expanding collapse in partially submerged granular soil slopes

Expanding collapse in partially submergedgranular soil slopes

Radoslaw L. Michalowski

Abstract: The traditional approach to stability analysis of granular slopes leads to the safety factor that is associated witha planar failure surface approaching the slope face, whether the slope is ‘‘dry’’ or submerged. However, for partially sub-merged slopes, a more critical, nonplanar failure surface can be formed. A family of geometrically similar surfaces can befound that is characterized by the same safety factor. If the safety factor drops down to unity and the slope becomes unsta-ble, then a mechanism of any size can form. Alternatively, the failure may start at some small region and then the volumeof the mechanism of failure can expand, giving rise to a progressive failure of a different kind that is typically associatedwith slopes. This progression has the character of a ‘‘disturbance’’ or a shock-like kinematic discontinuity propagating intothe soil at rest. A quantitative analysis is presented and it is demonstrated that the soil dilates while the mechanism ex-pands, leaving the slope weakened and susceptible to a deep failure. This is a plausible mode of failure of partially sub-merged slopes, the type that is most likely responsible for large subaqueous landslides, and is similar to the well-documented instability propagation in ‘‘quick clay.’’

Key words: slopes, limit state analysis, progressive failure, submerged slopes.

Resume : L’approche traditionnelle pour l’analyse de stabilite des pentes faites de materiaux granulaires permet d’obtenirun facteur de securite associe a une surface de rupture plane, pres de la surface de la pente, que ce soit pour une pente« seche » ou submergee. Cependant, dans le cas des pentes partiellement submergees, une surface de rupture non plane, etplus critique, peut etre formee. Une famille de surface ayant une geometrie similaire, et caracterisee par le meme facteurde securite, peut etre formee. Si le facteur de securite descend a une valeur de 1 et que la pente devient instable, alorsn’importe quelle taille de mecanisme peut se former. D’un autre cote, la rupture peut debuter dans une petite region, en-suite le volume du mecanisme de rupture peut prendre de l’expansion, ce qui amene a un type different de rupture progres-sive qui est normalement associee aux pentes. Cette progression ressemble a une perturbation ou une discontinuitecinetique de type onde de choc qui se propage a travers le sol au repos. Une analyse quantitative est presentee, et demon-tre que le sol se dilate a mesure que le mecanisme prend de l’expansion, laissant ainsi la pente affaiblie et susceptible aune rupture en profondeur. Ce mode de rupture est plausible pour des pentes partiellement submergees et est probablementresponsable des importants glissements de terrains sous-marins, en plus d’etre similaire a la propagation des instabilitesdans l’ « argile rapide » qui elle est bien documentee.

Mots-cles : pentes, analyse de l’etat limite, rupture progressive, pentes submergees.

[Traduit par la Redaction]

Introduction

Most stability considerations of slopes built of frictionalsoils associate a collapse with a planar failure surface ap-proaching the slope face, and the safety factor for granularslopes is expressed as F ¼ tanf=tanb, where f is the internalfriction angle of the soil and b is the slope inclination angle.This safety factor is independent of whether the slope issubmerged or not (Fig. 1a). This generally accepted percep-tion was put to the test by Baker et al. (2005), who foundthat for partially submerged granular slopes a mechanismmay form with a nonplanar failure surface that produces asafety factor lower than the one produced using the tradi-tional approach. Baker et al. concluded that the most criticalpool level is about the mid point of the slope height. In amore recent note (Michalowski 2009) it was pointed out

that a family of geometrically similar failure surfaces canbe identified (of the type shown in Fig. 1b), each surface in-dependent of the slope height and characterized with thesame safety factor. Each of these surfaces intersects theslope face above and below the water table, as shown inFig. 2.

An argument will be made that the mechanism of the typeshown in Fig. 1b can form in a small region close to thewater table, and then expand into the soil at rest. Such aprogressive (expanding) failure of the partially submergedslope has a boundary that has the nature of a ‘‘disturbance’’moving into the intact soil. This is contrary to the traditionalperception (and analyses) where a simultaneous triggering ofthe failure in the entire landslide is considered or where theprogression of failure is limited to a well-defined volumewith gradually increasing distortions.

Received 20 December 2007. Accepted 19 May 2009. Published on the NRC Research Press Web site at cgj.nrc.ca on 20 November2009.

