# Robust Static Structural System Identification Using Rotations
## 4 Using Redundant Rotations for Parameter Estimation
### Verification for a High-Rise Frame Structure  
Strategy 2 cannot be applied in frame structures as the compatibility condition is not available in such structures. Hence, strategy 4 was applied in the identification of parameters of a frame structure simulated by a FEM [49] studied previously (see Figure 11a). In that study, the observability of the parameters was studied symbolically, while the numerical analysis and the effect of measurement errors were not included. In this paper, the focus is the identification of the bending stiffness of the floor slab in the middle right of the third floor. Figure 11b shows the FEM for this part. It is parameterized by three bending stiffnesses, \(EI_{9}\)-\(EI_{11}\). The parameterization of the remaining part is the same as that in [49]. The bending stiffnesses \(EI_{9}\) = \(EI_{10}\) = 1.75 \(\times\) 10\({}^{8}\) N m\({}^{2}\). Damage to the right side of the target floor slab was simulated by a 30% reduction in bending stiffness \(EI_{11}\), i.e., \(EI_{11}=1.225\times 10^{8}\) N\(\cdot\)m\({}^{2}\). An overload of 40 kN/m was applied to this floor slab to simulate a static load test. Due to the unsatisfactory result of using deflections, 10 rotations (\(w_{32}\), \(w_{72}\), \(w_{74}\), \(w_{76}\), \(w_{78}\), \(w_{80}\), \(w_{82}\), \(w_{84}\), \(w_{86}\) and \(w_{46}\)) were measured to identify \(EI_{9}\)-\(EI_{11}\). The increments in the rotations due to this load were calculated by direct analysis. Five error levels (1-5%) were studied. For each error level, 100 measurement sets were generated using Equation (10). To identify \(EI_{9}\)-\(EI_{11}\), two rotations were required for each parameter. Taking two rotations from (\(w_{32}\), \(w_{72}\), \(w_{74}\), \(w_{76}\)), (\(w_{76}\), \(w_{78}\), \(w_{80}\), \(w_{82}\)) and (\(w_{82}\), \(w_{84}\), \(w_{86}\), \(w_{46}\)), respectively, \(EI_{9}\)-\(EI_{11}\) led to 117 essential sets composed of six rotations. Thus, 117 estimations were obtained for each measurement set and the outliers in these estimations were ruled out. Figure 11c presents the error bar of the final estimations of \(EI_{9}\)-\(EI_{11}\) under different error levels. The centerlines indicate the mean of estimations for each set. The vertical error bars cover two standard deviations of the estimations. Hence, the length of these bars indicates the extent of dispersion in the estimations.  
Figure 11: (**a**) The FEM of the high-rise building (adapted from [49]). (**b**) The FEM for the floor slab to be identified. (**c**) Error bar plot of the estimations of \(EI_{9}\)-\(EI_{11}\).  
From Figure 11, it is seen that: (1) The estimations are unbiased, since the centerline is located at 1. (2) The estimations of \(EI_{10}\) always have the least variation. (3) The increase in variations in the estimations follows a linear relationship with error levels. (4) The sensitivity of the increase of variation with error levels is the lowest for \(EI_{10}\). In fact, the better estimations in \(EI_{10}\) are due to the higher curvature of the zones parameterized by \(EI_{10}\) than the remaining zones, which agrees with the result in the previous study [24].  
In the current study, only rotations were used in the measurement sets. The possibility of using the combination of deflections and rotations will be investigated in future study.