Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions APPEDIX 1. FLAC MODELS AD DEIATIOS A1.1 Applied models for FLAC ode A1.1.1 Model for he veril sress omprison beween he FLAC ubiquious joins model nd he heoreil developmen in Jeger nd Cook (1979) ile Compressive srengh of shle speimen wih plne of wekness g 5 10 se mess off def hsol loop k (0,18) be90.0*(18.0-k)/18.0 lf90-be ommnd mo null mo ubi pro den 700 bulk 4.5e9 she.3e9 fri 19 o 1.4e5 en 3.5e5 pro jo 1e5 jfri 8 jng lf jen 1e6 fix y j 1 fix y j ini yvel -1e-7 j ini yvel 1e-7 j 1 se s_dmp omb sep 4000 prin be prin sigmv prin nl end_ommnd end ; def sigmv sum0.0 loop i (1,igp) sumsum+yfore(i,jgp) sigmvsum/(x(igp,jgp)-x(1,jgp)) end def ve ve(ydisp(3,1)-ydisp(3,))/(y(3,)-y(3,1)) end 18
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions ; def nl mohesion(1,1) mfifriion(1,1)*degrd jjohesion(1,1) jfijfriion(1,1)*degrd sm.0*m*os(mfi)/(1.0-(mfi)) if be90*in(be/90) hen sj-1 else divsj((1.0- n(jfi)*n(be*degrd))*(.0*be*degrd)) if divsj0.0 hen sj-1 else sj.0*j/divsj end_if end_if if sj<0 hen nlsm else nlmin(sj,sm) end_if end his nsep 100 his unbl his sigmv his nl his be his ve his yv i 1 j 1 hsol sve UCT.sv plo hold grid plo hold his 3 ross vs 4 begin 4000 skip 40 reurn A1.1. Model for homogeneous sndsone profile wih unduled ground surfe 15 0 inlinion he limb s surfe g 50,100 m m prop s5.e9 b5.9e9 d600 fri1 oh1e10 en1e10 def mon rj1.0/jzones sum0.0 loop i (130,35) y_hnge-1.1*(igp*degrd) y(i,1)y(i-1,1)+0.9*y_hnge 19
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions sumsum+(y(i-1,1)-y(1,1)) y(i,1)y(i,1)-0.*sum/i loop j (,jgp-1) y(i,j)y(i,1) + (y(i,jgp)- y(i,1))*(j-1)*rj end mon fix x i1 fix x i51 fix x y j1 his unbl se lrge solve ; ; ile k.0; 15 deg inl. prop s5.e9 b5.9e9 en5.5e6 oh7e5 fri1 d600 def k0_se loop i (1,izones) loop j (1,jzones) sxx(i,j).0*syy(i,j) end k0_se ; se grv9.81 ini xdis0 ydis0 solve sve k15-ss.sv A1.1.3 FLAC model for m hik embedded shle lyer 30m deph nd 5 0 inlinion he limb s surfe g 50,100 m m j 1 70 m u j 71 7 m m j 73 100 prop s5.e9 b5.9e9 d600 fri1 oh1e10 en1e10 j1,70 prop s.3e9 b4.5e9 d700 fri14 oh1e10 en1e10 j71,7 prop j0 j1e10 jf8 j1e10 j71,7 prop s5.e9 b5.9e9 d600 fri1 oh1e10 en1e10 j73,100 def mon rj1.0/jzones sum0.0 loop i (130,35) 0
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions y_hnge-1.1*(igp*degrd) y(i,1)y(i-1,1)+0.3*y_hnge sumsum+(y(i-1,1)-y(1,1)) y(i,1)y(i,1)-0.*sum/i loop j (,jgp-1) y(i,j)y(i,1) + (y(i,jgp)- y(i,1))*(j-1)*rj end mon se grv9.81 fix x i1 fix x i51 fix x y j1 his ydis i76 j100 solve ; ; ile k.0; m shle; 05 deg prop s5.e9 b5.9e9 en5.5e6 oh7e5 fri1 d600 j1 70 ;SST prop s.3e9 b4.5e9 en3.5e6 oh4.4e5 fri14 d700 j71,7 ;Shle prop j0 j1e5 jf8 j1e6 i36 10 j71,7 prop s5.e9 b5.9e9 en5.5e6 oh7e5 fri1 d600 j73,100;sst def k0_se loop i (1,izones) loop j (1,jzones) sxx(i,j).0*syy(i,j) end k0_se ini xdis0 ydis0 se lrge solve A1. Sress nlysis Mny uhors, suh s Singh (1979) nd Fed (199), disuss sher sress s he only sress h riggers slope filure. Kulhwy e l. (1973) seem o be he firs o menion he differene beween he Mohr- Coulomb sher filure rierion nd he sress filure rierion developed uhors. Ug finie elemen 1
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions nlysis, hese luled he sfey for bsed on sress level fer he ssumpion h he rok is brough o filure by inreg he vlue of one of he prinipl sresses 1, while holding he oher, 3, onsn. Figure A1.1 shows flw or miro frure in wodimensionl Cresin oordine sysem. Le us ssume h pir of sresses s on he flw (presened by horizonl nd veril sress omponens). Their resul is he sress norml o he flw s plne. Sine he flw is no olliner wih one of he prinipl sress direions, here is some sher sress he flw s plne s well. These sresses in he virgin sress ondiions re in equilibrium he flw, so here is no flw exension, propgion or olesene wih he neighbouring ones. These sress ondiions re known s virgin sress se nd ould be denoed s virgin horizonl ( ), virgin veril ( ) nd virgin sher ( ) sress omponens. In his se we n denoe he sress norml o he flw s plne s. XY XX YY mining iviies (Figure A1.1b) bring bou hnge in he sress se, known s resuln sress se. These sresses ould be denoed s resuln horizonl XX sress ( ), resuln veril sress ( ), nd resuln sher sress ( ). The firs wo resuln sress omponens (horizonl nd veril) will form new resuln se, norml o sress of he flw s plne ( ). XY YY
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions Y ) X YY XY XY XX XX YY Y b) X YY XY XY XX XX YY Figure A1.1 Sress se in he infiniesiml flw, in virgin ondiions, nd b) fer exvion Therefore, we n sy h ny possible hnges in he flw pern (exension, new flw propgion, olesene beween mirorks in he rok or even filure) will resul from he differene beween hose wo loding ondiions. Hene, i follows h: Δ XX XX XX (A1.1) Δ YY YY YY (A1.) 3
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions Δ XY XY XY (A1.3) where Δ XX, Δ nd Δ re he sress differenes YY XY for he horizonl, veril, nd sher sress omponens. Olson (1993) uses his priniple o lule sress hnges used by eoni irregulriies. One n esily see h he sress differene in Equions A1.1 A1.3 hs negive sign in he se when he meril relxes from he virgin sress se or posiive sign in he se of inresed loding when ug rok mehnis sign onversion. orml fore o he filure plne is in use s bsl elemen for he limi equilibrium mehods in slope sbiliy nlyses. As menioned in Chper 1, observed filure plnes wih embedded nisoropi weker lyers re minly prllel o he sedimenion. Aording o he ype of horizonl nd veril sresses (virgin nd resuln), he norml o he filure plne indued sress (fer Equions A1.1- A1.3) n be ombinion of wo ompressive sresses, ombinion of wo ensile sresses or ombinion of ensile nd ompressive sress. Figure A1. shows he se of bixil ension pplied o he meril plne of wekness. In mrix noion, sress rnsforms s follows [ ] [ λ][ ][ λ] T (A1.4) 4
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions where [ λ ] is direionl oe mrix. For he ngle α shown in Figure A1. osα α α [ λ] osα (A1.5) Applying Equion A1.4, we ould wrie Y Y Δ XX Δ Y'Y' X Δ XX X Δ YY Figure A1. Sress se pplied o he plne of wekness nd remoe bixil loding λ λ + λ λ + λ λ + λ os α + α osα + α osα + 1 1 1 1 1 1 1 λ 1 α (A1.6) λ λ + λ λ + λ λ + λ 1 1 1 α osα + 1 1 1 1 os α 1 1 1 λ α + α osα (A1.7) λ λ + λ λ + λ λ + λ 1 1 1 α α osα α osα + 1 1 1 1 1 λ os α (A1.8) 5
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions Hene, for he new se of he o-ordine sysem (X Y ), he norml sress differene hve he form of Δ Y' Y' Δ will Δ Y' Y' Δ Δ XX α Δ α osα + Δ os α XY YY (A1.9) where Δ XX, Δ YY nd Δ XY re he sress differenes beween he sress se fer slope exvion nd he virgin sress se for he horizonl, veril nd sher sress omponens respeively. See Equions A1.1 o A1.3. A1.3 Equions in Chper 4 A1.3.1 Equion 4.16 K D I 1 ξ ξ n os 1 dξ (A1.10) ξ eple u ξ. Then du os dξ nd Equion A1.10 hs he form: K D I 1 n du u (A1.) 6
Appendix 1. FLAC models nd derivions (A1.1) u r n 1 (A1.13) r r n 1 1 (A1.14) r n 1 (A1.15) r n 1 A1.3. Equion 4.19 (A1.16) 0 1 Δ r n n Dividing boh sides on Equion A1.16 by n : (A1.17) 0 1 Δ r 7 Universiy of Preori ed Krprov, K (007)
