# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 5 Numerical Example: Synthesis of a Ten-Bar Truss Cantilever
### The Superstructure and Topological Alternatives  
The truss cantilever is composed of 6 joints and 10 joint inter-connections--bars. The basis of the superstructure is composed from identical sets \(i\) and \(j\), \(i\in I\), \(j\in I\), where  
Figure 5: The ten-bar truss cantilever. \(I\equiv J=\{\)1,2,3,4,5,6\(\}\). Regarding the defined bars in Figure 5, the inter-connection matrix **EL** is:  
\[\textbf{EL}=[el_{i,j}]=\left[ \begin{array}{cccccc}0&0&1&1&0&0\\ 0&0&1&1&0&0\\ 0&0&0&1&1&1\\ 0&0&0&0&1&1\\ 0&0&0&0&0&1\\ 0&0&0&0&0&1\\ 0&0&0&0&0&0\end{array} \right]\]  
Since each nodal load represents a separate load case, there are two load cases defined, \(k\)\(\in\)\(K\) = {1,2}. According to the MINLP-TS formulation, the load vectors are defined as \(\mathbf{p}_{\text{x,1,}i}\)\(\equiv\)\(\mathbf{p}_{\text{x,2,}i}\) = {0,0,0,0,0,0}, \(\mathbf{p}_{\text{y,1,}i}\) = {0,0,-P,0,0,0} and \(\mathbf{p}_{\text{y,2,}i}\) = {0,0,0,0,-P,0}. Joints 1 and 2 are supported in both \(X\) and \(Y\) directions; thus, the supporting vectors are \(\mathbf{s}_{\text{x,}i}\)\(\equiv\)\(\mathbf{s}_{\text{y,}i}\) = {1,1,0,0,0,0}. A single topological binary variable \(y_{i,i}^{\text{top}}\) is defined for each of the six joints \((y_{1,1}^{\text{top}},y_{2,2}^{\text{top}},y_{3,3}^{\text{top}},\)\(y_{4,4}^{\text{top}},y_{5,5}^{\text{top}},y_{6,6}^{\text{top}})\), and a single topological binary variable \(y_{i,j}^{\text{top}}\) is subjected to each of the ten bars, i.e., to each pair of joints \(i\) and \(j\), for which the inter-connection coefficient \(el_{i,j}=1\) (\(y_{1,3}^{\text{top}},y_{1,4}^{\text{top}},y_{2,3}^{\text{top}},y_{2,4}^{\text {top}},y_{3,4}^{\text{top}},y_{3,5}^{\text{top}},y_{3,6}^{\text{top}},y_{4,5}^ {\text{top}},y_{4,6}^{\text{top}},y_{5,6}^{\text{top}})\). The remaining topological binary variables \(y_{i,j}^{\text{top}}\) are not active and are set to zero values. Binary variables are also defined for discrete sizing alternatives. Regarding the 16 different alternative discrete values from the defined vector \(\mathbf{q}_{i,j,l}\), an extra binary variable \(y_{i,j,l}^{\text{gt}}\) is defined for each \(l\)-th discrete value for each \(i\)\(\rightarrow\)\(j\) bar. For the 10 defined bars, the considered problem thus involves 10 \(\times\) 16 = 160 binary variables for discrete dimensions. In total, 6 + 10 + 160 = 176 binary variables are defined. With respect to the supporting and loading of the structure as well as the arrangement of bars, joints 1 to 5 are fixed joints and will be included in all possible topology alternatives, while joint 6 is an alternative joint, since it may be included in or removed from the superstructure. In addition, bars 1\(\rightarrow\)3, 3\(\rightarrow\)5 and 2\(\rightarrow\)4 are fixed bars, while all other bars represent alternative structural elements. Regarding the requirement for kinematical stability of the structure, a set of inter-connection logical constraints (18) for the present example can be written using Equations (18a-d) as follows:  
\[y_{1,A}^{\text{top}}+y_{2,3}^{\text{top}}+y_{3,4}^{\text{top}}-2\geq 0 \tag{18a}\] \[y_{3,6}^{\text{top}}+y_{4,5}^{\text{top}}+y_{5,6}^{\text{top}}-y_{6,6}^{\text {top}}-1\geq 0\] (18b) \[y_{i,6}^{\text{top}}-y_{6,6}^{\text{top}}\leq 0,\;i\in I,\;\forall\;(i,6) \;\Big{|}\;el_{i,6}=1\] (18c) \[y_{4,6}^{\text{top}}-y_{6,6}^{\text{top}}=0 \tag{18d}\]  
Equation (18a) concerns the panel 1-2-3-4 in which at least two bracing bars, i.e., one of the diagonals (1\(\rightarrow\)4 or 2\(\rightarrow\)3) and the vertical (3\(\rightarrow\)4), or just both diagonals or all three of them have to be selected in order to provide the kinematical stability. The possible arrangements thus define four different topological alternatives for panel 1-2-3-4. In panel 3-4-5-6, joint 6 can be removed; therefore, according to Equation (18b), at least two bracing members have to be present when joint 6 is selected (\(y_{6,6}^{\text{top}}=1\)), and a single bracing member is determined if joint 6 is rejected (\(y_{6,6}^{\text{top}}=0\)).