High-Order FEM Formulation for 3-D Slope Instability

Transcription

1 Appld Mathmatcs, 013, 4, Publshd Onln May 013 ( Hgh-Ordr FEM Formulaton for 3-D Slop Instablty Twar Ram Chandra 1, Bhandary tra Prakash, Yatab Ryuch 1 Go-Dsastr Laboratory, Graduat School of Scnc and Engnrng, Ehm Unvrsty, Matsuyama, Japan Graduat School of Scnc and Engnrng, Ehm Unvrsty, Matsuyama, Japan Emal: rctwar1975@gmal.com Rcvd Fbruary 7, 013; rvsd March 13, 013; accptd March 0, 013 Copyrght 013 Twar Ram Chandra t al. Ths s an opn accss artcl dstrbutd undr th Cratv Commons Attrbuton Lcns, whch prmts unrstrctd us, dstrbuton, and rproducton n any mdum, provdd th orgnal work s proprly ctd. ABSTRACT Hgh-ordr fnt lmnt mthod (FEM) formulaton also rfrrd to as spctral lmnt mthod (SEM) formulaton s currntly mplmntd n ths papr for 3-dmnsonal (3-D) lasto-plastc problms n stablty assssmnt of largscal slops (vgtatd and barrn slops) n dffrnt nstablty condtons such as ssmc and saturaton. W hav rvwd th SEM formulaton, and hav sought ts applcablty for vgtatd slops. Utlzng p (hgh-ordr polynomal dgr or spctral dgrs) and h (msh opraton for qualty mshng n rqurd lmntal budgts) rfnng tchnqus n th xstng FEM, th complxty of problm doman can b wll addrssd n gratr numrcal stablty. Unlk th xstng FEM formulaton, ths hgh-ordr FEM mploys th sam ntgraton and ntrpolaton ponts to achv a progrssv rspons of th nstablty, whch drastcally rducs th computatonal costs (formaton of dagonalzd mass matrx) and offrs sgnfcant bnfts to slop nstablty computatons for sral and paralll mplmntatons. Wth ths formulaton, w hav achvd th followng thr qualts n slop nstablty modlng: 1) gomtrc flxblty of th fnt lmnts, ) hgh computatonal ffcncy, and 3) rlabl spctral accuracy. A sampl problm has also bn prsntd n ths papr, whch has accommodatd all aforsad numrcal qualts. Kywords: Fnt Elmnt Mthod; Spctral Elmnt Mthod; Slop Instablty; Vgtatd and Barrn Slops; Paralll Algorthm 1. Introducton Th hgh-ordr fnt lmnt mthod (FEM) s also rfrrd to as spctral lmnt mthod (SEM), whch s orgnally appld n flud mchancs, but s now bng appld by many rsarchrs n many flds. Bcaus of ts capacty to dlvr low numrcal dsprson wth rspct to xstng fnt lmnt mthods, many rsarchrs hav adoptd SEM mthods n ssmc wav propagaton analyss (.g., [1-10]). Th SEM mthod combns th flxblty of FEM wth th accuracy of a spctral approach, adoptng th hxahdral lmnts satsfactorly, whch rprsnts th complxts of problm doman. Th doman of th hxahdral lmnts s dscrtzd usng hgh-dgr Lagrang ntrpolants, and ntgraton ovr an lmnt s accomplshd basd on th Gauss- Lobatto-Lgndr (GLL) ntgraton rul. A combnaton of dscrtzaton and ntgraton ffort rsults n a dagonalzd mass matrx, whch drastcally rducs th computaton ffort, and supports paralll mplmntaton, whch dmonstrats th most ffctv and ffcnt mthod among all mthods currntly n us for slop nstablty computatons. In ths contxt, a us of SEM n slop nstablty has bn mad for th frst tm by Ghart t al. [11], n lasto-plastc framwork (.., Spcfm 3d Gotch, an opn sourc packag). Ths papr furthr mplmnts th SEM packag for vgtatd slop stablty ncorporatng root rnforcmnt formulaton. Wth ths formulaton, th rol of vgtaton n sol slop stablty can b ustfd analytcally and numrcally. Th advantag of SEM ovr xstng FEM s n th us of hgh-ordr bass functons. Hgh-ordr lmnts ar wll stablshd n gotchncal FEM (.g., 15 nod trangls), but th computatonal burdn s not addrssd by any xstng FEM mthods. Th advantag of formng a dagonalzd mass matrx for a psudo-statc analyss s, n fact, not suffcnt. Howvr, ths papr fnds a broadr scop for psudo-statc applcatons for vgtatd and barrn slop nstablty, and prdcts rlabl valus of safty factors through crtan rfnmnt tchnqus,.., p-ncras n spctral dgr and h-ncras n lmntal budgts, and nsurng th qualty msh n th SEM. It nvstgats th applcaton of SEM to 3-D slop nstablty analyss n lasto-plastc framwork. As an applca- Copyrght 013 ScRs.

