Run-out of landslides in brittle soils. A MPM analysis

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1 Run-out of landslides in bittle soils. A MPM analysis A. Yeo 1 & E.E. Alonso 1 & N.M. Pinyol 1,2 1 Deatment of Geotechnical Engineeing and Geosciences, UPC, Bacelona, Sain 2 Intenational Cente of Numeical Methods in Engineeing ABSTRACT: In this ae a sloe stability oblem due to an incease of the oe essue is analysed by means of the mateial oint method (MPM). A stain softening model is imlemented and the ogessive failue is examined in tems of mobilized shea stain. A aametic study is efomed vaying the stength decay but maintaining a common eak enveloe. The influence of the bittleness of the mateial, evaluated in tems of the bittleness index (I B ) oosed by Bisho (1967), in un-out and velocity of unstable sloes is examined. The initiation of motion in the defined sloe is obseved fo values of I B >0.5 and a well coelation between un-outs and bittleness is found. It can be concluded that MPM is able to simulate both the initiation of the failue and the ost-failue stages. 1 INTRODUCION Landslide isk analysis and satial identification of vulneable aeas equie estimations of the slide un-out and the velocity of the unstable mass. When esevois, lakes o fjods ae otentially affected by landslides on thei magins, the calculation of tsunami effects equie infomation on the dislaced soil/ock mass and its exected velocity. Ideally, a suitable comutational method should be able to edict the onset of failue, stating at some known initial stable configuation, and the esonse in subsequent times eoducing the motion of the unstable mass. The tansition fom stable to unstable conditions involves a majo distotion of the soil and a change in constitutive behaviou. As a esult, the entie evolution of the sloe becomes a comlex ocess. The classical limit equilibium methods (LEMs) and the existing finite element methods (FEMs) have been used in numeous studies in ode to edict the failue mechanism in sloe stability analysis (Cascini et al., 2010; Conte et al., 2010). Howeve, they ae unable to simulate ost failue esonses because of the limitations of these methods in eoducing lage dislacements. As altenatives, some advanced numeical methods ae being develoed, such as the smoothed aticle hydodynamics (SPH) (Bui & Fukagawa, 2008; Bui et al., 2011) and the mateial oint method (MPM), among othes. In this wok, the MPM (Sulsky, Zhou, & Scheye, 1995) was used to simulate the whole unstable ocess. It combines featues of continuum and aticle methods discetizing the media in two diffeent ways: a set of mateial oints (Lagangian mesh) which move though a fixed finite element gid (Euleian mesh). This dual descition events mesh distotion oblems and contacts between diffeent bodies ae automatically solved. The fomulation is develoed in dynamic tems and inetia effects ae natually accounted fo. A fully couled hydomechanical mateial oint code was develoed fo satuated soils. A stain softening elasto-lastic constitutive law was imlemented with the uose of analysing ogessive failue henomena. In this ae, the model is alied to exloe the behaviou of unstable sloes unde satuated conditions. The analysis focuses on the detemination of un-out distances and velocities of the unstable mass. Seveal cases ae analyzed chaacteized by a stain softening law defined in tems of bittleness. The esults ae discussed with the aim of deiving actical conclusions. 2 BASIS OF MPM FORMULATION The MPM (Sulsky et al., 1994) discetizes the continuum as a set of subdomains. The mass of each subdomain is consideed to be concentated in a oint so-called mateial oint (Fig. 1). Then, the density of the mixtue can be exessed as N x, t m x x (1) 1

2 in which m and x ae the mass and the osition of the th mateial oint, δ(x) is the Diac delta function and N is the total numbe of mateial oints. Othe oeties such as velocities, stains and stesses, ae also caied by the mateial oints. On the othe hand, the govening equations ae solved at nodes of a suot comutational mesh, which coves the full domain of the oblem and emains fixed duing the calculation. The standad linea shae functions ovide the elationshi between mateial oints and nodes of any oint of the domain. The MPM fomulation fo a mechanical oblem was esented in Sulsky & Scheye (1996) whee the equation of dynamic momentum balance is discetized. Diffeent authos have also extended the MPM to solve couled hydo-mechanical oblems such as Zabala & Alonso, (2011), Jassim (2012) and Abe et al (2014). In this wok, the fomulation esented by Jassim (2012) has been used. An additional aamete η is also necessay in ode to contol the ate of stength decease. The effect of η in a tiaxial test simulation is shown in Figue 2. Vetical stain is escibed and the confining stess was 10 kpa. The esults show that high values of η lead to faste degadation of the soil. Figue 2. Evolution of vetical stess in tiaxial tests fo diffeent values of η. 4 SLOPE INSTABILITY PROBLEM Figue 1. Scheme of the satial discetization used in MPM fomulation. 