Optmal Topology Dsgn for Rplacabl of Rtculatd Shll Basd on Snstvty Analyss Yang Yang Dpartmnt of Naval Archtctur, Dalan Unvrsty of Tchnology, Laonng, CN Ma Hu Collg of Rsourc and Cvl Engnrng, Northastrn Unvrsty,, Laonng, CN H Zhng Dpartmnt of Cvl Engnrng, Dalan Unvrsty of Tchnology, Laonng, CN SUMMARY: Th control approach takn n ths papr s to rplac slctd bars n rtculatd shll wth passv vscolastc damprs. Snstvty mthod s proposd to dtrmn th optmal topology of damprs n th shll. Basd on th gnvalu prturbaton and arthquak spctrum concpt, th snstvty of shll s calculatd. Contrast th snstvts of all lmnts; manwhl, consdrng th symmtry topology of lmnts, th rasonabl topology of dampr s slctd. Th optmal shll s analyzd undr arthquak actons. Th rsults show that: th snstvty mthod s ffctv for gttng optmal topology. Th dsplacmnt control ffct of optmal topology dynamc rsponss rachs to 5%-3% and th axal forc control ffct rachs to 16%-45%. kywords: Topology optmzaton, Rtculatd shll, Fnt lmnt, Snstvty analyss, Rplacabl lmnts 1. GENERAL INSTRUCTIONS Th trnd n dvlopmnt of long-span shlls has bn towards hghr, longr and mor laboratd structural confguratons. As a rsult, nw challngs hav arsn to nsur th safty and prformanc for ths shll structurs whn subctd to strong arthquaks and svr wnds (Cao and Zhang, 2). Varous vbraton control mthods hav bn proposd for shll control, ncludng rubbr-barng solators (Shngn and Nk, 21), tund mass and tun lqud damprs (Shn and Lan, 21), vscous damprs (Fan t al., 25; Kasa t al., 21), controllabl fluds dvcs (Onoda t al., 1996; Xu t al., 21; Oh and Onoda, 22), and rplacabl damprs (N, 21; Yang t al., 211). Four prototyp vscous damprs ar usd for rtculatd cylndrcal shll for laboratory tstng by Lang (Lang t al., 23) proposd th ffctv topologs and paramtrs cods for rplacabl dampr of cylndrcal shll. To valuat th paramtr ffcts of dampr topologs and dampng coffcnt for rtculatd sphrcal shll, varous dampr topologs and xampls wr smulatd by Yang (Yang t al., 211). Th locaton and amount of damprs ar mportant paramtrs n rsarch of rplacabl bar-typ dampr n rtculatd shll. Th sutabl locaton of damprs can obtan bttr ffcts whl th amount of damprs s fxd or obtan bttr ffcts usng fwr damprs. Agrawal and Yang (Agrawal and Yang, 1999) dvlopd a nw combn algorthm for optmal locaton rsarch of passv damprs n spac structurs suffr arthquak acton or wnd load. Ths algorthm s usful for spac structur such as rtculatd shll, but th algorthm s dffcult to b ralzd. Gntc algorthm s usd by Sngh and Morsch (Sngh and Morsch, 22) to dtrmn th optmal locaton and sz of damprs n structur. Som rsarchrs hav consdrd snstvty mthod n optmal locaton study of damprs. N (N, 21) uss ths concpt to dtrmn th locaton and amount of rplacabl damprs n doubl-layr rtculatd cylndrcal shll. Th rsults ndcat that t s th most ffctv locaton to rplac th lmnts whr snstvts of shll natural frquncy and dformaton can b affctd asly. Furthrmor, th authors dvlop th optmal rul corrspondng to that mthod.
Structural dsgn snstvty analyss concrns th rlatonshp btwn dsgn varabls avalabl to th dsgn ngnr and structural rsponss dtrmnd by th laws of mchancs. Chn (Chn, 1991) dvlops th snstvty thors for structur vbraton analyss. Kyung and Nam (Kyung and Nam, 25) dvlop th snstvty analyss thory for structural optmzaton and ntroducd analyss mthods dply. Habb (Harbb t al., 27) adapts th snstvty mthod to solv th optmum shap dsgn for shll structurs. Amn and Ghadr (Amn and Ghadr, 211) dvlop a nw algorthm n optmal topology study for structur wth MR damprs, th ffctv s dmonstratd n th analyss. In ths study, snstvty s usd to dtrmn th topology of control damprs n doubl-layr rtculatd sphrcal shll. To calculat th snstvty of structural natural frquncy varous, gnratd by small prturbaton of vry lmnt scton, gnvalu prturbaton and arthquak spctrum concpt ar usd. Contrastng th lmnts snstvts, thn consdrng th symmtry topology, th rasonabl locaton of dampr s slctd. Bas on thory rsarch and formula dducton, ANSYS Paramtrc Dsgn Languag (APDL) ncorporatd n th ANSYS fnt lmnt s usd for th work. Morovr, paramtr ffcts of snstvty analyss ar th rqurd rsults for optmal dsgn. 2. SENSITIVITY THEORY 2.1. Egnvalu Dsgn Snstvty Analyss Dsgn snstvty analyss s commonly usd to rprsnt a structural paramtr that can affct th rsults of th analyss. Whn th cross-sctonal ara of a truss componnt changs, th dynamc rsults vary for th appld vbraton load bcaus th stffnss matrx changs. In such cass, th locaton and th rlatd paramtrs of th truss componnt can b a dsgn. Th natural frquncy of vbraton load s gnvalu of a gnralzd gnvalu problm; hnc, t dpnds on th dsgn. In ths papr, th xpanson nto powr srs s usd to obtan drvatvs of such gnvalus n whch rpatd gnvalus appar as an ffcnt soluton to th optmal locaton dsgn problm. For a dscrt systm such as rtculatd shll, th frdom s N, { q } s th gnralzd coordnat matrx. [ K ] and [ M ] ar th stffnss and mass matrxs accordng to matrx {} q. ω s th natural frquncy of structur, whch s dfnd as functon 2 λ = ω. Th oscllaton quaton of MDOF undampd structur s shown as: [ M ]{} q& + [ K]{ q} = {} & (2.1) whr {} q& & s th gnralzd acclraton vctor of MOD structur. Th natural vbraton of structur s harmonc oscllaton, th functon s gvn as: {} = {} u ( ωt φ) q cos (2.2) whr {} u s th vbraton mod matrx. Aftr substtutng Eq. (2.2) nto Eq. (2.1), th drvatv of Eq. (2.1) wth rspct to th gnvalu problm of dscrt systm s dscrbd as: [ K]{} u λ[ M ]{} u = (2.3) Th varatons of structur ar gvn by stffnss and mass matrxs. Th altrnatv stffnss and mass matrxs ar shown.
[ M ] [ ] + ε[ ] = M M 1 (2.4a) [ K ] [ ] + ε[ ] = K K 1 (2.4b) whr ε s a prturb dsgn paramtr. Th systm s orgnal systm accordng wth = and [ M ] as and mass matrxs of orgnal systm. [ K 1 ] stffnss and mass matrxs of systm. Furthrmor, whl [ ] and [ M ] approach [ K ] and [ M ] sparatly. ε. [ K ] ε and ε [ M 1 ] ar th varatons for ε and ε [ ] approach zro, [ K ] In th drvatvs, t s assumd that all gnvalus ar smpl and not rpatd. Undr ths ε condtons, th gnvalus and vbraton mods ar subtl changd wth a small numbr of [ K 1 ] and ε [ M 1 ]. 