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Wind loads on free-standing canopy roofs: Part 2 overall wind forces
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Journal of Wind Engineering and Industrial Aerodynamics 96 (2008) 1029–1042 www.elsevier.com/locate/jweia
Wind loads on free-standing canopy roofs: Part 2 overall wind forces Yasushi Uematsua,, Theodore Stathopoulosb, Eri Iizumic a New Industry Creation Hatchery Center, Tohoku University, Aoba-ku, Sendai 980-8579, Japan Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Canada H3G 1M8 c Department of Architecture and Building Science, Tohoku University, Aoba-ku, Sendai 980-8579, Japan
b
Available online 26 July 2007
Abstract Wind loads on free-standing canopy roofs have been studied in a wind tunnel. Three types of roof geometries, i.e. gable, troughed and mono-sloped roofs, with roof pitches ranging from 01 to 151, were tested. Wind pressures were measured simultaneously at many points both on the top and bottom surfaces of the roof model for various wind directions. The paper describes the characteristics of overall wind forces and moments acting on the roof with special attention to load combination. Correlation between wind force and moment coefficients is investigated and wind force coefficients for the design of main wind force resisting systems are proposed. The roof is assumed rigid and simply supported by four corner columns, whose axial forces appear as the most important load effect. Two loading patterns that cause the maximum tension and compression of columns are considered. The proposed values are also compared with the specifications of the Australian/New Zealand Standard [Standards Australia, 2002. Australian/New Zealand Standard, AS/NZS 1170.2]. r 2007 Elsevier Ltd. All rights reserved. Keywords: Free-standing canopy roof; Wind tunnel experiment; Overall wind force; Main wind force resisting system; Design wind load; Codification
1. Introduction The characteristics of local pressures on free-standing canopy roofs are described in a companion paper, Part 1 (Uematsu et al., 2007). This paper presents the experimental results on overall wind forces and moments acting on these roofs. Corresponding author. Tel./fax: +81 22 795 7875.
E-mail address: [email protected] (Y. Uematsu). 0167-6105/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2007.06.026
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Gumley (1984), Letchford and Ginger (1992) and Ginger and Letchford (1994) measured area-averaged pressures for various areas using a pneumatic averaging technique. Furthermore, Ginger and Letchford (1994) calculated the overall forces using the covariance integration technique. Holmes (2001) discusses the application of their results to load estimation. Recently, Letchford et al. (2000) measured the mean wind forces on solid and porous canopy roof models using a force balance technique. This is perhaps the first attempt to measure the overall forces on canopy roofs directly. More recently, Altman (2001) has made extensive measurements of the overall forces and moments using a high-frequency force balance developed at Clemson University, USA. In the present study, the wind forces and moments are computed from the time history of pressures measured simultaneously at many points both on the top and bottom surfaces. Such simultaneous pressure measurements leading to a reliable definition of realistic windinduced forces on canopy roofs have not been made so far, to the authors’ best knowledge. Correlation between the wind force and moment coefficients is also investigated. Design wind force coefficients are based on suitable load combinations for development of the unbalanced load distribution. The analysis assumes that the roof is rigid and supported by the four corner columns, and the axial forces induced in the columns are taken as the most important load effect for determining the wind force coefficients. 2. Definition of wind force and moment coefficients The notation and sign of the wind forces and moments are schematically illustrated in Fig. 1, together with the coordinate system. The wind force and moment coefficients are defined as follows: CNW ¼
NW , qH ðb l=2Þ
(1a)
CNL ¼
NL , qH ðb l=2Þ
(1b)
+ NW
+ NL b +L
y
+ My
θ = 0˚ (Duo-pitched roof) +N
+L + My θ = 0˚ (Mono-sloped roof)
+Mz
θ = 0˚
+ My
z
x
l
+Mx
θ = 90˚
Fig. 1. Definition of wind force and moment coefficients and coordinate system. (a) Cross section; (b) Plan view.
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CL ¼
L , qH ðblÞ
(1c)
CN ¼
N , qH ðb lÞ
(1d)
CMx ¼
Mx , qH ðbl 2 Þ
(1e)
CMy ¼
My , qH ðb2 lÞ
(1f)
CMz ¼
Mz , qH ðbl 2 Þ
(1g)
where Mx, My and Mz are moments about the x-, y- and z-axis, respectively; qH is the velocity pressure at the mean roof height H; and b* the actual length of the roof ( ¼ b/ cos b, with b being the roof pitch).
3. Wind tunnel experiments The experimental arrangement and procedures are described in Part 1. In the measurements, models with the regular tap arrangement (tap arrangement 1) are used. Using the pressure difference coefficients at 48 points on the roof model, the wind force and moment coefficients are computed. The statistics (mean, standard deviation, maximum and minimum) of the wind force and moment coefficients are computed by applying ensemble average to the nine consecutive runs.
4. Results and discussion 4.1. Mean wind force coefficients Fig. 2 shows the variation of the mean values of C N W and C N L (C N W and C N L ) with roof pitch b for gable roofs when y ¼ 01, in which the results of previous studies are also plotted for comparative purposes. The present results generally agree with those of Altman (2001). Gumley’s results (1984) for b ¼ 01 show a normal force coefficient approximately equal to 0.2. This is probably due to the effect of blockage induced by the windward four columns at eaves edge reducing the roof uplift force. It should be noted that the ratio of the column width to the distance between adjacent columns was as large as 0.18 in the model used in Gumley (1984). A similar comparison for the mean value of CN (C N ) on mono-sloped roofs is shown in Fig. 3. Again, the present results agree well with those of Altman (2001). In fact, the results of most previous experiments agree well when y ¼ 1801, for which the effect of the roof supporting system on the wind forces seems to be less significant.
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1.0
0.5 Present Gumley (1984) Letchford et al.(1992) Letchford et al.(2000) Altman (2001)
0.0
CNL (mean)
CNW (mean)
1.0
-0.5
Present Gumley (1984,) Letchford et al. (1992) Letchford et al. (2000) Altman (2001)
0.5 0.0 -0.5 -1.0
0
10
20 30 Roof pitch (deg)
0
40
10
20 30 Roof pitch (deg)
40
Fig. 2. Mean normal force coefficients on the windward and leeward halves of gable roofs (y ¼ 01). (a) Windward half; (b) Leeward half.
0.0
2.0
CN (mean)
CN (mean)
1.5 -0.5
-1.0
1.0 0.5
Present
0.0
Gumley (1984) Letchford et al. (2000) Altman (2001)
-1.5
-0.5 0
10
20 30 Roof pitch (deg)
40
0
10
20 30 Roof pitch (deg)
40
Fig. 3. Mean normal force coefficients on mono-sloped roofs for y ¼ 01 and 1801. (a) Wind direction: 01; (b) Wind direction: 1801.
