Study on Wind-Induced Response and Wind Vibration Coefficient of Tower Crane Under Different Wind Speed Spectra Excitation

Abstract

The wind-induced response and wind vibration coefficient of tower cranes under different wind speed spectra and wind direction angles are investigated to reveal the influence of wind speed distribution on the wind vibration characteristics of tower cranes. Firstly, the theoretical formula for the wind vibration coefficient based on random vibration theory is derived. Then, the APDL modeling method establishes a parametric finite element analysis model with the QTZ25 tower crane as a research object, and wind speed time history at various heights is simulated using the harmonic synthesis method. Finally, the wind speed is converted into wind loads, and the finite element model of the QTZ25 crane is subjected to random vibration analysis and spectra analysis under wind loads. The displacement time history response and displacement power spectral density at different heights are obtained, and the wind vibration coefficients are derived accordingly. The analysis results show that the wind vibration coefficient in descending order is Simiu, Davenport, and Harris wind speed spectra excitation (WSSE), respectively. The maximum deviation of the wind vibration coefficient is 12.01% at 26.09 m height with a 45° wind direction angle. In contrast, the maximum deviation of the wind vibration coefficient is 18.57% at the same height with a 0° wind direction angle. This study demonstrates that the wind-induced response of the QTZ25 tower crane remained the same under different wind speed spectra. Meanwhile, the wind vibration coefficient varied for different heights and wind direction angles, and the wind vibration coefficient was higher under Simiu WSSE and lower under Harris WSSE.

1 Introduction

Tower cranes are important lifting and transportation tools in the construction industry. With the deepening of urbanization, tower cranes are showing a trend of being taller, larger, and heavier. The slight vibrations arising from the truss structure of tower cranes under wind load can cause fatigue in local components and decrease their reliability. Therefore, the study on the wind-induced response and wind vibration coefficients of tower cranes under wind load excitation has significant theoretical and engineering value.

Due to the randomness of wind loads, accurately simulating wind loads is crucial for obtaining accurate structural wind-induced responses. Xu et al. [1] simulated pulsating wind loads using the MATLAB linear filtering method, performed wind vibration response analysis on an inner-suspended outer-tensioned holding pole, and revealed the wind vibration characteristics of the holding pole structure. Ke et al. [2, 3] used dynamic time-domain dynamic calculation methods to analyze the distribution of wind vibration coefficients and their values in chimney structures under different objectives. Based on the modal decoupling principle and detailed time history integration method, Pan et al. [4, 5] proposed a multi-mode detailed time history integration method to derive wind-induced response and wind vibration coefficient calculation formulas for steel pipe truss structures. Deng et al. [6] considered the contribution of the first four vibration modes of the mast structure to the wind-induced response, and a simplified formula for the wind vibration coefficient of the pole was proposed. Wang et al. [7] conducted wind tunnel tests on aerial elastic models to study the aerodynamic response for high-rise single-pole transmission towers under different wind speeds. Chu et al. [8, 9] studied the wind-induced response of wind turbine towers under different WSSE based on random vibration theory. Huang et al. [10] obtained the form factor coefficients of holding poles under different wind direction angles and arm postures through high-frequency balance force tests in the wind tunnel. Ma and Rezaiee-Pajand et al. [11, 12], combining the harmonic superposition method with the autoregressive method, developed a new wind time history hybrid simulation method and studied the wind-induced dynamic response of typical tower crane structures using the Newmark-β method. Luisa [13] based on the idea of gust factor, considering the cross-contribution of different turbulence conditions and wake excitation, a numerical method suitable for cranes is proposed to evaluate the wind-induced response of slender structures. Hamada and Chang et al. [14, 15] studied the aerodynamic characteristics and dynamic response of transmission tower line systems, indicating that wind-induced dynamic response plays a major role in the vibration characteristics of multi-span guyed transmission tower line systems.

In summary, current research mainly focuses on mast-like structures such as transmission towers and holding poles. Meanwhile, there is limited study on the wind-induced response of tower cranes under different wind speed spectra. This paper will use different wind speed spectra to simulate wind speeds, and then investigate the wind-induced response and wind vibration coefficient of tower cranes. It will reveal the impact of wind speed distribution on the wind vibration characteristics of tower cranes.

2 Theoretical Analysis of Wind Response of Tower Crane Structure

Based on the theory of structural random vibration, the structural motion equation of tower cranes can be expressed as:

$${\varvec{M}}\ddot{x}(t)+{\varvec{K}}x(t)+{\varvec{C}}\dot{x}(t)=P(t)$$

(1)

where

M, C, and K are the mass, damping, and stiffness matrix of the structure, respectively,

$\ddot{x}(t)$, $\dot{x}(t)$, $x(t)$ are the node’s acceleration, velocity, and displacement, respectively,

$P(t)$ is the Vector representation of pulsating wind force.

$${x}_{i}(t)=\sum\limits_{j=1}^{n}{\phi }_{ij}{q}_{j}(t)$$

(2)

where

${x}_{i}(t)$ is the displacement of node i,

${\phi }_{ij}$ is the jth mode shape coefficient of node i,

${q}_{j}(t)$ is the j-th mode shape.

The solution of substituting Eq. (2) into Eq. (1) is deduced as:

$${M}_{j}^{*}\{{\ddot{q}}_{j}(t)+2{\xi }_{j}{\omega }_{j}{\dot{q}}_{j}(t)+{\omega }_{j}^{2}{q}_{j}(t)\}={P}_{j}^{*}f(t)$$

(3)

where

${\omega }_{j}$, ${\xi }_{j}$ are the natural frequency and damping ratio of j-th mode shape,

$f(t)$ is the pulsating wind load,

${M}_{j}^{*}$, ${P}_{j}^{*}$ are the generalized mass and generalized force in mode shape coordinate system:

$${M}_{j}^{*}={\sum }_{i}{m}_{i}{\phi }_{ji}^{2}, {P}_{j}^{*}={\sum }_{i}{\kappa }_{i}{A}_{i}{\phi }_{ji}$$

where

${\kappa }_{i}$ is the pulsating surface force,

${A}_{i}$ is the windward area corresponding to node i,

${m}_{i}$ is the mass of the section of the structure.

According to the random vibration theory, the generalized integral open square of the output power spectral density (PSD) of the tower crane in the frequency spectra is the root variance ${\sigma }_{x}$ of the pulsation response with a mean of zero:

$$\begin{aligned}{\sigma }_{xi}& =\sqrt{\int\limits_{-\infty }^{\infty }{S}_{x}({z}_{i},\omega )d\omega }=\sqrt{\sum \limits_{j=1}^{n}{\sigma }_{xji}^{2}}\\ & =\sqrt{\sum\limits_{j=1}^{n}\frac{{\phi }_{ji}^{2}}{({M}_{j}^{*}{)}^{2}}\int\limits_{-\infty }^{\infty }\sum\limits_{i}\sum\limits_{k}{\kappa }_{i}{A}_{i}{\kappa }_{k}{A}_{k}{\rho }_{ik}(\omega ){\phi }_{ji}{\phi }_{jk}\sqrt{{S}_{f}({z}_{i},\omega ){S}_{f}({z}_{k},\omega )}{\left|{H}_{j}(i\omega )\right|}^{2}d\omega } \end{aligned}$$

(4)

where

x, j, $\omega$ are the response displacement, mode shape, and circular frequency of pulsating wind, respectively,

i, k are the node number,

${z}_{i}$ is the node height,

${\sigma }_{xi}$, ${\sigma }_{xji}$ are the root variance of the nodal displacement response, corresponding to the jth mode of vibration;

${S}_{x}({z}_{i},\omega )$ is the displacement response output PSD,

${S}_{f}({z}_{i},\omega )$ is the input excitation PSD,

${\rho }_{ik}(\omega )$ is the spatial correlation coefficient between nodes i and k.

The total displacement of the crane structure is expressed as:

$${x}_{i}={\beta }_{z}{\bar{x}}_{i}$$

(5)

$${\beta }_{z}=1+\frac{\mu {\sigma }_{yi}}{{\bar{x}}_{i}}$$

where

${x}_{i}$ is the total displacement response of the structure,

${\bar{x}}_{i}$ is the static displacement average,

$\mu$ is the peak factor under probabilistic conditions,

${\beta }_{z}$ is the wind vibration coefficient.

3 Establishment of a Parametric Finite Element Model for a Typical Tower Crane

The QTZ25 tower crane is selected as the typical research object. The parameterized finite element analysis model of the QTZ25 tower crane was established using ANSYS Parametric Design Language (APDL), as shown in Fig. 1. The tower shaft has ten standard segments, and the lifting arm has seven segments. The tower top height of the QTZ25 tower crane is 30,470 mm, and the tower shaft cross-section parameters are 1300 mm × 1300 mm. The lifting arm cross-section has a bottom edge length of 1000 mm and a height of 1000 mm. The balance arm has a cross-section of 300 mm × 100 mm, and the balance arm height above the ground is 25,000 mm. The balance arm length is 10,000 mm. The tower crane is divided into 12 parts along the height direction, and the middle position of each part is taken as the height value for the wind speed in that part, as shown in Fig. 1a.

Fig. 1

QTZ25 parametric finite element model

The wind direction angle θ is treated as a variable, and the finite element numerical simulations are performed for two calculation conditions with θ values of 0° and 45°, as shown in Fig. 1b. In these two calculation conditions, the shape factor ${\mu }_{s}$ is 1.628 and 2.04, respectively. In Fig. 1a, V represents the wind speed.

4 QTZ25 Tower Crane Wind Response Analysis and Wind Vibration Coefficient Calculation

4.1 Simulation of the Time History of Pulsating Wind Loads

Considering the excitation of different wind speed spectra, the formulas of various wind speed spectra are shown in Eqs. (6)–(8).

(1) Davenport WSSE

$$\frac{n{S}_{v}(n)}{{v}_{*}^{2}}=\frac{4{x}^{2}}{(1+{x}^{2}{)}^{4/3}},x=1200\frac{n}{{\bar{v}}_{10}}$$

(6)

(2) Harris WSSE

$$\frac{n{S}_{v}(n)}{{v}_{*}^{2}}=\frac{4x}{(2+{x}^{2}{)}^{5/6}},x=1800\frac{n}{{\bar{v}}_{10}}$$

(7)

(3) Simiu WSSE

$$\frac{n{S}_{v}(n)}{{v}_{*}^{2}}=\frac{200f}{(1+50f{)}^{5/3}},f=\frac{nz}{{\bar{v}}_{10}}$$

(8)

where

${v}_{*}$ is the shear flow rate: ${v}_{*}^{2}=K({\bar{v}}_{10}{)}^{2}$,

${S}_{v}(n)$ is the pulsating wind speed PSD,

n is the frequency,

K is the wind spectrum coefficient,

${\bar{v}}_{10}$ is the average wind speed at 10 m above the ground.

From the pulsating wind speed PSD, the pulsating wind pressure PSD is:

$${S}_{w}(n)=4{\left(\frac{\varpi }{{\bar{v}}_{10}}\right)}^{2}{S}_{v}(n)$$

(9)

where

$\varpi$ is the average wind pressure,

${S}_{w}(n)$ is the pulsating wind pressure PSD.

The wind pressure p at the windward face node of the tower crane can be obtained by multiplying the average wind pressure $\varpi$ by the body size factor ${\mu }_{s}$ and height factor ${\mu }_{z}$:

$$p={\mu }_{s}{\mu }_{z}\varpi$$

(10)

By selecting Class B landforms, the average wind speed ${\bar{v}}_{10}=10.0$ m/s at a height of 10 m, and the average wind pressure in the working state of the tower crane is 150 N/m². When the height above the ground is 10 m, the three pulsating wind speed PSD curves and pulsating wind pressure PSD curves are shown in Fig. 2.

Fig. 2

Wind speed PSD and wind pressure PSD curves with different wind speed distribution laws

The wind speed time history (WSTH) at different heights of the Tower crane is simulated by the harmonic synthesis method and considers the effect of different wind speed spectra.

The wind speed simulation at the height of 10.8 m and 23.85 m of the QTZ25 Tower crane is shown in Fig. 3, Fig. 4, and Fig. 5, respectively.

Fig. 3

The WSTH curve of Davenport WSSE

Fig. 4

The WSTH curve of Harris WSSE

Fig. 5

The WSTH curve of Simiu WSSE

The pulsating wind load of each node of the tower crane is calculated as follows:

$$P(z,t)=0.5{\mu }_{s}\rho A\hspace{0.33em}[V(z,t){]}^{2}$$

(11)

where

${\mu }_{s}$ is the body size factor,

$\rho$ is the air density, $\rho ={1.25\,{\text{kg}}/{\text{m}}}^{3}$,

$A$ is the member projection area,

$V(z,t)$ is the wind speed time history at height z.

