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Large deflection analysis of geometrically exact spatial beams under conservative and nonconservative loads using intrinsic equations

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Abstract

Large deflection analysis of geometrically exact beams under conservative and nonconservative loads is considered. A Chebyshev collocation method is used for the discretization of the governing equations of the spatial beam with intrinsic formulation. In the case of nonconservative (follower) loads, since the formulation is expressed in a deformed frame, the formulation is fully free from any displacement or rotational variables, and a direct solution can be achieved. In the case of conservative loads, the applied loads are functions of beam rotations, and in order to retain the intrinsic nature of the formulation, a direct integration of rotations is avoided and an iterative solution of the governing equations in addition to an update on the beam rotations as a post-processing step has been employed. A number of test cases have been considered and are compared to the existing experimental as well as numerical results in the literature. A very good agreement has been observed in the comparison cases, and it is shown that the intrinsic formulation in conjunction with a Chebyshev collocation method while exhibiting advantages in ease of implementation, computational cost and convergence characteristics over other methods can effectively capture the large deflection behavior of spatial beams under various load conditions.

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References

  1. Argyris J.H., Symeonidis S.: Nonlinear finite element analysis of elastic systems under nonconservative loading-natural formulation. Part I. Quasistatic problems. Comput. Methods Appl. Mech. Eng. 26, 75–123 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  2. Batista M.: Large deflections of shear-deformable cantilever beam subject to a tip follower force. Int. J. Mech. Sci. 75, 388–395 (2013)

    Article  Google Scholar 

  3. Bauchau O.A.: Flexible Multibody Dynamics. Springer, Dordrecht (2010)

    Google Scholar 

  4. Beléndez T., Neipp C., Ndez A.B.: Numerical and experimental analysis of a cantilever beam: A laboratory project to introduce geometric nonlinearity in mechanics of materials. Int. J. Eng. Educ. 19, 885–892 (2003)

    Google Scholar 

  5. Beléndez T., Pérez-Polo M., Neipp C., Beléndez A.: Numerical and experimental analysis of large deflections of cantilever beams under a combined load. Phys. Scr. 2005(T118), 61 (2005)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Bisshopp K.E., Drucker D.C.: Large deflection of cantilever beams. Q. Appl. Math. 3, 272–275 (1945)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Češarek P., Saje M., Zupan D.: Kinematically exact curved and twisted strain-based beam. Int. J. Solids Struct. 49, 1802–1817 (2012)

    Article  Google Scholar 

  10. Chang C.S., Hodges D.H.: Stability studies for curved beams. J. Mech. Mater. Struct. 4, 1257–1270 (2009)

    Article  Google Scholar 

  11. Chang C.S., Hodges D.H.: Vibration characteristics of curved beams. J. Mech. Mater. Struct 4, 675–692 (2009)

    Article  Google Scholar 

  12. Chen L.: An integral approach for large deflection cantilever beams. Int. J. Nonlinear Mech. 45, 301–305 (2010)

    Article  Google Scholar 

  13. Gruttmann F., Sauer R., Wagner W.: A geometrical nonlinear eccentric 3d-beam element with arbitrary cross-sections. Comput. Methods Appl. Mech. Eng. 160, 383–400 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hodges D.H.: A mixed variational formulation based on exact intrinsic equations for dynamics of moving beams. Int. J. Solids Struct. 26, 1253–1273 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hodges D.H.: Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams. AIAA J. 41, 1131–1137 (2003)

    Article  Google Scholar 

  16. Hodges D.H.: Nonlinear Composite Beam Theory. AIAA, Reston, VA (2006)

    Book  Google Scholar 

  17. Ibrahimbegović A.: On the choice of finite rotation parameters. Comput. Methods Appl. Mech. Eng. 149, 49–71 (1997)

    Article  MATH  Google Scholar 

  18. Iura M., Atluri S.N.: On a consistent theory, and variational formulation of finitely stretched and rotated 3-d space-curved beams. Comput. Mech. 4, 73–88 (1988)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  20. Jelenić G., Saje M.: A kinematically exact space finite strain beam model–finite element formulation by generalized virtual work principle. Comput. Methods Appl. Mech. Eng. 120, 131–161 (1995)

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  22. Karlson K.N., Leamy M.J.: Three-dimensional equilibria of nonlinear pre-curved beams using an intrinsic formulation and shooting. Int. J. Solids Struct. 50, 3491–3504 (2013)

    Article  Google Scholar 

  23. Khaneh Masjedi, P., Ovesy, H.R.: Chebyshev collocation method for static intrinsic equations of geometrically exact beams. Int. J. Solids Struct. (2014). doi:10.1016/j.ijsolstr.2014.10.016

  24. Kimiaeifar A., Domairry G., Mohebpour S.R., Sohouli A.R., Davodi A.G.: Analytical solution for large deflections of a cantilever beam under nonconservative load based on homotopy analysis method. Numer. Methods Partial Differ. Equ. 27, 541–553 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kondoh K., Atluri S.N.: Large-deformation, elasto-plastic analysis of frames under nonconservative loading, using explicitly derived tangent stiffnesses based on assumed stresses. Comput. Mech. 2, 1–25 (1987)

