Structural Engineering and Mechanics

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Direct calculation of interface warping functions for considering longitudinal discontinuities in beams

  • Lee, Dong-Hwa (Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology) ;
  • Kim, Hyo-Jin (Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology) ;
  • Lee, Phill-Seung (Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology)
  • Received : 2021.10.05
  • Accepted : 2021.10.31
  • Published : 2021.12.10
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Abstract

In this paper, we present a new method to calculate interface warping functions for the analysis of beams with geometric and material discontinuities in the longitudinal direction. The classical Saint Venant torsion theory is extended to a three-dimensional domain by considering the longitudinal direction. The interface warping is calculated by considering both adjacent cross-sections of a given interface. We also propose a finite element procedure to simultaneously calculate the interface warping function and the corresponding twisting center. The calculated interface warping functions are employed in the continuum-mechanics based beam formulation to analyze arbitrary shape cross-section beams with longitudinal discontinuities. Compared to the previous work by Yoon and Lee (2014a), both geometric and material discontinuities are considered with fewer degrees of freedom and higher accuracy in beam finite element analysis. Through various numerical examples, the effectiveness of the proposed interface warping function is demonstrated.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2018R1A2B3005328). This work was also supported by the "Human Resources Program in Energy Technology" of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), granted financial resource from the Ministry of Trade, Industry & Energy, Republic of Korea (No. 20184030202000). We would like to acknowledge the technical support from ANSYS Korea.

References

  1. Alsafadie, R., Hjiaj, M. and Battini, J.M. (2011), "Three-dimensional formulation of a mixed corotational thin-walled beam element incorporating shear and warping deformation", Thin Wall. Struct., 49, 523-533. https://doi.org/10.1016/j.tws.2010.12.002.
  2. ANSYS (2018), ANSYS Mechanical APDL Theory Reference, ANSYS Inc.
  3. Barretta, R., Luciano, R. and Willis, J.R. (2015), "On torsion of random composite beams", Compos. Struct., 132, 915-922. https://doi.org/10.1016/j.compstruct.2015.06.069.
  4. Bathe, K.J. (1996), Finite Element Procedures, Prentice Hall, New York.
  5. Batoz, J.L. and Dhatt, G. (1990), Modelisation des Structures par Elements Finis: Solides Elastiques, Hermes Sciences.
  6. Battini, J.M. and Pacoste, C. (2002), "Co-rotational beam elements with warping effects in instability problems", Comput. Meth. Appl. Mech. Eng., 191, 1755-1789. https://doi.org/10.1016/S0045-7825(01)00352-8.
  7. Benscoter, S.U. (1954), "A theory of torsion bending for multicell beams", J. Appl. Mech., 21, 25-34. https://doi.org/10.1115/1.4010814.
  8. Cardona, A. and Geradin, M. (1988), "A beam finite element non-linear theory with finite rotations", Int. J. Numer. Meth. Eng., 26, 2403-2438. https://doi.org/10.1002/nme.1620261105.
  9. Carrera, E., Giunta, G., Nali, P. and Petrolo, M. (2010), "Refined beam elements with arbitrary cross-section geometries", Comput. Struct., 88, 283-293. https://doi.org/10.1016/j.compstruc.2009.11.002.
  10. Dvorkin, E.N., Celentano, D., Cuitino, A. and Gioia, G. (1989), "A Vlasov beam element", Comput. Struct., 33, 187-196. https://doi.org/10.1016/0045-7949(89)90140-5.
  11. El Fatmi, R. (2007a), "Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory", Int. J. Solid. Struct., 44, 5912-5929. https://doi.org/10.1016/j.ijsolstr.2007.02.006.
  12. El Fatmi, R. (2007b), "Non-uniform warping including the effects of torsion and shear forces. Part II: Analytical and numerical applications", Int. J. Solid. Struct., 44, 5930-5952. https://doi.org/10.1016/j.ijsolstr.2007.02.005.
  13. El Fatmi, R. and Ghazouani, N. (2011a), "Higher order composite beam theory built on Saint-Venant's solution. Part-I: Theoretical developments", Compos. Struct., 93, 557-566. https://doi.org/10.1016/j.compstruct.2010.08.024.
  14. Genoese, A., Genoese, A., Bilotta, A. and Garcea, G. (2013), "A mixed beam model with non-uniform warpings derived from the Saint Venant rod", Comput. Struct., 121, 87-98. https://doi.org/10.1016/j.compstruc.2013.03.017.
  15. Genoese, A., Genoese, A., Bilotta, A. and Garcea, G. (2014), "A geometrically exact beam model with non-uniform warping coherently derived from the Saint Venant rod", Eng. Struct., 68, 33-46. https://doi.org/10.1016/j.engstruct.2014.02.024.
  16. Ghazouani, N. and El Fatmi, R. (2011b), "Higher order composite beam theory built on Saint-Venant's solution. Part-II: Built-in effects influence on the behavior of end-loaded cantilever beams", Compos. Struct., 93, 567-581. https://doi.org/10.1016/j.compstruct.2010.08.024.
  17. Gjelsvik, A. (1981), The Theory of Thin Walled Bars, Wiley, New York.
  18. GonC alves, R., Ritto-Correa, M. and Camotim, D. (2010), "A large displacement and finite rotation thin-walled beam formulation including cross-section deformation", Comput. Meth. Appl. Mech. Eng., 199, 1627-1643. https://doi.org/10.1016/j.cma.2010.01.006.
  19. Gruttmann, F., Sauer, R. and Wagner, W. (1999), "Shear stresses in prismatic beams with arbitrary cross-sections", Int. J. Numer. Meth. Eng., 45, 865-889. https://doi.org/10.1002/(SICI)1097-0207(19990710)45:7<865::AID-NME609>3.0.CO;2-3.
  20. Hughes, T.J.R. (2000), The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications.
  21. Ibrahimbegovic, A. (1995), "On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements", Comput. Meth. Appl. Mech. Eng., 122, 11-26. https://doi.org/10.1016/0045-7825(95)00724-F.
  22. Ibrahimbegovic, A. (1997), "On the choice of finite rotation parameters", Comput. Meth. Appl. Mech. Eng., 149, 49-71. https://doi.org/10.1016/S0045-7825(97)00059-5.
  23. Iesan, D. (2008), Classical and Generalized Models of Elastic Rods, Chapman and Hall/CRC, New York.
  24. Kim, H.J., Lee, D.H., Yoon, K. and Lee, P.S. (2021), "A multi-director continuum beam finite element for efficient analysis of multi-layer strand cables", Comput. Struct., 256, 106621. https://doi.org/10.1016/j.compstruc.2021.106621.
