Probabilistic modelling of maximum wind pressure on structures

Journal of Wind Engineering and Industrial Aerodynamics 74—76 (1998) 1111—1121

Probabilistic modelling of maximum wind pressure on structures Evelia Schettini, Giovanni Solari* DISEG, Department of Structural and Geotechnical Engineering, University of Genova, Via Montallegro 1, 16145 Genova, Italy

Abstract This paper formulates a probabilistic model which is able to represent the maximum equivalent pressure applied by the wind on a structural surface. Unlike the classical methods, where the randomness is circumscribed within the maximum mean loading component, this model relies also on the randomness of the maximum fluctuating action. Furthermore, it takes into account the presence of quadratic pressure terms. The application of Taylor series expansions retaining up to the first order derivative terms gives the 1st and 2nd order statistical moments in closed form. The comparison between the results obtained and those supplied by classical methods points out the central role of the turbulence intensity. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: FOSM (First-Order Second-Moments); Gust factor; Maximum wind pressure; Quadratic pressure terms; Turbulence intensity; Wind velocity

1. Introduction The maximum equivalent pressure is the pressure that statically applied on a structural surface produces the maximum force due to the actual wind field. It is usually calculated by two different methods [1]. When applying the first method, it is associated with the maximum wind velocity given as the product of the mean wind velocity by the related gust factor. This factor is a function of the duration of the gust peak which depends on the exposed surface according to suitable spatial-temporal equivalence criteria. The second method expresses the maximum value of the equivalent pressure as the product of the mean static pressure by the related gust factor. This factor takes into

* Corresponding author. E-mail: [email protected]. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 1 0 2 - 0

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account the coherence of the fluctuating component using the theory of multivariate processes. Both methods circumscribe the randomness of the physical phenomenon within the maximum values of mean velocity and pressure. The respective gust factors synthesize the role of the maximum turbulent fluctuations, which are random variables, according to the pseudo-deterministic principles formulated by Davenport [2,3]. As a consequence, they preclude a full probabilistic representation of the maximum pressure over a target lifetime, excluding the possibility of carrying out rigorous reliability analyses. This paper proposes a probabilistic model which is able to represent the maximum equivalent pressure applied by the wind on a structural surface, taking into account the randomness of the maximum turbulent components and the quadratic pressure terms. Using the quasi-static theory [2] and the equivalent wind spectrum technique [4], the maximum summation method formulated in Ref. [5] is initially generalized from the velocity to the equivalent pressure. Based on classical probability theorems, the probability distribution of its maximum value is calculated. Taylor series expansions are applied to derive closed-form expressions of the first and second statistical moments of this quantity. Comparisons between the results of this method and those provided by classical procedures are finally discussed enhancing the central role of the turbulence intensity.

2. Formulation of the problem Neglecting the dependence on height above the ground, the instantaneous wind velocity »(t) is schematized as a stochastic stationary process given by [5]: »(t)"» (t)[1#I »I @(t)], (1) 0 V where t is the time; » (t) is the mean wind velocity averaged on subsequent time 0 intervals *¹, ranging between 10 min and 1 h, corresponding to the macro-meteorological peak [6]; I "p (t)/» (t) is the turbulence intensity, assumed herein as V V{ 0 deterministic and independent of time; »I @(t)"»@(t)/p (t) is the reduced turbulence; V{ »@(t) is the atmospheric turbulence related to the micro-meteorological peak [6]; p (t) V{ is the turbulence standard deviation over subsequent *¹ intervals. » (t) and p (t) 0 V{ vary so slowly with time that be considered as constant in *¹; »@(t) and »I @(t) rapidly vary with time [5]. The maximum instantaneous wind velocity » during a time interval ¹A*¹ is M determined by applying a probabilistic model referred to as the maximum summation method [5]. Since turbulent fluctuations are large when mean wind velocity is large, it is assumed that » occurs concomitant with the maximum reduced turbulence in M a suitable neighborhood of the maximum mean wind velocity » in ¹ (Fig. 1). 0M Therefore it follows that: » "» (1#I »I @ ), M 0MR V 0.R » "B » , 0MR 0M

(2) (3)

E. Schettini, G. Solari/J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 1111–1121

1113

Fig. 1. Maximum instantaneous velocity.

