# Research on Equivalent Static Load of High-Rise/Towering Structures Based on Wind-Induced Responses

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## Abstract

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## 1. Introduction

## 2. Refinement of Structural Wind-Induced Response Assessment Method

_{j}(t) is the jth modal generalized coordinate time course, ζ

_{j}is the jth modal damping ratio, ω

_{j}is the jth modal natural vibration angular frequency, F

_{j}(t) is the jth modal generalized load, ϕ

_{ji}is the vibration displacement component of the ith node of the jth vibration mode, ρ is the air density, generally taken as 1.25 kg/m

^{3}, C

_{D}is the quasi-constant wind pressure coefficient, A

_{i}is the windward area of the ith node, $\overline{U}({z}_{i})$ is the average wind speed of incoming flow at height z

_{i}, and u(z

_{i}, t) is the time course of incoming pulsating wind speed at height z

_{i}.

#### 2.1. The Full Three-Component Expressions in the Frequency Domain

_{FiFj}(f) can be expressed as:

_{FiFj}(f) is known, then the total pulsation response spectrum S

_{qj}(f) of the modal generalized coordinates q

_{j}can be expressed as Equation (7), and then using the modal superposition principle (Equation (3)), the response spectrum S

_{yi}(f) at node i of the structure can be obtained, as in Equation (8).

_{k}(f) is the kth modal complex frequency response function, and ${H}_{j}^{*}(f)$ is the conjugate of ${H}_{j}(f)$. The response spectra of the bending moment and shear force can be obtained by replacing the vibration displacement ϕ in Equation (8) with bending moment or shear force.

_{Ti}can be calculated according to Equation (10), and it contains three components: the background component σ

_{B}, the resonance component σ

_{R,}and the background–resonance coupling component σ

_{BR}. Because of the complicated nature of the calculation for σ

_{T}, researchers often choose to determine each component, and then calculate σ

_{T}according to Equation (11). The expressions for each component are as follows.

_{1}, z

_{1}) and (x

_{2}, z

_{2}), g

_{u}is the peak wind speed factor, generally taken as 3.5, g

_{R}is the peak factor of the resonant response, and ${\rho}_{BR}$ is the background resonant coupled response mutual relationship number [32].

_{R}and σ

_{BR}are quite complicated, and are not convenient for engineering applications. As high and flexible structures such as high-rise buildings and towering structures have a sparse natural frequency, the wind vibration response of such structures is mainly related to the first few modes, so the simplification of the calculation of such buildings can be carried out by ignoring the contribution of the coupling term between each structural vibration mode and the coupling terms of background and resonance. At this point, the total pulsation response of the structure can be expressed via the following equation.

#### 2.2. The Full Three-Component Expressions in Time Domain

#### 2.3. Comparison of Time–Frequency Domain Results for the Full Three Components

_{10}= 35.8 m/s, wind profile power index α = 0.15, turbulence I

_{10}= 0.14, and air density ρ = 1.25 kg/m

^{3}. The section drag coefficient C

_{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 ζ = 0.01.

_{T}, σ

_{B}and σ

_{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 σ

_{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 σ

_{BR}in the total pulsation response for each variable is less than 2%, so the effect of neglecting σ

_{BR}on the total pulsation response is small.

## 3. The Assessment Method for the Equivalent Static Wind Load of Arbitrary Structural Response

#### Theoretical Analysis

_{B}(z) and G

_{R}(z) at any position on the structure.

_{B}(z) of the structure’s response when under wind load at any location is

_{Rj}(z) of the structure, in terms of all responses at any location under wind load, is

_{B}(z) and G

_{R}(z) can be determined by the height z and the influence function i(z, z′).

## 4. Example and Analysis

_{B}and G

_{Rj}of the same structure are calculated according to the method given in this paper. The physical parameters of the structure are: dimensions H × W × D = 200 × 50 × 40 m, natural frequency f

_{1}= 0.22 Hz; damping ratio ξ = 0.01; first order vibration mode φ

_{1}(z) = (z/H)

^{β}; mass distribution m(z) = m

_{0}(1 − λ(z/H)); m

_{0}= 5.5 × 10

^{5}kg/m; section drag coefficient C

_{D}= 1.3. The wind environment parameters are: fundamental wind speed ${\overline{U}}_{10}=30\mathrm{m}/\mathrm{s}$; wind profile power index $\alpha =0.15$; and turbulence I

_{10}= 0.2, while the wind spectrum type is Davenport wind spectrum, and the coherence function parameter C

_{X}= C

_{Z}= 11.5. The following four operating conditions are obtained by adjusting the values of the vibration index β and the mass discount factor λ, respectively (Table 1).

_{B}and G

_{R1}via the DGLF method proposed by Davenport, as well as the QGLF method, and the MGLF method, which was proposed by him. The DGLF method directly takes the first-order vibration mode as linear, i.e., the vibration index β = 1 (of course, the DGLF method can also obtain results when β ≠ 1), but the DGLF method itself cannot consider the mass discount of the structure along with the height. Therefore, Zhou could only derive the result of one of the above four conditions using the DGLF method. The MGLF method also divides the structure ${\sigma}_{T}$ into two parts: the ${\sigma}_{B}$ is calculated directly based on the wind load and influence function, and the ${\sigma}_{R}$ is calculated by the indirect method, which distinguishes the influence of β and λ, so the results of the four conditions are different. The results are presented in reference [20].

_{B}and G

_{Rj}values at arbitrary positions are calculated, and when the height is taken as 0 or H, the GLF, G

_{B}and G

_{R1}values at these positions can also be obtained—the results are shown in Table 2. The values in parentheses are the results relative to the results of Zhou Yin [20]. It can be seen that the GLF values of the base bending moments obtained by the method proposed in this paper are identical to those obtained by Zhou Yin [20]. The values of base shear force GLF and top displacement GLF deviate slightly, but the maximum deviation is only 0.6%, meaning they are consistent with each other. The expressions of the basal bending moment GLF and top displacement GLF obtained according to the method of this paper are identical to those of Zhou Yin [20]. The deviations in the result may be caused by numerical integration. Due to the different values derived for the respective integration step and the upper integration limit when numerical integration is performed, the integration results will be somewhat deviant. The specific values of these parameters are not given by Zhou Yin [20]. The first-order frequency of this high-rise building is f

_{1}= 0.22 Hz; the trapezoidal integration formula is used in this paper for the numerical integration calculation, and the frequency integration step is df = 1/1000 Hz, while the upper integration limit is f

_{MAX}= 6 Hz. Other values are also taken (f

_{MAX}= 1 Hz, 3 Hz and 9 Hz; df = 1/500 Hz and 1/2000 Hz) for verification to ensure the calculation’s accuracy. The results show that the resulting G

_{B}decreases slightly when df = 1/500 Hz compared to df = 1/1000 Hz, but it remains unchanged when df = 1/2000 Hz, and the resulting G

_{B}decreases slightly when f

_{MAX}= 1 Hz and 3 Hz, compared to f

_{MAX}= 6 Hz, while it remains unchanged when f

_{MAX}= 9 Hz, which indicates that the df and f

_{MAX}used in the numerical integration are reasonable.

_{YB}and G

_{YR}, obtained under the four working conditions, are the results for condition 1. Since the effects of β and λ can all be considered when calculating the displacement response using the method in this paper, different results can be obtained under each of the four conditions, and because of that, the results under conditions 2, 3 and 4 will not be compared.

_{B}are not affected by β and λ, and G

_{B}is constant under all four conditions. Further, the base bending moment G

_{B}is slightly smaller than the base shear force G

_{B}, since the displacement influence function is obtained according to the assumption of the first-order vibration mode, and G

_{YB}increases with the increase of β. Since the ${\sigma}_{R}$ values of the base bending moment and shear force are obtained according to the resonant displacement inertia force ${\widehat{P}}_{R}({z}^{\prime})$ and the influence function (Equation (26)), and ${\widehat{P}}_{R}({z}^{\prime})$ contains the effects of β and λ, the ${\sigma}_{R}$ response of each condition is not same, and both G

_{R}values decrease with the increase in β. For the displacement of ${\sigma}_{R}$, although both ${\widehat{P}}_{R}({z}^{\prime})$ and the influence function contain the parameters β and λ, the parameter λ can be removed, so G

_{YR}is not affected by parameter λ, and only increases with the increase in β. The G

_{MR}decreases with the increase in β, but remains unchanged or increases when λ increases from 0 to 0.2, which is mainly because λ can be approximately removed when β = 1. When only the first-order vibration mode is considered, the G

_{QR}of base shear force decreases with the increase in β, and increases with the increase in λ.

## 5. Conclusions

_{BR}in the total pulsation response of each response is within 2%, so the influence of neglecting σBR on the total pulsation response is small;

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Table 1.**The values of the parameters for the four working conditions [20].

Condition | Vibration Index β | Mass Discount Factor λ |
---|---|---|

1 | 1.0 | 0.0 |

2 | 1.6 | 0.0 |

3 | 1.0 | 0.2 |

4 | 1.6 | 0.2 |

Top Displacement | Basal Bend Moment | Basal Shear Force | |||||||
---|---|---|---|---|---|---|---|---|---|

Condition | G_{YB} | G_{YR} | DGLF | G_{MB} | G_{MR} | MGLF | G_{QB} | G_{QR} | QGLF |

1 | 0.6520 (1.000) | 0.9761 (0.998) | 2.1738 (0.999) | 0.6520 (1.000) | 0.9761 (1.000) | 2.1738 (1.000) | 0.6560 (0.994) | 0.8275 (1.002) | 2.0560 (1.000) |

2 | 0.6591 | 1.0302 | 2.2230 | 0.6520 (1.000) | 0.9532 (1.000) | 2.1549 (1.000) | 0.6560 (0.994) | 0.7460 (1.003) | 1.9934 (0.999) |

3 | 0.6520 | 0.9761 | 2.1738 | 0.6520 (1.000) | 0.9761 (1.000) | 2.1738 (1.000) | 0.6560 (0.994) | 0.8438 (1.001) | 2.0688 (0.999) |

4 | 0.6591 | 1.0302 | 2.2230 | 0.6520 (1.000) | 0.9589 (1.000) | 2.1596 (1.000) | 0.6560 (0.994) | 0.7612 (1.002) | 2.0049 (1.000) |

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**MDPI and ACS Style**

Yang, J.; Zhang, J.; Li, C.
Research on Equivalent Static Load of High-Rise/Towering Structures Based on Wind-Induced Responses. *Appl. Sci.* **2022**, *12*, 3729.
https://doi.org/10.3390/app12083729

**AMA Style**

Yang J, Zhang J, Li C.
Research on Equivalent Static Load of High-Rise/Towering Structures Based on Wind-Induced Responses. *Applied Sciences*. 2022; 12(8):3729.
https://doi.org/10.3390/app12083729

**Chicago/Turabian Style**

Yang, Junhui, Junfeng Zhang, and Chao Li.
2022. "Research on Equivalent Static Load of High-Rise/Towering Structures Based on Wind-Induced Responses" *Applied Sciences* 12, no. 8: 3729.
https://doi.org/10.3390/app12083729