Skip to main content
SpringerLink
Log in

Carrera unified formulation (CUF) for shells of revolution. I. Higher-order theory

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Here, higher order models of elastic shells of revolution are developed using the variational principle of virtual power for 3-D equations of the linear theory of elasticity and generalized series in the coordinates of the shell thickness. Following the Unified Carrera Formula (CUF), the stress and strain tensors, as well as the displacement vector, are expanded into series in terms of the coordinates of the shell thickness. As a result, all the equations of the theory of elasticity are transformed into the corresponding equations for the expansion coefficients in a series in terms of the coordinates of the shell thickness. All equations for shells of revolution of higher order are developed and presented here for cases whose middle surfaces can be represented analytically. The resulting equations can be used for theoretical analysis and calculation of the stress–strain state, as well as for modeling thin-walled structures used in science, engineering, and technology.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Al-Khatib, O.J., Buchanan, G.R.: Free vibration of a paraboloidal shell of revolution including shear deformation and rotary inertia. Thin-Walled Struct. 48, 223–232 (2010)

    Article  Google Scholar 

  2. Carrera, E.: A class of two-dimensional theories for anisotropic multilayered plates analysis. In: Atti Della Accademia Delle Scienze di Torino. Classe di Scienze Fisiche Matematiche e Naturali, vol. 19–20, pp. 1–39 (1995)

  3. Carrera, E.: Multilayered shell theories that account for a layer-wise mixed description. Part II: numerical evaluations. AIAA J. 37, 1117–1124 (1999)

    Article  Google Scholar 

  4. Carrera, E.: Multilayered shell theories that account for a layer-wise mixed description. Part I: governing equations. AIAA J. 37, 1107–1116 (1999)

    Article  Google Scholar 

  5. Carrera, E.: Developments, ideas and evaluations based upon the Reissner’s mixed variational theorem in the modeling of multilayered plates and shells. Appl. Mech. Rev. 54, 301–329 (2001)

    Article  Google Scholar 

  6. Carrera, E.: Theories and finite elements for multilayered plates and shells. Arch. Comput. Methods Eng. 9(2), 87–140 (2002)

    Article  MATH  Google Scholar 

  7. Carrera, E.: Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch. Comput. Methods Eng. 10(3), 215–296 (2003)

    Article  MATH  Google Scholar 

  8. Carrera, E., Zozulya, V.V.: Carrera unified formulation for the micropolar plates. Mech. Adv. Mater. Struct. 29(22), 3163–3186 (2022). https://doi.org/10.1080/15376494.2021.1889726

    Article  Google Scholar 

  9. Carrera, E., Zozulya, V.V.: Closed-form solution for the micropolar plates: Carrera unified formulation (CUF) approach. Arch. Appl. Mech. 91, 91–116 (2021)

    Article  Google Scholar 

  10. Carrera, E., Zozulya, V.V.: Carrera unified formulation (CUF) for the micropolar beams: analytical solutions. Mech. Adv. Mater. Struct. 28(6), 583–607 (2021)

    Article  Google Scholar 

  11. Carrera, E., Zozulya, V.V.: Carrera unified formulation (CUF) for the micropolar plates and shells. I. Higher order theory. Mech. Adv. Mater. Struct. 29(6), 773–795 (2022)

    Article  Google Scholar 

  12. Carrera, E., Zozulya, V.V.: Carrera unified formulation (CUF) for the micropolar plates and shells. II. Complete linear expansion case. Mech. Adv. Mater. Struct. 29(6), 796–815 (2022)

    Article  Google Scholar 

  13. Carrera, E., Elishakoff, I., Petrolo, M.: Who needs refined structural theories? Compos. Struct. 264, 113671 (2021)

    Article  Google Scholar 

  14. Cauchy, L.: Sur l’equilibre et le mouvernent d’une plaque solide. Exerc. Mat. 3, 328–355 (1828)

    Google Scholar 

  15. Chernobryvko, M.V., Avramov, K.V.: Natural vibrations of parabolic shells. J. Math. Sci. 217, 229–238 (2016)

    Article  MATH  Google Scholar 

  16. Chernobryvko, M.V., Avramov, K.V., Romanenko, V.N., Batutina, T.J., Tonkonogenko, A.M.: Free linear vibrations of thin axisymmetric parabolic shells. Meccanica 49, 2839–2845 (2014)