where » is the maximum reduced mean velocity; B3 [0, 1] is a random variable, 0MR called the noncontemporaneity factor, taking into account the noncontemporaneity of maxima; »I @ is the maximum reduced turbulence in the temporal interval *¹ 0.R 0MR where the mean wind velocity assumes the value » . The distributions and the 0MR and » are evaluated and discussed in Ref. [5]. statistical moments of » , B, »I @ 0.R M 0M Consider now a rectangular surface orthogonal to the mean wind direction, where A is the width and H is the height. Applying the quasi-static theory [2] to Eq. (2), the maximum local pressure P is M associated with » by the formula M P "1oC »2 B2(1#I »I @ )2, (4) M 2 p 0M V 0.R where o is the air density and C is the mean pressure coefficient assumed as uniform P over the surface. Both o and C are treated here as deterministic quantities. 1 The equivalent wind spectrum technique [4] allows one to estimate the maximum equivalent pressure P applied by the wind over the surface in ¹ with high precision %2M and simplicity. It is demonstrated that the aerodynamic action due to the real multivariate wind velocity (Eq. (1)) is the same due to an equivalent wind characterized by a perfectly coherent velocity field: (5) » (t)"» (t)[1#I »I @ (t)], %2 0 V %2 where »I @ , defined as the reduced equivalent velocity fluctuation, is a stationary %2 stochastic process characterized by a power spectral density function (psdf ) S I , V{%2 called the reduced equivalent wind spectrum, given by [4]: nC A nC H C 0.4 z sMnnqN, (6) S I (n)"S I (n)C 0.4 x V{ V{%2 » » 0MR 0MR 1 1 (1!e~2g), CM0N"1, (7) CMgN" ! g 2g2 sin2 d , sM0N"1, (8) sMdN" d2

G

H G

H

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where n is the frequency; S I is the psdf of »I @ ; C , C are the horizontal and vertical V{ x z exponential decay coefficients [7]; q is the duration of the gust peak [1]. As a consequence, the maximum equivalent pressure is given by )2, P "1oC »2 B2(1#I »I @ V %20.R %2M 2 P 0M

(9)

is the maximum equivalent reduced fluctuation in *¹ . Its cumulative where »I @ %20.R 0MR distribution function (cdf ), mean and standard deviation are given by [3]

G

C

DH

1 vJ @2 FI (vJ @ )"exp !l I *¹ exp ! %20.R V{%20.R %20.R V{%2 2 p2I V{%2

G

H

,

(10)

0.5772 kI "p I J2 ln(*¹l I )# , V{%20.R V{%2 V{%2 J2 ln(*¹l I ) V{%2

(11)

pI p V{%2 " pI V{%20.R J6 J2 ln(*¹l ) VI {%2

(12)

in which vJ @ is the state variable of »I @ ; p and l I are the standard deviation %20.R %20.R VI {%2 V{%2 and the expected frequency of »I @ in *¹ . Applying the procedure formulated in %2 0MR Ref. [1]:

S

1 pI " , V{%2 1#0.56qJ 0.74#0.29¸I 0.63 0 » l I " 0MR % 2 V{ ¸ V

S

1 , 31.25qJ 1.44#1.23¸I 1.23 1

(13)

(14)

where ¸I "0.42(A#HI )#0.16JAHI , 0

(15)

¸I "0.04(A#HI )#0.92JAHI . 1

(16)

A"C N/¸ ; HI "C H/¸ ; qJ "q» /¸ ; ¸ is the integral length scale of the x V z V 0MR V V turbulence. When A, H tend towards 0, the maximum equivalent pressure almost coincides with the maximum local pressure. Admitting, on the safe side, that » is concomitant with » , i.e. assuming B"1, M 0M P is given by %2M P "1oC »2 (1#I »I @ )2, %2M 2 P 0M V %20.

(17)

where »I @ is the maximum reduced equivalent turbulence which occurs in the %20. temporal interval *¹ where the maximum wind velocity is » . 0M 0M

E. Schettini, G. Solari/J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 1111–1121

»I @

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Neglecting the randomness of the maximum equivalent turbulence, i.e. assigning "k I , k I , being the mean value of »I @ , Eq. (17) becomes %20. V{%20. V{%20. %20. P "1oC »2 (1#I k I )2 V V{%20M %2M 2 P 0M

(18)

in which k I has the clear significance of a peak factor [1]. It is given by Eqs. (11) V{%20. and (14) setting » "» . 0MR 0M Assuming that turbulence is small, coherently with classical methods it follows that P "P G , %2M 0M 1

(19)

P "1oC »2 , 0M 2 1 0M

(20)

G "1#2I k I , 1 V V{%20.

(21)

where P is the maximum value of the mean pressure in ¹; G is the linearized 0M P pressure gust factor [1]. The comparison between Eq. (9) and Eqs. (19)—(21) points out the evolution of the method proposed with respect to the classical theory.

3. Probability distribution Applying classical probability theorems and assuming » and »I @ as statist0MR %20.R ically independent, the cdf of the maximum equivalent pressure is given by the formula JL

p%2M

F (p )" P%2M %2M

P 0

f

A

B

JpL 1 %2M ! (v )F dv , 0MR V0MR 0MR VI {%20.R I v I V 0MR V

(22)

is where pL "2p /oC ; p and v are the state variables of P and » ; f %2M %2M 1 %2M 0MR %2M 0MR V0MR the probability density function (pdf ) of » ; F I is the cdf of »I @ (Eq. (10)). 0MR V{%20.R %20.R Considering B and » as statistically independent: 0M 1

P

A B

f (b) f B V0M

f (v )" V0MR 0MR

0

v

1 0MR db b b

(23)

where b is the state variable of B; f and f are the pdf of B and » . 0M B V0M Assuming, on the safe side, B"1, i.e. defining f (b)"d(b!1), d( ) being the Dirac B function, Eq. (22) assumes the form: JL

p%2M

F (p )" P%2M %2M

P 0

where v

0M

f» (v 0M

0M

)F»I %{20M

A

B

JpL 1 %2M! dv 0M I v I V 0M V

is the cdf of »I @ . is the state variable of » ; F I %20M 0M V{%20M

(24)

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Neglecting the randomness of the maximum turbulence and assuming that turbulence intensity is small, finally leads to the formula implicit in the classical methods:

AS B

F (p )"F P%2M %2M V0M

pL %2M G P%2

(25)

where F is the cdf of the maximum mean wind velocity in ¹. V0M 4. Statistical moments Based on Eq. (9) the maximum equivalent pressure may be expressed as P "F(X , X , X ), where X "» , X "B and X "»I @ are statistically %2M 1 2 3 1 0M 2 3 %20.R independent random variables whose distributions and statistical moments are known. By virtue of these properties, the 1st and 2nd order statistical moments of P may be conveniently evaluated by the FOSM (First-Order Second-Moment) %2M method [8—11]. Expanding P in Taylor series around the mean values l of X (i"1,2,3) %2M Xi i retaining up to the first-order derivative terms, the mean and the variance of P are %2M given by k "1oC k2 k2(1#I k I { )2, P%2M 2 P V0M B V V%20.R

C

(26)

D

p2 p2 I2 p I V0M# B# V V{%20.R p2 "4k2 , P%2M k2 P%2M k2 (1#I k I )2 B V0M V V{%20.R

(27)

where k and p2 are the mean and the variance of » ; k and p 2 are the mean V0M 0M B B V0M and p2I are and the variance of B, given in Ref. [5] as functions of I and ¹; k I V V{%20.R V{%20.R given by Eqs. (11)—(14) assigning » "kv k . The simplicity and expressivity with 0MR 0M B which Eqs. (26) and (27) clarify the single roles of different parameters is apparent. As well as for the velocity [5], the precision of Eqs. (26) and (27) is usually so high that do not require the use of second-order approaches [11]. It is relevant to note that despite the convolution integrals in Eqs. (22) and (23), the pdf of P usually retains the characteristic shape of the » extreme distribution. As %2M 0M such, especially due to the added randomness of B and »I @ , a lognormal model %20.R often represents a suitable approximation of the actual distribution [11]. Eqs. (26) and (27) offer simple and sufficient elements for its derivation. It is also meaningful to observe that classical formulae provided by the literature can be easily obtained from Eqs. (26) and (27) imposing B"1, i.e. l "1, r2"0, B B neglecting the randomness of the maximum equivalent reduced fluctuation, i.e. p2I "0, and assuming I @1. It follows that V V{20.R k "1oC k2 G , P%2M 2 P V0M P

(28)

p2 V0M , p2 "4 k2 P%2M P%2Mk2 V0M

(29)

E. Schettini, G. Solari/J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 1111–1121

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where k "o C k2 /2 is the mean value of the maximum mean pressure P in ¹; P0M P V0M 0M G is the linearized pressure gust factor defined by Eq. (21). P The comparison between Eqs. (26), (27) and Eqs. (28), (29) highlights the evolution of the method proposed with respect to the classical theory. The role played by B is usually quantitatively small. The incremental effect due to quadratic pressure terms obviously increases with increasing I as previously noted by several authors [12—15]. V The addendum associated with the randomness of the maximum value of the fluctuations (Eq. (27)) contributes to increase the variance of the maximum equivalent pressure to an extent which increases with increasing I and which, in any case, is not V negligible.

5. Applications The procedure described in this paper is used to evaluate the maximum equivalent pressure applied by the wind on a point-like square screen orthogonal to the mean wind direction. Referring to the symbols introduced above, A"H"1 m; the barycentre of the screen lies at 30 m height over the ground; C "1, o"1.25 kg/m3. P Applying the process analysis [16,17], the cdf of the maximum mean wind velocity in ¹ is given by F (v )"expM!j f (v )¹N, V0M 0M 0 V0 0M v k k v k~1 0 exp ! 0 , f (v )"F d(v )#(1!F ) 0 0 0 c c V0 0 c

