Tangent moduli of hot-rolled I-shaped axial members considering various residual stress distributions Seungjun Kim a , Taek Hee Han b , Deok Hee Won b,c , Young Jong Kang c,n a Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843, USA b Coastal Development & Ocean Energy Research Division, Korea Institute of Ocean Science and Technology, Ansan 426-744, Republic of Korea c School of Architectural, Civil and Environmental Engineering, Korea University, Seoul 136-713, South Korea article info Article history: Received 12 March 2013 Accepted 15 November 2013 Available online 7 December 2013 Keywords: Residual stress Tangent modulus Hot-rolled section Plastic hinge method Inelastic analysis abstract This paper presents an equation for the effective tangent moduli for steel axial members of hot-rolled I-shaped section subjected to various residual stress distributions. Because of the existence of residual stresses, the cross section yields gradually even when the member is subjected to uniform axial stresses. In the elasto-plastic stage, the structural response can be easily traced using rational tangent modulus of the member. In this study, the equations for rational tangent moduli for hot-rolled I-shaped steel members in the elasto-plastic stage were derived based on the general principle of force-equilibrium. For practical purpose, the equations for the tangent modulus were presented for conventional patterns of the residual stress distribution of hot-rolled I-shaped steel members. Through a series of material nonlinear analyses for steel axial members modeled by shell elements, the derived equations were numerically verified, and the presented equations were compared with the CRC tangent modulus equation, the most frequently used equation so far. The comparative study shows that the presented equations are extremely effective for accurately analyzing elasto-plastic behavior of the axially loaded members in a simple manner without using complex shell element models. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction In general, hot-rolled steel members are not free from residual stresses distributed in various patterns owing to the characteristics of the manufacturing method. The existence of residual stresses may induce gradual yielding in the section of steel members even when external axial loads are applied. To predict the elasto-plastic response in the gradual yield stage of the axially loaded steel members, the tangent modulus has been widely used for its simplicity. The CRC (Column Research Council) tangent modulus suggested by Galambos [1] has been the most widely used one. In particular, the modulus was used in the plastic hinge method and refined plastic hinge method to consider gradual yielding of the I-shaped member that has specific residual stress distributions by Chen and Lui [2], Liew et al. [3], Chen and Liew [4], Kim and Chen [5], Kim et al. [6], Kim et al. [7], and Kim [8]. But the CRC tangent modulus, which was basically derived via Euler's elastic buckling equation with assumption of constant maximum compressive residual stress, has limitations in applic- ability where different various residual stress distributions are to be considered. As shown in Eq. (1), various effective factors, such as maximum residual stress values, patterns and the sectional shapes of members, cannot be considered in the CRC tangent modulus. E t ¼ 1:0E P r0:5P y ð1Þ E t ¼ 4 P P y E 1  P P y   P Z0:5P y ð2Þ In this study, rational new tangent moduli are derived based on the general principle of force-equilibrium. First of all, the linear residual stress distributions suggested by Galambos and Ketter [9], and by ECCS [10], respectively, are considered as the general residual stress patterns of hot-rolled steel members. In addition, the parabolic stress distribution suggested by Szali and Papp [11] is also taken into account. Including the effect of the maximum or minimum residual stress value, the presented equations for the tangent moduli produce clearer and more accurate computation of the elasto-plastic behavior. The derived equations are verified based on the results of nonlinear finite element analyses for axially loaded steel members modeled by shell elements with residual stress distributions aforementioned. The accuracy and the verification of the presented tangent modulus equations are evidently shown by comparing the load–displacement curves of nonlinear finite element analyses with those of present study (Fig. 1). Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/tws Thin-Walled Structures 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.11.005 n Corresponding author. Tel.: þ82 2 3290 3317; fax: þ82 2 921 5166. E-mail address: yjkang@korea.ac.kr (Y.J. Kang). Thin-Walled Structures 76 (2014) 77–91