## 5 Benchmark Numerical Example
### Initial Optimization Results
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Stiffener** & **Type** & **Top** & **Position** & **Both** & **No** \\ \hline \multirow{4}{*}{ \begin{tabular}{c} End-plate rib \\ stiffener \\ \end{tabular}  } & \multirow{4}{*}{Extended} & \  
The evolution of fitness values is illustrated in Figure 10. It can be observed that the minimum fitness values begin to stabilize after the seventh generation, a trend that is similarly reflected in the average fitness values. It is also evident that the best-performing configurations in the third, fourth, and fifth generations slightly exceeded the allowable joint rotation limit. As a result, their fitness values include the penalty term \(p(\theta)\) in addition to the objective function \(f(V)\). Nevertheless, the final solution satisfies all structural criteria, yielding zero penalty function values while simultaneously minimizing the connection volume. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Minimum Thickness** & **Maximum Thickness** & **Step Size** \\  & \(t_{ep,min}\) **(mm)** & \(t_{ep,max}\) **(mm)** & **(mm)** \\ \hline Flush & 18 & 25 & 1 \\ Extended & 18 & 25 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: End-plate thickness options defining a subset of the design space. Figure 10: Fitness evolution during the optimization process: (**a**) minimum fitness value in each generation and (**b**) average fitness value in each generation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Cl** & **Bolt Configuration** & **Bolt Position** & **Diameter of** \\  & **Cl** & **Cl2** & **Cl3** & **\(b\) (mm)** & **\(a\) (mm)** & **Bolt \(d_{b}\) (mm)** \\ \hline Flush & 16 & 17 & 18 & 25 & 1 \\ Extended & 18 & 25 & 11 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: End-plate thickness options defining a subset of the design space. Notably, compared to other relevant and previously validated connection configurations capable of sustaining the load level of \(F=221.56\) kN used during the design, the final optimized solution demonstrates the best performance with the lowest fitness value, as shown in Table 8. It should be noted that among the validated models, configuration SC4 achieved the lowest volume ratio; however, it exceeded the allowable joint rotation limit of 0.025 rad at the specified load level. In contrast, the optimized solution results in a slightly higher volume but satisfies all structural criteria, yielding the most favorable compliant design with a 10.4% improvement in structural efficiency compared to SC4. Additionally, when compared to SC2--the best-performing configuration that did not trigger penalization--the optimized solution achieves a 39.3% reduction in volume while maintaining full compliance with all design constraints. In addition to the previous comparison, Table 9 presents the equivalent plastic strain values for the key structural components of both the validated and optimized connection configurations. It is worth noting that compared to SC3, which exhibited the highest level of inelastic deformations among the validated models, the optimized solution achieved an approximate 53.6% reduction in plastic strain. These results further confirm that the proposed optimization framework effectively ensures structural integrity while simultaneously reducing the material volume of the connection. Finally, the moment-rotation relationships are illustrated in Figure 11. It can be seen that the optimized configuration satisfies the maximum joint rotation criterion, as the joint rotation at the predefined load level remains within the limit, with a value of \(\theta=0.024\) rad. Overall, the proposed method facilitates the development of a design that meets structural safety requirements while enhancing material efficiency. Figure 11: Comparison of the moment–rotation response between the optimized and validated connection configurations.