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Journal of Wind Engineeringand Industrial Aerodynamics 92 (2004) 1173–1190

Quasi-steady theory and point pressures on acubic building

Peter J Richards a, , Roger P Hoxey b

aMechanical Engineering, University of Auckland, Private Bag 92019, Auckland, New Zealand b Silsoe Research Institute, Wrest park, Silsoe, Bedford MK45 4HS, UK

Received 10 December 2003; received in revised form 10 July 2004; accepted 14 July 2004Available online 11 September 2004

Abstract

A quasi-steady method which uses observed mean pressure coefcients to predict theexpected peak positive or negative pressures is developed. It is shown that in the case of wall

pressures this involves calculating the joint probability of instantaneous wind direction andgust dynamic pressure. With roof pressures the situation is more complex since the pressuresare also sensitive to elevation angles and so the joint probability also includes this angle.Comparison of these predictions with observed data from the Silsoe 6m cube show reasonableagreement.r 2004 Elsevier Ltd All rights reserved.

Keywords: Quasi-steady; Peak pressures; Cube; Wind loads; Full-scale

1. Introduction

Many wind loading standards, such as AS/NZS 1170.2:2002 [1], employ a quasi-steady approach for the design of static structures. Cook [2] comments ‘‘The quasi-steady approach is a compromise which assumes that all the uctuations of load aredue to the gusts of the boundary layer; thus the contribution from the building-generated turbulence is suppressed by this method. This leads to a design approach

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0167-6105/$- see front matter r 2004 Elsevier Ltd All rights reserved.doi:10.1016/j.jweia.2004.07.003

Corresponding author. Tel. +64-9-3737999; fax: +64-9-3737479.E-mail address: [email protected] (P.J. Richards).

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called the equivalent-steady-gust method. In situations where the contribution fromthe building is not large, for overall forces and moments for example, the accuracy of this approach is quite good. For local forces on cladding, however, particularly inregions of separated ow near the periphery of the roof, the accuracy of theapproach is poor. Codes make special provision for local forces, hence their accuracyis reasonable.’’ With this approach the design peak pressure is given by

^

pðyÞ ¼ 12

r ^

V 2C pðyÞ; ð1Þ

where r is the air density, ^

V is the expected peak wind speed and

C pðyÞ is the meanpressure coefcient, which is taken to be a function of mean wind direction alone.Cook [2] further shows that the equivalent-steady-gust model is a severesimplication of the quasi-steady vector model where the instantaneous pressure isgiven by

pðtÞ ¼ 12

r V 2ðtÞC pðy; bÞ; ð2Þwith the magnitude of the wind speed vector given by

V 2ðtÞ ¼ ð

U þ uÞ2

þ v2 þ w2 ð3Þin terms of the mean wind speed

U and the uctuating turbulence components u, vand w.

The uctuating azimuth and elevation angles are given by

y0 ¼ y

y ¼ arctan ðv=ð

U þ uÞÞ and b ¼ arctan ðw=ð

U þ uÞÞ: ð4ÞThe instantaneous pressure coefcient C p is assumed to be a function of theinstantaneous azimuth angle y and elevation angle b.

While the quasi-steady vector model has its limitations and cannot be expected toperfectly predict the pressure on a building, it can be shown that it is thesimplications that are used in reducing the quasi-steady vector model to theequivalent-steady-gust model that introduces many errors. Further it is suggestedthat the quasi-steady vector model is particularly useful in showing what effects canbe attributed to variations in the onset ow and what contribution originates frombuilding generated turbulence or other sources.

2. Simplications of the quasi-steady vector model

The primary reason for simplifying the quasi-steady vector model (Eq. (2)) intothe equivalent-steady-gust model (Eq. (1)) is the difculty of obtaining a relationshipbetween the pressure coefcient and the instantaneous azimuth and elevation angles.Wind tunnel or full-scale studies can easily provide data for the mean pressure

coefcient variation with mean wind direction, but more detailed information is hardto obtain. This is a problem which has been encountered by both the developers of wind standards and researchers who have used quasi-steady methods in the analysis

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P.J. Richards, R.P. Hoxey / J. Wind Eng. Ind. Aerodyn. 92 (2004) 1173–11901174

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of experimental data. In either case there are a number of approximations which arecommonly made. These include:

