## 5 Numerical Examples
### The 25-Bar Planar Truss
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Design** & **SCPSO** & **GA** & **GA** & **MHS** & **FA** & **D-ICDE** & **ABC** & **MLA** & \multirow{2}{*}{**CSBO**} & **NCSBO** \\  & **Variables** & [56] & [65] & [57] & [60] & [58] & [61] & [62] & [63] & & \\ \hline
1 & A\({}_{1}\) (in\({}^{2}\)) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & **0.1** \\
2 & A\({}_{2}\) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & **0.1** \\
3 & A\({}_{3}\) & 1 & 1.1 & 1.1 & 1 & 1 & 0.9 & 1 & 1.0 & 1 & **1** \\
4 & A\({}_{4}\) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & **0.1** \\
5 & A\({}_{5}\) & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & **0.1** \\
6 & A\({}_{6}\) & 0.1 & 0.1 & 0.2 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & **0.1** \\
7 & A\({}_{7}\) & 0.1 & 0.1 & 0.2 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & **0.1** \\
8 & A\({}_{8}\) & 0.9 & 1 & 0.7 & 1 & 1 & 1.0 & 0.9 & 0.9 & 0.9 & **0.9** \\
9 & A\({}_{4}\) (in\({}^{2}\)) & 36.952 & 36.230 & 35.470 & 37.820 & 37.320 & 36.830 & 36.211 & 37.759 & 37.498 & 37.6768 \\
10 & Y\({}_{4}\) & 54.5786 & 58.560 & 60.370 & 55.485 & 55.740 & 58.530 & 54.637 & 54.625 & 54.672 & **54.50722** \\
11 & Z\({}_{4}\) & 129.9758 & 115.590 & 129.070 & 128.730 & 126.620 & 122.670 & 129.965 & 129.625 & 130 & **130** \\
12 & X\({}_{8}\) & 51.7317 & 46.460 & 45.060 & 52.068 & 50.140 & 49.210 & 52.074 & 51.927 & 51.880 & **51.87948** \\
13 & Y\({}_{8}\) & 139.5316 & 127.950 & 137.040 & 139.590 & 136.400 & 136.740 & 139.980 & 139.678 & 139.4944 & **139.5099** \\ \multicolumn{2}{c}{Best weight} & & & & & & & & & \\ (lb) & 117.227 & 124.0015 & 124.940 & 117.380 & 118.830 & 118.760 & 117.333 & 117.316 & 117.264 & **117.257** \\ \multicolumn{2}{c}{Mean weight} & & & & & & & & & & \\ (lb) & 122.876 & N/A & N/A & N/A & N/A & N/A & N/A & 123.728 & 119.702 & **117.336** \\ \multicolumn{2}{c}{Worst weight} & & & & & & & & & & & \\ (lb) & 132.672 & N/A & N/A & N/A & N/A & N/A & N/A & 127.482 & 124.665 & **117.647** \\ ODNSA & 4500 & N/A & 6000 & 6000 & 6000 & 6000 & 5100 & 30,000 & **5310** & **4830** \\ TNSA & 4500 & N/A & 6000 & 6000 & 6000 & 6000 & 18,000 & 30,000 & **18,000** & **18,000** \\ STD (lb) & 3.671 & N/A & N/A & N/A & N/A & N/A & 2.22 & 2.384 & 0.352 & **0.142** \\ CVP (\%) & 0.527 & None & 0.008 & None & None & 0.17 & None & None & None & None \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of optimized designs for the 25-bar truss. For the optimal designs obtained via NCSBO and CSBO, all stress constraints for structural members and displacement limits for nodes have been effectively satisfied, as illustrated in Figures 13 and 14, respectively. The color map in Figures 15 and 16, representing the optimum configuration with member stress capacity, shows all members in light blue, indicating a considerable margin below maximum capacity. Figure 14 demonstrates that the displacement constraints in the \(x\) and \(y\) directions, particularly at nodes 1 and 2, reach their limits at \(\pm\)0.349999 inches in both optimal designs. Examining these figures reveals that displacement constraints serve as the primary governing factor in the optimization process of the 25-bar truss. Figure 14: Comparison of nodal displacements at the optimized designs of the 25-bar truss by (**a**) CSBO and (**b**) NCSBO. Figure 15: _Cont._  
Figure 15: The optimized shapes of the 25-bar truss by CSBO with a color map of stress in members: (**a**) 3D view, (**b**) top view, (**c**) XZ view, and (**d**) YZ view. In the optimization of the 25-bar truss, the convergence curve of the proposed NCSBO is compared with that of CSBO, as shown in Figure 17. The final optimal solutions are achieved after 4830 and 5310 analyses for NCSBO and CSBO, respectively. It is evident from the figure that NCSBO exhibits faster convergence, particularly in the early stages of the optimization process. In both cases, the optimization process continued until the maximum number of structural analyses (18,000) was reached, even without convergence.