Peru Gheibie Duzgun-Rock Slope

8/6/2019 Peru Gheibie Duzgun-Rock Slope

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7th South American Congress on Rock Mechanics, December 2 to 4, 2010.

Comparacin del ndice de fiabilidad de un talud de roca obtenidaspor equilibrio lmite y mtodos de modelacin numrica

Comparison of reliability index of a rock slope derived by limit

equilibrium and numerical modeling methods

Gheibie S., S.H.B Duzgun

Mining Engineering Department, Middle East Technical University 06531, Ankara, Turkey

BriefThis paper deals with probabilistic analysis of rock slopes through numerical approaches as well as comparing

the results with probabilistic limit sate formulation of the same slope. It is found that these two probabilistic

approaches yield different safety conditions where the probabilistic limit equilibrium approach provides safe

slope condition, whereas the probabilistic numerical analysis approach gives unstable slope condition. The

main reason for such a difference would be different parameters used in the two probabilistic stability analyses

AbstractRock slope stability analysis is considered one of the important issues in rock engineering. The engineers always

try to have the best design in terms of economical and safety aspects. Nowadays, it is tried to consider the

uncertainty as influential concept in rock structure designs. In most of the cases the limit equilibrium method is

used to model the rock slopes probabilistically to include the uncertainty effects. However, it was known that this

classical method does not consider some main facts in rock slope stability. Thus, it is offered to use advanced

numerical modeling methods in analyzing the rock slopes. In this paper, the probabilistic slope stability analysis

was combined by Distinct Element Numerical method to compare the probability of failure of a sliding rock

slope. Results of this study shows that there is a considerable difference between numerical and classical

approaches, the slope modeled by classical limit equilibrium method does not fail even in one pair of friction

angle and cohesion and the reliability index is 3.57 which shows the safety level of the slope. However, in

numerical method the reliability index is about 0.235 which shows that the slope is highly unsafe. This is

because, the classical method does not include the joints mechanical factors in detail, however, distinct element

method also considers Joint Shear Stiffness which is believed to have the most important effect on joints shear

behavior.

1. INTRODUCTION

Numerous numbers of lives and properties werelost all over the world due to slope failures

although stability analyses are carried out. Mostof these analyses are based on the deterministicmethods which do not consider the effect of

uncertainty associated with parameters likeground water pressure, rock mass, anddiscontinuities shear strength. Suchuncertainties cause variation in failureprobability of slopes that have the same factor

of safety. As a result, the use of probabilisticanalysis techniques that take into account suchkind of uncertainties became more common in


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recent years. According to Newtons basicmechanics, it is clear that there are two load andresistance concepts that will lead to stability orinstability of a block on a slope. If theresistance is lower than the exerting load theblock will slide. Blocks mass, earthquake and

water pressure due to existing water in jointsare considered as load parameters and shearstrength (cohesion and friction angle) of theinterface between block and the slope are calledresisting parameters. To analyze the stability ofthe rock blocks on a slope, it is needed todetermine both the load and resistanceparameters. However, most of the parametersare varying due to time and position. For anexample, earthquake and rain fall are timevarying parameters or cohesion and frictionangle are varying spatially. Due to varyingnature of the rock mass, most of the time,engineers collect data from different parts of arock mass. Usually, engineers use the meanvalue of parameters or the most critical data tobe sure of their designs reliability. However,these days it is clear that this type of design willover/under estimate the most economicaldesign. Fortunately, in reliability engineeringthere are methods that quantifies theuncertainties and can determine the reliabilityof a design in more rational ways. Therefore,

the application of reliability methods hasconsiderably increased in recent decades.However, in rock mechanics, especially in rockslope stability, due the complexity of media incomparison with other engineering media, theyare rarely applied in practice. In recent decades,considerable works have been done inprobabilistic modeling of rock slope; however,they have just used classical mechanicalmodeling such as Dzgn et al. (1995), Low(1997), Park and West (2000), Dzgn andBhasin (2008), Rodriguez et al. (2006),

Jimenez-Rodriguez and Sitar (2007), Dianqinget al. (2009) and Tatone and Grasselli (2010)Some of the most widely used probabilisticmethods areMonte Carlo simulation technique, Rosenblueth point estimate method, andreliability index methods. Among these theFirst Order Reliability Method (FORM)proposed by Hasofer and Lind (1974) is mostwidely used one as it considers the uncertaintyand variability of the parameters involved aswell as their correlation structure.

