ARTICLE IN PRESSJournal of Wind Engineering and Industrial Aerodynamics 94 (2006) 833–844 www.elsevier.com/locate/jweia Wind loads on curved roofs P.A. BlackmoreÃ, E. Tsokri Building Research Establishment Ltd, Garston, Watford WD25 9XX, UK Available online 16 October 2006 Abstract Curved roofed buildings are increasingly used in the modern built environment because they offer aerodynamically efficient shapes and provide architects and designers with an alternative to regular rectangular building forms. However, there is little information available on the wind loads on these roof forms. The Eurocode for wind actions (EN1991-1-4) includes pressure coefficients for a limited range of aspect ratio cylindrical roofs from measurements in low-turbulence conditions but only for wind blowing normal to the eaves. There is some concern regarding the reliability of these data, consequently EN1991-1-4 allows National Choice (National Determined Parameter) for wind loads on these roofs. This paper describes a series of parametric wind tunnel studies undertaken at BRE to measure wind pressures on a wide range of curved roof models in a properly scaled atmospheric boundary layer simulation and gives an alternative to the EN1991-1-4 recommended procedure. r 2006 Elsevier Ltd. All rights reserved. Keywords: Wind tunnel testing; Wind pressures; Wind loads; Curved roofs; EN1991-1-4 1. Introduction The Eurocode for wind actions (EN1991-1-4, 2004) gives recommended values for external pressure coefficients on curved roofs. These data were measured in low-turbulence conditions, hence there is some concern regarding their reliability. EN1991-1-4 therefore allows National Choice (National Determined Parameter) for wind loads on these roofs. Each Member State must therefore decide whether to adopt the recommended procedure or to specify an alternative procedure in their National Annexes to EN1991-1-4. There is however surprisingly little information available on the pressure coefficients on these roof ÃCorresponding author. E-mail address: [email protected] (P.A. Blackmore). 0167-6105/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2006.06.006 08. In general there is a factor of 2 or more between the smallest and largest values. No data are given for wind blowing along the ridge.5. and the reference height varies. 0.05 to 0. 0.2 and 0.6 for structures with side walls (no information is given regarding the effect of side wall height) and without side walls.05. Figs.10) on curved roofs with rise height to span ratios (f/d) from 0. For greenhouses without side walls one data set is given for all height to span ratios (h/s).5. the size of the areas over which the pressure coefficients are averaged are different. the Canadian code (User’s Guide—NBC. 1) is used. f/d ¼ 0. but in general these are not in a form suitable for codification or they lack essential experimental details which are necessary for codification purposes. EN13031 has data for greenhouses with and without side walls. 94 (2006) 833–844 forms. Comparison of published data The Eurocode for wind actions (EN1991-1-4) gives external pressure coefficients (Cpe. Fig. Blackmore. Some other National wind codes. The Australian and New Zealand code gives external pressure coefficients (Cpe) on curved roofs for rise height to span ratios (r/d) of 0. The US (ASCE) code has data for rise height to span ratios (r) from 0 to 0. with side wall height to rise height ratio (h/r)p2. Tsokri / J.5.3 for side wall height to width ratios (H/W) of 0. Comparison of these data is not straightforward because they are presented as functions of different rise height parameters. hereafter the nomenclature given in EN1991-1-4 (Fig.4 and hr/sX0. Aerodyn. 