4.2 Wind-Induced Response Calculation and Analysis

For the wind-induced response of the Tower crane under different WSSE, Eq. (11) is applied to the finite element model to obtain the displacement time history response of the QTZ25 Tower crane at different heights under different wind directions as shown in Figs. 6 and 7.

Fig. 6

Displacement time history at different heights at 0° wind angle

Fig. 7

Displacement time history at different heights at the 45° wind angle

It can be seen from the analysis of displacement time history response in Fig. 6 and Fig. 7 that under the excitation of different wind speed spectra when the wind direction angle is 0° and 45°, respectively, the displacement response amplitude increases with the increase of height. The displacement time history of the QTZ25 Tower crane structure at typical heights of 10.8 and 23.85 m is consistent, indicating that the wind-induced displacement response of the Tower crane under the excitation of different wind speed spectra is almost the same.

4.3 Calculation and Analysis of Wind Vibration Coefficient

The displacement PSD of the QTZ25 Tower crane at different wind directions and heights under different wind speed spectral excitation is shown in Fig. 8 and Fig. 9, respectively.

Fig. 8

Displacement PSD at different heights at 0° wind angle

Fig. 9

Displacement PSD at different heights at a 45° wind angle

It can be seen from the analysis in Figs. 8 and 9 that with the increase of excitation frequency, the displacement PSD at different heights changes significantly, and the displacement PSD values at the same height at different wind directions are also quite different. At the same height, the displacement PSD value under the excitation of the Simiu wind speed spectrum is the largest, followed by the Davenport wind speed spectrum, and the displacement PSD value under the action of the Harris wind speed spectrum is the smallest. It can also be seen from Figs. 8 and 9 that under the excitation of different wind speed spectra, the QTZ25 Tower crane is prone to resonance at the first three natural frequency ranges, producing more significant displacement.

According to the national reliability index regulations, consider the value of the peak factor μ under a certain probability of crossing is 3.24¹. According to the displacement response and displacement PSD of the QTZ25 Tower crane at different wind directions and heights under the Davenport, Harris, and Simiu WSSE, the comparison of structural wind vibration coefficients is shown in Fig. 10.

Fig. 10

Wind vibration coefficient of QTZ25 structure at different heights

It can be seen from Fig. 10 that with the increase of height, the wind vibration coefficient of the QTZ25 Tower crane structure under different WSSE gradually increases. At the same height, the wind vibration coefficient in descending order is Simiu, Davenport, and Harris WSSE, respectively. Compared with the Simiu wind speed spectrum, The wind vibration coefficient values under Davenport WSSE are closer to those under Harris WSSE because the Simiu wind speed spectrum considers the impact of height changes on wind speed. The higher the height, the greater the wind speed. It can also be seen from Fig. 10 that different wind directions have a specific impact on the wind vibration coefficient of the QTZ25 Tower crane structure. Under the excitation of the three wind speed spectra, when the wind direction angle is 0°, the maximum deviation of the wind vibration coefficient at 26.09 m height is 18.57%. When the wind direction angle is 45°, the maximum deviation of the wind vibration coefficient at 26.09 m height is 12.01%.

5 Conclusion

Based on the random vibration theory, the theoretical formula of the wind vibration coefficient of the Tower crane structure is derived. Taking the QTZ25 Tower crane as the research object, the parametric finite element analysis model of its beam structure is established. The wind speed distribution law is simulated by the Davenport, Harris, and Simiu wind speed spectra. The wind-induced response and spectra analysis of the QTZ25 Tower crane are carried out accordingly. The displacement response and displacement PSD at different wind directions and heights are obtained. Then the wind vibration coefficient formula is used to obtain the wind vibration coefficient at different heights and wind directions under three wind speed spectral excitations. The specific conclusions are as follows:

• (1) Under the excitation of different wind speed spectra, the displacement response of Tower cranes at different heights is consistent, which means nearly the same displacement time response can be obtained with different wind speed distribution laws.

• (2) Under the excitation of different wind speed spectra and wind direction angles, the value of the wind vibration coefficient at the same height of the Tower crane is different. The wind vibration coefficient in descending order is Simiu, Davenport, and Harris WSSE, respectively.

• (3) Under different wind speed spectral excitations, the maximum deviation of the wind vibration coefficient at 26.09 m is 12.01% at a 45° wind direction angle and 18.57% at a 0° wind direction angle.