# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 5 Numerical Example: Synthesis of a Ten-Bar Truss Cantilever
### Discussion of Results
Considering these solutions, one can gain information about the influence of changeable topology in the final result with respect to different optimization constraints (with/without shape optimization and with/without buckling constraints). The optimal discrete sizing variables and continuous shape variables are therefore presented for both the initial and the optimal topologies of the four performed syntheses. The results are given in Tables 6 and 7 and graphically presented in Figures 9-12. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Example TC10a**} & \multicolumn{2}{c}{**Example TC10b**} \\ \cline{2-5}  & **Initial Topology** & **Optimal Topology** & **Initial Topology** & **Optimal Topology** \\ \hline \(A_{1,3}\) (cm\({}^{2}\)) & 96.77 & 96.77 & 141.94 & 141.94 \\ \(A_{1,4}\) (cm\({}^{2}\)) & 109.68 & 109.68 & 64.51 & 64.51 \\ \(A_{2,3}\) (cm\({}^{2}\)) & 19.35 & 19.35 & 51.61 & 51.61 \\ \(A_{2,4}\) (cm\({}^{2}\)) & 167.74 & 180.64 & 141.94 & 141.94 \\ \(A_{3,4}\) (cm\({}^{2}\)) & 19.35 & 19.35 & 6.45 & - \\ \(A_{3,5}\) (cm\({}^{2}\)) & 96.77 & 96.77 & 51.61 & 51.61 \\ \(A_{3,6}\) (cm\({}^{2}\)) & 6.45 & - & 51.61 & 51.61 \\ \(A_{4,5}\) (cm\({}^{2}\)) & 141.94 & 141.94 & 6.45 & - \\ \(A_{4,6}\) (cm\({}^{2}\)) & 6.45 & - & 109.68 & 109.68 \\ \(A_{5,6}\) (cm\({}^{2}\)) & 6.45 & - & 96.77 & 96.77 \\ \hline \({}^{c}\)x\({}_{4}\) (cm) & & & 827.2361 & 909.1875 \\ \({}^{c}\)y\({}_{4}\) (cm) & & & 833.5299 & 801.3623 \\ \({}^{c}\)x\({}_{6}\) (cm) & & & 1566.7881 & 1580.4308 \\ \({}^{c}\)y\({}_{6}\) (cm) & & & 432.9369 & 417.6378 \\ \hline WEIGHT (N) & 19,492.7981 & **19,266.5406** & 17,168.1830 & **16,936.2314** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimal results, examples without buckling constraints TC10a (topology optimization) and TC10b (topology and shape optimization). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Example TC10c**} & \multicolumn{2}{c}{**Example TC10d**} \\ \cline{2-5}  & **Initial Topology** & **Optimal Topology** & **Initial Topology** & **Optimal Topology** \\ \hline \(A_{1,3}\) (cm\({}^{2}\)) & 387.06 & 387.06 & 387.06 & 387.06 \\ \(A_{1,4}\) (cm\({}^{2}\)) & 180.64 & - & 77.42 & - \\ \(A_{2,3}\) (cm\({}^{2}\)) & 109.68 & 109.68 & 51.61 & 51.61 \\ \(A_{2,4}\) (cm\({}^{2}\)) & 67.74 & 77.42 & 109.68 & 109.68 \\ \(A_{3,4}\) (cm\({}^{2}\)) & 6.45 & 270.94 & 6.45 & 154.84 \\ \(A_{3,5}\) (cm\({}^{2}\)) & 96.77 & 270.94 & 270.94 & 270.94 \\ \(A_{3,6}\) (cm\({}^{2}\)) & 419.25 & - & 167.74 & - \\ \(A_{4,5}\) (cm\({}^{2}\)) & 6.45 & 96.77 & 6.45 & 77.42 \\ \(A_{4,6}\) (cm\({}^{2}\)) & 64.51 & - & 96.77 & - \\ \(A_{5,6}\) (cm\({}^{2}\)) & 51.61 & - & 77.42 & - \\ \hline \({}^{c}\)x\({}_{4}\) (cm) & & & 303.1833 & 1311.2365 \\ \({}^{c}\)y\({}_{4}\) (cm) & & & 829.4580 & 484.7189 \\ \({}^{c}\)x\({}_{6}\) (cm) & & & 1327.1457 & - \\ \({}^{c}\)y\({}_{6}\) (cm) & & & 486.3984 & - \\ \hline WEIGHT & 41,838.6690 & **32,204.1317** & 28,553.4737 & **26,357.4575** \\ (N) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Optimal results, examples with buckling constraints TC10c (topology optimization) and TC10d (topology and shape optimization). Figure 10: Optimal solutions, example TC10b (topology and shape optimization, without buckling constraints). (**a**) Initial topology; (**b**) optimal topology. Figure 9: Optimal solutions, example TC10a (topology optimization, without buckling constraints). (**a**) Initial topology; (**b**) optimal topology. Table 6 comprises the results of the optimization without buckling constraints. The results of example TC10a can be regarded as the solutions of discrete sizing optimization at the fixed shape and topology (initial topology) and at fixed shape and changeable topology (optimal topology). Similarly, the results of example TC10b can be regarded as the solutions of the shape and discrete sizing optimization at the fixed topology (initial topology) and at the changeable topology (optimal topology).