0. for wind blowing on to the eaves. This paper compares the EN1991-1-4 data with other published data and with data measured from a series of parametric wind tunnel studies undertaken at BRE. 1996) give external pressure coefficients for curved roofs. Notwithstanding these issues.25 and 0.A.ARTICLE IN PRESS 834 P. although in some cases even the sign of the pressure coefficient varies between different data sets. 1995) and the American (ASCE) code (ASCE.3. From these figures it can be seen that there is a wide spread of Cpe values. Pressure coefficients are also given by Cook (1995) and in EN13031-1:2001 (BS EN13031-1. and side wall height to span width ratio of 0. especially on the windward section of the roof. For wind on parallel to the ridge the data for pitched roof is used. for greenhouses with side walls data are given for hr/sp0. E. for wind blowing on to the eaves.1 and h/d ¼ 0. Wind Eng. An alternative procedure is proposed to replace the EN1991-1-4 recommended procedure in the UK National Annex.2 and 0.5. 2001) for commercial greenhouses (where human occupancy levels are restricted). Ind. f/d ¼ 0.5 for side wall height to span ratios (h/d) of 0 and 0. It should be noted that each of the above references uses different nomenclature. The Canadian code has data for a single curved roof of rise height to span ratio of 0. b) for rise height to width ratios (R/W) of 0. Data are given for wind onto the eaves and parallel with the ridge.6. 2. Cook presents data from Blessmann (1987a. 2 and 3 show comparisons for the cases of h/d ¼ 0. 1 shows the data given in EN1991-1-4. such as the Australian and New Zealand code (Australian/New Zealand Standard. 2002).1. There are other sources of pressure coefficients on curved roofs such as research papers and commercial wind tunnel studies. therefore it is not easy for Member States to make an informed decision on the choice of pressure coefficients. These large differences cannot be .17. Comparison between Cpe values for curved roofs with h/d ¼ 0.A.0 B -1.4 -0.5) -0.3 0.4 0.1 0.2 -0.5 -1 -1. Blackmore.10 0. Ind. Wind Eng. Aerodyn.6 0. EN1991-1-4 data for curved roofs. etc. Tsokri / J.2 0 -0.1.05 0.ARTICLE IN PRESS P. 1.2 d (h/ >0 .2 A (h/d > 0.8 -1.5) Fig. 94 (2006) 833–844 835 A B C f h cpe.5 f/d C C B A (h/d > 0.4 0.6 I d A( h= 0) A 0. E.5.5 0 Cpe -0. 2. accounted for by the differences in area. f/d ¼ 0. reference height.5 Roof zone Fig. 1 Windward zone A Middle zone B Leeward zone C EN1991-1-4 ANZ (h/r=1) ANZ (h/r=2) ASCE Cook EN13031 0. . It is clear that there is wide variability between the data sets and further experimental data were required in order to advise on the appropriate data to be used in the UK National Annex to EN1991-1-4.5) 0.8 0. R4 and R5 models had 46 pressure taps. These studies also included a range of building length/width (L/d) ratios from 1 to 10 to examine the effect of two-dimensional flow at L/d ¼ 10 and three-dimensional flow at L/d ¼ 1.4 0.05 to 0. Module type ZOC33B was used for this study. The reference height for deriving pressure coefficients was taken as the ridge height (h+r) of the model. Experimental details A parametric study was carried out in the BRE no. These sensors are piezoresistive differential sensors with a sampling rate of up to 20 kHz and a full-scale pressure range of 1250 Pa. Ind. The sand grain size.8 -1 -1. This had plan dimensions of 100 Â 100 mm. Pressures were measured simultaneously using Scanivalve miniature pressure transducers. Mean and peak pressure coefficients (for an averaging period.2 0 -0. these results are reported in Breeze et al. Sand grains were attached to the roofs of the models to promote supercritical flow separation. 3.ARTICLE IN PRESS 836 P.4 -0.06 to 1. The R3. f/d ¼ 0. To investigate the effect of building length. giving k/(h+f) ratios from 5 Â 10À2 to 4 Â 10À3.2 Cpe -0.4 Middle zone B Leeward zone C EN1991-1-4 ANZ (h/r=1) ANZ (h/r=2) Can ASCE Cook EN13031 Roof zone Fig. was 0. (2004). 