Plates and Shells

  • Konstantin NaumenkoEmail author
  • Holm Altenbach
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 112)


Thin and moderately thick shell structures are designed as structural components in many engineering applications because of light weight and high load-carrying capacity. In many cases they are subjected to high temperature environment and mechanical loadings, such that inelastic material behavior must be taken into account. Examples of high-temperature shell components include pressure vessels, boiler tubes, steam transfer lines, thin coatings, etc. A steam transfer line under long-term operation considering creep-damage material behavior is discussed in Naumenko and Altenbach (2016, Chap. 1). Chapter 5 presents examples of inelastic structural analysis of plates and shells. Section 5.1 gives an overview of modeling approaches including various theories of plates and shells as well as various constitutive models of inelastic material behavior. Governing equations of the first order shear deformation theory of plates are presented in Sect. 5.2. An emphasis is placed on the direct formulation of inelastic constitutive laws. Section 5.3 illustrates examples of steady-state creep analysis of circular plates. Advanced constitutive models with internal state variables, such as the damage parameter require the use of advanced plate theories to consider edge effects. Section 5.4 illustrates an example of a rectangular plate with different types of boundary conditions. The results based on the plate theory are compared with the results according to the three-dimensional theory. Section 5.5 presents governing equations and the solution procedure for the creep behavior of a thin-walled pipe subjected to the internal pressure and the bending moment.


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  1. Altenbach H, Naumenko K (1997) Creep bending of thin-walled shells and plates by consideration of finite deflections. Computational Mechanics 19:490 – 495CrossRefGoogle Scholar
  2. Altenbach H, Naumenko K (2002) Shear correction factors in creep-damage analysis of beams, plates and shells. JSME International Journal Series A, Solid Mechanics and Material Engineering 45:77 – 83CrossRefGoogle Scholar
  3. Altenbach H, Zhilin PA (2004) The theory of simple elastic shells. In: Kienzler R, Altenbach H, Ott I (eds) Theories of Plates and Shells. Critical Review and New Applications, Springer, Berlin, pp 1 – 12zbMATHGoogle Scholar
  4. Altenbach H, Morachkovsky O, Naumenko K, Sichov A (1996) Zum Kriechen dünner Rotationsschalen unter Einbeziehung geometrischer Nichtlinearität sowie der Asymmetrie der Werkstoffeigenschaften. Forschung im Ingenieurwesen 62(6):47 – 57CrossRefGoogle Scholar
  5. Altenbach H, Altenbach J, Naumenko K (1997a) On the prediction of creep damage by bending of thin-walled structures. Mechanics of Time Dependent Materials 1:181 – 193CrossRefGoogle Scholar
  6. Altenbach H,Morachkovsky O, Naumenko K, Sychov A (1997b) Geometrically nonlinear bending of thin-walled shells and plates under creep-damage conditions. Archive of Applied Mechanics 67:339 – 352CrossRefGoogle Scholar
  7. Altenbach H, Breslavsky D, Morachkovsky O, Naumenko K (2000a) Cyclic creep damage in thinwalled structures. The Journal of Strain Analysis for Engineering Design 35(1):1 – 11Google Scholar
  8. Altenbach H, Kolarow G, Morachkovsky O, Naumenko K (2000b) On the accuracy of creepdamage predictions in thinwalled structures using the finite element method. Computational Mechanics 25:87 – 98CrossRefGoogle Scholar
  9. Altenbach H, Kushnevsky V, Naumenko K (2001) On the use of solid- and shell-type finite elements in creep-damage predictions of thinwalled structures. Archive of Applied Mechanics 71:164 – 181CrossRefGoogle Scholar
  10. Altenbach H, Huang C, Naumenko K (2002) Creep damage predictions in thin-walled structures by use of isotropic and anisotropic damage models. The Journal of Strain Analysis for Engineering Design 37(3):265 – 275CrossRefGoogle Scholar