On the other hand, numerical methods are

considered the most efficient methods ofanalyzing the stability of rock slope. Thismethod is extensively used in research and

industrial approaches. However, it has not usedwell in probabilistic methods. It is estimatedthat if the powerful numerical modeling methodcan be applied probabilistically, the morerealistic designs can be performed. In this studyprobabilistic analysis of rock slopes through

numerical approaches as well as comparing theresults with probabilistic limit sate formulationof the same slope is presented.

2. FORM APPROACH

2.1 THE PERFORMANCE FUNCTIONThe reliability assessment of an engineeringstructure usually involves the consideration ofmany variables. In particular, the supply and thedemand generally depend on several othervariables. In the FORM approach the supplyand demand concepts should be generalized.The level of performance of a structureobviously depends on the properties of supplyand demand formulation. For the purpose ofgeneralized formulation, it is necessary todefine a performance function or a statefunction as shown below (Duzgun, 1994):

( ) ( , , ,..., )1 2 3~g x g x x x xn= (1)

Where, ( ) ( )1 2~ , ,..., nx x x x= is a vector of basicvariables. The function ( )g x determines the

performance or the state of the structure.Accordingly, the limiting performance isdefined as

~( ) 0g x = which is the limit-state of

the system. When~

( )g x >0, system is in safestate and

~( )g x


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( )' *' 01 '1 *

ng

x xixi i

=

=

(2)

In which the partial derivatives'

*

g

x i

are

evaluated at the ( )'* '*,...,1x xn .Thus the minimum distance from the tangentplane to the origin of the reduced variates is thereliability index . This minimum distance tothe tangent plane on the failure surface isdetermined through the Lagrange multiplier.The computation procedure is summarized in(Ang and Tang 1984). The other importantfactor in this procedure is the computation ofdirection cosines i as

12

1

_

2

ii

n

k

g

z

i

g

z

=

=

By considering that *i ix = .

Clearly, Figure (1) illustrate that the point withthe minimum distance to the origin of thereduced variates is the most probable failurepoint.

Figure 1 Tangent Plane to g(x) = 0 at x*(After Ang and Tang, 1984)

3. PROBABILISTIC MODELING OF

SLOPE BY UDEC

For the cases where rock mass is discontinuousmedia, Distinct Element Modeling is one of themost suitable approch. The Universal DistinctElement Code (UDEC) which is a two-dimensional numerical program based on thedistinct element method for discontinuum

modeling was used in this study.A rock slope shown in figure (2) wasconstructed in UDEC media, the slope has a joint inclined 30

0toward the out of slope, and

that the angel of slope itself is about 540. It wasassumed that the rock material is in elastic stageand will not yield; this assumption was done toassess the reliability of rock slope easily, since,if rock material falls in plastic it is also neededto assess its reliability which will be morecomplicated. Moreover, usually rock slopestability problems involve low normal loads,where rock mass usually behaves elastically.The constitutive law was Mohr-Coulomb in thenumerical analysis.

The advantage of numerical modeling againstlimit state modeling is that, it is possible toformulate the process much realistic than limitstate modeling. For example, the Joint Shear

and Normal Stiffness are not considered in limitstate modeling, which were proved to have themost controlling effect in joint behavior (Bandis1990).The first step in assessing the reliability is tomodel the performance function, which wasassumed that when the sample reaches its peakstrength it is called failure. Thus it was tried toconsider the failure function a state in which therock joint had reached its peak value.Barton (1972) described joint shear stiffness(Ks) as the average gradient of the

shear stress-shear displacement curve for thesection of the curve below peak strength. Shearstiffness can be estimated from direct shear

(3)

Figure 2 the geometry of rock slope

540

30 m

300

30 m


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testing results, and its value depends on thesizeof a sample tested and generally increases withan increased in normal stress. Barton andChoubey (1977) suggested the followingequation for the estimation of the peak shearstiffness (MPa/m):

[ ]Rn

nsJCSLogJRC

LK

+

= 10.tan..