2 T. R. CHADRA ET AL. 9 ton, ths papr utlzs th sam sourc packag to dmonstrat th stablty aspcts of vgtatd and barrn slop n dffrnt nstablty condtons such as ssmc and saturaton.. Mathmatcal Foundaton.1. Hgh-Ordr FEM Formulaton Tracton or strss vctor t m can b wrttn n tnsor notaton as pr Cauchy s formula as follows. t nˆ (1) whr, m and nˆ ar known rspctvly as th Cauchy strss tnsor and unt outward normal to th boundary. Accordng to wton s Law of consrvaton of lnar momntum, th tm rat of chang of lnar momntum of partcls quals th nt forc xrtd on thm, whch s xprssd as follows. d mv F () dt whr, mv, and F ar rspctvly th mass of th partcl, ts vlocty, and nt forc actng on th partcl. For an arbtral (sub) doman of a sold contnuum of 3 dnsty kg/m subctd to body forcs (pr unt volum) f /m 3, and surfac forcs (pr unt ara) t /m actng on th boundary, th prncpl of consrvaton of lnar momntum can b wrttn as follows. u d f d td t (3) whr, u m s known as th dsplacmnt vctor. Usng th Cauchy s formula, th Equaton (3) can b wrttn as both tnsor form and gnral laboratd form. ; f u n xx x xy yxz z fx ux (4) x y z f u y yx yy yz y y x y z f u zx zx whr, u s known as th partcl acclraton. W us a smcolon (;) for covarant dffrntaton. Th gnralzd Hook s Law can b wrttn n th followng form. C (5) kl kl whr, Ckl Wm C s known as lastcty tnm kl n sor for lnarly lastc sotropc matral,.., n r m rckl 1 rckl, C W C W C W C kl m kl r kl s kl r zz z z s and W s known as mass dnsty,.., W Wrr Wss arr 1 ars. hr, W and ar known rspctvly as th wght and dnsty functon of sol matral [1]. Ths two xprssons nclud root rnforcmnt ffct of vgtaton, whr Wr, W s, r, s, and a r ar known rspctvly as wght functon of roots, wght functon of sol, dnsty of roots, dnsty of sols, and root ara rato (RAR). RAR dnots th fracton of sol cross-scton occupd by roots AR A. An addtonal cohson du to th prsnc of roots can b calculatd by two maor charactrstcs of root systms T R (avrag tnsl strngth of root fbrs), and RAR. Both T R and RAR ar nfluncd by spcs and st factors such as local clmat, sol typ, sason, root typ and sz as wll as root archtctur (.g., [13,14]). Usng prpndcular root rnforcmnt modl, th addtonal root cohson C r can b computd by th followng rlaton as follows. Cr kt r (6) whr, k and t r ar known rspctvly as th common coffcnt factor and a moblzd tnsl strngth of root fbrs as follows. k sn costan (7) tr TRar whr, and ar known rspctvly as th angl of shar dstorton n shar zon and angl of ntrnal frcton of sol. Th common valu of k can b takn as 1.15 [15] or 1. [16]. To account th varablty of root damtr, xprsson C r can b furthr wrttn as follows. cr k Trar (8) 1 whr, T and r a r ar known rspctvly as th tnsl strngth of root and RAR, both spcfd pr damtr class, and s th numbr of class consdrd [17]. Sol s consdrd as ansotropc matral (sol matral proprts vars n all drcton). Th symmtry of Cauchy s strss tnsor and th gnralzd xprsson of Hook s Law (Equaton (5)) mpls that Ckl Ckl. In th abov xprsson (5), C kl /m and kl m ar rspctvly known as fourth-ordr lastcty tnsor and Cauchy stran tnsor. In tnsor form, th Cauchy stran can b furthr wrttn as follows. 1 kl uk; l ul; k (9) Furthr, th Equaton (4) frst xprsss to wghtd ntgral form and thn follows ntgraton by parts and Copyrght 013 ScRs.

