## 4 Numerical Example
### Experimental Results  
The experimental analysis [15] shows the displacement record during the load path. The results have been reported here for the sake of clarity: in Table 8, the maximum applied load, and the corresponding residual displacement of the point P\({}_{3}\) have been reported. The measures have been recorded when, after some cycles of loading, the displacement did cease increasing, and the load was removed, \(\hat{\varphi}_{\mathbf{P_{3}}}^{0}\). The measure has been repeated when the displacement, \(\hat{\varphi}_{\mathbf{P_{3}}}^{\infty}\), reduced, completing its a recovery. Finally, under the loads indicated by the (*), the displacement did not stabilize. In this case, the experiment finished after many cycles, and the residual displacement has been recorded. Even in this case, a new measure has been recorded after time to keep the memory of the recovering effect. Table 8 contains the experimental results, as reported in [15]. The experimental results denote that the actual residual displacements are greater than the calculated through step by step procedure. Indeed, in the analysis, the average limit bending moment has been used, the experimental report describes that the qualification tests to evaluate the mechanical \begin{table}
\begin{tabular}{c c} \hline \hline \(m\) & \(\mathbf{v_{P_{3}}}\)**Upper Bound** \\ \hline
1.01 & 4.09 \\
1.03 & 4.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Calculated displacement upper bound. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**F[kN]** & \(\mathbf{v_{P_{3}}}\)**[\(\mathbf{m}\)]** & \(\mathbf{\Delta\varphi_{3}}\) at P\({}_{3}\) and dissipated energy \(\mathcal{E}_{\text{D}}\). \\ \hline
2.03 & \(6.52*10^{-4}\) & \(5.21*10^{-3}\) & \(1.70*10^{-3}\) \\
2.06 & \(2.074*10^{-3}\) & \(0.0166\) & \(5.44*10^{-3}\) \\
2.065 & \(2.34*10^{-3}\) & \(0.0187\) & \(6.14*10^{-3}\) \\
2.07 & \(2.578*10^{-3}\) & \(0.0206\) & \(6.76*10^{-3}\) \\
2.08 & \(3.052*10^{-3}\) & \(0.0244\) & \(8.00*10^{-3}\) \\
2.09 & \(3.52*10^{-3}\) & \(0.0282\) & \(9.24*10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Step by step results: Displacement \(\mathbf{v_{P_{3}}}\), rotation \(\Delta\varphi_{3}\) at P\({}_{3}\) and dissipated energy \(\mathcal{E}_{\text{D}}\). \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{F[kN]}\) & \(\mathbf{\overset{\alpha}{v_{P_{3}}}}\)**[mm]** & \(\mathbf{\overset{\alpha}{v_{P_{3}}}}\)**[mm]** \\ \hline
2.032 & 4.07 & 3.61 \\
2.069 & 4.67 & 4.32 \\
2.099 & 5.48 & 5.25 \\
2.139 * & 7.06 & 6.72 \\
2.180 * & 11.42 & 11.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Experimental residual displacements, before recovery \(\hat{\varphi}_{\mathbf{P_{3}}}^{0}\), complete recovery \(\hat{\varphi}_{\mathbf{P_{3}}}^{\infty}\); (*) indicates ratcheting load. properties of the specimens, produced non-uniform results and the average reported value, as shown in Figure 6, is affected by a meaningful standard deviation. In conclusion, during the experiment, the structure underwent to ratcheting for loads greater than 2.099 kN. The recorded displacement rate did not vanish, and the measured values refer to the end of the experiment. For loading less to 2.099, the permanent displacement increment vanishes definitely, i.e., it tends to remain constant in time after two or three cycles; hence, the shakedown has occurred. The proposed method has been applied to the structure; the program Equation (21) has been used to calculate the shakedown multiplier and the residual rotations. Moreover, the resulting Melan's residuals have been introduced to formulate the program Equation (27), where the displacement in the middle of the right span has been calculated in terms of the residual rotation. The upper bound of the displacement has been calculated for two different values of the factor \(m\). Table 9 contained the comparison between the experimentally recorded displacement and the upper bound calculated through the present proposed procedure.