Steady-state and transient wind characteristics of low-rise building roofs with openings in vulnerable areas

Abstract

A strong wind would cause roof openings on a low-rise building and bring further structural damage. Previous related studies focused on different shapes of openings on flat roofs. Little study has been done on the sloped roofs with openings in vulnerable roof areas. In this study, wind tunnel tests were carried out to investigate the steady-state and transient internal pressure characteristics due to opening at the vulnerable roof areas of a low-rise building. The tests considered both steady-state and transient openings of roof. The experimental results indicated that the steady-state internal pressure distribution tends to be uniform and that the internal pressure induced by leeward roof ridge opening is obviously lower than that induced by the windward one. The fluctuation effect around the orifice area is apparent with the skewed wind direction and the combined effect of internal and external pressures on the unopened roof side is significantly smaller than that on the opened side. Current design provision of China for internal pressure evaluation is found to be unconservative. The transient overshoot is closely related to the opening location in the vulnerable roof area and is more pronounced when the opening is on the leeward side. Among the internal pressure coefficients commonly adopted in design, the extreme net wind pressure coefficient is most important, which is affected most at the orifice near eaves, roof corners and tails, and leeward roof ridge.

Keywords

extreme net wind pressure low-rise building roof opening transient overshoot wind-induced internal pressure

Introduction

Wind-induced internal pressure is generally caused by the damage of doors, windows, and roof of a low-rise building in windstorm condition. Eaves, roof ridges, and roof corners are deemed as vulnerable areas for gable-roofed low-rise buildings according to previous investigation. Once a damage occurs in these vulnerable areas, the interior of building will be exposed to strong winds resulting in the combination action of internal and external pressure on the roof, which may exacerbate the whole failure of the cladding. The joint action of internal and external pressure on roof due to dominant opening on building envelope has been generally noticed.

Extensive studies have been done about response characteristic of internal pressure induced by wall dominant opening and different nonlinear governing equations applied to predicted internal pressure (Chen et al., 2012; Guha et al., 2012; Holmes and Ginger, 2012; Tecle, 2011; Oh et al., 2007; Quan et al., 2012a, 2012b). The internal pressure response characteristic mainly depend on external excitation around the opening, and other influence factors including opening size ratio, opening geometries and location, background leakage, roof flexibility, and building volume, which is verified by many researchers (Guha et al., 2011; Kopp et al., 2008). Ginger et al. (1997, 2008, 2010) conducted extensive internal pressure studies by means of wind tunnel experiments, field measurement, and theoretical analysis. The test results showed that theoretical equation could well estimate fluctuating internal pressure response with different dominant opening sizes and building volumes. Helmholtz resonance of internal pressure driven by upstream turbulent flow enlarges fluctuating internal pressure obviously. The internal pressure fluctuation will be magnified in large magnitude if the frequency of vortex-shedding-induced resonance determined by openings characteristic matches to Helmholtz frequency. Sharma and Richards (2003) compared Helmholtz resonance excited by oblique wind flows with that produced by normal onset flow, which indicated that tangential flows are more inclined to produce stronger internal pressure response. Similar results were also concluded by Guha et al. (2012). It has been noticed that roofs of low-rise building and large-span structure are wind-sensitive component, sustaining extremely large suctions in severe wind events, which is likely damaged to produce openings. However, wind-induced internal pressure due to roof local failure has been less studied. Wang and Li (2015) studied the characteristics of internal and external pressure, as well as their interaction caused by different openings’ geometries on roof corner of a flat roof low-rise building. Xu and Lou (2018) analyzed the internal pressure characteristics induced by opening on a hemi-ellipsoidal roof and compared with the situation of opening on the wall. They concluded that internal suctions due to roof failure may cause more serious cladding damage than door or window failure. Besides the understanding of internal pressure response characteristic by means of experimental investigation and theoretical analysis, what is more concerned in engineering is the design loading, such as net loading fluctuating characteristic caused by joint action of external and internal pressure (Ginger and Letchford, 1999; Sharma and Richards, 2005). Recently, Habte et al. (2017) studied frame forces influenced by wind-induced internal pressure using a low-rise building model with multiple opening on the wall. They found that the frames near the opening sustain larger forces, and that ASCE 7–10 (American Society of Civil Engineers (ASCE), 2010) under-estimated frame forces compared with experimental results.

