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


Beams are structural members that are designed to support lateral forces and bending moments. Beams can be also subjected to combined bending, torsion as well as axial tensile or compressive loads. In the case of linear elasticity the laterally loaded beams, rods subjected to torque as well as axially loaded rods can be analyzed separately and the superposition principle can be applied to establish the resultant stress and deformation states. For nonlinear material behavior such a superposition is not possible and combined loadings should be considered. Furthermore, inelastic material response may be different for tensile and compressive loadings leading to a shift of the neutral plane under pure bending. Beams are also important in testing of materials. Three or four point bending tests are frequently used to analyze inelastic behavior experimentally. Examples are presented in Chuang (1986); Scholz et al (2008); Xu et al (2007) for homogeneous beams, in Weps et al (2013) for laminated beams and in Nordmann et al (2018) for beams with coatings. Beams are discussed in monographs and textbooks on creep mechanics (Boyle and Spence, 1983; Hult, 1966; Kachanov, 1986; Kraus, 1980; Malinin, 1975, 1981; Odqvist, 1974; Penny and Mariott, 1995; Skrzypek, 1993), where the Bernoulli-Euler beam theory and elementary constitutive equations, such as the Norton-Bailey constitutive law for steady-state creep are applied.

Chapter 3 presents examples of inelastic structural analysis for beams. In Sect 3.1 the classical Bernoulli-Euler beam theory is introduced. Governing equations and variational formulations for inelastic analysis are introduced. Closed-form solutions and approximate analytical solutions are derived for beams from materials that exhibit power law creep and stress regime dependent creep. Numerical solutions by the Ritz and finite element methods are discussed in detail. Creep and creep-damage constitutivemodels are applied to illustrate basic features of stress redistribution and damage evolution in beams. Furthermore, several benchmark problems for beams are introduced. The reference solutions for these problems obtained by the Ritz method are applied to verify user-defined creep-damage material subroutines and the general purpose finite element codes.

