POD-Based Analysis of Dynamic Wind Load Effects on a Large Span Roof

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1 POD-Based Aalysis of Dyamic Wid Load Effects o a Large Spa Roof Xi-yag Ji, Yi Tag ad Hai Ji 3 Professor, Wid Egieerig Research Ceter, Chia Academy of Buildig Research, Beiig 3, Chia, ixiyag@cabrtech.com Associate Professor,Wid Egieerig Research Ceter, Chia Academy of Buildig Research, Beiig 3, Chia, yitag_hua@63.com 3 Egieer,Wid Egieerig Research Ceter, Chia Academy of Buildig Research, Beiig 3, Chia, ABSTRACT Based o the Proper Orthogoal Decompositio (POD) method, wid load ad wid excited dyamic resposes of Guagzhou New Railway Statio, which is a typical wid sesitive large-spa roof structure, are aalyzed. Multi-correlated statioary radom wid pressure measured i wid tuel test are decomposed ito a set of sub-processes by diagoalizig their covariace or cross power spectral desity. The physically meaigful likage betwee the wid loads ad the attedat backgroud ad resoat respose of structures i the POD framework is also aalyzed. Fially, the accuracy ad effectiveess of the wid-excited respose aalysis usig reduced order model of wid load are ivestigated. KEYWORDS: LARGE-SPAN ROOF, PROPER ORTHOGONAL DECOMPOSITION (POD), BACKGROUND RESPONSE, RESONANT RESPONSE Itroductio The roof spa used i stadiums, airport termial ad comprehesive trasportatio hub is gettig larger ad larger to fulfill the demad of uique architectural desig ad multiply service fuctio. These structures are sesitive to wid iduced vibratio because of its log atural periods ad light self-weight, therefore the effects of wid load, especially of dyamic wid load, are sigificat ad become the domiate load i structural desig. It is ot easy to determie efficietly the dyamic wid load effects for large spa roof cosiderig the characteristic of o-gaussia wid load iput ad desely distributio of structural vibratio modes. Researchers have to measure the distributed fluctuatig wid pressures i wid tuel test ad the take great efforts to calculate the wid excited resposes with cosiderig a huge umber of structure modes. Recetly, proper orthogoal decompositio (POD) method becomes a popular methods to look ito the mechaism of wid load [Gilliam(4) ad Tamura(999)] ad to simplify wid resposes calculatio by usig the liks betwee wid load ad structure eigemodes. Geerally speakig, POD has two forms. The first form, here referred to as the covariace proper trasformatio (CPT), trasforms the process i order to make its covariace matrix diagoal. The secod, referred to as the spectral proper trasformatio (SPT), is defied ad decomposed cocerig that the orthogoality at the zero time lag of the pricipal compoets may be ot sufficiet. The SPT trasforms a multivariate process i order to make its cross-power spectral desity matrix diagoal. Carassale (,) ad Che (5) et al. developed ad discussed a method usig CPT ad SPT oitly to make meaigful likage betwee wid loads ad the attedat backgroud ad resoat respose of structures. But their work maily cocer the liear structures such as high-rise buildigs for which the umber of structural modes are limited ad the wid load decomposed results are quite simple. I this paper, a caopy of ew Guagzhou railway statio is preseted as a example

2 to itroduce POD usage i large spa structure. Physically meaigful likage betwee wid loads ad the subsequet backgroud ad resoat respose of structures i the POD framework is aalyzed. By comparig with the stochastic respose usig traditioal CQC method, the accuracy ad effectiveess i estimatig wid-excited respose o large spa roof with reduced-order modelig are ivestigated. Wid tuel test New Guagzhou Railway Statio, located i Payu Guagzhou, is oe of the four largest railway ad city trasportatio hub i Chia with 4m 45m architectural scale o the groud plae. Figure shows its bird view of the layout. The wid tuel test was coducted i the boudary layer wid tuel TJ-3 i the school of Civil Egieerig, Togi Uiversity. Experimet model with geometric scale of : was made of orgaic glass as show i Figure. The typical boudary layer was simulated for suburba terrai category B i accordace with the Chiese Code. The model was istrumeted with pressure ports at varyig locatios of the roof ad each glass curtai wall which were coected to a pressure trasducer system. Figure 3 shows the locatios of pressure ports o oe of the caopies. Ad Figure 4 shows its RMS pressure coefficiet distributio at degree wid directio (across the railway). The detailed POD aalysis i this paper is maily cocered o this caopy at degree wid directio. Fig. New Guagzhou Railway Statio Fig. Experimet model i wid tuel o Fig.3 Pressure ports o the south caopy Fig.4 RMS pressure coefficiets of the south caopy 5.45 Dyamic respose aalysis based o POD Cosiderig a M degree-of-freedom structure described i terms of its first q pricipal coordiate X(t) uder exteral loadig P(t), based o radom vibratio theory, the spectral matrix of X(t) is give by S Η Ψ S ΨΗ () T * X( f ) = ( f) P( f) ( f)

3 where Ψ is structural mode matrix; Η ( f ) = diag[ H( f ),..., HM ( f)] ; ad H ( f ) is the th complex frequecy respose fuctio ( f ) = ( =,..., ) 4π f f i f f () H M + ξ where Η * ( f ) is cougate traspose of Η ( f ); ξ is the th structural dampig ratio. By usig POD results of cross spectral matrix, it follows that where ( f ) = [ ( f ) * * S ( ) = Η( ) χ( ) Γ( ) χ ( ) Η ( ) (3) X f f f f f f χ χ ] is M N matrix defied as cross-modal participatio matrix T χ ( f ) = ΨΘ( f ) (4) where Θ( f ) = [ θ ( f ),..., θ ( f )] is N N square spectral modal matrix. Whe equatios of N structural motios i terms of modal coordiate are ucoupled, ad the correlatio amog the modal respose compoets is egligible, the mea square of X i mode ca be expressed σ + N X γ = = H ( f) χ ( f) ( f) df ( =,..., M) (5) where γ ( f ) is the th eigevalue of the cross spectral matrix. The wid-excited structural respose is covetioally separated ito a backgroud compoet ad a resoat compoet. I case that the structure is lightly damped ad its atural frequecies are well separated, resoat compoet ca be approximated by N σ = 3 χ ( ) ( ) (,..., ) Xr f γ f = M (6) 8( π f ) ξ = M M Ψ N σ = 3 γ = Xr = 8( π f ) ξ = σ Ψ χ ( ) ( ) (7) = Yr f f where σ ad σ = mea square respose of resoat compoet i th pricipal ad Xr Yr Lagragia coordiate system respectively; Ψ is the th structural mode vector. Cosiderig the quasi-static characteristic, backgroud compoet of respose icludig all structural modal cotributio ca be expressed i the form N T Yb = = λ σ bb (8) where σ is mea square value of backgroud respose i Lagragia coordiate system. Yb b = ΑΦ ; Α = [ A ] i is ifluece matrix, A i =ith compoet of the Lagragia displacemet vector Y, Y i, uder uit th compoet of load P, P =; λ is the th covariace eigevalue; Φ is the th covariace modal eigevector. Usig (7) ad (8), total respose ca be expressed i the form Results ad discussio + + N T M Ψ N = Y Yb Yr = λ f γ f = = 8( π f ) ξ = 3 σ σ σ bb χ ( ) ( ) (9) Wid load modes Fig.5 shows the load mode cotributios to the total wid load eergy (i.e. λ / λ ). = The first wid loadig mode with domiat eigevalue carries about 3% total eergy of the loadig process. Ad eigevalues of the first five load modes carry about 84% total eergy. A plot of the eigevalues of the cross spectral matrix at varyig frequecies are show

4 i Fig.6 (a), ad Fig.6 (b), (c) at the first ad secod structural mode frequecies respectively. The first wid loadig mode carries about % total eergy of the process at the first structural mode frequecy, ad about 39% at the secod structural mode frequecy. It is oted that at the lower or higher structural frequecy rage the first eigevalues are all domiat ad are of the same order. That is quite differet from the results preseted by Che (5) which is maily cocerig the alog-wid load of high-rise buildigs. Fig. 7 shows the first three wid loadig modes. Fig. 8 ad 9 show the first three wid loadig mode shapes at first ad secod structural modal frequecies respectively..4 λ / λ = Wid load mode umber Fig.5 Eigevalues of wid loadig based o covariace matrix Wid excited respose based o POD Oly the dyamic resposes at degree wid directio are discussed here. Basic wid ad structural parameters for wid excited respose calculatio are: () mi averaged wid speed at m height: 8.3m/s; () structural dampig ratio:.; (3) The frequecy rage for the itegratio is f [.5,.87] (Hz). Withi the itegratio frequecy rage, 44 structural modes are take ito accout ad the atural frequecies varyig from.77 to.85 Hz correspodig to the first ad 44 th mode respectively. As defied i (4), χ represets a measure of the orthogoality betwee the loadig modal shapes ad structural modes. Whe absolute value of cross-modal participatio coefficiets χ =, we say that the th load mode is orthogoal to the th structural mode. Figs. [a-b] show χ at first two structural modal frequecies. The o-zero cross-modal participatio coefficiets are scattered i the matrix ad the maximum value maily distributes aroud the bad of structural modes, which is ot as arrow or diagoal as preseted by Carassale () for wid load of high-rise buildigs. It is idicated that, with regard to large spa roof structure, it should be cautious to simplify the calculatio of large roof structure respose by usig the orthogoality characteristic of loadig modal shapes ad structural modes. Table presets the cotributios of wid loadig modes based o the covariace matrix to the backgroud compoets of the max vertical displacemet. Though the first wid loadig mode cotaiig 3% total eergy of the loadig process, it cotributed oly about 7% to the mea square displacemet. Cotrarily, the third wid loadig mode which cotais 4% total eergy cotributes more tha 4% to the vertical displacemet respose. It is illustrated clearly that the cotributio of the wid loadig mode is ot oly depedet o its eigevalue, but also o its coefficiet vector b, which is the proectio of the wid loadig mode shape o the ifluece fuctio. O the other had, the backgroud respose of the max vertical displacemet ca be cosidered a local respose i certai exted, ad the lower

5 loadig mode is maily cotributed to the global respose. So, it is exhibited that the cotributio ratio of first load mode to higher modes is comparatively less, ad cosequetly the additioal loadig modes are required to estimate the respose preciously. Table shows the cotributio of wid loadig modes based o the cross spectral matrix to the resoat respose i the first ad secod structural modes. As metioed above, the loadig modes which deped o the spatial variatio of the fluctuatig wid pressure field do ot ecessarily esure orthogoality to the structural modes. I this specific example, the mea square resoat respose of the first structural mode is domiated by the respose cotributed by the first load mode (early 5%). However, for the resoat respose of secod mode, ot oly the first wid loadig mode(early 3%), but the secod ad fourth wid loadig modes also have otable cotributios of about 9% ad 5% respectively. For the sake of compariso, RMS vertical displacemets are also calculated usig CQC method, which is regarded as precise oe, show i Fig. a. Fig.b shows the results calculated by root sum square of backgroud ad resoat compoet based o POD expressed i (9), with cosiderig the first th ad first 7th loadig modes respectively. It is showed that the RMS values of vertical displacemet calculated usig the two methods are agree well at most places γ(f) γ(f ) γ(f ) Frequecy wid load mode umber wid load mode umber (a) At differet frequecy (b) At the first structural frequecy (c) At the secod structural frequecy Fig.6 Eigevalues of wid loadig based o cross spectral matrix (a) The first mode (b) The secod mode (c) The third mode Fig.7 Wid loadig mode shapes based o covariace matrix (a) The first mode (b) The secod mode (c) The third mode Fig.8 Wid loadig mode shapes based o cross spectral matrix at the first structural. -.

6 frequecy (a) The first mode (b) The secod mode (c) The third mode Fig.9 Wid loadig mode shapes based o cross spectral matrix at the secod structural frequecy Load mode umber (b) At the first structural frequecy (c) At the secod structural frequecy Fig. Absolute values of cross-modal participatio coefficiets Table : Cotributios of wid loadig modes to backgroud resposes λ ( 4 ) σ ( -6 ) Yb Load mode umber Table : Cotributios of wid loadig modes to resoat resposes First structural modal respose γ ( f ) R k ( f ) k σ Xr Load mode umber Secod structural modal respose γ ( f ) R k ( f ) k σ Xr

7 Coclusio (a) Results usig CQC method (b) Results based o POD method Fig. RMS value of vertical displacemets Based o the simultaeously measuremet of wid pressure o the large-spa roof of New Guagzhou railway statio i wid tuel test, the huge radom loadig field were decomposed by the POD techique o both the covariace ad corss-spectral matrices. The dyamic wid respose of max vertical displacemet of a caopy is calculated by usig POD based method ad traditioal CQC method respectively. The o-zero cross-modal participatio coefficiets o the large roof structure, which are scattered maily aroud the bad of structural mode, showed that it is ot as arrow or diagoal as preseted i high-rise buildigs. It should be cautious to simplify the calculatio of large roof structure respose by usig the orthogoality characteristic of wid load ad structural modes. Results idicate that the first th wid load modes of covariace matrix cotai about 9% eergy ad cotribute more tha 99.8% to the backgroud compoet of the mea square displacemet. The mea square resoat respose of the first structural mode is domiated by the respose cotributed by the first load mode (early 5%). However, for resoat respose of secod structure mode, ot oly the first wid loadig mode(3%), but the secod ad fourth wid loadig modes also have otable cotributio(9% ad 5%). Refereces Carassale L, Piccardo G, Solari G. Double modal trasformatio ad wid egieerig applicatios [J]. Joural of Egieerig Mechaics,, 7(5): Carassale L,Solari G. Wid modes for structural dyamics: A cotiuous approach[j]. Probabilistic Egieerig Mechaics,, 7: Che X,Kareem A. Proper Orthogoal Decompositio-Based Modelig, Aalysis, ad Simulatio of Dyamic Wid Load Effects o Structures[J]. Joural of Egieerig Mechaics, 5, 3(4): Gilliam X, Duyak J P, Smith D A, Wu F. Usig proectio pursuit ad proper orthogoal decompositio to idetify idepedet flow mechaisms [J]. Joural of wid egieerig ad idustrial aerodyamics, 4, 9: Tamura Y, Sugauma S, Kikuchi H, Hibi K. proper orthogoal decompositio of radom wid pressure field [J]. Joural of Fluids ad Structures, 999, 3: