System-reliability-based design and topology optimization of structures under constraints on first-passage probability

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1 Srucural Safy 76 (09) 8 94 Cos lss avalabl a SccDrc Srucural Safy o u r a l h o m p a g : w w w. l s v r. c o m / l o c a / s r u s a f Sysm-rlably-basd dsg ad opology opmzao of srucurs udr cosras o frs-passag probably Juho Chu a, Juho Sog b,, Glauco H. Paulo c a School of Archcur, Syracus Uvrsy, Syracus, NY, Ud Sas b Dparm of Cvl ad Evromal Egrg, Soul Naoal Uvrsy, Soul, Rpublc of Kora c School of Cvl ad Evromal Egrg, Gorga Isu of chology, Alaa, GA, Ud Sas A R I C L E I N F O Kywords: Rlably-basd dsg opmzao Rlably-basd opology opmzao Sochasc xcao Squal compoudg mhod Paramr ssvy Sysm rlably Frs-passag probably A B S R A C For h purpos of rlably assssm of a srucur subc o sochasc xcaos, h probably of h occurrc of a las o falur v ovr a m rval,.. h frs-passag probably, of ds o b valuad. I hs papr, a w mhod s proposd o corpora cosras o h frs-passag probably o rlably-basd opmzao of srucural dsg or opology. For ffc valuaos of frs-passag probably durg h opmzao, h falur v s dscrbd as a srs sysm v cossg of saaous falur vs d fd a dscr m pos. h probably of h srs sysm v s h compud by us of a sysm rlably aalyss mhod rmd as h squal compoudg mhod. h ado ssvy formulao s drvd for calculag h paramr ssvy of h frs-passag probably o facla h us of ffc grad-basd opmzao algorhms. h proposd mhod s succssfully dmosrad by umrcal xampls of a spac russ ad buldg srucurs subcd o sochasc arhqua groud moos.. Iroduco Fdg h opmal dsg of a srucural sysm wh rgard o safy, cos or prformac s o of h mos ssal ass srucural grg pracc. h opmal dsg should achv maor dsg obcvs rprsg rlabl oprao ad safy v udr sochasc xcaos causd by aural hazards such as arhquas ad wd loads. Du o hr radomss aural dsasrs, howvr, sgfca ucras may xs h sy ad char- acrscs of h xcaos. hrfor, h prformac of such srucural sysms ds o b assssd probablscally durg h opmzao procss. o dal wh ucras ffcvly srucural dsg/opology opmzao, varous opmzao algorhms ad framwors wr dvlopd rcly. For sac, h so-calld robus dsg/opology opmzao algorhms [ 3 ] am o rduc h ssvy of h opmal prformac of a srucur wh rspc o h radomss of rs. By coras, Rlably-basd dsg/opology opmzao [4 0] ams o fd opmal soluos sasfyg h probablsc co- sras o h srucural prformac dcaors. So far, hs suds hav b maly focusg o accoug for ucras sac loads rprsg ypcal load pars of h srucur. Rc suds o srucural opmzao cosdrg dyamc xcaos mployd a small umbr of drmsc m hsors rprsg possbl fuur ralzaos [, ], or focusd o paral dscrpors of h dyamc rsposs such as mod frqucs [3]. hs approachs hav rsc lmaos bcaus () a sgl or small umbr of sampl m hsors may o rprs all possbl ralzaos of sochasc xcaos, ad () s praccally mpossbl o assss h probabls ha h srucural dsg dos o sasfy h cosras o prformacs,.. falur probabls usg hs approach. hrfor, h probablsc prdco of srucural rsposs basd o radom vbrao aalyss s dd h procss for opmal dsg. o ovrcom hs chcal challg, h auhors rcly proposd a w mhod for opology opmzao of srucurs udr sochasc xcaos [4]. I h proposd mhod, a ff c radom vbrao aalyss mhod basd o h us of h dscr rprsao mhod [5] ad srucural rlably hors (s [6] for a rvw) wr grad wh a sa-of-h-ar opology opmzao framwor. h auhors also dvlopd a sysm rlably-basd opology opmzao framwor udr sochasc xcaos [7] o cop wh sysm falur vs cossg of sascal dpd compo vs usg h marx-basd sysm rlably mhod [8]. h dvlopd mhod hlps sasfy probablsc cosras o a sysm Corrspodg auhor. E-mal addrss: uhosog@su.ac.r (J. Sog). hps://do.org/0.06/.srusaf Rcvd 5 Spmbr 07; Accpd 7 Ju / 08 Elsvr Ld. All rghs rsrvd.

2 J. Chu al. Srucural Safy 76 (09) 8 94 falur v, whch cosss of mulpl lm-sas dfd rms of dffr locaos, falur mods or m pos as opmzs a srucural sysm. I hs suds by h auhors, h saaous falur prob- abls of h srucur wr valuad a dscr m pos. Howvr, o promo applcaos of dsg/opology opmzao o grg dsg pracc, h frs-passag probably,.. h probably of a las o occurrc of h falur ovr a m rval, ds o b smad durg h opmzao procss. Spc al. [9] proposd a framwor for RBDO of lar sysms cosrad o h frs-passag probably. hs approach dcoupls h sd r- lably aalyss loop from h opmzao loop by solvg sub-opmzao problm formulad from smulao rsuls. Bobby al. [0] prsd a smulao-basd framwor for opology opmzao of wd-xcd buldg srucurs wh h cosdrao of h frs- passag probably. h frs-passag probably hlps promo h us of h proposd sochasc opmzao framwor for h dsg of h laral loadrssg sysm or szg srucural lms udr sochasc xcaos wh a f durao such as arhqua xcaos. o hs d, hs papr roducs a sochasc dsg ad opology opmzao mhod ha ca hadl probablsc cosras o h frs-passag probably, ad dmosras h mhod usg umrcal xampls.. Radom vbrao aalyss usg dscr rprsao mhod I h aformod rlably-basd dsg opmzao framwor udr sochasc xcaos [ 4, 7], h auhors proposd o prform radom vbrao aalyss by us of h dscr rprsao mhod [5] ordr o compu h saaous falur probably of h sochasc rspos a dscr m pos. I h proposd approach, for xampl, a zro-ma saoary Gaussa pu xcao procss f( ) s dscrzd as f ( ) v s ( ) s( ) v whr s ( ) ( [ s ( ),, s ( )] ) s a vcor of drmsc fucos ha dscrb h spcral characrscs of h procss, ad v [v, v,, v ] s a vcor of ucorrlad sadard ormal radom varabls. Amog xsg mhods avalabl o dvlop a dscr rprsao modl Eq. (), a popular o for groud xcao modlg s usg a flr rprsg h characrsc of h sol mdum ad a radom puls ra. For xampl, f a flrd wh os s usd, h modl Eq. () s cosrucd as 0 f ( ) h ( τ) W ( τ) dτ s( ) v f π Φ 0/Δ h ( τ ) dτ < <,,, s ( ) f 0 whch W( τ) dos h wh os procss whos powr spcral dsy fuco s Φ WW (ω) Φ 0, h f ( ) s h mpuls rspos fuco of h flr, Δ, ad dos h umbr of h m - rvals roducd for h gv m prod (0, ). h dals of h drvao of Eq. () ar avalabl Chu al. [4]... Rspos of lar sysm udr sochasc xcaos h rsposs of lar sysms o sochasc xcao ca b drmd by h covoluo gral cossg of hr mpuls rspos fuco ad h dscrzd pu procss Eq. (). ha s, a rspos m hsory u( ) of h lar sysm subcd o h so- chasc xcao f( ) s drvd as () () s s u ( ) f ( τ) h ( τ) dτ v s ( τ) h ( τ ) dτ v a ( ) a( ) v 0 0 whr h s ( ) s h mpuls rspos fuco of h lar srucural sysm, ad a( ) dos a vcor of drmsc bass fucos a ( ) s( τ) hs( τ) dτ,,, 0 (4) Drvg h mpuls rspos fuco a f lm sg ca b compuaoally challgg or cumbrsom. o facla h procss, h auhors proposd ovl umrcal procdurs Chu al. [4]... Isaaous falur probably of lar sysm udr sochasc xcaos I srucural rlably aalyss, h probably ha h oucom of a radom vcor X s locad sd h falur doma Ω f,.. h falur probably, s compud by a gral Pf fx ( x ) dx Ω f (5) whr f X ( x) s h o probably dsy fuco (PDF) of h radom vcor X. h falur doma s dfd by h ara whr h lm-sa fuco g( x),.g. capacy mus dmad, as h gav sg. I gral, compug h mul-fold gral Eq. (5) s orval or compuaoally challgg. Srucural rlably mhods such as FORM ad SORM (s [6] for a rvw) rasform h spac of h radom varabl x o h ucorrlad sadard ormal spac v. h, h lm-sa fuco s approxmad by a lar (FORM) or quadrac fuco (SORM) a h dsg po, of alravly rmd as h mos probabl falur po (MPP). For xampl, FORM, h falur probably s approxmad as Pf Φ[ β] (6) whr β s h rlably dx,.. h shors dsac from h org of h sadard ormal spac o h larzd falur surfac, ad Φ[ ] dos h cumulav dsrbuo fuco (CDF) of h sadard ormal dsrbuo. Usg h dscr rprsao mhod dscrbd abov, lm-sa fucos dfd for dsplacm or ohr srucural rsposs ca b dscrbd h spac of sadard ormal radom varabl v. For xampl, h saaous falur v E f dfd for a lar srucur subcd o h Gaussa pu procss Eq. () s gv by E (, u, v) { g(, u, v ) 0}, whr g(, u, v) u u ( ) f u a( ) v whr u 0 s h prscrbd hrshold o h dsplacm rspos. I hs cas, h rlably dx β s compud from h gomrc - rprao of h lm-sa surfac as a closd form xprsso [5] u0 β(, u 0) a( ) I s od ha h lm-sa fuco Eq. (7) s lar hs cas, ad hus h falur probably by Eq. (6),.. P f Φ[ β(,u 0 )] dos o roduc rrors causd by fuco approxmao or rqur olar opmzao o fd h dsg po. If h srucur bhavs olarly or h pu procss s o-gaussa, o ds o us rlably mhods such as FORM or SORM o compu h falur probably approxmaly. Usg hs dscr rprsao mhod, o ca rduc h compuaoal cos of h radom vbrao aalyss, whch should b rpvly prformd durg h opmzao procsss o compu h saaous falur probably a ach updad s of dsg varabls. 0 (3) (7) (8) 8

3 J. Chu al. Srucural Safy 76 (09) Frs-passag probably of lar sysm udr sochasc xcaos h frs-passag probably s commoly ulzd o fd h probably of h falur v dscrbd wh a m rval [ 3]. O of h avalabl approachs for formulag h frs- passag probably P fp dfs h problm as a srs sysm problm,.. < > fp ( 0 max ( )) { ( ) 0} 0< < P P u u P u u (9) Usg h dscr rprsao, h frs-passag probably of a sysm wh c lm-sa fucos (df d for dffr falur mods or locaos) s dscrbd as c c sys f (,, ) 0 P fp P E P E u v E sys (0) whr dos h frs-passag falur v rgardg h -h cosra, E ( ) dos h saaous falur v of h -h lm- sa fuco a m, ad s h oal umbr of dscrzd m pos. o compu h f rs-passag probably Eq. (0), s r- qurd o valua h falur probabls of h compo vs a ach m po wh a rval. Morovr, a ffc, rlabl ad robus algorhm s rqurd o valua h sysm falur probably wh a propr cosdrao of sascal dpdcy bw h compo vs. I s also dsrabl o compu h paramr ssvy of h srs sysm falur probabls Eqs. (9) ad (0) o abl h us of ffc grad-basd opmzrs. o addrss hs rqurms, h squal compoudg mhod (SCM; [4]) ad h Chu-Sog-Paulo (CSP; [5]) mhod ar adopd hs sudy. 3. Opmzao of srucurs subcd o sochasc xcao udr rs-passag probably cosras 3.. Srucural dsg opmzao Rlably Basd Dsg Opmzao (RBDO) of a srucur ams o achv h opmal dsg udr probablsc cosras o ucra prformac, arsg from ucras maral proprs or loads. h RBDO problm of a srucur udr frs-passag probably co- sras ca b formulad as a Sysm Rlably Basd Dsg Opmzao (SRBDO) problm [9],.. m f ( d) d ob f d s. P ( E sys) P E (, ) arg P { g (, d) 0} P f,,, c uppr dlowr d d wh M( d) u (, d ) + C( d) u (, d ) + K( d) u(, d ) f(, d) () whr f ob ( d) dos h obcv fuco of h dsg, d lowr ad d uppr ar h lowr ad uppr bouds of h vcor of dsg varabls d, rspcvly. g ( ) rprss h lm-sa fuco whos gav sg dcas h volao of a gv cosra, c s h umbr of h cosras, P( g ( ) 0) s h probably of h falur v, ad P arg f s h arg falur probably. M, C, ad K rprs h global mass, dampg ad sffss ma- rcs of h srucur, rspcvly, ad ü, u, u, ad f ar h accl- rao, vlocy, dsplacm ad forc vcors a m, rspcvly. A proporoal dampg modl ow as Raylgh dampg [6] s usd hroughou hs papr. I hs approach, h dampg marx s drmd as a lar combao of h sffss ad mass marx, ha s C κ 0 M + κ K. h coffcs κ 0 ad κ h Raylgh dampg modl ar drmd so as o hav cra modal dampg facors. For arhqua groud xcaos, h forc vcor Eq. () s drmd by a vcor of ffcv arhqua forcs,.. f(, d) M( d) l u g ( ) M( d) lf ( ) () whr l rprss h drcoal dsrbuo of mass wh uy r- sulg from a u groud dsplacm ad ü g s h groud acclrao m hsory. 3.. Srucural opology opmzao opology opmzao (s [7] for a rvw) ams o fd h opmal maral dsrbuos a dsg doma subcd o racos ad dsplacm boudary codos whl sasfyg gv dsg cosras. hus, vry po of a dsg doma s xpcd o rprs hr a xsc of maral or a vod rgo. h Sold Isoropc Maral wh Palzao (SIMP; [8] ) modl, whch s adopd hs sudy, cosdrs a couous maral dsy a dsg varabl usg h powr fuco rprsao,.. ψ ( x) x p (3) whr p s h palzao facor ad x s a dsy assocad wh lm h f lm mhod sg. h opology opm- zao soluos usg h SIMP, or rlad modls, may suffr from chcrboard pars ad msh-dpdcy [9]. o ovrcom hs problms, varous mhods hav b proposd (.g. [ 30 33] ). I hs sudy, a proco mhod [30] s mplmd o oba a flrd dsy basd o lm dsg varabls wh h ghborhood such as: w ( r ) d rm r N r r ρ w r w ( r ), ( ) f r m m 0 ohrws N (4) whr d dos dsg varabl of lm, N rprss h s of lms wh h radus r m of lm, w( r ) s h wghg fuco, ad r s h dsac bw h crods of lm ad. Usg h SIMP modl, h sffss ad mass marx of lm ad hr drvavs wh rspc o a lm dsy ar obad as follows h lm-basd compuaoal framwor [7]: p K ρ ρ K M ρ q ( ), ( ) ρ M 0 0 K ρ ( ) 0 ( ) 0 ρ p M ρ q pρ K, qρ M ρ (5) whr q s h palzao paramr, ad K 0 ad M 0 ar compud by 0 K B D BΩ, M N ρm NdΩ Ω Ω (6) 0 0 whr B dos h sra dsplacm marx of h shap fuco drvavs h doma Ω of lm, m rprss h mass dsy of h maral ad N s h shap fuco of lm. D 0 s h lascy sor of h sold maral, whr h dsy s. opology opmzao of a srucur udr sochasc xcao wh cosras o h frs-passag probably ca b formulad as m f ( ρ ~ ) d ob P E f ρ s. P ( E ) (, ~ sys ) P { g (, ρ ~ arg ) 0} Pf,,, c 0 < ε ρ Ω wh M( ρ ~ ) u (, ρ ~ ) + C( ρ ~ ) u (, ρ ~ ) + K ( ρ ~ ) u (, ρ ~ ) f (, ~ ρ) (7) whr d dos h vcor of dsg varabls, Ω s a s of f lm dcs ad ρ ~ s h vcor of flrd dss dfd as: ρ ~ Pd (8) whr P rprss h flrg marx whos lm s drmd by 83

