Three-dimensional back-analysis of saturated rock slopes ... · 1971; Hoek & Bray, 1981; (b) ......

23
Three-dimensional back-analysis of saturated rock slopes in discontinuous rock—a case study Y. H. HATZOR and R. E. GOODMAN { Two historic block failures on the left abutment of Pacoima Dam, California, are back-analysed using three-dimensional limit equilibrium analy- sis. Water pressures within the boundary joints are introduced by parametric addition of vec- tors using enhanced stereographic projection. The magnitude and orientation of the resultant necessary to cause failure are established, as a function of the available friction angle. The influence of joint friction on the joint water pressure required to initiate block sliding and on the failure mode is discussed. It is shown that with increasing joint water pressure the orientation of the resultant dictates a changing failure mode, in the cases studied, from double plane to single plane sliding. The actual failure mode depends on the available friction angle of the joints. The influence of increasing joint water pressure on the factor of safety is also discussed. It is shown that the rotation of the resultant force by the addition of joint water forces rapidly reduces the factor of safety. The rate of change of the factor of safety with respect to joint water pressure in the base joints is shown to be higher for double plane sliding and lower for single plane sliding for the same block, when both failure modes are investigated. Block theory analysis is used to assess the stability of the right abutment of Pacoima Dam, presently concealed by a thick layer of gunite, and the critical blocks in the abutment are found using the procedure of Hatzor. It is demonstrated that in contrast to the unstable left abutment, the right abutment is perfectly safe, by virtue of kinematics alone. This theo- retical result is strongly validated by the field performance of the right abutment, which experienced several episodes of strong ground motions in the past. KEYWORDS: case history; dams; design; limit state design=analysis; slopes. L’article pre ´sente une re ´tro-analyse de deux ruptures de bloc de la cule ´e gauche du barrage de Pacoima en Californie. Cette analyse repose sur un examen de l’e ´quilibre limite a ` trois dimensions. Les pressions de l’eau dans les joints limites sont introduites par l’addition parame ´trique de vecteurs a ` l’aide d’une projec- tion ste ´re ´ographique ame ´liore ´e. La grandeur et l’orientation de la re ´sultante ne ´cessaire a ` la rupture sont e ´tablies en fonction de l’angle de frottement disponible. Les auteurs examinent l’influence du frottement des joints sur la pression d’eau ne ´cessaire a ` de ´clencher le glisse- ment des blocs, ainsi que sur le mode de rupture. Ils montrent que, dans les cas e ´tudie ´s, quand la pression de l’eau sur les joints augmente, l’orientation de la re ´sultante dicte un changement du mode de rupture: un glisse- ment dans un plan se substitue au glissement dans deux plans. Le mode de rupture de ´pend en fait de l’angle de frottement des joints. Les auteurs examinent aussi la fac ¸on dont l’aug- mentation de la pression de l’eau sur les joints modifie le coefficient de se ´curite ´. Ils montrent que ce coefficient baisse rapidement sous l’effet de la rotation de la re ´sultante provoque ´e par l’augmentation de la pression sur les joints. Quand ils ont examine ´ les deux modes de rupture, soit le glissement dans deux plans et le glissement dans un plan provoque ´s par la pression de l’eau sur les joints de base, ils ont constate ´ que, pour un me ˆme bloc, le coefficient de se ´curite ´ changeait plus rapidement dans un glissement dans deux plans que dans un glissement dans un plan. Les auteurs ont re- cours a ` l’analyse de la the ´orie des blocs pour e ´valuer le stabilite ´ de la cule ´e droite du barrage de Pacoima, actuellement dissimule ´e sous une e ´paisse couche de gunite, et a ` la me ´thode de Hatzor pour identifier les blocs critiques dans la cule ´e. Ils de ´montrent que, contrairement a ` la cule ´e gauche, qui est instable, la cule ´e droite, rien que du fait de la cine ´matique, ne pre ´sente aucun danger. Ce re ´sultat the ´orique est forte- ment valide ´ par le comportement de la cule ´e droite, qui, par le passe ´, a subi plusieurs grands mouvements de sol. Hatzor, Y. H. & Goodman, R. E. (1997). Ge ´otechnique 47, No. 4, 817–839 817

Transcript of Three-dimensional back-analysis of saturated rock slopes ... · 1971; Hoek & Bray, 1981; (b) ......

Three-dimensional back-analysis of saturated rock slopes indiscontinuous rockÐa case study

Y. H. HATZOR � and R. E. GOODMAN{

Two historic block failures on the left abutmentof Pacoima Dam, California, are back-analysedusing three-dimensional limit equilibrium analy-sis. Water pressures within the boundary jointsare introduced by parametric addition of vec-tors using enhanced stereographic projection.The magnitude and orientation of the resultantnecessary to cause failure are established, as afunction of the available friction angle. Thein¯uence of joint friction on the joint waterpressure required to initiate block sliding andon the failure mode is discussed. It is shownthat with increasing joint water pressure theorientation of the resultant dictates a changingfailure mode, in the cases studied, from doubleplane to single plane sliding. The actual failuremode depends on the available friction angle ofthe joints. The in¯uence of increasing jointwater pressure on the factor of safety is alsodiscussed. It is shown that the rotation of theresultant force by the addition of joint waterforces rapidly reduces the factor of safety. Therate of change of the factor of safety withrespect to joint water pressure in the base jointsis shown to be higher for double plane slidingand lower for single plane sliding for the sameblock, when both failure modes are investigated.Block theory analysis is used to assess thestability of the right abutment of PacoimaDam, presently concealed by a thick layer ofgunite, and the critical blocks in the abutmentare found using the procedure of Hatzor. It isdemonstrated that in contrast to the unstableleft abutment, the right abutment is perfectlysafe, by virtue of kinematics alone. This theo-retical result is strongly validated by the ®eldperformance of the right abutment, whichexperienced several episodes of strong groundmotions in the past.