3 CONSTITUTIVE MODELLING This ae extends the basic non-associated Moh- Coulomb law intoducing stain softening lasticity with the aim of modelling a stength loss afte eak stength conditions. Moeove, in ode to educe the singulaities of Moh-Coulomb yield suface, the modifications oosed by Abbo & Sloan (1995) have been imlemented. The softening behaviou is accounted fo by educing the stength aametes (fiction angle φ, and cohesion c ) with the accumulated equivalent lastic stains ɛ eq accoding to the following softening ules: c e eq c c c (2) eq e (3) The model equies the secification of eak (c,φ ) and esidual (c,φ ) stength aametes. The instability of a sloe, 6 m high and 37º stee, was analysed. The sloe failue was tiggeed by inceasing the oe wate essue at the lowe bounday simulating a heatic level ise. This is a lane stain simulation, the lowe bounday was fixed and hoizontal dislacements along vetical contous wee evented. The wate essue is zeo along the sloe suface, the lateal contous ae imemeable and satuated conditions ae consideed duing all the calculation. The mesh was efined in the egion whee the failue is exected in ode to get moe accuate esults and otimise the comutational cost. The Moh-Coulomb stain softening constitutive model esented in the evious section was used to simulate the bittle behaviou of a soil. The oeties of the mateial ae esented in Table1. Table 1. Soil aametes of the sloe. Soil aametes Value Poosity (-) 0.2 Intinsic emeability (m 2 ) Dy unit weight (kn/m 3 ) 20 Young modulus (kpa) Poisson s coefficient 0.33 Peak cohesion (kpa) 5 Residual cohesion (kpa) 0.5 Peak fiction angle (º) 35 Residual fiction angle (º) 25 Rate of stength softening aamete η (-) 500 The calculation stats with the alication of an incease in oe essue along the lowe bounday (duing 1 second, u to 40 kpa). Afte this, the wate essue on the bounday is maintained constant duing all the simulation.

3 The incease of oe essue educes the effective stesses in the sloe. This fact leads to some oints to each the eak conditions and the stain softening effect, imlicit in the constitutive model, deceases the stength aametes of the lastic zones down to the esidual yield suface. As a esult, the gavitational stesses ae sufficient to induce a ogessive failue. Figue 3 esents the time evolution of the excess oe essue, the cohesion and the dislacement exeienced by a mateial oint, P, indicated also in the figue. Initially, the oint is stable and the oe essue inceases due to the ise of the heatic level. At 15 seconds, 4 kpa of excess oe essue is enough to lastify the oint, tiggeing a sudden do of the cohesion (contolled by η), fom eak to esidual value. 1 second afte (time equied to fom the failue mechanism) the sloe becomes unstable and the oint initiates the movement. At 22 seconds the oint is stable. The oe essue esents numeical instabilities duing the un-out that aeas when the mateial oints coss fom one element to a neighbouing one. Duing the stable eiods, oe essue esents small oscillations due to dynamic fomulation and bounday effects. Failue develoment and un-out of the sloe is illustated in Figue 5. Contous of the shea stain ovide the evolution of the shea band. At the beginning of the instability the shea band is localized; at the end of the movement it seads and becomes a wide shea zone. In this case, the maximum dislacement calculated is 7.5 m. Figue 4. Distibution of the mobilized shea stength (nomalized) along the initial shea band at diffeent times. Figue 3. Evolution of excess oe essue, cohesion and dislacement of a mobilized mateial oint P. In Figue 4 the ogessive failue henomenon is eesented. The evolution of the mobilized shea stength along the shea band shows that the degadation of the mateial initiates at the foot of the sloe and oagates uwads. The mobilized shea stength (nomalized) can be undestood in tems of a mobilized fiction angle ˆ as: q sin ˆ (4) c tan whee q ; 3 (5) 2 2 σ 1 and σ 3 being the maximum and minimum effective incial stesses. Figue 5. Distibution of the shea stain at diffeent times. Note the diffeent scales of the shea stain. 5 PARAMETRIC STUDY OF THE POST- FAILURE BEHAVIOUR In geotechnical engineeing fields, bittleness can be undestood as a mateial condition chaacteized by a eduction of the caability to cay load as the stain inceases. This may deend on many factos, such as eak and esidual stengths, ate of degadation of the mateial, natual heteogeneities of the soil o loading conditions. In this wok, the bittleness of the soil has been defined in tems of the bittleness index (I B ) oosed by Bisho (1967). It is a measue of the decease of the stength fom a eak value (τ ) to a esidual one (τ ) (Eq.7) and it anges fom 0 to 1.

