The Seventh Asia-Pacific Conference on Wind Engineering, November 8-1, 009, Taiei, Taiwan A PEAK FACTOR FOR PREDICTING NON-GAUSSIAN PEAK RESULTANT RESPONSE OF WIND-EXCITED TALL BUILDINGS M.F. Huang 1, C.M. Chan, K.C.S. Kwok 3 and Wenjuan Lou 4 1 Postdoctoral Researcher, Institute of Structural Engineering, Zhejiang University, Hangzhou 31007, P.R.China. E-mail: hmfust@gmail.com Associate Professor, Det. of Civil and Environmental Engineering, The Hong Kong Univ. of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. E-mail: cecmchan@ust.hk 3 Professor, School of Engineering, University of Western Sydney, NSW, Australia, E-mail: k.kwok@uws.edu.au 4 Professor, Institute of Structural Engineering, Zhejiang University, Hangzhou 31007, P.R.China. E-mail: Louwj@zju.edu.cn ABSTRACT In the structural design of tall buildings, eak factors have been widely used to redict mean extreme resonses of tall buildings under wind excitations. Using the out-crossing theory for the first-assage roblem, a robabilistic eak factor, which is directly related to an exlicit measure of structural reliability against a Gaussian resonse rocess, can be obtained for structural design. Based on the asymtotic theory of statistical extremes, a new closed-form eak factor, the so-called Gamma eak factor, can be also obtained for a non- Gaussian resultant resonse, characterized by a Rayleigh distribution rocess. Using the Gamma eak factor, a combined eak factor method has been develoed for redicting the exected maximum resultant resonses of a building undergoing lateral-torsional motion. Utilizing the wind tunnel data derived from synchronous multiressure measurements, the time history resonse analysis has been carried out for the 45-story CAARC standard tall building. The Gamma eak factor has been alied to redict the eak resultant acceleration of the building. Results of the building examle indicate that the Gamma eak factor rovides accurate rediction of the mean extreme resultant acceleration resonse for the dynamic serviceability erformance design of modern tall buildings. KEWORDS: LEVEL-CROSSING RATE (LCR); TIME-VARIANT RELIABILIT; MEAN EXTREME RESPONSE; PEAK FACTOR METHOD; DNAMIC SERVICEABILIT Introduction Considering inherent random characteristics of wind and uncertainties in the material roerties and structural erformance, much research work has also been carried out on the robabilistic evaluation of wind-induced resonse of structures with uncertain arameters [Kareem (1987), Solari (1997), Hong et al. 001]. These studies about wind effects on structures emloyed the classical work of Davenort to estimate the exected or mean value of the largest eak resonse. Davenort (1964) has shown that, if the underlying arent distribution of a resonse rocess is Gaussian, then the extreme values of the rocess will asymtotically follow a Gumbel distribution. For a zero-mean resonse rocess, the so-called eak factor, can be defined as the ratio of the largest eak resonse to the standard deviation value of the resonse. In general, the Davenort s eak factor rovides satisfactory estimates of the maximum eak resonse for wide-band resonse rocesses; but it may yield conservative estimates for narrow-band resonse rocesses [Kareem (1987), Gurley et al. (1997)].
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-1, 009, Taiei, Taiwan The aroach using the Davenort s eak factor for estimating the exected maximum eak resonses of tall buildings to wind is based on the assumtion that the underlying stochastic resonse is Gaussian. Such an assumtion is valid for many general wind engineering alications. However, non-gaussian wind effects may arise from secific but imortant situations, such as resonses of a non-linear building system, turbulence-induced local ressure fluctuations on building surfaces, and combined resultant acceleration resonses of a tall building for dynamic serviceability design [Melbourne and Palmer 199, Isyumov et al. (199), Chan et al. (009)]. Secific efforts have been made to modify the Davenort s eak factor for redicting non-gaussian gust and extreme effects either using mathematical series reresentation of distribution functions or using statistical aroach [Gurley et al. (1997), Sadek and Simiu (00), Holmes and Cochran (003), Tieleman et al. (007)]. This aer firstly resents an analytical exression of the robabilistic eak factor, which is directly related to an exlicit measure of reliability (or the robability of random eak resonses without exceeding a certain threshold value). Secondly, based on the asymtotic theory of statistical extremes, the so-called Gamma eak factor can be analytically obtained for a non-gaussian resultant resonse, characterized by a Rayleigh distribution rocess. Using the Gamma eak factor, a combined eak factor method can then be develoed for redicting the exected maximum resultant acceleration resonses of tall buildings under wind excitations. Finally, the eak resultant acceleration resonses of the CAARC building were calculated using the Gamma eak factor, and comared with the time history resonse data derived from wind tunnel based synchronous multi-ressure measurements. Probabilistic Peak factor The largest eak resonse over a given time duration τ can be defined as a new random variable = max ( t) ;0 t τ (1) Using the time-variant reliability ( ) τ { } where b indicates a chosen resonse magnitude; ( ) 0 / R t from the Poisson model [Vanmarcke (1975)], for a fixed time duration τ, the cumulative distribution function (CDF) of τ can be exressed as b F ( b) = R ( τ ) = exv0τ τ () σ v = σ & πσ, reresenting the mean zerocrossing rate of the rocess ( t ). For a rescribed robability of the largest eak resonse being within the secific threshold over the time eriod τ, the corresonding resonse threshold denoted as b τ, can be obtained by solving the following equation, b τ, P ( τ bτ, ) = = exv0τ ex (3) σ By taking logarithm transform twice at both sides of Eq. (3) and rearranging terms, one obtains bτ, v0τ ln σ = ln 1/ (4) b τ ( ) The ratio of, / σ given in Eq. (4) can be regarded as a eak factor. Based on the fact that the largest extreme values of a Gaussian rocess asymtotically follow the Gumbel distribution, Davenort (1964) develoed the following eak factor for ractical use
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-1, 009, Taiei, Taiwan g = ln v τ + γ / ln v τ (5) f 0 0 where the Euler s constantγ = 0.577. For a narrow-band resonant resonse, the mean zerocrossing rate v0 can be simly aroximated by the natural frequency of a building, and the observation time duration τ may be normally taken as 600s or 3600s in wind engineering ractice. It is worth noting that the Davenort s eak factor is indeendent of sectral bandwidth arameter. Based on extreme value theory, the robability of the largest eak resonse not exceeding the exected maximum eak resonse can be evaluated by using the Tye I extreme value distribution (or the Gumbel distribution) as γ e P( g σ ) = e = 0.5704 (6) τ f The Davenort s eak factor was develoed under the assumtion that the outcrossings constitute a Poisson model, which has been found to be too conservative when the resonse ( t) is a narrowband rocess and the threshold level b is not high enough with resect to the standard deviation value of the resonse. Furthermore, the consecutive outcrossings of the resonse ( t ), cannot be realistically assumed as indeendent events, as they tend to occur generally in clums. Vanmarcke (1975) develoed a corrected mean outcrossing rate based on a modified Poisson model accounting for the deendence among the crossing events as π 1. b 1 q σ ηb = v b (7) b 1 σ where q 1 λ / ( λ λ ) = =shae factor that characterizes the bandwidth of the rocess, in 1 0 which the sectral moments λ m can be defined as m λm = ω G ( ) ; 0,1,, 4 0 ω dω m = (8) where G ( ω ) =one-sided ower sectral density function of the rocess and one can show 0 4 that λ = σ, λ = σ & and λ = σ &&. Using the Vanmarcke s corrected mean out-crossing rate of level b in Eq. (7), the CDF of τ can be rewritten as π 1. b 1 q σ F ( b) = ex v τ 0τ b (9) 1 σ Given a secific robability of no exceedance of the largest eak resonse being within the secific threshold b τ over the time eriod τ, Eq. (9) can be exressed as, π 1. 1 q g = ex v0τ g ex 1 (10)