Ignoring the effects of the elevation angle : With this approximation the term C p(y,b) isreduced to C p(y). This approximation is appropriate for pressures on the walls of many buildings since these are relatively insensitive to the elevation angle. Datacollected on the Silsoe 6 m Cube showed that tilting the cube by up to 5 1 had verylittle effect on the windward face vertical centreline pressures. Sharma [3] obtainedsimilar results in the University of Auckland wind tunnel by tilting a 1:50 scale modelof the Texas Tech Building. These experiments showed that the pressures in the centreof the windward wall were insensitive to tilting although some points nearer the edgedid show some sensitivity, with values for |d C p /d b| as high as 0.5 rad

1. Even thesevalues are small in comparison with the derivatives obtained by the same method forroof pressures. Sharma and Richards [4] quote roof pressure coefcient derivative ashigh as 5.73 rad

1 and show that retaining the relationship between pressurecoefcient and elevation angle can be important in the analysis of roof pressurespectra. Letchford and Marwood [5] used a similar tilting model technique in theUniversity of Oxford Low Speed Wind Tunnel and obtained values of |d C p /d b|greater than 10 rad 1 near the roof corner. They also found that incorporating the wcomponent turbulence term improved the comparison of quasi-steady predicted rmspressures with measured values. It can therefore be concluded that where possible theeffects of elevation angle should be retained, particularly with roof pressures, howeverit is recognised that retaining this term is often difcult since determining thesensitivity of the pressure coefcient to elevation angles is often impossible.

Linearising the pressure coefcient-azimuth angle relationship : Many authors,including Kawai [6], linearise the pressure coefcient-azimuth angle function.While this is convenient it is very inaccurate near regions of maximum orminimum pressure. In fact since the standard deviation of wind direction is oftenof the order of 10 1, a linear range of 7 301 is needed for this assumption to beaccurate, but this seldom occurs (see for example Fig. 3 later in this paper).Richards et. al. [7] have suggested using a short Fourier series to represent thisfunction in order to avoid the need for linearisation.

Approximating the instantaneous pressure coefcient function by the mean pressurecoefcient : Richards et. al. [7] show that if an instantaneous function of the formof Eq. (2) exists then the observed mean pressures have lower extreme values.Hence if this function is approximated by measured mean pressure coefcientsthen the expected maximum and minimum pressures will under-predict the likelyvalues. Methods such as those used by Richards et al. [7] or Banks and Meroney[8], which seek to nd a function which when combined with direction variationswould lead to the observed mean values, can be used to give a more consistentestimate of the instantaneous function.

While it is difcult to avoid such simplication and approximations the followinganalysis seeks to minimise these, although some are still necessary due to limitedinformation.

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3. Quasi-steady prediction of peak wall pressures

3.1. Quasi-steady predictions

As discussed in Section 2, wall pressures are relatively insensitive to elevationangles and so Eq. (2) may be simplied to

pðtÞ ¼ 12

r V 2ðtÞC pðyÞ; ð5Þwhich can be further simplied in form to

pðtÞ ¼ qðtÞC pðyÞ; ð6Þwhere q(t) is the instantaneous dynamic pressure. The form of C p (y), which is theinstantaneous variation of pressure coefcient with instantaneous direction, cannotbe easily determined since direct measurement requires q and y to be constant, butdoing this would remove the turbulent stresses that affect the ow eld. In practicemost standards assume that the instantaneous coefcient is equal to the meancoefcient:

C pðyÞ ¼

C pðyÞ: ð7ÞWhile this approach is simple and is a good rst order approximation, it is limited

and as pointed out in the previous section will tend to under-predict extreme values.Eq. (7) is even less accurate when applied to the prediction of expected peakpressures. This occurs because an extreme pressure may occur when an extremedynamic pressure

^

q combines with a high pressure coefcient which occurs at anangle near, but not necessarily at,

y: Predicting the expected peak value is therefore a joint probability problem. If Eq. (6) is assumed to apply then for peak positivepressures the probability of exceeding a particular threshold p+ is given by

Qð p4 pþÞ ¼

X

y¼360

y¼0Q q4

pþC pðyÞ P ðyÞDy; ð8Þ

where Q q4 pþC pðyÞ is the probability that the dynamic pressure will be strong enoughto produce the required pressure when combined with the instantaneous pressurecoefcient for that direction, P (y) is the probability density of wind directions whichwill depend on the mean and standard deviation of wind directions during anyobservation period and Dy is a narrow band of wind angles. Since the application of Eq. (6) means that positive pressures can only occur with a positive C p it is taken that

if C p (y)o 0 then Q q4 pþC pðyÞ ¼ 0:The parallel expression for peak negative pressures is

Qð po p Þ ¼Xy¼360

y¼0Q q4

pC pðyÞ P ðyÞDy: ð9Þ

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By determining Q( p4 p+ ) and Q( po p ) for a range of thresholds it is possible todetermine the peak positive

^

p and peak negative

p pressures that have a particularprobability of being exceeded.

3.2. Peak wall pressures on the Silsoe cube

The Silsoe 6 m cube, Fig. 1(a) , has a plain smooth surface nish and has beeninstrumented with surface tapping points on a vertical and on a horizontal centrelinesection with additional tappings on one quarter of the roof. Simultaneousmeasurements have been made of 32 pressures and of the simultaneous winddynamic pressure and direction derived from a sonic anemometer positionedupstream of the building at roof height. Tapping points are constructed of simple7 mm diameter holes (a size sufcient to prevent water blocking the tapping points)and the pressure signals transmitted pneumatically, using 6 mm internal diameterplastic tube to transducers mounted centrally. Tube lengths of up to 10 m are used inthis system giving a frequency response of 3 dB down at 8 Hz. While data wascollected from a ring of 16 taps located at the mid-height of the cube, only data fromthe ve taps shown in Fig. 1(b) will be presented in this section. These taps werelocated 0.4 m (0.066 h) from the vertical edges of the cube. In addition the data hasbeen processed in order to be presented as equivalent data at Tap 17. The velocityprole at the Silsoe Research Institute site, with southwest to west winds, has beenmeasured at various times and the recent measurements are well matched by a simple

logarithmic prole with a roughness length z0=0.006–0.01 m. This means that thecube has a Jensen number ( h/z0) of 600–1000. The longitudinal turbulence intensityat roof height is typically 19–20%. The cube is supported so that it may be rotatedrelative to the wind. This facility was used in order to provide a variety of approachwind angles while using only winds from a limited range of directions. In this way thesurrounding terrain was fairly homogeneous for all tests.

For the basic data recording simultaneous measurements of the pressures weremade at a rate of 4.167 samples per second (3000 samples in 12 min) together withthe three components of the wind speed. A 12-min record length was used. Therecords were processed to give mean, peak and uctuating properties. Data has been

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(a) (b)

Taps 22

and 29

WindDirection

θ

ReferenceMast(1.0h high,1.04h to theside of cubecentre)

X

Y

0.066h

3.48h

Tap 17

Taps 28

and 23

Taps 9

Fig. 1. (a) The Silsoe 6 m cube and (b) the position of the mid-height and roof pressure tappings.


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collected for the ve pressure taps shown in Fig. 1(b) with a range of 12-min meanwind directions and strengths. During some periods the cube was normal to theprevailing wind while at other times the cube was rotated through 45 1. A total of 32812-min blocks of data were recorded, however only those with mean dynamicpressures greater than 20 Pa were used since the lower speed blocks tend to produceunreliable results.

The data was processed in order to provide mean, standard deviation, maximumand minimum values for the pressures, reference wind dynamic pressure anddirection. The pressure data has been reduced to coefcient form in the followingmanner:

C p ¼

pq

; ^

C p ¼ ^

p^

q and

^

C p ¼ ^

p^

q: ð10Þ

In this process the peak values ^

p;

p and ^

q are the single most extreme value observedduring the particular 12-min period. As a result they represent an estimate of thevalue with a 1 in 3000 chance of being exceeded. By using the ratio of peak values allthe coefcients are expected to be of the same order. The data from taps 22, 23, 28and 29 have been transposed to give equivalent data for tap 17 but at the appropriatemean wind angle. For taps 23 and 29 this simply means that the equivalent angle is y-901 and y-180 1, respectively, whereas for taps 22 and 28 the equivalent angles are180 1-y and 270 1-y. This resulted in 1360 data points, which cover most anglesbetween 0 1 and 360 1.

A Fourier series of the form

C pðyÞ ¼X6

k ¼0ak cosðk yÞ þ

bk sinðk yÞ ð11Þhas been tted to the mean pressure coefcient data by using a least squares method.The rms error was 0.045 in C p .

Analysis of wind records at the site show that the direction variations areapproximately normally distributed about the mean angle such that during each 12-min block

P ðyÞ ¼ 1

s y ffiffiffiffiffiffi2pp exp ðy

yÞ22s 2y !: ð12ÞDuring the runs the standard deviation of wind directions s y ranged from71(0.122 rad) to 18 1 (0.314 rad) with an average of 10 1(0.174 rad). Fig. 2 shows atypical example of the distribution of wind directions about the mean.

The instantaneous pressure coefcient function has therefore been constructed byfollowing the method in Richards et al. [7], where

C pðyÞ ¼X6

k ¼0 ak cosðk yÞ þ bk sinðk yÞ; ð13Þ

with ak ¼

ak expð12 k 2s 2yÞ; bk ¼

bk expð12 k 2s 2yÞ and s y = p/18 rad (10 1).

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Fig. 3 shows the variation of mean pressure coefcient with mean wind direction.It may be observed that the combination of pressures from the ve tappings creates asingle consistent curve with relatively little scatter. The dashed line is the t to thedata (Eq. (11)) while the solid line is the corresponding estimated instantaneousvariation (Eq. (13)). The instantaneous curve reaches more extreme values in both

the positive and negative directions by about 0.1 in C p .In order to evaluate the probability of exceedence functions for the pressure in (8)

and (9) it is necessary to determine the probability of exceedence function for the

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0

0.5

1

1.5

2

2.5

3

3.5

4

-1 -0.8 -0.6 -0.4 -0.2 0 10.2 0.4 0.6 0.8

Instantaneous - Mean Wind Direction (radians)

p d f

Measured dataNormal

Fig. 2. Probability density function for the wind direction variations around the mean.

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 45 90 135 180 225 270 315 360

Mean Wind Direction (degrees)

C p

Tap 17

Tap 23 rotated

Tap 29 rotated

Tap 22 mirrored

Tap 28 mirrored & rotated

Cp meanCp instant

Fig. 3. Variation of mean and instantaneous pressure coefcients direction for Tap 17.


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wind dynamic pressures. In other quasi-steady analyses, such as Banks and Meroney[8], the wind speed has often been assumed to be normally distributed. Although theoverall wind speed statistics are reasonably matched by a normal distribution,analysis of data recorded at Silsoe at a height of 6 m shows that the distribution of wind speeds is slightly skewed towards increased wind speed. This means that if anormal distribution model is used the probability of occurrence of the high windsspeeds, which are relevant for the prediction of extreme pressures, will beunderestimated. This is quite clear in Fig. 4 , which shows a typical probabilitydensity function for the wind speed on a log scale. A normal distribution can be seento match the central data but does not match either the low or high speed regions. Asa result a modied normal distribution was used.

This took the form

P ðC V Þ ¼ C V ð1 bÞ þ ab

C bþ1V c ffiffiffiffiffiffi2pp exp ðC V aÞ

2

2C 2bV c2 !; ð14Þwhere V is the wind speed, C V ¼ V =

V ; a 1 bc2; b is a shape factor in the range0o b o 1 and c s V =

V the turbulence intensity. The Silsoe data indicated thatb E 0.23.

Eq. (14) was used since it has the following characteristics:

The probability tends to zero as C V tends towards either 0 or N

. The integral of the probability from 0 to N is unity. The shape can be adjusted to match the observed data.

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0.0001

0.001

0.01

0.1

1

10

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Wind Speed Coefficient Cv

p d f

Measured dataNormalModified Normal

Fig. 4. Probability density function of the wind speed.


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As illustrated in Fig. 5 , it is essentially a mapping of a normal distribution of the variable y which lies in the range - N o yo N and has zero mean onto the

semi-innite range 0o

C V o N

through the relationship

y ¼ C V ac ffiffiffi2p C

bV

: ð15ÞThe corresponding probability density function for the dynamic wind pressure isgiven by

P ðqÞ ¼ P ðC V Þ

ffiffiffiffiffiffiffiffiffiffiffiffi2r

V 2q

q

; with C V ¼ ffiffiffiffiffiffiffiffiffi2qr V 2s ð16Þand the probability of exceedence obtained by numerically integration,

Qðq4 qÞ ¼ 1 Z q

0

P ðqÞdq: ð17Þ

Fig. 6 shows the resulting probability of exceedence values in terms of the dynamicpressure coefcient C q , which is the ratio of the particular dynamic pressure to themean dynamic pressure ðC q ¼ q=qÞ:With b E 0.23, the turbulence intensity c=0.19 and a probability of exceedence of 1 in 3000 ( Q (4 C q)=0.00033), Eqs. (14), (16) and (17) show that the expected peakto mean ratio for the dynamic pressure is 2.9, which is close to the average observedratio of 2.88.

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Fig. 5. Mapping of a normal distribution in y onto the semi-innite space for the velocity coefcient.


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An Excel spreadsheet has been developed which uses the above equations topredict the maximum and minimum pressure coefcients expected at Tap 17 with aprobability of exceedence of 1 in 3000 for each mean wind direction. Thesecalculations are shown as lines in Fig. 7 along with the observed mean, peak positive

(max) and peak negative (min) pressure coefcients. It should be noted that thesecurves are derived from tting the mean pressure coefcient data and do not rely onany measurements of peak values.

Fig. 7 shows that while the mean data points are clustered around the tted line,the peak data show much greater scatter. Nevertheless it can be seen that in generalboth the maximum and minimum data follow the trend suggested by the quasi-steady predictions. It may be noted that if the mean pressure coefcient is negativefor a range of angles around that being considered (for example between 220 1 and360 1) then the maximum positive pressure is near zero and the minimum pressurecoefcients are near to the mean values. Similarly if the mean pressure coefcient is

positive over a range around a particular direction (80 1 to 100 1) then the minimumpressure is near zero and the maximum pressure coefcient is close to the mean valuefor that direction. These observations support the assumptions made earlier, inassociation with the evaluation of Eq. (8) and (9), regarding the peak values when themean pressure coefcient is of opposite sign. Fig. 7 also shows that at times themaximum and minimum values are signicantly different from the mean value. Themost extreme example of this occurs with a mean wind direction of 35 1, at whichpoint the mean pressure coefcient is near zero, however the maximum pressurecoefcient are in the range 0.5–0.9 and the minimum pressure coefcients are in therange

0.75 to

1.5. It is reasonable to assume that the peak positive pressures

occur during strong gusts while the instantaneous wind direction has swung aroundto about 55 1, where the instantaneous function gives a coefcient of 0.74, and thatthe peak negative coefcients occur during strong gust and instantaneous wind

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0.0001

0.001

0.01

0.1

1

10

0 0.5 1 1.5 2 2.5 3 3.5 4

Dynamic Pressure Coefficient Cq

P r o

b a

b i l i t y o

f E x c e e

d e n c e

Q ( > C q

)

Measured dataNormalModified NormalQ(>Cq)=0.00033Cq=2.9 2

Fig. 6. Probability of exceedence values for various dynamic pressure levels.


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directions around 151, where the instantaneous pressure coefcient is 0.9. Theextreme values hence depend on the joint probability of strong gusts and appropriate

wind directions.The most noticeable difference between the expected and measured peak values

occurs with minimum pressures between 0 1 and 60 1 and around 180 1. At these anglesTap 17 is on the side of the cube and is affected by the separating and reattachingow. It is thought that these lower minimum peak pressures are the result of thedynamic behaviour of this ow, which results in the periodic formation of intensevortices, which are attached to the leading edge of the cube for a short time and thenshed into the general ow. This observation re-emphasis the point made at the end of

Section 1, that a quasi-steady model cannot be expected to account for every effect,but if applied in a systematic manner can show what observations can be attributedto quasi-steady processes and what should be attributed to other processes such asbuilding generated turbulence.

4. Quasi-steady prediction of peak roof pressures

Although the pressure at some positions on a building may not be sensitive toelevation angles, other positions will be. Fig. 8(a) shows the changes in mean

pressure coefcient on a vertical centreline plane as the cube is tilted into the wind. Inthis case a wind direction of 90 1 is parallel to the plane. It may be seen that thewindward wall pressures hardly change, while those in the centre of the roof change

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-3

-2

-1

0

1

2

3

0 60 120 180 240 300 360

Mean Wind Direction (degrees)

P r e s s u r e

C o e

f f i c i e n

t

maxmean

minCp maxCp meanCp min

Fig. 7. A comparison of measured maximum, minimum and mean pressure coefcients (symbols)with those predicted by a quasi-steady model (lines) for a point on the cube sidewall at mid height andx/h=0.066.


