# Robust Static Structural System Identification Using Rotations
## 4 Using Redundant Rotations for Parameter Estimation
### Verification for a Simply Supported Bridge
#### 4.2.2 Case 2: Parameter Estimation for the Whole Structure  
In this section, the target parameters are the three bending stiffnesses \(EI_{1}\)-\(EI_{3}\). Two load cases were studied. In the load case 1 (Figure 10a), a concentrated load \(V_{5}=100\) kN was applied at one-third point (node 5) of the structure, while in load case two (Figure 10b), a uniformly distributed load \(q=20\) kN\(/\)m was applied over the whole span. For both load cases, seven rotations were measured and their locations are indicated in Figure 10a,b. According to the observability analysis, there were 32 essential measurement sets able to identify the parameters of this structure. Hence, the estimations of the bending stiffnesses for strategies 3 and 4 were calculated by averaging the estimations from these  
Figure 8: Box plot of the estimations of the bending stiffness \(EI_{2}\) using different strategies and essential sets with (**a**) error level = 5%; (**b**) error level = 10%. (Strat abbreviated for strategy.) Figure 9: The probability density curve of the estimations using strategies 2 (compatibility conditions), 3 (averaging) and 4 (averaging with outliers ruled out). 32 essential sets. To check the performance of different strategies, 200 measurement sets were generated using Equation (10). The 200 estimations for \(EI_{1}\)-\(EI_{3}\) using strategies 2-4 are illustrated as the nine columns of boxplots in Figure 10c,d. The corresponding mean and c.o.v.s are presented in Table 2.  
The parameters and strategies are indicated by the subscripts of the x labels in these figures. For instance, \(EI_{2,3}\) denotes the estimation of the bending stiffness \(EI_{2}\) using strategy 3. As observed in Table 2 and Figure 10c, all the estimations from different strategies were unbiased as the mean was always located at 1. Regarding the c.o.v., imposing the compatibility condition (strategy 2) achieved the best performance for all parameters in both load cases 1 and 2. The extent of dispersion in the estimations was significant for those obtained from strategy 3. After strategy 4 was adopted, the c.o.v.s for estimations using strategy 3 reduced as outliers were ruled out. A decrease in c.o.v.s for \(EI_{3}\) in load case 1 from 0.123 to 0.055 was observed. From the result for load case 2, similar conclusions can be drawn. In addition, the performance of using strategy 4 was very close to that of using strategy 2. The comparison between the results from load cases 1 and 2 shows the effect of load case on the estimation accuracy. The poor estimation in \(EI_{3}\) is due to the less excited flexural behavior of this region in load case 1. In load case 2, the mid span region was well excited and the zones adjacent to the supports were less excited under the external loads. Hence, the dispersion of the estimations for \(EI_{2}\) was low while that for \(EI_{1}\) and \(EI_{2}\) was larger. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multicolumn{2}{c}{**Load Case 1**} & \multicolumn{2}{c}{**Load Case 2**} \\ \cline{2-5}  & **Mean** & **c.o.v.** & **Mean** & **c.o.v.** \\ \hline \(EI_{1,2}\) & 1.001 & 0.027 & 1.001 & 0.030 \\ \(EI_{1,3}\) & 1.003 & 0.038 & 1.003 & 0.043 \\ \(EI_{1,4}\) & 1.002 & 0.035 & 1.002 & 0.031 \\ \(EI_{2,2}\) & 1.000 & 0.017 & 1.000 & 0.014 \\ \(EI_{2,3}\) & 1.000 & 0.018 & 1.001 & 0.017 \\ \(EI_{3,4}\) & 1.000 & 0.017 & 1.000 & 0.014 \\ \(EI_{3,2}\) & 1.003 & 0.050 & 1.001 & 0.030 \\ \(EI_{3,3}\) & 1.014 & 0.123 & 1.003 & 0.043 \\ \(EI_{3,4}\) & 1.002 & 0.055 & 1.002 & 0.031 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Statistical summary of the estimations using different strategies. Figure 10: (**a**) Load case 1 (a concentrated load on the one-third point) and the measurement location; (**b**) load case 2 (a uniform load over the span) and the measurement location; (**c**) boxplots of the estimations under load case 1; (**d**) boxplots of the estimations using under load case 2 (note: \(EI_{ij}\) means the estimation of \(EI_{i}\) using strategy \(j\)).