Vortex-induced vibrations of structures

Transcription

1 Struturl Engineers World Congress 7, Noeber -7, 7. Bnglore, Indi. Vortex-indued ibrtions of strutures Send Ole Hnsen ABSTRACT Vortex-indued ibrtions y our on slender strutures suh s hineys, towers nd bridge deks. A full nlytil desription of the phenoenon is still not ilble, nd the proedures used to predit ortex-indued ibrtions of strutures re still rther rude. The different opinions on how to represent the phenoenon thetilly re leted in the riety of proedures used in prtil designs. For instne, the present Euroode on wind tions inludes both the spetrl odel nd the ortex-resonne odel for prediting ortex-indued ibrtions of strutures. The present pper desribes the ost iportnt flow nd struturl hrteristis goerning ortex-indued ibrtions of strutures. The different thetil pprohes presently used to odel the phenoenon re desribed nd their odelings of the different physil spets re disussed. The influene of ir turbulene nd Reynolds nuber re soe of the physil spets foused on. The results obtined in series of wind tunnel tests reently rried out with different ross setions, suh s irulr ylinders, shrp-edged setions, otgons nd bridge deks, re used s bsis for seleting the ost urte thetil odels to predit ortexindued ibrtions of strutures. The oprison of the wind tunnel test results with the preditions obtined fro the different thetil odels shows ler piture of their ury nd shortoings. Thus, the results presented in this pper will be useful bsis for the future seletion of n urte thetil odel for prediting ortex-indued ibrtions in struturl odes. EYWORDS Vortex shedding, ortex-indued ibrtions, otion-indued wind lods, turbulene, spetrl odel, ortex-resonne odel, wind odes. INTRODUCTION Although gret del of effort hs been de during reent dedes to iproe the nlytil odels used for prediting ibrtions due to ortex shedding, the nlytil odels ilble re still rther rude. The ross-wind foring ehniss he proed to be so oplex tht there is no generl nlytil ethod ilble to lulte ross-wind struturl response. The in physil preters inoled in the foring ehniss he been lrified, but the bsi dt used in full-sle preditions he not rehed generl greeent ong reserhers. Espeilly, the ethods they use to tke eroelsti effets, i.e. otion-indued wind lods, into ount differ onsiderbly. Two priry design spets should be foused on in design proedure:

2 . Rre, extree eents ourring sy one in -5 yers. These eents re iportnt in onnetion with seriebility nd their ontribution to ftigue should be nlysed.. Frequent eents ourring ny ties during the expeted lifetie. These eents will often gie the jor ontributions to the ftigue dge lulted. It is essentil tht both spets entioned boe re inluded in design odels for ortex shedding. Euroode EN 99--4:5 () proposes two pprohes for prediting ortex-indued ibrtions of strutures. Approh bsed on the ortex-resonne odel proides response estites, whih re lrger thn the frequent eents nd lower thn the rre eents. This ws pointed out by Dyrbye nd Hnsen () in their disussion of the influene of different eteorologil onditions for the otion-indued wind lods deeloped in ortex-indued ibrtions. Approh bsed on the spetrl odel tkes rre s well s frequent eents into ount by inluding the influene of turbulene in the ibrtion plitudes predited. This enbles tht the spetrl odel does not underestite the rre eent response nor oerestites the frequent eent response. The spetrl odel is lso used s bsis for the speifitions of ortex-indued ibrtions in the Cndin nd the Dnish wind ode, nd in the CICIND odel ode for hineys. The design proedure foused on in this pper is bsed on the spetrl odel originlly suggested by Vikery nd Clrk (3) nd further ined by Vikery nd Bsu (4). Their dt inly inludes irulr ross setions. The design proedure presented in this pper extends the sope to shrp-edged ross setions s well. Furtherore, this pper fouses on the influene of turbulene in the ibrtion plitude preditions, nd this hnges the preditions onsiderbly towrds the behiour obsered for ny full-sle strutures. It is hoped tht this pper ould be used in the efforts to diret the different reserh groups towrd oon understnding of ortex-indued ibrtions of line-like strutures. None of the strutures onsidered in the pper re prt of group. Group effets re, theore, not onsidered. BASIC PHYSICS OF VORTEX-INDUCED VIBRATIONS The bsi physis of ortex-indued ibrtions re desribed below. Resonne wind eloity Vortex-indued ibrtions y our when orties re shed lterntely fro opposite sides of struture. This gies rise to flututing lod perpendiulr to the wind diretion. As the orties re shed lterntely first fro one side then the other, hronilly rying lterl lod with the se frequeny s the frequeny of the ortex shedding is fored. The frequeny n s (z) of the lterl lod used by ortex shedding t lotion z is: ( z) ns ( z) St () b( z) in whih St is the Strouhl nuber, is the en eloity of the pprohing wind, nd b is the ross-wind diension of the struture onsidered. Signifint ibrtions y

3 our if the dointing frequeny of ortex shedding, n s, is the se s the nturl frequeny, n e, for the struture ibrting in ode in the ross-wind diretion. Theore, the resonne wind eloity r defined by n s ne is equl to: neb r () St in whih n e is the nturl frequeny, b is the erene ross-wind width nd St is Strouhl nuber. Of ourse, the ortex-indued ibrtions lulted do not depend on the hoie of erene quntities, e.g. the erene width. The erene quntities hosen often er to the point, t whih the struturl ibrtions re lrgest. Sruton nuber The Sruton nuber S beoes one of the ruil preters for the ortex-indued ibrtions of strutures, see e.g. Sruton (5). It is proportionl to the struturl dping nd to the rtio between the ibrting ss nd the ss of the ir displed by the struture, nd it is defined s: δ S (3) ρb s e in whih ρ is the ir density, δ s quntifies the struturl dping by the logrithi dereent pproxitely equl to δ s πζ s, where ζ s is the struturl dping rtio, nd the effetie ss e per unit length is gien by: e h g M ξ ( z) dz (4) in whih ξ (z) is the ode shpe, the integrl in the denointor is tken oer the struturl prt with length h exposed to ortex shedding fores, nd M g is the odl ss, whih for line-like struture of length L y be expressed s: M g L ( z) ξ ( z) dz (5) where (z) is the ibrting ss per unit length. Motion-indued wind lods Struturl otion indues feedbk to the ir flow generting the ross-wind lod on the struture. For flexible strutures these otion-indued wind lods re signifint. The onept of eroelstiity oering these lod ontributions is disussed below.

4 Struturl otion interts with the wind field in suh wy tht the dointing ortex shedding frequeny synhronises with the struture s nturl frequeny. This phenoenon is lled lok-in. Mny experients he been de in order to deterine the influene of struturl otions on the orreltions of the ross-wind loding. The results of these experients show tht inresing ibrtion plitudes use n inrese of orreltion length. The boe-entioned properties of lok-in nd otion-dependent orreltions re relted to the erodyni ross-wind loding used by the struturl ibrtions. In its ost siple for, the otion-indued wind lod F onsists of n inerti lod proportionl to the elertions of the struture nd n erodyni dping proportionl to the eloity of the struture: F h & ξ & ξ (6) def def ξ def & ξ nd & ξ def re the ross-flow defletions, eloities nd elertions, respetiely,, def of the struture. For ost iil engineering strutures in ir, the dded ss of ir h is sll reltie to the struturl ss, nd y thus be disregrded. Howeer, the erodyni dping fore & ξ def will redue the effetie dping of the ibrtions when is negtie. As this ours for wind eloities lose to the ritil wind eloity it is ery essentil preter when lulting the response. The liner dependene between loding nd eloity indited in eqution (6) is suffiiently urte for sll ibrtions of up to pprox. 5-% of the struturl width. For lrger ibrtion plitudes non-liner dping ters beoe iportnt. The erodyni dping preter illustrted in figure deterines the liner ter of the otion-indued wind lods on the struture. The figure shows tht the erodyni dping y beoe negtie for redued wind eloities lrger thn or of the order of the Strouhl nuber reiprol. Influene of turbulene The effet of turbulene on ortex-indued ibrtions hs been onsidered experientlly by Vikery (6), nd renk nd Nielsen (7) inlude turbulene in their theoretil liftosilltor odel. The desription below extrts the bsi influene of ir turbulene. Figure - shows the inrese in plitude for fixed ylinder suddenly relesed in wind tunnel with low turbulene flow. The nturl frequeny of the ylinder ibrtions is.8 Hz. The upper figure ers to wind eloity pprox. 4% lower thn the resonne wind eloity r, i.e.. 96r, the iddle figure to r nd the lower figure to. 5r. The figure shows slowly inrese in plitude where xiu plitudes re rehed fter pprox. nturl ibrtion periods. Furtherore, the xiu plitude rehed depends strongly on the rtio between the tul wind eloity nd the resonne wind eloity. Lrge-sle turbulene in the tosphere y be interpreted s slowly rying en wind eloity. Looking t the results shown in Figure -, it is not surprising tht lrge

