stability analysis of jointed/weathered rock slopes using the hoek-brown failure criterion

This article was downloaded by: [University of Newcastle (Australia)]On: 04 October 2014, At: 23:11Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Geosystem EngineeringPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tges20

Stability Analysis of Jointed/Weathered Rock SlopesUsing the Hoek-Brown Failure CriterionKwang-Ho You a , Yeon-Jun Park a & Ethan M. Dawson ba Dept. of Civil Eng. , The University of Suwonb URS Corporation , San FranciscoPublished online: 04 Sep 2012.

To cite this article: Kwang-Ho You , Yeon-Jun Park & Ethan M. Dawson (2000) Stability Analysis of Jointed/Weathered RockSlopes Using the Hoek-Brown Failure Criterion, Geosystem Engineering, 3:3, 90-97, DOI: 10.1080/12269328.2000.10541157

To link to this article: http://dx.doi.org/10.1080/12269328.2000.10541157

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shallnot be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/loi/tges20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/12269328.2000.10541157

http://dx.doi.org/10.1080/12269328.2000.10541157

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

Geosystem Eng., 3(3), 90-97, (September 2000)

90

Stability Analysis of Jointed/Weathered Rock Slopes Using the

Hoek-Brown Failure Criterion

Kwang-Ho You1), Yeon-Jun Park1) and Ethan M. Dawson2)

1)Dept. of Civil Eng., The University of Suwon2)URS Corporation, San Francisco

(Received October 6, 2000 ; Accepted December 4, 2000)

ABSTRACT : The failure of rock slopes usually occursalong the discontinuities. But for weathered or highlyjointed rock mass, failure surface is often curved as insoil slopes. To assess the stability of jointed/weatheredrock slopes, shear strength reduction technique is adaptedfor use with the non-linear Hoek-Brown failure criterion,an empirical approach to estimating rock mass strength.Hoek-Brown strength parameters can be estimated usingrock mass classification schemes such as RMR or GSI.Numerical results obtained by strength reduction are com-pared with limit analysis upper bound solutions derivedusing a series of linear failure surfaces tangent to theactual non-linear failure surface. Results are presented ingraphs of normalized slope height versus RMR that couldbe used as stability charts.

Key words : stability, rock slope, jointed rock, weatheredrock, strength reduction, limit analysis, stability number

INTRODUCTION

The shear strength reduction technique is an increasinglypopular method of performing slope stability analysisusing a finite element or finite difference program(Zienk-iewicz et al., 1975; Donald and Giam, 1988; Matsui andSan, 1992; Ugai, 1988; Kobayashi, 1990; Ugai and Lesh-chinsky, 1995). Stability analysis is performed by reduc-ing the shear strength of the rock mass in stages untilcollapse occurs. The factor of safety is defined as theratio of the actual shear strength of rock mass to thereduced shear strength at failure. The technique has anumber of advantages over slope stability analysis withthe method of slices. For instance, the critical failure sur-face is found automatically.

The strength reduction technique is used for slope sta-bility analysis using the non-linear Hoek-Brown failurecriterion (Hoek, 1983), an empirical approach to estimat-ing rock mass strength that has been found useful inengineering practice. Hoek-Brown strength parameters canbe estimated using rock mass classification schemes suchas the Rock Mass Rating (RMR) of Bieniawski (1976),or the more recent Geological Strength Index (GSI) of

Hoek (1994).One difficulty in adapting the strength reduction tech-

nique for use with the Hoek-Brown failure criterion isthat there are no analytical slope stability solutionsagainst which numerical results can be verified. Ofcourse, results can be checked against method-of-slicesprograms, but these programs themselves have not beenverified for non-linear failure criteria. A solution to thisdilemma was proposed by Drescher and Christopoulos(1988), who demonstrated a simple technique for comput-ing limit analysis solutions for slope stability problemswith non-linear failure criteria. They showed how upperbound limit analysis solutions can be computed using aseries of linear failure surfaces, tangent to the non-linearsurface. The upper bound is minimized by testing manydifferent tangents.

In this paper, the strength reduction technique, using theHoek-Brown failure criterion is implemented using theexplicit-finite-difference code FLAC (Itasca, 1998). Nu-merical simulations are performed for a wide range ofslope angles and rock strength properties. The results arecompared to upper bound limit analysis solutions ob-tained using the technique of Drescher and Christopoulosand they are presented in graphs of normalized slopeheight versus RMR for easy application to real problemsby field engineers.

THE HOEK-BROWN FAILURE CRITERION

One of the most difficult aspects of stability analysis ofrock slopes is choosing the appropriate shear strength.When the joint spacing is close, or block size is small incomparison to the height of the slope, and when slidingis not controlled by one or two dominant discontinuitiesor discontinuity sets, the Hoek-Brown failure criterion canbe used. The generalized Hoek-Brown failure criterion forjointed rock masses is given by

(1)

where σ1 and σ3 are the major and minor principal com-

σ1 σ3 σci mb

σ3

σci

------- s+⎝ ⎠⎛ ⎞

12---

+=

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ewca

stle

(A

ustr

alia

)] a

t 23:

11 0

4 O

ctob

er 2

014

Stability Analysis of Jointed/Weathered Rock Slopes Using the Hoek-Brown Failure Criterion 91

pressive stresses and where σci is the unconfined compres-sive strength of the intact rock. The dimensionlessconstants mb and s depend on the characteristics of therock mass. Various empirical formulas have been pro-posed for estimating these parameters based on rock massclassification schemes such as RMR or GSI. Theseempirical formulas also require as input the Hoek-Brownconstant mi for the intact rock pieces.

We have used the RMR-based formulas presented byHoek and Brown (1988), assuming 'disturbed rock mass'conditions as recommended by Sjöberg (1999a, 1999b).The later GSI-based formulas suggested by Hoek andBrown (1997) lead to unrealistically large slope heights.The RMR-based formulas are perhaps more appropriatefor estimating the peak strength of a rock mass, ratherthan the residual strength required for large-scale slopestability problems.

For disturbed rock mass conditions Hoek and Brown(1988) suggest the formulas:

(2)

(3)

MOHR-COULOMB SURFACE TANGENTTO HOEK-BROWN SURFACE

The limit analysis solution described in the next sectionrequires Mohr-Coulomb failure envelopes tangent to theHoek-Brown failure envelope at a given value of σ3. TheMohr-Coulomb failure criterion, expressed in terms of theprincipal stresses is

(4)

where

(5)

is the slope of the σ1 versus σ3 line and

(6)

is the unconfined compressive strength. φ is the frictionangle and c is the cohesion (see Figure 1).

If we use σ3 to denote the value of the minor principalstress at which the two surfaces are tangent, the frictionangle φ ′ for the Mohr-Coulomb failure surface is

(7)

where

(8)

The cohesion c' for the Mohr-Coulomb surface tangentto the Hoek-Brown surface is

(9)

where

(10)

LIMIT ANALYSIS SOLUTION

For the linear Mohr-Coulomb failure criterion, an upperbound limit analysis solution for a homogenous slope wasderived by Chen (1975). This solution has been used toverify strength reduction implementations as well asmethod of slices programs. Developing a limit analysissolution for a non-linear criterion is more difficult,although certainly possible. For instance, Zhang and Chen(1987) present a numerical procedure for deriving upperbound solutions to plane strain slope stability problemsusing a generalized Mohr-Coulomb type nonlinear yieldcondition.

A simpler technique was proposed by Drescher andChristopoulos (1988). Instead of using the non-linear cri-terion directly, a series of linear (Mohr-Coulomb) yieldsurfaces tangent to the non-linear surface are employed.With this technique Drescher and Christopoulos obtainedupper bounds within 1% of the upper bounds computedwith the much more involved procedure of Zhang andChen. The technique is based on the following theoremof limit analysis:

"A limit load computed from a convex yield surfacewhich circumscribes the actual surface will be an upperbound on the actual limit load." Chen (1975)

These conditions are satisfied by any Mohr-Coulomb

mb miRMR 100–

14----------------------------⎝ ⎠

⎛ ⎞exp=

s RMR 100–6

----------------------------⎝ ⎠⎛ ⎞exp=

σ1 σcm kσ3+=

k1 φsin+1 φsin–--------------------=

σcm 2c k=

φ′sink′ 1–k′ 1+-------------=

k′ 1mb

2------+

mb

σci------- σ3 s+⎝ ⎠

⎛ ⎞12---–

=

c′σ′cm

2 k′------------=

σ′cm 1 k′–( ) σ3 σci+mb

σci------- σ3 s+⎝ ⎠

⎛ ⎞12---

=

Fig. 1. Mohr-Coulomb failure surface tangent to a Hoek-Brownfailure surface.

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ewca

stle

(A

ustr

alia

)] a

t 23:

11 0

4 O

ctob

er 2

014

92 Kwang-Ho You, Yeon-Jun Park and Ethan M. Dawson

yield surface tangent to the Hoek-Brown surface (see Fig-ure 1). Thus, a limit load computed for the tangent Mohr-Coulomb surface will be an upper bound to the limit loadfor the Hoek-Brown surface. A least upper bound can beobtained by minimizing the limit load with respect to thepoint of tangency. This can be achieved by finding thetangent Mohr-Coulomb surface which gives the lowestlimit load.

This technique allows us to use Chen's solution(1975)to derive a limit analysis solution for the non-linear Hoek-Brown failure surface. Chen's solution assumes a logspiral failure surface with the location of the center ofrotation found by numerical minimization. The solutionsare presented in the form of dimensionless stability num-bers NMC defined as

(11)

where H is the critical height for a slope with unit weightγ and cohesion c. An analogous stability number NHB canbe defined for the Hoek-Brown yield surface as

(12)

Both stability numbers can be interpreted as normal-ized slope heights. The stability number for the Hoek-Brown yield surface can be related to that of a tangentMohr-Coulomb surface by

(13)

A least upper bound for NHB can be computed by mini-mizing expression (13) with respect to the point of tan-gency.

Chen (1975) tabulates values of NMC for various valuesof the slope angle (from the horizontal) and the frictionangle. For this study additional values of NMC wererequired for higher frictional angles than those consid-ered by Chen. These were obtained by numerically mini-mizing the expressions provided by Chen using theprogram MathCad (Math Soft, 1998).

Stability numbers were computed for various values ofRMR, mi and slope angles. For each value of RMR, theempirical formulas (2) and (3) were used to computevalues of the Hoek-Brown constants mb and s. A seriesof MohrCoulomb surfaces tangent to the resulting Hoek-Brown surface were then computed using equations (7)through (10). For each MohrCoulomb surface, the corre-sponding stability number NMC was obtained from the tab-ulated values. For friction angles intermediate to thosetabulated, stability numbers were estimated by cubic-splineinterpolation. Equation (13) was then used to compute theHoek-Brown stability number NHB for each tangent failuresurface. The minimum of these values provides a leastupper bound for the normalized slope height. The numer-

ical minimization was performed using MathCad (MathSoft, 1998).

The stability numbers obtained in this way for slopeangles β (from the horizontal) of 30o, 40o, 50o and 60o

for mi = 5, 10, 15, 20 and 30 are listed up in Table 1 andplotted in Figure 2 through 6. Stability numbers for mi =5, 10, 15, 20 and 30 with a slope angle β = 40o areshown in Figure 7. It can be seen that the normalizedslope height increases dramatically with increasing RMR.

NMCγHc

-------=

NHBγHσci-------=

NHBc

σci-------NMC=

Fig. 3. FLAC stability numbers as a function of RMR for slopesof various angles. No groundwater, mi = 10.


Fig. 4. Limit analysis and FLAC stability numbers as a functionof RMR for slopes of various angles. No groundwater, mi = 15

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ewca

stle

(A

ustr

alia

)] a

t 23:

11 0

4 O

ctob

er 2

014


Table 1. Hoek-Brown Stability Numbers (γH/σci)

mi = 5 slope angleRMR

β = 30o β = 40

o β = 50o β = 60

o

10 0.0223 0.0129 0.0075 0.004715 0.0320 0.0184 0.0109 0.006820 0.0457 0.0264 0.0157 0.009525 0.0655 0.0380 0.0226 0.013930 0.0938 0.0547 0.0326 0.020235 0.1344 0.0786 0.0478 0.029640 0.1925 0.1131 0.0698 0.043145 0.2759 0.1622 0.0999 0.063250 0.3949 0.2337 0.1446 0.092655 0.5669 0.3360 0.2073 0.135660 0.8131 0.4845 0.3036 0.199165 1.1653 0.6998 0.4420 0.293570 1.6749 1.0095 0.6443 0.434575 2.4062 1.4574 0.9384 0.638380 3.4578 2.1058 1.3754 0.9420


β = 30o β = 40

o β = 50o β = 60

o

10 0.0443 0.0251 0.0143 0.002515 0.0633 0.0359 0.0204 0.003620 0.0905 0.0514 0.0292 0.016225 0.1294 0.0736 0.0419 0.023430 0.1850 0.1053 0.0601 0.033735 0.2646 0.1507 0.0862 0.048740 0.3788 0.2156 0.1239 0.070445 0.5415 0.3088 0.1778 0.101950 0.7745 0.4424 0.2549 0.147455 1.1083 0.6337 0.3686 0.213760 1.5843 0.9078 0.5297 0.309065 2.2659 1.3009 0.7628 0.450770 3.2425 1.8687 1.0979 0.655675 4.6440 2.6882 1.5846 0.955080 6.6548 3.8561 2.2852 1.3886


β = 30o β = 40

o β = 50o β = 60

o

10 0.0663 0.0375 0.0210 0.011315 0.0958 0.0536 0.0301 0.016220 0.1350 0.0766 0.0431 0.023325 0.1936 0.1096 0.0616 0.033430 0.2769 0.1567 0.0883 0.048035 0.3961 0.2239 0.1266 0.069040 0.5661 0.3199 0.1812 0.099345 0.8096 0.4585 0.2599 0.142950 1.1570 0.6567 0.3724 0.203855 1.6542 0.9395 0.5339 0.296660 2.3655 1.3427 0.7655 0.427865 3.3819 1.9214 1.0981 0.617070 4.8360 2.7528 1.5773 0.892175 6.9155 3.9448 2.2644 1.288980 9.8894 5.6478 3.2557 1.8652

Table 1. Hoek-Brown Stability Numbers (γH/σci) (continued)


β = 30o β = 40

o β = 50o β = 60

o

10 0.0884 0.0499 0.0280 0.001315 0.1263 0.0713 0.0401 0.001820 0.1806 0.1019 0.0579 0.030425 0.2581 0.1458 0.0819 0.043630 0.3691 0.2084 0.1171 0.062335 0.5274 0.2979 0.1675 0.089340 0.7542 0.4259 0.2398 0.131645 1.0779 0.6095 0.3428 0.178250 1.5413 0.8715 0.4944 0.265655 2.2033 1.2463 0.7069 0.382560 3.1499 1.7811 1.0111 0.548665 4.5018 2.5493 1.4462 0.794970 6.4378 3.6464 2.0684 1.138175 9.1934 5.2162 2.9668 1.639080 13.146 7.4683 4.2517 2.3622


β = 30o β = 40

o β = 50o β = 60

o

10 0.1326 0.0747 0.0420 0.000915 0.1896 0.1069 0.0603 0.001320 0.2617 0.1527 0.0860 0.045125 0.3872 0.2183 0.1239 0.064530 0.5535 0.3121 0.1762 0.092335 0.7912 0.4460 0.2504 0.132140 1.1307 0.6377 0.3585 0.189345 1.6157 0.9116 0.5109 0.271150 2.3098 1.3031 0.7324 0.388255 3.3011 1.8629 1.0488 0.555160 4.7170 2.6633 1.4994 0.796965 6.7446 3.8078 2.1404 1.139370 9.6401 5.4482 3.0612 1.681675 13.784 7.7840 4.3834 2.380280 19.701 11.224 6.3184 3.4388


Dow

nloa

ded

by [

Uni

vers

ity o

f N

ewca

stle

(A

ustr

alia

)] a

t 23:

11 0

4 O

ctob

er 2

014


From these charts, one can determine slope height orslope angle when RMR, γ and σci are known.

Extension of the limit analysis solution to slopes withgroundwater is difficult because Chen's solution (1975)applies only to dry slopes. Chen's solution was extendedto account for groundwater in terms of the pore pressurecoefficient ru by Michalowski (1995a, 1995b). However,the pore pressure coefficient does not seem to be a goodway to describe pore pressure within rock slopes.

FLAC

The strength reduction technique was implementedusing the explicit-finite-difference code, FLAC (ItascaConsulting Group, 1998). While, strictly speaking, FLACis a finite difference code, it is not very different fromelement-by-element finite element codes.

For given element shape functions, the set of algebraicequations solved by FLAC is identical to that solved withthe finite element method. However, in FLAC, this set ofequations is solved using dynamic relaxation (Otter, Cas-sell & Hobbs 1966), an explicit, time-marching proce-dure in which the full dynamic equations of motion are

integrated step by step. Static solutions are obtained byincluding damping terms that gradually remove kineticenergy from the system.

The convergence criterion for FLAC is the nodal unbal-anced force, the sum of forces acting on a node from itsneighboring elements. If a node is in equilibrium, theseforces should sum to zero. For this study, the unbalancedforce of each node was normalized by the gravitationalbody force acting on that node. A simulation was consid-ered to have converged when the normalized unbalancedforce of every node in the mesh was less than 10-3.

STRENGTH REDUCTION

For slopes, the factor of safety F is traditionally definedas the ratio of the actual shear strength to the minimumshear strength required to prevent failure (Bishop, 1955).As Duncan (1996) pointed out, F is the factor by whichthe shear strength of the rock mass must be divided tobring the slope to the verge of failure. An obvious wayof computing F with a finite element or finite differenceprogram is simply to reduce the shear strength of therock mass until collapse occurs. The resulting factor ofsafety is the ratio of the actual shear strength of the rockmass to the reduced shear strength at failure. This tech-nique was used as early as 1975 by Zienkiewicz,Humpheson & Lewis (1975), and has since been appliedby Naylor (1982), Donald & Giam (1988), Matsui & San(1992), Ugai (1989), Ugai & Leshchinsky (1995) andothers.

A detailed discussion of strength reduction using FLACcan be found in Dawson et al. (1999).

For this study we are interested in the stability numberNHB rather than the factor of safety. Stability numbers canbe computed by adjusting σci (see equation (13)) untilcollapse occurs. This critical value of σci is found bybracketing and bisection. First, upper and lower bracketsare established. The initial lower bracket is any value ofσci for which the slope simulation converges. The initialupper bracket is any σci for which the simulation doesnot converge. Next, a point midway between the upperand lower brackets is tested. If the simulation converges,the lower bracket is replaced by this new value. If thesimulation does not converge, the upper bracket isreplaced. The process is repeated until the differencebetween upper and lower brackets is less than a speci-fied tolerance.

A Hoek-Brown plasticity model, written in FLAC’sbuilt-in programming language FISH, is supplied withFLAC. However, this constitutive model runs at only athird of the speed of FLAC’s built-in models. To speed upcomputations, FLAC’s built-in MohrCoulomb model isused to approximate a Hoek-Brown model. The approxi-mation is done on an element by element basis such thateach element in the mesh has its own friction angle andcohesion, depending on the element's stress state. Specifi-cally, the minor principal stress σ3 for each element is


Fig. 7. Limit analysis and FLAC stability numbers as a functionof RMR for various values of mi . No groundwater, β = 40o.

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ewca

stle

(A

ustr

alia

)] a

t 23:

11 0

4 O

ctob

er 2

014


used in equations (7) through (10) to compute an effec-tive friction angle and cohesion for the element. Duringthe bracketing and bisection procedure, the effective fric-tion angle and cohesion are updated each time a simula-tion converges to equilibrium.

EXAMPLE STABILITY ANALYSIS

The two stability analysis procedures described above,limit analysis and strength reduction, will be illustratedthrough a detailed example. Consider a 300-m high 40o

slope with RMR = 50, mi = 15 and unit weight γ = 25 kN/m3. The height of the slope is rather high for a cut slopebut determined as such to induce failure when the rockhas high RMR. If the slope is not high enough, failure isinduced at very low strength and lose some precision.

Using equations (2) and (3), the Hoek-Brown parame-ters for the rock mass are mb = 0.422 and s = 2.404 10-4.The Hoek-Brown failure surface for these parameters isplotted in Figure 8, along with the critical MohrCoulombtangent surface for the slope angle 40.

For this MohrCoulomb surface (φ = 22.02 and c = 0.317Mpa) the stability number computed using the solution ofChen (1975) is NMC = 23.645. The corresponding Hoek-

Brown stability number, from equation (13), is NHB =0.7046. Thus, according to the limit analysis solution, theslope should be stable for an unconfined compressivestrength (for the intact rock pieces) of σci 10.64 Mpa.

The FLAC numerical mesh for the slope, shown in Fig-ure 9, is 25 elements wide and 25 elements high. Hori-zontal displacements are fixed for nodes along the leftand right boundaries while both horizontal and verticaldisplacements are fixed along the bottom boundary. Theslope is simulated in plane strain, using small-strain mode(the coordinates of the nodes are not updated accordingto the computed nodal displacements). The rock mass ismodeled as a linear elastic-perfectly plastic material witha Hoek-Brown yield condition and an associated flowrule.

Using the bracketing and bisection procedure, a stabil-ity number of NHB = 0.6817 is computed, approximately3% lower than the limit analysis upper bound solution.If the mesh size is increased to 60 by 60 elements, thestability number computed reduces to NHB = 0.6567,approximately 7% less than the limit analysis solution.The velocity field at collapse (for the 25 by 25 elementmesh) is shown in Figure 10. Superimposed on the veloc-ity field is the critical log spiral slip surface for the tan-gent MohrCoulomb criterion. The failure mechanism forthe FLAC simulation appears to be almost identical tothat assumed for the limit analysis solution. The princi-pal stresses at failure for each element in the FLAC simu-lation are plotted in Figure 4, along with the Hoek-Brownand tangent MohrCoulomb failure criteria.

STRENGTH REDUCTION-LIMITANALYSIS COMPARISON

For comparison with the limit analysis solution, a seriesof strength reduction computations was conducted using a60 by 60 element mesh. Using FLAC with this mesh size,Dawson et al. (1999) found that strength reduction stabil-ity numbers for the Mohr Coulomb criterion were gener-ally within 1 percent of the limit analysis solution ofChen (1975). Hoek-Brown strength reduction results areFig. 8. Hoek-Brown failure surface, critical MohrCoulomb fail-

ure surface and FLAC element stresses at collapse for examplestability analysis.

Fig. 9. FLAC numerical mesh for example stability analysis.Fig. 10. FLAC velocity field at failure, along with critical Mohr-Coulomb log spiral failure surface.

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ewca

stle

(A

ustr

alia

)] a

t 23:

11 0

4 O

ctob

er 2

014


compared to limit analysis solutions in Figure 4 for slopeangles of 30o, 40o, 50o and 60o for mi = 15. Results of formi = 5, 10, 15, 20 and 30 with a slope angle β = 40o areshown in Figure 7. As one would expect, the strength-reduction stability numbers are slightly below the upperbound limit analysis solution. For the steeper slope angles,the difference in stability number is 2 to 3 percent, butfor the gentler slopes the difference is as much as 7 percent.

CONCLUSION

The strength reduction technique has been adapted foruse with the nonlinear Hoek-Brown failure criterion toassess the stability of jointed/weathered rock slopes. Toverify the numerical results obtained, upper bound limitanalysis solutions were developed using a method pro-posed by Drescher and Christopoulos (1988). This methodemploys a series of linear yield conditions which are tan-gent to and exceed the actual nonlinear yield condition. Aleast upper bound is obtained by testing many differenttangent yield surfaces.

Numerical and limit analysis results are presented ingraphs of normalized slope height versus RMR that mightbe useful as stability charts. However, these charts areonly as reliable as the empirical formulas for estimatingHoek-Brown parameters on which they are based. Theresults of the study confirm the value of the techniqueproposed by Drescher and Christopoulos for obtaininglimit analysis solutions for non-linear yield conditions.

REFERENCES

Bieniawski, Z.T., 1976, Rock mass classifications in rockengineering. Proceedings of the Symposium on Explora-tion for Rock Engineering, Z.T. Bieniawski, Z.T., ed. Vol.1, A.A. Balkema, Rotterdam, Holland. pp. 97-106.

Bishop, A.W., 1955, The use of the slip circle in the stabil-ity analysis of slopes. Géotechnique 5, pp. 7-17.

Chen, W.F., 1975, Limit Analysis and Soil Plasticity. Amster-dam, Elsevier.

Dawson, E.M. and Roth, W.H., 1999, Slope stability analy-sis with FLAC, Proceedings, FLAC and Numerical Mod-eling in Geomechanics, Detournay, C. and Hart, R. eds.Minneapolis, September 1999, Balkema, pp. 3-9.

Dawson, E.M., Roth, W.H. & Drescher, A., 1999, Slope sta-bility Analysis by strength reduction. Géotechnique 49,No. 6, pp. 835-840.

Donald, I.B. & Giam, S.K., 1988, Application of the nodaldisplacement method to slope stability analysis. Proc.Fifth Australia-New Zealand Conf. on Geomech., Sydney,Australia, pp. 456-460.

Drescher, A. & Christopoulos, C., 1988, Limit analysis slopestability with nonlinear yield condition. InternationalJournal for Numerical and Analytical Methods in Geome-chanics. 12. pp 341-345.

Duncan, J.M., 1996, State of the art: limit equilibrium andfinite-element analysis of slopes. J. Geotech. Engng. Div.Am. Soc. Civ. Engrs. 122, No. 7, pp. 577-596.

Hoek, E., 1983, Strength of jointed rock masses, 1983 Rank-ine Lecture, Géotechnique 33, No. 4, pp. 187-223.

Hoek, E. and Brown, E.T., 1988, The Hoek-Brown failurecriterion - a 1988 update. In Rock Engineering for Under-ground Excavtions, Proc. 15

thCanadian Rock Mech Symp,

(Curran, J.C. ed.) pp. 18-31. Dept. of Civil Engineering,University of Toronto.

Hoek, E., 1994, Strength of rock and rock masses. ISRMNew Journal 2, No. 2 pp. 4-16.

Hoek, E. and Brown, E.T., 1997, Practical estimates of rockmass strength. Int. J. Rock Mech. Min Sci., 34, No. 8, pp.1165-1186.

Itasca Consulting Group, 1998, FLAC, Fast Lagrangian Anal-ysis of Continua, Users Guide. Itasca Consulting Group,Minneapolis, Minnesota, USA.

Kobayashi, M., 1990, A study on application of finite ele-ment method to stability and settlement analysis in geo-technical engineering. Technical Note of PHRIMT, Japan,No. 1 (in Japanese).

MathSoft, 1998, Mathcad 8 User's Guide. MathSoft, Inc.,Cambridge Massachusetts.

Matsui, T. & San, K.C., 1992, Finite element slope stabilityanalysis by shear strength reduction technique. Soils andFound. 32, No. 1, pp. 59-70.

Michalowski, R.L., 1995a, Stability of slopes: limit analysisapproach. Rev. Eng. Geol. 10, pp. 51-62.

Michalowski, R.L., 1995b, Slope stability analysis: a kine-matical approach. Géotechnique 45, No. 2, pp. 283-293.

Naylor, D.J., 1982, Finite elements and slope stability. Numer.Meth. in Geomech., Proc. NATO Advanced Study Insti-tute. Lisbon, Portugal, 1981, pp. 229-244.

Otter, J.R.H., Cassell, A.C. & Hobbs, R.E., 1966, Dynamicrelaxation (Paper No. 6986). Proc. Inst. Civil Eng. 35, pp.633-656.

San, K.C., Leshchinsky, D. & Matsui, T., 1994, Geosyn-thetic reinforced slopes: limit equilibrium and finite ele-ment analyses. Soils and Found. 34, No. 2, pp. 79-85.

Sjöberg, J., 1999a, Analysis of Large-Scale Rock Slopes. Doc-toral thesis, 1999:01, Luleå University of Technology.

Sjöberg, J., 1999b, Analysis of the Aznalcollar pit slope fail-ures - A Case Study, Proceedings, FLAC and NumericalModeling in Geomechanics, Detournay, C. and Hart, R.(eds.) Minneapolis, September 1999, Balkema, pp. 63-70.

Ugai, K., 1989, A method of calculation of total factor ofsafety of slopes by elasto-plastic FEM. Soils and Founda-tions 29, No. 2, pp. 190-195 (in Japanese).

Ugai, K. & Leshchinsky, D., 1995, Three-dimensional limitequilibrium and finite element analyses: a comparison ofresults, Soils and Foundations 35, No. 4, pp. 1-7.

Zienkiewicz, O.C., Humpheson, C. & Lewis, R.W., 1975,Associated and non-associated visco-plasticity and plastic-ity in soil mechanics. Géotechnique 25, No. 4, pp. 671-689.

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ewca

stle

(A

ustr

alia

)] a

t 23:

11 0

4 O

ctob

er 2

014


Kwang-Ho You, received his BS(1984) and MS(1986) fromYonsei University and Ph.D.(1992) from the University ofMinnesota. He had done his research on numerical modelingfor slopes and underground structures such as tunnels andcaverns at the research institute of Samsung construction com-pany as a senior researcher(1993-1997). He is now an assis-tant professor at civil engineering department in University ofSuwon and interested in geostatistics, numerical modelling ofgeotechnical structures.([email protected])

Yeon-Jun Park, received his BS(1981) and MS(1983) fromSeoul National University and Ph.D.(1992) from the Univer-sity of Minnesota. He did research on rock mechanics androck engineering at Korea Institute of Geology, Mining &Materials(1983-1985, 1992-1997). He is now an assistant pro-

fessor at civil engineering department in University of Suwon.His main interest is laboratory testing of rocks and rockjoints, numerical modelling of ground structures includingrock slopes, tunnels and large underground structures.([email protected])

Ethan M. Dawson, received his BS(1984) from PrincetonUniversity, MS(1991) and Ph.D.(1995) from the University ofMinnesota. He worked for Itasca Consulting Group(1990-1992) while he was a graduate student, then joined Damesand Moore(1995-2000). He is currently working for URS as asenior engineer. His main research interest is numerical mod-eling of geotechnical engineering problems using finite ele-ment, finite difference and distinct element methods.([email protected])

Kwang-Ho You Yeon-Jun Park Ethan M. Dawson

Dow

nloa

ded

by [

Uni

vers

ity o

f N

ewca

stle

(A

ustr

alia

)] a

t 23:

11 0

4 O

ctob

er 2

014

stability analysis of jointed/weathered rock slopes using the hoek-brown failure criterion

Documents