## 3 Results
### Direct Analysis  
This section describes how the proposed methodology was used to calculate the shear and bending rotations of three academic examples to give a hint of the importance of shear rotation for deep beams. To do so, a simply supported beam with a concentrated load, a simply supported beam with a distributed load and a cantilever with a concentrated load were analyzed. The cross-section of all these structures corresponded with that presented in Figure 2b and the mechanical properties with those in Table 1. To evaluate the effects of the shear rotations, parametric analyses of the beam slenderness were carried out. In these analyses, the lengths of the beams (\(L\)) varied from 1 to 15 m while their heights (\(h\)) remained constant (with a value of 5 m). The results of the simply supported beam with a concentrated load of 100 kN at mid-span (\(V\) = 100 kN) are summarized in Figure 6. Figure 6a includes the bending (\(w_{b,1}\)) and shear (\(w_{s,1}\)) rotations obtained at the left support for different beam slenderness values (\(L/h\)). Figure 6 presents the percentage error that would result if the shear rotations were neglected. To do so, the shear rotations at node 1 were divided by the total rotations at that node (\(w_{b,1}+w_{s,1}\)). For illustrative purposes, the slenderness limits for deep beams proposed by the Eurocode EN [41] and by the ACI Committee 318 [42] are highlighted in this figure, together with the error limits of 2% and 5%. As expected, Figure 6 illustrates how the slenderer the beam, the less significant the effects of the shear rotations. On the one hand, Figure 6a shows that the shear rotation at the support (\(w_{s,1}\)) did not depend on the beam length, as its value (0.005 rad) remained constant for all the analyzed slenderness values. This was not the case for the bending rotations (\(w_{b,1}\)), as they increased nonlinearly with the length and with the slenderness ratio for a beam of constant height. In fact, these rotations varied from 0.0025 rad (\(L/h=1\)) to 0.5625 rad (\(L/h=15\)). On the other hand, Figure 6b shows that the smaller the \(L/h\) ratio, the higher the error of neglecting shear rotations. In fact, shear rotations represented 68.2% of the total rotations when \(L/h=1\). This figure also shows that a percentage error of 2% was reached for an \(L/h\) ratio of 9.9. This error was increased to 5% for an \(L/h\) ratio of 6.2. The geometrical definitions of deep beams in the Eurocode EN and ACI Committee 318 led to errors of 18.2% and 11.12%, respectively, when shear rotations were neglected in this example. The results of the simply supported beam with a uniform load of 100/L kN/m applied alongside the beam are summarized in Figure 7. Figure 7a includes the bending (\(w_{b,1}\)) and shear (\(w_{s,1}\)) rotations obtained at the left support for different beam slenderness values (\(L/h\)). Figure 7b presents the percentage of the rotations represented by the shear rotation. This figure also includes the geometrical boundary for deep beams proposed by the Eurocode EN and by the ACI Committee 318, as well as the error limits of 2% and 5%. The effects of the beam slenderness on the rotations of the simply supported beam with a uniform load were similar to those obtained for a concentrated force. In fact, as presented in Figure 7a, the shear rotation did not depend on the \(L/h\) ratio, as it remained constant with a value of 0.005 rad. The bending rotations varied from 0.001 to 0.409 when the \(L/h\) ratios were increased from 1 to 15. Figure 7b shows that in this example, shear rotations represented 73.1% of the rotations when \(L/h=1\). This figure also shows that for the geometrical limitations of the deep beams presented in the Eurocode EN and  
Figure 6: Parametric analysis of the slenderness ratio on example 1: (**a**) bending (\(w_{b,1}\)) and shear (\(w_{s,1}\)) rotations at node 1 and (**b**) percentage error of rotations in node 1 when ws,1 was neglected.