Wind-Induced Dynamic Performance Evaluation of Tall Buildings Considering Future Wind Climate

Abstract

The ongoing impacts of climate change, driven by both anthropogenic and global warming, significantly influence wind characteristics, resulting in increased wind speeds. Consequently, buildings that currently satisfy safety and serviceability standards may face challenges in the future. Despite extensive studies on wind-induced responses of tall buildings, there is a notable lack of comparative analyses assessing their performance under both historical and projected future wind conditions influenced by climate change. This study investigates the wind-induced performance of a 151 m tall building located in Suzhou, China, employing time history generation based on power spectral density functions. The analysis evaluates the acceleration responses of the building under both historical and projected future wind scenarios across different return periods and compares the responses to identify the potential changes in the building’s performance due to changing wind conditions. The structural acceleration responses are projected to rise significantly under future wind conditions. Furthermore, this study uses a time-domain Monte Carlo simulation framework to conduct a fragility analysis of the case study building, assessing the comfort of human occupants and the likelihood of exceeding performance thresholds under various wind scenarios. The fragility curve for the case study building is plotted for human occupant comfort as a function of mean wind speed. A substantial increase in the building’s fragility concerning occupant comfort is observed. The future wind climate will significantly impact the performance of tall buildings, necessitating proactive measures to address increased wind-induced effects and ensure long-term safety and habitability.

1. Introduction

The rapid growth of urban populations and limited land availability have led to an increase in tall buildings. These buildings are designed for multiple purposes, such as residential, office, retail, and other occupancy, which encourages a more complex and slender design of buildings. Their slender and flexible designs make them inherently vulnerable to dynamic forces, particularly wind loads [1]. Moreover, anthropogenic effects such as global warming are rapidly changing the climate, which is expected to have a significant impact on wind patterns and extremes, potentially leading to profound consequences for the built environment [2]. Furthermore, research by Kim et al. [3] underscores the importance of considering both mean wind speeds and turbulence intensity when evaluating building performance under changing climatic conditions.

Global warming is often claimed to significantly impact the frequency and intensity of windstorms, especially tropical cyclones. There is consensus on a thermodynamically driven future increase in global and regional maximum wind speeds and on the global incidence of high-intensity tropical cyclones [4,5,6]. Webster et al. [7] observed a significant upward trend in tropical cyclones of Category 4 and 5 globally between 1995 and 2004. Klotzbach [8] expanded the analysis, focusing on more reliable post-1986 data, and observed only a modest rise in the frequency of Category 4 and 5 storms in the North Atlantic and Northwest Pacific over a twenty-year period. However, the current increase in strong cyclones in the North Atlantic and Caribbean may signify a trend attributable to climate change. Emanuel [9] forecasted that future warming might amplify the destructive capacity of tropical cyclones, and considering the growing coastal population, there may be a significant increase in hurricane-related losses in the twenty-first century. Holland and Bruyere [10] developed the Anthropogenic Climate Change Index (ACCI) to assess the influence of human-induced warming on tropical cyclone activity and found that there was a significant increase in the frequency of Category 4 and 5 storms, which was counterbalanced by the reduction in the frequency of Category 1 and 2 storms. They also concluded that there has been a 25–30% regional and worldwide increase in Category 4–5 storms per °C of anthropogenic global warming since 1975. Knutson et al. [11] further stated that greenhouse warming would result in a worldwide increase in the severity of tropical cyclones, with projected intensity rises of 2–11% by 2100.

The Intergovernmental Panel on Climate Change (IPCC) Sixth Assessment Report (AR6) 2021 [12] observed a rise in the worldwide percentage of intense tropical cyclones (Category 3–5) during the last forty years. The report forecasted a global increase in the frequency of major tropical cyclones (Category 4–5) and the maximum wind speeds of the major tropical cyclones due to escalating global warming. Patricola and Wehner [13] looked at how humans affect climate change by simulating 15 major tropical cyclone events around the world in pre-industrial and future climates. They found that future anthropogenic warming would strongly raise wind speeds. Yamaguchi et al. [14] examined the future translation speed of tropical cyclones by performing high-resolution large-ensemble simulations for both the present and future climates spanning 2051–2110. Modeling results from the present and future climates showed that the average global translation speeds from 1951 to 2011 were 17.5 km/h, and from 2051 to 2110 they were 18.0 km/h. This suggests that climate change could cause future translation speeds to rise.

The coastal cities in southeastern China are especially vulnerable to intense winds, storm surges, and substantial rainfall caused by typhoons [15]. Mei and Xie [16] noted that typhoons impacting East and Southeast Asia have strengthened by 12–15% (~5 m/s) during the last 37 years (1977–2014), highlighting a significant increase in both the frequency and intensity of major storms (category 4 and 5). Kang and Elsner [17] observed an increasing number of rapidly intensifying tropical cyclones (RI-TCs) among the total number of tropical cyclones from 1986 to 2015 as a consequence of global warming. Song et al. [18] examined the long-term trend in the annual mean of lifetime maximum intensity (LMI) of RI-TCs across the Western North Pacific Ocean (WNP) and identified an increase in the average from 1970 to 2019, indicating an increase in typhoon strength. Collectively, these research studies suggest that climate change will increase the risk of tropical cyclones and their accompanying wind speeds.

A tall building structure must securely withstand the strong winds it will encounter over its anticipated lifespan. While wind-induced vibrations may not compromise structural integrity, they can cause discomfort to occupants. Various standards, such as those from the Architectural Institute of Japan (AIJ) [19] and ISO 10137 [20], provide criteria for evaluating building motion to ensure occupant comfort. However, these changing climatic conditions can significantly alter the wind loads that buildings experience, potentially posing challenges to buildings that currently meet safety and serviceability standards in the future. Teran [21] investigated the effects of climate change on wind loading for tall buildings, particularly in urban areas like Toronto. The study proposed a multidisciplinary framework that utilizes computational fluid dynamics simulations to evaluate a building’s response under current and projected future wind conditions. The results suggested the current design of the building may be insufficient to withstand wind loads in future wind scenarios. Recently, Zhu et al. [22] investigated the effects of climate change on the wind-induced responses of high-rise buildings, focusing on safety and serviceability. Their results indicated a significant increase in predicted wind speeds by 2100 compared to current conditions. The study also found a significant increase in the building’s acceleration response under climate change scenarios, potentially impacting occupant comfort, especially on higher floors.

Despite numerous researchers having investigated the wind-induced responses of tall buildings [23,24,25,26], there has not been enough comprehensive comparative study of these responses under wind loads throughout history and in the future due to climate change. This study aims to investigate the impact of climate change on the performance of tall building structures by conducting wind analysis under historical and projected future wind conditions. This study employs the time history analysis method to conduct a dynamic analysis of a building under along-wind vibrations. The numerical vibration analysis method is validated by comparing the numerically simulated response of the building with the response from the wind tunnel test. Moreover, this study presents a time-domain Monte Carlo simulation framework for probabilistic vibration exceedance assessment of the building, which aims to estimate the failure probability against human occupant comfort criteria, considering historical and future wind conditions. Furthermore, the fragility curve is plotted as a function of mean wind speed. This study provides a basis for variation in the performance of buildings under historical and future wind speeds due to climate change, which can help quantify the change in acceleration response with change in wind speed in the coming decades to ensure the building’s safety and serviceability throughout its lifespan.

2. Basic Settings for the Simulations

2.1. Building Information and Prevailing Wind Conditions

The case study building is a 49-story residential and office building having a prismatic floor plan up to the higher residential floor, with lateral frames serving as the main wind-force-resisting system. The overall dimensions of the building are about 17 m (depth) × 56 m (width) × 152 m (height). Figure 1 displays the finite element model (FEM) of the case study building. This model primarily includes beams, columns, and shear walls. The beams and columns were modeled using frame sections, while the wall elements were modeled using shell sections. In total, the model consists of 6715 rectangular beam members of various sizes, 210 column members (13 rectangular and 110 circular), and 5862 shear wall elements of thickness ranging from 400 mm to 1000 mm. The structural system incorporates 187 distinct member classifications, with each group defined by unique cross-sectional properties and material strength characteristics, as detailed in

Table 1. Information on structural elements of the building.

Element Type	Section Type	Classification	Concrete Strength	Section Size	Quantity
Beams	Rectangular	96 types	30–60 Mpa	Various section sizes Width: 100 mm to 1350 mm Depth: 100 mm to 750 mm	6715
Columns	Rectangular	3 types	30 Mpa	500 mm × 500 mm	7
				600 mm × 500 mm	3
				600 mm × 600 mm	3
	Circular	9 types	30–60 Mpa	Diameter 800 mm	43
				Diameter 850 mm	67
Walls	Thickness	79 types	30–60 Mpa	Various thickness: from 400 mm to 1000 mm	5862

. The foundation is modeled with fixed support conditions. All structural elements are interconnected with rigid joints for design purposes [27].

Table 2. Structural dynamic characteristics of the building.

Mode	Time Period (s)	Frequency (Hz)	Direction	Damping Ratio
1	3.3841	0.2955	Translation in the Y-direction	2%
2	3.2268	0.3099	Translation in the X-direction
3	2.7755	0.3603	Rz Torsion

displays the structural dynamic characteristics of the case study building. The frequencies of the first three modes of the structure are 0.295 Hz, 0.310 Hz, and 0.360 Hz, respectively. For serviceability assessment under wind loading, a 2% damping ratio has been adopted for reinforced concrete buildings [27,28,29]. Therefore, this study also used a 2% classical damping as a modal damping ratio to assess dynamic performance and occupant comfort.

The case study building is located in Suzhou, a coastal city in China, which is particularly vulnerable to the strong winds associated with typhoons, as tropical cyclones originating from the WNP Ocean frequently impact southeast China [30]. Huang et al. [15] examined the effects of climate change on wind speed, using historical data alongside SSP245 and SSP585 scenario analyses. They performed full-track typhoon simulations to examine wind hazards in coastal areas under changing climatic conditions. They developed a custom random forest typhoon track model to generate future typhoon scenarios in the Northwest Pacific, using the sea surface temperature (SST) changes projected by CMIP6 under the SSP245 emission model. This research utilizes the estimated wind speed data for Suzhou by the same model, as shown in

Table 3. Historical and future estimated mean wind speed of Suzhou, m/s.

Case	Return Period	10-Year	50-Year	100-Year
Historical climate	Historical (1979–2015)	24.9	30.6	32.2
Future climate	Mid-future (2019–2055)	25.3	31.6	33.3
	Late-future (2064–2100)	26.4	32.5	34.4

[15].

2.2. Performance Evaluation of the Building Under Wind Loads

Time-history analysis is a reliable method to evaluate the aerodynamic performance of tall buildings under strong wind conditions. Although it is possible to efficiently carry out the dynamic analysis of tall buildings under wind loads in the frequency domain [31], the time history analysis technique offers a more detailed understanding of their wind-induced responses [32]. This approach not only provides an alternative for calculating statistical measures of wind-induced responses but also gives valuable insights into response distribution, aiding in estimating exceedance probability within performance-based design contexts [33]. In the time history analysis of a wind-excited tall building, the initial task involves generating fluctuating wind forces, which the quasi-steady assumption could link to the wind velocity [34].

Researchers have recognized the need for random vibration, spectral-based, and peak-estimation approaches in wind engineering since the beginning of tall building research due to the presence of random turbulence in the load and the dynamic vibration of the structure [35]. Hwang et al. [36] conducted a study on the generation of time-history wind loads from power spectral density (PSD) functions to evaluate habitability. Jeong et al. [37] employed the same approach to perform a performance-based wind design for tall buildings. In the same way, Micheli et al. [38] studied high-performance control systems in wind-excited tall buildings by simulating wind loads using the time history method based on the PSD function.

This paper conducts a wind analysis of the case study building using the time history generation method, which incorporates a quasi-steady approximation. Figure 2 outlines a framework for the overall methodology. The research framework consists of two primary components: dynamic analysis (Part I) and fragility assessment (Part II). The dynamic analysis involves simulating wind loads and evaluating the corresponding acceleration responses of the buildings. By comparing the acceleration responses under different scenarios, this study identifies potential changes in building performance due to evolving wind conditions. The fragility analysis assesses the probability of exceeding performance thresholds related to human occupant comfort using a Monte Carlo simulation framework.

2.2.1. Generation of Wind Speed Time History

The time-varying wind speed is calculated by summing the mean and time-varying component from the mean using Equation (1):

$$U_z\left(t\right)={\overline{U}}_z+u_z\left(t\right)$$

(1)

where U_z(t) is time-varying wind speed, Ū_z is the mean wind speed, and u_z(t) is the fluctuating wind speed component due to turbulence at z height of the building.

The wind speed changes from zero to a maximum at a certain height from the ground surface. The maximum wind speed height can be different depending on the roughness of the ground surface. This process generally treats wind speeds in the atmospheric boundary layer as stationary random processes [34]. The case study building was divided into 20 representative floors for wind load simulation. Mean wind speed at the representative floors of the building, Ū_z, was computed from the power law using Equation (2):

$${\overline{U}}_z={\overline{U}}_{10}({\displaystyle \frac{z}{10}}{)}^{\alpha }$$

(2)

where Ū₁₀ is reference speed, which is generally measured at the height of 10 m from the ground surface. α is a dimensionless power-law exponent that depends on the terrain roughness. In accordance with the provisions of the Code for Loading of Building Structures (GB50009-2012) [39], the case study building falls under the Class B landform, with a value of approximately 0.15.

Turbulence intensity (I) is calculated by using Equation (3):

$$I_z={\displaystyle \frac{{\sigma }_z}{{\overline{U}}_z}}$$

(3)

where σ_z is the standard deviation, given by Equation (4):

$${\sigma }_z=\sqrt{6u_{\ast }}$$

(4)

where $u_{\ast }$ represents the frictional velocity of the wind flow.

Mean wind speed and turbulence intensity profiles are shown in Figure 3 and Figure 4, respectively. Figure 3a–c represent the comparison of mean wind speed profiles for historic, mid-future, and late-future wind speeds of return periods of 10 years, 50 years, and 100 years, respectively.

Shinozuka and Deodatis [40] and Deodatis [41] suggested a method for the generation of time history using the power spectral density function. A power spectral density matrix S(ω) for an n-story building with natural frequency ω is given by Equation (5):

$$\mathbf{S}(\omega )=\left(\begin{array}{ccc}{S}_{11}\left(\omega \right)& \dots & S_{1n}\left(\omega \right)\\ ⋮& \ddots & ⋮\\ S_{n1}\left(\omega \right)& \cdots & S_{nn}\left(\omega \right)\end{array}\right)$$

(5)

The elements of the S(ω) matrix are calculated by Equation (6):

$$s_{ji}=\left\{\begin{array}{ll}{S}_j\left(\omega \right),& j=i\\ \sqrt{S_j\left(\omega \right)S_i\left(\omega \right)}{\gamma }_{ji}\left(\omega \right),& j\ne i\end{array}\right.$$

(6)

where S_j(ω) is the Kaimal power spectral density function for the jth level, given by Equation (7):

$$S_j(\omega )={\displaystyle \frac{200}{2\pi }}{u}_{\ast }^2{\displaystyle \frac{z}{{\overline{U}}_j}}{\left(1+50{\displaystyle \frac{\omega z}{2\pi {\overline{U}}_j}}\right)}^{{\displaystyle \frac{-5}{3}}}$$

(7)

and $u_{\ast }$ can be computed by using Equation (8):

$$u_{\ast }={\displaystyle \frac{0.4{\overline{U}}_j}{\mathit{ln}(z/z_0)}}$$

(8)

where z represents the height of the respective floors and z₀ is the terrain roughness height. The value of z₀ is 0.05 m for terrain roughness category B as per China wind loading code GB50009-2012 [39]. γ_ji(ω) is the coherence function between the wind speed turbulence at two different floors, jth and ith, at heights z_j and z_i, given by Equation (9):

$${\gamma }_{ji}(\omega )=\mathit{exp}\left[-{\displaystyle \frac{\omega }{2\pi }}{\displaystyle \frac{C_z|z_j-z_j|}{\frac{1}{2}({\overline{U}}_j+{\overline{U}}_i)}}\right]$$

(9)

where C_z is a constant called a correlation coefficient that can be taken as 10 for structural design purposes.

The power spectral density matrix S(ω) is decomposed by the Cholesky factorization to obtain H(ω), and the fluctuating wind velocity time history at the jth level can be generated with weighted amplitude harmonic wave superposition. The Cholesky decomposition method is carried out for the decomposition of the PSD matrix S(ω) as follows:

$$\mathbf{S}\left(\omega \right)=\mathbf{H}\left(\omega {)}^{T\ast }\mathbf{H}\right(\omega )$$

(10)

where H(ω) is a lower triangular matrix as depicted in Equation (11), whose diagonal elements are real and non-negative functions of natural frequency (ω). The superscript T indicates the matrix transpose, while the asterisk represents the complex conjugate.

$$\mathbf{H}(\omega )=\left(\begin{array}{ccc}{H}_{11}\left(\omega \right)& \dots & 0\\ ⋮& \ddots & ⋮\\ H_{n1}\left(\omega \right)& \cdots & H_{nn}\left(\omega \right)\end{array}\right)$$

(11)

