## 2 Method  
The optimisation of the I-girder involves the minimisation of the cross-sectional area, while maintaining the critical lateral-torsional buckling load \(M_{cr}\) and the ultimate post-buckling shear stress \(\tau_{u}\) above certain predetermined values. While, in practice, plate girders are often designed by trial and error, the application of optimisation techniques can lead to better structural performance and more economical designs. The parameters that can be varied in the process of optimisation are the flange thickness \(t_{f}\), flange width \(b_{f}\), web plate thickness \(t_{w}\), depth of the web plate \(D\), and unbraced length between transverse stiffeners \(a\). To start this process, certain constraints need to be placed on the design parameters and the load-carrying capacity of the structure. These constraints ensure that the critical lateral-torsional buckling load and the ultimate post-buckling stress obtained from the optimisation process are greater than certain threshold values. In this study, these threshold values are chosen to be the values of the bending moment capacity and the ultimate shear stress of the girder beam tested by [PERSON] et al. [14]. The main finding of the tests carried out by [PERSON] et al. [14] was that the ultimate shear stress equations of Eurocode 3 are not conservative for certain ranges of web plate slenderness. In the next step, the design parameters for which constraints are necessary are identified. One of these design parameters is the slenderness of the web plate, which has a major impact on the ultimate post-buckling shear strength of the girder plate. The effect of this parameter on the ultimate post-buckling shear resistance of plate girders has been studied extensively both experimentally and numerically [23; 24; 25]. According to their slenderness ratio, plate girders are classified as compact, noncompact, or slender; note that the slenderness of both the web and flange needs to be considered [26]. Compact sections are defined as sections that allow the full development of the plastic moment prior to the local buckling of the flange or the web, whereas the local buckling of a non-compact section can occur before the full development of the plastic moment. Slender sections are sections that would fail due to local buckling before reaching the yield stress. Figure 4 shows the classification of the beam sections according to their slenderness, together with the variation of the corresponding nominal moment. In this study, the girder beam section is optimised under different slenderness constraints. In the first optimisation attempt, the slenderness of the section is chosen according to the girder tested by [PERSON] et al. [14]. To apply this constraint, the limits of the slenderness as per AISC [19] are used. The slenderness limits for the flange and web are, respectively,  
\[\lambda_{f}=\frac{b_{f}}{2t_{f}}\leq\sqrt{\frac{E}{\sigma_{y}}}, \tag{7}\]  
\[\lambda_{w}=\frac{D}{t_{w}}\leq 5.7\sqrt{\frac{E}{\sigma_{y}}}. \tag{8}\]  
A section is classified as slender if any one of these limits is violated. According to these slenderness limits, the girder used as a reference in this study has a slender web, but its flange is not slender. In the second optimisation attempt, both the flange and the web are constrained to be noncompact. Furthermore, the stiffener spacing is constrained to be not less than 750 mm, which is equal to the stiffener spacing considered in the experimental study of [PERSON] et al. [14]. Further constraints are imposed on the lateral-torsional buckling load and the ultimate shear stress capacity of the section. Based on the experimental studies of the girder plates conducted by [PERSON] et al. [14], the \(M_{cr}\) and \(\tau_{u}\) values of the section are not allowed to be less than \(11.83\times 10^{10}\) Nmm and 64 MPa, respectively. Under these conditions, the best and most optimised combination of the flange and web dimensions is obtained through the harmony search algorithm.