4. of 1 s full scale) were determined at each roof tap location using the extreme value methodology of Mayne and Cook (1979) and Cook (1982). Comparison between Cpe values for curved roofs for h/d ¼ 0. This meant that pressures could only be measured simultaneously over each 100 mm long section of model at any one time. Blackmore. and the R6 models had 55. k.6 -0.25 Hz at full scale. The models tested had rise height/width ratios (f/d) from 0. Aerodyn. Wind Eng.6 mm. Tsokri / J. maximum and minimum external peak gust pressure coefficients on a range of curved roof models. 3. The reference wind velocity was 10 m/s at a model scale height of 200 mm.2 -1. Table 1 shows the full range of models tested. E.3. The boundary layer simulation was representative of open country terrain with an integral length scale of approximately 1:250. equivalent to 0. Data were sampled at 400 Hz. Measurements were also made without sand roughened surfaces to investigate the effect of subcritical flow. L. 3 boundary layer wind tunnel to measure mean. t. Measurements were made at 151 increments of .0—see definition sketch in Fig. The model linear scale was 1:250. Only one pressure tapped model section was constructed for each roof shape. nonpressure tapped dummy models of the same shape were used.A.5 and wall height/width ratios (h/d) from 0. 94 (2006) 833–844 Windward zone A 0. Ind.1. Blackmore. Wind Eng. 94 (2006) 833–844 837 Fig. Area-averaged pressures were obtained for each of the zones shown in Fig.0. Wind normal to eaves 4.ARTICLE IN PRESS P. 4. wind direction. 4.1. Tsokri / J. Aerodyn. Pressures on the windward zone ‘a’ Fig. E. 4. 4. Definition sketch with key to loaded areas. Results The results presented here are limited to area averaged pressures over the zones shown in Fig. 5 shows the measured worst case positive pressure coefficients averaged over zone ‘a’ for wind directions 07451 for L/b ¼ 1. 4 by averaging the area weighted pressure-time histories at the tap locations within each zone and performing an extreme value analysis on the averaged pressure signal. Also included for comparison are data from . The peak pressure coefficients were then converted to pseudo-steady coefficients by dividing by ^ ¯ K2 (where K is the gust factor ¼ V ðt ¼ 1Þ=V ) given in Table 1.A.1. 20 1.0. Ind. 5.5) ASCE windward (h=0) ASCE windward (h/d>0) Cook zone a (h=0 0.687 2.0.12 1.20 10. There is a weak dependence on h/d in the measured data but for practical purposes this is so small that it can be disregarded.931 1.35 rise height ratio (r/d) 0.00 0.50 1.15 0.3 0.6 Cpe 0.2 0 0 0. which gives a high degree of confidence in the measurements.05 0.05 0.5) Measured zone a (h/d=1) EN zone A (h=0) EN zone A (h/d=0. The data presented in Fig.675 1. Blackmore.5 h/f 1. .67 3. The measured data tend to agree closely with the EN1991-1-4 and ASCE data for h/d ¼ 0.50 1. Aerodyn.06 0. but is quite different from EN1991-1-4 and ASCE which both have a strong dependency on h/d.5 0.2 0. E.00 2. This h/d independence is consistent with the ANZ code. 5 are for L/b ¼ 1.838 1.834 1.00 20.05 0. for codification purposes.5 Fig.00 f/d 0.00 0. other published sources. Therefore.50 1.00 2. based on the experimental data it seems to be reasonable.50 1.633 1.699 1.ARTICLE IN PRESS 838 P.33 0.0.06 0.05 0. The measured data (not presented here) are also similar for L/b ¼ 2. Wind Eng. Tsokri / J.06 0.5 0.3 0.724 1. Positive pressure coefficients on windward zone ‘a’ for wind dir 07451. 4.819 1.06) Measured zone a (h/d =0.00 0.45 0.170 1. The measured data also agree quite closely with the data from Cook for h/d ¼ 0.1 0.4 0.A.00 0.06 0.0 and 10.1 0.3 0.4 0.60 5.8 0. 94 (2006) 833–844 Table 1 Model configurations tested (all dimensions in mm) Model number Dimensions h R3B1 R3B2 R3B3 R4B1 R4B2 R4B3 R5B1 R5B2 R5B3 R6B1 R6B2 R6B3 6 50 100 6 50 100 6 50 100 6 50 100 d 100 100 100 100 100 100 100 100 100 100 100 100 f 5 5 5 10 10 10 30 30 30 50 50 50 Ratios h/d 0.597 Gust factor K 1 Measured zone a (h=0. to assume that positive pressures in windward zone ‘a’ can be considered to be independent of h/d and L/d.1 0.3 0.00 0.098 1.25 0.00 10.1 0. 