×

Variational finite difference methods in linear and nonlinear problems of the deformation of metallic and composite shells (review). (English. Russian original) Zbl 1295.74103

Int. Appl. Mech. 48, No. 6, 613-687 (2012); translation from Prikl. Mekh. Kiev 48, No. 6, 3-80 (2012).
Summary: Variational finite difference methods of solving linear and nonlinear problems for thin and nonthin shells (plates) made of homogeneous isotropic (metallic) and orthotropic (composite) materials are analyzed and their classification principles and structure are discussed. Scalar and vector variational finite-difference methods that implement the Kirchhoff-Love hypotheses analytically or algorithmically using Lagrange multipliers are outlined. The Timoshenko hypotheses are implemented in a traditional way, i.e., analytically. The stress-strain state of metallic and composite shells of complex geometry is analyzed numerically. The numerical results are presented in the form of graphs and tables and used to assess the efficiency of using the variational finite-difference methods to solve linear and nonlinear problems of the statics of shells (plates).

MSC:

74S20 Finite difference methods applied to problems in solid mechanics
74K25 Shells
74E30 Composite and mixture properties
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids

Software:

FDEM
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N. P. Abovskii, N. P. Andreev, and A. P. Deruga, Variational Principles in the Theories of Elasticity and Shells [in Russian], Nauka, Moscow (1978).
[2] N. P. Abovskii, N. P. Andreev, A. P. Deruga, L. V. Endzhievskii, V. B. Idel’son, and N. I. Marchuk, ”Development and application of extreme variational principles to nonlinear design of complex shell systems,” in: Space Structures in the Krasnoyarsk Region [in Russian], KPI, Krasnoyarsk (1986), pp. 3–17.
[3] N. P. Abovskii, N. P. Andreev, A. P. Deruga, and V. I. Savchenkov, Numerical Methods in the Theories of Elasticity and Shells [in Russian], Izd. Krasnoyarsk. Univ., Krasnoyarsk (1986).
[4] N. P. Abovskii and A. P. Deruga, ”Basic results and areas of development of the variational finite-difference method in the design of complex-shaped space frames,” Space Structures in the Krasnoyarsk Region [in Russian], KPI, Krasnoyarsk (1989), pp. 13–26.
[5] V. G. Bazhenov, A. G. Ugodchikov, and A. P. Shinkarenko, ”Numerical analysis of the elastic-plastic deformation of shells with curved openings under impulsive loading,” Int. Appl. Mech., 15, No. 5, 398–402 (1979). · Zbl 0438.73070
[6] V. G. Bazhenov and D. T. Chekmarev, Variational Finite-Difference Schemes in Nonstationary Wave Dynamic Problems for Plates and Shells [in Russian], Izd. Nizhegorod. Univ., Nizhny Novgorod (1992). · Zbl 0925.73493
[7] V. G. Bazhenov and D. T. Chekmarev, Variational Finite-Difference Method in the Nonstationary Dynamics of Plates and Shells: Problem Solving Manual [in Russian], Izd. NNGU, Nizhny Novgorod (2000). · Zbl 0925.73493
[8] V. G. Bazhenov and A. P. Shinkarenko, ”Variational finite-difference method for solving two-dimensional problems of the dynamics of elastoplastic shells,” in: Applied Problems of Strength and Plasticity [in Russian], Issue 3, Gor’k. Univ., Gorky (1976), pp. 61–69.
[9] V. N. Barashkov, ”Variational finite-difference algorithm for solving problems of elasticity and plasticity,” Part 1, Izv. Tomsk. Politekhn. Univ., 306, No. 3, 23–28 (2003).
[10] S. V. Bosakov and O. V. Kozunova, ”Variational finite-difference approach to the nonlinear design of foundations plates on a stratified elastic bed,” in: Methods for Solving Applied Problems of Solid Mechanics [in Ukrainian], Issue 10, Dnipropetr. Nats. Univ., Dnipropetrivsk (2009), pp. 34–40.
[11] D. V. Vainberg, P. P. Voroshko, and A. L. Sinyavskii, ”Numerical solution of a spatial problem of elasticity,” in: Design of Space Structures [in Russian], Issue 12, Stroiizdat, Moscow (1969), pp. 4–26.
[12] D. V. Vainberg, V. M. Gerashchenko, I. Z. Roitfarb, and A. L. Sinyavskii, ”Using the variational method to derive finite-difference equations of the bending of plates,” in: Strength of Materials and Structural Theory [in Russian], Issue 1, Budivel’nyk, Kyiv (1965), pp. 23–33.
[13] P. M. Varvak, New Methods for Solving Problems of Strength of Materials [in Russian], Vyshcha Shkola, Kyiv (1977).
[14] K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, Oxford (1975). · Zbl 0339.73035
[15] P. P. Voroshko, ”Variational finite-difference method for solving a three-dimensional problem of elasticity,” in: Numerical Methods for Design of Space Structures [in Russian], Izd. KISI, Kyiv (1969), pp. 45–48.
[16] K. Z. Galimov, Fundamentals of the Nonlinear Theory of Thin Shells [in Russian], Izd. Kazan. Univ., Kazan (1975).
[17] A. I. Golovanov, O. N. Tyuleneva, and A. F. Shigabutdinov, Finite-Element Method in the Statics and Dynamics of Thin-Walled Structures [in Russian], Fizmatlit, Moscow (2006).
[18] E. A. Gotsulyak, V. N. Ermishev, and N. T. Zhadrasinov, ”Convergence of the curvilinear-mesh method in problems of the theory of shells,” in: Strength of Materials and Theory of Structures [in Russian], Issue 39, Budivel’nyk, Kyiv (1981), pp. 80–84.
[19] Ya. M. Grigorenko and A. P. Mukoed, Computer Solution of Problems in the Theory of Shells [in Russian], Vyshcha Shkola, Kyiv (1979).
[20] D. S. Griffin and R. B. Kellogg, ”A numerical solution for axially symmetrical and plane elasticity problems,” Int. J. Solids Struct., 3, No. 5, 781–794 (1967). · Zbl 0152.42801 · doi:10.1016/0020-7683(67)90053-4
[21] O. G. Gurtovyi and S. O. Tinchuk, ”Variational finite-difference implementation of refined models in problems of the deformation of multilayer coatings,” in: Resource-Saving Materials, Structures, and Buildings [in Ukrainian], Issue 16, NUVGP, Rivne (2007), pp. 170–177.
[22] Yu. A. Gusman and L. A. Oganesyan, ”Convergence analysis of finite-difference schemes for degenerate elliptic equations,” Zh. Vych. Mat. Mat. Fiz., 5, No. 2, 351–357 (1965).
[23] A. P. Deruga, ”Variational finite-difference schemes based on superconvergence,” in: Proc. 4th All-Russia Seminar on Problems of Optimal Structural Design [in Russian], NGASU, Novosibirsk (2002), pp. 118–130.
[24] A. P. Deruga, ”A feature of the variational finite-difference scheme for inhomogeneous shells,” in: Space Structures in the Krasnoyarsk Region [in Russian], KISI, Krasnoyarsk (1993), pp. 173–179.
[25] Zh. S. Erzhanov and T. D. Karinbaev, Finite-Element Method in Problems of Rock Mechanics [in Russian], Nauka Kaz. SSR, Alma-Ata (1975).
[26] I. P. Ermakovskaya, V. A. Maksimyuk, and I. S. Chernyshenko, ”Nonlinear elastic two-dimensional problems of the statics of orthotropic thin shells and a method for solving them,” Red. Zh. Prikl. Mekh., Kyiv (1988), No. 7526-V 88 dep. at VINITI 10/19/88, Annot. Prikl. Mekh., 25, No. 2, 129 (1989).
[27] A. A. Il’yushin, Plasticity [in Russian], Gostekhizdat, Moscow–Leningrad (1948).
[28] B. Ya. Kantor and E. V. Eseleva, ”Vector variational finite-difference method in the analysis of the deformation of flexible Timoshenko shells,” in: Proc. 17th Int. Conf. on the Theory of Shells and Plates (September 15–20, 1995) [in Russian], Vol.2, Izd. Kazan. Gos. Univ., Kazan (1996), pp. 130–133.
