Experimental study of wind loads on gable roofs of low-rise buildings with overhangs

Abstract

Gable roofs with overhangs (eaves) are the common constructions of low-rise buildings on the southeastern coast of China, and they were vulnerable to typhoons from experience. The wind pressure distributions on gable roofs of low-rise buildings are investigated by a series of wind tunnel tests which consist of 99 test cases with various roof pitches, height-depth ratios and width-depth ratios. The block pressure coefficients and worst negative (block) pressure coefficients on different roof regions of low-rise buildings are proposed for the main structure and building envelope, respectively. The effects of roof pitch, height-depth ratio, and width-depth ratio on the pressure coefficients of each region are analyzed in detail. In addition, the pressure coefficients on the roofs for the main structure and building envelope are fitted according to roof pitch, height-depth ratio and width-depth ratio of the low-rise building. Meanwhile, the rationality of the fitting formulas is verified by comparing the fitting results with the codes of different countries. Lastly, the block pressure coefficients and worst negative pressure coefficients are recommended to guide the design of low-rise buildings in typhoon area and act as references for the future’s modification of wind load codes.

Keywords

low-rise building / gable roof / wind loads / wind tunnel test / block pressure coefficient / load code

Introduction

Recent natural calamity investigations show that wind-induced disasters cause large economic losses and many casualties worldwide each year. Most of these are related to the damage in residential, industrial and other low-rise buildings [¹]. Research on wind-induced disasters shows that the surfaces of low-rise buildings are often toppled during a typhoon. Therefore, the study of wind loads on roof coverings and other building envelopes is of great significance.

Over the past decades, wind tunnel tests on low-rise buildings have been extensively performed to study the parameters influencing wind loads on low-rise gable-roofed buildings, such as roof pitch, height-depth ratio, width-depth ratio, and eaves geometry. Many researches have demonstrated that the roof pitch plays a significant role in wind pressures on gable roofs of low-rise buildings [²–⁸]. Meanwhile, the researches by Gerhardt and Kramer [⁹], Holmes [³], Krishna [¹⁰] and Ginger and Holmes [¹¹] indicate that the suctions on roofs have rough grow tendency with the increase of height-depth ratio.

However, there have been relatively fewer studies concerned on the influence of width-depth ratio (aspect ratio), except for the work of Kanda and Maruta [¹²], Ginger and Holmes [¹¹,¹³] and Ho et al. [¹⁴]. They found that the increase of width-depth ratio can lead to the variation of wind loads on windward and leeward sloped roof with oblique winds.

Gable roof is the typical roof type on southeastern coast of China. As southeastern China receives plenty of rain every year, the roof is usually equipped with overhangs (eaves) for drainage in the architectural design. Only a few researchers have drawn their attentions on how eaves affect the wind loads on roofs [¹⁵–²⁰]. These studies assessed the effect of geometry dimension and eaves shape on wind pressure distribution of the roofs by the methods of wind tunnel test and field measurement.

The wind load codes and standards in many countries [²¹–²⁴] have provided specifications about the wind loads on roofs of low-rise buildings with considering the effect of roof pitch, height-depth ratio, width-depth ratio, and so on. However, there are considerable differences between the specifications in these codes, which may owe to the various details of low-rise building in each country. Meanwhile, the provisions about the wind pressure coefficient on roofs for main structure and building envelope in Chinese load code [²⁵] are quite simple, by contrast, it can not cover the real situations of low-rise buildings in China.

In this present study, a cluster of wind tunnel tests which consist of 99 gable-roofed low-rise buildings with overhangs were carried out to determine the external wind pressure distributions on the roofs. The 99 buildings have 11 different roof pitches, 3 different height-depth ratios, and 3 different width-depth ratios. The experimental apparatus includes dimensions of the 99 buildings, layout of the taps, wind field type, wind speed and wind direction are introduced in Section 2. Section 3 displays the data processing method of block pressure coefficients for main structure and worst negative (block) pressure coefficients for building envelope. In Section 4, it analyzes the effects of roof pitch, height-depth ratio and width-depth ratio on mean and worst negative pressure coefficients. Then, the fitting formulas of pressure coefficients with three geometric parameters on each roof region are shown in Section 5. Section 6 compares the fitted results with the codes of different countries and Section 7 proposes the values of block pressure coefficients and worst negative pressure coefficients for wind load code.

