# Research on the Scope of Spectral Width Parameter of Frequency Domain Methods in Random Fatigue
## 5 Example Analysis  
This article will take the canopy of a stadium as the research object. ANSYS, a large-scale general finite element analysis (FEA) software developed by American ANSYS, is used to model and perform random vibration simulation analysis, and the Matlab software is used to perform random fatigue life analysis on the random fatigue response stress power spectral density to verify the results obtained in this paper. The concrete structure and finite element model of the stadium stand canopy are shown in Figure 9.  
A cantilever steel truss structure is used for the grandstand canopy. The canopy area is 1729 m\({}^{2}\) and the amount of steel used is 36.5 kg/m\({}^{2}\). The basic members are round steel pipes and the structure is divided into nine steel trusses with a spacing of 7.8 m. Each steel truss consists of two parallel circular arc steel pipes 2.6 m apart to form its upper chord plane, and two circular arc steel pipes joined together form its lower chord, which is connected by a web bar in the middle. Nine steel trusses are connected to the foundation by nine reinforced concrete columns. The steel truss is joined to the column by hinge joints. In order to ensure its geometrical stability, two tension rods extending from the bottom of the column are connected with two upper chord rods, respectively, and then each steel truss is connected into a stable whole by the purlin and the support. According to the requirements of force and function, five sizes of round steel pipe are used in the structure: upper chordal longitudinal bar \(\Phi 168\times 10\), lower chordal longitudinal bar \(2\Phi 168\times 10\), upper chordal transverse bar, diagonal bar and web bar (excluding support) \(\Phi 89\times 4\), support web \(\Phi 133\times 8\), and cable-stayed bar \(\Phi 168\times 6\). The two lower chords are combined to form a triangular truss system. The modal analysis of the above finite element model is carried out. First, the dynamic characteristics of the stadium stand canopy model are solved by the modal solution method, and the multi-step natural frequencies and modes are obtained. The first 10 natural frequencies are shown in Table 2.  
To analyze the stress situation of the structure, this paper mainly considers the effect of wind load. A surface load is applied above the stand shed. To cover the pressure power spectral density of the first 260 natural frequencies, the load frequency range is 0-100 Hz, as shown in Figure 10. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Mode Shape** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline
**Frequency/Hz** & 2.37 & 3.25 & 3.68 & 4.07 & 4.40 & 4.66 & 4.85 & 5.47 & 6.30 & 7.12 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculated natural frequency of structure. Figure 9: Stadium stand hood model: (**a**) Real image; (**b**) finite element model. Random vibration spectrum analysis is performed on the stand shed model, and the axial stress and displacement solutions of the stand shed are extracted, as shown in Figure 11.  
The stress power spectral density of the maximum stress point of the response axis is obtained by finite element calculation, as shown in Figure 12.  
Figure 11: Finite element results: (**a**) Axial stress; (**b**) relative displacement. Figure 10: Pressure load spectrum. It can be seen from Figure 12 that the first peak of the stress response spectrum occurs at 3.2576 Hz, that is, the second natural frequency of the structure, which corresponds to the mode of the structure. For the stress response, the first peak occupies the main part, and the latter is very small. The random fatigue life analysis of the structure of the bleacher tent was carried out. First, the axial stress at the maximum stress of the structure is 106.3 MPa.