# The MINLP Approach to Topology, Shape and Discrete Sizing Optimization of Trusses
## 5 Numerical Example: Synthesis of a Ten-Bar Truss Cantilever
### The MINLP Syntheses  
Using the developed MINLP-TS formulation, the following examples of synthesis of the ten-bar truss cantilever were performed:  
1. Example **TC10a**: _Topology and discrete dimension optimization without buckling constraints_. The structure is optimized at a fixed shape (at fixed nodal coordinates) under stress and displacement constraints, while the buckling constraints Equation (15) are excluded. The independent variables are the cross-sectional areas of bars \(A_{i,j}\), with the lower/upper bounds defined to be equal to the smallest and the largest discrete value of vector \(q_{i,j,l}\) (\(A_{i,j}^{\text{exLO}}\) = 6.45 cm\({}^{2}\), \(A_{i,j}^{\text{UP}}\) = 225.81 cm\({}^{2}\)). 2. Example **TC10b**: _Topology, shape and discrete dimension optimization without buckling constraints_. Alongside the sizing variables \(A_{i,j}\), the joint coordinates (\({}^{\text{c}}x_{i}\), \({}^{\text{c}}y_{i}\)), too, are now the independent variables. Considering the supporting and loading conditions, the coordinates of joints 4 and 6 are changeable, while the other joints are fixed. The bounds on shape variables are: \({}^{\text{c}}x_{\text{LO}}^{\text{LO}}\) = 114 cm, \({}^{\text{c}}x_{\text{UP}}^{\text{LP}}\) = 1914 cm, \({}^{\text{c}}y_{\text{4}}^{\text{LO}}\) = \({}^{\text{c}}y_{\text{6}}^{\text{LO}}\) = 214 cm, \({}^{\text{c}}y_{\text{4}}^{\text{UP}}\) = \({}^{\text{c}}y_{\text{6}}^{\text{UP}}\) = 1214 cm, \({}^{\text{c}}x_{\text{6}}^{\text{LO}}\) = 1028 cm, \({}^{\text{c}}x_{\text{6}}^{\text{UP}}\) = 1928 cm. Figure 6: Fixed and alternative structural elements. Example **TC10c**: _Topology and discrete dimension optimization including buckling constraints._ Buckling constraints Equation (15) are added to the example _TC10a_. The limit buckling stress of a compressed element is considered to be equal to the Euler buckling stress for circular cross-sections; thus, \(\sigma_{i,j}^{\rm cr}=A_{i,j}\pi\mathrm{E}/(4s_{i,j}^{\rm sf2})\). The buckling lengths of elements are considered to be equal to the system lengths for both in-plane and out-of-plane buckling. Taking into consideration the buckling constraints, the cross-sectional areas of bars are expected to take higher values. The vector of discrete values is thus expanded to:  
The new vector contains 24 alternative discrete values, and the total number of active binary variables is 10 \(\times\) 24 + 8 = 248. Example **TC10d**: _Topology, shape and discrete dimension optimization including buckling constraints._ The bounds on shape variables are the same as defined in example TC10b, while the buckling constraints and the vector of discrete cross-sectional areas are the same as in example TC10c. All four examples of truss synthesis were carried out using both the MTP and the STP strategies in order to find and estimate the advantages of each MINLP strategy. In all the performed optimizations, the process was terminated after the theoretical result \(z_{n}^{\rm t}\) of topology optimization with continuous sizing variables had not improved in five successive MINLP iterations; thus, \(n_{\rm fin}=5\). When pre-screening of binary variables for discrete dimensions was used, two neighboring (one lower and one upper) discrete values to the corresponding continuous value obtained in the previous continuous sizing optimization were active in the successive discrete sizing optimization. All other discrete values were temporarily deactivated, i.e., their associated binary variables were given zero value. The tests showed that in the case of discrete sizing optimization at fixed topology, the first feasible solution always represented the optimal result, while all following feasible solutions represented worse results. Thus, each discrete sizing optimization phase was concluded when the first feasible solution was obtained. As an interface for mathematical modeling and data inputs/outputs GAMS (General Algebraic Modeling System) [81] was used.