Prediction of Sloshing and Seismic Response of a Floating Roof in a Cylindrical Liquid Storage Tank by the Response Spectrum Method

Transcription

1 Predictio of Soshig ad Seismic Respose of a Foatig Roof i a Cyidrica Liquid Storage Tak by the Respose Spectrum Method Tetsuya Matsui Professor, Departmet of Architecture, Meijo Uiversity, Nagoya , Japa. E-mai: matsuite@ccmfs.meijo-u.ac.jp Abstract A respose spectrum method is proposed to predict the soshig respose of a foatig roof i a cyidrica iquid storage tak uder seismic excitatio. The iquid is assumed to be iviscid, icompressibe, ad irrotatioa, whie the foatig roof is cosidered as a orthotropic eastic pate of variabe thickess. The dyamic iteractio betwee the iquid ad the foatig roof is take ito accout exacty withi the framework of iear potetia theory. The aaysis is based o expadig the respose of the foatig roof ito the free-vibratio modes i iquid to derive the diagoaised equatios of motio for the couped iquid-foatig roof system. The moda aaysis based o the respose spectrum is the carried out to predict the soshig respose. Numerica resuts are preseted to iustrate the appicabiity of the proposed approach. Keywords: iquid storage tak, foatig roof, soshig, og-period groud motio, potetia theory, moda aaysis, respose spectrum.

2 . INTRODUCTION Soshig of cotaied iquid is oe of the major cosideratios i the desig of iquid storage taks. I the past major earthquakes may taks have bee subjected to serious damages which may be attributed to iquid soshig. Especiay durig the September 3 Tokachi-oki Earthquake, Japa, seve oi storage taks with a sige-deck type foatig roof ocated at Tomakomai were seriousy damaged (Hatayama, 5). Damages icude the sikig of the foatig roof which ed oe of the seve taks to the whoe surface fire, as observed aso i the 999 Kocaei Earthquake, Turkey (The Japa Society of Civi Egieers, 999). Athough the faiure of the foatig roof ad the fire of oi storage taks have bee observed frequety, e.g. durig the 964 Niigata Earthquake ad the 983 Nihokai-chubu Earthquake, the sikig of the foatig roof caused by soshig has ever bee experieced so far i Japa. This was a very dagerous situatio that the oi surface was directy exposed to the air. Aroud the potoos of the suke roofs the damages due to oca buckig ca be observed, from which it is presumed that the damaged potoo was progressivey fied with oi, ost its buoyacy, ad was ed to sikig. However, the detai of the mechaism of sikig of the foatig roof is ot fuy uderstood. After the 3 Tokachi-oki Earthquake, the Fire ad Disaster Maagemet Agecy of Japa (5) has issued the ameded Notificatio of the Fire Defese Law i which the desig spectrum for soshig was icreased to amost twice the spectrum i the past Notificatio. I additio, the stadard for the seismic desig of foatig roofs uder og-period groud motio, which has ever bee icuded i the past Notificatio, has bee ewy reguated. This requires the evauatio of earthquake-resistace capacity of the foatig roof of may existig as we as ewy desiged taks, ad raisig it up to the eve requested by the ameded Notificatio. Thus there is a rapidy icreasig demad for predictig the soshig respose of foatig roofs uder og-period seismic excitatio. I the earier papers by the author, aaytica soutios have bee preseted for the soshig of a foatig roof i a cyidrica iquid storage tak uder seismic excitatio. The soutios which are exact withi the framework of iear potetia theory have bee derived i expicit forms for a foatig roof ideaized as a uiform isotropic eastic pate (Matsui, 6), ad a sige-deck type foatig roof composed of a ier deck ad a outer potoo (Matsui, 7a). A computatioa method based o couped fiite eemet ad aaytica soutios has aso bee proposed to dea with a foatig roof composed of a orthotropic eastic pate of variabe thickess (Matsui, 7b). These earier studies have bee based o expadig the respose of the foatig roof ito the free vibratio modes i air (dry modes). Due to the presece of the coupig terms, the resutig equatios of motio for the couped iquid-foatig roof system caot be diagoaised. I the preset paper, with the purpose of appyig the respose spectrum method, the approach based o expadig the respose ito the free vibratio modes i iquid (wet modes) is empoyed to derive the diagoaised equatios of motio. The moda aaysis based o the respose spectrum is the carried out to predict the soshig respose. Numerica resuts are preseted to iustrate the appicabiity of the proposed approach.

3 . BOUNDARY-VALUE PROBLEM The soshig i a cyidrica iquid storage tak of radius R with fat bottom is cosidered here. The tak is partiay fied with iquid to a height H. The iquid surface is covered by a foatig roof which is assumed to move aways i cotact to the iquid. The tak wa may be assumed to be rigid with reasoabe accuracy because the atura periods of she vibratio modes are much shorter tha the atura periods of soshig modes. A cyidrica coordiate system ( r, θ, z) is defied, as show i Figure, with the origi at the ceter of the tak bottom ad the z -axis verticay upwards. The tak is subjected to a horizota groud acceeratio x () t i the directio θ =. g The iquid is assumed to be iviscid, Figure Tak geometry ad icompressibe ad irrotatioa. The the iquid coordiate system motio may be competey defied by a veocity potetia fuctio φ, for which the boudary-vaue probem ca be defied as φ = i iquid (a) φ = x g ( t)cosθ at the tak wa r = R (b) r φ = at the tak bottom z = z φ = w at the iquid surface z = H (d) z where w deotes the vertica dispacemet of the foatig roof, ad the dot deotes the derivative with respect to time t. The vaidity of the superpositio pricipe is assumed for both the dispacemet of the foatig roof ad the iquid motio. The the dispacemet of the foatig roof w ca be represeted as a iear superpositio of the free vibratio modes Z ( r)cosθ i air N wr (, θ,) t = ξ tz()cos r θ () = where ξ ( t ) deotes the moda dispacemet, ad N is the adopted umber of modes. The free vibratio modes i air ca be evauated aayticay for a uiform isotropic pate (Matsui, 6), ad a sige-deck type foatig roof (Matsui, 7a). For more geera cases such as a orthotropic eastic pate of variabe thickess, the fiite eemet method ca be empoyed to evauate the free vibratio modes (Matsui, 7b). (c)

4 Herei the foatig roof is assumed to be a orthotropic eastic pate obeyig the foowig costitutive aws (Timosheko ad Woiowsky-Krieger, 959) w w w Mr = Dr + D + r r r r θ w w w Mθ = Dθ + D + r r r θ r w w Mr θ = Dr θ r r r θ θ where M r, M ad M deote the bedig ad twistig momets, ad θ rθ Dr, Dθ, D ad D θ deote the bedig ad twistig rigidities. r Negectig the dampig term*, the equatio of motio of the foatig roof ca be expressed i the form N ξ() + ω ξ() cosθ = (4) = m t t Z p where m deotes the mass desity of the foatig roof per uit surface area, ad ω the atura circuar frequecy of the foatig roof i air. The hydrodyamic pressure p o the foatig roof ca be evauated from the iearised Beoui s equatio φ p = ρ ρ gw (5) t z= H where ρ deotes the mass desity of iquid, ad g deotes the acceeratio of gravity. (3) The equatios ()-(5) costitute the compete set of equatios which determies the respose of the couped iquid-foatig roof system i a rigid cyidrica tak. 3. LIQUID-FLOATING ROOF COUPLING ANALYSIS BASED ON DRY- MODE EXPANTION The soutio for the potetia φ that satisfies the Lapace equatio (a), the wa coditio (b), the bottom coditio (c), ad the iquid-surface kiematic coditio (d) is give by (Matsui, 7b) ( εi ) ( H R) ( εi ) J ( ) I N g cosh zr J rr φ = x g () t r+ () cos a iξ t θ (6) i= Ωi εi cosh εi εi = where I is the adopted umber of free surface modes, J deotes the Besse fuctio of the first kid of order oe, ε i deotes the i th positive root of J ( ε i ) =, Ω deotes the atura circuar frequecy of free surface mode of i th order give by i * The dampig term is icuded ater.

5 ad Ω = g i i tah ( i H R) R ε ε (7) ε a = Z J ( ε u) udu (8) i i i J ( εi ) The ukow moda dispacemets ξ ( t ) ivoved i (6) ca be determied by sovig the equatio of motio (4) of the foatig roof. O substitutio of () ad (6) ito (5) ad the ito (4), the equatio of motio of the foatig roof ca be writte as N m Z ξ() t + ωξ() t = ( εi r R) J ( ε ) I N N g J = ρ xg() tr+ a () () iξ t ρg Zξ t (9) i= Ωi εi i = = O mutipyig rz o the both sides of (9), itegratig with respect to r over r R, ad makig use of the orthogoaity reatio of the free vibratio modes ead to the equatio of motio i terms of the moda dispacemet ξ ( t ) δm + μ ξ () t + δmω + K ξ () t = γ xg () t ( =,,, N) () = ( ) ( ) where δ deotes the Kroecker deta defied by δ = ( ), δ = ( = ), ad R M = m Z rdr () g μ ρr aa () I = i= Ωi εi i i i R ( ε ) K = ρg Z Z rdr (3) R Z rdr γ = ρ (4) I the equatio of motio (), M deotes the effective moda mass, μ the added mass coefficiet due to iquid, K the hydrostatic restorig force coefficiet due to chage of buoyacy, ad γ the participatio factor. The itegratios i (8), (), (3) ad (4) are evauated aayticay for a uiform isotropic pate (Matsui, 6) ad a sige-deck type foatig roof (Matsui, 7a). For more geera cases these ca be evauated umericay, e.g., by appyig Gaussia quadrature formua (Matsui, 7b).. 4. LIQUID-FLOATING ROOF COUPLING ANALYSIS BASED ON WET- MODE EXPANTION As discussed i the precedig sectio, the approach based o the dry-mode expasio eads to the o-diagoaised equatios of motio () for the couped iquid-foatig

6 roof system. I order to appy the respose spectrum method, it is preferabe to empoy the wet-mode expasio to derive the diagoaised equatios of motio. Let deote the atura circuar frequecy of th order of the foatig roof i iquid by ˆ ω ad the correspodig atura modes by Λ, which are evauated by sovig the homogeeous equatio of motio () with vaishig the forcig term. The the moda dispacemet ξ ( t ) associated with the dry-mode expasio ca be expressed as a superpositio of the wet mode Λ L = ξ () t = Λ ζ () t (5) where ζ ( t) deotes the moda dispacemet associated with the wet-mode expasio, ad L( N) is the adopted umber of wet modes. Substitutig (5) ito (), premutipyig Λ o the both sides, ad makig use of the orthogoaity reatio of the free vibratio modes ead to the diagoaised equatios of motio i terms of the moda dispacemet ζ ( t) associated with the wet-mode expasio ˆ ζ () t + h ˆ ωζ () t + ˆ ω ζ () t = ˆ γ x () t ( =,,, L ) (6) where g N γ = ˆ =, Mˆ + ˆ μ γ Λ N N N (7) Mˆ = Λ M Λ, ˆ μ = Λ μ Λ m m = = m= I the above equatios, ˆ γ deotes the participatio factor, M ˆ the effective moda mass, ad ˆ μ the added mass coefficiet, respectivey, associated with the wet-mode expasio, ad h ˆ deotes the dampig ratio of th mode added to cosider the dampig effects. Oce the moda dispacemets ζ ( t ) have bee obtaied by sovig the equatio of motio (6), these ca be substituted ito (5) ad the ito () ad (3) to evauate the roof dispacemet w, ad the bedig ad twistig momets M r, M θ, M θ o the foatig roof r L = w= ζ () t Zˆ ( r)cosθ (8) L dzˆ ˆ ˆ dz Z Mr = ζ () t Dr + D cosθ = dr r dr r L dzˆ ˆ ˆ Z d Z Mθ = ζ() t Dθ D + cosθ = r dr r (9) dr L dzˆ ˆ Z Mr θ = ζ () t Dr θ siθ = r dr r where

7 N Zˆ () r = Λ Z () r () = 5. MODAL ANALYSIS BASED ON RESPONSE SPECTRUM With the diagoaised equatios of motio (6) i had, the moda aaysis based o the respose spectrum ca ow be carried out to predict the maximum resposes. The maximum moda resposes of the roof dispacemet w, ad the bedig ad twistig momets M r, M θ, M r θ o the foatig roof i th mode ca be predicted from ( w) = ˆ γ Zˆ cos θ S ( Tˆ, hˆ ) () D dzˆ ˆ ˆ dz Z ( M ) ˆ ˆ ˆ r = γ Dr + D cos θ S (, ) DTh dr r dr r dzˆ ˆ ˆ Z d Z ˆ ˆ ˆ θ = γ θ + θ D ( M ) D D cos S ( T, h) r dr r dr dzˆ ˆ Z ( M ) ˆ si ( ˆ, ˆ rθ = γdr θ θ S ) D T h r dr r where SD ( T, h ) deotes the dispacemet respose spectrum of the iput groud motio, ad Tˆ = π ˆ ω deotes the atura period of the foatig roof i iquid. There are two methods most commoy-used for superposig the maximum moda resposes to predict the maximum tota respose. Oe is the square-root-of-sum-ofsquares (SRSS) method, ad the other is the compete-quadratic-combiatio (CQC) method, which is effective for the system with cosey-spaced atura periods. For exampe, the maximum tota respose of roof dispacemet w ca be predicted from max L = () w = ( w) (3) by the SRSS method, or from L L w = c ( w) ( w) (4) max = = by the CQC method, where c deotes the correatio coefficiet betwee the th ad th modes, give by c 8 hh ( h + β h ) β = ( β ) + 4 β ( + β ) + 4( + ) β 3 hh h h (5)

8 with β ˆ ˆ = ω ω (Der Kiureghia, 98). The simiar expressios ca be obtaied for the maximum tota resposes of bedig ad twistig momets. 6. RESULTS AND DISCUSSION Numerica studies have bee performed to iustrate the appicabiity of the proposed method. Tabe shows the pricipa parameters of the tak mode aayzed here. This is the mode of typica oi-storage tak of,m 3 capacity with a sige-deck type foatig roof, for which a aaytica soutio has bee preseted (Matsui, 7a). The EW compoet of the 3 Tokachi-oki earthquake recorded at Tomakomai K-NET statio (HKD939645EW) was used as a iput groud motio, of which the time history of acceeratio ad the veocity respose spectrum are show i Figure. Figure 3 shows the maximum resposes predicted with varyig the adopted umber of modes. The stiffess-proportioa dampig was assumed with the dampig ratio show i Tabe for the fudameta mode. Both the resuts predicted by the SRSS ad CQC methods are preseted. It ca be oted that oy 3 modes are sufficiet to obtai the coverged soutios for the roof dispacemet, whie more tha 3 modes are required to obtai the reasoabe predictios for the bedig stresses. This is because a arger umber of modes are eeded to express precisey the bedig stresses i the sige-deck type foatig roof, which chage rapidy i the eighbourhood of the coectio betwee the deck ad the potoo (Matsui, 7a). Figure 4 shows the predicted maximum resposes aog θ = (which is parae to the directio of groud motio). The predictios based o the respose spectrum method are compared with the resuts of time history aaysis. Both the SRSS ad CQC methods are foud to provide Tabe Pricipa parameters of tak modes Radius 4 m Liquid depth m Mass desity of iquid 85 kg/m 3 Thickess of deck 4.5 mm Width of potoo sectio 5 m Height of potoo sectio 8 mm Thickess of upper ad ower decks of potoo 4.5 mm Thickess of ier ad outer rims of potoo mm Bedig rigidity of deck.79 kn-m Bedig rigidity of potoo.67 6 kn-m Twistig rigidity of potoo.59 6 kn-m Effective cross-sectioa coefficiet of potoo 577 mm 3 Mass desity of ier deck (per uit area) 58 kg/m Mass of potoo (per uit egth) 89 kg/m Mass momet of iertia of potoo (per uit egth) 85 kg-m /m Youg's moduus 6 GPa Poisso's ratio.3 Dampig ratio (for the fudameta mode).5