6 shows the measured worst case negative pressure coefficients averaged over zones ‘a+b’.3. 4.15 0.2. the EN1991-1-4 Cpe values are the same as those at f/d ¼ 0.0 and 10.5) Measured data zone a+b (h/d = 0. also supported by the data from Cook. Ind.2 -0.5) Measured data zone a+b (h/d = 1) Cook zones a+b (h/d = 0. Wind Eng. At f/d ¼ 0.ARTICLE IN PRESS P.2 is probably incorrect.45 0. Negative pressure coefficients on windward zones ‘a+b’ for wind direction 07451. Suctions on the windward zones ‘a+b’ Fig. Suctions on the leeward zones ‘e+f’ Fig. 4. so it appears to be a real effect. this trend is not supported by any other data. The measured suctions on the windward zones ‘a+b’ reduce with increasing f/d. the measured suctions generally seem to be independent of f/d up to f/d ¼ 0. 4. For larger f/d the measured suctions show a sharp increase.25) Cook zones a+b (h/d = 0. The ASCE values show the opposite trend of decreasing suction with increasing f/d. In the central zone ‘c+d’.2 -1. for L/d ¼ 1. There is a clear dependency on h/d. .4 0 0.2. The data from Cook for f/dX0. Tsokri / J. 7 shows the measured worst case negative pressure coefficients averaged over central zones ‘c+d’.0.A. for L/d ¼ 1.3 0.05 0. Aerodyn.5.4 0. Suctions on the central zones ‘c+d’ Fig. the measured suctions tend to increase with f/d.1. but it is consistent with measurements made on other models with L/d ¼ 2.2.1 0. whereas EN1991-1-4 has a single curve for all h/dX0. 94 (2006) 833–844 839 4. although the measured data are of the order of 50% smaller than the EN values. However.0.6 -0. so it is speculated that the large suction given in EN1991-1-4 at f/d ¼ 0. A similar trend is also observed with the EN1991-1-4 data. In the leeward zones ‘e+f’.1.8 -1 -1. Blackmore.35 0. for wind directions 07451.2.0.4. 8 shows the measured worst case negative pressure coefficients averaged over the leeward zones ‘e+f’ for wind directions 07451. 0 -0. It is not clear at this stage what the cause of this.0. the trends in the measured data and the data from other sources agree quite well at f/d ¼ 0.0. Unfortunately this cannot be validated from the BRE measurements because no data were measured for f/d ¼ 0.1.4 -0. This is generally similar to the EN behaviour. These zones are broadly comparable with those used in EN1991-1-4.25 are similar to the EN values.5 Cpe EN zone A (h/d > = 0.5) ASCE windward (h>0) Rise height ratio (r/d) Fig. for wind directions 07451 for L/d ¼ 1.3.1. 6. E.25 0.2 0. 5 -0. 2.5) EN zone C Cook (h/d=0) Cook (h/d=0. Blackmore.2 0.0 and 10.45 0.e. Aerodyn. Negative pressure coefficients on leeward zones ‘e+f’ for wind dir 07451. The suctions measured on windward zones ‘a+b’ are generally fairly similar at all positions along the length of the roof.0 and 10.25 0.0 to 2.35 0. Wind Eng. Ind.05 0.4 -0.5) -0.6 -0.6 Measured datazone e+f (h/d=0. 9 and 10 show wind tunnel measurements over windward zones ‘a+b’ and central zones ‘c+d’. 7.0 it was found that the positive pressures on the windward zone ‘a’ and the suctions in leeward zones ‘e+f’ appear to be independent of L/d. 94 (2006) 833–844 0 0 -0.15 0.0 to 4.2 0.2 Cpe -0. From Fig.05 0. .A. for L/d of 1.25 to account for increasing L/d. However.35 0.0.8 -1 -1.4 0. 4. E. 4.1 0. 8. Figs.25) Measured datazone e+f (h/d=0.06) Measured datazone e+f (h/d=1) ASCE Leewardzone Cook (h/d=0. From measurements on models with L/d ¼ 2. 0 0 0.0.5 Measured data zone c+d (h/d=0.0. respectively.ARTICLE IN PRESS 840 P.0. the suctions in windward zones ‘a+b’ and the central zones ‘c+d’ increase with increasing L/d.25 0.0.0.4 0. 5–8 and the above discussion is based on models with L/d ¼ 1. i.3 0. Tsokri / J. Negative pressure coefficients on middle zones ‘c+d’ for wind dir 07451.8 Rise height ratio (r/d) Fig.4 0.3 0.45 0.5) Rise height ratio (r/d) Fig.5) EN zone B Cook (h/d=0) Cook (h/d=.15 0.06) Measured data zone c+d (h/d=1) ASCE Middle zone Cook (h/d=.1 0. there is a non-linear increase in suction with L/d. Figs.4 Cpe -0.2 -0.2 -1.25) Measured data zone c+d (h/d=0. The ANZ code applies a factor ¼ (L/d)0. 