Rock Engineering and Rock Mechanics: Structures in and onRock Masses Alejano, Perucho, Olalla & Jimnez (Eds)

2014 Taylor & Francis Group, London, 978-1-138-00149-7

A new approach of limit equilibrium models for rock joints

Manuel J.A. Leal-GomesUniversity of Trs-os-Montes e Alto Douro, Vila Real, Portugal

Carlos Dinis da GamaIST, University of Lisbon, Lisbon, Portugal

ABSTRACT: This paper refers the limitations and drawbacks of the main classical rock mass joint limitequilibrium models of forces. Their main hindrance is due to the terms in which they are approached by forcedimensions [LMT2]. Consequently linear parameters are transformed in dimensionless variables, like roughnessslope and there is a lack of information in these models. A new original limit equilibrium model of strain energiesthat integrates linear parameters as roughness amplitude is introduced. It solves most problems of the classicalmodels because it uses energies that are richer in dimensions [L2MT2] and in information. This model considersstrength and deformation features to achieve the proposed aim.

1 INTRODUCTION

The classical rock mass joint equilibrium limit mod-els have serious drawbacks because of the termsin which they are approached by force dimensions[LMT2]. Actually they contain insufficient physi-cal dimensions and low level information to allowa complete approach to the joint sliding problem.The full integration of geometrical variables such asroughness amplitude and length dimensions is not pos-sible, except when they are transformed in dimension-less variables like roughness inclination. Furthermore,in accordance with classical models (Patton 1966,Barton & Choubey 1977) two homothetical asperitieshaving the same slope but different amplitudes h haveexactly the same strength. In effect, the asperity havinglarger amplitude is stronger (Fig. 1).

Figure 1. The same i is obtained for different h and b.

This paper approaches those aspects and otherinconsistencies of the classical models and introducesa new and original limit equilibrium model of strainenergies. Energies have more dimensions [L2MT2]than forces and by integration of strength and deforma-bility features of joints a more accurate description ofjoint shear yielding is possible.

2 THE EFFECT OF ROUGHNESS

In accordance with Muralha (1995) two parameters arenecessary for adequate description of joint roughness.A textural parameter related with slope and an ampli-tude parameter related with distance between the topand the base of the asperities. But in equations of limitequilibrium models of forces only there is place for oneroughness parameter. Under the influence of Pattonand because these equations are relations betweentangential and normal forces, a textural parameter istaken. Difficulties in the association of images, mor-phology, physical phenomena and basic definitionsappear in this field. Naf problems like to decide ifa sinusoidal surface is rougher than another homo-thetical sinusoidal surface having lower maximumamplitude of roughness (Rmax) appear in spite of itsgreater sliding strength. Some indexes like Rmax agreewith this hypothesis. Other indexes, like the fractaldimension D and asperity slope i of profiles, deny thishypothesis. Actually, in accordance with Patton, theamplitude of roughness is completely ignored and onlythe roughness inclination is considered in roughnessdescriptions.

However Leal-Gomes (1998) verified in tests per-formed on matched samples from a large artificial joint

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in porphyritic granite that greater strengths correspondto sample profiles having larger roughness amplitude.The increase of shear strength with Rmax probably isdue to a peculiar mechanism initially not considered:more intense tangential forces must concentrate ontaller and stronger asperities of joint walls than overless conspicuous asperities.

Leal-Gomes (2000) discusses this effect referringthat it seemed that the upper wall sliding was blockedagainst the taller asperities of lower wall. So the testsshow greater peak shear strengths than those deducedfrom measurement of roughness slope i applying Pat-tons model. So it probably is conservative. The energyin excess to overcome irregular asperities having largeramplitudes, which is reflected in higher shear forces,is transformed in increase of dilation angles withoutmorphological correspondence.

Once more is referred that using the factor tan(r + i) (where r = residual friction angle) and sub-stituting h/b (where b=width of the asperity base)(dimensions of both h and b [L]) by tan i vital infor-mation is lost. The same tan i is achieved for differentproportional h and b (Fig. 1). When forces are con-sidered in slidings only in a very insufficient andambiguous way the amplitudes may be representedby dimensionless coefficients as demanded by forceequilibriums. Sliding models of forces only rudimentaland unintentionally integrate linear parameters relatedwith scale problems. And there isnt any meaning-ful advance on this matter while this hindrance is notovercame.

3 THE EFFECT OF n

If Pattons model is considered we may obtain

for low n and

for highn, where c= fictitious cohesion;n = averagenormal pressure in joints; = shear strength. R has thebehavior showed in Figure 2. In accordance with Pat-ton R decreases hyperbolically when n is high puttingmost of concrete problems of joint strength estimationon the safe side of engineering because this modeldoes those strengths lower than they effectively are.Actually when roughness are more strongly and tightlymatched on account of larger n, Figure 2 suggeststhat the asperity shear contribution is less meaning-ful and it contributes less for the global joint strength(decreasing of c/n) which obviously is absurd. SoPattons model must be conservative, also by this rea-son, for tension fields involved in works (Leal-Gomes2010). But besides these general considerations, it isimpossible to say how much conservative it is.

Bartons model

Figure 2. Hyperbolical decreasing of /n with n.

where JRC= Joint Roughness Coefficient; JCS=JointStrength Coefficient of joint walls, already implicitlyintegrates some of these considerations. For instancethe Barton & Choubeys (1977) typical profiles hav-ing higher JRC, also have higher Rmax and are stronger.But the consideration of roughness amplitude is unin-tentional like in Pattons model. Although Bartonsmodel is an improvement in comparison with Pat-tons model, it doesnt solve the basic obstacles abovepointed out. For instance, for very highn, log(JCS/n)tends toward zero as n is closer and closer JCS (theparallel hyperbolical decreasing of R in Pattons modelmust be observed). So, when the asperity shear effectis more important and notorious, being the roughnessof the two joint walls strongly imbricate and preventedthe dilation, it disappears from the cited models equa-tions that are reduced to residual parameters (r). Sothe shear strength obtained is situated so much on thesafe side of engineering that it is almost useless.

There is other formalizations for the same problemsthat use force equilibriums without clear advances inspite of several refinements on their parameters andconceptions.

4 A LIMIT EQUILIBRIUM MODEL OF STRAINENERGIES FOR ROCK JOINTS

Almost all the paradoxes, incongruities and limitationsof models of limit equilibrium of forces for rock jointsare solved when the strain energies involved in theirslidings are considered.

Leal-Gomes et al. (2013) studying the work neededfor upper wall of a matched horizontal joint to over-come regular asperities on the lower wall having slopei and amplitude h demonstrated that the strain energyW necessary for the achievement of limit equilibriumis provided by

where = friction angle; N= normal force in the dis-continuity; c = asperity shear strength; as = asperitysheared area percentage at a joint having an area A;

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dp = peak displacement; (as.A.c)= force required forjoint asperity failure; and

If the joint is tilted the corresponding angle must be subtracted or added to i in accordance withunfavorable or favorable positions.

Experimental checking of equation (4) is impossi-ble because joint strain energies can not be measuredeither in laboratory or in field. They only can becalculated. Because this is less appealing than con-ducting experimentation, very little research has beendevoted to the adequate quantification of strain ener-gies included in the sliding process of rock joints.However, these calculations of strain energies fordetermining joint yielding are very useful becausethey allow better joint stability evaluations and providetrustworthy indications about joint stabilities.

The meaning of variables dp, c, as and i or tan1 V(dilation under a considerable n) that are containedin equation (4) is necessary for understanding thisapproach.

dp is related with roughness amplitude h throughequation (5). In fact, dp is the displacement to be con-sidered because it is the parameter that assures jointpeak strength failure and therefore that both dilationand asperity failures are completely achieved. How-ever, determining the value of dp sometimes is difficultunder high n due to the ductile behavior of slides. dpalso depends on discontinuity scale.

Variable h is easily measured at geological sites,but it is not the most adequate parameter in a strainenergy formulation connected with slide assessments,because they are always complex. More conspicuousasperities are first to be sheared and the roughnessdeformations along joints under high n must be con-sidered. Therefore the maximum roughness amplitudeh does not interfere directly in the energy value assess-ment, but it is intrinsically connected to the slidingprocess through dp.

For dp estimating there is mainly the Barton (1990)formula

where L= discontinuity length in meters. And theAsadollahi et al. (2010) formula

where in opposition to Bartons equation (6), dp dimin-ishes as JRC increases. But equation (6) does notconsider explicitly the intervention of n (which hasthe greatest importance in this phenomenon) likeequation (7) does.

Equation (6) seems to overestimate dp values forsound, rough and matched rock joints. For granitesound samples having between 10.5 and 16 cm inlength dp values provided by equation (7) are closer the

experimental data (Leal-Gomes et al. 2013). But forsmall joint samples (L= 5 cm) equation (6) is ratheraccurate.

About the asperity shear strength c, its value maybe around JCS/2, although this involves some overesti-mation. However, the asperity confinement producedby the increase of average normal stress in matcheddiscontinuities, may put this c obtained by this wayon the safe side of engineering. Therefore this proce-dure seems adequate. JCS is very easily obtained in thefield with Schmidt hammer tests (Ulusay & Hudson2007).

Dilation angle i under very low normal stresscorresponds to the morphological slope of asperities.But as n increases, asperity wearing and deforma-tions occur in joint slides, resulting in a reduction ofthe actual dilation angle (tan1 V).

Ladanyi & Archambault (1970) studied this prob-lem and proposed the function

where V= tangent of the dilation angle under a sig-nificant n; c uniaxial compressive strength of therock walls. In accordance with Ladanyi & Archam-bault k2 is a parameter with a value near 4. Howeverk2 must be larger than 4 to have suitability of equation8) to the data published by these authors. With respectto data published by Leal-Gomes (1998), k2 must behigher than 100, for providing an adequate descriptionof dilation variation as n increases. Actually, when nis very low (1 ( n/c)) becomes very close to 1 andthe suitability of V obtained through (8) to experimen-tal data using low k2 is not achieved. Little attentionhas been paid to studies of dilation refinements underseveral orders of n and for different scales.

Morphological dilation i was often used in modelsunder high n and this behavior is completely unsuit-able because roughness deforms and it is smashed astangential forces act, so the global effect is the reduc-tion of dilation. On the other hand,V is apparently littleaffected by scale factors (Leal-Gomes 1998).

Barton & Choubey (1977), Bandis (1980),Grasselli & Egger (2003), Leal-Gomes & Dinis-da-Gama (2009), Marache et al. (2008), Belem et al.(2009) considered the problem of estimating the asper-ity sheared area (as.A). ButYoshinaka et al. (1993) andLadanyi &Archambault (1970) were who provided themost expeditious tools to approach this problem.

The experimental work of Yoshinaka et al. (1993)on joint samples of Inada granite having from 400 cm2

up to 1600 cm2 in area A, under n between 0.5 and2 MPa, allowed that Leal-Gomes et al. (2013) draw thegraph of Figure 3. And in accordance with Ladanyi &Archambault (1970) the phenomenon is ruled by theexpression

where k1 is about 1.5.

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Figure 3. Evolution of as as n increases, for severalconstant A.

5 CONCLUSIONS

A new rock joint stability criterion is proposed by com-parison between energy W values obtained throughequation (4) and the increase of position energy G thatis necessary to override lower wall roughness ampli-tude h. In accordance with this criterion, if G is lowerthan W, the discontinuity is stable and otherwise it isnot stable.

The greatest hindrance of this limit equilibriummodel of strain energies for rock joints is found inlow W values obtained through equation (4) mainly forsmall samples. For large samples, W values obtainedare acceptable. W values for small samples are ofthe order of few Joules while common sense sug-gests that actual W in these slides is higher. This factis due to insufficient control on parameters, graphsand equations supporting equation (4), as well as tounattended losses of energy by heat, plastic defor-mation processes, reduction of broken asperities tosmaller fragments and dust between joint sliding wallsand several inefficiencies of the applied shear forcesystems.

The assessment of rock joint stability through tra-ditional limit equilibrium models of forces has beenconducted in a partial way and a non quantifiableextent. Actually they only consider non dimensionalparameters of roughness. But the consideration ofroughness amplitude (a parameter with linear dimen-sions) is essential for a larger and complete approach.This is only possible through the limit equilibriummodels of strain energies that are proposed in this arti-cle. With classical models it is impossible to decide thepercentage of strength that must be ascribed to dilationand what percentage of that strength must be ascribedto roughness amplitude in slidings.

In effect, the proposed joint stability assessmentmethod through strain energies integrates deformabil-ity features with conventional strength criteria. And ituses energies that are richer in physical dimensions[L2MT2] and therefore in intrinsic information thanforces [LMT2].

The reason to detect low number of accidentswith those incipient models dealing with forces isdue to the conservative approach of the Pattonsand Bartons models. In effect, these models ana-lyze partially the joint sliding phenomena and mostengineers prefer the safe solutions they produce. Forexample, the calculation of tan(+ i) leads to valuesgreater than 1, and so conservative evaluations areobtained. Actually tan(+ i) rises rapidly and so shearstrengths because they are underestimated in calcula-tions involving parameters which consider parameterswith lower values than the effective ones.

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