Among the damage types of low-rise buildings, the sudden collapse of roof is most serious. What researchers focused on transient internal pressure study concerned about was whether or not the sudden increasing in internal pressure may exceed steady-state internal pressure peaks (Stathopoulos, 1989). With a transient opening, the internal pressure of building would increase suddenly and induces the transient overshooting effect, which is first verified analytically by Liu and Saathoff (1981). Sharma and Richards (1997) compared the transient response of a building with sudden openings using the results of computational solutions and model-scale wind tunnel experiment. They verified the validation of the proposed computer model in predicting Helmholtz frequency and analyzing discharge characteristics of flows passing through openings in different wind field conditions. Similar works were also conducted by Guha et al. (2013). Tecle et al. (2013) used solenoid valves to simulate the abrupt openings on doors and windows and obtained lower peak transient internal pressures than subsequent steady-state peak pressure at varying wind directions and opening sizes. Duan et al. (2012) simulated the abrupt opening process of doors and windows and studied the effects of opening ratio and background porosity factor on peak transient internal pressure.

Few previous studies focused on wind-induced internal pressure caused by roof local failure. In view of this, extensive wind tunnel tests were carried out to study the wind load characteristic caused by openings on different vulnerable roof areas of a low-rise building. The internal pressure characteristics of roof, the correlation between internal and external pressures, and the transient overshoot effect considering different opening locations and terrains were investigated. The wind load characteristics of external and internal pressures under different opening conditions were also analyzed.

Experimental details

The test model is a gable roof of low-rise building with a roof slope of 18.4°. The model scale is 1:40 with the geometric size of 300 mm (length) × 200 mm (width) × 233.3 mm (ridge height) and internal dimension of 280 mm × 180 mm × 213 mm. The blocking rate is less than 5%. The most unfavorable roof pressure condition, that is, unopened roof (Peng, 2016), was assumed to determine the opening location, shape, and ratio of the test model as summarized in Table 1. The opening area ratio was defined as opening area size to the whole roof size. The model is made of double-layer plexiglass to meet the stiffness requirement. The measuring points are evenly arranged on the inner and outer surfaces of the double layer. Figure 1(a) shows the measuring point arrangement for opening Cases 1, 2, and 3. In Cases 4 and 5, cover plates were adopted to change opening shapes based on Case 3. Numerical compensation was employed to correct the tubing effects before data processing to ensure actual internal pressure fluctuation (Li et al., 2019).

Table 1.

Test conditions at roof openings.

Cases	Opening location	Opening shapeand dimensions(mm)	Openingarearatio (%)
Case 1	Windward corner square		5.7
Case 2	Leeward roof L-shaped		5.7
Case 3	Windward corner L-shaped	As above	5.7
Case 4	Windward corner gable		2.8
Case 5	Windward corner eaves		4.3

Figure 1.

Model schematic: schematic diagram of (a) measuring points and (b) wind directions.

According to the similarity formula (equation (1)) proposed by Holmes (1980), the internal pressure test requires volume expansion. Based on the geometrical dimensions of the model, the internal pressure fluctuation characteristics were correctly simulated by connecting the cavities at the bottom openings (as illustrated in Figure 2(b)). The after-compensation building volume is approximately 0.043 m³ (corresponding full-scale internal volume of 633 m³)

\frac{V_{m}}{V_{p}} = \frac{{(l_{m} / l_{f})}^{3}}{{(U_{m} / U_{f})}^{2}}

(1)

In this equation, $V_{m}$ , $l_{m}$ , $U_{m}$ were internal volume and geometric length of the model, and reference wind speed, respectively. $V_{f}$ , $l_{f}$ , $U_{f}$ were internal volume and geometric length of the prototype building, and reference wind speed in practice boundary layer, respectively.

Figure 2.

Model layout and volume compensation illustration: (a) test model and (b) schematic diagram of internal volume compensation.

