Estimates of extreme wind effects and wind load factors:influence of knowledge uncertainties

by

Fabio Minciarelli and Massimiliano GioffrèUniversity of Perugia

Perugia, ITALY

Mircea GrigoriuCornell University

Ithaca, NY 14853 USA

and

E. SimiuBuilding and Fire Research Laboratory

National Institute of Standards and TechnologyGaithersburg, MD 20899 USA

Reprinted from Probabilistic Engineering Mechanics, Vol. 16, No. 4, 331-340, October2001.

NOTE: This paper is a contribution of the National Institute of Standards andTechnology and is not subject to copyright.

PROBABILISTICENGINEERINGMECHANICS

ELSEVIER Probabilistic Engineering Mechanics 16 (2001) 331-340

www.elsevier.com/locate/probengmech

Estimates of extreme wind effects and wind load factors:influence of knowledge uncertainties

Fabio Minciarellia,b, Massimiliano Gioffrec, Mircea Grigoriud, Emil Simiub,*

'University of Perugia, Perugia, ItalybBuilding and Fire Research Laboratory, National Institute of Standards and Technology, Building 226, Gaithersburg, MD 20899-8611, USA

cUniversity of Perugia, Perugia, ItalydComell University, Ithaca, NY 14853, USA

Received 7 ApriI2000; revised 19 ApriI2001; accepted 19 Apri12001

Abstract

We propose a probabilistic methodology for developing improved load factors in standard provisions for wind loads, and use it to examine:the cause of the discrepancy noted in the 1980s between estimates of safety indices for wind and gravity loads; the relative magnitude of loadfactors for hurricane and non-hurricane regions; and the effect of the length of wind tunnel pressure records on the estimation of peak windeffects. According to our calculations, (1) the discrepancy between estimates of safety indices for gravity and wind loads is an artifact that canbe removed by using current knowledge on probability distributions of extreme wind speeds; (2) the disregard in the ASCE 7-98 Standard of(a) knowledge uncertainties, and (b) errors inherent in the limited number of climatological data on which hurricane wind speed simulationsare based, leads to incorrect wind load factor estimates; and (3) increasing beyond 30 min or even 20 min the length of pressure records usedfor the estimation offtuctuation peaks appears to have a relatively small effect on estimates of overall wind effects. We outline future researchon wind directionality, sampling errors in the estimation of peak wind effects, and the use of probabilistic descriptions of wind effects andstructural capacity to estimate probabilities of occurrence of nonlinear limit states, including structural collapse. @ 2001 Elsevier ScienceLtd. All rights reserved.

Keywords: Building technology; Hurricanes; Load factors; Monte Carlo simulation; Reliability; Structural engineering; Wind effects

1. Introduction

The estimation of wind loads and of the reliability ofwind-excited structures has been investigated extensivelyduring the last few decades. However, a number of impor-tant issues remain unsolved and warrant examination anddebate:

to statistical analyses in which estimation uncertaintieswere not accounted for, risk consistency requires thatwind load factors be significantly larger for hurricanewinds than for extratropical winds [2]. This finding wastaken into account in the ASCE Standard 7-98 [3].However, in that Standard load factors are calculated asthe ratio between point estimates of 500 and 50-yr windloads, without accounting for knowledge uncertainties.In this paper, we perform estimates of load factors thattake such uncertainties into account.

3. Wind load peaks are based in the ASCE 7-98 Standardand its predecessors on records taken in the wind tunnelover periods equivalent to I h in full scale. Wind tunneloperators sometimes specify peaks based on recordslonger than I h -in some instances as long as tens ofhours. However, it may be the case that record lengths ofI h or even less, say 30 or 20 min, are sufficient, at leastfor codification purposes. It is therefore in order to exam-ine the extent to which the record length affects the esti-

* ColTesponding author. Tel.: + 1-301-975-6076; fax: + 1-301-869-6275. mates of wind effects when knowledge uncertainties areE-mail address: simiu@nist.gov (E. Simiu). duly accounted for.

0266-8920/01/$ -see front matter @ 2001 Elsevier Science Ltd. All rights reserved.Po: S0266-8920(0 1 )00024-8

1. According to the methods used by Ellingwood et at. [1],estimates of safety indices for structures designed inaccordance with the ASCE 7 Standard and its predeces-sors are considerably lower for wind than for gravityloading. One of the aims of this paper is to considerthis issue and propose a correction to those methods.

2. In the ASCE 7-95 Standard and its predecessors, windload factors specified for hurricane effects are the same asthose specified for extratropical storm effects. According

332 F. Minciarelli et al. I Probabilistic Engineering Mechanics 16 (2001) 331-340

discussed in Section 6). We use the model

Fpk(T,N) = tpCpJT)abcV2(T,Haero,Zo,N) (1)

where p is the air density, which is sufficiently well knownthat no uncertainty is usually associated with its value;CpJ T) is the peak factor for the fluctuating wind effect(e.g. bending moment, shear force, axial force, displace-ment) induced by the storm. It is associated with the largestpeak wind effect occurring during time T. The peak factor isa random variable that will be discussed subsequently;V(T,Haero,Zo,N) is the wind speed at reference heightHaero, averaged over the time interval T. The nominalmean recurrence interval of the wind speed is N, in years,and the roughness of the terrain that characterizes the siteupwind of the building is Zoo Haero is commonly, but notnecessarily, chosen to be the top of the building or theeave height; a, b, c are random variables associated withimperfect knowledge ('epistemic uncertainties'), asopposed to randomness inherent in a variable ("aleatoryuncertainties'). For example, the terrain rougb:ness isusually known. with some uncertainty, whereas the windspeed and the peak value of a fluctuating random processare inherently random variables. Unless otherwise indi-cated, all uncertainty variables, including a, b, c will beassumed to have truncated normal distributions with meanunity. The parameters a, b and c are defined as follows:

