Table of Contents
ISRN Civil Engineering
Volume 2013, Article ID 916581, 39 pages
http://dx.doi.org/10.1155/2013/916581
Review Article

Bars under Torsional Loading: A Generalized Beam Theory Approach

Department of Civil Engineering, National Technical University, Zografou Campus, 157 80 Athens, Greece

Received 11 July 2012; Accepted 23 December 2012

Academic Editors: P. J. S. Cruz and L. Gambarotta

Copyright © 2013 Evangelos J. Sapountzakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. B. Saint-Venant, “Memoire sur la torsion des prismes,” Memoires des Savants Etrangers, vol. 14, pp. 233–560, 1855. View at Google Scholar
  2. K. Marguerre, “Torsion von voll-und hohlquerschnitten,” Der Bauingenieur, vol. 21, pp. 317–322, 1940. View at Google Scholar
  3. J. F. Ely and O. C. Zienkiewicz, “Torsion of compound bars—A relaxation solution,” International Journal of Mechanical Sciences, vol. 1, no. 4, pp. 356–365, 1960. View at Google Scholar · View at Scopus
  4. G. Haberl and F. Och, “Finite elements solution for torsional rigidity and shear center of any cross section,” Zeitschrift für Flugwissenschaften, vol. 22, no. 4, pp. 115–119, 1974. View at Google Scholar · View at Scopus
  5. H. Dankert and J. und Dankert, Technische Mechanik—Computerunterstützt, B.G. Teubner, Stuttgart, Germany, 1995.
  6. M. A. Jaswon and A. R. S. Ponter, “An intergral equation solution of the torsion problem,” Proceedings of the Royal Society of London A, vol. 273, no. 1353, pp. 237–246, 1963. View at Google Scholar
  7. E. Sauer, Schub Und Torsion Bei Elastischen Prismatischen Balke, vol. 29, Mitteilungen aus dem Institut für Massivbau der Technischen Hochschule Darmstadt, Wilhelm Ernst & Sohn, Berlin, Germany, 1980.
  8. J. T. Katsikadelis and E. J. Sapountzakis, “Torsion of composite bars by boundary element method,” Journal of Engineering Mechanics, vol. 111, no. 9, pp. 1197–1210, 1985. View at Google Scholar · View at Scopus
  9. W. Cornelius, Über Den Einfluss der Torsionssteifigkeit Auf Die Verdrehung Von Tragwerken, MAN-Forschungsheft, 1951.
  10. F. W. Bornscheuer, “Beispiel und formelsammlung zur spannungsberechnung dünnwandiger stäbe mit wölbbehindertem querschnitt,” Der Stahlbau, vol. 21, no. 12, pp. 225–232, 1952. View at Google Scholar
  11. F. W. Bornscheuer, “Beispiel und formelsammlung zur spannungsberechnung dünnwandiger stäbe mit wölbbehindertem querschnitt,” Der Stahlbau, vol. 22, no. 2, pp. 32–44, 1953. View at Google Scholar
  12. C. F. Kollbrunner and K. und Basler, Torsion, Springer, Berlin, Germany, 1966.
  13. K. H. Roik, J. Carl, and J. und Linder, Biegetorsionsprobleme Gerade Dünnwandiger Stäbe, Wilhelm Ernst & Sohn, Berlin, Germany, 1972.
  14. H. Friemann, Schub Und Torsion in Gerade Stäben, Werner, Düsseldorf, Germany, 1993.
  15. K. H. Roik, Vorlesungen Über Stahlbau (Grundlagen), Wilhelm Ernst & Sohn, Berlin, Germany, 1978.
  16. E. Ramm and T. J. und Hofmann, Der Ingenieurbau: Grundwissen, -Baustatik/BaudynamikErnst & Sohn, Berlin, Germany, 1995.
  17. F. Gruttmann, W. Wagner, and R. und Sauer, “Zur berechnung von wölbfunktion und torsionskennwerten beliebiger stabquerschnitte mit der methode der finiten elemente,” Bauingenieur, vol. 73, no. 3, pp. 138–143, 1998. View at Google Scholar
  18. E. J. Sapountzakis, “Solution of non-uniform torsion of bars by an integral equation method,” Computers and Structures, vol. 77, no. 6, pp. 659–667, 2000. View at Publisher · View at Google Scholar · View at Scopus
  19. E. J. Sapountzakis and V. G. Mokos, “Nonuniform torsion of composite bars by boundary element method,” Journal of Engineering Mechanics, vol. 127, no. 9, pp. 945–954, 2001. View at Publisher · View at Google Scholar · View at Scopus
  20. E. J. Sapountzakis, “Nonuniform torsion of multi-material composite bars by the boundary element method,” Computers and Structures, vol. 79, no. 32, pp. 2805–2816, 2001. View at Publisher · View at Google Scholar · View at Scopus
  21. E. J. Sapountzakis and V. G. Mokos, “Warping shear stresses in nonuniform torsion by BEM,” Computational Mechanics, vol. 30, no. 2, pp. 131–142, 2003. View at Publisher · View at Google Scholar · View at Scopus
  22. E. J. Sapountzakis and V. G. Mokos, “Warping shear stresses in nonuniform torsion of composite bars by BEM,” Computer Methods in Applied Mechanics and Engineering, vol. 192, no. 39-40, pp. 4337–4353, 2003. View at Publisher · View at Google Scholar · View at Scopus
  23. J. T. Katsikadelis, Boundary Elements: Theory and Applications, Elsevier, Amsterdam, The Netherlands, 2002.
  24. V. G. Mokos, Nonuniform torsion of bars by boundary element method [Postgraduate Thesis], National Technical University of Athens, Civil Engineering Department, 2001.
  25. R. Schardt, “Eine erweiterung der technischen biegelehre für die berechung biegesteifer prismatischer faltwerke,” Der Stahlbau, vol. 35, no. 161–171, p. 384, 1966. View at Google Scholar
  26. J. Murin and V. Kutis, “3D-beam element with continuous variation of the cross-sectional area,” Computers and Structures, vol. 80, pp. 329–338, 2002. View at Google Scholar
  27. J. Murin, “3D beam element with changing cross sectional area,” Engineering Mechanics, vol. 1, pp. 25–35, 1999. View at Google Scholar
  28. Z. Cywinski, “Torsion des dünnwandigen Stabes mit veränderlichem, einfach symmetrischem, offenem Querschnitt,” Der Stahlbau, vol. 10, pp. 301–307, 1964. View at Google Scholar
  29. J. W. Wekezer, “Elastic torsion of thin walled bars of variable cross sections,” Computers and Structures, vol. 19, no. 3, pp. 401–407, 1984. View at Google Scholar · View at Scopus
  30. M. Eisenberger, “Nonuniform torsional analysis of variable and open cross-section bars,” Thin-Walled Structures, vol. 21, no. 2, pp. 93–105, 1995. View at Google Scholar · View at Scopus
  31. G. Mehlhorn, Der Ingenieurbau, Ernst & Sohn, Berlin, Germany, 1996.
  32. E. J. Sapountzakis and V. G. Mokos, “Nonuniform torsion of bars of variable cross section,” Computers and Structures, vol. 82, no. 9-10, pp. 703–715, 2004. View at Publisher · View at Google Scholar · View at Scopus
  33. E. J. Sapountzakis and V. G. Mokos, “Nonuniform torsion of composite bars of variable thickness by BEM,” International Journal of Solids and Structures, vol. 41, no. 7, pp. 1753–1771, 2004. View at Publisher · View at Google Scholar · View at Scopus
  34. J. Murin, “Beam element with varying cross-section satisfying local and global equilibrium conditions,” Mechanical Engineering, vol. 49, no. 3, pp. 208–223, 1998. View at Google Scholar
  35. R. J. Reilly, “Stiffness analysis of grids including warping,” ASCE Journal of the Structural Division, vol. 98, no. 7, pp. 1511–1523, 1972. View at Google Scholar · View at Scopus
  36. R. S. Barsoum and R. H. Gallagher, “Finite element analysis of torsional-flexural stability problems,” International Journal For Numerical Methods in Engineering, vol. 2, pp. 335–352, 1970. View at Google Scholar
  37. P. Waldron, “Elastic analysis of curved thin-walled girders including the effects of warping restraint,” Engineering Structures, vol. 7, no. 2, pp. 93–104, 1985. View at Google Scholar · View at Scopus
  38. P. Waldron, “Stiffness analysis of thin-walled girders,” ASCE Journal of the Structural Division, vol. 6, pp. 1366–1384, 1986. View at Google Scholar
  39. Y. B. Yang and W. McGuire, “Procedure for analysing space frames with partial warping restraint,” International Journal for Numerical Methods in Engineering, vol. 20, no. 8, pp. 1377–1398, 1984. View at Google Scholar · View at Scopus
  40. M. Z. Ahmed and F. E. Weisgerber, “Torsion constant for matrix analysis of structures including warping effect,” International Journal of Solids and Structures, vol. 33, no. 3, pp. 361–374, 1996. View at Publisher · View at Google Scholar · View at Scopus
  41. C. E. Augarde, “Generation of shape functions for straight beam elements,” Computers and Structures, vol. 68, no. 6, pp. 555–560, 1998. View at Google Scholar · View at Scopus
