# Robust Static Structural System Identification Using Rotations
## 3 Theoretical Motivation for Measuring Rotations
### Statistical Analysis of a Simply Supported Bridge
At low error levels, the positive mode occurs close to one and the value of probability density is high for all three sets. This is no longer true as error levels increase. Figure 4b,c provide the probability density curves for the estimations of \(El_{2}\) using set 1 and set 3 with 20% error. For set 1, the biased positive mode x = 0.535 is related to a small density of 0.971. In addition, the negative mode x = \(-\)1.152 is not negligible. On the contrary, for set 3, the positive mode x = 0.990 is related to a much higher probability density of 5.664, while the negative mode x = \(-\)99.771 is negligible and hence not shown. This implies that for high error levels, the estimation using deflections leads to underestimations, and even negative estimations. In addition, the severe right skewness of the probability density curve for set 1 indicates a much larger variation in the estimations than in the case of using rotations (set 3). \begin{table}
\begin{tabular}{c c c} \hline
**Displacements** & **Values** & **Unit** \\ \hline \(v_{5}\) & \(-\)0.010971 & m \\ \(v_{7}\) & \(-\)0.011829 & m \\ \(v_{9}\) & \(-\)0.009600 & m \\ \(w_{5}\) & \(-\)0.001829 & rad \\ \(w_{9}\) & 0.002286 & rad \\ \hline \end{tabular}
\end{table} 
Table 1: Increments in displacements for example 1 due to the concentrated load \(V_{5}=-\)100 kN. Figure 4: Probability density curves for the estimations of \(El_{2}\) using (**a**) set 1 (\(v_{5}\), \(v_{7}\) and \(v_{9}\)) with 5% error (theoretical distribution and inferred from data), (**b**) set 1 (\(v_{5}\), \(v_{7}\) and \(v_{9}\)) with 20% measurement error and (**c**) set 3 (\(w_{5}\), \(w_{9}\)) with 20% measurement error (shaded area for 2.5% and 97.5% percentiles). (KDE for kernel density estimation). The 95% confidence intervals of the estimations of \(EI_{2}\) using set 3 with error levels ranging from 0% to 20% are depicted in Figure 5. The lower and upper bounds of the confidence interval are the cutting points of the 2.5% regions on the left and right sides of the probability density curve (shaded in Figure 4c). Figure 5 shows that the bounds of these confidence intervals satisfy a linear relation with the error level. In addition, these intervals are bounded by \([1-E_{level},1+E_{level}]\) for each error level. Figure 6 provides the variations inj the positive mode, the mean and the coefficient of variation (c.o.v.) for the estimations of \(EI_{2}\) using different sets at different error levels. The median is always one for set 3 at any error level. The mean and the c.o.v. were analyzed with the interval [0, 2]. The remarkable consistency between the mode and the mean for set 3 justifies the unbiasedness of the estimations using rotations (Figure 6a). With 20% error, the deviation in the mean is 0.24% and the related c.o.v. is 0.046. On the contrary, the deviations in the mean increase rapidly with errors for set 1 and set 2. The positive mode for set 1 was close to zero when the error level was high. This indicates a large c.o.v., as seen in Figure 6b, and a right skewness. When more deflections were used, larger c.o.v.s were observed as the positions of the c.o.v. curves became higher. This analysis shows that: (1) The statistical analysis is capable of analyzing the distribution of the estimations. (2) When the error level is high, the estimations using deflections might be negative and the distribution is severely right skewed. (3) Unlike the case of using deflections, the estimations using rotations are robust and insensitive to error levels, since the confidence intervals of the estimations are well bounded. (4) The higher the number of deflections used in the measurements, the higher the sensitivity of the estimations to errors. Figure 5: Confidence interval of the estimation for \(EI_{2}\) under different error levels. This is because the measurements appear in the denominator of the analytical solution of stiffnesses. As demonstrated before, the value of the stiffness \(EI_{2}\) is affected by the values of the term \(w_{5}-w_{9}\).