Numerical Simulation of Wind-Induced Vibration Response Characteristics of High-Rise Buildings with Podiums

Abstract

High-rise building structures with podiums are widely present, and establishing a fast numerical prediction method to evaluate their wind-induced response characteristics is of great significance for engineering applications. This article proposes a process algorithm based on the AR (autoregression) method to solve the time history of fluctuating wind speed and determine fluctuating wind load. The simulated fluctuating wind speed spectrum obtained through this algorithm matches the target wind speed spectrum, and the wind-induced response characteristics of high-rise buildings with podiums were studied using MIDAS GEN (2021) structural analysis software. In order to evaluate the influence of different parameters on the wind-induced vibration response of high-rise buildings with podiums, a total of 11 comparative conditions were set, including the presence or absence of podiums, podium height, podium area, and podium layout conditions. A comprehensive time history analysis was conducted on the displacement, acceleration, shear force, and overturning moment of the wind-induced vibration response of high-rise buildings with podiums. The results indicate that in high-rise buildings with podiums, adding podiums and increasing their height and area can help suppress the inter-story displacement of the main building and the inter-story acceleration, inter-story shear force, and intra-story overturning moment of the middle and lower floors, which is beneficial for the safety and stability of the high-rise building structure. The layout of the podium has an impact on the wind-induced vibration response of the main building. When the podium and main building are symmetrically arranged in the downwind direction, the maximum displacement of each floor is small, while the maximum displacement curve of buildings with asymmetric layout at the junction of the podium and main building is not smooth. The design of the central layout of the podium and main building can effectively reduce the maximum shear force and maximum overturning moment of the higher floors of the building, but the effect is opposite at lower floors.

1. Introduction

As the name suggests, wind-induced vibration is the term used to describe the shaking of structures. The effect of wind on building vibration was not immediately apparent in an era when buildings were typically not very tall. It was only later, with the development of high-rise structures and wide-span bridges, etc., that wind-induced vibration events started to become more common and even resulted in wind-damaged engineering structures such as buildings and bridges, like the Tacoma Suspension Bridge in the United States in 1940 [1], that people started to investigate this effect. Numerous academics have investigated wind vibration in buildings. To study the wind loads of supertall multi-tower conjoined buildings from the perspectives of mean and pulsating wind pressure characteristics and vortex volume distribution, researchers Shitang Ke and Hao Wang [2,3] performed wind tunnel tests based on large vortex simulation for a supertall three-tower conjoined building. In order to study the spatial distribution characteristics of downwind pulsating wind loads and pulsating wind speed coherence functions, Zeng Jadong and Li Mingshui et al. [4] conducted surface pressure simultaneous measurement experiments on a rectangular rigid model. They then proposed a model expression for the downwind pulsating wind load coherence function of a rectangular building that can involve flow field and structural parameters like wind field category, vertical spacing, and others. In order to study the interrelationships reflecting the statistical coupling between modal responses under spatially–temporally varying dynamic wind excitation, M.F. Huang and C.M. Chan et al. [5] use a 60-story asymmetric full-size steel–concrete hybrid building as an example. They then propose a framework for analyzing lateral torsional wind vibration forces in high-rise buildings based on three-dimensional modal vibration patterns. In order to study wind-induced torsional loads on rectangular high-rise buildings, Vladimir Guzmán-Sols and Adrian Pozos-Estrada et al. [6] conducted wind tunnel tests on five rigid models with the same dimensions and various aspect ratios. Based on the results, they proposed a parameter on the variation of foundation torque coefficients with aspect ratio and wind direction for rectangular high-rise buildings that agrees well with the experimental results equations. A popular area of study in recent years has been the damage theory analysis of high-rise building structures. The numerical simulation study of these structures’ wind vibration response characteristics has some value for engineering [7,8,9].

A survey of existing domestic and foreign research shows that few studies of this kind have been carried out, despite the fact that the main podium one-piece high-rise building structure is a common structure in a class of projects. In this study, the pulsating wind velocity at various heights is solved using the AR approach [10], and the pulsating wind load is calculated using a formula. Finally, the software simulates the building’s response to wind vibration under pulsing wind stress.

2. Overview of AR Method

The pulsating wind speed time range must be resolved first in order to obtain the building pulsating wind load time range by self-programming. The wind tunnel test, harmonic synthesis method, and linear filtering method are the current three methods for determining the pulsing wind speed. A series of artificially generated random numbers with zero mean and a white spectrum are input into the filter for processing and are then output as a random process with a specific spectrum. This technique is known as the linear filtering method, also known as the white noise filtering method. The linear filter method can be further broken down into the autoregressive (AR), sliding average, etc. Due to its benefits of low computational effort, quick computation, and high accuracy in determining the pulsing wind speed of buildings, the AR method has been extensively employed by a significant number of academics [11,12,13].

