Flexural-torsional buckling strength of I-girders with discrete torsional braces under various loading conditions Canh Tuan Nguyen a , Hyun-Sung Joo a , Jiho Moon b , Hak-Eun Lee a,⇑ a Civil, Environmental and Architectural Engineering, Korea University, 5-1, Anam-dong, Sungbuk-gu, Seoul 136-701, South Korea b Civil and Environmental Engineering, University of Washington, Seattle, WA 98195-2700, USA article info Article history: Received 6 June 2011 Revised 7 December 2011 Accepted 8 December 2011 Available online 20 January 2012 Keywords: Flexural-torsional buckling Bifurcation analysis Inelastic buckling Torsional restraint Torsional bracing Stiffness requirement Equivalent moment factor abstract Torsional bracing systems have been used widely in I-girder bridges to increase the flexural-torsional buckling strength and distribute the load to the adjacent girders. To evaluate the required stiffness of the bracing and the flexural-torsional buckling strength of the I-girder with a multiple torsional bracing system, the equivalent continuous torsional bracing concept is often adopted regardless of the type of torsional bracing system. However, a previous study on the I-girder with discrete torsional bracings under uniform bending reported that the equivalent continuous torsional bracing concept has certain limitations on discrete torsional bracing problems and it needs to be investigated for general loading cases. This article presents an analytical solution for the elastic flexural-torsional buckling strength and stiffness requirements of I-girders with discrete torsional bracings under various loading conditions. First, a review was performed of the previous study on the elastic critical buckling moment and torsional stiffness requirement of the I-girder with discrete torsional bracings under uniform bending. This solu- tion was then extended for various loading conditions. From the derived analytical solutions, the equiv- alent moment factor was proposed for practical engineering purposes and the proposed solutions were verified by comparing them with the results of finite element analysis and those of other previous stud- ies. Finally, non-linear finite element analyses including the effects of the initial imperfection and resid- ual stresses were conducted to examine the inelastic buckling strengths of I-girders with discrete torsional bracings under various loading conditions. The results showed that the buckling curves from the current design specification provide reasonably conservative flexural-torsional buckling strengths of the I-girder with discrete torsional bracings when the proposed elastic solutions are applied to obtain the buckling parameters. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Torsional bracing systems are often used to connect the sepa- rated girders, as shown in Fig. 1. The flexural-torsional buckling strengths can be improved by introducing an adequate bracing sys- tem, and the torsional bracing system allows load transfer to the adjacent girders. A concrete slab or deck attached to the girders is categorized into a continuous bracing system, as shown in Fig. 1. For this bracing system, Taylor and Ojalvo [1] derived the exact solu- tion for the elastic flexural-torsional buckling strength of the beam under uniform bending. The cross beams or frames are typical exam- ples of a discrete torsional bracing system, which are commonly used in I-girder systems. In this study, the focus was made on the I-girder with a discrete torsional bracing system. Several studies have been presented on the flexural-torsional buckling of the I-girder with a discrete torsional bracing system. Wakabayashi and Nakamura [2] conducted experimental study on H-shaped beams braced by purlins or a sub-beam at central span. Tong and Chen [3,4] derived the elastic flexural-torsional buckling strength and stiffness requirement for a beam with a tor- sional bracing at mid-span under uniform bending. Trahair [5] pro- posed equations for the elastic buckling moment and the stiffness requirement for beams with a torsional restraint at the mid-span based on numerical approximations. Valentino et al. [6] extended Trahair’s work [5] to the inelastic flexural-torsional buckling strength. Tong and Chen [4] and Yura [7] proposed analytical solu- tions based on the concept of continuous torsional bracing. They assumed that the discrete bracings can be considered as a contin- uous bracing by summing the stiffness of each brace and dividing by the beam length. This value is called the equivalent continuous bracing stiffness [4,7]. The effects of the moment gradient and transverse loading on the flexural-torsional buckling strength of the I-girder with a torsional restraint at mid-span have also been investigated by several researchers [8–10]. A literature review shows that most studies are limited to beams with torsional bracing at the mid-span and elastic 0141-0296/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.12.022 ⇑ Corresponding author. Tel.: +82 2 3290 3315; fax: +82 2 928 5217. E-mail address: helee@korea.ac.kr (H.-E. Lee). Engineering Structures 36 (2012) 337–350 Contents lists available at SciVerse ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct