Computers and Geotechnics

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1 Computers and Geotens 36 (2009) Contents lsts avalable at SeneDret Computers and Geotens journal omepage: A system relablty approa for evaluatng stablty of rok wedges wt orrelated falure modes Danqng L *, Cuangbng Zou, Wenbo Lu, Qngu Jang State Key Laboratory of Water Resoures and Hydropower Engneerng Sene, Wuan Unversty, 8 Dongu Sout Road, Wuan , PR Cna artle nfo abstrat Artle story: Reeved 13 January 2009 Reeved n revsed form 5 Aprl 2009 Aepted 14 May 2009 Avalable onlne 13 June 2009 Keywords: Rok wedge System relablty N-dmensonal equvalent metod Probablst fault tree Correlated falure modes Ts paper proposes a system relablty approa for evaluatng te stabltes of rok wedgeonsderng multple orrelated falure modes. A probablst fault tree s employed to model te system aspets of te problem. Te system relablty analyss s performed usng an N-dmensonal equvalent metod takng nto aount orrelatons between dfferent falure modes. Relablty senstvty analyses at tree dfferent levels, namely, sngle lmt state funton level, sngle falure mode level, and system relablty level, were arred out to study te effet of anges n varables on te stablty of te wedge. An example ase was analysed to llustrate te proposed approa. Te stablty of te wedge an be evaluated effently usng te proposed system relablty approa n a more systemat and quanttatve way. Te probabltes of falure of te wedge from te N-dmensonal equvalent metod are farly onsstent wt tose from te Monte Carlo smulaton metod. Te results demonstrate tat te probablty of falure wll be overestmated f te orrelatons between dfferent falure modes of te wedge are not taken nto aount. Tey also demonstrate tat te relatve mportane of dfferent falure modes to te system relablty of te wedge an dffer onsderably and be treated systematally and quanttatvely by te proposed approa. Te senstvty results are gly dependent on te seleted senstvty analyss level. Ó 2009 Elsever Ltd. All rgts reserved. 1. Introduton Wedge falures are te most frequently observed rok slope falures, and an our over a wde range of geologal and geometral ondtons. Aordngly, te study of wedge stablty s an mportant omponent for rok slope engneerng. Wedge falures are often governed by te nterseton of at least two rok dontnuty sets, w requre a soluton of fores n tree-dmensonal spae [20,15]. Te meansms leadng to wedge falures n rok slopes ave been extensvely studed n te lterature [17,31,10,11,25,30]. Te metods used nlude a stereograp projeton tenque, engneerng graps, and vetor analyss. Also, Low [21] proposed ompat losed-form equatons for te fator of safety of two-jont tetraedral wedges. Most of tese analyses are based on determnst metods w do not reflet te unertanty of te underlyng parameters. It s wdely aepted tat rok wedge stablty analyss often ontans many unertantes due to nadequate nformaton from ste araterzaton, and nerent varablty and measurement errors n te geologal and geotenal parameters. Terefore, relablty-based approaes w allow te systemat and quanttatve treatment * Correspondng autor. E-mal addresses: danqng@wu.edu.n (D. L), bzou@wu.edu.n (C. Zou), wblu@wu.edu.n (W. Lu), jq1972@yaoo.om.n (Q. Jang). of tese unertantes ave beome a top of nreasng nterest for rok slope engneerng [19,6,23]. In te lterature, Low [21] proposed losed-form equatons for te alulatons of te fator of safety for te wedge stablty n rok slopes wt an nlned upper ground surfae tat dps n te same dreton as te slope fae. Te system relablty of te wedge onsderng four falure modes s evaluated usng Cornell s bound metod [4] and Monte Carlo smulaton. Low [22] furter nvestgated te system relablty of te wedge n w te versatle four-parameter beta dstrbutons are used for derbng te bas random varables n te rok wedge stablty model. Based on Low [21], Jmenez-Rodrguez and Star [18] explored a dsjont ut-set formulaton to model te system relablty of te wedge n w ea ut-set orresponds to a falure mode of te wedge. Fadlelmula et al. [8] ompared te falure probabltes of a wedge wen te Coulomb lnear sear falure rteron and te Barton Bands non-lnear sear falure rteron were appled. However, te dsjont ut-set formulaton annot aount for te orrelatons between dfferent falure modes of te wedge. Altoug su orrelatons may be taken nto aount n Monte Carlo smulatons, t s too tme onsumng to be of pratal nterest to engneers, and te senstvty analyss at dfferent relablty levels as not been nvestgated suffently. Te objetve of ts paper s to propose a system relablty approa for evaluatng te stablty of rok wedges wt multple X/$ - see front matter Ó 2009 Elsever Ltd. All rgts reserved. do: /j.ompgeo

2 D. L et al. / Computers and Geotens 36 (2009) orrelated falure modes. Frst, lmt state funtons for wedge falures are formulated. Ten te system aspets of te wedge stablty analyss usng lmt equlbrum metods are represented by a probablst fault tree [2]. Te versatle four-parameter beta dstrbutons are used to derbe te bas random varables defned n te wedge stablty model. An N-dmensonal equvalent metod s proposed to perform te relablty analyss of te wedge. Fnally, an example s presented to llustrate te proposed metod. Te mportane of onsderng orrelatons between dfferent falure modes and te advantages of system relablty analyss over te tradtonal determnst approaes are also demonstrated. Relablty senstvty analyses at tree dfferent levels, namely, sngle lmt state funton level, sngle falure mode level, and system relablty level, are arred out to evaluate te effet of anges n varables on te stablty of te wedges. 2. Formulaton of lmt state funton for wedge falure Te frst step n te system relablty analyss of a wedge s to dentfy te relevant falure modes based on nformaton of wedge geometry and fores atng on te wedge. Tese falure modes provde a bass for te formulaton of lmt state funtons so te problem of stablty of ndvdual wedges n rok slopes onsdered. A tetraedral wedge may be formed by two ntersetng dontnutes (Fg. 1), wt a typal wedge geometry were H s te egt of slope and s te egt of wedge. Te symbols of a, X and e are te nlnaton angles of te slope fae, te upper slope surfae, and te nterseton lne of te two dontnuty planes, respetvely. d 1 and d 2 denote te dps of te dontnuty planes 1 and 2, respetvely, and 1 and 2 are te two angles n te orzontal trangular BDC w are related to strkes of te jonts [21]. Sne te presene of tenson raks, external fores due to water pressure, tensoned anors, and sesm aeleratons wll sgnfantly nrease te omplexty of te equatons for te fator of safety [15], a wedge tat s only subjeted to fores due to frton, oeson and water pressure onsdered for smplty. For te tetraedral wedge sown n Fg. 1, four dfferent falure modes may our as follows [10,21]: sldng along te lne of nterseton of two dontnuty planes formng te blok (Falure Mode 1, also alled bplane sldng); sldng along dontnuty plane 1 only (Falure Mode 2); sldng along dontnuty plane 2 only (Falure Mode 3); and a floatng falure (Falure Mode 4) w ould be ndued by g water pressure or n stu stresses, or appled fores, or bot. Note tat su falure modes represent only a lmted set of falure possbltes of rok wedges [12]. Wt te dentfed falure modes, lmt state funtons for te wedge an be formulated as follows: let X denote a vetor of all random varables tat sould be taken nto aount to evaluate te wedge stablty. Te random varables nlude orentatons, oesons and frton angles of te dontnutes, and te loadng ondtons. Te lmt state funton for te wedge stablty an be expressed as: gðxþ ¼F S 1 were F S s te fator of safety. If g(x) s less tan zero, te wedge s n te falure doman. Oterwse t s n te safe doman. Funton g(x) = 0 represents te lmt state surfae. Te formulatons of F S are dussed below. Te losed-form equatons proposed by Low and Ensten [24], Low [21], and Low [22] for te stablty of tetraedral wedges n rok slopes wt an nlned upper ground surfae are adopted for te alulaton of te fator of safety. Te equatons are explt funtons of dontnuty orentaton, wedge egt, nlnaton angles of slope fae and upper ground surfae, water pressure parameters, and frton angle and oeson of dontnutes. Te fator of safety for a bplane sldng mode s: F S ¼ a 1 b 1G w1 1 þ 3b 1 þ 3b 2 tan / 1 þ 2 a 2 b 2G w2 tan / 2 were 1, 2 and / 1 ; / 2 are te oesons and frton angles of dontnutes 1 and 2; G w1 and G w2 are te water pressure parameters; s te unt wegt of rok; = / w s te spef densty of rok were w s te unt wegt of water; a 1, a 2, b 1, b 2 are parameters dependng on te geometry of te wedge, w an be alulated by: a 1 ¼ ½sn d 2 ot d 1 os d 2 osð 1 þ 2 ÞŠ sn w snð 1 þ 2 Þ a 2 ¼ ½sn d 1 ot d 2 os d 1 osð 1 þ 2 ÞŠ sn w snð 1 þ 2 Þ b 1 ¼ a 0 sn 2 sn d 2 b 2 ¼ a 0 sn 1 sn d 1 wt: qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff sn w ¼ f1 ½sn d 1 sn d 2 osð 1 þ 2 Þþos d 1 os d 2 Š 2 g sn w a 0 ¼ ½snð 1 þ 2 Þ sn d 1 sn d 2 Š 2 ðot e ot aþ snð e 1 þ 2 Þ ¼ artan sn 1 ot d 2 þ sn 2 ot d 1 ð1þ ð2þ ð3þ ð4þ ð5þ ð6þ ð7þ ð8þ ð9þ For a pyramdal pressure dstrbuton as sown n Fg. 2 [15], one an obtan: G w1 ¼ G w2 ¼ 0:5j j ¼ H ¼ tan X 1 1 tan X tan a tan e ð10þ ð11þ Note tat H and beome equal wen X s equal to zero. If only te lengt of DC, as sown n Fg. 1, s known an be obtaned from: DC ¼ ðot e ot aþðot 1 þ ot 2 Þ ð12þ Fg. 1. Tetraedral wedge stablty model of rok slope. Note tat Eq. (2) s vald wen te followng ondtons are met:

3 1300 D. L et al. / Computers and Geotens 36 (2009) a 2 b 2G w2 b 1G w1 a 1 Z tan / 2 þ 3b 2 2 F S2 ¼ rffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2 1 þ b 1G w1 a 1 sn w ð19þ H were all symbols are as defned prevously. Eq. (19) s vald only wen: 8 a 1 b 1G w1 >< < 0 a 2 b 2G w2 b 1G w1 ð20þ a 1 Z > 0 >: 8 >< a 1 b 1G w1 > 0 >: a 2 b 2G w2 > 0 ð13þ Substtutng Eq. (2) nto Eq. (1), te lmt state funton an be obtaned as: gðxþ ¼ a 1 b 1G w1 tan / s 1 þ a 2 b 2G w2 tan / s þ 3b 1 þ 3b 2 1 ð14þ Te bas random varables n ts lmt state funton are 1, 2, G w1, G w2, d 1, d 2, / 1, / 2, 1, and 2, w are te same as tose for all te oter falure modes. Symbols a, X,, and n Eq. (13) are determnst pysal parameters. Wen Eq. (13) s not vald, several falure modes (sldng along dontnuty plane 1 only, sldng along dontnuty plane 2 only, or floatng falure) sould be onsdered. Te fator of safety, F S1, for sldng along dontnuty plane 1 only s gven by: a 1 b 1G w1 b 2G w2 a 2 Z F S1 ¼ rffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2 1 þ b 2G w2 a 2 sn w were Z ¼ os d 1 os d 2 þ sn d 1 sn d 2 osð 1 þ 2 Þ tan / 1 þ 3b 1 1 ð15þ ð16þ All oter symbols n Eqs. (15) and (16) are as defned prevously. Eq. (15) s vald only wen te followng ondtons are satsfed: 8 >< a 2 b 2G w2 < 0 >: Lne of nterseton of two planes Fg. 2. Vew normal to te lne of nterseton for pyramdal water pressure dstrbuton (After [15]). a 1 b 1G w1 b 2G w2 a 2 Z > 0 ð17þ Substtutng Eq. (15) nto Eq. (1), te lmt state funton an be obtaned: a 1 b 1G w1 b 2G w2 a 2 Z tan/ 1 þ 3b 1 1 gðxþ ¼ r ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 1 ð18þ 2 1 þ b 2G w2 a 2 snw Smlarly, te fator of safety, F S2, for sldng along dontnuty plane 2 only s obtaned as: 1 2 H Floatng falure ould be ndued by g water pressures or n stu stresses as prevously dussed. Ts enaro ours wen te followng ondtons are met: a 1 b 1G w1 b 2G w2 a 2 Z < 0 ð21þ a 2 b 2G w2 b 1G w1 a 1 Z < 0 ð22þ In addton to tese falure modes, te knematal onstrant for te formaton of a tetraedral wedge must be fulflled, and s gven by te followng nequalty: X < e < a ð23þ If te ondton represented by nequalty (23) s not satsfed, a tetraedral wedge meansm s not possble and te slope s safe as far as te wedge falure onerned. It sould be noted tat te Eqs. (1) (23) were ntally derved by Low and Ensten [24], Low [21], and Low [22], and are used n ts study for te system relablty analyss of te wedge. 3. System relablty approa Many pysal systems tat are omposed of multple omponents an be lassfed as eter seres systems, parallel systems, or ombned systems. A system of sngle omponents s a seres system f t s n a state of falure wenever any one of tomponents fals. Su a system s also alled a weakest lnk system. In oter words, te relablty or safety of te system requres tat none of te omponents fal. A system of sngle omponents s a parallel system f t s n a state of falure wen all of tomponents fal. In oter words, f any one of te omponents survves, te system remans safe. Many struturean be onsdered as a ombnaton of seres and parallel systems. Su systems are referred to aombned systems. More detaled nformaton an be found n Ang and Tang [1]. In te most general ase n te feld of system relablty evaluaton, te probablty of falure of a strutural system an be modelled by a seres of parallel systems su as falure modes. Tombned system may be defned n te form: P f ¼ P [ k \ ðg j ðxþ 0Þ j2c k ð24þ were g j (X) s te jt lmt state funton; C k denotes te kt subset representng a set of lmt states wose jont exesonsttutes te falure of te system, and te unon s over all of te subsets. In Eq. (24), C k ontans a sngle element for ea subset, k, for te speal lass of a seres system, and a sngle subset, k = 1, for te speal lass of a parallel system. Te relablty of te above general strutural system s evaluated by te followng steps: Frst, te probablty of falure of ea parallel system s evaluated. Seondly, te evaluaton of te orrelatons between te parallel systems due to ommon varables or orrelated varables s performed. Fnally, te probablty of falure for te seres system s evaluated on te bass of te results

4 D. L et al. / Computers and Geotens 36 (2009) obtaned from te frst two steps. Evaluaton of te orrelaton between a par of parallel systeman be easly arred out f te safety margns for te parallel systems are lnear. However, n general ts s not te ase. Terefore an alternatve s to nvestgate te possblty of ntrodung an equvalent lnear safety margn for ea parallel system, w wll be dussed later. Jmenez- Rodrguez et al. [19], and Jmenez-Rodrguez and Star [18] used a dsjont subset formulaton n w te performane of te system s modelled as a seres assembly of dsjonted parallel sub-systems. Te total probablty of falure of te system may be obtaned as te sum of te ndvdual falure modes. Tat s, te orrelatons between dfferent falure modes are not onsdered, w leads to an overestmaton of te system probablty of falure for te wedge. To evaluate te orrelatons between dfferent falure modes effently, a probablst fault tree s developed to solve te general system problem as derbed below Probablst fault tree analyss In te feld of system relablty evaluaton, fault tree analyss (FTA) provdes an organzed means for dentfyng soures of strutural system falure and ter nteratons tat may lead to one or more falure pats. Te fault tree provdes a rsk assessment tool by w a omplated strutural system an be managed systematally and quanttatvely. It s not pratal to present detaled dusson on ow to dentfy falure events and pats n ts paper and nterested readeran onsult Ang and Tang [1] for more nformaton. We fous ere on ow tese falure pats are modelled usng a probablst fault tree [2], w provdes a systemat way to manage multple falure modes. A probablst fault tree as tree major aratersts: bottom events, ombnaton gates, and te onnetvty between te bottom events and gates. Only AND and OR gates are urrently nluded n te probablst fault tree approa. Te AND gate s used to model a parallel system wle te OR gate s used to model a seres system. Te lmt state funtons are defned n te bottom events so tat te orrelatons between dfferent falure modes represented by te lmt state funtonan be onsdered. In a onventonal fault tree approa, owever, probablty values are frst assgned to all bottom events tat are assumed to be ndependent. Next, tey are propagated troug te log gates of te fault tree to alulate te probablty of falure. Troug a probablst fault tree, all te falure modean be defned, and te orrelatons between dfferent falure modean be taken nto onsderaton n a more ratonal way. Note tat a falure mode an nvolve one or more lmt states. By ombnng all te falure modes and te orrespondng lmt states, a system lmt state surfae an be onstruted pee by pee. Based on te four falure modes of a wedge, te system relablty of te wedge an be modelled by a probablst fault tree as sown n Fg. 3. Note tat te performane of te system s modelled as a seres of all parallel systems (falure modes). Tat s, te falure of te overall system (wedge) wll our wen any falure mode ours. For ea falure mode nvolvng several omponents, falure wll our only wen all omponents n te orrespondng parallel subsystem fal. Te performane of ea omponent s defned by a lmt state funton. Table 1 lsts te pysal nterpretaton of te lmt state funtons and te defntons of te lmt state funtonorrespondng to ea falure mode. Havng onstruted a probablst fault tree for a system, te relablty of te system an be alulated usng te system relablty approa. For relablty evaluaton of a system, te relablty of ea omponent sould be omputed frst as a probablty of falure P f, gven by: Z P f ¼ PðgðXÞ 0Þ ¼ f ðxþdx ð25þ gðxþ0 were f(x) s te probablty densty funton of te varables assoated wt te rok wedge stablty model. Te probablty of falure n Eq. (25) an be easly omputed by te omponent relablty metods, su as te FORM [13,26], te SORM [3] and te Monte Carlo smulaton metod (e.g., [1]). Te system relablty modelled by te probablst fault tree an be alulated usng dfferent system relablty evaluaton metods, su as te narrow relablty bounds metod [5], te adaptve mportane samplng metod [32], and te frst order multnormal (FOMN) metod proposed by Hoenbler and Rakwtz [16]. Te N-dmensonal equvalent metod s, owever, used to perform te system relablty analyss due to ts auray and effeny as demonstrated below N-dmensonal equvalent metod Consder te ase were a seres system omposed of n omponents. A useful frst step s to transform all safety margns assoated wt te n omponents to ter standardzed forms usng te FORM as follows: Z ¼a * T Y * þb ¼ 1; 2;...; n ð26þ Fg. 3. Probablst fault tree model for system relablty of wedge stablty.

5 1302 D. L et al. / Computers and Geotens 36 (2009) Table 1 Pysal nterpretaton of lmt state funtons for system relablty of wedge. Lmt state funton Interpretaton Eqs. g 1 ¼ F S 1 0 Wedge sldng on bot planes (2) g 2 ¼ a 1 b 1G w1 0 Wedge ontat on plane 2 (20) g 3 ¼ a 2 b2gw2 0 Wedge ontat on plane 1 (17) g 4 ¼ F S1 1 0 Wedge sldng along plane 1 (15) only g 5 ¼ a 1 b1gw1 b2gw2 Wedge floatng ondton on a 2 Z 0 s plane 2 (21) g 6 ¼ F S2 1 0 Wedge sldng along plane 1 only g 7 ¼ a 2 b2gw2 b1gw1 Wedge floatng ondton on a 1 Z 0 s plane 1 g 8 ¼ X e 0 Knemat admssblty (23) g 9 ¼ e a 0 Knemat admssblty (23) (19) (22) Let te vetor Y * of te bas varables be nreased by an nrement represented by vetor D Y *. Ten te orrespondng probablty of falure for te seres system s: " # P fs ðd Y * Þ¼1P \n ½a * T ðy* þd Y * Þþb P 0Š ¼1 ¼ 1 U n ðb * ½aŠ T D Y * ; ½qŠÞ ð32þ were [a] T s te matrx onsstng of te vetor of dreton osnes for ea falure surfae. Based on Eq. (32), te orrespondng nomnal relablty ndex nrement for Y * wt an nrement vetor D Y * an be gven by: b S ðd Y * Þ¼U 1 ½p fs ðd Y * ÞŠ ¼ U 1 ½1 U n ðb * ½aŠ T D Y * ; ½qŠÞŠ ð33þ Te safety margn of equvalent falure surfae for Y * wt an nrease of D Y * s: Z e ðd Y * Þ¼a * et Y * a * et D Y * þb e and te orrespondng equvalent relablty ndex s: ð34þ were * a ¼½a 1 ; a 2...; a n Š T s a unt vetor; * Y s a vetor of standard normal varables. b s te t relablty ndex of te t omponent. Generally, te probablty of falure for su a seres system s defned by: P f ¼ 1 U n ðb * ; ½qŠÞ ð27þ were ðb * ¼ b 1 ; b 2 ;...; b n Þ s a vetor n w te omponents are te relablty ndes of te falure elements; [q] s te orrelaton matrx for te lnear and normally dstrbuted safety margns of te falure elements; U n () s te N-dmensonal standard multnormal ntegral wt orrelaton oeffent matrx [q]. Te orrespondng relablty ndex b s s: b S ¼U 1 ½1 U n ðb * ; ½qŠÞŠ ð28þ To develop te N-dmensonal equvalent metod, te equvalent omponent onept proposed by Gollwtzer and Rakwtz [9] s used. Gollwtzer and Rakwtz [9] supposed tat tere s an equvalent lnear safety margn representng a seres or a parallel system. Its standard safety margn Z e an be expressed as: Z e ¼a * et Y * þb e ð29þ were b e s te equvalent relablty ndex for te onsdered seres or parallel system, and *e a ¼½a e 1 ; ae 2 ;...; ae m ŠT s a unt vetor. Te equvalent lnear safety margn sould meet two ondtons. Frstly, te equvalent relablty ndex b e s equal to te nomnal relablty ndex b S, namely: b e ¼ b S ð30þ Seondly, te senstvtes of b e and b S wt respet to anges n te bas random varables reman te same, e ðd Y * ¼ S ðd Y * Þ k ð31þ * * k * * D Y ¼0 D Y ¼0 n w b e ðd Y * Þ and b S ðd Y * Þ represent te equvalent relablty ndex and nomnal relablty ndex for te vetor Y * wt a small nrease vetor D Y * ; and l s a proportonal oeffent. Te vetor ~a e must be evaluated numerally. However, Gollwtzer and Rakwtz [9] dd not provde te detaled equatons to alulate ~a e. Based on te equatons (30) and (31) proposed by Gollwtzer and Rakwtz [9], te vetor ~a e an be determned usng te proposed approa as follows. b e ðd Y * Þ¼a e 1 DY 1 a e 2 DY 2 a e m DY m þ b e ð35þ were a e s equal to te dervatve of k be ðd Y * Þ wt respet to Y k, expressed as: a e ðd Y * Þ k ¼ S ðd Y * Þ k k ¼ 1; 2;...; m ð36þ * * k * * D Y ¼0 D Y ¼0 were b S ðd Y * Þ s te nomnal relablty ndex obtaned from Eq. (33). Te proportonal oeffent, l, s used to ensure tat *e a s a unt vetor, namely: vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2 3 X l ¼ s ðd Y * 2 u Þ t 4 ð37þ k k¼1 D Y * ¼0 * Te equvalent lnear safety margn representng a parallel system an also be onstruted usng a smlar metod. In general, te probablty of falure for a parallel seres system s: P f ¼ 1 U n ð b * ; ½qŠÞ ð38þ Te equvalent lnear safety margn for a parallel system an be determned by replang Eqs. (28), (32), and (33) wt te followng equatons, respetvely: b S ¼U 1 ½U n ð * b ;½qŠÞŠ P f ðdy * Þ¼P \n a * T ðy* þdy * Þþb 0 ¼ U n ð * b þ½aš T DY * ;½qŠÞ ¼1 b S ðdy * Þ¼U 1 U n ð * b þ½aš T DY * ;½qŠÞ ð39þ ð40þ ð41þ Te remander of te dervaton steps are dental wt tose for a seres system. After obtanng te equvalent lnear safety margns usng te above metod, te orrelaton oeffents, q j, between te equvalent safety margnan be gven by: q j ¼ a T a j ð42þ Te standard multnormal ntegral U n ðb * ; ½qŠÞ n te above equatons an be alulated usng te FOMN approa [16]. Experene sows tat n most ases te FOMN approa s very aurate [27] and effent wen a omputer program s used.

6 D. L et al. / Computers and Geotens 36 (2009) Numeral mplementaton for relablty omputaton In ts study, a C-language based omputer program WHUREL (WuHan Unversty Relablty omputer program for rok slopes) was developed for alulatng te relablty ndex b and te most probable falure ponts X *. WHUREL apable of alulatng te relablty of a omponent usng FORM as well as te system relablty of a seres system or a parallel system, usng te proposed N-dmensonal equvalent metod. Te Cornell s bound metod and te narrow relablty bounds metod [5], for relablty analyss of a seres system, are also nluded n WHUREL. In addton, WHUREL an perform bot te omponent relablty senstvty analyses and system relablty senstvty analyses for te bas random varables. A major advantage of WHUREL s tat te lmt state funtons, defned by te bottom events n te probablst fault tree as sown n Fg. 2, an be expressed n te form of a set of user-defned subroutnes. 4. An llustratve example 4.1. Case geometry and materal propertes As an example, te rok wedge stablty model sown n Fg. 1 s nvestgated. Te followng determnst parameters are adopted for te analyses: a =70 ( ), X =0, w = 9.8 (kn/m 3 ), and = 2.6 [21,18]. Te egt of wedge,, rangng from 10 to 30 m, s used to aount for te effet of on te system relablty of te wedge. Random varables n te wedge stablty model are assumed to be ndependent of ea oter. However, oeson and frton angle / are onsdered to be negatvely orrelated. Correlaton oeffents of q 1;/1 ¼ q 2;/2 ¼0:5 are used to model ommon sear test results n w te oeson generally dereases as te frton angle nreases and ve versa [22,23,14]. Note tat for rok dontnutes, wt te development of sear, te oeson wll beome zero and ts wll not be reversble,.e. ts loss of oeson annot be reovered. Terefore, n a forward and reversed sear pat, te oeson sould not be reoverable. Ts may not be applable for a wedge falure, but one may need to onsder su possbltes n oter applatons. Te statstal parameters for te nput varables n te wedge stablty model are lsted n Table 2. All varables n Table 2 follow a beta dstrbuton. It sould be noted tat, nstead of te normal dstrbutons or lognormal dstrbutons used n Low [21], Duzgun et al. [7], and Jmenez-Rodrguez and Star [18], all varables are modelled by te four-parameter (q, r, a, b) beta dstrbutons n w te frst two parameters are sape parameters, wle te last two parameters defne te lower and upper lmts of te range. Ts s beause te normal dstrbuton s symmetral and, teoretally, as a range from 1 to +1. For a parameter tat admts only postve values, te probablty of enroang nto te negatve realm s very small f te oeffent of varaton of te parameter s 0.25 or less [22]. Sne te lognormal dstrbuton exludes negatve values and faltates te matematal dervatons, t as been suggested n leu of te normal dstrbuton. Te lognormal dstrbuton s wtn te range of (0, +1). Bot normal dstrbutons and lognormal dstrbutons are not sutable beause all te onsdered varables n te wedge stablty model admt only postve values, and are bounded. Compared wt te normal or lognormal dstrbutons, te four-parameter beta dstrbuton s more versatle as demonstrated by Low [22], Low [23], w an be symmetral f q = r or nonsymmetral f q r. Te mean, l X, and standard devaton, r X, of random varable X followng a beta dstrbuton are gven by (e.g., [1]): l X ¼ a þ q ðb aþ q þ r ð43þ sffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff qr r X ¼ ðb ðq þ rþ 2 aþ2 ð44þ ðq þ r þ 1Þ In Table 2, t s assumed tat 1 = 2, and / 1 = / 2. Tese assumptons, wle not essental, are adopted for smplty. Te proposed metod an andle mu more omplated enaros Analyss results WHUREL s employed to perform te relablty analyses of te onsdered wedge stablty. To llustrate te omputatons of te system relablty of te wedge usng te N-dmensonal equvalent metod, we frst take te wedge wt a egt of 20 m. Aan be seen from Fg. 3, te system of te wedge stablty omposed of four falure modes wt a seres-onneted relatonsp between tem. Among of tem, te Falure Mode 1 onssts of fve elements wt a parallel-onneted relatonsp between tem. Te relablty ndes for sngle elements are alulated usng te FORM, w are 1.613, 4.296, 1.388, 4.298, and for g 1 6 0, g 2 6 0, g 3 6 0, g 8 6 0, and g 9 6 0, respetvely. Te orrelaton matrx an be determned usng Eq. (42): :133 0:654 0:212 0:179 0: :111 0: : ½qŠ¼ 0:654 0: :343 0: :212 0: : : :179 0: :535 0: ð45þ Ten, te relablty ndex for te Falure Mode 1 an be alulated as usng Eq. (39), and te orrespondng probablty of falure s usng Eq. (38). Applyng te N-dmensonal equvalent metod, te equvalent lnear safety margn Z 1 for te parallel system orrespondng to te Falure Mode 1 an be gven by: Z 1 ¼ 0: : :36/ 1 0:428/ 2 þ 0:215d 1 0:0215d 2 0: :106 2 þ 0:447G w þ 1:905 ð46þ Table 2 Summary statsts of bas random varables n te wedge stablty model. Varable Mean, l Standard devaton, r Lower bound, a Upper bound, b 1 (kpa) (kpa) / 1 ( ) / 2 ( ) d 1 ( ) d 2 ( ) ( ) ( ) G w Smlarly, te relablty ndes for te Falure Modes 2 4 are 1.388, 4.362, and 5.563, respetvely. In addton, te equvalent lnear safety margns Z 2 Z 4 for te parallel systemorrespondng to te Falure Modes 2 4 as sown n Fg. 3 an be obtaned. Based on te equvalent lnear safety margns Z 1 Z 4, te orrelaton matrx for Z 1 Z 4 an be alulated usng Eq. (42) as follows: :355 0:269 0:0214 0: :0694 0:535 ½qŠ ¼6 7 ð47þ 4 0:269 0: : :0214 0:535 0:0208 1

7 1304 D. L et al. / Computers and Geotens 36 (2009) Te system relablty ndex of te wedge s usng Eq. (28), and te orrespondng system probablty of falure s usng Eq. (27). Te orrespondng equvalent lnear safety margn for te seres system onsstng of te above four falure modes as sown n Fg. 3 an be expressed as: Z ¼0: þ 0:190 2 þ 0:109/ 1 þ 0:129/ 2 þ 0:0296d 1 0:449d 2 þ 0:119 1 þ 0: :783G w þ 1:263 ð48þ To ondut te verfaton and valdaton for te N-dmensonal equvalent metod, te results obtaned from te N-dmensonal equvalent metod are ompared wt tose obtaned from te Monte Carlo smulaton metod. For smplty and llustratve purpose, all te varables n Table 2 are tentatvely assumed to be lognormal dstrbutons wt te means and standard devatons as sown n te table. Moreover, te orrelaton oeffents of q 1;/1 ¼ q 2;/2 ¼0:5 are used agan. Table 3 sows te system relablty between te Monte Carlo smulaton metod and te N- dmensonal equvalent metod. Note tat farly good agreement was obtaned between te results usng te N-dmensonal equvalent metod and te exat solutons usng te Monte Carlo smulaton metod wen te wedge egt was vared from 10 m to 30 m. For nstane, te resultng relatve errors n te system relablty ndex and te system probablty of falure are smaller tan 4% and 2%, respetvely. Tus, te N-dmensonal equvalent metod an be used to ompute system relablty effently and aurately. Applyng te proedure demonstrated earler, te relablty ndes and te orrespondng probabltes of falure for varous wedge egtan be determned. To reflet te ontrbuton of ea falure mode to te system probablty of falure for te wedge, te probabltes of falure on a log ale for ea falure mode of te wedge are sown n Fg. 4 based on te data n Table 2. Te ablty to ompute su ontrbutons s a valuable feature of te proposed metod sne t provdes quanttatve nformaton for mprovng te desgn proess for rok slopes wt potental wedge falure problems. It an be seen from Fg. 4 tat te relatve nfluene of four falure modes on te system relablty an dffer onsderably. Falure Modes 1 (Bplane falure) and 2 (Plane 1 falure only) are te falure modeontrbutng most to te system probablty of falure. For nstane, for a wedge wt a egt of 20 m, te probabltes of falure are , , and for Falure Modes 1 to 4, respetvely. In addton, Falure Mode 1 beomes te most lkely falure mode wt nreasng wedge egt. For example, te probablty of falure for te wedge wt a egt of 30 m assoated wt Falure Mode 1 s 0.37 w s sgnfantly ger tan te assoated wt Falure Mode 2. Su useful nformaton for rok slope desgn would not ave been revealed n a onventonal determnst desgn approa or n a sngle falure mode relablty analyss. Te Table 3 Comparson of system relablty between N-dmensonal equvalent metod and Monte Carlo smulaton metod. (m) N-dmensonal equvalent metod Monte Carlo smulaton metod % Errors n b and P f b P f b 0 P f0 b b 0 / b 0 100% Note: Te sample sze of Monte Carlo smulaton s 100,000. P f P f0 / P f0 100% Probablty of falure E-3 1E-4 1E-5 1E-6 1E-7 1E-8 1E-9 Falure Mode 1 (Bplane falure) Falure Mode 2 (Plane 1 falure) 1E-10 Falure Mode 3 (Plane 2 falure) Falure Mode 4 (Floatng falure) 1E Hegt of rok wedge (m) Fg. 4. Comparson between probabltes of falure for dfferent falure modes. probablty of falure for Falure Mode 1 s gly senstve to te wedge egt, wle te wedge egt as almost no nfluene on te probablty of falure for Falure Modes 2 and 4. For example, wen te wedge egt vares from 10 to 30 m, te probablty of falure for Falure Mode 1 nreases from to It an be seen tat te ontrbuton of ea falure mode to te system probablty of falure depends on te geometry of te wedge, and te oeson and frton angle of te dontnutes formng te wedge. It sould be noted, owever, tat wen te egt of te rok wedge nreases, te areas of planes 1 and 2 also nrease. Based on te onepts of spatal varaton [29], we sould, n teory, use te redued varanes for varables su aoeson and frton angle. For smplty, te spatal varaton s not taken nto onsderaton n te present study. However, ts sould be nvestgated furter so tat te omparson between system relablty for varous wedge egtan be onduted n a more ratonal way. As ndated earler, Falure Modes 1 and 2 are te falure modes ontrbutng most to te overall probablty of falure. Terefore, te orrelaton between Falure Modes 1 and 2 sould be nvestgated. Fg. 5 sows te orrelaton oeffent between Falure Modes 1 and 2 as a funton of te wedge egt. It an be seen tat te orrelaton oeffents vary from to wt nreasng wedge egt, w mples tat Falure Modes 1 and 2 are orrelated falure modes rater tan ndependent falure modes. Torrelaton wll sgnfantly affet te stablty of te wedge and sould be properly taken nto aount. Negletng te orrelatons between dfferent falure modes of te wedge wll lead to an overestmated system probablty of falure as demonstrated n Fg. 6. In addton, due to te nrease n te probabltes of falure assoated wt Falure Modes 1 and 2 wt nreasng wedge egt, te orrelaton between Falure Modes 1 and 2 beomes stronger. To ompare te system probabltes of te falure of te wedge usng dfferent metods, Fg. 6 sows te system probabltes of te falure of te wedge as a funton of wedge egt usng Cornell s bound metod [4], Jmenez-Rodrguez and Star s metod [18], and te N-dmensonal equvalent metod. Note tat te system probablty bounds usng Cornell s bound metod are too wde to be of any pratal nterest. However, wen te probabltes of falure for all falure modes are very small for small wedge egts, te system probablty bounds beome mu narrower, and ene suffently aurate. Te system probablty bounds usng te proposed N-dmensonal equvalent metod are fully wtn te omputed Cornell s bounds, w also ndate tat te proposed N-

8 D. L et al. / Computers and Geotens 36 (2009) Correlaton oeffent System probablty of falure Hegt of rok wedge (m) Fg. 5. Correlaton oeffent between Falure Modes 1 and 2 as a funton of wedge egt. Lower bound of Cornell's metod Te N-dmensonal equvalent metod Upper bound of Cornell's metod Jmenez-Rodrguez and Star's metod Hegt of rok wedge (m) Fg. 6. Comparson between system probabltes of falure usng dfferent metods. dmensonal equvalent metod s vald. On te oter and, te system probablty bounds usng te Jmenez-Rodrguez and Star s metod are ger tan te Cornell upper bounds beause Jmenez-Rodrguez and Star [18] assumed tat te system probablty of falure for te wedge an be taken as te sum of probabltes of falure assoated wt te onsdered four falure modes. Aordngly, te system probablty of falure of te wedge wll be overestmated, espeally for larger wedge egts. For te wedge wt a egt of 30 m, te system probabltes of falure usng te proposed metod and te Jmenez-Rodrguez and Star s metod are and 0.453, respetvely. It sould be noted, owever, tat te dfferene n te system probablty of falure among te above tree metods beomes very small wen te wedge egt s relatvely small. For example, for a wedge wt a egt of 15 m, te system probabltes of falure usng te N-dmensonal equvalent metod, te Jmenez-Rodrguez and Star s metod, and Cornell s bound metod are , , and (0.0825, ), respetvely. As expeted, te system probabltes of falure appear to be smlar. In prate, te negatve orrelaton between oeson and frton angle s often not modelled, to smplfy omputatons. To aount for te effet of negatve orrelaton between te oeson and frton angle, te varaton of relablty ndex wt te orrelaton oeffent q between and / s sown n Fg. 7. Note tat te system relablty ndex slgtly nreases wt q. For nstane, wen q vares from 0 to 0.99, te system relablty ndex only nreases from to Te system relablty of te wedge wll be underestmated f te orrelaton between and / s not taken nto onsderaton Senstvty analyss An analyss s performed to assess te senstvty of te random varables to te relablty of te wedge for ea falure mode as well as to te overall system performane. Su a senstvty analyss ould be useful n ost analyss and desgn plannng. For example, f te senstvty of a varable s low tere s lttle need to be very aurate about te determnaton of ts varable. Also, f neessary, te varable mgt well be treated as a determnst rater tan a random varable, w wll redue te dmensonalty of te spae of random varables. Su analyses are arred out usng WHUREL at tree levels, namely, te sngle lmt state funton level, te sngle falure mode level, and te system relablty level, w are dussed below. Wen te bas random varables are not ndependent, te senstvty oeffents defned n ts study are not nformatve n relaton to te bas random varables beause of te transformaton to ndependent standardzed spae. For ts reason, te oeson and frton angle are assumed to be ndependent, even toug te statsts of bas random varables wt beta dstrbutons n Table 2 are used agan. It sould be noted tat all senstvty oeffents defned and omputed n ts paper are dmensonless. Table 4 sows te FORM results for te ase of sngle lmt state funtons assoated wt te wedge stablty. Te results nlude te senstvty oeffents a * representng te senstvty of te omputed relablty results to anges n te random varables as defned n FORM, togeter wt te omputed desgn ponts X * orrespondng to te most lkely falure pont transformed bak to te orgnal spae. A postve sgn s often used wen te orrespondng bas varable s a load varable, wle a negatve sgn s often used wen t s a resstane varable [28]. Note tat te larger s te absolute value of te t omponent of a * orrespondng to random varable x, te ger s te senstvty wt respet to te t random varable x. At te sngle lmt state funton level, te relablty assoated wt g n Falure Mode 1 s manly senstve to anges n 2, System relablty ndex, β System relablty ndex for =20 (m) ρ 1,φ 1 =ρ 2,φ Correlaton oeffent,ρ Fg. 7. Varaton of system relablty ndex wt te orrelaton oeffent.

9 1306 D. L et al. / Computers and Geotens 36 (2009) Table 4 Senstvty oeffents and desgn ponts of bas random varables. Varable g g g X * a * X * a * X * a * 1 (kpa) 3.87E E E E E+01 2 (kpa) 3.38E E E E E01 / 1 ( ) 3.18E E E E E+01 / 2 ( ) 3.43E E E E E01 d 1 ( ) 5.04E E E E E E02 d 2 ( ) 4.84E E E E E E01 1 ( ) 6.12E E E E E E02 2 ( ) 1.91E E E E E E02 G w 6.13E E E E E E01 Note: a * and X * represent te senstvty oeffents and te desgn ponts, respetvely. Te symbol represents tat te varable s not nluded n te lmt state funton. G w and / 1, wle / 2 s te least sgnfant random varable. For g n Falure Mode 2, / 1, G w and d 1 are sgnfant random varables wt g senstvty oeffents, wle d 2 s te least sgnfant random varable. Smlarly, te relablty assoated wt g n Falure Mode 3 s qute senstve to anges n 2, G w and / 2, wle t s nsenstve to d 1 and 1. Based on tese results, t an be seen tat te anges n te oeson and frton angle ave a sgnfant nfluene on te omputed relablty for all te tree lmt state funtons. Aordngly, te determnaton of oeson and frton angle wt suffent auray s of paramount mportane for an adequate assessment of wedge stablty. Te senstvty oeffent for G w also as a g postve value for all tree lmt state funtons, w ndates tat a good dranage system for te slope an nrease te wedge stablty. Tese fndngs orrespond postvely wt ommon understandng of rok slope engneerng prate and rok means prnples. For te senstvty analyses at te sngle falure mode level, Fg. 8 presents te senstvty oeffents of random varables for Falure Mode 1 and te ase for =20m.InFg. 8, te gest senstvty oeffent orresponds to 2, wle te lowest orresponds to d 2. Ts ndates tat anges n 2 ave a sgnfant nfluene on te relablty for Falure Mode 1, wle anges n d 2 appear to be te least sgnfant fator. In addton, te relablty for Falure Mode 1 s also senstve to G w, / 1 and 1. Compared wt te senstvty analyss results for g at te sngle lmt state funton level, te sgnfant random varables reman te same. However, te least sgnfant random varable s very dfferent from tat for g Furter study sows te system relablty senstvty oeffents for varous wedge egts (Fg. 9). It an be observed from Fg. 9 tat G w s te most sgnfant varable wt te gest senstvty oeffent regardless of te anges n te wedge egt. Tat s, te ange n G w as a sgnfant nfluene on te system relablty of te wedge. Terefore, desgnng a good dranage system for te slope s an effetve way to mprove te wedge stablty. Tonluson onsstent wt te onlusons drawn from te results at bot te sngle lmt state funton level and te sngle falure mode level. In addton, te system relablty of te wedge s qute senstve to d 2, espeally for low egts of te wedge, w ndates tat te wedge geometry s a key fator n te wedge stablty analyss, empaszng te mportane of a good geologal nvestgaton of dontnutes n te rok mass. However, d 2 s te least sgnfant fator n te sngle lmt state funton g as sown n Table 4, and te senstvty oeffent of d 2 n Falure Mode 1 as ndated n Fg. 8 s almost equal to zero. Note tat te senstvtes of relablty results n bas random varables at dfferent levelan dffer onsderably. Ts gly depends on te seleted senstvty analyss level. It an also be seen from Fg. 9 tat G w, d 2 and 2 ave larger senstvty oeffents wen te wedge egt s below 20 m, wle 2 and d 1 beome sgnfant varables wen te wedge egt s above 20 m. In general, 1, / 2 and 1 ave lttle nfluene on te system relablty of te wedge for wedge egts between 10 and 30 m Falure Mode 1, =20 (m) Senstvty oeffent φ 1 φ 2 δ 2 θ 1 θ 2 δ 1 Gw Senstvty oeffents φ 1 φ 2 δ 1 δ 2 θ 1 θ 2 G w Hegt of rok wedge (m) Fg. 8. Comparson between senstvty oeffents of random varables for Falure Mode 1. Fg. 9. Comparson of system relablty senstvty oeffents for random varables.