3 10 T. R. CHADRA ET AL. rarrangs th xprssons smultanously n th followng form [18]. wu d w d w fd wtd (10) ; Th Equaton (10) s th wak form of th govrnng quaton, whr, and ar known rspctvly as th volum of th doman and boundary of th doman. Th nxt stp would b th us of Lagrang ntrpolaton functon and fnd out th dsplacmnt fld as pr spctral lmnt approach. Intrpolaton functons n both local (.g., x 1 dnots for orgn and x dnots for crtan postv x l dstanc), and natural (.g., dnots for orgn and 1 for postv x coordnat, x and 1 for ngatv x coordnat, x 1 ) coordnats ar as follows. n xx n, 1 x x, 1 (11) W hav usd Gauss-Lgndr-Lobatto (GLL) ntrpolaton ponts of polynomal of dgr n by th rlaton, n1 n ach drcton of lmnt. Th xpanson of any lmnt s accomplshd basd on Lagrang polynomals of sutabl dgr n constructd for n 1 ntrpolaton nods. Th total numbr of ntrpolaton ponts s th product of th numbr of GLL 3 ponts along ach drcton,. Th work n- 1 trpolaton s carrd out usng Lagrang polynomals dfnd on th GLL ponts to obtan th progrssv rspons of dsplacmnt fld. It nds to valuat th dsplacmnt, ts spatal drvatvs, and ntgrals ncountrd n th wak formulaton n thos nods. Wth ths tchnqu, ach lmnt of th msh contans n 1 3 GLL ponts, whr grd ponts that l on th sds, dgs, or cornrs of an lmnt ar shard amongst nghborng lmnts of th doman (.g., n 3, total GLL ponts = 64 os.). Th dsplacmnt fld (.., computatons of dsplacmnt) as pr th SEM can b xprssd as follows (.., smlar dsplacmnt functon as uss n FEM formulaton,.., u u ). 1 u u (1) whr, th ntrpolaton functon n natural coordnats s, dtrmnd by th tnsor product of 1-D Lagrang polynomals as follows. 3 (13) 1, 1 whr, s th ndx of a GLL pont locatd at,,. 3 For ths numrcal ntgraton, a pont 1 x x n a dformd lmnt s mappd to a pont n th natural lmnt as follows. g 1 x x (14) whr, and g ar known rspctvly as a shap functon, and g s th numbr of gomtrcal nods x of an lmnt. Th GLL ponts ar usd as quadratur ponts (ntgraton ponts) for ths numrcal ntgraton. A quadratur rul s an approxmaton of th dfnt ntgral of a functon, usually statd as a wghtd sum of functon valus at spcfd ponts wthn th doman of ntgraton. Th most mportant ssu of SEM s th us of GLL quadratur for spatal ntgraton, usng th sam ponts for ntrpolaton and ntgraton, and formaton of dagonalzd mass matrx. Th SEM s a contnuous Galrkn mthod, n whch th ntrpolaton functon s takn as th tst functon w. Substtut- ng th valu of w n th Equaton (1), w obtan th followng rlaton [18]. u u w 1 (15) Substtutng th valu of u n th wak formulaton, ths quaton turns as follows [17]. M U KU F (16) In th Equaton (16), U and U ar known rspctvly as th acclraton and dsplacmnt vctors. Smlarly, M, K, and F ar known rspctvly as th mass matrx, stffnss matrx and forc vctor of an lmnt, and ar gvn as follows. M K T T B CB d, d, d T T F fd t whr, T and ar known rspctvly as transpos and volum of an lmnt. Othr notatons such as,, B and C ar known rspctvly as th ntrpolaton functon matrx, th stran-dsplacmnt matrx, and lastcty matrx. Wth ths consdratons, th lmntal mass matrx can b furthr wrttn as follows [18]. d M x x x (17) In ths rlaton, and vary from 1 to. Th lmntal mass matrx can b furthr xprssd wth th ntgraton basd on GLL quadratur ovr th GLL ponts as follows [18]. Copyrght 013 ScRs.

4 T. R. CHADRA ET AL M w J (18) whr, w and J ar known rspctvly as th ntgraton wghts and th dtrmnants of th Jacoban matrx valuatd at th th ntgraton pont. Th lmntal mass matrx can b furthr xprssd wth usng th orthogonalty of th ntrpolaton functon n th followng rlaton [18]. M w J (19) 1 sd th lmnts n whch th flds ar dscrbd. Dsplacmnt functon s xprssd n ach lmnt n trms of hgh-dgr Lagrang ntrpolants (polynomals). Th ntgrals ar thn composd basd on GLL quadratur ponts, whch ar usd to form an xact dagonal matrx, and ar thrfor smplfd th algorthm. In ths formulaton, both ntgraton pont and ntrpolaton pont l n th sam pont that nsurs th rducton of ntrpolaton ffort for th dsplacmnt and strss computaton (Fgur 1), and lads to fast and xact soluton wth gratr numrcal stablty. To addrss th dffrnt dgr of sol saturaton and por-watr prssur, th followng rlaton can b appld n aforsad formulaton [18]. In th abov xprsson, rprsnts th Kronckr dlta 1, ; 0, that smplfs th mathmatcal problm. Thrfor, th lmntal mass matrx s dagonal, whch s tru for global mass matrx. Ths fa- P () cltats an ffctv tm-marchng schm, whch s a whr, P whw dnots th watr prssur computd sgnfcant advantag of th SEM ovr th xstng FEM. from smpl hydrostatc rlaton ( w : unt wght of wa- A st of global quaton can b obtand by assmblng tr, h : dpth of watr column). th lmntal quatons [18]. w MU KU F (0).. Strngth Rducton Tchnqu whr, U, K K, and F F ar known r- Th shar strngth rducton tchnqu s frst proposd by Znkwcz t al. [19], and s furthr xtndd to spctvly as global dsplacmnt vctor, global stffnss achv th safty factor for th slop nstablty assssvctor and global forc vctor ( K : lmntal stffnss mnt by dffrnt rsarchrs (.g., [0-5]). In ths tchmatrx, F : lmntal forc matrx). Ths formulaton nqu, an applcaton of gravty loadng s followd by a can only addrss th tm-dpndnt lasto-plastc prob- systmatc rducton n sol strngth untl falur occurs, lms rlvant to slop nstablty n th followng form. whch s achvd usng a strngth rducton factor (SRF), KU F (1) to th frctonal and cohsv componnts of strngth n Th scop of SEM s n dynamc slop nstablty problms; howvr, t offrs sgnfcant bnfts to statc slop nstablty from computatonal pont of vw. Th SEM s an lgant formulaton of th FEM wth a hgh dgr pcws polynomal bass (hgh-ordr FEM). Th maor dffrnc btwn th xstng FEM and SEM s n th choc of th bass (form) functons n- th form of factord frctonal f and cohsv componnt c n th basc quaton of tan c. f f arctan tan SRF (3) cf c SRF At crtan stag of computaton, Gauss ponts undrgo plastc dformaton, whch rqurs a larg numbr of Fgur 1. Dagrammatcal ntrprtaton of soluton procdur for FEM and SEM. Copyrght 013 ScRs.

