# Size and Shape Optimization of a Guyed Mast Structure under Wind, Ice and Seismic Loading
## 5 Structural Optimization
#### Constraints Involved in the Structural Optimization Problem  
The optimization problem statement is reported in (4) and the constraints were treated with the penalty-based approach illustrated in (5), by converting the constrained problem into an equivalent unconstrained one. The resolution of the optimization task considers the structural design assessment required by national and international codes to ensure the safety of constructions. In particular, the structural verifications derive from Eurocode 3  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Measure** & **Value** \\ \hline \(\Phi_{0}\) & [mm] & 168.3 \\ \(t_{0}\) & [mm] & 12.5 \\ L & [mm] & 6000 \\ Mass & [kg] & 288 \\ no elements & [-] & 5 \\ Total Mass & [kg] & 1440 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Total mass of the main pole.  
Figure 10: Parametric study on the design variables involved and representation of the different scenarios described in Table 5.  
(EN 1993-1-1: 2005) and are referred to the ultimate limit state (ULS). The design verifications include tensile, compression, and buckling verification, and a combined assessment, such as the interaction capacity according to Anne B of the Eurocode 3:  
\[\frac{D}{C} = \frac{N_{Ed}}{\frac{\chi_{y}Af_{yk}}{\gamma_{M1}}}+k_{yy}\frac{M_{ y,Ed}}{\frac{\chi_{LT}W_{pl,y}f_{yk}}{\gamma_{M1}}}+k_{yz}\frac{M_{z,Ed}}{\frac{W_{pl,z}f_{yk}}{ \gamma_{M1}}}\leq 1 \tag{8}\] \[\frac{D}{C} = \frac{N_{Ed}}{\frac{\chi_{z}Af_{yk}}{\gamma_{M1}}}+k_{zy}\frac{M_ {y,Ed}}{\frac{\chi_{LT}W_{pl,y}f_{yk}}{\gamma_{M1}}}+k_{zz}\frac{M_{z,Ed}}{ \frac{W_{pl,z}f_{yk}}{\gamma_{M1}}}\leq 1 \tag{9}\]  
where \(D\) stands for the demand and \(C\) stands for the capacity of the structure. Specifically, \(N_{Ed}\) is the acting axial force, whereas \(M_{y,Ed}\) and \(M_{z,Ed}\) represent the acting bending moments in the two principal directions of a planar local reference system centered on the cross section center of gravity. \(A\) is the cross section area of the pole, \(W_{pl,y}\) and \(W_{pl,z}\) are the plastic section modulus in the two principal directions, \(f_{yk}\) is the yielding strength of the steel, whereas \(\gamma_{M1}\) is the partial safety factor for instability conditions, equal to 1.05 from the Italian National Anne. \(\chi_{LT}\) is the reduction factor for lateral-torsional buckling, whereas _kyy_, _kyz_, _kzy_, and _kzz_ are interaction factors whose values are derived according to two alternative approaches based on Anne A (method 1)and Anne B (method 2). The global structural deformation referred to the service limit state (SLS) has also been considered by verifying the top displacement of the mast. Specific recommendations for guyed mast structures are missing in national and international codes. Therefore, the authors adopted the suggestions defined in the Italian Technical Code NTC2018 (D.M.17/01/2018) reported in Chapter 4.2.4.2.2 Table 4.2.XIII related to limitations of lateral displacements of steel multi-storey frame structures. These limitations express a threshold condition in terms of the total height of the structure \(H\):  
\[\delta_{\text{SLS,top}}\leq\delta_{\text{SLS,top,lim}}=\frac{H}{500}=\frac{3000 0\text{ mm}}{500}=60\text{ mm} \tag{10}\]  
Since this condition is specific for steel multi-storey frame structures, the authors will assume this value as a reasonable choice to ensure service life assessment and preservation of working conditions of the telecommunication guyed mast tower. In the next section, a discussion on the results is carried out.