Experimental apparatus

The experiment was carried out in TJ-2 Boundary Layer Wind Tunnel at Tongji University. The 99 gable roof low-rise buildings with overhangs have been tested in the wind tunnel. The roof pitches (b) of the models are 0°, 4.8°, 9.4°, 14°, 18.4°, 21.8°, 26.7°, 30°, 35°, 45°, and 60°. The height-depth ratios (H₀/D) are 0.55, 0.8, and 1.05 with width-depth ratios (B/D) of 0.6, 1.2, and 1.8. H₀ is the building eave height, i.e., the dimension between the ground and the upside of eave strut and H is the mean roof height. B and D are the width and the depth of the building, respectively. The definitions of these parameters are presented in Fig. 1, and the test cases are listed in Table 1. The eave bottom is at the same height with ring beams and the height differences between eave bottom and top of eave struts are keeping a constant of 0.4 m for all cases (i.e., the height of the ring beam is 0.4 m). The horizontal length (De) of the overhang is 0.6 m when the roof pitch is between 0° and 30°, then it changes to 0.57, 0.4, and 0.23 m when the roof pitch is 35°, 45°, and 60°, respectively. The rigid model of pressure measurement test is shown in Fig. 2. The model is made by Perspex and Acrylonitrile Butadiene Styrene (ABS) to keep sufficient rigidity when the wind speed reaches 12 m/s at the height of 1 m in the wind tunnel. The geometric scale of the rigid model is 1/40 with the architecture prototype very close to the typical ones. As shown in Fig. 3, the roof of the biggest model has 290 taps on it. The thick dash line in Fig. 3 represents the boundary of model combinations of different small parts.

The velocity scale of this test is 1/3 and the time ratio is 3/40. The wind directions range from 0° to 360° with an increment of 15° clockwise. The atmospheric boundary layer flow in the experiment is over open terrain (B category in Chinese code). The Reynolds number of the test ranges from 6.7×10⁶ to 1.37×10⁷. The simulated wind profile is shown in Fig. 4. The mean velocity profile index (a) is 0.15, and the turbulence intensity at the top of the model (10 m in full scale) is approximately 22%. The sampling time is 57.6 s at a sampling rate of 312.5 Hz, which represents a period of 12.8 min in full scale.

Data processing method

Pressure coefficient

The instantaneous pressure coefficient at the i^th tap in the q wind direction is defined as

(1a)C_p(i,\theta ,t)={\displaystyle \frac{p(i,\theta ,t)-p_{\mathrm{ref}}}{\mathrm{0}.\mathrm{5}\rho V_H{}^{\mathrm{2}}}},

where C_p(i, q, t) and p(i, q, t) are the instantaneous pressure coefficient and instantaneous pressure at the i^th tap in the q wind direction, respectively; p_ref is the static pressure at the reference height; H is the reference height which corresponding to the mean roof height in this formula; V_H is the mean wind speed at the reference height and r is the air density. For the overhang, the instantaneous pressure coefficient is defined as

(1b)C_p(i,\theta ,t)={\displaystyle \frac{p_{up}(i,\theta ,t)-p_{lo}(i,\theta ,t)}{\mathrm{0}.\mathrm{5}\rho V_H{}^{\mathrm{2}}}},

where p_up(i, q, t) and p_lo(i, q, t) are the instantaneous pressure on the upper and lower surfaces of the overhangs, respectively.

The mean wind pressure coefficient is the average of instantaneous pressure coefficient history (N is equal to 18,000),

(2)C_p{,}_{mean}(i,\theta )={\displaystyle \sum _{j=\mathrm{1}}^N{C}_p(i,\theta ,t)/N}.

Block pressure coefficients for main structure

Division of the roof

The pressure coefficients of the roof vary drastically in different wind direction and it various at different area of the roof even in the same wind direction. Based on the distribution regularities of mean wind pressure coefficient of the roofs as well as American Load Specification [²²] and Japanese Load Code [²¹], the roof is divided into different parts (blocks) to consider the wind loads on it. As shown in Fig. 5, the roof includes the windward overhang (O_u), windward side (R_u), and leeward side (R_l) when the wind direction is perpendicular to the ridge. When the wind direction is parallel to the ridge, the roof includes the surface adjacent to the approaching flow (R_a), middle surface of the roof (R_b), and surface apart from the flow (R_c). The parameter L in Fig. 5(b) represents the smaller value of 4H and B.

Computational method of the block pressure coefficients for main structure

For wind directions perpendicular to the ridge, the wind angles in considering range from -45° (315°) to 45°. The actual wind directions are between 0° and 45° as the symmetry of the building. Similarly, for wind directions parallel to the ridge, the actual wind angles are ranging from 45° to 90°. The block pressure coefficient on each area is the minimum of area-weighted average of the mean pressure coefficients obtained from related taps under all the calculated wind directions. The formula is as follows:

(3)C_p(j)=\underset{\phi }{\min }\left({\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^n{C}_{p,mean}(i,\theta )A_i}}{{\displaystyle \sum _{i=\mathrm{1}}^n{A}_i}}}\right),

where C_p(j) is the block pressure coefficient in area j, j is the range of wind direction, C_p,mean(i,q) is the mean pressure coefficient of the i^th tap in the q wind direction, A_i represents the tributary area of tap i, and n is the total tap number in area j. Regarding the positive pressure coefficient occurring at a high roof pitch, the “min” in this formula should be changed into “max”.

Worst negative (block) pressure coefficients for building envelope

Roof division of building envelope

The extreme negative pressure coefficients on different parts of the roof surface of a low-rise building are quite different. The local extreme pressures on the ridge, the edges and the corners of a gabled roof are significantly larger than that on the middle of the roof. In the research of Wang [²⁶], with a roof pitch of 14°, the extreme negative pressure coefficient can reach -17.0 at the corner of the roof, whereas that is merely -3.0 in the middle of the roof. Therefore, the roof is divided into many independent parts based on the layout of the pressure taps (Fig. 4) to analyze the distribution characteristics of the pressure coefficient in detail. As illustrated in Fig. 6, these parts include: 1) the interior of the roof (Zone 1); 2) the edges of the roof including the edge near the eave (Zone 2), the edge near the ridge (Zone 3) and the edge near the gable wall (Zone 4); and 3) the corners of the roof including the corner near the ridge (Zone 5) and the corner near the eave (Zone 6).

Computational method of worst negative pressure coefficients for building envelope

To match the specific size of the building envelope, the time history of local pressure coefficient was examined by the TVL method [²⁷,²⁸]. Lawson [²⁷] considered the maximum external load on a cladding panel is due to the maximum pressure which can be correlated over the whole panel. In turn, correlation over a given distance can be related to the averaging time of the pressure measurement, either through the concept of eddy size or by considering coherence functions. Their relationship is shown in Eq. (4):

(4)T=K{L}_e/V,

where T is the moving average time span, V is the mean velocity of the flow, L_e is the characteristic length of the structure (which usually means the diagonal length of the envelope building), and K is a constant relevant to the coherence function.

The key point of this method is the value of K. Lawson [²⁷] believed that K is most respectively by 4.5. However, Holmes [²⁹] suggested K as 1.0. Uematsu and Isyumov [³⁰] studied the area-averaged pressure on different zone of the roofs of low-rise buildings and proposed the value of K based on different cases. Combining all these situations and referring to the results of Uematsu and Isyumov [³⁰], this study employed K is equal to 17 as the case is open country terrain.

Local pressure coefficient time series C_p(i, q), which has 1 m characteristic length and 1 m² area of the local building envelope, was obtained through a 0.065 s moving average of the pressure coefficient time series C_p(i, q, t). The moving average time was determined by Eq. (4). After the moving average, the minimum of pressure time series {\stackrel{\vee }{C}}_p(i,\theta ) was obtained through Sadek and Simiu [³¹] method. The mean extreme negative pressure coefficient of each zone was the weighted average of the pressure time series as shown in Eq. (5),

(5){\stackrel{\vee }{C}}_p(\theta )={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^n{A}_i\cdot {\stackrel{\vee }{C}}_p(i,\theta )}}{{\displaystyle \sum _{i=\mathrm{1}}^n{A}_i}}},

where A_i is the tributary area of the i^th tap and n is number of taps in each zone, {\stackrel{\vee }{C}}_p(\theta ) is the area-averaged extreme negative pressure coefficients. The smallest {\stackrel{\vee }{C}}_p(\theta ) in each zone under all the wind directions then become the worst negative pressure coefficients (C_p,min) as shown in Eq. (6).

(6)C_{p,\min }=\underset{\theta =\mathrm{0}°~\mathrm{360}°}{\min }({\stackrel{\vee }{C}}_p(\theta )).

(6)

Effects of roof pitch, height-depth ratio, and width-depth ratio on pressure coefficients

In this section, we aim to explore the effect of shape parameter such as roof pitch, height-depth ratio, and width-depth ratio on pressure coefficients of low-rise building. Therefore, the block pressure coefficients for main structure and worst negative pressure coefficients for building envelope under different geometric parameters are analyzed in this part.

The low-rise building with H₀/D= 0.55 and B/D= 1.8 is the most representative dwelling house in the world. As the limitations of space and large number of cases are considered in this paper, the B/D is set to 1.8 when study how the height-depth ratio (H₀/D) affects wind pressures. Similarly, the H₀/D is set to 0.55 to study the effect of the width-depth ratio (B/D). For the convenience of expression, a negative pressure coefficient is described as ‘large’ or ‘small’ with respect to its absolute value.

Block pressure coefficients for main structure (mean pressure)

The effects of roof pitch (b), height-depth ratio (H₀/D), and width-depth ratio (B/D) on the block pressure coefficient are illustrated in Fig. 7.

