# Estimation of Structural Deformed Configuration for Bridges Using Multi-Response Measurement Data
## 4 Validation of Results
### Effectiveness of Using Multi-Response Data
#### 4.3.1 Beam Model  
The multi-response sensors are placed in the order of displacement, slope, and strain using the EI-DPR-distance method. Due to the fact that there are too many combinations of sensor locations, the sensor locations for the case of using only two-response sensors are shown in Tables 6 and 7. For the beam mode, the slope and strain data are derived from the node and element, respectively. Examples of EDSs based on the combinations of sensors are shown in Figure 19. The red, blue, and green circles indicate the displacement, slope, and strain, respectively. As shown in Figure 19, the estimation accuracy using slope and strain data was more accurate than that using only displacement data.  
The estimation errors for each RDS according to the combination of response data are shown in Figures 20-23. The maximum estimation errors were 16.99% for RDS1, 19.89% for RDS2, 24.95% for RDS3, and 27.53% for RDS4. For all RDS, the estimation errors were reduced according to the additional number of slope and strain data. In addition, estimation accuracies using the slope and strain data were more improved than those using the slope and slope data, respectively. Particularly in RDS3, the estimation error was  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Displacement (Node)** & \multicolumn{6}{c}{**Number of Slope Data (Node)**} \\ \hline
**Number of Data** & **[**20**]** & \(\varnothing\)**1** & \(\varnothing\)**2** & \(\varnothing\)**3** & \(\varnothing\)**4** & \(\varnothing\)**5** & \(\varnothing\)**6** \\ \hline \(\omega\)1 & 8 &  
reduced by approximately 20%, with only one additional data point because the second sensor was properly placed in a location where the deformed shape could not be captured through the first sensor.  
Figure 19: Example of estimated deformed shape according to the sensor number and combination (RDS4): (**a**) number of data (\(\omega=2\)); (**b**) number of data (\(\omega=2\), \(\varnothing=4\)); (**c**) number of data (\(\omega=2\), \(\varepsilon=2\)); (**d**) number of data (\(\omega=2\), \(\varnothing=1\), \(\varepsilon=1\)).  
Figure 20: _Cont._  
Figure 21: _Cont._  
Figure 20: NMAPE according to the number and combination of sensors (RDS1): (**a**) displacement + slope, (**b**) displacement + strain, (**c**) displacement + slope + strain.  
Figure 21: NMAPE according to the number and combination of sensors (RDS2): (**a**) displacement + slope, (**b**) displacement + strain, (**c**) displacement + slope + strain.  
Figure 22: NMAPE according to the number and combination of sensors (RDS3): (**a**) displacement + slope, (**b**) displacement + strain, (**c**) displacement + slope + strain.  
Figure 23: NMAPE according to the number and combination of sensors (RDS4): (a) displacement + slope, (b) displacement + strain, (c) displacement + slope + strain.  
Figure 24: Cont.  
The average errors for the displacement data number are presented in Figure 22 to evaluate the effect of using multi-response data more clearly. For all RDS, the trend of improving accuracy owing to the additional slope and strain data can be clearly seen in Figure 24. In particular, in the case of using slope and strain data together, a reduced level of estimation error was the highest. Therefore, the algorithm developed in this study can be utilized to efficiently estimate the deformed shape of the beam model by comprehensively using data on the slope, strain, and displacement.