4.2. Peak wind force coefficients Fig. 4 depicts the maximum and minimum peak values of C N W and C N L on gable roofs for a wind direction range from y ¼ 01 to 451. The estimated values from the results of Ginger and Letchford (1994), who measured maximum and minimum peak values of the area-averaged pressure over six panel areas, are also plotted. Using these results, the present study estimated the maximum and minimum peak normal force coefficients on the windward and leeward halves of the roof by calculating the ensemble average of the peak pressure coefficients on three panel areas. The estimation is based on the assumption that the peak area-averaged pressures on these three panel areas occur simultaneously; this of course may result in an overestimate of the actual peak values. Indeed, the estimated normal force coefficients are generally larger in magnitude than the present results. This is also due to differences in flow conditions (open country exposure in the present study and suburban exposure in the study of Ginger and Letchford, 1994). 4.3. Correlation between wind force and moment coefficients Fig. 5 shows a phase-plane representation of the C N W C N L relation for a gable roof. The dashed lines represent the mean values and the circle represents the condition where
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1 CNL (max & min)
3 CNW (max & min)
1033
2 1 0
0 -1 Present (max) Present (min)
-2
Ginger & Letchford (1994) (max) Ginger & Letchford (1994) (min)
-3
-1 0
10
20 30 Roof pitch (deg)
0
40
10
20 30 Roof pitch (deg)
40
Fig. 4. Maximum and minimum peak wind force coefficients on the windward and leeward halves of gable roof (y ¼ 0–451). (a) Windward half; (b) Leeward half.
0.1
CNL
0.0 -0.1 -0.2 -0.3 -0.4 -0.8
-0.4
0.0 CNW
0.4
0.8
Fig. 5. Phase-plane representation of the C N W C N L relation: gable roof (b ¼ 51, y ¼ 01).
the C N W value becomes an extreme in each run. The result indicates a poor correlation between C N W and C N L . Similar features were observed for all cases tested. Therefore, a combination of the peak values of C N W and C N L , which is often used in code provisions, may overestimate the design wind loads, because they are not induced simultaneously. The variation of CL, C M x and C M y (maximum, mean and minimum values) with wind direction y is plotted in Fig. 6 for a gable roof and in Fig. 7 for a mono-sloped roof, both with b ¼ 101. When yE01, the peak values of CL and C M y are generally large in magnitude, while the value of jC M x j is relatively small. With an increase in y, the magnitude of the peak values of CL and C M y decreases, while that of jC M x j increases. In the mono-sloped roof case, the value of |CL(min)| and C M y ðmaxÞ are large when yE1801. Fig. 8 shows the coefficients of correlation (R) between the wind force and moment coefficients, plotted as a function of y. When the wind direction is nearly normal to the eaves (yE01 or 1801), the correlation between CL and C M y is relatively high and that between CL and C M x is rather low. The coefficient of correlation between CL and C M y is low for oblique winds. Fig. 9 shows the variation of the maximum and minimum axial forces induced in the columns with wind direction y for three roof geometries, where the axial force is reduced
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0.05
1.0
0.00
mean
max
0.0
mean
CMy
CL
0.5
CMx
max
0.1
max
-0.05 min
0.0
mean
-0.1
-0.10
min
min
-0.5
-0.2
-0.15 0
30 60 90 Wind direction (deg)
0
30 60 90 Wind direction (deg)
0
30 60 90 Wind direction (deg)
1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5
0.2
Maximum Mean Minimum
0.4
Maximum Mean Minimum
0.0 -0.1
60 120 180 Wind direction (deg)
0.2 0.1 0.0
-0.2 0
Maximum Mean Minimum
0.3 CMy
0.1 CMx
CL
Fig. 6. Variation of the statistics of CL, C M x and C M y with wind direction y: gable roof (b ¼ 101). (a) CL; (b) CMx; (c) CMy.
-0.1 0
60 120 180 Wind direction (deg)
0
60 120 180 Wind direction (deg)
Fig. 7. Variation of the statistics of CL, C M x and C M y with wind direction y: mono-sloped roof (b ¼ 101). (a) CL; (b) CMx; (c) CMy.
1.0
CL-CMy
0.5 R
0.5 R
1.0 CL-CMy
0.0
0.0
CL-CMx
CL-CMx
0
30 60 90 Wind direction (deg)
CL-CMx
-1.0
-1.0
-1.0
0.0 -0.5
-0.5
-0.5
CL-CMy
0.5 R
1.0
0
90 30 60 Wind direction (deg)
0
60 120 180 Wind direction (deg)
Fig. 8. Coefficient of correlation between the wind force and moment coefficients. (a) Gable roof (b ¼ 51); (b) Gable roof (b ¼ 101); (c) Mono-sloped roof (b ¼ 51).
by qH(bl/4). Note that the axial forces induced in the columns are easily computed from the time history of pressures and their influence coefficients for the axial forces, because the roof is assumed rigid and supported by four corner columns, as already mentioned. In the case of gable and troughed roofs, the maximum axial force (maximum tension) occurs at a wind direction ranging from 01 to approximately 451 and the minimum axial force (maximum compression) occurs at yE01. In the mono-sloped roof case, the behavior in the wind direction range from 01 to 901 is similar to that for the gable and troughed roofs. The
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max
0.5 0 min
-0.5 -1 0
1
Reduced axial force
1
Reduced axial force
Reduced axial force
Y. Uematsu et al. / J. Wind Eng. Ind. Aerodyn. 96 (2008) 1029–1042
max
0.5 0 min
-0.5
30 60 90 Wind direction (deg)
-1 0
2.0
1035
max
1.0 min
0.0 -1.0 -2.0
30 60 90 Wind direction (deg)
0
60 120 180 Wind direction (deg)
Fig. 9. Variation of the maximum and minimum axial forces with wind direction. (a) Gable roof (b ¼ 101); (b) Troughed roof (b ¼ 101); (c) Mono-sloped roof (b ¼ 101).
+
=
WIND
WIND
WIND
Wind force distribution due to lift L
Wind force distribution due to moment My
Wind force distribution due to L and My
Fig. 10. Equivalent pressure distribution on the windward and leeward halves caused by CL and C M y .
maximum compression is induced at yE1801. It is interesting to note that the minimum axial force at y ¼ 01 and the maximum axial force at y ¼ 1801 are both small in magnitude, nearly equal to zero. 4.4. Design wind force coefficients Based on the above findings, the design wind force coefficients can be evaluated according to the following procedure: Step 1: The basic values of the wind force coefficients C N W and C N L , denoted as C N W 0 and C N L 0 , are determined from a combination of CL and C M y that produces the maximum load effect for winds normal to the eaves (y ¼ 01 or 1801). The values of C N W 0 and C N L 0 are calculated as follows (see Fig. 10): C N W 0 ¼ C L 4C M y ,
(2a)
C N L 0 ¼ C L þ 4C M y .
(2b)
Step 2: In order to consider the effects of C M x and wind direction on the axial forces induced in the columns, a correction factor g, which is defined as the ratio of the actual peak force for y ¼ 01–451 to that computed from C N W 0 and C N L 0 , is introduced.
