## 2 Materials and Methods
### Calculation of Total Rotations with Structural Software  
A new methodology is presented in this section to calculate total rotations in structures using [PERSON]'s stiffness matrix method with first-order shear deformation. This methodology uses [PERSON]'s beam theory to calculate the shear rotations to be added to the bending rotations calculated using the software. This is done to highlight those beam elements that converge at a given node that might have different rotations at each beam element in the same node. The bending rotations have to be the same for all the converging elements in the common node but this does not apply to the shear rotations. The proposed method is summarized as the following three steps:  
* **Step 1**: calculate the bending rotations (\(w_{b,ij}\)) and the shear forces (\(S_{i,j}\)) at each node (\(i\)) for each element (\(j\)) with the simulation software.
* **Step 2**: apply [PERSON]'s beam theory to calculate the shear rotations (\(w_{s,ij}\)) at each node (\(i\)) for each element (\(j\)). According to this theory, the shear rotations can be calculated from \(G\), \(A_{v}\) and \(S_{i,j}\). The former two terms can be directly obtained from the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & \(v_{b,2}\) **(mm)** & \(v_{s,2}\) **(mm)** & \(w_{b,1}\) **(rad)** & \(w_{s,1}\) **(rad)** \\ \hline [PERSON]’s beam theory & 33.34 & 20.00 & 0.010 & 0.005 \\ Midas/Civil without shear & 33.34 & 0.00 & 0.010 & 0.000 \\ Midas/Civil with shear & 33.34 & 20.00 & 0.010 & 0.000 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Properties of the FEM of the simply supported beam.  
mechanical properties of the structure, while \(S_{i,j}\) corresponds with the shear forces calculated in step 1.
* **Step 3**: obtain the real rotations at the \(i\)th node at element \(j\) by summing up the bending (\(w_{b,j}\)) and the shear (\(w_{s,ij}\)) rotations calculated in steps 1 and 2, respectively.  
A summary of the procedure is shown in the flow chart in Figure 4. The shear rotations at node \(i\), element \(j\) are only calculated if the shear force \(S_{i,j}\) is not null. Steps 2 and 3 of this flow chart can be easily implemented and programmed.  
As shown in step 2, the shear rotations are proportional to the forces in the shear diagram. Hence, discontinuities in this diagram produced by concentrated loads or reactions might lead to different rotations at the same node. In addition, in accordance with [PERSON]'s theory, shear rotations might appear, even at clamped supports, as they are not affected by the boundary conditions. This is a great difference from other methods [19]. Moreover, unlike other methods in the literature [35; 36; 37; 38], the new method enables the analysis of complex structures, including redundant ones with different imposed loads. The proposed method assumes a constant value for the material's Young's modulus. Therefore, plastic rotations cannot be calculated. In future research, the method will be developed to include the effects of plastic shear rotations.  
To illustrate the application of the proposed methodology in a statically redundant structure, the results of analyzing a continuous beam with a concentrated load are given below.