[29] A. V. Karmishin, V. I. Myachenkov, and A. A. Repin, ”Variational method for deriving finite-difference equations for orthotropic plates,” Nekot. Vopr. Prochn. Konstr., 3, 63–67 (1967).
[30] V. A. Koldunov, A. N. Kudinov, and O. I. Cherepanov, ”Variational finite-difference analysis of the stress–strain state and stability of toroidal structural members,” Vestn. TvGU, Ser. Prikl. Mat., 2, 45–50 (2005).
[31] A. N. Guz, A. S. Kosmodamianskii, V. P. Shevchenko, et al., Stress Concentration, Vol. 7 of the 12-volume series Mechanics of Composite Materials [in Russian], A.S.K., Kyiv (1998).
[32] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers. Definitions, Theorems, and Formulas for Reference and Review, McGraw-Hill, New York (1974). · Zbl 0187.00101
[33] V. V. Kuznetsov and S. V. Levyakov, ”Finite-variation method in the nonlinear mechanics of shells,” Sib. Zh. Vych. Mat., 11, No. 3, 329–340 (2008). · Zbl 1201.74263
[34] I. S. Kuz, ”Numerical implementation of the variational finite-difference method for domains with curved boundary,” in: Numerical Analysis, Mathematical Simulation, and Their Application in Mechanics [in Russian], Izd. MGU, Moscow (1988), pp. 59–63.
[35] R. Courant, K. Friedrichs, and H. Lewy, ”On the partial difference equations of mathematical physics,” IBM J. Res. Develop., 11, No. 2, 215–234 (1967). · Zbl 0145.40402 · doi:10.1147/rd.112.0215
[36] V. A. Lomakin, ”On the theory of anisotropic plasticity,” Vestn. MGU, Ser. Mat. Mekh., No. 4, 49–53 (1964).
[37] V. A. Lomakin and M. A. Yumashev, ”Stress–strain relationships with nonlinear deformation of orthotropic glass-reinforced plastics,” Mech. Comp. Mater., 1, No. 4, 15–18 (1965).
[38] R. R. Mavlyutov, Stress Concentration in Airframe Elements [in Russian], Nauka, Moscow (1981).
[39] V. A. Maksimyuk, ”Using Lagrangian multipliers in static problems for composite shells,” Dop. NAN Ukrainy, No. 11, 75–79 (1998).
[40] V. A. Maksimyuk and I. S. Chernyshenko, ”Numerical solution of boundary-value problems in the theory of thin shells and plates with a curved hole,” Teor. Prikl. Mekh., 30, 117–126 (1999).
[41] I. E. Mileikovskii and S. I. Trushin, Design of Thin-Walled Structures [in Russian], Stroiizdat, Moscow (1989).
[42] S. G. Mikhlin, ”Variational methods for solving problems of mathematical physics,” Usp. Mat. Nauk, 5, No. 6, 3–51 (1950).
[43] V. P. Mushchanov and O. I. Demidov, ”Variational finite-difference design of a sandwich plate on a Winkler foundation,” Such. Promysl. Tsyviln. Budivn., 6, No. 2, 77–91 (2010).
[44] V. F. Mushchanov and A. I. Demidov, ”Elastoplastic stress state of circular conical shells of variable and constant thickness with a hole,” Metal. Konstr., 14, No. 3, 125–142 (2008).
[45] V. F. Mushchanov and A. I. Demidov, ”Elastoplastic state of a circular toroidal shell with a rectangular hole,” Such. Promysl. Tsyviln. Budivn., 3, No. 2, 67–77 (2007).
[46] W. F. Noh, ”CEL: A time-dependent, two-space-dimensional, coupled Euler–Lagrange code,” Meth. Comp. Phys., 3, 117–179 (1964).
[47] L. A. Oganesyan and L. A. Rukhovets, ”Variational finite-difference schemes for linear elliptic equations of the second order in a two-dimensional domain with piecewise-smooth boundary,” Zh. Vych. Mat. Mat. Fiz., 8, No. 1, 97–114 (1968). · Zbl 0253.65062
[48] L. A. Oganesyan, ”Numerical design of plates,” in: Computer Solution of Engineering Problems [in Russian], Proc. Conf. on Mechanization and Automation of Engineering and Management, Leningrad, June (1963), pp. 84–97.
[49] L. A. Oganesyan and L. A. Rukhovets, Variational Finite-Difference Methods for Solving Elliptic Equations [in Russian], Izd. AN Arm. SSR, Yerevan (1979). · Zbl 0496.65053
[50] I. B. Rudenko, ”Equilibrium of an elastoplastic spherical shell with two different circular holes,” in: Problems of Computatonal Mechanics and Strength of Structures [in Ukrainian], Issue 15, Lira, Dnipropetrovsk (2011), pp. 156–161.
[51] G. N. Savin, Stress Distribution around Holes [in Russian], Naukova Dumka, Kyiv (1968).
[52] A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York (2001). · Zbl 0971.65076
[53] E. A. Storozhuk, I. S. Chernyshenko, and I. B. Rudenko, ”Elastoplastic state of spherical shells with cyclically symmetric circular holes,” Int. Appl. Mech., 48, No. 5, 573–582 (2012). · Zbl 1295.74067 · doi:10.1007/s10778-012-0539-5
[54] E. A. Storozhuk, ”Vector variational finite-difference method in nonlinear problems of the theory of thin shells with curved holes,” in: Sistems Technologies. Mathematical Problems of Technical Mechanics [in Ukrainian], Issue 3 (62), Dnipropetrovsk (2009), pp. 149–156.
[55] E. A. Storozhuk, I. S. Chernyshenko, and I. B. Rudenko, ”Influence of plastic deformation on the stress concentration in a multiply connected spherical shell,” in: Problems of Computational Mechanics and Structural Strength [in Ukrainian], Issue 18, Vyd. DNU, Dnipropetrovsk (2012), pp. 83–188.
[56] E. A. Storozhuk, I. S. Chernyshenko, and I. B. Rudenko, ”Vector variational finite-difference method for solving elastoplastic problems for multiply connected spherical shells,” in: Problems of Computational Mechanics and Structural Strength [in Ukrainian], Issue 13, IMA-Pres, Dnipropetrovsk (2009), pp. 228–234.
[57] A. N. Guz, I. S. Chernyshenko, V. N. Chekhov, et al., Theory of Thin Shells Weakened by Holes, Vol. 1 of the five-volume series Methods of Shell Design [in Russian], Naukova Dumka, Kyiv (1980). · Zbl 0524.73072
[58] D. T. Chekmarev, Generation and Analysis of Finite-Difference Representation of Variational Finite-Difference and FE Schemes: Information Technology and Computer Simulation in Mathematics and Mechanics (Educational Materials for Qualification Program) [in Russian], Nizhny Novgorod (2007).
[59] K. F. Chernykh, Linear Theory of Shells [in Russian], Izd. Leningr. Univ., Leningrad (1965).
[60] V. I. Akimova and N. A. Shul’ga, ”Design of thick slabs by the variational-difference method,” Int. Appl. Mech., 27, No. 6, 558–563 (1991).
[61] R. S. Atkatsh, M. L. Baron, and M. P. Bieniek, ”A finite difference variational method for bending of plates,” Comp. Struct., 11, No. 6, 573–577 (1980). · Zbl 0437.73063 · doi:10.1016/0045-7949(80)90063-2
[62] D. V. Babich and L. P. Khoroshun, ”Stability and natural vibrations of shells with variable geometric and mechanical parameters,” Int. Appl. Mech., 37, No. 7, 837–869 (2001). · Zbl 1051.74613 · doi:10.1023/A:1012503024244
[63] D. V. Babich and N. Ya. Martynova, ”Variational-difference method of calculating the critical loads for shells of revolution,” Strength of Materials, 15, No. 6, 800–802 (1983). · doi:10.1007/BF01524767
[64] V. D. Barve and S. S. Dey, ”Isoparametric finite difference energy method for plate bending problems,” Comp. Struct., 17, No. 3, 459–465 (1983). · Zbl 0512.73080 · doi:10.1016/0045-7949(83)90137-2
[65] T. Belytschko, J. S.-J. Ong, W. K. Liu, and J. M. Kennedy, ”Hourglass control in linear and nonlinear problems,” Comp. Meth. Appl. Mech. Eng., 43, 251–276 (1984). · Zbl 0522.73063 · doi:10.1016/0045-7825(84)90067-7
[66] D. Bushnell, ”Computerized analysis of shells-governing equations,” Comp. Struct., 18, No. 3, 471–536 (1984). · Zbl 0528.73076 · doi:10.1016/0045-7949(84)90068-3
[67] D. Bushnell, ”Finite-difference energy models versus finite-element models: Two variational approaches on one computer program,” in: S. J. Fenves et al. (eds.), Numerical and Computer Methods in Structural Mechanics, Academic Press, New York (1973), pp. 291–336.
[68] D. Bushnell, B. O. Almroth, and F. Brogan, ”Finite-difference energy method for nonlinear shell analysis,” Comp. Struct., 1, No. 3, 361–387 (1971). · Zbl 0231.73021 · doi:10.1016/0045-7949(71)90020-4
[69] I. S. Chernyshenko and V. A. Maksimyuk, ”On the stress–strain state of toroidal shells of elliptical cross section formed from nonlinear elastic orthotropic materials,” Int. Appl. Mech., 36, No. 1, 90–97 (2000). · Zbl 1002.74062 · doi:10.1007/BF02681963
[70] R. Courant, K. Friedrichs, and H. Lewy, ”On the partial difference equations of mathematical physics,” IBM J. Res. Develop., 11, 215–234 (1967). · Zbl 0145.40402 · doi:10.1147/rd.112.0215
[71] R. Courant, K. Friedrichs, and H. Lewy, ”Uber die partiellen Differenzengleichungen der mathematischen Physik,” Math. Ann., 100, 32–74 (1928). · JFM 54.0486.01
[72] R. Courant, ”Variational methods for the solution of problems of equilibrium and vibrations,” Bull. Amer. Math. Soc., 49, 1–23 (1943). · Zbl 0063.00985 · doi:10.1090/S0002-9904-1943-07818-4
[73] E. U. Dadamukhamedov, V. A. Maksimyuk, I. S. Chernyshenko, ”The influence of the stiffness of reinforcing elements on the deformed state of shells with a hole,” J. Math. Sci., 74, No. 4, 1170–1172 (1995). · Zbl 0850.73171 · doi:10.1007/BF02431086
[74] N. Eratli and A. Gursoy, ”Variational derivative for buckling loads of beams and frames,” Int. Appl. Mech., 41, No. 7, 825–830 (2005). · Zbl 1089.74591 · doi:10.1007/s10778-005-0151-z
[75] L. M. Fielding, S. F. Villaca, and L. F. T. Garcia, ”Energetic finite difference with arbitrary meshes to plate-bending problems,” Appl. Math. Model., 21, No. 11, 691–698 (1997). · Zbl 0898.73073 · doi:10.1016/S0307-904X(97)00099-1
[76] V. F. Godzula and K. I. Shnerenko, ”Stressed state of a composite cylindrical shell with an elliptical hole,” Int. Appl. Mech., 29, No. 11, 925–929 (1993). · doi:10.1007/BF00848276
[77] V. F. Godzula and K. I. Shnerenko, ”Stress–strain analysis of a composite truncated conical shell,” Int. Appl. Mech., 43, No. 7, 761–766 (2007). · Zbl 1164.74018 · doi:10.1007/s10778-007-0075-x
[78] L. I. Golub, V. A. Maksimyuk, and I. S. Chernyshenko, ”Numerical nonlinear elastic analysis of orthotropic spherical shells with an elliptic cutout,” Int. Appl. Mech., 38, No. 2, 203–208 (2002). · Zbl 1094.74521 · doi:10.1023/A:1015769128796
[79] D. S. Griffin and R. S. Varga, ”Numerical solution of plane elasticity problems,” J. Soc. Indust. Appl. Math., 11, 1046–1062 (1963). · Zbl 0122.18903 · doi:10.1137/0111077
[80] K. K. Gupta and J. L. Meek, ”A brief history of the beginning of the finite element method,” Int. J. Numer. Meth. Eng., 39, 3761–3774 (1996). · Zbl 0884.73067 · doi:10.1002/(SICI)1097-0207(19961130)39:22<3761::AID-NME22>3.0.CO;2-5
[81] A. N. Guz, I. S. Chernyshenko, and K. I. Shnerenko, ”Stress concentration near openings in composite shells,” Int. Appl. Mech., 37, No. 2, 139–181 (2001). · Zbl 1010.74518 · doi:10.1023/A:1011316421387
[82] A. N. Guz, E. A. Storozhuk, and I. S. Chernyshenko, ”Nonlinear two-dimensional static problems for thin shells with reinforced curvilinear holes,” Int. Appl. Mech., 45, No. 12, 1269–1300 (2009). · Zbl 1272.74426 · doi:10.1007/s10778-010-0268-6
[83] K. S. Havne and E. L. Stanton, ”On energy-derived difference equations in thermal stress problems,” J. Franklin Inst., 284, No. 2, 127–143 (1967). · Zbl 0218.73014 · doi:10.1016/0016-0032(67)90585-6
[84] J. C. Houbolt, A Study of Several Aerothermoelastic Problems of Aircraft Structure in High-Speed Flight, Verlag Leemann, Zurich (1958).
[85] F. F. Ihlenburg, ”Plate bending analysis with variational finite difference methods on general grid,” Comp. Struct., 48, No. 1, 141–151 (1993). · Zbl 0800.73508 · doi:10.1016/0045-7949(93)90465-P
[86] D. E. Johnson, ”A difference based variational method for shells,” Int. J. Solids Struct., 6, 699–724 (1970). · Zbl 0212.57502 · doi:10.1016/0020-7683(70)90012-0
[87] Yu. V. Kokhanenko, ”Numerical analysis of edge effects in laminated composites under uniaxial compression,” Int. Appl. Mech., 47, No. 6, 700–706 (2011). · Zbl 1295.74025 · doi:10.1007/s10778-011-0494-6
[88] A. L. Kravchuk, E. A. Storozhuk, and I. S. Chernyshenko, ”Stress distribution in flexible cylindrical shells with a circular cut beyond the elastic limit,” Int. Appl. Mech., 24, No. 12, 1179–1182 (1988). · Zbl 0729.73549
[89] L. Lusternik, ”Über einige Anwendungen der direkten Methoden in Variations Rechnung,” Mat. Sb., 33, No. 2, 173–202 (1926).
[90] V. A. Maksimyuk, ”Physically nonlinear problems of the theory of orthotropic composite shells with a curvilinear opening,” Int. Appl. Mech., 34, No. 9, 835–839 (1998). · doi:10.1007/BF02700840
[91] V. A. Maksimyuk, ”Solution of physically nonlinear problems of the theory of orthotropic shells using mixed functionals,” Int. Appl. Mech., 36, No. 10, 1349–1354 (2000). · Zbl 1130.74422 · doi:10.1023/A:1009490201405
[92] V. A. Maksimyuk, ”Study of the nonlinearly elastic state of an orthotropic cylindrical shell with a hole, using mixed functionals,” Int. Appl. Mech., 37, No. 12, 1602–1606 (2001). · doi:10.1023/A:1014849713889
[93] V. A. Maksimyuk and I. S. Chernyshenko, ”Mixed functionals in the theory of nonlinearly elastic shells,” Int. Appl. Mech., 40, No. 11, 1226–1262 (2004). · Zbl 1122.74329 · doi:10.1007/s10778-005-0032-5
[94] V. A. Maksimyuk and I. S. Chernyshenko, ”Nonlinear elastic state of thin-walled toroidal shells made of orthotropic composites,” Int. Appl. Mech., 35, No. 12, 1238–1245 (1999). · Zbl 1049.74691 · doi:10.1007/BF02682397
[95] V. A. Maksimyuk, E. A. Storozhuk, and I. S. Chernyshenko, ”Using mesh-based methods to solve nonlinear problems of statics for thin shells,” Int. Appl. Mech., 45, No. 1, 32–56 (2009). · Zbl 1183.74154 · doi:10.1007/s10778-009-0166-y
[96] V. A. Maksimyuk and I. S. Chernyshenko, ”Numerical solution of problems for shells of variable stiffness with allowance for transverse shears,” Int. Appl. Mech., 27, No. 3, 275–278 (1991). · Zbl 0729.73939
[97] K. Mallikarjuna Rao and U. Shrinivasa, ”A set of pathological tests to validate new finite elements,” Sadhana, 26, 549–590 (2001). · Zbl 1023.74047 · doi:10.1007/BF02703459
[98] D. Mijuca, ”On hexahedral finite element HC8/27 in elasticity,” Comp. Mech., 33, No. 6, 466–480 (2004). · Zbl 1115.74372 · doi:10.1007/s00466-003-0546-9
[99] P. K. Mishra and S. S. Dey, ”Discrete energy method for the analysis of cylindrical shells,” Comp. Struct., 27, No. 6, 753–762 (1987). · Zbl 0626.73081 · doi:10.1016/0045-7949(87)90288-4
[100] W. F. Noh, ”CEL: A time-dependent, two space dimensional, coupled Eulerian–Lagrange code,” Meth. Comp. Phys., No. 3, 117–179 (1964).
[101] A. Ozutok and A. Akin, ”The solution of Euler–Bernoulli beams using variational derivative method,” Sci. Res. Ess., 5, No. 9, 1019–1024 (2010).
[102] M. D. Pazhooh, M. A. Dokainish, and S. Ziada, ”Finite element modal analysis of an inflatable, self-rigidizing toroidal satellite component,” Exper. Appl. Mech., 6, No. 1, 281–288 (2011).
[103] V. G. Piskunov, Yu. M. Fedorenko, and A. E. Stepanova, ”Variational-difference method in the problem of the vibrations of laminated plates,” Int. Appl. Mech., 28, No. 8, 506–511 (1992). · doi:10.1007/BF00847068
[104] E. J. Ruggiero, A. Jha, G. Park, and D. J. Inman, ”A literature review of ultra-light and inflated toroidal satellite components,” Shock Vibr. Digest, 35, No. 3, 171–181 (2003). · doi:10.1177/0583102403035003001
[105] H. G. Schaeffer, Newton–Raphson Conjugate-Gradient Technique for Calculation of Axisymmetric Buckling of Shallow Spherical Shells with Variable Edge Constraint, NASA-TN-D-5664 (1970).
[106] H. G. Schaeffer and W. L. Heard, Jr., Evaluation of an Energy Method Using Finite Differences for Determining Thermal Midplane Stresses in Plates, NASA TN D-2439 (1964).
[107] W. Schonauer and T. Adolph, ”FDEM: How we make theFDMmore flexible than the FEM,” J. Comp. Appl. Math., 158, 157–167 (2003). · Zbl 1027.65151 · doi:10.1016/S0377-0427(03)00461-8
[108] A. C. Scordelis and K. S. Lo, ”Computer analysis of cylindrical shells,” Amer. Concr. Ins. J., 61, No. 5, 539–560 (1964).
[109] N. P. Semenyuk, V. M. Trach and V. V. Merzlyuk, ”On the canonical equations of Kirchhoff–Love theory of shells,” Int. Appl. Mech., 43, No. 10, 1149–1156 (2007). · doi:10.1007/s10778-007-0115-6
[110] Yu. N. Shevchenko and A. Z. Galishin, ”Particularizing the constitutive equations of elastoplasticity of a transversely isotropic material,” Int. Appl. Mech., 47, No. 6, 662–669 (2011). · Zbl 1295.74016 · doi:10.1007/s10778-011-0489-3
[111] K. I. Shnerenko and V. F. Godzula, ”Stress concentration near an opening in a composite shell with account of material inhomogeneity,” Mech. Comp. Mater., 39, No. 5, 433–438 (2003). · doi:10.1023/B:MOCM.0000003293.93478.e7
[112] K. I. Shnerenko and V. F. Godzula, ”Stress state of a cylindrical composite panel weakened by a circular hole,” Int. Appl. Mech., 42, No. 5, 555–559 (2006). · doi:10.1007/s10778-006-0120-1
[113] J. P. Singh and S. S. Dey, ”Variational finite difference method for free vibration of sector plates,” J. Sound Vibr., 136, 91–104 (1990). · doi:10.1016/0022-460X(90)90940-2
[114] J. P. Singh and S. S. Dey, ”Variational finite difference approach to buckling of plates of variable stiffness,” Comp. Struct., 36, No. 1, 39–45 (1990). · Zbl 0709.73092 · doi:10.1016/0045-7949(90)90172-X
[115] M. L. Stein and J. Q. Sanders, Jr., A Method for Deflection Analysis of Thin Low-Aspect-Ratio Wings, NASA TN 3640 (1956).
[116] E. A. Storozhuk and I. S. Chernyshenko, ”Elastic-plastic state of cylindrical shells with a reinforced hole under combined loading,” Int. Appl. Mech., 24, No. 9, 895–897 (1988). · Zbl 0705.73132
[117] E. A. Storozhuk and I. S. Chernyshenko, ”Elastoplastic axially asymmetric deformation of shells with curvilinear openings,” Int. Appl. Mech., 22, No. 7, 644–649 (1986). · Zbl 0618.73051
[118] E. A. Storozhuk and I. S. Chernyshenko, ”Reinforcement of the contour of a hole in an inelastic shell,” Int. Appl. Mech., 24, No. 11, 1064–1068 (1988). · Zbl 0711.73158
[119] E. A. Storozhuk, I. S. Chernyshenko, and V. L. Yaskovets, ”Elastoplastic state of spherical shells in the region of an elliptical hole,” Int. Appl. Mech., 25, No. 7, 667–672 (1989). · Zbl 0708.73022
[120] H. Y. Sun, Y. N. He, and X. L. Feng, ”On error estimates of the penalty method for the unsteady conduction-convention problems 1: Time discretization,” Int. J. Numer. Anal. Model., 9, No. 4, 876–891 (2012). · Zbl 1263.76051
[121] E. N. Troyak, E. A. Storozhuk, and I. S. Chernyshenko, ”Elastoplastic state of a conical shell with a circular hole on the lateral surface,” Int. Appl. Mech., 24, No. 1, 65–69 (1988). · Zbl 0711.73069
[122] A. K. Verma and S. S. Dey, ”Integrated analysis of curved bridge superstructures by variational finite difference method,” Comp. Struct., 38, No. 5–6, 597–603 (1991). · Zbl 0825.73929 · doi:10.1016/0045-7949(91)90011-A
[123] W. C. Walton, Application of a General Finite-Difference Method for Calculating Bending Deformations of Solid Plates, NASA TN D-536 (1960).
[124] W. Wunderlich and W. D. Pilkey, Mechanics of Structures: Variational and Computational Methods, CRC Press, Boca Raton, FL (2003). · Zbl 1030.74001
[125] Y. Xing, B. Liu, and G. Liu, ”A differential quadrature finite element method,” Int. J. Appl. Mech., 1, 207–227 (2010). · doi:10.1142/S1758825110000470
[126] Y. F. Xing and B. Liu, ”High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain,” Int. J. Numer. Meth. Eng., 80, 1718–1742 (2009). · Zbl 1183.74328 · doi:10.1002/nme.2685
[127] V. L. Yaskovets, E. A. Storozhuk, and I. S. Chernyshenko, ”Elastoplastic equilibrium of a spherical shell in the form of an eccentric ring,” Int. Appl. Mech., 26, No. 1, 56–61 (1990). · Zbl 0737.73090
[128] L. Zu, ”Stability of fiber trajectories for winding toroidal pressure vessels,” Comp. Struct., 94, No. 5, 1855–1860 (2012). · doi:10.1016/j.compstruct.2011.11.027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.