2.2 Dynamc Snstvty Analyss Hr, for rtculatd shll, s th numbr of mod and s th numbr of lmnt. { u } mod and gnvalu of mod. Thy mt th quaton blow: [ ] [ M ]{ ) u } K 1 M 1 and λ ar th ( K λ = (2.5) Commonly, stffnss and mass ar sldom xprssd as th dsplay functon of varabls. In ths study, for th rason of programmng asr, th dffrntal formulas of snstvty ar xprssd as prturbaton form. [ ΔK ] and[ ΔM ] ar rspctvly th ncrmnt of stffnss matrx and mass matrx Δ. λ gnvalu and gnvctor rspctvly. Th modal snstvty gnratd by th small prturbaton n th dsgn varabl [ ] b λ and { } u to varabl b (=1,2,,L), can b xprssd as follows. Δ and { }, Δ ar th ncrmnt of u λ and { } u,, th drvatvs of whr λ Δ = λ Δb λ, (2.6) { } { Δu } u =, (2.7) Δb Δ and { Δ } u can b calculatd by frst-ordr prturbaton quaton. In fnt lmnt analyss problms, [ ΔK ] and [ ΔM ] mass matrx rspctvly, shown as follows. [ Δ ] = [ Δ K ] ar th total ncrmnt of lmnt stffnss and K (2.8) [ Δ ] = [ Δ M ] M (2.9) Thrfor, th modal snstvty quatons ar transformd to fnt lmnt snstvty quatons. Thus, λ and{ u, } can b rducd as:, λ, = λ, (2.1)
whr λ and { }, { } { u } u,, = (2.11) u, ar th snstvty of λ and{ u } of lmnt. Th snstvty of structural natural frquncy s st as th optmal obctv functon. Th lmnts at th locaton whr th varty of structural natural frquncy s max, whl th shll suffrs a small prturbaton, ar rplacd by th bar-typs damprs. Th structural dynamc snstvty ndcats th ffcts of structural dsgn varabl to structural dynamc charactrstcs. Th ffct s mor rmarkabl at th locaton whr th snstvty s bggr. In ths study, th scton ara of lmnt s st as dsgn varabl. Th lmnts at th locaton whr th snstvts of natural frquncy ar bggr than th othrs ar rplacd by th bar-typs damprs, whr scton ara s suffrd from a small prturbaton. Th optmal obctv functon of damprs locaton can b wrttn as whr rspons spctrum. J n ( ) = = 1 α (2.12) λ, α s th wght paramtr, whch can b st as th gnvalu corrspondng to arthquak s th snstvty of structural -ordr natural frquncy gnratd by th λ, scton ara prturbaton of lmnt. ( ) vbraton mod. J ndcats th nflunc dgr of lmnt to th 3. MODEL GENERATION 3.1 Dsgn Paramtrs of Shll Fgur 1. Plan and lvaton vw of doubl-layr rtculatd shll Th Kwtt-8 sphrcal rtculatd sphrcal shll s commonly usd n ngnrng. Th modl shown n Fgur 1 s 4m span and th hght s 8m. Th thcknss of shll btwn uppr and lowr grd s.8m. Th numbr of nods s 289. Th numbr of lmnts s 936. In calculaton, th dstrbutd mass s 2kg/m 2 and modld by MASS21 ELEMENT n ANSYS. Nods n lowr layr hav no xtrnal load. Th bar lmnt of shll s mad by stl pp and modld by PIPE2 ELEMENT n ANSYS. PIPE2 s Plastc Straght Pp Elmnt. Th lmnt has sx dgrs of frdom at ach nod: translatons n th nodal x, y, and z drctons and rotatons about th nodal x, y, and z axs. Strss stffnng and larg dflcton capablts ar ncludd. Th bars n th uppr and lowr shll hav an nnr damtr of 123.5mm and an outr damtr of 127mm whch stffnss s about 4 1 7 N/m. Th bars n-btwn uppr and lowr shlls hav an nnr damtr of 118mm and an