5 1 T. R. CHADRA ET AL. tratons for th convrgnc of th rsults. W consdr t as falur stag, and consdr factor of safty (FOS) to that stag of SRF. 3. Spctral Elmnt Dscrtzaton SEM works prmarly on hxahdral lmnts. Each hxlmnt contans at last 0 nods n 6 facs for nonlnar problms, howvr, w hav workd on symmtrc nod numbrs (GLL ponts,.., n 1, whr n rprsnts th polynomal dgr) of, 3, 4, 5 tc., n ach drcton (.., X, Y, and Z) of th problm doman that rspctvly quvalnt to 8, 7, 64, 15 by a smpl rlatonshp n 1 3. Th govrnng quatons of non-lnar problms contan hgh-dgr of polynomals or powr trms to captur th complxts of th problm doman. Th non-lnar soluton thus rqurd also dmands hug numbr of traton. SEM dscrtzaton, as shown n Fgur, has grat nflunc on numrcal stablty. Poor qualty msh crats numrcal nstablty (.., ncras n th computatonal cost, lack of convrgnc, and th naccuracy of th rsults). Dffrnt rsarchrs hav workd on hxahdral mshng and hav mphaszd on qualty mshng for gratr numrcal stablty (.g., [6-9]). Th succssful applcaton of th SEM ncluds th ffctv msh opraton, whch ncluds dscrtzaton and mappng. SEM dscrtzaton follows fv stps: 1) th doman s splt nto hxahdral, ) ach sub-doman s mappd nto a rfrnc lmnt, 3) GL nods ar thn ntroducd, 4) spctral grd ponts ar thn mappd back nto th doman, and 5) whol doman s mappd wth spctral grd ponts (Fgur ). 4. An Applcaton to Larg-Scal Slop Instablty Modlng As an applcaton to larg-scal slop nstablty, w hav usd a nwly rlasd opn-sourc program SPEC- FEM3D_GEOTECH along wth msh gnratng toolkt, CUBIT [30] and rsult vsualzaton tool, Paravw [31]. Wth ths mplmntaton, w hav nsurd th rlabl rsults n lss and lss computatonal burdn usng h- and p-rfnmnt tchnqus. W hav found a rlabl rsult at 3 GLL ponts and 10,000 hxahdron lmnts wth qualty mshng opraton followd as pr th SEM tchnqu. Th rsults show that ths packag works ffcntly to larg and complx slop nstablty problms of dffrnt dgr of complxts. Th program ffcntly uss fully unstructurd hxahdral mshs, and shows capacty of dscrtzng th problm doman. W hav succssfully usd th program n paralll mplmntaton (usng Mssag passng ntrfac, MPI [3] and parttonng tool, SCOTCH [33]) for Lnux platform of 4- procssor computr, and hav dalt wth partally and fully saturatd sol slop condton along wth psudostatc ssmc condton for vgtatd and barrn sol slops. In ths formulaton, w hav mployd ntal and fnal watr tabl to th modl so that th program countd partally saturatd slops, and watr prssur actd n th wt rgon of th slop along wth gravty Fgur. Spctral lmnt dscrtzaton tchnqu. Copyrght 013 ScRs.

6 T. R. CHADRA ET AL. 13 load that rducd th sol strngth paramtrs. It consdrs a homognous sol layr blow and abov th watr tabl, whch consdrd th sam matral proprts wth or wthout consdrng th sol saturaton ffcts. Th mshng ffort accommodats th submrgd mshs whl calculatng th hydrostatc prssur. It asly dntfs submrgd nods of unform mshng szs; howvr, t consdrs naccuratly whn th watr tabl touchd th msh of sharp cornrs. Wth ths accommodaton, w hav obsrvd a complt ground watr fluctuaton ffct on sol slop nstablty, whch w hav usd to valuat th stablty of slops n possbl slop stablty nhancmnt procts. Fgur 3(a) shows a typcal modl for larg-scal slop nstablty of slop layr hght, H 5 (m), slop lngth, L 00 (m), slop bradth, B 100 (m), and slop angl 6.6 (dgrs). W hav consdrd root dpth of r 3 (m), and hav assumd th watr tabl d postons to b at vry sngl mtr hght of th slop layr (.., hw 1 to 5 m). W hav consdrd horzontal ssmc coffcnt, K h of 0.10 g for a svr arthquak vnt as pr Trzagh [34]. Fgur 3(b) and Fgur 3(c) rprsnt rspctvly th hxahdron mshng modl n 3D, and doman dcomposton nto 4 subdomans. Fgur 3(d) rprsnts th typcal dsplacmnt fld of 4 rprsntatv safty factors of 1.05, 1.10, 1.15, and 1.0 n cas of slop matral modl of ML along wth root-cohson of 0 k/m, and watr tabl poston of hw 1 m. Th rsults show that thr s dstnct chang n dsplacmnt at SRF of 1.15 and 1.0, whch ndcats th possbl safty factor for that computaton. Computatons procds n ach cas of slop nstablty prpard for dffrnt sol slop matral condtons as n Tabl 1. Ths computatonal framwork adopts modulus of lastcty E and possons rato rspctvly of k/m and 0.3 for slty sand, many fns (SM-ML) Fgur 3. Typcal slop nstablty modlng: (a) schmatc dagram of slop modl; (b) hxahdral mshng n CUBIT; (c) doman dcomposton to 4 sub-domans, and (d) dsplacmnt fld of sol matral modl ML (as pr USCS catgory) at SRF 1.05, 1.10, 1.15, and 1.0. Copyrght 013 ScRs.

7 14 T. R. CHADRA ET AL. Tabl 1. Avrag gotchncal sol paramtr of 8 sol typs (as pr USCS catgory) adoptd by Krahnbuhl and Wagnr [35]. Matral Unt wght (k/m 3 ) Cohson (k/ m ) Frctonal angl ( 0 ) Slty sand, many fns (SM-ML) Slty to clayy sand (SM-SC) Clayy sand, many fns (SC-CL) Clayy sand, wth hgh plastc fns (SC-CH) Slt (ML) Slt to clayy sol (CL-ML) Clayy slt (CL) Clay (CH) to slt (ML), and rspctvly of 8500 k/m and 0.35 for slt to clayy sol (CL-ML) to clay (CH), as pr th common practc of sol paramtrs followd n gotchncal FEM. Fgur 4(a) shows th comparatv nstablty condtons wth 8 sols as pr USCS catgory at watr tabl poston 1 m from th bottom of sol layr (., hw 1 m). Rsults show rspctvly that 3 sol typs, such as CL-ML, CL, and CH prform hghr safty factors of 1.4, 1.45, and 1.60; sol typs SM-SC and SC-CH prform avrag safty factors of 1.10 and 1.15; sol typs SC-CL and ML prform crtcal safty factor of 1.0, and rmanng sol typ SM-ML prforms unstabl safty factor of 0.85 wth th slop modl consdrd (Tabl ). Fgur 4(b) shows 4 nstablty cass (.., C 1 - C 4 ) of slop modl for sol typ of ML. Rsults confrm that slop modl prforms hghly unstabl condton n cas of dry watr tabl and dry and wt sason (partal and full saturaton) ssmc condton as wll as wt sason statc condton, howvr, slop prforms farly stabl rsult wth dry statc and dry sason statc cass (Tabl 3). Computatons show that fully saturatd ssmc