The paper outlines the fundamentals of the method of solving static problems of geometrically nonlinear deformation, buckling, and postbuckling behavior of thin thermoelastic inhomogeneous shells with complex-shaped midsurface, geometrical features throughout the thickness, or multilayer structure under complex thermomechanical loading. The method is based on the geometrically nonlinear equations of three-dimensional thermoelasticity and the moment finite-element scheme. The method is justified numerically. Results of practical importance are obtained in analyzing poorely studied classes of inhomogeneous shells. These results provide an insight into the nonlinear deformation and buckling of shells under various combinations of thermomechanical loads
Similar content being viewed by others
References
N. A. Alfutov, Fundamentals of Stability Analysis of Elastic Systems [in Russian], Mashinostroenie, Moscow (1978).
I. Ya. Amiro, O. A. Grachev, V. A. Zarutskii, et al., Stability of Ribbed Shells of Revolution [in Russian], Naukova Dumka, Kyiv (1987).
D. V. Babich, “Stability of thermosensitive shells nonuniformly heated throughout the thickness,” Dokl. AN Ukrainy, No. 4, 41–45 (1993).
V. A. Bazhenov, E. S. Dekhtyaryuk, N. A. Solovei, and O. P. Krivenko, “Generating finite-element models of complex shells,” in: Proc. Int. Sci. Conf. on Architecture of Shells and Strength Design of Thin-Walled and Engineering Structures of Complex Shape [in Russian], Izd. RUDN, Moscow (2001), pp. 30–34.
V. A. Bazhenov, O. P. Krivenko, and M. O. Solovei, “Effect of thermomechanical loading conditions on the stability and postbuckling behavior of shells with constant and stepwise-varying thickness,” Opir Mater. Teor. Sporud, 77, 30–42 (2005).
V. A. Bazhenov, O. P. Krivenko, and M. O. Solovei, “Stability of conical shells with linearly varying thickness,” Opir Mater. Teor. Sporud, 78, 46–51 (2006).
V. A. Bazhenov, O. P. Krivenko, and M. O. Solovei, “Analyzing solutions of buckling problems for shells with temperature-dependent material properties,” Opir Mater. Teor. Sporud, 79, 73–81 (2006).
V. A. Bazhenov, O. P. Krivenko, and M. O. Solovei, “Convergence and accuracy of solutions for a spatial finite element in problems of nonuniform heating of rods and beams,” Opir Mater. Teor. Sporud, 80, 54–65 (2006).
V. A. Bazhenov, O. P. Krivenko, and N. A. Solovei, “Assessment of the curvature effect on the stability and postbuckling behavior of ribbed panels,” Strength of Materials, 39, No. 6, 658–662 (2007).
V. A. Bazhenov, A. S. Sakharov, N. A. Solovei, O. P. Krivenko, and N. Ayat, “Moment scheme of the finite-element method in problems of the strength and stability of flexible shells subjected to the action of forces and thermal factors,” Strength of Materials, 31, No. 5, 499–504 (1999).
V. A. Bazhenov, M. O. Solovei, and O. P. Krivenko, “Nonlinear equations of deformation of ribbed thin multilayer shells under thermomechanical loading,” Opir Mater. Teor. Sporud, 64, 116–127 (1998).
V. A. Bazhenov, M. O. Solovei, O. P. Krivenko, and N. Ayat, “Stability of flexible shells under combined thermomechanical loading,” Opir Mater. Teor. Sporud, 65, 75–90 (1999).
V. A. Bazhenov, M. O. Solovei, and O. P. Krivenko, “Equations of the moment finite-element scheme in buckling problems for inhomogeneous shells under thermomechanical loading,” Opir Mater. Teor. Sporud, 66, 22–25 (1999).
V. A. Bazhenov, M. O. Solovei, and O. P. Krivenko, “Stability of smooth, ribbed, and cracked flexible shallow panels,” Opir Mater. Teor. Sporud, 67, 92–103 (2000).
