A PEAK FACTOR FOR PREDICTING NON-GAUSSIAN PEAK

RESULTANT RESPONSE OF WIND-EXCITED TALL BUILDINGS

M.F. Huang1, C.M. Chan

2, K.C.S. Kwok

3 and Wenjuan Lou

4

1Postdoctoral Researcher, Institute of Structural Engineering, Zhejiang University,

Hangzhou 310027, P.R.China. E-mail: hmfust@gmail.com 2Associate Professor, Dept. of Civil and Environmental Engineering, The Hong Kong Univ. of

Science and Technology, Clear Water Bay, Kowloon, Hong Kong. E-mail: cecmchan@ust.hk 3Professor, School of Engineering, University of Western Sydney, NSW, Australia, E-mail:

k.kwok@uws.edu.au 4Professor, Institute of Structural Engineering, Zhejiang University, Hangzhou 310027,

P.R.China. E-mail: Louwj@zju.edu.cn

ABSTRACT

In the structural design of tall buildings, peak factors have been widely used to predict mean extreme responses

of tall buildings under wind excitations. Using the out-crossing theory for the first-passage problem, a

probabilistic peak factor, which is directly related to an explicit measure of structural reliability against a

Gaussian response process, can be obtained for structural design. Based on the asymptotic theory of statistical

extremes, a new closed-form peak factor, the so-called Gamma peak factor, can be also obtained for a non-

Gaussian resultant response, characterized by a Rayleigh distribution process. Using the Gamma peak factor, a

combined peak factor method has been developed for predicting the expected maximum resultant responses of a

building undergoing lateral-torsional motion. Utilizing the wind tunnel data derived from synchronous multi-

pressure measurements, the time history response analysis has been carried out for the 45-story CAARC

standard tall building. The Gamma peak factor has been applied to predict the peak resultant acceleration of the

building. Results of the building example indicate that the Gamma peak factor provides accurate prediction of

the mean extreme resultant acceleration response for the dynamic serviceability performance design of modern

tall buildings.

KEYWORDS: LEVEL-CROSSING RATE (LCR); TIME-VARIANT RELIABILITY; MEAN EXTREME

RESPONSE; PEAK FACTOR METHOD; DYNAMIC SERVICEABILITY

Introduction

Considering inherent random characteristics of wind and uncertainties in the material

properties and structural performance, much research work has also been carried out on the

probabilistic evaluation of wind-induced response of structures with uncertain parameters

[Kareem (1987), Solari (1997), Hong et al. 2001]. These studies about wind effects on

structures employed the classical work of Davenport to estimate the expected or mean value

of the largest peak response. Davenport (1964) has shown that, if the underlying parent

distribution of a response process is Gaussian, then the extreme values of the process will

asymptotically follow a Gumbel distribution. For a zero-mean response process, the so-called

peak factor, can be defined as the ratio of the largest peak response to the standard deviation

value of the response. In general, the Davenport’s peak factor provides satisfactory estimates

of the maximum peak response for wide-band response processes; but it may yield

conservative estimates for narrow-band response processes [Kareem (1987), Gurley et al.

(1997)].

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

The approach using the Davenport’s peak factor for estimating the expected maximum

peak responses of tall buildings to wind is based on the assumption that the underlying

stochastic response is Gaussian. Such an assumption is valid for many general wind

engineering applications. However, non-Gaussian wind effects may arise from specific but

important situations, such as responses of a non-linear building system, turbulence-induced

local pressure fluctuations on building surfaces, and combined resultant acceleration

responses of a tall building for dynamic serviceability design [Melbourne and Palmer 1992,

Isyumov et al. (1992), Chan et al. (2009)]. Specific efforts have been made to modify the

Davenport’s peak factor for predicting non-Gaussian gust and extreme effects either using

mathematical series representation of distribution functions or using statistical approach

[Gurley et al. (1997), Sadek and Simiu (2002), Holmes and Cochran (2003), Tieleman et al.

(2007)].

This paper firstly presents an analytical expression of the probabilistic peak factor,

which is directly related to an explicit measure of reliability (or the probability of random

peak responses without exceeding a certain threshold value). Secondly, based on the

asymptotic theory of statistical extremes, the so-called Gamma peak factor can be analytically

obtained for a non-Gaussian resultant response, characterized by a Rayleigh distribution

process. Using the Gamma peak factor, a combined peak factor method can then be developed

for predicting the expected maximum resultant acceleration responses of tall buildings under

wind excitations. Finally, the peak resultant acceleration responses of the CAARC building

were calculated using the Gamma peak factor, and compared with the time history response

data derived from wind tunnel based synchronous multi-pressure measurements.