4 J. Chu al. Srucural Safy 76 (09) 8 94 ( P) l wl w w r w ( r ), ( ) f Nl l 0 ohrws Nl (9) A flowchar for opology opmzao of a srucur cosrad by frs-passag probably s provdd Appdx A. Varous grg cosras ca b corporad o h abov formulaos of rlably-basd dsg opmzao ad opology opmzao udr frs-passag probably. o promo applcaos of h proposd mhod o russ ad buldg srucurs, grg cosras o srss h bar ad r-sory drf rao ar drvd blow Frs-passag probably cosras o srss bar lms Cosdr a bar a russ srucur wh h local od umbrs ad dog h d pos of h bar as show Fg.. A u vcor pog from od o od s dfd as cosθ sθ (0) h global dsplacm vcors of h d ods of bar ar wr as u u u g g g u, x, whr u u,, y u u u g, x, y () h srss ( ) a russ lm udr sochasc xcaos ca b compud from h srss sra rlaoshp basd o Hoo s law as follows: D D D σ (, d ) L u g d u g d L B u ( (, ) (, )) ( L u l (, d ) u l (, d )) () whr D dos Youg s modulus, L s h lgh of h lm, u l ( ) ad u l ( ) ar d dsplacms alog h russ axs, ad B [ ] (3) h logao Eq. () ca b dscrbd by usg h dscr rprsao form Eq. (3),.. D σ (, d) ( a(, d) v a (, d) v) L Fg.. Bar gomry. (4) h saaous falur probably a m s xprssd rms of srss h russ lm as P ( E (, d)) P ( g (, d ) 0) P ( σ σ (, d ) 0) Φ[ β (, d)] f 0 σ (5) whr 0 dos h hrshold valu of srss. From h gomrc rprsao assocad wh h falur v of lm, h r- lably dx a m s compud as L σ0 L σ0 β σ (, d) D a(, d) a (, d) D b (, d) (6) h frs-passag probably of h srss lm sa fuco s h compud as P fp _ σ ( Esys ) P Efσ (, d) P { σ0 σ (, d) 0} Φ [β (, d), β (, d ),,β (, d); ρ, ρ,, ρ ] σ σ σ,,3, Φ [ β, R] σ (7) whr Φ dos h mulvara ormal CDF,, rprss h corrlao coffc bw h ormal radom varabls rprsg falur v ad, ad ad R ar h vcors of h rlably - dcs ad h corrlao coffc marx, rspcvly. h corrla- o coffc marx R s cosrucd as ρ, ρ, R, ρ, l α( ) α( l) ρ ρ,, (8) whr ( ) a( )/ a( ) dos h gav ormalzd grad vcor of h lm-sa fuco valuad a h dsg po whch s obad by u 0 a( )/ a( ). h mulvara CDF Eq. (7) ad hos h followg Scos 3.4 ad 3.5 ar compud by SCM [4]. h CSP mhod [5] whos dals ar prsd Sco 5 s usd o compu h ssvy of h mulvara CDF Frs-passag probably cosra o r-sory drf rao h frs-passag probably ca b compud rms of h r- sory drf rao, whch s o of h sgfca dsg crra srucural grg, dfd as a(, d) v for Δ (, d) H H v ( a(, d ) a(, d) ) for 3, 4,, s H (9) whr Δ dos h sory drf a floor lvl, H rprss h sory hgh blow lvl, ad s s h umbr of sory lvls. h saaous falur probably rms of h r sory-drf raos s P ( E (, d)) P ( g (, d) 0) P u Δ fδ 0Δ Φ[ β (, d)],,, Δ Δ (, d) 0 H s (30) whr u 0Δ dos a hrshold valu of h r-sory drf rao, ad β Δ rprss h rlably dx whch ca b compud as Hu0Δ a(, d) β Δ (, d) H u0δ a(, d) a(, d) for for 3, 4,, s Fally, h frs-passag probably s Δ (, d) sys f d Δ 0Δ Δ H P fp _ ( E ) P E (, ) P u 0 Φ [β (, d), β (, d ),,β (, d); ρ, ρ,, ρ ] Δ Δ Δ,,3, Φ [ β, R] Δ { } 4. Calculag ssvy of rs-passag probably (3) (3) o us ffc grad-basd opmzao algorhms for RBDO, s ssal o calcula h ssvy of h falur probably wh rspc o varous dsg paramrs. I hs papr, a ssvy formulao mployg h ado mhod [34] s drvd for h frs- passag probably of a lar srucur basd o h dscr 84

5 J. Chu al. Srucural Safy 76 (09) 8 94 abl opology opmzao problm ( Fg. ): flrg paramrs for groud xcaos ad a hrshold valu of h probablsc cosra. Φ 0 ωf ζ f (s) Δ (s) u 0Δ 00 5 π /400 rprsao mhod. I s od ha h ssvy of h sysm falur probably wh rspc o a paramr θ s obad by a cha rul,.. Pf ( E sys) Pf ( Esys) β (θ) θ β (θ) θ (33) Rcly, Chu al. [5] proposd h CSP mhod o compu h drvavs of h sysm falur probably wh rspc o h rlably dx basd o h us of h SCM. h CSP mhod compus ssvs of paralll ad srs sysms, as wll as gral sysms wh rspc o rlably dcs ffcly ad accuraly. 4.. Ssvy of sysm rlably usg SCM h CSP mhod compus paramrc ssvy of h sysm rlably basd o h SCM mhod. h ma da of h CSP mhod s carryg ou ssvy aalyss afr h sysm falur v s smplfd usg h SCM. hs da s brfly xplad usg a srs sysm xampl formulad as a -fold gral h corrlad sadard ormal spac,.. P ( E srs ) P ( E E E ) P (β Z ) φ ( z; R) d z Ω f (34) Suppos h -h compo s compoudd a h las sp,.. compoudd wh h supr-compo E S, whch dos h uo of all h compo vs xcp h -h o. Ulzg h formula for b-vara ormal CDF [35], h ssvy of h srs sysm falur probably wh rspc o β s obad as [5] P ( Esrs ) Φ [ φ β, β S ; ρ ], S ( β ) β β (35) ρ S, whr s h updad corrlao co ff c bw E ad E S obad durg h squal compoudg [4], ad βs Φ[ P ( E S )] Φ[ P ( E )] p p S (36) whr S dos h dx s of h compos E S. Smlarly, h paramr ssvs of paralll ad cu-s sysms wr drvd by Chu al. [5]. 