DYNAMIC ANALYSIS OF CONCRETE RECTANGULAR LIQUID STORAGE TANKS

Transcription

1 The 4 th World Coferece o Earthquake Egeerg October 2-7, 28, Bejg, Cha DYNAMIC ANAYSIS OF CONCRETE RECTANGUAR IQUID STORAGE TANKS J.Z. Che, A.R. Ghaemmagham 2 ad M.R. Kaoush 3 Structural Egeer, C2M I Caada, Toroto, Otaro, Caada. E-mal: Jaso.Che@ch2m.com 2 Ph.D. Studet, Departmet of Cvl Egeerg, Ryerso Uversty, Toroto, Otaro. E-mal: aghaemma@ryerso.ca 3 Professor, Departmet of Cvl Egeerg, Ryerso Uversty, Toroto, Otaro, Caada. E-mal: kaoush@ryerso.ca ABSTRACT : A structural model usg the geeralzed sgle degree of freedom (SDF) system s proposed for sesmc desg of cocrete rectagular qud Cotag Structures (CS). The proposed model cosders the effect of fleblty of tak wall o hydrodyamc pressures ad uses the cosstet mass approach. The proposed model s compared wth the results obtaed usg the curret practce as well as the fte elemet method. It s cocluded that the curret approach desg codes ad stadards does ot truly represet the behavor of CS. The proposed model usg the geeralzed SDF system ca be smply used sesmc desg of CS. KEYWORDS: Reforced cocrete, lqud cotag, rectagular tak, sesmc, dyamc aalyss. INTRODUCTION qud cotag structures (CS) as part of evrometal egeerg facltes are prmarly used for water ad sewage treatmet plats ad other dustral wastes. Normally, they are costructed of reforced cocrete the form of rectagular or crcular cofguratos. Curretly there are few codes ad stadards avalable for sesmc desg of CS North Amerca. I almost all of codes ad stadards, the ouser s model (ouser, 963) has bee adopted for dyamc aalyss of CS. Ths model appromates the effect of hydrodyamc pressure for a two fold-symmetrc-flud cotaer subjected to horzotal accelerato as show Fgure (a). The hydrodyamc pressures duced by earthquakes are separated to two parts of mpulsve ad covectve compoets whch are appromated by the lumped added masses. The added mass terms of mpulsve pressure s assumed rgdly coected to the tak wall ad the added mass terms of covectve pressure s assumed coected to the tak wall usg fleble sprgs to smulate the effect of sloshg moto. I ths model, the boudary codto the calculato of hydrodyamc pressures s treated as rgd. Although the ouser s model has bee appled the sesmc desg of CS the past, recet studes show that due to the assumpto of the lumped added mass ad the rgd tak wall, ths method leads to overly coservatve results. Che ad Kaoush (25) developed a procedure referred to as the sequetal method for computg hydrodyamc pressures based o a two-dmesoal model for rectagular taks whch the effect of fleblty of tak wall was take to cosderato. ater Kaoush et al. (26) ad Ghaema et al. (25) appled the staggered method to solve the coupled lqud storage tak problems three-dmesoal space. Compared to the ouser s model, these results show that most cases the lumped mass approach overestmates the base shear ad base momet sgfcatly. Che ad Kaoush (27) proposed a geeralzed sgle degree of freedom (SDF) system for dyamc aalyss of CS. The cosstet mass approach ad the effect of fleblty of tak wall o hydrodyamc pressures were cosdered. The prescrbed vbrato shape fuctos represetg the mode shapes for the catlever wall boudary codto were valdated. I ths paper, the proposed structural model usg the geeralzed SDF system s compared wth the ouser s Model adopted the curret desg codes ad stadards. The desg charts for the added mass of

2 The 4 th World Coferece o Earthquake Egeerg October 2-7, 28, Bejg, Cha lqud due to mpulsve hydrodyamc pressure ad the correspodg effectve heght are preseted. The cotrbuto of hgher modes to the dyamc respose of CS s cluded the proposed model. The square root of sum of square (SRSS) method s used for the combato of the frst two modes. A case study represetg a tall tak s preseted. The results are compared wth those obtaed usg the ouser s model as well as the fte elemet method. It s recommeded that the curret desg approach eed to be modfed. The proposed structural model usg the geeralzed SDF system ca be cosdered as smple model to overcome the curret defceces desg of CS. 2. GENERAIZED SDF SYSTEM FOR DYNAMIC ANAYSIS OF CS 2. Aalyss Model ad Equato of Moto The ouser s model (963) s show Fgure (a). Fgure (b) shows a catlever tak wall wth the dstrbuted mass m( ad stffess EI( per ut heght subjected to the earthquake groud accelerato ü g (t). The wall ehbts a fte umber of degrees of freedom for fleural mode of respose. If there are some predetermed shapes to appromate the vbrato of the system, the the moto of the system ca be descrbed by a sgle varable, or geeralzed coordate whch oly oe DOF ests. The system dealzed ths maer s referred to as geeralzed SDF systems. I ths study, the geeralzed SDF system s appled to solve the dyamc respose of lqud storage taks subjected to earthquakes. The equato of moto for a geeralzed SDF system s that: m ~ ~ u& + c~ u& + k u = ~ p (2.) Where m ~, c ~, k ~, p ~ are defed as the geeralzed system of mass, dampg, stffess ad force respectvely. (a) ouser s Model Fgure Aalyss Model (b) Geeralzed SDF System For smplcty, the prescrbed vbrato shape fucto SF represetg the frst mode shape for the catlever wall boudary codto ca be used dyamc aalyss as follows: y y SF(= ψ ( = (2.2) W W

3 The 4 th World Coferece o Earthquake Egeerg October 2-7, 28, Bejg, Cha The valdty of the shape fucto SF was verfed ad dscussed the prevous study (Che ad Kaoush, 27). The drect couplg method s used the dyamc aalyss. The teracto betwee lqud ad tak wall s solved drectly the equato of moto usg the added mass method. 2.2 Added Mass of qud The hydrodyamc pressure ca be solved usg the separato of varables method whch satsfes the boudary codtos. The hydrodyamc pressure dstrbuto o the fleble wall codto ca be epressed as follows: 2 ρ l tah( λ ) p = cos( λ cos( u&& ( t) dy λ (2.3) λ = Where λ = (2-)π/2. As the seres the above equato covergece very fast, oly the frst three terms of the seres are used for practcal applcatos. Whe usg the geeralzed SDF system the dyamc aalyss of CS, the hydrodyamc pressure s corporated to the couplg aalyss through the added mass. The geeralzed ad effectve added mass of lqud due to mpulsve hydrodyamc pressure, m ~ ad m, ca be calculated usg Eqs. 2.4 ad 2.5 respectvely. m~ m = = 2 ρl tah( λ )[ = λ + 2 ( ) ρl tah( λ 2 ) = λ cos( λ cos( λ ψ ( dy] 2 ψ ( dy (2.4) (2.5) Based o the ouser s model, the rato of the effectve added mass of lqud due to mpulsve hydrodyamc pressure M to the total mass of lqud the cotamet M s epressed as: M tah[.866( / )] = (2.6) M.866( / ) Smlarly for the geeralzed SDF system, the rato of geeralzed ad effectve added mass of lqud due to hydrodyamc pressure for the prescrbed mode shape to the half mass of lqud CS,.e. m ~ / M ad m / M, ca be calculated. It s worth otg that compared to the total mass of lqud the ouser s model, oly half the mass of lqud s cosdered the geeralzed SDF system based o the two-fold symmetrc flud structural model. Fgures 2(a) ad 2(b) show the added mass of lqud due to mpulsve pressure based o the ouser s model ad the geeralzed SDF system usg shape fucto ψ ( = whch are both correspodg to a rgd tak wall, ad the ratos of m ~ / M ad m M / as fuctos of the rato of wdth of tak to depth of lqud /. The shape fucto SF s used for the frst mode cosderg the fleblty of tak wall dyamc aalyss. It s worth otg that Fgure 2 s oly for the full tak codto,.e. = W.

4 The 4 th World Coferece o Earthquake Egeerg October 2-7, 28, Bejg, Cha Fgure 2 shows that the tred of curves for the ouser s model ad Shape fucto SF for the frst mode shape usg the geeralzed SDF system s smlar. owever, the results obtaed usg the ouser s model are more tha two tmes of those obtaed usg ψ ( = for the geeralzed SDF system cosderg the fold-symmetrc-flud structural model. The reaso for the dfferece respose s due to the dfferet methods used the calculato of hydrodyamc pressure. I ths study, the hydrodyamc pressure s calculated usg the velocty potetal method. It s assumed that the lqud s deal, whch s compressble ad vscd. owever, a smple Newtoa vscous shear model s used the ouser model whch may also result stff respose. m ~ M st Mode 2d Mode ouser Rgd m M st Mode 2d Mode ouser Rgd (a) Geeralzed Added Mass (b) Effectve Added Mass Fgure 2 Rato of Added Mass of qud due to Impulsve ydrodyamc Pressure vs. / Rato ( = W ) 2.3 Effectve eght I the curret desg practce, the ertal mass of cocrete wall ad the added mass of lqud due to hydrodyamc pressure are lumped at defed effectve heghts. The ertal mass of cocrete tak wall s lumped at the ceter of gravty of the tak wall. If the tak wall s uform, the total ertal mass of tak wall s lumped at the md-heght of the wall. The added mass of lqud due to mpulsve hydrodyamc pressure s lumped at the cetrod of the mpulsve lateral force. Ths heght ca be calculated usg Eqs. 2.7 ad 2.8 as follows (ACI 35.3, 26): For taks wth <. 333 For taks wth. 333 h, = ( ) (2.7) h, =. 375 (2.8) I the geeralzed SDF system, the effectve heghts at whch the effectve added mass of lqud due to hydrodyamc pressure s appled, h, ca be calculated as follows: h = = λ = 2 ρ λ l 2 ρ l tah( λ tah( λ ) cos( λ ) cos( λ cos( λ cos( λ ψ ( dy y dy ψ ( dy dy (2.9)

5 The 4 th World Coferece o Earthquake Egeerg October 2-7, 28, Bejg, Cha Fgure 3 shows the ormalzed effectve heght at whch the hydrodyamc pressure s appled as fucto of the rato of half wdth of tak to lqud depth / for the full tak codto.e. = W. The fgure shows the frst two modes, the rgd wall boudary codtoψ ( = ad the ouser s model. It ca be foud that the effectve heghts h obtaed from the ouser s model ad the rgd wall boudary codto ψ( = are smlar. For lqud cotag structures, the effectve heght at whch the total dyamc lateral force s appled ca be calculated usg Eq. 2.. Ths epresso cludes both the effects of ertal mass of tak wall ad the added mass of lqud due to hydrodyamc pressure. h m h + m h w w = (2.) mw + m It s worth otg that cosderg the fleblty of tak wall, the effectve heght h at whch the overall lateral dyamc force s appled s hgher tha that obtaed from the rgd wall codto. h/ st Mode 2d Mode Rgd ouser / Fgure 3 Effectve eght Factors for Impulsve ydrodyamc Pressure vs. / Rato ( = W ) 2.4 Effect of gher Modes The smlar method used dyamc aalyss of CS for frst mode ca be appled o the dyamc aalyss for hgher modes. The square root of sum of square (SRSS) method ca be used for the combato of hgher modes. Geerally, the cluso of the frst two modes should provde suffcetly accurate results for desg purposes. 3. FINITE EEMENT IMPEMENTATION FOR DYNAMIC ANAYSIS OF CS I ths study, a dfferet FEM procedure based o the fact that hydrodyamc pressure dstrbuto s govered by wave equato lqud doma s used to verfy results. Assumg that water s compressble ad eglectg ts vscosty, the small-ampltude rrotatoal moto of water s govered by the two-dmesoal wave equato: 2 P (, y, t) = (3.)

6 The 4 th World Coferece o Earthquake Egeerg October 2-7, 28, Bejg, Cha I the couplg system of lqud structure the pressures are appled to the structure surface as the loads o the cotaer walls. The geeral equato of flud structure ca be wrtte the followg form: [ M ]{ U&& } + [ C]{ U& } + [ K]{ U} = { f } [ M ]{ U&& g } + [ Q]{ P} = { F } + [ Q]{ P} [ G]{} P&& T T + [ C ]{} P& + [ ]{ P} = { F} ρ[ Q] ({ U&& } + { U&& }) = { F } ρ[ Q] { U&& } g 2 (3.2) I whch, [ M ], [ C ] ad [ K ] are mass, dampg ad stffess matrces of structure whle [ ] ad [ G] are represetg stffess ad mass for lqud doma. The term [ C ] s the dampg matr of lqud whch s depedet o the vscosty of lqud ad wave absorpto lqud doma ad boudares. The matr [ Q ] trasfers the lqud pressure to the structure as well as structural accelerato to the lqud doma. A 8-ode soparametrc elemet wth two traslatos degree of freedom each ode s used to model the tak walls ad foudato. The lqud doma s modeled usg four-ode soparametrc flud elemets wth pressure degree of freedom each ode. The fte elemet model s used to vestgate the behavor of a tall tak as dscussed the followg desg eample. 4. DESIGN EXAMPE I ths vestgato, a tall tak studed prevously s used as a desg eample. The dmesos ad propertes of the tak are as follows: =9.8m, w =2.3m, =.2m, t w =.2m, E c = MPa, ρ w = 23 kg/m 3, ρ l = kg/m 3, ν=.7 The desg respose spectrum based o ASCE 7-5 s used to obta the respose spectral accelerato. The ste s assumed to be located the West Coast of US Washgto State ad the parameters for the desg respose spectrum are that: () Short perod mamum spectral respose accelerato: S s =.25 (2) -secod mamum spectral respose accelerato: S =.6 (3) Ste class B The calculatos usg the fte elemet method (FEM) ad the ACI 35.3 code are also preseted ths study. The fte elemet method proposed the prevous study (Che ad Kaoush, 25) for Model 4 s used for verfcato. The cosstet mass for both tak wall ad added mass of lqud due to mpulsve hydrodyamc pressure are cosdered the FEM. Also, the results obtaed usg the proposed FEM procedure are compared to those assocated wth added mass FEM method ad ACI code. The horzotal compoet recorded for 94 El-Cetro s used as ectato of the system. The horzotal compoet was scaled such a way that peak groud accelerato reaches.4g. The model cofgurato s depcted Fgure 4. ACI 35.3 (26) Code cosders the effect of ductlty through the respose modfcato factor R. It s worth otg that the respose modfcato factor R ad the mportace factor I, are ot cosdered ths study (.e. R ad I are assumed as ut. Therefore, the comparso betwee the proposed model ad ACI 35.3 Code s o the bass of elastc aalyss.

7 The 4 th World Coferece o Earthquake Egeerg October 2-7, 28, Bejg, Cha Fgure 4 Fte Elemet Mesh Cofgurato The calculato results are summarzed Table for the tall tak. The comparso of the results obtaed usg both FEM procedures ad the proposed model shows good agreemet. owever, the base shear obtaed usg ACI 35.3 Code s about.85 tmes hgher tha that obtaed usg the proposed geeralzed SDF system. The base momet for ACI 35.3 Code as compared to the proposed geeralzed SDF system s about.36 tmes hgher. It s cocluded that the desg usg the ouser s model adopted the curret desg stadards ad codes s overly coservatve. Method W FEM (Proposed Method) Table Summares of Dyamc Respose Proposed Model FEM (added mass) st Mode 2d mode SRSS ACI 35.3 m ~ ( 3 kg) m ( 3 kg) W h w (m) m ~ ( 3 kg) m ( 3 kg) h (m) h (m) k ~ ( 3 kn/m) T (sec) A a (m/sec 2 ) g.65g -.833g d ma (mm) V B (kn) M B (knm) P (kn) M (knm) CONCUSIONS I ths paper, a structural aalyss model usg the geeralzed SDF system s proposed for sesmc desg of CS. The proposed model ca cosder the cosstet mass ad the effect of fleblty of tak wall desg. The coceptual procedure for ths methodology s smlar to that of the ouser s model adopted the curret desg codes ad stadards. owever, the geeralzed ad effectve added mass of lqud due to mpulsve hydrodyamc pressure ad the correspodg effectve heght are troduced the proposed model. The curves for the added mass of lqud due to mpulsve hydrodyamc pressure ad the correspodg effectve heght are preseted ad compared wth those adopted the curret desg codes ad stadards.

8 The 4 th World Coferece o Earthquake Egeerg October 2-7, 28, Bejg, Cha The calculato for a tall tak s preseted ad compared wth the results obtaed usg the curret practce ad the fte elemet method. The comparso shows that the results obtaed from FEM ad the proposed model are good agreemet. owever, the results obtaed usg the curret practce are overly coservatve. It s recommeded to use the geeralzed SDF system for sesmc desg of cocrete rectagular CS. The proposed model ca provde farly accurate results for the structural desg whle stll matag the smplcty. REFERENCES Amerca Cocrete Isttute, ACI (26). Sesmc desg of qud Cotag Cocrete structures, Farmgto lls, MI, U.S. Amerca Socety of Cvl Egeers (ASCE). (25). Mmum Desg oads for Buldgs ad Other Structures, ASCE 7, ASCE Stadard, SEI/ASCE 7-2, Resto, VA, U.S. Che, J. Z. ad Kaoush, M. R. (27). Geeralzed SDF System for Aalyss of Cocrete Rectagular qud Storage Taks. Proceedgs of the 9th Caada Coferece o Earthquake Egeerg, Ottawa, Otaro Che, J. Z. ad Kaoush, M. R. (25). Sesmc Respose of Cocrete Rectagular Taks for qud Cotag Structures. Caada Joural of Cvl Egeerg 32, Ghaema, M., Kaoush, M. R. ad Mrzabozorg,. (25). Tme Doma Dyamc Aalyss of Rectagular qud Cotaers Three-Dmesoal Space. Joural of Europea Earthquake Egeerg XIX:2, 3-9. ouser, G. W. (963). The Dyamc Behavor of Water Taks. Bullet of the Sesmologcal Socety of Amerca 53:2. Kaoush, M. R., Mrzabozorg,. ad Ghaema, M. (26). Dyamc Aalyss of Rectagular qud Cotaers Three-Dmeso Space. Caada Joural of Cvl Egeerg 33, 5-57.