100 (4)

Where L is the joint length in meters, n isnormal stress acting on joint, JRC is jointroughness coefficient, JCS is joint compressivestrength and r is the residual friction angle of a joint. Barton and Bakhtar (1983) revealed thatthe peak shear displacement is reached whenthe joint has displaced 0.98% of its length. Thecrack was assumed to have smooth surface, thismeans the JRC value is zero, thus the above

equation can be rewritten as:

[ ]RnsL

K tan..100

= (5)

Joint length is 60 m, r is a random variableand n was calculated by writing codes inUDEC. The other joint parameter is JointNormal Stiffness. However, as the joint surfaceis smooth, according to the applied joint normalstiffness, it is estimated that there will not beconsiderable movement in normal direction.

Then, it is considered almost fix. Thus, in pairsof friction angle and cohesion the blockassumed to fail when it has displacement of0.98% of its length. The random variables inthis study were joint friction angle and cohesionof discontinuity, which were assumed to havethe normal distribution. Table (1) shows themean and standard deviation for both ofvariables.

Table 1 the mean and STD for random variablesStandard Deviation Mean Min Max c.o.v

Cohesion (Pa) 3e4 12e4 2e4 18e4 0.25

Friction Angle 8.50

350

200

400

0.24

After a wide range of model running, figure(3) shows the failure surface and also the safeand non-safe regions.

Failure Surface

30

32

34

36

38

1.00E+04 6.00E+04 1.10E+05 1.60E+05 2.10E+05

Cohesion

FrictionAngle

Mean of cohesion

and friction angle

Safe Region

Unsafe Region

Figure 3 the failure surface

As can be seen in the figure (4) and figure (5)the failure surfaces can be derived by curvefitting through regression analysis. To have aprecise regression, the curve was divided intotwo different domains then the relation of frictionangle and cohesion was derived for each domain.The equations 6.1 and 6.2 show the failurefunction derived by UDEC:

For 410418.2 exe (Figure 4)

( ) += 95.360001.0106 210 CCxg (6.1)

And for 418410 exe (Figure 5)

( ) ++= 59.3210510 6211 CCxg (6.2)

Performance Function 1

y = 6E-10x2 - 0.0001x + 36.946

R2 = 0.9852

32

32.5

33

33.5

34

34.5

35

35.5

0.00E+00 5.00E+04 1.00E+05 1.50E+05 2.00E+05

Cohesion

Friction

angle

Figure 4 the failure function for domain 1


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Performance Function 2

y = -1E-11x2 + 5E-06x + 32.594

R2 = 0.9841

32.95

33

33.05

33.1

33.15

33.2

0.00E+00 5.00E+04 1.00E+05 1.50E+05 2.00E+05

Cohesion

Friction

angl

e

Figure 5 the failure function for domain 2

By applying the FORM to the derived equationsthe design point or the closest point on the failuresurface from mean of random variable is(C=12e4, = 33.18) and the reliability index is0.235.

The variability of pairs of random variables onthe failure surface was calculated, where c.o.v.for is 0.02 and for C is 0.51.

4. PROBABILISTIC MODELING OF THESLOPE BY LIMIT EQUILIBRIUM METHOD

Figure (2) shows the geometry of a slope whichwas studied. The, first step in probabilistic limitequilibrium analysis is to define the performancefunction. The usual method to model theperformance of a rock slope is to use the limitstate equation. Most of the researches have usedthis method to define the performance function.It is generally defined as given in Eq. 4. (Duzgun,1994):

( )g x R Df f= (7)

Then, for a rock slope lying on a crack:

( )

pp

pp

CosVSinW

SinVUCosWcAxg

+=

tan(8)

Where,

=W Weight of the sliding block (ton/m)=V Force due to water pressure in the

tension crack (ton/m)=U Uplift force due to pressure on the

sliding surface (ton/m)=f Dip of slope face (radians)

=p Dip of discontinuity plane (radians)= Base area of the sliding block (m

2/m)

=c Cohesion (ton/m2)

= Friction angle

The same random variables in table (1) were

also used in this formulation. For computationof the RocPlane software and spreadsheet byFadlemula (2007) were utilized. The computedprobability of failure is zero, since there is nopair of friction angle and cohesion in domain bywhich the slope fails and the corresponding is3.57.