Appendix 1. FLAC models nd derivions (A1.18) Δ r (A1.19) ( ) Δ r (A1.0) ( ) Δ (A1.1) Δ 5 0 5 0.. (A1.) Δ 0.5 os r (A1.3) Δ 5 0. os r epling wih l + 8 Universiy of Preori ed Krprov, K (007)
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions l r 0. 5Δ os (A1.4) A1.3.3 Equion 4.1 l ( / ) ( 0. 5 Δ ) r os / (A1.5) ( l + ) ( / ) r os( 0. 5 Δ / ) (A1.6) [ ( l + ) / ] os ( / ) ( 0. 5 Δ / ) (A1.7) os ( 0. 5 Δ / ) ( / ) [ ( l + ) / ] (A1.8) 0. 5 Δ / ros ( / ) [ ( l + ) / ] (A1.9) Δ P ros ( / ) [ ( l + ) / ] (A1.30) A1.3.4 Equion 4.4 Subsiuing Equion 4.19 ino Equion 4.3: 9
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions ( / ) ( 0. 5Δ ) + r os / (A1.31) nd ( / ) ( 0. 5Δ ) r os / 1 (A1.3) os ( / ) ( 0. 5Δ / ) 1 (A1.33) Δ ( ) / os (A1.34) e n reple he lef hnd side of he Equion A1.34 wih os ( f ), whih is shown in Equions A1.35 nd A1.35b below. os + or os (A1.35) (A1.35b) Le us firs ombine Equions A1.34 nd A1.35. Then we hve: Δ os + os (A1.36) 30
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions As he os υ os( υ), hen he Equion A1.36 hs four possible soluions. If he boh rgumens re wih he sme sign ( + or - ), hen fer pplying he inverse oe funion he Equion A1.36 n be wrien s: + Δ (A1.37) Dividing boh sides of Equion A1.37 by / we obin wih furher mnipulion: + Δ (A1.38) I is seen h Equion A1.38 is n impossible soluion beuse he lef-hnd side of he equion lwys will hve negive vlue ( nd re rel posiive numbers), while he righ-hnd side lwys will be posiive. If we ssume h boh rgumens of oe funions in Equion A1.36 re wih opposie signs, hen we will hve: Δ +. (A1.39) nd similr o Equion A1.38 we n wrie: + Δ (A1.40) I is lso seen h Equion A1.40 is impossible soluion beuse lef-hnd side of he equion lwys will hve vlue bigger hn one, while he righ-hnd side lwys will hve vlue lower hn one. 31
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions Le us now ombine Equions A1.34 nd A1.35b. Then we hve gin wo opions: boh oe rgumens re eiher he sme or opposie sign. Le us firs onsider he se wih he sme sign rgumens. Hene, we will hve equion in he form of: os Therefore, Δ os (A1.41) Δ (A1.4) Dividing boh sides of Equion A1.40 by /, we obin fer furher mnipulion: Δ. (A1.43) Equion A1.43 hs only mening if > 0 (priulrly, when >.) nd n be used in he furher beuse is boh sides re wih he sme sign. Afer ssuming h is smller hn, boh sides of Equion A1.43 re posiive nd smller hn one. If we ssume h he oe rgumens in Equion A1.41 hve differen signs, hen we n wrie: Δ (A1.44) nd Δ. (A1.45) I n be seen h Equion A1.45 is lso possible soluion beuse boh sides of he equion re he 3
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions sme sign nd smller hn one in ses where On he oher hnd he ondiion >. > does no omply wih Equion 4.3 nd Figure 4.9, Chper 4. Therefore, he only possible soluion is he equion A1.43. If, hen we will hve pre-exising ensile frure propgion nd onsequenly, Δ 0. Hene we n wrie P (A1.46) A1.4 Equions in Chper 5 Equion 5.16 l nδ nφ + osδ Muliply boh sides of Equion A1.47 by hve l δ nφ osδ + (A1.47) os δ, we will (A1.48) Subsiuing osδ 1 δ in Equion A1.48: δ nφ 1 δ + l l If we rnsfer he oeffiien (A1.49) from righ-hnd side of Equion A1.48 o he lef-hnd side nd squre boh sides: l l δ δ + n φ δ n φ (A1.50) 33
Universiy of Preori ed Krprov, K (007) Appendix 1. FLAC models nd derivions Equion A1.50 beomes fer simplifiion: δ l l ( 1 + n φ ) δ + n φ 0 (A1.51) This is qudri equion in δ in whih here is rel soluion only if: ( ) l φ l 4 4 1 + n n φ 0. A1.5) Equion A1.51 simplifies o se φ l 0 (A1.53) where seφ l (A1.54) nd 1 l φ se (A1.55) 34