# Estimation of Structural Deformed Configuration for Bridges Using Multi-Response Measurement Data
## 4 Validation of Results
### Sensor Placement Method
#### 4.2.2 Comparison of Results  
As previously mentioned, the estimation results obtained using the SSF are generally more accurate than those obtained using the MSF. Therefore, in this study, the EI-DPR-distance method that can be applied to SSFs with \(N_{sf}~{}>~{}N_{md}\) is proposed and compared with the EI and EI-DPR methods. Due to the fact that the EI and EI-DPR methods cannot be applied when all SSFs are used, the SSFs that included the maximum value for each response, which correspond to the number of sensors, were selected and applied. A truss model with a relatively complex geometric shape was used for comparison of the sensor placement method, and slope only (case 1) and strain with one displacement (case 2) were considered for the sensor combination. In case 2, the position of the displacement sensor was fixed at the 22 th node in Figure 4a. The positions of the sensors selected by each sensor placement method are shown in Tables 4 and 5, and the selection procedure of sensor locations for case 2 is shown in Figure 15. In Figure 15, the normalized \(E_{D}\) was used to compare the \(E_{D}\) value of each sensor placement method under the same conditions. In the EI and EI-DPR methods, the nodes with the smallest \(E_{D}\) are removed one by one for each iteration until the number of remaining nodes is the same as the number of sensors. In addition, the influence of the previously selected displacement sensor is not considered in the EI and EI-DPR methods. By contrast, in the EI-DPR-distance method proposed in this study, the node with the highest \(E_{D}\) is selected as the sensor location for each iteration, and the location of the next sensor is determined by recalculating the distance coefficient considering the recently selected sensor. Therefore, the EI-DPR-distance method is a more effective sensor placement method when using multi-response data because the influence of different response sensors previously selected can be considered. For cases 1 and 2, the shape estimation results according to the number of sensors are shown in Figures 16 and 17, respectively. Case 2, which additionally used one displacement data, generally showed more accurate results for all RDSs except RDS1. In case 1, the estimation error corresponding to the EI-DPR-distance method was the lowest for RDS4, and the estimation errors corresponding to all sensor placement methods were similar for RDS1-3. In case 2, the estimation errors corresponding to EI-DPR and EI-DPR-distance for all RDS were similar, but the estimation errors for EI depended on the RDS and number of sensors. In particular, for RDS2 and RDS3, the estimation errors increased despite the increase in the number of sensors. This phenomenon occurred for RDS4 in Case 1, even when EI-DPR was used. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Optimization Method**} & \multicolumn{5}{c}{**Number of Slope Measured Data (\(N_{md}\))**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline EI & 1 & 12 & 14 & 10 & 21 & 3 \\ EI-DPR & 1 & 12 & 13 & 11 & 22 & 2 \\ EI-DPR-distance & 1 & 12 & 17 & 7 & 18 & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Sensor location in the FEM model used for estimation (Case 1: slope only). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Optimization Method**} & \multicolumn{5}{c}{**Number of Strain Measured Data (\(N_{md}\))**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline EI & 46 & 36 & 32 & 3 & 42 & 10 \\ EI-DPR & 23 & 34 & 55 & 3 & 44 & 10 \\ EI-DPR-distance & 23 & 39 & 55 & 4 & 14 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Sensor location in the FEM model used for estimation (Case 2: strain with one displacement). Figure 16: _Cont._  
Figure 15: Procedure example of each sensor placement method (Case 2: strain with one displacement): (**a**) EI method, (**b**) EI-DPR method, (**c**) EI-DPR-distance method.