KEYWORDS: case history; dams; design; limit statedesign=analysis; slopes.

L'article preÂsente une reÂtro-analyse de deuxruptures de bloc de la culeÂe gauche du barragede Pacoima en Californie. Cette analyse reposesur un examen de l'eÂquilibre limite aÁ troisdimensions. Les pressions de l'eau dans lesjoints limites sont introduites par l'additionparameÂtrique de vecteurs aÁ l'aide d'une projec-tion steÂreÂographique ameÂlioreÂe. La grandeur etl'orientation de la reÂsultante neÂcessaire aÁ larupture sont eÂtablies en fonction de l'angle defrottement disponible. Les auteurs examinentl'in¯uence du frottement des joints sur lapression d'eau neÂcessaire aÁ deÂclencher le glisse-ment des blocs, ainsi que sur le mode derupture. Ils montrent que, dans les cas eÂtudieÂs,quand la pression de l'eau sur les jointsaugmente, l'orientation de la reÂsultante dicteun changement du mode de rupture: un glisse-ment dans un plan se substitue au glissementdans deux plans. Le mode de rupture deÂpend enfait de l'angle de frottement des joints. Lesauteurs examinent aussi la facËon dont l'aug-mentation de la pression de l'eau sur les jointsmodi®e le coef®cient de seÂcuriteÂ. Ils montrentque ce coef®cient baisse rapidement sous l'effetde la rotation de la reÂsultante provoqueÂe parl'augmentation de la pression sur les joints.Quand ils ont examine les deux modes derupture, soit le glissement dans deux plans etle glissement dans un plan provoqueÂs par lapression de l'eau sur les joints de base, ils ontconstate que, pour un meÃme bloc, le coef®cientde seÂcurite changeait plus rapidement dans unglissement dans deux plans que dans unglissement dans un plan. Les auteurs ont re-cours aÁ l'analyse de la theÂorie des blocs poureÂvaluer le stabilite de la culeÂe droite du barragede Pacoima, actuellement dissimuleÂe sous uneeÂpaisse couche de gunite, et aÁ la meÂthode deHatzor pour identi®er les blocs critiques dans laculeÂe. Ils deÂmontrent que, contrairement aÁ laculeÂe gauche, qui est instable, la culeÂe droite,rien que du fait de la cineÂmatique, ne preÂsenteaucun danger. Ce reÂsultat theÂorique est forte-ment valide par le comportement de la culeÂedroite, qui, par le passeÂ, a subi plusieurs grandsmouvements de sol.

Hatzor, Y. H. & Goodman, R. E. (1997). GeÂotechnique 47, No. 4, 817±839

817

INTRODUCTION

Rock slope stability analysis must address rockdiscontinuities. In a pervasively fractured hardrock, conventional soil mechanics solutions are notrelevant. Engineers have devised three differentapproaches to evaluate the stability of jointed rockslopes

(a) adaptation of soil mechanics solutions usingthe theory of elasticity and statics (e.g.Terzaghi, 1962; Bray, 1966, 1967; Jaeger,1971; Hoek & Bray, 1981;

(b) limit equilibrium analysis formulated in threedimensions using vector methods (e.g. Hoek &Bray, 1981; Londe et al., 1969; Bray & Brown,1976)

(c) limit equilibrium and kinematic methods usingstereographic projection and vector analysis(e.g. Hoek & Bray, 1981; Londe et al., 1969,1970; John, 1968; Goodman, 1976; Goodman& Shi, 1985).

Each of the three methods above has been usedquite extensively by rock engineers over the pastthree decades. The ®rst approach has been typi-cally limited to a two-dimensional treament ofsimple joint patterns. Terzaghi (1962), for example,used soil mechanics techniques to analyse thestability of jointed rock slopes containing one po-tential surface of sliding; he strongly emphasizedthe adverse effect of joint water pressures on thestability of jointed rock slopes. Bray (1966, 1967)extended typical soil mechanics solutions and elas-ticity concepts to fractured and jointed rocks; hederived limit equilibrium equations suitable forsome simple cases of joint patterns under a two-dimensional state of stress, but ignored the effectof joint water pressure. Jaeger (1971) pointed outthat the fundamental dif®culty in rock slope analy-sis is uncertainty in the law of friction for lowvalues of normal stress. He used an elementarytwo-dimensional theory of stability of rock slopesfor single and double plane sliding to show howthe assumed friction law affects the factor ofsafety.

Solution methods based on vector analysis havebeen typically limited to the case of a tetrahedralwedge. The solutions (e.g. Bray & Brown, 1976)are nevertheless useful because they address thecases of single and double plane sliding, the onlytwo possible modes of sliding failure, and theyallow for water pressures inside the discontinuities.

Incorporation of stereographic projection techni-ques provides a fully three-dimensional graphicalpresentation of the problem and helps one visualizecomplex three-dimensional entities. It is necessaryto emphasize, however, that all constructions onthe stereographic projection can be performed bysimple methods of vector analysis. Various workers(Hoek & Bray, 1981; Londe et al., 1969, 1970;John, 1968; Goodman, 1976) have shown howvector analysis operations can be performed usingthe stereographic projection, and thus how limitequilibrium analysis can be achieved. Practical ad-vance was introduced with block theory (Goodman& Shi, 1985). Founded on a rigorous mathematicalbase and incorporating concepts from topology andset theory, block theory provides kinematical testsfor the removability of a block bounded by anarbitrary number of surfaces, in addition to ®ndingthe applicable failure mode and the state of staticequilibrium. Using relatively simple block theoryanalyses, one can now achieve a comprehensivestability evaluation of jointed rock slopes, providedthe geometry of the problem is clearly de®ned, andall active forces are known.

All methods discussed above assume a de®nedand predetermined rock mass geometry. Often inpractice the engineer must extrapolate from anavailable set of data to a slope where the structuremay not be exposed. This is especially crucialwhen a re-evaluation of a rock slope stability isnecessary after the rock face has been concealedby a layer of shotcrete, for example in analysingthe impact of a probable maximum ¯ood (PMF)on the stability of existing dam abutments. Thispaper discusses the incorporation of statisticalmethods in block theory analysis in order toobtain critical key blocks, following the work byHatzor (1993). In order to evaluate the sensitivityof critical block stability to water pressures, twohistoric block failures in the left abutment and thespillway of the Pacoima concrete gravity dam areback-analysed to establish the required water pres-sure to induce failure. The failures occurred dur-ing an intense rainstorm in 1938. The structuralintegrity of the right abutment, which remainedintact during the same rainstorm and during the6´6 magnitude San Fernando earthquake in 1971(Swanson & Sharma, 1979) is shown to be aresult of its favourable orientation with respect tothe rock mass geometry, evident by removabilityconsiderations alone.

Block theory fundamentals are presented byGoodman & Shi (1985), where proofs for alltheorems can be found, and therefore will not bediscussed here. A very helpful review of blocktheory is given by Goodman (1989) and recentdevelopments in the application of block theory aresummarized by Goodman (1995). Some essentialconcepts are brie¯y discussed below.

Manuscript received 12 January 1995; revised manuscriptaccepted 31 May 1996.Discussion on this paper closes 1 December 1997; forfurther details see p. ii.� Ben-Gurion University of the Negev, Beer Sheva.{ University of California, Berkeley.

818 HATZOR AND GOODMAN

Block theory assumptionsThe application of block theory here assumes

that discontinuity surfaces are perfectly planar, andextend entirely through the volume of interest;namely, discontinuities do not terminate within theregion of a key block. New cracking is not consid-ered and block deformation is neglected.

Mathematical conceptsThe real power of block theory stems from its

ability to distinguish between removable and non-removable half-space combinations. A block is theregion of intersection of half-spaces formed by thediscontinuities that form the block faces. A blockis ®nite and removable if the conditions of Shi'stheorem (Goodman & Shi, 1985) are met. In arock mass with n non-parallel joint planes thereare 2n unique half-space intersections (joint pyra-mids, JPs), yet only a few are removable from afree face excavated through the system. Shi's theo-rem provides the solution for the removability of ablock formed with a particular JP.

Mechanical conceptsWe distinguish between two principal modes of

failure:

(a) lifting or falling, when the block loses contacton all faces, and

(b) sliding, which can occur on any face indivi-dually or on two non-parallel faces simulta-neously along their line of intersection. In thelifting mode, the direction of initial blockmotion coincides with the direction of theresultant force acting on the block. The slidingmode is divided into single and double facesliding. In single face sliding, the normal tothe plane of sliding, the resultant force and thedirection of sliding are all coplanar. In doubleface sliding, the direction of sliding is parallelto the line of intersection of the two boundaryplanes on which the block slides simulta-neously.

Forces that are frequently considered in engi-neering are represented in block theory using vec-tor analysis as well as stereographic projection. Ifthe components of any force vector F are(X , Y , Z), then the magnitude of F is jFj � (X 2 �Y 2 � Z2)1=2 and the direction cosines of F are

f � X

jFj ,Y

jFj ,Z

jFj� �

(1)

such that F � jFj f .A series of coincident forces F1, F2, . . ., Fn is

usually replaced in block theory applications withits resultant R:

R �Xn

i�1

Fi �Xn

i�1

X i,Xn

i�1

Yi,Xn

i�1

Zi

!(2)

Friction force is a reaction to sliding, and henceacts in the opposite direction to the direction ofsliding. If Ni, i � 1, . . ., n are the magnitudes ofthe normal reaction forces from each sliding plane,then the resultant friction force is

Rf � ÿXn

i�1

(Ni tanöi)s (3)

where öi is the friction angle on plane i and s isthe sliding direction, assuming a Mohr±Coulombcriterion with zero cohesion for the shear strengthof the sliding plane.

Water pressures acting on the faces of a blockproduce a water force (Rw) in the direction of theinward normal (ÿ ni) of each face that experiencesthe corresponding water head (a block theory con-vention is that the positive direction of a jointnormal points out of the block and into the rock):

Rw �Xn

i�1

Si(ÿ ni) (4)

where Si is the integral of water pressure timesarea over the whole submerged portion of face i.

THREE-DIMENSIONAL LIMIT EQUILIBRIUM

ANALYSIS

When the rock mass structure is clearly de®nedand the block geometry is known, a block-theory-based rock slope stability analysis can be per-formed using the following steps:

(a) block theory removability and mode analysis(b) limit equilibrium analysis(c) support dimensioning.

The stability analysis utilizes the `friction circle'concept and the three-dimensional solution for theslip of a tetrahedral wedge using vector analysis(Wittke, 1965) or the stereographic projection(Londe et al., 1969, 1970; John, 1968; Goodman,1976). The basic components of this solution areshown in Fig. 1 using a lower hemisphere stereo-graphic projection.