  25. Kim, H.J., Yoon, K. and Lee, P.S. (2020), "Continuum mechanics based beam elements for linear and nonlinear analyses of multilayered composite beams with interlayer slips", Compos. Struct., 235, 111740. https://doi.org/10.1016/j.compstruct.2019.111740.
  26. Lee, P.S. and McClure, G. (2006), "A general three-dimensional L-section beam finite element for elastoplastic large deformation analysis", Comput. Struct., 84, 215-229. https://doi.org/10.1016/j.compstruc.2005.09.013.
  27. Mancusi, G. and Feo, L. (2013), "Non-linear pre-buckling behavior of shear deformable thin-walled composite beams with open cross-section", Compos. B. Eng., 47, 379-390. https://doi.org/10.1016/j.compositesb.2012.11.003.
  28. Pacoste, C. and Eriksson, A. (1977), "Beam elements in instability problems", Comput. Meth. Appl. Mech. Eng., 144, 163-197. https://doi.org/10.1016/S0045-7825(96)01165-6.
  29. Petrov, E. and Geradin, M. (1998a), "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. Meth. Appl. Mech. Eng., 165, 43-92. https://doi.org/10.1016/S0045-7825(98)00061-9.
  30. Petrov, E. and Geradin, M. (1998b), "Finite element theory for curved and twisted beams based on exact solutions for three-dimensional solids. Part 2: Anisotropic and advanced beam models", Comput. Meth. Appl. Mech. Eng., 165, 93-127. https://doi.org/10.1016/S0045-7825(98)00060-7.
  31. Pi, Y.L., Bradford, M.A. and Uy, B. (2005), "Nonlinear analysis of members curved in space with warping and Wagner effects", Int. J. Solid. Struct., 42, 3147-3169. https://doi.org/10.1016/j.ijsolstr.2004.10.012.
  32. Qureshi, MAM. and Ganga Rao, H. (2014), "Torsional response of closed FRP composite sections", Compos. B. Eng., 61, 254-266. https://doi.org/10.1016/j.compositesb.2013.12.065
  33. Sapountzakis, E.J. and Mokos, V.G. (2003), "Warping shear stresses in nonuniform torsion of composite bars by BEM", Comput. Meth. Appl. Mech. Eng., 192, 4337-4353. https://doi.org/10.1016/S0045-7825(03)00417-1.
  34. Sapountzakis, E.J. and Mokos, V.G. (2004), "Nonuniform torsion of composite bars of variable thickness by BEM", Int. J. Solid. Struct., 41, 1753-1771. https://doi.org/10.1016/j.ijsolstr.2003.11.025.
  35. Timoshenko, S.P. and Goodier, J.N. (1970), Theory of Elasticity, McGraw Hill.
  36. Vlasov, V.Z. (1961), Thin-walled Elastic Beams, Israel Program for Scientific Translations, Jerusalem.
  37. Wagner, W. and Gruttmann, F. (2001), "Finite element analysis of Saint-Venant torsion problem with exact integration of the elastic-plastic constitutive equations", Comput. Meth. Appl. Mech. Eng., 190, 3831-3848. https://doi.org/10.1016/S0045-7825(00)00302-9.
  38. Wagner, W. and Gruttmann, F. (2002), "A displacement method for the analysis of flexural shear stresses in thin-walled isotropic composite beams", Comput. Struct., 80, 1843-1851. https://doi.org/10.1016/S0045-7949(02)00223-7.
  39. Yoon, K. and Lee, P.S. (2014a), "Modeling the warping displacements for discontinuously varying arbitrary cross-section beams", Comput. Struct., 131, 56-69. https://doi.org/10.1016/j.compstruc.2013.10.013.
  40. Yoon, K. and Lee, P.S. (2014b), "Nonlinear performance of continuum mechanics based beam elements focusing on large twisting behaviors", Comput. Meth. Appl. Mech. Eng., 281, 106-30. https://doi.org/10.1016/j.cma.2014.07.023.
  41. Yoon, K., Kim, D.N. and Lee, P.S. (2017b), "Nonlinear torsional analysis of 3D composite beams using the extended St. Venant solution", Struct. Eng. Mech., 62, 33-42. https://doi.org/10.12989/sem.2017.62.1.033.
  42. Yoon, K., Lee, P.S. and Kim, D.N. (2015), "Geometrically nonlinear finite element analysis of functionally graded 3D beams considering warping effects", Compos. Struct., 132, 1231-1247. https://doi.org/10.1016/j.compstruct.2015.07.024.
  43. Yoon, K., Lee, P.S. and Kim, D.N. (2017a), "An efficient warping model for elastoplastic torsional analysis of composite beams", Compos. Struct., 178, 37-49. https://doi.org/10.1016/j.compstruct.2017.07.041.
  44. Yoon, K., Lee, Y. and Lee, P.S. (2012), "A continuum mechanics based 3-D beam finite element with warping displacements and its modeling capabilities", Struct. Eng. Mech., 43, 411-437. https://doi.org/10.12989/sem.2012.43.4.411.