Probabilistic Peak factor

The largest peak response over a given time duration τ can be defined as a new

random variable

{ }max ( ) ;0Y Y t tτ τ= ≤ ≤ (1)

Using the time-variant reliability ( )R t from the Poisson model [Vanmarcke (1975)], for a

fixed time duration τ , the cumulative distribution function (CDF) of Yτ can be expressed as

( ) ( )2

0 2exp exp

2Y

Y

bF b R v

ττ τ

σ

= = − −

(2)

where b indicates a chosen response magnitude; ( )0 / YYv σ πσ= & , representing the mean zero-

crossing rate of the process ( )Y t . For a prescribed probability p of the largest peak response

being within the specific threshold over the time period τ , the corresponding response

threshold denoted as , pbτ , can be obtained by solving the following equation

( )2,

, 0 2exp exp

2

p

p

Y

bP Y b p v

ττ τ τ

σ

≤ = = − −

(3)

By taking logarithm transform twice at both sides of Eq. (3) and rearranging terms, one

obtains

( )

, 02lnln 1/

p

Y

b v

p

τ τ

σ= (4)

The ratio of , /p Ybτ σ given in Eq. (4) can be regarded as a peak factor. Based on the fact that

the largest extreme values of a Gaussian process asymptotically follow the Gumbel

distribution, Davenport (1964) developed the following peak factor for practical use

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

0 02ln / 2 lnfg v vτ γ τ= + (5)

where the Euler’s constant 0.5772γ = . For a narrow-band resonant response, the mean zero-

crossing rate 0v can be simply approximated by the natural frequency of a building, and the

observation time duration τ may be normally taken as 600s or 3600s in wind engineering

practice. It is worth noting that the Davenport’s peak factor is independent of spectral

bandwidth parameter. Based on extreme value theory, the probability of the largest peak

response not exceeding the expected maximum peak response can be evaluated by using the

Type I extreme value distribution (or the Gumbel distribution) as

( ) 0.5704ef YP Y g e

γ

τ σ−−≤ = = (6)

The Davenport’s peak factor was developed under the assumption that the out-

crossings constitute a Poisson model, which has been found to be too conservative when the

response ( )Y t is a narrowband process and the threshold level b is not high enough with

respect to the standard deviation value of the response. Furthermore, the consecutive out-

crossings of the response ( )Y t , cannot be realistically assumed as independent events, as they

tend to occur generally in clumps. Vanmarcke (1975) developed a corrected mean out-

crossing rate based on a modified Poisson model accounting for the dependence among the

crossing events as

1.2

2

2

1 exp2

1 exp2

Y

b b

Y

bq

vb

π

ση

σ

− −

=

− −

(7)

where ( )21 0 21 /q λ λ λ= − =shape factor that characterizes the bandwidth of the process, in

which the spectral moments mλ can be defined as

0

( ) ; 0,1, 2, 4mm YG d mλ ω ω ω

∞= =∫ (8)

where ( )YG ω =one-sided power spectral density function of the process and one can show

that 20 Yλ σ= , 2

2 Yλ σ= & and 2

4 Yλ σ= && .

Using the Vanmarcke’s corrected mean out-crossing rate of level b in Eq. (7), the CDF

of Yτ can be rewritten as

( )

1.2

0 2

2

1 exp2

exp

exp 12

Y

Y

Y

bq

F b vb

τ

π

στ

σ

− −

= − −

(9)

Given a specific probability of no exceedance p of the largest peak response being within the

specific threshold , pbτ over the time period τ , Eq. (9) can be expressed as

1.2

0 2

1 exp2

exp

exp 12

p

p

q g

p vg

π

τ

− − = −

−

(10)

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

where pg is the so called probabilistic peak factor, which can be defined as the ratio of

, /p Ybτ σ . Taking logarithm at both sides of (10) and rearranging terms, one obtains

2

1.20 1 exp exp 1ln(1/ ) 2 2

p

p

gvq g

p

τ π − − = −

(11)

In light of the fact that ( )2exp / 2 1pg >> , Eq. (11) can be first rewritten as

2

1.20 1 exp expln(1/ ) 2 2

p

p

gvq g

p

τ π − − ≈

(12)

Using Eq. (4) as a first approximation of pg and substituting it into the left side of Eq. (12),

the probabilistic peak factor pg can be explicitly expressed in terms of p, 0v , τ and q as

