## 4 Applications
### Nine-Bar Truss  
A nine-bar 2D truss structure is shown in Figure 5. The height and span of the truss structure are 3 m and 6 m, respectively. The total number of elements is nine, and the total number of nodes is six. The horizontal and vertical displacements of node 1 and node 5 are constrained. The vertical loads shown in Figure 5 are applied to nodes 2, 4, and 6. In this example, the Young's modulus and cross-sectional area of each bar, as well as the load, are considered as independent interval parameters. Therefore, they are defined according to Equation (58):  
\[ \begin{split}& E_{i}^{I}=[70\big{(}1-\mu_{E_{i}}\big{)},70\big{(}1+ \mu_{E_{i}}\big{)}]\text{GPa},\\ & A_{i}^{I}=[300\big{(}1-\mu_{A_{i}}\big{)},300\big{(}1+\mu_{A_{i} }\big{)}]\text{mm}^{2},\\ & P^{I}=[5000\big{(}1-\mu_{P}\big{)},5000\big{(}1+\mu_{P}\big{)}] \text{N}\end{split}  \tag{58}\]  
where \(\mu_{E_{i}}\) and \(\mu_{A_{i}}\) are the uncertainty levels for [PERSON]'s modulus and the cross-sectional area of each rod (\(i=1,2,\cdots,9\)), and \(\mu_{P}\) is the uncertainty level of the load \(P\). Three cases are considered and the level of uncertainty in each case is shown in Table 1. Simulations of this nine-bar truss are carried out using MATLAB R2018b on a computer with a 2.90 GHz AMD Ryzen 7 CPU and 16.0 GB RAM. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Uncertainty Levels** & **Case1** & **Case2** & **Case3** \\ \hline \(\mu_{E_{i}},\mu_{A_{i}}(i=1,2,\cdots,9),\mu_{P}\) & 10\% & 20\% & 30\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Uncertainty levels of interval parameters for the nine-bar truss in three cases. Figure 5: Nine-bar truss structure. The levels of uncertainty for all interval variables in the first two cases are 10% and 20%, respectively. Then, the level of uncertainty is set to 30% in case 3 to illustrate the advantages of the proposed method when dealing with large levels of uncertainty. In each case, the upper and lower bounds of the displacement are solved using VM, SDAM, STM, and the PM. The results of VM are used as the reference solution. In order to investigate the influence of \([\varepsilon]\) on the accuracy and efficiency of the proposed method, different convergence values (\([\varepsilon]\)=1%, 5%, 1%) were considered for the calculations. The upper and lower bounds and errors of the node displacements (\(x\) displacement \(u_{2}\) for node 2 and \(y\) displacement \(v_{4}\) for node 4) obtained using different methods are presented in Tables 2 and 3. The calculation times for the different methods at the 30% uncertainty level are shown in Table 4. Iterative error curves for structural displacements at the 10%, 20%, and 30% uncertainty levels are given in Figure 6. \begin{table}
\begin{tabular}{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)** & **Error** \\ \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. Figure 6: Iterative error curves of overall structural displacements of the nine-bar truss for the three cases.