The advantage of the friction circle concept isthat it de®nes the three-dimensional locus of `safe'resultants acting on the block. The in¯uence of anyadditional force can thus be examined by correctlyrotating the resultant force, using stereographicprojection procedures or vector operations. An ex-ample is shown in Figs 2 and 3, where a waterforce (Uk) is acting on joint Jk . The originalresultant (W) is rotated to a new orientation (R0)along a great circle connecting W and Uk by the

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 819

angle á, the angle between the weight vector andthe water force measured in their common plane(Fig. 3). It can be seen that the new orientation ofthe resultant force now plots inside the `unsafe'locus, corresponding to sliding in the direction ofthe intersection I ij.

This procedure can be used to study the effectof water pressure inside a discontinuity (or tensioncrack) on the overall block stability. If it is foundthat the new resultant force is plotted within theunsafe zone for the analysed JP, no further analysisis required, as it can be stated that ®lling of adiscontinuity with water is suf®cient to drop thefactor of safety against sliding below 1´0. If, how-ever, the block remains safe, it is necessary toinvestigate the in¯uence of water pressure that maydevelop inside the base joints.

To study the in¯uence of water pressure insidethe base joints, similar procedures are applied. Fig.4 shows a two-dimensional pressure distributionthat is assumed for the case where both of the baseplanes and the tension crack are ®lled with water(although the force on plane J j is not shown). To®nd the new resultant force, the resultant waterforces from all planes must be added: R � W �

Uk � U i � U j. In the absence of data regardingthe exact position of the phreatic surface, it isimpossible to know in advance the exact pressuredistribution in each joint, and therefore the exactvalues of Ui and U j. In the analyses performedhere a planar phreatic surface of uniform gradientabove both joints is assumed, and the water headsabove the centroid of each base plane, hc, areassumed to be equal. This assumption permits asolution, calibrated for increasing heads, in bothplanes. A solution based on the above assumptionswas proposed by Londe et al. (1970), who haveused the fact that three vectors can be addedin any desired order to reach the same ®nalresultant:

A� B � Rab

A� C � Rac

Rab � C � Rac � B � Rabc (5)

This vector addition can be performed on thestereographic projection, using great circles to con-nect between two vectors that are added, and bymeasuring the angles between them in the correctdirection. A construction corresponding to equation

Fig. 1. Schematic presentation of the generalized friction circle for double face sliding(mode ij) of a tetrahedral block in¯uenced by gravity loading (W) only (L.H., lowerhemisphere)

820 HATZOR AND GOODMAN

(5) is shown in Fig. 5. This approach can be usedto solve for resultants corresponding to increasingheads inside the base planes. The solution is basedon the parametric addition of three vectors sug-gested by Londe et al. (1969, 1970) and discussedby Goodman (1976). Let R0 be the resultant of theblock weight W and water force in the tensioncrack Uk . Then we can say that

R0 � xUi � yU j � R0 � yU j � xUi � R (6)

where x and y are variables corresponding to somehead level in each joint. Using force polygons forR0 � xUi and for R0 � yU j, the angle of rotationof R0 corresponding to each head level can befound. These angles can be scaled on the greatcircle connecting R0 and the corresponding waterforce, using the stereographic projections. A net ofresultant paths emerges, and one can select thedesired one to study the advance of the resultant atdifferent loading conditions. A schematic illustra-tion of the procedure is shown in Fig. 6, where thedemonstrated resultant path (heavy line) corre-sponds to increasing head levels, of equal magni-tudes, in both base planes simultaneously.

The factor of safety is de®ned as follows (Londeet al., 1970):

FS � tanöavailable

tanörequired

(7)

A factor of safety of 1´0 represents limit equili-brium, where the resultant force plots on the boun-dary of the safe zone. Using the friction circleconcept and the boundary between safe and unsafezones, it is possible to read from the stereographicprojection the required friction angle for stabilityunder the given forces. This is the angle betweenthe centre of the friction circle and the resultant,measured along a great circle that connects the twopoints. The available friction angle, a materialproperty, is the angle between the centre of thefriction circle and its circumference. It can be seen,therefore, that when the resultant plots outside thefriction circle, the required friction angle is greaterthan the available friction angle. By equation (7),this leads to a factor of safety smaller than 1´0.

BACK-ANALYSIS OF TWO WEDGE FAILURES AT

PACOIMA DAM

In this section an analysis of two catastrophicblock failures which occurred at Pacoima Damnear Los Angeles, California, is described. The

Fig. 2. Schematic cross-section parallel to Iij showing a block with joint water force (Uk) inside a `tension crack'(Jk)

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 821

Fig. 3. In¯uence of the water force (Uk) on the orientation of the ®nal resultant (R0) andon the overall stability of the block

Fig. 4. Schematic cross-section parallel to Iij showing assumed joint-water pressure distribution inside Jk and Ji

822 HATZOR AND GOODMAN

input data are based on geological, structural andmechanical data collected in the ®eld. The analysisis performed using the techniques discussed above.Some striking conclusions emerge, with respect tothe in¯uence of pressure head in joints, joint fric-tion and block geometry on overall block stability.

Historical documentation of the failuresAn eye-witness account of the case study is

quoted below (Kuess, 1966).

