Reliability Evaluation of Slopes Using Particle Swarm Optimization

Transcription

1 atoal Uvrsty of Malaysa From th lctdworks of Mohammad Khajhzadh 20 Rlablty Evaluato of lops Usg Partcl warm Optmzato Mohammad Khajhzadh Mohd Raha Taha hmd El-shaf valabl at:

2 t 2 IC 20 Hotl Equatoral Bag-Putrajaya, Malaysa, 4-5 Jauary 20 L E P U T PER J O R TI D I OCI O E I UIV UDET E T R I TI I K E E B DO G I M L Y I l a o t a r I c tf c C o IC 20 f r c 0 Procdg of th Itratoal Cofrc o dvacd cc, Egrg ad Iformato Tchology 20 Hotl Equatoral Bag-Putrajaya, Malaysa, 4-5 Jauary 20 ICEIT 20 Procdg of th Itratoal Cofrc o dvacd cc, Egrg ad Iformato Tchology Cuttg Edg ccs for Futur ustaablty IB Orgazd by Idosa tudts ssocato Uvrst Kbagsaa Malaysa Rlablty Evaluato of lops Usg Partcl warm Optmzato Mohammad Khajhzadh, Mohd. Raha Taha, hmd El-shaf Dpartmt of Cvl ad tructural Egrg, Uvrsty Kbagsaa Malaysa Bag, 43600, Malaysa E-mal: mkhajzadh2000@yahoo.com bstract Th objctv of ths rsarch s to dvlop a umrcal procdur to rlablty valuato of arth slop ad locatg th crtcal probablstc slp surfac. Th prformac fucto s formulatd usg smplfd Bshop s lmt qulbrum mthod to calculat th rlablty dx. Th rlablty dx dfd by Hasofr ad Ld s usd as a dx of safty masur. archg th crtcal probablstc surfac that s assocatd wth th lowst rlablty dx wll b formulatd as a optmzato problm. I ths papr, partcl swarm optmzato s appld to calculat th mmum Hasofr ad Ld rlablty dx ad crtcal probablstc falur surfac. To dmostrat th applcablty ad to vstgat th ffctvss of th algorthm, two umrcal xampls from ltratur ar llustratd. Rsults show that th proposd mthod s capabl to achv bttr solutos for rlablty aalyss of slop f compard wth thos rportd th ltratur. Kywords lop stablty, rlablty valuato, partcl swarm optmzato. I. ITRODUCTIO lop stablty problms play a crucal rol both mg ad gotchcal grg flds, whch prmarly dal wth arth structurs. Th stablty assssmt of slops, whr thr stablty wll caus major damag to th surroudg vromts, should b carrd out usg sutabl tchqu. Most sol slop stablty aalyss ar basd o th dtrmstc mthods whch sol layrs ar assumd to b uform ad avrag sol proprts ar usd. I dtrmstc mthods a factor of safty s commoly usd to xprss th safty of a slop. Th factor of safty s a rato of som xprsso of rsstac to som corrspodg xprsso for factors causg stablty of th slop. I gral, th factor of safty s ot a cosstt masur of rsk. lops wth th sam valus of th factor of safty may xst at dffrt rsk lvls dpdg o th varablty sol proprts. It s mpossbl to quatfy how much safr a slop bcoms as th factor of safty s crasd. Ths dcats a d for mor objctvly structurd ad quattatv approach toward hadlg ucrtats volvd th problms. Th probablstc approach s a atural choc for ths typ of aalyss, bcaus t allows for th drct corporato of ucrtats to th aalytcal modl. I rct yars, svral attmpts hav b do to dvlop a probablstc slop stablty aalyss [-5]. Th rsults of probablstc aalyss may b xprssd as a probablty of falur or rlablty dx. Hasofr ad Ld [6] proposd a varat dfto of th rlablty dx. Thy dfd th rlablty dx β as th mmum dstac from th org th stadard ormal spac to th lmt stat surfac. To apply th probablstc aalyss usg Hasofr- Ld rlablty dx (β HL ) t s cssary to solv a costrat optmzato problm to fd th mmum rlablty dx or maxmum probablty of falur utlz th approprat optmzato tchqu. s a wly dvlopd subst of volutoary algorthm optmzato, th partcl swarm optmzato has dmostratd ts may advatags ad robust atur rct dcads. It s drvd from socal psychology ad th smulato of th socal bhavor of brd flocks partcular. Isprd by th swarm tllgc thory, Kdy cratd a modl whch Ebrhart th xtdd to formulat th practcal optmzato mthod kow as partcl swarm optmzato (PO)[7]. Th PO algorthm has som advatags compard wth othr optmzato algorthms. It s a smpl algorthm wth oly a fw paramtrs to b 63

3 adjustd durg th optmzato procss, rdrg t compatbl wth ay modr computr laguag. It s also a vry powrful algorthm bcaus ts applcato s vrtually ulmtd. I ths papr, w propos a partcl swarm optmzato (PO) for mmzg th Hasofr-Ld rlablty dx (β HL ) ad dtrm th crtcal probablstc slp surfac of arth slop. II. PROBBILITIC LOPE TBILITY LYI Th problm of th probablstc aalyss s formulatd by a vctor, X=[X,X 2,X 3,,X ], rprstg a st of radom varabls. From th ucrta varabls, a prformac fucto g(x) s formulatd to dscrb th lmt stat th spac of X. Th prformac fucto dvds th vctor spac X to two dstct rgos. Th safty rgo for g(x)>0 ad th falur rgo g(x)<0, whl th lmt stat surfac s g(x)=0. Th prformac fucto for th slop stablty s a fucto of th factor of safty (F) usually dfd as: g(x) = F- () I th abov quato, F s a factor of safty ad ca b valuatd usg ay lmt qulbrum mthod. I ths papr a smplfd Bshop s mthod s usd to calculat th safty factor. Bshop [8] cosdrd oly crcular slp surfacs for aalyss. I Bshop s mthod, th safty factor s dtrmd by tral ad rrors procdur, bcaus th factor of safty appars both sds of Eq. (). I hs mthod, th tr slc shar forcs ar gord, ad oly th ormal forcs ar usd to df th tr slc forcs. Th dtals of forcs actg o a typcal slc ar show Fg.. Th quato for th factor of safty s drvd from th momt qulbrum as follows: F = = cb taα [ cb sc α + ( mα [ W ub. ]ta ϕ )] F ha Ws α+ kh W(cos α ) R = = ll th paramtrs of Eq. (2) ar dfd Fg. ad m α s dfd as: sα taφ mα = /[cosα + ] F (3) Fg. Forcs actg o a typcal slc Bshop s mthod (2) Th probablty of falur of th slop ca b xprssd trms of th prformac fucto by th followg tgral: Pf = P[ g( X 0)] (4) Th most ffctv applcatos of probablty thory to th aalyss of slop stablty hav statd th ucrtats th form of a rlablty dx (β). Th rlablty dx provds mor formato ad s a bttr dcato of th stablty of a slop tha th factor of safty alo bcaus t corporats formato of th ucrtaty th valus of th prformac fucto. It also provds a good comparatv masur of safty; slops wth hghr β ar cosdrd safr tha slops wth lowr β. Dpd o th form of th prformac fucto svral dftos of th rlablty dx xst. Hasofr ad Ld [6] proposd a varat dfto of th rlablty dx as th mmum dstac from th org th stadard ormal spac to th lmt stat surfac. Ths dstac s dfd as β HL, ca b dscrbd as ad Fg. 2: β /2 = m( U T. U ) (5) X F Fg.2 Th gomtrcal rprstato of th dfto of th rlablty dx whr X s a vctor rprstg th st of radom varabls x, F s th falur doma. To dtrm th H-L rlablty dx (β HL ), all th radom varabls X should b trasformd to a stadard ormal spac U, by a orthogoal trasformato such that: x μ u = (6) σ s mtod bfor, th H-L rlablty dx (β HL ) s dfd as th mmum dstac from th org of th axs th stadard ormal spac to th lmt stat surfac. To valuat β HL th followg costrad optmzato problm should b solvd: Mmz β HL (7) ubjct to g( U ) = 0 olv Eq. (7) s quvalt to solv th rlaxd form obtad by palty mthod as: Mmz β ( ) l HL + r g U (8) Th paramtrs r ad l ar problm dpdt, ad r should b a sutably larg postv costat. I th prst study, th valus st for r ad l wr 000 ad 2, rspctvly. Th soluto of th abov optmzato problm s th dsg pot or MPP th stadardzd ormal varabls spac. vral algorthms hav b rcommdd for th 64

4 soluto of optmzato problm Eq. (8). I th currt study, a partcl swarm optmzato s proposd for th soluto. III. PRTICLE WRM OPTIMIZTIO Th orgal partcl swarm optmzato algorthm troducd by Kdy ad Ebrhart 995 [7]. Th PO s drvd from a smplfd vrso of th flock smulato. It also has faturs that ar basd upo huma socal bhavour. PO cotas a umbr of partcls whch calld th swarm. Th partcls ar talzd radomly th mult dmsoal sarch spac of a objctv fucto. Each partcl rprsts a pottal soluto of th optmzato problm. Th partcls fly through th sarch spac ad thr postos ar updatd basd o ach partcl s prsoal bst posto as wll as th bst posto foud by th swarm. Th objctv fucto s valuatd for ach partcl durg tratos, ad th ftss valu s usd to dtrm whch posto th sarch spac s bttr tha th othrs. t vry trato, th updat movs a partcl by addg a k + k chag vlocty V to th currt posto X as llustratd th followg quato [9]: k+ k k+ X = X + V =,2,3,..., (9) Th vlocty s a combato of thr cotrbutg factors: () prvous vlocty V k, (2) movmt th drcto of th local bst P k, ad (3) movmt th drcto of th global bst P k g. Th mathmatcal formulato s xprssd as [4]: k k k k k k V + = w V + c r ( P X ) + c2 r2 ( P g X ) (0) whr w s a rta wght to cotrol th fluc of th prvous vlocty; r ad r 2 ar two radom umbrs uformly dstrbutd th rag of (0, ); c ad c 2 ar two acclrato costats usually cosdrd qual 2; P k s th bst posto of th th partcl up to trato k ad P k g s th bst posto amog all partcls th swarm up to trato k. Th rta wghtg fucto Eq. (0) s usually calculatd usg followg quato: w= wmax ( wmax wm ) k / G () whr w max ad w m ar maxmum ad mmum valus of w, G s th maxmum umbr of tratos ad k s th currt trato umbr. Fgur 3 shows th posto updat of a partcl PO. IV. UMERICL EXMPLE Ths scto vstgats th valdty ad ffctvss of th proposd algorthm to probablstc slop stablty aalyss. To vrfy ad assss th applcablty of th proposd two bchmark problms wr slctd from th ltratur. Th procdur has b carrd out usg a computr program was dvlopd by MTLB. Th program sarchs for th most crtcal dtrmstc ad probablstc slp surfac. Basd o abov xplaato, th mplmtato procdur of th proposd mthod for th rlablty aalyss of th arth slop s costructd as follows:. Italz a st of partcls postos ad vlocts radomly dstrbutd throughout th dsg spac boudd by spcfd lmts. 2. Evaluat th objctv fucto valus usg Eq. (8) for ach partcl th swarm. 3. Updat th optmum partcl posto at currt trato ad global optmum partcl posto. 4. Updat th vlocty vctor as spcfd Eq. (0) ad updat th posto of ach partcl accordg to Eq. (9). 5. Rpat stps 2 4 utl th stoppg crtra s mt. To calculat th mmum valu of β HL usg PO th paramtrs of th algorthm should b adoptd accuratly. I our study, propr f tug of ths paramtrs was obtad utlzg svral xprmtal studs xamg th ffct of ach paramtr o th fal soluto ad covrgc of th algorthm. s a rsult, a populato of 40 dvduals was usd; w max ad w m wr chos as 0.95 ad 0.45 rspctvly; ad th valus of th acclrato costats (c ad c 2 ) wr slctd qual to 2.Fally, a fxd umbr of maxmum trato (G) of 3000 was appld. Th optmzato procdur was trmatd wh o of th followg stoppg crtra was mt: () th maxmum umbr of gratos s rachd; () aftr a gv umbr of tratos, thr s o sgfcat mprovmt of th soluto.. Exampl Fgur 4 shows th gomtry of a slop homogous sol. Th paramtrs cosdrd as radom varabls th probablstc aalyss ar: th ffctv frcto agl, ffctv cohso, ut wght ad por watr prssur rato. Tabl I prsts th ma valus ad stadard dvato assocatd wth ach radom varabl. Fg. 3 Posto updat of partcl PO Fg. 4 Cross scto of homogous slop-xampl 65

