Advertisement

An objective and path-independent 3D finite-strain beam with least-squares assumed-strain formulation

  • P. Areias
  • M. Pires
  • N. Vu Bac
  • Timon RabczukEmail author
Original Paper
  • 23 Downloads

Abstract

An all-encompassing finite-strain representation of rods, shells and continuum can share a common kinematic/constitutive framework where specific conditions for strain, stress and constitutive updating are applied. In this work, finite strain beams are under examination, with several classical requirements met by cooperative techniques judiciously applied. Specifically: the use of a continuum constitutive law is possible due to the relative strain formulation previously introduced, the rotation singularity problem is absent due to the use of a consistent (quadratic) updated Lagrangian technique. Objectiveness and path-independence of director interpolation are satisfied due to the use of a Löwdin frame. These properties are proved in this work. Moreover, high coarse-mesh accuracy is introduced by the least-squares assumed-strain technique, here specialized for a beam. Examples show the accuracy and robustness of the formulation.

Keywords

Geometrically exact beams Assumed strains Least-squares Nonlinear Constitutive laws 

Notes

References

  1. 1.
    Antman SS (2005) Nonlinear problems of elasticity, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  2. 2.
    Antman SS, Marlow RS (1991) Material constraints, lagrange multipliers, and compatibility. Arch Ration Mech Anal 116:257–299CrossRefzbMATHGoogle Scholar
  3. 3.
    Antman SS, Schuricht F (2003) The critical role of the base curve for the qualitative behavior of shearable rods. Math Mech Solids 8:75–102MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Areias P Simplas. Portuguese Software Association (ASSOFT) registry number 2281/D/17 http://www.simplas-software.com. Accessed 30 Dec 2018
  5. 5.
    Areias P, Rabczuk T, César de Sá J, Natal Jorge R (2015) A semi-implicit finite strain shell algorithm using in-plane strains based on least-squares. Comput Mech 55(4):673–696MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bathe KJ, Bolourchi S (1979) Large displacement analysis of three-dimensional beam structures. Int J Numer Methods Eng 14:961–986CrossRefzbMATHGoogle Scholar
  7. 7.
    Battini J-M (2002) Co-rotational beam elements in instability problems. Technical Report, Royal Institute of Technology, Department of Mechanics, SE-100 44 Stockholm, Sweden, JanuaryGoogle Scholar
  8. 8.
    Betsch P, Stein E (1999) Numerical implementation of multiplicative elasto-plasticity into assumed strain elements with application to shells at large strains. Comput Method Appl Mech Eng 179:215–245MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cardona A, Géradin M (1988) A beam finite element non-linear theory with finite rotations. Int J Numer Methods Eng 26:2403–2438CrossRefzbMATHGoogle Scholar
  10. 10.
    Cowper GR (1966) The shear coefficient in Timoshenko’s beam theory. J Appl Mech 33(2):335–340CrossRefzbMATHGoogle Scholar
  11. 11.
    Crisfield MA, Jelenić G (1999) Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proc R Soc Lond A 455:1125–1147MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eugster SR, Hesch C, Betsch P, Glocker Ch (2014) Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. Int J Numer Methods Eng 97:111–129MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gruttmann F, Wagner W (2001) Shear correction factors in Timoshenko’s beam theory for arbitrary shaped cross-sections. Comput Mech 27:199–207CrossRefzbMATHGoogle Scholar
  14. 14.
    Hughes TJR (2000) The finite element method. Dover Publications, Mineola (Reprint of Prentice-Hall edition, 1987)zbMATHGoogle Scholar
  15. 15.
    Hutchinson JR (2001) Shear coefficients for Timoshenko beam theory. ASME J Appl Mech 68:87–92CrossRefzbMATHGoogle Scholar
  16. 16.
    Ibrahimbegović A (1995) On finite element implementation of geometrically nonlinear Reissner’s beam theory: three-dimensional curved beam elements. Comput Method Appl Mech Eng 122:11–26CrossRefzbMATHGoogle Scholar
  17. 17.
    Ibrahimbegovicć A, Frey F, Kozar I (1995) Computational aspects of vector-like parametrization of three-dimensional finite rotations. Int J Numer Methods Eng 38:3653–3673MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jelenić G, Crisfield MA (1999) Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput Methods Appl Mech Eng 171:141–171MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Klinkel S, Govindjee S (2002) Using finite strain \(3D\)-material models in beam and shell elements. Eng Comput 19(8):909–921zbMATHGoogle Scholar
  20. 20.
    Korelc J (2002) Multi-language and multi-environment generation of nonlinear finite element codes. Eng Comput 18(4):312–327CrossRefGoogle Scholar