V. A. Bazhenov, M. O. Solovei, and O. P. Krivenko, “Effect of the parameters of ribs on the stability of finite panels,” Opir Mater. Teor. Sporud, 69, 18–24 (2001).
V. A. Bazhenov, M. O. Solovei, and O. P. Krivenko, “Stability of flexible shallow panels with stepwise-varying thickness,” in: System Technologies: Mathematical Problems of Engineering Mechanics [in Ukrainian], Issue 2, Dnepropetrovsk (2001), pp. 7–11.
V. A. Bazhenov, N. A. Solovei, and O. P. Krivenko, “Effect of variable thickness on the stability of shallow panels under uniform pressure,” in: System Technologies: Mathematical Problems of Engineering Mechanics [in Ukrainian], Issue 4, Dnepropetrovsk (2003), pp. 15–20.
V. A. Bazhenov, N. A. Solovei, and O. P. Krivenko, “Stability of shallow shells of revolution with linearly varying thickness,” Aviats.-Kosmich. Tekh. Tekhnol., No. 2, 18–25 (2004).
V. A. Bazhenov, A. S. Sakharov, and V. K. Tsykhanovskii, “The moment finite-element scheme in problems of nonlinear continuum mechanics,” Int. Appl. Mech., 38, No. 6, 658–692 (2002).
A. M. Belostotskii, “Finite-element models of spatial plates, shells, and solids: Creation, program implementation, and research,” Sb. Nauch. Trudov Gidroproekta, 100, 24–35 (1985).
V. V. Bolotin, “Nonlinear theory of elasticity and stability in large,” Rasch. Prochn., 3, 310–354 (1958).
D. Bushnell and S. Smith, “Stress and buckling of nonuniformly heated cylindrical and conical shells,” AIAA J., 9, No. 12, 2314–2321 (1971).
D. V. Vainberg, E. A. Gotsulyak, and V. I. Gulyaev, “Thermomechanical instability of a deformable medium,” Sopr. Mater. Teor. Sooruzh., 16, 153–156 (1972).
N. V. Valishvili, Methods for Computer Design of Shells of Revolution [in Russian], Mashinostroenie, Moscow (1976).
P. M. Varvak, I. M. Buzun, A. S. Gorodetskii, et al., Finite-Element Method [in Russian], Vysshaya Shkola,
L. F. Vakhlaeva and V. A. Krys’ko, “Stability of flexible sloping shells in a temperature field,” Int. Appl. Mech., No. 1, 12–18 (1983).
A. S. Vol’mir, “Modern problems in the theory of plates and shells in aircraft,” in: Important Problems of Aviation Science and Technology [in Russian], Mashinostroenie, Moscow (1984), pp. 77–87.
A. S. Vol’mir, Stability of Deformable Systems [in Russian], Nauka, Moscow (1967).
K. S. Galiev, L. A. Gordon, and L. A. Rozin, “Forming a universal stiffness matrix in the FEM,” Izv. VNIIG, 105, 174–188 (1974).
K. Z. Galimov, Fundamentals of the Nonlinear Theory of Thin Shells [in Russian], Izd. KGU, Kazan (1975).
K. Z. Galimov and Kh. M. Mushtari, “Some issues of strength and stability of shells in nonuniform temperature field,” Tr. Fiz.-Tekh. Inst. KF AN SSSR, 1, 3–12 (1954).
V. V. Gnatyuk, V. V. Ulitin, and A. N. Snitko, “Experimental-theoretical study of the influence of imperfections on the stability of cylindrical panels subject to heating,” Strout. Mekh. Rasch. Sooruzh., No. 6, 35–38 (1987).
A. I. Golovanov and M. S. Kornishin, “Introduction to the finite-element method in statics of thin shells,” Kazan. Fiz.-Tekh. Inst. KF AN SSSR, Kazan (1990).
A. V. Gondlyakh, “Iterative analytical theory of deformation of multilayer shells,” Sopr. Mater. Teor. Sooruzh., 53, 33–37 (1988).
