## 3 Results
### Inverse Analysis  
This section describes how the observability method (OM) proposed by [PERSON] et al. [43], which is an inverse analysis method for models made out of Timoshenko beam elements, was applied to calculate the mechanical properties of academic examples to illustrate the importance of shear rotations in deep beams. The method was able to calculate the mechanical properties of beam elements from the deflections and rotations of the structure. It is important to highlight that the OM is based on [PERSON]'s stiffness matrix and, therefore, the system of equations does not include the effects of the shear rotations. In fact, in this inverse analysis method, the shear deformation effects are only considered in the vertical deflections.  
To evaluate the effects of the shear rotations in the inverse analysis, the three beams described in the preceding section (a simply supported beam with a concentrated load, a simply supported beam with a uniform load and a cantilever beam with a concentrated load) were calculated. It is important to highlight that introducing the shear rotations into  
Figure 8: Parametric analysis of the effect of the slenderness ratio on a cantilever beam with a concentrated load: (**a**) bending (\(w_{b,3}\)) and shear (\(w_{s,3}\)) rotations at node 3 and (**b**) percentage error of rotations in node 3 when ws,3 was neglected.  
the OM represents a modeling error that affects the accuracy in the estimation of the beam element properties. Obviously, this error depends, to a great extent, on the magnitude of the shear rotations and, therefore, on the beam slenderness. To evaluate the effect of the slenderness of these beams, a set of parametric analyses were carried out. In these analyses, the lengths of the beams varied from 1 to 15 m while their cross-section remained constant (height, 5 m; width, 0.2 m). For the inverse analysis of each beam, two unknown parameters (the beam element inertia and the shear area) were considered. The following measurement data were considered: w1 and v2 for the simply supported beams and w3 and v3 for the cantilever beam. The different rotations introduced into the measurement data were simulated using the procedure proposed in Section 3 (as they try to simulate the rotations that would be measured on site).  
Figure 9 presents the errors in the inertias estimated for the three analyzed beams for different slenderness (\(L/h\)) ratios. These errors were calculated as the percentage error of the estimated inertia \(\overline{I}\) with respect to the actual inertia \(I\) (2.083 m\({}^{4}\)). For illustrative purposes, the slenderness limits for deep beams proposed by the Eurocode EN and by the ACI Committee 318 are highlighted in this figure.  
Analysis of Figure 9 shows that shear rotations play an important role in the inverse analysis of deep beams. In fact, the lower the \(L/h\) ratio, the higher the errors in the estimated inertias. For example, for the simply supported beam, the errors when \(L/h=1\) were as high as 73.2% for the uniform load and 67.9% for the concentrated one. The error reached 32.0% for the cantilever beam with the concentrated load. At the boundary of the Eurocode for deep beams, the error in the inertia estimation for the simply supported beam with the concentrated load, the simply supported beam with the distributed load, and the cantilever beam with a concentrated force were 18.2%, 23.4% and 5.3%, respectively. For the ACI geometry boundary for deep beams, the errors in these structures were reduced to 11.1%, 14.7% and 3.0%, respectively. These results illustrate that for deep beams, the effects of shear rotations should not be neglected. In contrast, in slender beams, the errors in inertia estimation were significantly lower. For example, when \(L/h=15\), the errors were below 1% in the three analyzed structures.