Further thoughts on the design of cladding

Building and Environment, Vol. 12, 39--41. Pergamon Press 1977. Printed in Great Britain

Further Thoughts on the Design of Cladding T. W. EVERETT* T. V. L A W S O N * In an earlier note[l] we suggested that two aspects in the design of cladding ought to be considered separately. The first was the use of a design wind speed with the correct return period and the second was the evaluation of a value of pressure coefficient from a wind tunnel investigation, the two procedures being entirely independent. The first evaluation is carried out by meteorologists using a Fisher Tippett type I or Gumbel distribution. For the second, we suggested two further separate considerations be applied--the choice of the correct averaging time, dependent upon the size of the cladding element, and the choice of a suitable frequency of occurrence of the pressure coefficient in the wind tunnel investigation. Although the note received favourable comments from most readers, several pointed out that the part concerning frequency of occurrence of the pressure coefficient in the wind tunnel investigation did not conform to strict statistical principles. This we readily conceded; the purpose of the note was to suggest a very simple approach which gave values o f about the right magnitude. In this present paper we would like to examine the extreme value analyses in a more rigorous manner.

The design wind speed for any return period and any probability of occurrence is given by

APPROACHING WIND AS I N THE original paper, it is assumed that in the wind tunnel investigation the approaching wind is modelled correctly and has velocity profiles, turbulence profiles, spectra and integral lengths in conformity with reference [2]. The magnitude of the hourly average wind speed at a given location can be estimated from a Gumbel analysis in which

17 = a + b { -

In [ { - In (1-P(10)}/N]}

and the probability of a given wind speed being exceeded is P ( V ) = 1 - e x p { - N e -y} which can be differentiated to give

= a+b'yo

where

ae(V)_

17"is the design wind speed

dV

aP(V) ~y ~y

. . . . ~V

[Ne -y exp { - N e - ~ } ] / b

(1)

a and b are constants depending upon the location

PROBABILITY DENSITY FUNCTION FOR PRESSURE M E A S U R E M E N T S

yo is the reduced variate = - In { - [ In (1 P(V))]/N}

Several authors have shown (i.e. reference [4])that if the mean value of the pressure coefficient is positive, the probability density distribution is approximately Gaussian, and if the mean pressure coefficient is less than - 0 . 2 5 the distribution approximates to an exponential curve for values of standard deviation greater than about 3: between these two values intermediate distributions apply. To simplify matters, let us assume that the value of mean pressure coefficient of -0.25 separates the two distributions. The overall probability of occurrence of extreme values, P(P), can be related to the parent probability density distribution by the expression (from reference [5]) P(P) = 1 - exp { - vTv'(2rc)apx(x) } (2)

where P ( V ) is the probability of the wind speed V being exceeded once in N years. F o r most locations the value of a and b are not presented, but for locations in which the value of wind speed likely to be exceeded for a range of numbers of years (i.e., reference [3]) these constants can be evaluated. If the probability P(IO is taken as 0.637 for the 10 and 100 y return periods Vao = a + b y l o 1~1oo = a+byloo and ylo and yloo are respectively 2.31 and 4.59, so that a = 217"1o- ¢,oo

where

and b = (1~1oo- 17"~o)/2.28.

v = {S~ n2S(n) dn/S~ S(n) dn }112 T = time interval in question px(x) = probability density distribution of the

*Department of Aeronautical Engineering, University of Bristol, Queen's Building, Bristol.

original sample. 39

T. IV. Everett and T. V. Lawson

40 I

I I I II1

I

l

I

i

U Ill

I

~

Pressure

1400

!

I

I

I i IIn[

n

i

i

I I

_l]

coefficients based on meteorological standard reference pressure Pressures averaged over 2s

1200

1

o 09 xx

* ~ O S S m ~ .¢l|e LO¢O?Io~W.D ~ 1• 2x 30 4(} 5A 6& 7D

Q_ 4 0 0

200 IOO 0

I 30 66 0 7 4 120 54 0 49 9C 7 7 120 7 4 120 ref. [ 6 ] I

I I

c~fflcken*= I PressurQ I% 0.06 u/9

-I.56 -I.48 -IA7 -I.30 1.37 t.75 0.66 ref, I

-I.19 -464 -0.66-493 -0.13 -454 -0.36 -458 0.75 440 1.06 551 0.13 268 [61 ref. Eli

I I Ill

QO00l

~

Croydon wind dora assumed

i

l

i i l lt~l

0.001 Probability

~

=

,

,

I,=,l

0,01

I

I

I

I I I

'0.1

of pressure being exceeded once in 5 0 y r

Fig. 1. Variation of design pressure with probability of occurance. A value for T which represents the duration of the height of the storm has to be assumed; a value of 10 m i n is suggested.

