## 2 Materials and Methods
### Optimization of the RC Cross-Section  
In the previous sections it has been shown that the majority of combinations designed to optimise the structural efficiency of the original geometry invariably result in a RC cross-section imbalance. This, in turn, compromises the structure's ability to withstand applied loads and increases the risk of obtaining failure modes where the reinforcement is in the elastic regime. In order to address this challenge, two optimisation approaches were explored, where the variation of the initial cross-section size was allowed. The first was aimed at minimising modifications to the cross-section, albeit with certain trade-offs. The second approach allowed the free resizing of the cross-section to meet the most demanding scenario. Considering the variable geometry of the cross-section, a minimum and maximum diameter will be defined for the smallest and largest cross-sections of the reef branch, respectively. In **approach 1**, a diameter of 196.5 mm was considered to the cross-section, corresponding to the largest cross-section found the initial design. The \(\mathrm{A_{s,min}}\) was then calculated, and an initial solution was determined (1\(\mathrm{\SIUnitSymbolO}\)). This resulted in a real steel area (\(\mathrm{A_{s,real}}\)), which served as a reference for establishing a maximum limit for the section diameter (\(\mathrm{D_{max}}\)). For the flexural analysis, which was critical when the smaller cross-section was considered, the minimum viable diameter (\(\mathrm{D_{min}}\)) was calculated to ensure that \(\mathrm{A_{s,pl}}\) (flexural) matched \(\mathrm{A_{s,real}}\) (axial). For each configuration, the flexural (\(\mathrm{M_{Pl,Rd}}\)) and axial (\(\mathrm{N_{pl,Rd}}\)) strengths were obtained. Figure 13 presents the concrete and reinforcement combinations that meet the constraints of this approach for the RC cross-section. The upper graph illustrates the changes in minimum (\(\mathrm{D_{min}}\)) and maximum (\(\mathrm{D_{max}}\)) cross-sectional diameters correlated with different reinforcement configurations and concrete strengths. The lower graph displays the corresponding \(\mathrm{M_{Pl,Rd}}\) and \(\mathrm{M_{Ed}}\). The plotting colour-code in the graphs distinguished the steel class. Figure 12: Shear strength (\(\mathrm{V_{Rd}}\)) versus concrete strength for cross-section types (**a**) RC and (**b**) CS. The results show that the alteration of the cross-section dimensions resulted in a significant improvement of M\({}_{\rm pl,Rd}\). Most of the material combinations exceeded the requirements of the load combination ULS(B)_Ewcm,mastorm, but none met the abnormal combination requirements, ALS_Ewcme. A common trend across most combinations is the need to increase the diameter of the initial cross-section to optimize performance. The analysis identifies combinations that result in minimal changes (\(-7\%\) to \(7\%\)) in the cross-sectional diameters, maintaining the original design's geometric effect. This reduction in material contributes to a more efficient and economical structural solution. The maximum value found for D\({}_{\rm min}\) was 208 mm, occurring in the C35/45 + DIN 8.8 combination. This represents a 38.7% increase compared to the initial section (150 mm). On the other hand, the minimum value found for D\({}_{\rm min}\) was 140.19 mm, obtained for the C90/105 + 4.8 combination. D\({}_{\rm max}\) varied between 196.5 mm, for the C60/75 + DIN 5.8  
Figure 13: Correlation between the minimum (D\({}_{\rm min}\)) and maximum (D\({}_{\rm max}\)) cross-section diameters and the flexural strength, M\({}_{\rm pl,Rd}\), as a function of concrete compressive strength, \(\ell_{\rm c}\), and reinforcement configuration (approach 1). combination, and 241.44 mm, for the C50/60 + S400 combination. The latter combination showed a 22.8% increase compared to the baseline design. In **approach 2**, a free optimization was carried out to find the smallest cross-section and structural combination that guarantees the most challenging scenario.