# Observing Material Properties in Composite Structures from Actual Rotations
## 3 Results
### Example 3: Application to a Simplified Model of a Composite Bridge
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Elements 1 and 7** & **Elements 2 and 6** & **Elements 3 and 5** & **Element 4** \\ \hline area [m\({}^{2}\)] & 12.385 & 12.520 & 12.670 & 12.75 \\ shear area [m\({}^{2}\)] & 8.17 & 9.29 & 10.62 & 11.37 \\ inertia [m\({}^{4}\)] & 38.346 & 48.347 & 67.98 & 60.921 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Properties of the Finite Element Model of the bridge. Figure 6: Composite bridge on-site [47]. Figure 7: Definition of the load case: (**a**,**b**) Stage i. (**c**,**d**) Stage i + 1. (**e**) Load case used for the inverse analysis. The simplified FEM of this structure is presented in Figure 8. This FEM includes seven elements and six concentrated loads. In this example, we exclusively analyzed the flexural and shear responses. Notably, the axial resistance mechanism was omitted since no axial forces were introduced in the applied loading scenario. Consequently, the values of the cross-sectional areas are not observed, given that they have no impact on the inverse analysis. As shear stiffness and bending stiffness from elements 2, 3, 4, 5 and 6 are supposed to be observed from the previous construction steps, such properties are considered as known. Hence, the shear and bending stiffness of the newly built elements (elements 1 and 7) are the targets of the analysis. This is to say I\({}_{1}\), I\({}_{7}\) and A\({}_{v1}\), A\({}_{v7}\) are assumed as unknowns. In this example, there are four unknowns (it is to say EI\({}_{1}\), GA\({}_{v1}\) and EI\({}_{7}\), GA\({}_{v7}\)). In order to identify the four unknown mechanical properties, a set of at least four measurements is required. The analysis of the OM proves that the COM is not able to observe any structural properties by only measuring the rotations in this example. To observe the structural parameters with the proposed methodology, the variation of deflection and rotation should be taken into account (that is to say w\({}_{1}\), v\({}_{2}\) and w\({}_{7}\), v\({}_{6}\)). The value of the theoretical rotation and vertical deflection of node 1 are presented in Figure 8b,c. [PERSON]'s beam theory is used to calculate the values of measured variables in node number 1 (the value at node number 7 is the same). The initial value coefficients of the unknown inertia and shear area (I\({}_{1}\), I\({}_{7}\) and A\({}_{v1}\), A\({}_{v7}\)) were randomly chosen between a ratio of 0.8 and 1.2 of the theoretical values. The 50 analyzed random initial values are performed. Also, no matter what set of the initial value is chosen, in all circumstances the obtained values of the two-iteration processes have proper convergence to the real values of the inertia. The normalized average, standard  
Figure 8: Example 3. (**a**) FEM of the bridge with load case used for inverse analysis. (**b**) Deformed shape with the value of bending vertical deformation and with bending and shear vertical deformation at node 1. (**c**) Deformed shape with bending rotation and with bending and shear rotation at node 1.  
deviation, and coefficient of variation of the remaining results as well as the mode are presented in Table 5.  
These results prove the adequate convergence of the proposed methodology in a bridge structure for the random initial values of the unknown parameters. These results prove the efficiency of the proposed COM to estimate the inertia when the effects of shear rotations are considered, in the case that the shear area cannot be observed by the proposed method. It is to say that the traditional COM is not able to identify the value of the mechanical properties and it lacks physical meaning when the theoretical rotations are measured. In addition, it is important to highlight that, as far as the authors know, no other SSI method based on SMM in the literature is able to include the effects of these rotations in the inverse simulation.