## 4 Numerical Example
### Numerical Results
Since the formulation in terms of generalized vectors of rotations and bending moments, the operator assumes the form of a square matrix here called the influence matrix. The influence matrix can be calculated directly in the case of one-dimensional structures applying the definition given in Equation (11). The particularization of Equation (11) to the case of beams in bending gives the following representation of the residual bending moment in terms of \(\mathbf{Z}\):  
\[\mathbf{M}^{r}=\mathbf{Z}\Delta\mathbf{\varphi} \tag{22}\]  
where the element \(Z_{ij}\) of \(\mathbf{Z}\) is of the bending moment at \(P_{i}\) due to the rotation \(\Delta\varphi_{j}\) at \(P_{j}\). The optimization program (21) has been solved with the data of the reported example. As a result, together with the load multiplier \(s_{\alpha}\), a set of rotation \(\Delta\widetilde{\mathbf{\varphi}}\) is furnished, this represents the Melan residual:  
\[\mathbf{M}^{r}_{m}=\mathbf{Z}\Delta\widetilde{\mathbf{\varphi}} \tag{23}\]  
Through the known Melan residual, the upper bound of the dissipation in Equation (15) is calculated as  
\[\mathcal{E}_{\text{D}}\leq\frac{m}{2(m-1)}\Delta\widetilde{\mathbf{\varphi}}^{T} \mathbf{Z}\Delta\widetilde{\mathbf{\varphi}} \tag{24}\]  
To calculate an upper bound of the residual displacements on the structure at shakedown, the program Equation (19) has been rewritten accordingly to Positions (20) and (21). The dissipation, in the actual case of bending, can be calculated as the sum of the partial dissipation due to the positive and negative parts of the rotations \(\Delta\mathbf{\varphi}\), namely \(\Delta\mathbf{\varphi}=\Delta\mathbf{\varphi}^{+}-\Delta\mathbf{\varphi}^{-}\). The objective function of the second optimization program can be assumed to be one of the desired permanent displacement in the form of a linear combination of dislocations through a constant parameters vector \(\mathbf{c}\). \[u=\mathbf{c}^{T}(\Delta\mathbf{\varphi}^{+}-\Delta\mathbf{\varphi}^{-}) \tag{25}\]  
From this position, a lower bound of the dissipated energy, \(\mathcal{E}_{\text{D}}\), assumes the form  
\[\mathcal{E}_{\text{D}}\geq-\mathbf{M}^{-}_{y}\cdot\Delta\mathbf{\varphi}^{-}+\mathbf{M}^{ +}_{y}\cdot\Delta\mathbf{\varphi}^{+} \tag{26}\]  
\[\sup_{(\Delta\mathbf{\varphi}^{+},\Delta\mathbf{\varphi}^{-})}\mathbf{u}\left| \begin{array} []{c}-\mathbf{Z}(\Delta\mathbf{\varphi}^{+}-\Delta\mathbf{\varphi}^{-})-\frac{s_{\alpha }}{m}\mathbf{M}^{min}<\left|\mathbf{M}^{+}_{y}\right|\\ Z(\Delta\mathbf{\varphi}^{+}-\Delta\mathbf{\varphi}^{-})+\frac{s_{\alpha}}{m}\mathbf{M}^{ max}<\left|\mathbf{M}^{+}_{y}\right|\\ -\mathbf{M}^{-}_{y}\cdot\Delta\mathbf{\varphi}^{-}+\mathbf{M}^{+}_{y}\cdot\Delta\mathbf{ \varphi}^{+}\leq\frac{m}{2(m-1)}\Delta\widetilde{\mathbf{\varphi}}^{T}\mathbf{Z}\Delta \widetilde{\mathbf{\varphi}}\end{array} \right. \tag{27}\]  
For the proposed example, the resulting \(\mathbf{Z}\), obtained using the assigned data, was calculated and is  
Figure 5: Sample point numbering and position. \[Z=\]  
\[Z=\]  
\[ \begin{array}{|c|cccccccc|}0&0&0&0&0&0&0&0\\ 0&-4.18\times 10^{5}&-4.18\times 10^{5}&-8.36\times 10^{5}&-8.36\times 10^{5}&-4.18 \times 10^{5}&-4.18\times 10^{5}&0\\ 0&-4.18\times 10^{5}&-4.18\times 10^{5}&-8.36\times 10^{5}&-8.36\times 10^{5}&-4.18 \times 10^{5}&-4.18\times 10^{5}&0\\ 0&-8.36\times 10^{5}&-8.36\times 10^{5}&-1.67\times 10^{6}&-1.67\times 10^{6}&-8.36 \times 10^{5}&-8.36\times 10^{5}&0\\ 0&-8.36\times 10^{5}&-8.36\times 10^{5}&-1.67\times 10^{6}&-1.67\times 10^{6}&-8.36 \times 10^{5}&-8.36\times 10^{5}&0\\ 0&-4.18\times 10^{5}&-4.18\times 10^{5}&-8.36\times 10^{5}&-8.36\times 10^{5}&-4.18 \times 10^{5}&-4.18\times 10^{5}&0\\ 0&-4.18\times 10^{5}&-4.18\times 10^{5}&-8.36\times 10^{5}&-8.36\times 10^{5}&-4.18 \times 10^{5}&-4.18\times 10^{5}&0\\ 0&0&0&0&0&0&0&0\\ \end{array}