5 sle turbulene will he pronouned effet on ortex-indued ibrtions. When the en wind eloity for short period of tie is equl to the resonne wind eloity, the plitudes will grow slowly, but s soon s the en wind eloity hs hnged wy fro the resonne wind eloity, lrge plitudes will not grow up. The tul plitudes will be of stohsti nture, i.e. inrese when the wind eloity is lose to the resonne wind eloity nd redue when this is not the se. The obsertions desribed boe er to strutures not hing extreely low Sruton nubers. At low Sruton nubers lrge ibrtions y deelop een in turbulent flow. The influene of lrge sle turbulene y be estited pproxitely by integrting the erodyni dping preter esured for different en wind eloities nd weighed with Gussin distribution desribing the rition of the longitudinl turbulent oponent. A ore urte pproh will be to nlyse the differentil eqution desribing the ritions of the lift oeffiient in tie. Mesureents of erodyni dping ters in turbulent flow y lso be used. The erodyni dping depends on turbulene intensity nd not on the bsolute ritions of the wind eloity. The ritil Sruton nuber, t whih the jup fro sll to lrge ibrtions ours, depends strongly on the low frequeny turbulene with lrge sles, but not on the high frequeny turbulene with sll sles. VORTEX-INDUCED VIBRATIONS BASED ON THE SPECTRAL MODEL Originlly, Vikery nd Clrk (3) proposed the spetrl odel used to predit ortexindued ibrtions of line-like strutures. During the lst pprox. 3 yers the forultion nd erodyni preters used in the odel he been nlysed in seerl ppers nd text books, see e.g. Vikery nd Bsu (4), Vikery (6) nd Dyrbye nd Hnsen (). The otion-indued wind lods in the spetrl odel re tken into ount by erodyni dping of the for y& by& 3, where the first, liner ter introdues negtie erodyni dping nd the lst, non-liner ter gies positie dping ensuring tht the response is self-liiting. For sll plitudes of up to pprox. 5-% of the struturl width, the erodyni dping is desribed suffiiently urte by the first, liner ter. The erodyni dping rtio ζ is gien by, see Vikery nd Bsu (4): ρb σ y, ζ, γ (7) e γ LL, b in whih the onstnts γ nd γ L re defined in eqution () nd () below. Assuing tht the erene eloity pressure q / St ρb ne, in whih ρ is the ir density, the stndrd deition σ y of the struturl defletion is gien by, see Dyrbye nd Hnsen (), eqution (7.4.8)

6 σ ( z) y b γ CC, ρb b ξ ( z) (8) St e h ξ S σ y,, γ 4π γ LL, b The stndrd deition of the struturl defletion follows the ode shpe. The erodyni onstnt C, depends on the ross-setionl shpe, nd for irulr ylinder lso of the Reynolds nuber. It is gien by: C, ~ 4 π CL, λ (9) 8 π B in whih C ~ L is the lift oeffiient, λ is the lod orreltion length, nd B is the spetrl bndwidth, ll non-diensionl preters desribing the ortex-indued lod on nonibrting strutures. C ~ L nd B, nd thereby the onstnt C,, re funtions of turbulene intensity, nd for irulr ross-setions lso of Reynolds nuber, see e.g. Vikery (6). The orretion ftors γ C, γ nd γ L depend on the ortex-shedding fores on the struture in obintion with the ode shpe long the exposed length h. They re expressed by: h g ( z, ne ) dz h γ C () h ξ ( z) dz h ξ h, h ( z) b ( z) ξ ( z) dz b γ () h ξ ( z) dz ξ ( z) dz ξ γ L () h 4 ( z) ξ ( z) dz ξ, 4 nd the funtion g introdued in eqution () is equl to, see Dyrbye nd Hnsen (), eqution (7.4.4)

7 g( z, n) ~ q( z) b( z) C ( z) ξ( z) B n L ( ) s ~ exp qbc Lξ B( z) ns ( z) B( z) n / n ( z) (3) where the ortex-shedding frequeny n ( z) St ( z) / b( z). s The orretion ftor γ C in eqution () depends pririly on the ode shpe nd to soe extent lso on the eloity profile nd dieter rition with height. Assuing unifor ode shpe nd eloity profile, height independent width b, lift oeffiient C ~ L nd spetrl bndwidth B, the funtion gzn (, e ) beoes equl to when the ortexshedding frequeny ns( z) is equl to the nturl frequeny n e for ll heights z. It is often good pproxition to ssue tht ( z) is independent of height z. For strutures with onstnt width b, the orretion ftor γ beoes equl to nd the orretion ftor γ L only depends on the ode shpe. The orretion ftor γ L beoes equl to nd 9 / 5.34 for unifor nd prboli ode shpes, respetiely. Prboli ode shpes y, theore, he lrger liiting plitudes thn unifor ode shpes. This is not surprising sine the stbilising non-liner dping fore ts long shorter struturl length, when the ode shpe beoes ore oplex. The generl expressions boe y be siplified ssuing tht the ross-wind diension nd ll erodyni preters re onstnt long the struture. The erodyni preters inlude the en wind eloity nd eloity pressure q, the lift oeffiient C ~ L, the spetrl bndwidth B, nd the erodyni dping preter The ssuption y see to be ery liiting. Howeer, this is not the sitution sine the ortex shedding fores re ost iportnt long reltiely short distne on the struture, where the struturl ibrtions re lrgest. Thus, using erene preters representtie for the lotions where the ode shpe hs its xiu defletion will norlly gie urte estites of ortex-indued ibrtions. The orretion ftor γ C for the erodyni preter. C beoes equl to: γ C (4) h ξ ( z) dz ξ h The orretion ftor for the erodyni dping preter beoes γ, nd the orretion ftor for the liiting plitude preter follows diretly fro eqution () h ξ ( z) dz ξ γ L h (5) 4 ξ ( z) dz ξ 4 These orretion ftors re ssued in the ode design proedure desribed below.

8 CODE DESIGN PROCEDURE The influene of turbulene hs been inluded in the present design proedure in order to tke obsertions on full-sle strutures duly into ount. The ode design proedure presented oers generl ode shpes nd is not restrited to ode shpes with non-hnging signs. Vortex-indued response The effet of ortex-indued ibrtions y be lulted fro the effet of the inerti fore F I (z) per unit length ting perpendiulr to the wind diretion nd gien by: ( πn ) ξ( z) e k pσ y,x ξx F ( z) ( z) (6) I in whih (z) is the ibrting ss per unit length, n e is the nturl frequeny, k p is the pek-ftor, nd σ y, x is the stndrd deition of the struturl defletion t the point where the ode shpe ξ (z) hs its lrgest defletion ξ x. σ y, x is gien by: σ y,x b C ρb b (7) St e h S σ y,x 4π Lb in whih the erodyni onstnt C γ CC,, the erodyni dping preter is ssued to be equl to,, nd the norlised liiting plitude L γ LL,, see eqution (8) boe. The orretion ftors should be lulted using ξ ξx. The stndrd deition of the struturl defletion y be deterined by soling eqution (7). The solution is gien by: σ y,x b + + (8) where the onstnts nd re equl to: S 4π L ρb L C b (9) 4 St h e The orretion ftors C γ nd L γ re gien in tble for fie different siple ode shpes, four with non-hnging sign nd one with hnging sign.

9 In the present siplified nd pproxite pproh, the erodyni dping preter is estited for sooth flow ses nd funtion of longitudinl turbulene intensity, I, gies the redution in turbulent flow, i.e.: ( Re, I ), x (Re) ( I ) () The funtion y pproxitely be deterined by 3I for I. 5 nd ( I ).5 for I >. 5. This siplified odel tkes turbulene into ount in rther rude wy nd it gies only rough inditions not underestiting the response. Further studies re needed to lrify the influene of turbulene ore urtely. The xiu plitude y x is lulted by ultiplying the stndrd deition σ y, x gien by eqution (7) with pek-ftor k p, i.e. yx k p σ y. For sll plitudes below pprox. -% of the ross-wind diension, the pek-ftor is pprox depending on the nturl frequeny of the struture. For lrge plitudes, the pek-ftor is equl to nd for interedite plitudes, the pek-ftor inreses grdully with deresing plitude. The following siplified expression y be used: k 4 ( +. rtn(.75( S /(4π )) )) () p The expression in eqution () ws originlly proposed by Rusheweyh nd Sedlek (8). Ftigue lultion Ftigue lultions ould be bsed on the design proedure proposed using the probbility of different turbulene intensities t wind eloities lose to the ritil wind eloity. The ourrene of different tospheri stbility onditions y beoe iportnt. The onstnts C nd depend on the wind eloity rtio / r, where is the en wind eloity nd r is the resonne wind eloity. They re pproxitely gien by: 3/ r / C C,x exp () r B (Re) ( I ) f ( / ) (3), x r The funtion f hs its xiu lue of for r, nd it y, s rough pproxition, be ssued to derese linerly to the lue of for r. The funtion f is roughly equl to for < r nd for > r. The ourrene of en wind eloities up to pprox. 5- /s ould be bsed on infortion inluded in the Europen Wind Atls. Originlly, these infortion ws intended to be used in onnetion with preditions of wind energy prodution fro wind turbines, but the Europen Wind Atls will in ost ses lso proide urte wind dt input used for lulting ftigue dge due to ortex shedding.

10 The ourrene of different turbulene intensities s funtion of en wind eloity hs not been inestigted thoroughly in the pst. In onnetion with the Dnish wind ode, the results fro two wind esuring sttions were used to estblish distributions of turbulene intensities. Both sttions re hrterised by wind oer lnd with roughness lengths z of between. nd.5. For the two sttions nlysed, the ourrene of different turbulene intensities I (z) t height z y pproxitely be gien by Gussin distribution with : en lue orresponding to neutrl tospheri onditions, i.e. I z) / ln( z / z ) ( stndrd deition grdully deresing fro pprox..6 t en wind eloities below pprox. 5 /s to pprox..3 for en wind eloities of pprox. /s. The probbility onneted with negtie rguents y be ssued to orrespond to zero turbulene intensity. The probbilities of turbulene intensities bsed on the Dnish esureents will probbly oerestite low turbulene situtions in ost prts of Europe thereby leding to oerestited ftigue dge of the struture. Exple three spn bridge with sinusoidl ode shpe see figure 3 The ortex-indued response is deterined for the first ode of full-sle bridge with box-girder ross setion s shown in figure. The bridge is three spn bridge with siple supports. The ss per unit length of the bridge is 4 kg/ nd the length of eh spn is 5. The ertil ross wind diension is b The nturl frequeny is equl to n e.73 Hz, nd the struturl dping is ssued to be δ s. 4. Two spns re ssued to be wind exposed nd one spn not wind exposed, see figure 3. This indites tht the wind-exposed length is h, nd the totl length is L 5. The resonne wind eloity gien in eqution () is lulted to be r.8 /s using the Strouhl nuber of St. deterined in the tests rried out. For this resonne wind eloity, low turbulene situtions do not our inditing tht the redution of the erodyni dping preter due to turbulene y be tken into ount. The erodyni preter lues re: C,. 9,,. 9 nd L,. 85 s deterined in the wind tunnel test, see figure 4. The erodyni preters C nd L re deterined to C γ CC, nd L γ LL, using the orretion ftors of γ C.4 nd γ L.6 deterined in ordne with eqution (4) nd (5), see lso tble. The erodyni dping preter beoes I ssuing turbulene intensity of 3%. ( ) ( ) 6, The effetie ss per unit length beoes, see eqution (4): e L h () z ( ξ () z ) ( ξ () z ) dz dz 3 6 kg/

