Effect of Structural Forms on Wind-Induced Response of Tall Buildings: A Finite Element Approach

Abstract

Tall buildings are vulnerable to wind loads, which can cause significant displacements that can affect their stability, strength, and serviceability. Their structural configuration can significantly influence their behavior to wind loads. There are not enough comparative studies in the literature examining the effects of wind loads on different structural configurations. This study examines the response of tall buildings to wind loads by varying their structural forms. Twelve models of tall buildings of different heights and structural configurations were analyzed using the finite element method. Wind loads were applied to the models as equivalent static forces, according to existing codes. The maximum displacements were calculated for each model, and the results were compared. It was found that a considerable reduction in the response was achieved by including shear walls at specific locations in the building’s layout, thereby identifying the optimal location. However, the effectiveness of the different configurations converges at building heights greater than 120 m. In addition, the maximum displacement on the same floor in buildings with the same structural form may vary depending on the building’s total height. An increase in wind velocity results in an almost linear increase in the maximum displacements of the buildings. The findings of this study can assist designers in optimizing shear wall placement in tall building designs.

1. Introduction

The effect of the wind on a building can cause damage or total destruction. Several cases of damage or destruction due to wind loads have been reported. For example, the JP Morgan Chase Tower in Houston suffered facade damage, the Capital One Tower in Lake Charles, Louisiana, was severely damaged during a hurricane, a 37-story building in Belém, Brazil, collapsed [1], etc. The building must be able to withstand the forces of the wind. In particular, (i) the strength of the building must be sufficient to withstand the wind forces, and (ii) the stiffness of the building must ensure comfort and meet the functional criteria. The effect of wind loads on any structure is based on these two characteristics [2].

The wind phenomenon is quite complex due to its interaction with structures, which leads to various flow cases. It consists of a set of vortices of varying dimensions and orbital characteristics, which are entrained in a general air current moving along the Earth’s surface, giving the wind a gusty or turbulent nature.

A consequence of the turbulence is that the dynamic load of a building is a function of the eddies. Large eddies, which have dimensions commensurate with the structure, result in well-related pressures as they enclose the structure. In contrast, small vortices cause pressures at various points that are essentially unrelated [3].

Tall structures, in particular, respond dynamically to wind loads. The phenomena that create oscillations are turbulence, fiber splitting, instabilities and the combination of bending and torsion. Thin structures are the most vulnerable to wind direction due to vorticity.

A very basic issue related to the sway of a building is the influence of its motion on humans. Human sensitivity is such that even small deformations may cause discomfort. Consequently, in most cases of tall buildings, functionality is as important as strength.

A similarly complex phenomenon is the current that forms around a building due to variations in average velocity, flow separation and the creation of vortices. These phenomena are related to the strong pressure variations that occur at the surface of the building, which in turn create large aerodynamic loads on the structural system and localized variable forces on the facade. The oscillation that occurs is either translational or torsional, and its amplitude depends on the dynamic properties of the building as well as the wind forces [3].

Research on wind action in tall buildings varies depending on (i) the material of the building and its structural form and (ii) the way the wind forces are approximated based on experiments or regulations and standards. Tall timber buildings were examined by Reynolds et al. [4] according to Eurocode 1, which highlighted the importance of damping in reducing the response; by Cao and Stamatopoulos [5] based on Eurocode 1 and on time history wind loads that emphasized the differentiation of the parameters of mass, damping, and stiffness; and by Tjernberg [6] according to Eurocode and ISO standards, where the dynamic response of various structural systems was evaluated. Tall steel buildings were investigated by Aly [7] through an aerodynamic tunnel experiment discussing methodologies and different design standards that can validate Eurocode-based wind estimations. Chen [8], based on a wind load spectrum, computed the wind response of tall buildings in urban environments, assisting in understanding the limits of using a static approach for wind force application. Htun [9] proposes a simplified approach for estimating the wind response of tall buildings, while Kumar and Rai [10] and Yuvaraj et al. [11] estimate the wind response of tall buildings according to IS standards. Tall reinforced concrete buildings were studied, using code methods and, more specifically, IS standards, by Gaikwad et al. [12], investigating the effectiveness of outrigger system and shear walls, Verma et al. [13] comparing structures of different aspect ratios, Jamaluddeen and Banerjee [14] for different shapes and heights of buildings, indicating the most effective, and Rajendra and Ghugal [15] providing the importance of wind pressures due to gusts, while Gherbi and Belgasmia [16] used nonlinear time history analysis under the influence of dampers in improving the dynamic response of the building. A comparative analysis between tall reinforced concrete, steel and composite buildings was carried out by Abdo [17] under the actions of wind and earthquake, based on IS standards. On the structural framework of tall buildings, Memon et al. [18] reviewed, in detail, the fundamental characteristics of six of the world’s tall buildings and reported on the challenges encountered in their modelling, analysis, design and construction. Ali and Moon [19] focused on the evolution of tall building structural systems and the technological drivers behind their development. Kim and Shin [20] studied examples of technological innovations in tall building design, illustrating the interaction between architectural form and design tools. Moon [21] reported on the structural performance of diagrid systems for complex-form tall buildings. Elnimeiri and Gupta [22] focused on sustainable construction through tall buildings, reporting the elements that compose such a structure, the strategies that confirm this approach and the structure–environment interaction.

More recently, studies have approached wind action through the computational fluid dynamics (CFD) method. Meena et al. [23] investigated and compared the effect of wind on regular- and irregular-shape models by modifying the corner patterns. The importance of the influence of the shape of the cross-section of a building, keeping the floor area constant, was highlighted. Yadav and Roy [24] studied the aerodynamic behavior and dynamic response of prismatic and tapered tall buildings. The significant influence of different wind angles of incidence on the aerodynamic forces acting on the models was shown. The importance of the non-steady inlet flow for the accurate estimation of the peak response was also emphasized, especially for the wake flow, where vortices greatly affect the aerodynamic loads. Aguirre-López et al. [25] researched the control of wind flow around the perimeter of a tall building by adding triangular- and rectangular-shaped balconies as a type of passive control by rotating the two types at different angles. It emerged that the aerodynamics of the building is varied by the existence of balconies. Sometimes, the forces are moderated, while in other cases, there is some concentration of stresses.

The thorough design of tall buildings against wind loads is of great importance, since their actions are sudden and rapidly evolving. The literature does not provide a systematic analysis that practicing engineers can follow as a guideline for selecting the most appropriate structural form to minimize wind effects based on the building’s plan geometry. One of the goals of this study is to provide practicing engineers with a practical guideline for applying wind loads to buildings exposed to wind actions in accordance with the Eurocode.

Past research on the dynamic behavior of tall buildings subjected to wind loads has prioritized the optimization of the building’s external form [26,27,28] over its internal configurations, as the building’s aerodynamic characteristics considerably influence its response. The lateral stiffness of these structures is often provided by core-outrigger systems rather than solely by shear walls. Therefore, while the internal layout is less critical, it remains important to examine. The placement of shear walls has not been systematically studied in the context of wind design for buildings. This study sheds light on the optimization of shear wall placement in the design of tall buildings subjected to wind loads. Understanding how the shear wall configuration affects the dynamic behavior, especially as building height increases, is essential for improving structural resilience, optimizing material usage, and ensuring occupant safety. Addressing this gap is crucial for advancing both design practices and the theoretical understanding of tall building performance under wind loads.

Using a simplified methodology that assumes a basic building geometry, fixed base support and linear elastic behavior, various structural configurations can be evaluated as a preliminary design tool to aid decision making in selecting the most suitable structural form for a given project. This methodology was applied using the provisions of Eurocode 1—Part 4 to determine the wind effect on tall structures of different heights and structural forms. The maximum peak displacement was calculated for each structure along the wind direction at different heights of tall concrete buildings and for different structural forms and the results were compared.

2. Wind Characteristics

2.1. General

The term “wind” refers to air that is in horizontal motion, while the vertical or nearly vertical motion of air is called a “current”. The motion near the earth’s surface is three-dimensional, with the horizontal motion being much more intense than the vertical motion. Vertical air motion is important for meteorology, but is of small importance near the ground surface, in contrast with the horizontal motion. In addition, the gradual slowing of wind speed and high turbulence that occurs near the ground surface are very important for the structure.

A common observation is that the wind flow is quite complex and turbulent in nature, even though it is not visible. For example, walking on a stormy day, you will notice the continuous flow of wind, with sudden gusts at intervals. These sudden variations in speed, called gusts or turbulence, have a key role in the oscillations of structures [29].

2.2. Design Elements

When designing a building for wind loads, its environment, such as neighboring buildings and topography, must be taken into account, as they may significantly affect the response of the structure. For example, the oscillation of the top of a tall building due to wind may not be visible to a passerby but may cause concern and annoyance to the occupants of the top floors. In a few cases, winds, other than tornadoes or hurricanes, have caused significant structural damage to new buildings. However, modern skyscrapers with lightweight curtain walls, dry partitions and high-strength materials are more vulnerable than older skyscrapers because the older ones have the advantage of the weight of masonry partitions, heavy stone facades and large structural members [29].

Important aspects to consider when designing for wind loads are the following [29]:

• Strength and stability.
• The weakening of structural members and connections due to variations in wind loads.
• Excessive lateral deformation that can cause cracks in internal partitions and exterior cladding, misalignment of mechanical systems and possible permanent deformations of non-structural elements.
• The frequency and amplitude of oscillation that may cause discomfort to occupants of tall and flexible buildings.
• The possible buffeting that may increase the magnitude of wind speeds in neighboring buildings.
• The discomfort on walkways due to strong surface winds.
• The annoying acoustic disturbances.
• The coordination of building oscillations with vibrations of elevator hoist ropes.

3. Wind Actions

3.1. Regulation Framework

Eurocode 1—Part 4 is a European standard for calculating wind actions in the structural design of buildings and other civil engineering structures up to a height of 200 m. It applies either to the entire structure or to parts of it [30]. Figure 1 summarizes the procedure for calculating wind action and its corresponding parameters.

3.2. Dynamic Effect

Wind is characterized as a phenomenon of a dynamic nature. It is a mass of moving air that, when it impacts a structure, transfers kinetic energy. The kinetic energy is converted into dynamic energy, causing the structure to deform. However, if the structure is flexible, a part of energy sets it in motion. The wind action can be simplistically considered as generating static loads in the structure. However, in reality, when the frequency of the excitation coincides with the natural frequencies of the building, then the wind causes a dynamic load [31]. In addition, loads associated with gusts or turbulence that change quickly and abruptly, creating impacts much greater than if they were applied gradually, must be studied as if they were dynamic in nature. The intensity of a wind load depends on how quickly it changes and on the response of the structure. Therefore, if the stresses on a building due to a wind gust, which initially increases and then decreases, are considered dynamic or static depends largely on the dynamic response of the structure to which it is applied [29].

4. Analysis of Tall Buildings

The wind effect on tall structures of different heights and structural forms was studied considering tall concrete buildings of different heights. The maximum peak displacement was calculated for each structure in the wind’s direction using the finite element software ANSYS Mechanical 2021 R2. The geometry of the buildings and the procedure followed to obtain their response to wind loads are presented in the following sections.

4.1. Description

The buildings under consideration had the same typical orthogonal plan but different heights. Their structural form consisted either of frames or of frames with shear walls. The shear walls were placed: (i) at the corners of the perimeter, (ii) symmetrically on the perimeter and (iii) in the center creating a core.

Table 1. Building details.

Length in X	40 m	Length in Y	20 m	Height of story	4 m
Total heigh of buildings		100 m	120 m	140 m
Number of stories		25	30	35
Beam size		0.30 × 0.60 m	0.35 × 0.65 m	0.40 × 0.70 m
Column size		0.80 × 0.80 m	0.90 × 0.90 m	1.00 × 1.00 m
Thickness of Shear wall		0.30 m	0.35 m	0.40 m

presents the details of these buildings, and Figure 2 shows the different configurations of the structural forms considered in the analysis. To simplify the analysis and reduce computational demands, floors (which were assumed to act as rigid diaphragms) were excluded. The buildings were assumed to be fixed at their base. The material used was concrete, whose properties are listed in

Table 2. Material properties.

Compressive strength	50 MPa
Tensile strength	5 MPa
Young’s Modules	37 GPa
Poisson’s ratio	0.18
Density	2400 kg/m3

. The cross-sections of the structural elements were kept constant along the height of the building to simplify the analysis and enable a comparison between different structural configurations. The behavior of the structures was assumed to be linear elastic. The wind direction was considered to be parallel to the shorter side of the buildings, as they had the least resistance in that direction. The Y-axis was aligned parallel to the wind direction, while the X-axis was perpendicular to it.

For each model tested, a code name was assigned in the form of two numbers separated by a dash (e.g., 100-1). The first number represents the building’s height, and the second number after the dash indicates the structural form configuration. The following structural forms and their assigned number were considered: frames (1), frames with shear walls located at the perimeter corners (2), frames with shear walls located at the perimeter center (3), and frames with shear walls forming a central core (4). Figure 2 presents the code names corresponding to buildings with a height of 100 m, along with their plan views and 3D structural forms.

The finite element models that were used consisted of beam and shell elements (Figure 3). Columns and beams were represented by two-node 3D beam elements (BEAM 188) with six degrees of freedom per node and shear walls by four-node shell elements (SHELL 181) with six degrees of freedom per node. The use of beam and shell elements with compatible degrees of freedom enabled proper force and moment transfer through shared nodes in the finite element mesh. ANSYS automatically enforced continuity at these connections, ensuring consistent and accurate interaction between elements without requiring special interface components.

The total number of nodes and elements used was 54,714 and 29,280, respectively. Figure 4 illustrates the procedure followed to model the buildings.

4.2. Modal Analysis

Initially, modal analysis was performed in the buildings to find their natural frequencies. The fundamental frequencies (mode 1) and the corresponding periods of the buildings are presented in

Table 3. Dynamic features of models.

Model	f (Hz)	T (s)	Model	f (Hz)	T (s)	Model	f (Hz)	T (s)
100-1	0.3908	2.5589	120-1	0.3276	3.0525	140-1	0.2885	3.4662
100-2	0.4656	2.1478	120-2	0.3886	2.5733	140-2	0.3333	3.0000
100-3	0.4704	2.1259	120-3	0.3893	2.5687	140-3	0.3343	2.9913
100-4	0.5298	1.8875	120-4	0.4245	2.3557	140-4	0.3494	2.8620

. The first 12 natural frequencies of the models are depicted in Figure 5.

The largest displacements occur in buildings with low natural frequencies. For all examined cases, the lowest natural frequencies were observed in the frame system when no shear walls were present (Figure 5: red curves). The addition of shear walls increased the stiffness. The highest natural frequencies were observed when the shear walls were positioned in the center of the buildings (Figure 5: blue curves).

4.3. Calculation of Wind Forces

The wind acted across the largest dimension b, as shown in Figure 6, and the forces were calculated according to Eurocode 1—Part 4 [30].

The procedure applied for the determination of the wind forces is presented for the 120-4 building model, which had a height of H = 120 m and central shear walls. The following parameters were calculated: (i) the peak velocity pressure q_b(z), (ii) the external w_e and internal w_i pressures, (iii) the structural factor c_sc_d, (iv) the total pressures w_D, w_E and (v) the forces F_D, F_E.

