. Rostovtsev N.A. K teorii uprugosti neodnorodnykh tel [To the Theory of Elasticity of Inhomogeneous Bodies]. Prikladnaya matematika i mekhanika [Applied Mathematics and Mechanics]. 1964, vol. 28, no. 4, pp. 601—611. (In Russian)
  11. Nowinski J. Axisymmetric Problem of the Steady-State Thermal-Dependent Properties. Applied Scientific Research. 1964, vol. 12, no. 4—5, pp. 349—377. DOI: http://dx.doi.org/10.1007/BF03185007.
  12. Olszak W., Urbanovski W., Rychlewski J. Sprężysto-plastyczny gruboscienny walec niejednorodny pod działaniem parcia wewnetrznego i siły podłużnej. Arch. mech. stos. 1955, vol. VII, no. 3, pp. 315—336.
  13. Olszak W., Urbanowski W. Sprężysto-plastyczna gruboscienna powłoka kulista z materiału niejednorodnego poddana działaniu cisnienia wewnetrznego i zewnetrznego. Rozprawy inżynierskie. 1956, vol. IV, no. 1, pp. 23—41.
  14. Andreev V.I. Ravnovesie tolstostennogo shara iz nelineynogo neodnorodnogo materiala [Equilibrium of a Thick-Walled Sphere Made of Nonlinear Inhomogeneous Material]. Stroitel’naya mekhanika i raschet sooruzheniy [Structural Mechanics and Calculation of Structures]. 1983, no. 2, pp. 24—27. (In Russian)
  15. Andreev V.I. Nekotorye zadachi i metody mekhaniki neodnorodnykh tel [Some Problems and Methods of Inhomogeneous Bodies Mechanics]. Moscow, ASV Publ., 2002, 288 p. (In Russian)
  16. Vasilenko A.T., Grigorenko Ya.M., Pankratova N.D. Napryazhennoe sostoyanie tolstostennykh neodnorodnykh sfericheskikh obolochek pri nesimmetrichnykh nagruzkakh [The Stress State of Thick-Walled Non-Uniform Spherical Shells]. Prikladnaya mekhanika [Applied Mechanics]. 1982, vol. XVIII, no. 4, pp. 22—28. (In Russian)
  17. Grigorenko Ya.M., Vasilenko A.T., Pankratova N.D. O reshenii zadach statiki sloistykh obolochek v trekhmernoy postanovke [On the Solution of Statics Problems of Layered Shells in Three-Dimensional Statement]. Vychislitel’naya i prikladnaya matematika [Computational and Applied Mathematics]. 1981, no. 43, pp. 123—132. (In Russian)
  18. Andreev V.I. About the Unloading in Elastoplastic Inhomogeneous Bodies. Applied Mechanics and Materials. 2013, vols. 353—356, pp. 1267—1270. DOI: http://dx.doi.org/10.4028/www.scientific.net/AMM.353-356.1267.
  19. Lukash P.A. Osnovy nelineynoy stroitel’noy mekhaniki [Fundamentals of Nonlinear Structural Mechanics]. Moscow, Stroyizdat Publ., 1978, 208 p. (In Russian)
  20. Andreev V.I. Equilibrium of a Thick-Walled Sphere of Inhomogeneous Nonlinear-Elastic Material. Applied Mechanics and Materials. 2013, vols. 423—426, pp. 1670—1674. DOI: http://dx.doi.org/10.4028/www.scientific.net/AMM.423-426.1670.

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