## 1 Introduction  
Overhanging beams and continuous beams are common structures in buildings and bridges, which are directly subjected to transverse loadings, resulting in negative moments occurring at interior supports, as shown in Figure 1. For flexural steel members, the loss of stability of lateral-torsional buckling (LTB) is a common failure phenomenon. Moreover, elastic moment determination to predict the buckling strength values of steel beams must be performed and checked as one ultimate limit state considered to be one of the critical standards, which can be found in several current design standards and guidelines [1, 2, 3, 4, 5, 6]. However, so far, many researchers have proven that the recommendations of some standards and guidelines are too conservative or even unsafe under specified conditions, which has to be investigated in detail under particular circumstances to ensure the economic and safety indicators. Based on many previous studies, a buckling resistance evaluation greatly depends on two factors, bending moment distribution and restriction at supports, which have been widely numerically investigated by many researchers under several specified investigated factors. Theoretical studies on the elastic critical buckling of steel beams have been carried out in many works in the literature, wherein studies on the elastic buckling of simply supported steel beams with a monosymmetric cross-section subjected to transverse loadings or end moments were presented in [7, 8, 9]. Here, [PERSON] and [PERSON] [7], [PERSON] et al. [8]and [PERSON] and [PERSON] [9] revealed that the proposed design graphs are different from current design solutions in which the moment gradient factor (C\({}_{b}\)) is employed. In addition, [PERSON] et al. [10] also investigated the elastic lateral-torsional buckling utilizing an I-beam and applying the Bubnov-Galerkin method [11] and the finite element method. The authors developed alternative equations concerning the C\({}_{b}\) correction factor and the end constraint boundary conditions. As a result, in three end restraint cases, more precise solutions were attained with greater safety and less conservative values over the entire selection of linear moment gradients in the three above-mentioned cases. In addition, studies on beams subjected to simultaneous transverse loadings and end moments were also carried out in [12; 13; 14; 15]. As a result, the moment gradient factors based on the bending diagram were proposed. The equivalent uniform moment factor (EUMF) method was investigated by [PERSON] et al. [14]. The results indicated, in the case of simply supported beams with very conservative values, additionally, non-conservative values occurred when support types were used to limit lateral bending and warping. Furthermore, the investigation of [PERSON] et al. [15] also revealed unacceptable results for a procedure from a design standard [16], for most cases taking into account common bending moment distributions considering the moment gradient factor values observed from twelve moment distribution comparisons. Moreover, a proposed equation was proven to be appropriate for most cases of typical applications however, the equation does not show good agreement when concentrated loads occur, as with all quarter-point moment methods. In terms of the impact of the loading position in cross-sectional depth on the elastic critical buckling of the beam, it has been also studied in [17; 18; 19; 20]. Generally, these studies have considered the shape of the cross-section, including singly and doubly symmetric cross-sections. The effect of loading conditions, including beams subjected to end moments and transverse loadings, as well as the effect of load heights on the elastic critical LTB were also taken into account. The study of [PERSON] et al. [19] compared the results from the SSRC Guide [6] with the results of the FEA method, leading to a new design method restricted in cases of monosymmetric I-beams with a monosymmetric degree ranging between 0.1 and 0.9.