Comparative Deterministic and Probabilistic Analysis of Two Unsaturated Soil Slope Models after Rainfall Infiltration

Transcription

1 Jordan Journal of Cvl Engneerng, Volume 11, No. 1, 2017 Comparatve Determnstc and Probablstc Analyss of Two Unsaturated Sol Slope Models after Ranfall Infltraton Manoj Kr. Sahs 1) and Partha Pratm Bswas 2) 1) Jadavpur Unversty, Inda. E-Mal: 2) Professor, Jadavpur Unversty, Inda. E-Mal: ABSTRACT The prme am of ths paper s to develop a computatonal scheme for the relablty analyss of two unsaturated natural slope models after ranfall nfltraton, consderng the uncertan system parameters as random varables. Analyss has been conducted wthn the framework of the well known nfnte slope lmt equlbrum model, sutably modfed to take nto account the shear strength for a partally saturated sol. The pore pressures n the shear strength equaton may be postve or negatve, dependng on the poston of the potental falure plane n relaton to the water table or phreatc surface, and, more mportantly, n relaton to the wettng front at any gven tme. Analyss of the tme dependent process of nfltraton and the progresson of the wettng front s based on the wdely known Green-Ampt model. The developed model s based on an approxmate, yet smple method; namely Mean Value Frst Order Second Moment (MVFOSM) method of relablty analyss, treatng the basc geotechncal parameters as random varables, where the random varables are assumed to be normally dstrbuted and uncorrelated. The analyss clearly brngs out the mportance of carryng out a relablty analyss under the probablstc framework. KEYWORDS: Relablty analyss, Unsaturated slope models, Ranfall nfltraton, Green-Ampt model. INTRODUCTION As we are well nformed about the nstant of ranfall, nduced slope falure s nowadays becomng a vulnerable event. Innumerable research has been conducted on ths topc, but t s stll not properly understood and has remaned unpredctable for many years. Several nstants are there, where steep slopes reman stable for many years, whle some gentle slopes fal durng or mmedately after ranfall. The basc mechansm of ranfall nduced slope falure due to subsequent development of postve pore water Receved on 28/8/2015. Accepted for Publcaton on 15/11/2016. pressure though ntally negatve pore water pressure (matrc sucton) plays a crucal role (Fredlund and Rahardjo, 1993). Ths postve pore water pressure reduces the shear strength, causng sol slopes to become unstable and then fal. To fnd the nfltraton depth, generally onedmensonal nfltraton models, such as Green-Ampt model (Chow et al., 1988), are used. Green-Ampt model s the frst physcally based nfltraton model to gve n nteracton wth nfltraton capacty of a partcular sol slope. It s well known that slope stablty s a geotechncal problem whch s surrounded by uncertanty. So, t s customary to consder all the parameters assocated wth slope stablty as random JUST. All Rghts Reserved.