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Step 3: The design wind force coefficients C N W and C N L , which give equivalent static wind loads, are provided as follows: C N W ¼
gC N W 0 , Gf
(3a)
C N L ¼
gC N L 0 , Gf
(3b)
CMy 0.0
0.0
-0.1
-0.1
0.25
CMy
0.1
0.1
0.20
CMy
where Gf represents a gust effect factor, which should be determined based on the load effect. Fig. 11 shows a phase-plane representation of the C L C M y relation for low roof pitches and y ¼ 01. The circles represent the maximum and minimum peak values of CL during each of the nine runs. The correlation between CL and C M y is generally high for mono-sloped roofs. The value of C M y at the instant when the maximum CL (CL max) occurs is nearly equal to the maximum C M y (C M y max ). This feature implies that the maximum load effect is given by a combination of these peak values, i.e. the maximum tension is given by ‘C L max þ C M y max ’. Similarly, the maximum compression is given by ‘C L min þ C M y min ’; the suffix ‘min’ represents the minimum value of the coefficient under consideration. When the roof pitch is relatively high, such as b ¼ 151, for example, compression is no longer induced in any column. In the case of y ¼ 1801, the value of C M y at the instant when CL min occurs is nearly equal to C M y max . In the gable and troughed roof cases, on the other hand, the C L C M y correlation is relatively low and becomes lower as the roof pitch increases. The ‘peak+peak’ combination does not always give the maximum load effect. The envelope of the C L C M y trajectory is approximated by a hexagon shown in Fig. 12. The critical condition producing the maximum load effect may be given by one of the apexes of the hexagon. In order to investigate the load combination effects, the axial forces induced in the columns are computed for the six combinations of CL and C M y (Points 1–6) in Fig. 12. Table 1 summarizes the conditions that give the maximum load effect for each case. Load cases ‘A’ and ‘B’ represent the conditions producing the maximum tension and compression in the columns, respectively. Note that the column subjected to the maximum tension or the maximum compression depends on the roof geometry. Substituting the CL and C M y values (for the cases provided in Table 1) into Eq. (2), the values of C N W 0 and C N L 0 for each case are obtained.
0.15 0.10
CL -0.2 -0.2 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.2
0.05 CL 0.0
0.2
0.4
0.00
-0.05 -0.5 0.6
CL 0.0
0.5
1.0
Fig. 11. Phase-plane representation of the CL–C M y relation (y ¼ 01). (a) Gable roof (b ¼ 51); (b) Gable roof (b ¼ 101); (c) Mono-sloped roof (b ¼ 51).
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Fig. 12. Model of the envelope of the trajectory.
Table 1 Combination of CL and C M y values producing the maximum load effect for gable and troughed roofs Roof
Pitch (deg)
Load case A
B
0
C L max þ C M y max
C L min þ C M y min
Gable
5 10 15
C L max þ C M y max C L max þ C M y mean C L max þ C M y mean
C L min þ C M y min C L min þ C M y min C L min þ C M y min
Troughed
5 10 15
C L max þ C M y max C L max þ C M y max C L max þ C M y max
C L min þ C M y mean C L min þ C M y mean C L min þ C M y mean
Flat
The correction factor g in Eq. (3) is obtained by calculating the ratio of the actual maximum or the minimum axial force to the predicted value from C N W 0 and C N L 0 . Fig. 13 shows the gust effect factor Gf, defined as the ratio of the maximum or the minimum axial force to the mean value induced in the columns, plotted against the mean reduced axial force N mean for y ¼ 01–451 (W.D.1) and y ¼ 1351–1801 (W.D.2). When the value of jN mean j is small, Gf exhibits a large value. However, as jN mean j increases, the values of Gf collapse into a narrow range around Gf ¼ 2.0 (dashed line), which corresponds to a peak factor of gv ¼ 3.0, based on the quasi-steady assumption, i.e. GfE(1+gvIuH)2 ¼ (1+3.0 0.14)2 ¼ 2.0. Therefore, Gf ¼ 2.0 is used for evaluating the wind force coefficients hereafter. Plotted on Figs. 14–16 are the estimated values of C N W and C N L for gable and monosloped roofs (triangles). For comparative purposes, the maximum and minimum peak values of C N W and C N L are plotted by circles; these values are divided by Gf to make a direct comparison with C N W and C N L . Note that they are not induced simultaneously. Furthermore, the specification of the Australian/New Zealand (AS/NZ) Standard (Standards Australia, 2002) is also shown by the dashed lines. The wind force coefficients on the windward half for the two load cases are consistent with the maximum and minimum peak values. When yE01 (W.D.1), the values of C N W for load cases A and B are nearly equal to those of C N W min and C N W max and close to the AS/NZ specification.
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10 Flat (W.D. 1)
8
Gable (W.D. 1) Troughed (W.D. 1)
6 Gf
Mono-sloped (W.D. 1) Mono-sloped (W.D. 2)
4
Gf = 2.0
2 0 0
0.1
0.2
0.3
|N*mean| Fig. 13. Gust effect factor based on the load effect.
0.5
1.0 AS/NZ (positive)
0.0
0.0 AS/NZ (positive)
CNL*
CNW*
0.5
AS/NZ (negative)
AS/NZ (negative)
-0.5 -0.5
Load case A
Load case B
Load case A
Load case B
Maximum
Minimum
Maximum
Minimum
-1.0
-1.0 0
5
10 15 Roof pitch (deg)
20
0
5
10 15 Roof pitch (deg)
20
Fig. 14. Equivalent static wind force coefficients C N W and C N L (gable roof, W.D.1). (a) Windward half; (b) Leeward half.
0.5
0.5 AS/NZ (positive)
AS/NZ (positive)
0.0
-0.5
CNL*
CNW*
0.0 AS/NZ (negative)
-1.0
AS/NZ(negative)
-0.5
-1.5
Load case A
Load case B
Load case A
Load case B
Maximum
Minimum
Maximum
Minimum
-1.0
-2.0 0
5
10 15 Roof pitch (deg)
20
0
5
10 15 Roof pitch (deg)
20
Fig. 15. Equivalent static wind force coefficients C N W and C N L (mono-sloped roof, W.D.1). (a) Windward half; (b) Leeward half.
Similarly, when yE1801 (W.D.2), the values of C N L of mono-sloped roofs for load cases A and B are nearly equal to those of C N L min and C N L max . Note that C N L , not C N W , represents the wind force coefficient on the windward half when yE1801. Regarding the
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0.5
1.5
1039
AS/NZ (positive)
0.0
0.5
CNW*
CNL*
1.0 AS/NZ (positive)
0.0
-0.5
AS/NZ (negative)
-0.5
AS/NZ (negative)
Load case A
Load case B
Load case A
Load case B
Maximum
Minimum
Maximum
Minimum
-1.0
-1.0 0
5
10 15 Roof pitch (deg)
0
20
5
10 15 Roof pitch (deg)
20
Fig. 16. Equivalent static wind force coefficients C N W and C N L (mono-sloped roof, W.D.2). (a) Windward half; (b) Leeward half.