IIi this paper, we discuss these issues within a prob-abilistic framework. It is not our intention to presentdefinitive results: some parameters and models requireadditional research and should be based on a consensusyet to be established. Rather, we wish to develop an inves-tigation tool -with an attendant set of computer programs-that is transparent and simple to use, and to presentresults that, in Our opinion, shed useful light on reliabilityissues related to wind loading. The main motivation forthis work is to achieve standard provisions for windloads that are more realistic and risk-consistent than current

provisions.Our investigation is limited to structures that do not

experience significant wind-induced dynamic amplificationeffects. IIi Section 2 we describe the physical and probabil-istic framework used in our investigation. We then use thisframework for the following purposes. First, we wish toexamine the finding by Ellingwood et at. [1] that their esti-mates of the safety indices inherent in the ASCE Standardand its predecessors are significantly lower for wind than forgravity .loads. We do so by investigating the role of theassumed distribution of the extreme wind speeds in the esti-mation of load factors. Second, we report estimates ofhurri-cane wind load factors that take knowledge uncertaintiesinto account, and show that they reinforce earlier findingsthat, to ensure risk-consistency, wind load factors must besignificantly larger for hurricane-prone regions than forregions not subjected to hurricanes. Third,. we examine therole of the length of the wind tunnel record of fluctuatingwind effects on the estimated magnitude of estimated peakwind effects. Fourth, we examine the sensitivity of our loadfactor estimates to changes in the variabilities of the uncer-tainty parameters on which the estimates are based. Fifth,we briefly discuss three topics for which further research isneeded: the use of simulations to estimate peaks andsampling errors in the estimation of peaks, the role ofwind directionality, and the use of probability distributionsof wind effects and structural capacities to estimate prob-abilities of structural collapse and other limit states invol-ving structural nonlinearities. The paper ends with a set ofconclusions.

a reflects aerodynamic errors inherent in the wind tunnelbeing used (results of wind tunnel tests may depend uponthe specific wind tunnel facility in which the tests areconducted);b reflects aerodynamic errors due to (a) the use of windtunnel rather than full-scale measurements, and (b) theim~rfect calibration of wind tunnel to full-scale data;c reflects errors in the transformation of aerodynamiceffects into stresses or other structural effects.

2.2. Wind speed model

The wind speed is estimated by using:

.measured or simulated wind speed data at the standardmeteorological elevation Hme! (e.g. 10 m above ground)in open terrain with standard roughness Zol (e.g.Zol = 0.05 m). The averaging time T for the wind speeddata varies. In the ASCE 7-98 Standard the interval T =

3 s is used. Simulated hurricane wind speed data avail-able in NIST public files (accessible as indicated inRef. [4]) and based on the work of Batts et al. [5] areaveraged over one minute. From these observed orsimulated wind speed data, point estimates can bemade of wind speed data with various mean recurrenceintervals. The estimation methods are described sub-sequently, as are estimation methods for the samplingerrors corresponding to winds with N-yr mean recurrenceintervals. The estimates are also affected by observation

2. Probabilistic modeling of extreme wind effects

2.1. Wind effect model

We denote by Fpk(T,N) the peak wind load effect with anN-yr nominal mean recurrence interval, occurring in theprototype subjected to the action of a storm with durationT. The time interval T can be, say, 20, 30 min, 1 h or longer,and the storm is assumed to induce wind loads that may beassumed to be a statistically stationary process. The nominalmean recurrence interval can be, say, 50, 500 or 10,000 yr,and corresponds to wind speeds estimated without account-ing for wind directionality effects (these are briefly

F. Minciarelli et al. I Probabilistic Engineering Mechanics 16 (2001) 331-340

333

sound information, for example, information obtained fromwind tunnels that meet standard performance criteria andare calibrated against dependable full-scale test results; orinformation obtained from certified weather stations. Grosserrors or information based on substandard sources of infor-matio~ are not accounted for in our uncertainty model.

2.3. Estimation of wind speeds V(T,Hmet,Zol,N) with amean recurrence interval N

errors, modeled by the uncertainty factor q, which, asnoted earlier, is assumed to be normally distributedwith mean unity.