9 Figure (a) Time history of acceeratio, ad (b) veocity respose spectrum of EW compoet of the 3 Tokachi-oki earthquake recorded at Tomakomai K-NET statio Figure 3 (a) Maximum roof dispacemet aog the tak wa, (b) maximum radia bedig stress i the deck aog the coectio with the potoo, ad (c) maximum circumferetia bedig stress o the potoo predicted with varyig umber of modes Figure 4 (a) Maximum roof dispacemet, (b) maximum radia bedig stress, ad (c) maximum circumferetia bedig stress aog θ= compared with time history aaysis reasoabe predictios for the roof dispacemet, whereas the CQC method gives better predictios for the bedig stresses tha the SRSS method, which are cose to the resuts of time history aaysis. 7. CONCLUSIONS A respose spectrum method has bee proposed to predict the soshig respose of a foatig roof i a cyidrica iquid storage tak uder seismic excitatio. The vaidity of the proposed method has bee cofirmed by compariso with the resuts of time history aaysis. It ca be cocuded that the CQC method is preferabe tha the SRSS method because the former gives better predictios for both the roof dispacemet ad the bedig stresses, which are cose to the resuts of time history aaysis.

10 8. ACKNOWLEDGEMENTS This work has bee carried out as a part of research project for upgradig seismic performace of structures based o ew cocepts such as seismic cotro, which is i progress i the Advaced Research Ceter for Seismic Experimets ad Computatios, Meijo Uiversity. The fiacia supports by the Grat-i-Aid for Scietific Research from the Japa Society of Promotio of Sciece (Grat No ) ad from Meijo Uiversity Research Istitute are aso gratefuy ackowedged. The strog motio data was provided by the K-NET of Natioa Istitute for Earth Sciece ad Disaster Prevetio, Japa. 9. REFERENCES Der Kiureghia, A. (98) A respose spectrum method for radom vibratio aaysis of MDOF systems, Earthq Egg Struct Dy, Vo 9, pp The Fire ad Disaster Maagemet Agecy of Japa (5) O the eforcemet of the miisteria ordiace which ameds a part of the rue cocerig the cotro of hazardous materias. Notificatio 4, 5 (i Japaese). Hatayama, K., Zama, S., Nishi, H., Yamada, M., Hirokawa, M. ad Ioue, R. (5) The damages of oi storage taks durig the 3 Tokachi-oki earthquake ad the og period groud motios. Proc. of the JSCE-AIJ Joit Symposium o Huge Subductio Earthquakes -Wide Area Strog Groud Motio Predictio-, pp7-8 (i Japaese). The Japa Society of Civi Egieers (999) The 999 Kocaei earthquake, Turkey- Ivestigatio ito damage to civi egieerig structures-. Matsui, T. (6) Soshig i a cyidrica iquid storage tak with a foatig roof uder seismic excitatio, Proc of the ASME Pressure Vesses ad Pipig Divisio Coferece (PVP6-ICPVT-9353), Vacouver, Caada, Juy 3-7. Matsui, T. (7a) Soshig i a cyidrica iquid storage tak with a sige-deck type foatig roof uder seismic excitatio, Proc of the ASME Pressure Vesses ad Pipig Divisio Coferece (PVP7-649), Sa Atoio, Texas, U.S.A., Juy -6. Matsui, T. (7b) Soshig ad dyamic iteractio betwee iquid ad foatig roof i a cyidrica iquid storage tak subjected to seismic excitatio, Proc of ECCOMAS Thematic Coferece o Computatioa Methods i Structura Dyamics ad Earthquake Egieerig, Rethymo, Crete, Greece, Jue 3-6. Timosheko, S.P. ad Woiowsky-Krieger, P. (959) Theory of Pates ad Shes, d ed, McGraw-Hi, New York.