2 Jordan Journal of Cvl Engneerng, Volume 11, No. 1, 2017 varables, whch necesstates requrements of a probablstc analyss. The fact that falure has been reported for computed safety factors larger than 1 ndcates that no matter what number s found by analytcal means, there s always some chance of falure. Ths gves rse to development of probablstc approach. In probablstc approach, the am s to determne the probablty of falure correspondng to the relablty ndex, takng dfferent uncertantes nto consderaton. FORMULATION Infltraton Model Green-Ampt (1911) model proposed the smplfed process of nfltraton theory. Water s ponded to a small depth on the sol surface; so, all the water whch the sol can nfltrate s avalable at the surface. However, durng a ranfall, water wll pond on the surface only f the ranfall ntensty s greater than the nfltraton capacty of the sol. The ncpent pondng tme T p s the elapsed tme between the tme ranfall begns and the tme water begns to pond on the sol surface (Chow et al., 1988). If ranfall begns on dry sol, the vertcal mosture profle n the sol may appear. Pror to pondng tme (T < T p ), the ranfall ntensty s less than the potental nfltraton rate and the sol surface s unsaturated. Pondng begns when the ranfall ntensty exceeds the potental nfltraton rate. At ths tme (T = T p ), the sol surface s saturated. As ranfall contnues (T > T p ), the saturated zone extends deeper nto the sol and overflow occurs from the ponded water. The analyss of the tme dependent process of nfltraton and the progresson of wettng front can be calculated based on the followng four assumptons. (1) The sol surface s mantaned constantly wet by water pondng; (2) A sharp wetted front exsts; (3) The hydraulc conductvty s constant throughout the sol; and (4) The sol matrc sucton at the wettng front remans constant. The followng Eqs. (1) through (3) are used to calculate the depth of the wettng front at ncpent pondng (z p ), the cumulatve nfltraton at ncpent pondng (F p ) and the ncpent pondng tme (T p ). These equatons correspond to the condtons that the ranfall ntensty s greater than the mnmum nfltraton capacty of the sol and that the ranfall duraton s greater than T p (Cho and Lee, 2002; Xe et al., 2004; Muntohar and Lao, 2008). z P K S K Ks f Fp zp ( K ) T p Fp K sf ( K ) f s s s (1) (2) (3) where, = the ntal mosture defct (m 3 /m 3 ), f = the sucton head at the wettng front (m) and K s = the coeffcent of saturated hydraulc conductvty. The nfltraton rate (f) and the cumulatve nfltraton (FF) and ts correspondng depth of wettng front (z w ) at any tme T can be calculated usng the followng equatons. FF f FF FpK s(t T p) f In F pf (4) f f K s 1 FF (5) z FF (6) w Determnstc Approach - Factor of Safety Once the depth of wettng front (whch s also the depth of assumed potental slp surface) wth tme has been determned, the modfed nfnte slope model enables the calculaton of the factor of safety (FS) at each depth

3 Comparatve Determnstc and Manoj Kr. Sahs and Partha Pratm Bswas Model-I: Only ground surface data s obtaned. The model descrpton s shown n Fg. 1. In ths case, the possble slp surface s the wettng front. Case-I: For slp surface along the wettng front: (Z w <H) C ' cos 2 tan ' W satzw uw w SF (8) Z sn cos sat W Case-II: For slp surface along the rocked surface: (Z w H) C Hcos u tan SF H sncos ' 2 ' W sat w w sat (9) Fgure (1): The nfnte slope model used n the study for Model-I The safety factor s calculated usng Eq. (7) (Crosta, 1998; Cho and Lee, 2002, Collns and Zndarcc, 2004). c ' cos 2 tan ' w satzw uw w FS (7) sat zw sn cos Model-II: A rocked dstrbuton s avalable. The model descrpton s shown n Fg. 2. In ths case, the possble slp surface s ether the wettng front or the rocked nterface. So, three lmt equlbrum equatons may arse for three dfferent cases. The safety factor s calculated usng Eqs. (8-10) (Rahardjo et al., 1995; Crosta, 1998). Case-III: For slp surface along the rocked surface: (Z w H) ' 2 2 ' C W H cos FFW cos tan SF H FF sncos w (10) The value of u w n Eqs. (7-10) depends upon the exact locaton of the assumed falure plane. Thus, three stuatons may arse, as follows. Case-I: Potental falure plane s just above the wettng front. In ths case, u w = γ w z w cos 2 β Case-II: Potental falure plane s just below the wettng front. In ths case, u w = -ψ f γ w. Case-III: When pore pressure s zero. u w = 0. The two models mentoned have been used as nfnte slope models n ths study as shown n Fg. 1 and Fg. 2, wheren the potental slp surface s the wettng front for model-i and for model-ii as s gven by Eq. (8) and Eq. (9). Fgure (2): Rocked dstrbuton s avalable for Model-II Probablstc Approach For probablstc analyss, the smple Mean-Value Frst-Order Second-Moment method (MVFOSM) has been used. Takng the performance functon and lmt state as FS-1 = 0, the relablty ndex based on the MVFOSM model s gven by:

4 Jordan Journal of Cvl Engneerng, Volume 11, No. 1, 2017 β FS σfs EFS 1 X 2 n n FS 2 FS FS σ X 2 ρσxσ X j 1 X,j1 X X j FS μ 1 (11) where n = number of sol parameters (c' w, w, γ t, K s ); E[FS] = expected value of FS; σ[fs] = standard devaton of FS; μ x = mean value of sol parameter X ; σ[x ] = standard devaton of X and ρ = correlaton coeffcent between X and X j. Calculaton of Probablty of Falure Correspondng to the value of the relablty ndex FS assumng normal dstrbuton, the probablty of falure P f (Ang and Tang, 1975) s calculated by the followng equaton. P f = Φ (- FS ) (12) where Φ() s the cumulatve standard normal dstrbuton functon. ILLUSTRATIVE EXAMPLE Descrpton For the sake of numercal study, an nfnte slope secton prevously analyzed by Xe et al. (2004) havng the followng data s consdered n ths paper: ntal mosture defct ( ) = 0.28; sucton head at the wettng front (ψ f ) = 55 cm; coheson (c' w ) = 3 kpa; saturated unt weght ( sat ) = 18 kn/m 3 ; angle of nternal frcton ( w) = 25. The ranfall data are as shown n Table 1. The analyses have been carred out consderng the value of hydraulc conductvty (Ks) = 1.87 cm/hr. To study the effect of varaton of slope nclnatons, the analyses have been carred out for three dfferent slope nclnatons, = 30, 40 and 45. Table 1. Ranfall data for the llustratve example Tme (hr) Ranfal l Intensty (cm/hr) Tme (hr) Ranfall Intensty (cm/hr) RESULTS AND DISCUSSION Determnstc Analyss The worst possble pore pressure has been consdered and the factor of safety has been calculated usng the developed computer program RISSA (Ranfall Induced Slope Stablty Analyss). Fg. 3 shows that the factor of safety decreases wth the ncrease n the depth of wettng front for each slope nclnaton. It s qute obvous that pore water pressure ncreases due to the ncrease n depth of wettng front. Ths causes a reducton n the effectve stress, whch, n turn, reduces shear strength of sol leadng to slope nstablty. For example, for a partcular varaton of wettng front depth, such as from to , the factor of safety vares from % to % and % respectvely, for 30, 40 and 45, respectvely. Moreover, t can also be stated that beyond a certan depth (1.5m), there s a sudden drop n the factor of safety rrespectve of the consdered slope angle

5 Comparatve Determnstc and Manoj Kr. Sahs and Partha Pratm Bswas Fg. 5 shows that due to the presencee of rockbed, the factor of safety becomes less n comparson wth Model-I and that beyond a certan depth (say 150cm), t becomes sgnfcantly much less as the parameters of the gven slope are of concern. As n case of model-ii, the decrement rate ranges from 0.196% to 41%. Fgure (3): Varaton of factor of safety wth Fgure (4): Varaton of factor of safety wth wettng front depth for steep and flat slopes Probablstc Analyss As already stated, n vew of the fact that the geomechancal parameters nvolved n the stablty analyss are uncertan, an attempt has been made n ths paper to determne the probablty of falure of the slope complementary to the conventonal factor of safety. For ths purpose, a computer program; RRISSA (Relablty Analyss for Ranfall Induced Slope Stablty Analyss) has been developed usng the Mean Value Frst Order Second Moment (MVFOSM) method, assumng the random varables to be normally dstrbuted and uncorrelated. The developed computer program has been used to determne the relablty ndex and the assocated probablty of falure for varous depths of wettng front and for a unform c.o.v.. of 0.2, as presented n Fgs. (6-10). Fg. 4 presents a plot of factor of safety versus, where n case of steep slope the varaton of factor of safety s less than that of flat slope as depth of wettng front s ncreased. Fgure (6): Varaton of relablty ndex wth Fgure (5): Varaton of FOS wth wettng front depth for model-i and Model-II Fg. 6 presents a plot of relablty ndex versuss depth of wettng front based on the values obtaned by the developed program. From Fg. 6, t can be observed that relablty ndex decreases wth ncreasng depth of wettng front, as expected. It s also observed thatt relablty ndex decreases wth ncrement of slope steepness as per expectaton