.conversion factors r to account for wind speed averagingtime. To obtain wind speeds averaged over time 7, windspeeds averaged over time T are multiplied by the factorr(T, 7). While, like all our uncertainty variables, weassume r to be normally distributed, its mean is notunity. Rather, the means of the variable r(T ,7 = I h)are obtained from [6] as functions of T (see also Ref.[7], Chapter 2). For example, for T = 3 s, Mean[r(T =3 s, t = I h)] = 0.65, that is, the l-h average is 0.65 timesthe 3 s peak. If 7 = 20 min and 30 min, the informationprovided by Durst yields the following means of r:

Mean[r(T = 3 s, t = 20 min)] = 0.66

Denote the mean and the standard deviation of the sampleof extreme wind speeds by E(X) and s(X), respectively. TheExtreme Value Type I distribution and its inverse (the vari-ate corresponding to a mean recurrence interval N in years)are

F1(x) = exp{ -exp[(x -.u)/u]},

(3)Mean[r(T = 3s, t = 30 miD)] :=:: 0.655u = [(6) l/2/1T]S(X), .u = E(X) 0.57720-.

.Factors converting wind speeds averaged over 1 h (or,e.g. 30 or 20 min) say, over open terrain with roughnessZol at the standard meteorological elevation Hme! to meanhourly wind speeds at the aerodynamic reference eleva-tion Haem over the terrain with roughness Zo upwind of thebuilding. These factors are yielded by (a) the logarithmiclaw and (b) the relation between wind speeds in differentroughness regimes ([7], pp. 42, 48), and are functions ofthe roughness lengths Zol and Zo. (Effects of terrain fetchthat may be insufficient for the full development of theboundary layer upwind of the building, or of escarpmentsor hills, are not dealt with in this paper). We denote by uand s the random variables, again assumed to be normallydistributed with mean unity, that reflect the uncertaintieswith respect to the actual values of Zol and ZOo The follow-ing approximate expression is obtained:

V( T, Haem' Zo, N)

xI(N) = JL -0" In( -In(1 -I/N)). (4)

The reverse Weibull distribution and its inverse are

Fm(x) = exp{ -[(JL -x)IO"] 'Y}, JL = E(X) + 0"['(1 + 1/1);

u = us(X)/[I'(1 + 2/1) -F(1 + 1/1)]1/2 'Y = -IlK. (5)

XIII(N) = JL -u[ln(l -I/N)] I/y. (6)

In Eq. (5), 'Y (or K) is the distribution's tail length

parameter.

2.4. Sampling errors in the estimation of extreme windspeeds

= UH[(SZo)/(UZOl)]6qr(T, T)V(T, Hmet' zol,N)

x {In Haero/(UZol) }/{In Hmet/(szo)} (2)

where the exponent 15 = 0.0706 in the relation between

wind speeds in different roughness regimes may beassumed to have negligible variability, andV(T,Hmet'Zol,N) denotes the estimated wind speedwith a mean recurrence interval N, averaged over timeT, at elevation Hmet above open terrain with roughnesslength Zol. The factor uH in Eq. (2) reflects the uncertaintywith respect to the applicability of this model to hurri-cane wind speeds, which differ to some extent fromextratropical wind speeds. For extratropical stormwinds the factor UH is, by definition, unity; for hurricanewind speeds its mean and coefficient of variation areassumed to be unity and 0.05, respectively.

The resulting estimate of the wind speed of average returnperiod N yr is

iN = E(x) + a(N)s(x). (8a)

where

a(N) = -0.779697In(-ln(1 -IIN)) -0.450053 (8b)

for the E. V. Type I distribution, and

a(N) = 1'(1 + l/y)ln(1 -I/N)] 1/')'/[1'(1 + 21y)

-l'(1 + 1/1))] 1/2 (8c)

for the reverse Weibul1 distribution (Eqs. (4) and (6).The value of iN is uncertain because the estimates

(E(x), s(x») of (E(x), s(x)) are random variables whosevariance depends on the sample size n. The mean and

It is assumed throughout that, the quantities involved inthe calculation of wind effects are estimated on the basis of

The mean E(x) and the standard deviation s(x) must beestimated from an n-yr record {XI,X2, ...,xn} of the extremewind speeds. For example,

334 F. Minciarelli et al. / Probabilistic Engineering Mechanics 16 (2001) 331-340

variance of XN can be approximated by [8,9]

Mean[XN] ~ E(X) + s(X)a(N)[(n -1)/n]1/2

The distribution of the peak value Cpk( 7) can be approxi-mated as follows:

(9)FIF,..(X) = P( Cpk(t) :5 x) = exp( -J.(x)r), (13)

+ ..fiY3a(N)[[(1 l/n)( 14 -1)/2 + l/n)]/[ 14S[XN]. =l.(x) = [ui/(21T) exp[ -(1/2)(g(X)-1)1 (14)

2/(n l)llT a(N)2[(I- l/n)( 'Y4 -1)/2

+

1/n)}/2}1/2s(X)/nl/2 {lO)

where 'Y3 and 'Y4 denote the coefficients of skewness andkurtosis of the distribution F1 or Fm (Eq. (3) or Eq. (5)).

For hurricane-prone regions, the estimates of extremewind speeds are obtained by using the peaks over thresholdapproach. Sampling errors are estimated on the basis of thenumbers of data that exceed the respective thresholds.However, it should be remembered that the data themselvesare obtained by Monte Carlo simulation on the basis ofinformation on relevant climatological parameters: pressuredifferences between the edge and the center of the hurricane,the radius where hurricane wind speeds are highest and thetranslation velocity of the hurricane. This climatologicalinformation is based on records of about 100 yr, correspond-ing to a total number of hurricanes at a particular location ofabout 50 (depending upon geographical location). There-fore, the climatological information on which the simulatedhurricane data is based, is itself subjected to samplingerrors. The effect of these climatological sampling errorson the estimated extreme wind speeds based on simulateddata was studied in Ref. [10]. According to those studies, thecoefficient of variation of the sampling errors in the estima-tion of the 50-yr wind speeds is typically of the order of0.10. For 500- yr wind speeds, we assume that the coefficientof variation of the climatological sampling errors is 0.125.The total sampling error therefore depends upon the thresh-old, and is obtained in all cases by noting that the totalvariance of the sampling error is equal to the sum of thevariance associated with the size of the simulated datasample used in the analysis, and the variance associatedwith climatological parameter uncertainties.

is the upcrossing rate of the threshold x per unit time, O"i isthe standard deviation of the process z, the symbol. denotesdifferentiation with respect to time, and -1 denotes 'inverseof. The distribution FIF,T can be used to calculate the meanand variance of Cp'"

If X(t) represents an internal force (e.g. the bendingmoment at a particular cross-section of a frame), itsmarginal distribution and covariance function can be esti-mated from a wind-tunnel record of that force. These twoproperties define compJetely the model of Eq. (11), sincethey determine the memoryless transformation g(Z(t» andthe covariance function (w) ([9], pp. 44-53).

Because the peak load effect given by Eqs. (12)-(14)depends on the marginal distribution and the covariancefunction of X(t), Cpk will depend on the particular sampleconsidered in the analysis and on its size. In general, noclosed form expressions are available for the estimation ofthe sampling errors in the estimation of Cpk' It is, however,possible to estimate these errors from numerical simula-tions. This topic will be briefly discussed in Section 6.

The calculations performed in this paper were based on a42-h record of wind-induced bending moments at the kneeof a typical low-rise industrial building steel frame,provided by Dr M. Kasperski of Bochum University. Thelargest peak normalized with respect to the r.m.s. for the 42-h record was 6.5. The normalized mean of the peaks for theI-h records was 5.00, with a c.O.V. = 0.09. For the 30 and20-min records, the mean and c.O.V. were 4.6, 0.10 and 4.4,0.11, respectively.

3. Wind load factors

All results in this section were obtained by Monte Carlosimulation, the number of samples in each simulation beingns = 2,000.

2.5. Peak factors Cpk

Consider a non-Gaussian process3.1. Regions not subjected to hurricane winds

X(t) = FWl«p(Z(t))) = g(Z(t)), 0 :5 t $ T. (11) The wind load factor is defined in the ASCE 7-98 Stan-dard as the square of the ratio between the point estimate(the 50 percentile) of the 500-yr wind speed and the pointestimate of the 50- yr wind speed, both estimates being basedon the assumption that the extreme wind speeds are bestfitted by the Extreme Value Type I distribution. TheASCE 7-98 Standard further specifies that the structuralmember experience the state associated with strength designunder the wind effect consisting of the 50-yr wind effecttimes the load factor so defined (see ASCE 7-98 Standard

Commentary, p. 114).

where F IF denotes the marginal distribution of the process,l/J(Z) denotes the distribution of a Gaussian variablewith zero mean and unit variance and Z(t) is a stationaryGaussian process with zero mean, uriit variance and covar-iance function (w) = E[Z(T)Z(T + w)].

Denote the peak value of. the internal force dUring theinterval (0, T) by

(12)Cpk(t) = max X(t)OStSr

F. Minciarelli et al. / Probabilistic Engineering Mechanics 16 (ZOO1) 331-340 335

Table IBasic set of means and standard deviations of the uncertainty parameters,assumed to be nonnally distributed

bUncert. paramo a q

MeanCoeff. of VaT.

1 1 1 1 1 1 See Section 20.05 0.05 0.025 0.1 0.1 0.025 0.05

conducted following the development in the 1970s ofmodem extreme value theory and approaches, including

peak-over-threshold methods, strongly suggests thatextreme wind speeds are better fitted by Type ill distribu-tions of the largest values (reverse Weibull distributions)which, unlike the Type I distribution, have bounded uppertails [11-13]. We performed Monte Carlo simulations forestimating load factors LF by using first the Extreme ValueType I distribution and then the reverse Weibull distribu-tion, all other assumptions and parameters being the same.Sampling errors in the estimation of the extreme windspeeds, and peak values and sampling errors in their estima-tion, were estimated as indicated in Sections 2.4 and 2.5.The basic set of parameter values used in the calculations islisted in Table 1.

In addition, in our basic calculations it was assumed thatthe roughness lengths are Zo = 1.00 m and Zol = 0.07 m

(For other values used in the calculations, see Section 5.)In our opinion, the assumption that the uncertainty para-

meters are normally distributed, though not exact, is reason-able, as are the values of the means and standard deviationslisted in Table 1. A calibration of these values against thevalue of the load factor believed to be reasonable onthe basis of past practice can be performed, depending onthe results of the calculations.

First, we discuss briefly the influence on the results of thecalculations of the assumed probabilistic of the extremewind speeds. The results showed that estimated load factorsare significantly larger if an infinite-tailed model (EVI),rather than a finite-tailed model (RW), is used for the distri-bution of the largest wind speeds. For example, for theparameters of Table I, if the sample coefficient of variationof the extreme wind speed data is C.o. v = 0.15, then LF =1.55 under the assumption that the reverse Weibull distribu-tion with tail length parameter K = -0.2 is valid, and LF=1.90 under the assumption that the Extreme Value Type Idistribution is valid. (The tail length parameter K = -0.2,while not universally valid, is usually a reasonably conser-vative approximation for most stations -see Ref. [12].)These results are typical.

Under the assumption that the extreme wind speeds havea reverse Weibull distribution (see Refs. [11-13]), it followsfrom our results that load factors based on the Type IExtreme Value distribution are significantly overestimated.Therefore, in our opinion, the reason for the low safetyindex estimates for wind loading obtained by the procedureused in [1] is the use in that procedure of the assumption thatextreme wind speeds are best fitted by the Extreme ValueType I distribution -an assumption consistent with the1970s state of the art in extreme value applications towind speeds. We note that, using this assumption for theestimation of wind speeds with relatively short mean recur-rence intervals (50 yr, say) is, by and large, acceptable. Theassumption becomes onerous if it is used for long meanrecurrence intervals, such as those associated with strengthdesign or ultimate structural capacity.

336 F. Minciarelliefal. / Probabilistic Engineering Mechanics.16 (2001) 331-340

Note that the load factor specified in the ASCE 7-95Standard for non-hurricane winds is 1.3, after multiplicationby the wind direction reduction factor, assumed in the Stan-dard to be 0.85; before reduction the load factor is 1.3/0.85 = 1.53 (i.e. close to our calculated value correspondingto the reverse Weibull assumption, LF = 1.55). The ASCE

7-95 value of the load factor was based on engineeringjudgment and experience, rather than on safety index calcu-lations reported by Ellingwood et al. [1] (as was mentionedearlier, these were performed before modern extremevalue distribution theory became widely applied -see, e.g.Ref. [14]). We note the agreement of our estimated valuewith the ASCE 7-95 value. We also note that the value 1.53was augmented in the ASCE 7-98 Standard (p.. 4, Section2.3.2) to 1.36/0.85 = 1.6 (rather than being 1.5, as is

indicated erroneously in the Commentary to the Standard,see ASCE 7-98, p. 114). This augmented value is stillapproximately consistent with our calculated value LF =

1.55. The reason for this consistency lies in our choice ofuncertainty parameters, which as we suggested earlier,appear to be reasonable. However, should it be considerednecessary to use different uncertainty parameters, theestimated value of the load factor would change to someextent. The sensitivity of the results to parameter changes isdiscussed in Section 5.

same set of uncertainty parameters, for extratropical windspeeds with coefficient of variation cov = 0.15, assumed tobe best fitted by the reverse Weibull distribution with K =

-0.20. (Note that the Extreme Value Type I distribution,which as indicated earlier, has been assumed to be an appro-priate distributional model for wind speeds in non-hurricaneregions, has to our knowledge never been considered applic-able to hurricane wind speeds.) As was indicated earlier, theresult that load factors for hurricane-prone regions are largerthan for non-hurricane regions is not new, and was takeninto account in the development of the ASCE 7-98 Standard.The methodology presented in this paper allows, in ouropinion, a more realistic estimation of the load factors,which takes into account the various uncertainties discussedin this section. As is the case for non-hurricane regions,uncertainty parameters that differ from those chosen herecan be used in the estimates.

The mean and standard deviation of the data sets thatexceed the 20.7 m/s threshold are 25.3 and 4.87 m/s; forthe 22.8, 24.87, 26.95 and 29 m/s thresholds, they are (27,4.5 m/s), (28.3, 4.25 m/s), (29.8, 3.9 m/s) and (30.9, 3.8 m/s). Results of sensitivity studies are presented in Section 6.

4. Inflnence of length 'T of time series of pressuresrecorded in the wind tunnel

3.2.

Regions subjected to hurricane winds

We performed simulations with the basic set of uncertaintyparameters of Table 1, using the 999 simulated T = 1 min

hurricane wind speeds for a northwestern Florida coast loca-tion, developed by Batts et al. [5] and available in a publicelectronic file (the identifier of the data set in the electronicfile is 'milepost 1100'). Simiu and Heckert [12] and Simiu,Heckert and Whalen [2] reported that hurricane wind speedsare best fitted by reverse Weibull distributions for which it isreasonable to assume tail length parameter K "'" -0.2, say,

and that estimates of coastal hurricane wind speeds withmean recurrence intervals of 50 to 2,000 yr based on thisassumption are consistent with estimates obtained indepen-dently by other authors (e.g. Refs. [7], p. 117. For this reason,in our simulations, the distribution fitted to the extreme windspeeds was reverse Weibull with tail length parameter K =

-0.20. The distribution parameters were found by using Eq.(5) and the peaks over threshold method. Sampling errors inthe estimation of the extreme wind speeds, and peak valuesand sampling errors in their estimation, were estimated asindicated in Sections 2.4 and 2.5, respectively. The basicset of parameter values used in the calculations is listed inTable 1.

