Modeling of the Fluid Solid Interaction during Seismic Event

Transcription

1 Journal o Material cience and Enineerin A 5 (3-4) (015) doi: / / D DAVID PIHING Modelin o the luid olid Interaction durin eimic Event Jan Vachulka * tevenon and Aociate, Vejprnicka 56, Pilen, Czech epublic Abtract: he luid olid interaction belon to one o the important topic in tructural dein in nuclear ield. In the pat decade many reearcher uch[1-3], [6-8] have dedicated lare eort to ind epecially modal hape and tranient repone o the cylindrical luid illed tank. e eort wa dedicated to rectanular tank [3, 4]. he main aim o thi paper i to preent acceptable method or determinin luid-tructure interaction durin eimic event uin tandard inite element code without implemented luid inite element. he method i baed on aumption that in-vacuum modal hape o the tructure are known and alo the modal hape o the ree liquid urace are known. he mode hape in vacuum are determined by tandard inite element code and ree urace mode are derived analytically or uin boundary element method. he boundary condition on the luid-tructure interace i obtained emi-analytically in orm o ourier or eel-ourier erie or imple domain or uin boundary element method or complicated luid domain. he luid i aumed to be irrotational and incompreible. Applyin Galerkin method the ytem o i obtained. he ytem o equation i olved in time domain uin Newmark interation cheme. he eimic repone o the liquid-illed tank i calculated. he calculated example how very ood areement with the previouly publihed. he method can be ued in nuclear technoloy dein. Key word: Interaction, luid, potential, inite element, boundary element. 1. Introduction he incompreible luid olid interaction problem can be decribed uin ollowin et o equation: a-place equation in luid domain 0 ree urace condition o the luid 0 z t luid-olid interace equation rad n u n olid domain equation (Elaticity equation) where i luid potential, u i olid body diplacement, n i outer normal o luid-olid interace he et o thee equation can be olved uin EM, combination o the EM and EM or emianalitical method in orm o ininite erie or imple domain can be ued. Correpondin author: Jan Vachulka. vachulka@tevenon.cz.. Decription o Propoed Method he propoed method i baed on the aumption, that we do know the olution o the aplace equation and the boundary condition will be atiied approximately uin ininite erie or EM. he method i baed on oriinal aumption o luid potential decompoition. he primary unknown i the convective part o the luid potential and deormation o the wall on the luid-olid interace. he convective part o the luid potential atiie the riid wall boundary condition and can be obtained in cloed orm or imple domain or uin EM or complicated domain. he deormation o the olid wall can be expreed a the linear combination o the uitable in-vacuum mode. Hence convective part o potential unction and in vacuum modal hape are ull vector pace. ummin up the above aumption we have:, 0 (1) where i convective part o luid potential

2 17 Modelin o the luid olid Interaction durin eimic Event 0 on () 0 on (3) n he impulive part o the potential atiie (4), (5), (6): 0 on (4) rad n u n on, (5) n a 0 on (6) iidly impulive part o the luid potential atiie (7), (8), (9): 0 on (7) & rad r r r n u n on (8) n 0 on (9) All part o potential are bound on the ree urace by ollowin relationhip (10) 1 ( ) 1 0 (10) n n In Eq. (1)-(10) u mean the temporal derivative o the olid part diplacement, u i the velocity o the eimic motion, n i the outer normal o luid-olid interace, and,,, are viible in i. 1.1 Numerical Approach-Weihted eidual Method Applyin weihted reiduum approach, virtual work principle and relation (1)-(10) we et the equation or luid domain (11) and olid domain (1). i. 1 luid and olid domain. 1 n n d n d d d (11) 0 d u n d u ( u u ) d (1) Introducin the approximation o the olid and luid reion in orm: u r N r( t ) (13) ( t ) (14) N r( t ) (15) u N u (t) r (16) u ( t ) r n (17) n ubtitution into Eq. (11) and (1) uin Green theorem yield in Eq. (18) K 0 r C r 0 K - C M MA 0 r 0 0 M 0 N (18) u u he particle o Eq. (18) are ollowin: K N N d (19) n which i tine matrix o the luid, M N N d (0) i the ma matrix o the luid N N d n (1) i the load vector o the luid

