# Implementation of Data from Wind Tunnel Tests in the Design of a Tall Building with an Elliptic Ground Plan

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Building

^{2}was applied on the whole building.

## 3. The Analysis of the Effects of Wind on the Elliptical Cylinder

#### 3.1. The Reduced-Scale Model and the Wind Tunnel

^{5}for fully turbulent flow should be larger than 5 × 10

^{5}(for an angular building shape, larger than 5 × 10

^{4}). “Transient flow” is defined by a critical Reynolds number. Laminar flow is defined by the Reynolds number for laminar wind flow. All these Reynolds numbers can be determined by wind tunnel tests or CFD simulations. The Re of an experiment and CFD was from 3.2 × 10 × 10

^{4}to 8.1 × 10 × 10

^{4}, depending on the wind direction and thus the characteristic dimension of the model.

_{pe}are lower for the modified model due to flow separation and applied roughness on the surface. More information about the Reynolds number of the model with the smooth/rough surface and the critical value of the Reynolds number can be found in [27].

_{pe}were calculated by using the following expression:

_{WT}is the external wind pressure on the surface of the model measured in a wind tunnel (Pa) and v

_{ref}is the reference wind velocity at the height of the top of the structure (m/s). The air density ρ (kg/m

^{3}) is a function of measured air temperature T in ° and barometric pressure BP in Pa.

_{pe}for all four levels and for three wind directions are presented in Figure 6, Figure 7 and Figure 8.

_{pe}values were approximately the same for all levels. Therefore, simplification by using the “envelope of the data” was adopted, and maximum positive/negative pressures were considered in the further calculations. The values of c

_{pe}are listed in Table 1.

_{pe}were necessary for the estimation of the most unfavorable wind direction with respect to the shape of the structure. Firstly, the peak velocity pressures (Figure 9) were calculated for all four levels according to [20,21] by using Equations (3) and (4). Then, these values were used for the calculation of the external wind pressures using Equation (5). After that, the resultant forces R (kN) and torsional moments M

_{r}(kNm) were determined for all considered wind directions. The maximum value of resultant force and maximum value of torsional moment gave the information on which wind direction is the most dangerous for the investigated structure.

_{e}is the external wind pressure (Pa), q

_{p}(z

_{e}) is the peak velocity pressure at the height z

_{e}(m) in Pa, c

_{pe}is the external pressure coefficient (-), l

_{v}(ze) is the turbulence intensity (-), ρ is the air density (kg/m

^{3}), and ν

_{m}(z

_{e}) is the mean wind velocity at the height z

_{e}(m/s). The turbulence intensity represents the fluctuation part of wind flow and is a function of turbulence factor k

_{1}, orography factor c

_{o}, and roughness length z

_{0}. c

_{r}(z) is the roughness factor, and v

_{b}is the basic wind velocity (m/s) calculated by directional factor c

_{dir}(-), seasonal factor c

_{season}(-), and fundamental value of the basic wind velocity v

_{b}

_{,0}(m/s). These parameters are defined in [20,21].

#### 3.2. CFD Simulation in the Program ANSYS FLUENT

_{pe}values calculated for three roof alternatives—a totally flat roof, a roof with a roof parapet of 500 mm, and a roof with a roof parapet of 2500 mm—are compared in Figure 14. The last alternative should be used for the design of modern green fully useful roofs (sport and relaxation zones with vegetation).

#### 3.3. The Simplification of the Elliptic Shape

_{e}, defined as follows:

^{−7}m

^{2}/s), v(z

_{e}) is the peak wind velocity (m/s) at height z

_{e}(m), q

_{p}(z

_{e}) is the peak wind pressure (Pa) calculated by using Equation (3), and ρ is the air density (1.25 kg/m

^{3}). In our case, b = 24.965 m, q

_{p}(z

_{e}) = 1103.26 Pa, v(z

_{e}) = 42.01 m/s. The resultant Reynolds number was 6.99 × 10

^{7}.

_{p}

_{,0}is the external pressure coefficient without free-end flow depending on Reynolds number (depicted in Graph 7.27 in [20]). ψ

_{λα}is the end-effect factor defined as follows:

_{A}is the position of the flow separation (°), α

_{min}is the position of the minimum pressure (°), and ψ

_{λ}is the end-effect factor (determined from Graph 7.36 in [20]). Two parameters had to be calculated: the solidity ratio φ and the slenderness λ. The solidity ratio was calculated using the following equation:

^{2}) and A

_{c}is the overall envelope area (m

^{2}).

_{p}(ze) = 1103.26 Pa, v(z

_{e}) = 42.01 m/s. The resultant Reynolds number was 8.85 × 10

^{7}. Also, the effective slenderness λ was recalculated.

#### 3.4. The Application of the Results to the 3D Model of the Building Used for Static and Dynamic Analysis

_{pe}in all four levels determined for a particular wind direction were the same (the differences were very small). So, the external pressure coefficients were considered with the same values for all levels (Table 1—“Envelope of the data”). Wind load (kN) was applied on the surface of the structure in a horizontal direction—perpendicularly to the tangent line at a considered point. It was calculated by using the peak wind pressures at the considered height multiplied by the load width and load height at a given point. In our case, the load panels as a special element in the SCIA ENGINEER were not used because of computing capacity. Wind load was applied as a force load at 16 points on the perimeter of the ground plan in the places of all horizontal slabs. The following wind directions were considered for static analysis: 0° (it is parallel with the x-axis), 20° (it caused the maximum torsional moment), and 90° (it caused the maximum resultant force and is parallel with the y-axis).

## 4. Static Analysis

#### 4.1. Applied Loads

^{2}(underground floors), 1.66 kN/m

^{2}(commercial spaces, technical floor, and the flats), 0.69–1.25 kN/m

^{2}(the stairs), and 3.85 kN/m

^{2}(the roof). The characteristic values of variable loads were considered according to the utilization of the floors [28,29]: 2.5 kN/m

^{2}(underground floors), 3 kN/m

^{2}(commercial spaces), 2 kN/m

^{2}(the flats), 3 kN/m

^{2}(the stairs), 4 kN/m

^{2}(technical floor), 0.75 kN/m

^{2}(the roof). Partition walls were considered as variable loads (with a characteristic value equal to 1.2 kN/m

^{2}). The wind load was taken from the previous analysis (see Section 3.4). The snow load was calculated according to [30,31]. The resultant characteristic value of snow load on the ground was equal to 0.48 kN/m

^{2}(altitude of the site 138 m, zone III in the snow map of Slovakia, roof snow load shape coefficient 1.0, thermal coefficient 1.0, flat roof). Advising was given for the most unfavorable combination of applied loads for the building.

#### 4.2. The Subsoil Stiffness Coefficient

^{3}) was calculated as follows:

^{2}) and s is soil displacement calculated from the soil parameters ascertained by a hydrogeological survey or from the literature (m).

^{3}.

^{3}, m = 0.2, E

_{def}= 50,000 kPa, Poisson’s ratio ν = 0.2, conversion coefficient between the values of E

_{def}and E

_{oed}is β = 0.9). The resultant value of the soil displacement calculated using Equation (14) was 5.2 mm. It was lower than the limit value defined for multi-story buildings with wall structural systems made of monolithic reinforced concrete w

_{lim}= 60 mm [33]. The calculated subsoil stiffness coefficient was 40,420 kN/m

^{3}.

_{zi}is the vertical component of increment stress (kN/m

^{2}) in the middle of the layer with the thickness of h

_{i}(m), m

_{i}is the correction coefficient (-), σ

_{ri}is the vertical component of the original geostatic stress (kN/m

^{2}) in the middle of the layer, E

_{oedi}is the oedometric modulus of deformation (kPa) calculated as E

_{def}/β (E

_{def}is the deformation modulus (kPa) and β is the conversion coefficient), and ν is Poisson’s ratio (-).

_{de}is extreme design contact stress on the bottom side of the foundation plate (kPa) with the consideration of short-term loads and R

_{d}is the design bearing capacity of foundation soil (kPa).

_{d}is the design value of cohesion of soil (kPa). γ

_{1}and γ

_{2}are the bulk density of soil above/under the foundation (kN/m

^{3}). b

_{eff}is the effective width of the foundation (m). N

_{cd}, N

_{dd}, and N

_{bd}are design bearing parameters depending on the design value of the angle of internal friction φ

_{d}(°). s

_{c}, s

_{d}, and s

_{b}are parameters depending on the shape of the foundation (-). d

_{c}, d

_{d}, and d

_{b}are parameters depending on the depth of the foundation. i

_{c}, i

_{d}, and i

_{b}are parameters expressing the slope of applied loads. The parameters g

_{c}, g

_{d}, and g

_{b}express the slope of the building site. Effective values of soil parameters were considered. More information can be found in [33,34,35]. The resultant value of R

_{d}was 7660 kN/m

^{2}, and σ

_{de}= 290 kPa.

#### 4.3. The Limit Values

_{max}without the inclination of the footing bottom is defined as follows:

_{cal}is the calculated horizontal displacement of the top story in the x-direction or y-direction and H is the height of the horizontal slab of the top story measured from the footing bottom. In this case, H was 80.25 m and u

_{max}was 40 mm.

_{max}of a reinforced concrete horizontal slab was considered as L

_{max}/250, where L

_{max}is the maximum span. In our case, L

_{max}was 6 m, and v

_{max}= 24 mm.

## 5. Results

_{z}equal to 50 MN/m

^{3}, and the wind directions of 0°, 20°, and 90° are depicted in Figure 23, Figure 24 and Figure 25.

## 6. Discussion

^{3}), in both the x- and y-directions. In the x-direction (parallel with the major axis), a significant value was calculated only for the wind direction of 90° (5.2–6.7 mm). In the y-direction (parallel with the minor axis), both the wind direction of 20° and the wind direction of 90° caused large values of displacements (in the ranges of 10.8–12.2 mm and 9.3–11.0 mm).

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Investigated building dimensions in mm; (

**b**) typical floor; (

**c**) underground part—parking spots; (

**d**) the ground plan—dimensions in mm.

**Figure 4.**Reduced-scale model—dimensions are in mm: (

**a**) measuring taps (1–16) and wind directions (0°–90°); (

**b**) measured levels—A, B, C, D.

**Figure 13.**Mesh grid and streamlines for wind direction 45°. There were no measuring taps on the roof of test model. Therefore, CFD simulation was a useful tool for the determination of c

_{pe}. The roof of the model was totally flat (Figure 5). In the case of a real structure, it should be enclosed by a roof parapet. In static and dynamic analysis, a roof parapet with a height of 1 m was considered. The model created for the CFD had a roof parapet with a height of 0.5 m. The explanation is as follows: In the model used for static and dynamic analysis, the layers of the roof are considered only as the area load. But in the case of CFD analysis, the model is created as “an envelope of the structure”.

**Figure 19.**Alternative No. 2 (the comparison with the rectangle)—(

**a**) wind direction 0°; (

**b**) wind direction 90°.

**Figure 28.**Three-dimensional deformation u

_{tot}calculated for k

_{z}= 40,421 kN/m

^{3}, wind direction 0°.

**Figure 29.**Three-dimensional deformation u

_{tot}calculated for k

_{z}= 40,421 kN/m

^{3}, wind direction 20°.

**Figure 30.**Three-dimensional deformation u

_{tot}calculated for k

_{z}= 40,421 kN/m

^{3}, wind direction 90°.

Measuring Point (-) | Mean Values | “Envelope of the Data” | ||||
---|---|---|---|---|---|---|

0° | 90° | 20° | 0° | 90° | 20° | |

1 | 0.57 | −0.46 | −0.11 | 0.68 | −0.49 | −0.11 |

2 | −0.23 | −0.02 | 0.46 | −0.37 | −0.02 | 0.61 |

3 | −0.37 | 0.40 | 0.17 | −0.42 | 0.47 | 0.20 |

4 | −0.43 | 0.60 | −0.02 | −0.48 | 0.67 | −0.02 |

5 | −0.46 | 0.67 | −0.33 | −0.55 | 0.72 | −0.36 |

6 | −0.35 | 0.63 | −0.27 | −0.39 | 0.69 | −0.29 |

7 | −0.32 | 0.43 | −0.58 | −0.37 | 0.53 | −0.70 |

8 | −0.09 | −0.01 | −0.18 | −0.12 | −0.02 | −0.36 |

9 | −0.02 | −0.47 | −0.16 | −0.08 | −0.51 | −0.17 |

10 | −0.09 | −0.45 | −0.19 | −0.11 | −0.46 | −0.19 |

11 | −0.30 | −0.13 | 0.09 | −0.35 | −0.14 | 0.10 |

12 | −0.35 | −0.44 | −0.24 | −0.41 | −0.48 | −0.26 |

13 | −0.40 | −0.42 | −0.35 | −0.55 | −0.48 | −0.39 |

14 | −0.42 | −0.41 | −0.49 | −0.50 | −0.47 | −0.57 |

15 | −0.44 | −0.13 | −0.83 | −0.49 | −0.14 | −0.89 |

16 | −0.32 | −0.45 | −1.18 | −0.37 | −0.47 | −1.22 |

Wind Direction (°) | Resultant Force (kN) | Force Arm (m) | Angle of the Force (°) | Torsional Moment (kNm) |
---|---|---|---|---|

0 | 0.34 | −77.48 | 147 | −26.48 |

20 | 27.55 | −29.74 | 275 | −819.35 |

90 | 60.95 | 0.44 | 270 | 26.94 |

Wind Direction (°) | Resultant Force (kN) | Force Arm (m) | Angle of the Force (°) | Torsional Moment (kNm) |
---|---|---|---|---|

0 | 8.65 | −15.00 | 188 | −129.77 |

20 | 51.76 | −31.02 | 272 | −1605.58 |

90 | 126.42 | 0.61 | 271 | 77.20 |

Wind Direction (°) | Resultant Force (kN) | Force Arm (m) | Angle of the Force (°) | Torsional Moment (kNm) |
---|---|---|---|---|

0 | 10.99 | −0.04 | 180 | 0.47 |

10 | 30.47 | −33.43 | 265 | −1018.50 |

20 | 59.81 | −29.11 | 275 | −1741.24 |

30 | 71.47 | −22.85 | 270 | −1632.84 |

40 | 83.78 | −17.42 | 266 | −1459.28 |

50 | 104.23 | −12.69 | 264 | −1322.31 |

60 | 124.84 | −9.16 | 265 | −1143.94 |

70 | 133.43 | −6.46 | 266 | −862.02 |

80 | 149.04 | −3.35 | 268 | −499.85 |

90 | 142.39 | 0.22 | 270 | 32.00 |

Wind Direction (°) | Resultant Force (kN) | Force Arm (m) | Angle of the Force (°) | Torsional Moment (kNm) |
---|---|---|---|---|

0 | 11.05 | −13.25 | 202 | −146.36 |

20 | 77.43 | −19.64 | 281 | −1520.86 |

90 | 171.19 | 0.52 | 270 | 88.96 |

Subsoil Stiffness Coefficient (MN/m^{3}) | Wind Direction | Limit Value (mm) | |||||
---|---|---|---|---|---|---|---|

0° | 20° | 90° | |||||

25 | −0.5 | 0.2 | −3.6 | −2.6 | 0.4 | 6.7 | |

40.421 | −0.5 | 0.2 | −3.6 | −2.9 | 0.4 | 6.1 | |

50 | −0.5 | 0.1 | −3.6 | −2.9 | 0.3 | 5.8 | 40 |

100 | −0.5 | 0 | −3.6 | −3.2 | 0.3 | 5.2 |

Subsoil Stiffness Coefficient (MN/m^{3}) | Wind Direction | Limit Value (mm) | |||||
---|---|---|---|---|---|---|---|

0° | 20° | 90° | |||||

25 | −0.3 | 0.5 | 2.3 | 12.2 | −1.4 | 11.0 | |

40.421 | −0.3 | 0.4 | 2.2 | 11.7 | −1.4 | 10.4 | |

50 | −0.4 | 0.4 | 2.2 | 11.5 | −1.4 | 10.1 | 40 |

100 | −0.4 | 0.3 | 2.1 | 10.8 | −1.4 | 9.3 |

Slab No. | Structural Height | The Limit Value SH/1500 | Horizontal Displacement u_{up} | Horizontal Displacement u_{down} | Difference u _{up} − u_{down} | Advisement |
---|---|---|---|---|---|---|

(-) | (mm) | (mm) | (mm) | (mm) | (mm) | |

21 | 3200 | 2.13 | 2.2 | 2.1 | 0.1 | satisfied |

20 | 3200 | 2.13 | 2.1 | 2 | 0.1 | satisfied |

19 | 3200 | 2.13 | 2 | 1.9 | 0.1 | satisfied |

18 | 3200 | 2.13 | 1.9 | 1.8 | 0.1 | satisfied |

17 | 3200 | 2.13 | 1.8 | 1.7 | 0.1 | satisfied |

16 | 3200 | 2.13 | 1.7 | 1.6 | 0.1 | satisfied |

15 | 3200 | 2.13 | 1.6 | 1.5 | 0.1 | satisfied |

14 | 3200 | 2.13 | 1.5 | 1.4 | 0.1 | satisfied |

13 | 3200 | 2.13 | 1.4 | 1.3 | 0.1 | satisfied |

12 | 3200 | 2.13 | 1.3 | 1.2 | 0.1 | satisfied |

11 | 3200 | 2.13 | 1.2 | 1.1 | 0.1 | satisfied |

10 | 3200 | 2.13 | 1.1 | 1 | 0.1 | satisfied |

9 | 3200 | 2.13 | 1 | 0.9 | 0.1 | satisfied |

8 | 3200 | 2.13 | 0.9 | 0.8 | 0.1 | satisfied |

7 | 3200 | 2.13 | 0.8 | 0.7 | 0.1 | satisfied |

6 | 3200 | 2.13 | 0.7 | 0.6 | 0.1 | satisfied |

5 | 3200 | 2.13 | 0.6 | 0.5 | 0.1 | satisfied |

4 | 3200 | 2.13 | 0.5 | 0.3 | 0.2 | satisfied |

3 | 3200 | 2.13 | 0.4 | 0.2 | 0.2 | satisfied |

2 | 3200 | 2.13 | 0.3 | 0.1 | 0.2 | satisfied |

1 | 3950 | 2.63 | 0.2 | 0 | 0.2 | satisfied |

Slab No. | Structural Height | The Limit Value SH/1500 | Horizontal Displacement u_{up} | Horizontal Displacement u_{down} | Difference u _{up} − u_{down} | Advisement |
---|---|---|---|---|---|---|

(-) | (mm) | (mm) | (mm) | (mm) | (mm) | |

21 | 3200 | 2.13 | 11.7 | 11.3 | 0.4 | satisfied |

20 | 3200 | 2.13 | 11.3 | 10.9 | 0.4 | satisfied |

19 | 3200 | 2.13 | 10.9 | 10.5 | 0.4 | satisfied |

18 | 3200 | 2.13 | 10.5 | 10.1 | 0.4 | satisfied |

17 | 3200 | 2.13 | 10.1 | 9.6 | 0.5 | satisfied |

16 | 3200 | 2.13 | 9.6 | 9.1 | 0.5 | satisfied |

15 | 3200 | 2.13 | 9.1 | 8.6 | 0.5 | satisfied |

14 | 3200 | 2.13 | 8.6 | 8.1 | 0.5 | satisfied |

13 | 3200 | 2.13 | 8.1 | 7.5 | 0.6 | satisfied |

12 | 3200 | 2.13 | 7.5 | 7 | 0.5 | satisfied |

11 | 3200 | 2.13 | 7 | 6.4 | 0.6 | satisfied |

10 | 3200 | 2.13 | 6.4 | 5.8 | 0.6 | satisfied |

9 | 3200 | 2.13 | 5.8 | 5.2 | 0.6 | satisfied |

8 | 3200 | 2.13 | 5.2 | 4.6 | 0.6 | satisfied |

7 | 3200 | 2.13 | 4.6 | 4 | 0.6 | satisfied |

6 | 3200 | 2.13 | 4 | 3.4 | 0.6 | satisfied |

5 | 3200 | 2.13 | 3.4 | 2.8 | 0.6 | satisfied |

4 | 3200 | 2.13 | 2.8 | 2.2 | 0.6 | satisfied |

3 | 3200 | 2.13 | 2.2 | 1.6 | 0.6 | satisfied |

2 | 3200 | 2.13 | 1.6 | 1 | 0.6 | satisfied |

1 | 3950 | 2.63 | 1 | 0 | 1 | satisfied |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hubová, O.; Macák, M.; Franek, M.; Ivánková, O.; Konečná, L.B.
Implementation of Data from Wind Tunnel Tests in the Design of a Tall Building with an Elliptic Ground Plan. *Buildings* **2023**, *13*, 2732.
https://doi.org/10.3390/buildings13112732

**AMA Style**

Hubová O, Macák M, Franek M, Ivánková O, Konečná LB.
Implementation of Data from Wind Tunnel Tests in the Design of a Tall Building with an Elliptic Ground Plan. *Buildings*. 2023; 13(11):2732.
https://doi.org/10.3390/buildings13112732

**Chicago/Turabian Style**

Hubová, Oľga, Marek Macák, Michal Franek, Oľga Ivánková, and Lenka Bujdáková Konečná.
2023. "Implementation of Data from Wind Tunnel Tests in the Design of a Tall Building with an Elliptic Ground Plan" *Buildings* 13, no. 11: 2732.
https://doi.org/10.3390/buildings13112732