For the mean hourly wind speed thresholds 19.4, 21.4,23.3,25.3 and 27.2 m/s (sample sizes 233, 172, 136, 100and 77, respectively), the load factors obtained by simula-tion were, respectively, LF = 2.14, 2.13, 2.13, 2.15, and2.21. The hurricane load factor LF "'" 2.14, say, should becompared with the load factor LF "'" 1.55 obtained, with the

Estimates of wind effects depend upon the length 'T of thetime series of the pressures recorded in the wind tunnel. Thisdependence is of interest insofar as unduly long recordingperiods might create data storage problems for aerodynamicdatabases on buildings with a large number (say, hundreds)of pressure taps -see Ref. [2]. On the one hand it isdesirable that the length of the time series 'T of the pressuresrecorded in the wind tunnel be as short as possible. Thiswould be convenient for both data storage and data proces-sing reasons. On the other hand, a length 'T that would be tooshort would cause unacceptable sampling errors in the esti-mation of extreme wind effects.

We performed simulations for three cases: 'T = 1 h, 'T =30 min and 'T = 20 min. As mentioned in Section 2.5, therespective means and standard deviations of the peaks weretaken to be equal to their averages over 42 1 h sets, 8430 min sets and 126 20 min sets of time histories extractedfrom the 42-h long records of the wind-induced fluctuatingmoment at a frame knee mentioned earlier. For 'T = 1 h, 'T =30 min and 'T = 20 min the mean and standard deviation ofthe peak factor were (in non-dimensional units), 5.00 and0.45,4.6 and 0.46,4.4 and 0.48, respectively. We conductedtwo sets of calculations for wind speeds in non-hurricaneregions. In the first set, it was assumed that the probabilitydistribution of the peaks is normal. In the second set, it wasassumed that the distribution of the peaks is Extreme Value

Type I.For non-hurricane extreme winds with cov = 0.15, and

the basic set of uncertainty parameters (Table 1), under the

F. Minciarelli et at. / Probabilistic Engineering Mechanics 16 (2001) 331-340 337

addition, we assume that the sample size of the set ofextreme wind speed data is n = 40, and that the averagingtime of the recorded extreme wind speeds is T = 3 s.

Influence of coefficient of variation of the extreme windspeeds for non-hurricane winds assumed to have a reverseWeibull distribution with tail length parameter K = -0.2,

the load factor corresponding to a coefficient of variationcov = 0.15 of the sample of extreme wind data is LF =1.55. For a cov = 0.12, the corresponding estimated loadfactor is LF = 1.5. If the extreme wind speeds are assumedto be best fitted by an Extreme Value Type I distribution, therespective values are LF = 1.9 and LF = 1.8. Henceforth,we list in this section only load factors for cov = 0.15.

Influence of variabilities associated with (1) testing indifferent wind tunnels, (2) wind tunnel testing versus full-scale testing, (3) terrain roughness, (4) observation errors,(5) conversion of aerodynamic pressures into structuraleffects, (6) and conversion from wind speed averaged overinterval T to mean hourly, 30, or 20-min wind speeds.Results on calculations based on various values of the stan-dard deviations of the uncertainty parameters listed in Table1 are shown in Table 2. In all calculations, the samplingerrors in the estimation of extreme wind speeds weretaken into account by assuming the sample size to be n =

40. We noted earlier that the calculations were performed byassuming the terrain roughness lengths to be Zo = 0.07 m,Zol = 1.00 m. In addition, we performed calculations, inwhich, we assumed Zo = 0.05 m, Zol = 1.00 m, and Zo =0.05 m, Zol = 0.35 m, and Zo = 0.07 m, Zol = 0.35 m.

While, as expected, the influence of these assumptions onthe 50 and 500-yr load estimates can be significant, theinfluence on the load factors was found to be negligible

(about 1%).Additional information on the effect of various uncertain-

ties is contained in Table 2.Table 2 shows, for example, that if all variabilities of the

uncertainty parameters listed in Table 1 are zero, rather thanhaving the values of Table 1, assuming the wind speedshave a reverse Weibull distribution, the load factor is LF =1.36 (case 2), rather than LF = 1.54 (case 1), and the esti-mate of F pk(500, 0.90) is 698, rather than 805. (Assumingthe wind speeds have an Extreme Value Type I distribution,the load factor is LF= 1.75, rather than LF= 1.90). Weconclude that neglecting all these uncertainties, as has beendone for the development of load factors in the ASCE 7-98Standard, would be inappropriate.

Table 2 also shows that, if instead of the variabilitieslisted in Table 1, we use variabilities augmented by 50percent with respect to those of Table 1, the load factorsincrease by about 10 percent for the reverse Weibullcase, and slightly less for the Extreme Value Type I case