9 it can be seen that the suction on zones ‘a+b’ increases by about 10% for each increase in L/d from 1. It appears therefore that the EN values for zone B are more appropriate for long buildings with two-dimensional flow rather than shorter buildings which generate three-dimensional flow. 10). This is evidence that the flow is changing in nature from three-dimensional flow on the end sections to twodimensional flow in the central section.35.L/d=2 Measured data.6 -1.3 0.4 -0.5) Measured data.ARTICLE IN PRESS P. are similar to the wind tunnel measurements made at l ¼ 0.4 -1. For example.L/d=4 Measured data. 10 shows that the suction on central zones ‘c+d’ generally increases with increasing L/d. . Ind.05 0. Fig.e.35 0.6 Cpe -0. 9.6 Cpe -0. 10. L=1000 Rise height ratio (f/d) Fig.1 0.35 0. L=200 Measured data.1L (measurement P1 in Fig.05 0.4 0. 94 (2006) 833–844 841 0 0 -0.45 0. are highly dependent on the position along the roof. at the gable end of the roof at section l ¼ 0.5L.L/d=10 (P5) P3 P4 P5 Rise height ratio (r/d) Fig.2 0. Wind Eng.15 0.3 0.5L.4 0. Effect of building length on pressure coefficients on central zones ‘c+d’. The EN1991-1-4 zone B values shown in Fig.A. Aerodyn. l.15 0.2 -0. Effect of building length on pressure coefficients on windward zones ‘a+b’. 10) Cpe ¼ À1.5 EN zone A (h/d>=0. 10.8 -1 -1.L/d=10 (P2) Measured data. L=100 Measured data.1 0. The suctions on the central roof zones ‘c+d’.8 0. 0 -0.5L on the central zone ‘a+b’.5 P1 P2 EN zone A (h/d >=0.L/d=10 (P4) Measured data. i.45 0.4 -0.2 0.L/d=10 (P1) Measured data.1L to 0.L/d=10 (P3) Measured data. the suction more than doubles as l increases from 0.2 -1.25 0.L/d=1 Measured data. Blackmore.8 -1 -1. whereas halfway along the roof at the mid-section where l ¼ 0.25 0. Cpe ¼ À0. Tsokri / J.2 -1. E.2 -0. (measurement P5 in Fig.5) Measured data.4 0. of the 1000 mm long model. again the increase is non-linear.62. L=400 Measured data. e.5d).5d). therefore for the curved roof data the worse case values for all wall heights have been used (the measured data are actually only very weakly dependent on h so this seems to be a reasonable assumption).A. wind direction 907451.2 -1. of the curved roofs has been determined by drawing a straight line from eaves to ridge. Wind Eng. 4. i.2 Zone I -0. a.4 -1. for the purposes of codification of wind pressures on curved roof buildings. with data given in EN1991-1-4 for duo-pitch roofs. 11 shows a comparison between measurements made on the curved roof models for L/d ¼ 1. it is judged. The pitch angle a of the curved roofs should be taken as a ¼ tanÀ1(f/0. Tsokri / J.8 Cpe. This is probably caused by vortices shed from the ridge of the duopitch roofs which does not occur on curved roofs.0. However.2.ARTICLE IN PRESS 842 P. Blackmore.4 -0. some codes.e. Comparison between curved roof measurements and duo-pitch data for wind direction 901. Here the duopitch suctions become increasing larger than the measurements as the pitch angle increases.8 0 5 Zone F solid symbols . The codified duo-pitch data are not given as a function of side wall height h.0. such as ANZ treat curved roofs for this wind direction as if they were duo-pitch roofs. Wind parallel to the eaves EN1991-1-4 does not give any guidance for wind blowing onto the eaves of curved roofed buildings.measurements on curved roofs 10 15 20 25 30 35 40 45 Roof pitch (average roof pitch for curved roofs) e/4 wind F H G = 90° G H e/4 F I ridge or trough I 50 Key to roof zones Fig.6 -0. with minor changes. The appropriate pitch angle. 