  42. K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, USA, 1996.
  43. E. J. Sapountzakis and V. G. Mokos, “3-D beam element of variable composite cross section including warping effect,” Acta Mechanica, vol. 171, no. 3-4, pp. 151–169, 2004. View at Publisher · View at Google Scholar · View at Scopus
  44. E. J. Sapountzakis and V. G. Mokos, “3-D elastic beam element of composite or homogeneous cross section including warping effect with applications to spatial structures,” Chronika, Scientific Journal of the TCG, Section I, Civil Engineering, Rural and Surveying Engineering, vol. 24, no. 1-3, pp. 115–139, 2004. View at Google Scholar
  45. E. J. Sapountzakis and V. G. Mokos, “3-D beam element of composite cross section including warping and shear deformation effects,” Computers and Structures, vol. 85, no. 1-2, pp. 102–116, 2007. View at Publisher · View at Google Scholar · View at Scopus
  46. D. J. Gorman, Free Vibration Analysis of Beams and Shafts, Wiley, New York, NY, USA, 1975.
  47. R. D. Belvins, Formulas for Natural Frequency and Mode Shape, D. Van Nostrand, New York, NY, USA, 1979.
  48. C. Kameswara Rao, “Torsional frequencies and mode shapes of generally constrained shafts and piping,” Journal of Sound and Vibration, vol. 125, no. 1, pp. 115–121, 1988. View at Google Scholar · View at Scopus
  49. J. M. Gere, “Torsional vibrations of beams of thin walled open cross section,” Journal of Applied Mechanics, vol. 21, pp. 381–387, 1954. View at Google Scholar
  50. J. B. Carr, “Torsional vibration of uniform thin-walled beams of open section,” Aeronautical Journal, vol. 73, no. 704, pp. 672–674, 1969. View at Google Scholar · View at Scopus
  51. P. Christiano and L. Salmela, “Frequencies of beams with elastic warping restraint,” Journal of the Structural Division, ASCE, vol. 97, pp. 1835–1840, 1971. View at Google Scholar
  52. J. W. Wekezer, “Vibrational analysis of thin-walled bass with open cross sections,” ASCE Journal of the Structural Engineering, vol. 115, pp. 2965–2978, 1989. View at Google Scholar
  53. A. M. Abdel-Ghaffar, “Free torsional vibrations of suspension bridges,” ASCE Journal of the Structural Division, vol. 105, no. 4, pp. 767–788, 1979. View at Google Scholar · View at Scopus
  54. D. Krajcinovic, “A consistent discrete elements technique for thinwalled assemblages,” International Journal of Solids and Structures, vol. 5, no. 7, pp. 639–662, 1969. View at Google Scholar · View at Scopus
  55. D. V. Mallick and R. Dungar, “Dynamic characteristics of core wall structures subjected to torsion and bending,” Structural Engineer, vol. 55, no. 6, pp. 251–261, 1977. View at Google Scholar · View at Scopus
  56. J. R. Banerjee, S. Guo, and W. P. Howson, “Exact dynamic stiffness matrix of a bending-torsion coupled beam including warping,” Computers and Structures, vol. 59, no. 4, pp. 613–621, 1996. View at Publisher · View at Google Scholar · View at Scopus
  57. Y. Matsui and T. Hayashikawa, “Dynamic stiffness analysis for torsional vibration of continuous beams with thin-walled cross-section,” Journal of Sound and Vibration, vol. 243, no. 2, pp. 301–316, 2001. View at Publisher · View at Google Scholar · View at Scopus
  58. C. Kameswara Rao and A. Appala Satyam, “Torsional vibrations and stability of thin-walled beams on continuous elastic foundation,” AIAA Journal, vol. 13, no. 2, pp. 232–234, 1975. View at Google Scholar · View at Scopus
  59. C. Kameswara Rao and S. Mirza, “Torsional vibrations and buckling of thin-walled beams on elastic foundation,” Thin-Walled Structures, vol. 7, no. 1, pp. 73–82, 1989. View at Google Scholar · View at Scopus
  60. Z. Zhang and S. Chen, “A new method for the vibration of thin-walled beams,” Computers and Structures, vol. 39, no. 6, pp. 597–601, 1991. View at Google Scholar · View at Scopus
  61. M. Eisenberger, “Torsional vibrations of open and variable cross-section bars,” Thin-Walled Structures, vol. 28, no. 3-4, pp. 269–278, 1997. View at Google Scholar · View at Scopus
  62. J. Lee and S. E. Kim, “Flexural-torsional coupled vibration of thin-walled composite beams with channel sections,” Computers and Structures, vol. 80, no. 2, pp. 133–144, 2002. View at Publisher · View at Google Scholar · View at Scopus
  63. L. P. Kollár, “Flexural-torsional vibration of open section composite beams with shear deformation,” International Journal of Solids and Structures, vol. 38, no. 42-43, pp. 7543–7558, 2001. View at Publisher · View at Google Scholar · View at Scopus
  64. M. Ganapathi, B. P. Patel, T. S. Kumar, and M. Touratier, “Torsional vibration and damping analysis of sandwich beams,” Journal of Reinforced Plastics and Composites, vol. 18, no. 2, pp. 96–117, 1999. View at Google Scholar · View at Scopus
  65. E. J. Sapountzakis, “Torsional vibrations of composite bars by BEM,” Composite Structures, vol. 70, no. 2, pp. 229–239, 2005. View at Publisher · View at Google Scholar · View at Scopus
  66. E. J. Sapountzakis, “Torsional vibrations of composite bars of variable cross-section by BEM,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 18–20, pp. 2127–2145, 2005. View at Publisher · View at Google Scholar · View at Scopus
  67. E. J. Sapountzakis and V. G. Mokos, “Dynamic analysis of 3-D beam elements including warping and shear deformation effects,” International Journal of Solids and Structures, vol. 43, no. 22-23, pp. 6707–6726, 2006. View at Publisher · View at Google Scholar · View at Scopus
  68. V. Z. Vlasov, Thin-Walled Elastic Beams, Israel Program for Scientific Translations, Jerusalem, Israel, 1961.
  69. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, McGraw−Hill Book Company, New York, NY, USA, 2nd edition, 1961.
  70. C. F. Kollbrunner and K. Basler, Torsion in Structures: An Engineering Approach, Springer, Berlin, Germany, 1969.
  71. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw−Hill Book Company, New York, NY, USA, 3rd edition, 1984.
  72. M. Schulz and F. C. Filippou, “Generalized warping torsion formulation,” Journal of Engineering Mechanics, ASCE, vol. 124, no. 3, pp. 339–347, 1998. View at Google Scholar
  73. S. W. Reagan and W. D. Pilkey, “Constrained torsion of prismatic bars,” Finite Elements in Analysis and Design, vol. 38, no. 10, pp. 909–919, 2002. View at Publisher · View at Google Scholar · View at Scopus
  74. M. Kraus, Computerorientierte Berechnungsmethoden Für Beliebige Stabquerschnitte Des Stahlbaus, Von der Fakultät für Bauingenieurwesen der Ruhr-Universität Bochum genehmigte Dissertation zur Erlangung des Grades Doktor-Ingenieur, 2005.
  75. R. El Fatmi, “Non-uniform warping including the effects of torsion and shear forces. Part II: analytical and numerical applications,” International Journal of Solids and Structures, vol. 44, no. 18-19, pp. 5930–5952, 2007. View at Publisher · View at Google Scholar · View at Scopus
  76. R. Heilig, “Der schubverformzugseinfluss auf die wölbkrafttorsion von stäben mit offenem profil,” Stahlbau, vol. 30, no. 4, p. 97, 1961. View at Google Scholar
  77. R. Heilig, “Beitrag zur theorie der kastenträger beliebiger querschnittsform,” Stahlbau, vol. 30, no. 11, p. 333, 1961. View at Google Scholar
  78. K. Roik and G. Sedlacek, “Theorie der Wölbkrafttorsion unter Berücksichtigung der sekundären Schubverformungen-Analogiebetrachtung zur Berechnung des querbelasteten Zugstabes,” Stahlbau, vol. 35, no. 2, pp. 43–52, 1966, H. 5, S. 160. View at Google Scholar
  79. D. Schade, “Zur Wölbkrafttorsion von Stäben mit dünnwandigem Querschnitt,” Ingenieur-Archiv, vol. 38, pp. 25–34, 1969. View at Google Scholar
  80. H. Rubin, “Wölbkrafttorsion von durchlaufträgern mit konstantem querschnitt unter berücksichtigung sekundärer schubverformungen,” Stahlbau, vol. 74, pp. 826–842, 2005. View at Google Scholar
  81. T. M. Roberts and H. Al-Ubaidi, “Influence of shear deformation on restrained torsional warping of pultruded FRP bars of open cross section,” Thin-Walled Structures, vol. 39, pp. 395–414, 2001. View at Google Scholar
  82. N. I. Kim and M. Y. Kim, “Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects,” Thin-Walled Structures, vol. 43, pp. 701–734, 2005. View at Google Scholar
  83. J. Murín and V. Kutiš, “An effective finite element for torsion of constant cross-sections including warping with secondary torsional moment deformation effect,” Engineering Structures, vol. 30, pp. 2716–2723, 2008. View at Google Scholar
  84. M. Kraus, Computerorientierte Bestimmung Der Schubkorrekturfaktoren Gewalzter I-Profile, Festschrift Rolf Kindmann, Shaker, Aachen, Germany, 2007.
