Modeling of the Fluid Solid Interaction during Seismic Event

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1 Jounal o Mateial cience and Enineein A 5 (3-4) (015) doi: / / D DAVID PIHING Modelin o the luid olid Inteaction duin eimic Event Jan Vachulka * tevenon and Aociate, Vejpnicka 56, Pilen, Czech epublic Abtact: he luid olid inteaction belon to one o the impotant topic in tuctual dein in nuclea ield. In the pat decade many eeache uch[1-6 have dedicated lae eot to ind epecially modal hape and tanient epone o the cylindical luid illed tank. e eot wa dedicated to ectanula tank [3, 7. he main aim o thi pape i to peent acceptable method o deteminin luid-tuctue inteaction duin eimic event uin tandad inite element code without implemented luid inite element. he method i baed on aumption that in-vacuum modal hape o the tuctue ae known and alo the modal hape o the ee liquid uace ae known. he mode hape in vacuum ae detemined by tandad inite element code and ee uace mode ae deived analytically o uin bounday element method. he bounday condition on the luid-tuctue inteace i obtained emi-analytically in om o ouie o eel-ouie eie o imple domain o uin bounday element method o complicated luid domain. he luid i aumed to be iotational and incompeible. Applyin Galekin method the ytem o i obtained. he ytem o equation i olved in time domain uin Newmak inteation cheme. he eimic epone o the liquid-illed tank i calculated. he calculated example how vey ood aeement with the peviouly publihed. he method can be ued in nuclea technoloy dein. Key wod: Inteaction, luid, potential, inite element, bounday element. 1. Intoduction he incompeible luid olid inteaction poblem can be decibed uin ollowin et o equation: a-place equation in luid domain Δ ϕ = 0 ee uace condition o the luid ϕ + = 0 z t luid-olid inteace equation ad ( ϕ ) n = u& n olid domain equation (Elaticity equation) whee ϕ i luid potential, u i olid body diplacement, n i oute nomal o luid-olid inteace he et o thee equation can be olved uin EM, combination o the EM and EM o emianalitical method in om o ininite eie o imple domain can be ued. Coepondin autho: Jan Vachulka. vachulka@tevenon.cz.. Deciption o Popoed Method he popoed method i baed on the aumption, that we do know the olution o the aplace equation and the bounday condition will be atiied appoximately uin ininite eie o EM. he method i baed on oiinal aumption o luid potential decompoition. he pimay unknown i the convective pat o the luid potential and deomation o the wall on the luid-olid inteace. he convective pat o the luid potential atiie the iid wall bounday condition and can be obtained in cloed om o imple domain o uin EM o complicated domain. he deomation o the olid wall can be expeed a the linea combination o the uitable in-vacuum mode. Hence convective pat o potential unction and in vacuum modal hape ae ull vecto pace. ummin up the above aumption we have: ϕ = ϕ, Δ ϕ = Δ ( ϕ ) = 0 (1) whee ϕ i convective pat o luid potential

2 167 Modelin o the luid olid Inteaction duin eimic Event Δ ϕ = 0 on () = 0 on (3) he impulive pat o the potential ϕ atiie (4), (5), (6): Δ ϕ = 0 on (4) = ad ( ϕ ) = u& on, (5) a ϕ = 0 on (6) iidly impulive pat o the luid potential ϕ atiie (7), (8), (9): Δ ϕ = 0 on (7) & ad ( ) = ϕ = u on (8) ϕ = 0 on (9) All pat o potential ae bound on the ee uace by ollowin elationhip (10) 1 ( ϕ ) 1 + & ϕ = + && ϕ = 0 (10) In Eq. (1)-(10) u & mean the tempoal deivative o the olid pat diplacement, u & i the velocity o the eimic motion, n i the oute nomal o luid-olid inteace, and,,, ae viible in i. 1.1 Numeical Appoach-Weihted eidual Method Applyin weihted eiduum appoach, vitual wok pinciple and elation (1)-(10) we et the equation o luid domain (11) and olid domain (1). i. 1 luid and olid domain. 1 ϕ δϕ δϕ && ϕ δϕ d + d = d + δϕ d + (11) d δu ρ ( u&& + u&& ) d 0 ( & & & ) δε σd = ρ δu n ϕ (1) Intoducin the appoximation o the olid and luid eion in om: u = N ( t ) (13) [ { } [ { ( t )} ϕ = (14) N [ { ( & t )} ϕ = (15) u N = u (t) (16) φ ϕ = φ u& ( t ) = n (17) ubtitution into Eq. (11) and (1) uin Geen theoem yield in Eq. (18) [ K [ 0 { } [ C [ { & } + [ 0 [ K {} [- [ C {} & [ M [ MA [ 0 {} && { } [ 0 [ 0 [ M {} && [ 0 { N} (18) + u&& + = u& he paticle o Eq. (18) ae ollowin: [ K = [ N [ N ρ ρ d (19) which i tine matix o the luid, ρ [ M = [ N [ N d (0) i the ma matix o the luid φ { N} = ρ [ N d (1) i the load vecto o the luid