    Article  Google Scholar 

  17. Clark, R.A.: On the theory of thin elastic toroidal shells. J. Math. Phys. 29, 146–178 (1950)

    Article  MATH  Google Scholar 

  18. Clark, R.A.: Asymptotic solutions of toroidal shell problems. Q. Appl. Math. 16, 47–60 (1958)

    Article  MATH  Google Scholar 

  19. Czekanski, A., Zozulya, V.V.: Dynamics of vibrating beams using first-order theory based on Legendre polynomial expansion. Arch. Appl. Mech. 90, 789–814 (2020)

    Article  Google Scholar 

  20. El-Raheb, M., Wagner, P.: Harmonic response of cylindrical and toroidal shells to an internal acoustic field. Part II. Results. J. Acoust. Soc. Am. 78(2), 747–757 (1985)

    Article  Google Scholar 

  21. El-Raheb, M., Wagner, P.: Harmonic response of cylindrical and toroidal shells to an internal acoustic field. Part I. Theory. J. Acoust. Soc. Am. 78(2), 738–746 (1985)

    Article  Google Scholar 

  22. Galimov, K.Z., Paimushin, V.N.: Theory of shells of complex geometry, p. 164. Kazan University Press, Kazan (1985)

    Google Scholar 

  23. Gil-Oulbé, M., Ndomilep, I.J.A., Ngandu, P.: Pseudospheric shells in the construction. RUDN J. Eng. Res. 22(1), 84–99 (2021)

    Google Scholar 

  24. Gray, A., Abbena, E., Salamon, S.: Modern differential geometry of curves and surfaces with mathematica, 3rd edn., p. 1016. Chapman and Hall/CRC, New York (2006)

    MATH  Google Scholar 

  25. Guliaev, V.I., Bazhenov, V.A., Lizunov, P.P.: Non-Classical Shell Theory and its Application to Solving of Engineering Problems. L’vov, Vyscha Shkola, 192p (1978)

  26. Sun, B. (ed.): Toroidal Shells. Nova Science Publishers Inc., 186 p (2012)

  27. Kang, J.H., Leissa, A.W.: Three-dimensional vibration analysis of thick Paraboidal shells. Int. J. JSME 45(1), 2–7 (2002)

    Article  Google Scholar 

  28. Kang, J.-H., Leissa, A.W.: Free vibration analysis of complete paraboloidal shells of revolution with variable thickness and solid paraboloids from a three-dimensional theory. Comput. Struct. 83, 2594–2608 (2005)

    Article  Google Scholar 

  29. Kang, J.H., Leissa, A.W.: Three-dimensional vibration analysis of thick hyperboloidal shells of revolution. J. Sound Vib. 282, 277–296 (2005)

    Article  MATH  Google Scholar 

  30. Khoma, I.Y.: Generalized Theory of Anisotropic Shells. Naukova Dumka, Kiev, 172p (1987) (in Russian)

  31. Kil’chevskiy, N.A.: Fundamentals of the Analytical Mechanics of Shells. NASA TT, F-292, Washington, 361p (1965)

  32. Klochkov, Y.V., Nikolaev, A.P., Kiseleva, T.A., Marchenko, S.S.: Comparative analysis of the results of finite element calculations based on an ellipsoidal shell. J. Mach. Manuf. Reliab. 45(4), 328–336 (2016)

    Article  Google Scholar 

  33. Klochkov, Y.V., Nikolaev, A.P., Sobolevskaya, T.A., Vakhnina, O.V., Klochkov, MYu.: Comparative analysis of the results of finite element calculations based on an ellipsoidal shell. Lobach. J. Math. 41(3), 373–381 (2020)

    Article  MATH  Google Scholar 

  34. Klochkov, Y.V., Nikolaev, A.P., Ishchanov, T.R., Andreev, A.S.: Comparative analysis of the results of finite element calculations based on an ellipsoidal shell. J. Mach. Manuf. Reliab. 49(4), 301–307 (2020)

    Article  Google Scholar 

  35. Korjakin, A., RikardS, R., Altenbach, H., Chate, A.: Free damped vibrations of sandwich shells of revolution. J. Sandwich Struct. Mater. 3, 171–196 (2001)

    Article  Google Scholar 

  36. Kornishin, M.S., Paimushin, V.N., Snigirev, V.F.: Computational Geometry in Problems of Shell Mechanics, p. 208. Nauka, Moscow (1989)

    Google Scholar 

  37. Kovarik, V.: Stresses in Layered Shells of Revolution, p. 432. Elsevier, Amsterdam (1989)

    MATH  Google Scholar 

  38. Krawczyk, J.: Infinitesimal isometric deformations of a pseudospherical shell. J. Math. Sci. 109, 1312–1320 (2002)

    Article  Google Scholar 

  39. Krivoshapko, S.N.: Static, vibration, and buckling analyses and applications to one-sheet hyperboloidal shells of revolution. Appl. Mech. Rev. 55(3), 241–270 (2002)

    Article  Google Scholar 

  40. Krivoshapko, S.N.: Research on general and axisymmetric ellipsoidal shells used as domes, pressure vessels, and tanks. Appl. Mech. Rev. 60(11), 336–355 (2007)

    Article  Google Scholar 

  41. Krivoshapko, S.N.: On application of parabolic shells of revolution in civil engineering in 2000–2017. Struct. Mech. Eng. Constr. Build. 4, 4–14 (2017)

    Article  Google Scholar 

  42. Krivoshapko, S.N., Ivanov, V.N.: Encyclopedia of Analytical Surfaces, p. 761. Springer, New York (2015)

    Book  MATH  Google Scholar 

  43. Krivoshapko, S.N., Ivanov, V.N.: Pseudospherical shells in building industry. Build. Reconstr. 2, 32–40 (2018)

    Google Scholar 

  44. Kuhnel, W.: Differential Geometry: Curves, Surfaces, Manifolds, 3rd edn., p. 418. American Mathematical Society, Providence (2016)

    Google Scholar 

  45. Leung, A.Y.T., Kwok, N.T.C.: Free vibration analysis of a toroidal shell. Thin-Walled Struct. 18, 317–332 (1994)

    Article  Google Scholar 

  46. Lutskaya, I.V., Maximuk, V.A., Chernyshenko, I.S.: Modeling the deformation of orthotropic toroidal shells with elliptical cross-section based on mixed functionals. Int. Appl. Mech. 54, 660–665 (2018). https://doi.org/10.1007/s10778-018-0920-0

    Article  Google Scholar 

  47. Meish, V.F.: Numerical solution of dynamic problems for reinforced ellipsoidal shells under nonstationary loads. Int. Appl. Mech. 41(4), 386–391 (2005)

    Article  Google Scholar 

  48. Meish, V.F., Maiborodina, N.V.: Stress state of discretely stiffened ellipsoidal shells under a nonstationary normal load. Int. Appl. Mech. 54(6), 675–686 (2018)

    Article  Google Scholar 

  49. Ming, R.S., Pan, J., Norton, M.P.: Free vibrations of elastic circular toroidal shells. Appl. Acoust. 63, 513–528 (2002)

    Article  Google Scholar 

  50. Naboulsi, S.K., Palazotto, A.N., Greer, J.M.: Static-dynamic analyses of toroidal shells. J. Aerosp. Eng. 13(3), 110–121 (2000)

    Article  Google Scholar 

  51. Pelekh, B.L., Lazko, V.A.: Laminated Anisotropic Plates and Shells with Stress Concentrators. Naukova Dumka, Kiev, 296p (1982)

  52. Pelekh, B.L., Sukhorol'skii, M.A.: Contact Problems of the Theory of Elastic Anisotropic Shells. Naukova Dumka, Kiev, 216p (1980) (in Russian)

  53. Carrera, E., Cinefra, M., Petrolo, M., Zappino, E.: Finite Element Analysis of Structures through Unified Formulation. Wiley, New Delhi, p 385 (2014)

    Book  MATH  Google Scholar 

  54. Poissons, D.: Memoire sur l'equilibre et le mouvement des corps elastique. In: Memoires de l'Academie Royale des Sciences, vol. VIII, pp. 357–570, 623–627 (1829)

  55. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edn. CRC Press LLC, 855p (2004)

  56. Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells, 2nd edn., p. 561. CRC Press LLC, New York (2006)

    Book  Google Scholar 

  57. Rekach, V.G., Krivoshapko, S.N.: Calculation of Shells of Complex Geometry, p. 176. UDN University Press, Moscow (1988)

    MATH  Google Scholar 

  58. Senjanovic, I., Alujevic, N., Catipovic, I.: Cakmak D, Vladimir N Vibration analysis of rotating toroidal shell by the Rayleigh–Ritz method and Fourier series. Eng. Struct. 173, 870–891 (2018)

    Article  Google Scholar 

  59. Senjanovic, I., Alujevic, N., Catipovic, I., Cakmak, D., Vladimir, N., Cho, D.-S.: Natural vibration analysis of pressurised and rotating toroidal shell segment by Rayleigh–Ritz. Eng. Model. 2(4), 57–81 (2019)

    Google Scholar 

  60. Sun, B.: Deformation and stress analysis of catenary shell of revolution. Preprints (2021). https://doi.org/10.20944/preprints202104.0494.v1

  61. Sun, B.: Closed-form solution of axisymmetric slender elastic toroidal shells. J. Eng. Mech. ASCE 136(10), 1281–1288 (2010)

    Article  Google Scholar 

  62. Sutcliffe, W.J.: Stress analysis of toroidal shells of elliptical cross-section. Int. J. Mech. Sci. 13(11), 951–958 (1971)

    Article  MATH  Google Scholar 

  63. Tangbanjongkij, C., Chucheepsakul, S., Jiammeepreecha, W.: Analytical and numerical analyses for a variety of submerged hemi-ellipsoidal shells. J. Eng. Mech. ASCE 146(7), 1–15 (2020)

    Article  Google Scholar 

  64. Timoshenko, S., Woinowsky-Krieger, S.: Theory of Plates and Shells, 2nd edn., p. 611. McGraw-Hill Book Company, Paris (1959)

    MATH  Google Scholar 

  65. Vekua, I.N.: Shell Theory, General Methods of Construction, p. 287. Pitman Advanced Publishing Program, Boston (1986)

    Google Scholar 

  66. Vlasov, V.Z.: General Theory of Shells and Its Application in Engineering. Published by NASA-TT-F-99, 913p (1964)

  67. von Seggern, D.H.: CRC Standard Curves and Surfaces with Mathematica, 2nd edn., p. 660. CRC Press Taylor & Francis Group, Boca Raton (2016)

    Book  MATH  Google Scholar 

  68. Wan, F.Y.M., Weinitschke, H.J.: On shells of revolution with the Love–Kirchhoff hypotheses. J. Eng. Math. 22, 285–334 (1988)

    Article  MATH  Google Scholar 

  69. Washizu, K.: Variational Methods in Elasticity and Plasticity, 3rd edn., p. 630. Pergamon Press, New York (1982)

    MATH  Google Scholar 

  70. Wenmin, R., Wenguo, L., Wei, Z.: A survey of works on the theory of toroidal shells and curved tubes. Acta Mech. Sin. (English Ser.) 15(3), 225–234 (1999)

    Article  Google Scholar 

  71. Xie, X., Zheng, H., Jin, G.: Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions. Compos. B 77, 59–73 (2015)

    Article  Google Scholar 

  72. Zingoni, A., et al.: Equatorial bending of an elliptic toroidal shell. Thin-Walled Struct. 96, 286–294 (2015)

    Article  Google Scholar 

  73. Zozulya, V.V.: A high order theory for linear thermoelastic shells: comparison with classical theories. J. Eng. (2013)

  74. Zozulya, V.V.: The combines problem of thermoelastic contact between two plates through a heat conducting layer. J. Appl. Math. Mech. 53(5), 622–627 (1989)

    Article  MATH  Google Scholar 

  75. Zozulya, V.V.: Contact cylindrical shell with a rigid body through the heat-conducting layer in transitional temperature field. Mech. Solids 2, 160–165 (1991)

    Google Scholar 

  76. Zozulya, V.V.: Laminated shells with debonding between laminas in temperature field. Int. Appl. Mech. 42(7), 842–848 (2006)

    Article  Google Scholar 

  77. Zozulya, V.V.: Mathematical modeling of pencil-thin nuclear fuel rods. In: Gupta, A. (ed.) Structural Mechanics in Reactor Technology, pp. C04-C12. Canada, Toronto (2007)

    Google Scholar 

  78. Zozulya, V.V.: A high-order theory for functionally graded axially symmetric cylindrical shells. Arch. Appl. Mech. 83(3), 331–343 (2013)

    Article  MATH  Google Scholar 

  79. Zozulya, V.V.: A higher order theory for shells, plates and rods. Int. J. Mech. Sci. 103, 40–54 (2015)

    Article  Google Scholar 

  80. Zozulya, V.V.: Nonlocal theory of curved rods. 2-D, high order, Timoshenko’s and Euler–Bernoulli models. Curve. Layer. Struct. 4, 221–236 (2017)

    Article  Google Scholar 

  81. Zozulya, V.V.: Couple stress theory of curved rods. 2-D, high order, Timoshenko’s and Euler–Bernoulli models. Curve. Layer. Struct. 4, 1192–2132 (2017)

    Google Scholar 

  82. Zozulya, V.V.: Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler–Bernoulli models. Curve. Layer. Struct. 4, 104–118 (2017)

    Article  Google Scholar 

  83. Zozulya, V.V.: Higher order theory of micropolar curved rods. In: Altenbach, H., Öchsner, A. (eds.) Encyclopedia of continuum mechanics, pp. 1–11. Springer, Berlin, Heidelberg (2018)

    Google Scholar 

  84. Zozulya, V.V.: Higher order couple stress theory of plates and shells. J. Appl. Math. Mech. (ZAMM) 98(10), 1834–1863 (2018)

    Article  Google Scholar 

  85. Zozulya, V.V.: Higher order theory of micropolar plates and shells. J. Appl. Math. Mech. (ZAMM) 98(6), 886–918 (2018)

    Article  Google Scholar 

  86. Zozulya, V.V., Carrera, E.: Carrera unified formulation (CUF) for the micropolar plates and shells. III. Classical models. Mech. Adv. Mater. Struct. (2021). https://doi.org/10.1080/15376494.2021.1975855

  87. Zozulya, V.V., Saez, A.: High-order theory for arched structures and its application for the study of the electrostatically actuated MEMS devices. Arch. Appl. Mech. 84(7), 1037–1055 (2014)

    Article  MATH  Google Scholar 

  88. Zozulya, V.V., Saez, A.: A high order theory of a thermo elastic beams and its application to the MEMS/NEMS analysis and simulations. Arch. Appl. Mech. 86(7), 1255–1272 (2015)

    Article  Google Scholar 

  89. Zozulya, V.V., Zhang, Ch.: A high order theory for functionally graded axisymmetric cylindrical shells. Int. J. Mech. Sci. 60(1), 12–22 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the visiting professor grants provided by Politecnico di Torino Research Excellence 2021, and the Committee of Science and Technology of Mexico (CONASYT), which are gratefully acknowledged, which are gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Aeronautics and Aerospace Engineering, Politecnico Di TorinoCorso Duca Degli Abruzzi, 24, 10129, Turin, Italy

    E. Carrera & V. V. Zozulya

  2. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Nesterov Str. 3, Kyiv, 252680, Ukraine

    V. V. Zozulya

  3. Department of Mechanical Engineering, College of Engineering, Prince Mohammad Bin Fahd University, Al Khobar, Kingdom of Saudi Arabia

    E. Carrera

Authors
  1. E. Carrera
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Zozulya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Zozulya.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carrera, E., Zozulya, V.V. Carrera unified formulation (CUF) for shells of revolution. I. Higher-order theory. Acta Mech 234, 109–136 (2023). https://doi.org/10.1007/s00707-022-03372-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-022-03372-7

Search

Navigation