4.3.1. Peak Velocity Pressure

Peak velocity pressure is the pressure of the wind’s peak velocity and includes the fluctuations in mean and short-term velocity. In our study, the calculations extended to the top of the tallest building under consideration, which had a height of 140 m.

• Basic wind velocity [30]

The basic wind velocity v_b, is given by:

$$v_b=c_{dir}\times c_{season}\times v_{b,0}=1\times 1\times 33=33 \mathrm{m}/\mathrm{s}$$

(1)

where:

• c_dir: is the directional factor (with a recommended value of 1);
• c_season: is the season factor (with a recommended value of 1);
• v_b,0: is the fundamental basic wind velocity, taken from the National Appendix 33 m/s [33].

• Basic velocity pressure [30]

The basic velocity pressure q_b, is given by:

$$q_b={\displaystyle \frac{1}{2}}\times \rho \times v_b^2={\displaystyle \frac{1}{2}}\times 1.25\times {33}^2=0.681 \mathrm{K}\mathrm{N}/{\mathrm{m}}^2$$

(2)

where:

• ρ: is the air density (recommended value 1.25 kg/m³).

• Terrain roughness [30]:

The roughness factor c_r(z) at height z is calculated as follows:

$$c_r\left(z\right)=\left\{\begin{array}{c}{k}_r\times \mathit{ln}\left({\displaystyle \frac{z}{z_0}}\right), \gamma \iota \alpha z_{min}\le z\le z_{max} \\ c_r\left(z_{min}\right), \gamma \iota \alpha z<z_{min}\end{array}\right.=\left\{\begin{array}{c}0.156\times \mathit{ln}\left({\displaystyle \frac{z}{0.003}}\right), \gamma \iota \alpha 1\le z\le 200 \\ c_r\left(1\right), \gamma \iota \alpha z<1\end{array}\right.$$

(3)

where:

• k_r: is the terrain coefficient given by the following equation [30]:

$$k_r=0.19\times {\left({\displaystyle \frac{z_0}{z_{0,II}}}\right)}^{0.07}=0.19\times {\left({\displaystyle \frac{0.003}{0.05}}\right)}^{0.07}=0.15$$

(4)

• z₀: is the roughness length, which for terrain category 0 is equal to 0.003 m;
• z_min: is the minimum height, equals to 1 m for terrain category 0;
• z_0,II: is the roughness length, equals to 0.05 m for terrain category ΙΙ;
• z_max: is the maximum height, taken equal to 200 m.

Figure 7 presents the height distribution of the roughness factor.

• Mean wind velocity [30]:

The mean wind velocity v_m(z) at height z is determined from the following equation:

$$v_m\left(z\right)=c_r\left(z\right)\times c_o\left(z\right)\times v_b=c_r\left(z\right)\times 1\times 33=c_r\left(z\right)\times 33$$

(5)

where:

• c_o(z): is the orography factor, assumed equal to 1.

Figure 8 presents the height distribution of the mean wind velocity.

• Turbulence intensity [30]:

The turbulence intensity I_v(z), at height z, is calculated by:

$$I_v\left(z\right)={\displaystyle \frac{{\sigma }_v}{v_m\left(z\right)}}={\displaystyle \frac{5.149}{v_m\left(z\right)}}$$

(6)

where:

• σ_v: is the turbulence standard deviation and is given by:

$${\sigma }_v=k_r\times v_b\times k_I=0.156\times 33\times 1=5.149 \mathrm{m}/\mathrm{s}$$

(7)

where:

• k_I: is the turbulence factor. The recommended value is 1.

Figure 9 presents the height distribution of the turbulence intensity.

• Peak velocity pressure [30]:

The peak velocity pressure q_p(z) at height z is given by:

$$q_p\left(z\right)=\left[1+7\times I_v\left(z\right)\right]\times {\displaystyle \frac{1}{2}}\times \rho \times v_m^2\left(z\right)=\left[1+7\times I_v\left(z\right)\right]\times 0.625\times v_m^2\left(z\right)$$

(8)

Figure 10 presents the height distribution of the peak velocity pressure.

4.3.2. Pressure on Surfaces

To determine the pressures on the building surfaces, only the general geometry (height, plan dimensions) is required, not the structural form (walls, columns, etc.).

• Reference heights [30]:

For windward walls of rectangular buildings, the reference height z_e depends on the ratio of the height h to the side b, perpendicular to the wind direction. Figure 11 shows the reference heights for buildings of 120 m height.

For h > 2b = 120 > 80 m, $z_e=\left\{\begin{array}{c}40 \mathrm{m}\\ 80 \mathrm{m}\\ 120 \mathrm{m}\end{array}\right.$ with corresponding peak pressures, $\left\{\begin{array}{c}{q}_p\left(40\right)=2.596\mathrm{KN}/{\mathrm{m}}^2\\ q_p\left(80\right)=2.902\mathrm{KN}/{\mathrm{m}}^2\\ q_p\left(120\right)=3.089\mathrm{KN}/{\mathrm{m}}^2\end{array}\right.$ .