In this study, wind tunnel tests were carried out at the Wind Engineering Research Center of Hunan University of Science and Technology (China). The cross section of the wind tunnel is 4.0 m width × 3.0 m height with test section length of 21 m. A pressure test was also performed using the three-dimensional fluctuating anemometer (TFI) and pressure scanning valve system (PSI). The sampling frequency in the tests was 333.2 Hz. Varying wind directions/angles (θ) between 0° and 360° with the intervals of 10°, plus 45°, 135°, 225°, and 315°, were considered in the data acquisition for steady-state internal pressure (Figure 1(b)). According to the extreme pressure distribution of unopened roof, θ = 30° was found to represent the most unfavorable condition. To study the transient internal pressure characteristics, extensive transient opening tests were carried out using a designed device (Figure 3) at wind direction of 30°. Figure 2(a) shows the layout of the test model. According to the Chinese Code for Building Loads (GB 50009-2012, 2012), wind field of terrain categories A, B, and C with a scale ratio of 1:40 were determined for simulation, respectively. The empirical formula for calculating wind speed and turbulence intensity are $U (z) = U_{r} (Z / Z_{r})^{α}$ and $I_{u} (z) = I_{10} (Z / 10)^{- α}$ , respectively, herein, $Z_{r}$ is reference height (i.e. ridge height, $Z_{r} = 233.3 mm$ ), $U_{r}$ and $I_{10}$ are reference wind speed and turbulence intensity, α = 0.12, 0.14, and 0.23 for terrain categories A, B, and C, respectively. The experimental mean wind velocity and turbulence intensity at reference height are 10.7 m/s and 12.5%, 10.8 m/s and 14.9%, and 8.3 m/s and 26.4% for the three terrain categories’ wind field, respectively. The simulated wind field profiles are shown in Figure 4. Figure 5 shows wind speed spectrum at ridge height. In these figures, n denotes frequency, $S_{v} (n)$ is spectrum density, $σ^{2}$ is the variance of fluctuating wind velocity, z is characteristic dimension of building (z = 0.2), and u is longitudinal wind velocity at reference height. The turbulence integral scale of three categories wind field is 2.9, 2.0, and 0.79, respectively. The Reynolds number based on roof ridge height and wind speed at this height is found to be 1.66 × 10⁵, 1.68 × 10⁵, and 1.29 × 10⁵.

Figure 3.

Transient opening device illustration.

Figure 4.

Wind speed and turbulence profile: (a) terrain category A, (b) terrain category B, and (c) terrain category C.

Figure 5.

Experimental and theoretical spectra of approach wind velocity at 233.3 mm elevation: (a) terrain category A, (b) terrain category B, and (c) terrain category C.

Steady-state openings on vulnerable roof areas

Wind-induced internal pressure

Five different roof opening configurations with 0° wind direction (i.e. perpendicular to the longer building side; Figure 1(b)), as shown in Figure 6, were considered to study the internal pressure characteristics due to failure of vulnerable roof areas.

Figure 6.

Distribution of the internal pressure coefficients at 0° wind direction: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, and (e) Case 5.

The distribution of wind-induced internal pressures depends on the location of opening. In each opening configuration, the distribution of internal pressures is relatively uniform with no obvious high-pressure area (Figure 6). The internal pressure coefficients at windward openings, namely, Cases 1, 3, and 5, are approximately equal (i.e. −0.6, −0.53, and −0.58, respectively) (Figure 6(a), (c), and (e)). This is due to the similar roof orifice lengths and opening ratios. The internal pressure for Case 4 having the smallest orifice length and smallest opening ratio is the smallest at −0.48. Clearly, the wind-induced internal pressure on the vulnerable roof area is related to the orifice length on the windward side. With the leeward roof opened, the internal pressure is −0.3, which is obviously smaller in magnitude than that at a windward opening (i.e. Case 1, 3, 4, or 5). The air flow enters directly into the building interior through the windward roof opening, while it enters the building interior in form of vortex through the leeward roof ridge opening. As such, the leeward opening has a weakening effect on the internal pressure of roof.

Amplification of fluctuating internal pressure

The measuring points at the orifice are numbered from the roof gable to eave as E1–E6 (Figure 1(a)). The fluctuation characteristics of steady-state internal pressure on a vulnerable roof area were studied and expressed using the fluctuating internal pressure amplification factor (FIPAF) $γ_{C_{pi} / C_{pw}}$ defined as

γ_{C_{pi} / C_{pw}} = \frac{σ_{C_{pi}}}{σ_{C_{pw}}}

(2)

where $σ_{C_{pw}}$ is the root mean square of external pressures and $σ_{C_{pi}}$ is the root mean square of internal pressures. The calculated FIPAFs for six typical wind angles of 0°, 45°, 90°, 120°, 135°, and 180° are shown in Figure 7.

Figure 7.

Fluctuating internal pressure amplification factors (FIPAFs) at typical wind directions: (a) measurement point E1, (b) measurement point E2, (c) measurement point E3, (d) measurement point E4, (e) measurement point E5, and (f) measurement point E6.

At location E1 being near the roof gable area (Figure 1(a)), the variation of the FIPAFs of windward roof openings is insignificant, while the FIPAFs gradually decrease on the leeward openings with θ (Figure 7(a)). At location E2 (near the ridge), the FIPAFs gradually decrease with θ in general, but jump to 1.31 suddenly with θ = 135° at the leeward opening as the length of contact between the incoming flow and the orifice is the smallest with this wind angle. At E3 (corner of the orifice), the FIPAFs vary from 0.44 to 1.19 under Case 2, while the FIPAFs for the other opening conditions change slightly. At E4 and E5 (middle of the roof), Cases 1, 3, and 5 have a small variation of FIPAFs, while Case 2 reaches a peak FIPAF at θ = 135° and Case 4 gradually decreases FIPAFs after reaching the peak value at θ = 45°. At E6 (near the roof eave), all five opening conditions show a gradual increase in FIPAFs, with the leeward opening having significantly larger values than the windward one. Obviously, this variation of FIPAFs is related to the distance between the orifice and the incoming flow under different wind directions.

In addition, when the windward roof is opened, the FIPAF at measurement points E3, E4, and E5 (middle of the roof) is greater than 1 at θ = 45° and less than 1 at other locations and wind directions. When the leeward roof is opened, except the gable area, the FIPAF at θ≈ 135° exceeds 1. This indicates that the fluctuation of internal pressures greatly influences the roof. In designing a low-rise building, the local fluctuation effect at the orifice under a skewed wind direction deserves an attention.

Correlation of internal and external pressures

The combined effect of internal and external pressures can be assessed using a correlation coefficient $ρ_{ei}$ defined as

ρ_{ei} = \frac{E [C_{P_{e}} (t) C_{P_{i}} (t)] - {\bar{C}}_{P_{e}} {\bar{C}}_{P_{i}}}{{\tilde{C}}_{P_{e}} {\tilde{C}}_{P_{i}}}

(3)

where $C_{P_{e}} (t)$ and $C_{P_{i}} (t)$ are, respectively, the external pressure and internal pressure coefficients, ${\bar{C}}_{P_{e}}$ and ${\tilde{C}}_{P_{e}}$ are, respectively, the mean and the root mean square wind pressure coefficients of external pressure, and ${\bar{C}}_{P_{i}}$ and ${\tilde{C}}_{P_{i}}$ are, respectively, the mean and the root mean square wind pressure coefficients of internal pressure. Figure 8 shows the spatial distributions of $ρ_{ei}$ on the roof under the five opening conditions.

Figure 8.

Distributions of correlation coefficients between the internal and external pressures of roof $(ρ_{ei})$ : (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, and (e) Case 5.

$ρ_{ei}$ for entire roof of varying opening conditions is ranged 0.1–0.8 positively, combined with the same sign of external and internal pressure, indicating a mutual offsetting effect between the internal and external pressures on building roof. In addition, the $ρ_{ei}$ along an opened side is obviously larger than that on an unopened side, especially at the orifice edge up to 0.5–0.8, which indicated that the unopened side is more affected by the combination action of internal and external wind pressures due to a weak offsetting effect, representing an unfavorable condition for low-rise building roof. For the windward roof opening conditions (Figure 8(a), (c), (d), and (e)), the $ρ_{ei}$ in the roof tail region on the opened side and in the tail region near the ridge area on the unopened side are significantly smaller than other areas. With the leeward roof opened, the smaller $ρ_{ei}$ values appear in the front and tail areas on the unopened side. This indicates that $ρ_{ei}$ values tend to decrease with the increasing distance from the orifice due to the decreasing fluctuating external pressure. In other words, these areas are more adversely affected by the combined effect of internal and external pressures, thus also deserving an attention.

Comparisons with GB 50009-2012 provisions

The Chinese Code for Building Loads (GB 50009-2012, 2012) only simply stipulates the average internal pressure coefficient $({\bar{C}}_{P_{i}})$ of a building envelope as ±0.2 for an full enclosed building, 0.4 times the external pressure at the orifice $(P_{e, o})$ with the opening area ratio ( $γ$ , defined as the ratio of opening area size to the whole wall area size) = 2%–10%, 0.6 times $P_{e, o}$ with $γ = 10 % - 30 %$ ; and 0.6 times $P_{e, o}$ with $γ > 30 %$ . This provision is applicable to buildings with dominant wall openings. There is no code specification for non-dominant roof openings. This article discusses the differences between the experimental and provision-determined ${\bar{C}}_{P_{i}}$ values for the five roof opening configurations and further makes some reasonable suggestions. Representative ${\bar{C}}_{P_{i}}$ results at 0° wind direction are shown in Table 2.

Table 2.

Comparison of test and code-determined ${\bar{C}}_{P_{i}}$ values of openings.

Orifice measuring point no. ${\bar{C}}_{P_{i}}$		1	2	3	4	5	6
Case 1	Code	−0.214	−0.214	−0.368	−0.267	−0.247	−0.198
	Test	−0.626	−0.586	−0.617	−0.595	−0.635	−0.588
	Error (%)	65.740	63.530	40.317	55.146	61.092	66.261
Case 2	Code	−0.132	−0.118	−0.128	−0.127	−0.132	−0.155
	Test	−0.301	−0.309	−0.317	−0.311	−0.306	−0.306
	Error (%)	56.189	61.648	59.554	59.122	56.785	49.444
Case 3	Code	−0.153	−0.171	−0.218	−0.228	−0.213	−0.280
	Test	−0.549	−0.527	−0.537	−0.520	−0.506	−0.508
	Error (%)	72.241	67.518	59.351	56.070	57.838	44.877
Case 4	Code	−0.206	−0.199	−0.208	−0.193	−0.215	−0.217
	Test	−0.521	−0.500	−0.502	−0.480	−0.482	−0.463
	Error (%)	60.466	60.190	58.634	59.875	55.382	53.188
Case 5	Code	−0.234	−0.195	−0.238	−0.232	−0.307	−0.256
	Test	−0.584	−0.571	−0.588	−0.588	−0.575	−0.591
	Error (%)	59.955	65.889	59.443	60.611	46.689	56.722

As indicated in Table 2, the code ${\bar{C}}_{P_{i}}$ values are generally much lower than the test ones, depending on the locations of opening. The difference between the two sets of results is 49.4%–61.6% for leeward openings (Case 2, Table 2), while for windward openings, except those at the orifice, the average difference is approximately 50% (range = 40.3%–72.2%). In addition, from the viewpoint of the locations of opening measuring points, the corner of orifice of windward roof openings and the orifice near the gable of leeward roof openings have less error in code-calculated ${\bar{C}}_{P_{i}}$ values. Therefore, the accuracy of code-calculated ${\bar{C}}_{P_{i}}$ values depends on whether or not the air flow can directly pass through the orifice.

The design code (GB 50009-2012, 2012) also defines the ratio of internal pressure to external pressure. The scattered ratios at the orifice points of the five opening conditions under the full range of wind angles (θ = 0°–360° with an interval of 10°, plus 45°, 135°, 225°, and 315°, totally 41 wind angles, Figure 1(b)) are shown in Figure 9. As seen, the ratios range from 0.35 to 1.71 with the majority clustered between 0.7 and 1.0. Therefore, when the low-rise building has only one dominant opening on the roof and the opening ratio is 0.02–0.1, the internal pressure may be taken as 0.7–1.0 times the external pressure coefficient of the orifice, which is much larger than the specified value in provisions.

Figure 9.

Scatter plot of internal-to-external pressure ratios with full wind angles.

Transient wind characteristics of openings in vulnerable roof areas

Effect of opening location on transient overshoot

When the vulnerable sheathings of roof are turned over abruptly by windstorms, the internal pressure increases instantaneously due to the transient wind pressure. The transient overshoot ratio R is used to express the instantaneous increasing phenomenon of peak internal pressure, which is defined as

R = \frac{{\hat{C}}_{pi, i}}{{\hat{C}}_{pi, s}}

(4)

{\hat{C}}_{pi, s} = {\bar{C}}_{pi} + g {\tilde{C}}_{pi}

(5)

where ${\hat{C}}_{pi, i}$ is the peak transient internal pressure determined from the internal pressure time history curve (Figure 10), ${\hat{C}}_{pi, s}$ is the peak steady-state internal pressure, ${\bar{C}}_{pi}$ is the average steady-state internal pressure, ${\tilde{C}}_{pi}$ is the steady-state fluctuating internal pressure, and g is a peak factor taken as 3.2 because of the Gaussian characteristic of internal pressure (Wang and Li, 2015). In Figure 10, t₀ controlled below 0.2 s denotes duration time from starting opening to internal pressure reaching stabilization. Taking the shorter roof side as the X-axis, the windward longer roof side as the Y1-axis, and the leeward longer roof side as the Y2-axis (Figure 11), R distributions along the three directions are shown in Figure 12.

Figure 10.

Time series of the internal pressure coefficient during opening process.

Figure 11.

Axis direction diagram.

Figure 12.

Roof overshoot ratio distributions in the three directions: (a) X-axis, (b) Y1-axis, and (c) Y2-axis.

As seen, the variation of overshoot ratios along the X-axis depends on opening locations (Figure 12). For opening Case 1, R value of the windward roof virtually does not change with distance from orifice, while that of the leeward side increases gradually. For opening Case 4 and Case 5 (having relatively small opening area sizes), R value decreases with the increasing distance from the eave and increases gradually toward the leeward roof. For opening Case 3, the variation trend of R values is completely opposite and the distribution of R values is separated by the windward and leeward roofs where R increases first and then decreases. When the leeward roof ridge is opened (opening Case 2), R is much larger than other opening cases along X-axis and Y-axis and increases with the increasing distance from the eave. This indicates that leeward roof ridge failure may cause further transient damage on roof. Observing the distribution of R values along the Y1-axis (Figure 12), with the increasing distance from the gable, R values increase gradually when the windward roof corner is opened and gradually stabilizes near the tail of the roof. When the leeward roof ridge is opened, R gradually reduces. According to the distribution of R values along the Y2-axis (Figure 12), when the windward roof corner is opened, R steadily increases from the roof front to tail area. When the leeward roof ridge is opened, R is smaller at the orifice and becomes more uniform at other locations.

Effect of terrain category on transient overshoot

Uniform flow field and Terrain Categories A, B, and C (wind field profiles as shown in Figures 4 and 5) were considered to study the transient peak internal pressure characteristics of openings on vulnerable roof areas. The corresponding turbulence intensity of the four wind fields are 3.5%, 11.7%, 14.2%, and 24.2%, respectively (tested at reference height, i.e., roof ridge height). The transient overshoot ratios (R) of the measured points under opening Case 1, Case 2, and Case 3 are weighted with area to evaluate the action of transient overshoot effect to the entire roof, as listed in Table 3.

Table 3.

Transient overshoot ratios under different terrain categories.

Terrain category(turbulence)	Opening case	Overshoot ratio (R)
Uniform flow field (3.5%)	1	1.31
	2	1.42
	3	1.21
Category A (11.7%)	1	1.37
	2	1.66
	3	1.32
Category B (14.2%)	1	1.48
	2	1.75
	3	1.44
Category C (24.2%)	1	1.59
	2	1.96
	3	1.56

R of the openings in varying vulnerable roof areas increases with the increasing of upstream turbulence. The difference of R values is not significant between Case 1 and Case 3 for windward roof openings, while it is obviously larger for leeward roof openings.

To quantify the correlation, only Case 1 and Case 2 were considered since the transient overshoot ratios of Case 3 are similar to that of Case 1. The following two correlation formulas were derived:

For transient overshoot ratios on the windward roof

R_{1} = 0.01434 I_{u} + 1.2462

(6)

For transient overshoot ratios on the leeward roof

R_{2} = 0.02597 I_{u} + 1.3482

(7)

where I_u is the turbulence of upstream flow in varying terrain categories.

The precision of equations (6) and (7) is indicated in Table 4, showing good correlation particularly for leeward roof openings (Case 2). Therefore, equations (6) and (7) may be used to estimate the peak transient internal pressure.

Table 4.

Precision of the correlation formulas—equations (5) and (6).

Cases	Sum of residuals	Correlation coefficient
Case 1	0.00267	0.91545
Case 2	0.00186	0.98125

Comparison of internal pressure coefficients

In wind load calculations of low-rise buildings with roof openings, the following three internal pressure coefficients have been frequently used (Yu et al., 2007): the steady-state internal pressure coefficient, the transient internal pressure coefficient for sudden openings, and the extreme net wind pressure coefficient under the combined effect of internal and external pressures, calculated by

C_{net} = {\bar{C}}_{pe} - {\bar{C}}_{pi} \pm {\hat{C}}_{pei}

(8)

{\hat{C}}_{pei} = \sqrt{g_{e}^{2} {\tilde{C}}_{pe}^{2} + g_{i}^{2} {\tilde{C}}_{pi}^{2} - 2 ρ g_{e} g_{i} {\tilde{C}}_{pe} {\tilde{C}}_{pi}}

(9)

where ${\bar{C}}_{pe}$ is the mean pressure coefficients of external, ${\bar{C}}_{pi}$ is the mean pressure coefficient of internal pressures, ${\hat{C}}_{pei}$ is the net wind pressure fluctuation, ${\tilde{C}}_{pe}$ is the root mean square (RMS) pressure coefficient, and $g_{e}$ is the peak factor. The following measuring points were selected: the orifice (E1), middle (E13), and tail (E28) on the opened roof side, and the top (F1), middle (F18), and tail (F35) on the unopened roof side (Figure 1(a)). They are numbered as 1–6, respectively. The three internal pressure coefficients under the five opening conditions are listed in Table 5 for studying the variation of the coefficients at different roof locations.

Table 5.

Comparison of the three internal pressure coefficients at different roof locations.

Location no.		1	2	3	4	5	6
Case 1	Steady-state	−1.003	−1.024	−1.001	−0.978	−0.983	−1.023
	Transient	−1.019	−1.044	−1.065	−1.006	−1.054	−1.001
	Combined effect	−0.873	−0.451	−1.176	−1.649	−0.677	−0.651
Case 2	Steady-state	−0.875	−0.887	−0.874	−0.862	−0.872	−0.889
	Transient	−1.203	−1.245	−1.211	−1.234	−1.227	−1.188
	Combined effect	−0.229	−0.454	−0.572	−1.927	−1.033	−0.641
Case 3	Steady-state	−1.038	−1.026	−1.040	−1.026	−1.044	−1.028
	Transient	−1.276	−1.301	−1.321	−1.272	−1.280	−1.279
	Combined effect	−0.356	−0.649	−0.515	−1.775	−0.437	−0.537
Case 4	Steady-state	−0.879	−0.886	−0.918	−0.882	−0.903	−0.903
	Transient	−0.993	−1.020	−1.015	−0.961	−1.007	−1.028
	Combined effect	−0.804	−0.971	−0.678	−2.076	−0.625	−0.710
Case 5	Steady-state	−1.115	−1.121	−1.123	−1.116	−1.132	−1.138
	Transient	−1.314	−1.326	−1.361	−1.327	−1.334	−1.342
	Combined effect	−1.027	−0.628	−1.130	−1.655	−0.904	−0.661

As indicated in Table 5, with the change of the location of roof measuring point, the steady-state and transient internal pressure coefficients vary slightly, while the change of extreme net wind pressure coefficient is directly related to the location of measuring point. As can be seen from the table, when the windward side is opened, the extreme net wind pressure coefficient is larger than the maximum steady-state internal pressure coefficient at the orifice edge or top of unopened side. It is also greater than the transient internal pressure coefficient at the upper corner of unopened side as the perpendicular distance there to the incoming flow is the shortest with the transient opening oriented at 30° with respect to short side of the roof (Figure 1(b)), thus subjected to the intense combined effect of internal and external pressures. When the leeward roof ridge is opened, the extreme net pressure coefficients at the top and middle of unopened side are larger than the maximum steady-state internal pressure coefficients and the combined effect in the upper portion of unopened side is significant, alerting a special attention when designing a low-rise building.

In addition, the maximum value of extreme net pressure coefficients is much larger than the counterpart of steady-state or transient internal pressure coefficient for each opening condition, especially Case 4 (having the smallest opening ratio) which has the value of −2. This indicates that the extreme net pressure is significantly affected by the combined action of internal and external pressures. If considering the steady-state internal pressure only, the extreme net wind pressure will be smaller than the transient internal pressure and the induced error would be as high as 135.4%. Therefore, in designing an envelope structure, the influence of transient opening and the combined effect of internal and external pressures after local roof sheathing failure should be considered better.

