# Robust Static Structural System Identification Using Rotations
## 3 Theoretical Motivation for Measuring Rotations
### Statistical Analysis of a Simply Supported Bridge  
Example 1 was a simply supported bridge measuring \(3L\) long. Statistical analysis was carried out for this structure using different measurement sets. The parameterization of the FEM for this structure is shown in Figure 3. The target parameter was \(EI_{2}\). OM (observability method) indicates that any two rotations, or one rotation plus two deflections, or three deflections, in nodes 5-9 are qualified to identify \(EI_{2}\). Without loss of generality, three measurement sets, set 1 (\(v_{5}\), \(v_{7}\), \(v_{9}\)), set 2 (\(v_{5}\), \(v_{7}\), \(w_{9}\)) and set 3 (\(w_{5}\), \(w_{9}\)), were studied, where \(v_{i}\) and \(w_{i}\) denote the measured deflection and rotation, respectively. Employing structural system identification using COM with set 3 (\(w_{5}\), \(w_{9}\)), the analytical expression of \(EI_{2}\) (as shown in Equation (13)) can be obtained by solving the observability equation symbolically. \[EI_{2}^{\text{set3}}=\frac{\text{L}}{32(w_{5}-w_{9})}\cdot\left\{16\sum\limits_ {i=1}^{5}(M_{i}-M_{14-i})+8M_{6}-8M_{8}+4\sum\limits_{i=2}^{5}[(i-1)V_{i}L+(i- 1)V_{14-i}L]+19V_{6}L+20V_{7}L+19V_{8}L\right\} \tag{13}\]  
In Equation (13), the external loads are collected in the numerator while the measurements (\(w_{5}\), \(w_{9}\)) are collected in the denominator. The estimation of \(EI_{2}\) depends on the loading case and the measured displacements, not related to \(EI_{1}\) or \(EI_{3}\). Assume that \(L\) = 3 m, and the depth and width of the cross section are 0.5 m and 0.3 m. The inertia and the elastic modulus are 3.125 \(\times\) 10\({}^{-3}\) m\({}^{4}\) and 3.5 \(\times\) 10\({}^{7}\) kN/m\({}^{2}\), respectively. A vertical concentrated load \(V_{5}\) = 100 kN is applied at the one-third point. The ratio of the maximum deflection to the span is 1/760. The increments in the measured displacements obtained from FEM are listed in Table 1. Since \(V_{5}\) = \(-\)100 kN and all the other loads are null, Equation (13) can be simplified as Equation (14). \[EI_{2}^{\text{set3}}=\frac{V_{5}L^{2}}{2(w_{5}-w_{9})}=\frac{1}{\frac{2(w_{5}- w_{9})}{V_{5}L^{2}}} \tag{14}\]  
Figure 3: FEM for a simply supported bridge. As can be observed in Equation (14), the analytical equation of stiffnesses includes the term \(w_{5}-w_{9}\). Hence, the sign of the analytical equation can be affected by the sign of the term \(w_{5}-w_{9}\). When errors are introduced into the measurements, the sign of the term \(w_{5}-w_{9}\) might be reversed. Consequently, negative \(EI\) values might appear. The advantage of this analytical method is that if such values appear, it is easy to locate the source of error and either to discard the results, to improve the measurement data or to change the measurement points to avoid close-to-zero values. With Table 1, Equations (10)-(12), the distributions of \(w_{5}-w_{9}\) and \(\frac{2(w_{5}-w_{9})}{V_{5}L^{2}}\) as well as \(El_{2}^{set3}\) can be obtained. The distributions for \(El_{2}^{set1}\)(\(v_{5}\), \(v_{7}\) and \(v_{9}\)) and \(El_{2}^{set2}\)(\(v_{5}\), \(v_{7}\) and \(w_{9}\)) were obtained in the same way. To validate the statistical analysis, 2000 samples of measurement set (\(v_{5}\), \(v_{7}\) and \(v_{9}\)) were generated by Equation (10) with an error level of 5%. Two thousand estimations of \(El_{2}\) were obtained. All estimations were normalized by their nominal values. The probability density curve of the estimation of \(El_{2}\) was obtained from the 2000 estimations using kernel density estimation, which was carried out by the ksdensity command in Matlab. The validity of the proposed statistical analysis is justified by the complete agreement between the estimated probability density curve and the theoretical one from Equation (11) in Figure 4a. The distribution of \(El_{2}\) using set 1 (\(v_{5}\), \(v_{7}\) and \(v_{9}\)) is severely right skewed, characterized by the extremely long right tail.