5. CONCLUSIONS

The stability of a rock slope potential to slidewas studied in this article. Joints are playingimportant roles in controlling the slide of a rockblocks. Joints shear stiffness is an importantparameter which is not considered in limitequilibrium method, thus a factor which affectsthe shearing of a rock joint is not taken intoaccount in computations. Thus, it seems thatanalyzing the rock slopes stability withoutconsidering this parameter may produce errorsin designs.

In this article, a rock block sliding on a jointplane was modeled by both limit equilibrium

and distinct element methods. Comparison ofthe results between the two methods revealedthe importance of the way a rock slope ismodeled. For limit equilibrium methodRocPlane software and a spreadsheet providedby Fadlemula (2007) were used, the analysisshow that for the given random variables intable (1), the slope in figure (1) never will failand the probability of failure is zero and thecorresponding reliability index is 3.57.

However, results obtained form UDEC provesthat the rock slope has a critical circumstance

which may be considered as a failing structure.The reliability index has been calculated as0.235 which is showing a low safety.

This difference mainly comes from the factthat the limit equilibrium modeling does not payattention to the facts of rock joints shearbehavior like joint shear stiffness. Also, thereare some differences between the calculatednormal stress by numerical modeling and limitequilibrium method which affects the shearingphenomena.

By considering the fact that numerical methods

are more reliable than limit equilibrium ones, itcan be concluded that for more reliable designs


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the engineers should combine the probabilisticmethod by numerical modeling.

REFERENCESAng, A.H.S., and Tang, W.H., (1984). "Probability

Concepts in Engineering Planning and Design" Vol.2.Decision, risk, and reliability. John Wiley and Sons.Bandis, S.C. (1990). "Mechanical properties of rock

joints" in Proceeding of the international symposiumon rock joints, Loen, Norway, 4-6 June: 125-140.

Barton, N: (1972). "A model study of rock-jointdeformation" Int. J Rock Mech. Min Sci. Geomech.Abstr. (9): 579602.

Barton N.and V. Choubey, (1977). "The shearstrength of rock joints in theory and practice" RockMech. (10): 154.

Duzgun HSB, Bhasin RK. (2008). "Probabilisticstability evaluation of Oppstadhornet rock slope"Norway, Rock Mech. Rock Eng doi: 10.1007/s00603-

008-0011-3.Duzgun, H. S. B., Bozdag, T. and

Pasamehmetoglu, A.G., (1995). "A ReliabilityApproach to Wedge Stability Analysis" Proc. of 8thISRM Congress: 389-392.

Duzgun, H. S. B., (1994). "Plane failure analysis ofrock slopes: A Reliability Approach" M.Sc. Thesis,Middle East Technical University, Turkey.

Fadlelmula, M. M (2007). Probabilistic Modeling ofFailure in Rock Slopes M.Sc. Thesis, Middle EastTechnical University, Turkey.

Hasofer, A.M. and Lind, N.C., (1974). "Exact andinvariant second-moment code format" J.Engrg.Mech., ASCE, 100(1):111-121.

Jimenez-Rodriguez, R., Sitar, N., and Chacon, J.,(2006). "System reliability approach to rock slopestability" Int. J. Rock Mech. Min. Sci. 43 (6):847859.

Jimenez-Rodriguez R, Sitar N. (2007). "Rock wedgestability analysis using system reliability methods"Rock Mech. Rock Eng. 40(4):41927.

Low, B. K., (1997). "Reliability analysis of rockwedges J. Geotech. Geoenviron. Engng. 123(6):498505.

Park, H.J., West, T.R., (2001). "Development of aprobabilistic approach for rock wedge failure Eng.Geol. (59): 233 251.

Shinozuka, M., (1983). "Basic Analysis of StructuralSafety" J. of Structural Division, ASCE (3): 109.

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Peru Gheibie Duzgun-Rock Slope