# Size and Shape Optimization of a Guyed Mast Structure under Wind, Ice and Seismic Loading
## 4 Finite Element Modeling  
The structural model was developed using two different element types: beams and cables. Beam elements model the main pole and all structural elements except for the cables. They possess the geometric and material properties of structural elements. The beam elements are used to model the main pole and secondary elements. Moreover, except for the main pole, rotation releases are applied at the ends in order to consider no flexural rigidity, as occurring for trussed structures. Cable elements are used to simulate the steel ropes. Cable elements undergo large displacements that give rise to geometric nonlinearities. Therefore, the equilibrium of the cables is considered in the deformed configuration using SAP2000. As a result, the structural behaviour of guyed towers can be highly nonlinear, especially for low pre-tension cables, which are prone to large displacements. On the contrary, the nonlinear behaviour becomes less pronounced by increasing the pre-tension, resulting in high compression levels and minor instability effects. This paper considers the envelope of the maximum and minimum responses associated with each load condition. Figure 5 plots the three modes with a higher mass participation ratio. These are the 10 th, 11 th, and 12 th modes obtained from the dynamic analysis of the mast structure with the dead loads. On the contrary, the first modes arising from the dynamic analysis have lower mass participation factors and are characterized by local deformation of the structural elements. The 10 th, 11 th, and 12 th modes are the first modes exhibiting the global deformation of the mast structure. X and Y identify the in-plane orthogonal directions. The 10 th and 11 th modes have an approximate 26% mass participation factor in the Y and X directions, respectively. The natural period is very low and at approximately 0.4 s. The 13 th mode is mainly torsional with nearly a 7 and 4% mass participation factor in the X and Y directions. Figure 6 shows the positive (in dark green and purple) and negative (in red and light green) maximum and minimum envelopes of the axial, shear forces, and bending moments acting on the structural elements. Figure 7 plots the performance ratios of all structural elements except for the cables. The performance ratio is the ratio between the maximum stress in the structural element and the yielding stress. The performance ratios are defined by the colour map next to Figure 7. The plots highlight the presence of a structural element  
Figure 4: Elastic response spectrum corresponding to the service limit state (SLV), where \(Sa\) is the spectral acceleration. in the first half of the central pole with a high-performance ratio, depicted in yellow. The first section of the central pole has a low performance ratio, lower than 0.25. After the section with a performance ratio in the range 0.4-0.65, the following sections fall in the range 0.25-0.4 and are coloured in green. The top sections of the central pole are not significantly stressed, with a performance ratio of 0-0.25. The bracings have low stress, plotted in cyan, with performance ratios of 0-0.25. Figure 8 shows the maximum displacements in the X (\(u_{1}\)), Y (\(u_{2}\)) directions and their combination (\(u_{t}\)) at the service limit state. The maximum displacement is located at the top of the tower, in particular at joint 6 (z = 30.00 m), with a maximum displacement equal to \(u_{t}\) = 18.7 mm. Figure 5: (**a**) Mode 10 th—Ts = 0.437 s—mass participation ratio X = 9.6%, Y = 26.2%; (**b**) Mode 11 th—Ts = 0.434 s—mass participation ratio X = 26.4% Y = 9.2%; (**c**) Mode 12—Ts = 0.206 s—mass participation ratio X = 7.2% Y = 4.4%. Figure 6: (**a**) Axial force, (**b**) shear force (\(V_{2}\)), (**c**) bending moment (\(M_{2}\)), (**d**) shear force (\(V_{3}\)), (**e**) bending moment (\(M_{3}\)). Figure 7: Performance ratios of the pole before optimization. Cables are depicted with magenta colour because their performance ratios are not included in the current representation.