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Wind direction effects on cladding and structural loads
Wind direction effects on cladding and structural loads E m i l Simiu
Center for Building Technology, National Bureau of Standards, Washington, DC, USA J a m e s J. Filliben
Center for Applied Mathematics, National Bureau of Standards, Washington, DC, USA (Received February 1980; revised August 1980)
A simple procedure is proposed for estimating wind loads corresponding to various return periods, which takes into account directional information on both wind speeds and aerodynamic response. Examples of the application of the procedure are given, which show that cladding loads calculated without taking directional information on extreme wind speeds into account may in certain cases be larger than the actual loads by a factor of two or more. It is also shoran that it is not appropriate, in general, to account for wind direction effects by multiplying loads determined without regard for these effects by a reduction factor of 0.8, as has been suggested in the literature. In its present form, the procedure is applicable to cladding panels and to members of relatively rigid structures in regions not subjected to hurricane winds. Introduction Wind direction is, in general, an important design factor for both climatological and aerodynamic reasons. As far as the climatology of extreme winds is concerned, it is known that as a result of basic atmospheric circulation patterns the distribution of extreme wind speeds with respect to direction is in general nonuniform. Aerodynamically, the effect of a wind flow associated with a given gradient speed depends (a), upon the nature of the terrain - which may vary with direction and (b) upon the orientation of the structure with respect to the mean flow. Information on the directional characteristics of the extreme wind climate in the United States can be obtained for a large number of stations from records available at the National Climatic Center (National Oceanic and Atmospheric Administration). Information on the dependence of aerodynamic forces upon direction can be obtained, for various types of structure, from full-scale or laboratory experiments. The purpose of this paper is to present a simple procedure for estimating design wind loads corresponding to any specified mean recurrence interval by combining these two types of information. The procedure is applicable to building components and structural members that do not exhibit any significant aeroelastic effects or resonant amplification of the response to the wind load, e.g., cladding panels, exterior walls, rigid roof systems, and structural members of low- and medium-rise buildings or other relatively rigid structures. Examples of the application of the procedure presented
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here will then be given for a few cases of interest in design practice. The numerical results obtained in these examples will be used in an attempt to assess various approaches to the question of how wind direction effects should be accounted for in developing specifications on design wind loads. In particular, it will be shown that the approach employed in reference 2 may lead in certain cases to significant overestimates of the wind loads corresponding to any specified mean recurrence interval. It will also be shown that, on the other hand, it is legitimate to question the appropriateness of accounting for wind directionality by using a blanket reduction factor of the order of 0.8, as suggested in references 3 and 5. The procedure presented in this paper is based on the description of the wind climate by series of recorded largest annual speeds blowing from various directions at any given station. While such a description is useful at stations where the wind climate is well-behaved, it may be misleading in regions that may experience hurricane winds or winds associated with other types of exceptional storms. In these regions a modified form of the procedure presented here must be employed. Wind d i r e c t i o n a n d e x t r e m e w i n d speeds Data published by the National Oceanic and Atmospheric Administration (NOAA) do not include sufficient information to allow the description of the extreme wind speed climate as a function of direction. However, such information can be obtained for a large number of stations from
Eng. Struct., 1981, Vol. 3, July 181
Wind direction effects: E, Simiu and J. J. Filliben Table 1 Largest annual fastest-mile wind speeds at 10 m above ground in open terrain Sheridan, Wyoming, 1958-1977
Huron, South Dakota, 1958-1977
Toledo, Ohio, 1959-1977
N
NE
E
SE
S
SW W
NW
N
NE
E
SE
S
SW W
NW
N
NE
E
SE
S
SW W
NW
28 41 36 25 22 31 22 33 36 44 36 28 28 33 23 28 24 22 31 44
20 25 21 18 23 14 15 31 21 14 19 16 13 15 19 23 28 22 24 20
23 19 16 30 22 23 18 20 19 16 19 15 20 22 26 19 19 19 28 19
50 29 34 36 36 33 34 33 34 40 35 36 35 31 36 32 37 27 33 40
23 25 26 27 16 36 19 18 14 36 21 22 37 22 37 18 25 28 38 36
50 40 43 47 47 63 54 67 51 51 39 54 61 49 55 46 57 39 47 34
51 38 45 38 52 48 54 43 45 42 40 34 37 31 44 39 49 33 33 44
70 61 61 60 61 57 61 55 61 62 47 67 54 47 47 65 56 51 47 56
34 56 41 34 32 36 38 38 55 36 44 55 36 39 43 47 39 35 46 44
40 25 33 47 37 39 35 35 32 51 49 42 30 37 27 37 48 28 36 49
37 25 35 26 51 48 35 25 34 30 27 26 26 37 25 34 28 36 27 28
32 31 63 42 46 40 38 39 36 42 39 38 36 37 34 49 46 61 36 39
48 49 45 41 51 44 46 38 37 42 42 38 46 49 47 38 39 37 44 38
31 45 29 63 44 28 56 46 37 38 37 37 39 42 34 34 77 46 59 49
56 55 52 45 71 65 62 51 59 62 79 61 46 47 55 50 47 57 48 49
37 25 34 33 28 28 32 33 38 34 26 23 22 25 30 23 28 27 37
31 30 30 38 26 37 35 34 28 36 33 39 25 30 23 30 29 26 29
32 32 31 24 22 28 26 30 27 25 18 24 25 26 26 20 21 26 23
34 23 23 21 39 32 18 21 31 25 35 26 17 28 26 24 35 18 39
20 35 22 28 33 31 24 27 25 23 27 24 29 20 35 25 26 32 26
42 38 39 43 46 40 45 34 61 44 36 44 46 36 44 41 44 42 39
39 37 51 39 42 39 51 36 42 42 51 36 48 48 39 39 43 39 43
44 38 32 56 38 37 42 38 34 34 54 55 43 39 39 39 51 34 28
Vma x
31 7.0 44
20 4.7 31
21 3.9 30
35 4.8 50
26 7.9 38
50 8.4 67
42 6.7 54
57 6.8 70
41 7.2 56
38 7.9 51
32 7.4 51
41 8.4 63
43 4.6 51
44 38 12.6 9.7 77 66
56 8.7 79
30 4.9 38
31 4.1 37
26 3.7 32
27 6.8 39
27 4.5 35
42 5.8 61
42 5.1 51
41 7.9 56
All directions
V = 59;s(V) = 6.5
s(V)
~'= 60; s(V) = 8.8
unpublished NOAA records. Given in Table 1 are largest annual fastest-mile wind speeds blowing from each octant at 10 m above ground in open terrain in mph (1 mph = 0.447 m/s) obtained for three stations from NOAA records. Also given in Table i are summary statistics including the sample mean, the sample standard deviation, and the maximum value of the largest annual fastest-mile wind speeds blowing from each octant, V, s(V), and Vmax, respectively. The last line of Table l includes sample statistics for the samples of data, Via, ( / = 1, 2 . . . . , n where n = sample size), defined as follows: V~ = max[ Vj(ai)]
(1)
in which Vi(ai) = largest annual fastest-mile wind speed blowing from octant i (i = 1, 2 , . . . , 8), i.e., for any given j, Vfl is the largest of the speeds Vj(ai). The wind speeds V~ (] = 1, 2 . . . . , n) are referred to as the set of largest annual speeds from any direction. For example, at Huron, the largest annual speeds from any direction for the years 1958-1961, which correspond to the first four lines of data in Table I, are 56, 56, 63, and 63 mph. Note that the characteristics of the extreme winds depend strongly upon direction at each of the three stations of Table 1. For example, east winds are considerably weaker tha northwest or southwest winds in all three cases. If it can be determined that the terrain roughness depends upon direction at the anemometer site, the data should be corrected accordingly. This can in principle be done on the basis of the short-term wind profile measurements, or from comparisons of short-term records of wind speeds at upper elevations with the surface data. 6 The dependence of the extreme wind speeds upon direction can in principle be described by fitting a joint probability distribution of the wind speeds and directions to data such as those of Table 1. Research on the appropriate functional form of such joint distributions is, to date, quite scant. Indeed, to the writers' knowledge, no model
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Eng. S t r u c t . , 1981, V o l . 3, J u l y
29 44 30 41 66 54 36 24 49 37 33 36 36 40 38 37 26 37 32 32
Q = 47;s(V) = 6.5
for the joint distribution of wind speeds and directions is currently available that would satisfy the requirements of (a), providing an adequate fit to actual data and (b), yielding, by integration with respect to the azimuthal parameter, a marginal distribution consistent with accepted probabilistic models of the extreme wind speeds from any direction (e.g., the extreme value type I or the Weibull distribution). It will be shown subsequently that a rigorous statistical analysis of wind forces on cladding, exterior walls, rigid roofs, and structural members of low- and medium-rise structures or of other relatively rigid structures, can be carried out without any recourse to joint probability distribution models of the wind speeds and directions.
Wind direction and a e r o d y n a m i c forces As previously mentioned, the aerodynamic forces induced on a structure by a flow associated with a given gradient speed will in most cases depend upon the flow direction, owing to the following two factors: (1) The oncoming flow in the atmospheric boundary layer depends significantly upon the terrain features upwind of the structure (i.e., the terrain roughness, or the possible presence of obstructions). If the terrain features vary with direction, so will the oncoming flow and, consequently, the aerodynamic forces on the structure. (2) Even if the terrain features around the structure are the same for all azimuths, the magnitude of the aerodynamic forces will depend upon the orientation of the structure with respect to the mean flow, except in the case in which the structure has a circular shape in plan. With few exceptions, aerodynamic forces on structures and their dependence upon direction cannot be determined analytically and must therefore be obtained by measurements. The peak aerodynamic force (or pressure) acting on a component or member, and induced by a flow with mean
Wind direction effects: E. Simiu and J. J. Filliben 0.158
speed blowing from octant i (i = 1, 2 . . . . . 8) can be expressed as:
p~(~i) aCp(~i) v~(~i)
where Cp(cq) = peak force (or pressure) cofficient, Vt((~i) = mean wind speed at top of boundary layer, and a = factor with the dimensions ofpp/V~. Coefficients Cp(~i) are given in Figures I to 4 for the following four cases. Case I [1, p. F19] (Figure 1) corresponds to the uplift force induced by wind on one column of an interior symmetrical frame in a building surrounded on all sides by open terrain. (Building dimensions: 30.5 m long, 24.4 m wide, 4.9 m eave elevation, 1 : 12 roof slope.) Note that, owing to the unsteady nature of the flow around the building, the wind induces an uplift force regardless of the direction from which it blows. Case II (Figure 2) corresponds, for the same building as in case I, to the larger of the uplift forces acting on either of the two columns of the frame. Case III (Figure 3) corresponds, approximately, to the compression force on the upper horizontal member of an end braced frame in a building with the same dimensions as the building of case I, but in which all lateral loads are transmitted to braced frames located at the end walls [reference 1, p. A4.1, columns Cp9 and Cplo]. Case IV (Figure 4) corresponds to the absolute values of the sections on the exterior wall at a point (identified as tap 127 in reference 2, p. 26) located 33.7 m above ground near the corner of a 101 m tall building in urban terrain. (For a description of the building and its surroundings, see reference 2). The coefficients shown in Figure 4 represent
0.3
0
0
,~
/
T
0158
0.162
l
0.41
~ T a p 127
2.90
1.47
\
T \
083 Figure 1 Coefficients Cp(ai) for case I
/
0.79
T \
Figure 2 Coefficients Cp(ai) for case II
Coefficients
Cp(c~i) for case IV
an envelope of coefficients measured for 24 mean wind directions (reference 2, pp. 72-114). 066
065
"
0 83
0.72
Estimation of aerodynamic forces corresponding to a specific r e t u r n period
083
0 52 - - ~
O. 65
T
05, ~
Figure 4
065
f
1.21
03
.'#
/
0.158
042
130
065
\
Figure 3 Coefficients Cp(c~i) for case III
/ 0.52
0.158
(2)
=
0.3
0.162
0.52
0 65
Current approaches Two approaches for estimating aerodynamic forces corresponding to a specified return period are documented in the literature. In one of these approaches, 2 it is assumed that the conditional probability distribution of the extreme winds, given that the wind blows from a specified octant, is the same as the probability distribution of the extreme winds from any direction (as defined in this paper immediately following equation (1)). The second approach requires the fitting of a joint probability distribution to the extreme wind data corresponding to various azimuth angles. From this joint distribution, and from aerodynamic information on the direction-dependent response, a joint probability distribution of response and direction is obtained. The probability distribution of the response is then, by definition, equal to the marginal distribution calculated by inte-
Eng. S t r u c t . , 1 9 8 1 , V o l . 3, J u l y
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W i n d d i r e c t i o n effects: E. S i m i u a n d J. J. F i l l i b e n
grating this joint distribution with respect to the azimuthal parameter.6 Proposed approach While the approach presented in this paper does incorportate directional information on extreme wind speeds, it does not require the estimation of joint probability distributions of wind speeds - or wind-induced effects and directions. Rather, a random variable is defined which consists of the maximum annual wind effect of concern (e.g., pressure, force, stress, and so forth), or of a function thereof suitably chosen for computational convenience. A cumulative probability distribution function (CDF) is then fitted to the data consisting of the values taken on by this random variable in a sufficiently large number of consecutive years. The wind effect with a N-year return period corresponds to the value of the variate for which the ordinate of the CDF is 1 - I/N. Application: estimated 50-yr pressure at a point on exterior wall o f a high-rise building To illustrate the application of the procedure, the 50-yr pressure will be sought at the exterior wall location (tap 127 in reference 2) for which the pressure coefficients are given in Figure 4. The following random variable will be used: rdir = max [C~/2 (ai) V(ai)] (3) It will be assumed that the structure is located in Huron, S.D. (see Table 1), and that the building of Figure 4 is so oriented that its short side is parallel to the north direction and the point under consideration is northeast of the centre of the building. The factors Cp(ai) and C~/2(~i) will then have the absolute values given in Table 2. Using these values in conjunction with the extreme wind speeds at Huron listed in Table 1, the following set of values of the random variable Ut i t is obtained : 62, 54, 59, 71,87, 82, 64, 52, 60, 56, 54, 46, 45, 63, 46, 58, 87, 61,67, 56 These data were subjected to a statistical analysis aimed at determining the best fitting Weibull, extreme value type I, or extreme value type II distribution and its parameters. The computer program used in this analysis was similar to that described in reference 4. It was found that the data were best fitted by an extreme type I distribution, which yielded d ~ r : 93.5 mph (41.8 m/s), where ~ r = estimated value of rair corresponding to a probability of exceedance 0.02 in any one year (i.e., a return period N = 50 yr). A random variable will now be defined, which is consistent with the procedure of reference 2, i.e., which, in contradistinction to r ctir, does not reflect climatological wind direction effects: ~od = max [C~/2(ai)] V~o
bution, it follows from the data of Table 1 that V~o = 82.5 mph (36.9 m/s), so that, with max[C~/2(a)] = 1.7 (Table 2), equation (4) yields rnso a = 140.2 mph (62.7 m/s). The ratio of the 50-yr pressure estimated by ignoring climatological wind direction effects, p~oa, to the 50-yr pressure estimated by taking these effects into account, dir Pso, is then, in this example: n d _ (140.2 Pso air \ ~ . 5 - ! Pso
nd P20 dir P20
Values of
- 2.34
P air 20e
= a max [Cp(ai) V2max(~i)]
(7)
it follows from Tables 1 and 2 that: nd P20 dir P 20e
= 2.22
(8)
This ratio is seen to differ rather insignificantly from the _ndt_dir ratio p:o/p2o - This is the case because the difference between the estimated 20-yr pressure, pd~r, and the extreme dir pressure, P2oe, that would actually have been recorded in the 20-yr period covered by Table 1, is only of the order of 5%. If we now define : P2oend = d
max [Cp (ai)] {max [ Vmax(ai)] }2
(9)
where pn2de = pressure calculated by ignoring climatological wind direction effects and based upon the maximum wind speed from any diiection actually recorded in the 20-yr period covered by Table 1, max[Vmax(C~i)], rather than upon the estimated value V~o, we obtain the result: nd
P 2oe dir
- 2.37
(10)
P 2oe i.e., this ratio differs by less than 2% from the ratio given by equation (6). This shows that, in this example, deviations from actual data that are inherent in the estimates p dir A 20 un the one hand and P2o na on the other hand are of the same order of magnitude.
Additional case studies: dir
al-
I Cp(ai)land C~2(ai )
N
NE
E
SE
S
SW
W
NW
ICp(ai)l C~/Z(ai)
0.42
1.21
2.90
0_72
0,79
1.30
1.47
0.41
0.65
1.10
1.70
0.85
0,89
1.14
1,21
0.64
Pso nd P5o
dir P20
na
a3 = a4 -
P 2oe nd P2o dir P20e
na
P2oe
Eng. Struct., 1981, Vol. 3, July
(6)
(lla) (1 lb)
P2o dir
ai
184
(5)
ncl It is of interest to compare the 20-yr pressure, P2o, estimated by ignoring climatological wind direction effects, to the maximum pressure, pa~e , that would actually have occurred during the 20-yr period covered by Table 1. Since:
a2Table 2
= 2.25
i.e., if the procedure of reference 2 - which ignores climatological wind direction effects - were used, the 50-yr pressure would be overestimated by 225%. It can be verified that for a 20-yr return period:
(4)
where V~o = estimated value of the 50-yr wind speed from any direction. Assuming that the extreme winds from any direction are described probabilistically by a type I distri-
12
(1 lC) (11d)
W i n d d i r e c t i o n e f f e c t s : E. S i m i u a n d J. J. F i l l i b e n
Ratios al to a4 (for the Toledo station subscript 20 must be replaced by subscript 19) were calculated in the cases previously designated as I, II, III and IV (Figures 1, 2, 3 and 4, respectively) for various building orientations and wind climates. The results of the calculations are given in
into account the directional characteristics of the wind
climate is clearly quite important for at least five of the eight possible orientations of the building, even if allowance is made for possible sampling errors. This effect is seen to be much less pronounced in cases II and III, and to some extent in case I, owing to the less peaked character of the variation of Cp with direction in these cases. As previously mentioned, it was suggested 3,s that it is appropriate to account for wind direction effects - particulady for the purpose of developing building code wind loading provisions - by multiplying loads determined without regard for these effects by a reduction factor of about 0.8. The results of Table 3 suggest that this approach is in general not justified. For example, if this approach were adopted, the loads corresponding to case II (Sheridan) would be underestimated by 20% for all buildings with a SE (or NW) orientation of the short side and by about 12% for all buildings with a NE (or SW) orientation. Thus, if a uniform distribution of the building orientations were assumed, the use of a reduction factor of 0.8 would lead to the underestimation of the wind loads on half of the buildings of the aerodynamic type represented in Figure 2. Clearly, the underestimation would be even more severe in case III. Table 3 also shows averages of the quantities al to a4 for all directions. In this connection it should be emphasized that the use of such averaged reduction factors could well be unacceptable as far as the safety of buildings with certain unfavourable orientations is concerned. For example, in case IV, if the average reduction factor 0.65 were used, the cladding load would be severely underestimated for all buildings oriented with the short side in
Table 3.
Interpretation of r e s u l t s Note that in all cases the differences between the values o f al and a2 are small for practical purposes. However, the differences between the values of a2 and a3 are in certain cases significant (e.g., case II, NE; case III, NE). The differences between the values of a2 and a4 are somewhat less than those between a2 and a3, but are still relatively large in a few cases. The significance of the differences between a2 and a 3 is the following. If a2 < 1, it would in theory be permissible to reduce the load obtained as in reference 2 (i.e., ignoring climatological wind direction effects) by using the reduction factor a2. However, if a3 > a2, the load so reduced would be smaller than the actual load that would have occurred during the period under consideration. In view of the errors inherent in the statistical handling of the data it may therefore be prudent, judging from the examples of Table 3, to use for the design of structures with known orientation reduction factors that are larger than the estimated ratios a2. In this context, the need for a study of sampling errors in the estimation ofp~ r for various values of N is obvious. Such a study would have to be conducted for a large variety of aerodynamic and climatic situations. Note, however, that in the case of cladding loads at a specified point (case IV of Table 3) the effect of taking Table3
Ratiosa~,a2, a 3 a n d a 4 Direction of short side of building N
NE
E
SE
Sa
SW a
Wa
NW a
Average for all directions
0.75 0.74 0.85 0.81
0.59 0.59 0.67 0.64
0.53 0.53 0.53 0.51
0.64 0.62 0.67 0.63
0.72 0.71 0.80 0.75
Case I (Figure I) Huron
aI a2 a~ a4
0.66 0.65 0.79 0.75
0.84 0.81 1.00 0.95
0.80 0.79 0.79 0.75
0.94 0.94 1.07 1.00
Case II (Figure 2) Toledo
a~ a2 a~ a,
0.72 0.76 0.87 0.80
0.78 0.83 1.09 1.00
0.82 0.86 0.87 0.80
0.86 0.89 0.91 0.84
0.80 0.84 0.94 0.86
Case II (Figure 2) Sheridan
al a2 a3 a4
0.79 0.80 0.78 0.79
0.85 0.84 0.89 0.91
0.79 0.80 0.78 0.79
1.00 0.98 0.98 1.00
0.86 0.86 0.86 0.88
Case I I (Figure 2) Huron
a~ a2 a3 a4
0.77 0.78 0.85 0.79
0.82 0.79 1.00 0.95
0.82 0.80 0.85 0.79
0.97 0.95 1.07 1.00
0.85 0.83 0.94 0.88
Case I I I (Figure 3) Huron
ai a2 a3 a4
0.94 0.94 1.00 0.94
0.80 0.79 1.00 0.94
0.96 0.95 1.00 0.94
0.96 0.94 1.07 1.00
0.92 0.91 1.02 0.96
Case I V b (Figure 4) Huron
a1 a2 a3 a4
0.44 0.43 0.45 0.42
0.62 0.60 0.67 0.63
0.48 0.46 0.45 0.42
0.75 0.82 1.00 0.94
0.58 0.60 0.74 0.72
0.90 0.96 1.06 1.00
0.52 0.53 0.53 0.50
0.54 0.57 0.60 0.56
0.60 0.62 0.69 0.65
a If no entry is shown, values are same for S and N, SW and NE, W and E, NW and SE, respectively b Point under consideration (pressure tap 127, see reference 2) is in the NE-E octant when short side is parallel to north direction, in E-SE direction when short side is parallel to northeast direction, etc.
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Wind direction effects: E. Simiu and d. J. Fill/ben
the SE or SW direction, with the possible consequence that, for these buildings, the number of cladding failures due to wind loading would be unacceptably large.
Conclusions A simple procedure was proposed for estimating wind loads corresponding to various return periods, which takes into account directional information on both wind speeds and aerodynamic response. Examples of the application of the procedure were given, which show that (1), cladding loads calculated without taking directional information on extreme wind speeds into account may in certain cases be larger than the actual loads by a factor of two or more, and (2), it is not appropriate, in general, to account for wind direction effects by multiplying loads determined without regard for these effects by a reduction factor of about 0.8, as suggested elsewhere. 3,s The procedure presented in the paper is applicable to building components and structural members that do not exhibit any significant aeroelastic effects or resonant amplification of the response to the wind load, e.g., cladding panels, exterior walls, rigid roofs, and structural members of low- and medium-rise buildings or other relatively rigid structures. Also, the application of the procedure in its present form is limited to well-behaved wind climates, i.e., to regions where hurricane or other extraordinary winds may not be expected to occur or are not taken into account in design. (A modified version of the procedure, applicable to hurricane-prone regions, is currently being developed by the writers.) Information on the dependence of extreme wind speeds upon direction can be obtained for a large number of US weather stations from unpublished records of the National Oceanic and Atmospheric Administration. The writers believe that it would be useful (1), to extract and publish such information, and (2), to undertake on its basis studies of sampling errors in the estimation of the dependence of wind effects upon direction.
Acknowledgements The writers wish to thank M. J. Changery, of the National Climatic Center, National Oceanic and Atmospheric Administration, who provided the data used in Table 1, and M. E. Butts of the Center for Building Technology, National Bureau of Standards, for useful exchanges and valuable contributions to this paper. The work presented here was partly supported by the National Science Foundation under agreement No. ENV-7716113 and the Department of Energy, Office of Assistant Secretary, Conservation and Solar Applications. Any opinion, findings and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation or the Department of Energy.
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References 1 Davenport, A. G. et al. "Wind loads on tow-rise buildings: Final report of phases I and II, parts 1 and 2, BLWT-558-1977', Boundary Layer Wind Tunnel Laboratory, The University of Western Ontario, London, Ontario, Canada, 1977 2 Peterka, J. A. and Cermak, J. E. 'Wind tunnel study of Atlanta office building', Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, 1978 3 Ravindra, M. K. et al. 'Wind and snow load factors for use in LFRD', J. Struct. Div., ASCE, 14, (ST9), Prec. Paper 14006, September, 1978, pp. 1443-1457 4 Simiu, E. and Fill/ben, J. J. 'Probability distribution of extreme wind speeds', J. Struct. Div., ASCE, 102, (ST9), Prec. Paper 12381, September, 1976, pp. 1861-1877 5 Stathopoulos, T. Discussion of Paper, "Wind and snow load factors for use in LRFD', J. Struct. Div.. ASCE, 105, (ST10), October, 1975, pp. 2132-2133 6 Surry, D. and Davenport, A. G. 'Modeling the wind climate: an overview', Prec., Workshop on Wind Climate, Asheville, North Carolina, November 12-13, 1979, Texas Tech. University, Institute for Disaster Research, Lubbock, Texas, 1980
Notations a, d = factors in equations (2), (7) and (9) ax, a2, a3, a4 = ratios defined by equations (11) Cp = peak force or pressure coefficient dir = superscript indicating that climatological wind directionality effects are taken into account max[ ] = denotes maximum value nd = superscript indicating that climatological wind directionality effects are ignored N = return period p~r, p~va = wind effect (e.g., pressure, force) with an N-year return period, estimated by taking into account and by ignoring climatological wind directionality effect, respectively pdir _nd . . Ne, PNe = wlnd effect that would have actually occurred in given N-year record, calculated by taking into account and by ignoring wind directionality effect, respectively pp = peak aerodynamic force or pressure r air, r~ zr = random variable defined by equation (3), value of r air corresponding to a N-year return period, respectively r ha, r~ a = random variable defined by equation (4), value of r na corresponding to a N-year return period, respectively s(V) = sample standard deviation of V V, V, Vmax= wind speed, sample mean of V, sample maximum value of V, respectively Vi(ai) = largest annual fastest-mile wind speed observed in jth year of record from octant i Vja, V} = largest annual fastest-mile wind speed blowing in ]th year of record from any direction, fastest-mile with N-year return period blowing from any direction, respectively Vt = wind speed at top of boundary layer a i = denotes octant i (i = 1, 2 . . . . . 8)
Center for Building Technology, National Bureau of Standards, Washington, DC, USA J a m e s J. Filliben
Center for Applied Mathematics, National Bureau of Standards, Washington, DC, USA (Received February 1980; revised August 1980)
A simple procedure is proposed for estimating wind loads corresponding to various return periods, which takes into account directional information on both wind speeds and aerodynamic response. Examples of the application of the procedure are given, which show that cladding loads calculated without taking directional information on extreme wind speeds into account may in certain cases be larger than the actual loads by a factor of two or more. It is also shoran that it is not appropriate, in general, to account for wind direction effects by multiplying loads determined without regard for these effects by a reduction factor of 0.8, as has been suggested in the literature. In its present form, the procedure is applicable to cladding panels and to members of relatively rigid structures in regions not subjected to hurricane winds. Introduction Wind direction is, in general, an important design factor for both climatological and aerodynamic reasons. As far as the climatology of extreme winds is concerned, it is known that as a result of basic atmospheric circulation patterns the distribution of extreme wind speeds with respect to direction is in general nonuniform. Aerodynamically, the effect of a wind flow associated with a given gradient speed depends (a), upon the nature of the terrain - which may vary with direction and (b) upon the orientation of the structure with respect to the mean flow. Information on the directional characteristics of the extreme wind climate in the United States can be obtained for a large number of stations from records available at the National Climatic Center (National Oceanic and Atmospheric Administration). Information on the dependence of aerodynamic forces upon direction can be obtained, for various types of structure, from full-scale or laboratory experiments. The purpose of this paper is to present a simple procedure for estimating design wind loads corresponding to any specified mean recurrence interval by combining these two types of information. The procedure is applicable to building components and structural members that do not exhibit any significant aeroelastic effects or resonant amplification of the response to the wind load, e.g., cladding panels, exterior walls, rigid roof systems, and structural members of low- and medium-rise buildings or other relatively rigid structures. Examples of the application of the procedure presented
0141-0296/81/030181-06/$02.00 © IPC Business Press
here will then be given for a few cases of interest in design practice. The numerical results obtained in these examples will be used in an attempt to assess various approaches to the question of how wind direction effects should be accounted for in developing specifications on design wind loads. In particular, it will be shown that the approach employed in reference 2 may lead in certain cases to significant overestimates of the wind loads corresponding to any specified mean recurrence interval. It will also be shown that, on the other hand, it is legitimate to question the appropriateness of accounting for wind directionality by using a blanket reduction factor of the order of 0.8, as suggested in references 3 and 5. The procedure presented in this paper is based on the description of the wind climate by series of recorded largest annual speeds blowing from various directions at any given station. While such a description is useful at stations where the wind climate is well-behaved, it may be misleading in regions that may experience hurricane winds or winds associated with other types of exceptional storms. In these regions a modified form of the procedure presented here must be employed. Wind d i r e c t i o n a n d e x t r e m e w i n d speeds Data published by the National Oceanic and Atmospheric Administration (NOAA) do not include sufficient information to allow the description of the extreme wind speed climate as a function of direction. However, such information can be obtained for a large number of stations from
Eng. Struct., 1981, Vol. 3, July 181
Wind direction effects: E, Simiu and J. J. Filliben Table 1 Largest annual fastest-mile wind speeds at 10 m above ground in open terrain Sheridan, Wyoming, 1958-1977
Huron, South Dakota, 1958-1977
Toledo, Ohio, 1959-1977
N
NE
E
SE
S
SW W
NW
N
NE
E
SE
S
SW W
NW
N
NE
E
SE
S
SW W
NW
28 41 36 25 22 31 22 33 36 44 36 28 28 33 23 28 24 22 31 44
20 25 21 18 23 14 15 31 21 14 19 16 13 15 19 23 28 22 24 20
23 19 16 30 22 23 18 20 19 16 19 15 20 22 26 19 19 19 28 19
50 29 34 36 36 33 34 33 34 40 35 36 35 31 36 32 37 27 33 40
23 25 26 27 16 36 19 18 14 36 21 22 37 22 37 18 25 28 38 36
50 40 43 47 47 63 54 67 51 51 39 54 61 49 55 46 57 39 47 34
51 38 45 38 52 48 54 43 45 42 40 34 37 31 44 39 49 33 33 44
70 61 61 60 61 57 61 55 61 62 47 67 54 47 47 65 56 51 47 56
34 56 41 34 32 36 38 38 55 36 44 55 36 39 43 47 39 35 46 44
40 25 33 47 37 39 35 35 32 51 49 42 30 37 27 37 48 28 36 49
37 25 35 26 51 48 35 25 34 30 27 26 26 37 25 34 28 36 27 28
32 31 63 42 46 40 38 39 36 42 39 38 36 37 34 49 46 61 36 39
48 49 45 41 51 44 46 38 37 42 42 38 46 49 47 38 39 37 44 38
31 45 29 63 44 28 56 46 37 38 37 37 39 42 34 34 77 46 59 49
56 55 52 45 71 65 62 51 59 62 79 61 46 47 55 50 47 57 48 49
37 25 34 33 28 28 32 33 38 34 26 23 22 25 30 23 28 27 37
31 30 30 38 26 37 35 34 28 36 33 39 25 30 23 30 29 26 29
32 32 31 24 22 28 26 30 27 25 18 24 25 26 26 20 21 26 23
34 23 23 21 39 32 18 21 31 25 35 26 17 28 26 24 35 18 39
20 35 22 28 33 31 24 27 25 23 27 24 29 20 35 25 26 32 26
42 38 39 43 46 40 45 34 61 44 36 44 46 36 44 41 44 42 39
39 37 51 39 42 39 51 36 42 42 51 36 48 48 39 39 43 39 43
44 38 32 56 38 37 42 38 34 34 54 55 43 39 39 39 51 34 28
Vma x
31 7.0 44
20 4.7 31
21 3.9 30
35 4.8 50
26 7.9 38
50 8.4 67
42 6.7 54
57 6.8 70
41 7.2 56
38 7.9 51
32 7.4 51
41 8.4 63
43 4.6 51
44 38 12.6 9.7 77 66
56 8.7 79
30 4.9 38
31 4.1 37
26 3.7 32
27 6.8 39
27 4.5 35
42 5.8 61
42 5.1 51
41 7.9 56
All directions
V = 59;s(V) = 6.5
s(V)
~'= 60; s(V) = 8.8
unpublished NOAA records. Given in Table 1 are largest annual fastest-mile wind speeds blowing from each octant at 10 m above ground in open terrain in mph (1 mph = 0.447 m/s) obtained for three stations from NOAA records. Also given in Table i are summary statistics including the sample mean, the sample standard deviation, and the maximum value of the largest annual fastest-mile wind speeds blowing from each octant, V, s(V), and Vmax, respectively. The last line of Table l includes sample statistics for the samples of data, Via, ( / = 1, 2 . . . . , n where n = sample size), defined as follows: V~ = max[ Vj(ai)]
(1)
in which Vi(ai) = largest annual fastest-mile wind speed blowing from octant i (i = 1, 2 , . . . , 8), i.e., for any given j, Vfl is the largest of the speeds Vj(ai). The wind speeds V~ (] = 1, 2 . . . . , n) are referred to as the set of largest annual speeds from any direction. For example, at Huron, the largest annual speeds from any direction for the years 1958-1961, which correspond to the first four lines of data in Table I, are 56, 56, 63, and 63 mph. Note that the characteristics of the extreme winds depend strongly upon direction at each of the three stations of Table 1. For example, east winds are considerably weaker tha northwest or southwest winds in all three cases. If it can be determined that the terrain roughness depends upon direction at the anemometer site, the data should be corrected accordingly. This can in principle be done on the basis of the short-term wind profile measurements, or from comparisons of short-term records of wind speeds at upper elevations with the surface data. 6 The dependence of the extreme wind speeds upon direction can in principle be described by fitting a joint probability distribution of the wind speeds and directions to data such as those of Table 1. Research on the appropriate functional form of such joint distributions is, to date, quite scant. Indeed, to the writers' knowledge, no model
182
Eng. S t r u c t . , 1981, V o l . 3, J u l y
29 44 30 41 66 54 36 24 49 37 33 36 36 40 38 37 26 37 32 32
Q = 47;s(V) = 6.5
for the joint distribution of wind speeds and directions is currently available that would satisfy the requirements of (a), providing an adequate fit to actual data and (b), yielding, by integration with respect to the azimuthal parameter, a marginal distribution consistent with accepted probabilistic models of the extreme wind speeds from any direction (e.g., the extreme value type I or the Weibull distribution). It will be shown subsequently that a rigorous statistical analysis of wind forces on cladding, exterior walls, rigid roofs, and structural members of low- and medium-rise structures or of other relatively rigid structures, can be carried out without any recourse to joint probability distribution models of the wind speeds and directions.
