## 5 Numerical Examples
### The 47-Bar Planar Truss  
The sizing and layout optimization of a 47-bar planar truss tower (Figure 18) aimed to simulate scenarios involving power lines and their potential failures. Mass density and Young's modulus of elasticity are assigned as 0.3 lb/in\({}^{3}\) and 3 \(\times\) 10\({}^{4}\) ksi. The tower is subjected to three independent loading conditions, which are as follows. Figure 16: The optimized shapes of the 25-bar truss by NCSBO with a color map of stress in members: (**a**) 3D view, (**b**) top view, (**c**) XZ view, and (**d**) YZ view. Figure 17: Comparison of convergence curves for the 25-bar truss. Figure 18: The 47-bar planar truss tower. In the first case, a concentrated force of 6000 lbf in the positive X-direction and a concentrated force of 14,000 lbf in the negative Y-direction are applied at nodes 17 and 22. In the second loading condition, a concentrated force of 6000 lbf in the positive X-direction and a concentrated force of 14,000 lbf in the negative Y-direction is applied only at node 17. The third condition provides a concentrated force of 6000 lbf in the positive X-direction and a concentrated force of 14,000 lbf in the negative Y-direction at node 22 only. The cross-sectional areas of the tower's members were grouped into 27 sizing design variables, ranging from 0.1 to 5.0 in 2, which allowed for a comprehensive exploration of design possibilities. Additionally, 17 layout variables were considered, governing the positioning of nodes within the truss tower structure. These variables were intricately linked, ensuring the coherence and stability of the tower design. \[ \begin{array}{ll}&\text{A}_{3}=\text{A}_{1};\text{A}_{4}=\text{A}_{2};\text{ A}_{5}=\text{A}_{6};\text{A}_{7};\text{A}_{8}=\text{A}_{9};\text{A}_{10};\text{A}_{12}= \text{A}_{11};\text{A}_{14}=\text{A}_{13};\\ &\text{A}_{15}=\text{A}_{16};\text{A}_{18}=\text{A}_{17};\text{A}_{20}=\text{ A}_{19};\text{A}_{22}=\text{A}_{21};\text{A}_{24}=\text{A}_{23};\text{A}_{26}=\text{A}_{25}; \\ &\text{A}_{27};\text{A}_{28};\text{A}_{30}=\text{A}_{29};\text{A}_{31}=\text{ A}_{32};\text{A}_{33};\text{A}_{35}=\text{A}_{34};\text{A}_{36}=\text{A}_{37};\text{A}_{38}; \\ &\text{A}_{40}=\text{A}_{39};\text{A}_{41}=\text{A}_{42};\text{A}_{43};\text{ A}_{45}=\text{A}_{44};\text{A}_{46}=\text{A}_{47}\\ &\text{x}_{2}=-\text{x}_{1};\text{x}_{4}=-\text{x}_{3};\text{y}_{4}=\text{y}_ {3};\text{x}_{6}=-\text{x}_{5};\text{y}_{6}=\text{y}_{5};\text{x}_{8}=-\text{ x}_{7};\text{y}_{8}=\text{y}_{7};\\ &\text{x}_{10}=-\text{x}_{9};\text{y}_{10}=\text{y}_{9};\text{x}_{12}=-\text{ x}_{11};\text{y}_{12}=\text{y}_{11};\text{x}_{14}=-\text{x}_{13};\text{y}_{14}= \text{y}_{13};\\ &\text{x}_{20}=-\text{x}_{19};\text{y}_{20}=\text{y}_{19};\text{x}_{21}=-\text{ x}_{18};\text{y}_{21}=\text{y}_{18}\\ \end{array} \]  
Critical constraints were imposed to ensure the tower's structural integrity under various loading conditions. Tension and compression stress limits were set at 20 ksi and 15 ksi, respectively. Moreover, the Euler buckling strength of each member was accounted for, with the buckling strength defined as 3.96 \(\times\) EA/L2. The optimization of the 47-bar truss structure is particularly challenging due to multiple loading conditions and a high number of design variables, making it significantly more complicated than previous benchmark examples. A review of the existing literature, summarized in Table 8, reveals that NCSBO produced the second lightest design after SCPSO, weighing 1864.7753 lb compared to 1864.10 lb. However, NCSBO achieved an optimal design weighing 1864.7753 lb after 7580 analyses, while SCPSO found a slightly lighter design at 1864.10 lb (0.036% lighter) after 25,000 analyses. While D-ICDE results in a lower weight of 1744.80 lb, it fails to meet stress constraints. Similarly, the results yielded by the ABC and BBM violate stress constraints, as indicated in Table 8. Therefore, NCSBO demonstrates superiority among almost all optimization methods reported in Table 8 in achieving feasible and optimal outcomes for this test structure.