Windward overhang (O_u)

Figure 7(a) shows that the negative block pressure coefficients of windward overhang (O_u) increase with roof pitch when b is between 0° and 9.5°. However, for b is larger than 9.5°, the block pressure coefficients linearly decrease with the roof pitch. The most unfavorable block pressure coefficient occurs at a roof pitch of 9.5°, which is -2.41 when B/D= 1.8 and is -2.29 when H₀/D= 0.55, respectively.

When B/D= 1.8 and the roof pitch remains the same, the block pressure coefficients of the windward overhang roughly grow with the H₀/D. For H₀/D= 0.55, it has a same regularity for various B/D, except when b is 0° or 60°.

In short, the roof pitch affects the block pressure coefficient of the windward overhang obviously than the height-depth and width-depth ratios. Meanwhile, the wind loads on windward overhang are proportional to both height-depth ratio (H₀/D) and width-depth ratio (B/D).

Windward side (R_u)

The roof pitches smaller than 10° are usually regarded as flat roofs, and the roof pitches larger than 10° are defined as slope roofs. The analysis of windward side and leeward side for a gable (double-pitch) roof of low-rise building is only considering the cases of roof pitch larger than 10°.

Figure 7(b) displays that the block pressure coefficients of the windward side are all negative, and it decreases noticeably with b when b is between 14° and 45°. However, it becomes positive when b is equal to 60°.

When B/D= 1.8 and b does not change, the block pressure coefficients of the windward side increase with the height-depth ratio. When H₀/D= 0.55 and b is 14°–30°, the block pressure coefficients of the windward side increase with the width-depth ratio. The influence of the height-depth ratio is not obvious when the roof pitch is larger than 30°.

In summary, the block pressure coefficients of windward side have the same regularity with windward overhang when consider the parameters of roof pitch, height-depth and width-depth ratios.

Leeward side (R_l)

Figure 7(c) illustrates that the block pressure coefficients of the leeward side (R_l) become roughly concave with the variation of the roof pitch. The block pressure coefficients are relatively large at roof pitches of 21.8° to 30°.

When the B/D is equal to 1.8, the influence of the H₀/D is not apparent on the block pressure coefficients of the leeward side. The most unfavorable block pressure coefficient is -0.97 at H₀/D= 0.80 and b = 21.8°.

When the H₀/D is equal to 0.55, the block pressure coefficients of the leeward side are increase steadily with the B/D, except for the roof pitch of 14°. The most unfavorable block pressure coefficient is -0.95 which occurs at a roof pitch of 21.8° and B/D= 0.6.

Surface adjacent to the approaching flow (R_a)

Figure 7(d) shows that the roof pitch does not obviously affect the pressures of the roof surface adjacent to the approaching flow (R_a) when the wind direction is parallel to the ridge. It also shows that the block pressure coefficients of R_a increase gradually with the height-width ratio (H₀/D) in most cases and increase with the depth-width ratio (B/D) when the roof pitches are less than 30°.

In short, the greater the height-depth ratio (H₀/D) or the width-depth ratio (B/D) is, the more unfavorable the wind load on the roof surface adjacent to the approaching flow is in most cases.

Middle surface (R_b)

As illustrated in Fig. 7(e), the effects of roof pitch, height-depth ratio, and width-depth ratio on the pressures of the middle surface (R_b) and the roof surface adjacent to the approaching flow (R_a) are similar.

When B/D= 1.8 and H₀/D= 0.55–1.05, the pressure coefficients of the middle surface (R_b) become roughly concave with various roof pitches. The block pressure coefficients are relatively large at a roof pitch of 9.5° to 21.8°. The most unfavorable block pressure coefficient occurs at a roof pitch of 9.5°, which is -0.77 at H₀/D= 1.05.

When H₀/D= 0.55 and the b remains the same, the block pressure coefficients of the R_b increases with the H₀/D except when the roof pitches are 0° and bigger than 30°.

Surface apart from the approaching flow (R_c)

According to the division rule of roof surfaces, the roof surface apart from the approaching flow (R_c) exits only when the width of the building (B) is bigger than 1.5L (Fig. 5(b)). Therefore, the influences of b and H₀/D on the block pressure coefficients for the roof surface apart from the approaching flow are discussed only when the B/D is equal to 1.8.

Figure 7(f) shows that the influence of the roof pitch is not distinct on the pressures of R_c when the wind direction is parallel to the ridge. On the other hand, the block pressure coefficients of R_c increase with the H₀/D when the roof pitches are less than 45°. The most unfavorable block pressure coefficient occurs at a roof pitch of 9.5°, which is -0.65 at H₀/D= 1.05.

Overview

In general, the roof pitch plays a significant role in the block pressure coefficients of the roof surfaces. Meanwhile, the greater the height-depth ratio (H₀/D) or the width-depth ratio (B/D) is, the more unfavorable the wind load on the roof surfaces is in most cases.

Worst negative pressure coefficients for building envelope

As we know, the worst negative pressure coefficients are utilized to evaluate the wind load on building envelope. Figure 8 demonstrates how roof pitch, height-depth ratio and width-depth ratio affect the worst negative pressure coefficients on different zones.

Zone 1 (the interior of the roof)

Figure 8(a) shows that the influence of the roof pitch is not obvious on the pressures of the interior of roof (Zone 1).

When the B/D is equal to 1.8, the worst negative pressure coefficients of Zone 1 are relatively large as b = 4.8° –14°. The largest worst negative pressure mostly occurs at a roof pitch of 9.5°, which is –2.71 at H₀/D= 1.05.

When the H₀/D is equal to 0.55, the maximum worst negative pressure coefficients of Zone 1 occur at B/D= 0.6, which are -2.99 at b = 9.5° and -2.94 at b = 30°, respectively. While for the B/D being 1.2 and 1.8, the values of the worst negative pressure coefficients are close in most cases.

Zone 2 (the edge near the eave)

Figure 8(b) demonstrates that the roof pitch has a remarkable influence on the pressures of the edge near the roof (Zone 2).

When the B/D is equal to 1.8, the worst negative pressure coefficients grow with the increase of height-depth ratio at a roof pitch less than or equal to 14°, and the largest worst negative pressure coefficient is -2.76 at H₀/D= 1.05 and b = 0°. The pressures exhibit small changes at other roof pitches.

When the H₀/D is equal to 0.55, the worst negative pressure coefficients with B/D= 1.2 are significantly larger than those with B/D= 0.6 and 1.8. The largest value is -3.23 occurring at B/D= 1.2 and b = 18.4°. While for the B/D being 0.6 and 1.8, the values of the worst negative pressure coefficients are close when the b is between 9.5°–60°.

Zone 3 (the edge near the ridge)

Figure 8(c) illustrates that the worst negative pressure coefficients of the edge near the ridge (Zone 3) become roughly concave with the variation of roof pitch. The negative pressure is relatively large at roof pitch of 14°–30°.

When the B/D is equal to 1.8, the influence of the height-depth ratio is little. The largest worst negative pressure coefficient is -4.00 when H₀/D= 0.55 at a roof pitch of 26.6°.

With H₀/D being equal to 0.55, the worst negative pressure coefficients decrease with the width-depth ratio when the roof pitch is less than or equal to 9.5°. However, a converse trend occurs when the roof pitch is between 26.6°–45°. The largest worst negative pressure coefficient is -4.10 when B/D= 1.2 at a roof pitch of 21.8°.

Zone 4 (the edge near the gable wall)

Figure 8(d) shows that the worst negative pressure coefficients of the edge near the gable wall (Zone 4) increase with the roof pitch when b is between 0° and 9.5°. By contrast, it decreases steadily with the roof pitch when b is larger than 9.5°.

When B/D= 1.8 and the roof pitch remains the same, the worst negative pressure coefficients decrease roughly with the H₀/D for most cases. When H₀/D= 0.55 and the roof pitch does not change, the worst negative pressure coefficients also decrease with the B/D when the b is less than 26.6°. For these two conditions, the maximum worst negative pressure coefficients are both -5.37, occurring at a roof pitch of 9.5°, B/D= 1.8 and H₀/D= 0.55.

Zone 5 (the corner near the ridge)

Figure 8(e) demonstrates that the worst negative pressure coefficients of the corner near the ridge (Zone 5) have the same changing tendency with that of the edge near the ridge (Zone 3). The negative pressure is relatively large at roof pitch of 18.4°–30°.

Both height-depth ratio and width-depth ratio have little influence on the maximum worst negative pressure coefficient. The maximum worst negative pressure coefficient is -6.11 when H₀/D= 0.8 at a roof pitch of 21.8° and it is -5.97 when B/D= 0.55 at a roof pitch of 26.6°.

Zone 6 (the corner near the eave)

Figure 8(f) illustrates that the roof pitch has significant influence on the pressures of the corner near the eave (Zone 6), while the influence of the height-depth ratio or width-depth ratio is relatively little.

When the B/D is equal to 1.8, the worst negative pressure coefficients are first decrease then increase and finally decrease with the roof pitch and the two inflection points are appearing at 14° and 26.6°, respectively. The largest worst negative pressure coefficient is -5.54 when H₀/D= 1.05 at a roof pitch of 0°.

When the H₀/D is equal to 0.55, the worst negative pressure coefficients have a similar change tendency with that of various B/D and it has a largest value of -5.28 when B/D= 1.2 and b = 26.6°.

Summary

In general, the roof pitch plays a significant role in both block pressure coefficients and worst negative pressure coefficients on roof surfaces. Meanwhile, the height-depth ratio (H₀/D) and the width-depth ratio (B/D) of the building affect the pressures on the roof in a same degree.

Fitting formulas of pressure coefficients on each roof region

In this section, the nonlinear least square method is utilized to fit the block pressure coefficients and worst negative pressure coefficient of each region based on the experimental results. Furthermore, the fitting quality of each formula is evaluated by the correlation coefficient (r) and the quality coefficient (QC).

Fitting formulas of block pressure coefficients for main structure

The fitting formulas of the block pressure coefficients on the roof for the main structure are shown below.

Windward eave (O_u)

(7)C_p(\mathrm{Ou})=\mathrm{0.042}{p}_{\mathrm{1}}-\mathrm{0.381}{p}_{\mathrm{2}}-\mathrm{0.172}{p}_{\mathrm{3}}-\mathrm{1.741.}

Windward side (R_u)

(8)C_p(\mathrm{Ru})=\mathrm{1.159}{p}_{\mathrm{1}}{}^{\mathrm{0.235}}-\mathrm{0.201}{p}_{\mathrm{2}}-\mathrm{0.061}{p}_{\mathrm{3}}-\mathrm{2.62.}

Leeward side (R_l)

(9)C_p(\mathrm{Rl})=-\mathrm{1.072}{p}_{\mathrm{1}}{}^{\mathrm{0.223}}+\mathrm{0.017}{p}_{\mathrm{1}}-\mathrm{0.045}{p}_{\mathrm{2}}-\mathrm{0.091}{p}_{\mathrm{3}}+\mathrm{1.09.}

Roof close to the approaching flow (R_a)

(10)C_p(\mathrm{Ra})=\mathrm{0.002}{p}_{\mathrm{1}}{p}_{\mathrm{2}}{p}_{\mathrm{3}}-\mathrm{0.032}{p}_{\mathrm{2}}{p}_{\mathrm{3}}-\mathrm{0.002}p{}_{\mathrm{1}}-\mathrm{0.217}{p}_{\mathrm{2}}-\mathrm{0.053}{p}_{\mathrm{3}}-\mathrm{0.578.}

Middle roof (R_b)

(11)C_p(\mathrm{Rb})=\mathrm{0.139}{p}_{\mathrm{1}}{}^{\mathrm{0.976}}{p}_{\mathrm{2}}{}^{-\mathrm{0.01}}{p}_{\mathrm{3}}{}^{-\mathrm{0.029}}+\mathrm{0.127}p{}_{\mathrm{1}}-\mathrm{0.201}{p}_{\mathrm{2}}-\mathrm{0.167}{p}_{\mathrm{3}}-\mathrm{0.14.}

Roof apart from the approaching flow (R_c)

(12)C_p(\mathrm{Rc})=-\mathrm{0.026}p{}_{\mathrm{1}}{}^{\mathrm{0.567}}{p}_{\mathrm{2}}{}^{\mathrm{0.628}}+\mathrm{0.005}p{}_{\mathrm{1}}p{}_{\mathrm{2}}-\mathrm{0.12}{p}_{\mathrm{2}}-\mathrm{0.358.}

The formulas above are expressed by three geometric parameters, p₁ is the roof pitch b (in radians, i.e., b·p/180), p₂ is the height-depth ratio H₀/D, and p₃ is the width-depth ratio B/D. As the results showed in the section 4, the three dimension parameters have different degrees of influence on the regional pressure coefficient. The power such as 0.235 in Eq. (8) shows the influence of roof pitch on the block pressure coefficient of windward side (R_u).

The accuracy analysis of the abovementioned fitting formulas is illustrated in Fig. 9. The correlation coefficient (r) and the quality coefficient (QC) for the pressure coefficients of the roof are listed in Table 2. The equation of the r and QC are shown as below:

(13)r={\displaystyle \frac{{\displaystyle \sum (X_i-\bar{X})(Y_i-\bar{Y})}}{\sqrt{{\displaystyle \sum {(X_i-\bar{X})}^{\mathrm{2}}{\displaystyle \sum {(Y_i-\bar{Y})}^{\mathrm{2}}}}}}},

(14)QC=\sqrt{{\displaystyle \frac{{\displaystyle \sum {\left({\displaystyle \frac{X_i-Y_i}{\bar{X}}}\right)}^{\mathrm{2}}}}{n-\mathrm{1}}}},

where Y_i and X_i are the fitted value and test value of the pressure coefficients, respectively, \bar{Y} and \bar{X} are the averaged fitted value and test value of the pressure coefficients, respectively.

Figure 9 demonstrates that the fitted values of the block pressure coefficients on each surface agree well with the test values. The largest and smallest discrepancies between the test and fitted values occur at the surface of leeward side (R_l) and windward side (R_u), respectively. As illustrated in Table 2, the correlation coefficients (r) of the test and fitted values are 0.97 and 0.98 at the surface of windward overhang (O_u) and windward side (Ru), respectively. However, the correlation coefficient drops to 0.73 at the leeward side (R_l). The correlation coefficients for the surface adjacent to the approaching flow (R_a), middle surface of the roof (R_b), and surface apart from the flow (R_c) are 0.88, 0.85, and 0.86, respectively. The possible reason may be that the block pressures for the leeward side are complex and uncertain due to the effects of the ridges when the wind direction is perpendicular to the ridge.

The quality coefficients (QC) of the pressure coefficients for the surface of windward side (R_u) and the surface adjacent to the approaching flow (R_a) are smaller than 5%, and those for the rest of the surfaces are smaller than 10%. It demonstrates that the fitting formulas have high accuracy.

Fitting formulas of worst negative pressure coefficients for building envelope

The fitting formulas of the worst negative pressure coefficient on the roof for the building envelope are shown below.

Zone 1

(15)C_{p,\min }(\mathrm{1})=\mathrm{0.03}{p}_{\mathrm{1}}{}^{\mathrm{2}}+\mathrm{1.65}{p}_{\mathrm{2}}{}^2-\mathrm{0.62}{p}_{\mathrm{3}}{}^2-\mathrm{2.7}3p_{\mathrm{2}}+\mathrm{1.83}{p}_{\mathrm{3}}-2\mathrm{.34}.

Zone 2

(16)C_{p,\min }(\mathrm{2})=-\mathrm{0.63}{p}_{\mathrm{1}}{}^{\mathrm{3}}+\sqrt{p_{\mathrm{1}}}+\mathrm{0.0}9p_{\mathrm{3}}/p_{\mathrm{2}}+\mathrm{0.3}\tan p_{\mathrm{1}}-3\mathrm{.1}.

Zone 3

(17)C_{p,\min }\mathrm{(3)=}-\mathrm{19.6}{p}_{\mathrm{1}}{}^4+\mathrm{30.5}{p}_{\mathrm{1}}{}^3-\mathrm{6.5}{p}_{\mathrm{1}}{}^2+\mathrm{0.}\mathrm{187}{p}_{\mathrm{2}}{p}_{\mathrm{3}}-\mathrm{4.9}{p}_{\mathrm{1}}-\mathrm{2.}

Zone 4

(18)C_{p,\min }(\mathrm{4})=-\mathrm{0.13}{p}_{\mathrm{1}}{}^2-\mathrm{0.}\mathrm{13}{p}_{\mathrm{3}}{}^2+\mathrm{2.32}p\mathrm{1}-\mathrm{0.008}{p}_{\mathrm{3}}{}^{-2}{p}_{\mathrm{2}}{}^{-2}-\mathrm{4.4.}

Zone 5

(19)C_{p,\min }(\mathrm{5})=-\mathrm{21}{p}_{\mathrm{1}}{}^{\mathrm{3}}+\mathrm{38.2}{p}_{\mathrm{1}}{}^2-\mathrm{17.7}{p}_{\mathrm{1}}+\mathrm{0.12}{p}_{\mathrm{2}}{p}_{\mathrm{3}}{}^2-0\mathrm{.4}{p}_{\mathrm{3}}-\mathrm{2.4.}

Zone 6

(20)C_{p,\min }(\mathrm{6})=-\mathrm{0.}\mathrm{145}{p}_{\mathrm{3}}{}^2+\mathrm{1.42}{p}_{\mathrm{1}}+\mathrm{0.0}2p_{\mathrm{2}}{}^{-2}-\mathrm{0.}\mathrm{21}{p}_{\mathrm{3}}{}^{-2}-4\mathrm{.68}.

The definitions of each parameter in the abovementioned formulas are the same with those in Section 5.1. It is reasonable for the fitting formulas of Eq. (16) include tan function, as the values of p₁ range from 0 to 2p/3. The accuracy analysis of the fitting formula of each zone is illustrated in Fig. 10. The correlation coefficient (r) and the quality coefficient (QC) for the worst negative pressure coefficients of the roof are listed in Table 3. Figure 10 shows that the deviations of the fitted values and the test values for the worst negative pressure coefficients are larger than that of the block pressure coefficients. From Table 3, it displays that the correlation coefficients (r) for the fitting formulas of six Zones are 0.72, 0.86, 0.89, 0.93, 0.92, and 0.86, respectively. The quality coefficients (QC) for Zone 1, Zone 4, Zone 5, and Zone 6 are smaller than 10%, whereas, the quality coefficients for Zone 2 and Zone 3 are slightly larger than 10%. The correlation coefficient and the quality coefficient for the central of the roof (Zone 1) are obvious smaller than those of the rest zones. The possible reason may be that Zone 1 is in the central area of the roof, the diversity of the worst negative pressure coefficients on this zone is less than that of on the rest zones.

Comparison with the specifications of different codes

The fitted values in Section 5 are compared with the American load specification [²²], the Japanese load code [²¹], and the Chinese load code [²⁵]. As the definitions of pressure coefficients (relevant reference wind speed, averaging times, etc.) differ widely in different codes, the pressure coefficients in American load specification [²²] and Japanese load code [²¹] are converted to those in the Chinese load code [²⁵] according to the method of St. Pierre et al. [³²].

The corresponding comparison results of various test cases of block pressure coefficients on different roof surfaces are illustrated in Fig. 11. It shows that the negative pressures (suctions) on the leeward side of the roof in this study are larger than that of other codes, which is probably because the test models in this study include the overhangs. The rest of the block pressure coefficients on each surface are all within the range of the three codes.

Figure 12 displays the comparison of worst negative pressure coefficients on different roof zones. The worst negative pressure coefficients of the building envelope in Chinese load code [²⁵] are in most cases smaller than those in the American load specification [²²], the Japanese load code [²¹], and the fitted results in this study. The fitted results in this study range between the values of the American load specifications and the Japanese load code. It implies that the proposed values in Chinese load code are relatively unsafe.

Proposals of pressure coefficients for wind load code

The provisions about the wind pressure coefficient on roof for the main structure and the building envelope in Chinese load code [²⁵] are quite simple, which can not cover the real situations of low-rise buildings in China.

Eqs. (7)–(12) provide the block pressure coefficients on roof surface for the main structure. The mean values of pressure coefficients of similar cases are calculated to utilize as the proposed values of the block pressure coefficients as references for the future’s modification of wind load codes, which are listed in Tables 4–7.

Table 4 shows that the roof pitch has great effect on the block pressure coefficient on overhang for main structure. The negative block pressure coefficients are decreased from -2.0 at the roof pitch of 0°–10° to a positive block pressure coefficient of 0.1 at roof pitch of 60°.

In Tables 5‒7, it is evident that the block pressure coefficients on roof for main structure have been less affected by the building size than by the roof pitch. The differences of block pressure coefficients on roof with various aspect ratios are only about 0.1. For the effect of the roof pitch, the block pressure coefficients on windward roof decrease from -0.8 at roof pitch of 14° to -0.15 at roof pitch of 35°, while the pressure coefficients on leeward roof are at the range of -0.7--0.9, for the wind direction perpendicular to the ridge (W₁). The block pressure coefficients on windward roof at the region of Ra, Rb and Rc range from -0.5--0.9 for the wind direction parallel to the ridge (W₂).

Eqs. (15)–(20) provide the worst negative pressure coefficients on each roof zone for the building envelope. The envelopment values of worst negative pressure coefficients of similar cases are proposed as references for the future’s modification of wind load codes, which are listed in Table 8. It shows that the roof pitch has little impact on the worst negative pressure coefficients for building envelope in the middle area of the roof (Zone 1, Zone 2, and Zone 3). However, the worst negative pressure coefficients vary a lot in the edge and corner (Zone 4, Zone 5, and Zone 6). The most unfavorable worst negative pressure coefficients are -6.2 and -5.8 on Zone 5, occurring at the roof pitch of 20° and 25°, respectively.

Conclusions

This study examined the wind pressure test results of 99 gable-roofed low-rise buildings with overhangs that consist of 11 roof pitches, 3 height-depth ratios, and 3 width-depth ratios. The block pressure coefficient on each roof surface for main structure and the worst negative pressure coefficient on each roof region for building envelope were calculated through a series of wind tunnel tests. The conclusions are as follows:

1) The roof pitch has great effect on the block pressure coefficient on overhang and windward roof for main structure. The negative block pressure coefficients on overhang are decreased from -2.0 at the roof pitch of 0° ~ 10° to a positive block pressure coefficient of 0.1 at the roof pitch of 60°. The block pressure coefficients on windward roof are decrease from -0.8 at roof pitch of 14° to -0.15 at roof pitch of 35° for the wind direction perpendicular to the ridge (W₁). Meanwhile, the building size has less influence on the block pressure coefficients on roof, it only have a difference of about 0.1 with various aspect ratios.

2) The roof pitch has little impact on the worst negative pressure coefficients for building envelope in the middle area of the roof (Zones 1, 2, 3). However, the worst negative pressure coefficients vary a lot in the edge and corner (Zones 4, 5, 6). The most unfavorable worst negative pressure coefficients are -6.2 and -5.8 on Zone 5, occurring at the roof pitch of 20° and 25°, respectively.

3) The block pressure coefficient for main structure and the worst negative pressure coefficient for building envelope are recommended for the roofs of low-rise buildings based on the test values and different countries’ codes, which can be used as references for the future’s modification of Chinese wind load codes.

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