4.. Ssvy of rs-passag probably RBDO o facla h us of a grad-basd opmzr, h ssvy of h frs-passag probably RBDO s compud usg h cha rul,.. Pfp ( Esys ) ( Φ [ β, R ]) ( Φ [ β, R ]) β ( d) d d β d β ( d) c d (37) whr c ( Φ[, R])/ β ca b compud usg Eq. (35). h paral drvav β / d s obad by β ( d) d a (, d) (, d) d C a cs a (, d) 3/ (38) whr C cs s h coffc drmd dpdg o h cosra usd opmzao,.g. C cs Lσ 0 / D : srss bar Hu0Δ : drf rao H u 0Δ : r-sory drf rao (39) Wh a uform m sp sz s usd,.. Δ,,,, ad 0, Eq. (38) ca b rwr from Eq. (4) as follows (s mor dals of h drvao Appdx of Chu al. [4]): Fg.. Ssvy comparso: (a) gomry, loadg codo, ad locaos whr ssvy s rpord ( abl ), (b) ssvs from h ado mhod (AJM), ad (c) ssvs from h f dffrc mhod (FDM). 85

6 J. Chu al. Srucural Safy 76 (09) 8 94 abl Ssvy comparso of frs-passag probably o a dsplacm cosra opology opmzao. Δd FDM AJM P f /d A P f /d B P f/d C P f/d A P f /d B P f /d C β ( d) d a (, d) (, d) d + C a cs + a (, d) 3/ (40) Furhrmor, h uform sp sz lads o h followgs for h paramr ssvy Eq. (37): Pfp ( Esys ) d whr β ( d) a l + (, d) c C χ a cs l l + (, d) d d l as (, d) κ (, d) s d s 3/ (4) χl c/ a (, d), κ s(, d ) C cs χ s+ as(, d) l + (4) Fg. 3. Compuaoal m comparso for ssvy aalyss by h FDM ad h AJM Ssvy of rs-passag probably by ado varabl mhod h ssvy Eq. (4) cluds h mplcly dfd drvav rm of a s (, d)/ d, s,,. hos mplc drvavs ca b compud usg h drc dffrao mhod (DDM), h f dff rc mhod (FDM) or h ado varabl mhod (AJM) [34]. Fg. 4. A spac russ dom xampl: (a) prspcv vw of h dom, (b) pla vw ad drcos of appld groud acclraos ad (c) lm umbrg chocs. 86

7 J. Chu al. Srucural Safy 76 (09) 8 94 Fg. 5. Gomry of a spac russ dom: (a) basc grd, (b) sco vw alog grd l 5, ad (c) sco vw alog grd l 3 7. abl 3 Paramrs for flrs of groud moo modls ad cosras opmzao (spac russ dom opmzao xampl). Φ 0_g Φ 0_g ω f ζ f (s) Δ (s) Ial cross-sco aras (m ) Chu al. [4] drvd a approach of ssvy calculaos assocad wh a s (, d)/ d usg h ado varabl mhod. h umrcal ss cofrmd supror prformac of AJM compard o DDM ad FDM. Basd o h AJM drvao, h ssvy of h frs- passag probably Eq. (4) s rwr as Pfp ( E sys) A( d) f(, d) λ + u(, d) η (Δ ) d d d f(, d ) f(, d) (0.5 + γ η )(Δ ) (0.5 γ + η)(δ ) d d + B ( d + E ( d) ) u(, d) u(, d) d d + u (0, d) u(, d) λ B d + E d ( ) ( ) d d + u (0, d) λ E( d) d hrshold valu π u 0Δ x /800 u 0Δ y /800 (43) whr λ + dos h ado varabl vcor. A( d), B( d), ad E( d) rprs followgs rspcvly: A( d ) M( d ) + γδ C( d ) + η (Δ ) K( d) B ( d ) M ( d ) + ( γ )Δ C ( d ) + (0.5 + γ η )(Δ ) K ( d) E ( d ) M ( d ) + ( γ )Δ C ( d ) + (0.5 γ + η )(Δ ) K ( d) (44) 4.4. Ssvy aalyss of rs-passag probably RBO Ssvy aalyss of frs-passag probably sochasc opology opmzao s smlar o h drvao for RBDO dscrbd abov. h ma dffrc of ssvy aalyss opology opmzao coms from h proco mhod o oba h flrd dsy as show blow. P E β R β R ρ ~ fp ( sys) ( Φ [, ]) ( Φ [, ]) β ( ) d d β d ( Φ [ β, R]) β ( ρ ~ ) ρ l β l ρl d β R ~ ( Φ [, ]) β ( ρ) ( P) h β row ρ~ ( Φ [ β, R]) β ( ρ ~ ) ( P) h row ρ β ~ (45) Fg. 6. Opmzd spac russ dom corrspodg o d ffr agls of groud acclraos: (a) θ g 0, θ g 30, (b) θ g 0, θ g 60, (c) θ g 0, θ g 90 (Color lgds: A A m gr, 0.0 m < A 0. m blu, 0. m < A 0.4 m brow, 0.4 m < A rd). (For rprao of h rfrcs o color hs fgur lgd, h radr s rfrrd o h wb vrso of hs arcl.) 87

8 J. Chu al. Srucural Safy 76 (09) 8 94 Fg. 7. Opmzd cross-scoal aras of russ lms corrspodg o h groud acclraos appld a d ffr agls. Fg. 8. Covrgc hsory: (a) volum ad (b) frs-passag probably. Comparso bw dyamc rsposs by h al srucur ad h opmzd srucurs: (c) radomly grad groud acclraos (θ g 0, θ g 30 ), (d) drf rao h x-drco ad () drf rao h y-drco. 88

9 J. Chu al. Srucural Safy 76 (09) 8 94 Fg. 9. (a) Dsg doma ad loadg codo, (b) od of rs for a p drf rao cosra, ad (c) ods of rs for r-sory drf raos. abl 4 Paramrs for groud moo flr modl ad cosras of opology op- mzao (opology opmzao xampl). whr P dos h raspos of h f lrg marx Eq. (9) ad ( P) h row s h -h row vcor of P. hus, P Φ 0 ω f ζ f (s) Δ (s) I. dsy Colum sz hrs. valu 7.5 5π m 0.6 m u 0Δ /50 ( E β R ρ ~ sys ) ( Φ [, ]) β ( ) P d ρ β ~ (46) fp whr h paral drvav of β ( ) wh rspc o a lm dsy ca b compud as xplad Scos 4. ad Vr cao of calculad ssvy h ado ssvy mhod drvd for h frs-passag prob- ably cosras s sd for h opology opmzao problm Fg. (a) hrough comparso wh h f dffrc mhod o