## 2 Background and Previous Research  
[PERSON] and [PERSON] [29] provided the equation to calculate the elastic critical LTB for a doubly symmetric I-beam under equal end moments as follows:  
\[M_{ocr}=\frac{\pi}{L}\sqrt{EI_{y}GJ+\left(\frac{\pi E}{L}\right)^{2}I_{y}C_{w}} \tag{1}\]  
where \(L\) is the unbraced length; \(E\) and \(G\) are the Young's modulus and the shear modulus, respectively; \(I_{y}\) is the inertia moment of the cross-section about the minor axis; \(J\) and \(C_{w}\) are the torsional and warping constants, respectively.  
In order to take into account the influence of varying bending moments within the laterally unbraced length, a moment gradient factor (\(C_{b}\)) was proposed. Moreover, the elastic critical LTB strength of beams subjected to general loading conditions (\(M_{cr}\)) is also calculated by multiplying \(M_{ocr}\) by \(C_{b}\) based on the study of [PERSON] [30]. The author offered a moment modification factor that has been also incorporated into the American Institute of Steel Construction (AiSC) Specifications (1986) [5]:  
\[C_{b}=1.75+1.05\bigg{(}\frac{M_{2}}{M_{1}}\bigg{)}+0.3\bigg{(}\frac{M_{2}}{M _{1}}\bigg{)}^{2} \tag{2}\]  
in which \(M_{1}\) and \(M_{2}\) are the greater and smaller end moments, respectively. In addition, the value of the ratio \(M_{2}\)/\(M_{1}\) is taken as negative for end moments triggering single-curvature bending and positive for end moments causing double-curvature bending. It should be noted that Equation (2) was developed for the case of beams subjected to end moments without transverse loadings.  
The moment gradient factor was established by [PERSON] [12], which has been also incorporated into the AISC Specifications (2016) [1]:  
\[C_{b}=\frac{12.5M_{max}}{2.5M_{max}+3M_{A}+4M_{B}+3M_{c}} \tag{3}\]  
where \(M_{max}\) is the absolute value of the maximum moment; \(M_{A}\), \(M_{B}\) and \(M_{C}\) are the absolute values of the moments at the quarter point, the center point and the three-quarter point, respectively. It should be mentioned that this factor can be applied to bending moment diagrams of any shape. Moreover, the loads are imposed at the shear center of the cross-section.  
BS 5950 [4] provides the equation for the moment correction factor given by  
\[C_{b}=\frac{M_{max}}{0.2M_{max}+0.15M_{A}+0.5M_{B}+0.15M_{c}}\leq 2.273 \tag{4}\]  
The equation of \(C_{b}\) used in the AS 4100 [2] is taken as  
\[C_{b}=\frac{1.7M_{max}}{\sqrt{M_{A}^{2}+M_{B}^{2}+M_{C}^{2}}}\leq 2.5 \tag{5}\]  
The position of the transverse loading in the cross-sectional depth also influences the elastic critical LTB strength of the beam. [PERSON] et al. [17] suggested a simplified formula for the moment gradient factor of simply supported beams, as presented in Equation (6), which has been also mentioned in [PERSON] [6]:  
\[C_{b}^{*}=(1.4^{2y/h})C_{b} \tag{6}\]  
where \(C_{b}\) is determined using Equation (3); \(h\) is the beam height; \(y\) is the distance from the midheight of the cross-section to the transverse loading position. Furthermore, the value of \(y\) is negative for transverse loading above the midheight and positive for transverse loading below the midheight.  
Moreover, there are also studies on the moment gradient factor for the elastic critical LTB of steel beams, which were performed by [PERSON] et al. [14] and [PERSON] [15]. In their studies, the factor \(C_{b}\) was also used only for the case of transverse loadings imposed at the shear center of the cross-section. For this study, the factor \(C_{b}\) for steel beams subjected to transverse loadings on top flanges with negative end moments is followed, according to the recommendations from the investigation of [PERSON] et al. [17] and [PERSON] [6].