  85. V. G. Mokos and E. J. Sapountzakis, “Secondary torsional moment deformation effect by BEM,” International Journal of Mechanical Sciences, vol. 53, pp. 897–909, 2011. View at Google Scholar
  86. S. H. Lo, “A new mesh generation scheme for arbitrary planar domains,” International Journal for Numerical Methods in Engineering, vol. l21, pp. 1403–1426, 1985. View at Google Scholar
  87. S. P. Timoshenko, “On the correction for shear of the differential equation for transverse vibrations of prismatic bars,” Philosophical Magazine, vol. 41, pp. 744–746, 1921. View at Google Scholar
  88. S. P. Timoshenko, “On the transverse vibrations of bars of uniform cross section,” Philosophical Magazine, vol. 43, pp. 125–131, 1922. View at Google Scholar
  89. D. G. Ashwell, “The axis of distortion of a twist prism,” Philosophical Magazine, vol. 42, pp. 820–832, 1951. View at Google Scholar
  90. M. Gregory, “The bending and shortening effect of pure torque,” Australian Journal of Applied Science, vol. 11, pp. 209–216, 1960. View at Google Scholar
  91. A. A. Ghobarah and W. K. Tso, “A non-linear thin-walled beam theory,” International Journal of Mechanical Sciences, vol. 13, no. 12, pp. 1025–1038, 1971. View at Google Scholar · View at Scopus
  92. M. M. Attard, “Nonlinear theory of non-uniform torsion of thin-walled open beams,” Thin-Walled Structures, vol. 4, no. 2, pp. 101–134, 1986. View at Google Scholar · View at Scopus
  93. M. M. Attard and I. J. Somervaille, “Non-linear analysis of thin-walled, open beams,” Computers and Structures, vol. 25, no. 3, pp. 437–443, 1987. View at Google Scholar · View at Scopus
  94. N. S. Trahair, “Nonlinear elastic nonuniform torsion,” Journal of Structural Engineering, vol. 131, no. 7, pp. 1135–1142, 2005. View at Google Scholar
  95. F. Mohri, N. Damil, and M. Potier Ferry, “Large torsion finite element model for thin-walled beams,” Computers and Structures, vol. 86, no. 7-8, pp. 671–683, 2008. View at Publisher · View at Google Scholar · View at Scopus
  96. E. J. Sapountzakis and V. J. Tsipiras, “Non-linear elastic non-uniform torsion of bars of arbitrary cross-section by BEM,” International Journal of Non-Linear Mechanics, vol. 45, no. 1, pp. 63–74, 2010. View at Publisher · View at Google Scholar · View at Scopus
  97. E. J. Sapountzakis and V. J. Tsipiras, “Composite bars of arbitrary cross section in nonlinear Elastic nonuniform torsion by BEM,” Journal of Engineering Mechanics, vol. 135, no. 12, pp. 1354–1367, 2009. View at Publisher · View at Google Scholar · View at Scopus
  98. J. T. Katsikadelis, “The analog equation method. A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies,” Theoretical and Applied Mechanics, vol. 27, pp. 13–38, 2002. View at Google Scholar
  99. G. Chen and N. S. Trahair, “Inelastic nonuniform torsion of steel I-beams,” Journal of Constructional Steel Research, vol. 23, pp. 189–207, 1992. View at Google Scholar