É. I. Grigolyuk and V. V. Kabanov, Stability of Shells [in Russian], Nauka, Moscow (1978).
É. I. Grigolyuk and V. I. Shalashilin, Problems of Nonlinear Deformation: Parameter Continuation Method in Nonlinear Problems of Solid Mechanics, Nauka, Moscow (1988).
Ya. M. Grigorenko and A. T. Vasilenko, Theory of Shells of Variable Stiffness, Vol. 4 of the five-volume series Methods of Shell Design [in Russian], Naukova Dumka, Kyiv (1981).
Ya. M. Grigorenko and V. I. Gulyaev, “Nonlinear problems of shell theory and their solution methods (review),” Int. Appl. Mech., 27, No. 10, 929–947 (1991).
Ya. M. Grigorenko and A. P. Mukoed, Solving Nonlinear Problems in the Theory of Shells on a Computer [in Russian], Vyshcha Shkola, Kyiv (1983).
A. N. Guz and I. Yu. Babich, Three-Dimensional Theory of Stability of Rods, Plates, and Shells [in Russian], Vyshcha Shkola, Kyiv (1980).
V. I. Gulyaev, V. A. Bazhenov, and E. A. Gotsulyak, Stability of Nonlinear Mechanical Systems [in Russian], Vyshcha Shkola, Lviv (1982).
J. Descloux, Méthode des Éléments Finis, Suisse, Lausanne (1973).
M. W. Johnson, Jr. and R. W. McLay, “Convergence of the finite element method in the theory of elasticity,” ASME, J. Appl. Mech., 35, No. 2, 274–278 (1968).
V. S. Ekel’chik, “Bending of rectangular orthotropic plates elastically fixed along the boundary,” TsNIITS, 36, 32–48 Moscow (1962).
N. B. Erofeeva, “Bending of an orthotropic rectangular plate clamped along the boundary,” Izv. VUZov, Stroit. Arkhitekt., No. 7, 52–55 (1970).
G. N. Zamula, K. M. Ierusalimskii, and G. S. Karpova, “Analysis of stability and heat resistance of complex reinforced structures,” Uch. Zap. TsAGI, 20, No. 4, 84–87 (1989).
O. C. Zienkiewicz, The Finite-Element Method in Engineering Science, McGraw-Hill, New York (1971).
O. C. Zienkiewicz, B. M. Irons, F. C. Scott, and J. S. Campbell, “Three-dimensional stress analysis,” in: B. F. de Veubeke (ed.), High Speed Computing of Elastic Structures, Universite de Liege (1971).
V. P. Il’in and V. V. Karpov, Stability of Ribbed Shells against Large Displacements [in Russian], Stroiizdat, Leningrad (1986).
B. Ya. Kantor, Nonlinear Problems in the Theory of Inhomogeneous Shallow Shells [in Russian], Naukova Dumka, Kyiv (1974).
V. V. Kirichevskii and A. S. Sakharov, Nonlinear Problems in the Thermomechanics of Structures Made of Weakly Compressible Elastomers [in Russian], Budivel’nyk, Kyiv (1992).
A. P. Kiselev, Development of the Finite-Element Method in Studies of the Linear and Nonlinear Deformation of Two and Three-Dimensional Elastic Bodies [in Russian], Author’s Abstract of DSc Thesis, Volgograd Gos. Sel’khoz. Akad., Volgograd (2007).
V. N. Kislookii, N. V. Koval’chuk, A. D. Legostaev, and N. A. Solovei, “Stability analysis of reinforced slightly conical shells with large holes in geometrically nonlinear formalism,” Int. Appl. Mech., 20, No. 11, 1037–1042 (1984).
V. N. Kislookii, A. S. Sakharov, and N. A. Solovei, “Moment scheme of the finite-element method in geometrically nonlinear problems regarding the strength and stability of shells,” Strength of Materials, 9, No. 7, 808–817 (1977).