3 Modelin o the luid olid Inteaction duin eimic Event 168 [ K = [ [ D[ d () i the tine matix o the olid. N n N d [ = ρ [ { } { } [ i the couplin matix o the lohin and bulin pat o luid potential M = ρ N n N d (3) [ [ { } { } [ A i added ma matix M = [ [ N [ N { } = [ N { } { n} ρ (4) i the ma matix o the olid 1 ρ φ d (5) load vecto o caued by luid peue = ρ N (6) { } [ { } { } d load vecto o caued by olid inetia {} = { 1 } + { } (7) i the total load vecto [ C i dampin matix o the olid C i dampin matix o the luid [ 3. Pactical e o the Popoed Method I we et input data in the om o the in vacuum modal hape o the olid pat and lohin potential o the luid in uounded by a iid olid, then applyin the bounday condition (5) and (6) the value o ϕ can be obtained. imilaly the value o ϕ can be et om bounday condition (8) and (9). he input modal hape can be the unction (epecially polynomial) o the nodal diplacement (i we ue a EM code). he luid domain can be imple (ectanula, cicula, pheical) and the value o ϕ and ϕ can be ot in cloed om o o complicated domain uin bounday inteal technique. he poblem lead to ytem o linea equation i EM i applied. he value o ϕ o iid olid domain can be ot in cloed om o imple domain o uin bounday inteal technique o complicated domain. he poblem lead to eien-value olution. hi eult om Eq. (), (3) and (10) aume that no eimic motion and iid wall o the olid pat. Havin all the pat o luid potential and in vacuum modal hape the luid-tuctual matice can be aembled and eimic I poblem can be olved. he method can be ued by eninee olvin I poblem havin tandad EM code without luid element implemented. he method i in act the itz method with in-vacuum modal hape a bae unction o olid and bulin potential and alo the lohin hape o the luid in peence o iid olid a the bae unction o the luid. 4. he eult and Compaion Peviouly publihed eimic epone o the tall adial toae tank with iid bottom wa choen a an example. he adial time hitoie o the top o the tank wee compaed. he tank wa loaded only in hoizontal diection, input motion wa E-Cento. he diamete o tank wa m, heiht m, thickne o the hell 5.4mm. he tank wa made o teel and ull o the wate. elative dampin atio % wa conideed. it ive membe o ouie-eel-eie wee choen o appoximation o the luid potential. he eect o lohin wa not taken into account. he EM model wa contucted uin the hell element in ode to calculate in vacuum modal hape and can be een on i.. i. inite element model o the toae tank modelled uin linea thin-walled hell element.

4 169 Modelin o the luid olid Inteaction duin eimic Event i. 3 it thee in-vacuum modal hape ued in analyi (equencie:19.9 Hz, Hz, 9.59 Hz), the total diplacement. i. 4 ime hitoy o acceleation o the top o tall toae tank publihed by Haoun [3. a a i. 5 ime hitoy o acceleation o the top o tall toae tank calculated uin popoed method. i. 6 it thee modal hape o the luid illed tank (equencie: 5.40 Hz, Hz, 4.76 Hz).

5 Modelin o the luid olid Inteaction duin eimic Event 170 able 1 Compaion o the epone o the top o the tank. epone adial diplacement (mm) adial acceleation (m - ) Popoed method Haoun [ Dieence (%) able able compaion o undamental equencie. equency Popoed method Haoun [3 Dieence (%) Concluion he popoed method howed ood aeement with peviouly publihed eult. he dieence in undamental equencie and elected epone i acceptable in technical calculation. he dieence i mainly caued by the act that Haoun (1980) had ued the cloed om olution o cicula hell, but the popoed method ue mode calculated by inite element method. he popoed method can be ued by eninee who have to olve the luid tuctue inteaction poblem uin tandad commecial otwae without implemented luid inite element. hi pocedue can be alo ued o olvin o dynamic epone o the immeed tuctue in maine enineein. he limitation o popoed method i actually to linea poblem, nevethele the linea appoximation i acceptable in many technical topic. eeence [1 Amabili, M., Paidoui, M. P., and aki, A. A Vibation o Patially illed Cicula ank with in tiene and lexible ottom. Jounal o ound and Vibation 13: [ Amabili, M ee Vibation o Patially illed Hoizontal Cylindical hell. Jounal o ound and Vibation 191: [3 Haoun, M. A Dynamic Analyi o iquid toae ank. Paaneda Calionia, [4 Goncalve, P.., and amo, N ee Vibation Analyi o Cylindical ake Patially illed with iquid. Jounal o ound and Vibation 195: [5 Gupta,. K ee Vibation o Patially illed Cylindical ank. Enineein tuctue 17: [6 Gupta,. K., and Hutchinon, G Eect o Wall lexibility on the Dynamic epone o iquid toae ank. Enineein tuctue 13: [7 Jeona, K. H., Yoo, G. H., and ee,. C Hydoelatic Vibation o wo Identical ectanula Plate. Jounal o ound and Vibation 7: