## 2 Materials and Methods
### Example 1: Simply Supported Beam with a Concentrated Load  
Consider a 10 m long and 0.2 m wide simply supported beam modeled with three nodes and two beam elements depicted in Figure 2a. This structure can be considered a deep beam according to the Eurocode and the ACI criteria, and the shear effects are not negligible. This beam had a constant cross-section and the value of its Young's modulus \(E\), shear area \(A_{v}\), cross-sectional area A and inertia I of the beam are 30 GPa, 0.833 m\({}^{2}\), 1 m\({}^{2}\) and 2.083 m\({}^{4}\), respectively. Dimensions of the beam cross-section are presented in Figure 2b and its mechanical properties are listed in Table 1. The boundary conditions of the structure were horizontal and vertical displacements restricted in node 1 and vertical displacement restricted in node 3 (this is to say \(u_{1}=v_{1}=v_{3}=0\)). The beam was subjected to a concentrated vertical force at mid-span (node 2) of 100 kN (\(V_{2}=100\) kN). \begin{table}
\begin{tabular}{c c} \hline \hline
**Properties (Unit)** & **Value** \\ \hline Area (m\({}^{2}\)) & 1.000 \\ Shear area (m\({}^{2}\)) & 0.833 \\ Inertia (m\({}^{4}\)) & 2.083 \\ Young’s modulus (GPa) & 30.000 \\ Poisson’s ratio \(v\) & 0.250 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the FEM of the simply supported beam. The structure was analyzed with Midas/Civil [37] with and without considering the shear deformation effects. The obtained vertical deflections due to the bending/Bernoulli model (\(v_{b}\)) and due to the bending plus shear/Thimoshenko model (\(v_{b}+v_{s}\)) are summarized in Figure 3a, while the rotations for such models are presented in Figure 3b. Please note that in Figure 3b, the terms \(w_{(b)}\) and \(w_{(b+s)}\) stand for the rotations obtained with the Bernoulli and Timoshenko models of the structural software, respectively. Figure 3b also includes the rotations that were numerically calculated using the derivation of the vertical total deflections, including shear effects (\(v_{b}+v_{s}\)) throughout the beam direction (axis \(x\)). This information is referenced as \(\mathrm{d}(v_{b}+v_{s})/\mathrm{d}x\) in the figure. As would be expected, analysis of Figure 3a shows that higher deflections were obtained when the shear deformation was considered. Nevertheless, this was not the case when the shear rotations were analyzed, as the same rotations were obtained using the commercial software with and without considering the shear deformation effects. These results are illustrated in Figure 3b, where it is shown that the rotations obtained using Midas/Civil were the same independent of whether the shear deformation effects were considered or not. The analysis of this figure shows that in this example, the commercial software failed in the calculation of the rotations, as the values obtained using the program did not correspond with those obtained using a numerical derivation of the vertical deflections. This is to highlight that these total rotations were not a continuous function, as jumps in the rotation laws can be present if point loads exist. Figure 3: Example 1: (**a**) vertical deflections and (**b**) rotations. Figure 2: Example 1: simply supported structure: (**a**) finite element model and (**b**) cross-section. [PERSON]'s theory can be used to calculate the deformations and rotations in the analyzed simply supported beam.