4 t Internatonal Conferene on Eartquake Geotenal Engneerng June 25-28, 2007 Paper No. 25 SLOPE STABILITY ANALYSIS IN SEISMIC AREAS BY A RELIABILITY APPROACH Dala YOUSSE ABDEL MASSIH, Abdul-Hamd SOUBRA 2 and Ela EL-HACHEM 3 ABSTRACT A relablty-baed analy of te tablty of lope n em area performed n t paper. Qua-tat repreentaton of eartquake effet ung te em oeffent onept adopted. Two determnt approae are ued. Te frt one provde upper-bound oluton n te framework of te knematal metod of te lmt analy teory. Te eond one baed on numeral modelng of te lope tablty ung te Lagrangan explt fnte dfferene ode LAC 3D. Te random varable ondered n te analy are te ol ear trengt parameter and and te orzontal em oeffent. Te repone urfae metodology ued to fnd an approxmaton of te analytally-unknown performane funton and te orrepondng relablty ndex wle ung te numeral LAC 3D mulaton. However, dret mnmzaton of te relablty ndex performed wle ung te lmt analy approa. Te numeral probablt reult obtaned from tee metod are preented and dued. eyword: lope, eartquake load, lmt analy, mulaton, relablty, repone urfae. INTRODUCTION Te tablty analy of lope n em area by a peudo-tat approa a been extenvely nvetgated n lterature ung determnt approae. In tee metod, average value of te nput parameter (angle of nternal frton, oeon and em oeffent) are ued and a global afety fator alulated (ee for ntane Cen & Sawada, 983; Lenky & San, 994; Malowk 2002 and Loukd et al., 2003 among oter). A relablty-baed approa for te lope tablty analy n em area more ratonal tan te tradtonal determnt metod ne t take nto aount te nerent unertanty of ea nput varable. Te relablty teory wa ntrodued by everal autor n te analy of lope tablty but wtout takng nto onderaton te em loadng (ee for ntane Vanmarke, 977; Cowdury & Xu, 993; Crtan et al., 994; Low et al., 998; Huen Malkaw et al., 2000; Auvnet & Gonzale, 2000; El-Ramly et al., 2002; and Svakumar Babu & Muke, 2004 among oter). Mot of tee tude are baed on approxmate determnt model a te lmt equlbrum metod w founded on a pror aumpton onernng () te form of te lp urfae and () te normal tre dtrbuton along t urfae or te nter-le fore. PD Student, Unverty of Nante & Lebanee Unverty, BP -547, Berut, Lebanon. E-mal: Dala.Youef@unv-nante.fr 2 Profeor, Unverty of Nante, Inttut de Reere en Géne Cvl et Méanque, UMR CNRS 683, Bd. de l unverté, BP 52, 44603 Sant-Nazare edex, rane (Correpondng autor). E- mal: Abed.Soubra@unv-nante.fr 3 Atant Profeor, Lebanee Unverty, aulty of Engneerng, Roume, Lebanon. E-mal: Elaaem@ul.edu.lb
Te applaton of te relablty teory to te lope tablty problem takng nto aount te peudotat em loadng mu le nvetgated (ee for ntane Crtan & Urzua, 998 and Al- Hamoud & Tatamon, 2000). In t paper, a relablty-baed analy of te tablty of eart lope n em area performed. Qua-tat repreentaton of eartquake effet ung te em oeffent onept adopted were te em loadng repreented by orzontal nertal fore. Two determnt model are ued for te aement of te relablty ndex of te lope. Te frt one baed on te upper-bound metod of te lmt analy teory ung rotatonal log-pral falure meanm (Cen, 975). Te eond one baed on numeral modelng ung te Lagrangan explt fnte dfferene ode LAC 3D. Te Haofer-Lnd relablty ndex adopted ere for te aement of te lope relablty. Te random varable ondered n te analy are te ol ear trengt parameter and and te orzontal em oeffent. Te repone urfae metodology ued to fnd an approxmaton of te analytally-unknown performane funton and te orrepondng relablty ndex wle ung te numeral LAC 3D mulaton. However, dret mnmaton of te relablty ndex performed wle ung te lmt analy approa. After a bref derpton of te ba relablty onept, te lmt analy falure meanm and te numeral LAC 3D ode are preented. Ten, te probablt analy and te orrepondng numeral reult are preented and dued. BASIC RELIABILITY CONCEPTS Two dfferent meaure are ommonly ued n lterature to derbe te relablty of a truture: Te relablty ndex and te falure probablty. Te relablty ndex of a geotenal truture a meaure of te afety tat take nto aount te nerent unertante of te nput varable. Te wdely ued relablty ndex te one defned by Haofer and Lnd (974). It matrx formulaton gven by: β T ( x ) C ( x ) = mn () x n w x te vetor repreentng te n random varable, te vetor of ter mean value, C ter ovarane matrx and te falure regon. Te mnmaton of equaton () performed ubjet to te ontrant G ( x) 0 were te lmt tate urfae G ( x) = 0, eparate te n dmenonal doman of random varable nto two regon: a falure regon repreented by G ( x) 0 and a afe regon gven by G ( x) > 0. Te laal approa for omputng β relablty ndex by equaton () baed on te tranformaton of te lmt tate urfae nto te pae of tandard normal unorrelated varate. Te ortet dtane from te tranformed falure urfae to te orgn of te redued varate te relablty ndex β. An ntutve nterpretaton of te relablty ndex wa uggeted n Low and Tang (997) were te onept of an expandng ellpe led to a mple metod of omputng te Haofer-Lnd relablty ndex n te orgnal pae of te random varable. Tee autor tated tat te mnmzaton of te relablty ndex equvalent to fnd te mallet dperon ellpod tat tangent to te lmt tate urfae. Wen te random varable are nonnormal and orrelated, te optmaton approa ue te Rakwtz-eler equvalent normal tranformaton wtout te need to dagonalze te orrelaton matrx a own n Low (2005). Te N N omputaton of te equvalent normal mean and equvalent normal tandard devaton σ for ea tral degn pont are automatally found durng te ontraned optmzaton ear. Te metod of omputaton of te relablty ndex ung te onept of an expandng ellpe uggeted by Low and Tang (997) ued n t paper. rom te rt Order Relablty Metod ORM and te Haofer-Lnd relablty ndex β, one an approxmate te falure probablty a: Pf Φ( β ), were Φ () te umulatve dtrbuton funton of a tandard normal varable. In t metod, te
lmt tate funton approxmated by a yperplane tangent to te lmt tate urfae at te degn pont. DETERMINISTIC MODELS OR SLOPE STABILITY ANALYSIS A peudo-tat approa adopted n t paper for te mulaton of te em loadng. Only orzontal em nerta fore are appled everywere n te ol ma, te vertal em fore often beng dregarded. Te ol ondered n te analy araterzed by t oeon, t angle of nternal frton and t unt wegt γ. Te fator of afety defned a te rato of te avalable ear trengt of te ol to tat requred to mantan equlbrum. It te fator by w te avalable ear trengt parameter need to be redued to brng te lope to falure. Lmt analy model A rgd rotatonal log-pral falure meanm tat pae below te toe of te lope ondered ere n te framework of te knematal metod of te lmt analy teory (f. gure ). Te meanm pang below te toe may lead n ertan ae to more rtal reult tan te one pang troug te toe. It redue to te meanm pang troug te toe wen β = β. or furter detal on te teoretal formulaton of te log-pral lmt analy model, te reader may refer to Cen (975). θ 0 θ θ r 0 α H Logartm Logartm pral Spral falure falure urfae ' β β V ( θ ) gure : alure meanm Numeral modelng ung te Lagrangan explt fnte dfferene ode LAC 3D LAC 3D (at Lagrangan Analy of Contnua) a ommerally avalable tree-dmenonal fnte dfferene ode n w an explt Lagrangan alulaton eme and a mxed dretzaton zonng tenque are ued. It ould be mentoned tat LAC 3D nlude an nternal programmng opton (ISH) w enable te uer to add own ubroutne. Te ol doman dvded by te uer nto a 3D fnte dfferene me of polyedral zone. Contant tran-rate element of tetraedral ape woe verte are te node of te me are ued. T enable te veloty feld to be lnear nde te tetraedron. Ea element beave aordng to a prerbed lnear or nonlnear tre/tran law n repone to appled fore or boundary retrant. Several onttutve model are avalable. In t ode, altoug a peudo-tat (.e. non-dynam) meanal analy requred, te equaton of moton are ued. Te oluton to a tat problem obtaned troug te dampng of a dynam proe by nludng dampng term tat gradually remove te knet energy from te ytem. or furter detal, one an refer to LAC 3D manual. Te teoretal bakground of te LAC 3D ode an be ummarzed a follow: Te equaton of moton of an equvalent tat problem nvolvng nertal term were wrtten n a drete form at te dfferent node of te dretzed medum. T enable one to tranform te equaton of moton of
te ontnuum nto a et of Newton law at te node of te me. Te later equaton onttute a ytem of dfferental equaton w are olved ung te Lagrangan explt fnte dfferene eme n tme. Te new dealzed medum an be vewed a an aembly of pont mae loated at te node of te me and onneted by lnear prng ne te ytem of ordnary dfferental equaton obtaned mlar to tat derbng te moton of a ma-prng ytem. Te analogy wt te dealzed medum mmedate f one nterpret te tatally equvalent nodal fore of all ontrbutng tetraedra and nodal appled load (alled ereafter out-of-balane fore) a te reultant of prng reaton and external appled fore n te ma-prng ytem. Te out-of-balane fore of all node are equal to zero wen te medum a reaed equlbrum. In te preent numeral model, te nertal term are ued a mean to rea, n a numerally table manner, te teady tate of tat equlbrum or plat flow. T performed by replang te ma nvolved n te nertal term by a fttou nodal ma woe value enure numeral tablty of te ytem on t route to teady tate. Te alulaton eme nvoke equaton of moton n ter dretzed form (.e. Newton law at te dfferent node) to derve new velote and dplaement from tree and fore. Ten, tran rate are derved from velote, and new tree from tran rate. Te tree and deformaton are alulated at everal mall tmetep (alled ereafter yle) untl a teady tate of tat equlbrum or plat flow aeved. Te onvergene to t tate may be ontrolled by a maxmal prerbed value of te unbalaned fore for all element of te model. It ould be mentoned tat n LAC 3D, te applaton of prerbed dplaement or tree on te ol and/or te truture reate unbalaned fore n te ytem. Dampng ntrodued n order to remove tee fore or to redue tem to very mall value ompared to te ntal one. In t paper, LAC 3D ued for te determnaton of te lope afety fator determned by te trengt reduton metod. T metod mplemented n LAC 3D troug te "SOLVE fo" ommand. T ommand enable an automat ear for fator of afety ung te braketng approa, a derbed n Dawon et al. (999): Smulaton are run for a ere of tral value of te tral afety fator. Te oeon and te angle of nternal frton at ea tral are adjuted aordng to te equaton: tral = (2) tral tral = artan tan (3) tral tral Te value of at w falure our found ung a braketng and beton approa. Upper and lower braket are frt etabled. Te ntal lower braket orrepond to any tral fator of afety for w te ytem table. Te ntal upper braket orrepond to any tral fator of afety for w te ytem untable. Next, a new value, mdway between te upper and lower braket, teted. If te ytem table for t mdway value, te lower braket replaed by t new tral fator of afety. If te ytem doe not rea equlbrum, te upper braket ten replaed by te mdway value. Te proe repeated untl te dfferene between upper and lower braket le tan a pef tolerane. RELIABILITY ANALYSIS O SLOPE STABILITY Te am of t paper to perform a relablty-baed analy of lope tablty n a em area. Te falure or unatfatory performane mode ondered n te analy nvolve te ultmate lmt tate and empa on te lope tablty falure. Te two determnt model preented above are ued. Te repone urfae metodology utlzed to fnd an analytal approxmaton of te unknown
performane funton and te orrepondng rtal Haofer-Lnd relablty ndex wle ung te numeral LAC 3D mulaton. However, dret mnmzaton of te Haofer-Lnd relablty ndex performed wle ung te lmt analy model ne te loed form oluton of te performane funton known n t ae. Due to unertante n ol ear trengt parameter and orzontal em oeffent, te oeon, te angle of nternal frton, and te em oeffent are ondered a random varable. or te probablty dtrbuton of te random varable, tree ae are tuded. In te frt ae, referred to a normal varable,, and are ondered a normal varable. In te eond ae referred to a non-normal (EVD) varable, aumed to be lognormally dtrbuted and aumed to be bounded and a beta dtrbuton ued (enton and Grfft 2003). Te parameter of te beta dtrbuton are determned from te mean value and tandard devaton of (Haldar & Maadevan, 2000). or te em oeffent an Extreme Value type II Dtrbuton (EVD) (Haldar & Maadevan, 2000) ued. Te trd ae, named a non-normal (Exp D) varable mlar to te eond one exept tat an Exponental Dtrbuton (Exp D) (Haldar & Maadevan, 2000) ondered for te em oeffent. or normal and non-normal dtrbuton, bot orrelated and unorrelated varable are ondered. After a bref derpton of te performane funton ued n te preent analy, te repone urfae metodology and t numeral mplementaton are preented. Ten, te probablt numeral reult are preented and dued. Performane funton Te performane funton orrepondng to te unatfatory performane mode (lope tablty falure) ued n t relablty analy gven a follow: were G = (4) te afety fator alulated by te lmt analy model or LAC 3D mulaton. Repone urfae metod If te performane funton an explt funton of te random varable, te relablty ndex an be alulated ealy. In LAC 3D model, te loed form oluton of te performane funton not avalable and te determnaton of te relablty ndex ten not tragtforward. Terefore, an algortm baed on te repone urfae metodology propoed by Tandjra et al. (2000) ued n t paper n te am to alulate te relablty ndex and te orrepondng degn pont. Te ba dea of t metod to approxmate te performane funton by an explt funton of te random varable, and to mprove te approxmaton va teraton. Te approxmate performane funton ued n t tudy a a quadrat form. It ue a eond order polynomal wt quared term but no ro term. Te expreon of t approxmaton gven by: G n n 2 0 b. x = = ( x) = a + a x +. (5) were x are te random varable, n te number of te random varable and, a, b are te oeffent to be determned. In t paper, tree random varable are ondered (.e. n = 3 ). Tey are araterzed by ter mean value and ter oeffent of varaton σ. A bref explanaton of te ued algortm a follow: - Evaluate te performane funton G ( x) at te mean value pont and te 2n pont ea at ± kσ were k = n t paper.
2- Te above 2 + x a, b. T obtan a tentatve repone urfae funton w baed on te value of te 2 n + ampled pont near te mean value pont. 3- Solve equaton () to obtan a tentatve degn pont and a tentatve β ubjet to te ontrant tat te tentatve repone urfae funton of tep 2 be equal to zero. 4- Repeat tep to 3 untl onvergene. Ea tme tep repeated, te 2 n + ampled pont are entred at te new tentatve degn pont of tep 3. n value of G ( ) an be ued to olve equaton (5) for te oeffent ( ) Numeral mplementaton of te repone urfae metod A derbed n te prevou eton, te determnaton of te Haofer-Lnd relablty ndex requre a, of te tentatve repone urfae va te reoluton of () te determnaton of te oeffent ( ) b equaton (5) for te 2n+ ampled pont and () te mnmaton of te Haofer-Lnd relablty ndex ubjet to te ontrant tat te tentatve repone urfae funton of tep 2 be equal to zero. Tee two operaton w onttute a ngle teraton were done ung te optmzaton toolbox avalable n Matlab 7.0 oftware. Several teraton are performed untl onvergene of te Haofer- Lnd relablty ndex. Note tat te determnaton of te performane funton at te 2n+ ampled pont performed ung determnt LAC 3D alulaton. Te reult of tee omputaton onttute te nput data for te determnaton of te oeffent of te tentatve repone urfae a, ung Matlab 7.0. Alo, te value of te degn pont determned ung te mnmzaton ( ) b proedure n Matlab 7.0 an nput data for te determnaton of te performane funton at te 2n+ ampled pont n LAC 3D. Terefore, an exange of data between LAC 3D and Matlab 7.0 n bot dreton wa neeary to enable an automat reoluton of te teratve algortm for te determnaton of te Haofer-Lnd relablty ndex. Te lnk between LAC 3D and Matlab 7.0 wa performed ung text fle and ISH language ommand. NUMERICAL RESULTS Te numeral reult preented n t paper onder te ae of a lope wt an angle equal to 63.4 o (2V to H lope). Te ol a a unt wegt of 20 kn/m 3. Te llutratve value ued for te tattal moment (.e. mean and oeffent of varaton COV ) of te ear trengt parameter o and ter oeffent of orrelaton ρ, are gven a follow: = 20 kpa, = 30, COV = 20%, COV = 0% and ρ, = 0. 5. Lmt analy reult Te determnaton of te relablty ndex performed by mnmzaton of equaton () not only wt repet to te random varable (,, ), but alo wt repet to te geometral parameter of te falure meanm ( θ θ β 0,, ) own n fgure (). Te obtaned urfae orrepondng to te mnmum relablty ndex referred to ere a te rtal probablt urfae. Te relablty ndex obtaned ung t urfae maller (.e. more rtal) tan te one alulated by ung te rtal determnt urfae. Relablty ndex, degn pont and falure probablty or a orzontal em oeffent equal to 0.3, te rtal determnt lope egt tat lead to falure wa found equal to 7.278 m. T value wa ued n all ubequent alulaton. Table () preent te Haofer-Lnd relablty ndex and te orrepondng degn pont for dfferent value of te mean of for normal and non-normal (EVD) varable wen COV = 30%. Bot orrelated and unorrelated ear trengt parameter are ondered. Te relablty ndex dereae
wt te nreae of te mean of (.e. wt te dereae of te lope afety fator S ) untl t vane wen te ultmate determnt tate of falure (.e. S = ) reaed. T ultmate tate te one for w te degn pont equal to te mean pont for normal varable and equvalent normal mean pont for non-normal varable. Te orrepondng falure probablty equal to 50%. Table : Relablty reult for dfferent value of te mean of a) Normal varable Unorrelated ear trengt parameter (kpa) ( o) β P (%) f γ 0.05 2.65 28.0 0.052.96 2.48.58.09.06 0. 4.8 28.37 0.0.59 5.6.4.07.0 0.5 5.83 28.80 0.68.8.8.26.05.2 0.2 7.43 29.24 0.220 0.77 22.03.5.03.0 0.25 8.85 29.65 0.265 0.37 35.55.06.0.06 0.3 20.00 30.00 0.3 0.00 50.00.00.00.00 b) Normal varable Correlated ear trengt parameter o β P (%) γ (kpa) ( ) 0.05.6 3.26 0.054 2.36 0.9.79 0.95.08 0.0 3.09 30.89 0.4.9 2.8.53 0.97.4 0.5 5.8 30.55 0.75.4 7.90.32 0.98.7 0.20 7.6 30.28 0.228 0.90 8.28.7 0.99.4 0.25 8.79 30. 0.270 0.43 33.50.06.00.08 0.30 20.00 30.00 0.300 0.00 50.00.00.00.00 ) Non-normal (EVD) varable for COV = 30% Unorrelated ear trengt parameter o β P (%) (kpa) ( ) f f γ 0.05 3.4 26.76 0.06 2.22.3.49.9.6 0. 4.86 27.8 0.2.66 4.84.35.5.23 0.5 6.38 28.67 0.8.2 3.6.22.09.2 0.2 7.78 29.32 0.23 0.62 26.69.2.06.6 0.25 9.0 29.84 0.27 0.7 43.3.05.03.0 0.3 9.6 30.0 0.32 0.00 50.00.02.0.08 d) Non-normal (EVD) varable for COV = 30% Correlated ear trengt parameter o β P (%) (kpa) ( ) f γ 0.05 2.46 29.8 0.07 2.92 0.8.60.07.32 0. 4.69 29.90 0.5 2.06.96.36.03.50 0.5 6.3 30.07 0.20.34 8.97.23.00.34 0.2 7.7 30.0 0.24 0.73 23.4.3.00.20 0.25 9.07 30.04 0.28 0.20 42.20.05.00. 0.3 9.6 30.0 0.32 0.00 50.00.02.00.08 Te value (, and ) of te degn pont orrepondng to dfferent mean value of te orzontal em oeffent an gve nformaton about te retane and load fator of te dfferent random varable a follow: =, = tan( ) tan, γ =. or unorrelated ear trengt parameter, te value of and at te degn pont are maller tan ter repetve mean value and nreae wt te nreae of te orzontal em load. Tey tend
to te equvalent normal mean value wen = 0. 3. However, found to be ger tan t mean value ne te orzontal em oeffent a drvng parameter, and t tend to te equvalent normal mean wen = 0. 3. Te retane fator and dereae wt te nreae of te em oeffent. However, te load fator γ frt nreae and ten dereae wt te nreae of. Te later reult an be explaned a follow: for mall value of te em oeffent, te lope tablty not gly affeted by te em oeffent and tu a mall load fator obtaned. Note fnally tat for negatvely orrelated ear trengt parameter, lgtly exeed te mean for ome value of te em oeffent. gure (2) ow te varaton of te Haofer-Lnd relablty ndex β wt te afety fator S for te ae of normal and non-normal (EVD) varable wen COV = 30%. Te reult are preented for bot orrelated and unorrelated ear trengt parameter. Te relablty ndex orrepondng to unorrelated ear trengt parameter wa found maller tan te one of negatvely orrelated varable for bot type of dtrbuton. One an onlude tat aumng unorrelated ear trengt parameter onervatve n omparon to aumng negatvely orrelated ear trengt parameter (ee Motyn & L, 993). rom te ame fgure, t an be own tat te relablty ndex of normal varable ger tan tat of non-normal (EVD) varable for value of te afety fator varyng from to about.23. or ger value of te afety fator, te relablty ndex of normal varable beome maller. A a onluon, te aumpton of normal varable onervatve for g value of te afety fator. 3.50 3.00 Normal varable for unorrelated ear trengt parameter Normal varable for orrelated ear trengt parameter Non-normal (EVD) varable for unorrelated ear trengt parameter Non-normal (EVD) varable for orrelated ear trengt parameter β 2.50 2.00.50.00 0.50 0.00.00.0.20.30 gure 2: Relablty ndex veru afety fator gure (3) ow tat te aumpton made on te probablty dtrbuton of te em oeffent may largely affet te probablty dtrbuton of te lope afety fator. A larger dperon of t fator noted wen one onder an exponental dtrbuton. A a onluon, te aumpton of an Exp D for te em oeffent gly onervatve n omparon to te EVD. Effet of te varablty of ea random varable on te CD of te afety fator gure (4) ow te effet of a ange n te oeffent of varaton of ea random varable on te CD of te afety fator. Te mean value of te em oeffent et equal to 0.. Te aumpton of normal varable and orrelated ear trengt parameter ondered ere. our ae are analyzed. Te frt ae, referred to a "referene ae", onder te value of te oeffent of varaton a gven n te ntroduton of te eton named "Numeral reult" wt COV = 30%. Te oter ae orrepond to an nreae by 0% of te oeffent of varaton of ea varable. It an be een tat a mall varaton n te oeffent of varaton of gly affet te
CD urve of te afety fator. However, te CD le entve to te varablty of. or, one an note tat te varaton of t unertante doe not gnfantly affet te falure probablty. CD 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 EVD Exp D 0 0.5.5 2 2.5 gure 3: Effet of te probablty dtrbuton of on te CD of CD 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 referene ae COV = 30% COV = 20% COV = 40% 0.5 0.7 0.9..3.5.7.9 gure 4: Effet of te varablty of te random varable on te CD of LAC 3D reult Determnt reult T eton foue on te determnaton of te afety fator defned wt repet to te ol ear trengt aratert and tan of a lope n te preene of peudo-tat eartquake loadng ung numeral LAC 3D mulaton. Te rtal determnt lope egt found n lmt analy for a em oeffent of 0.3 ued ere (.e. H = 7. 284m ). A a reult of everal verfaton run, te ol doman and te me own n fgure (5) are ued n te analy. Te regon wa dvded orzontally nto 30 zone and vertally nto 20 zone. 0 m 7.284 m 2 Logartm pral falure meanm 5 m Z Z 3 m Y X Y X gure 5: Slope geometry and me ued n LAC 3D gure 6: Logartm pral falure meanm and LAC 3D veloty feld Sne t a 2D ae, all dplaement n te y dreton are fxed. or te dplaement boundary ondton n te ( X, Z ) plan, te bottom boundary wa aumed to be fxed and te rgt and left vertal boundare were ontraned n moton n te orzontal dreton. A onventonal elatperfetly plat model baed on te Mor-Coulomb falure rteron ued to repreent te ol. Te ol elat properte ued are te ear modulu G = 00MPa and te bulk modulu = 33MPa
(for w te equvalent Young modulu and Poon rato are repetvely E = 240MPa and ν = 0.2 ). Te value of te ol ear trengt parameter ued n te analy are: = 20kPa and o =ψ = 30. Te orrepondng afety fator wa found equal to 0.96 w lgtly lower tan tat of te lmt analy reult (.e. = ). T good agreement between te two oluton may be explaned a follow: te ol ma n moton nearly mlar n bot elato-plat (fnte dfferene) and rgd-plat (lmt analy) approae (f. gure 6). It wa eked tat a more refned me mprove te reult by only % (.e. only a % reduton n te afety fator) wt a very g nreae n te alulaton tme (by 285%). Tu, te me preented above wll be ued n all ubequent alulaton. Relablty ndex, degn pont and falure probablty Table (2) preent te Haofer-Lnd relablty ndex and te orrepondng degn pont for dfferent value of te mean of ung te repone urfae metodology preented above. Te ae of orrelated and unorrelated ear trengt parameter for normal dtrbuton are ondered n t eton. Te ame onluon found n lmt analy reman vald ere. By omparng te preent reult to te lmt analy one [f. table a) and b)], one an ee tat te relablty ndex alulated by LAC 3D mulaton ger tan tat determned by te lmt analy model for mall value of. or ger value, te relablty ndex alulated ung LAC 3D mulaton beome maller (.e. more onervatve) tan tat of te log-pral meanm. Table 2: Relablty reult for dfferent value of te mean of a) Normal varable Unorrelated ear trengt parameter (kpa) ( o) β P (%) f γ 0.05 2.24 27.48 0.054 2.3.67.63..07 0. 4.3 28.09 0.7.66 4.85.40.08.7 0.2 7.90 29.8 0.2226 0.70 24.9.2.03. 0.3 20.00 30.00 0.3000 0.00 50.00.00.00.00 b) Normal varable Correlated ear trengt parameter o β P (%) (kpa) ( ) f γ 0.05 0.39 3.00 0.056 2.63 0.43.93 0.96. 