# Estimation of Structural Deformed Configuration for Bridges Using Multi-Response Measurement Data
## 4 Validation of Results
### Effect of Shape Function Type
#### 4.1.1 Beam Model  
Figures 7 and 8 show examples of the MSF and SSF for the four-span beam model. The total number of used MSFs is six, and the total number of used SSFs is 56. The SSF is derived by applying a unit load on all nodes except the boundary points. The response data of displacement, slope, and strain were used individually to analyze the effect of the shape function type on the estimation result for each response. The number of response data is 1-6 for the displacement and slope, and 1-5 for the slope. In addition, the sensor location, as shown in Table 2, is arbitrarily selected as the point at which the largest response can occur.  
Figure 6: Load conditions for real deformed shape assumed by FEM model (truss): (**a**) RDS1, (**b**) RDS2, (**c**) RDS3, (**d**) RDS4.  
The estimation error for EDS of RDS1-4 using each response data is shown in Figure 9. However, because the trend of the results according to MSF and SSF does not appear clearly, the number of sensors required to estimate the deformed shape within a 5% error is additionally shown in Figure 10. In Figure 10, the transparent bar indicates that more than six sensors are required to secure an NMAPE within 5%. In the case of using displacement  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Measured Data**} & \multicolumn{5}{c}{**Number of Measured Data (\(N_{md}\))**} \\ \cline{2-7}  & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline Displacement (Node) & 8 & 54 & 23 & 39 & 11 & 51 \\ Slope (Node) & 1 & 61 & 31 & 16 & 46 & - \\ Strain (Element) & 8 & 54 & 23 & 38 & 16 & 45 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Sensor location in the FEM model used for estimation (beam).  
Figure 8: Examples of structural shape function (SSF) derived by static analysis (beam): (**a**) SSF4, (**b**) SSF22, (**c**) SSF37, (**d**) SSF46.  
Figure 7: Examples of mode shape function (MSF) derived by frequency analysis (beam): (**a**) MSF1, (**b**) MSF2, (**c**) MSF3, (**d**) MSF4.  
data, there is a difference in the accuracy trend according to the number of sensors, but the overall results show that SSF is more useful for RDS1, RDS3, and RDS4. When slope data were used, the estimation results corresponding to the MSF were found to be more accurate for all RDSs except RDS3. In Figures 9c and 10c, which represent the estimation results obtained using strain data, the SSF is generally useful for all RDSs except RDS2. As a result of the deformed shape estimation according to the shape function type for the beam model, the specific shape function type is not always useful for estimation. However, when MSF is used, an additional process for calculating the optimal number of MSFs is required to ensure an appropriate estimation accuracy, because the high-order mode shapes are considerably different from the deformed shape that can be caused by the load. Therefore, the SSF is more useful for estimating the deformed shape of the beam model in general owing to its convenience of use.  
Figure 9: Normalized mean absolute percent error (NMAPE) by MSF and SSF according to the number of sensors: (**a**) displacement, (**b**) slope, (**c**) strain.