The curve is therefore assumed to be pn(r/) = 0.40 exp {-1.35r/}

(5)

and when this value is inserted into equation (2)

FOR C~ MEAN > - 0 . 2 5 G A U S S I A N DISTRIBUTION

P(P) = 1 - exp { - exp [ - 1.35(r/- u)] } where

F o r a Gaussian distribution

u = [In {vT}]/1.35

p.~(x) -- [exp {-t/2/2}]/(~/(ETr)a)

(3)

where

dP(P)_ OP(P) Or~ __.m = [1.35 exp { - 1.35(r/-u) dC, or~ ecp

r~ = ( x - . 7 ) l a

so that equation (2) can be written

- exp [ - 1.35(r/- u)l }l/a

P(P) = 1 - e x p [ - v T e x p {-r/2/2}]

dCp

(7)

and the value of design pressure coefficient can be evaluated from either equation (4) or (5).

and therefore, on differentiation

dP(P)

and therefore, on differentiation

OP(P) Or~ Or/ "0-C, = [ - v T r / e x p { - r / 2 / 2

JOINT PROBABILITY

- vTexp ( - r/2/2) }]/a. The value of Cp design can now be calculated as Cpdeslg:l

=

Cp....

-1- r/a

(4)

I n studies at Bristol the value of a is not presented, only the value which is exceeded for 1% of the time; if this is written as C,~ and the distribution is Gaussian

Cz'aes,v~ = (r//2"32)C,1 + (1 - r//2.32)Cp . . . .

(5)

F O R Cp. . . . < - 0 . 2 5 E X P O N E N T I A L DISTRIBUTION F r o m probability density curves presented in reference [4], which accord with those measured at Bristol, an approximation to the curve can be obtained if it is assumed that the exponential curve goes through the Gaussian curve at r/ = 2.70 and passes through the p o i n t p ( x ) = 0.40 at r/ = 0.

The probability of a given pressure occurring is equal to the integral of the product of the probability of all combinations of wind speed and pressure coefficient which give the required pressure. In mathematical terms, this means Pressure = C~ x ½pV 2 a n d the probability P is given by P (Pressure) = SSp(Cp) x p ( V ) dCp d V and the probability that a given value will be exceeded can be obtained u p o n integration. A computer programme has been written which does this integration numerically and the variation of cumulative probability with the value of pressure is shown in Fig. 1 for seven locations on a wind tunnel investigation described in reference [6]. The values determined by the simple approach of reference [1] are also shown on the Figure for comparison. The

Further Thoughts on the Design o f Cladding

excellent agreement would suggest that results based upon the simple method should be used in most cases where values with low probabilities are not required.

41

Checks can be made when loading of buildings of unusual shape or environment are being studied that agreement between the two metl3ods still exists.

REFERENCES I. T.V. Lawson, The design of cladding. Build. Environ. 11, 37 (1976). 2. Characteristics of wind speed in the lower layers of the atmosphere near the ground: strong winds (neutral atmosphere). E.S.D.U. Data Item 72026, Amend. B (1974); Characteristics of atmospheric turbulence near the ground. E.S.D.U. Data Item 74031, Amend A (1975) (Available from E.S.D.U. Ltd., 251-259 Regent Street, London WlR 7AD). 3. C. E. Hardman, N. C. Helliwell & J. S. Hopkins, Extreme winds over the U.K. for periods ending 1971, Climatological Memorandum No. 50A, H.M.S.O. (1973). 4. J. A. Peterka & J. E. Cermak, Wind pressures on buildings--probabilitydensities Paper No. 11373 J. Struct. Div. Proc. Am. Soc. cir. Engrs 101, ST6 (1975). 5. A. G. Davenport, Discussion of Reference 4 ibid. 6. T. W. Everett & T. V. Lawson, Pressure measurements on the proposed redevelopment of the old secretariat, Lagos, Nigeria. Department of Aeronautical Engineering, University of Bristol Report No. TVL 7611 (1976) (Confidential).