Engineering Structures 31 (2009) 758–767 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct Buckling modeling of reinforcing bars with imperfections Leonardo M. Massone a,∗ , Daniel Moroder b a Department of Civil Engineering, University of Chile, Blanco Encalada 2002, Santiago, Chile b Civil Engineering, University of Bologna, Viale Risorgimento 2, 40136, Bologna, Italy article info Article history: Received 29 April 2008 Received in revised form 3 November 2008 Accepted 20 November 2008 Available online 3 January 2009 Keywords: Reinforcing bar Buckling Reinforced concrete Model abstract Reinforced concrete columns in seismic zones are subjected to combined actions, resulting in axial loads in longitudinal reinforcing bars. Thus, knowing the bar’s response, especially when it is subjected to important axial compressive forces that might lead to buckling, is important. A bar buckling model based on concentrated plasticity and with the capability of introducing an initial imperfection is described. The initial imperfection is imposed by bending the bar with a transversely applied nonpermanent force. Additionally, a comprehensive study of the monotonic tensile response beyond the peak stress point and a simple cyclic rule, complete the physical approach of the model. Comparisons of the model with experimental results reveal that peak capacity (average axial stress) is well captured, as well as the post- peak response shape (average axial stress versus strain), with differences observed basically in the peak capacity for specimens with high bar imperfection-to-diameter ratio, and in the shape of the post-peak response for specimens with low bar length-to-diameter ratio. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction Reinforced concrete columns in seismic zones are subjected to combined actions that include mainly axial, moment and shear forces. Longitudinal reinforcing bars act as members that resist axial loads, which also contribute in maintaining the column’s moment. Thus, the axial response of longitudinal bar becomes relevant. In absence of buckling effects the axial response can be associated with the monotonic or cyclic response of bars. That situation, although ideal, might not represent all cases. Reinforced concrete columns under cyclic lateral displacement, which represents a seismic action, would remain elastic under small displacements. Under severe loading, lateral displacement would increase, and in combination with compressive axial forces, deterioration of cover concrete that ends with spalling would reveal part of the longitudinal bars which are laterally supported by stirrups. A large distance between stirrups would trigger buckling at lower loads, which also affects the column’s response. Therefore, a buckling modeling is required to establish a good understanding of column behavior, especially when the longitudinal bar response may be affected by relatively large stirrup separation. The study of buckling begins with Euler in the 18th century. He developed a simple equation to calculate the critical load for the elastic case. More recent developments have included material inelasticity. The application to reinforced concrete modeling ∗ Corresponding author. Tel.: +56 2 9784984; fax: +56 2 6892833. E-mail address: lmassone@ing.uchile.cl (L.M. Massone). appeared with Bresler and Gilbert [1], providing the information about tie spacing requirements and the buckling behavior of longitudinal reinforcement steel in compressed concrete members based on critical load estimation at yielding. Further efforts have been done by researchers in order to capture not only the buckling capacity of longitudinal bars, but also to describe their monotonic, as well as their axial cyclic response. Numerical simulations using fiber discretization of beam-column elements with distributed plasticity have been introduced (e.g., [2,3]), characterizing in part the monotonic response. Cyclic response has also been estimated based on calibration of monotonic experimental response of bars subjected to buckling (e.g., [3,4]), allowing the introduction of cyclic constitutive material laws for steel including buckling into beam and column analysis. Other authors have adopted different modeling approaches to introduce the buckling behavior to beam and columns responses, such as introducing concentrated plasticity models based on steel material constitutive laws (e.g., [5,6]). 2. Research significance The model described in this paper considers a similar approach adopted by Restrepo [6] but with the capability of introducing an initial imperfection, imposed by bending the bar with a trans- versely applied nonpermanent force. Additionally, a comprehen- sive study of the monotonic response pointed out the need for defining the tensile response beyond the peak stress point, and a consistent point of fracture. Altogether, with a reliable compres- sive constitutive law for the steel material and a simple cyclic rule, 0141-0296/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2008.11.019