  11. Altenbach H, Naumenko K, Zhilin PA (2005) A direct approach to the formulation of constitutive equations for rods and shells. In: Pietraszkiewicz W, Szymczak C (eds) Shell Structures: Theory and Applications, Taylor & Francis, Leiden, pp 87 – 90Google Scholar
  12. Altenbach H, Altenbach J, Naumenko K (2016) Ebene Flächentragwerke. Springer, BerlinCrossRefGoogle Scholar
  13. Altenbach J, Altenbach H, Naumenko K (1997c) Lebensdauerabschätzung dünnwandiger Flächentragwerke auf der Grundlage phänomenologischer Materialmodelle für Kriechen und Schädigung. Technische Mechanik 17(4):353 – 364Google Scholar
  14. Altenbach J, Altenbach H, Naumenko K (2004) Egde effects inmoderately thick plates under creep damage conditions. Technische Mechanik 24(3 - 4):254 – 263Google Scholar
  15. Bagheri B, Schulze SH, Naumenko K, Altenbach H (2019) Identification of traction-separation curves for self-adhesive polymeric films based on non-linear theory of beams and digital images of t-peeling. Composite Structures 216:222 – 227CrossRefGoogle Scholar
  16. Betten J, Borrmann M (1987) Stationäres Kriechverhalten innendruckbelasteter dünnwandiger Kreiszylinderschalen unter Berücksichtigung des orthotropen Werkstoffverhaltens und des CSD - Effektes. Forschung im Ingenieurwesen 53(3):75 – 82CrossRefGoogle Scholar
  17. Betten J, Butters T (1990) Rotationssymmetrisches Kriechbeulen dünnwandiger Kreiszylinderschalen im primären Kriechbereich. Forschung im Ingenieurwesen 56(3):84 – 89CrossRefGoogle Scholar
  18. Bialkiewicz J, Kuna H (1996) Shear effect in rupture mechanics of middle-thick plates plates. Engng Fracture Mechanics 54(3):361 – 370CrossRefGoogle Scholar
  19. Bodnar A, Chrzanowski M (2001) Cracking of creeping structures described by means of cdm. In: Murakami S, Ohno N (eds) IUTAM Symposium on Creep in Structures, Kluwer, Dordrecht, pp 189 – 196CrossRefGoogle Scholar
  20. Boyle JT, Spence J (1983) Stress Analysis for Creep. Butterworth, LondonCrossRefGoogle Scholar
  21. Breslavsky D, Morachkovsky O, Tatarinova O (2014) Creep and damage in shells of revolution under cyclic loading and heating. International Journal of Non-Linear Mechanics 66:87–95CrossRefGoogle Scholar
  22. Burlakov AV, Lvov GI, Morachkovsky OK (1977) Polzuchest’ tonkikh obolochek (Creep of thin shells, in Russ.). Kharkov State Univ. Publ., KharkovGoogle Scholar
  23. Burlakov AV, Lvov GI, Morachkovsky OK (1981) Dlitel’naya prochnost’ obolochek (Long-term strength of shells, in Russ.). Vyshcha shkola, KharkovGoogle Scholar
  24. Byrne TP, Mackenzie AC (1966) Secondary creep of a cylindrical thin shell subject to axisymmetric loading. J Mech Eng Sci 8(2):215 – 225CrossRefGoogle Scholar
  25. Carrera E (2003) Historical review of Zig-Zag theories for multilayered plates and shells. Appl Mech Rev 56(2):287 – 308MathSciNetCrossRefGoogle Scholar
  26. Chen S, Zang M,Wang D, Zheng Z, Zhao C (2016) Finite element modelling of impact damage in polyvinyl butyral laminated glass. Composite Structures 138:1–11CrossRefGoogle Scholar
  27. Combescure A, Jullien JF (2017) Creep buckling of cylinders under uniform external pressure: Finite element simulation of buckling tests. International Journal of Solids and Structures 124:14–25CrossRefGoogle Scholar
  28. Eisenträger J, Naumenko K, Altenbach H, Köppe, H (2015a) Application of the first-order shear deformation theory to the analysis of laminated glasses and photovoltaic panels. International Journal of Mechanical Sciences 96:163–171CrossRefGoogle Scholar
  29. Eisenträger J, Naumenko K, Altenbach H, Meenen J (2015b) A user-defined finite element for laminated glass panels and photovoltaic modules based on a layer-wise theory. Composite Structures 133:265–277CrossRefGoogle Scholar
  30. Fessler H, Hyde TH (1994) The use of model materials to simulate creep behavior. The Journal of Strain Analysis for Engineering Design 29(3):193 – 200CrossRefGoogle Scholar
  31. FilippiM, Carrera E, Valvano S (2018) Analysis of multilayered structures embedding viscoelastic layers by higher-order, and zig-zag plate elements. Composites Part B: Engineering 154:77 – 89CrossRefGoogle Scholar
