# Size and Shape Optimization of a Guyed Mast Structure under Wind, Ice and Seismic Loading
## 5 Structural Optimization  
In optimization problems, the main goal is to find the best conditions in terms of the optimal set of design parameters collected in the design vector \(\mathbf{x}\), which minimizes an objective function (OF) \(f(\mathbf{x})\)[45, 46, 47]. These problems can be categorized into single-objective or multi-objective based on the number of OFs involved, and a further classification is based on the presence (or not) of constraints [48, 49, 50]. In the structural optimization field, it is common to deal with constrained optimization, whose general statement is [51]:  
\[\min_{\mathbf{x}\in\Omega}\{f(\mathbf{x})\}\] (4) s.t. \[g_{q}(\mathbf{x})\leq 0\quad\forall q=1,\ldots,n_{q}\] \[h_{r}(\mathbf{x})=0\quad\forall r=1,\ldots,n_{r}\]  
where \(\mathbf{x}=\{x_{1},\ldots,x_{j},\ldots,x_{n}\}^{T}\) is the design vector to be optimized, whose terms are limited into a hyper-rectangular multidimensional box-type search space domain of interest denoted as \(\Omega\), given by the Cartesian product of the range of interest of each \(j\)-th of each design variable bounded in \([x_{j}^{l},x_{j}^{u}]\), \(\Omega=[x_{1}^{l},x_{1}^{u}]\times\ldots\times[x_{j}^{u},x_{j}^{u}]\times \ldots\times[x_{n}^{l},x_{n}^{u}]\). The term \(g_{q}(\mathbf{x})\) in (4) denotes inequality constraints whereas \(h_{r}(\mathbf{x})\) are equality ones, which further reduce the feasible search space inside \(\Omega\). In structural optimization, it is typical to deal with inequality constraints, and a common goal is to minimize the global cost of the structure. Since this involves many terms, the main attempt is minimizing the self-weight of the structure, indirectly connected to material cost, i.e., material usage and natural resources consumption [51]. Several strategies have been developed over the years to handle constraints [52, 53, 54]. In the present work, the penalty function-based approach was implemented due to its simplicity, allowing converting the problem with OF \(f(\mathbf{x})\) into a new unconstrained version \(\phi(\mathbf{x})\):  
\[\min_{\mathbf{x}\in\Omega}\{\phi(\mathbf{x}))\}=\min_{\mathbf{x}\in\Omega}\{f(\mathbf{x})+H( \mathbf{x})\} \tag{5}\]  
where \(H(\mathbf{x})\) is the penalty function. Adopting a static-penalty strategy, \(H(\mathbf{x})\), assume this form [55, 56]  
\[H_{s}(\mathbf{x})=w_{1}H_{NVC}(\mathbf{x})+w_{2}H_{SVC}(\mathbf{x}) \tag{6}\]  
where \(H_{NVC}\) is the number of violated constraints and \(H_{SVC}\) is the sum of all violations:  
\[H_{SVC}(\mathbf{x})=\sum_{p=1}^{n_{p}}\max\{0,g_{p}(\mathbf{x})\} \tag{7}\]  
\(w_{1}\) and \(w_{2}\) are the violation control parameters, whose numerical values are assumed equal to \(w_{1}=w_{2}=100\) following [55]. In the current study, the authors carried out a parametric study on the design variables of the guyed mast. This fact has led to eight different scenarios, summarized in Table 5. In addition, the starting initial values of the design parameter are listed in Table 6, while the general optimization workflow is illustrated in Figure 9. To compare the results, the focus is related only to the performance ratios PR of the central pole of the guyed radio mast, being the pole the most stressed element. It consists of five segments 6.00 m long with the same cross-section. Thus, starting from the ground level:  
1. Pole\({}_{1}\) (0.00 to 6.00 m);
2. Pole\({}_{2}\) (6.00 to 12.00 m);
3. Pole\({}_{3}\) (12.00 to 18.00 m);
4. Pole\({}_{4}\) (18.00 to 24.00 m);
5. Pole\({}_{5}\) (24.00 to 30.00 m). Starting with a constant diameter of the cross-section for the pole, at the end of the optimization, it is advisable to find a tapered solution following a linear relationship withthe height, as represented in Figure 10f. Accordingly, it is possible to shape the pole cross-section with two design variables described by the bottom \(\Phi_{i}\) and top \(\Phi_{f}\) diameters.