  100. K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, Oxford, UK, 1975.
  101. B. Rozmarynowski and C. Szymczak, “Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section,” Journal of Sound and Vibration, vol. 97, no. 1, pp. 145–152, 1984. View at Google Scholar · View at Scopus
  102. M. R. M. C. Da Silva, “Non-linear flexural-flexural-torsional-extensional dynamics of beams—I. Formulation,” International Journal of Solids and Structures, vol. 24, no. 12, pp. 1225–1234, 1988. View at Google Scholar · View at Scopus
  103. M. R. M. C. Da Silva, “Non-linear flexural-flexural-torsional-extensional dynamics of beams—II. Response analysis,” International Journal of Solids and Structures, vol. 24, no. 12, pp. 1235–1242, 1988. View at Google Scholar · View at Scopus
  104. P. F. Pai and A. H. Nayfeh, “Three-dimensional nonlinear vibrations of composite beams—I. Equations of motion,” Nonlinear Dynamics, vol. 1, no. 6, pp. 477–502, 1990. View at Publisher · View at Google Scholar · View at Scopus
  105. P. F. Pai and A. H. Nayfeh, “Three-dimensional nonlinear vibrations of composite beams—II. flapwise excitations,” Nonlinear Dynamics, vol. 2, no. 1, pp. 1–34, 1991. View at Publisher · View at Google Scholar · View at Scopus
  106. P. F. Pai and A. H. Nayfeh, “Three-dimensional nonlinear vibrations of composite beams—III. Chordwise excitations,” Nonlinear Dynamics, vol. 2, no. 2, pp. 137–156, 1991. View at Publisher · View at Google Scholar · View at Scopus
  107. A. Di Egidio, A. Luongo, and F. Vestroni, “A non-linear model for the dynamics of open cross-section thin-walled beams—part I: formulation,” International Journal of Non-Linear Mechanics, vol. 38, no. 7, pp. 1067–1081, 2003. View at Publisher · View at Google Scholar · View at Scopus
  108. A. Di Egidio, A. Luongo, and F. Vestroni, “A non-linear model for the dynamics of open cross-section thin-walled beams—part II: forced motion,” International Journal of Non-Linear Mechanics, vol. 38, no. 7, pp. 1083–1094, 2003. View at Publisher · View at Google Scholar · View at Scopus
  109. J. C. Simo and L. Vu-Quoc, “A geometrically-exact rod model incorporating shear and torsion-warping deformation,” International Journal of Solids and Structures, vol. 27, no. 3, pp. 371–393, 1991. View at Google Scholar · View at Scopus
  110. P. F. Pai and A. H. Nayfeh, “A fully nonlinear theory of curved and twisted composite rotor blades accounting for warpings and three-dimensional stress effects,” International Journal of Solids and Structures, vol. 31, no. 9, pp. 1309–1340, 1994. View at Google Scholar · View at Scopus
  111. E. J. Sapountzakis and V. J. Tsipiras, “Nonlinear nonuniform torsional vibrations of bars by the boundary element method,” Journal of Sound and Vibration, vol. 329, no. 10, pp. 1853–1874, 2010. View at Publisher · View at Google Scholar · View at Scopus
  112. E. J. Sapountzakis and V. J. Tsipiras, “Warping shear stresses in nonlinear nonuniform torsional vibrations of bars by BEM,” Engineering Structures, vol. 32, no. 3, pp. 741–752, 2010. View at Publisher · View at Google Scholar · View at Scopus
  113. E. Lutz, W. Ye, and S. Mukherjee, “Elimination of rigid body modes from discretized boundary integral equations,” International Journal of Solids and Structures, vol. 35, no. 33, pp. 4427–4436, 1998. View at Google Scholar · View at Scopus
  114. P. K. V. V. Nukala and D. W. White, “A mixed finite element for three-dimensional nonlinear analysis of steel frames,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 23–26, pp. 2507–2545, 2004. View at Publisher · View at Google Scholar · View at Scopus
  115. L. H. Teh and M. J. Clarke, “Plastic-zone analysis of 3D steel frames using beam elements,” Journal of Structural Engineering, vol. 125, no. 11, pp. 1328–1337, 1999. View at Google Scholar · View at Scopus
  116. A. Saritas and F. C. Filippou, “Frame element for metallic shear-yielding members under cyclic loading,” Journal of Structural Engineering, vol. 135, no. 9, pp. 1115–1123, 2009. View at Publisher · View at Google Scholar · View at Scopus
  117. M. R. Attalla, G. G. Deierlein, and W. McGuire, “Spread of plasticity: quasi-plastic-hinge approach,” Journal of Structural Engineering, vol. 120, no. 8, pp. 2451–2473, 1994. View at Google Scholar · View at Scopus