  32. Galishin A, Zolochevskii A, Sklepus S (2017) Feasibility of shell models for determining stress–strain state and creep damage of cylindrical shells. International Applied Mechanics 53(4):398–406MathSciNetCrossRefGoogle Scholar
  33. Ganczarski A, Skrzypek J (2000) Damage effect on thermo-mechanical fields in a mid-thick plate. J Theor Appl Mech 38(2):271 – 284Google Scholar
  34. Ganczarski A, Skrzypek J (2004) Anisotropic thermo-creep-damage in 3d thick plate vs. reissner’s approach. In: Kienzler R, Altenbach H, Ott I (eds) Theories of Plates and Shells. Critical Review and new Applications, Springer, Berlin, pp 39 – 44CrossRefGoogle Scholar
  35. Jones D (2004) Creep failures of overheated boiler, superheater and reformer tubes. Engineering Failure Analysis 11(6):873–893CrossRefGoogle Scholar
  36. von Kármán T (1911) Festigkeitsprobleme im Maschinenbau. In: Encyklop. d. math. Wissensch. IV/2, Teubner, Leipzig, pp 311 – 385CrossRefGoogle Scholar
  37. Kashkoli M, Tahan KN, Nejad M (2017) Time-dependent creep analysis for life assessment of cylindrical vessels using first order shear deformation theory. Journal of Mechanics 33(4):461–474CrossRefGoogle Scholar
  38. Koundy V, Forgeron T, Naour FL (1997) Modeling of multiaxial creep behavior for incoloy 800 tubes under internal pressure. Trans ASME J Pressure Vessel & Technology 119:313 – 318Google Scholar
  39. Krieg R (1999) Reactor Pressure Vessel Under Severe Accident Loading. Final Report of EU-Project Contract FI4S-CT95-0002. Tech. rep., Forschungszentrum Karlsruhe, KarlsruheGoogle Scholar
  40. Le May I, da Silveria TL, Cheung-Mak SKP (1994) Uncertainties in the evaluations of high temperature damage in power stations and petrochemical plant. International Journal of Pressure Vessels and Piping 59:335 – 343CrossRefGoogle Scholar
  41. Lebedev LP, Cloud MJ, Eremeyev VA (2010) Tensor Analysis with Applications in Mechanics. World ScientificGoogle Scholar
  42. Libai A, Simmonds JG (1998) The Nonlinear Theory of Elastic Shells. Cambridge University Press, CambridgeGoogle Scholar
  43. Lin TH (1962) Bending of a plate with nonlinear strain hardening creep. In: Hoff NJ (ed) Creep in Structures, Springer, Berlin, pp 215 – 228CrossRefGoogle Scholar
  44. Liu Y, Murakami S, Kageyama Y (1994) Mesh-dependence and stress singularity in finite element analysis of creep crack growth by continuum damage mechanics approach. European Journal of Mechanics A Solids 35(3):147 – 158Google Scholar
  45. Lo KH, Christensen RM, Wu EM (1977) A high – order theory of plate deformation. Part I: Homogeneous plates. Trans ASME J Appl Mech 44(4):663 – 668Google Scholar
  46. Miyazaki N (1987) Creep buckling analyses of circular cylindrical shells under axial compression-bifurcation buckling analysis by the finite element method. Trans ASME J Pressure Vessel & Technol 109:179 – 183Google Scholar
  47. Miyazaki N (1988) Creep buckling analyses of circular cylindrical shell under both axial compression and internal or external pressure. Computers & Struct 28:437 – 441Google Scholar
  48. Miyazaki N, Hagihara S (2015) Creep buckling of shell structures. Mechanical Engineering Reviews 2(2):14–00,522CrossRefGoogle Scholar
  49. Murakami S, Suzuki K (1971) On the creep analysis of pressurized circular cylindrical shells. International Journal of Non-Linear Mechanics 6:377 – 392CrossRefGoogle Scholar
  50. Murakami S, Suzuki K (1973) Application of the extended newton method to the creep analysis of shells of revolution. Ingenieur-Archiv 42:194 – 207CrossRefGoogle Scholar
  51. Nase M, Rennert M, Naumenko K, Eremeyev VA (2016) Identifying traction–separation behavior of self-adhesive polymeric films from in situ digital images under t-peeling. Journal of the Mechanics and Physics of Solids 91:40–55MathSciNetCrossRefGoogle Scholar
  52. Naumenko K, Altenbach H (2016) Modeling High Temperature Materials Behavior for Structural Analysis: Part I: ContinuumMechanics Foundations and ConstitutiveModels, Advanced Structured Materials, vol 28. SpringerGoogle Scholar