The basic idea behind the AR method’s steps is to first locate the stochastic process, then determine the regression coefficients given the stochastic process. The M-dimensional AR model can be used to represent the pulsing wind velocity field.

$$\mathit{v}\left(t\right)=-{\displaystyle \sum }_{k=1}^q{\mathit{\psi }}_k\mathit{v}\left(t-k∆t\right)+\mathit{N}\left(t\right)$$

(1)

where q is the order of the M-dimensional AR model [14]; ${\mathit{\psi }}_k$ is the regression coefficient matrix of order M × M; p is the order of the AR model simulation; $∆t$ is the time step of the wind speed time history of the AR model; and $\mathit{N}\left(t\right)$ is a random vector with zero mean for a given variance. The matrix of autoregressive coefficients can be written as follows:

$$\mathit{R}=-{\mathit{R}}^{*}{\mathit{\psi }}_k$$

(2)

The expression is where $-{\mathit{R}}^{*}$ is the autocorrelation matrix of order I × I (I = q × M), and the elements in $-{\mathit{R}}^{*}$ can be obtained using the Wiener–Khinchin formula:

$${\mathit{R}}_{ij}={{\displaystyle \int }}_0^{\infty }{S}_{ij}\cos \left(2\pi nj∆t\right)dn,i,j=1,2\dots ,M$$

(3)

where $S_{ij}$ is the pulsating wind speed mutual spectrum density function; $n$ stands for the pulsating wind speed frequency; The $S_{ij}$ can be obtained from the known fluctuating wind speed spectrum S and spatial correlation coefficient $\mathrm{coh}\left(r,n\right)$. The following section will introduce a number of readily available pulsing wind speed spectra. The expression for the spatial correlation coefficient $\mathrm{coh}\left(r,n\right)$ is:

$$\mathrm{coh}\left(r,n\right)=\exp \left\{{\displaystyle \frac{-2n\sqrt{C_x^2{\left(x_i-x_j\right)}^2+C_y^2{\left(y_i-y_j\right)}^2+C_z^2{\left(z_i-z_j\right)}^2}}{v_{z_i}+v_{z_j}}}\right\}$$

(4)

In Equation (4), the variable n represents the frequency of fluctuating wind speed; the variable r represents a spatial vector; $v_{z_i}$ and $v_{z_j}$ signify the average wind speed at $z_i$ and $z_j$ in the space; and $C_x$, $C_y$, and $C_z$ denote the left–right, front–back, and top–bottom attenuation coefficients of a space, respectively.

By multiplying both sides of Equation (1) simultaneously right by v(t) = [v₁ (t), v₂ (t), …, v_m (t)] and then substituting the derived autoregressive coefficient matrix ψ_k, we can obtain the expression:

$${\mathit{R}}_N={\displaystyle \sum }_{k=1}^q{\mathit{\psi }}_k\mathit{R}\left(k∆t\right)+\mathit{R}\left(0\right)$$

(5)

where ${\mathit{R}}_N$ is subject to Cholesky decomposition, and the decomposition expression is as follows:

$${\mathit{R}}_N=\mathit{L}{\mathit{L}}^T$$

(6)

Thus, the random vector N(t) with a zero mean given the variance may be calculated from L, which stands for the lower triangular matrix. The expression is as follows:

$$\mathit{N}\left(t\right)=\mathit{L}\mathit{N}\left(t\right)$$

(7)

By inserting the resulting $\mathit{N}\left(t\right)$ into Equation (1), it is possible to find the pulsing wind velocity process vector with time interval t and spatial correlation.

The expression employed in this study, which uses the Davenport [15] pulsating wind speed spectrum, is as follows:

$$S_v\left(n\right)={\displaystyle \frac{4K{x}^2{v}_{10}^2}{n{\left(1+x^2\right)}^{\frac{4}{3}}}}$$

(8)

$$x={\displaystyle \frac{1200n}{v_{10}}}$$

(9)

where $v_{10}$ defines the reference wind speed at 10 m height; $K$ denotes the ground roughness coefficient; $S_v\left(n\right)$ denotes the pulsing wind speed power spectrum; and $n$ denotes the pulsating wind frequency.

3. Pulsating Wind Load Solution

The program is created using the AR method, imported into the MATLAB (R2024a) software, and run to output the pulsating wind speed time history data and the pulsating wind speed time history graph at each point. Finally, the calculated simulated spectrum curve is compared with the target spectrum curve to confirm the accuracy of the simulated wind speed time history data.

3.1. Computational Model

The study conditions are shown in

Table 1. Simulated working condition Table.

Working Condition	Computational Model	Podium Height	Podium Area
With podium or without	High-rise with podium	24 m	80 m × 80 m
	High-rise without podium	0	0
Podium height condition	High-rise with podium	10 m	80 m × 80 m
		17 m	80 m × 80 m
		24 m	80 m × 80 m
Podium area condition	High-rise with podium	24 m	70 m × 70 m
		24 m	80 m × 80 m
		24 m	90 m × 90 m

. The main building is 40 m in length, width, and height. The model size is shown in Figure 1. Measurement point locations are 10, 28, 56, 84, 112, 140, 168, and 196 m on the center line of the main building’s windward side. The measurement point map is shown in Figure 2. The podium area working condition measurement point distribution and 24 m podium working condition are the same.

3.2. Time History Simulation of Fluctuating Wind Speed

Wang Xiuqiong [16] conducted a pertinent study regarding the Davenport wind speed spectrum and provided more reasonable values for the ground roughness coefficient k according to different roughness types, including 0.00129 for Class A landforms, 0.00215 for Class B landforms, 0.00464 for Class C landforms, and 0.01291 for Class D landforms (in the Code for Structural Loads of Buildings (GB50009-2012) [17]. The ground roughness is divided into four categories, i.e., A, B, C, and D. Since B is the studied landform type, 0.00215 is used as the k value. Only height correlation is taken into account for spatial correlation; hence, $C_z=10$ is the only number used (empirical value). Equation (4) uses the wind speed exponential law to calculate the variables $v_{z_i}$ and $v_{z_j}$. The wind speed exponential law’s expression is as follows:

$$V_z=V_0{\left({\displaystyle \frac{Z}{Z_0}}\right)}^{\alpha }$$

(10)

where $Z_0$ is the standard height; $V_0$ is the standard wind speed at $Z_0$; $V_z$ is the average wind speed at $Z$ above the ground; and $\alpha$ is the ground roughness index.

The reference wind pressure in this study is 0.25 at a height of 10 m, based on the following equation between wind pressure and wind speed:

$$p={\displaystyle \frac{v^2}{1600}}$$

(11)

Moreover, 20 m/s can be calculated as the reference wind speed at 10 m, and the other necessary values are established as given in

Table 2. Parameter setting of fluctuating wind speed time history simulation.

Project	Parameter Setting
Density of pulsating wind medium	ρ = 1.225 kg/m3
Time history simulation order	4th order
Total simulation time	200 s
Simulation interval	0.1 s
Total sampling points	2048
Initial frequency	0.01 Hz
Frequency increment	0.01 Hz
Cut-off frequency	10 Hz

.

According to the calculation of the MATLAB program, the fluctuating wind speed at each height is shown in Figure 3.

According to Figure 3, each height measurement point’s peak pulsing wind speed is maintained at roughly 10 m/s, and they all oscillate up and down at 0. The simulated spectrum curve (black straight line on the right figure) produced by MATLAB software for each height measurement point is more consistent with the target spectrum curve (red dashed line on the right figure), as can be seen from the power spectrum comparison graph, demonstrating that there is some degree of confidence in the simulated pulsating wind speed time range data in this section.

The pulsating wind pressure time range data can be calculated using the obtained pulsating wind speed time range data, and the pulsating wind load time range data can be calculated using a different formula. The wind pressure includes both average and pulsing wind pressure since the natural wind in the atmospheric environment is made up of both average and pulsating wind. The equation for the link between wind speed and wind pressure is:

$$P={\displaystyle \frac{1}{2}}\rho V^2$$

(12)

where $\rho$ is the air density and $V$ is the total wind speed. Since the total wind speed is composed of average wind speed and fluctuating wind speed, Equation (12) can be expressed as follows:

$$P={\displaystyle \frac{1}{2}}\rho {\left(\overline{v}+v\right)}^2$$

(13)

where $\overline{v}$ is the average wind speed and $v$ is the fluctuating wind speed. The wind speed exponential rule is used to obtain the average wind speed at each location.

Table 3. Summary of average wind speed at each height measuring point.

Measurement point height	10 m	28 m	56 m	84 m	112 m	140 m	168 m	196 m
average wind speed	20 m/s	23.58 m/s	26.35 m/s	28.11 m/s	29.44 m/s	30.51 m/s	31.41 m/s	32.2 m/s

displays the average wind speed at each height measurement point, and Figure 4 displays the pulsing wind pressure time range at each height measurement point that was determined by Equation (13).

Based on the calculated fluctuating wind pressure time history data, the fluctuating wind load time history can be directly calculated through the following formula:

$$F=PA{\mu }_s$$

(14)

In this study, the windward area A of the simulated height measurement point is calculated as the product of the measurement point’s distance from two neighboring measurement sites divided by half, then multiplied by the breadth of the windward surface. ${\mu }_s$ indicates the building wind carrier type coefficient, which is specified in the Code for Structural Loads of Buildings (GB50009-2012) [17].

To be conservative, the wind load coefficient on the windward side and the wind load coefficient on the leeward side are typically superimposed to take 1.4 when designing and calculating the wind resistance performance of the building. This is because in the wind resistance design of the building structure, the wind pressure on the windward side of the building and the wind suction force on the leeward side of the building should be in the same direction. The wind load coefficient on the windward side is taken to be 1.25 in this study, and Figure 5 displays the wind load time range for each height measurement point derived from Equation (14).

4. Verification of Numerical Simulation Results of Wind-Induced Vibration Response

The computational model and wind tunnel test data for the verification examples are sourced from reference [18]. The wind tunnel test model is a single-story high-rise building model with a length, width, and height of 30 m × 15 m × 240 m. Reference [18] obtained through wind tunnel tests that the maximum displacement of the top layer is 0.438 m. The numerical simulation conditions in this article are consistent with those in reference [18]. The incoming wind direction is perpendicular to the long side, and the ground roughness type is Class B with a roughness of 0.15. The initial wind speed at a height of 10 m is 28.5 m, and the damping ratio is set to 0.02.

By using Midas Gen (2021) software to perform time history analysis on the model and output displacement data for each layer of the model. Due to space limitations, this article only compares the mean results of displacement response at the top level. The average maximum displacement of the top layer obtained through numerical simulation is about 0.53 m, while the average maximum displacement of the top layer obtained from wind tunnel experiments in reference [16] is about 0.438 m. Considering that the model used in the original wind tunnel experiment was a scaled-down model and other factors, the results were relatively small, and the differences in the results were within an acceptable range. Therefore, the numerical calculation method for wind vibration response results in this paper is feasible.

5. Wind-Induced Vibration Analysis

For the convenience of comparison, the damping calculation method for high-rise building structures in this article adopts the modal damping method, and the damping ratio is uniformly set to 0.05. Regarding the setting of boundary conditions in modal simulation of high-rise building structures, the bottom nodes of the podium of high-rise buildings are set to general elastic support boundary conditions. All other boundary nodes are free and have sliding boundary conditions. The boundary conditions for other operating conditions are also set in the same way.

5.1. Comparison of Wind Vibration Results Between High-Rise Buildings with and Without Podiums

Prior to performing the time analysis, the modal analysis of the finite element model’s first 16 orders of vibration patterns is performed using the eigenvalue analysis method in Midas Gen, which may guarantee the model’s plausibility. Under the condition of calculating the vibration mode to the 16th order,

Table 4. The first six vibration modes of the finite element model for high-rise buildings with podiums (10 m).

Category	The First Six Vibration Modes
Vibration mode diagram
Order	First order	Second order	Third order	Fourth order	Fifth order	Sixth order
Vibration type	the translational vibration mode in the x-axis direction	the translational vibration mode in the y-axis direction	the torsional vibration mode around the z-axis	the flexural vibration mode in the x-axis direction	the flexural vibration mode in the y-axis direction	the translational torsional coupling vibration mode

shows the first six orders of the vibration mode diagram for the high building without a podium.

The requirements for vibration types of high-rise buildings in the Code for Vibration Analysis of Buildings mainly include basic vibration modes such as translation, torsion, bending, and planar torsional coupling. Modal analysis was conducted on the finite element models of high-rise buildings with podium heights of 10 m, 17 m, and 24 m, respectively. The analysis results for podium heights of 10 m are shown in Table 4. According to Table 4, the vibration types of each mode basically meet the requirements of the first six vibration types of high-rise buildings in the Technical Code for Concrete Structures of Tall Buildings [19], indicating that the finite element model is properly set.

The numerical simulation results of wind-induced vibration response for two working conditions (high-rise buildings with and without podiums) are visualized and compared through post-processing, as shown in Figure 6, Figure 7, Figure 8 and Figure 9. Figure 6 shows the variation of top displacement over time and the comparison of maximum displacement for each floor. Figure 7 shows the comparison of the highest acceleration of each floor and the variation of the top floor acceleration over time. Figure 8 shows the comparison of maximum shear forces on each floor and the variation of bottom shear force over time. Figure 9 depicts the time history of the bottom overturning moment and the comparison of the maximum overturning moment on each floor.

When looking at Figure 6a, it is clear that there are not many differences between the two working conditions, as evidenced by the displacement time history curves of the top level of the building with and without a podium. Even still, the peak of the curve shows that the curve without a podium has a slightly bigger overall numerical magnitude than the one with a podium, showing that the displacement pulsation of the top floor of the building without a podium is larger.

In both operating situations, the maximum inter-story displacement can be seen in Figure 6b to significantly increase as the floor height increases. This is in keeping with the general rule of wind load on high-rise buildings since as a building’s height increases, so does its flexibility, making it more susceptible to wind stress. When the displacement curves for the two working conditions are compared, it is clear that the maximum displacement between each floor in the working condition without a podium is greater than that in the working condition with a podium, and the difference between the two gradually grows as the floor increases.

In general, adding a podium at the base of a high-rise building helps to reduce the wind-induced vibration displacement response of the entire structure because the podium structure at the base increases the building’s bottom stiffness to some extent, increasing the building’s overall wind resistance.

Figure 7a demonstrates that the acceleration time history curves of the top level of the building with and without the podium are still very comparable in both situations, much like the displacement time history scenario. The maximum acceleration difference between the two working conditions does not increase with the height of the building but instead begins at zero and increases until it reaches its maximum value around the tenth floor, after which it decreases until the two curves essentially coincide, and finally, the acceleration of the working condition without a podium is lower than the acceleration of the working condition with a podium, as can be seen from the detailed diagram in the lower right corner of Figure 7b. The stiffness difference is what drives the curve’s development tendency. The inter-story acceleration of a building drops as its stiffness rises, and the inter-story acceleration of a building with a podium condition is lower from the bottom level than it would be without a podium condition because the podium makes the bottom of the building stiffer. The influence of the bottom podium on the building’s inter-story stiffness then steadily reduces as the number of floors rises. Overall, the podium predominantly suppresses inter-story acceleration in the middle and lower floors, and the strength of the suppression weakens as the number of stories rises.