A. D. Kovalenko, Thermoelasticity [in Russian], Vyshcha Shkola, Kyiv (1975).
N. V. Koval’chuk and N. A. Solovei, “The stress-strain state and stability of conical shells,” Int. Appl. Mech., 24, No. 5, 484–488 (1988).
W. T. Koiter, “Elastic stability and postbuckling behavior,” in: R. E. Langee (ed.), Proc. Symp. on Nonlinear Problems, Univ. Press, Madison (1963), pp. 257–275.
V. A. Koldunov, A. N. Kudinov, and O. I. Cherepanov, “Three-dimensional stability analysis of shells,” in: Proc. 6th Int. Sci. Symp. on Modern Problems of Plasticity and Stability in Solid Mechanics (Tver, March 1–3, 2006) [in Russian], TGTU, Tver (2006), pp. 31–39.
V. A. Koldunov and O. I. Cherepanov, “Numerical model for design of shells and shell structures using three-dimensional nonlinear theory of elasticity,” in: Complex Systems: Data Processing, Modeling, and Optimization [in Russian], TvGU, Tver (2002), pp. 48–59.
O. P. Krivenko, Stability of Flexible Shells with Variable Thickness Parameters under Thermomechanical Loading [in Ukrainian], Author’s Abstract of PhD Thesis, Kyiv. Nats. Univ. Budivn. Arkhitekt., Kyiv (2005).
O. P. Krivenko, “Influence of combined boundary conditions on the stability of shallow shells of revolution,” Opir. Mater. Teor. Sooruzh., 81, 84– 90 (2007).
O. P. Krivenko, “Influence of combined boundary conditions on the stability of shallow panels with square planform,” Opir. Mater. Teor. Sooruzh., 82, 65–70 (2008).
V. I. Kucheryuk, A. D. Dorogin, and V. P. Bochagov, “Design of multilayer plates by an experimental-theoretical method,” Sroit. Mekh. Rasch. Sooruzh., No. 2, 72–74 (1983).
S. G. Lekhnitskii, Anisotropic Plates [in Russian], Gostekhizdat, Moscow (1957).
C.-L. Liao and J. N. Reddy, “Analysis of anisotropic stiffened, composite laminates using a continuum-based shell element,” Comput. Struct., 34, No. 6, 805–815 (1990).
S. S. Marchenko, Octagonal Three-Dimensional Finite Element with Vector Approximation of Displacement Fields for Analysis of the Deformation of Shells of Revolution [in Russian], Author’s Abstract of PhD Thesis, Volgogr. Gos. Arkhitekt.-Stroit. Univ., Volgograd (2004).
V. V. Gorev, B. Yu. Uvarov, V. V. Filippov, et al., Elements of Steel Structures, Vol. 1 of the three-volume series Metal Structures [in Russian], Vyssh. Shk., Moscow (1997).
À. S. Sakharov, V. N. Kislookii, V. V. Kirichevskii, et al., Finite-Element Method in Solid Mechanics [in Russian], Vyshcha Shkola, Kyiv (1982).
K. H. Murray, “Comments on the convergence of finite element solutions,” AIAA J., 8, No. 4 (1970).
V. A. Bazhenov, A. S. Sakharov, A. V. Gondlyakh, and S. L. Mel’nikov, Nonlinear Problems in the Mechanics of Multilayer Shells [in Russian], NII Strout. Mekh., Kyiv (1994).
A. P. Nikolaev and A. P. Kiselev, “Using the three-dimensional theory to design shells,” in: Proc. Int. Sci. Conf. on Architecture of Shells and Strength Analysis of Thin-Walled Building and Engineering Structures of Complex Shape [in Russian], Izd. RUDN, Moscow (2001), pp. 29–30.
A. P. Nikolaev and A. P. Kiselev, “Design of shells based on three-dimensional finite elements in the form of a triangular prism and octagon,” in: Proc. Int. Sci. Conf. on Architecture of Shells and Strength Analysis of Thin-Walled Building and Engineering Structures of Complex Shape [in Russian], Izd. RUDN, Moscow (2001), pp. 319–323.