    MATH  Google Scholar 

  26. Lee K.: Large deflections of cantilever beams of non-linear elastic material under a combined loading. Int. J. Nonlinear Mech. 37, 439–443 (2002)

    Article  MATH  Google Scholar 

  27. Mason J.C., Handscomb D.C.: Chebyshev Polynomials. CRC Press LLC, Boca Raton (2003)

    MATH  Google Scholar 

  28. Nallathambi A.K., Lakshmana Rao C., Srinivasan S.M.: Large deflection of constant curvature cantilever beam under follower load. Int. J. Mech. Sci. 52, 440–445 (2010)

    Article  Google Scholar 

  29. Pai P.F.: Three kinematic representations for modeling of highly flexible beams and their applications. Int. J. Solids Struct. 48, 2764–2777 (2011)

    Article  Google Scholar 

  30. Pai P.F., Anderson T.J., Wheater E.A.: Large-deformation tests and total-Lagrangian finite-element analyses of flexible beams. Int. J. Solids Struct. 37, 2951–2980 (2000)

    Article  MATH  Google Scholar 

  31. Pai P.F., Palazotto A.N.: Large-deformation analysis of flexible beams. Int. J. Solids Struct. 33, 1335–1353 (1996)

    Article  MATH  Google Scholar 

  32. Petrov E., Geradin M.: Finite element theory for curved and twisted beams based on exact solutions for three-dimensional solids part 1: Beam concept and geometrically exact nonlinear formulation. Comput. Methods Appl. Mech. Eng. 165, 43–92 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  33. Rao B.N., Rao G.V.: On the large deflection of cantilever beams with end rotational load. ZAMM Z. Angew. Math. Mech. 66, 507–509 (1986)

    Article  MATH  Google Scholar 

  34. Reissner E.: On one-dimensional finite-strain beam theory: the plane problem. Z. Angew. Math. Phys. 23, 795–804 (1972)

    Article  MATH  Google Scholar 

  35. Reissner E.: On one-dimensional large-displacement finite-strain beam theory. Stud. Appl. Math. 52, 87–95 (1973)

    MATH  Google Scholar 

  36. Reissner E.: On finite deformations of space-curved beams. Z. Angew. Math. Phys. 32, 734–744 (1981)

    Article  MATH  Google Scholar 

  37. Schillinger D., Evans J.A., Reali A., Scott M.A., Hughes T.J.: Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations. Comput. Methods Appl. Mech. Eng. 267, 170–232 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  38. Shvartsman B.S.: Large deflections of a cantilever beam subjected to a follower force. J. Sound Vib. 304, 969–973 (2007)

    Article  Google Scholar 

  39. Shvartsman B.S.: Analysis of large deflections of a curved cantilever subjected to a tip-concentrated follower force. Int. J. Nonlinear Mech. 50, 75–80 (2013)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  41. Simo J.C., Vu-Quoc L.: A three-dimensional finite-strain rod model. Part ii: Computational aspects. Comput. Methods Appl. Mech. Eng. 58, 79–116 (1986)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  43. Sotoudeh Z., Hodges D.H., Chang C.S.: Validation studies for aeroelastic trim and stability of highly flexible aircraft. J. Aircr. 47, 1240–1247 (2010)

    Article  Google Scholar 

  44. Srpčič S., Saje M.: Large deformations of thin curved plane beam of constant initial curvature. Int. J. Mech. Sci. 28, 275–287 (1986)

    Article  MATH  Google Scholar 

  45. Wackerfuβ J., Gruttmann F.: A mixed hybrid finite beam element with an interface to arbitrary three-dimensional material models. Comput. Methods Appl. Mech. Eng. 198, 2053–2066 (2009)

    Article  Google Scholar 

  46. Wackerfuβ J., Gruttmann F.: A nonlinear Hu–Washizu variational formulation and related finite-element implementation for spatial beams with arbitrary moderate thick cross-sections. Comput. Methods Appl. Mech. Eng. 200, 1671–1690 (2011)

    Article  Google Scholar 

  47. Zakharov Y.V., Okhotkin K.G., Skorobogatov A.D.: Bending of bars under a follower load. J. Appl. Mech. Tech. Phys. 45, 756–763 (2004)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Aerospace Engineering and Center of Excellence in Computational Aerospace Engineering, Amirkabir University of Technology, Tehran, Iran

    Pedram Khaneh Masjedi & Hamid Reza Ovesy

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  1. Pedram Khaneh Masjedi
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  2. Hamid Reza Ovesy
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Correspondence to Pedram Khaneh Masjedi.

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Khaneh Masjedi, P., Ovesy, H.R. Large deflection analysis of geometrically exact spatial beams under conservative and nonconservative loads using intrinsic equations. Acta Mech 226, 1689–1706 (2015). https://doi.org/10.1007/s00707-014-1281-3

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  • DOI: https://doi.org/10.1007/s00707-014-1281-3

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