0.5
CNW*(Load case A) CNW*(Load case B) CNW_Nmax/Gf CNW_Nmin/Gf
0.5
CNL*, CNL_Npeak/Gf
CNW*, CNW_Npeak/Gf
1.0
0.0
-0.5
CNL*(Load case A) CNL*(Load case B) CNL_Nmax/Gf CNL_Nmin/Gf
0.0
-0.5 0
5
10 15 Roof pitch (deg)
20
0
5
10 15 Roof pitch (deg)
20
Fig. 17. Comparison of C N W and C N L with the actual wind force coefficients producing the maximum and minimum axial forces (gable roof, W.D.1). (a) Windward half; (b) Leeward half.
leeward half, on the other hand, the wind force coefficients for the two load cases are similar to each other and close to either of the maximum or the minimum peak value. The values are significantly different from the AS/NZ specification. The results for troughed roofs were found quite similar to those of the gable roofs. eN and C eN producing the maximum tension and The actual wind force coefficients C W L compression in the columns are obtained from the time history analysis. The results are compared with the C N W and C N L values for the two load cases in Fig. 17; the values of eN and C eN are obtained by applying ensemble average to the results of nine runs and C W L then dividing by Gf. Results are consistent with each other. Therefore, the proposed values of C N W and C N L correspond approximately to the pressure distribution that causes the maximum load effect. The maximum shear force induced in the columns is obtained from the time history analysis and compared with the predicted values from C N W and C N L together with Gf ¼ 2.0. The calculation of the shear force considers the torsional moment Mz as well as the drag. The results for gable and troughed roofs are shown in Fig. 18, which indicates that the predicted values from C N W and C N L capture the actual maximum shear force
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0.4 Reduced shear force
Reduced shear force
0.4 Load case A
0.3
Load case B
0.2
Time history
0.1 0 -0.1
Load case A
0.3
Load case B
0.2
Time history
0.1 0 -0.1
0
5
10 15 Roof pitch (deg)
20
0
5
10 15 Roof pitch (deg)
20
Fig. 18. Maximum reduced shear forces induced in the columns (W.D.1). (a) Gable roof; (b) Troughed roof.
1.5
3.0 Present (Max)
Present (Max)
1.0
2.0
0.5
N*/Gf
N*/Gf
Present (Min)
AS/NZ (max)
0.0 AS/NZ (min)
AS/NZ (max)
Present (Min)
1.0 0.0
-0.5
AS/NZ (min)
-1.0
-1.0 0
5
10 15 20 Roof pitch (deg)
25
30
0
5
10 15 20 Roof pitch (deg)
25
30
Fig. 19. Comparison for axial forces between the predicted values from C N W and C N L and from the AS/NZ specifications (W.D.1). (a) Troughed roof; (b) Mono-sloped roof.
reasonably well. In fact, the maximum shear force obtained from the time history analysis lies between the two predicted values corresponding to the load cases A and B. Finally, the axial forces induced in the columns are computed by using C N W and C N L for the two load cases and compared with those predicted from the AS/NZ Standard. Sample results are shown in Fig. 19. The AS/NZ Standard generally provides two values of the wind force coefficients for each of the windward and leeward halves and this results in four combinations of C N W and C N L . The maximum and minimum axial forces among the four values from the AS/NZ Standard are generally consistent with the results for the load cases A and B, respectively, in spite of the difference in the wind force coefficients (see Figs. 14–16). Consequently, the proposed values of C N W and C N L can be used for design purposes. Table 2 summarizes the proposed wind force coefficients (tentative), which are obtained from a simplified model of the variation of the wind force coefficients with roof pitch b. In the present study, it is assumed that the roof is supported by four corner columns, and the axial forces induced in the columns are taken as the load effect for discussing the wind force coefficients. These coefficients can also be used for the cases where the roof is supported by more columns, say six, because the critical condition may be given when
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Table 2 Wind force coefficients C N W and C N L Roof
Pitch (deg)
Wind direction
Load case A C N W
Flat
Load case B C N L
C N W
C N L
0
W.D.1
0.40
0.10
0.20
0.10
Gable
5 10 15
W.D.1 W.D.1 W.D.1
0.40 0.20 0
0.12 0.33 0.55
0.35 0.50 0.65
0.08 0.27 0.45
Troughed
5 10 15
W.D.1 W.D.1 W.D.1
0.49 0.58 0.68
0.21 0.33 0.44
0.13 0.07 0
0.27 0.43 0.60
Mono-sloped
5 10 15
W.D.1 W.D.1 W.D.1
0.65 0.90 1.15
0.07 0.03 0
0.13 0.07 0
0.07 0.03 0
Mono-sloped
5 10 15
W.D.2 W.D.2 W.D.2
0.07 0.03 0
0.27 0.13 0
0.13 0.17 0.20
0.47 0.73 1.00
yE01 or 1801 irrespective of the number of columns, and the values of C N W 0 and C N L 0 are determined based on a combination of CL and C M y when y ¼ 01 or 1801. Furthermore, the effect of C M x on the axial forces, which is considered by the correction factor g, becomes largest when the roof is supported by four columns. Therefore, the wind force coefficients proposed in the present study will be conservative in so far as they may somewhat overestimate the wind-induced loads in the columns when the number of columns is more than four. 5. Concluding remarks The overall wind forces and moments acting on free-standing canopy roofs have been investigated based on a series of wind tunnel experiments. The study provides basic data for the establishment of design wind loads for this type of roofs. Two load cases yielding maximum tension and compression in the columns are considered. Based on the combination of CL and C M y , design wind force coefficients C N W and C N L , which give equivalent static wind loads, are proposed as a function of roof pitch from 01 to 151. Axial forces produced by using these coefficients are consistent with those obtained from the AS/NZ Standard. Further research is necessary to establish design wind force coefficients for higher roof pitches. Acknowledgments This study was made during the first author’s appointment at the Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Quebec, Canada, as a visiting professor, supported by the Ministry of Education, Culture, Sports,
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Science and Technology, Japan, for the period May 2002–February 2003. The authors are much indebted to Mr. Kai Wang, graduate student of Concordia University, for his assistance with the experiments. References Altman, D.R., 2001. Wind uplift forces on roof canopies. M.Sc. Thesis, Clemson University, NC, USA. Ginger, J.D., Letchford, C.W., 1994. Wind loads on planar canopy roofs, Part 2: fluctuating pressure distributions and correlations. J. Wind Eng. Ind. Aerodyn. 51, 353–370. Gumley, S.J., 1984. A parametric study of extreme pressures for the static design of canopy structures. J. Wind Eng. Ind. Aerodyn. 16, 43–56. Holmes, J.D., 2001. Wind Loading of Structures. Spon Press. Letchford, C.W., Ginger, J.D., 1992. Wind loads on planar canopy roofs, Part 1: mean pressure distributions. J. Wind Eng. Ind. Aerodyn. 45, 25–45. Letchford, C.W., Row, A., Vitale, A., Wolbers, J., 2000. Mean wind loads on porous canopy roofs. J. Wind Eng. Ind. Aerodyn. 84, 197–213. Standards Australia, 2002. Australian/New Zealand Standard, AS/NZS 1170.2. Uematsu, Y., Stathopoulos, T., Iizumi, E., 2007. Wind loads on free-standing canopy roofs, Part 1: local pressures, J. Wind Eng. Ind. Aerodyn., to appear.
