# Truss Structure Optimization with Subset Simulation and Augmented Lagrangian Multiplier Method
## 4 Test Problems and Optimization Results
### Spatial 72-Bar Truss Structure  
The third test example regards the optimal design of the spatial 72-bar truss structure shown in Figure 6. The modulus of elasticity and material density are the same as in the previous test cases. The 72 members are organized into 16 groups, as follows: (1) \(A_{1}\)-\(A_{4}\), (2) \(A_{5}\)-\(A_{12}\), (3) \(A_{13}\)-\(A_{16}\), (4) \(A_{17}\)-\(A_{18}\), (5) \(A_{19}\)-\(A_{22}\), (6) \(A_{23}\)-\(A_{30}\), (7) \(A_{31}\)-\(A_{34}\), (8) \(A_{35}\)-\(A_{36}\), (9) \(A_{37}\)-\(A_{40}\), (10) \(A_{41}\)-\(A_{48}\), (11) \(A_{49}\)-\(A_{52}\), (12) \(A_{53}\)-\(A_{54}\), (13) \(A_{55}\)-\(A_{58}\), (14) \(A_{59}\)-\(A_{66}\), (15) \(A_{67}\)-\(A_{70}\), and (16) \(A_{71}\)-\(A_{72}\). The truss structure is subject to 160 constraints on stress and displacement. A displacement limit within \(\pm 0.25\) inch is imposed on the top nodes in both \(x\) and \(y\) directions, and all truss elements have an axial stress limit within \(\pm 25\) ksi. The truss is subject to the two independent loading conditions listed in Table 7. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \(N\) & **Best** & **Mean** & **Worst** & **SD** & **NSA** \\ \hline \multirow{4}{*}{ALSSO} & 100 & 545.1241 & 545.2569 & 545.7793 & 0.135161 & 27,009 (23,170) \\  & 200 & 545.1254 & 545.205 & 545.4292 & 0.067821 & 38,301 (32,580) \\  & 500 & 545.1057 & 545.185 & 545.2819 & 0.044924 & 86,490 (90,500) \\ \hline ABC-AP [36] & & 545.19 & N/A & 545.28 & N/A & 300,000 \\ SAHS [31] & & 545.12 & 545.94 & 546.6 & 0.91 & 9051 \\ TLBO [34] & & 545.09 & 545.41 & 546.33 & 0.42 & 15,318 \\ MSPSO [27] & & 545.172 & 546.03 & 548.78 & 0.8 & 10,800 \\ CA [37] & & 545.05 & 545.93 & N/A & 1.55 & 9380 \\ HPSSO [41] & & 545.164 & 545.556 & 546.99 & 0.432 & 13,326 \\ TLBO [35] & & 545.175 & 545.483 & N/A & 0.306 & 12,199 \\ FPA [38] & & 545.159 & 545.73 & N/A & 0.59 & 8149 \\ WEO [39] & & 545.166 & 545.226 & 545.592 & 0.083 & 19,750 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of statistical performance in the 25-bar truss problem. Figure 5: Convergence curve obtained for the 25-bar truss problem with \(N=500\). This truss problem has been previously studied using deterministic global methods [3], a GA-based method [9], HS [28], ACO [16], PSO [23], ALPSO [25], BB-BC [33], SAHS [27], TLBO [34], CA [37], and WEO [39]. Their best designs are compared against that obtained by the proposed method, and are shown in Table 8. The proposed method produced a new best design with a weight of 379.5922 lb. Among the deterministic global methods, the DIRECT-I algorithm found the best design for this test case (Table 8). However, the corresponding structural weight is larger than the structural weight obtained by the proposed method. It should be noted that the maximum displacements of GA, HS, ALPSO, BB-BC, and FPA exceed the \(x\)- and \(y\)-displacement limits on node 17. In particular, the optimum design found by HS also violates the compressive stress limits of bar members 55, 56, 57, and 58 under load case 2. The optimum design found by the proposed method satisfies both stress limits and displacement limits and is better than SAHS, TLBO, and CA. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Node**} & \multicolumn{3}{c}{**Case 1 (kips)**} & \multicolumn{3}{c}{**Case 2 (kips)**} \\ \cline{2-7}  & \(P_{x}\) & \(P_{y}\) & \(P_{z}\) & \(P_{x}\) & \(P_{y}\) & \(P_{z}\) \\ \hline
17 & 5.0 & 5.0 & \(-5.0\) & 0.0 & 0.0 & \(-5.0\) \\
18 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \(-5.0\) \\
19 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \(-5.0\) \\
20 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \(-5.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Loading conditions acting on the 72-bar truss. Figure 6: Schematic of the spatial 72-bar truss structure. Statistical data listed in Table 9 prove that the present algorithm was also very robust for this test problem. Figure 7 shows the convergence curve obtained for the last optimization run with \(N=500\).