4 I B (7) whee tan (8) c n tan (9) c n σ n being the effective nomal stess in a aticula oint on the sliding suface. A aametic study of 24 simulations was caied out with the aim of studying the sloe stability and the ost-failue behaviou as a function of the soil bittleness. The initial geomety and bounday conditions ae the same fo all simulations and identical to the one descibed in evious section. Mateial oeties wee given in Table 1 excet fo the esidual cohesion (c ) and the esidual fiction angle (φ ) which wee changed accoding to Table 2. Note that simulation numbe 14 coincides with the case esented above. A eesentative value of I B is calculated fo each simulation that becomes unstable (Table 2). Fo those simulations whee the sloe emains stable, detemining a aticula I B is not ossible because thee is not a sliding suface. Note that I B deends on σ n. Accoding with Equations (8) and (9), σ n contols the contibution of the stength aametes to the bittleness of the mateial. Fo examle, fo the aticula case in which σ n = 0, the value of I B will deend entiely of the cohesion decease. Theefoe, diffeent values of effective nomal stess may leads to diffeent values of bittleness. In each case σ n has been calculated as the aveage of the nomal stesses of 15 mateial oints distibuted along the initial failue mechanism. Table 2. Simulations efomed defined by diffeent esidual stength aametes and I B. Simulation c (kpa) φ (º) I B Stabililty Stable Stable Stable Stable Unstable Stable Unstable Unstable Unstable Unstable Stable Unstable Stable Unstable Unstable Unstable Unstable Stable Unstable Unstable Unstable Unstable Unstable Unstable In Figue 6, a comaison between diffeent initial failue mechanisms obtained with MPM and with LEM (Mogensten-Pice) is shown. The shae of the failue sufaces is vey simila to LEM s ediction. Howeve, the deth of the failue suface deends on the case simulated. The esidual cohesion lays a significant ole. The highe is c, the deee is the failue suface. Figue 7 shows the coelation between un-outs and I B. Run-out is defined hee as the distance between the toe of the initial sloe and the toe of the sloe afte failue once equilibium has been established. Note that this definition is not necessaily equivalent to the maximum dislacement achieved by any oint of the sloe. The esults indicate that a theshold (I B *) can be defined. Sloes with highe values of I B become unstable. In this study, the theshold I B * is aound 0.5. Just above this value, the un-outs incease suddenly to 7 metes. Run-out inceases nonlinealy with I B in the manne indicated in the figue. To fully undestand Figue 7, it is imotant to highlight that when I B exceeds 0.83, the toe of the mobilized mass each the bounday of the domain, with is located at 26 m of the initial toe. Theefoe, the maximum un-out does not exceed 26 m. Moeove, when I B >0.75, mobilized mateial oints abandon the dense calculation mesh and ente the oughe mesh. In these cases the integation becomes less accuate and esults may be slightly less eliable. Figue 6. Comaison between the initial failue mechanisms obtained in the MPM simulations and in a LEM analysis. In Figue 8 the stable geometies afte failue ae illustated fo simulations 13 to 17 which have been assigned the same c = 0.5 kpa. Theefoe I B is contolled by the esidual fiction angle. The same atten of dislacements is obseved in all the cases. Howeve, the un-out inceases and the sloe becomes flatte with inceasing values of bittleness.

5 Figue 7. Relationshi between un-out (m) and the coesonding I B fo all the simulations. Figue 9 shows the final geometies of two simulations (5 and 21) chaacteized by the same value of I B. Desite having a simila value of the un-out, the distibution of final dislacements is vey diffeent. This is mainly a consequence of the diffeent values of the esidual cohesion. Mateials having a low esidual cohesion c lead to shallow failues unlike the dee failues associated with highe esidual cohesion. In the final at of this section simulations 13, 14, 15, 16 and 17 ae analysed in moe detail in ode to highlight the dynamics of the failue. Figues 10 and 11 show the evolution of the dislacement and the velocity of a mateial oint initially located just above the initial shea band. These lots illustate the diffeent hases of an instability ocess. Initially, and fo a given eiod of time, the sloe is stable and dislacements and velocities ae zeo. At this stage, the ogessive failue takes lace. When the initial failue mechanism develos, the velocity inceases aidly and the mobilized mass moves fowad. Afte a eak value of the velocity (in the cases analysed it vaies fom 1 to 2 m/s), the velocity deceases until the sloe eaches a new stable ofile. Figue 9. Final geomety fo simulations 5 and 21, esectively. The dislacements of the mateial oints ae indicated in the colou scales. Figue 8. Final geomety fo simulations 13, 14, 15, 16 and 17, esectively. c =0.5 kpa in all of them. The dislacements of the mateial oints ae indicated in the indicated colou scale. Figue 10. Dislacement of the mateial oint P fo the simulations 13, 14, 15, 16 and 17.

6 - The maximum slide acceleation is calculated at the ealy stages of the motion. Maximum velocities incease with deceasing esidual fiction and theefoe with inceasing I B. The highe the I B, the longe the time inteval to each a final stable geomety. - The final stable geomety afte failue is contolled by the esidual stength. Lowe esidual fictional angles leads to flatte final sloes. 7 REFERENCES Figue 11. Velocity of the mateial oint P fo the simulations 14, 15, 16 and 17. Examining Figues 10 and 11, it can be concluded that in sloes with lage values of I B : (1) the instability occus ealie; (2) the velocity inceases moe suddenly; (3) the eak velocities each lage values; (4) it is necessay moe time to each the stable geomety; and (5) the final un-out is longe. 6 CONCLUSIONS Sloe stability and ost-failue behaviou have been analysed by means of MPM. The method is able to simulate both the initiation of the failue, which involves small stains, and the ost-failue stage, geneally chaacteized by lage dislacements. Some examles involving a egula geomety and a homogeneous sloe ae analysed in the ae. The tiggeing mechanism in all the cases is an imosed incease in wate essue on the lowe bounday of the domain. The chaacteization of the sloe mateial by means of a stain softening elastolastic constitutive law allows the simulation of the ogessive failue henomenon. Ievesible shea stain contous show the oagation of the failue suface fom the foot of the sloe to the ue at. The un-out has been eesented in tems of the bittleness index, I B, oosed by Bisho (1967). In all calculations a common eak stength enveloe is adoted. Altenative combinations of esidual cohesion and fiction have been selected and a degee of bittleness has been assigned to the unstable cases. In the defined sloe, failue (in the sense of the initiation of motion) is obseved fo values of I B >0.5. Run-outs, defined as the distance between the initial toe of the sloe and the toe of the final stable ofile, coelate well with bittleness, ovided I B is calculated fo the aveage value of the nomal effective stess acting on the initial failue suface. Additional esults obtained can be summaized as follows: Abbo, A., & Sloan, S. (1995). A smooth hyebolic aoximation to the Moh-Coulomb yield citeion. Comutes & Stuctues, 54(3), Abe, K., Soga, K., & Bandaa, S. (2014). Mateial Point Method fo Couled Hydomechanical Poblems. Jounal of Geotechnical and Geoenv, doi: /(asce)gt Bisho, A. W. (1967). Pogessive failue with secial efeence to the mechanism causing it. In Poceedings of the Geotechnical Confeence ( ). Bui, H., & Fukagawa, R. (2008). Lagangian meshfee aticles method (SPH) fo lage defomation and failue flows of geomateial using elastic lastic soil constitutive model. Intenational Jounal fo Numeical and Analytical Methods in Geomechanics, 32, doi: /nag Bui, H. H., Fukagawa, R., Sako, K., & Wells, J. C. (2011). Sloe stability analysis and discontinuous sloe failue simulation by elasto-lastic smoothed aticle hydodynamics (SPH). Géotechnique, 61(7), doi: /geot.9.p.046 Cascini, L., Cuomo, S., PASTOR, M., & Sobino, G. (2010). Modeling of ainfall-induced shallow landslides of the flow-tye. Jounal of Geotechnical and Geoenvionmental Engineeing, 136 (1),, doi: / ASCE GT Conte, E., Silvesti, F., & Toncone, A. (2010). Stability analysis of sloes in soils with stain-softening behaviou. Comutes and Geotechnics, 37(5), doi: /j.comgeo Jassim, I., Stolle, D., & Vemee, P. (2012). Two- hase dynamic analysis by mateial oint method. Intenational Jounal fo Numeical and Analytical Methods in Geomechanics. doi: /nag Sulsky, D., Chen, Z., & Scheye, H. L. (1994). A aticle method fo histoy-deendent mateials. Comute Methods in Alied Mechanics and Engineeing, 118(1 2), doi: / (94) Sulsky, D., & Scheye, H. L. (1996). Axisymmetic fom of the mateial oint method with alications to usetting and Taylo imact oblems. Comute Methods in Alied Mechanics and Engineeing, 139(1 4), doi: /s (96) Zabala, F., & Alonso, E. E. (2011). Pogessive failue of Aznalcólla dam using the mateial oint method. Géotechnique, 61(9), doi: /geot.9.p.134