(case 3).The influence of changing individual, pairs of, and three

variabilities is also shown in Table 2. We note that, accord-ing to Table 2, the single most important variability amongthose listed in Table 1 is the variability of the ratio r between

simplifying assumption that the peaks are normally distrib-uted, and the assumption that the extreme wind speeds arebest fitted by a reverse Weibull distribution with tail lengthparameter K = -0.2, we obtained, for T = 60, 30 and20 min, respectively, load factors LF = 1.54, 1.55 and1.57, and estimates of Fpk(N = 500,0.90) of 805, 800,and 793 (non-dimensional units), respectively; under theassumption that the wind speeds have an Extreme ValueType I distribution, the results were LF = 1.90, 1.90,1.93; and of Fpk(N = 500,0.90) = 1084,1077,1067. If itwas assumed that, rather than being normally distributed,the peaks have Extreme Value Type I distribution, theresults obtained differed insignificantly from those corre-sponding to a normal distribution of the peaks. The reasonfor these small differences lies in the 'drowning' of thelarger extreme values of the peak factor Cpk yielded by theType I distribution in the numerous errors and uncertaintiesinvolved in estimation of Fpk'

Therefore, for peak statistics comparable to those used inour calculations, that is, statistics typical of internal forces inframes of low-rise buildings, 30-min records would beadequate for codification purposes. However, in our opinionthis issue should be further investigated by consideringother statistics of fluctuating time histories.

Wind tunnel operators commonly conduct tests over peri-ods longer than 1 h and recommend for design the largestpeak recorded during those periods. To evaluate this prac-tice we consider the largest peak of the 42-hBochum record.We assume that the variability of this peak is negligible, thatis, that largest peaks of 42-h records would differ little fromrecord to record. For the 42-h record, the largest non-dimen-sional peak is 6.0; using the basic set of uncertainty para-meters (Table 1), a coefficient of variation of the extremespeeds cov = 0.15, and the assumption that the reverseWeibull distribution is an appropriate model of the extremewind speeds, we obtain LF = 1.50, instead of the valueLF = 1.55 obtained by using the statistics for the 1-h record,and Fpk(N = 500,0.90) = 914 instead of 805. Using thestatistics corresponding to a 42-h record instead of a l-hrecord thus results in a reduction of the estimated load factorby about 3 percent. However, the estimate of Fpk(N =500,0.90) increases by almost 15 percent. Adopting thislarger value of F pk(N = 500,0.90) would be unrealisticallyconservative, however, since powerful storms whosehighest mean speeds may be assumed to be uniform overdurations exceeding 15-60 min rarely occur in nature.

5. Estimates of load factors with and withoutconsideration of knowledge uncertainties: sensitivitystudies

5.1. Non-hurricane wind regions

Having investigated in Section 4 the role of the recordlength 'T we assume throughout this section that 'T = 1 h. In

338 F. Minciarelli et at. / Probabilistic Engineering Mechanics 16 (2001) 331-340

Table .2Calculated values of load factor LF corresponding to various sets of uncertainty parameters a, b, C, S, U, q, r. (The symbols sa,sb,sc,sq,sr,ss,su used in the firstcolumn of the table denote standard deviations of the variates a, b, c, q, r, s, u, respectively)

1.541.361.691.481.391.511.371.411.521.531.541.441.511.421.541.451.541.59

1.91.752.061.851.761.861.751.771.871.891.891.811.871.791.891.821.911.96

1. Basic set (see Table 1)2. sa= sb=sc= su= sq=sr= ss=O3. Basic set increased by 50%4. sa= sb= sc = su = sq = ss =0; sr=0.055. sa=sb=sc=sq =sr=su=O; ss=O.I6. sa = sb = sc = su = sq = O;sr = 0.05, ss= 0.17. sa= sb= sc = sq = sr= 0; ss= su= 0.18. sa=sb=sc=sq=O; su=ss=O.19. Basic set, except ss = 010. Basic set, except su = 0II. Basic set, except sq = 012. Basic set, except sr = 013. Basic set, except ss = su = 014. Basic set, except ss = sr = 015. Basic set, except sa = sb = sc = 016. Basic set, except sr = 0.02517. Basic set except su = 0.218. Basic set except ss = 0.2

the peak 3-s gust speed and the corresponding mean hourlyspeed (cases 12 and 16 versus case 1). We also note that theinfluence of the terrain roughness upwind of our low-risestructure is modest (cases 9 and 18 versus case 1). Accord-ing to our results, it appears that seeking the greatest possi-ble precision in the determination of that roughness may notbe worthwhile from a designer's viewpoint. This is evenmore so the case for the terrain roughness at the meteoro-logical site (cases 10 and 17 versus case 1).

Note in Table 3 that if the basic set of parameters is takeninto account the load factor is LF = 2.14, whereas if thoseparameters are disregarded LF = 1.99. This suggests that

the ASCE 7-98 failure to account for those parametersappears to be in need of correction. According to Table 3,the contribution to the sampling errors due to the uncertaintywith respect to the climatological parameters (i.e. to thelimited number of data on the basis of which hurricanewind speeds are simulated) is also significant. If this uncer-tainty is taken into account, the basic set of parametersyields a load factor LF = 2.14; if it is disregarded, thatsame set yields LF = 1.96. Because the number of simu-

lated wind speed data used in the analysis is relatively.I.~ge,the contribution of the sampling errors associated with thewind speed sample size is negligible. Therefore, it appearsthat simulating large numbers of data (e.g. 10,000), ratherthan, say, about 1000 (as in Refs. [5,10]) has limited, if any,usefulness for practical purposes. Finally, we note that thedependence of the load factors upon the wind speed thresh-old is weak (about 3% for the cases of Table 3; similarresults were obtained for other cases).