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noticeably, particularly at Tap 9. Fig. 8(b) shows the changes in mean pressurecoefcient at Tap 9 for a range of wind directions around 90 1. The results in thisgure suggest that for this point

@C pðy; bÞ@b 0:077

C pðy; 0Þdeg1 3:14

C pðy; 0Þrad1

ð18ÞWhere the elevation effects are signicant then Eq. (9) becomes

Qð po p Þ ¼Xy¼360

y¼0 Xb¼3s b

b¼ 3s bQ q4

pC pðy; bÞ P ðbÞDbP ðyÞDy; ð19Þwith the range of b set at three standard deviations either side of the mean, which is

assumed to be zero, in order to include all likely elevation angles. Data collectedat a height of 6 m suggested that the standard deviation of b is about 4.5 1 (0.078 rad).Fig. 9 shows that the elevation angles are approximately normally distributedaround a mean of 0 1. Although it may be observed that the elevation angles may beslightly skewed towards positive (upward angles) the use of a modied distribution inthis case is not warranted since the angles of primary interest are those around zerorather than the extremes, as was the case with the dynamic pressure.

Evaluation of Eq. (19) is further complicated by the effects of the Reynolds shearstress which means that the vertical velocity, and hence elevation angle, is partiallycorrelated with the wind speed, as illustrated in Fig. 10 .

The data shown in Fig. 10 had the following statistics:

Mean wind speed

U ¼ 9:58 m =s: Standard deviation of streamwise velocity s u=2.04 m/s. Standard deviation of vertical velocity s w=0.75 m/s. Directly calculated Reynolds shear stress uw ¼ u2n ¼ 0:293m 2=s

2:

A linear t to the data in Fig. 10 has a gradient of 0.0704, which is also the ratioof uw to the variance of the streamwise velocity s u 2. It may also be noted that if thefriction velocity, u , is calculated from the mean wind speed and a roughness length

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(a) (b)

-1.5

-1.0

-0.5

0.0

0.5

1.0

0 6 12 18position (m)

m e a n

C p

zero pitch

2.5 deg pitch

5 deg pitch

windward roof leeward-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

60 75 90 105 120wind angle

m e a n

C p

zero pitch2.5 deg pitch5 deg pitch

Tap 9

Fig. 8. (a) The effect of tilting the Silsoe 6 m cube on vertical centreline plane mean pressure coefcientsfor a wind direction parallel to the plane (90 1) and (b) the variation of mean pressure coefcients, over arange of wind directions, for Tap 9 which is 3.5m from the windward edge of the roof at 90 1.


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of 6 mm, then a simple logarithmic prole gives the friction velocity as 0.57 m/s andthe expected Reynolds stress as u2n ¼ 0:308m 2=s

2; which is close to the directlymeasured value.

The instantaneous elevation angle is

b ¼ arctan wU þ u

wU u if b is small : ð20Þ

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Fig. 10. UW scatter plot for 1 h of wind data at a height of 6 m.

0

1

2

3

4

5

6

7

-0 .5 -0 .4 -0 .3 -0.2 -0 .1 0 0 .1 0.2 0.3 0.4 0 .5

Elevation Angle (radians)

p d f

Measured dataNormal

Fig. 9. Probability density function for the elevation angle in the free stream at a height of 6 m.


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This means that the variation of the elevation angle which is correlated with thestreamwise uctuations, b , can be estimated from

b n ¼ u2n

s 2uu

U þ u u2n

s 2uV

U V : ð21ÞSince the standard deviation of b is small and the range of elevation angles is alwayscentred on zero, then linearising the relationship between instantaneous pressurecoefcient and elevation angle is far more justied than it would be for the azimuthangle y. While this linearisation is more justied, it is recognised that there may besituations where it is not sufciently accurate. In more complex environments wherethe variation of b is large, possibly due to upstream buildings, or the variation of C pwith b is highly nonlinear, this method will be very approximate. Including these

approximations leads to the following expression for the quasi-steady minimumpressure:

p ¼ C pðy; 0Þ þ @C pðy; bÞ

@bu2n

s 2u

V

U V þ @C pðy; bÞ@b b0 r2 V 2; ð22Þ

where b0 is the random variation of elevation angle and is assumed to be normallydistributed.