vrfy accuracy ad ffccy. h sochasc ssmc acclrao f( ) s modld as a flrd wh-os procss usg h Kaa-am flr modl wh h sy Φ 0 [6,3]. h u-mpuls rspos fuco of h flr ad h ffcv forc vcor causd by h arhqua xcao ar drmd as follows, rspcvly: ( ζ f ) ωf K hf ( ) xp( ζf ωf ) s( ωf ζ ) f ζ f ζ f ω f cos( ω f ζ ) f K ( f 0 ) f( d, ) M( d) l f ( ) M( d) l h ( τ) W ( τ) dτ (47) (48) abl summarzs h Kaa-am flr paramrs of doma frqucy ω f ad badwdh ζ f, h m rval of rs, ad h hrshold valu of h drf rao a ach m po. h ssvs of h frs-passag probably wh rspc o h dsg varabls locad a h hr pos A, B ad C Fg. (a) ar compud by h proposd ado mhod ad h FDM, rspcvly. h srucural colums rprsd by wo vrcal ls Fg. (a) ar modld by fram lms. Youg s modulus E,000 MPa ad mass dsy ρ m,400 g/m 3 ar usd as maral proprs for boh h quadrlaral ad fram lms. h saaous falur v a a dscrzd m po s cosdrd rms of a avragd drf rao valuad a wo ods of rs. hus, h frs-passag v s dfd as + E E ρ ~ ( a( ρ ~ a ρ ~, ) Lf (, ) Rgh) v sys f (, ) u 0 Δ 0Δ H (49) h ssvs of h frs-passag probably cosra Eq. (49) ar show Fg. (b) ad (c), whch show good agrms. h ssvs by h FDM mployg a rag of prurbaos (from 0 o 0 ) ar abulad abl for comparso wh h rsuls by h AJM, ad h fluc of h prurbao sz o h rsuls by h 89

10 J. Chu al. Srucural Safy 76 (09) 8 94 Fg. 0. opology opmzao rsuls from h four-sory buldg xampl cosrad by h frs-passag probably rm of h p drf rao cosra: (a) β arg.5, P arg f 6.68%, (b) β arg.5, P arg f 0.6%, ad (c) β arg 3.0, P arg f 0.3%. Fg.. opology opmzao rsuls from h four-sory buldg xampl cosrad by h frs-passag probabls rms of r-sory drf rao: (a) β arg.5, P arg f 6.68%, (b) β arg.5, P arg f 0.6%, ad (c) β arg 3.0, P arg f 0.3%. 90

11 J. Chu al. Srucural Safy 76 (09) 8 94 FDM. h compuaoal coss of h wo mhods ar compard Fg. 3 whl varyg h umbr of lms h problm. h compuaoal coss ar ormalzd by ha of h AJM for h 00- lm cas. I s od ha h proposd AJM rqurs dramacally lss compuaoal m ha h FDM. I should also b od ha ul FDM, AJM dos o rqur drmg h prurbao sz, for whch a opmal choc s grally o ow a pror. 5. Numrcal applcaos 5.. Spac russ dom opmzao I hs xampl, h wgh of a asymmrc spac russ dom composd of 04 lms ( Fg. 4) s mmzd udr cosras o h frs-passag probably of h drf rao valuad a h od of rs,.. h hghs lvao. h basc grd of h srucur, pla vw, ad sco vws ar provdd Fg. 5. Fg. 4(c) shows h lm umbrg chocs of h spac russ dom. A ach od of h srucur, addoal masss (0,000 g) rprsg o-srucural masss such as claddgs ar qually appld. Youg s modulus E 0 GPa ad mass dsy m 7,850 g ar usd as maral proprs for ach russ lm. h groud acclrao s grad by usg h Kaa-am flr. h flr ad opmzao paramrs ar prsd abl 3. h probablsc cosra s dfd rms of h p drf rao valuad a h op ( z 5 m). For a loadg scaro, wo drco compos of arhqua groud xcaos a agls (θ g, θ g ) show Fg. 4(b) ar cosdrd smulaously ad appld o h srucur. h arg falur probably ad a lowr boud of dsg varabls ar s o P arg f (β arg arg f Φ[ P f ] 3.0) ad 0.0 m. h opmzao formulao cosdrg mulpl groud acclraos wh cosras o drf raos boh x- ad y-drcos s dvlopd as m f ( d) ob d arg s. P fp, x dr ( E f (, d): g (, d) 0) P x x f x dr arg Pfp, y dr ( E f (, d): g (, d) 0) P y y f y dr 0.0m d.5m wh M( d) u (, d ) + C ( d) u (, d ) + K ( d) u (, d ) M ( d) l ( θ ) f ( ) M ( d) l ( θ ) f ( ) g g g g (50) Fg. 6 shows ha h spac russ doms opmzd wh fxd θ g whl varyg θ g o hr dffr agls. Opmal rsuls from h cas of h appld groud acclrao wh θ g 0, θ g 90 show Fg.. Covrgc hsory of h four-sory buldg. Frs-passag probabls of r-sory drf rao cosras: (a) volum ad (b) frs-passag probably of ach r-sory drf rao. 9

12 J. Chu al. Srucural Safy 76 (09) 8 94 ha h cross-scoal aras of bracgs ad vrcal lms, spcally a a lowr lvl, ar crasd o rduc dsplacms boh x- ad y-drcos ad falur probabls. Also, as h agl θ g bcoms closr o θ g, h opmal volum crass. By chagg θ g from 90 o 60 (or 30 ), h cras h falur probably h x-drco s much hghr ha h dcras h falur probably h y-drco. h opmzd ara of ach lm s plod Fg. 7. A s xpcd, russ mmbrs, whch ar closly algd o h appld groud acclraos ar largd spcally for lowr lvls. hus, russ mmbrs ar szd mor symmrcally boh x- ad y- drcos for groud acclraos wh θ g 0, θ g 90 compard o θ g 30 or θ g 60. Fg. 8(a) ad (b) show covrgc hsors of h volum ad h frs-passag probably. h proposd mhod abls achvg h arg falur probably wh rducd volums. h comparso of dyamc rsposs of h al srucur ad h opmzd srucur s show Fg. 8(c) hrough () udr radomly grad sampls