sol slop condton s th worst scnaro of slop nstablty and slop modl ncrass FOS sgnfcantly n th cas of lowrng th watr tabl to a sgnfcant dpth (Fgur 4(c), and corrspondng Tabl 4). Fgur 5(a) shows th ffct of root-rnforcmnt on FOS. Th computaton suggsts th slop prforms consdrabl stablty wth ncras n root cohson n cas of low lvl watr tabl poston (.g., hw 1 m). Th contrbuton of root-rnforcmnt n slop stablty can only b achvd wth farly stabl sol slop condtons. Th contrbuton of root cohson can only b ffctv n dry sason statc condtons of th slop modl consdrd (Tabl 5). Thortcally, t prforms th postv ffct on slop stablty whatvr th dgr of nstablty xstd wthn th modld doman; howvr, th stablty gratly dpnds on gomtry of th slop modl as compard to possbl contrbuton by root-cohson. Fg- ur 5(b) ndcats lastc and plastc tratons for lastoplastc modlng. It shows that lastc and plastc parts of th curv rspctvly, whch rqur lss and hgh numbr of tratons for th convrgnc of th rsults. W obsrv FOS rspctvly of 1.10 n frst root-cohsons (.., c 0 k/m and c 10 k/m ) and 1.15 n cas of rmanng root cohson valus (.., c 0 k/m and c 30 k/m ), whch ndcat th nflunc of slop gomtry and sol matral proprts on slop stablty than wth root-rnforcmnt, howvr, ts ffct on slop stablty s no longr xcludd n th analyss bcaus th contrbuton of root-rnforcmnt solly dpnds on as modl. Wth ths accommodaton, th safty factors thus obtand n ach possbl cas hav bcom th bst possbl masurmnts to valuat th stablty of varous stags of landslds. Wth th applcaton of h (.., msh, lmntal budgts) and p (.., spctral dgr, dgr of ntrpolaton, GLL ponts) rfnmnt tchnqus, SEM prforms to b an ffcnt mthod ovr th xstng FEM. 5. Concluson SEM formulaton allows th smulaton of th complcatd strss-stran bhavor of sols, whch can cop wth rrgular gomtrs, complx boundary condtons, and por-watr prssur rgms n larg-scal problm doman of vgtatd and barrn slops. Wth applcaton of h- (.., msh, lmntal budgts), and p (.., spctral dgr, dgr of ntrpolaton, GLL ponts) rfnmnt tchnqus, SEM prforms to b an ffcnt mthod ovr th xstng FEM. It rlats wth hgh-ordr FEM that can captur th complxty of th problm doman, and forms th dagonalzd mass matrx, whch sgnfcantly rducs th computaton burdn. As an applcaton of SEM, w hav utlzd a rcntly rlasd opn sourc program SPECFEM3D_GEOTECH, whch has capacty of smulatng progrssv falur n 3D for larg-scal Copyrght 013 ScRs.

8 T. R. CHADRA ET AL. 15 Fgur 4. Dsplacmnt vrsus SRFs: (a) 8 sol matral modls from SM-ML to CH (as pr USCS sol catgory) n cas of h w = 1 m; (b) ssmc (K x = 0.1 g) and saturaton (h w = 1 m) condton of sol matral modl ML (as pr USCS catgory), and (c) ground watr fluctuatons ffct on slop nstablty at watr tabl postons, h w, of 1.0 m,.0 m, 3.0 m, 4.0 m, and 5.0 m from th bottom of sol surfac of sam matral modl of ML. Tabl. Summary of th FOS n dffrnt sol matral modls as pr USCS catgory. SM-ML SM-SC SC-CL SC-CH ML CL-ML CL CH Tabl 3. Summary of FOS n dffrnt nstablty cass. Sol typ Dry watr tabl (dry) Dry sason watr tabl (partally saturatd h w = 1 m) Wt sason watr tabl (fully saturatd h w = 5 m) (USCS) C 1 C C 3 C 4 C 5 C 6 ML Tabl 4. Summary of th FOS for dffrnt watr tabl postons n statc condton. Watr tabl postons for dry and wt sasons (m) Sol typ (USCS) h w = 1 h w = h w = 3 h w = 4 h w = 5 ML Copyrght 013 ScRs.

