# Fundamental Characteristics of Wind Loading on Vaulted-Free Roofs

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## Abstract

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## 1. Introduction

## 2. Experimental Arrangement and Procedure

#### 2.1. Investigated Building

_{top}) and eaves (h) depend on f/B. The range of f/B and L/B, which may affect the wind flow around the roof and the wind pressures on the roof significantly, almost covers the values of practical buildings constructed so far in Japan.

#### 2.2. Wind Tunnel Model

_{L}) of 1/100. Figure 2 shows an example of an instrumented model with f/B = 0.3 and L/B = 1. Both the roof and the columns, including the tubing system, were integrally formed. The span B and the length L of the instrumented models are both 150 mm. The roof thickness is 2 mm, and the outer diameter of the columns is 6.5 mm. Each instrumented model has seven pressure taps of 0.6 mm diameter both on the top and bottom surfaces along each of the two arcs, Lines C and E, as shown in Figure 3. In the figure, a local coordinate system (ξ,η) is defined. Because the net wind force on a free roof is estimated using the difference between wind pressures on the top and bottom surfaces, it is desirable to set pressure taps at the same location on the top and bottom surfaces. In practice, however, such an instrumentation is impossible for 2 mm thick models. Then, the pressure tap on the bottom surface was located at a point 2 mm away from that on the top surface along the arc, considering that the spatial variation of wind pressures on the bottom surface is smaller than that on the top surface.

#### 2.3. Wind Tunnel Flow

_{z}and turbulence intensity I

_{z}of the flow at the model’s center without the model in place. In the figure, U

_{z}is normalized by the value U

_{600}at a reference height of z = 600 mm, where a pitot static tube was placed. The profile of U

_{z}approximately followed the power law with an exponent of α = 0.27. The value of I

_{z}at the mean roof height H (=80 mm) was approximately 0.2. The normalized power spectrum of wind speed fluctuation at a height of z = 100 mm is shown in Figure 5b. The power spectrum corresponds to the Karman-type spectrum well (an integral length scale of L

_{x}≈ 0.2 m). According to the AIJ Recommendations for Loads on Buildings [6], the value of L

_{x}is estimated as 58 m at full scale, which corresponds to 0.58 m at model scale, assuming that the geometric scale of wind tunnel flow is 1/100. The value of L

_{x}for the wind tunnel flow is as small as about 1/3 of the target value. However, according to Tieleman et al. [20,21,22], who discussed the similarity of wind tunnel flow required for the wind tunnel experiments investigating the roof wind pressures on low-rise buildings, the L

_{x}value of wind tunnel flow should be larger than 0.2 times the target value and the maximum length of the wind tunnel model. The wind tunnel flow used in the present study almost satisfies Tieleman et al.’s criteria.

_{H}at the mean roof height H was set to 9 m/s. The Reynolds number Re, defined by $Re={U}_{H}H/\nu $, with $\nu $ being the kinematic viscosity of air, was about 4.8 × 10

^{5}. The blockage ratio Br, defined as the ratio of the model’s vertical cross-section to the wind tunnel’s cross-section, was about 2% at most (in the case of f/B = 0.4 and L/B = 3). As for the values of Re and Br in the present experiment, the requirements of the ASCE Wind Tunnel Testing for Buildings and Other Structures [23], i.e., Br < 5% and Re > 1.1 × 10

^{4}, were satisfied.

_{H}and twice the radius of curvature (R) of the roof; the same definition of Re was adopted by Natalini et al. [15]. The reason why they used 2R instead of H as the characteristic length for defining Re was to relate the flow around a vaulted-free roof with that around a circular cylinder, in which the diameter (=twice the radius) of the cylinder is usually used for defining Re. They found that the distribution of mean wind pressure coefficients on the roof was almost independent of Re when Re > 1.0 × 10

^{5}. Such a feature is consistent with a finding by Macdonald et al. [24] for cylindrical structures; they investigated the effect of Re on the wind pressure distributions on circular cylindrical structures by comparing the wind tunnel results with those obtained from a full-scale measurement of actual silos in natural winds. Macdonald et al. [24] indicated that the wind tunnel experiments conducted at Re > 1.0 × 10

^{5}in a turbulent boundary layer might reasonably reproduce a practical full-scale situation. Recently, a similar result was obtained by Liu et al. [25]. In the present experiment, the values of Re defined in terms of U

_{H}and 2R are about 2.53 × 10

^{5}, 1.41 × 10

^{5}, 1.09 × 10

^{5}, and 1.01 × 10

^{5}for f/B = 0.1, 0.2, 0.3, and 0.4, respectively. Accordingly, it is expected that the results obtained in the present experiment may reproduce practical conditions almost satisfactorily.

#### 2.4. Experimental Procedure of Pressure Measurements

_{H}at the mean roof height H (=8 m) was determined based on the specifications in the AIJ Recommendations for Loads on Buildings [5]. The ‘Basic wind speed’, U

_{0}, was assumed 35 m/s as a representative value for the Main Island of Japan. Note that U

_{0}corresponds to the 10 min mean wind speed at 10 m above the ground for flat open terrain for the 100-year recurrence. Assuming that the return period is 100 years and the ‘Terrain Category’ is III, which corresponds to suburban exposure, U

_{H}is calculated as 21.3 m/s. Because the mean wind speed at the mean roof height H (=8 cm) in the wind tunnel experiment was 9 m/s, the velocity scale λ

_{V}of wind tunnel flow is calculated as 1/2.4, resulting in the time scale of λ

_{T}(=λ

_{L}/λ

_{V}) = 1/42.

_{p}as follows:

## 3. CFD Simulation

#### 3.1. Computational Model and Domain

#### 3.2. Computational and Boundary Conditions

_{s}is assumed to be 0.15. The PISO (pressure implicit with the splitting of operators) algorithm is adopted for the pressure–velocity coupling. As for the spatial discretization, the second-order centered difference scheme is used. The second-order implicit scheme is adopted for the time derivative. The computational time T is 15 s, which is equal to 10 min at full scale. The time step is set to $\Delta T=1\times {10}^{-4}$ s.

_{x}/U

_{H}> 0.1, with L

_{x}being the integral scale of turbulence, are smaller than those of the Karman-type spectrum, due to a filtering effect. In the lower reduced frequency range, the shape of the reduced power spectrum, which is important for discussing the wind loads on buildings, may be generally consistent with that of the Karman-type spectrum. Based on a comparison between the wind tunnel experiment and CFD simulation for the wind pressure coefficient distributions on the roofs (as shown in Figure 11), it is expected that the inflow turbulence generated by the preliminary computation provides an appropriate inflow condition.

#### 3.3. Validation of the CFD Analysis

_{max}. It can be seen that the agreement between the experiment and CFD analysis is generally good (error < 10%), indicating that the LES used in this study is appropriate.

## 4. Comparison with Previous Results

_{max}> 0.4 may be due to a difference in the turbulence intensity of the wind tunnel flow and/or some experimental errors; the turbulence intensity at the mean roof height was about 0.2 in the present experiment, while it was about 0.25 in the Natalini et al.’s experiment [15].

## 5. Results and Discussion

#### 5.1. Mean Wind Pressure and Force Coefficients

#### 5.1.1. General Features

#### 5.1.2. Effects of f/B and L/B on the Mean Wind Pressure and Force Coefficients

_{max}≈ 0.2, while negative pressures act widely on the middle and leeward areas, s/s

_{max}> 0.2. The minimum ${\overline{C}}_{pt}$ (maximum |${\overline{C}}_{pt}$|) value generally occurs near the top of the roof, i.e., at s/s

_{max}≈ 0.5, where the wind speed along the roof becomes the maximum. Note that the increase in wind speed along a convex surface produces a decrease in pressure on the surface according to Bernoulli’s equation. The magnitude of the minimum ${\overline{C}}_{pt}$ increases with an increase in f/B. When f/B = 0.1, the variation of ${\overline{C}}_{pt}$ is relatively small. When f/B = 0.3 and 0.4, on the other hand, the variation is relatively large, and the f/B curve has an inflection point at s/s

_{max}≈ 0.8 and 0.7 for f/B = 0.3 and 0.4, respectively. Figure 16a,b, respectively, show the distributions of time-averaged streamlines around the roofs with f/B = 0.1 and 0.4 in the (x-z) plane at the mid-span. It can be seen from Figure 16a that the wind flows smoothly along the top surface of the roof without separation when f/B = 0.1. On the other hand, the flow separates from the top surface in the case of f/B = 0.4, as shown in Figure 16b. The point of flow separation roughly corresponds to the inflection point of the ${\overline{C}}_{pt}$ curve. As for the distribution of ${\overline{C}}_{pb}$ (Figure 15b), the ${\overline{C}}_{pb}$ value changes from negative to positive to leeward when f/B = 0.1 and 0.2. It can be seen from Figure 16a that the flow separates downward at the leading edge, resulting in large suctions near the leading edge. The separated flow reattaches on the bottom surface at s/s

_{max}$\approx $ 0.5 when f/B = 0.1. The point of reattachment of the separated flow roughly corresponds to the inflection point of the ${\overline{C}}_{pb}$ curve. The reattached flow generates positive pressures on the bottom surface. In contrast, the separated flow does not reattach on the bottom surface when f/B = 0.4, generating suctions of moderate magnitude over the whole area. It is found that the distribution of ${\overline{C}}_{f}$ (Figure 15c) is similar to that of ${\overline{C}}_{pt}$ (Figure 15a). This is because the variation of ${\overline{C}}_{pb}$ is so small that ${\overline{C}}_{pt}$ dictates the ${\overline{C}}_{f}$ distribution. Positive wind forces (downward) act on the windward area, s/s

_{max}< 0.2–0.3, while negative wind forces (upward) act on the middle and leeward areas, s/s

_{max}> 0.2–0.3. The maximum ${\overline{C}}_{f}$ value occurs at s/s

_{max}$\approx $ 0. The minimum ${\overline{C}}_{f}$ value occurs in a range of s/s

_{max}= 0.5–0.7, the location of which depends on the f/B ratio.

#### 5.1.3. Effect of Wind Direction θ on the Mean Wind Pressure and Wind Force Coefficients

_{max}= 0.6–0.8), as shown in Figure 19a,d. This is because a conical vortex is generated above the top surface due to the flow separation at the windward roof edge (verge). The minimum value of ${\overline{C}}_{pt}$ increases in magnitude as the f/B ratio increases. The value for f/B = 0.4 is about twice that for f/B = 0.1. In such oblique winds, large suctions are generated on the bottom surface in the windward area due to the flow separation at the windward edge. At the same time, positive ${\overline{C}}_{pb}$ values are generated on the bottom surface in the leeward area due to the impinging flow, as shown in Figure 19b,e. The positive ${\overline{C}}_{pb}$ values are rather large when f/B = 0.4. Consequently, large magnitude positive (downward) and negative (upward) wind forces are generated in the windward and leeward half areas, respectively (see Figure 19c,f). The higher the f/B ratio, the larger the magnitude is.

#### 5.1.4. Empirical Formulas for the Mean Wind Force Coefficient Distributions

#### 5.2. Maximum and Minimum Peak Wind Force Coefficients

#### 5.2.1. General Features

_{max}= 0–0.2), which are induced using a combination of large negative pressures owing to flow separation on the bottom surface (see Figure 16a) and large positive pressures owing to flow impinging on the top surface at θ ≈ 0°. On the other hand, ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$ becomes very large in magnitude near the top of the roof (s/s

_{max}= 0.5–0.8). The magnitude increases with an increase in f/B. This is due to large suctions induced on the top surface by the flow acceleration at θ ≈ 0° (see Figure 16). The results for Line E are significantly different from those for Line C. The values of ${\widehat{C}}_{f,\mathrm{cr}}$ are almost the same except for an area of s/s

_{max}= 0.8–0.1, irrespective of f/B. The values of ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$ range from about −3 to −5 at many pressure taps. Very large absolute values of ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$ occur at s/s

_{max}= 0.6–0.7, which are induced using a combination of large suctions on the top surface due to the generation of the conical vortex and large positive pressures due to the flow impinging on the bottom surface in an oblique wind (see Figure 20 and Figure 21). These features of ${\widehat{C}}_{f,\mathrm{cr}}$ and ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$ correspond well to those for the mean wind force coefficients described in Section 5.1.2.

_{H}at the mean roof height H was about 0.16, approximately 20% smaller than that of the present experiment. Figure 27 and Figure 28 show comparisons between the present and previous experiments for ${\widehat{C}}_{f,\mathrm{cr}}$ and ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$ on Lines C and E, respectively. It is found that the difference is relatively small for Line C. In contrast, we can see a significant difference for Line E; the values of ${\widehat{C}}_{f,\mathrm{cr}}$ and ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$ obtained from the present experiment are generally larger in magnitude than those obtained from the previous experiment. Therefore, it may be concluded that the turbulence intensity affects the peak suctions on the top surface due to the generation of a conical vortex and the positive pressures on the bottom surface due to the flow impinging in an oblique wind more significantly.

#### 5.2.2. Estimation of the Maximum and Minimum Peak Wind Force Coefficients Based on a Peak Factor Approach

_{tb}is the correlation coefficient between wind pressure fluctuations on the top and bottom surfaces. As mentioned above, large positive wind forces are generated near the windward eaves due to the combination of large positive pressures on the top surface and large suctions on the bottom surface at θ ≈ 0°. Similarly, large peak suctions are generated near the windward verge due to the combination of large positive pressures on the bottom surface and large suctions on the top surface at θ = 30–40°. In such cases, it is thought that the correlation between wind pressures on the top and bottom surfaces is high; that is, large positive pressures on one side and large suctions on the other side occur almost simultaneously, resulting in large wind forces. Thus, it is assumed for simplicity that R

_{tb}= −1.

## 6. Concluding Remarks

_{n}may be dependent on such factors as the turbulence intensity of the approach flow. In order to generate the results, the range of parameters should be expanded. This is the subject of our future study.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 11.**Comparison between the wind tunnel experiment (EXP) and CFD for the mean wind pressure coefficients, ${\overline{C}}_{pt}$ and ${\overline{C}}_{pb}$, when θ = 0°.

**Figure 12.**Comparison between the present and previous experiments (Natalini et al. [15]) for the mean wind pressure coefficient distributions along the centerline (f/B = 0.2, L/B = 2).

**Figure 15.**Effect of f/B on the mean wind pressure and force coefficients on Line C (L/B = 1, θ = 0°).

**Figure 17.**Effect of L/B on the distributions of mean wind pressure and force coefficients along Lines C and E (f/B = 0. 1, θ = 0°).

**Figure 20.**Distributions of time-averaged streamlines in the (x, z)-plane (f/B = 0.1, L/B = 1, θ = 45°). (

**a**) Section (a); (

**b**) Section (b); (

**c**) Section (c).

**Figure 21.**Distributions of time-averaged streamlines in the (x, z)-plane (f/B = 0.4, L/B = 1, θ = 45°). (

**a**) Section (a); (

**b**) Section (b); (

**c**) Section (c).

**Figure 25.**Most critical values of the maximum and minimum peak wind force coefficients, irrespective of wind direction, ${\widehat{C}}_{f,\mathrm{cr}}$ and ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$, on Line C.

**Figure 26.**Most critical values of the maximum and minimum peak wind force coefficients, irrespective of wind direction, ${\widehat{C}}_{f,\mathrm{cr}}$ and ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$, on Line E.

**Figure 27.**Comparison between the present and previous experiments for ${\widehat{C}}_{f,\mathrm{cr}}$ and ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$ on Line C.

**Figure 28.**Comparison between the present and previous experiments for ${\widehat{C}}_{f,\mathrm{cr}}$ and ${\stackrel{\u02c7}{C}}_{f,\mathrm{cr}}$ on Line E.

**Figure 29.**Comparison between experiment and estimation for the maximum and minimum peak wind force coefficients on Lines C and E.

f/B | L/B | f (m) | B (m) | L (m) | h_{top} (m) |
---|---|---|---|---|---|

0.1 | 1, 2, 3 | 1.5 | 15 | 15, 30, 45 | 8.8 |

0.2 | 1, 2, 3 | 3.0 | 15 | 15, 30, 45 | 9.5 |

0.3 | 1, 2, 3 | 4.5 | 15 | 15, 30, 45 | 10.3 |

0.4 | 1, 2, 3 | 6.0 | 15 | 15, 30, 45 | 11.0 |

Inlet boundary | Inflow turbulence (Sampling data from preliminary computation) |

Outlet boundary | Advective outflow condition |

Side and top boundaries | Free-slip condition |

Surfaces of ground and roof | No-slip condition |

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**MDPI and ACS Style**

Ding, W.; Uematsu, Y.; Wen, L.
Fundamental Characteristics of Wind Loading on Vaulted-Free Roofs. *Wind* **2023**, *3*, 394-417.
https://doi.org/10.3390/wind3040023

**AMA Style**

Ding W, Uematsu Y, Wen L.
Fundamental Characteristics of Wind Loading on Vaulted-Free Roofs. *Wind*. 2023; 3(4):394-417.
https://doi.org/10.3390/wind3040023

**Chicago/Turabian Style**

Ding, Wei, Yasushi Uematsu, and Lizhi Wen.
2023. "Fundamental Characteristics of Wind Loading on Vaulted-Free Roofs" *Wind* 3, no. 4: 394-417.
https://doi.org/10.3390/wind3040023