outr damtr of 121mm, th stffnss s about 3 1 7 N/m. Th lngth of lattudnal bar on uppr layr s 2.6-2.8m. Th lngth of slantng bar on uppr layr s 3.7-4.4m. Th lngth of costal bar on uppr layr s 3.7m. Th lngth of bar on lowr layr s 2.8-4.4m. Th lngth of bar btwn uppr and lowr layr s 2.-2.4m. Th shll s hngd at th support, whch s modld by NODE ELEMENT wth 3 rotaton dgr of frdom n ANSYS. Th lmnt onts of shll ar rgd, whch s also for nod and dampr. Th lmnts n lowr and mddl layr of doubl layr rtculatd shll hav only axal prssur but no bndng momnt or th bndng momnt s vry small. So w can rplac ths lmnts wth damprs. Bcaus of th complxty of shll wth support column or othr support structurs, whch could affct th control rsponss, th shll studd hr s placd on th ground. Th controlld shll wth substructur wll b studd n th follow-up work. Th dampng coffcnt of global shll fts th Raylgh Dampng thory and th dampng rato s.2. Th stl s Q235 typ whch has a modulus of lastcty of 2 1 11 N/m 2. Th yld strngthσ of stl s 2.35 1 8 N/m 2. Th bucklng cod of stl fts th Von Mss Isotropc Hardnng Cod. 3.2 Dsgn Paramtrs of Rplacabl Bar-Dampr Th dampng coffcnt of VE dampr s 1 1 5 Ns/m and th stffnss s 6 1 4 N/m, whch s modld by COMBIN14 ELEMENT n ANSYS. Th modl sktch of rplacabl bar-dampr s shown n Fgur 3. Th dgr of lmnt COMBIN14 s accordng to th amount of damprs. Evry shll bar s sparatd nto thr fnt lmnt parts and vry dampr s on FEM lmnt. Th larg dsplacmnt gomtrc non-lnarty analyss mthod s adaptd n calculaton. Th rplacabl VE damprs ar modld as a lnar Klvn-Vogt lmnt,.., lnar stffnss and vscous dampng F = k u + cu& (3.1) whr c s th dampng coffcnt, k th stffnss of dampr, u th rlatv dformaton of dampr, and u& th rlatv vlocty of dampr. Th matral usd n dampr s pzolctrc and stl, whch dtal charactrstc of dampr s prsntd n Natonal Scnc Foundaton rport (Yang Y. 29). Th prlmnary stp prncpal and tst of dampr s fnshd. Ths knd of dampr can provd th VE charactrstc for th adustabl fatur of pzolctrc. Th paramtrs of damprs s: k s qual to 6 1 4 N/m, and c s qual to 1 1 5 Ns/m. Th lasto-plastc charactrstc of stl usd n shll s : th modulus of lastcty s 2 1 11 N/m 2 and th yld strngthσ of stl s 2.35 1 8 N/m 2. Th bucklng cod of stl fts th Mss Isotropc Hardnng Cod. Th quaton of moton of th orgnal rtculatd shll s gvn by th followng scond-ordr dffrntal quaton [ M ]{} u& + [ C]{ u& } + [ K ]{} u = P & (3.2) whr [ M ] s th mass matrx, [ K ] s th stffnss matrx, [ C] α [ M ] + β[ K ] matrx, {} u th nodal dsplacmnt vctor, P th nput load vctor and = s th dampng α 2ξω ω 1 2 =, ω1 + ω2 2ξ β =. ω + ω Th Raylgh Dampng s calculatd by th frst two frquncs of th shll. Th dampng rato ξ s.1. α and β ar usd n ANSYS to modl Raylgh Dampng. For th rtculatd shll wth rplacabl damprs, th quaton of moton s gvn by [ ]{} u& + ([ C ] + [ ΔC] ){} u& + [ K ] + [ ΔK ] ( ){} u P M & = (3.3) 1 2