Study on extreme net wind pressure

Net wind pressure of building roofs received extensive concern (Dai et al., 2020). The distribution of maximum value of extreme net pressure coefficient in terms of wind direction/angle (θ) is shown in Figure 13 where the corresponding measuring points are shown in Figure 14. A comprehensive analysis considering all possible wind directions was made to find the pattern of the most unfavorable extreme net wind pressure coefficient under the combined action of internal and external pressures.

Figure 13.

Distribution of the maximum values of extreme net wind pressures coefficients.

Figure 14.

Measuring locations of the maximum of extreme net wind pressure.

It is found that the variation pattern is consistent for all opening conditions on the windward roof, while the trend for leeward roof openings is slightly different but remains consistent overall. For θ = 30°–140°, the extreme net wind pressure coefficient decreases first and then increases. The minimum coefficient occurs at θ≈80° for windward roof openings and θ≈100° for leeward roof openings, and the corresponding coefficient values are only approximately −0.6 and −0.65, respectively. This represents the most favorable wind blowing direction considering the combined effect of internal and external pressures. Withθ > 140°, the change of extreme net pressures stabilizes. The maximum coefficient appears at θ = 320° for windward roof openings (about −2.45) and atθ = 140° and 225° for leeward roof openings (about −2.1 and −2.15), representing the worst combined pressure effect for an envelope structure.

According to the distribution of measuring locations (Figure 14), when the windward roof is opened, the maximum values of extreme net wind pressures are mainly distributed around the orifices at eaves, the tail of windward roof, and the gable area of leeward roof. While the maximum values in leeward roof opening condition (Case 2) are mainly distributed at the corner area of windward roof and the eave area of leeward roof. Under the combined effect of internal and external pressures, these areas in a low-rise building are deemed vulnerable to have further serious damage.

Conclusion

The internal pressure characteristics of transient opening process and steady-state stage induced by failure of local vulnerable roof areas of low-rise buildings were studied. Extensive wind tunnel tests of varying opening configurations at vulnerable areas were carried out to investigate the characteristics of internal pressure distribution, the correlation between internal and external pressures, the transient overshoot effects, and the coefficient of extreme net wind pressure. The results of this study reveal the following major findings:

The distribution of internal pressures caused by local roof sheathing failure is directly related to the opening locations and relatively uniform for each opening condition considered. The leeward roof ridge opening significantly reduces the internal pressure of the roof. The wind-induced internal pressure fluctuation effect of low-rise building under skewed wind directions needs more attention.

The correlation coefficient between the internal and external pressures of an opening on a vulnerable roof area of low-rise building is positive, indicating an offsetting pressure effect. The $ρ_{ei}$ value of an opened roof side is significantly larger than that on an unopened side, especially at the orifice edge. The $ρ_{ei}$ value of varying areas is related to their distance from the orifice.

Current design code gives significantly lower internal pressure coefficients for an envelope structure than the test values and is thus not conservative. It is suggested that with only one dominant opening on the roof vulnerable area and the opening ratio γ = 2%–10%, the internal pressure coefficient can be taken as 0.7–1.0 times the external pressure coefficient of orifice.

The transient overshoot of low-rise building roofs is affected by the opening location and is more pronounced when the leeward roof ridge is opened. The overshoot ratio generally increases linearly with the turbulence. The derived fitting formulas (equations (5) and (6)) can be used to estimate the overshoot ratios of roof openings under different incoming flow turbulence with reasonable accuracy.

In designing a low-rise building envelope, among the commonly used internal pressure coefficients, the extreme net wind pressure coefficient considering the combined effect of internal and external pressures is most important. The extent of wind effects on different roof areas varies, but most apparent at the orifices of eaves, the corners of roof, and the ridge of leeward roof.

Footnotes

Acknowledgements

Mr Kui Guo and Miss Shu Jiang are acknowledged for their help in the wind tunnel tests and relevant studies.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by the National Nature Science Foundation of China (project nos 51578237 and 51708207), the Degree & Postgraduate Education Reform Project of Hunan Province (2019JG ZD063), Hunan Taught Information (2018, NO. 436, 1017), and Hunan Education Department of Science Research Project (19A168) to which the authors are very grateful.

ORCID iDs

Yangjin Yuan

Yi Li

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