3 Modelin o the luid olid Interaction durin eimic Event 173 K D d () i the tine matrix o the olid. N r n N d i the couplin matrix o the lohin and bulin part o luid potential M N r n N A d (3) i added ma matrix M N N i the ma matrix o the olid N r n (4) 1 d (5) load vector o caued by luid preure N r r (6) d load vector o caued by olid inertia 1 (7) i the total load vector C i dampin matrix o the olid C i dampin matrix o the luid 3. Practical e o the Propoed Method I we et input data in the orm o the in vacuum modal hape o the olid part and lohin potential o the luid in urrounded by a riid olid, then applyin the boundary condition (5) and (6) the value o can be obtained. imilarly the value o can be et rom boundary condition (8) and (9). he input modal hape can be the unction (epecially polynomial) or the nodal diplacement (i we ue a EM code). he luid domain can be imple (rectanular, circular, pherical) and the value or and can be ot in cloed orm or or complicated domain uin boundary interal technique. he problem lead to ytem o linear equation i EM i applied. he value o or riid olid domain can be ot in cloed orm or imple domain or uin boundary interal technique or complicated domain. he problem lead to eien-value olution. hi reult rom Eq. (), (3) and (10) aume that no eimic motion and riid wall o the olid part. Havin all the part o luid potential and in vacuum modal hape the luid-tructural matrice can be aembled and eimic I problem can be olved. he method can be ued by enineer olvin I problem havin tandard 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 preence o riid olid a the bae unction o the luid. 4. he eult and Comparion Previouly publihed eimic repone o the tall radial torae tank with riid bottom wa choen a an example. he radial time hitorie o the top o the tank were compared. he tank wa loaded only in horizontal direction, input motion wa E-Centro. he diameter o tank wa m, heiht m, thickne o the hell 5.4mm. he tank wa made o teel and ull o the water. elative dampin ratio % wa conidered. irt ive member o ourier-eel-erie were choen or approximation o the luid potential. he eect o lohin wa not taken into account. he EM model wa contructed uin the hell element in order to calculate in vacuum modal hape and can be een on i.. i. inite element model o the torae tank modelled uin linear thin-walled hell element.

4 174 Modelin o the luid olid Interaction durin eimic Event i. 3 irt three in-vacuum modal hape ued in analyi (requencie:19.9 Hz, Hz, 9.59 Hz), the total diplacement. i. 4 ime hitory o acceleration o the top o tall torae tank publihed by Haroun [3]. ar ar i. 5 ime hitory o acceleration o the top o tall torae tank calculated uin propoed method. i. 6 irt three modal hape o the luid illed tank (requencie: 5.40 Hz, Hz, 4.76 Hz).

5 Modelin o the luid olid Interaction durin eimic Event 175 able 1 Comparion o the repone o the top o the tank. epone adial diplacement (mm) adial acceleration (m - ) Propoed method Haroun [3] Dierence (%) able able comparion o undamental requencie. requency Propoed method Haroun [3] Dierence (%) Concluion he propoed method howed ood areement with previouly publihed reult. he dierence in undamental requencie and elected repone i acceptable in technical calculation. he dierence i mainly caued by the act that Haroun (1980) had ued the cloed orm olution or circular hell, but the propoed method ue mode calculated by inite element method. he propoed method can be ued by enineer who have to olve the luid tructure interaction problem uin tandard commercial otware without implemented luid inite element. hi procedure can be alo ued or olvin o dynamic repone o the immered tructure in marine enineerin. he limitation o propoed method i actually to linear problem, neverthele the linear approximation i acceptable in many technical topic. eerence [1] Amabili, M., Paidoui, M. P., and aki, A. A Vibration o Partially illed Circular ank with in tiener and lexible ottom. Journal o ound and Vibration 13: [] Amabili, M ree Vibration o Partially illed Horizontal Cylindrical hell. Journal o ound and Vibration 191: [3] Haroun, M. A Dynamic Analyi o iquid torae ank. Paaneda Caliornia, [4] Jeona, K. H., Yoo, G. H., and ee,. C Hydroelatic Vibration o wo Identical ectanular Plate. Journal o ound and Vibration 7: [5] Zhou, D., and Chan, Y. K Vibration o Vertical ectanular Plate in Contact with Water on One ide. Earthquake Enineerin and tructural Dynamic 9: [6] Goncalve, P.., and amo, N ree Vibration Analyi o Cylindrical ake Partially illed with iquid. Journal o ound and Vibration 195: [7] Gupta,. K ree Vibration o Partially illed Cylindrical ank. Enineerin tructure 17: [8] Gupta,. K., and Hutchinon, G Eect o Wall lexibility on the Dynamic epone o iquid torae ank. Enineerin tructure 13: