# Efficient Sizing and Layout Optimization of Truss Benchmark Structures Using ISRES Algorithm
## 4 Benchmark Problems
### 72-Bar Truss Structure  
The schematic in Figure 15 shows 72-bar benchmark spatial truss structure. The structure consists of 16 independent design variables (as seen in Table 5) which are grouped together due to its structural symmetry. It is considered to be made out of material with a mass density of 0.1 lb./in\({}^{3}\) and an elastic modulus of 10,000 ksi. Under both of the loading conditions, the truss tower has 320 non-linear design constraints (72 tension/compression and 16 displacement). The maximum design variable (Amax) is set to 4.0 in\({}^{2}\) and the minimum design variable (Amin) is set to 0.1 in\({}^{2}\) for case 1 and 0.01 in\({}^{2}\) for case 2. The displacement in the topmost nodes (17, 18, 19, and 20) are limited to \(\pm\)0.25 in in the x and y direction, and the normal member stresses are limited to 25 ksi for both tension and compression, as given in Table 6. Multiple runs of the ISRES algorithm are conducted to generate diverse candidate solutions. The best-performing solutions are then selected based on the objective function, aiming to minimize the truss weight while satisfying all the imposed constraints. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Element Group** & **Members** & **Element Group** & **Members** \\ \hline A\({}_{1}\) & 1,2,3,4 & A\({}_{9}\) & 37,38,39,40 \\ A\({}_{2}\) & 5,6,7,8,9,10,11,12 & A\({}_{10}\) & 41,42,43,44,45,46,47,48 \\ A\({}_{3}\) & 13,14,1  
For comprehensive evaluation, the results obtained from the ISRES algorithm are compared with solutions achieved through various other optimization methods reported in the literature. The comparison encompasses a wide range of metaheuristic algorithms, gradient-based techniques, and other state-of-the-art approaches. The comparison considers the quality of the solutions and the computational efficiency of each method. Furthermore, the optimized truss designs undergo a rigorous finite element analysis to validate their structural integrity and stability. The structural analysis evaluates stress distributions, member sizes, and overall performance under both loading conditions. By conducting the comparison and verification, the study contributes valuable insights into the effectiveness of the ISRES algorithm in handling complex spatial truss-optimization problems. The findings inform engineers and designers of the algorithm's applicability and performance, particularly in challenging structural design scenarios. In conclusion, the study successfully demonstrates the efficacy of the ISRES algorithm in optimizing the sizing and layout of the 72-bar benchmark spatial truss structure. The \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Node** & \multicolumn{3}{c}{**Condition I**} & \multicolumn{3}{c}{**Condition II**} \\ \cline{2-7}  & \(\mathbf{P_{x}}\) & \(\mathbf{P_{y}}\) & \(\mathbf{P_{z}}\) & \(\mathbf{P_{x}}\) & \(\mathbf{P_{y}}\) & \(\mathbf{P_{z}}\) \\ \hline
17 & 5.0 & 5.0 & \(-\)5.0 & 0 & 0 & \(-\)5.0 \\
18 & 0 & 0 & 0 & 0 & 0 & \(-\)5.0 \\
19 & 0 & 0 & 0 & 0 & 0 & \(-\)5.0 \\
20 & 0 & 0 & 0 & 0 & 0 & \(-\)5.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Loading conditions for the 72-bar truss structure. Figure 15: Schematic of the 72-bar structure. comparison with other optimization methods and the structural analysis validation bolster the confidence in the optimized truss designs. These results have significant implications for engineering and architectural applications, guiding practitioners in selecting appropriate optimization strategies for complex spatial truss-design challenges. The research advances the field of optimization techniques in structural engineering, enabling the creation of efficient and reliable truss designs for a diverse range of real-world applications. Figure 16 provides a comprehensive representation of the allowable displacements for every element in the 72-bar structure, along with the allowable limits for each element, for both load case 1 and load case 2.