5.2. Hurricane wind regions

Results of calculations on the influence of uncertaintiesaffecting the estimation of load factors and of extreme windspeeds for hurricane winds are summarized in Table 3.

Table 3Estimated load factors LF for various wind speed thresholds under consid.eration of various types of errors

Threshold (m/s) Clim. error" Sampl. Error suhb Basic errorsc LF

0.05 y0 n0.05 y0 n

2.141.991.961.65

19.419.419.419.4

yynn

yyyy 6. Future research

2.131.981.961.64

0.05 y0 n0.05 y0 n

To complement the research reported in this paper thefollowing research topics are suggested:

19.419.419.419.4

yynn

nnnn

00

1.71.69

27.227.2

yn

a See Section 2.4 following Eq. (10)b suh = standard deviation of parameter UH. see Eq. (2).C Y indicates that all the errors from the basic set (Table 1) are accounted

for; n indicates that all those errors are disregarded.

1. The development of a catalog of estimates of the meanand standard deviation of the peak Cpk' A possibleapproach to this task is to simulate time series with typi-cal marginal distributions and spectral densities, andobtain the requisite statistics on peaks from the timeseries.

2. Accounting for directionality effects. Recent research

F. Minciarelli et at. / Probabiiis.tic Engineering Mechanics 16 (2001) 331-.,340 339

(Rigato et al. [4]) has shown that the directionality reduc-tion factor Cdir, that is, the ratio between wind effectswith a mean recurrence interval N estimated by takingwind direction into account, and wind effects with anominal mean recurrence interval N estimated withoutaccounting for wind direction, increases as N increases.For example, for N = 50 yr, it is reasonable to assumeCdir = 0.85. However, for N = 500 yr, the value of thereduction factor can be Cdir = 0.95. This variation is not

currently accounted for in the ASCE 7-98 Standard.3. Estimating probability distributions of wind effects

covering very long mean recurrence intervals. This canbe achieved by using the computer program developedfor this paper. The availability of such distributions willmake it possible to estimate failure probabilities forstructures with nonlinear behavior, including structuresclose to the collapse limit state.

simulations on which current estimates of hurricanewind speeds are based. These errors far outweigh errorsdue to the limited size of the simulated hurricane windspeed data sets.

3. Earlier estimates of load factors and reliability indexes,according to which structures calculated in accordancewith conventional standard provisions are less reliableunder wind than under gravity loads, depended uponthe assumption that extreme wind speed distributionshave infinitely long tails. Recent statistical analyses ofextreme wind speeds strongly suggest that this assump-tion is not warranted, and that extreme wind speeds arebetter modeled by reverse Weibull distributions of thelargest values, which have finite upper tails. The use ofthis distribution in lieu of the Extreme Value Type Idistribution results in a considerable reduction of themagnitude of the estimated load factor correspondingto a nominal 500-yr wind effect, and helps to bring inline reliabilities (safety indices, load factors) under windloads and under gravity loads.

4. For any given mean and standard deviation of the peaksof the wind effect fluctuations, the assumed probabilitydistribution of the peaks appears to have a negligibleeffect upon the overall estimated wind effects.

5. For low-rise buildings, increasing wind tunnel pressurerecord lengths beyond20 or 30 min to obtain better esti-mates of peaks does not appear to improve significantlyestimates of overall wind effects.

7. Conclusions

The computer programs used in this work are to beassembled and made public by NlST and the Departmentof Civil Engineering, Texas Tech University.

Acknowledgements

M. Grigoriu acknowledges partial support by Texas TechUniversity (TTU) under the NISTfTTU CooperativeAgreement on Windstorm Mitigation Initiative, and fromthe Building and Fire Research Laboratory, NationalInstitute of Standards and Technology (BFRL/NIST). F.Minciarelli's work was performed during his tenure as aGuest Researcher at BFRL. He and M. Gioffre were alsopartially supported by BFRUNIST.

In this paper we presented a probabilistic framework forthe estimation of load factors that accounts for knowledgeuncertainties. Our estimates showed that reasonableassumptions and estimation procedures lead to load factorsLF = 1.55, which are comparable to those validated byexperience for regions not subjected to hurricanes. Theseload factors may be reduced for wind directionality effects,which remain to be incorporated in the calculations. Loadfactors for hurricane-prone regions should be considerablylarger, that is, LF= 2.15, again before reduction for winddirectionality effects. These values are tentative, and theiruse would depend upon consensus on the appropriatedescriptions of the uncertainty parameters used in thecalculations.

It was assumed throughout the paper that the quantitiesinvolved in the calculation of wind effects are estimated onthe basis of sound information, for example, informationobtained from wind tunnels that meet standard performancecriteria, and calibrated against dependable full-scale testresults; or information obtained from certified weatherstations. Gross errors or information based on substandardsources of information are not accounted for in ouruncertainty model.

Based on the calculations presented in the paper, weconclude that:

References1. Parameter uncertainties that are neglected in the ASCE

7-98 Standard can contribute significantly to the esti-mated magnitude of the load factor.

2. Estimates of load factors for hurricane winds mustaccount for sampling errors in the estimation of clim-atological parameters such as pressure defect, radiusof maximum hurricane wind speeds, and hurricanetranslation velocity, which are used in Monte Carlo

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