9 16 T. R. CHADRA ET AL. Fgur 5. Root-rnforcmnt ffct (sol slop modl of USCS sol ML at h w = 1 m): (a) dsplacmnt (m) vrsus SRFs, and (b) non-lnar traton (os.) vrsus SRFs. Tabl 5. Summary of th FOS for dry sason watr tabl (.., h w = 1.0 m) n dffrnt root-cohson. Sol typ (USCS) Wt sason watr tabl (partally saturatd,.., h w = 1.0 m n dffrnt root cohson (k/m ) c = 0 c = 10 c = 0 c = 30 ML problm doman n varous cass of nstablty. Th computatons hav carrd out from svral possbl strngth rducton factors (SRFs) to fnd out th rlabl FOS from tral SRFs on th bass of abrupt chang n dsplacmnt. W hav succssfully prformd th rol of followng paramtrs on slop stablty: 1) matral proprts; ) ssmc and saturaton condtons; 3) watr tabl postons; and 4) root-rnforcmnts. Computatons suggst that root-rnforcmnt ffct contrbuts sgnfcantly on dry sason slop stablty (statc condton). Th gomtry of problm doman along wth sol matral proprts hav gratr rol on slop stablty. Wth ths smpl mplmntaton, t has vdncd that th bnft of SEM ovr FEM approach can b wll xamnd n larg and complx modlng domans of vgtatd and barrn slops too. 6. Acknowldgmnts Th authors would lk to acknowldg th valuabl commnts and suggstons provdd by Prof. Padma Bahadur Khadka, Trbhuvan Unvrsty pal. REFERECES [1] P. Cupllard, E. Dlavaud, G. Burgos, G. Fsta, J. P. Vlott, Y. Capdvll and J. P. Montagnr, RgSEM: A Vrsatl Cod Basd on th Spctral Elmnt Mthod to Comput Ssmc Wav Propagaton at th Rgonal Scal, Gophyscal Journal Intrnatonal, Vol. 188, o. 3, 01, pp do: / x x [] D. Jonas, D. Basab and M. K. Sn, Stablty of th Hgh-Ordr Fnt Elmnts for Acoustc or Elastc Wav Propagaton wth Hgh-Ordr Tm Stppng, Gophyscal Journal Intrnatonal, Vol. 181, o. 1, 010, pp do: / x x [3] I. Khan and B. H. V. Toppng, Paralll Fnt Elmnt Analyss Usng Jacob-35 Condtond Conugat Gradnt Algorthm, Advancs n Engnrng Softwar, Vol. 5, o. -3, 1996, pp do: / (95) [4] J. Y. Km and S. R. L, An Improvd Sarch Stratgy for th Crtcal Slp Surfac Usng Fnt Strss Flds, Computrs and Gotchncs, Vol. 1, o. 4, 1997, pp do: /s066-35x(97)0007-x [5] D. Komattsch and J. Tromp, Introducton to th Spctral Elmnt Mthod for Thr-Dmnsonal Ssmc Wav Propagaton, Gophyscal Journal Intrnatonal, Vol. 139, o. 3, 1999, pp do: / x x [6] D. Komattsch and J. Tromp, Spctral-Elmnt Smulatons of Global Ssmc Wav Propagaton I. Valdaton, Gophyscal Journal Intrnatonal, Vol. 149, o., 00, pp do: / x x [7] D. Komattsch, S. Tsubo and J. Tromp, Th Spctral- Elmnt Mthod n Ssmology. Ssmc Earth: Array Analyss of Broadband Ssmograms, Gophyscal Monograph Srs, Vol. 157, 005, pp Copyrght 013 ScRs.

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