of groud xcaos wh h flr paramrs r- pord abl 3. Ovrall rducos h drf rao h opmzd srucur ar obsrvd, whch aurally rducs h llhood of xcdac of h hrshold valu durg h xcao. 5.. Opmzao of a bracg sysm usg opology opmzao h prvous umrcal applcao of h bracg sysm s cosdrd as sz opmzao for a gv srucural layou. By coras, opology opmzao ca dfy h opmal bracg layou of a srucur. o dmosra hs opmzao udr frs-passag prob- ably cosras, h proposd mhod s appld o h dsg doma udr arhqua xcaos as show Fg. 9(a). Durg h opmzao for mmzg volum, h frs-passag falurs ar d- fd rms of r-sory drf raos a ach lvl, ad a p drf rao a h buldg hgh (s Fg. 9(b) ad (c)). h srucural colums rprsd by wo vrcal ls show Fg. 9(a) ar modld by fram lms whos dss rma uchagd hroughou h opmzao procss. Youg s modulus E,000 MPa ad mass dsy m,400 g/m 3 ar usd as maral proprs for boh h quadrlaral ad fram lms. h addoal mass of 4,000 g s cosdrd a ach floor lvl as show Fg. 9(a). h dampg ma- rx s cosrucd usg a Raylgh dampg modl wh a % dampg rao. abl 4 summarzs h Kaa-am flr paramrs of doma frqucy ω f ad badwdh ζ f, colum sz, h m rval of rs, ad h hrshold valu u 0 of h avrag drf rao a ach m po. h flrg radus r s 0.5 m, ad a prscrbd dsy 0.7 s appld uformly hroughou h msh. opology opmzao rsuls ar show from dffr arg falur probabls of h r-sory drf raos, ad h p-drf rao cosras ar show Fgs. 0 ad. For h p drf rao cosra, h cras h hcsss lowr lvls ad addoal brachs of maral dsrbuos ar obsrvd as h arg falur probabls dcras, whras bracg pos ad opologs rma rlavly h sam for all hr cass udr h p drf rao cosra Fg. 0. O h ohr had, Fg. shows ha cocos of opologs o ach floor lvl ca b chcd for r-sory drf rao cosras xcp h lows lvl. As h arg falur probably s rducd, h scod lows rsco po of bracg ad colum also dcrass lvao, such ha h cas of Fg. (c), h rsco po s o h scod lvl. Irsco pos of bracgs for uppr wo lvls boh cosrad opmzao problms ar a h mdpo of wo fl oors so ha X shaps of bracgs wh 90 ar obsrvd. A lowr lvls, h bracg rsco pos bcom hghr. I addo, hr s a sgfca cras maral dsrbuo lowr lvls of h p dsplacm cosra, whras ovrall hcsss of bracgs hroughou h buldg hgh ar crasd for h r-sory drf rao cosra. hus, opmzao rsuls show ha rforcg lowr rgos wll b a ffc approach o corol h p dsplacm whras adusg ach bracg modul wll lad o succssful dsgs of srucurs fulfllg r-sory drf rao crra (s Fg. ). For cosrucably ad ashc aspc of archcur, a par rpo cosra [ 33, 36] ca b mplmd h proposd framwor. hrfor, dffr chocs rgardg h umbr of pars or sz of h prmary rgo opmzao wll rsul varous opologs, whch ca provd dvrs opos of soluos for h archcural ad grg schmac dsg procss. 6. Summary ad cocludg rmars I hs papr, a opmzao framwor s proposd o corpora h frs-passag probably o sz opmzao ad opology op- mzao of srucurs. Usg h dscr rprsao mhod ad hors of srucural sysm rlably, h f rs-passag probably s compud ffcly durg h opmzao procss wh a propr cosdrao of h sascal dpdc bw compo falur vs. Paramr ssvy formulao of h probablsc cosra o h frs-passag probably s also drvd basd o h ado mhod ad squal compoudg mhod o facla h usag of ffc opmzao algorhms. h dvlopd mhod s succssfully appld o h laral bracg sysm of srucurs subcd o sochasc groud moos o dfy opmal mmbr szs udr grg cosras assocad wh srucural dsg crra such as h srss, h dsplacm as wll as h r-sory drf rao. I h umrcal applcao of h spac russ dom subc o smulaous mulpl arhqua groud moos, h proposd opmzao framwor provds rlabl srucural soluos for varous loadg scaros. Furhrmor, h proposd mhod ca b furhr xdd o cosdr h frs-passag probably cosra cosrucd by com- bg dffr yps of falur vs such as dffr m pos ad locaos as wll as mulpl dsg crra. h opmzd sysm ca whsad fuur ralzao of sochasc procsss wh a dsrd lvl of rlably. I addo, umrcal xampls show ha h proposd opology opmzao framwor ca provd a ffc way for srucural grs o oba opmal dsg soluos ha sasfy probablsc cosras o h sochasc rspos h cocpual (ad schmac) dsg procss. h proposd mhod s basd o h assumpo of a saoary procss for h arhqua groud moos. h sochasc xcao grad by aural hazards (.g. arhquas, hurrcas) ca b osaoary ad/or o-gaussa. hus, fuur rsarch could focus o dvlopms of opmzao framwors udr o-saoary sochasc procsss h m doma as wll as frqucy doma. Acowldgms h auhors grafully acowldg fudg provdd by h Naoal Scc Foudao (NSF) hrough procs 3443 ad h scod auhor acowldgs h suppor from h Isu of Egrg Rsarch a Soul Naoal Uvrsy. h hrd auhor acowldgs suppor from h Raymod All Jos Char a h Gorga Isu of chology. h formao provdd hs papr s h sol opo of h auhors ad dos o cssarly rflc h vws of h sposorg agcs. 9

