DISTINCT ELEMENT ANALYSIS FOR PROGRESSIVE FAILURE IN …

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i) Associate Professor, Graduate School of Engineering, Tottori University, Tottori, Japan (tnishi＠cv.tottori-u.ac.jp).ii) Civil Engineer, Geoscience Research Laboratory, Tokyo, Japan.iii) Former Graduate Student, Graduate School of Engineering, Tottori University, Tottori, Japan.

The manuscript for this paper was received for review on August 13, 2009; approved on April 5, 2010.Written discussions on this paper should be submitted before March 1, 2011 to the Japanese Geotechnical Society, 4-38-2, Sengoku, Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

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SOILS AND FOUNDATIONS Vol. 50, No. 4, 505–513, Aug. 2010Japanese Geotechnical Society

DISTINCT ELEMENT ANALYSIS FORPROGRESSIVE FAILURE IN ROCK SLOPE

TSUYOSHI NISHIMURAi), TSUYOSHI FUKUDAii) and KOUJI TSUJINOiii)

ABSTRACT

This paper investigates a numerical modeling of progressive failure in rock mass using the distinct element analysis.The numerical modeling consists of two processes. The ˆrst is to get the mechanical properties of synthetic specimenscomposed of circular rigid elements with the bonded eŠect between elements. The second is to analyze deformation ofa reduced-scale rock slope under the gravity increased condition. With the calibrated properties of the synthetic speci-men unchanged, an approach to initialize stress state in the slope model is studied. Evolution of displacements and theresulting initiation of failure surface in the slope model are displayed.

Key words: distinct element method, gravity increase procedure, progressive failure, reduced-scale model, rock slope(IGC: G6/G13)

INTRODUCTION

The Purpose of This PaperSlope instability occurs in many parts of urban and

rural Japan and causes damage to housing, roads, rail-ways and other facilities. Slope engineering has always in-volved some form of risk management and this has led todevelopments in identiˆcation and the characterizationprocesses for potential slope failure as well as in methodsto evaluate the frequency of occurrence. An essential partof the hazard (slope failure) identiˆcation is the ability topredict the character of failure (type, volume), the post-failure motion (travel distance, velocity) and the state ofthe activity (Fell et al., 2008). The literature on slope sta-bility analysis using the limit equilibrium method (LEM)and the ˆnite element method (FEM) were reviewed byDuncan (1996), who pointed out the advantages and limi-tations of the various methods for practical use in en-gineering problems. Jing and Hudson (2002) thenpresented a review of the techniques, recent advances,problems and future development directions in numericalmodeling for rock mechanics and rock engineering. In hisexpanded version of the brief review in 2003, Jing sug-gested that computer methods available today are inade-quate when facing the challenge of practical problems, es-pecially when the representation of rock fracture systemsand fracture behavior are a pre-condition for successfulmodeling. Despite of all the advances in both continuumand discontinuum approaches, the development of ad-vanced numerical methods is a key issue of importance.This paper attempts to develop a numerical procedure by

providing a means to analyze the kinetic failure processand the state of stability as a function of a trial gravita-tional acceleration to a reduced-scale slope model. Thisprocedure is also able to explain the possible depth andvolume of failure at the site.

Progressive Failure in SlopeIt is a given that the failure surface does not develop at

the same instant throughout the slope, regardless ofwhether the slope is rock or soil. There must be a progres-sive event of failure development to form a distinctivefailure surface. The mechanism for progressive failure inslopes is that the peak stress exceeds the strength at onepoint in the slope, resulting in stress redistribution due tothe lower residual strength of the material (Fig. 1). Thismechanism was originally addressed by Bjerrum (1967) toexplain discrepancies between average shear stressesback-calculated along failure surfaces in an over-consoli-dated clay slope and the shear strength of the same soil ina laboratory test. Failure development has been di‹cultto quantify even for homogeneous soil. Progressivefailure is deˆned in this paper as the successive develop-ment of a failure surface in a slope through stress concen-tration, loss of shear strength and stress redistribution(Fig. 1). Although the process of the progressive failurecould develop over time due to stress concentrations andstrength degradation, the term ``progressive failure''used here does not refer to time dependent properties ofthe materials. The modeling of time-dependent behaviorof rock and soil is somewhat more challenging work asthe model needs to incorporate chemical reaction-rate

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Fig. 1. Failure surface initiation and its propagation (after Bjerrum, 1967)

506 NISHIMURA ET AL.

theory (Potyondy, 2007). The progressive mechanism ismodeled using the kinetics of packed non uniform-sizedcircular elements bonded together at their contact points.Their mechanical behavior was simulated by the distinctelement method (DEM) (Cundall, 1971).

MODELING OF THE FAILURE SURFACE IN ROCKSLOPE

Failure Surface in Rock SlopeMost rock slopes are inhomogeneous structures com-

prised of anisotropic layers of rock characterized bydiŠerent material properties, and they are often discon-tinuous because of jointing, bedding and faults. In rockslope stability analyses, the failure surface is often prede-ˆned as a persistent plane or a series of interconnectedplanes, where the planes are ˆtted to the surfaces basedon the structural observation. Such assumptions are part-ly due to the constraints of the analytical technique em-ployed (e.g., the LEM, the DEM, etc.) and can be valid incases where the response of a single discontinuity or asmall number of discontinuities is of critical importanceto the stability. However, especially on large scale slopes,it is highly unlikely that such a system of fully persistentdiscontinuous planes exists a priori to form the failuresurface. Instead, the persistence of the key discontinuitiesmay be limited and a complex interaction between thepreexisting ‰aws, the stress concentration and the result-ing crack generation is required to bring the slope tofailure (Einstein et al., 1983). On small engineered slopes,excavation leads to signiˆcant changes in the stress distri-bution of slopes and may generate fully persistent planeswhich keep propagating with stress re-distribution. Larg-er natural rock slopes seldom experience such a distur-bance and have existed with relatively stable features overthe period of thousands of years. This does not implythat in natural rock slopes a system of discontinuitiesmay not be interconnected developing the portion ofwhere the failure surface will be formed. Strength degra-

dation may occur in rock mass in a time-dependent man-ner and drive the slope to an unstable state. Thus, theproblem of rock slope instability requires progressivefailure modeling to drive the slope to catastrophic events.

Methodologies and Techniques for the Development of aFailure Surface

At the tip of the joints, stresses increase and subse-quent failure in rock occurs. Progressive failure in rockmass involves the failure of intact rock as its strength isexceeded. Based on the coulomb shearing strengthcriterion, both the shearing resistance depending on thenormal stress value (i.e., the frictional strength) and thecohesion of intact rock between discontinuous jointsresist shear failure. Rock slope analyses relating to thecriterion have been reported, most of which have been fo-cused on modeling strength degradation, and adopting anapparent cohesion loss.

Kaneko et al. (1997) used the displacement-discontinui-ty method (DDM) and fracture mechanics' principles tomodel the progressive development of shear crack in rockslope. In their analysis, rock material was assumed to behomogeneous and no preexisting cracks were considered.They compared the DDM results with the conventionallimit equilibrium method (LEM) and discussed the allow-able slope heights under the given strength parametersand slope angles.

Eberhardt et al. (2004) discussed some aspects of themodeling of the progressive failure of surface develop-ment linking initiation and degradation to eventual catas-trophic slope failure, using a hybrid method that com-bines both continuum and discontinuum numerical tech-niques to model fracture propagation. They concludedthat the use of the hybrid modeling technique helped toprovide important insight into the underlying mechan-ism, focusing on the example of a rockslide in the SwissAlps.

Recent developments in the ˆeld of rock slope analysiswere reviewed by Stead et al. (2006). They attempted to il-

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Fig. 2. Modeling of crack generation: (a) element discritization andfailure plane, (b) continuous discritization of elements along thedirection of new cracks, (c) a priori placement of crack elementalong the possible direction of the crack propagation (after Owen etal., 2007)

Fig. 3. Force transmission between elements

507PROGRESSIVE FAILURE IN ROCK SLOPE

lustrate the wide range of tools available with particularemphasis on emerging powerful modeling of hybrid tech-niques that allow for realistic simulations of rock slopefailure. Generally, there are two choices of hybrid tech-nique; one involving a hybrid continuum-/discrete-element method and another technique based on theDEM. In the ˆrst choice, the fracture plane is aligned intwo ways, as shown in Fig. 2.

(1) a process known as intra-element fracturing where aseries of new nodal points and elements are systematicallycreated, as shown in Fig. 2(b).

(2) a process known as inter-element fracturing where aseries of new nodal points are systematically created butno new element is generated, as shown in Fig. 2(c). Thisprocess is usually preferred from the computationalstandpoint and the discrete fracture orientation is alignedwith the best oriented element boundary attached to the

node considered.The presence of contact phenomena during post frac-

turing requires an eŠective contact simulation procedure.One fundamental advantage of the DEM is that thephenomena can be incorporated into the method by therigid body contact with the spring-dashpot system. Thismethod has been used to investigate a wide variety ofrock slope failure mechanisms (e.g., Fisher andEberhardt, 2007; Franz et al., 2007). Jiang et al. (2008)proposed an expanded distinct element method forsimulating crack generation and propagation due mainlyto stress re-distribution. In their modeling, cracks onlypropagate with a predetermined path in the plastic zonesobtained a priori elasto-plastic analysis, which has beencarried out to estimate the direction of principal stressesand the extent of plastic zones.

A review of numerical models in discontinuous mediawas presented by Bobet et al. (2009). In the recent de-velopment of the DEM, bonded particle models are in-troduced to simulate the mechanical behavior of rockgoverned by growth and the eventual interaction ofcracks (Potyondy and Cundall, 2004). This model hasbeen used for a wide range of applications within geo-technical engineering (Wang et al., 2003; Utili and Nova,2008).

In this paper, a numerical modeling of progressivefailure in rock mass using the DEM is presented. Theshortcoming of the DEM related to contact detection ispartly reduced by a modeling using a circular element. Asystem composed of many circular elements with a bond-ed eŠect at their contact point reproduces many featuresof rock behavior, including elasticity, fracturing anddamage accumulation. The simulation of crack genera-tion and propagation can be accomplished by continu-

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Fig. 4. Reproduction of in situ stress in slope simulation under thegravity increased condition


ously bond breaking under a given condition, as is shownin Fig. 3. The width of the bonded section is the diameterD of the smaller element joining the two bonded ele-ments. Bond breaking occurs along the element surfaceand then the cracks are oriented perpendicular to the lineconnecting the center of the two previously bonded ele-ments. The incremental crack length and distribution de-pends on the packing of the elements and their size. Parti-cle size is not a free parameter: it is related to the propa-gation of cracks and the fracture of rock material. Othermodel parameters (such as contact stiŠness and frictioncoe‹cient) are based on micro mechanical properties, notbased on physical properties measured by laboratorytests, thereby a calibration process for the modelparameters to reproduce behavior of rock materials is re-quired through simulations of laboratory tests. With thecalibrated properties unchanged, an approach to initial-ize stress state in the slope model was studied.

DISTINCT ELEMENT MODEL IMPLEMENTATION

Microparameters and MacroparametersFor a continuum analysis, the input parameters (such

as modulus and strength) can be given by an experimentperformed on laboratory size specimens, e.g., the uniaxi-al test and the Brazilian test. For the DEM, which mimicsthe macroscale behavior of rock material using a simplepacking assembly, the input parameters are not known apriori. The relation of model parameters based on micromechanical properties and commonly measured macro-properties can be ascertained by conducting a calibrationanalysis. Potyondy and Cundall (2004) performed ananalysis of a cubic packing of disks joined by contactbonds subjected to an extension strain. The results oftheir analysis were equivalent to those of a material stud-ied by Linear Elastic Fracture Analysis (LEFM) in termsof both observable behavior and mathematical descrip-tion. They also suggested that modulus and strength mustbe independence of particle size if there are su‹cient par-ticles across the specimen to obtain a representativevolume of the material response. While none of themodels available is complete or fully veriˆable, a com-parison between the behaviour of the model and themeasurable responses given by laboratory test demon-strates the capability of the model.

A Synthetic Specimen and a Reduce-scale Slope Modelunder the Gravity Increased Condition

In the numerical attempts using the DEM, prior to theslope failure simulation, the eŠects of the micro-proper-ties on the macroproperties were analyzed using the syn-thetic specimens mentioned in the previous section.Nevertheless, the same sized elements as those used in thespecimen analysis can not be adopted because currentcomputing power limits the number of elements torepresent slope proˆle over hundreds meters in height. Aslope modeling that reproduces the macro-properties ofthe synthetic specimens should be created and executed inreasonable computation time on a standard desktop-type

computer, providing the slope proˆle and the same levelof stresses in the slope. In Fig. 4, the in situ stress value inslope with a full-scale model and a reduced-scale model isillustrated. The deformation and the failure of thereduced-scale slope model was analyzed in a procedurewhich involved increasing the gravitational accelerationfrom G (＝9.8 m/s2) to nG (nÆ1). The relations betweenthe reduced-scale slope model and the prototype is writ-ten by:

Length:

Area:

Volume:

Lp＝nLm

Ap＝n2Am

Vp＝n3Vm

(1a)

(1b)

(1c)

The unit weight of the rock material in the reduced-scaleslope model and the overburden pressure at depth zm

(＝zp/n) is be given by:

gm＝ngp

sm＝gmzm＝(ngp)(zp/n)＝sp

(2a)

(2b)

The slope model was able to reproduce the same mag-nitude of the overburden pressure to the prototype. Thematerial set-up procedure was as follows.

(1) Particle generationCircular elements were placed in a rectangular area so

there was no overlap between particle-particle and theparticle-wall.

(2) Particle network buildingThrough this stage, an assembly is made. In the assem-

bly, all particles have contact points with neighboring ele-ments. This is done by the introduction of accelerationfor a speciˆc direction such as the gravitational ˆeld.Throughout the assembly, the bond eŠect is installed be-tween all the elements in contact to set up an initial bond-ed network (original assembly). The contact forces due tothe bonded eŠects do not have a signiˆcant eŠect on be-

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Table 1. Microparameters of synthetic rock material for rock slopemodel

Element

density r 2650 (kg/m3)radius r 1.0, 0.7, 0.5 (cm)contact stiŠness kn 100 MN/m

ks 25 MN/mcoe‹cient of friction m 0.577 (tan－1 m＝309)

Bond

stiŠness En 100 MN/m2

Es 25 MN/m2

shear strength tc 1.0 MN/m2

tensile strength sc 1.0 MN/m2

Fig. 5. Specimen simulation of synthetic rock material and thedamage patterns during (a) uniaxial test and (b) biaxial test

Fig. 6. Specimen simulation of synthetic rock material and thedamage patterns during the Brazilan test

Table 2. Analytical condition and macroparameters of synthetic rockmaterial for rock slope model

Analytical condition

time increment: Dt＝1.0×10－5 (sec)upper boundary displacement: Du＝1.0×10－6 (cm)the ratio of the number of element: r10 : r07 : r05＝45 : 17 : 38

Macro-parameters

uniaxial strength qu (MN/m2) 1.440tensile strength st (MN/m2) 0.156young's modulus E (MN/m2) 145.0poisson's ratio n 0.295cohesion c (MN/m2) 0.359angle of internal friction q (9) 36.9


havior of the assembly when the bonding force is muchsmaller than the setting bond strength and the magnitudeof applied external forces.

(3) TrimmingModels for specimen and slope failure simulations are

shaped by deleting elements in the original assembly.The microproperties in Table 1 with three radii of 1

cm, 0.7 cm and 0.5 cm were used. Then a rectangularspecimen (120×60 cm2) for biaxial test and a circle speci-men for the Brazilian test (diameter: 60 cm) were formedas stage (3), as shown in Figs. 5 and 6. It is well estab-lished that when the number of particles exceeds 1000, itis enough to obtain meaningful biaxial test results (Utiliand Nova, 2008). In our study, approximately 3000 parti-cles were used in the biaxial test run and around 1000 inthe Brazilian test run. No gravity was applied to the speci-mens. Table 2 lists the analytical conditions and the ratioof the number of element generated in the stage (1). In thetable, the lower su‹ces, e.g., 10 of r10, mean radius of cir-cular element. The program used here was originally de-veloped based on the concept of the DEM with the bond-ed eŠect. The damping forces applied to each particlewere formulated as follows: (1) the damping eŠect wassimilar to hysteretic damping, in which the energy loss percycle was independent on the rate at which the cycle iscomputed. The damping forces was given by a factorproportional to the unbalanced forces acting on each par-ticle while the particle had a bonded contact point, (2) thedamping eŠect is dependent on the rate, i.e., the viscousdashpot is assumed at contact point when the particle hadrigid body contact and no bonded contact.

MACROPROPERTIES OF THE SYNTHETICSPECIMEN AND SLOPE FAILURE SIMULATION

Macroproperties of the Synthetic Specimen and the SlopeModel Set-up

Figures 5 and 6 show the specimens near the peak load.The linkage of the bond break cuts across the specimensin which the dots represent tension-induced or shear-induced failure. The macroparameters obtained fromboth simulations are listed in Table 2.

Figure 7 shows the slope model which was formed bydeleting particles on the left side of the original assembly.The height of ground h was 200 cm and the width w was200 cm. Three slopes with toe angles b of 609, 709and809were used, and the height of the three slopes was con-stant at 100 cm. As seen in Eq. (2), the overburden pres-sure sm in the slope model corresponded to the pressure atheight nhm, e.g., for n＝100, the height of 100 cm in the

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Fig. 7. Rock slope model with constant inclination using the assemblyof circular element

Fig. 8. Step-wise increase of n-value to get the allowable height of the slope using the reduced-scale slope model

Fig. 9. Initiation of failure at the toe of slope and the static state of theslope b＝809for n＝140


model corresponds to a height of 100 m. An assemblycomposed of un-bonded particles, i.e., the strength is setsc＝tc＝0, produced a frictional cohesionless materialand it meant that the slope model cannot be formed usingthe un-bonded model according to the stability factor2c/g, where c is cohesion and g is unit weight. To simu-late a frictional cohesive material, the bonded modelshown in Fig. 3 was adopted.

Reduced-scale Slope Model under the Gravity IncreasedCondition

Crack generation and slope failure were analyzed withthe increase of gravity. The increase was explained withthe value of n in nG where G is gravitational acceleration.The value of n was increased in a step-wise manner, as isshown in Fig. 8. In this simulation, the initial value of n(n0) and the increment of n (Dn) were assumed to be 100and 5. If increased gravity gives little damage to the slopemodels, the sum of incremental displacement in one timestep (Dt ) will gradually decrease and then the slopereaches a quasi-static state by the damping eŠects men-tioned in A Synthetic Specimen and a Reduce-scale Slope

Model under the Gravity Increased Condition, and asseen in Fig. 9 for b＝809, n＝140. This means that theenergy introduced by the increased gravity is stored in theassembly in the form of strain energy in both the particle-particle contact points and the bond material. When thestress values in the bond material reach either strength,failure at the contact point will occur, and a resultingfailure plane will propagate. This propagation drives theslope model to large deformation and catastrophic slopefailure, as shown in Figs. 10, 11 and 12. In these threeˆgures, the upper ˆgures show the contact points wherethe bond broke and the lower ˆgures show the particledisplacement. The value of iteration of the time step Dt,which gives the computational elapsed time since theslope models have been set under the given gravity ac-celeration, is shown just beneath each ˆgure. Thesevalues can be used only as a relative rate of deformationunder the gravity increased condition.

Needless to say, no such increase in gravity occurs onthe surface of the earth, but when the balance of forcesacting on the slope is disturbed, the slope will fail. Slopewill fail due to a triggering event and the degradation of

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Fig. 10. Propagation of failure point and displacement of elements forming under the acceleration of 145G (The inclination of the slope b＝809)



the components of the strengths with time. As such, thevalue of n could be treated as a parameter which indicatesthe stability of the slope investigated. The slope of b＝

809reached the quasi-static state with damage at n＝140,as shown in Fig. 9, and then fails at n＝145 in Fig. 10. Itis then possible to calculate the allowable height Hm

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Fig. 13. Comparison of the allowable slope height by the DEM andthe LEM


(＝nhm) of the slope to be higher than 140 m and lowerthan 145 m. If the height Hp of a slope at the site is 100 m,the ratio of Hm/Hp can be used as the factor to explain thestate of stability for the slope.

The question of where failure initiates and howpropagates are of importance. From these three ˆgures, itis observed that failure occurs at the toe of the slope andthe resulting failure propagation can be two types, (1)failure propagation that curves from the toe to the upperface of the slope but does not reach the crest of the slope,and (2) failure propagation that curves from the toe to theupper face of the slope and the following propagation tothe crest with tension cracks occurring at the slope crest.Although in many cases the ˆrst sign of instability inslopes are often tension cracks at the slope crest, this doesnot imply that failure initiates at the point. When a sub-stantial part of the shear strength in materials is fromfriction, failure will occur at the toe of the slope. It is thenplausible that the failure progresses from the toe towardthe crest of the slope. The failure surface can be fully de-veloped before any tension cracks occur at the top.

Failure initiation in soil slope is dependent upon thegroundwater condition. When pore water pressure isadded, failure will occur within the slope, where un-drained condition prevail, and propagate toward the toeor the crest. On the other hand, if the slope is in thedrained condition, failure is more likely to initiate at thetoe or the crest and propagate inward.

COMPARISON OF THE NUMERICAL RESULTSWITH THE CONVENTIONAL LEF METHOD

Figure 13 shows the relation between the allowableslope height Hm (the upper value, see Reduced-scale SlopeModel under the Gravity Increased Condition) and theslope toe angle. The height for the slope of b＝809is thelowest. This could result in that the slope of 809is themost unstable slope. This ˆgure shows also the relationbased on the limit equilibrium method (LEM) (Hoek andBray, 1977). The LEM calculation was conducted usingthe mechanical properties (the cohesion and the internalfriction angle) obtained from the synthetic specimenanalysis described in the sections A Synthetic Specimen


and a Reduce-scale Slope Model under the Gravity In-creased Condition and Macroproperties of the SyntheticSpecimen and the Slope Model Set-up, assuming that theslope of the ground is homogenous and the unit weightcan be expressed by g＝2500 kN/m3. This value of g is areference value of rock and is not calculated using thevalue of the porosity in the assembly, as seen in Fig. 7.

The comparison indicates that for angles smaller thanb＝709, the allowable height by the LEM is lower thanthat given by numerical results, and for angles greaterthan b＝709, the LEM produces a greater allowableheight. Numerical results may be less sensitive to changesin the slope toe angle. This could be due to the diŠerencesin both methods: (1) the numerical method includesprogressive failure modeling, (2) the circular failure enve-lope is assumed in the LEM but in the numerical method,the envelope is not circular.

CONCLUSION

Various numerical methods (continuum and discon-tinuum methods, and hybrid methods which combineboth continuum and discontinuum techniques to simulatefracturing process) have been applied to demonstrate theevolution of failure in rock slope. In this paper, a numeri-cal modeling of progressive failure in rock mass using thedistinct element method with the bonded eŠect element isintroduced to modeling the rock slope failure. Themodeling can give a possible failure volume of rock slopebased on the geometrical data and strength properties,such as cohesion and the internal friction angle. A com-parison of our results with LEM results was given interms of the allowable slope height. No decisive state-ment about the eŠect of the mechanical parameters canbe made because the analysis was performed using limitedinput values. Ongoing work should be done to incor-porate the eŠects of the macro parameters and the initialstress condition.

ACKNOWLEDGEMENTS

This work is partly supported by Japanese Society forthe Promotion of Sciences (JSPS), Grant-in-Aid forScience Research, No. 20260462.

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