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-1, 009, Taiei, Taiwan where g is the so called robabilistic eak factor, which can be defined as the ratio of σ. Taking logarithm at both sides of (10) and rearranging terms, one obtains b τ, / In light of the fact that ( g ) v0τ π 1. g 1 ex q g = ex 1 ln(1/ ) ex / >> 1, Eq. (11) can be first rewritten as v0τ π 1. g 1 ex q g ex ln(1/ ) (1) Using Eq. (4) as a first aroximation of g and substituting it into the left side of Eq. (1), the robabilistic eak factor g can be exlicitly exressed in terms of, v 0, τ and q as v0τ 1. v0τ g ln 1 ex q π ln (13) ln(1/ ) ln ( 1/ ) The eak factors, calculated according to the Davenort s eak factor given in Eq. (5) and the robabilistic eak factor of Eq. (13), are lotted as a function of v 0 within the range of 0.1-1.1 Hz for tyical multi-story building structures with several chosen values of bandwidth shae factor q as shown in Figure 1. For the sake of comarison with the conventional Davenort s eak factor, the robabilistic eak factor has been given based on Eq. (13) with the robability of no exceedance =0.5704 and an excitation duration time τ =3600s. It is evidently shown in Figure 1 that the Davenort s eak factor is indeendent of the sectral bandwidth arameter q, and always gives more conservative results, articularly for a narrowband rocess with a smaller value of q. For a wide-band rocess with the value of q aroaching to 1, the robabilistic eak factor aroaches to the value of the Davenort s eak factor. 4.3 (11) 4.1 Peak factors 3.9 3.7 3.5 Davenort's eak factor (=57%) Probabilistic eak factor (=57%; q=0.7) Probabilistic eak factor (=57%; q=0.4) Probabilistic eak factor (=57%; q=0.) 3.3 0.0 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1. mean zero-crossing rate ν 0 (Hz) Figure 1: Peak factors with sectral bandwidth arameter q Peak distribution for a non-gaussian combined random rocess Under wind excitation, a building may vibrate in a lateral-torsional manner such that the maximum resultant resonse may involve several comonent resonses in a 3-D manner. Assuming that the corner of a building exeriences two erendicular translational comonent resonses, X(t) and (t). Then the combined resultant rocess can be written as
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-1, 009, Taiei, Taiwan ( ) ( ) A( t) = X t + t (14) Assuming that X(t) and (t) follow a Gaussian distribution with a zero-mean and a common standard deviation of σ X = σ, then the resultant rocess A(t) follows a Rayleigh distribution, which can be given as a a f A ( a) = ex (15) σ A σ A where σ A denotes the mode value of A(t). Since it is assumed that the two comonent rocesses have the same standard deviation, the mode value of A(t) coincides with the comonent standard deviation such that σ A = σ X = σ. As can be observed from full-scale measurement data and recently demonstrated by means of a numerical study [Chen and Huang (009)], the actual eak resultant resonse attains the most deviation from the SRSS combination of two individual eak comonents whenσ X = σ. Since the most significant joint action between two comonent resonses X and occurs when σ X = σ, it becomes necessary and meaningful to first investigate the joint action effects under the condition of two random comonent rocesses with equal fluctuating variation (i.e. σ X = σ ) and then later to extend to more general cases in which σ X σ. The eak distribution of the non-gaussian combined random rocess A(t) can be determined by its mean level-crossing rate (LCR). For a stationary combined rocess A(t), the robabilistic distribution function (PDF) of eaks of the combined rocess can be obtained as the derivative of the frequentist definition of robability + 1 dvb f A ( b) = (16) m + V db where A m denotes a random variable, which reresents the eak values arising from a combined rocess A(t). For a narrowband rocess, each ucrossing event can ossibly lead to a corresonding eak. The exected number of eaks above the given threshold level b er second can then be aroximated by the level ucrossing rate v + b. If the desired eaks are counted by the eakover-threshold aroach, the exected number of total eaks er second V + can be well estimated from the mean ucrossing rate of a sufficiently small threshold level b such as mode + + value σ A, i.e., V v b. The robability eak distribution of a non-gaussian combined b = σ A rocess given in Eq. (16) can then be related to the mean LCR as + 1 dvb f A ( b) (17) m + db Considering the Rayleigh resultant rocess A(t) with the known mean LCRv + b [Huang (008)], the robability eak distribution of the non-gaussian combined rocess A(t) can be obtained from Eq. (17) as 1 b 1 b f A ( b) 1 ex 1, b σ m A (18) σ A σ A σ A By introducing the intermediate resonse threshold level v σ A c = b / σ A 1 associated with a eak-deendent intermediate random variable C = Am / σ A 1, which is referred to as the intermediate eak variable, the elementary robability of the event for an occurring eak with
