Journal of Constructional Steel Research 65 (2009) 1075–1086 Contents lists available at ScienceDirect Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr Extended second order approximate analysis of frames with sway-braced column interaction Jostein Hellesland * Mechanics Division, Department of Mathematics, University of Oslo, P.O. Box 1053 – Blindern, NO-0316 Oslo, Norway article info Article history: Received 3 June 2008 Accepted 19 August 2008 Keywords: Frames Columns Storey-based buckling Critical load Effective lengths Storey magnifier method abstract Present approximate second order methods for the analysis of frames with sway are not capable of reflecting the transition from sway to partially braced, and nearly fully braced behaviour of individual columns in the frames. The main aim of the paper is to extend the approximate storey magnifier approach to account for such a transition. The key to this is in the manner local second order effects are reflected. A high order shear relationship is proposed, and general sway magnifier, critical load and effective length formulations are presented both in terms of first order lateral storey stiffness and critical, free-sway column loads. Their interrelationship, and simplifications leading to existing approaches, and adaptations in present codes and standards, are discussed. Comparisons are made with exact critical loads, sway and moment magnifiers for nearly unbraced, partially braced and nearly fully braced systems. The proposed, extended approach provides predictions that generally are in very good agreement with exact results at all axial load levels. The more simplified approaches provide good agreement for low to moderate load levels for some column end restraint combinations, and up to relatively high load levels for other combinations. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction In approximate second order analyses of frames with sidesway, the so-called N –Δ type methods have found extensive use. The basic concept of the initial form of such methods is that the drift and sway moments produced by the vertical (gravity) loads, can be accounted for by equivalent, fictitious lateral loads acting at the beam (floor) levels. The method accounts for the interaction between laterally stiff and flexible columns on the same level, and can be applied to both unbraced or partially braced frames. The method can be applied in an iterative manner by computing total load effects through successive corrections of the first order sidesway displacements. Alternatively, in particular for individual stories, it may be applied in a non-iterative manner based on closed form equations obtained by considering the reduction in lateral column stiffness due to the axial loads. Such storey-based applications are of interest here. A number of treaties involving the application of various forms of this method to elastic and inelastic structures have been published. Early work (1965–85), dealing with critical loads or second order sway magnification effects, or both, include Rosenblueth [1], Fey [2], Parme [3], Stevens [4], Rubin [5], Horne [6], Wood et al. [7], Hellesland [8], LeMessurier [9], MacGregor and * Tel.: +47 2285 5950; fax: +47 2285 4349. E-mail address: josteinh@math.uio.no. Hage [10] and Lai and MacGregor [11]. Related, more recent work includes Aristizabal-Ochoa [12], Lui [13], Xu and Liu [14], Girgin et al. [15], and others, with emphasis on critical load analysis. A most valuable asset of approximate methods is their transparency with respect to the important variables. This also applies to the manner in which second order effects are reflected. In frames with sidesway, this concerns (1) overall, global (‘‘N Δ’’) effects, due vertical loads acting on the sidesway of the frame system as such, (2) individual, local (‘‘N δ’’) effects, due to axial member loads acting on the deflections away from the chord between member ends, and thus causing nonlinear (curved) moment distributions along the members, and (3) local effects due to changing restraint stiffness at member ends due to vertical, inter-storey column interaction. The global second order effects are well taken care of in these approaches. This is, to some extent, also the case for local second order (N δ) effects. The latter are generally reflected through a factor with labels such as ‘‘bending shape factor’’ [5], ‘‘flexibility factor’’ [8] or ‘‘stiffness reduction factor’’ [9]. According to the reviewed literature, and textbooks, e.g., [16], it is generally, but incorrectly, stated that the flexibility factor varies between 1 and 1.22 (1.2). This range is normally appropriate for columns in common, regular unbraced frames with columns having similar stiffness and axial load level, with relatively small local second order effects, but it may not be adequate for columns in irregular unbraced frames, or partially braced frames. In such frames, one or more columns may be significantly more flexible than the 0143-974X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2008.08.008