5 TBLE I TTITICL PROPERTIE OF OIL PRMETER- EXMPLE Radom Ma tadard Dstrbuto varabl dvato c' 8.0k/m 2 3.6k/m 2 Log-ormal ta φ' ta Log-ormal γ 8.0k/m 3 0.9k/m 3 Log-ormal r u Log-ormal Th problm was prvously solvd by L ad Lumb [0], Hassa ad Wolff [] ad Bhattacharya t al. [2]. Th rsults of th proposd mthod ad prvous studs ar summarzd Tabl II. I Tabl II, F m ad β F ar th mmum factor of safty ad th rlablty dx assocatd wth th crtcal dtrmstc slp surfac, rspctvly, ad β m s th mmum rlablty dx corrspodg to th crtcal probablstc slp surfac. ccordg to aalysg th rsults of ths tabl, t ca b obsrvd that, th mmum rlablty dx calculatd usg prstd mthod s 2.22, whch s lowr tha th valus rportd by L ad Lumb (2.5) Hassa ad Wolff (2.293), Bhattacharya t al. (2.239). Furthr, th mmum factor of safty calculatd from a dtrmstc aalyss basd o th ma valus of th sol proprts obtad by PO s.309, whch s lowr tha.326 rportd by Bhattacharya t al. [2]. Th corrspodg crtcal dtrmstc ad th crtcal probablstc slp surfacs ar also prstd Fg. 4. s t ca b s, two surfacs ar locatd rasoably clos to ach othr as xpctd a homogous slop. It s bcaus of th proxmty of th valus of β F ad β m prstd Tabl II. Th falur surfacs rportd by prvous rsarchrs ar also smlarly locatd. TBLE II REULT COMPRIO- EXMPLE Mthod β F β m F m L ad Lumb [0] Hassa ad Wolff [] Bhattacharya t al [2] Prst study (PO) B. Exampl 2 Fgur 5 shows th cross scto ad gomtry of a two layrd slop clay boudd by a hard layr blow ad paralll to th groud surfac. Th sol strgth paramtrs that ar rlatd to th stablty of slop, cludg frcto agl φ, ad cohso c, ar cosdrd as radom varabls. Th statstcal momts (ma valu ad stadard dvato) of th paramtrs ar summarzd Tabl III. Ths xampl was also solvd prvously by Hassa ad Wolff [] ad Bhattacharya t al [2] trms of F m, β F ad β m. Th rsults obtad from currt study togthr wth a comparso of thos rportd by prvous rsarchrs ar summarzd Tabl IV. For th rsults show ths tabl, t ca b cosdrd that th mmum rlablty dx valuatd usg PO s 2.77, whch s almost lowr tha thos rportd by Hassa ad Wolff [] ad Bhattacharya t al. [2]. Bsds, th mmum factor of safty obtad by PO s foud to b smallr tha th othrs. Th corrspodg crtcal dtrmstc ad th crtcal probablstc slp surfacs ar prstd Fg. 5. I accordac wth th dffrc th valus of β F ad β m prstd Tabl IV, th two surfacs ar locatd sgfcatly sparat. Fg. 5 Cross scto of o homogous slop-xampl 2 TBLE III TTITICL PROPERTIE OF OIL PRMETER- EXMPLE 2 Matral Paramtr Ma tadard Dstrbuto dvato ol c 38.3k/m k/m 2 Log-ormal φ 0 - Log-ormal ol 2 c k/m k/m 2 Log-ormal φ Log-ormal TBLE IV REULT COMPRIO- EXMPLE 2 Mthod β F β m F m Hassa ad Wolff [] Bhattacharya t al [2] Prst study (PO) V. COCLUIO Ths papr outls a procdur of probablstc aalyss of arth slop. Th Hasofr-Ld rlablty dx (β HL ) s usd stad of th covtoal rlablty dx β. Th problm of sarchg th crtcal probablstc surfac wth th mmum rlablty dx, β m, ca b formulatd as a optmzato problm ad a modfd partcl swarm optmzato s proposd for th soluto. Th dscrbd framwork has b codd MTLB ad usd to carry out paramtrc studs for th umrcal problms. Th applcablty of th proposd mthodology dvlopd hr, has b xamd o two slop stablty problms from th ltratur. comparso of rsults show that, th rsults obtad th prst study usg MPO has valuatd valus of mmum rlablty dx that ar rasoably lowr tha thos rportd th ltratur ad s capabl to dtfy th falur squc. Furthr as llustratd trough th tst problms; th crtcal probablstc ad dtrmstc slp 66

6 surfac s almost clos for slop a homogous sol whras ths surfacs ar locatd qut sparat for o homogous slops. REFERECE []. Hassa, ad T. Wolff, arch algorthm for mmum rlablty dx of arth slops, J. Gotch. Govro. Vol. 25, pp , 999. [2] G. Bhattacharya, D. Jaa,. Ojha ad. Chakraborty, Drct sarch for mmum rlablty dx of arth slops, Comput. Gotch. Vol. 30, o. 6, pp , [3] D. Tobutt, ad E. Rchards, Th rlablty of arth slops, It. J. umr.al. mt. Vol. 3 o. 4, pp , [4]. Cho, Effcts of spatal varablty of sol proprts o slop stablty, Eg. Gol. Vol. 92 o. 3-4, pp , [5] G. X, J. Zhag, ad J. L, daptd Gtc lgorthm ppld to lop Rlablty alyss, Fourth Itratoal Cofrc o atural Computato, pp , [6]. Hasofr, ad. Ld, Exact ad varat scod-momt cod format, J. Eg. Mch-CE. Vol. 00, o., pp. -2, 974. [7] J. Kdy, ad R. Ebrhart, Partcl swarm optmzato, I Proc. IEEE Itratoal Cofrc o ural tworks, 995, pp [8]. W. Bshop, Th us of th slp crcl th stablty aalyss of arth slops,. Gotchqu, vol. 5, o., pp. 7-7, 955. [9] Y. h, ad R. Ebrhart, modfd partcl swarm optmzr, I Proc. IEEE World Cogrss o Evolutoary Computato, 998, p [0] K. L, ad P. Lumb, Probablstc dsg of slops, Ca. Gotch. J. Vol. 24, o. 4, pp ,