The spring of 1938 was noted by high intensityrainfall and a heavy runoff which producedspillway ¯ow and large valve releases. Thesedischarges eroded rock and created vibration inthe area that resulted in two signi®cant failures.Approximately 50 ft. downstream from the baseof the dam, the ®rst rock fall extends from streambed upward some 200 ft. The breakaway occurredalong one of the vertical ESE striking joints andwas triggered by undercutting and vibrationcaused by valve release. The second failureoccurred north and adjacent to the spillway.Thousands of yards of rock were released fromthe stream bed and up to near the crest of theridge, to a point less than 100 ft. from the

buttress. Water debouched from the spillway halfway up the slope back in 1938, so that erosionand vibration were profound while spillway ¯owwas in progress. The breakaway plane wasapparently along one of the NW dipping jointswith the crown controlled by the vertical shears.

General settingPacoima Dam is located in Pacoima Wash on

the southern side of the San Gabriel Mountains,approximately 7 km north-east of San Fernando,California. The rocks of the area are derived fromintrusion of dioritic magma into a sedimentaryrock series. Schistose rocks in the area are themetamorphosed remnants of the sediments. Thepredominant rock type is quartz diorite gneiss, ordiorite gneiss. Associated with the diorite gneissare minor amounts of quartz monzonite, granite,gabbro, quartz veins, pegmatite and altered rock.

Rock structureIn order to characterize the principal joint sets,

three scan-lines were performed in which a total of173 joints were measured. The overall joint struc-

Fig. 5. Schematic diagram showing addition of three vectors by the intersectionof three great circles, performed on the stereographic projection

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 823

ture is illustrated in the upper hemisphere projec-tion of poles (Fig. 7), which shows the clusteringof joint orientations. Four joint sets are recognizedand their structural attributes are shown in Table 1,where the frequency is corrected for bias usingTerzaghi's correction (Terzaghi, 1965).

Joint shear strengthMost joints in the ®eld are tight and free of

®lling material, except for a few where a cohesion-less ®lling material was sampled. The joint planesare therefore assumed to be non-cohesive. A fric-tion angle of 308 was estimated for all joint sets,on the basis of tilt tests and characteristic rough-ness pro®les. The sensitivity of the stability analy-sis to the value of the friction angle is studied indepth below.

Analysis of spillway failureDuring the ¯ood of 1938 the water that dis-

charged from the spillway chute caused a largerock-slide in the valley in which the spillway islocated. A photograph from 1940 shows the cavity

of the block that was released during that event(Fig. 8(b) and (c)). The block was formed by acombination of a lower half-space of a member ofthe foliation set (J1), an upper half-space of amember of the face maker joint (J2) and an upperhalf-space of a member of the NW dipping jointset (J3) (Fig. 9). The removability of the block isdemonstrated by means of block theory removabil-ity analysis in Fig. 10, using an upper hemispherestereographic projection, where the joints areshown in solid lines and the free face is dashed.The region corresponding to the block is shown(JP 100). A limit equilibrium analysis for thisblock is shown in Fig. 11, assuming a 308 frictionangle on both joint surfaces. Note that a signi®cantrotation of the resultant force (W ) is necessary forthe block to become unstable and slide on the lineof intersection of J2 and J3 (I23). When the activeresultant force is parallel to gravity (W in Fig. 11),it plots within the safe zone. When the tensioncrack, in this case a member of the foliation planes(see Fig. 9) is ®lled with water, the new resultantforce (R0) still plots within the safe zone. In Fig.12, contour lines of equal friction are superim-posed on the limit equilibrium plot. It can be seen

N

Ro

R

hc = 2 m

hc = 1 m

hc = 2 m

hc = 3 m

hc = 1 m

hc = 3 m

hc = 4 mhc = 4 m

–n j –n i

WE

S

L.H. projection

Fig. 6. Schematic diagram showing parametric addition of three vectors using thestereographic projection, assuming equal heads above the centroid of the base joints

824 HATZOR AND GOODMAN

that the friction angle required to maintain stabilityof the block with the tension crack ®lled is 208.Since the available friction angle is 308, the factorof safety for this case is 1´59 (by equation (7)).

Because the block did slide, water pressuresinside the base planes must be considered as well.An equal head for both base planes is assumed.Using the procedure discussed above, a solution iscalibrated for increasing pressure heads from 0´1 m(0´98 kN=m2) to 1 m (9´8 kPa) in 0´1 m increments.

For each pressure level, the rotation of the resultantforce is calculated using pressures from bothplanes, and the path that describes the rotation ofthe resultant force is obtained (Fig. 13). It can beseen that a pressure head of 0´25 m (2´45 kPa) issuf®cient to rotate the resultant force from a safeposition into the unsafe zone where sliding on theline of intersection takes place.

Note that with increasing head the resultant posi-tion dictates double face sliding in mode I23. If the

Fig. 7. Upper hemisphere projection of discontinuity pole distribution as measured at PacoimaDam

Table 1. Principal joint set attributes

Joint set No. Discontinuity type and name Mean attitude (dip=dipdirection)

Frequency: 1=m Mean spacing: m

1 Foliation 84=78 0´22 4´6

2 Joints `Face maker' 82=170 0´5 1´98

3 Shears 44=264 0´33 3

4 Faults `Shallow dipping' 28=65 0´2 4´9

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 825

head level is raised above 0´6 m, however, thesliding mode is changed. The mode change isfurther studied in Fig. 12(a), an enlarged plot of theanalysis shown in Fig. 11 with superimposed con-tours of equal friction angle values (28 intervals).Limit equilibrium is attained when the resultantplots on the contour of the available friction angle.Thus for each resultant position the friction anglecontour on which it plots indicates the requiredfriction angle for limit equilibrium. The requiredfriction angle values for each head level increase

are plotted in Fig. 12(b). It can be seen that afriction angle of 748 is required for a pressure headof 0´6 m to develop. If that pressure does develop,the resultant will dictate sliding in mode 3, that is,single face sliding on J3. Furthermore, if theoreti-cally the friction angle was as high as 908, apressure head of 1 m would have failed the block inlifting, namely the block would separate from allboundary planes and ¯oat, out of its position.

The in¯uence of pressure head increase on thefactor of safety, assuming a friction angle of 308, is

Fig. 8. (a) general view of Pacoima Dam; note the spillway at the left abutmentreleasing water after the 1938 ¯ood; (b) side view of the spillway after the failure; (c)top view of the spillway block failure; photographs courtesy of Los Angeles CountyDepartment of Public Works

826 HATZOR AND GOODMAN

studied in Fig. 12(c). It can be seen that the rate ofchange of safety factor with respect to the pressurehead in the base planes decreases when the failuremode changes from double face sliding to singleface sliding. The results of this analysis thereforesuggest that double face sliding is more easilytriggered than single face sliding, for a given block

geometry. This ®nding is supported by the resultsof another back-analysisÐthe study of the canyonwall failure in the left abutment.

Analysis of left abutment failureAnother large block failure developed as a result

of the 1938 ¯ood in the canyon downstream fromthe dam. The block slid out of the left abutmentrock, and the resulting debris pile can still be seenin the river bed today. The debris and the surfacefrom which the block was separated are shown inFig. 14. The block is formed by the intersection ofthe upper half-space of the foliation planes (J1),the lower half-space of the face maker joint set(J2) and the upper half-space of the shears (J3)(Fig. 15). Since the joint surfaces are not accessi-ble at present, the mean orientations of the jointsets were used for analysis. The removability ofthe block from the slope of the canyon is demon-strated in Fig. 16 (JP 010). As in the case of thespillway block, we ®rst evaluate the stability of theblock under gravitational force alone (Fig. 17). Itcan be seen that the block is stable under its ownweight (W) and under the new resultant (R0) withwater pressure in a ®lled tension crack (J2). Usinga contour plot of lines of equal friction super-imposed on the diagram in Fig. 17, it can be seenthat R0 plots on the contour of ö � 108 (Fig.18(a)). With an available friction angle of 308, thefactor of safety for that condition is 3´27.

To ®nd the level of water pressure that musthave developed inside the base joints to initiatesliding, equal heads inside both base planes (J1 andJ3) were assumed. The pressure head was thenincreased from 2 m (19´6 kPa) to 20 m (196 kPa)

Fig. 8. (Continued)

Fig. 9. Exact geometry of the spillway block

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 827

Fig. 10. Block theory removability analysis of the spillway block (JP 100) (U.H., upper hemisphere)

Fig. 11. Limit equilibrium analysis of the spillway block for a water-®lled`tension crack' (J1)

828 HATZOR AND GOODMAN

in steps of 2 m, and the rotation of the resultantforce in each interval was calculated and plotted(Fig. 19). When the pressure head reached 8 m(78 kPa) the resultant force (R) rotated into theunsafe region.

The analysis shown in Fig. 18 allows inspectionof the change in failure mode as loading of theblock progresses. The region of failure mode I13 isde®ned by the locus bounded by the great circlesconnecting ÿn1, ÿn3 and ÿI13 and the locus ofmode 3 is the spherical triangle connecting ÿI13,I23 and n3 (I23 is not shown). The particular

geometry of the block dictates the path of theresultant force under the applied pressures. It canbe seen that owing to the given geometry theresultant travels within the safe zone up to a pres-sure head of 8 m. When that pressure is exceeded,the resultant force emerges in the mode 3 zone,similar in essence to the behaviour of the spillwayblock. In this case, however, the analysis is notmerely academic but of practical signi®cance be-cause the required friction angle for a modechange is only 308, beyond which failure com-mences by sliding on plane 3 (Fig. 18(b)).

Fig. 12. (a), (b) sensitivity analysis showing in¯uence of available friction angle on the critical water head at whichsliding ensues; contours represent iso-friction values in 28 intervals; location of resultant force vector (R) underincreasing water heads is shown by points representing head increase steps of 0´1 m; (c) sensitivity analysisshowing in¯uence of head increase on factor of safety

1·2

0·8

0·4

00·1 0·2 0·3 0·4 0·5 0·6 0·7 0·8 0·9 1·0

Head in base joints: m

Fac

tor

of s

afet

y

(c)

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 829

Investigation of the sensitivity of the factor ofsafety to the critical pressure head (Fig. 18(c))validates the result suggested by the analysis of thespillway failure. It is again shown that the rate ofchange of the factor of safety decreases when thesliding mode changes from double face to singleface sliding.

The mechanical response of the blocks to pres-sure head loading explored in this section isentirely dictated by the geometry of the boundaryplanes, because water pressures are, by nature,always normal to the discontinuity surface, and aredirected into the block interior. Only a fully three-dimensional analysis could reveal this complexfailure mechanism. Another three-dimensional as-pect of block stability, its removability, is exploredin the following section.

BLOCK THEORY STABILITY ANALYSIS OF RIGHT

ABUTMENT

Historical backgroundThroughout the service life of Pacoima Dam,

from its completion in 1929 to the present, therehave been few stability problems associated withthe right abutment. Two signi®cant historical events

attest to that, the 1938 ¯ood discussed above andthe 1971 San Fernando earthquake. During themassive ¯ood of 1938, all the damage was concen-trated at the left abutment. Two large rock-slides,triggered by the ¯ood, were analysed above. Theright abutment, however, remained intact. Further-more, the source of the 1971 San Fernando earth-quake, which registered 6´6 on the Richter scale,was located almost directly below Pacoima Dam(Gere & Shah, 1984). Extremely large ground ac-celerations, over 1´0g, were recorded at the top ofthe dam, and the canyon below was subjected tonumerous small landslides. There were indicationsthat the base-rock acceleration might have been inthe range of 0´6 to 0´8g (Swanson & Sharma,1979). Following the earthquake, signi®cant grounddisplacements of two large rock masses were mea-sured in the left abutment. The right abutment,however, performed very well during that event,and ground displacements were not noticed(Swanson & Sharma, 1979).

The signi®cance of free face orientationThe relationship between rock geometry and

free face orientation may be considered as the

Fig. 13. Travel path of the resultant force acting on the spillway block starting from a®lled tension crack (R0) through pressure steps of 0´981 kPa in both side planes(equivalent to 0´1 m water head above centroids in both base planes)

830 HATZOR AND GOODMAN

single, most signi®cant factor affecting rock slopestability in competent±discontinuous rock masses.In cases where a joint plane or a line of intersec-tion of two joint planes dips into the free face and`daylight', plane failures or wedge failures respec-tively may be expected. This has been demon-strated using the two cases analysed above, whereslippage occurred. A three-dimensional analysis, inthis case block theory analysis, allows detection ofsuch failure mechanisms and timely installation ofappropriate support. The same analysis methodmay also prove that a given slope is perfectly safe,owing to the removability consideration. The studyof Pacoima Dam provides an example of such acase where favourable orientation of the rock masswith respect to the right abutment put that abut-ment in a safe category, unlike the opposite leftabutment, as will be shown below.

AnalysisThe right abutment stability can be explained by

means of the critical-key-block method of analysisdeveloped by Hatzor (1992) and discussed byHatzor & Goodman (1992, 1993). In this methodwe look at all possible joint combinations (JCs)and ®nd the removable JP of each, with respect tothe analysed free face. In the case of PacoimaDam there are four joint sets and therefore fourjoint combinations must be analysed, assuming themost likely blocks are tetrahedral (Hatzor, 1992).Fig. 14. Site of left abutment failure

Fig. 15. Exact geometry of the left abutment block

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 831

Fig. 16. Block theory removability analysis for the left abutment block (JP 010)

Fig. 17. Limit equilibrium analysis for the left abutment block

832 HATZOR AND GOODMAN

The results of removability analysis are shown inTable 2 and in Fig. 20.

Every joint combination forms one JP which isremovable from the right abutment free surface.The critical-key-block method of analysis assigns arelative block failure likelihood value for eachremovable JP. The block failure likelihood P(B) isa product of three independent parameters: jointcombination probability (P(JC)), joint pyramidshape parameter (K) and the JP instability para-meter (F). The mathematical derivation of thethree parameters is developed by Hatzor (1992)

and reviewed by Hatzor & Goodman (1993). Thepredictive capability of P(B) is demonstrated,using two tunnelling case studies, by Hatzor &Goodman (1992), and the goodness of ®t for eachindividual parameter (P(JC), K and F) is studiedby Hatzor (1993).

The results of the critical-key-block analysisperformed for all removable JPs are shown inTable 3 and in Fig. 21. Fig. 21 clearly shows thatthere are only two JPs of concern: JP 100 of JC(1)and JP 001 of JC(4). Fig. 20 shows the remova-bility analysis for each joint combination for the

Fig. 18. (a), (b) sensitivity showing in¯uence of available friction angle on the critical water head atwhich sliding ensues (water head is increased by 2´0 m steps starting from 2 m up to 20 m; (c)sensitivity analysis showing in¯uence of increasing head on factor of safety

02 4 6 8 10 12 14 16 18 20

Head in base joints: m

Fac

tor

of s

afet

y

3

2

1

(c)

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 833

right abutment free face (dashed). JC(1) is analysedin Fig. 20(a), where two JPs plot inside the spacepyramid: JP 100 and JP 000 (unlabelled). JP 000has no mode and therefore is not removable evenif the friction angle on the joints is zero degrees.JP 100 is not removable because it has one edge(I23) which plots on the contour of the free sur-face. This is a rare case producing non-harzardousblocks, as discussed by Hatzor & Goodman (1993).This same line of intersection appears in JC(4) and

therefore a similar result is obtained there (Fig.20(d), Table 2). JC(2) and JC(3) both produceremovable blocks which have no mode: JP 100 inJC(2) and JP 001 in JC(3) (Figs 20(b) and (c),Table 2). Thus it can be seen that owing to theparticular interaction between free face orientationand rock mass geometry in the right abutment, thatabutment is safe against block sliding. This resultis obtained as a consequence of the occurrence ofnon-hazardous blocks which have a common edgewith the free face.

Figure 22(a) shows a synthetic generation ofjoint traces on the right abutment created byJC(1)Ðthe JC with the highest relative intersectionprobability. The trace map generation procedureused here follows the techniques of Shi et al.(1985) and Shi & Goodman (1989). Fig. 22(b)shows the traces of JP 100 of JC(1) after all thetrees and branches are cut off so that only remova-ble areas remain (Shi & Goodman, 1989). It canbe seen that even if the realization of the joints inthe right abutment deviates slightly from the idealorientations so that removable rather than non-hazardous JPs form (where all lines of intersectionsplot within the space pyramid), the removable areawould only be a small proportion (7´75%) of the

Fig. 19. Travel path of the resultant force vector starting from a ®lled tension crack (R0)through pressure steps of 19´62 kPa (equivalent to 2 m water head above the centroid of eachbase joint)