11 The Sruton nuber y now be found s, see eqution (3): δ s S ρb e 7.3 The onstnts nd in eqution (9) re equl to: L S 4 π 4π.6.43 b C b ρ L 4 St h.6 6. e nd eqution (8) gies the xiu stndrd deition of the defletion: σ y,x b (.43) The pek-ftor in eqution () beoes k p The inerti fore F w (z) in eqution (6) is illustrted in figure 3. CIRCULAR CYLINDERS Full-sle obsertions nd odified erodyni preters re desribed below. Full-sle obsertions Full-sle obsertions of steel hineys he been reported extensiely, see e.g. Prithrd (9) nd Dly (). Soe of the hrteristi fetures re foused on below in onnetion with the hineys onsidered by Dyrbye nd Hnsen (). During the winter in 995/96 four Dnish hineys he ll experiened uneptble lrge ibrtions. All four hineys were onstruted during the period fro 97 to 98, nd no serious ibrtions he been reported until the lrge ibrtions ourred during the winter in 995/96, i.e. pprox. yers fter onstrution. The lrge ibrtions of ll 4 hineys were obsered during periods of old wether with tepertures of pprox. -ºC to -5ºC. These ibrtions ourred pririly in the orning nd / or in the eening, inditing tht the ir flow y be hrterised by extreely low turbulene leels due to stble strtifition of the tosphere. The obsertions de during the winter 995/96 were not unique. Siilr obsertions he been de in ny ountries. It should be ephsised tht the lrge ibrtions desribed boe re not used by group effets or orrosion probles. Furtherore, the lrge ibrtions re not judged to originte fro struturl or foundtion hnges due to the old wether. Rre eents with lrge ibrtions he lso ourred during wether onditions with tepertures well boe ºC.

12 Howeer, the probbility of low turbulene situtions in old wether is lrger thn in norl wether situtions. Aording to the results shown, under speil onditions, e.g. ertin eteorologil situtions with old nd sooth ir flow oer reltiely long period of tie, sy of pproxitely hour, soe slender steel strutures y experiene lrger ibrtions thn predited by the ortex-resonne odel of Euroode EN 99--4:5. In onlusion, the ortex-resonne odel of Euroode soeties oerestites nd soeties underestites the response used by ortex shedding. This drwbk is not present in the spetrl odel of Euroode, in whih the ir turbulene introdues the ribility of struturl response obsered for full-sle strutures. Codified erodyni preters spetrl odel of Euroode The erodyni preters speified in the spetrl odel of Euroode re gien in tble, nd the erodyni dping preter is illustrted in figure 4. The lues speified re bsed on the lods nd ibrtions esured on lrge nuber of strutures, see Hnsen (). The influene of the ode shpes in for of orretion ftors he not been extrted fro the esureents. This ineent y be rried out t lter stge when the Euroode is reised. Inserting, x in eqution (7) gies the ibrtion plitudes illustrted in Figure 5 for sooth flow t different Reynolds nubers. Figure 5b-d show the ortex-indued ibrtions s funtion of turbulene intensity for the Reynolds nubers onsidered in figure 5. The influene of turbulene is seen to be pronouned for Sruton nubers of ny typil strutures. The lue of, x is ruil in order to estite relible results in the rre eents of eteorologil onditions with sooth ir flow nd t en wind eloity equl to the resonne wind eloity. The librtions rried out show tht the ode proedure proposed is ble to predit ll lrge ibrtions obsered on the strutures onsidered, see Hnsen () for further detils. SHARP-EDGED CROSS SECTIONS The erodyni preters for unifor ode shpes re desribed below for different shrp-edged ross setions, see lso tble 3. They he been deterined in series of setion odel tests rried out in our boundry lyer wind tunnel. Sooth flow is the bsi set-up in the tunnel. The turbulene intensity is pprox. % nd the turbulene hs sll sles. 3 spires loted t the inlet generte dditionl turbulene. Chnging the rottion speed of the entiltor genertes low frequeny turbulene. The setion odels were ounted horizontlly in the tunnel. By using end pltes the flow round the setion odels is pproxitely -diensionl. Otgon ross setion The otgon ross setion is illustrted in figure 6. The Sruton nuber of the odel is pprox., nd the turbulene intensity of the ir flow is pprox. 7%.

