## 4 Conclusions  
The lateral-torsional buckling and shear capacity of plate girders are significant factors in the design of plate girders. The buckling of plate girders used during the construction stage of the replacement project of the 102 Avenue over Groot Road Bridge in Edmonton, Alberta, Canada (Figure 1) is a recent example of the impact of these structural failure modes on infrastructures. In addition to satisfy the serviceability requirements, structures should also be designed in an economical way. In this study, it was demonstrated that by adequately dimensioning the cross-section of a girder beam, its cost can be significantly reduced without compromising the structural performance. The harmony search algorithm was utilised to optimise the cross-sectional dimensions under two different sets of slenderness constraints. In the first set, the flange slenderness was kept in the noncompact range, while the web was allowed to be slender. In the second set, the web slenderness was constrained in the noncompact range. In addition to satisfying certain slenderness requirements, the cross-sections were constrained to have buckling moments and ultimate shear stresses at least as high as the moment and ultimate shear stress capacity of the specimen tested by [PERSON] et al. [14].  
The optimisation under the first set of slenderness constraints resulted in a cross-sectional area corresponding to 63% of the cross-sectional area of the experimental specimen. Despite this favourable outcome, the 1-mm-thick web of this configuration was not practical. The following optimisation attempt with the second set of slenderness constraints resulted in a cross-sectional area approximately 84% of the cross-sectional area of the experimental specimen. These two outcomes show that a suitably optimised cross-section can lead to great savings in material and costs. However, without a rigorous optimisation procedure, it is difficult to design a cross-section with suitable dimensions; it was observed that either increasing or decreasing the slenderness of the web can lead to a better design. Further research should be conducted to better understand the effects of the flange and web slenderness on the structural performance. Furthermore, additional constraints should be placed on the design variables to satisfy the requirements for the span-to-depth ratio.  
Figure 12: The variation of the ultimate shear stress with respect to the angle of the panel diagonal.  
**Author Contributions:** Methodology, G.B.; formal analysis (coding), C.C.; writing--original draft preparation, C.C. and G.B.; writing--review & editing, S.K. and Z.W.G.; visualisation, C.C.; supervision, G.B., S.K. and Z.W.G.; funding acquisition, Z.W.G. All authors have read and approved the final manuscript.  
**Funding:** This work was supported by the National Research Foundation of Korea (NRF) through a grant funded by the Korea government (MSIT) (2020R1A2C1A01011131) and by the Energy Cloud R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT (2019M3F2A1073164).  
**Conflicts of Interest:** The authors declare no conflict of interest.  
Appendix A Torsional section properties of a doubly symmetric I-section [49]. St. Venant's torsion constant:  
\[J=\frac{2b\mu_{f}^{3}+\left(D+t_{f}\right)t_{w}^{3}}{3}\]  
Warping constant:  
\[C_{w}=\frac{\left(D+t_{f}\right)^{2}b_{f}^{2}t_{f}}{24}\]