• Pressure zones [30]:

At the perimeter of the building, the windward zone D, the leeward zone E, and the side zones A and B are created, which are calculated based on the parameter e, where:

$$e=\mathit{min}\left(b, 2h\right)=40 \mathrm{m}$$

(9)

$$\mathrm{z}\mathrm{o}\mathrm{n}\mathrm{e} \mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h} A=e/5=40/5=8 \mathrm{m}$$

(10)

$$\mathrm{z}\mathrm{o}\mathrm{n}\mathrm{e} \mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h} B=d-e/5=20-40/5=12 \mathrm{m}$$

(11)

Figure 12 shows the pressure zones A, B, D, E in view (a) and plan (b). The shapes are not scaled.

• External pressure coefficients [30]:

To find the external pressures, the external pressure coefficients c_pe are needed in order to multiply them by the corresponding peak pressures. They depend on the size of the loadable surface of the structure and the ratio h/d.

Table 4. The external pressure coefficients for each zone.

Zone	Zone Area (m2)		cpe,10
A	960	>10 m2	−1.2
B	1440		−0.8
D	4800		0.8
E	4800		−0.7

shows the zone areas and the corresponding external pressure coefficient.

Internal pressure coefficient [30]:

The internal pressure coefficient c_pi is determined by the size and distribution of openings in the building envelope. For h/d > 1 and opening ratio μ = 0.80 (assumption), the coefficient c_pi = −0.3 is obtained.

• External pressures [30]:

The external pressures are calculated as follows:

$$w_e=q_p\left(z_e\right)\times c_{pe}$$

(12)

Table 5. The values of external pressures in each zone and reference height.

Reference Height (m)	A (KN/m2)	B (KN/m2)	D (KN/m2)	E (KN/m2)
40	−3.115	−2.077	2.077	−1.817
80	−3.482	−2.322	2.322	−2.031
120	−3.707	−2.471	2.471	−2.162

presents the external pressures for each zone and reference height, and they are depicted in Figure 13.

• Internal pressures [30]:

The internal pressures vary only in height and are not divided around the perimeter like the external pressures. They are calculated as follows:

$$w_i=q_p\left(z_i\right)\times c_{pi}$$

(13)

Table 6. The values of internal pressures.

Reference Height (m)	wi (KN/m2)
40	−0.779
80	−0.871
120	−0.927

lists the internal pressures for each reference height, which are also depicted in Figure 14.

4.3.3. Structural Factor

The structural factor c_sc_d combines the reducing effect of the non-simultaneous development of maximum surface pressures (size factor c_s) and the increasing effect of building vibrations due to turbulence (dynamic factor c_d). Each building has its own coefficient because one of the parameters for its calculation is the fundamental frequency. The following calculations are for a 120 m tall building with shear walls located at the center (120-4).

• Reference height [30]:

The reference height $z_s$ for determining the coefficient of the building is given by:

$$z_s=0.60\times h=0.60\times 120=72 \mathrm{m}$$

(14)

• Turbulent length scale [30]:

The turbulent length scale $L\left(z\right)$ expresses the average gust size for heights up to 200 m.

$$\mathrm{F}\mathrm{o}\mathrm{r} z \ge z_{min},\hspace{1em}\hspace{1em}L\left(z\right)=L_t\times {\left({\displaystyle \frac{z}{z_t}}\right)}^{\alpha }=>L\left(72\right)=300\times {\left({\displaystyle \frac{72}{200}}\right)}^{0.38}=203.6 \mathrm{m}$$

(15)

where:

$$\alpha =0.67+0.05\times \mathit{ln}\left(z_0\right)=0.67+0.05\times \mathit{ln}\left(0.003\right)=0.38$$

(16)

• z_t: is the reference height. It is taken as equal to 200 m;
• L_t: is the reference length scale. It is taken as equal to 300 m;
• z_min: is the minimum height. 1 m is obtained for terrain category 0;
• z₀: is the roughness length. 0.003 m is obtained for terrain category 0.

• Power spectral density function [30]:

The wind distribution expresses the variation in the turbulence, and, in the frequency domain, it is represented by the non-dimensional power spectral density function $S_L\left(z,n\right)$. This function describes the distribution of wind energy at different heights and frequencies and is given by:

$$S_L\left(z,n\right)={\displaystyle \frac{6.8\times f_L\left(z,n\right)}{{\left(1+10.2\times f_L\left(z,n\right)\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}={\displaystyle \frac{6.8\times 1.666}{{\left(1+10.2\times 1.666\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}=0.092$$

(17)

where:

$$f_L\left(z,n\right)={\displaystyle \frac{n\times L\left(z\right)}{v_m\left(z\right)}}={\displaystyle \frac{0.425\times 203.6}{51.92}}=1.666 \mathrm{is} \mathrm{non-dimensional} \mathrm{frequency}$$

(18)

and n = n_1,x = 0.425 Hz, is the natural frequency of the building obtained from the modal analysis performed in the building (Section 4.2).

• Background factor [30]:

The background factor B² takes into account the lack of correlation of the pressure on the surface of the structure. For safety reasons, B² is taken as equal to 1.