The solution procedure involves solving Eq. (22), which is a quadratic in V , for agiven p , y and b0 combination, and then calculating the corresponding dynamic

pressure and hence obtaining the probability of exceedence for that situation. Thevalue of p with a probability of exceedence of 0.00033 is then extracted from thecalculations. The results of this procedure are shown in Fig. 11 , where the meanpressure data for Tap 9 is shown along with the mean pressure coefcient tted curveand the corresponding instantaneous curve. Also shown are three estimates for theexpected variation of minimum pressure coefcient with direction. The threeestimates have been obtained by using only the rst term in Eq. (22), the rst twoterms and then all terms. Including only the rst term creates a situation where theexpected minimum curve is atter than the mean pressure coefcient curve. Thisoccurs because in regions where the mean pressure coefcient is less negative a more

negative pressure can be generated at an instant when a gust combines with a changein wind direction that provides a more negative instantaneous coefcient. Inclusionof the Reynolds stress term generally reduces the level of the expected minimum.This occurs because a high wind speed is associated with a negative vertical velocityand hence a negative elevation angle.

Fig. 8(a) shows that for Tap 9 the pressure coefcient became less negative as thecube was tilted forward, this is equivalent to a negative elevation angle. Hence theexpectation is that high wind speeds will be correlated with lower coefcients andhence the expected peak pressure is less negative. Including the nal term in Eq. (22)results in the expected minimum becoming more negative. This may be slightly

surprising since the elevation angle is approximately normally distributed with a zeromean. This means that the angle is just as likely to be positive as it is negative. Hencewith a linear variation the magnitude of the pressure coefcient is increased as often

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as it is reduced. However, the probability distribution of the peak wind speed ishighly nonlinear and so any increase in the magnitude of the pressure coefcientmeans that a lower peak wind speed is needed in order to produce a given suction,and so this event becomes more likely.

Fig. 12 compares the observed minimum pressure coefcients with three methods

of estimating these. The rst of these is the reasonably common approach of simplyusing the mean pressure coefcient. This tends to provide a lower bound for the databut does not adequately match the variation. The second method is to allow for thevariation in wind direction (Eq. (9)), but ignore the effects of elevation. This providesa better estimate but it still tends to underestimate the general trend. The thirdmethod is to include both the azimuth and elevation variations (Eq. (19)). Thisprovides an estimate that does appear to match the general trend of the data,however, it is recognised that there is signicant scatter around even this estimate. Inaddition it is important to realise that in general data relating to the variation of pressure coefcients with elevation angle is difcult, if not impossible, to obtain and

so these terms may need to be estimated.Fig. 13 shows the same data as a ratio of the minimum pressure coefcient

(minimum pressure divided by maximum wind dynamic pressure) to the meanpressure coefcient (mean pressure divided by the mean wind dynamic pressure).This shows that the full quasi-steady analysis is correctly predicting the situationswhere the minimum pressure coefcient is greater than the mean pressure coefcient.

In terms of design wind loads on buildings it is reassuring to note that both theobserved and quasi-steady data suggest that the pressure coefcient ratio is nearestto one when the pressure coefcients have high magnitudes, particularly in the range250 1 –290 1. Hence using the combination of the mean pressure coefcient and a gust

wind pressure to estimate the peak pressure appears to be most accurate in the mostsevere situations. However, even in these situations both the data and quasi-steadyanalysis suggest that this approach underestimates the extreme pressures by at least

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-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

00 45 90 135 180 225 270 315 360

Wind direction (degrees)

P r e s s u r e

C o e

f f i c i e n t

Mean Cp dataMean Cp curveInstataneous CpCp min with no elevation termsCp min with Reynolds stress termCp min with full equation

Fig. 11. Mean pressure coefcient data for Tap 9 and the expected minimum pressure coefcients derivedfrom this.


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15% and possibly more, with some data in the range 250 1 –2901 showing a ratio ashigh as 2. Hoxey et al. [9,10] have discussed the sources of the uncertainty in suchmeasured data and argue that the primary source is the measurement of the peak

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0

1

2

3

4

5

0 45 90 135 180 225 270 315 360

Wind Direction (degrees)

C p m

i n / C p m e a n

dataQ-S Full equation

Fig. 13. A comparison of the observed minimum to mean pressure coefcient ratio at roof Tap 9 on theSilsoe cube with the expected values predicted by the full quasi-steady analysis.