11 that there is reasonable agreement between the measured data and the EN duo-pitch data for all roof zones except zone G. i. a ¼ tanÀ1(f/0. Ind. It can be seen from Fig. . Fig. that it is reasonable to use the EN duopitch data for the wind parallel to the ridge. the EN1991-1-4 values give a safe envelop to the wind tunnel measurements for L/dp1. 11.10 Zone H -1 Zone G -1.EN1991-1-4 duopitch data open symbols. Aerodyn. 94 (2006) 833–844 -0. 5. Proposed variation to the EN1991-1-4 recommended method From the results presented it can be seen that in most cases. E.6 -1. 25 0. The suctions on zones A and B increase with increasing L/d.15 0. 1987a. E. References ASCE Standard.05 Zone A for all h/d Cpe. Aerodyn.2:2002. 12 is for L/dp1. Fig.4 0. Fig. Structural design actions. AS/NZS 1170.2 0 -0.8 -1 -1.35. 7. For zone B this effect can be accounted for by multiplying the values for L/d ¼ 1 by (L/dÀ2)0. The effective pitch angle a of the curved roofs should be taken as a ¼ tanÀ1(f/0. Blackmore. June 2002. For zone A this can be accounted for by subtracting the factor (1À(L/d)0. Conclusions A series of parametric measurements have been carried out at BRE to measure wind pressures on curved roofs. although in some cases. the EN1991-1-4 recommended procedure requires some modifications in order to give safe and economical design values of wind pressure. Ind. for the purposes of codification the EN1991-1-4 data for duo-pitch roofs may be used to determine wind pressures on curved roofs. . Part 2: wind actions.5 Zone C Zone B 0. For wind angle 907451 (wind parallel to the ridge). for the purposes of codification a simple revised version of the EN approach might be more appropriate. Blessmann.35 Rise to width ratio (r/d) 0. Wind Eng. Tsokri / J.3 0. 12 shows a proposed modified version of EN Fig. 6.2 0. ANSI/ASCE 7-95.2 -1.5d). Proposed revision to EN199-1-4 Fig. However. Caderno Tecnico CT-86.11 based on the wind tunnel measurements.4 0.6 -0. Significant reductions in wind loads could be made by developing a more sophisticated model based on f/d and h/d ratios.45 0. 1a Parte. American Society of Civil Engineers.5 Fig.1 0.11 for curved roofs.0. J. particularly for the zone B values these values can be very conservative. 94 (2006) 833–844 843 1 0. The positive values in zone A and the suctions in zone C do not require correction for L/d.10 Zone A for h/d≥ 0.6 0. Minimum design loads for buildings and other structures.8 0.A. Acao do vento em coberturas curvas. The proposed revision is shown in Fig.2 -0.4 -0. Universidade Federale do Ria Grande do Sul.4 0. Porto Alegre.ARTICLE IN PRESS P. 12.. June 1996. 7. 12.25) from the values for L/d ¼ 1. Australian/New Zealand Standard. The main conclusions that can be drawn from this study are: For wind angle 07451 (wind normal to the eaves). CO. Cook. Universidade Federale do Ria Grande do Sul. J. Cranfield. Porto Alegre.ARTICLE IN PRESS 844 P.J.. 94 (2006) 833–844 Blessmann. N. Cook.J. September 2004. E. Part 2.. J. P. Latest draft..R. G. BS EN13031-1. 2004. Aero. Caderno Tecnico CT-88. Tsokri / J. In: Proceedings of Fifth International Conference on Wind Engineering. J. Ind. analysis and application of wind loading data. 10. NRC-CNRC. Mayne. N. Calibration of the quasi-static and peak-factor approaches to the assessment of wind loads against the method of Cook and Mayne.. 1987b. Greenhouses—design and construction: Part 1: Commercial production greenhouses. Tsokri. September 2004. Wind Eng. Butterworths. 6th UK Wind Engineering Society Conference. E. Canadian Commission on Building and Fire Codes. 1995. Vento em coberturas curves—pavilhoes vizinhos. The designers guide to wind loading of building structures.. Eurocode 1: Actions on structures—Part 1–4: General Actions—wind actions. Blackmore. Acquisition. . Ind. N. April 2002...J. Aerodyn. User’s Guide—NBC 1995 Structural commentaries (Part 4). USA. 1979.. 1982.A. Blackmore. Wind Eng. Wind tunnel tests on low-rise cylindrical roofs. National Research Council of Canada. Breeze. EN1991-1-4. 2001. CEN. 315–341. Cook.