The time history curves for bottom shear in the two operating circumstances are essentially the same, as can be seen in Figure 8a. The difference between the peaks of the time history curves for the cases with and without podiums demonstrates that the effect of the podium on the building’s bottom shear is not immediately apparent. The maximum acceleration between floors in both working circumstances drops noticeably with an increase in floor height, as shown in Figure 8b. The difference in maximum acceleration between floors is primarily visible in the lower floors, and the inter-story shear force without podium condition is greater than that with podium condition. However, the difference in shear force between the two does not increase with the height of the building; instead, it starts at zero until the maximum value is reached around the tenth floor, then it gradually decreases until the two curves essentially coincide. Overall, the middle and low levels still represent the majority of the podium’s effect on the building’s inter-story shear, and the strength of the impact will continue to diminish as the number of stories rises.

Because there is a transformation relationship between the shear force and the overturning moment, it can be seen from Figure 9a that the shape of the bottom overturning moment time history curve under the two working conditions is essentially the same as the bottom shear time history curve in Figure 8a. This indicates that the podium’s influence on the building’s bottom overturning moment is as minimal as the bottom shear force.

Figure 9b demonstrates that the amplitude and variety of the two curves’ differences are still comparable to Figure 8b. This means that rather than increasing as a building’s height rises, the difference between the two overturning moments instead increases from 0 until it reaches its greatest value around the 10th story, then reduces until the two curves almost converge. The scale in Figure 9 is too tiny, and the overturning moment values are quite great, which reduces the distance between the two, causing the curves to be very similar.

Overall, the middle and lower floors still reflect the majority of the podium’s shear stress on the overturning moment between the building’s floors, and the severity of the effect reduces as the number of stories rises.

In order to better analyze and compare the numerical simulation results, the key node layers in the building model were selected for quantitative comparison, highlighting the influence of the presence of the podium on the wind-induced vibration response results. The data comparison is shown in

Table 5. Comparison of key data with or without podium working conditions.

	Buildings Without Podium	Buildings With Podium
Average displacement of layers	0.0692 m	0.0614 m
Average layer acceleration	0.0398 m/s2	0.0388 m/s2
Average layer shear force	4927.4704 kN	4860.7483 kN
Average bending moment of each layer	376,259.8879 kN·m	372,937.091 kN·m
Intermediate displacement	0.0693 m	0.0620 m
Intermediate layer acceleration	0.0406 m/s2	0.0392 m/s2
Intermediate layer shear force	5629.6551 kN	5581.7768 kN
Intermediate bending moment	313,499.17 kN·m	312,401.4622 kN·m
Top displacement	0.1356 m	0.1266 m
Top acceleration	0.0795 m/s2	0.08 m/s2
Top-level shear force	236.5530 kN	237.2933 kN
Top bending moment	946.2118 kN·m	949.1731 kN·m

. From Table 5, it can be seen that under the same conditions, the average values of various parameters in the middle floors of the main body of high-rise buildings with podiums are lower than those in the corresponding positions of buildings without podiums. However, at the top level of the main body of high-rise buildings, except for the top level with a podium where the displacement is smaller than that without a podium, all other top-level response parameters with a podium are higher than those without a podium. It can be inferred that one of the reasons for this phenomenon is the presence of podiums, which leads to a relative increase in the bottom stiffness and a relative decrease in the upper stiffness effect of the building.

5.2. The Influence of Podium Height on Wind-Induced Vibration Results

Figure 10 shows the time history of the top floor displacement and the maximum displacement of each floor under the three working conditions. Figure 11 shows the time history of the top floor acceleration and the maximum acceleration of each floor. Figure 12 shows the time history of the bottom shear force and the maximum inter-story shear force of each floor. Figure 13 shows the time history of the bottom overturning moment and the maximum overturning moment of each floor.

It can be seen from Figure 10a that with the increase in the podium height, the overall value of the displacement time history curve of the top floor is gradually decreasing, and from the peak value of the curve under the three working conditions, the overall amplitude of the curve is also decreasing. This shows that the higher the podium, the smaller the displacement fluctuation amplitude of the top floor of the building. By observing Figure 10b, it can be seen that with the increase in the number of building floors, the maximum inter-floor displacement under the three working conditions shows an obvious increasing trend. Secondly, comparing the maximum displacement curves between the floors under the three working conditions, it can be clearly seen that the maximum displacement between the floors of the building gradually decreases with the increase in the podium height, and the curve difference between the three working conditions gradually increases with the increase in the floors.

The podium structure at the bottom of the high-rise building can boost the bottom stiffness of the building to a certain level, improving the building’s total wind resistance, according to the research of the working conditions with and without the podium. Similar to this, the research of the podium height condition revealed that as the podium height increased, the stiffness of the building’s main body increased. As a result, the building’s inter-story displacement decreased and the displacement pulsation amplitude increased.

As the podium’s height increases, it is evident from Figure 11a that both the amplitude and period of the pulsation of the acceleration time history curve are gradually diminishing. As can be observed from Figure 11b, the acceleration rises noticeably as the floor height rises, and the maximum acceleration curve between floors follows the expected trend. The acceleration difference between each curve, which starts at zero and increases until it reaches its maximum value around the tenth floor, then gradually decreases until the two curves essentially coincide, can be seen in Figure 11b. During this time, the maximum acceleration magnitude between building floors gradually decreases with the increase in podium height. The acceleration numbers are in the opposite order of magnitude as before, starting at about the 45th story of the building, and the difference in acceleration is growing. The detailed image in Figure 12b’s lower right corner makes it easy to determine the magnitude difference between the acceleration values. Following analysis of the podium height scenario, it was discovered that elevating the podium can successfully reduce inter-story acceleration in the building’s middle and lower floors, similar to the cases with and without a podium. Reaching the middle and upper stories, this suppression effect is totally reversed as the floors rise higher.

The bottom shear time history curves for the three working conditions are extremely similar, as can be shown in Figure 12a. The three have different pulsation amplitudes, but there is no noticeable difference between them, so it can be concluded that altering the podium’s height does not significantly affect the building’s bottom shear.

As the floors rise higher, it is clear from Figure 12b that the maximum inter-story shear forces for the three operating conditions only differ marginally at the lower floors. The shear force difference between the curves in the figure is not as noticeable as it would be in the case with and without the podium because the value of the shear force at the base of the building does not vary significantly as a result of the little increase in podium height. However, as can be seen from the detailed diagram in the lower right corner of Figure 12b, the maximum inter-story shear force close to the intersection level of the main structure of the building and the podium gradually reduces with an increase in podium height.

Figure 13a reveals that the bottom overturning moment time history curve’s trend and period for the three working conditions are essentially identical to those of Figure 11a. This is due to the fact that shear force and overturning moment have a transformation connection that is comparable to the situation in operating conditions with and without a podium. Because the overturning moment values in Figure 13b are quite high and the shear forces between the layers for the three working conditions have been somewhat similar. The three curves’ differences are thus barely discernible and even close to overlapping. The inter-story overturning moment of the middle and lower floors of the building does, however, gradually decrease with an increase in podium height, as shown by the detailed diagram in Figure 13b’s right corner.

Similarly, in order to better analyze the data, important location nodes in the building model were selected for quantitative comparison to highlight the effect of podium height. The data comparison is shown in

Table 6. Comparison of key data of podium height working conditions.

Podium Height	10 m	17 m	24 m
Average displacement of layers	0.0667 m	0.0644 m	0.0614 m
Average layer acceleration	0.0396 m/s2	0.0392 m/s2	0.0388 m/s2
Average layer shear force	4912.7981 kN	4890.8438 kN	4860.7483 kN
Average bending moment of each layer	375,582.9025 kN·m	374,580.4379 kN·m	372,937.091 kN·m
Intermediate displacement	0.0679 m	0.0654 m	0.0620 m
Intermediate layer acceleration	0.04031 m/s2	0.0398 m/s2	0.0392 m/s2
Intermediate layer shear force	5620.1740 kN	5606.0419 kN	5581.7768 kN
Intermediate bending moment	313,279.6078 kN·m	313,044.7544 kN·m	312,401.4622 kN·m
Top displacement	0.1340 m	0.1309 m	0.1266 m
Top acceleration	0.0796 m/s2	0.0798 m/s2	0.08 m/s2
Top-level shear force	236.7026 kN	237.0792 kN	237.2933 kN
Top bending moment	946.8102 kN·m	948.3168 kN·m	949.1731 kN·m

.

From Table 6, it can be seen that as the height of the podium increases, the average values of various parameters and the parameters of the middle floor decrease. But at the top level, except for displacement, all other parameters increase with the height of the podium building. It can be inferred that one of the reasons for this phenomenon is still the relative weakening of the stiffness effect on the upper part of the building.

5.3. The Influence of Podium Area on Wind-Induced Vibration Results

The finite element models of the three podium area working conditions perform modal analysis. The specific settings for the three area conditions are shown in Table 1. Perform wind vibration response calculation and post-processing analysis on finite element models under three types of podium floor area conditions.

Following the time history analysis, Figure 14, Figure 15 and Figure 16 show the time history of the top floor displacement and the maximum displacement of each floor, the time history of the top floor acceleration and the maximum acceleration of each floor under the three working conditions, the time history of the bottom shear force and the maximum inter-story shear force of each floor under the three working conditions, and Figure 17 shows the time history of the bottom overturning bending moment and maximum overturning moment between floors under podium area condition under podium area condition.

The displacement time history curve for the top floor of the building under the three operating conditions has a very similar trend and period, as can be shown in Figure 14a. The overall value of the displacement time history curve of the top level, however, gradually declines as the podium’s surface area grows. Additionally, under the three operating conditions, the overall amplitude of the curve is likewise falling from the apex of the curve. This suggests that the displacement pulsation amplitude of the building’s top floor decreases as the podium’s area increases. The maximum displacement between floors in all three sets of working situations grows dramatically with the addition of building floors, as can be seen by first looking at Figure 14b. Second, when comparing the three curves, it is clear that as the height of the podium rises, the maximum displacement between floors of the building progressively reduces, and as floors rise, the disparity in the curves between the three groups of working circumstances gradually grows.

Combining the two sets of displacement diagrams, it is clear that expanding the podium area in the main podium one-story high-rise building is a good way to reduce the building’s overall response to wind-induced vibration displacement. This is because the inter-story displacement of the building will decrease and the displacement pulsation amplitude will weaken as the size of the podium increases, increasing the stiffness of the main body of the building. This situation is comparable to the former working condition, and the outcomes are as anticipated.