AB

C A BD

(30) (31)

in which f is the density function of » ; v is the state variable of » ; F is the V0 0 0 0 0 probability that » "0; j , c and k are model parameters. 0 0 Taking as reference values the mean wind data (*¹"600 s) measured at the meteorological station of Santa Caterina [18], the screen is assumed to be situated at two different sites characterized by roughness lengths z "0.1, 3 m. Table 1 lists the 0 main parameters of » . Table 2 summarizes the principal quantities associated with 0M the atmospheric turbulence [1]. The parameters related to the noncontemporaneity factor B, calculated according with methods proposed in Ref. [5], are given in Table 3. Tables 4 and 5 compare the mean values and the standard deviations of P de%2M rived from Eq. (22) (column 1) with the results supplied by the FOSM technique (Eqs. (26) and (27)) (column 2). The agreement is almost perfect independently of the values assumed by I and ¹. V Fig. 2 shows the distributions of the maximum equivalent pressure corresponding to different turbulence levels, for ¹"1, 100 yr. RM "1/(1!F ) is the mean return P%2M period. The solid lines correspond to the application of the model described in this paper; the dashed lines represent the results provided by the classical method; the dotted lines correspond to a lognormal distribution with mean and variance calculated by the FOSM technique. The classical method leads to results whose precision decreases with increasing turbulence intensity I , the period ¹ over which the maximum is calculated and the V

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E. Schettini, G. Solari/J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 1111–1121 Table 1 Parameters of the mean velocity z (m) 0

0.1

3

c (m/s) k F 0 k (m/s2) 0 k 0M (m/s) V p 0M (m/s) V

5.253 1.445 0.0045 6.38]10~4 25.61 2.31

2.696 1.445 0.0045 3.27]10~4 13.14 1.17

z (m) 0 I V

0.1 0.178

3 0.327

¸ (m) V C,C x z q (s) pI V{%2 l I %2 V{ k I %20.R V{ pI V{%20.R

154 11.5 1 0.91 0.106 2.80 0.406

86 11.5 1 0.90 0.101 2.71 0.402

Table 2 Turbulence parameters

Table 3 Parameters of the noncontemporaneity factor I V

0.178

0.327

k B p2 B

0.9703 0.0014

0.9457 0.0026

Table 4 Mean values and standard deviations of P

%2M

for ¹"1 yr

I V Method

0.178 (1)

(2)

0.327 (1)

(2)

k (N/m2) P%2M p (N/m2) P%2M

874 195

869 191

355 91

352 89

mean return period RM . The extremely high errors produced by the application of the classical method for high I values largely depend on the lack of the quadratic part of V the turbulence in the gust factor. Lognormal models based on the FOSM technique give rise to results whose superposition with rigorous diagrams enhances the precision of this approach.

E. Schettini, G. Solari/J. Wind Eng. Ind. Aerodyn. 74—76 (1998) 1111–1121 Table 5 Mean values and standard deviations of P

%2M

for ¹"100 y

I V Method

0.178 (1)

(2)

0.327 (1)

(2)

k (N/m2) P%2M p (N/m2) P%2M

1492 254

1491 257

606 126

605 128

Fig. 2. Distributions of the maximum equivalent pressure: (a) I "0.178; (b) I "0.327. V V

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6. Conclusions This paper formulates a simple probabilistic model aimed at representing the extreme values of the maximum equivalent pressure removing several simplified hypotheses used in the literature. The systematic application of the equivalent spectrum technique and of probability theorems provides a maximum distribution which correctly takes into account the maximum fluctuating components and the quadratic part of the turbulence. The use of the FOSM technique, favoured by the mathematical structure of the model, leads to a closed form expression of the mean value and of the variance of the maximum equivalent pressure. These formulae, characterized by great simplicity and maximum generality, clarify the role and the importance of single parameters. The classical method may be deduced from these expressions as a particular case. The differences between the results provided by the classical method and those obtained through the procedure proposed herein increase with increasing atmospheric turbulence, time interval in which the maximum is calculated and mean return period. It is implicit that reliability analyses based on traditional procedures are fully unreliable in the presence of high turbulence intensities.

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[15] A. Kareem, M.A. Tognarelli, K.R. Gurley, Modeling and analysis of quadratic term in the wind effects on structures, Proc., 2nd Europ. and African Conf. on Wind Engineering, Genova, 1997, pp. 1901—1910. [16] L. Gomes, B.J. Vichery, On the prediction of extreme wind speeds from the parent distribution, J. Wind Engng. Ind. Aerodyn. 2 (1977) 21—36. [17] G. Solari, Statistical analysis of extreme wind speeds, in: D.P. Lalas, C.F. Ratto (Eds.), Modelling of Atmospheric Flow Fields, World Scientific, Singapore, 1996, pp. 659—678. [18] G. Ballio, S. Lagomarsino, G. Piccardo, G. Solari, Probabilistic analysis of Italian extreme winds: Reference velocity and return criterion, J. Wind Engng. Ind. Aerodyn., submitted for publication.