( )

1.20 02ln 1 exp lnln(1/ ) ln 1/

p

v vg q

p p

τ τπ

≈ − −

(13)

The peak factors, calculated according to the Davenport’s peak factor given in Eq. (5)

and the probabilistic peak factor of Eq. (13), are plotted as a function of 0v within the range

of 0.1-1.1 Hz for typical multi-story building structures with several chosen values of

bandwidth shape factor q as shown in Figure 1. For the sake of comparison with the

conventional Davenport’s peak factor, the probabilistic peak factor has been given based on

Eq. (13) with the probability of no exceedance p=0.5704 and an excitation duration

time τ =3600s. It is evidently shown in Figure 1 that the Davenport’s peak factor is

independent of the spectral bandwidth parameter q, and always gives more conservative

results, particularly for a narrowband process with a smaller value of q. For a wide-band

process with the value of q approaching to 1, the probabilistic peak factor approaches to the

value of the Davenport’s peak factor.

3.3

3.5

3.7

3.9

4.1

4.3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

mean zero-crossing rate νννν 0 (Hz)

Pea

k f

act

ors

Davenport's peak factor (p=57%)

Probabilistic peak factor (p=57%; q=0.7)

Probabilistic peak factor (p=57%; q=0.4)

Probabilistic peak factor (p=57%; q=0.2)

Figure 1: Peak factors with spectral bandwidth parameter q

Peak distribution for a non-Gaussian combined random process

Under wind excitation, a building may vibrate in a lateral-torsional manner such that

the maximum resultant response may involve several component responses in a 3-D manner.

Assuming that the corner of a building experiences two perpendicular translational component

responses, X(t) and Y(t). Then the combined resultant process can be written as

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

( ) ( )2 2( )A t X t Y t= + (14)

Assuming that X(t) and Y(t) follow a Gaussian distribution with a zero-mean and a common

standard deviation of Xσ = Yσ , then the resultant process A(t) follows a Rayleigh distribution,

which can be given as

( )2

2 2exp

2A

A A

a af a

σ σ

= −

(15)

where Aσ denotes the mode value of A(t). Since it is assumed that the two component

processes have the same standard deviation, the mode value of A(t) coincides with the

component standard deviation such that A X Yσ σ σ= = . As can be observed from full-scale

measurement data and recently demonstrated by means of a numerical study [Chen and

Huang (2009)], the actual peak resultant response attains the most deviation from the SRSS

combination of two individual peak components when X Yσ σ= . Since the most significant

joint action between two component responses X and Y occurs when X Yσ σ= , it becomes

necessary and meaningful to first investigate the joint action effects under the condition of

two random component processes with equal fluctuating variation (i.e. X Yσ σ= ) and then

later to extend to more general cases in which X Yσ σ≠ .

The peak distribution of the non-Gaussian combined random process A(t) can be

determined by its mean level-crossing rate (LCR). For a stationary combined process A(t), the

probabilistic distribution function (PDF) of peaks of the combined process can be obtained as

the derivative of the frequentist definition of probability

1

( )m

bA

dVf b

dbV

+

+−∞

= − (16)

where Am denotes a random variable, which represents the peak values arising from a

combined process A(t). For a narrowband process, each upcrossing event can possibly lead to a corresponding

peak. The expected number of peaks above the given threshold level b per second can then be

approximated by the level upcrossing rate bv+ . If the desired peaks are counted by the peak-

over-threshold approach, the expected number of total peaks per second V +−∞ can be well

estimated from the mean upcrossing rate of a sufficiently small threshold level b such as mode

value Aσ , i.e., A

bb

V vσ

+ +−∞

=≈ . The probability peak distribution of a non-Gaussian combined

process given in Eq. (16) can then be related to the mean LCR as

1

( )m

A

bA

dvf b

dbvσ

+

+≈ − (17)

Considering the Rayleigh resultant process A(t) with the known mean LCR bv+ [Huang

(2008)], the probability peak distribution of the non-Gaussian combined process A(t) can be

obtained from Eq. (17) as

2 2

2 2

1 1( ) 1 exp 1 ,

2mA A

A A A

b bf b b σ

σ σ σ

≈ − − − ≥

(18)

By introducing the intermediate response threshold level 2 2/ 1Ac b σ= − associated with a

peak-dependent intermediate random variable 2 2/ 1m AC A σ= − , which is referred to as the

intermediate peak variable, the elementary probability of the event for an occurring peak with