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A hybrid-mixed finite element formulation for the geometrically exact analysis of three-dimensional framed structures

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Abstract

This paper addresses the development of a hybrid-mixed finite element formulation for the quasi-static geometrically exact analysis of three-dimensional framed structures with linear elastic behavior. The formulation is based on a modified principle of stationary total complementary energy, involving, as independent variables, the generalized vectors of stress-resultants and displacements and, in addition, a set of Lagrange multipliers defined on the element boundaries. The finite element discretization scheme adopted within the framework of the proposed formulation leads to numerical solutions that strongly satisfy the equilibrium differential equations in the elements, as well as the equilibrium boundary conditions. This formulation consists, therefore, in a true equilibrium formulation for large displacements and rotations in space. Furthermore, this formulation is objective, as it ensures invariance of the strain measures under superposed rigid body rotations, and is not affected by the so-called shear-locking phenomenon. Also, the proposed formulation produces numerical solutions which are independent of the path of deformation. To validate and assess the accuracy of the proposed formulation, some benchmark problems are analyzed and their solutions compared with those obtained using the standard two-node displacement/ rotation-based formulation.

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References

  1. Reissner E (1973) On one-dimensional large-displacement finite-strain beam theory. Stud Appl Math 11: 87–95

    Google Scholar 

  2. Reissner E (1981) On finite deformations of space-curved beams. J Appl Math Phys 32: 734–744

    Article  MATH  Google Scholar 

  3. Simo J (1985) A finite strain beam formulation: the three-dimensional dynamic problem, part I. Comput Methods Appl Mech Eng 49: 55–70

    Article  MATH  Google Scholar 

  4. Simo J, Vu-Quoc L (1986) A three-dimensional finite-strain rod model, part II: computational aspects. Comput Methods Appl Mech Eng 58: 79–116

    Article  MATH  Google Scholar 

  5. Cardona A, Geradin M (1988) A beam finite element non-linear theory with finite rotations. Int J Numer Methods Eng 26: 2403–2438

    Article  MathSciNet  MATH  Google Scholar 

  6. Simo J, Vu-Quoc L (1991) A geometrically exact rod model incorporating shear and torsion-warping deformation. Int J Solids Struct 27: 371–393

    Article  MathSciNet  MATH  Google Scholar 

  7. Pimenta P, Yojo T (1993) Geometrically exact analysis of spatial frames. Appl Mech Rev 46(11): S118–S128

    Article  Google Scholar 

  8. Ibrahimbegovic A, Frey F, Kozar I (1995) Computational aspects of vector-like parametrization of three-dimensional finite rotations. Int J Numer Methods Eng 38: 3653–3673

    Article  MathSciNet  MATH  Google Scholar 

  9. Crisfield M, Jelenic G (1999) Objectivity of strain measures in the geometrically exact three-dimensional beam theory and its finite-element implementation. Proc R Soc 455: 1125–1147

    Article  MathSciNet  MATH  Google Scholar 

  10. Jelenic G, Crisfield M (1999) Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput Methods Appl Mech Eng 171: 141–171

    Article  MathSciNet  MATH  Google Scholar 

  11. Iura M, Atluri S (1988) Dynamic analysis of finitely stretched and rotated three-dimensional space-curved beams. Comput Struct 29(5): 875–889

    Article  MATH  Google Scholar 

  12. Iura M, Atluri S (1989) On a consistent theory and variational formulation of finitely stretched and rotated 3-D space-curved beams. Comput Mech 4(3): 73–88

    Google Scholar 

  13. Quadrelli B, Atluri S (1996) Primal and mixed variational principles for dynamics of spatial beams. AIAA J 34(11): 2395–2401

    Article  MATH  Google Scholar 

  14. Quadrelli B, Atluri S (1998) Analysis of flexible multibody systems with spatial beams using mixed variational principles. Int J Numer Methods Eng 42: 1071–1090

    Article  MathSciNet  MATH  Google Scholar 

  15. Atluri S, Iura M, Vasudevan S (2001) A consistent theory of finite stretches and finite rotations, in space curved beams of arbitrary cross-section. Comput Mech 27: 271–281

    Article  MATH  Google Scholar 

  16. Spring K (1986) Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: a review. Mech Mach Theory 21: 365–373

    Article  Google Scholar 

  17. Zupan E, Saje M, Zupan D (2009) The quaternion-based three-dimensional beam theory. Comput Methods Appl Mech Eng 198: 3944–3956

    Article  MathSciNet  Google Scholar 

  18. Bufler H (1993) Conservative systems, potential operators and tangent stiffness: reconsideration and generalization. Arch Appl Mech 63: 51–58

    Article  MATH  Google Scholar 

  19. Géradin M, Cardona A (2001) Flexible multibody dynamics: a finite element approach. Wiley, West Sussex

    Google Scholar 

  20. Ritto-Corrêa M, Camotim D (2002) On the differentiation of the Rodrigues formula and its significance for the vector-like parametrizarion of Reissner-Simo beam theory. Int J Numer Methods Eng 55(9): 1005–1032

    Article  MATH  Google Scholar 

  21. Ibrahimbegovic A (1995) On finite element implementation of geometrically nonlinear Reissner’s beam theory: three-dimensional curved beam elements. Comput Methods Appl Mech Eng 122: 11–26

    Article  MATH  Google Scholar 

  22. Pimenta PM (1996) Geometrically exact analysis of initially curved rods. Advances in computational techniques for structural engineering, pp 99–108

  23. Saje M, Turk G, Kalagasidu A, Vratanar B (1998) A kinematically exact finite element formulation of elastic-plastic curved beams. Comput Struct 67: 197–214

    Article  MATH  Google Scholar 

  24. Zupan D, Saje M (2003) The three-dimensional beam theory: finite element formulation based on curvature. Comput Struct 81: 1875–1888

    Article  Google Scholar 

  25. Kapania R, Li J (2003) On a geometrically exact curved/twisted beam theory under rigid cross-section assumption. Comput Mech 30: 428–443

    Article  MATH  Google Scholar 

  26. Kapania R, Li J (2003) A formulation and implementation of geometrically exact curved beam elements incorporating finite strains and finite rotations. Comput Mech 30: 444–459

    Article  MATH  Google Scholar 

  27. Mata P, Oller S, Barbat A (2007) Static analysis of beam structures under nonlinear geometric and constitutive behavior. Comput Methods Appl Mech Eng 196: 4458–4478

    Article  MathSciNet  MATH  Google Scholar 

  28. Planinc I, Saje M (1999) A quadratically convergent algorithm for the computation of stability points: the application of the determinant of the tangent stiffness matrix. Comput Methods Appl Mech Eng 169: 89–105

    Article  MATH  Google Scholar 

  29. Lens E, Cardona A (2008) A nonlinear beam element formulation in the framework of an energy preserving time integration scheme for constrained multibody systems dynamics. Comput Struct 86: 47–63

    Article  Google Scholar 

  30. Ibrahimbegovic A, Taylor R (2002) On the role of frame-invariance in structural mechanics models at finite rotations. Comput Methods Appl Mech Eng 191: 5159–5176

    Article  MathSciNet  MATH  Google Scholar 

  31. Betsch P, Steinmann P (2002) Frame-indifferent beam finite element based upon the geometrically exact beam theory. Int J Numer Methods Eng 54: 1775–1788

    Article  MathSciNet  MATH  Google Scholar 

  32. Romero I (2004) The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput Mech 34: 121–133

    Article  MathSciNet  MATH  Google Scholar 

  33. Makinen J (2007) Total Lagrangian Reissner’s geometrically exact beam element without singularities. Int J Numer Methods Eng 70: 1009–1048

    Article  MathSciNet  Google Scholar 

  34. Ghosh S, Roy D (2008) Consistent quaternion interpolation for objective finite element approximation of geometrically exact beam. Comput Methods Appl Mech Eng 198: 555–571

    Article  MathSciNet  Google Scholar 

  35. Jakobsen B (1994) The Sleipner accident and its causes. Eng Fail Anal 1(3): 193–199

    Article  Google Scholar 

  36. de Veubeke BF (1965) Stress analysis. Displacement and equilibrium models in the finite element method. Wiley, New York, pp 145–197

    Google Scholar 

  37. Debongnie J, Zhong H, Beckers P (1995) Dual analysis with general boundary conditions. Comput Methods Appl Mech Eng 122: 183–192

