# Research on Equivalent Static Load of High-Rise/Towering Structures Based on Wind-Induced Responses
## 2 Refinement of Structural Wind-Induced Response Assessment Method
### Comparison of Time-Frequency Domain Results for the Full Three Components  
In this section, the wind-induced vibration response of a single-pole lightning rod structure will be studied to compare the frequency domain method and time domain method. The study will include displacement, bending moment and shear force. A lightning rod is a typical towering structure with a small natural frequency and significant wind-induced vibration effect, which often causes engineering accidents [36]. The physical parameters of the lightning rod investigated in this section are: total height 60 m, tower tip height 2 m, tower height 58 m. It is composed of seven sections of variable-thickness thin-walled steel pipe, with a diameter from 1.5 m tapered to 0.15 m, a wall thickness from 16 mm tapered to 5 mm; the material is Q345 steel, and the steel pipes are connected by plug-in adder flange. The steel pipe size is shown in Figure 1b. The simulation was carried out in the ANSYS finite element software, using Beam188 unit, and each of the lower six sections of steel pipe were divided into five units; each upper section was equivalent to 1 unit, and a total of 31 units were used in the model. In order to simplify the calculation, it does not include the tip of the tower, and does not consider the overlap areas between the steel pipes; the steel cylinder is only the interface connection and rigid connection. The five orders of vibration pattern for structure X-direction are shown in Figure 1c, with frequencies of 0.75, 2.09, 4.31, 7.46 and 11.66 Hz. Only the along-wind wind-induced response is analyzed in the simulation.  
The lightning rod operates in a Class B site with a basic wind speed \(U_{10}=35.8\) m/s, wind profile power index \(\alpha=0.15\), turbulence \(I_{10}=0.14\), and air density \(\rho=1.25\) kg/m\({}^{3}\). The section drag coefficient \(C_{\rm D}\) takes the value of 0.6 according to the load code [37], the Davenport wind spectrum was considered in this case, and the Davenport coherence function exponential decay coefficient is \(C_{Z}=7\), \(C_{X}=8\). The harmonic superposition method is used to simulate the pulsating wind load time course, and the other parameters are selected according to the specifications. Rayleigh damping is used for the simulation and the modal damping ratio is taken as \(\zeta=0.01\).  
Figure 2 shows the curve of the displacements, bending moments and shear forces (\(\sigma_{\text{T}}\), \(\sigma_{\text{B}}\) and \(\sigma_{\text{R}}\), respectively), obtained by the frequency domain method and the time domain method, as well as the percentage of the coupled components in the total pulsation response obtained by the time domain method. As shown in the picture, the \(\sigma_{\text{T}}\) values obtained from the frequency domain and time domain methods are in good agreement, and the deviations in the displacement of the tower tip, the base bending moment and the base shear force are all within 1.5%. As regards the displacement and bending moment responses, the background response obtained by the frequency domain method is smaller, and the resonance response is larger, than those obtained by the time domain method. The percentage of \(\sigma_{\text{BR}}\) in the total pulsation response for each variable is less than 2%, so the effect of neglecting \(\sigma_{\text{BR}}\) on the total pulsation response is small.  
Figure 1: Geometry parameters and the FE model of the lightning rod.  
Figure 2: Comparison of time/frequency domain results and the percentage of coupling components.