W. Nowacki, Theory of Elasticity [in Polish], PWN, Warsaw (1970).
V. V. Novozhilov, Theory of Thin Shells [in Russian], Sudpromgiz, Leningrad (1962).
P. M. Ogibalov and V. F. Gribanov, Thermal Stability of Plates and Shells [in Russian], Izd. MGU, Moscow (1968).
J. T. Oden, Finite Elements of Nonlinear Continua, McGraw-Hill, New York (1971).
A. V. Perel’muter and V. I. Slivker, Design Models of Structures and Possibility to Analyze Them [in Russian], Izd. Stal’, Kyiv (2002).
V. G. Piskunov and V. E. Verizhenko, Linear and Nonlinear Problems for Layered Structures [in Russian], Budivel’nyk, Kyiv (1986).
Ya. S. Podstrigach and R. N. Shvets, Thermoelasticity of Thin Shells [in Russian], Naukova Dumka, Kyiv (1983).
Problems of Stability in Structural Mechanics [in Russian], Stroiizdat, Moscow (1965).
V. F. Gribanov, I. A. Krokhin, N. G. Panichkin, V. M. Sannikov, and Yu. I. Fomitchev, Strength, Stability, and Vibrations of Thermostressed Shell Structures [in Russian], Mashinostroenie, Moscow (1990).
V. N. Kislookii, A. D. Legostaev, A. S. Sakharov, N. A. Solovei, E. N. Il’chenko, and V. N. Kritskii, Prochnost’-75. Three-Dimensional Structures Design Software System, Part III, Section Distos: Statics and Dynamics of Shell Structures [in Russian], Manual Inv. No. 5759 dep. at RFAP IK AN USSR, Kyiv (1980).
A. O. Rasskazov, I. I. Sokolovskaya, and N. A. Shul’ga, Theory and Design of Layered Orthotropic Plates and Shells [in Russian], Naukova Dumka, Kyiv (1986).
R. B. Rikards, Finite-Element Method in the Theory of Shells and Plates [in Russian], Zinatne, Riga (1988).
A. S. Sakharov, “A moment finite-element scheme (MFES) that allows for rigid-body displacements,” Sopr. Mater. Teor. Sooruzh., 24, 147–156 (1974).
A. S. Sakharov, “Moment theory of the finite-element method,” in: Proc. VII Int. Congr. on Application of Mathematics and Engineering Sciences [in Russian], Weimar, 22, No. 2 (1975), pp. 218–227.
A. S. Sakharov and N. A. Solovei, “Convergence analysis of the finite-element method in problems of plates and shells,” in: Spatial Structures of Buildings and Installations [in Russian], Issue 3, Stroiizdat, Moscow (1977), pp. 10–15.
M. O. Solovei, “Modeling the thermoelastic properties of multilayer materials in buckling problems for inhomogeneous shells,” Opir. Mater. Teor. Sporud, 73, 17–30 (2003).
M. O. Solovei, “A modified three-dimensional finite element for modeling thin inhomogeneous shells,” Opir. Mater. Teor. Sporud, 80, 96–113 (2006).
M. O. Solovei, Stability and Postbuckling Behavior of Elastic Inhomogeneous Shells under Thermomechanical Loading [in Russian], Author’s Abstract of DSc Thesis, Kyivs’k. Nats. Univ. Budivn. Arkhitekt, Kyiv (2008).
N. A. Solovei, “Analysis of the stress–strain and stability of plates and shells with stepwise-variable stiffness using a modified finite element,” Sopr. Mater. Teor. Sooruzh, 43, 30–35 (1983).
N. A. Solovei, “Implementation of an algorithm for solving systems of nonlinear equations in buckling problems for shells in PROCHNOST’-75,” Vych. Prikl. Math., 62, 39–51 (1987).