6 Jordan Journal of Cvl Engneerng, Volume 11, No. 1,, 2017 Fg. 7 presents a plot of probablty of falure versus depth of wettng front usng the developed program. From detaled results relevant to Fgs. 6 and 7, t s seen that whle correspondng to a depth of wettng front of 171 cm for the 30 degree slope, the value of factor of safety s 1.22 and the correspondng probablty of falure s about 20% whch s rather hgh. Ths observaton clearly demonstrates the necessty and mportance of relablty analyss under a probablstc framework. Further, as expected, the probablty of falure ncreases wth the ncrease n wettng front depth for each slope angle. For example, f the depth of wettng front ncreases from 249 cm to 268 cm, the probablty of falure ncreases from about 9% to 51%, respectvely, for 30 degreee slope and 45 degree slope. Fgure (9): Varaton of relablty ndex wth Fg. 8 represents a plot of probablty of falure versus, whch shows that n case of steep slope, the varaton probablty of falure s neglgble, as expected. Fg. 9 represents a plot of relablty ndex versus, whchh shows that n case of steep slope the relablty ndex decreases as expected. Fgure (7): Varaton of falure probablty wth Fgure (10): Varaton of relablty ndex wth for model-i and model-ii CONCLUSIONS Fgure (8): Varaton of falure probablty wth The followng concludng remarks can be drawn on the bass of the observatons made n the study undertaken n ths paper. 1. The developed computer program for (MRISSA) determnstc analyss and (MRRISSA) for

7 Comparatve Determnstc and Manoj Kr. Sahs and Partha Pratm Bswas probablstc analyss of ranfall nduced slope nstablty have been found to be useful computng tools for the evaluaton of the status of stablty of slopes subjected to ranfall nfltraton. The programs are developed on the bass of nfnte slopes of unsaturated sols, consderng the wdely used Green-Ampt model of nfltraton and unsteady ranfall. 2. When ranfall nduced nstablty s modeled such that the wettng front s assumed as a potental falure plane, two dstnctly dfferent possbltes arse n respect of magntude of the developed pore water pressure: one when the falure plane s located just above the wettng front, and the other when t s just below the wettng front. In the former case, the falure plane beng located n a saturated zone (as per Green-Ampt assumpton) s subjected to postve pore water pressure, whle n the latter, the falure plane beng located n an unsaturated zone s subjected to negatve pore water pressure or sucton. In the former case, obvously, the value of safety factor s lower, and, therefore, ths case s crucal. 3. Falures of unsaturated slopes often occur as a consequence of water nfltraton after ntense or prolonged ranfall. Catastrophc falures can occur n slopes composed of loosely compacted sol. 4. Wth the help of developed computer program (MRISSA), a plot of FOS versus ranfall duraton can be easly obtaned, whch should be of great practcal value to keep and watch on the safety status as tme progresses. 5. The results obtaned from the probablstc analyss usng the developed computer programs based on MFOSM method depct the necessty and mportance of adoptng a probablstc approach. Even n that case n whch the values of the factor of safety (obtaned from the determnstc approach) are above unty ( ) (whch ndcates crtcal equlbrum), the correspondng propablty of falure values are found to be (nearly 41%). Ths smply demonstrates that a determnstc approach does not ndcate the true stablty status of the slopes. 6. The observatons noted n ths study are based on one slope case. Several such cases should be analyzed before arrvng at generalzed conclusons wth regard to the above mentoned ponts. REFERENCES Ang, A.H.S., and Tang, W.H. (1975). Probablty Concept n Engneerng Plannng and Desgn, 1, Cho, S.E., and Lee, S.R. (2002). Evaluaton of surfcal stablty for homogeneous slope consderng ranfall characterstcs. J. Geotech. Eng., 128 (9). Chow, V.T., Madment, D.R., and Mays L.W. (1988). Appled Hydology, Collns, B.C., and Zndarcc, D. (2004). Stablty analyses of ranfall nduced landsldes. J. Geotech. and Geoenvr. Engg., 130 (4), Fredlund, D.G., and Rahardjo, H. (1993). Sol mechancs for unsaturated sol. John Wley and Sons, New York, 517p. Green, W.H., and Ampt, G.A. (1911). Studes of sol physcs-i. The flow of ar and water through sols. J. Agrc. Sc., 4, Muntohar, A.S., and Lao, H.J. (2008). Analyss of ranfall-nduced nfnte slope falure durng typhoon usng a hydrologcal geotechncal mode. 7 February 2008, Sprnger Verlag. Xe, M., Esak, T., and Ca, M. (2004). A tme-spaced based approach for mappng ranfall nduced shallow landslde hazard. Envron. Geology, 45,