  53. Naumenko K, Eremeyev VA (2014) A layer-wise theory for laminated glass and photovoltaic panels. Composite Structures 112:283–291CrossRefGoogle Scholar
  54. Naumenko K, Eremeyev VA (2017) A layer-wise theory of shallow shells with thin soft core for laminated glass and photovoltaic applications. Composite Structures 178:434–446CrossRefGoogle Scholar
  55. Naumenko K, Altenbach J, Altenbach H, Naumenko VK (2001) Closed and approximate analytical solutions for rectangular Mindlin plates. Acta Mechanica 147:153 – 172Google Scholar
  56. Nordmann J, Thiem P, Cinca N, Naumenko K, Krüger M (2018) Analysis of iron aluminide coated beams under creep conditions in high-temperature four-point bending tests. The Journal of Strain Analysis for Engineering Design 53(4):255–265CrossRefGoogle Scholar
  57. Odqvist FKG (1962) Applicability of the elastic analogue to creep problems of plates, membranes and beams. In: Hoff NJ (ed) Creep in Structures, Springer, Berlin, pp 137 – D160CrossRefGoogle Scholar
  58. Paggi M, Kajari-Schröder S, Eitner U (2011) Thermomechanical deformations in photovoltaic laminates. The Journal of Strain Analysis for Engineering Design 46(8):772–782CrossRefGoogle Scholar
  59. Penny RK (1964) Axisymmetrical bending of the general shell of revolution during creep. J Mech Eng Sci 6:44 – 45CrossRefGoogle Scholar
  60. Podgorny AN, Bortovoj VV, Gontarovsky PP, Kolomak VD, Lvov GI, Matyukhin YJ, Morachkovsky OK (1984) Polzuchest’ elementov mashinostroitel’nykh konstrykcij (Creep of mashinery structural members, in Russ.). Naukova dumka, KievGoogle Scholar
  61. Psyllaki P, Pantazopoulos G, Lefakis H (2009) Metallurgical evaluation of creep-failed superheater tubes. Engineering Failure Analysis 16(5):1420–1431CrossRefGoogle Scholar
  62. Rabotnov YN (1969) Creep Problems in Structural Members. North-Holland, AmsterdamGoogle Scholar
  63. Reddy JN (1984) A simple higher-order theory for laminated composite plate. Trans ASME J Appl Mech 51:745 – 752CrossRefGoogle Scholar
  64. Roche RL, Townley CHA, Regis V, Hübel H Structural analysis and available knowledge. In: Larson LH (ed) High Temperature Structural Design, Mechanical Engineering Publ., London, pp 161 – 180Google Scholar
  65. Schulze S, Pander M, Naumenko K, Altenbach H (2012) Analysis of laminated glass beams for photovoltaic applications. International Journal of Solids and Structures 49(15 - 16):2027 – 2036CrossRefGoogle Scholar
  66. Spence J (1973) Creep of a straight pipe under combined bending and internal pressure. Nuclear Engineering and Design 24(1):88–104CrossRefGoogle Scholar
  67. Takezono S, Fujoka S (1981) The creep of moderately thick shells of revolution under axisymmetrical load. In: Ponter ARS, Hayhurst DR (eds) Creep in Structures, Springer-Verlag, Berlin, pp 128 – 143CrossRefGoogle Scholar
  68. Takezono S, Migita K, Hirakawa A (1988) Elastic/visco-plastic deformation of multi-layered shells of revolution. JSME, Ser 1 31(3):536 – 544CrossRefGoogle Scholar
  69. Timoshenko SP, Woinowsky-Krieger S (1959) Theory of Plates and Shells. McGraw-Hill, New YorkGoogle Scholar
  70. Weps M, Naumenko K, Altenbach H (2013) Unsymmetric three-layer laminate with soft core for photovoltaic modules. Composite Structures 105:332–339CrossRefGoogle Scholar
  71. Wriggers P (2008) Nonlinear Finite Element Methods. Springer, Berlin, HeidekbergGoogle Scholar
  72. Yang HTY, Saigal S, Masud A, Kapania RK (2000) A survey of recent shell finite elements. Int J Numer Meth Engng 47:101 – 127MathSciNetCrossRefGoogle Scholar
  73. Zhilin PA, Ivanova EA (1995) Modifitsirovannyi funktsional energii v teorii plastin tipa reissnera (a modified energy functional in the reissner type plate theory, in russ.). Izv RAS Mekhanika tverdogo tela 2:120 – 128Google Scholar
  74. Zienkiewicz OC, Taylor RL (1991) The Finite Element Method. McGraw-Hill, LondonGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut für MechanikFakultät für Maschinenbau, Otto-von-Guericke-UniversitätMagdeburgGermany

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