N. A. Solovei, “Solution of geometrically nonlinear buckling problems for ribbed multilayer shells based on the moment finite-element scheme,” Sopr. Mater. Teor. Sooruzh, 60, 110–117 (1992).
N. A. Solovei and O. P. Krivenko, “Influence of curvature on the stability of flexible shallow panels subject to nonuniform heating,” Opir. Mater. Teor. Sporud, 66, 18–21 (1999).
N. A. Solovei and O. P. Krivenko, “Comparative analysis of solutions to buckling problems for flexible shells subject to different laws of nonuniform heating,” Opir. Mater. Teor. Sporud, 70, 104–109 (2002).
N. A. Solovei and O. P. Krivenko, “Stability analysis of smooth (with linearly varying thickness) and faceted (with stepwise varying thickness) shallow spherical shells,” Opir. Mater. Teor. Sporud, 72, 83–96 (2003).
N. A. Solovei and O. P. Krivenko, “Influence of heating on the stability of smooth shallow spherical shells with linearly varying thickness,” Opir. Mater. Teor. Sporud, 74, 60–73 (2004).
N. A. Solovei and O. P. Krivenko, “Influence of heating on the stability of faceted shallow spherical shells,” Opir. Mater. Teor. Sporud, 75, 80–86 (2004).
N. A. Solovei, I. A. Tregubova, and O. A. Mazurkov, “Calculating the exact volume of curvilinear finite elements,” Prikl. Geometr. Inzhen. Grafika, Budivel’nyk, 48, 40–44 (1989).
G. Strang and G. J. Fix, An Analysis of the Finite-Element Method, Prentice-Hall, Englewood Cliffs (1973).
S. P. Timoshenko, Stability of Bars, Plates, and Shells [in Russian], Nauka, Moscow (1971).
S. P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York (1959).
J. W. Hutchison and W. T. Koiter, “Postbucking theory,” Appl. Mech. Rev., Vol. 23, 1353–1366 (1970).
V. N. Yushkin, Three-Dimensional Stress–Strain State of Articulated Cylindrical Shells based on the FEM [in Russian], Author’s Abstract of PhD Thesis, Volgogr. Gos. Arkhit.-Strout. Univ., Volgograd (2004).
ANSYS User’s Manual for revision 5.6. Vol. I. Procedure; Vol. II. Command; Vol. III. Elements; Vol. IV. Theory.
R. Hauptmann, S. Doll, M. Harnau, and K. Schweizerhof, “Solid-shell elements with linear and quadratic shapefunctuions at large deformations with nearly incompressible materials,” Comput. Struct., 79, No. 18, 1671–1685 (2001).
V. V. Kiricevskij, A. S. Sacharov, and N. A. Solovej, “FEM-Algorithmen fur geometrisch und physikalisch nichtlineare Aufgaben der Statik und Stabilitat raumlicher Konstruktionen,” Technische Mechanik, 8, No. 2, 63–70 (1987).
S. Klinkel, F. Gruttmann, and W. Wagner, “A continuum based three-dimensional shell element for laminated structures,” Comput. Struct., 71, No. 1, 43–62 (1999).
N. A. Solovei, “Geometrical modelling of shells with complex form by finite element system for strength analysis,” Prikl. Geometr. Inzhen. Grafika, 69, 245–251 (2001).
G. Zboinski and W. Ostachowicz, “An algorithm of a family of 3D-based, solid-to-shell transition, hpq/hp-adaptive finite elements,” J. Theor. Appl. Mech. (Poland), 38, No. 4, 791–806 (2000).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Mekhanika, Vol. 45, No. 9, pp. 3–40, September 2009.
Rights and permissions
About this article
Cite this article
Bazhenov, V.A., Solovei, N.A. Nonlinear deformation and buckling of elastic inhomogeneous shells under thermomechanical loads. Int Appl Mech 45, 923–953 (2009). https://doi.org/10.1007/s10778-010-0236-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10778-010-0236-1