For many materials inelastic behavior depends on the kind of stress state. Examples for stress state effects including different creep rates under tension, compression and torsion are discussed in Sect. 3.2. For such kind of material behavior, the classical beam theory may lead to errors in computed deformations and stresses. Section 3.3 presents a refined beam theory which includes the effect of transverse shear deformation (Timoshenko-type theory). Based on several examples, classical and refined theories are compared as they describe creep-damage processes in beams.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abaqus Benchmarks (2017) Benchmarks ManualGoogle Scholar
  2. Abaqus User’s Guide (2017) Abaqus Analysis User’s Guide. Volume III: MaterialsGoogle Scholar
  3. Altenbach H, Naumenko K (1997) Creep bending of thin-walled shells and plates by consideration of finite deflections. Computational Mechanics 19:490 – 495CrossRefGoogle Scholar
  4. 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
  5. 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
  6. Altenbach H, Kolarow G, Morachkovsky O, Naumenko K (2000) On the accuracy of creep-damage predictions in thinwalled structures using the finite element method. Computational Mechanics 25:87 – 98CrossRefGoogle Scholar
  7. 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
  8. 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
  9. Altenbach H, Eremeyev VA, Naumenko K (2015) On the use of the first order shear deformation plate theory for the analysis of three-layer plates with thin soft core layer. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 95(10):1004–1011MathSciNetCrossRefGoogle Scholar
  10. ANSYS (2001) Theory ManualGoogle Scholar
  11. Antman S (1995) Nonlinear Problems of Elasticity. Springer, BerlinCrossRefGoogle Scholar
  12. Boyle JT (2012) The creep behavior of simple structures with a stress range-dependent constitutive model. Archive of Applied Mechanics 82(4):495 – 514CrossRefGoogle Scholar
  13. Boyle JT, Spence J (1983) Stress Analysis for Creep. Butterworth, LondonCrossRefGoogle Scholar
  14. Chuang TJ (1986) Estimation of power-law creep parameters from bend test data. Journal of Materials Science 21(1):165–175CrossRefGoogle Scholar
  15. Eisenträger J, Naumenko K, Altenbach H, Köppe H (2015) 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
  16. Green AE, Naghdi PM, Wenner ML (1974) On the theory of rods. II. Developments by direct approach. Proceedings of the Royal Society of London A Mathematical and Physical Sciences 337(1611):485 – 507MathSciNetCrossRefGoogle Scholar
  17. Hosseini E, Holdsworth SR, Mazza E (2013) Stress regime-dependent creep constitutive model considerations in finite element continuum damage mechanics. International Journal of Damage Mechanics 22(8):1186 – 1205CrossRefGoogle Scholar
  18. Hult JA (1966) Creep in Engineering Structures. Blaisdell Publishing Company, WalthamGoogle Scholar
  19. Hutchinson JR (2001) Shear coefficients for timoshenko beam theory. Trans ASME J Appl Mech 68:87 – 92CrossRefGoogle Scholar
  20. Kachanov LM (1986) Introduction to Continuum Damage Mechanics. Martinus Nijhoff, DordrechtCrossRefGoogle Scholar
  21. Kaneko T (1975) On Timoshenko’s correction for shear in vibrating beams. J Phys D 8:1927 – 1936CrossRefGoogle Scholar
  22. Kowalewski ZL (1996) Creep rupture of copper under complex stress state at elevated temperature. In: Design and life assessment at high temperature, Mechanical Engineering Publ., London, pp 113 – 122Google Scholar
  23. Kraus H (1980) Creep Analysis. John Wiley & Sons, New YorkGoogle Scholar
  24. Levinson M (1981) A new rectangular beam theory. J Sound Vibr 74:81 – 87CrossRefGoogle Scholar
  25. 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
  26. Malinin NN (1975) Prikladnaya teoriya plastichnosti i polzuchesti (Applied Theory of Plasticity and Creep, in Russ.). Mashinostroenie, MoskvaGoogle Scholar
  27. Malinin NN (1981) Raschet na polzuchest’ konstrukcionnykh elementov (Creep Calculations of Structural Elements, in Russ.). Mashinostroenie, MoskvaGoogle Scholar
  28. Naumenko K (2000) On the use of the first order shear deformation models of beams, plates and shells in creep lifetime estimations. Technische Mechanik 20(3):215 – 226Google Scholar
  29. Naumenko K, Altenbach H (2007) Modelling of Creep for Structural Analysis. Springer, Berlin et al.CrossRefGoogle Scholar
  30. Naumenko K, Altenbach H (2016) Modeling High Temperature Materials Behavior for Structural Analysis: Part I: Continuum Mechanics Foundations and Constitutive Models, Advanced Structured Materials, vol 28. SpringerGoogle Scholar
  31. 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
  32. Naumenko K, Kostenko Y (2009) Structural analysis of a power plant component using a stress-range-dependent creep-damage constitutive model. Materials Science and Engineering A510-A511:169 – 174CrossRefGoogle Scholar
  33. Naumenko K, Altenbach H, Gorash Y (2009) Creep analysis with a stress range dependent constitutive model. Archive of Applied Mechanics 79:619 – 630CrossRefGoogle Scholar
  34. 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
  35. Odqvist FKG (1974) Mathematical Theory of Creep and Creep Rupture. Oxford University Press, OxfordGoogle Scholar
  36. Penny RK, Mariott DL (1995) Design for Creep. Chapman & Hall, LondonGoogle Scholar
  37. Rabotnov YN (1969) Creep Problems in Structural Members. North-Holland, AmsterdamGoogle Scholar
  38. Reddy JN (1984) A simple higher-order theory for laminated composite plate. Trans ASME J Appl Mech 51:745 – 752CrossRefGoogle Scholar
  39. Reddy JN (1997) Mechanics of Laminated Composite Plates: Theory and Analysis. CRC Press, Boca RatonGoogle Scholar
  40. Reissner E (1950) A variational theorem in elasticity. J Math Phys 29:90 – 95MathSciNetCrossRefGoogle Scholar
  41. Scholz A, Schmidt A, Walther HC, Schein M, Schwienheer M (2008) Experiences in the determination of TMF, LCF and creep life of CMSX-4 in four-point bending experiments. International Journal of Fatigue 30(2):357–362CrossRefGoogle Scholar
  42. 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
  43. Skrzypek JJ (1993) Plasticity and Creep. CRC Press, Boca RatonGoogle Scholar
  44. Weps M, Naumenko K, Altenbach H (2013) Unsymmetric three-layer laminate with soft core for photovoltaic modules. Composite Structures 105:332–339CrossRefGoogle Scholar
  45. Xu B, Yue Z, Eggeler G (2007) A numerical procedure for retrieving material creep properties from bending creep tests. Acta Materialia 55(18):6275–6283CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

Personalised recommendations