# Lateral-Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
## 1 Introduction  
Lateral-torsional buckling (LTB) is a common global instability phenomenon for slender structures, where structures that are bent in the plane of greatest flexural rigidity may buckle laterally and torsionally as the external load reaches the critical value. Therefore, it is necessary to consider LTB in the design of the beam when the flexural rigidity of the beam in the plane of bending is greater than the lateral flexural rigidity. Cantilevers are a common slender structure in civil engineering. Therefore, it is necessary to consider the limiting load of LTB in addition to deformation and stress analysis during the engineering design process. The American National Standard Institute/American Institute of Steel Construction (ANSI/AASC 360-16) proposed the lateral-torsional buckling modification factor, \(C_{\mathrm{b}}\), when calculating the LTB of simply supported beams. \(C_{\mathrm{b}}\) had an explicit formula but it was only recommended directly taking the value of 1.0 when calculating the \(C_{\mathrm{b}}\) of cantilever beams [1]; EN 1993-1-1:2005 also had a specific formulas for lateral-torsional buckling of simply supported beams, but it did not mention the instability calculation of cantilever [2]. As early as 1960, [PERSON] proposed a formula for calculating the critical moment (\(M_{\rm cr}\)) using \(C_{1}\), \(C_{2}\) and \(C_{3}\) factors [3]. Since then, the calculation method of \(C_{1}\), \(C_{2}\) and \(C_{3}\) has been increasingly refined. Many scholars have studied \(C_{1}\), \(C_{2}\) and \(C_{3}\) factors of lateral-torsional buckling of simply supported beams. [PERSON] et al. [4] studied the general formula of \(M_{\rm cr}\) under linear distributed moment. The study used dimensionless parameters and infinite series to calculate \(M_{\rm cr}\) and fitted the specific values of \(C_{1}\), \(C_{2}\) and \(C_{3}\). It was revealed that the more terms the buckling deformation function takes, the more accurate the calculation results will be. However, due to the complexity of the external load, the solution to \(M_{\rm cr}\) also needs to depend on other methods. [PERSON] et al. [5] further studied the expression of \(C_{\rm b}\) of simply supported beams under support moments, point loads, uniformly distributed loads and a combination thereof. [PERSON] et al. [6] presented a theory of stiffness reduction coefficient for the LTB design of welded web-tapered steel beams. [PERSON] et al. [7] conducted elastic LTB analysis on supported beams using ABAQUS and GBTul. [PERSON] et al. [8] assessed the LTB mode of plane frames by finite element theory, which greatly reduced the amount of modeling and calculation. So far, numerous problems about the elastic LTB of simply supported beams are being gradually solved. For cantilever steel beams, buckling deformation is more complex due to the characteristics of the boundary condition. Unlike simply supported beams, the maximum displacement and torsion angle of cantilever are located at the free end rather than near the mid-span. Therefore, the coefficients of \(C_{1}\), \(C_{2}\) and \(C_{3}\) are no longer fixed values. [PERSON] used infinite series to study the singly symmetric cross-section cantilever beam and obtained the critical load of the lateral-torsional buckling [9]. [PERSON] and [PERSON] [10; 11] carried out LTB experiments on four types of I-shaped cantilever steel beams with different cross-sections. [PERSON] et al. [12] extended the domain of application of \(C_{1}\), \(C_{2}\) and \(C_{3}\) factors to cantilevers. In addition, [PERSON] evaluated the performance of a 1D model of elastic LTB behavior [13]. Since then, many scholars have calculated \(M_{\rm cr}\)'s formula of cantilever beams with different sections under a large number of different loads. [PERSON] et al. [14] proposed a set of formulae to calculate \(M_{\rm cr}\) under concentrated load at the free end or uniformly distributed load. [PERSON] et al.