    Article  MathSciNet  MATH  Google Scholar 

  38. Washizu K (1982) Variational methods in elasticity and plasticity, 3rd edn. Pergamon Press, Oxford

    MATH  Google Scholar 

  39. Saje M (1990) A variational principle for finite planar deformation of straight slender elastic beams. Int J Solids Struct 26(8): 887–900

    Article  MATH  Google Scholar 

  40. Saje M (1991) Finite element formulation of finite planar deformation of curved elastic beams. Comput Struct 39: 327–337

    Article  MATH  Google Scholar 

  41. Santos H, de Almeida JM (2010) An equilibrium-based finite element formulation for the geometrically exact analysis of planar framed structures. J Eng Mech 136(12): 1474–1490

    Article  Google Scholar 

  42. Zupan D, Saje M (2003) Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures. Comput Methods Appl Mech Eng 192: 5209–5248

    Article  MathSciNet  Google Scholar 

  43. Zupan D, Saje M (2004) Rotational invariants in finite element formulation of three-dimensional beam theories. Comput Struct 82: 2027–2040

    Article  Google Scholar 

  44. Nukala P, White D (2004) A mixed finite element for three-dimensional nonlinear analysis of steel frames. Comput Methods Appl Mech Eng 193: 2507–2545

    Article  MATH  Google Scholar 

  45. Santos H, Pimenta P, de Almeida JM (2010) Hybrid and multi-field variational principles for geometrically exact three-dimensional beams. Int J Non-Linear Mech 45(8): 809–820

    Article  Google Scholar 

  46. Ritto-Corrêa M, Camotim D (2003) Work-conjugacy between rotation-dependent moments and finite rotations. Int J Solids Struct 40: 2851–2873

    Article  MATH  Google Scholar 

  47. Ghosh S, Roy D (2009) A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization. Comput Mech 44: 103–118

    Article  MATH  Google Scholar 

  48. Betsch P, Menzel A, Stein E (1998) On the parametrization of finite rotations in computational mechanics: a classification of concepts with application to smooth shells. Comput Methods Appl Mech Eng 155: 273–305

    Article  MathSciNet  MATH  Google Scholar 

  49. Ogden R (1977) Inequalities associated with the inversion of elastic stress-deformation relations and their implication. Math Proc Camb Philos Soc 81: 313–324

    Article  MathSciNet  MATH  Google Scholar 

  50. Sander G (1971) High speed computing of elastic structures, vol 61. Application of the dual analysis principle. Les Congres et Colloques de l’Universite de Liege, pp 167–207

  51. Maunder E (1986) A composite triangular equilibrium element for the flexural analysis of plates. Eng Struct 8(3): 159–168

    Article  Google Scholar 

  52. Almeida J, Freitas J (1991) Alternative approaches to the formulation of hybrid equilibrium finite elements. Comput Struct 40: 1043–1047

    Article  Google Scholar 

  53. Almeida J, Freitas J (1992) Continuity conditions for finite element analysis of solids. Int J Numer Methods Eng 33: 845–853

    Article  MATH  Google Scholar 

  54. Murakawa H, Atluri S (1977) On hybrid finite-element models in nonlinear solid mechanics. In: International conference on finite elements in nonlinear solid and structural mechanics

  55. Atluri S, Murakawa H (1977) Finite elements in nonlinear mechanics, vol 1. On hybrid finite element models in nonlinear solid mechanics. Tapir Press, Trondheim, pp 3–41

  56. Murakawa H, Atluri S (1978) Finite elasticity solutions using hybrid finite elements based on a complementary energy principle. ASME J Appl Mech 45(3): 539–547

    Article  MATH  Google Scholar 

  57. Murakawa H, Atluri S (1979) Finite elasticity solutions using hybrid finite elements based on a complementary energy principle. Part II: incompressible materials. ASME J Appl Mech 46: 71–78

    Article  MATH  Google Scholar 

  58. Murakawa H, Reed K, Atluri S, Rubenstein R (1981) Stability analysis of structures via a new complementary energy method. Comput Struct 13: 11–18

    Article  MathSciNet  MATH  Google Scholar 

  59. Seki W, Atluri S (1995) On newly developed assumed stress finite element formulations for geometrically and materially nonlinear problems. Finite Elem Anal Des 21: 75–110

