equivalent static wind loading on buildings - new perspective

7/28/2019 Equivalent Static Wind Loading on Buildings - New Perspective

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EquivalentStaticWindLoadingonBuildings:ANewPerspective

XinzhongChenandAhsanKareem

NatHazModelingLaboratory,UniversityofNotreDame,156FitzpatrickHall,

NotreDame,IN,USA

ABSTRACT:Aframeworkforevaluatingtheequivalentstaticwindload(ESWL)foranygivenpeakresponseofbuildingscharacterizedbyuncoupledmotionsinthethreeprimarydirectionsispresented.Thisincludesanewdescriptionofthebackgroundloadingbasedonthegustloadingenvelope, whereas the resonant component is described in terms of the inertial loading. TheESWLforthetotalpeakresponseis thenexpressedas alinearcombinationof thebackgroundand resonant components. A closed-form formulation of the ESWL based on this framework

utilizing ananalyticalwindloading modelispresented.The gust response factorsandESWLsfor various alongwind response components at different building elevations are discussed tohighlightadvantagesoftheproposedscheme.

KEYWORDS:Windloads,Windeffects,Gustresponsefactor,Buildings,Structuraldynamics,

Randomvibration.

1 INTRODUCTION

In current design practice, spatiotemporally varying wind loads on buildings are modeled as

equivalent static wind loads (ESWLs). Traditional gust response factor (GRF) approach

(Davenport1967)iswidelyusedinmostbuildingdesigncodesandstandardsforthealongwindresponsethatresultsinaloaddistributionsimilartothemeanwindload(e.g.,ZhouandKareem

2001). The GRF concept has been extended to the acrosswind and torsional response

components (Piccardo and Solari 1996; Kareem and Zhou 2002). However, the GRFs mayexhibit wide variations for different response components of a structure and may have

significantlydifferentvaluesforstructureswithsimilargeometricprofilesandassociatedwind

loadcharacteristics,butdifferentstructuralsystems.Fortheacrosswindandtorsionalresponses,

which are typically characterized by the low values of mean wind loading and associated

response, particularly, for symmetric buildings, the corresponding GRFs may not project the

samephysicalmeaningasthetraditionalGRFforthealongwindresponse.AnESWLdescriptionbasedonthepeakdynamicpressure/windload(includingthemean

load)hasbeenadoptedinsomebuildingdesigncodessuchasthedraftEurocode(ENV-1991),ASCE7-02 and the new Australian/New Zealand Standards (Holmes 2002). This formatdescribestheESWLasthepeakdynamicloadmultipliedbyaconstantcoefficientreferredtoasdynamicresponsefactor(DRF)(Holmes2002).InSolariandRepetto(2002),anidenticalESWLdistributionforallresponsecomponentswassuggested.Theyutilizedapolynomialexpansion,whichwasobtainedonthepremisethattheselectedESWLresultedinaccurateestimatesofalimitednumberofpre-selectedpeakresponsecomponents.

Separation of wind loads and their effects and the associated ESWLs into background(quasi-static) and resonant components provides not only an efficient response prediction


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framework but also a physically meaningful description of the loading (Davenport 1985;Kasperski1992;Holmes2002;Isyumov1999;Zhouetal.2000;ZhouandKareem2001;ChenandKareem2001).Accordingly,theresonantESWL(RESWL)canbeexpressedintermsoftheinertialload(e.g.,Davenport1985).WhereasthebackgroundESWL(BESWL)dependsontheexternal wind load characteristics and can be determined using a Load-Response-Correlation(LRC)approach(Kasperski1992),whichprovidesamostprobableloaddistribution(Kasperski1992 and Tamura et al. 2002). Based on the BESWL and RESWL, the corresponding peakresonantandbackgroundresponsescanbecalculatedusingasimplestaticanalysis.Thesearethencombinedusingthecompletequadraticcombination(CQC)approachorthesquarerootofthesumofsquares(SRSS)schemeforthetotalpeakresponse(excludingthemeancomponent).Alternatively,anESWLforthetotalpeakresponsecanbeexpressedasalinearcombinationofthebackgroundandresonantloadingcomponents(ChenandKareem2001;Holmes2002).

In this paper, a framework is presented for evaluating the ESWL for any given peakresponsecomponentofwind-excitedbuildingscharacterized byuncoupledmotionsinthethreeprimary directions. A new description of the BESWL is presented based on the gust loadingenvelope(peakdynamicloadingwithoutthemeancomponent).TheRESWLisgivenintermsofthe inertial load in each fundamental mode. The ESWL for the total peak response is then

expressed as a linear combination of the BESWL and RESWL. Based on this framework, aclosed-formformulationoftheESWLusingananalyticalwindloadingmodelispresented.TheGRFsandESWLsforvariousalongwindresponsecomponentsat different building elevationsarediscussedtohighlightadvantagesoftheproposedESWLdescription.

2GENERALFORMULATIONS

The response of a wind-excited building with one dimensionaluncoupled mode shapesin thetwo orthogonal translational and torsional directions at a given wind speed and direction isconsidered. The wind loads per unit height at elevation z above the ground have meancomponents of )(zPx , )(zPx and )(zP , and fluctuating components of ),( tzPx , ),( tzPyand ),( tzP

,inthetranslationalaxesxandyandabouttheverticalaxisz.Thediscussionhereis

focusedontheresponsewithonedimensionalinfluencefunctionsinthethreeprimarydirections.The uncoupled class of response in the three primary directions permits treatment of windloadingand building response ineach direction independently.Withoutloss ofgenerality, thefollowingdiscussionwillfocusontranslationalresponseinthexdirectionatagivenwindspeedandorientation;asimilartreatmentinotherdirectionsisimmediate.

For aspecific response ofinterest (displacement,bending moment,shearforceand othermemberforces)atabuildingelevationz0,R(z0, t),themean(static)andbackgroundcomponentscanbecalculatedbythestaticandquasi-staticanalysis.Theresonantcomponentcanbeanalyzedusingmodalanalysisrestrictedtothefundamentalmode.Themeanresponse,rootmeansquare(RMS)backgroundandresonantresponsesandthepeakdynamicresponse(excludingthemeanresponse)areexpressedas

==H H

PxxR

H

xx dzdzzzRzzdzzzPR xxb 0 0 2121210 ),()()(;)()( (1)

2222

max11

10

2

0 ;)(4)()(

)()()(

rbxr RrRbQH

x

H

xx

R ggRfSfdzzzm

dzzzzm

+=

=

(2)

=H H

PxxQ dzdzfzzSzzfS xxx 0 0 21,2121),()()()( (3)


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whereH=buildingheight;x(z)=influencefunctionindicatingtheresponseR(z0,t)underunit

load acting at the elevationz in x direction;x(z)= fundamental mode shape;f1 and 1=

fundamentalfrequencyanddampingratio(includingaerodynamicdamping),respectively; m(z)=

massperunitheight; ),( 21 zzR xxP and =),,( 21 fzzS xxP covarianceandcrosspowerspectraldensity

(XPSD)betweenPx(z1,t)andPx(z2,t); =)( fSxQ

powerspectraldensity(PSD)ofthegeneralized

modalforce; gband gr= peak factorsfor thebackground andresonantresponses, respectively,

typicallyranginginvaluebetween3and4.

FollowingtheLRCapproach(Kasperski1992),theBESWLforpeakbackground

response,bRb

g ,isgivenby

=H

Px

R

beR dzzzRz

gzF

xx

b

b 0111 ),()()(

(4)

whichdependsontheinfluencefunctionoftheresponseunderconsideration.Accordingly,theBESWLhasadifferentspatialdistributionfordifferentresponsecomponents,whichmaynotbeveryattractiveforcodeapplications.

For the purpose of simplifying the background load description, it is proposed here to

expresstheBESWLasthegustloading envelope(GLE), )()(' zRgzFxPbebx

= ,multipliedbya

backgroundfactor,Bz,

===H

ebxxRRRzebxzeR dzzFzBzFBzF bbbb 0'''' )()(;/);()( (5)

where ),()( zzRzRxxx PP

= ; 'bRb

g = peak background response under the loading envelope that

does notincludethe influence of loss inspatialcorrelation ofwind loadingover thebuilding

height;Bz represents the reduction effect with respect to the responseR(z0,t) due to loss of

correlation of wind loading. In cases where the wind loads are fully correlated, i.e.,

)()(),( 2121 zRzRzzR xxxx PPP = , BzbecomesunityandtheBESWLsbasedontheLRCandGLE

schemesconvergetothegustloadingenvelope, )('

zFebx .

TheRESWLforthepeakresonantresponse,rRr

g ,isgivenintermsoftheinertialload:

)(4)()(

)()()( 11

10

2fSf

dzzzm

zzmgzF

xQH

x

xrerx

= (6)

whichcanalsobeexpressedintermsofthedistributionofthepeakbasebendingmomentorbase shear force over the building height following the inertial load distribution. When thetorsionalresponseisunderconsideration,theRESWLisobtainedbydistributingthebasetorqueoverthebuildingheight.

The ESWL for the total peak dynamic response, Rmax , can be provided as a linearcombinationofthebackgroundandresonantloads(ChenandKareem 2000and2001;Holmes2002):

( ) maxmax' /;/;)()()( RgWRgWzFWzFBWzF

rb RrrRbberxrebxzbeR ==+= (7)

When the peak response includes the mean component, the ESWL is given

as )()( zFzP eRx .


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3CLOSED-FORMFORMULATION

Forthesakeofillustration,themassperunitheight,m(z),thefirstmodeshape,x(z),andtheinfluencefunctionoftheresponseR(z0,t),x(z),areexpressedas

equivalent static wind loading on buildings - new perspective

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