13 J. Chu al. Srucural Safy 76 (09) 8 94 Appdx A Flowchar of mplmao for RBO of srucurs cosrad by frs-passag probably. Fg. A Fg. A. Flowchar for opology opmzao of a srucur cosrad by frs-passag probably. Rfrcs [] Asadpour A, ooabo M, Gus J. Robus opology opmzao of srucurs wh ucras sff ss applcao o russ srucurs. Compu Sruc 0;89( ):3 4. [] Jas M, Lombar G, Schvls M. Robus opology opmzao of srucurs wh mprfc gomry basd o gomrc olar aalyss. Compu Mhods Appl Mch Eg 05;85: [3] Wag F, Lazarov B, Sgmud O. O proco mhods, covrgc ad robus formulaos opology opmzao. Sruc Muldscp Opm 0;43(6): [4] Mau K, Fragopol DM. Rlably-basd dsg of MEMS mchasms by opology opmzao. Compu Sruc 003;8(8 ):83 4. [5] Fragopol DM, Mau K. Rlably-basd opmzao of cvl ad arospac srucural sysms. Egrg Dsg Rlably Hadboo. Boca Rao, FL: CRC; 005. Chap. 4. [6] sompaas Y, Lagaros ND, Papadraas M. Srucural Dsg Opmzao Cosdrg Ucras. Lodo, UK: aylor & Fracs; 008. [7] Gus JK, Igusa. Srucural opmzao udr ucra loads ad odal locaos. Compu Mhods Appl Mch Eg 008;98():6 4. [8] Rozvay GIN. Exac aalycal soluos for bchmar problms probablsc opology opmzao. I: EgOp 008 Iraoal Cofrc o Egrg Opmzao, Ro d Jaro; 008. [9] Nguy H, Sog J, Paulo GH. Sgl-loop sysm rlably-basd opology opmzao cosdrg sascal dpdc bw lm-sas. Sruc Muldscp Opm 0;44(5): [0] Jalalpour M, Gus JK, Igusa. Rlably-basd opology opmzao of russs wh sochasc sff ss. Sruc Saf 03;43:4 9. [] Salaghh E, Hdar A. Opmum dsg of srucurs agas arhqua by wavl ural wor ad flr bas. Earhqua Eg Sruc Dy 005;34():67 8. [] Kavh A, Farzam M, Kalah M. m-hsory aalyss basd opmal dsg of spac russs: h CMA voluo sragy approach usg GRNN ad WA. Sruc Eg Mch 0;44(3): [3] Flpov E, Chu J, Paulo GH, Sog J. Polygoal mulrsoluo opology opmzao (PolyMOP) for srucural dyamcs. Sruc Muldscp Opm 06;53(4): [4] Chu J, Sog J, Paulo GH. Srucural opology opmzao udr cosras o saaous falur probably. Sruc Muldscp Opm 06;53(4): [5] Dr Kurgha A. h gomry of radom vbraos ad soluos by FORM ad SORM. Probab Eg Mch 000;5():8 90. [6] Dr Kurgha A. Frs- ad scod-ordr rlably mhods. I: Nolads E, Ghocl DM, Sghal S, dors. Chapr 4 Egrg Dsg Rlably Hadboo. Boca Rao, FL: CRC Prss;

14 J. Chu al. Srucural Safy 76 (09) 8 94 [7] Chu J, Sog J, Paulo GH. Sysm rlably basd opology opmzao of srucurs udr sochasc xcaos. I: h Iraoal Cofrc o Srucural Safy & Rlably, Nw Yor, NY; 03. [8] Sog J, Kag WH. Sysm rlably ad ssvy udr sascal dpdc by marx-basd sysm rlably mhod. Sruc Saf 009;3(): [9] Spc SMJ, Goffr M, Karm A. A ffc framwor for h rlably-basd dsg opmzao of larg-scal ucra ad sochasc lar sysms. Probab Eg Mch 06;44:74 8. [0] Bobby S, Spc SMJ, Karm A. Daa-drv prformac-basd opology opmzao of ucra wd-xcd all buldgs. Sruc Muldscp Opm 06;54: [] Vamarc EH. O h dsrbuo of h frs-passag m for ormal saoary radom procsss. J Appl Mch 975;4():5 0. [] Sog J, Dr Kurgha A. Jo frs-passag probably ad rlably of sysms udr sochasc xcao. J Eg Mch 006;3(): [3] Fumura K, Dr Kurgha A. al-quval larzao mhod for olar radom vbrao. Probab Eg Mch 007;(): [4] Kag WH, Sog J. Evaluao of mulvara ormal grals for gral sysms by squal compoudg. Sruc Saf 00;3():35 4. [5] Chu J, Sog J, Paulo GH. Paramr ssvy of sysm rlably usg squal compoudg mhod. Sruc Saf 05;55:6 36. [6] Clough R, Pz J. Dyamcs of Srucurs. Nw Yor: McGraw Hll; 993. [7] Bdsø MP, Sgmud O. opology Opmzao hory, Mhods ad Applcaos. scod d. Brl, Grmay: Egrg Ol Lbrary. Sprgr; 003. [8] Bdsø MP, Sgmud O. Maral rpolao schms opology opmzao. Arch Appl Mch 999;69(9): [9] Sgmud O, Prsso J. Numrcal sabls opology opmzao: a survy o procdurs dalg wh chcrboard, msh-dpdc ad local mma. Sruc Muldscp Opm 998;6: [30] Gus JK, Prvos JH, Blyscho. Achvg mmum lgh scal opology opmzao usg odal dsg varabls ad proco fucos. I J Numr Mh Eg 004;6(): [3] Brus E. A r-valuao of h SIMP mhod wh flrg ad a alrav formulao for sold-vod opology opmzao. Sruc Muldscp Opm 005;30: [3] Sgmud O. Morphology-basd blac ad wh flrs for opology opmzao. Sruc Muldscp Opm 007;33(4):40 4. [33] Almda SRM, Paulo GH, Slva ECN. A smpl ad ffcv vrs proco schm for vod dsrbuo corol opology opmzao. Sruc Muldscp Opm 009;39(4): [34] Cho K, Km N. Srucural Ssvy Aalyss ad Opmzao. Sprgr; 005. [35] Dlvs O, Mads HO. Srucural Rlably Mhods. Chchsr: Wly; 996. [36] Srombrg LL, Bgh A, Bar WF, Paulo GH. Applcao of layou ad opology opmzao usg par gradao for h cocpual dsg of buldgs. Sruc Muldscp Opm 0;43():