13 The ortex-indued response is shown in figure 7 nd 8, respetiely, for the ross setion without nd with the guide nes illustrted in figure 6. Guide nes re seen to suppress the ortex-indued ibrtions ery effetiely. The erodyni dping preter deterined in the wind tunnel tests is pprox..5 nd.64 for the ross setion with nd without the guiding nes. The erodyni preter C hs been estited to be pprox..3% of. The esured lue for of pprox..6 grees well with the esureents shown in figure for the otgon ross setion, both esureents rried out in turbulent flow. L-shped ross setion The L-shped ross setion is illustrted in figure 9. Figures nd show the results obtined in the wind tunnel tests rried out together with preditions bsed on the ortexresonne odel nd the spetrl odel, respetiely. The test onditions oer sooth flow nd flow with turbulene intensity of pprox. %. The preditions lulted using the spetrl odel gree well with the esureents rried out. Suh n greeent is not possible using the ortex-resonne odel. Bridge ross setion The bridge ross setion onsidered is illustrted in figure. Figures 3 nd 4 show the results obtined in the wind tunnel tests rried out together with preditions bsed on the ortex-resonne odel nd the spetrl odel, respetiely. The test onditions oer sooth flow nd flow with turbulene intensity of pprox. 3%. The preditions lulted using the spetrl odel gree well with the esureents rried out. Suh n greeent is not possible using the ortex-resonne odel. CONCLUSION Full-sle obsertions on ny steel hineys suggest tht lrge nd iolent ortexindued ibrtions soeties our s rre, extree eents nd tht the frequent ortexindued ibrtions our with uh sller plitudes. It is essentil tht both spets re inluded in design odels for ortex shedding. Euroode EN 99--4:5 proposes two pprohes for prediting ortex-indued ibrtions of strutures. Approh bsed on the ortex-resonne odel proides response estites, whih re lrger thn the frequent eents nd lower thn the rre eents. Approh bsed on the spetrl odel tkes rre s well s frequent eents into ount by inluding the influene of turbulene in the ibrtion plitudes predited. This enbles tht pproh does not underestite the rre eent response nor oerestites the frequent eent response. Approh of Euroode oers irulr ylinders. This pper extends pproh lso to oer different shrp-edged ross setions, nd the presented design proedure y be used for generl ode shpes hing hnging sign s well s onstnt sign. The influene of turbulene on the response is bsi prt of the design proedure foused on.