The wind turbulence time history, denoted as u_j(t) at the jth story level, is computed by the trigonometric function’s superposition with random phase angles, as formulated by Equation (12):

$$u_j(t)=2\sum _{q=1}^n\sum _{l=1}^{N_{\omega }}\left|H_{jq}\right({\omega }_{ql}\left)\right|\sqrt{\Delta \omega }\mathit{cos}\left[{\omega }_{ql}t-{\vartheta }_{ql}\left({\omega }_{ql}\right)+{\varphi }_{ql}\right]$$

(12)

where ∆ω is the frequency interval, defined as ω_u/N_ω, with ω_u being the cut-off frequency (set at 2 rad/s) and N_ω the number of divisions in the frequency domain (set to 500). The frequency ω_ql is computed by Equation (13):

$${\omega }_{ql}=l\Delta \omega -{\displaystyle \frac{q-1}{q}}\Delta \omega$$

(13)

The vector Φ_ql consists of random numbers drawn from a uniform distribution between [0, 2π], and ϑ_ql is given by Equation (14):

$${\vartheta }_{ql}(\omega )={\mathit{tan}}^{-1}\left\{{\displaystyle \frac{\mathit{Im}[H_{jq}\left({\omega }_{ql}\right)]}{\mathit{Re}[H_{jq}\left({\omega }_{ql}\right)]}}\right\}$$

(14)

where Im[H] and Re[H] indicate imaginary and real quantities of H, respectively.

The time-varying wind speeds at each representative floor of the building were simulated by summing up the mean wind (from Equation (2)) and turbulence component (from Equation (12)) as per Equation (1). Figure 5 compares the simulated historical, mid-future, and late-future wind speed time histories at the 44th floor of the building for 10-year, 50-year, and 100-year return period wind speed.

2.2.2. Wind Loads Simulation

Wind load time series can be obtained from the simulated wind velocity by quasi-steady approximation with aerodynamic coefficients. The along-wind load P_u(t) and across-wind load P_v(t) can be calculated by Equation (15) and Equation (16), respectively [34].

$$P_{u,z}(t)={\displaystyle \frac{1}{2}}\rho {\left({\overline{U}}_z+u_z(t)\right)}^2{C}_{d,z}{A}_{u,z}$$

(15)

$$P_{v,z}\left(t\right)={\displaystyle \frac{1}{2}}\rho v_z^2\left(t\right)C_{l,z}{A}_{v,z}$$

(16)

where ρ is the air density (ρ = 1.25 kg/m³); C_d,z and C_l,z are the drag coefficient and lift coefficient corresponding to height z; A_u,z and A_v,z are areas in along-wind and across-wind directions at the height of z. Lift coefficient C_l,z can be calculated by using Equation (17) [42], and drag coefficient C_d,z can be obtained by calculating the difference between the external pressure coefficient of the windward and leeward walls as given in Table 4 below by using Equation (18) [37]:

$$C_{l,z}=0.045{\left({\displaystyle \frac{D}{B}}\right)}^3-0.335{\left({\displaystyle \frac{D}{B}}\right)}^2+0.868\left({\displaystyle \frac{D}{B}}\right)-0.174$$

(17)

$$C_{d,z}=C_{pe1}-C_{pe2}$$

(18)

Table 4. External pressure coefficients, C_pe.

Parameters	Conditions	Equations
Windward wall, Cpe1	—	$C_{pe1}={0.8k}_z+0.03(D/B)$
Leeward wall, Cpe2	$D/B\le 1$	$C_{pe2}=-0.5$
	$D/B>1$	$C_{pe2}=-0.5+0.25ln{\left(D/B\right)}^{0.8}$
Pressure distribution coefficients for vertical profile, kz	$z\le z_b$	$k_z={\left(z_b/H\right)}^{2\alpha }$
	$z_b<z<0.8H$	$k_z={\left(z/H\right)}^{2\alpha }$
	$z\ge 0.8H$	$k_z={\left(z/H\right)}^{2\alpha }$

where B represents the breadth of the building (perpendicular to the along-wind direction) (m); D represents the depth of the building (perpendicular to the across-wind direction) (m); z represents the height from the ground (m); and z_b represents height above the ground surface starting at the atmospheric boundary layer (m). In this study, the along-wind direction is the Y-direction of the building. The value of the lift coefficient is 0.058.

Table 5. Aerodynamic coefficients of the representative floors of the building.

Location	Floor	Height (m)	Drag Coefficients, Cd
1	48	151.8	1.2572
2	47	148.8	1.2572
3	46	145.8	1.2572
4	45	142.3	1.2572
5	44	139.15	1.2572
6	43	136	1.2572
7	40	126.55	1.2572
8	37	117.1	1.2491
9	34	107.65	1.2307
10	31	98.2	1.211
11	28	88.7	1.1899
12	25	79.25	1.1673
13	23	72.95	1.1511
14	19	60.35	1.1156
15	16	50.9	1.0854
16	13	41.4	1.0508
17	10	31.95	1.0103
18	7	22.5	0.9602
19	4	13.05	0.8922
20	1	3.6	0.8628

lists drag coefficients at representative heights of the building.

Table 4. External pressure coefficients, C_pe.

Parameters	Conditions	Equations
Windward wall, Cpe1	—	$C_{pe1}={0.8k}_z+0.03(D/B)$
Leeward wall, Cpe2	$D/B\le 1$	$C_{pe2}=-0.5$
	$D/B>1$	$C_{pe2}=-0.5+0.25ln{\left(D/B\right)}^{0.8}$
Pressure distribution coefficients for vertical profile, kz	$z\le z_b$	$k_z={\left(z_b/H\right)}^{2\alpha }$
	$z_b<z<0.8H$	$k_z={\left(z/H\right)}^{2\alpha }$
	$z\ge 0.8H$	$k_z={\left(z/H\right)}^{2\alpha }$

where B represents the breadth of the building (perpendicular to the along-wind direction) (m); D represents the depth of the building (perpendicular to the across-wind direction) (m); z represents the height from the ground (m); and z_b represents height above the ground surface starting at the atmospheric boundary layer (m). In this study, the along-wind direction is the Y-direction of the building. The value of the lift coefficient is 0.058. Table 5 lists drag coefficients at representative heights of the building.

Figure 6 compares the time histories of the along-wind load acting on the building at the 44th floor for the same return periods (10-year, 50-year, and 100-year). Each sub-figure shows how wind loads fluctuate over time, illustrating the dynamic nature of wind forces.

2.2.3. Wind-Induced Dynamic Response Analysis of Tall Building

Dynamic response analysis for tall buildings involves studying structural behavior and performance under wind loads or other dynamic forces. This analysis helps assess the building’s stability, evaluate its response to wind-induced vibrations, and ensure its safety and functionality. The wind-induced response analysis of the tall building can be computed in the time domain. Many past studies have developed various methods and models. One approach involves a time-domain method for predicting wind-induced responses of tall buildings by analyzing the dynamic equations of motion numerically, taking into account the mass, stiffness, and damping properties of the building to determine the building response over time.

Generally, the floor diaphragm of the tall building structure is considered rigid. These multi-story structures, featuring rigid floor diaphragms, are represented by a concentrated mass at each floor level. Each floor’s movement is confined to three degrees of freedom, encompassing two translational movements and one rotational movement around the vertical axis. For a building with n stories, considered as a lumped mass system with 3n-degree of freedom, the equation of motion can be expressed in matrix form as in Equation (19):

$$\left[M\right]\left\{\ddot{x}\right\}+\left[C\right]\left\{\dot{x}\right\}+\left[K\right]\left\{x\right\}=\left[F\right]$$

(19)

where [M], [C], [K], and [F] are the mass matrix, damping coefficient matrix, stiffness matrix, and external forces of time. {x} is displacements, which is used generally to represent x, y displacements, and z rotation. Equation (20) represents the matrix form of Equation (20) as follows [43]:

$$\left(\begin{array}{ccc}M& 0& 0\\ 0& M& 0\\ 0& 0& I\end{array}\right)\left\{\begin{array}{c}\ddot{X}\\ \ddot{Y}\\ \ddot{\Theta }\end{array}\right\}+\left(\begin{array}{ccc}{C}_{XX}& 0& C_{X\Theta }\\ 0& C_{YY}& C_{Y\Theta }\\ C_{X\Theta }^T& C_{Y\Theta }^T& C_{\Theta \Theta }\end{array}\right)\left\{\begin{array}{c}\dot{X}\\ \dot{Y}\\ \dot{\Theta }\end{array}\right\}+\left(\begin{array}{ccc}{K}_{XX}& 0& K_{X\Theta }\\ 0& K_{YY}& K_{Y\Theta }\\ K_{X\Theta }^T& K_{Y\Theta }^T& K_{\Theta \Theta }\end{array}\right)\left\{\begin{array}{c}X\\ Y\\ Z\end{array}\right\}=\left\{\begin{array}{c}{F}_X\\ F_Y\\ F_{\Theta }\end{array}\right\}$$

(20)

where M = diag [m_i] is the mass submatrix, where m_i is the lumped mass at the ith floor. I = diag [I_i] is the mass moment of the inertia matrix of each floor’s diaphragm, where I_i = m_ir_i² and r_i is the radius of the gyration of each floor. X = (x₁, x₂, ……, x_n)^T, Y = (y₁, y₂, ……, y_n)^T, Θ = (θ₁,θ₂,..…,θ_n)^T represent the displacement response sub-vectors. The damping submatrices C_XX, C_YY, C_ΘΘ, C_XΘ, and C_YΘ and stiffness submatrices K_XX, K_YY, K_ΘΘ, K_XΘ, and K_YΘ define the dynamic properties of the building. The wind load forces in the X and Y directions and the torsional wind moment about the vertical axis are represented by the sub-vectors F_X = (F_x1, F_x2, ……, F_xn)^T, F_Y = (F_y1, F_y2, ……, F_yn)^T, and F_Q = (T_θ1, T_θ2, ……, T_θn)^T respectively.

The system of coupled differential equations can be transformed into a set of uncoupled equations using normal modes. By applying the orthogonality conditions of the mode shapes and assuming classical damping, the governing dynamic equations of motion can be transformed into a system of j = 1, 2, …., n uncoupled equations as shown in Equation (21):

$$\left(\begin{array}{ccc}{m}_1& 0& 0\\ 0& \ddots & 0\\ 0& 0& m_n\end{array}\right)\left\{\begin{array}{c}{\ddot{q}}_1\left(t\right)\\ ⋮\\ {\ddot{q}}_n\left(t\right)\end{array}\right\}+\left(\begin{array}{ccc}{c}_1& 0& 0\\ 0& \ddots & 0\\ 0& 0& c_n\end{array}\right)\left\{\begin{array}{c}{\dot{q}}_1\left(t\right)\\ ⋮\\ {\dot{q}}_n\left(t\right)\end{array}\right\}+\left(\begin{array}{ccc}{k}_1& 0& 0\\ 0& \ddots & 0\\ 0& 0& k_n\end{array}\right)\left\{\begin{array}{c}{q}_1\left(t\right)\\ ⋮\\ q_n\left(t\right)\end{array}\right\}=\left\{\begin{array}{c}{Q}_1\left(t\right)\\ ⋮\\ Q_n\left(t\right)\end{array}\right\}$$

(21)

$$m_j{\ddot{q}}_j+c_j{\dot{q}}_j+k_j{q}_j=Q_j$$

(22)

where m_j, c_j, k_j, and Q_j are the generalized mass, damping, stiffness, and forces, respectively, at the jth mode of the building system.

The finite element model (FEM)-based software SAP2000 v24.0.0 was employed to develop the case study building model. The linear modal time history analysis was conducted by applying the simulated wind load time histories to each representative floor of the building. By applying the wind loads as time-varying forces, the analysis predicts the building’s response at various points in time. It provides information on the structural response, including displacements, accelerations, and internal forces, as a function of time. These results can be used to assess the structural dynamic behavior of the structure under varying wind conditions. Subsequently, in this study, wind-induced acceleration responses are used to evaluate the dynamic characteristics of the building performance.

2.3. Simulation Results and Discussions

The acceleration responses of the building structure under historical, mid-future, and late-future wind conditions for different return periods (10-year, 50-year, and 100-year) are compared as shown in Figure 7 (along wind direction, Y-direction) and Figure 8 (across wind direction, X-direction). The acceleration responses at the 44th floor is compared as it is the highest residential floor of the building. From Figure 7, the acceleration responses at the 44th floor of the building in the along-wind direction are discussed as follows:

• For a 10-Year Return Period: The maximum acceleration for historical conditions was recorded at 0.1550 m/s², which increases to 0.1674 m/s² in mid-future conditions and further to 0.1708 m/s² in late-future conditions. This trend indicates an approximate increase of 8.0% and 10.0% for mid-future and late-future conditions, respectively, compared to historical data.
• For a 50-Year Return Period: The historical maximum acceleration was 0.2319 m/s², which rises to 0.2483 m/s² in mid-future and 0.2732 m/s² in late-future conditions. This represents increases of about 7.0% and 18.0%.
• For a 100-Year Return Period: The historical maximum acceleration of 0.2682 m/s² increases to 0.2850 m/s² and 0.3218 m/s² in mid-future and late-future conditions, respectively, reflecting increases of 6.0% and 20.0%.

Likewise, the peak acceleration values in the X-direction of the building follow the same pattern in the Y-direction.

Table 6. Acceleration responses at 44th floor of the building.

Return Period	Wind Cases	Accelerations (m/s2)
		amax,y	amin,y	amax,x	amin,x
10-year	Historical	0.1550	−0.1486	0.0072	−0.0071
	Mid-future	0.1674	−0.1690	0.0087	−0.0086
	Late-future	0.1708	−0.1695	0.0094	−0.0093
50-year	Historical	0.2319	−0.2188	0.0118	−0.0119
	Mid-future	0.2483	−0.2466	0.0137	−0.0137
	Late-future	0.2732	−0.2705	0.0153	−0.0151
100-year	Historical	0.2682	−0.2655	0.0150	−0.0148
	Mid-future	0.2850	−0.2792	0.0170	−0.0171
	Late-future	0.3218	−0.3016	0.0186	−0.0184

presents the peak acceleration value at the 44th floor of the building for different wind conditions and return periods in both directions of the building. The results indicate trends in acceleration response over time, highlighting the increasing impact of climate change on building performance.

2.4. Validation of Simulation Results with Wind Tunnel Test

This study conducts a wind tunnel test for the structural wind vibration response of the case study building. Figure 9 presents the wind tunnel test setup. The wind tunnel is equipped with a collectible automatic three-dimensional transfer frame system, using a multi-functional sharp split grid combination device and three rough elements, which can quickly simulate the atmospheric boundary layer airflow that matches the scale model and adapt to different site terrain categories. The high-frequency force balance (HFFB) test was conducted using the rigid building model in the wind tunnel, in which the geometric scale ratio of the wind tunnel model is 1:300. This test calculates the acceleration response of the building under different return periods and provides the peak acceleration of the highest residential floor.

The peak acceleration of the highest residential floor (at the height of 139.15 m) in the Y-direction (along-wind direction) of the building obtained from the wind tunnel test was 0.1302 m/s² for a basic wind speed of a 10-year return period of 21.9 m/s. Moreover, to validate the framework, a wind analysis of the building was performed for the same basic wind speed (21.9 m/s) using the proposed numerical method. Figure 10 presents the simulated acceleration response for the same basic wind speed used. The peak acceleration value obtained numerically is 0.1267 m/s², which shows a difference of only 2.68% compared to the wind tunnel result. This close correlation between the numerical and experimental results supports the validity of the proposed method.

3. Fragility Analysis for Human Occupant Comfort

Tall building vibrations can reach accelerations that may cause discomfort to occupants even if they satisfy strength requirements. The acceleration level and its correlation with human reaction and perception of the structure’s motion determine the occupant comfort criteria [44]. Monte Carlo simulations have been widely used to generate fragility curves by simulating numerous load scenarios that help to understand structure behavior under varied wind load conditions. Smith and Caracoglia [35] performed a fragility study of a tall structure subjected to turbulent wind loads, using a Monte Carlo simulation technique to model the along-wind response. Cimellaro et al. [45] computed fragility functions and developed an alternative methodology that incorporates several limit state parameters, including combinations of response variables for accelerations and inter-story drifts. Pozzuoli et al. [46] investigated the serviceability assessment of tall buildings under wind loads, including aeroelastic effects. The researchers conducted an extensive experimental campaign on a continuous equivalent aeroelastic model of regular square-section tall structures to assess the wind loads induced on the structure and its responses. Huang et al. [34] employed a Monte Carlo simulation approach to estimate the probability of vibration exceedance of a tall building under typhoon-induced vibration.

3.1. Occupant Comfort Criteria

While the evaluation of tall building vibration performance still lacks accepted vibration acceptability and occupant comfort standards [47], certain design codes have established peak vibration limits to ensure occupant comfort, even in the presence of strong winds. In 1991, the Architectural Institute of Japan (AIJ) published Guidelines for the Evaluation of Habitability for Building Vibration, which underwent revisions in 2004. This code applies the one-year-recurrence peak acceleration to evaluate the habitability of buildings to wind-induced vibrations. Based on the proportion of people who experience vibration sensations, the AIJ criterion establishes five levels of comfort performance: H-10, H-30, H-50, H-70, and H-90, corresponding to 10%, 30%, 50%, 70%, and 90% of people, respectively, experiencing vibrations without discomfort [19]. The Code of Practice on Wind Effects in Hong Kong 2019 [48] established acceptable occupant comfort levels for the peak accelerations for the return periods of 1 year and 10 years. ISO 10137 [20] offers comparable guidelines for assessing perception thresholds related to wind-induced vibration in tall buildings. The National Building Code of Canada, the Chinese Code, and the Hong Kong Code of Practice set peak acceleration limits of 15 and 25 milli-g for residential and commercial buildings, respectively, under strong wind conditions with a 10-year return period [34].

The ISO 6897 [49] specifies a vibration threshold dependent on frequency, serving as a criterion for occupant comfort. This threshold is based on the standard deviation of the acceleration response over a 10-min duration, corresponding to wind conditions with a 5-year return period, as given in Equation (23).

$${\sigma }_a^U=\mathit{exp}(-3.65-0.41\mathit{ln}f)$$

(23)

where f represents the building’s first modal frequency and ${\sigma }_a^U$ refers to the standard derivation criteria of acceleration. The vibration criteria based on the ISO 6987 [49] are simplified in terms of the peak acceleration as follows:

$${\widehat{a}}^U=g_f\mathit{exp}(-3.65-0.41\mathit{ln}f)$$

(24)

where g_f denotes the peak factor, which can be calculated for a Gaussian process as defined in Equation (25):

$$g_f=\sqrt{2\mathit{ln}{f}_p\tau }+{\displaystyle \frac{\gamma }{\sqrt{2\mathit{ln}{f}_p\tau }}}$$

(25)

where γ = 0.5772 is the Euler constant; τ is the duration of wind excitation of the building.

Moreover, Wang et al. [50] proposed six levels of occupant comfort based on peak acceleration responses, providing a classification of human subjective perception to wind-induced vibrations. The comfort levels, along with their corresponding peak acceleration limits, are summarized in

Table 7. Occupant comfort levels.

Level	Human Subjective Response	Peak Acceleration Limit (milli-g)
I	No sense	≤5
II	Low vibration sense	5–20
III	Medium vibration sense	20–35
IV	Trouble	35–50
V	Very Trouble	50–150
VI	Intolerable	≥150

.

3.2. Probabilistic Assessment of Vibration Analysis

This section presents a comprehensive fragility analysis of the tall building against human occupant comfort criteria, focusing on its response to historical, mid-future, and late-future wind conditions. The analysis aims to evaluate the probability of exceeding given performance thresholds, particularly regarding human occupant comfort. The modal peak acceleration is regarded as the perception threshold for the occupant comfort assessment. The flowchart for the dynamic performance assessment and failure probability evaluation is shown in Figure 2. The mean wind speeds are considered to evaluate the probability of failure using the Monte Carlo simulation. A performance function for the probabilistic assessment of vibration exceedance of a wind-induced tall building could be defined as follows:

$$G(U_r,\theta )=a^U-\mathit{max}[a(U_r,\tau ,\theta )]$$

(26)

where U_r represents the design wind speed and θ represents the modal property vector. a^U is the peak acceleration criterion, and a(U_r,τ,θ) refers to the wind-induced vibration of the building over a specified time duration τ. The time duration τ can be set to the values such as 10 min, 1 h, or more, and assuming that the wind loads and vibrations induced by storms or typhoons can be modeled as a stationary process. The design wind speed is averaged over the same time duration τ used for wind-induced vibration. Based on the performance function defined in Equation (23), the probability of failure due to the vibration exceedance is defined by the limit state G(Ur, θ) = 0, where G ≤ 0 indicates failure. Theoretically, the failure probability can be assessed through a multidimensional integral, as shown in Equation (27):

$$P_{f_p}=P\left(G_p\right(\mathrm{x},\mathrm{d}\left)\le 0\right)=\int G_p\le 0\cdots \int f_x(x_1,{\mathrm{x}}_2,\dots )d{x}_{1}d{x}_1\cdots d{x}_n$$

(27)

where G_p represents directional performance function G in the direction p. The vector x consists of random variables that quantify uncertainties in wind loads and structural properties for probabilistic exceedance evaluation, while d denotes the deterministic building parameters. f_x(x₁, x₂, …) represents the joint PDF of the random variables. The performance function in Equation (27) determines whether vibration exceedance leads to failure (G_p ≤ 0) or not (G_p > 0). By conducting a sufficiently large number of simulations, the failure probability due to vibration exceedance can be approximated using Equation (28):

$$P_{f_p}=P(G_p(\mathrm{x},\mathrm{d})\le 0)\approx {\tilde{P}}_{f_p}={\displaystyle \frac{1}{N}}\sum _{k=1}^N{\Pi }_{G_p}\left(x\right)$$

(28)

where Π_Gp is the so-called indicator function, which is equal to one when G_p ≤ 0 and zero otherwise.

3.3. Uncertainties in Occupant Comfort Problems

Significant uncertainties arise in wind-excited tall buildings due to aerodynamic wind loading characteristics or the properties of the building system. The statistical components of design wind speeds directly influence the uncertainty of wind loading characteristics. The design wind speed value is uncertain due to the sample mean and sample standard deviation being random variables, with their variation contingent upon the sample size. The uncertainty in estimated design wind speeds may be assessed by analyzing the mean and variance. The mass, stiffness, and damping of a structure all affect its dynamic properties, such as its natural frequency and the shapes of its vibration modes. A decrease in the uncertainty of wind-induced loads and corresponding responses cannot be achieved without addressing the uncertainty in structural damping, which is closely related to the predicted accelerations of a structure and significantly influences its ability to satisfy occupant comfort criteria. Consequently, structural damping, acknowledged as the most uncertain factor, is a significant consideration in the serviceability design of tall buildings. From a practical perspective, mass and stiffness exhibit far less uncertainty than damping and may therefore be regarded as deterministic variables [51]. In this study, while considering uncertainty in both mean wind speed and structural damping, the distributions of the random wind speed and damping were modeled as Gumbel and lognormal probability distribution functions, respectively.

Table 8. Statistical parameters for windspeed and damping.

Random Variables	Distribution Type	Wind Cases	Mean	Standard Deviation	COV
Wind speed (m/s)	Gumbel distribution	Historical 10-year	24.9	1.245	0.05
		Mid-future 10-year	25.3	1.265	0.05
		Late-future 10-year	26.4	1.320	0.05
		Historical 50-year	30.6	1.530	0.05
		Mid-future 50-year	31.6	1.580	0.05
		Late-future 50-year	32.5	1.625	0.05
		Historical 100-year	32.2	1.610	0.05
		Mid-future 100-year	33.3	1.665	0.05
		Late-future 100-year	34.4	1.720	0.05
Damping ratio	Lognormal distribution	All wind scenarios	0.02	0.003	0.15

represents the statistical parameters for wind speeds and damping ratios used to assess the probability of vibration exceedance of the tall building for occupant comfort.

3.4. Fragility Analysis Results and Discussions

In this study, the framework has considered the uncertainties of wind speed and damping in vibrations related to tall buildings occupant comfort problems. This study repeatedly carried out 1000 simulation cycles for each level of mean wind speed. The probability density distributions of the simulated peak acceleration responses of the building under historical, mid-future, and late-future wind loads for 10-year, 50-year, and 100-return periods are shown in Figure 11, Figure 12, and Figure 13, respectively. Three different types of distribution models were used to fit the simulated peak acceleration’s PDF: the Normal, Gumbel, and Weibull distributions. The simulated peak acceleration data for all wind scenarios seems to be best-fit by the normal distribution.

The simulated mean values of the peak acceleration data with the corresponding coefficient of variance (COV) from the normal distribution are listed in

Table 9. Mean value of peak acceleration responses and failure probability of the building.

Return Period	Wind Cases	Normal Distribution			Failure Probability (%)
		Mean (mlli-g)	Standard Deviation (mlli-g)	COV (%)
10-year	Historical	15.3973	1.8958	12.3125	0
	Mid-future	15.9956	1.8201	11.3787	0
	Late-future	17.4820	2.0793	11.8939	0
50-year	Historical	23.5594	2.7594	11.7125	25.50
	Mid-future	25.2051	3.2753	12.9945	46.00
	Late-future	26.6818	3.3870	12.6940	65.00
100-year	Historical	26.1526	3.0134	11.5223	58.50
	Mid-future	28.0376	3.6689	13.0856	81.00
	Late-future	29.7324	3.3611	11.3045	95.00

. For the 10-year return period, the mean value of maximum acceleration for historical conditions was 15.3973 milli-g, which increases to 15.9956 milli-g in mid-future conditions and further to 17.4820 milli-g in late-future conditions. For the 50-year return period, the mean value of maximum acceleration for historical conditions was 23.5594 milli-g, which increases to 25.2051 milli-g in mid-future conditions and further to 26.6818 milli-g in late-future conditions. For the 100-year return period, the mean value of maximum acceleration for historical conditions was 26.1526 milli-g, which increases to 28.0376 milli-g in mid-future conditions and further to 29.7324 milli-g in late-future conditions.

Table 9 also presents the failure probability of the building under different wind-loading scenarios. For the 10-year return period, the failure probability remains at 0% across all scenarios, indicating that the building is expected to perform adequately under frequent wind events. However, for the 50-year return period, we observe a significant increase in failure probability from 25.50% in historical conditions to 65% in the late-future scenario, highlighting a substantial risk as wind conditions become more severe. Likewise, for the 100-year return period, the failure probability escalates drastically from 58.50% historically to 95% in late-future conditions.

The fragility curve against occupant comfort criteria obtained for the case study building is shown in Figure 14. The curve also includes the failure probability values under different wind scenarios, which shows that the failure probability increases significantly from historical to future under 50-year and 100-year wind. The mean values of peak acceleration and mean wind speed relationships, along with occupant comfort levels, are presented in Figure 15. These results highlight a clear upward trend in acceleration responses and associated failure probability under projected future wind conditions as future wind conditions become increasingly severe due to climate change. This suggests that buildings designed to meet current safety standards may become inadequate as wind speeds continue to rise due to climate change.

4. Conclusions

This research evaluates the impact of climate change on the performance of a 151 m tall building by a comparative study under historical and projected future wind conditions. A probabilistic assessment of vibration was performed using a Monte Carlo simulation to estimate the probability of vibration exceedance for occupant comfort under historical, mid-future, and late-future conditions for 10-year, 50-year, and 100-year return period wind conditions.

The results indicate that structural acceleration responses could rise under future wind conditions. The maximum acceleration for the 10-year return period may increase by approximately 8% in mid-future conditions and 10% in late-future conditions compared to historical data. Similarly, for the 50-year return period, the acceleration may rise by 7% and 18%, and for the 100-year return period, by 6% and 20%, respectively.

A tendency toward increased fragility of the building concerning occupant comfort is observed. Under the 50-year return period, the probability of exceeding comfort thresholds could increase from 25.5% historically to 65% in late-future conditions. For the 100-year return period, this probability may surge from 58.5% historically to 95% in late-future scenarios.

This research mainly demonstrates that future wind climate may significantly influence the performance of tall buildings, necessitating proactive measures to address increased wind loads and ensure long-term safety and habitability. For future work, this comparative study can be extended to analyze across-wind vibrations and torsional effects, which play a significant role in building performance under extreme wind conditions. Additionally, while this study performed fragility analyses focusing on the serviceability limit state, which addresses the building’s performance to ensure occupant comfort and functionality, it is recommended to extend this analysis to incorporate the ultimate limit states, which consider the structure’s capacity to withstand extreme events without collapsing. Incorporating ultimate limit state analysis would provide a more comprehensive understanding of the building’s resilience to ensure it meets serviceability and safety requirements throughout its lifespan.