whr [ K ] s th stffnss matrx of shll not nclud damprs, [ ΔK ] th stffnss matrx assocatd wth th VE damprs, [ C ] th dampng matrx of shll not nclud damprs and [ ] = α [ M ] + [ K ], [ C] C 1 β1 Δ th dampng matrx assocatd wth th VE bar-typ dampr, whch s consdrd n COMBIN14 lmnt, α 1 and β 1 ar accordng to th frst two frquncs of th controlld shll. Th dampng matrx of structur wth VE dampr s consdrd to satsfy th orthogonal mods. Th quvalnt dampng rato of vbraton mod of controlld shll s shown as Ed ξ = 4πE (3.4) whr E d s th nrgy dsspaton of vbraton mod of controlld shll, whch s rlatd to th matral charactrstc of VE dampr and th vbraton mod of shll. E s th maxmum structural stran nrgy of vbraton mod, whch s rlatd to vbraton mod of shll and th quvalnt stffnss of shll. As a rsult of th dcras of dampr stffnss compard to orgnal shll bar, th dampr bar has bggr dformaton than shll bar. Th nrgy dsspaton of dampr s ncrasd wth th ncrasd dformaton. That wll rduc th dynamc rsponss of shll. 3.3 Earthquak Rcords.1.1.1 ( gx ms-2 ).5. -.5 ( gy ms-2 ).5. -.5 ( gz ms-2 ).5. -.5 -.1 2 4 6 8 1 12 14 16 18 2 -.1 2 4 6 8 1 12 14 16 18 2 -.1 2 4 6 8 1 12 14 16 18 2 (a) El Cntro arthquak ( gx ms-2 ).1.5. -.5 -.1 2 4 6 8 1 12 14 16 18 2 ( gy ms-2 ).1.5. -.5 -.1 2 4 6 8 1 12 14 16 18 2 ( gz ms-2 ).1.5. -.5 -.1 2 4 6 8 1 12 14 16 18 2 (b) Taft arthquak ( gx ms-2 ).1.5. -.5 -.1 2 4 6 8 1 12 14 16 18 2 ( gy ms-2 ).1.5. -.5 -.1 2 4 6 8 1 12 14 16 18 2 ( gz ms-2 ).1.5. -.5 -.1 2 4 6 8 1 12 14 16 18 2 (c) Northrdg arthquak Fgur 2. Ground acclraton rcord of arthquak actons
Thr nput arthquaks ar consdrd: El Cntro (N-S, 194), Taft (N-S, 1952) and Northrdg (N-S, 1995) as shown n Fgur 2 whr th pak valus of arthquak acclraton ar normalzd to.1m/s 2. Th normalzd valu.1 m/s 2 wll b tmd rlatv valu n analyss, for xampl, for frqunt arthquak analyss, th acclraton tm-hstory should b tmd 14; for svr arthquak analyss, t should b tmd 4. 4. OPTIMIZATION DESIGN 4.1 Controlld Shll Modl wthout Optmzaton To accss th ffct of dffrnt dampr topologs, fourtn dffrnt confguratons hav bn consdrd bfor, as Fgur 3. Topology 1 (36) Topology 2 (216) Topology 3 (144) Topology 4 (72) Topology 5 (6) Topology 6 (12) Topology 7 (96) Topology 8 (48) Topology 9 (192) Topology1 (12) Topology 11 (144) Topology 12 (12) Topology 13 (96) Topology 14 (112) Fgur 3. Topologs for bar-typ VE damprs (Th numbr n brackts s th amount of bar-typ damprs) Topology 1: All th dagonal and radal lmnts of th lowr and mddl layr ar rplacd wth bar-typ VE damprs. Topology 2, 3: All th lmnts of th lowr and mddl layr ar rplacd wth bar-typ VE damprs. Topology 4, 7, 8, 1, 13: All th prmtr lmnts ar rplacd wth bar-typ VE damprs. Topology 5: Th dscontnuous prmtr lmnts ar rplacd wth bar-typ VE damprs. Topology 6: Th radal lmnts of lowr and mddl layr and th topology 6 as wll as som has addtonal loop lmnts ar rplacd wth bar-typ VE damprs. Topology 6, 8, 12, 14: Th radal lmnts of lowr and mddl layrs ar rplacd wth bar-typ
VE damprs. Topology 11: Th dagonal lmnts of mddl layr ar rplacd wth bar-typ VE damprs. 4.2 Effct of Controlld Shlls wthout Optmzaton Control Effct of U x (%) 3 2 1-1 -2 El Cntro Northrdg Taft 1 2 3 4 5 6 7 8 9 1 11 12 13 14 Control Effct of U y (%) 3 25 2 15 1 5-5 -1-15 El Cntro Northrdg Taft 1 2 3 4 5 6 7 8 9 1 11 12 13 14 Topology modl Topology modl (a) Th control ffct of Ux (b) Th control ffct of Uy Control Effct of U z (%) 6 5 4 3 2 1 El Cntro Northrdg Taft 1 2 3 4 5 6 7 8 9 1 11 12 13 14 Topology modl Control Effct (%) 6 5 4 3 2 1 El Cntro Northrdg Taft 1 2 3 4 5 6 7 8 9 1 11 12 