14 Approh of Euroode is not ble to predit the ortex-indued ibrtions of the shrpedged ross setions onsidered in the pper. ACNOWLEDGEMENTS Optionsult, Mærsk Olie og Gs nd RAMBØLL de experientl results ilble for this pper. irstine Bk-ristensen, Sion Rex nd Mrtin Lollesgrd he prepred the wind tunnel tests rried out. All input re highly knowledged. AUTHOR AFFILIATIONS Send Ole Hnsen ApS, St. Jørgens Allé 5, D-65 Copenhgen, Denrk E-il: Phone: , Fx: REFERENCES. Euroode EN 99--4:5, Ations on strutures Prt -4: Generl tions Wind tions, Europen Stndrd EN 99--4:5.. C. Dyrbye nd S.O. Hnsen, Wind Lods on Strutures, John Wiley & Sons, B.J. Vikery nd A.W. Clrk, Lift or ross-wind response of tpered stks, Journl of Struturl Diision, ASCE, Vol. 98, pp. -, B.J. Vikery nd R.I. Bsu, Aross-wind ibrtions of strutures of irulr ross-setion. Prt. Deelopent of thetil odel for two-diensionl onditions, Journl of Wind Engineering nd Industril Aerodynis, Vol., pp , C. Sruton, An introdution to wind effets on strutures, Oxford Uniersity Press, Engineering Design Guides 4, B.J. Vikery, Wind lods & design riteri for hineys, CICIND report, Vol. 4, No., S. renk nd S.R.. Nielsen, Energy blned double osilltor odel for ortex-indued ibrtions, Journl of Engineering Mehnis, Vol. 5, pp. 63-7, H. Rusheweyh nd G. Sedlek, Crosswind ibrtions of steel stks ritil oprison between soe reently presented odes, Journl of Wind Engineering nd Industril Aerodynis, Vol. 3, pp B.N. Prithrd, Steel hiney osilltions: oprtie study of their reported perforne ersus preditions using existing tehniques, Eng. Strut., Vol. 6, Otober A.F. Dly, Elution of ethods of prediting the ross-wind response of hineys, CICIND report, Vol., S.O. Hnsen, Vortex indued ibrtions of line-like strutures, CICIND report, Vol. 5, No., B.J. Vikery, The response of hineys nd tower-like strutures to wind loding, Stte of the Art Volue, Ninth Interntionl Conferene on Wind Engineering, New Delhi, pp

15 TABLES Tble Corretions ftors s funtion of ode shpe Mode shpe ξ. ξ x. z / h. γ C, eqution (4) γ L, eqution (5) Unifor: ξ (z) Liner: ξ ( z ) z / h.73.9 Prboli: ξ ( z ) ( z / h).4.34 Sinusoidl: ξ ( z) sin( πz / h).4.6 Cntileer: ξ ( z) z / h.73.9 Tble Aerodyni preters for irulr ylinders in sooth flow. For interedite Reynolds nubers Re, the preters re ssued to ry linerly with the logrith of Re Aerodyni preter Re 5 Re 5 5 Re 6 C,x..5.,x.5 L Tble 3 Aerodyni preters for shrp-edged ross setions in sooth flow Cross setion Strouhl nuber C,, L, Otgon.6.5. *) - L-shpe Bridge *) Corretion of.64 esured for turbulene intensity of I 7%, see eqution () FIGURES Fig. Aerodyni dping esured in turbulent sher flow for irulr ylinder, n otgon nd squre. Dt fro Vikery ()

16 .75 y x / d.96 r r r Mesuring tie. [se.] Fig. - Aplitude built-up fter the ylinder hs been relesed in low turbulene flow Fig. 3 Inerti fore per unit length, see eqution (6)

17 Fig. 4 Aerodyni dping preter,x for sooth flow s funtion of Reynolds nuber.45.3 σ y / d Sooth flow, I % Re 5 Re 6 Re σ y / d Re 5 I % I % I % Sruton nuber, S. σ y / d Re Sruton nuber, S. σ Re 6 y / d I % I % I %.3.5 I % I % I % Sruton nuber, S Sruton nuber, S. Fig. 5-d Vortex-indued ibrtions s funtion of turbulene intensity nd Reynolds nuber. It is ssued tht e /ρd 5 nd h/d3, whih influene the low plitude prt of the ures shown

18 Fig. 6 Otgon ross setion. /b.9, /b.6. Strouhl nuber.6 Fig. 7 Vortex-indued response of otgon ross setion without guide nes Fig. 8 Vortex-indued response of otgon ross setion with guide nes

19 Fig. 9 L-shped ross setion. Strouhl nuber.5 Fig. Mesured response nd ibrtion preditions of L-shped ross setion bsed on the ortex-resonne odel with /(4π), w.53 for y x /b<., lt.65 Fig. Mesured response nd ibrtion preditions of L-shped ross setion bsed on the spetrl odel with C,.,,.5, L,.5, turb.76

20 Fig. Bridge ross setion. Strouhl nuber. Fig. 3 Mesured response nd ibrtion preditions of bridge ross setion bsed on the ortex-resonne odel with /(4π), w.44 for y x /b<., lt.4 Fig. 4 Mesured response nd ibrtion preditions of bridge ross setion bsed on the spetrl odel with C,.9,,.9, L,.85, turb.75