## 2 Materials and Methods
### Example 2: Continuous Beam with a Concentrated Load  
The analyzed beam included two 5 m long spans. The cross-section corresponded with that described in Figure 2b and the mechanical properties are described in Table 1. The model and loading case are depicted in Figure 5. The only external force applied is a concentrated vertical force at the first mid-span of 100 kN (\(V\) = 100 kN).  
Figure 4: Flow chart to calculate the actual rotations in beams.  
The results of the application of the proposed methodology are compared with those obtained by the computer software Midas/Civil in Figure 5. The first part of this figure (Figure 5a) includes the vertical deflections due to bending (\(v_{b}\)) and bending plus shear (\(v_{b}+v_{s}\)) obtained throughout the beam using Midas/Civil. Figure 5b,c include the shear diagram (\(S\)) and rotations (\(w\)) obtained using this software when bending (\(S_{(b)}\) and \(w_{(b)}\)) or bending plus shear strains (\(S_{(b+s)}\) and \(w_{(b+s)}\)) are considered. Finally, Figure 5d presents a comparison of the rotations calculated using Midas/Civil with those obtained using the proposed methodology when shear deformation effects were considered. This information is presented in the figure as the sum of the bending rotations obtained when the shear deformation was considered (\(w_{(b+s)}\)) plus the shear rotations obtained using the proposed methodology (\(w\)(Emadi's)). This figure also includes the rotations that were numerically calculated using the derivation of the vertical deflections, including shear effects throughout the beam directrix (axis \(x\)). These rotations are referenced as \(\mathrm{d}(v_{b}+v_{s})/\mathrm{d}x\).  
The analysis of Figure 5a shows that the vertical deformations at the first mid-span (1.19 mm) increased from 1.19 mm to 6.39 mm when shear deformation was considered.  
Figure 5: Example 2: continuous beam with and without shear effects: (**a**) vertical deflections, (**b**) shear diagrams, (**c**) rotations obtained with the commercial software, and (**d**) rotations including shear effects.  
Unlike isostatic structures, in statically redundant structures, the shear deformations affect the vertical reactions at the supports. This effect is displayed in Figure 5b, where the shear diagram of the beam with and without shear deformation is presented. For example, at the first part of the beam (from left bearing to point load), the shear force S increased by 9.8% (from \(-40.6\) MN to \(-44.6\) MN) when shear deformation was considered. These variations on the shear forces affected the bending moment diagrams of the structure and, therefore, the bending rotations of the beam were also modified. This effect is displayed in Figure 5c. The analysis of this figure shows that the increase in the shear forces due to the introduction of the shear deformation led to higher rotations in the Midas/Civil program. For example, rotation at the left bearing was increased by 25% (from 0.0008 to 0.0010 rad) when shear effects were considered. Nevertheless, it is important to highlight that in both cases (with and without shear effects), these rotations only took into account the bending effects. Finally, Figure 5d shows how the simulation results obtained using the computer software (\(w_{(b+s)}\)) did not correspond with those obtained from the derivation of the beam deflections throughout the beam directrix (\(x\)-axis). For example, the rotations obtained using the numerical derivation at the left support (0.0029 rad) were 266.76% higher than those obtained using the computer software (0.0011 rad). Figure 5d also shows that the rotations obtained using the numerical derivation corresponded exactly with those obtained using the proposed method, where the shear rotations (\(w\)(Emidi's)) were added to the bending rotations calculated using the computer software.