13 14 Topology modl (c) Th control ffct of Uz (d) Th control ffct of axal forc To assss th ffcts of th varous topologs shown n Fgur 3, th numrcal modl was subctd to th thr-dmnsonal arthquaks shown n Fgur 2, wth ach of th rcords normalzd to hav a maxmum acclraton of 4m/s 2. Th valus of th dampr paramtrs ar prsntd n scton 3. Th control ffcts of th maxmum dsplacmnt (UX, UY and UZ) ar prsntd n Fgur 4. From Fgur 4, obsrv that th rsults of nodal dsplacmnt rsponss ar sgnfcantly dffrnc n x-, y- and z- drcton. Most rsponss rducton n z-drcton rachs to 5%-5%. But th rsponss rducton s lss than 1% or vn lss n x- and y- drcton. Espcally for th dsplacmnt n x- and y- drcton subctd to Taft wav, th rsponss rducton s only 5% or lss than that. Th control ffct of axal forc s good xcpt for Taft wav. 4.3 Snstvty Analyss Fgur 4. Control ffct of th maxmum dsplacmnt of 14 knds of topologs (Control ffct = ((rspons of uncontrolld shll-controlld shll)/rspons of uncontrolld shll) 1%) In th shll modl, as mntond last part, to achv th dampr locaton optmzaton, dffrnt program such as structural analyss, automatc msh gnraton, snstvty analyss and mathmatcal programmng, ar ntr-rlatd. Program moduls ar dvlopd and communcatd by usng Matlab languag. Th snstvty curv and ts ascndng ordr arrangmnt curv of top 2 vbraton mods ar shown n Fgur 5.
Snstvty 3 25 2 15 1 5 2 4 6 8 1 lmnt ordr (a) Snstvty curv of lmnts Snstvty 3 25 2 15 1 5 2 4 6 8 1 ascndng ordr (b) Snstvty curv n ascndng ordr Fgur 5. Snstvty gnratd by scton ara prturbaton of top 2 vbraton mods 4.4 Control Effct of Topology Optmzaton Basd on th snstvty rsults, th optmal topology s slctd as Fgur 6. Th topology s th dagonal lmnts of mddl layr ar rplacd wth bar-typ VE damprs. Th numrcal modl of topology optmzaton s analyzd subctd to th thr-dmnsonal arthquaks. Th rsults ar shown Topology 12 (24) n Tabl 4.1. From Tabl 4.1, obsrv that th rsults of nodal dsplacmnt rsponss ar good n x-, y- and z- drcton. Rsponss rducton n z-drcton Fgur 6. Th optmal topology for bar-typ rachs to 17%-1%. Th rsponss rducton of El VE damprs(dgr of COMBIN14) Cntro and Northrdg wav s ovr 1% n x- and y- drcton and th rspons rducton of Taft wav rachs 4%-7%. Th ffcts ar bttr than th modl wthout optmzaton. Tabl 4.1 Th control ffct of optmal topology shll arthquak Ux(m) % Uy(m) % Uz(m) % F(1 5 N) % El Cntro uncontrol.16 --.1 --.34 -- 1.8219 -- control.13 18.75.9 1.23 32.35 1.548 42 Northrdg uncontrol.22 --.15 --.46 -- 3.13 -- control.2 9.9.12 2.38 17.39 1.7218 45 Taft uncontrol.22 --.28 --.62 -- 3.1887 -- control.21 4.55.26 7.14.45 27.42 2.6766 16 (Control ffct = ((rspons of uncontrolld shll-controlld shll)/rspons of uncontrolld shll) 1%) 5. CONCLUSIONS Rplacng th lmnts of shll wth bar-typ damprs s an attractv control mthod that offrs th rlabl control but nd not vary th grd form of shll. To tak full advantag of th optmal topology, th snstvty mthod s usd for dsgn and analyss. Kwtt-8 typ rtculatd shll modl has bn usd for control analyss. Manwhl, 14 knds of damprs topologs for ths rtculatd shll hav bn prsntd. Subsquntly, th ffct of damprs topologs s analyzd subctd to thr commonly usd 3D arthquak wavs. Th control ffct of 8 knds of damprs topologs s dffrnt and not good for vry cas. Aftr that, th snstvty analyss s usd to gt th optmal topology. Th snstvty ordrs ar calculatd basd on top 2 vbraton mod. Th optmal placmnt of damprs s slctd. Th numrcal modl of optmal topology s analyzd. Th control ffct s bttr than th othr topologs for vry cas. Rsponss rducton n z-drcton rachs to 17%-1%. Th