• Peak factor [30]:

The peak factor represents $k_p$, which is the ratio of the maximum value of the variable part of the response to its standard deviation. This ratio shall not be less than 3 and is given by:

$$k_p=\sqrt{2\times \mathit{ln}\left(v\times T\right)}+{\displaystyle \frac{0.6}{\sqrt{2\times \mathit{ln}\left(\nu \times {\rm T}\right)}}}=\sqrt{2\times \mathit{ln}\left(0.241\times 600\right)}+{\displaystyle \frac{0.6}{\sqrt{2\times \mathit{ln}0.241\times 600}}}=3.444$$

(19)

where:

• T represents the average time of mean wind velocity and is taken as 600 s.

The up-crossing frequency ν is given by:

$$\nu =n_{1,x}\times \sqrt{{\displaystyle \frac{R^2}{B^2+R^2}}}=0.425\times \sqrt{{\displaystyle \frac{0.474}{1+0.474}} }=0.241 \mathrm{H}\mathrm{z} \& \nu \ge 0.08 \mathrm{H}\mathrm{z}$$

(20)

The minimum value of 0.08 Hz corresponds to the maximum coefficient k_p = 3.

• Resonance response factor [30]:

The resonance response factor R² takes into account the turbulence in resonance and is calculated as:

$$R^2={\displaystyle \frac{{\pi }^2}{2\times \delta }}\times S_L\left(z_s,n_{1,x}\right)\times R_h\left({\eta }_h\right)\times R_b\left({\eta }_b\right)={\displaystyle \frac{{\pi }^2}{2\times 0.1}}\times 0.092\times 0.230\times 0.454=0.474$$

(21)

where the logarithmic reduction in damping for the fundamental bending mode δ is given by:

$$\delta ={\delta }_s+{\delta }_a+{\delta }_d=0.1+0+0=0.1$$

(22)

where:

• δ_s is the logarithmic reduction in structural damping, whose value is taken as equal to 0.1 for reinforced concrete buildings according to EN 1991-1-4 [30].
• δ_a is the logarithmic reduction in aerodynamic damping for the fundamental mode, usually estimated by the mass per unit area of the structure at the point of maximum amplitude of the mode shape. It is assumed to be 0.
• δ_d is the logarithmic reduction in damping by special energy absorption mechanisms (e.g., tuned mass dampers) calculated using appropriate techniques. It is assumed to be 0.

The aerodynamic admittance functions R_L and R_b are determined as follows:

$$R_h={\displaystyle \frac{1}{{\eta }_h}}-{\displaystyle \frac{1}{2\times {\eta }_h^2}}\times \left(1-e^{-2\times {\eta }_h}\right)={\displaystyle \frac{1}{3.765}}-{\displaystyle \frac{1}{2\times {3.765}^2}}\times \left(1-e^{-2\times 3.765}\right)=0.230$$

(23)

$$R_b={\displaystyle \frac{1}{{\eta }_b}}-{\displaystyle \frac{1}{2\times {\eta }_b^2}}\times \left(1-e^{-2\times {\eta }_b}\right)={\displaystyle \frac{1}{1.506}}-{\displaystyle \frac{1}{2\times {1.506}^2}}\times \left(1-e^{-2\times 1.506}\right)=0.454$$

(24)

where:

$${\eta }_h={\displaystyle \frac{4.6\times h}{L\left(z_s\right)}}\times f_L\left(z_s,n_{1,x}\right)={\displaystyle \frac{4.6\times 100}{203.6}}\times 1.666=3.765$$

(25)

$${\eta }_b={\displaystyle \frac{4.6\times b}{L\left(z_s\right)}}\times f_L\left(z_s,n_{1,x}\right)={\displaystyle \frac{4.6\times 40}{203.6}}\times 1.666=1.506$$

(26)

• Structural factor [30]:

$$c_s{c}_d={\displaystyle \frac{1+2\times k_p\times I_v\left(z_s\right)\times \sqrt{B^2+R^2}}{1+7\times I_v\left(z_s\right)}}={\displaystyle \frac{1+2\times 3.444\times 0.099\times \sqrt{1+0.474}}{1+7\times 0.099}}=1.07$$

(27)

4.3.4. Total Pressures

Having calculated the external and internal pressures and the structural factor, the total pressures on the surfaces can be calculated.

$$w_{total}=\left(c_s{c}_d\times w_e\right)-w_i$$

(28)

Table 7. The total pressures of surfaces D and E.

Reference Height (m)	Surface D	Surface E
	wD,total (KN/m2)	wE,total (KN/m2)
40	2.992	−1.157
80	3.345	−1.293
120	3.560	−1.377

shows the final pressures at the height of the windward surface D and leeward surface E, where the analyses were performed.

4.3.5. Forces

To determine the forces acting on the surfaces, the pressure was multiplied by the area over which it was applied (

Table 8. The forces on surfaces D and E.

FORCES SURFACE D
	Height (m)	b (m)	h (m)	A (m2)	wD,total (KN/m2)	FD (KN)
1	0–40	40	40	1600	2.992	4787.200
2	40–80	40	40	1600	3.345	5352.000
3	80–120	40	40	1600	3.560	5696.000
FORCES SURFACE E
	Height (m)	b (m)	h (m)	A (m2)	wE,total (KN/m2)	FE (KN)
1	0–40	40	40	1600	−1.157	−1851.200
2	40–80	40	40	1600	−1.293	−2068.800
3	80–120	40	40	1600	−1.377	−2203.200

).

Figure 15 provides an illustrative representation of the pressure distribution along the height and the point of application of forces at the center of the respective surfaces.

However, to achieve a better distribution of forces, an approximately uniform distributed load was calculated for each level, multiplying the corresponding pressure by the story height (

Table 9. The distributed loads on surfaces D and E.

DISTRIBUTED LOADS SURFACE D
Height (m)	Levels	hορ (m)	wD,total (KN/m2)	FD (KN/m)
0–40	1–10	4	2.992	11.968
40–80	11–20		3.345	13.380
80–120	21–30		3.560	14.240
DISTRIBUTED LOADS SURFACE E
Height (m)	Levels	hορ (m)	wE,total (KN/m2)	FE (KN/m)
0–40	1–10	4	−1.157	−4.628
40–80	11–20		−1.293	−5.172
80–120	21–30		−1.377	−5.508

), as shown in Figure 16.

4.4. Results

The uniform distributed loads that were calculated in Section 4.3 were actually the equivalent static forces of the wind loads. Static analysis was performed using the finite element method for all building models to determine the maximum displacements that would occur (

Table 10. Maximum model displacements [32].

Model	Max Displacement (m)	Model	Max Displacement (m)	Model	Max Displacement (m)
100-1	0.269	120-1	0.316	140-1	0.342
100-2	0.174	120-2	0.195	140-2	0.220
100-3	0.162	120-3	0.190	140-3	0.219
100-4	0.131	120-4	0.168	140-4	0.210

). Model displacements were computed using the Distributed Sparse Matrix Direct Solver, the default solver in ANSYS.

Initially, the analysis was conducted for buildings with frames (without shear walls). Then, the shear walls were added at different locations, as described in Section 4.1, significantly reducing the response. A graphical presentation of the displacements for building 120-4 is presented in Figure 17.

Comparing the response of the buildings with the same structural form, an increase in the response was observed with the increase in building height, as expected. Figure 18 presents this gradual increase, along with the corresponding percentage increase.

The optimal position, where the smallest displacements were observed, was when the walls were placed in the center, forming a core. In the diagram in Figure 19, the gradual reduction in the displacements is plotted, along with the corresponding reduction percentages relative to the building with only frames.

The maximum response of the buildings examined is shown in the aggregated diagram in Figure 20. The buildings without shear walls exhibited the highest response, which was significantly larger than the corresponding ones without shear walls. Furthermore, the variation in displacements was almost linear with increasing building height in all four structural forms.

In the aggregated diagrams in Figure 21, we observe that the change in maximum displacement with system alternation has approximately the same decreasing rate in all three cases of building height.

In addition, the maximum displacement at the same level (floor) and in buildings with the same structural form may vary depending on the building’s total height. In particular, in bare frames (Figure 22a), the lower levels exhibited greater displacement in the 100 m building, whereas from level 18 onward, the displacement increased in the 120 m building. A similar effect occurred when shear walls were added at the corners (Figure 22b) or at the center of the perimeter (Figure 23a), except that the displacement increased from level 23 onwards in the 140 m high building. In the frames with shear walls at the core (Figure 23b), the maximum displacements remained nearly the same up to level 8, after which they increased as the building’s height increased.

In the aggregate diagrams in Figure 24 and Figure 25, the maximum displacements are plotted against the building’s height for all structural forms, both separately for building height cases and collectively. It is observed that the position of the shear walls at the corner or center of the perimeter gave very close values of maximum displacements. In addition, the position of shear walls played a less important role in the maximum displacements at levels higher than 120 m.

Previous studies, although they have considered different materials and configurations of tall buildings while examining displacements under wind loads, have found that displacement increases with height, with values and rates of increase similar to those obtained in this study [34,35].

A sensitivity analysis was also performed by altering the fundamental basic wind velocity to investigate how this factor affects the results. This velocity is the first parameter defined for the determination of wind forces and is provided by the National Appendix. In Greece, for islands and coastal areas up to 10 km from the coast, the fundamental basic wind velocity is 33 m/s, while for the rest of the country, it is 27 m/s [33]. Initially, calculations were carried out for a wind speed of 33 m/s. However, for structural models with a height of 120 m, code-named 120-1, 120-2, 120-3, and 120-4, additional wind velocities of 27, 30, and 36 m/s were applied. The corresponding maximum displacements were calculated and are plotted in the diagrams in Figure 26. It is observed that increased wind velocity leads to an almost linear increase in the maximum displacements of the buildings, with the displacement increasing at a higher rate for the bare frames (120-1).

5. Discussion

Designing tall buildings is a great challenge that involves many decisions. One of the most important considerations in structural design is the choice of structural form. In the early stages, once the material and the overall building plan geometry have been determined, designers can carry out preliminary analyses to identify the most suitable structural form. In the early design stages, engineers must evaluate many design options. Simplifying the modelling process expedites the analysis by reducing computational time. The wind pressures and forces in this study were determined using Eurocode methodologies. Accordingly, a linear elastic analysis approach was adopted to remain consistent with the code-based framework. This approach is useful for wind load cases because wind effects are often serviceability-governed rather than ultimate-strength-governed, meaning that excessive deformation, vibrations, or occupant comfort issues usually control design before yielding or collapse does. In addition, the fixed-base assumption is an accepted simplification in the analysis of wind loads on tall buildings [5,27,28,34], particularly during the early stages of design or when following code-based procedures.

At early stages, designers can explore different placements of the structural elements that contribute most significantly to the building’s stiffness. A modal analysis should be performed to determine the structure’s fundamental frequency, which is essential for calculating wind loads using the Eurocode formulas. Subsequently, for buildings that are not sensitive to dynamic effects and not exceeding 200 m in height, these wind loads can be applied as static forces, and a static structural analysis can be performed to calculate maximum displacements. Various structural forms can then be considered until the most optimal configuration is identified.

Once the final geometry and structural form are selected, a more detailed analysis must be undertaken. This is necessary because tall buildings exposed to high wind loads may experience yielding and plastic deformations, which require the incorporation of material nonlinearities into the analysis. In addition, linear analysis assumes small displacements and ignores effects such as the P-Δ, which can be significant under large deformations. The flexibility of the foundation may also influence the overall behavior and natural frequencies of the building. In conclusion, after an initial simplified analysis used to determine the structural configuration, a more advanced analysis is required. This may involve advanced modeling techniques that account for material and geometric nonlinearities, computational fluid dynamics (CFD) for a more accurate simulation of wind effects and soil–structure interaction to better represent realistic boundary conditions. Vortices and wind dynamics can be more effectively analyzed using CFD simulations or wind tunnel testing. In addition, wind buffeting between adjacent buildings, generated due to flow and turbulence in narrow spaces, can be predicted by wind tunnel tests and CFD simulations.

This study provides guidelines for designers to conduct preliminary analyses of tall buildings subjected to wind loads. Although the building geometry was simple and the structural forms did not include all possible configurations of stiffness-contributing elements, the entire procedure that the designers should follow is presented step by step.

6. Conclusions

This study examined the response of twelve buildings under wind loads, varying their height and structural forms considering linear elastic analysis. The structural forms examined were symmetric about the axes or the center of the building’s plan and were placed either at the center, forming a core, or around the perimeter of the building. For these forms, buildings without shear walls exhibited the highest response, significantly greater than that of buildings with shear walls. Furthermore, the maximum displacement increased almost linearly with height, regardless of the structural form. The maximum displacements were nearly the same whether the shear walls were placed at the corners or at the center of the building’s perimeter.

It is worth noting that placing the shear walls at the center of the building, forming a core, resulted in the smallest structural response. This outcome can be attributed to the symmetry of the configuration, the presence of a clear load path, and the alignment of the center of gravity with the center of mass. Additionally, the mass was evenly distributed, with no eccentricities that could cause twisting under wind loads. In conclusion, while perimeter shear walls offer advantages in dynamic seismic design, primarily due to their increased torsional resistance, the behavior under simplified, statically applied lateral loads, such as code-based wind loads, favors the symmetry and stiffness concentration provided by a central core. This leads to a more efficient structural response under such loading conditions.

However, beyond a certain height, the position of the shear walls (for the cases examined) became less significant, as the maximum displacement values began to converge with increasing height.

The position of the shear walls played a less important role in the maximum displacements at heights greater than 120 m. In addition, the maximum displacements at each level increased almost linearly with building height across all structural forms. The maximum displacement at the same floor level and in buildings with the same structural form may vary depending on the building’s total height. In addition, an increase in wind velocity leads to an almost linear increase in the maximum displacements of the buildings, with the displacement increasing at a higher rate for the bare frames.

Comparing the results of the building with only frames to the one with shear walls placed in the optimal position, it was found that: (i) the natural frequency of the building increased from 21.1% to 35.6%, and (ii) the maximum displacement of the buildings decreased from 38.6% to 51.3%, depending on the height of the building. An increase in height reduced the effect.

The maximum displacement of the buildings increased with height. In the buildings with only frames, the increase in maximum displacement between 100 m and 140 m was 27.1%. In the buildings with the shear walls in the core, the increase in maximum displacement between 100m and 140 m was 60.3%.

In conclusion, practicing engineers should recognize that structural forms consisting of bare frames may be preferred for architectural reasons, such as maximizing space for interconnecting rooms; they are not suitable for high-rise buildings exceeding 60 to 80 m (for wind serviceability) unless additional stiffness is provided, for example, through the strategic placement of shear walls. This analysis showed that placing shear walls around the center creating a core is highly effective in reducing the structure’s dynamic response to wind loading.

In future work, the placement of shear walls will be further explored by considering locations beyond just the center or perimeter of the building. Buildings with irregular geometries or constructed from different materials will also be considered. The structural loading will be determined using both static analysis and computational fluid dynamics (CFD), allowing a comparison between the two approaches. Soil–structure interaction will also be considered to assess its effect on the natural frequencies, which, in turn, influence the wind-induced loads on the structure, and to more accurately represent realistic boundary conditions.