  21. 21.
    Kouhia R, Tuomala M (1993) Static and dynamic analysis of space frames using simple Timoshenko type elements. Int J Numer Methods Eng 36:1189–1221CrossRefGoogle Scholar
  22. 22.
    Liu I-Shih (2003) On the transformation property of the deformation gradient under a change of frame. J Elast 71(1):73–80MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Löwdin P-O (1950) On the nonorthogonality problem. Adv Quantum Chem 5:185–199CrossRefGoogle Scholar
  24. 24.
    MacNeal RH (1978) A simple quadrilateral shell element. Comput Struct 8:175–183CrossRefzbMATHGoogle Scholar
  25. 25.
    Macneal RH (1994) Finite elements: their design and performance, vol 10016. Marcel Dekker, New YorkGoogle Scholar
  26. 26.
    Mata P, Oller S, Barbat AH (2007) Static analysis of beam structures under nonlinear geometric and constitutive behavior. Comput Method Appl Mech Eng 196:4458–4478MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mathisen KM, Bazilevs Y, Haugen B, Helgedagsrud TA, Kvamsdal T, Okstad KM, Raknes SB (2017) A comparative study of beam element formulations for nonlinear analysis: corotational vs geometrically exact formulations. In: Skallerud B, Andersson HI (eds) 9th national Conference on Computational Mechanics MekIT 17, Trondheim, Norway, Department of Structural Engineering, NTNU, pp 245–272Google Scholar
  28. 28.
    Nukala PKVV, White DW (2004) A mixed finite element for three-dimensional nonlinear analysis of steel frames. Comput Methods Appl Mech Eng 193:2507–2545CrossRefzbMATHGoogle Scholar
  29. 29.
    Pai PF (1999) Shear correction factors and an energy-consistent beam theory. Int J Solids Struct 36:1523–1540CrossRefzbMATHGoogle Scholar
  30. 30.
    Pian THH, Tong P (1986) Relations between incompatible displacement model and hybrid stress model. Int J Numer Methods Eng 22(1):173–181MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pimenta PM, Campello EMB, Wriggers P (2008) An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: rods. Comput Mech 42:715–732MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Prathap G (1995) The variationally correct rate of convergence for a two-noded beam element, or whey residual bending flexibility correction is an extravariational trick. Commun Numer Methods Eng 11:403–407CrossRefzbMATHGoogle Scholar
  33. 33.
    Reissner E (1981) On finite deformations of space-curved beams. J Appl Math Phys 32:734–744zbMATHGoogle Scholar
  34. 34.
    Ritto-Correa M, Camotim D (2002) On the differentiation of the Rodrigues formula and its significance for the vector-like parameterization of the Reissner–Simo beam theory. Int J Numer Methods Eng 55:1005–1032MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Romero I (2004) The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput Mech 34:121–133MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Romero I (2008) A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations. Multibody Syst Dyn 20:51–68MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Shoemake K (1985) Animating rotation with quaternion curves. In: Proceedings of the 12th annual conference on computer graphics and interactive techniques, SIGGRAPH ’85, New York, NY, USA. ACM, pp 245–254Google Scholar
  38. 38.
    Simo JC (1985) A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput Method Appl Mech Eng 49:55–70CrossRefzbMATHGoogle Scholar
  39. 39.
    Simo JC, Vu-Quoc L (1986) A three-dimensional finite-strain rod model. Part II: computational aspects. Comput Methods Appl Mech Eng 58:79–116CrossRefzbMATHGoogle Scholar
  40. 40.
    Truesdell C, Noll W (2004) The non-linear field theories of mechanics, third edn. Springer, LondonCrossRefzbMATHGoogle Scholar
  41. 41.
    Wackerfuß J, Gruttmann F (2009) A mixed hybrid finite beam element with an interface to arbitrary three-dimensional material models. Comput Methods Appl Mech Eng 198:2053–2066MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Wolfram Research Inc. (2007) MathematicaGoogle Scholar
  43. 43.
    Zienkiewicz OC, Taylor RL, Zhu JZ (2013) The finite element method. Its basis & fundamentals, vol 1, 7th edn. Butterworth-Heinemann, The Boulevard, Langford Lane, Kidlington, OxfordGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • P. Areias
    • 1
    • 3
  • M. Pires
    • 2
  • N. Vu Bac
    • 4
  • Timon Rabczuk
    • 5
    Email author
  1. 1.Department of Physics, Colégio Luís António VerneyUniversity of ÉvoraÉvoraPortugal
  2. 2.Department of Mathematics, Colégio Luís António VerneyUniversity of ÉvoraÉvoraPortugal
  3. 3.CERIS/Instituto Superior TécnicoUniversity of LisbonLisbonPortugal
  4. 4.Institute of Structural MechanicsBauhaus-University WeimarWeimarGermany
  5. 5.Institute of Research & DevelopmentDuy Tan UniversityDanangViet Nam

Personalised recommendations