Wind direction and a e r o d y n a m i c forces As previously mentioned, the aerodynamic forces induced on a structure by a flow associated with a given gradient speed will in most cases depend upon the flow direction, owing to the following two factors: (1) The oncoming flow in the atmospheric boundary layer depends significantly upon the terrain features upwind of the structure (i.e., the terrain roughness, or the possible presence of obstructions). If the terrain features vary with direction, so will the oncoming flow and, consequently, the aerodynamic forces on the structure. (2) Even if the terrain features around the structure are the same for all azimuths, the magnitude of the aerodynamic forces will depend upon the orientation of the structure with respect to the mean flow, except in the case in which the structure has a circular shape in plan. With few exceptions, aerodynamic forces on structures and their dependence upon direction cannot be determined analytically and must therefore be obtained by measurements. The peak aerodynamic force (or pressure) acting on a component or member, and induced by a flow with mean
Wind direction effects: E. Simiu and J. J. Filliben 0.158
speed blowing from octant i (i = 1, 2 . . . . . 8) can be expressed as:
p~(~i) aCp(~i) v~(~i)
where Cp(cq) = peak force (or pressure) cofficient, Vt((~i) = mean wind speed at top of boundary layer, and a = factor with the dimensions ofpp/V~. Coefficients Cp(~i) are given in Figures I to 4 for the following four cases. Case I [1, p. F19] (Figure 1) corresponds to the uplift force induced by wind on one column of an interior symmetrical frame in a building surrounded on all sides by open terrain. (Building dimensions: 30.5 m long, 24.4 m wide, 4.9 m eave elevation, 1 : 12 roof slope.) Note that, owing to the unsteady nature of the flow around the building, the wind induces an uplift force regardless of the direction from which it blows. Case II (Figure 2) corresponds, for the same building as in case I, to the larger of the uplift forces acting on either of the two columns of the frame. Case III (Figure 3) corresponds, approximately, to the compression force on the upper horizontal member of an end braced frame in a building with the same dimensions as the building of case I, but in which all lateral loads are transmitted to braced frames located at the end walls [reference 1, p. A4.1, columns Cp9 and Cplo]. Case IV (Figure 4) corresponds to the absolute values of the sections on the exterior wall at a point (identified as tap 127 in reference 2, p. 26) located 33.7 m above ground near the corner of a 101 m tall building in urban terrain. (For a description of the building and its surroundings, see reference 2). The coefficients shown in Figure 4 represent
0.3
0
0
,~
/
T
0158
0.162
l
0.41
~ T a p 127
2.90
1.47
\
T \
083 Figure 1 Coefficients Cp(ai) for case I
/
0.79
T \
Figure 2 Coefficients Cp(ai) for case II
Coefficients
Cp(c~i) for case IV
an envelope of coefficients measured for 24 mean wind directions (reference 2, pp. 72-114). 066
065
"
0 83
0.72
Estimation of aerodynamic forces corresponding to a specific r e t u r n period
083
0 52 - - ~
O. 65
T
05, ~
Figure 4
065
f
1.21
03
.'#
/
0.158
042
130
065
\
Figure 3 Coefficients Cp(c~i) for case III
/ 0.52
0.158
(2)
=
0.3
0.162
0.52
0 65
Current approaches Two approaches for estimating aerodynamic forces corresponding to a specified return period are documented in the literature. In one of these approaches, 2 it is assumed that the conditional probability distribution of the extreme winds, given that the wind blows from a specified octant, is the same as the probability distribution of the extreme winds from any direction (as defined in this paper immediately following equation (1)). The second approach requires the fitting of a joint probability distribution to the extreme wind data corresponding to various azimuth angles. From this joint distribution, and from aerodynamic information on the direction-dependent response, a joint probability distribution of response and direction is obtained. The probability distribution of the response is then, by definition, equal to the marginal distribution calculated by inte-
Eng. S t r u c t . , 1 9 8 1 , V o l . 3, J u l y
183
W i n d d i r e c t i o n effects: E. S i m i u a n d J. J. F i l l i b e n
grating this joint distribution with respect to the azimuthal parameter.6 Proposed approach While the approach presented in this paper does incorportate directional information on extreme wind speeds, it does not require the estimation of joint probability distributions of wind speeds - or wind-induced effects and directions. Rather, a random variable is defined which consists of the maximum annual wind effect of concern (e.g., pressure, force, stress, and so forth), or of a function thereof suitably chosen for computational convenience. A cumulative probability distribution function (CDF) is then fitted to the data consisting of the values taken on by this random variable in a sufficiently large number of consecutive years. The wind effect with a N-year return period corresponds to the value of the variate for which the ordinate of the CDF is 1 - I/N. Application: estimated 50-yr pressure at a point on exterior wall o f a high-rise building To illustrate the application of the procedure, the 50-yr pressure will be sought at the exterior wall location (tap 127 in reference 2) for which the pressure coefficients are given in Figure 4. The following random variable will be used: rdir = max [C~/2 (ai) V(ai)] (3) It will be assumed that the structure is located in Huron, S.D. (see Table 1), and that the building of Figure 4 is so oriented that its short side is parallel to the north direction and the point under consideration is northeast of the centre of the building. The factors Cp(ai) and C~/2(~i) will then have the absolute values given in Table 2. Using these values in conjunction with the extreme wind speeds at Huron listed in Table 1, the following set of values of the random variable Ut i t is obtained : 62, 54, 59, 71,87, 82, 64, 52, 60, 56, 54, 46, 45, 63, 46, 58, 87, 61,67, 56 These data were subjected to a statistical analysis aimed at determining the best fitting Weibull, extreme value type I, or extreme value type II distribution and its parameters. The computer program used in this analysis was similar to that described in reference 4. It was found that the data were best fitted by an extreme type I distribution, which yielded d ~ r : 93.5 mph (41.8 m/s), where ~ r = estimated value of rair corresponding to a probability of exceedance 0.02 in any one year (i.e., a return period N = 50 yr). A random variable will now be defined, which is consistent with the procedure of reference 2, i.e., which, in contradistinction to r ctir, does not reflect climatological wind direction effects: ~od = max [C~/2(ai)] V~o
bution, it follows from the data of Table 1 that V~o = 82.5 mph (36.9 m/s), so that, with max[C~/2(a)] = 1.7 (Table 2), equation (4) yields rnso a = 140.2 mph (62.7 m/s). The ratio of the 50-yr pressure estimated by ignoring climatological wind direction effects, p~oa, to the 50-yr pressure estimated by taking these effects into account, dir Pso, is then, in this example: n d _ (140.2 Pso air \ ~ . 5 - ! Pso
nd P20 dir P20
Values of
- 2.34
P air 20e
= a max [Cp(ai) V2max(~i)]
(7)
it follows from Tables 1 and 2 that: nd P20 dir P 20e
= 2.22
(8)
This ratio is seen to differ rather insignificantly from the _ndt_dir ratio p:o/p2o - This is the case because the difference between the estimated 20-yr pressure, pd~r, and the extreme dir pressure, P2oe, that would actually have been recorded in the 20-yr period covered by Table 1, is only of the order of 5%. If we now define : P2oend = d
max [Cp (ai)] {max [ Vmax(ai)] }2
(9)
where pn2de = pressure calculated by ignoring climatological wind direction effects and based upon the maximum wind speed from any diiection actually recorded in the 20-yr period covered by Table 1, max[Vmax(C~i)], rather than upon the estimated value V~o, we obtain the result: nd
P 2oe dir
- 2.37
(10)
P 2oe i.e., this ratio differs by less than 2% from the ratio given by equation (6). This shows that, in this example, deviations from actual data that are inherent in the estimates p dir A 20 un the one hand and P2o na on the other hand are of the same order of magnitude.
Additional case studies: dir
al-
I Cp(ai)land C~2(ai )
N
NE
E
SE
S
SW
W
NW
ICp(ai)l C~/Z(ai)
0.42
1.21
2.90
0_72
0,79
1.30
1.47
0.41
0.65
1.10
1.70
0.85
0,89
1.14
1,21
0.64
Pso nd P5o
dir P20
na
a3 = a4 -
P 2oe nd P2o dir P20e
na
P2oe
Eng. Struct., 1981, Vol. 3, July
(6)
(lla) (1 lb)
P2o dir
ai
184
(5)
ncl It is of interest to compare the 20-yr pressure, P2o, estimated by ignoring climatological wind direction effects, to the maximum pressure, pa~e , that would actually have occurred during the 20-yr period covered by Table 1. Since:
a2Table 2
= 2.25
i.e., if the procedure of reference 2 - which ignores climatological wind direction effects - were used, the 50-yr pressure would be overestimated by 225%. It can be verified that for a 20-yr return period:
(4)
where V~o = estimated value of the 50-yr wind speed from any direction. Assuming that the extreme winds from any direction are described probabilistically by a type I distri-
12
(1 lC) (11d)
W i n d d i r e c t i o n e f f e c t s : E. S i m i u a n d J. J. F i l l i b e n
Ratios al to a4 (for the Toledo station subscript 20 must be replaced by subscript 19) were calculated in the cases previously designated as I, II, III and IV (Figures 1, 2, 3 and 4, respectively) for various building orientations and wind climates. The results of the calculations are given in
into account the directional characteristics of the wind
climate is clearly quite important for at least five of the eight possible orientations of the building, even if allowance is made for possible sampling errors. This effect is seen to be much less pronounced in cases II and III, and to some extent in case I, owing to the less peaked character of the variation of Cp with direction in these cases. As previously mentioned, it was suggested 3,s that it is appropriate to account for wind direction effects - particulady for the purpose of developing building code wind loading provisions - by multiplying loads determined without regard for these effects by a reduction factor of about 0.8. The results of Table 3 suggest that this approach is in general not justified. For example, if this approach were adopted, the loads corresponding to case II (Sheridan) would be underestimated by 20% for all buildings with a SE (or NW) orientation of the short side and by about 12% for all buildings with a NE (or SW) orientation. Thus, if a uniform distribution of the building orientations were assumed, the use of a reduction factor of 0.8 would lead to the underestimation of the wind loads on half of the buildings of the aerodynamic type represented in Figure 2. Clearly, the underestimation would be even more severe in case III. Table 3 also shows averages of the quantities al to a4 for all directions. In this connection it should be emphasized that the use of such averaged reduction factors could well be unacceptable as far as the safety of buildings with certain unfavourable orientations is concerned. For example, in case IV, if the average reduction factor 0.65 were used, the cladding load would be severely underestimated for all buildings oriented with the short side in
Table 3.
Interpretation of r e s u l t s Note that in all cases the differences between the values o f al and a2 are small for practical purposes. However, the differences between the values of a2 and a3 are in certain cases significant (e.g., case II, NE; case III, NE). The differences between the values of a2 and a4 are somewhat less than those between a2 and a3, but are still relatively large in a few cases. The significance of the differences between a2 and a 3 is the following. If a2 < 1, it would in theory be permissible to reduce the load obtained as in reference 2 (i.e., ignoring climatological wind direction effects) by using the reduction factor a2. However, if a3 > a2, the load so reduced would be smaller than the actual load that would have occurred during the period under consideration. In view of the errors inherent in the statistical handling of the data it may therefore be prudent, judging from the examples of Table 3, to use for the design of structures with known orientation reduction factors that are larger than the estimated ratios a2. In this context, the need for a study of sampling errors in the estimation ofp~ r for various values of N is obvious. Such a study would have to be conducted for a large variety of aerodynamic and climatic situations. Note, however, that in the case of cladding loads at a specified point (case IV of Table 3) the effect of taking Table3
Ratiosa~,a2, a 3 a n d a 4 Direction of short side of building N
NE
E
SE
Sa
SW a
Wa
NW a
Average for all directions
0.75 0.74 0.85 0.81
0.59 0.59 0.67 0.64
0.53 0.53 0.53 0.51
0.64 0.62 0.67 0.63
0.72 0.71 0.80 0.75
Case I (Figure I) Huron
aI a2 a~ a4
0.66 0.65 0.79 0.75
0.84 0.81 1.00 0.95
0.80 0.79 0.79 0.75
0.94 0.94 1.07 1.00
Case II (Figure 2) Toledo
a~ a2 a~ a,
0.72 0.76 0.87 0.80
0.78 0.83 1.09 1.00
0.82 0.86 0.87 0.80
0.86 0.89 0.91 0.84
0.80 0.84 0.94 0.86
Case II (Figure 2) Sheridan
al a2 a3 a4
0.79 0.80 0.78 0.79
0.85 0.84 0.89 0.91
0.79 0.80 0.78 0.79
1.00 0.98 0.98 1.00
0.86 0.86 0.86 0.88
Case I I (Figure 2) Huron
a~ a2 a3 a4
0.77 0.78 0.85 0.79
0.82 0.79 1.00 0.95
0.82 0.80 0.85 0.79
0.97 0.95 1.07 1.00
0.85 0.83 0.94 0.88
Case I I I (Figure 3) Huron
ai a2 a3 a4
0.94 0.94 1.00 0.94
0.80 0.79 1.00 0.94
0.96 0.95 1.00 0.94
0.96 0.94 1.07 1.00
0.92 0.91 1.02 0.96
Case I V b (Figure 4) Huron
a1 a2 a3 a4
0.44 0.43 0.45 0.42
0.62 0.60 0.67 0.63
0.48 0.46 0.45 0.42
0.75 0.82 1.00 0.94
0.58 0.60 0.74 0.72
0.90 0.96 1.06 1.00
0.52 0.53 0.53 0.50
0.54 0.57 0.60 0.56
0.60 0.62 0.69 0.65
a If no entry is shown, values are same for S and N, SW and NE, W and E, NW and SE, respectively b Point under consideration (pressure tap 127, see reference 2) is in the NE-E octant when short side is parallel to north direction, in E-SE direction when short side is parallel to northeast direction, etc.
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Wind direction effects: E. Simiu and d. J. Fill/ben
the SE or SW direction, with the possible consequence that, for these buildings, the number of cladding failures due to wind loading would be unacceptably large.
Conclusions A simple procedure was proposed for estimating wind loads corresponding to various return periods, which takes into account directional information on both wind speeds and aerodynamic response. Examples of the application of the procedure were given, which show that (1), cladding loads calculated without taking directional information on extreme wind speeds into account may in certain cases be larger than the actual loads by a factor of two or more, and (2), it is not appropriate, in general, to account for wind direction effects by multiplying loads determined without regard for these effects by a reduction factor of about 0.8, as suggested elsewhere. 3,s The procedure presented in the paper is applicable to building components and structural members that do not exhibit any significant aeroelastic effects or resonant amplification of the response to the wind load, e.g., cladding panels, exterior walls, rigid roofs, and structural members of low- and medium-rise buildings or other relatively rigid structures. Also, the application of the procedure in its present form is limited to well-behaved wind climates, i.e., to regions where hurricane or other extraordinary winds may not be expected to occur or are not taken into account in design. (A modified version of the procedure, applicable to hurricane-prone regions, is currently being developed by the writers.) Information on the dependence of extreme wind speeds upon direction can be obtained for a large number of US weather stations from unpublished records of the National Oceanic and Atmospheric Administration. The writers believe that it would be useful (1), to extract and publish such information, and (2), to undertake on its basis studies of sampling errors in the estimation of the dependence of wind effects upon direction.
Acknowledgements The writers wish to thank M. J. Changery, of the National Climatic Center, National Oceanic and Atmospheric Administration, who provided the data used in Table 1, and M. E. Butts of the Center for Building Technology, National Bureau of Standards, for useful exchanges and valuable contributions to this paper. The work presented here was partly supported by the National Science Foundation under agreement No. ENV-7716113 and the Department of Energy, Office of Assistant Secretary, Conservation and Solar Applications. Any opinion, findings and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation or the Department of Energy.
186
Eng. Struct., 1981, Vol. 3, July
References 1 Davenport, A. G. et al. "Wind loads on tow-rise buildings: Final report of phases I and II, parts 1 and 2, BLWT-558-1977', Boundary Layer Wind Tunnel Laboratory, The University of Western Ontario, London, Ontario, Canada, 1977 2 Peterka, J. A. and Cermak, J. E. 'Wind tunnel study of Atlanta office building', Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, 1978 3 Ravindra, M. K. et al. 'Wind and snow load factors for use in LFRD', J. Struct. Div., ASCE, 14, (ST9), Prec. Paper 14006, September, 1978, pp. 1443-1457 4 Simiu, E. and Fill/ben, J. J. 'Probability distribution of extreme wind speeds', J. Struct. Div., ASCE, 102, (ST9), Prec. Paper 12381, September, 1976, pp. 1861-1877 5 Stathopoulos, T. Discussion of Paper, "Wind and snow load factors for use in LRFD', J. Struct. Div.. ASCE, 105, (ST10), October, 1975, pp. 2132-2133 6 Surry, D. and Davenport, A. G. 'Modeling the wind climate: an overview', Prec., Workshop on Wind Climate, Asheville, North Carolina, November 12-13, 1979, Texas Tech. University, Institute for Disaster Research, Lubbock, Texas, 1980
Notations a, d = factors in equations (2), (7) and (9) ax, a2, a3, a4 = ratios defined by equations (11) Cp = peak force or pressure coefficient dir = superscript indicating that climatological wind directionality effects are taken into account max[ ] = denotes maximum value nd = superscript indicating that climatological wind directionality effects are ignored N = return period p~r, p~va = wind effect (e.g., pressure, force) with an N-year return period, estimated by taking into account and by ignoring climatological wind directionality effect, respectively pdir _nd . . Ne, PNe = wlnd effect that would have actually occurred in given N-year record, calculated by taking into account and by ignoring wind directionality effect, respectively pp = peak aerodynamic force or pressure r air, r~ zr = random variable defined by equation (3), value of r air corresponding to a N-year return period, respectively r ha, r~ a = random variable defined by equation (4), value of r na corresponding to a N-year return period, respectively s(V) = sample standard deviation of V V, V, Vmax= wind speed, sample mean of V, sample maximum value of V, respectively Vi(ai) = largest annual fastest-mile wind speed observed in jth year of record from octant i Vja, V} = largest annual fastest-mile wind speed blowing in ]th year of record from any direction, fastest-mile with N-year return period blowing from any direction, respectively Vt = wind speed at top of boundary layer a i = denotes octant i (i = 1, 2 . . . . . 8)