# 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
The results presented in Table 7 can be classified equivalently for the case of optimization with buckling constraints included. In this way, the achieved improvement in results at different optimization levels (topology and/or shape optimization) with regard to sizing optimization at fixed shape and topology (examples TC10a and TC10c at initial topology) can be estimated. The results of the comparative analysis are presented in Table 8.  
In the case of optimization with stress and displacement constraints only (buckling is not included), surprisingly little decrease in optimal weight was achieved by the topology optimization. The shape optimization (at the fixed topology) yielded a considerably better result. However, when the buckling constraints were included, the topology optimization led to a considerably higher improvement, while shape optimization still yielded a better result. As expected in both cases, the best result was obtained by combining the topology and shape optimizations, yet it is obvious that almost a three times higher improvement in result (decrease in weight) was achieved when buckling constraints were included when  
Figure 11: Optimal solutions, example TC10c (topology optimization, with buckling constraints). (a) Initial topology; (b) optimal topology. Figure 12: Optimal solutions, example TC10d (topology and shape optimization, with buckling constraints). (a) Initial topology; (b) optimal topology. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Buckling**} & \multicolumn{3}{c}{**Optimization Level**} \\ \cline{2-4}  & **Shape** & **Topology** & **Shape and Topology** \\ \hline Not included & 11.93 & 1.16 & 13.12 \\ Included & 31.75 & 23.03 & 37.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Improvement in final result (in %) at different optimization levels. compared to the case of the optimization with stress and displacement constraints only (37.00% vs. 13.12%). A more detailed analysis of the obtained results leads to a further finding. As has already been stated in the introductory sections, topology of truss structures was, in the past, often optimized by allowing very small (nonzero) lower bounds on cross-section areas of bars. When the cross-section areas of some bars were valued to the defined lower bounds, these bars were assumed to be redundant and were therefore excluded from the structure. Consequently, the optimal topology was adopted as the initial topology with the excluded >lower bound bars<. When it comes to the results of the proposed MINLP truss synthesis, this assumption proves to be correct when the buckling constraints are excluded. The obtained optimal topologies in examples TC10a and TC10b are simply the initial topologies, where the bars with the lowest discrete cross-section area (6.45 cm\({}^{2}\)) are excluded; see Table 6. This can be seen very clearly in Figures 9 and 10. On the other hand, when the buckling constraints are included, the discussed subject turns out to be different. In the optimal topologies of examples TC10c and TC10d, those bars are active, which, in the initial topology, took the lowest value. Excluding these bars would therefore lead to a non-optimal topology. Beside stress, the buckling of compressed bars, is one of the basic conditions that has to be accounted for in the design process of a truss structure. In the initial topology, the compressed bars are subjected to buckling constraints and therefore require larger areas of cross-sections compared to tension bars, which are subjected to stress constraints only. The method of topology optimization with the elimination the least stressed bars mentioned above is thus not appropriate for real engineering structures. Finally, the results of the proposed MINLP truss synthesis are compared to some results available in literature. The comparison of the optimal solution of example TC10a is presented in Table 9. The MINLP synthesis yielded the same result as the currently best solution presented in [49].