As demonstrated in Figure 15a, with an increase in podium floor area, the acceleration time history curves’ pulsation amplitude increases. As can be observed from Figure 15b, the maximum acceleration curves between floors grow significantly as the number of floors increases, and these changes essentially fulfill the expected requirements. The acceleration difference between each curve does not increase as the building height rises; instead, it increases from zero until it reaches its maximum value somewhere around the tenth floor, then it starts to decline until the two curves essentially coincide, during which point the acceleration magnitude decreases as the podium area grows. Beginning at the 45th level, the magnitude sequence of the acceleration values is reversed from what it was, followed by a growing discrepancy in the acceleration values, which lasts all the way to the building’s top floor. The detailed graphic shows the magnitude and sequence of the acceleration values close to the top floor. In general, expanding the podium’s area can successfully reduce the building’s middle and lower floors’ inter-story acceleration. By reaching the middle and upper floors, however, this suppression effect will be completely contrary to the prior one, while the difference is still minimal. Similar results to this one have been seen in previous working conditions.

The bottom shear time history curves for the three working conditions are quite similar to one another, as can be seen in Figure 16a. It can be seen that the podium height of a slight modification for the building bottom shear does not have a very visible impact since the overall difference in the pulsation amplitude of the three is not very great.

The maximum acceleration between floors under the three operating circumstances only minimally varies at the lower floors, as can be shown in Figure 16b, as the floor area increases. Due to the relatively slight change in the area of the podium, the change in the shear force at the base of the building is not obviously evident. However, as can also be seen from the detailed diagram in the lower left corner of Figure 16b, the value of the largest inter-story shear force in the area around the location where the building’s main structure and the podium meet diminishes as the area of the podium increases.

It is evident from Figure 17a that the bottom overturning moment time history curves under the three working conditions have a trend and period that are essentially the same as the bottom shear time history curves in Figure 16a due to a transformation relationship between the shear force and the overturning moment. The value of the overturning moment at the base of the building does not vary obviously because the change in the area of the podium is rather minor.

In Figure 17b, the distinctions between the three curves are not immediately apparent and have even come very close to overlapping since the shear forces between the layers in the three operating conditions have been extremely close and the values of the overturning moments have been quite large. The inter-story overturning moment of the building’s middle and lower floors, however, gradually reduces with an increase in podium area, as shown by the detailed subfigure in the right corner of Figure 17b, and the law is compatible with this change in the condition of the podium height.

The following is a summary of the research results on different podium area working conditions. Nodes from important parts of each model are selected for quantitative comparison, highlighting the podium effect. The data comparison is shown in

Table 7. Comparison of key data for podium area working conditions.

Podium Area	70 m × 70 m	80 m × 80 m	90 m × 90 m
Average displacement of layers	0.0641 m	0.0614 m	0.0593 m
Average layer acceleration	0.0393 m/s2	0.0388 m/s2	0.0384 m/s2
Average layer shear force	4891.9315 kN	4860.7483 kN	4836.6816 kN
Average bending moment of each layer	374,514.0384 kN·m	372,937.091 kN·m	371,780.4873 kN·m
Intermediate displacement	0.0651 m	0.0620 m	0.0596 m
Intermediate layer acceleration	0.0398 m/s2	0.0392 m/s2	0.0386 m/s2
Intermediate layer shear force	5604.7600 kN	5581.7768 kN	5564.9734 kN
Intermediate bending moment	312,944.1633 kN·m	312,401.4622 kN·m	312,093.2803 kN·m
Top displacement	0.1302 m	0.1266 m	0.1238 m
Top acceleration	0.0797 m/s2	0.08 m/s2	0.0802 m/s2
Top-level shear force	236.9437 kN	237.2933 kN	237.7068 kN
Top bending moment	947.7748 kN·m	949.1731 kN·m	950.8271 kN·m

.

From Table 7, it can be seen that as the area of the podium increases, the average values of various parameters and the parameters of the middle floor decrease. But at the top level, except for displacement, all other parameters increase with the increase in the podium area, which is similar to the first two working conditions, once again verifying the reliability of the simulation results. The three parameters of acceleration, shear force, and bending moment decrease with the increase in podium area at the lower level of the building but increase with the increase in podium area at the middle and upper levels, showing the opposite variation. It can be inferred that one of the reasons for this phenomenon is still the weakening of the stiffness effect of the upper part of the building.

6. The Influence of Podium Layout on Wind-Induced Vibration Results

6.1. Calculation Models for Several Podium Layouts

In all layout methods, the size of the podium is 60 m × 60 m × 10 m, and the cross-sectional size of the main body is the same as that of the building without a podium. The overall height of the building is 90 m. C1, C2, and C3 are symmetrical layouts, while C4, C5, and C6 are asymmetrical layouts. The model is shown in Figure 18.

The top views of the six models presented in this section are shown in Figure 19.

The loading method of pulsating wind load (nodal dynamic load) on the attached podium building is shown in Figure 20.

6.2. Analysis of Numerical Simulation Results of Wind-Induced Vibration Response

The maximum displacement of each layer under six working conditions is shown in Figure 21, and the values of the maximum displacement, average displacement, and the ratio of maximum displacement to average displacement of key layers are shown in

Table 8. Displacement of key floors in the conditions of podium layout.

Floor	The Maximum Displacement/m				Average Displacement/m
	C1/C3	C2	C4/C6	C5	C1/C3	C2	C4/C6	C5
top	0.0117	0.0117	0.0161	0.0162	0.0117	0.0117	0.0121	0.0123
10F	0.0044	0.0044	0.007	0.0071	0.0044	0.0044	0.0048	0.0048
7F	0.0024	0.0024	0.0042	0.0042	0.0024	0.0024	0.0028	0.0028
6F	0.0018	0.0019	0.0033	0.0034	0.0018	0.0019	0.0022	0.0022
5F	0.0013	0.0013	0.0025	0.0025	0.0013	0.0013	0.0016	0.0016
4F	0.0009	0.0009	0.0034	0.0036	0.0009	0.0009	0.002	0.002
3F	0.0005	0.0005	0.0022	0.0023	0.0005	0.0005	0.0013	0.0013
2F	0.0002	0.0002	0.0011	0.0011	0.0002	0.0002	0.0006	0.0006

. It can be seen that the maximum displacement of a symmetrical layout always increases with the height of the floor, while the displacement of an asymmetrical layout undergoes a sudden change at the junction of the podium and the main body. Moreover, the maximum displacement of an asymmetrical layout is generally greater than that of a symmetrical layout.

The maximum shear force and maximum overturning moment of each floor under six working conditions are shown in Figure 22 and Figure 23, respectively. The values of maximum shear force and overturning moment of key floors are shown in

Table 9. Shear force and overturning moment of key floors in the conditions of podium layout.

Floor	The Maximum Shear Force/kN				The Maximum Overturning Moment/kN·m
	C1/C3	C2	C4/C6	C5	C1/C3	C2	C4/C6	C5
19F	608.04	596.29	604.34	603.62	3040.22	2981.44	3021.69	3018.11
18F	981.71	952.30	976.29	975.32	7948.78	7742.95	7903.14	7894.73
17F	1339.33	1288.62	1332.92	1331.94	14645.43	14,150.09	14,567.76	14,554.41
12F	2875.35	2853.16	2873.61	2875.51	72,045.96	70,194.22	71,852.77	71,850.20
11F	3126.54	3130.81	3126.57	3129.30	87,678.68	85,696.84	87,485.63	87,496.71
10F	3356.46	3389.70	3357.97	3361.43	104,460.97	102,412.96	104,275.49	104,303.87
4F	4239.93	4413.46	4229.64	4231.34	222,719.75	224,429.28	222,504.72	222,637.92
3F	4512.38	4483.88	4536.85	4527.83	245,162.29	246,848.65	245,188.97	245,277.05
2F	4688.21	4527.78	4696.34	4677.32	268,405.76	269,487.54	268,670.66	268,663.66

. It can be seen that the maximum overturning moment and maximum shear force trend of the six working conditions are similar, but the difference is that the maximum shear force graph is convex on the top and concave on the bottom. Like the maximum displacement, the maximum shear force curves and maximum overturning moment curves of C1 and C3, C4 and C6 completely overlap. Except for the layout at the center, the maximum shear force curves of the other five layouts have small differences and are not smooth at the junction between the podium and the main body. The maximum shear force and maximum overturning moment of the layout at the center of the high-rise area are smaller than those of the other working conditions, while the values of the layout at the center of the low-rise area are relatively large.

From the above analysis, it can be concluded that the overall displacement of symmetrical layout is smaller than that of asymmetrical layout. It is speculated that this is due to the relative weakening of stiffness caused by eccentric layout, resulting in an increase in displacement. By observing the maximum shear force and maximum overturning moment of each floor, it can be concluded that the design of the central podium layout can appropriately reduce the maximum shear force and overturning moment of the higher floors of the building, but the effect is opposite in lower floors.

7. Conclusions

In order to study the wind-induced response characteristics of the main podium one-piece high-rise building structure, this paper presents a process algorithm for solving the pulsating wind speed time history and determining the pulsating wind load based on the AR method. It then applies this algorithm to the study of the wind-induced response characteristics of the structure, and the following conclusions are drawn:

(1)

It is discovered that the building’s response to wind-induced displacement is mostly impacted by the podium’s existence or absence. The middle and bottom floors of the building exhibit the most visible effects of the podium’s presence or absence on the acceleration response, bottom shear, and bottom overturning moment, and the influence diminishes with increasing building height.

(2)

As the height and area of the podium increase, the maximum displacement and average value of each floor of the building steadily decrease, and this effect is particularly evident in the upper floors. But as the height and area of the podium increase, the wind-induced vibration acceleration at the lower and middle levels of high-rise buildings decreases significantly, while the upper level is not obvious, and the shear force and overturning moment at the bottom of the building slightly decrease.

(3)

The layout of the podium has an impact on the wind-induced vibration response of the main building. When the podium and main building are symmetrically arranged in the downwind direction, the maximum displacement of each floor is small, while the maximum displacement curve of buildings with asymmetric layout at the junction of the podium and main building is not smooth. The design of the central layout of the podium and main building can effectively reduce the maximum shear force and maximum overturning moment of the higher floors of the building, but the effect is opposite at lower floors.