Efficient Line-Element Method for the Second-Order Analysis of Steel Members with Nonsymmetric Thick-Walled Cross Sections
Publication: Journal of Structural Engineering
Volume 150, Issue 2
Abstract
When designing steel members with nonsymmetric cross sections, it is essential to consider twisting effects when performing stability checks via second-order analysis according to the ANSI/AISC 360-22. Existing line-element formulations for nonsymmetric sections are mostly derived based on thin-walled assumptions, leading to an overestimation of member strength due to the inaccurate prediction of member behavior, especially when the cross sections have moderate wall thickness. This paper proposes an efficient line-element method for second-order analysis of steel members with nonsymmetric thick-walled sections considering the warping degree of freedom (DOF) and the twisting effects along with the transverse shear deformations. Additionally, a two-dimensional (2D) finite-element cross section analysis method employing the constant strain triangle (CST) element is developed to calculate the section properties for arbitrary cross sections, including the Wagner and the shear coefficients. The proposed method is implemented in the educational structural analysis software MASTAN2 and verified through two sets of examples.
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Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author by request.
Acknowledgments
The work described in this paper was partially supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU/21E/15203121) and a grant from the National Natural Science Foundation of China (No. 52008410).
References
Abdelrahman, A. H. A., L. Chen, S. W. Liu, and R. D. Ziemian. 2022. “Timoshenko line-element for stability analysis of tapered I-section steel members considering warping effects.” Thin-Walled Struct. 175 (Jun): 109198. https://doi.org/10.1016/j.tws.2022.109198.
Ahmad, M. G., and S. Hoa. 2016. “Flexural stiffness of thick walled composite tubes.” Compos. Struct. 149 (Aug): 125–133. https://doi.org/10.1016/j.compstruct.2016.03.050.
AISC. 2022. Specification for structural steel buildings. ANSI/AISC 360-22. Chicago: AISC.
Ali Faghidian, S. 2017. “Unified formulations of the shear coefficients in Timoshenko beam theory.” J. Eng. Mech. 143 (9): 06017013. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001297.
Allman, D. 1988. “Evaluation of the constant strain triangle with drilling rotations.” Int. J. Numer. Methods Eng. 26 (12): 2645–2655. https://doi.org/10.1002/nme.1620261205.
Arboleda-Monsalve, L. G., D. G. Zapata-Medina, and J. D. Aristizabal-Ochoa. 2008. “Timoshenko beam-column with generalized end conditions on elastic foundation: Dynamic-stiffness matrix and load vector.” J. Sound Vib. 310 (4–5): 1057–1079. https://doi.org/10.1016/j.jsv.2007.08.014.
Caillerie, D., P. Kotronis, and R. Cybulski. 2015. “A Timoshenko finite element straight beam with internal degrees of freedom.” Int. J. Numer. Anal. Methods Geomech. 39 (16): 1753–1773. https://doi.org/10.1002/nag.2367.
Chan, K. T., K. F. Lai, N. G. Stephen, and K. Young. 2011. “A new method to determine the shear coefficient of Timoshenko beam theory.” J. Sound Vib. 330 (14): 3488–3497. https://doi.org/10.1016/j.jsv.2011.02.012.
Chan, S. L., and S. Kitipornchai. 1987. “Geometric nonlinear analysis of asymmetric thin-walled beam-columns.” Eng. Struct. 9 (4): 243–254. https://doi.org/10.1016/0141-0296(87)90023-X.
Chen, L., A. H. A. Abdelrahman, S. W. Liu, and R. D. Ziemian. 2021. “Gaussian beam–column element formulation for large-deflection analysis of steel members with open sections subjected to torsion.” J. Struct. Eng. 147 (12): 04021206. https://doi.org/10.1061/(ASCE)ST.1943-541X.0003185.
Chen, L., W.-L. Gao, S.-W. Liu, R. D. Ziemian, and S.-L. Chan. 2022. “Geometric and material nonlinear analysis of steel members with nonsymmetric sections.” J. Constr. Steel Res. 198 (Aug): 107537. https://doi.org/10.1016/j.jcsr.2022.107537.
Cheng, X., H. Chen, Y. Gong, and Y. B. Yang. 2018. “Stocky thin-or thick-walled beams: Theory and analysis.” Eng. Struct. 159 (Jun): 55–65. https://doi.org/10.1016/j.engstruct.2017.12.027.
Couto, C., and P. V. Real. 2019. “Numerical investigation on the influence of imperfections in the local buckling of thin-walled I-shaped sections.” Thin-Walled Struct. 135 (May): 89–108. https://doi.org/10.1016/j.tws.2018.10.039.
Da Silva, R. G. L., A. C. C. Lavall, R. S. Costa, and H. F. Viana. 2018. “Formulation for second-order inelastic analysis of steel frames including shear deformation effect.” J. Constr. Steel Res. 151 (Feb): 216–227. https://doi.org/10.1016/j.jcsr.2018.09.011.
Davis, R., R. Henshell, and G. Warburton. 1972. “A Timoshenko beam element.” J. Sound Vib. 22 (4): 475–487. https://doi.org/10.1016/0022-460X(72)90457-9.
Edem, I. B. 2006. “The exact two-node Timoshenko beam finite element using analytical bending and shear rotation interdependent shape functions.” Int. J. Comput. Methods Eng. Sci. Mech. 7 (6): 425–431. https://doi.org/10.1080/15502280600826381.
Friedman, Z., and J. B. Kosmatka. 1993. “An improved two-node Timoshenko beam finite element.” Comput. Struct. 47 (3): 473–481. https://doi.org/10.1016/0045-7949(93)90243-7.
Gao, W. L., A. H. A. Abdelrahman, S. W. Liu, and R. D. Ziemian. 2021. “Second-order dynamic time-history analysis of beam-columns with nonsymmetrical thin-walled steel sections.” Thin-Walled Struct. 160 (Jun): 107367. https://doi.org/10.1016/j.tws.2020.107367.
Gere, J. M., and S. P. Timoshenko. 1991. Mechanics of materials. 3rd ed. Cheltenham, UK: Nelson Thornes.
Gruttmann, F., and W. Wagner. 2001. “Shear correction factors in Timoshenko’s beam theory for arbitrary shaped cross-sections.” Comput. Mech. 27 (3): 199–207. https://doi.org/10.1007/s004660100239.
Herrmann, L. R. 1965. “Elastic torsional analysis of irregular shapes.” J. Eng. Mech. Div. 91 (6): 11–19. https://doi.org/10.1061/JMCEA3.0000688.
Hsiao, K. M., and W. Y. Lin. 2000. “A co-rotational formulation for thin-walled beams with monosymmetric open section.” Comput. Methods Appl. Mech. Eng. 190 (8–10): 1163–1185. https://doi.org/10.1016/S0045-7825(99)00471-5.
Hutchinson, J. 2001. “Shear coefficients for Timoshenko beam theory.” J. Appl. Mech. 68 (1): 87–92. https://doi.org/10.1115/1.1349417.
Jensen, J. J. 1983. “On the shear coefficient in Timoshenko’s beam theory.” J. Sound Vib. 87 (4): 621–635. https://doi.org/10.1016/0022-460X(83)90511-4.
Kim, N. I., and M. Y. Kim. 2005. “Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects.” Thin-Walled Struct. 43 (5): 701–734. https://doi.org/10.1016/j.tws.2005.01.004.
Kindmann, R., and M. Kraus. 2011. Steel structures: Design using FEM. New York: Wiley.
Krahula, J. L., and G. F. Lauterbach. 1969. “A finite element solution for Saint-Venant torsion.” AIAA J. 7 (12): 2200–2203. https://doi.org/10.2514/3.5516.
Kuwamura, H. 2003. “Local buckling of thin-walled stainless steel members.” Steel Struct. 3 (3): 191–201.
Liu, S. W., W. L. Gao, and R. D. Ziemian. 2019. “Improved line-element formulations for the stability analysis of arbitrarily-shaped open-section beam-columns.” Thin-Walled Struct. 144 (Aug): 106290. https://doi.org/10.1016/j.tws.2019.106290.
Liu, S. W., R. D. Ziemian, L. Chen, and S. L. Chan. 2018. “Bifurcation and large-deflection analyses of thin-walled beam-columns with non-symmetric open-sections.” Thin-Walled Struct. 132 (May): 287–301. https://doi.org/10.1016/j.tws.2018.07.044.
Mason, W. E., Jr., and L. R. Herrmann. 1968. “Elastic shear analysis of general prismatic beams.” J. Eng. Mech. Div. 94 (4): 965–983. https://doi.org/10.1061/JMCEA3.0001004.
McGuire, W., R. H. Gallagher, and R. D. Ziemian. 2000. Matrix structural analysis. 2nd ed. New York: Wiley.
Mohri, F., N. Damil, and M. P. Ferry. 2008. “Large torsion finite element model for thin-walled beams.” Comput. Struct. 86 (7–8): 671–683. https://doi.org/10.1016/j.compstruc.2007.07.007.
Murín, J., M. Aminbaghai, V. Kutiš, V. Královič, T. Sedlár, V. Goga, and H. Mang. 2014. “A new 3D Timoshenko finite beam element including non-uniform torsion of open and closed cross sections.” Eng. Struct. 59 (Aug): 153–160. https://doi.org/10.1016/j.engstruct.2013.10.036.
Pignataro, M., N. Rizzi, G. Ruta, and V. Varano. 2010. “The effects of warping constraints on the buckling of thin-walled structures.” J. Mech. Mater. Struct. 4 (10): 1711–1727. https://doi.org/10.2140/jomms.2009.4.1711.
Pilkey, W. D. 2002. Analysis and design of elastic beams: Computational methods. New York: Wiley.
Prokić, A. 1993. “Thin-walled beams with open and closed cross-sections.” Comput. Struct. 47 (6): 1065–1070. https://doi.org/10.1016/0045-7949(93)90310-A.
Rao, S. S. 2017. The finite element method in engineering. Oxford, UK: Butterworth-Heinemann.
Reddy, J. N. 2019. Introduction to the finite element method. New York: McGraw-Hill Education.
Saadé, K., B. Espion, and G. Warzée. 2004. “Non-uniform torsional behavior and stability of thin-walled elastic beams with arbitrary cross sections.” Thin-Walled Struct. 42 (6): 857–881. https://doi.org/10.1016/j.tws.2003.12.003.
Schramm, U., L. Kitis, W. Kang, and W. D. Pilkey. 1994. “On the shear deformation coefficient in beam theory.” Finite Elem. Anal. Des. 16 (2): 141–162. https://doi.org/10.1016/0168-874X(94)00008-5.
Tang, Y. Q., Y. P. Liu, S. L. Chan, and E. F. Du. 2019. “An innovative co-rotational pointwise equilibrating polynomial element based on Timoshenko beam theory for second-order analysis.” Thin-Walled Struct. 141 (Feb): 15–27. https://doi.org/10.1016/j.tws.2019.04.001.
Timoshenko, S. P. 1945. “Theory of bending, torsion and buckling of thin-walled members of open cross section.” J. Franklin Inst. 239 (5): 343–361. https://doi.org/10.1016/0016-0032(45)90013-5.
Yang, Y. B., and W. McGuire. 1986. “Stiffness matrix for geometric nonlinear analysis.” J. Struct. Eng. 112 (4): 853–877. https://doi.org/10.1061/(ASCE)0733-9445(1986)112:4(853).
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Published In
Journal of Structural Engineering
Volume 150 • Issue 2 • February 2024
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Mar 2, 2023
Accepted: Sep 25, 2023
Published online: Nov 30, 2023
Published in print: Feb 1, 2024
Discussion open until: Apr 30, 2024
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