## 6 Conclusions  
In this paper, a comprehensive nonlinear optimization algorithm is proposed for the design of steel end-plate connections, in which the finite element method (FEM) is combined with a genetic algorithm (GA) to enable automated solution identification. The mathematical formulation of the optimization problem is presented, with the primary objective of achieving material efficiency by minimizing the volume of connection components while ensuring structural safety. To this end, three penalty functions are introduced to enforce key design constraints: limiting plastic deformation, controlling joint rotation, and maintaining structural stability. As part of the safety considerations, initial geometric \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Optimization Setup** & **1** & **Generation** & **10** & **15** \\ \hline OP1 & & & & \\ \(fitness=0.6005\) & \(fitness=0.5591\) & \(fitness=0.5313\) & - \\ OP2 & & & & \\ \(fitness=0.7679\) & \(fitness=0.5546\) & - & & \\ OP3 & & & & \\ \(fitness=0.9467\) & \(fitness=0.7636\) & \(fitness=0.7567\) & \(fitness=0.7567\) & \(fitness=0.7567\) \\ OP4 & & & & \\ \(fitness=0.6104\) & \(fitness=0.5990\) & \(fitness=0.5618\) & - \\ OP5 & & & & \\ \(fitness=0.7788\) & \(fitness=0.7153\) & \(fitness=0.6840\) & - \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Best solutions identified at selected generations for each optimization setup. imperfections--derived from the buckling mode shape of the investigated assemblies--are also incorporated into the analysis, with their corresponding mathematical formulations included within the optimization framework. The technical implementation is carried out using the PYTHON programming language in conjunction with the ABAQUS finite element software. The developed code systematically integrates the proposed optimization formulation and the genetic operators. The framework also includes a linear buckling analysis (LBA) to compute the critical buckling load factor and to generate imperfect geometries based on selected mode shapes. Subsequently, a geometrically and materially nonlinear analysis (GMNA) is performed to evaluate the plastic deformations and joint rotation, thereby establishing the objective function, represented by the fitness value, as a quantitative measure of the quality of each candidate solution. To investigate the operation and effectiveness of the proposed design algorithm, a benchmark numerical example involving a beam-to-column connection is presented. In this part of this study, eight connection configurations with varying parameters are validated against experimental results to ensure the reliability of the FE models. Subsequently, the optimization is carried out within the defined design space and under the prescribed constraint limits. The results of the optimization process are thoroughly analyzed and compared with the relevant validated connection configurations. The optimized solution achieves the best overall performance objective among all tested alternatives, demonstrating minimized material usage while fully satisfying all structural criteria. Specifically, the proposed approach leads to a 10.4% improvement in structural efficiency relative to the best-performing validated configuration, along with a 39.3% reduction in material volume and a 53.6% decrease in equivalent plastic strain when compared to the most efficient feasible benchmark and the configuration exhibiting the highest level of inelastic deformation, respectively. These findings highlight the effectiveness and practical potential of the proposed optimization algorithm in the context of steel connection design. In addition to the baseline optimization, a comprehensive sensitivity analysis is performed to assess the influence of key GA parameters. The results confirm that the most effective convergence behavior is observed in configurations that maintain a proper balance between exploration and exploitation--most notably in setup OP1.