rsponss rducton of El Cntro and Northrdg wav s ovr 1% n x- and y- drcton and th rspons rducton of Taft wav rachs 4%-7%. Th ffcts ar bttr than th modl wthout optmzaton. AKCNOWLEDGEMENT Ths rsarch s supportd by Natonal Scnc Foundaton of Chna Nos. 59836. REFERENCES Agrawal A K and Yang J N. (1999). Optmal Placmnt of Passv Damprs on Ssmc and Wnd-xctd Buldngs usng Combnatoral Optmzaton. Journal of Intllgnt Matral Systms and Structurs 1:12, 997-114. Cao Z and Zhang Y G. (2). A Study on th Ssmc Rspons of Lattcs Shlls. Intrnatonal Journal of Spac Structurs 15:3&4, 243-247. Chn Shuhuan. (1991). Vbraton Thory of Structurs wth Random Paramtrs. Jln Scnc and Tchnology Prss. (n Chns) Fan F, Shn S Z and Park G A R. (25). Study of th Dynamc Strngth of Rtculatd Doms undr Svr Earthquak Loadng. Intrnatonal Journal of Spac Structurs 2:4, 235-244. Habb Uysal, Rustm Gul and Umt Uzman. (27). Optmal Shap Dsgn of Shll Structurs. Engnrng Structurs Vol: 29, 8-87. Izuru Takwak. (1997). Optmal Dampr Placmnt for Mnmum Transfr Functons. Earthquak Engnrng and Structural Dynamcs Vol:26, 1113-1124. Kasa K, Motoyu S and Ook Y. (21). Vscolastc Dampr Modlng and Its Applcaton to Dynamc Analyss of Vsco-Elastcally Dampd Spac Frams. IASS Intrnatonal Symposum on Thory, dsgn and ralzaton of shll and spatal structurs, Nagoya, Japan, 266-267. Kyung K. Cho and Nam-Ho Km. (25). Structural Snstvty Analyss and Optmzaton I & II. Sprngr Publshr, London. Lang H T, Wu J Z and Zhang Y G. (23). Shakng Tabl xprmntal rsarch on passv control of doubl-layr rtculatd shll wth som bottom chords rplacd by damprs. Earthquak Engnrng and Engnrng Vbraton 23:4, 178-182. (n Chns) N L. (21). Thorcal Study on Sm-actv Control of Doubl-layr Cylndrcal Lattc Shll. Thss of Bng Unvrsty of Tchnology. (n Chns) Oh H U and Onoda J. (22). An Exprmntal Study of a Sm-actv MR Flud Varabl Dampr for Vbraton Supprsson of Truss Structurs. Smart Matrals and Structurs 11:1, 156-162. Onoda J, Oh H U and Mnsug K. (1996). Sm-actv Vbraton Supprsson of Truss Structurs by ER Flud Dampr. Structural Dynamcs and Matrals Confrnc Vol: 3, 1569-1577. Shn S. Z and Lan T T. (21). A Rvw of th Dvlopmnt of Spatal Structurs n Chna. Intrnatonal Journal of Spac Structurs 16:3, 157-171. Shngn K and Nk T. (21). A Study on Bas Isolatd Shll. IASS Intrnatonal Symposum on Thory, dsgn and ralzaton of shll and spatal structurs, Nagoya, Japan, 262-263. Sngh Mahndra P and Morsch Lus M. (22). Optmal Placmnt of Damprs for Passv Rspons Control. Earthquak Engnrng and Structural Dynamcs Vol: 31, 955-976. Yang Yang, B. F. Spncr, L Youmng and Shn Shzhao. (211). Ssmc Prformanc of Doubl-layr Sphrcal Rtculatd Shll wth Rplacabl Bar-typ Damprs. Intrnatonal Journal of Spac Structurs 26:1, 31-44.