R.L. Michalowski. Department of Civil And Environmental Engineering, University of Michigan, 2340 G.G. Brown Bldg., Ann Arbor,MI 48109, USA (e-mail: [email protected]).

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Can. Geotech. J. 46: 1371–1378 (2009) doi:10.1139/T09-064 Published by NRC Research Press

Background

It is a standard approach to consider the failure of a slopeas a simultaneous collapse of the entire region defined bythe boundaries of the failure mechanism. While this ap-

proach has been successfully used for practical estimates ofthe safety factor of earth slopes, there is evidence that land-slide mechanisms of subaqueous slopes do not start simulta-neously in their entire mass (Hampton et al. 1996). Onepossible mechanism entails a local failure triggered in asmall region and propagation of the failure front throughouta large mass of the sediment. Submarine landslides coveringlarge regions with volumes of thousands of cubic kilometreshave been reported in the literature (e.g., Hampton et al.1996). These landslides have not occurred simultaneously intheir entire volume. It will be argued that a similar mecha-nism is plausible in partially submerged slopes in granularsoils.

Two different mechanisms of progressive failure of suba-queous slopes have been documented (Van den Berg et al.2002): (i) where the soil liquefies and (ii) the breachingprocess. The first one is encountered in loose to medium-dense sands susceptible to compaction and, as a result, issusceptible to the increase in the pore-water pressure. Thesecond type of failure occurs in dense sands where thin slabsof sand ‘‘flake off’’ of the face of the slope. This process isgoverned by the shear of sand, which in the dense sand isassociated with dilation and suction in the granular material.The failure of a thin slab of the sand occurs once suction in

Fig. 1. Collapse mechanisms for partially submerged slopes in granular soils. (a) ‘‘shallow’’ translational failure; (b) rotational failure. H,height of the slope; h, height of the mechanism; L, water level below slope crest; l, water level below the mechanism’s uppermost point; r,log-spiral radius; r0, initial radius; v, velocity; v0, initial velocity associated with initial radius; q, coordinate angle; q0, initial coordinateangle for log-spiral surface; qh, terminal angle for log-spiral surface; u, angular velocity about the rotation center O.

Fig. 2. Equivalent failure surfaces for partially submerged slope. d,angle defining location of instantaneous centers of rotation.

1372 Can. Geotech. J. Vol. 46, 2009

Published by NRC Research Press

the slope is released. This process is also encountered indredging operations (Van den Berg et al. 2002) — at thistime, still awaiting its mathematical description.

Here, a progressive failure is considered where the localcollapse is initiated close to the pool level, and the collapseregion expands rapidly into the stationary soil. Applicationof the kinematic approach of limit analysis to granularslopes is described briefly in the next section, followed by adiscussion of a progressive failure mechanism. The massbalance equation is then used to relate the speed of the prop-agating boundary of the mechanism to the material veloc-ities in the dilating soil. Finally, a quantitative analysis ofan expanding failure mechanism is developed.

Limit analysis of granular slopesThe kinematic approach of limit analysis in soil mechan-

ics is well established (e.g., Chen 1975). Application of thekinematic approach to problems where pore-water pressureis involved was first considered in Michalowski (1995), andthe specific issues related to submerged slopes can be foundin the recent papers of Viratjandr and Michalowski (2006)and Michalowski (2009). Following the last reference in theprevious sentence, the balance of the work rate in a rota-tional mechanism with a log-spiral failure surface in granu-lar slopes leads to the following equation:

½1� gw

g¼ � f1 � f2 � f3 � f4

f5

where gw/g is the ratio of the unit weight of the pore fluid tothe unit weight of the soil (saturated). Coefficients fi are de-pendent on the geometry of the slope, geometry of the me-chanism, and the internal friction angle (they originate fromthe rate of work of the gravity forces and water pressure).Coefficients f1 through f4 can be found in Chen (1975).Coefficient f5 includes the influence of the pore- and sur-face-water pressure. This coefficient does not have a conve-nient analytical form and has to be evaluated numerically(see Viratjandr and Michalowski 2006).