Journal of Wind Engineering and Industrial Aerodynamics 96 (2008) 1029–1042 www.elsevier.com/locate/jweia
Wind loads on free-standing canopy roofs: Part 2 overall wind forces Yasushi Uematsua,, Theodore Stathopoulosb, Eri Iizumic a New Industry Creation Hatchery Center, Tohoku University, Aoba-ku, Sendai 980-8579, Japan Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Canada H3G 1M8 c Department of Architecture and Building Science, Tohoku University, Aoba-ku, Sendai 980-8579, Japan
b
Available online 26 July 2007
Abstract Wind loads on free-standing canopy roofs have been studied in a wind tunnel. Three types of roof geometries, i.e. gable, troughed and mono-sloped roofs, with roof pitches ranging from 01 to 151, were tested. Wind pressures were measured simultaneously at many points both on the top and bottom surfaces of the roof model for various wind directions. The paper describes the characteristics of overall wind forces and moments acting on the roof with special attention to load combination. Correlation between wind force and moment coefficients is investigated and wind force coefficients for the design of main wind force resisting systems are proposed. The roof is assumed rigid and simply supported by four corner columns, whose axial forces appear as the most important load effect. Two loading patterns that cause the maximum tension and compression of columns are considered. The proposed values are also compared with the specifications of the Australian/New Zealand Standard [Standards Australia, 2002. Australian/New Zealand Standard, AS/NZS 1170.2]. r 2007 Elsevier Ltd. All rights reserved. Keywords: Free-standing canopy roof; Wind tunnel experiment; Overall wind force; Main wind force resisting system; Design wind load; Codification
1. Introduction The characteristics of local pressures on free-standing canopy roofs are described in a companion paper, Part 1 (Uematsu et al., 2007). This paper presents the experimental results on overall wind forces and moments acting on these roofs. Corresponding author. Tel./fax: +81 22 795 7875.
E-mail address: [email protected] (Y. Uematsu). 0167-6105/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2007.06.026
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Gumley (1984), Letchford and Ginger (1992) and Ginger and Letchford (1994) measured area-averaged pressures for various areas using a pneumatic averaging technique. Furthermore, Ginger and Letchford (1994) calculated the overall forces using the covariance integration technique. Holmes (2001) discusses the application of their results to load estimation. Recently, Letchford et al. (2000) measured the mean wind forces on solid and porous canopy roof models using a force balance technique. This is perhaps the first attempt to measure the overall forces on canopy roofs directly. More recently, Altman (2001) has made extensive measurements of the overall forces and moments using a high-frequency force balance developed at Clemson University, USA. In the present study, the wind forces and moments are computed from the time history of pressures measured simultaneously at many points both on the top and bottom surfaces. Such simultaneous pressure measurements leading to a reliable definition of realistic windinduced forces on canopy roofs have not been made so far, to the authors’ best knowledge. Correlation between the wind force and moment coefficients is also investigated. Design wind force coefficients are based on suitable load combinations for development of the unbalanced load distribution. The analysis assumes that the roof is rigid and supported by the four corner columns, and the axial forces induced in the columns are taken as the most important load effect for determining the wind force coefficients. 2. Definition of wind force and moment coefficients The notation and sign of the wind forces and moments are schematically illustrated in Fig. 1, together with the coordinate system. The wind force and moment coefficients are defined as follows: CNW ¼
NW , qH ðb l=2Þ
(1a)
CNL ¼
NL , qH ðb l=2Þ
(1b)
+ NW
+ NL b +L
y
+ My
θ = 0˚ (Duo-pitched roof) +N
+L + My θ = 0˚ (Mono-sloped roof)
+Mz
θ = 0˚
+ My
z
x
l
+Mx
θ = 90˚
Fig. 1. Definition of wind force and moment coefficients and coordinate system. (a) Cross section; (b) Plan view.
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CL ¼
L , qH ðblÞ
(1c)
CN ¼
N , qH ðb lÞ
(1d)
CMx ¼
Mx , qH ðbl 2 Þ
(1e)
CMy ¼
My , qH ðb2 lÞ
(1f)
CMz ¼
Mz , qH ðbl 2 Þ
(1g)
where Mx, My and Mz are moments about the x-, y- and z-axis, respectively; qH is the velocity pressure at the mean roof height H; and b* the actual length of the roof ( ¼ b/ cos b, with b being the roof pitch).
3. Wind tunnel experiments The experimental arrangement and procedures are described in Part 1. In the measurements, models with the regular tap arrangement (tap arrangement 1) are used. Using the pressure difference coefficients at 48 points on the roof model, the wind force and moment coefficients are computed. The statistics (mean, standard deviation, maximum and minimum) of the wind force and moment coefficients are computed by applying ensemble average to the nine consecutive runs.
4. Results and discussion 4.1. Mean wind force coefficients Fig. 2 shows the variation of the mean values of C N W and C N L (C N W and C N L ) with roof pitch b for gable roofs when y ¼ 01, in which the results of previous studies are also plotted for comparative purposes. The present results generally agree with those of Altman (2001). Gumley’s results (1984) for b ¼ 01 show a normal force coefficient approximately equal to 0.2. This is probably due to the effect of blockage induced by the windward four columns at eaves edge reducing the roof uplift force. It should be noted that the ratio of the column width to the distance between adjacent columns was as large as 0.18 in the model used in Gumley (1984). A similar comparison for the mean value of CN (C N ) on mono-sloped roofs is shown in Fig. 3. Again, the present results agree well with those of Altman (2001). In fact, the results of most previous experiments agree well when y ¼ 1801, for which the effect of the roof supporting system on the wind forces seems to be less significant.
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1.0
0.5 Present Gumley (1984) Letchford et al.(1992) Letchford et al.(2000) Altman (2001)
0.0
CNL (mean)
CNW (mean)
1.0
-0.5
Present Gumley (1984,) Letchford et al. (1992) Letchford et al. (2000) Altman (2001)
0.5 0.0 -0.5 -1.0
0
10
20 30 Roof pitch (deg)
0
40
10
20 30 Roof pitch (deg)
40
Fig. 2. Mean normal force coefficients on the windward and leeward halves of gable roofs (y ¼ 01). (a) Windward half; (b) Leeward half.
0.0
2.0
CN (mean)
CN (mean)
1.5 -0.5
-1.0
1.0 0.5
Present
0.0
Gumley (1984) Letchford et al. (2000) Altman (2001)
-1.5
-0.5 0
10
20 30 Roof pitch (deg)
40
0
10
20 30 Roof pitch (deg)
40
Fig. 3. Mean normal force coefficients on mono-sloped roofs for y ¼ 01 and 1801. (a) Wind direction: 01; (b) Wind direction: 1801.
4.2. Peak wind force coefficients Fig. 4 depicts the maximum and minimum peak values of C N W and C N L on gable roofs for a wind direction range from y ¼ 01 to 451. The estimated values from the results of Ginger and Letchford (1994), who measured maximum and minimum peak values of the area-averaged pressure over six panel areas, are also plotted. Using these results, the present study estimated the maximum and minimum peak normal force coefficients on the windward and leeward halves of the roof by calculating the ensemble average of the peak pressure coefficients on three panel areas. The estimation is based on the assumption that the peak area-averaged pressures on these three panel areas occur simultaneously; this of course may result in an overestimate of the actual peak values. Indeed, the estimated normal force coefficients are generally larger in magnitude than the present results. This is also due to differences in flow conditions (open country exposure in the present study and suburban exposure in the study of Ginger and Letchford, 1994). 4.3. Correlation between wind force and moment coefficients Fig. 5 shows a phase-plane representation of the C N W C N L relation for a gable roof. The dashed lines represent the mean values and the circle represents the condition where
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3 CNW (max & min)
1033
2 1 0
0 -1 Present (max) Present (min)
-2
Ginger & Letchford (1994) (max) Ginger & Letchford (1994) (min)
-3
-1 0
10
20 30 Roof pitch (deg)
0
40
10
20 30 Roof pitch (deg)
40
Fig. 4. Maximum and minimum peak wind force coefficients on the windward and leeward halves of gable roof (y ¼ 0–451). (a) Windward half; (b) Leeward half.
0.1
CNL
0.0 -0.1 -0.2 -0.3 -0.4 -0.8
-0.4
0.0 CNW
0.4
0.8
Fig. 5. Phase-plane representation of the C N W C N L relation: gable roof (b ¼ 51, y ¼ 01).
the C N W value becomes an extreme in each run. The result indicates a poor correlation between C N W and C N L . Similar features were observed for all cases tested. Therefore, a combination of the peak values of C N W and C N L , which is often used in code provisions, may overestimate the design wind loads, because they are not induced simultaneously. The variation of CL, C M x and C M y (maximum, mean and minimum values) with wind direction y is plotted in Fig. 6 for a gable roof and in Fig. 7 for a mono-sloped roof, both with b ¼ 101. When yE01, the peak values of CL and C M y are generally large in magnitude, while the value of jC M x j is relatively small. With an increase in y, the magnitude of the peak values of CL and C M y decreases, while that of jC M x j increases. In the mono-sloped roof case, the value of |CL(min)| and C M y ðmaxÞ are large when yE1801. Fig. 8 shows the coefficients of correlation (R) between the wind force and moment coefficients, plotted as a function of y. When the wind direction is nearly normal to the eaves (yE01 or 1801), the correlation between CL and C M y is relatively high and that between CL and C M x is rather low. The coefficient of correlation between CL and C M y is low for oblique winds. Fig. 9 shows the variation of the maximum and minimum axial forces induced in the columns with wind direction y for three roof geometries, where the axial force is reduced
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0.05
1.0
0.00
mean
max
0.0
mean
CMy
CL
0.5
CMx
max
0.1
max
-0.05 min
0.0
mean
-0.1
-0.10
min
min
-0.5
-0.2
-0.15 0
30 60 90 Wind direction (deg)
0
30 60 90 Wind direction (deg)
0
30 60 90 Wind direction (deg)
1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5
0.2
Maximum Mean Minimum
0.4
Maximum Mean Minimum
0.0 -0.1
60 120 180 Wind direction (deg)
0.2 0.1 0.0
-0.2 0
Maximum Mean Minimum
0.3 CMy
0.1 CMx
CL
Fig. 6. Variation of the statistics of CL, C M x and C M y with wind direction y: gable roof (b ¼ 101). (a) CL; (b) CMx; (c) CMy.
-0.1 0
60 120 180 Wind direction (deg)
0
60 120 180 Wind direction (deg)
Fig. 7. Variation of the statistics of CL, C M x and C M y with wind direction y: mono-sloped roof (b ¼ 101). (a) CL; (b) CMx; (c) CMy.
1.0
CL-CMy
0.5 R
0.5 R
1.0 CL-CMy
0.0
0.0
CL-CMx
CL-CMx
0
30 60 90 Wind direction (deg)
CL-CMx
-1.0
-1.0
-1.0
0.0 -0.5
-0.5
-0.5
CL-CMy
0.5 R
1.0
0
90 30 60 Wind direction (deg)
0
60 120 180 Wind direction (deg)
Fig. 8. Coefficient of correlation between the wind force and moment coefficients. (a) Gable roof (b ¼ 51); (b) Gable roof (b ¼ 101); (c) Mono-sloped roof (b ¼ 51).
by qH(bl/4). Note that the axial forces induced in the columns are easily computed from the time history of pressures and their influence coefficients for the axial forces, because the roof is assumed rigid and supported by four corner columns, as already mentioned. In the case of gable and troughed roofs, the maximum axial force (maximum tension) occurs at a wind direction ranging from 01 to approximately 451 and the minimum axial force (maximum compression) occurs at yE01. In the mono-sloped roof case, the behavior in the wind direction range from 01 to 901 is similar to that for the gable and troughed roofs. The
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0.5 0 min
-0.5 -1 0
1
Reduced axial force
1
Reduced axial force
Reduced axial force
Y. Uematsu et al. / J. Wind Eng. Ind. Aerodyn. 96 (2008) 1029–1042
max
0.5 0 min
-0.5
30 60 90 Wind direction (deg)
-1 0
2.0
1035
max
1.0 min
0.0 -1.0 -2.0
30 60 90 Wind direction (deg)
0
60 120 180 Wind direction (deg)
Fig. 9. Variation of the maximum and minimum axial forces with wind direction. (a) Gable roof (b ¼ 101); (b) Troughed roof (b ¼ 101); (c) Mono-sloped roof (b ¼ 101).
+
=
WIND
WIND
WIND
Wind force distribution due to lift L
Wind force distribution due to moment My
Wind force distribution due to L and My
Fig. 10. Equivalent pressure distribution on the windward and leeward halves caused by CL and C M y .
maximum compression is induced at yE1801. It is interesting to note that the minimum axial force at y ¼ 01 and the maximum axial force at y ¼ 1801 are both small in magnitude, nearly equal to zero. 4.4. Design wind force coefficients Based on the above findings, the design wind force coefficients can be evaluated according to the following procedure: Step 1: The basic values of the wind force coefficients C N W and C N L , denoted as C N W 0 and C N L 0 , are determined from a combination of CL and C M y that produces the maximum load effect for winds normal to the eaves (y ¼ 01 or 1801). The values of C N W 0 and C N L 0 are calculated as follows (see Fig. 10): C N W 0 ¼ C L 4C M y ,
(2a)
C N L 0 ¼ C L þ 4C M y .
(2b)
Step 2: In order to consider the effects of C M x and wind direction on the axial forces induced in the columns, a correction factor g, which is defined as the ratio of the actual peak force for y ¼ 01–451 to that computed from C N W 0 and C N L 0 , is introduced.
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Step 3: The design wind force coefficients C N W and C N L , which give equivalent static wind loads, are provided as follows: C N W ¼
gC N W 0 , Gf
(3a)
C N L ¼
gC N L 0 , Gf
(3b)
CMy 0.0
0.0
-0.1
-0.1
0.25
CMy
0.1
0.1
0.20
CMy
where Gf represents a gust effect factor, which should be determined based on the load effect. Fig. 11 shows a phase-plane representation of the C L C M y relation for low roof pitches and y ¼ 01. The circles represent the maximum and minimum peak values of CL during each of the nine runs. The correlation between CL and C M y is generally high for mono-sloped roofs. The value of C M y at the instant when the maximum CL (CL max) occurs is nearly equal to the maximum C M y (C M y max ). This feature implies that the maximum load effect is given by a combination of these peak values, i.e. the maximum tension is given by ‘C L max þ C M y max ’. Similarly, the maximum compression is given by ‘C L min þ C M y min ’; the suffix ‘min’ represents the minimum value of the coefficient under consideration. When the roof pitch is relatively high, such as b ¼ 151, for example, compression is no longer induced in any column. In the case of y ¼ 1801, the value of C M y at the instant when CL min occurs is nearly equal to C M y max . In the gable and troughed roof cases, on the other hand, the C L C M y correlation is relatively low and becomes lower as the roof pitch increases. The ‘peak+peak’ combination does not always give the maximum load effect. The envelope of the C L C M y trajectory is approximated by a hexagon shown in Fig. 12. The critical condition producing the maximum load effect may be given by one of the apexes of the hexagon. In order to investigate the load combination effects, the axial forces induced in the columns are computed for the six combinations of CL and C M y (Points 1–6) in Fig. 12. Table 1 summarizes the conditions that give the maximum load effect for each case. Load cases ‘A’ and ‘B’ represent the conditions producing the maximum tension and compression in the columns, respectively. Note that the column subjected to the maximum tension or the maximum compression depends on the roof geometry. Substituting the CL and C M y values (for the cases provided in Table 1) into Eq. (2), the values of C N W 0 and C N L 0 for each case are obtained.
0.15 0.10
CL -0.2 -0.2 -0.4 -0.2 0.0 0.2 0.4 0.6 -0.2
0.05 CL 0.0
0.2
0.4
0.00
-0.05 -0.5 0.6
CL 0.0
0.5
1.0
Fig. 11. Phase-plane representation of the CL–C M y relation (y ¼ 01). (a) Gable roof (b ¼ 51); (b) Gable roof (b ¼ 101); (c) Mono-sloped roof (b ¼ 51).
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Fig. 12. Model of the envelope of the trajectory.
Table 1 Combination of CL and C M y values producing the maximum load effect for gable and troughed roofs Roof
Pitch (deg)
Load case A
B
0
C L max þ C M y max
C L min þ C M y min
Gable
5 10 15
C L max þ C M y max C L max þ C M y mean C L max þ C M y mean
C L min þ C M y min C L min þ C M y min C L min þ C M y min
Troughed
5 10 15
C L max þ C M y max C L max þ C M y max C L max þ C M y max
C L min þ C M y mean C L min þ C M y mean C L min þ C M y mean
Flat
The correction factor g in Eq. (3) is obtained by calculating the ratio of the actual maximum or the minimum axial force to the predicted value from C N W 0 and C N L 0 . Fig. 13 shows the gust effect factor Gf, defined as the ratio of the maximum or the minimum axial force to the mean value induced in the columns, plotted against the mean reduced axial force N mean for y ¼ 01–451 (W.D.1) and y ¼ 1351–1801 (W.D.2). When the value of jN mean j is small, Gf exhibits a large value. However, as jN mean j increases, the values of Gf collapse into a narrow range around Gf ¼ 2.0 (dashed line), which corresponds to a peak factor of gv ¼ 3.0, based on the quasi-steady assumption, i.e. GfE(1+gvIuH)2 ¼ (1+3.0 0.14)2 ¼ 2.0. Therefore, Gf ¼ 2.0 is used for evaluating the wind force coefficients hereafter. Plotted on Figs. 14–16 are the estimated values of C N W and C N L for gable and monosloped roofs (triangles). For comparative purposes, the maximum and minimum peak values of C N W and C N L are plotted by circles; these values are divided by Gf to make a direct comparison with C N W and C N L . Note that they are not induced simultaneously. Furthermore, the specification of the Australian/New Zealand (AS/NZ) Standard (Standards Australia, 2002) is also shown by the dashed lines. The wind force coefficients on the windward half for the two load cases are consistent with the maximum and minimum peak values. When yE01 (W.D.1), the values of C N W for load cases A and B are nearly equal to those of C N W min and C N W max and close to the AS/NZ specification.
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10 Flat (W.D. 1)
8
Gable (W.D. 1) Troughed (W.D. 1)
6 Gf
Mono-sloped (W.D. 1) Mono-sloped (W.D. 2)
4
Gf = 2.0
2 0 0
0.1
0.2
0.3
|N*mean| Fig. 13. Gust effect factor based on the load effect.
0.5
1.0 AS/NZ (positive)
0.0
0.0 AS/NZ (positive)
CNL*
CNW*
0.5
AS/NZ (negative)
AS/NZ (negative)
-0.5 -0.5
Load case A
Load case B
Load case A
Load case B
Maximum
Minimum
Maximum
Minimum
-1.0
-1.0 0
5
10 15 Roof pitch (deg)
20
0
5
10 15 Roof pitch (deg)
20
Fig. 14. Equivalent static wind force coefficients C N W and C N L (gable roof, W.D.1). (a) Windward half; (b) Leeward half.
0.5
0.5 AS/NZ (positive)
AS/NZ (positive)
0.0
-0.5
CNL*
CNW*
0.0 AS/NZ (negative)
-1.0
AS/NZ(negative)
-0.5
-1.5
Load case A
Load case B
Load case A
Load case B
Maximum
Minimum
Maximum
Minimum
-1.0
-2.0 0
5
10 15 Roof pitch (deg)
20
0
5
10 15 Roof pitch (deg)
20
Fig. 15. Equivalent static wind force coefficients C N W and C N L (mono-sloped roof, W.D.1). (a) Windward half; (b) Leeward half.
Similarly, when yE1801 (W.D.2), the values of C N L of mono-sloped roofs for load cases A and B are nearly equal to those of C N L min and C N L max . Note that C N L , not C N W , represents the wind force coefficient on the windward half when yE1801. Regarding the
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0.5
1.5
1039
AS/NZ (positive)
0.0
0.5
CNW*
CNL*
1.0 AS/NZ (positive)
0.0
-0.5
AS/NZ (negative)
-0.5
AS/NZ (negative)
Load case A
Load case B
Load case A
Load case B
Maximum
Minimum
Maximum
Minimum
-1.0
-1.0 0
5
10 15 Roof pitch (deg)
0
20
5
10 15 Roof pitch (deg)
20
Fig. 16. Equivalent static wind force coefficients C N W and C N L (mono-sloped roof, W.D.2). (a) Windward half; (b) Leeward half.