Table 2. Joint combinations and removable blocks forright abutment

JC (No.) Ji; J j; Jk JP � SPResults of removability analysis

1 123 100; 000Edge; no mode

2 124 100No mode

3 134 001No mode

4 234 001; 000Edge; no mode

834 HATZOR AND GOODMAN

Fig. 20. Block theory removability analysis for the four possible joint combinations in the rightabutment of Pacoima Dam: (a) JC1; (b) JC2; (c) JC3; (d) JC4

J1

J3

J2

I 23

Reference circle

Free face

JP100

JC1U.H. projection

(a)

J1

J2

Reference circle

Free face

JP100

JC2U.H. projection

J4

(b)

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 835

total abutment area. Thus, should removable blocksmaterialize in some locations at the right abutmentby joint combination 1, their in¯uence on overallstability would be negligible. Goodman (1995), in

his comprehensive review of block theory, alsodiscussed the results concerning the relatively lowrisk associated with the right abutment, using Fig.22. He has shown that this conclusion led to the

Fig. 20. (Continued)

J1 J3

Reference circle

Free face

JP001

JC3

U.H. projection

J4

(c)

J3

J2

Reference circle

Free face

JP001

JC4

U.H. projection

J4

I23

(d)

836 HATZOR AND GOODMAN

decision to construct a new supplementary chutespillway on the right abutment, with the dischargeconducted to avoid spillage against the sensitiveleft abutment.

The conclusion obtained here by means of blocktheory removability analysis is strongly supportedby ®eld evidence: the good performance the rightabutment exhibited along its history of turmoil,while visited by a severe ¯ood and several eventsof strong ground accelerations.

SUMMARY AND CONCLUSIONS

The application of analytical tools, particularlylimit equilibrium, mode and block removabilityanalysis, has been demonstrated, in the context ofblock failures at Pacoima Dam.

The application of block-theory-based analysiswith water pressures has been demonstrated usingtwo rock failures that occurred following a ¯ood in1938. Several lessons can be learned from theback-calculation of these failures:

(a) The orientation of the active resultant changesrapidly with increasing pressure heads that

develop inside the block boundary planes.Plotting the active resultant at each increaseof pressure revealed that pressure can changethe failure mode. In the two cases analysedhere the failure mode changed from doubleface to single face sliding as the water pressureincreased.

(b) The actual realization of mode change dependson the available friction angle which providesthe resistance to block sliding during pressurehead build-up. In one case (the so-calledspillway failure) it was shown that the modechange, from double to single face sliding,required a friction angle of 748, and a pressurehead of only 0´6 m above the centroid of thebase joints (after the tension crack at the backof the block had been completely ®lled). Inanother case (the left abutment failure), the

Table 3. Results of critical key block analysis (free face orientation 59=197)

JC No. Jijk JP code P(JC) K F P(B)

1 J1, J2, J3 100 7:11 3 10ÿ4 0´171 1´165 1:42 3 10ÿ4

2 J1, J2, J4 100 4:52 3 10ÿ4 0´077 No mode 03 J1, J3, J4 001 8:24 3 10ÿ6 3:3 3 10ÿ5 No mode 04 J2, J3, J4 001 8:49 3 10ÿ6 0´094 1´165 9´31 3 10ÿ5

Fig. 21. Block failure likelihood values for all jointcombinations

Fig. 22. Statistical joint trace generation of jointcombination 1 as it would appear on the rightabutment of Pacoima Dam (presently concealed bygunite): (a) joint trace map for JC(1); (b) trace mapof the critical key block area on the right abutment

J1J3

J2

(a)

(b)

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 837

same failure mode change occurs at a pressurehead of 8 m but requires a friction angle ofonly 308, the value assumed in the analysishere.

(c) The in¯uence of pressure head build-up on thefactor of safety was investigated as well. It wasshown graphically (Figs 12(c), 18(c)) how thefactor of safety decreases with increasingpressure head. It was found that the rate ofchange of the factor of safety with respectto pressure head decreases when the modechanges from double face to single facesliding. Therefore, for a given block geometry,a given increase of pressure head will reducethe factor of safety in the case of double facesliding more rapidly than in the case of singleface sliding.

(d) The single most important factor in¯uencingrock slope stability in competentÐdiscontin-uous rock masses is the interaction betweenthe rock mass geometry and the orientation ofthe free surface. When this relationship isunfavourable, rock failures should be expected,as was shown using the spillway and leftabutment failures of Pacoima Dam. When thisrelationship is favourable, the rock slope maybe assumed safe, as proved by the integrity ofthe right abutment of Pacoima Dam. Usingblock theory these conditions can be testedquite readily.

(e) The development of water pressures alongblock boundaries (discontinuity apertures)should be avoided as much as possible. It hasbeen shown that relatively small water pres-sures acting over a large area of discontinuitysurface mobilized sizeable blocks, which wereotherwise safe against sliding.

ACKNOWLEDGEMENTS

We would like to acknowledge the support ofWolfgang H. Roth of Dames and Moore, LosAngeles, and the assistance provided to us in the®eld by Michael Johnson of Los Angeles CountyDepartment of Public Works.

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Shi, G.-H. & Goodman, R. E. (1989). The key blocks ofunrolled joint traces in developed maps of tunnelwalls. Int J. Numer. Anal. Methods Geomech. 13,131±158.

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New Delhi, pp. 797±824.Terzaghi, K. (1962). Stability of steep slopes on hard

unweathered rock. GeÂotechnique 12, No. 4, 251±270.Terzaghi, R. D. (1965). Sources of error in joint surveys.

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rock slopes with and without additional loading. RockMech. Engng Geol., Supp. II. (In German).

THREE-DIMENSIONAL BACK-ANALYSIS OF ROCK SLOPES 839