  118. J. G. Orbison, W. McGuire, and J. F. Abel, “Yield surface applications in nonlinear steel frame analysis,” Computer Methods in Applied Mechanics and Engineering, vol. 33, no. 1–3, pp. 557–573, 1982. View at Google Scholar · View at Scopus
  119. C. Ngo-Huu, S. E. Kim, and J. R. Oh, “Nonlinear analysis of space steel frames using fiber plastic hinge concept,” Engineering Structures, vol. 29, no. 4, pp. 649–657, 2007. View at Publisher · View at Google Scholar · View at Scopus
  120. S. Baba and T. Kajita, “Plastic analysis of torsion of a prismatic beam,” International Journal for Numerical Methods in Engineering, vol. 18, no. 6, pp. 927–944, 1982. View at Google Scholar · View at Scopus
  121. Y. L. Pi and N. S. Trahair, “Inelastic torsion of steel I-beams,” Journal of Structural Engineering, ASCE, vol. 121, pp. 609–620, 1995. View at Google Scholar
  122. I. M. May and I. A. S. Al-Shaarbaf, “Elasto-plastic analysis of torsion using a three-dimensional finite element model,” Computers and Structures, vol. 33, no. 3, pp. 667–678, 1989. View at Google Scholar · View at Scopus
  123. K. J. Bathe and P. M. Wiener, “On elastic-plastic analysis of I-beams in bending and torsion,” Computers and Structures, vol. 17, no. 5-6, pp. 711–718, 1983. View at Google Scholar · View at Scopus
  124. W. Wunderlich, H. Obrecht, and V. Schroedter, “Nonlinear analysis and elastic-plastic load-carrying behaviour of thin-walled spatial beam structures with warping constraints,” International Journal for Numerical Methods in Engineering, vol. 22, no. 3, pp. 671–695, 1986. View at Google Scholar · View at Scopus
  125. B. A. Izzuddin and D. Lloyd Smith, “Large-displacement analysis of elastoplastic thin-walled frames. I: formulation and implementation,” Journal of Structural Engineering, vol. 122, no. 8, pp. 905–913, 1996. View at Google Scholar · View at Scopus
  126. B. A. Izzuddin and D. Lloyd Smith, “Large-displacement analysis of elastoplastic thin-walled frames. II: verification and application,” Journal of Structural Engineering, vol. 122, no. 8, pp. 915–925, 1996. View at Google Scholar · View at Scopus
  127. A. Billinghurst, J. R. L. Williams, G. Chen, and N. S. Trahair, “Inelastic uniform torsion of steel members,” Computers and Structures, vol. 42, no. 6, pp. 887–894, 1992. View at Google Scholar · View at Scopus
  128. G. J. Nie and Z. Zhong, “The elasto-plastic and geometrically nonlinear finite element model of space beam considering restraint torsion,” Key Engineering Materials, vol. 340-341, pp. 335–340, 2007. View at Google Scholar · View at Scopus
  129. X. F. Wang, Q. S. Yang, and Q. L. Zhang, “A new beam element for analyzing geometrical and physical nonlinearity,” Acta Mechanica Sinica/Lixue Xuebao, vol. 26, no. 4, pp. 605–615, 2010. View at Publisher · View at Google Scholar · View at Scopus
  130. J. M. Battini and C. Pacoste, “Plastic instability of beam structures using co-rotational elements,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 51-52, pp. 5811–5831, 2002. View at Publisher · View at Google Scholar · View at Scopus
  131. F. Gruttmann, R. Sauer, and W. Wagner, “Theory and numerics of three-dimensional beams with elastoplastic material behaviour,” International Journal for Numerical Methods in Engineering, vol. 48, no. 12, pp. 1675–1702, 2000. View at Google Scholar · View at Scopus
  132. J. C. Simo, K. D. Hjelmstad, and R. L. Taylor, “Numerical formulations of elasto-viscoplastic response of beams accounting for the effect of shear,” Computer Methods in Applied Mechanics and Engineering, vol. 42, no. 3, pp. 301–330, 1984. View at Google Scholar · View at Scopus
  133. F. Minghini, N. Tullini, and F. Laudiero, “Locking-free finite elements for shear deformable orthotropic thin-walled beams,” International Journal for Numerical Methods in Engineering, vol. 72, no. 7, pp. 808–834, 2007. View at Publisher · View at Google Scholar · View at Scopus
  134. E. J. Sapountzakis and V. J. Tsipiras, “Inelastic nonuniform torsion of bars of doubly symmetric cross section by BEM,” Computers and Structures, vol. 89, pp. 2388–2401, 2011. View at Google Scholar
  135. M. Ortiz and J. C. Simo, “Analysis of a new class of integration algorithms for elastoplastic constitutive relations,” International Journal for Numerical Methods in Engineering, vol. 23, no. 3, pp. 353–366, 1986. View at Google Scholar · View at Scopus
  136. A. E. Armenakas, Advanced Mechanics of Materials and Applied Elasticity, Taylor & Francis Group, New York, NY, USA, 2006.
  137. M. A. Crisfield, Non-Linear Finite Element Analysis of Solids and Structures, vol. 1 of Essentials, John Wiley and Sons, New York, NY, USA, 1991.
  138. W. Wagner and F. Gruttmann, “Finite element analysis of Saint-Venant torsion problem with exact integration of the elastic-plastic constitutive equations,” Computer Methods in Applied Mechanics and Engineering, vol. 190, pp. 3831–3848, 2001. View at Google Scholar
  139. E. J. Sapountzakis and V. J. Tsipiras, “Nonlinear inelastic uniform torsion of bars by BEM,” Computational Mechanics, vol. 42, no. 1, pp. 77–94, 2008. View at Publisher · View at Google Scholar · View at Scopus
  140. S. Y. Back and K. M. Will, “A shear-flexible element with warping for thin-walled open beams,” International Journal for Numerical Methods in Engineering, vol. 43, no. 7, pp. 1173–1191, 1998. View at Google Scholar · View at Scopus
  141. C. E. Massonnet, “A new approach (including shear lag) to elementary mechanics of materials,” International Journal of Solids and Structures, vol. 19, no. 1, pp. 33–54, 1983. View at Google Scholar · View at Scopus
  142. S. U. Benscoter, “A theory of torsion bending for multicell beams,” Journal of Applied Mechanics, vol. 21, pp. 25–34, 1954. View at Google Scholar
  143. R. El Fatmi, “Non-uniform warping including the effects of torsion and shear forces. Part I: a general beam theory,” International Journal of Solids and Structures, vol. 44, no. 18-19, pp. 5912–5929, 2007. View at Publisher · View at Google Scholar · View at Scopus
  144. C. Hong and G. E. Blandford, “C0 finite element formulation for thin-walled beams,” International Journal for Numerical Methods in Engineering, vol. 28, no. 10, pp. 2239–2255, 1989. View at Google Scholar · View at Scopus
  145. A. S. Gendy, A. F. Saleeb, and T. Y. P. Chang, “Generalized thin-walled beam models for flexural-torsional analysis,” Computers and Structures, vol. 42, no. 4, pp. 531–550, 1992. View at Google Scholar · View at Scopus
  146. Y. Hu, X. Jin, and B. Chen, “A finite element model for static and dynamic analysis of thin-walled beams with asymmetric cross-sections,” Computers and Structures, vol. 61, no. 5, pp. 897–908, 1996. View at Publisher · View at Google Scholar · View at Scopus
  147. H. Rubin, “Torsions-Querschnittswerte für rechteckige Hohlprofile nach EN, 10210–2:2006 und EN, 10219–2:2006,” Stahlbau, vol. 76, pp. 21–33, 2007. View at Google Scholar
  148. V. Le Corvec and F. C. Filippou, “Enhanced 3D Fiber Beam-Column Element With Warping Displacements,” in Proceedings of the 3rd ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Corfu, Greece, May 2011.
  149. J. Wackerfuß and F. Gruttmann, “A mixed hybrid finite beam element with an interface to arbitrary three-dimensional material models,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 27–29, pp. 2053–2066, 2009. View at Publisher · View at Google Scholar · View at Scopus
  150. J. Wackerfuß and F. Gruttmann, “A nonlinear Hu-Washizu variational formulation and related finite-element implementation for spatial beams with arbitrary moderate thick cross-sections,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 17–20, pp. 1671–1690, 2011. View at Publisher · View at Google Scholar · View at Scopus
  151. V. J. Tsipiras and E. J. Sapountzakis, “Secondary torsional moment deformation effect in inelastic nonuniform torsion of bars of doubly symmetric cross section by BEM,” International Journal of Non-Linear Mechanics, vol. 47, pp. 68–84, 2012. View at Google Scholar
  152. J. Navarro Gregori, P. Miguel Sosa, M. A. Fernández Prada, and F. C. Filippou, “A 3D numerical model for reinforced and prestressed concrete elements subjected to combined axial, bending, shear and torsion loading,” Engineering Structures, vol. 29, no. 12, pp. 3404–3419, 2007. View at Publisher · View at Google Scholar · View at Scopus