0.0 2.82 30.42 0.9 2.0.80.56 0.98.9 0.20 7.89 29.93 0.238 0.82 20.72..003.6 0.30 20.00 30.00 0.3000 0.00 50.00.00.00.00 Sentvty of te falure probablty to te varablty of ea random varable To tudy te effet of te varablty of te ol ear trengt parameter and te em oeffent on te falure probablty, gure 7 ow te ORM falure probablty veru te oeffent of varaton of, and. or ea urve, te oeffent of varaton of two parameter are et equal to te value gven by te "referene ae" and te oeffent of varaton of te trd parameter vared over te range 0% - 30%. Te ae of normal varable and orrelated ear trengt parameter ondered. Te mean value of te em oeffent taken equal to 0.. Te reult ow tat te falure probablty gly nfluened by te oeffent of varaton of te oeon. Te greater te atter n, te ger te falure probablty of te lope. T mean tat te aurate determnaton of te dtrbuton of t parameter very mportant n obtanng relable probablt reult. In ontrat, te falure probablty le entve to te varaton of COV. or, one an note tat te varaton of t unertante a a mnor effet on te falure probablty. Tee fndng are n onformty wt te reult of te lmt analy model.
Pf (%) 9 8 7 6 5 4 3 2 0 0 5 20 25 30 Coeffent of varaton (%) gure 3: alure probablty veru te oeffent of varaton CONCLUSION T paper preent a relablty-baed analy of te tablty of lope n em area. Qua-tat repreentaton of eartquake effet ung te em oeffent onept adopted. Two determnt approae are ued. Te frt one provde upper-bound oluton n te framework of te knematal metod of te lmt analy teory. Te eond one baed on numeral modelng of te lope tablty ung te Lagrangan explt fnte dfferene ode LAC 3D. Te random varable ondered n te analy are te ol ear trengt parameter and, and te orzontal em oeffent. Te repone urfae metodology wa ued to fnd an approxmaton of te analytally-unknown performane funton and te orrepondng relablty ndex wle ung LAC 3D mulaton. However, dret mnmzaton of te relablty ndex wa performed wle ung te lmt analy approa. Te man onluon of te paper an be ummarzed a follow: - Te relablty ndex obtaned ung te probablt urfae wa found maller (.e. more rtal) tan te one alulated by ung te rtal determnt urfae; - or normal and non-normal dtrbuton, orrelated and unorrelated ear trengt parameter, te relablty ndex dereae wt te nreae of te mean of (.e. wt te dereae of te lope afety fator S ). or unorrelated ear trengt parameter, te value of and at te degn pont were found maller tan ter repetve mean value (retng parameter). However, wa found to be ger tan t mean value ne a drvng parameter. Te retane fator and dereae wt te nreae of te em oeffent. However, te load fator γ frt nreae and ten dereae wt te nreae of. Te later reult an be explaned a follow: for mall value of te em oeffent, te lope tablty not gly affeted by te em oeffent and tu a mall load fator wa obtaned; - Te aumpton of unorrelated ear trengt parameter wa found onervatve n omparon to te aumpton of negatvely orrelated ear trengt parameter; - Te aumpton of normal varable wa found onervatve for g value of te afety fator; - Te aumpton of an Exp D for te em oeffent gly onervatve n omparon to te EVD; - A mall varaton n te oeffent of varaton of a a gnfant effet on te CD urve of te afety fator and onequently on te falure probablty; - Te omparon of LAC 3D reult wt te lmt analy one a own tat te relablty ndex alulated by LAC 3D mulaton wa ger tan tat determned by te lmt analy model for mall value of. However, for ger value of, te relablty ndex determned ung LAC 3D mulaton a beame maller (.e. more onervatve) tan tat of te log-pral meanm.
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