Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct Study of the stability of weightless thin-walled straight columns under centrically applied terminal compressive force by using equivalent forces L.M. Gil-Martín ⁎ , E. Hernández-Montes Department of Structural Mechanics, University of Granada, E.T.S. Ingenieros de Caminos, Campus Universitario de Fuentenueva, E-18072 Granada, Spain ARTICLEINFO Keywords: Equivalent forces Torsional buckling Flexural-torsional buckling ABSTRACT There are three possible modes of buckling of thin-walled straight steel columns: fexural buckling, torsional buckling, and fexural-torsional buckling. These modes of buckling are considered in the specifcation for the design of steel structures, such as Eurocode 3, which includes the particular case of torsional-fexural buckling of centrically loaded members with monosymmetric cross-sections. The system of diferential equations that governs the stability of centrically loaded weightless members was presented in the mid-twentieth century and has been widely addressed in both steel structures and instability books. In this work, a simpler way to obtain the diferential equations of stability for both torsional and fexural-torsional buckling modes by using equivalent forces is presented. The presented idea is especially useful in the academic context of civil engineering. Students and faculty members will appreciate the deduction of the instability equations governing the equilibrium in a few simple steps. 1. Introduction Elastic stability of beams and columns is one of the most important criteria in the design of structures. This topic is particularly important in the context of steel members due to the high slenderness and low torsional rigidity of these members [1,2]. Elastic stability has been widely studied in classical books [3–6]. Recently, more complicated studies regarding theory of stability con- sidering higher order beam theories [7] and more rigorous mathema- tical solution involving theoretically exact derivation for some stability problems [8] have been carried out. On the contrary, a simple and comprehensible procedure to address the study of the elastic stability of compressed members is presented here. It is known that thin-walled straight members subjected to axial terminal compression buckle in diferent modes depending on the type of deformation they sufer. If the compressed member bends but with no rotational deformation, it is called fexural bucking. This is the case of the double-symmetrical cross-sections commonly used in steel con- struction: the I-Shapes. Sometimes, the axial compressive force does not provoke the bending of the member but just the rotation of the cross-sections during buckling (i.e. the axis of the member is kept straight). This buckle mode is called torsional bucking and is usually studied for the particular case of short members with cruciform cross-sections. In a general case, the compressed member simultaneously sufers bending and torsion deformations during buckling. This mode of bucking is called fexural-torsional buckling and it happens in members with non-symmetrical cross-section in which the shear center and the centroid (gravity center) do not coincide. See Fig. 1. The calculation of the elastic critical load corresponding to the three modes of buckling is always calculated by imposing equilibrium of the member in the deformed state and considering both, nondeformable cross-sections and the assumption that displacements are very small. Traditionally, the three diferential equations governing the equili- brium of a deformed element in a torsional or fexural-torsional buck- ling mode have been obtained by: (i) considering the strut loaded by “fctitious” or equivalent transverse loads (load per unit length uni- formly distributed over the middle surface of the member) whose in- tensity is related to transverse bending of the beam [5]; and, (ii) ex- pressing the defection in the principal centroidal axes -at an arbitrary point of the cross-section- as a function of the shear center displace- ments and the rotation of the cross-section in its plane [3–6,9]. Besides, the concept of equivalent (transverse) forces have been widely used in Structural Mechanics. This concept has been commonly used in structures (e.g. Clough and Penzien [10]) to obtain an ap- proximation of the geometrical stifness matrix, K G , of truss and beams because it allows for accounting the geometrical second order efects in an easy way. Moreover, authors as Němec et al. [11] recommend the https://doi.org/10.1016/j.engstruct.2019.109726 Received 14 May 2019; Received in revised form 17 September 2019; Accepted 30 September 2019 ⁎ Corresponding author. E-mail addresses: mlgil@ugr.es (L.M. Gil-Martín), emontes@ugr.es (E. Hernández-Montes). Engineering Structures 200 (2019) 109726 0141-0296/ © 2019 Elsevier Ltd. All rights reserved. T