## 4 Numerical Example
### Numerical Results  
The aluminum beam described in the previous section was analyzed numerically with the proposed method. For the calculation of the structure, the formulation reported in the previous sections must be particularized to generalized stress and strain in bending beams. The generalized strain should be intended as the curvature of the beam's axis and the stress the bending moment, \(M\). The bending moment is decomposed into the pure elastic moment, \(M^{E}\) and the equivalent to \(\zeta_{ij}\), \(M^{r}\), residual bending moment. Finally, the permanent discontinuous strain, \(p_{hk}\), is the localized rotation discontinuity, \(\Delta\varphi_{i}\). The first step to perform is the calculation of the \(\mathbf{Z}\) matrix. It is pursued calculating the bending moment on the structure due to applied concentrated rotation at any point of it. To get numerical results, at first, one must choose a discrete set of points where apply the rotation and where read the moments. To get the discrete set of points, some reasoning must be done about the variability of the response of the structure, namely:  
In the proposed example, the moment varies linearly along with the structure, having discontinuous slope at the points where active or reactive forces act. Consequently, at these points, the moment is expected to have its local minimum or maximum. Those points, moreover, are a candidate to develop concentrated rotation and must be chosen as control points where one should enforce the constraints and where the dislocations must be applied. The control points are indicated with A, B, C, and D (Figure 4). The structure is divided into four elements at points A, B, C, D, as depicted in Figure 5. For the computational reason the calculation of all the relevant quantities involved in the routine, have been calculated at any endpoint of the elements \(P_{i}\), as well, concentrated rotations, \(\Delta\varphi_{i}\), have been applied on. Moreover, the constraints of optimization Program (14) have been collocated on \(P_{i}\), too. The maximum elastic moment discrete values for any point have been collected in a vector \(\mathbf{M}^{max}\) and the minimum in the vector \(\mathbf{M}^{min}\); in the case of one monotonically increasing load pattern, \(\mathbf{M}^{max}=\mathbf{M}^{min}\). In the same manner, the rotations \(\Delta\varphi_{i}\) have been collected into a vector \(\Delta\mathbf{\varphi}\), so that the Equation (14) transforms in a matrix form. \[\sup_{\Delta\mathbf{\varphi}}k\left| \begin{array}{c}-\mathbf{Z}\Delta\mathbf{\varphi}- k\mathbf{M}^{min}<\left|\mathbf{M}^{-}_{y}\\ \mathbf{Z}\Delta\mathbf{\varphi}+k\mathbf{M}^{max}<\left|\mathbf{M}^{+}_{y}\right|\end{array} \right.\right. \tag{21}\]  
In Equation (21), \(\mathbf{M}^{-}\) is the negative limit bending moment vector ordered as point \(P_{i}\), and \(M^{+}\) is the positive one. The choice of the control points is dependent on the prediction of the local extrema of the internal stress. It is essential to consider that if the stress violates the compatibility equation at points not \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cross-Section [mm]** & **Mechanical Constant** & **Range of Values** \\ \hline \hline \multirow{3}{*}{
\begin{tabular}{c} \\ \end{tabular} } & Young modulus E & \(6250-6400\)MPa \\  & Limit bending moment M\({}_{y}\) & \(0.3276-0.3324\)kNm \\  & Yield stress \(\sigma_{Y}\) & \(193.16-196.00\)MPa \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters. comprised in the control set, then the results overestimate the load multiplier. Consequently, one must consider setting up the procedure that converges from below in the domain of self-equilibrated stress but converges from above with the increase of the control points. In the proposed example, the position of the control points ensured that the compatibility was fulfilled everywhere in the structure. The mechanical parameters and the geometry of the beam have been introduced into the calculation of the operator \(\mathbf{Z}\).