## 4 Applications
### Nine-Bar Truss
\begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline \multirow{2}{*}{**Uncertainty Level**} & \multirow{2}{*}{**Bounds**} & \multicolumn{1}{c}{**SM**} & \multicolumn{3}{c}{**STM**} & \multicolumn{3}{c}{**SDAM**} & \multicolumn{3}{c}{**PM (\([\varepsilon]\) = 1\%)**} \\ \cline{3-8}  & & **Value** & **Value** & & & **Value** & **Value** & **(mm)** & **Error** \\  & & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline \multirow{2}{*}{10\%} & LB & \(-0.453\) & \(-0.404\) & \(10.91\%\) & \(-0.408\) & \(20.00\%\) & \(-0.452\) & \(0.28\%\) \\  & UB & \(0.453\) & \(0.404\) & \(-10.91\%\) & \(0.408\) & \(-20.00\%\) & \(0.453\) & \(0.00\%\) \\  & LB & \(-1.052\) & \(-0.808\) & \(23.21\%\) & \(-0.842\) & \(20.00\%\) & \(-1.049\) & \(0.26\%\) \\  & UB & \(1.052\) & \(0.808\) & \(-23.21\%\) & \(0.842\) & \(-20.00\%\) & \(1.052\) & \(0.00\%\) \\  & LB & \(-1.903\) & \(-1.212\) & \(36.31\%\) & \(-1.332\) & \(30.00\%\) & \(-1.900\) & \(0.18\%\) \\  & UB & \(1.903\) & \(1.212\) & \(-36.31\%\) & \(1.332\) & \(-30.00\%\) & \(1.903\) & \(0.00\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Bounds and errors of \(u_{2}\) for the three cases. By analyzing the results obtained from different methods, the following conclusions can be drawn:  
1. As can be seen from Table 4, the calculation time for the PM is much less than that of VM, despite the fact that the permissible error for the PM is as small as 1%. For VM, the equilibrium equation needs to be solved for all the possible combinations of intervals. Thus, the computational cost is exponentially related to the number of interval parameters. Such large computational costs are unacceptable, especially for structures containing many interval parameters. However, for the PM, the computational cost is much less than that of VM because the equilibrium equations only need to be solved at specific points. Therefore, the PM can be applied to interval finite element analysis with a large number of uncertain parameters and retains sufficient computational efficiency. 2. As expected, the errors for these interval analysis methods in Figure 7 all increase as the level of uncertainty rises. However, the calculations show that the displacement response bounds solved by the PM are more accurate than the other two methods. When the uncertainty level reaches 30%, the maximum errors for SDAM and STM reach 42.43% and 36.31%, respectively, while the maximum error for the PM is only 0.28%. Therefore, the PM is suitable for analyzing the displacement response of structures under high levels of uncertainty. 3. For all three cases, the iteration error gradually decreases. Until the iteration error decreases below the permissible error \([\epsilon]\), suitable computational results are obtained. Case 1 and case 2 require only five and six iterations, respectively, whereas case 3 requires seven iterations. This suggests that the number of iterations of the PM gradually increases as the uncertainty of the structure rises. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multirow{2}{*}{**VM**} & \multirow{2}{*}{**SDAM**} & \multirow{2}{*}{**STM**} & \multicolumn{2}{c}{**PM**} & \\ \cline{5-6}  & & & & \([\epsilon]\) = 1\% & \([\epsilon]\) = 5\% & \([\epsilon]\) = 1\% \\ \hline Calculation time(s) & 72.805 & 0.038 & 0.020 & 0.154 & 0.513 & 5.547 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The calculation times for the different methods at the 30% uncertainty level. Figure 7: Bounds of nodal displacements of the nine-bar truss obtained using the different methods: (a) \(x\)-displacement \(u_{2}\); (b) \(y\)-displacement \(v_{4}\).