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

00 45 90 135 180 225 270 315 360

Mean wind direction (degrees)

M i n i m u m

P r e s s u r e

C o e

f f i c i e n t

Cp min dataMean CpQ-S No elevation termsQ-S Full equation

Fig. 12. A comparison of the observed peak negative pressures at roof Tap 9 on the Silsoe cube with theexpected values predicted by the mean coefcients, a partial quasi-steady model (no elevation terms) and amore complete quasi-steady analysis.


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dynamic pressure at a location some distance from the building. With the referenceanemometer about 25 m from the centre of the building then measurements [9] haveshown that it may be expected that the standard deviation of the differences betweenmeasurements at the reference mast and at the centre of the building, but with thebuilding removed, would be about 10% of the typical dynamic pressure. Hence thepeak dynamic pressure measured at the reference mast may be much smaller thanthat actually affecting the building. In wind tunnel testing this uncertainty may beminimised by using extreme value analysis on multiple blocks of stationary data.However, in full-scale testing one is relying on nature to provide the wind and so notwo 12 min blocks are truly similar, not even in regard to wind direction, and soextreme value analysis would be questionable and is hence not attempted here.

In spite of these uncertainties, Fig. 14 shows that if the full quasi-steady pressurecoefcients are combined with the observed maximum wind dynamic pressures then

the expected minimum pressures at Tap 9 are well correlated with the measuredpressures although they are 6% too low.

5. Conclusions

A quasi-steady method which uses observed mean pressure coefcients to predictthe expected peak positive or peak negative pressures has been developed. It is shownthat in the case of wall pressures this involves calculating the joint probability of instantaneous wind direction and gust dynamic pressure. With roof pressures the

situation is more complex since the pressures are also sensitive to elevation anglesand so the joint probability also includes this angle. Comparison of these predictionswith observed data from the Silsoe 6 m cube show reasonable agreement. Although

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y = 0.9402xR2 = 0.8746

-800

-600

-400

-200

0-800 -600 -400 -200

0

Measured minimum pressure (Pa)

Q u a s

i - s

t e a

d y p r e s e u r e c o e

f f i c i e n t

* q m a x

( P a

)

Fig. 14. A comparison of the observed minimum pressure at roof Tap 9 on the Silsoe cube with theexpected values predicted by the full quasi-steady analysis.


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the data used in this study is derived from a single full-scale situation and hence maynot apply in general, it is believe by the authors that the principals outlined aregenerally applicable and are not unique to the particular study.

References

[1] AS/NZS 1170.2:2002, Structural design actions, Part 2: Wind actions, Standards Australia, 2002.[2] N.J. Cook, The designer’s guide to wind loading of building structures, Part 2: Static structures,

Butterworths, UK, 1990.[3] R.N. Sharma, The inuence of internal pressure on wind loading under tropical cyclone conditions,

Ph.D. Thesis, University of Auckland, New Zealand, 1996.[4] R.N. Sharma, P.J. Richards, The inuence of Reynolds stresses on roof pressure uctuations, J. Wind

Eng. Ind. Aerodyn. 83 (1999) 147–157.[5] C.W. Letchford, R. Marwood, On the inuence of v and w component turbulence on roof pressures

beneath conical vortices, J. Wind Eng. Ind. Aerodyn. 69–71 (1997) 567–577.[6] H. Kawai, Pressure uctuations on square prisms-application of strip and quasi-steady theories,

J. Wind Eng. Ind. Aerodyn. 13 (1983) 197–208.[7] P.J. Richards, R.P. Hoxey, B.S. Wanigaratne, The effect of directional variations on the observed

mean and rms pressure coefcients, J. Wind Eng. Ind. Aerodyn. 54/55 (1995) 359–367.[8] D. Banks, R.N. Meroney, The applicability of quasi-steady theory to pressure statistics beneath roof-

top vortices, J. Wind Eng. Ind. Aerodyn. 89 (2001) 569–598.[9] R.P. Hoxey, P.J. Richards, G.M. Richardson, A.P. Robertson, J.L. Short, The folly of using extreme-

value methods in full-scale experiments, J. Wind Eng. Ind. Aerodyn. 60 (1996) 109–122.[10] R.P. Hoxey, A.P. Robertson, P.J. Richards, How have full-scale measurements improved the

reliability of wind-loading codes? Dick Marshall’s contribution to full-scale measurements of wind

effects. 11th International Conference on Wind Engineering, Lubbock, Texas, USA, July 2003,pp. 29-48.

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Richards 2004