Because this consideration is based on the kinematic ap-proach of limit analysis, ratio gw/g in eq. [1] is the upperbound estimate of the value at which the slope loses itsstability (the pore-water pressure is considered an externalload and this external load increases with the increase ingw). This is not a very practical interpretation of this equa-tion, as ratio gw/g is typically known for a given slope.However, this equation can be used to find the lower-boundestimate to the internal friction angle necessary to avoid fail-ure of the slope when ratio gw/g is given. Because functionsfi in eq. [1] depend on the geometry of the failure mecha-nism and the internal friction angle (f), the procedure mustbe iterative (the geometric parameters are varied in thesearch for maximum f). The solution to maximum f re-vealed an interesting characteristic: the most critical mecha-nism is a log-spiral failure surface that intersects the slopeas shown in Fig. 1b.

The geometry of the mechanism at which the maximum f(best lower bound) was found is independent of the slopeheight. Because this solution is size-independent, a mecha-nism of any size will produce the same solution to maxi-mum f. The geometry of the mechanism is related to the

water table in the pool (l/h, Fig. 1b), but not to its specificlevel characterized by L/H. Consequently, the geometricallysimilar failure mechanisms for a granular slope illustrated inFig. 2 are all characterized by the same solution to maxi-mum f and, consequently, the same safety factor. Thissafety factor can be calculated as tanf=tanfm, where fm isthe iterative solution to maximum f from eq. [1]. The resultof calculations for slopes with different inclinations is pre-sented in Fig. 3 (for gw/g = 0.6). It is clear that the log-spiralmechanism for a partially submerged slope produces a lowersafety factor than the traditional formula tanf=tanb. Becauseeq. [1] yields a size-independent solution fm, the safety fac-tor tanf=tanfm is independent of the specific pool level, i.e.,the critical pool level is not well defined and small collapsemechanisms can form with very high or very low water ta-bles in the pool. This issue was a focus of the previous note(Michalowski 2009). Here it will be argued that the mecha-nism of collapse can expand, starting from a small failure inthe neighborhood of the water table and moving into the soilat rest, engaging a progressively larger volume of the soil.

Expanding failure mechanisms

Progressive and quasi-steady mechanismsWhereas most practical analyses of slope stability assume

a simultaneous trigger of the kinematic process in the entirefailure region, the true nature of a collapse is usually pro-gressive. At least two types of progressive failure in soilscan be distinguished: (i) a process where the features of themechanism progressively develop in the material to reachthe stage where a kinematic mechanism is formed and(ii) nucleation of a kinematic field (mechanism) in a smallregion and subsequent expansion of this region into the sta-tionary soil.

The first type often includes progressive development ofshear bands in the soil, such as those under a footing (e.g.,Michalowski and Shi 2003), or gradual development of the‘‘failure surface’’ in slopes (Palmer and Rice 1973). The sec-ond type entails an expanding region of failure with theboundaries moving into the stationary material. An exampleof such a process is wedge indentation (Drescher and Mi-chalowski 1984) or a three-dimensional problem of a pyra-mid indentation into the material at rest (Michalowski1985). The limit analysis of wedge indentation in a dilatingsoil is illustrated in Fig. 4. The mechanism of deformation isfully developed at every instant of the process, but it ex-pands as the wedge is being driven into the soil. BoundaryBCD is a kinematic discontinuity that propagates into thestationary soil, with the speed dependent on the speed ofwedge indentation. The inclination of the lip AB is thenpart of the solution and to conserve mass it must be so.

This mechanism of wedge indentation involves a propa-gating ‘‘disturbance’’ or a ‘‘failure front,’’ leading to expan-sion of the mechanism. A well-documented failure of ‘‘quickclay’’ (Gregersen and Loken 1979) also had the character ofa moving disturbance through the soil. Another phenomenonof a similar nature is propagation of the rarefaction wave instorage containers for granular material, which causes a dropin the material bulk density once the wave passes throughthe material (Michalowski 1987). This type of disturbancewill be referred to here as a ‘‘shock-like’’ discontinuity. It

Michalowski 1373


will be suggested that the failure of partially submergedslopes can be described by a nucleation of the collapse in asmall region, and subsequent expansion into the stationarysoil, similar to the process of wedge indentation in Fig. 4.

The velocity of the material on the two sides of a shock-like discontinuity and the discontinuity’s speed of propaga-tion are related through the principle of mass conservation.This relation will be considered in the next subsection.

Shock-like kinematic discontinuityThe wedge indentation (Fig. 4) is a process characterized

by a progressive mechanism (expansion of the failure re-gion). Such mechanisms often are described as self-similar(or quasi-steady), with geometric similarity at every stage(Hill et al. 1947). In such problems, the boundary of themechanism is considered as an ‘‘instability front’’ or a ‘‘dis-turbance’’ moving into the material at rest. Care is needed toassure that the mathematical description of such a process isconsistent with conservation of mass.

Stationary and shock-like velocity discontinuities in gran-ular media were considered earlier by Drescher and Micha-lowski (1984) and, for completeness, this is summarizedbelow. A continuity condition for a flux of matter across astationary discontinuity can be written as

½2� r1vn1 ¼ r2vn

2

where r is the density of the medium and vn is the velocity

component normal to the discontinuity; subscripts 1 and 2denote the density and velocity on one side or the other ofthe discontinuity (see Fig. 5a). Hence, any discontinuity inthe normal component of the material velocity needs to beassociated with a discontinuity in the mass density. In short,this condition can be written as zero mass flux increment

½3� ½rvn� ¼ 0

where square brackets denote the discontinuity (‘‘jump’’).The velocities in eq. [2] are measured with respect to thediscontinuity. Therefore, if the discontinuity is not station-ary, then eq. [2] has to be rewritten as

½4� r1ðvn1 � vn

pÞ ¼ r2ðvn2 � vn

pÞ

where vnp is the normal speed of propagation of the shock-

like discontinuity. We now set v1 = 0 on side 1 (Fig. 5c) torelate the shock propagation speed vn

p to the velocity of thematerial vn

2 when the shock moves into a stationary field

½5� vnp ¼ �

vn2

ðr1=r2Þ � 1

For a dilative process we have r1/r2 > 1, and a minussign indicates that the shock propagates in the direction op-posite to the velocity of the material on side 2. For incom-pressible materials (volume-preserving materials), r2 = r1and vn

2 ¼ vn1; thus, eq. [4] leads to the trivial statement vn

p ¼vn

p and eq. [5] is not applicable.

Pseudo-steady mechanism of slope collapseThe wedge indentation problem in Fig. 4 is special in that

the pattern of deformation remains geometrically similar atevery stage of the process (quasi-steady). It is now sug-gested that the failure of partially submerged slopes mayvery well be described with a quasi-steady mechanism. Be-cause any size of collapse of a partially submerged granularslope is equally realistic (as argued earlier, Fig. 2), themechanism may nucleate in some small region close to thepool level and then expand into the stationary soil, as illus-trated in Fig. 6a. Such a mechanism is plausible when thesafety factor for the slope is equal to unity, and the safetyfactor is equal to unity for all geometrically similar surfacesillustrated in Fig. 2. This mechanism could not occur inslopes with some cohesive component of strength as the ex-pansion of the mechanism in such slopes is related to thechange in the safety factor, with larger mechanisms havinga smaller safety factor. Hence, the initial mechanism insuch slopes is likely to include the entire height of the slope,as opposed to the partially submerged granular slopes.

One should point out a substantial difference between theexpanding mechanism due to wedge indentation and the ex-pansion of the slope failure. In the former, the velocities ofthe material and the propagation velocities of the shock-likediscontinuities are uniquely determined by the speed ofwedge penetration, v0. This is illustrated in the hodographin the center of Fig. 4. Consequently, the change in the soildensity during the mechanism expansion is also uniquely de-termined. In the slope case, Fig. 6a, the instantaneous speedof the soil particles along the moving log-spiral boundaryAB is determined by

Fig. 3. Safety factor for granular slopes as a function of slope in-clination and internal friction angle (adapted from Michalowski2009).

Fig. 4. Wedge indentation into dilatant soil (adapted from Drescherand Michalowski 1984).



½6� v ¼ vðqh�q0Þ tanf0

where v0 is the particle speed at q = q0 (point A in Fig. 1b).However, the speed of propagation of log-spiral AB into thestationary soil is not the material speed.

To describe the slope expanding failure mechanism quan-titatively, one needs to find the relation between the veloc-

ities of the material and the speed of the moving kinematicdiscontinuity into the soil at rest. Propagation of points Aand B up and down the slope (Fig. 6a) will now be relatedto material velocity, v0, through the relation in eq. [5]. Thiswill require that the change in density during the mechanismexpansion be estimated based on material properties (densesoils will dilate more than looser soils).

At the instant considered, the velocity of the material par-ticle at point A (Fig. 6) is v0 = ur0, where u is the angular ve-locity about the instant rotation center Oi and r0 is the radiusfrom Oi to point A. Velocity vn

2 in eq. [5] now becomes thecomponent of v0 normal to the discontinuity at point A, andthe geometrical relations in Fig. 6b lead to the propagation ve-locity of (nonmaterial) point A, vA, up the slope surface

½7� vA ¼ �v0

ðr1=r2ÞA � 1

sinf

cosðbþ q0 � fÞ

where (r1/r2)A is the ratio of the density of the stationarysoil at point A to the density just inside the failure mechan-ism at point A. Now, considering the velocity of the mate-rial at point B as resulting from eq. [6] and repeating thesteps as for point A, the nonmaterial velocity of point Bdown the slope becomes

½8� vB ¼v0

ðr1=r2ÞB � 1

sinf

cosðbþ qh � fÞ eðqh�q0Þ tanf

Both vA and vB are negative, indicating that the propaga-tion of the discontinuity is opposite to the normal compo-nents of the particle velocities. This mechanism ischaracterized by the well-defined ratio l/h that follows fromthe limit state analysis as discussed in an earlier section. Topreserve the geometrical similarity of the expanding mecha-nism, ratio vA/vB must be equal to l/(h – l). Therefore, thedensity will not change uniformly in the entire mechanism(see Appendix A). It may seem surprising at first that eventhough the dilatancy angle is constant along the entire fail-ure surface, the change in density is not uniform. Such isthe direct consequence of the principle of mass conservationand the self-similarity of the mechanism.

Velocity v0 in eqs. [7] and [8] is dependent on the angularspeed u and is equal to ur0, but r0 is increasing as themechanism expands. As the mechanism is geometricallysimilar, the ratio of radius r0 to the distance PA (Fig. 6a) isequal to the ratio of their rates, hence

½9� dr0

dt¼ �vA

sinðdþ bÞsinðd� q0Þ

where t is time, angle d is uniquely determined from thegeometrical relations in Fig. 6a (see Appendix A), and theminus sign is to assure a positive rate dr0/dt with negativespeed vA following from eq. [7]. Now, using the expressionin eq. [7] and considering that v0 = ur0, we have

½10� dr0

dt¼ ur0

ðr1=r2ÞA � 1

sinf

cosðbþ q0 � fÞsinðdþ bÞsinðd� q0Þ

¼ ur0

ðr1=r2ÞA � 1M

and

Fig. 5. (a) Velocities at stationary kinematic discontinuity, (b) ho-dograph, (c) velocities at shock-like discontinuity moving into sta-tionary region, and (d) normal components. vp, speed ofpropagation.

Fig. 6. (a) Propagation of the shock-like kinematic discontinuity(‘‘failure front’’) into the slope and (b) schematic for derivation ofvelocity of point A. Oi, instantaneous rotation center; rh, terminalradius of the log-spiral surface; vA, propagation velocity of point A(nonmaterial).

Michalowski 1375


½11�Zr0

r01

dr0

r0

¼ M

ðr1=r2ÞA � 1

Zt

0

uðtÞ dt

where M can be inferred from eq. [10] and r01 is the valueof r0 at t = 0.

Subscript 0 in r0 does not pertain to initial time, but to theradius at angle q0, and such a notation was adopted so that itis consistent with the original development of the log-spiralmechanism (Chen 1975). After integrating eq. [11], we ob-tain

½12� r0 ¼ r01 expM

ðr1=r2ÞA � 1Dq

� �

where Dq is the angular displacement since the beginning ofthe process (t = 0)

½13� Dq ¼Zt

0

uðtÞ dt

The minimum magnitude of r01 may be dependent on thecapillary rise, causing ‘‘apparent cohesion’’ and preventinga small initial size of the failure region. The size of the in-itial mechanism may also be dependent on other factors,such as the distribution of local imperfections.

The angular velocity as a function of time in eq. [11] can-not be easily assessed (unless a dynamic problem is solved),and the utility of eq. [12] is qualitative: the mechanism sizeexpands exponentially with the increase in angular displace-ment, but the speed of the mechanism propagation is not di-rectly involved in eq. [12]. This equation, however, confirmsthat the assumption of neglecting the change in the slope ge-ometry during the propagation of failure is reasonable.

It is practical to assume that the velocity of the soil at thefront of the mechanism expansion (for instance, v0 of theparticle at point A, Fig. 6a) is constant and to calculate therate of expansion of the mechanism (speed of the failurefront propagation) as a function of that material speed. Suchan assumption is intuitive but reasonable, and it implies thatthe angular rotation speed is a hyperbolic function of r0 (u =v0/r0), where r0 itself is increasing with time; integratingeq. [10] with constant v0 = ur0 yields

½14� r0 ¼ r01 þM

ðr1=r2ÞA � 1v0t

where t is time from the start of the process (v0t is not amaterial displacement as the particle at point A is subjectedto v0 only instantaneously, when the failure front passesthrough the particle at A).

Example and discussionConsider a partially submerged slope with an angle of in-

clination b = 308 and the ratio of the water unit weight to thesaturated soil unit weight gw/g = 0.5. Now, use eq. [1] to cal-culate the internal friction angle, f, necessary to maintainlimit equilibrium. Coefficients fi are functions of f, slope in-clination, and the geometry of the mechanism (q0, l/h). Equa-tion [1] was derived directly from the kinematic theorem oflimit analysis and it can be used to find the best lower bound

to f necessary to maintain limit equilibrium. Such calcula-tions are iterative, and they are described in more detail inMichalowski (2009). Parameters q0 and l/h were varied inthe process of maximizing f. The best estimate of f = 318was found, at q0 = 74.88 and l/h = 0.484 (qh = 105.58, butfor a mechanism intersecting the slope face, qh is not an in-dependent parameter; see Fig. 1b for q0, qh, l, and h). Now,let the failure start in a small region defined by vertical di-mension ht=0 = 2.0 m (and l = 0.968 m). Radius r0 isuniquely related to h through geometrical relations inFig. 6a, and the initial r0 = r01 = 5.48 m.

The problem is considered quasi-static, and the velocityof the mechanism expansion can only be assessed with re-spect to the given velocity boundary condition. The givenboundary velocity is v0 — the instantaneous velocity of amaterial particle at point A. Whereas the geometric locationof point A changes with the progress of the mechanism, v0is the velocity of the particle currently at point A. This ve-locity is likely to be of the order 10–1 m/s, and it is assumedthat v0 = 0.1 m/s and that it is not dependent on time. Thesand with f = 318 is fairly loose and will dilate very little,say 3% at point A, i.e., (r1/r2)A & 1.03. The dilation ratevaries along failure surface AB to reach about 4% at pointB (eq. [A2] in Appendix A). What is surprising is the speedat which the mechanism expands. While the particle at theadvancing front at point A moves at the assumed 0.1 m/s,the front propagates at a speed more than 60 times faster(6.15 and 6.56 m/s at points A and B, respectively, calcu-lated from eqs. [7] and [8], respectively). The height of thefailure zone reaches 65 m in 10 s.

If the soil were to dilate more, the expansion of the mech-anism would be slower. For instance, if the dilation was tobe 10% ((r1/r2)A & 1.11), the speed of point A on theshock propagating up the slope would be only 1.68 m/s,and the height of the mechanism would grow to about 20 mafter 10 s. In both examples, the speed of mechanism expan-sion is strongly dependent (proportional) on the assumedv0 = 0.1 m/s and on the change in the density caused by thepropagating shock.

The angular displacement (rotation) of the soil mass canbe calculated from eq. [12], as r0 follows from eq. [14] atevery instant of the process. The total rotation Dq after 10 sin the first example (3% dilation) is only 1.158, and it is2.748 in the second case. This angular displacement doesnot have a straightforward interpretation, as the rotation oc-curs about a moving center (center of instantaneous rota-tion). Roughly half of that rotation took place in the first2 s, as the rate of rotation drops off exponentially with anincrease in the size of the mechanism (maintaining constantv0 at moving point A). Only the material within the volumeof the originally nucleated mechanism (2 m in height) wassubjected to the total rotation calculated, whereas the rest ofthe material ‘‘participated’’ in the process for some fractionof the 10 s period.

These examples demonstrate that an expansion of themechanism in a partially submerged slope is not only plausi-ble, but it is a realistic mode of failure in granular soils. Thismode is consistent with mechanisms of subaqueous failures(Hampton et al. 1996). The true process of failure expansion,however, is modified by the rate effects caused by watermovement in the soil. Dilatancy of sand causes a temporary



drop in the pore-water pressure (suction) that momentarilystabilizes the slope, and the kinematic process continues oncethe suction is dissipated. This is an interpretation of the‘‘flaking off’’ of thin slabs of soil off the face of a slope in adredging process (Van den Berg et al. 2002). Here, the ap-pearance of the suction may temporarily impede the expan-sion of the dilated region, causing the process to be periodic.

As demonstrated in the examples, the rotation of the slopeduring the process of mechanism expansion is small, but thestrength of the sand in the entire region is decreased due todilation. Consequently, the ultimate failure will occur as adownward flow of the sand, rather than a small rotation.This is analogous to subaqueous failures, where the loss ofstability eventually leads to a turbidity current.

ConclusionsThe most critical failure mechanism found for partially

submerged granular slopes is a rotational mechanism with alog-spiral surface intersecting the face of the slope aboveand below the water table. This is contrary to popular beliefthat for slopes built of granular materials — dry or sub-merged — the most critical mechanism is a ‘‘shallow’’ col-lapse with the failure surface parallel to the slope. Thekinematic approach of limit analysis reveals that the solutionto the critical mechanism is independent of the slope height(size-independent solution), and one can identify a family ofdifferent-size failure surfaces that are geometrically similar,all characterized by the same safety factor. Once the safetyfactor drops down to unity, it is possible to construct a pro-gressive failure mechanism where the volume of the failingmass gradually increases. This expanding mechanism is geo-metrically similar at every stage of the process. The boun-dary of the mechanism has the nature of a shock-likekinematic discontinuity or a ‘‘disturbance’’ propagating intothe soil at rest. The dilatancy angle is constant along thepropagating boundary, but the density does not change uni-formly along that boundary. This is a direct consequence ofthe mass conservation and the quasi-steady (self-similar)character of the process.

Quantitative analysis indicated that the speed at which themechanism expands is governed mostly by the change in thedensity of the soil and it is one to two orders of magnitudelarger than the speed of material grains. The boundary of theexpanding mechanism has the nature of a disturbance mov-ing into the soil at rest and, once it passes through the field,the soil dilates and weakens. The conditions are then createdfor triggering a flow failure, analogous to turbidity currentsin subaqueous landslides. While no observational evidenceis documented at this time to support the theoretical find-ings, this is a plausible mode of failure of partially sub-merged slopes, of the type that is most likely responsiblefor large subaqueous landslides, and is similar to the well-documented instability propagation in ‘‘quick clay.’’

AcknowledgementsThe work presented in this paper was carried out while

the author was supported by the National Science Founda-tion, grant No. CMMI-0724022, and the Army Research Of-fice, grant No. W911NF-08–1-0376. This support is greatlyappreciated.

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Appendix ABecause the expansion of the failure region is geometri-

cally similar, point A and B on the failure surface, Fig. 6a,must be moving at the same rate as the rate of extension ofdistance l and h – l, therefore

½A1� vA

vB

¼ l

h� l

Substituting vA and vB in eq. [A1] with expressions ineqs. [7] and [8], one arrives at the density change at pointB relative to point A

Michalowski 1377


½A2� r1

r2

� �B

¼ 1� l

h� l

� r1

r2

� �A

� 1

� �cosðbþ q0 � fÞcosðbþ qh � fÞ e

ðqh�q0Þ tanf

In order for the mechanism to be quasi-steady, the centersof instantaneous rotation Oi at every instant of the processmust be located on one straight line passing through point P(intersection of the water table with the slope). Knowingthat the ratio of distance PA to PB in Fig. 6a must be the

same as the ratio l/(h – l), one can determine angle d of in-clination of line POi to the slope

½A3� tand ¼ sinq0 þ D sinqh

cosq0 þ D cosqh

where

½A4� D ¼ l

h� l

sinðq0 þ bÞsinðqh þ bÞ



expanding collapse in partially submerged granular soil slopes

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