# Lateral-Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
## 1 Introduction
[15; 16] proposed a closed-form expression for calculating elastic critical LTB loads and a design procedure for cantilevers. The design procedure considered elastic buckling and inelastic buckling and the calculation results of the procedure were highly consistent with Eurocode 3-2005 or A3C360-10. Because the boundary conditions of cantilever beams are more complex than those of simply supported beams, the analytical solutions of \(M_{\rm cr}\) of the cantilever are highly complex. Therefore, the solution to \(M_{\rm cr}\) requires more approaches, such as the finite difference [17], finite element [18; 19; 20], energy method [21; 22], finite integral [10] or the lateral-torsional buckling modification factor [1]. Among them, the energy method is one of the most common and basic methods. Therefore, the correct derivation or selection of total potential energy is the key to obtaining accurate calculation results. Additionally, [PERSON] et al. [23; 24] studied the inelastic LTB of cantilevers and continuous steel beams in order to provide better design methods than current specifications. Similarly, [PERSON] et al. [25] carried out inelastic buckling experiments on biaxial symmetric cantilever steel beams and proposed a numerical simulation method to accurately predict the displacement and inelastic critical buckling load of cantilever steel beams. In addition to the study of ordinary simply supported beams and cantilever beams, many scholars also considered other factors, such as pre-stressed beams, initial imperfections, flange-web interaction and material properties. [PERSON] et al. [26] studied the \(M_{\rm cr}\) of LTB of pre-stressed two-chord columns by experiment and numerical analysis. [PERSON] et al. [27; 28] presented an LTB theory of pre-stressed steel H-beams. This theory considered the initial rotation angle and it was applicable to simply supported beams and cantilever beams. [PERSON] used the energy method to obtain the buckling load of pre-stressed steel I-beams [29]. [PERSON] et al. [30] proposed analytical formulations for supported beams whose support was restrained from warping, and some scholars [31; 32; 33] have studied the LTB of special sections such as built-up sections. [PERSON] et al. [34] proposed a new theory to predict the shape and amplitude of unique global and local initial imperfections for the LTB of doubly symmetric I-section cantilevers. [PERSON] [35] and [PERSON] [36] studied the elastic LTB mode considering web-distortion or flange-web interaction. [PERSON] et al. [37; 38; 39] studied the elastic LTB mode of corrugated web beams. In addition, an increasing number of scholars have considered the influence of material properties on the LTB performance of beams [40; 41; 42]. For example, [PERSON] et al. [43; 44] studied the effect of chemical composition of materials on their strength by using neural networks and artificial intelligence technology. [PERSON] et al. [45] studied the LTB performance of cantilever beams using 3D printing technology. It could be seen from those studies on the lateral-torsional buckling of cantilever steel beams that most of their transverse loads were mainly a single load. The position of the load acting on the cross-section was restricted to the top flange (TF), bottom flange (BF) and the shear center (SC). The position of the load acting in the beam length direction was restricted to the free end. There was less research on lateral-torsional buckling of singly symmetric I-beams under CL or CUDL. In engineering practice, cantilever steel beams are usually affected by the concentrated load or the combination of concentrated load and uniformly distributed load, and the position of the concentrated load is not limited to the free end. For example, some cantilever beams will be subject to the concentrated load from the secondary beams and uniformly distributed load from the floor slabs. The position of the concentrated load may be in the middle of the span or other positions.