0.5
CNW*(Load case A) CNW*(Load case B) CNW_Nmax/Gf CNW_Nmin/Gf
0.5
CNL*, CNL_Npeak/Gf
CNW*, CNW_Npeak/Gf
1.0
0.0
-0.5
CNL*(Load case A) CNL*(Load case B) CNL_Nmax/Gf CNL_Nmin/Gf
0.0
-0.5 0
5
10 15 Roof pitch (deg)
20
0
5
10 15 Roof pitch (deg)
20
Fig. 17. Comparison of C N W and C N L with the actual wind force coefficients producing the maximum and minimum axial forces (gable roof, W.D.1). (a) Windward half; (b) Leeward half.
leeward half, on the other hand, the wind force coefficients for the two load cases are similar to each other and close to either of the maximum or the minimum peak value. The values are significantly different from the AS/NZ specification. The results for troughed roofs were found quite similar to those of the gable roofs. eN and C eN producing the maximum tension and The actual wind force coefficients C W L compression in the columns are obtained from the time history analysis. The results are compared with the C N W and C N L values for the two load cases in Fig. 17; the values of eN and C eN are obtained by applying ensemble average to the results of nine runs and C W L then dividing by Gf. Results are consistent with each other. Therefore, the proposed values of C N W and C N L correspond approximately to the pressure distribution that causes the maximum load effect. The maximum shear force induced in the columns is obtained from the time history analysis and compared with the predicted values from C N W and C N L together with Gf ¼ 2.0. The calculation of the shear force considers the torsional moment Mz as well as the drag. The results for gable and troughed roofs are shown in Fig. 18, which indicates that the predicted values from C N W and C N L capture the actual maximum shear force
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0.4 Reduced shear force
Reduced shear force
0.4 Load case A
0.3
Load case B
0.2
Time history
0.1 0 -0.1
Load case A
0.3
Load case B
0.2
Time history
0.1 0 -0.1
0
5
10 15 Roof pitch (deg)
20
0
5
10 15 Roof pitch (deg)
20
Fig. 18. Maximum reduced shear forces induced in the columns (W.D.1). (a) Gable roof; (b) Troughed roof.
1.5
3.0 Present (Max)
Present (Max)
1.0
2.0
0.5
N*/Gf
N*/Gf
Present (Min)
AS/NZ (max)
0.0 AS/NZ (min)
AS/NZ (max)
Present (Min)
1.0 0.0
-0.5
AS/NZ (min)
-1.0
-1.0 0
5
10 15 20 Roof pitch (deg)
25
30
0
5
10 15 20 Roof pitch (deg)
25
30
Fig. 19. Comparison for axial forces between the predicted values from C N W and C N L and from the AS/NZ specifications (W.D.1). (a) Troughed roof; (b) Mono-sloped roof.
reasonably well. In fact, the maximum shear force obtained from the time history analysis lies between the two predicted values corresponding to the load cases A and B. Finally, the axial forces induced in the columns are computed by using C N W and C N L for the two load cases and compared with those predicted from the AS/NZ Standard. Sample results are shown in Fig. 19. The AS/NZ Standard generally provides two values of the wind force coefficients for each of the windward and leeward halves and this results in four combinations of C N W and C N L . The maximum and minimum axial forces among the four values from the AS/NZ Standard are generally consistent with the results for the load cases A and B, respectively, in spite of the difference in the wind force coefficients (see Figs. 14–16). Consequently, the proposed values of C N W and C N L can be used for design purposes. Table 2 summarizes the proposed wind force coefficients (tentative), which are obtained from a simplified model of the variation of the wind force coefficients with roof pitch b. In the present study, it is assumed that the roof is supported by four corner columns, and the axial forces induced in the columns are taken as the load effect for discussing the wind force coefficients. These coefficients can also be used for the cases where the roof is supported by more columns, say six, because the critical condition may be given when
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Table 2 Wind force coefficients C N W and C N L Roof
Pitch (deg)
Wind direction
Load case A C N W
Flat
Load case B C N L
C N W
C N L
0
W.D.1
0.40
0.10
0.20
0.10
Gable
5 10 15
W.D.1 W.D.1 W.D.1
0.40 0.20 0
0.12 0.33 0.55
0.35 0.50 0.65
0.08 0.27 0.45
Troughed
5 10 15
W.D.1 W.D.1 W.D.1
0.49 0.58 0.68
0.21 0.33 0.44
0.13 0.07 0
0.27 0.43 0.60
Mono-sloped
5 10 15
W.D.1 W.D.1 W.D.1
0.65 0.90 1.15
0.07 0.03 0
0.13 0.07 0
0.07 0.03 0
Mono-sloped
5 10 15
W.D.2 W.D.2 W.D.2
0.07 0.03 0
0.27 0.13 0
0.13 0.17 0.20
0.47 0.73 1.00
yE01 or 1801 irrespective of the number of columns, and the values of C N W 0 and C N L 0 are determined based on a combination of CL and C M y when y ¼ 01 or 1801. Furthermore, the effect of C M x on the axial forces, which is considered by the correction factor g, becomes largest when the roof is supported by four columns. Therefore, the wind force coefficients proposed in the present study will be conservative in so far as they may somewhat overestimate the wind-induced loads in the columns when the number of columns is more than four. 5. Concluding remarks The overall wind forces and moments acting on free-standing canopy roofs have been investigated based on a series of wind tunnel experiments. The study provides basic data for the establishment of design wind loads for this type of roofs. Two load cases yielding maximum tension and compression in the columns are considered. Based on the combination of CL and C M y , design wind force coefficients C N W and C N L , which give equivalent static wind loads, are proposed as a function of roof pitch from 01 to 151. Axial forces produced by using these coefficients are consistent with those obtained from the AS/NZ Standard. Further research is necessary to establish design wind force coefficients for higher roof pitches. Acknowledgments This study was made during the first author’s appointment at the Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Quebec, Canada, as a visiting professor, supported by the Ministry of Education, Culture, Sports,
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Science and Technology, Japan, for the period May 2002–February 2003. The authors are much indebted to Mr. Kai Wang, graduate student of Concordia University, for his assistance with the experiments. References Altman, D.R., 2001. Wind uplift forces on roof canopies. M.Sc. Thesis, Clemson University, NC, USA. Ginger, J.D., Letchford, C.W., 1994. Wind loads on planar canopy roofs, Part 2: fluctuating pressure distributions and correlations. J. Wind Eng. Ind. Aerodyn. 51, 353–370. Gumley, S.J., 1984. A parametric study of extreme pressures for the static design of canopy structures. J. Wind Eng. Ind. Aerodyn. 16, 43–56. Holmes, J.D., 2001. Wind Loading of Structures. Spon Press. Letchford, C.W., Ginger, J.D., 1992. Wind loads on planar canopy roofs, Part 1: mean pressure distributions. J. Wind Eng. Ind. Aerodyn. 45, 25–45. Letchford, C.W., Row, A., Vitale, A., Wolbers, J., 2000. Mean wind loads on porous canopy roofs. J. Wind Eng. Ind. Aerodyn. 84, 197–213. Standards Australia, 2002. Australian/New Zealand Standard, AS/NZS 1170.2. Uematsu, Y., Stathopoulos, T., Iizumi, E., 2007. Wind loads on free-standing canopy roofs, Part 1: local pressures, J. Wind Eng. Ind. Aerodyn., to appear.