10 D. L et al. / Computers and Geotens 36 (2009) Summary and onlusons Ts paper as proposed a metodology for evaluaton of rok wedge stablty, usng a system relablty approa and onsderng multple orrelated falure modes. A probablst fault tree s used to model te system relablty of te wedge. Te proposed N-dmensonal equvalent metod s employed to perform te relablty omputaton. An example s presented to llustrate te proposed metodology. Several onlusonan be drawn from ts study: (1) Te stablty of a wedge an be evaluated effently usng te system relablty approa wt an N-dmensonal equvalent metod. Te system probabltes of falure for te wedge usng te N-dmensonal equvalent metod are farly onsstent wt tose obtaned from te Monte Carlo smulaton metod, w ndates tat te proposed N- dmensonal equvalent metod s vald. (2) Te system relablty of a wedge wt multple orrelated falure modean be modelled usng te probablst fault tree n a more ratonal way. Te orrelatons between te dfferent falure modean be taken nto onsderaton properly. Negletng su orrelatons wll result n an overestmated probablty of falure of te wedge. (3) Te most lkely falure mode an be dentfed usng te proposed metod, w ould be mportant for mprovng desgn and renforement measures for rok slope engneerng. Results sow tat te relatve mportane of dfferent falure modes to te system relablty an dffer onsderably. In te example, Falure Modes 1 (bplane falure) and 2 (plane 1 falure only) are sgnfantly more lkely tan Falure Modes 3 (plane 2 falure only) and 4 (floatng falure). Te ablty of te proposed approa to quantfy te relatve mportane of ea falure mode s a valuable feature tat an elp te desgner to establs prortes and deson makng for rok slope engneerng. It sould be noted tat te ontrbuton of ea falure mode to te system relablty s gly dependent on te wedge geometry, and te oeson and frton angle of te dontnutes formng te wedge. (4) Te system relablty of te wedge stablty nreases wt te negatve orrelaton between te oeson and frton angle. If su orrelaton s not taken nto aount, te system relablty of te wedge wll be underestmated. (5) Te senstvty analyss wt respet to bas random varables an be onduted at tree dfferent levels, namely, sngle lmt state funton level, sngle falure mode level, and system relablty level. Te senstvty results are gly dependent on te seleted senstvty analyss level. In te example, at te system relablty senstvty level, te water pressure parameter and te wedge geometry are sgnfant varables wt g senstvty oeffents. Terefore, to mprove te wedge stablty effetvely, a good dranage system for te partular slope sould be desgned and an adequate strutural araterzaton of te rok mass sould be onduted. Aknowledgments Ts work s supported by Natonal Natural Sene Funds for Dstngused Young Solars (Projet No ), Natonal Natural Sene Foundaton of Cna (Projet No ), and Program for New Century Exellent Talents n Unversty, Mnstry of Eduaton of Cna (Projet No. NCET ). Referenes [1] Ang HS, Tang WH. Probablty onepts n engneerng: empass on applatons to vl and envronmental engneerng. 2nd ed. New York: Jon Wley and Sons; [2] Ben H, Taker BH, Ra DS, et al. Probablst engneerng analyss usng te NESSUS software. Strut Safety 2006;28(1 2): [3] Bretung K. Asymptot approxmatons for probablty ntegrals. Probab Eng Me 1989;4(4): [4] Cornell CA. Bounds on te relablty of strutural systems. J Strut Dv (ASCE) 1967;93(1): [5] Dtlevsen O. Narrow relablty bounds for strutural systems. J Strut Me (ASCE) 1979;7(4): [6] Duzgun HSB, Basn RK. Probablst stablty evaluaton of Oppstadornet rok slope, Norway. Rok Me Rok Eng do: /s [7] Duzgun HSB, Yuemen MS, Karpuz C. A metodology for relablty-based desgn of rok slopes. Rok Me Rok Eng 2003;36(2): [8] Fadlelmula MM, Duzgun HSB, Karpuz C. Relablty-based modelng of wedge falure n rok slopes. In: Proeedngs of te 4t Asan-Paf symposum on strutural relablty and ts applatons. Hong Kong; p [9] Gollwtzer S, Rakwtz R. Equvalent omponents n frst-order system relablty. Relab Eng 1983(5): [10] Goodman RE. Introduton to rok means. 2nd ed. New York: Wley; [11] Goodman RE. Trty-fft Rankne leture: blok teory and ts applaton. Geotenque 1995;45(3): [12] Goodman RE, Keffer DS. Beavor of rok n slopes. J Geote Geoenvron Eng (ASCE) 2000;126(8): [13] Hasofer AM, Lnd NC. Exat and nvarant seond-moment ode format. J Eng Me (ASCE) 1974;100(1): [14] Hoek E. Pratal rok engneerng; <ttp:// PratalRokEngneerng.asp>, World Wde Web edton. [15] Hoek E, Bray J. Rok slope engneerng. 3rd ed. London: Insttuton of Mnng and Metallurgy; [16] Hoenbler H, Rakwtz R. Frst-order onepts n system relablty. Strut Safety 1983;1(3): [17] Jaeger JC. Frton of roks and te stablty of rok slopes. Geotenque 1971;21(2): [18] Jmenez-Rodrguez R, Star N. Rok wedge stablty analyss usng system relablty metods. Rok Me Rok Eng 2007;40(4): [19] Jmenez-Rodrguez R, Star N, Caon J. System relablty approa to rok slope stablty. Int J Rok Me Mn 2006;43(6): [20] Londe P, Vger G, Vormernger R. Te stablty of rok slopes, a treedmensonal study. J Sol Me Found Dv (ASCE) 1969;95(1): [21] Low BK. Relablty analyss of rok wedges. J Geote Geoenvron Eng (ASCE) 1997;123(6): [22] Low BK. Relablty analyss of rok slopes nvolvng orrelated nonnormals. Int J Rok Me Mn 2007;44(6): [23] Low BK. Effent probablst algortm llustrated for a rok slope. Rok Me Rok Eng 2008;41(5): [24] Low BK, Ensten. HH Smplfed relablty analyss for wedge meansms n rok slopes. In: Proeedngs of te 6t nternatonal symposum on landsldes. Rotterdam, Te Neterlands: A. A. Balkema; p [25] Natanal C. Knemat analyss of atve/passve wedge falure usng stereograp projeton. Int J Rok Me Mner S Geome Abstr 1996;33(4): [26] Rakwtz R, Fessler B. Strutural relablty under ombned random load sequene. Comput Strut 1978;9(5): [27] Tang LK, Melers RE. Improved approxmaton for multnormal ntegral. Strut Safety 1987;4(2): [28] Toft-Crstensen P, Baker MJ. Strutural relablty teory and ts applatons. Berln, Germany: Sprnger-Verlag; [29] Vanmarke E. Random felds: analyss and syntess. Cambrdge (MA): MIT Press; [30] Wang YJ, Yn JH. Wedge stablty analysonsderng dlatany of dontnutes. Rok Me Rok Eng 2002;35(2): [31] Warburton PM. Vetor stablty analyss of an arbtrary polyedral rok blok wt any number of free faes. Int J Rok Me Mner 1981;18(5): [32] Wu YT. Computatonal metods for effent strutural relablty and relablty senstvty analyss. AIAA J 1994;32(8):