## 1 Introduction
Moreover, [PERSON] [20] studied the analysis of unbraced, symmetric, prismatic beams of simply supported ones in terms of the lateral-torsional critical moment, proving an incorrect estimation in a previous study [21] in terms of the coefficient allowing for random ordinates of applied loadings. Moreover, the study also investigated the assumption of boundary conditions, non-linear bending moment distributions and the impact of the position of loading concerning the height of the cross-sectional element. On the other hand, [PERSON] et al. [22] and [PERSON] [23] numerically investigated the effect of the length-to-height ratio on the LTB of stepped beams with a continuously laterally  
Figure 1: Bending moment diagram. (**a**) Continuous beam, (**b**) Overhanging beam. -based top flange. In the studies, the loading conditions on stepped beam models were a concentrated load at midspan and a uniformly distributed load on the top flange with negative end moments. As a result, the data of these studies showed that the length-to-height ratio has a great influence on the LTB of stepped beams with continuous lateral bracing. Recently, cellular beams have been investigated and widely applied, with several advantages compared to conventional solid web steel beams. The numerical investigation of their elastic buckling behavior has been considered by several researchers [24; 25; 26; 27; 28], with the modified factors considering the presence of web perforations under specified conditions. In this case, the study by [PERSON] et al. [28] investigated the impact of load height on the C\({}_{b}\) factor under uniform load distribution for laterally unsupported I-beams. The C\({}_{b}\) factor attained from the numerical results was compared with the procedure calculations from SSRC guidance, leading to considerable variation between these results, which was due to serious web distortion, which is neglected in the equations in the SSRC guidance. Moreover, the application of this guidance is appropriate when the mode of lateral-torsional buckling is dominant for span beams that are longer than the investigated one. To date, there have been many standards and guidelines [1; 2; 3; 4; 5; 6] aimed at preventing the lateral-torsional buckling phenomenon; however, as mentioned above, these recommendations are either too conservative or unsafe in some circumstances. Moreover, although the elastic critical buckling of steel beams has been focused on and researched with many proposed models to analyze the related ultimate limit stage under specific circumstances, to the best of our knowledge, the effect of the length-to-height ratio with transverse loads at the top flange on the elastic critical LTB of prismatic steel beams has been studied with limitations in the literature. Therefore, this paper focuses on the investigation of the influence of the length-to-height ratio on the elastic critical buckling of prismatic steel I-beams. The simply supported beams were subjected to a uniformly distributed load or concentrated load on the top flange with negative end moments. In addition, a finite element program (ABAQUS) was employed to analyze the elastic critical buckling of beam models. Eventually, an equation was proposed, compared and verified with several standards and a previous study. As a result, the proposed equation indicated good agreement, with less conservative values compared to the previous study and the standards for most cases.