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-1, 009, Taiei, Taiwan { b A b db} m + is equal to the elementary robability of the event for the intermediate eak variable with{ c Am / σ A 1 c + dc} as c 1 f A m ( b) db = ex c dc, c 0 c + 1 (19) The PDF of the intermediate eak variable C can be further aroximated from Eq. (19) into a form of Gamma distribution, as c 1 fc ( c) ex c 4 (0) In general, the Gamma robability model of Eq. (0) results in a higher PDF value than that of the original PDF model at the tail range of intermediate eak variables, indicating a more conservative estimation of the crossing failure robability using the Gamma robability model given in Eq. (0). Gamma eak factor and combined eak factor method A closed-form exression of the eak factor for a combined resultant resonse rocess ( ) ( ) A( t) = X t + t withσ X = σ is develoed in this section. Based on the PDF of the intermediate eak variable C = Am / σ A 1 in the form of the Gamma distribution given in Eq. (0), the corresonding CDF of C can be analytically obtained as follows 1 1 FC ( c) = P( C c) = 1 1+ c ex c, c 0 (1) Since the extreme value arising from the Gamma PDF of C (i.e., Eq. (1)), converges asymtotically to the Tye I Gumbel distribution, the location arameter un and the scale arameter βn of the corresonding extreme value of the intermediate eak variable can be determined, resectively as follows, 1 1 1 1+ un ex un = () n 1 un 1 βn = [ nfc ( un) ] = n ex un 4 (3) Taking logarithm at both sides of Eq. (), one obtains un = ln n + ln(1 + 0.5 un) (4) When the samle size n is sufficiently large (e.g. when n>100), the characteristic largest value for the Gamma eak distribution can thus be aroximated as un ln n + ln ln n (5) The scale arameter or disersion of the largest eak value from the Gamma distribution can then be reduced from Eq. (3) to the following form as βn ln n /(ln n + ln ln n) (6) Based on Eqs. (5) and (6), the mean µ of the extreme value of the intermediate eak variable C can be given as follows C n γ ln n µ C = u ln ln ln n n + γβn = n + n + (7) ln n + ln ln n Using the relationshi of the intermediate eak variable C to the eaks of a Rayleigh rocess A m, i.e., C = Am / σ A 1, the exected eak factor of a combined resultant rocess with Rayleigh distribution can be written as 1
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-1, 009, Taiei, Taiwan γ ln n gg = µ C n + 1 = ln n + ln ln n + + 1 (8) ln n + ln ln n The above equation gives a closed-form formula for estimating the mean extreme resonse of a combined resultant rocess following the Rayleigh distribution. The exected eak factor determined by Eq. (8) is herein called as the Gamma eak factor, which is derived from the Gamma distribution given in Eq. (0). Since the eak distribution of the resultant rocess according to Eq. (17) is defined for those eaks over the threshold of mode valueσ A, the exected maximum resultant resonse can then be estimated in terms of the Gamma eak factor and the mode value of the resultant rocess A(t) as µ A = g n Gσ A (9) If two comonents do not have the same value of standard deviation, i.e., σ X < σ, the mean extreme resultant resonse of the combined rocess A(t) can then be aroximated as [Huang (008)] ( ) A g n G g f X g f µ σ + σ (30) The use of Eq. (30) for redicting the eak resultant acceleration resonse can be referred to as the combined eak factor (CPF) method. Illustrated Examle: The CAARC building The rototye CAARC building has an overall height of 180m and a rectangular floor lan dimension of 30 m by 45 m [Melbourne (1980)]. Aerodynamic wind forces acting on the 45-story CAARC building were measured by the synchronous multi-ressure measurement technique in the wind tunnel using a 1:400 scale rigid model. A 10-year return eriod hourly mean wind seed of 34.7m/s at the reference height of 90 m was used for calculating the acceleration resonses of the building. The 0-degree wind erendicular to the wide face acting in the short direction of the building was considered in the examle. Time history analysis was carried out by alying the measured time history wind forces in a comuterbased finite element model of the CAARC building. Table 1 resents the eak resultant acceleration results at the to corner of the CAARC building. The mean extreme resultant acceleration was calculated using the CPF method of Eq. (30) with two comonent standard deviation acceleration, which can be obtained from either frequency-domain aroach or time-domain aroach. The roosed combined eak factor method resulted in a eak resultant acceleration of 19.1 milli-g, slightly less than (-3%) that of the statistically evaluated value of 19.7 milli-g, which were calculated by averaging the 4 extreme resultant eaks over six samles of 10-minute resultant acceleration time history. Table 1: Peak resultant accelerations in the CAARC building under 0-dgree wind Peak resultant acceleration (milli-g) Used eak factors Time history analysis CPF method Eq. (30) g f g G 19.7 19.1 (-3%) 3.76 3.751 Conclusion In this aer, the robabilistic eak factor and the Gamma eak factor are develoed analytically by investigating the mean level-crossing rate and eak distribution of the resultant resonse rocess. The robabilistic eak factor is exlicitly exressed in terms of the random excitation duration, the non-exceedance robability and the sectral bandwidth shae arameter of the random acceleration resonse rocess. For a narrowband resonse rocess, 1/
The Seventh Asia-Pacific Conference on Wind Engineering, November 8-1, 009, Taiei, Taiwan the use of Davenort eak factor without accounting sectral bandwidth effects yields a conservative estimate of the mean extreme comonent resonse. The robabilistic eak factor has the advantage to redict extreme resonse at different levels of time-variant reliability or non-exceedance robability, which may be useful in the context of reliability erformancebased design of tall buildings. Based on the asymtotic theory of statistical extremes, the Gamma eak factor has been obtained for a non-gaussian combined resultant resonse, characterized by a Rayleigh distribution rocess. Using the Gamma eak factor, the combined eak factor method has been roosed for redicting the mean extreme resultant acceleration resonse of windsensitive tall buildings. The 45-story CAARC building tested in the wind tunnel was used to demonstrate the alicability of the combined eak factor method to evaluate the mean extreme resultant resonse. The wind tunnel derived acceleration time history results of the building verify that the combined eak factor method gives reasonably accurate rediction of the mean extreme resultant acceleration resonse of wind-excited tall buildings. References Chan C.M., Huang M.F., Kwok K.C.S. (009). Stiffness otimization for wind-induced dynamic serviceability design of tall buildings. Journal of Structural Engineering, ASCE, 135(8), 985-997. Chen X, Huang G. (009). Evaluation of eak resultant resonse for wind-excited tall buildings. Engineering Structures, 31, 858-868. Davenort A.G. (1964). Note on the distribution of the largest value of a random function with alication to gust loading. Proceedings, Institution of Civil Engineering, 8, 187-196. Gurley K.R., Tognarelli M.A., and Kareem A. (1997). Analysis and simulation tools for wind engineering. Probabilistic Engineering Mechanics, 1(1), 9-31. Holmes J.D, and Cochran L.S. (003). Probability distribution of extreme ressure coefficients. Journal of Wind Engineering and Industrial Aerodynamics, 91, 893-901. Hong H.P., Beadle S., and Escobar J.A. (001). Probabilistic assessment of wind-sensitive structures with uncertain arameters. Journal of Wind Engineering and Industrial Aerodynamics, 89, 893-910. Huang, M.F. (008). Performance-based serviceability design otimization of wind sensitive tall buildings. Ph.D. thesis, Hong Kong Univ. of Science and Technology, Hong Kong. Isyumov N., Fediw A.A., Colaco J. and Banavalkar P.V. (199). Performance of a tall building under wind action. Journal of Wind Engineering and Industrial Aerodynamics, 41-44, 1053-1064. Kareem A. (1987). Wind effects on structures: a robabilistic viewoint. Probabilistic Engineering Mechanics,, 166-00. Melbourne, W.H. (1980). Comarison of measurements of the CAARC standard tall building model in simulated model wind flows. Journal of Wind Engineering and Industrial Aerodynamics, 6, 78 88. Melbourne W.H., and Palmer T.R. (199). Accelerations and comfort criteria for buildings undergong comlex motions. Journal of Wind Engineering and Industrial Aerodynamics, 41-44: 105-116. Sadek F. and Simiu E. (00). Peak non-gaussian wind effects for database-assisted low-rise building design. Journal of Engineering Mechanics ASCE, 18(5), 530-539. Solari G. (1997). Wind-excited resonse of structures with uncertain arameters. Probabilistic Engineering Mechanics, 1(), 75-87. Vanmarcke E.H. (1975). On the distribution of the First-assage time for normal stationary random rocesses. Journal of Alied Mechanics, 4, 15-0. Tieleman H.W, Ge Z, and Hajj M.R. (007). Theoretically estimated eak wind loads. Journal of Wind Engineering and Industrial Aerodynamics, 95, 113-13.