    Article  MathSciNet  MATH  Google Scholar 

  60. Gao D (1996) Complementary finite-element method for finite deformation nonsmooth mechanics. J Eng Math 30: 339–353

    Article  MATH  Google Scholar 

  61. Peng Y, Liu Y (2009) Base force element method of complementary energy principle for large rotation problems. Acta Mech Sin 25: 507–515

    Article  MATH  Google Scholar 

  62. Sander G, Carnoy E (1978) Finite elements in nonlinear mechanics, vol. 1. Equilibrium and mixed formulations in stability analysis, Trondheim, pp 87–108

  63. Ibrahimbegovic A, Frey F (1995) Variational principles and membrane finite elements with drilling rotations for geometrically non-linear elasticity. Int J Numer Methods Eng 38: 1885–1900

    Article  MathSciNet  MATH  Google Scholar 

  64. Erkmen R, Mohareb M (2008) Buckling analysis of thin-walled open members—a finite element formulation. Thin-Walled Struct 46: 618–636

    Article  Google Scholar 

  65. Santos H (2009) Duality in the geometrically exact analysis of frame structures. PhD thesis, Universidade Técnica de Lisboa

  66. Santos H, de Almeida JM (2011) Dual extremum principles for geometrically exact finite strain beams. Int J Non-Linear Mech 46: 151–158

    Article  Google Scholar 

  67. Gao D, Strang G (1989) Dual extremum principles in finite deformation elastoplastic analysis. Acta Appl Math 17: 257–267

    Article  MathSciNet  MATH  Google Scholar 

  68. Gao D (2000) Duality principles in nonconvex systems: theory, methods and applications. Kluwer, London

    MATH  Google Scholar 

  69. Atluri S (1980) On some new general and complementary energy theorems for the rate problems in finite strain, classical elastoplasticity. J Struct Mech 8: 61–92

    Article  MathSciNet  Google Scholar 

  70. Gao D, Onat E (1990) Rate variational extremum principles for finite elastoplasticity. Appl Math Mech 11(7): 659–667

    Article  MathSciNet  MATH  Google Scholar 

  71. Prathap G, Bhashyam G (1982) Reduced integration and the shear flexible beam element. Int J Numer Methods Eng 18: 195–210

    Article  MATH  Google Scholar 

  72. Ibrahimbegovic A, Frey F (1993) Finite element analysis of linear and non-linear planar deformations of elastic initially curved beams. Int J Numer Methods Eng 36: 3239–3258

    Article  MathSciNet  MATH  Google Scholar 

  73. Romero I, Armero F (2002) An objective finite element formulation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics. Int J Numer Methods Eng 54: 1683–1716

    Article  MathSciNet  MATH  Google Scholar 

  74. Argyris J, Dunne P, Malejannakis G, Scharpf D (1978) On large displacement-small strain analysis of structures with rotational degrees od freedom. Comput Methods Appl Mech Eng 15(1): 99–135

    Article  MATH  Google Scholar 

  75. Argyris J, Balmer H, Doltsinis J, Dunne P, Haase M, Kleiber M, Malejannakis G, Mlejnek H, Muller M, Scharpf D (1979) Finite element method—the natural approach. Comput Methods Appl Mech Eng 17(18(1): 1–106

    Article  Google Scholar 

  76. Simo J, Vu-Quoc L (1986) On the dynamics of flexible beams under large overall motions-the plane case: part I. J Appl Mech 53: 849–863

    Article  MATH  Google Scholar 

  77. Ibrahimbegovic A, Shakourzadeh H, Batoz JL, Almikdad M, Guo YQ (1996) On the role of geometrically exact and second-order theories in buckling and post-buckling analysis of three-dimensional beam structures. Comput Struct 61(6): 1101–1114

    Article  MATH  Google Scholar 

  78. Timoshenko S, Gere J (1961) Theory of elastic stability, 2nd edn. McGraw-Hill, New York

    Google Scholar 

  79. Ziegler H (1968) Principles of structural stability. Blaisdell Publishing Company, Waltham

    Google Scholar 

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Santos, H.A.F.A., Pimenta, P.M. & Almeida, J.P.M. A hybrid-mixed finite element formulation for the geometrically exact analysis of three-dimensional framed structures. Comput Mech 48, 591–613 (2011). https://doi.org/10.1007/s00466-011-0608-3

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