Analysis of the Stability of Rock Slopes

Quarterly Journal of Engineering Geology and Hydrogeology

doi: 10.1144/GSL.QJEG.1973.006.01.04p93-124.

1973, v.6;Quarterly Journal of Engineering Geology and Hydrogeology

Pierre Londe

Analysis of the stability of rock slopes

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Notes

The Geological Society of London 2014

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Analysis of the stability of rock slopes

P IERRE LONDE

1. Introduction THE PROBLEMS entailed by the stability of rock slopes are among the most difficult with which the profession is faced. Yet they are not exclusively academic, as amply evidenced by disasters such as the Malpasset Dam abutment failure and the Vajont rock slide.

Owing to the scale effect, strength properties vary generally with specimen size, and little or nothing is known about the laws governing this variation. Fortunately, however, there is practically no scale effect when the residual shear strength of a continuous layer of soft material is considered. This is the case of fault gouge or sedimentary clayey joints which are responsible for most failures of natural rock slopes. In view of this, and in the present state of knowledge, only rock slopes with surfaces of separation of large extension can be satisfactorily analyzed for stability.

A three dimensional method of analysis of such slopes has been worked out. The principle of the method will be described first and then a few examples will be given of its application to dam abutments. It is obvious that the process is perfectly valid for rock slopes other than dam abutments provided there exist large surfaces of separation.

2. The principle of three-dimensional analysis Assumptions. The stability of the slope is analyzed in terms of the stability of a given rock volume (Fig. 1). The internal faces of the volume are plane: the surface (O, B, C) or plane P1; the surface (O, C, A) or plane P2; the surface (O, A, B) or plane/3. The volume is limited by the intersections of these three planes with the natural slope. The three planes P~ (i = 1, 2, 3) will be chosen according to the geological surfaces of separation: bedding, schistosity, joints, faults, etc. This assumption is conservative as the natural surfaces are generally irregular. The volume is indeformable. In other words, the geometry of the solid studied and of the surrounding massif may be considered, in practice, to be invariable (especially the directions of the contact planes) when the various forces are applied. It is also assumed that there can be no internal rupture in the rock volume. Cohesion and tensile strength are supposed nil along the contact planes. The shear strength along these planes therefore only depends on friction. It is supposed, moreover, that this friction is characterized by a single parameter for each face, namely the average effective angle of friction if1.

In a first stage of the analysis, the moments of the forces are assumed to have a negligible influence and are not taken into account for equilibrium. The influence of the moments are discussed in a second stage. This two stage process is justified a posteriori by

Q. Jl Engng Geol. Vol. 6, 1973, pp. 93-127, 42 figs. Printed in Great Britain.

at University of Colorado Boulder on October 2, 2014http://qjegh.lyellcollection.org/Downloaded from

P. Londe

the fact that it succeeds in most practical instances because the influence of the moments is generally small. Thus, only rupture by translation of the rock volume is considered.

The foregoing assumptions simplify the stabilityanalysis, as the following two deduc- tions can be made: (a) the stress distribution on each face does not come into play, and (b) only the direction of the resultant R of the applied forces intervenes in the stability and not its magnitude.

Forces and Rupture. The forces taken into account are as follows:

The external forces The total weight W, which comprises the weight W1 of the rock volume and the weight W2 of the part of any structure supported by this volume;

The thrust Q of any structure, reduced to a single force, as the effect of the moments is left out at the first stage;

The forces due to seepage water U1, U2 and U3 which are respectively applied to planes P1, P2 and P3;

Possibly the force due to earthquakes, G, on the debatable but still commonly accepted assumption that an earthquake can be represented by a static force and the force T of pre-stressed ties if any.

The reactions of the base planes. They are designated by R1, R2 and R3. These forces are compressive. They are, however, nil in the open planes, i.e. in those where

9 " - ' U} , . 9 . . / . " - . . 9

/ ~ ~ s " ? "

9 t , ( /. "," 9 " / .B -~-~/ ( - .d . . .

9 . / " . - j ~ .

; - : : . . /

9 " ? :..i::.i.i-: :.:: .., 9 . " 9

" . t " . . ' " ~ "

FIG. 1. Stability of a tetrahedral rock volume.

94


Stability of rock slopes

separation occurs. Each force R~ must form with the normal to the plane P~ an angle smaller than, or equal to, the friction angle 4'~.

Rupture. Every displacement of the solid, which is a manifestation of its instability, is called rupture.

Rupture can only occur with the separation of one, two or three of the faces; there is sliding on two of the base planes in the first case, on one in the second case and on none in the last case. Thus, there are seven types of rupture which are called Z1, Z2, Z3, Z12, Z23, Z3~, Zz23, as shown on Fig. 2. So as to cover all the possible cases, Zo has been added in which all the planes remain in contact

ZONE

Zo

Zi

NATURE OF SLIDING CONTACT OPEN vECTOR ~ FACES , FACES

i , 2 l l3

D~RECTION O~ 2 & 3

Z2 DIRECTION OB 3 IJ ~.

Z 3 DIRECTION OC ~ I I 2

ON PLANE 3

Z I. 2 DIRECTION 3 1 & 2 BETWEEN OA IL Oe

ON PLANE Z

Z 2 3 ~,RECT,~ I 2 a 3 BETWEEN 08 .& OC

ON PLANE Z ]

Z 3.1 OmeCTION 3 a Z BETWEEN OC a OA

~NSIDE i Z 1,2.3 TRtHEDRON ! 1,2113

F OA,OB, OC i

FIG. 2.

95

D | AGR AM

c

' C

8 ~ ~/~Otl

A , ,

B ~A


P. Londe

and which is that of perfect stability. The subscripts of Z show the faces which separate from their base at the onset of rupture.

Conditions of equilibrium--Construction on a sphere. As the moments of the forces are not taken into account, at least in the first stage of the analysis, stability is governed only by the direction of the resultant R of the external forces. By varying this direction, the different forms of rupture shown in Fig. 2 can be obtained.

Consider a sphere (S) having its centre at O and its radius equal to unity. To each direction in space of the forces -R (the vectorial sum of the reactions R~) passing through O, there corresponds a point r on the sphere and vice-versa. This representation in space was proposed for the first time by Vigier (Londe et al, 1970). Eight zones may be distinguished for point r on the sphere (S) each of them corresponding to one of the failure types Z o, Z1, Z2, Z3, Z12, Z23, Z31, Z123 (Fig. 3). These zones do not overlap and cover the whole of the sphere surface. The boundary curve between two zones is always a portion of a great circle.

The method of analysis basically consists of drawing first the eight zones on the unit sphere (S). These zones depend only on the geometry of the problem, i.e. on the geological data. Then point r is defined in terms of the forces, including uplift forces. Each point r thus corresponds to only one possible type of rupture. However, stability can be ensured

FIG. 3. FIG. 4.

96



provided the relevant angles of friction are above the values required for limit equilibrium. Each set of given r r r yields a closed curve on the sphere; point r has to be on this curve to obtain the condition of limit equilibrium.

It can be shown that the vertices of the spherical triangles forming the eight failure zones are the traces of two trihedrons centered at O (Fig. 4). The first, called (v0, is composed of the three half normals (directed toward the inside of the rock volume) to planes P1, P2, P3- The second, called (vu) is composed of the trihedron supplementary to the first; the edge 0" for instance is perpendicular to both edges i and j of trihedron (v0. The vertices l, 2, 3 of trihedron (v0 are directly determined from the strike and dip of planes P1, P2, P3 as obtained from the geological data. The vertices 1.2, 2.3, 3.1 of trihedron (v~j) are easily determined by making use of the fact that the two trihedrons are supplementary, thus side 1.2-2.3 for instance is a portion of the great circle having its pole at vertex 2. Conversely side 1-2 is a portion of the great circle having its pole at point 1.2.

Zone Z0, corresponding to stability whatever the friction angles, is the surface of triangle 1, 2, 3. Zone Z123, corresponding to instability, whatever the friction angles, is triangle I. 2, 2.3, 3.1. Zone Z1

, _ - corresponding to possible rupture which depends on the values of friction angles r

; ~ and r with sliding on planes P2 and P3 is / , - "~" ; ;~ / ~ triangle 2, 3, 2.3. It is easy to determine / j S-h ~,~.,~ \ .; / " ,~ J zones Zz and Z3 in the same manner.

" Boundary curve fo r

tan i~l = f l

tan r = f z tan ,~'a = f3

FIG. 5.

FIG. 6.

Upper po in t a

~ ~ o I t'o .plone_._PPi . /I

.Lower point o~/'

4JEG 97


P. Londe

Zone Z12, corresponding to possible rupture, which depends on the value of friction angle 4a with sliding on plane Pa, is triangle 3, 1.3, 2.3. It is again easy to determine zones Z23 and Zal in the same manner. Figure 3 shows the zoning of the sphere.

The friction angles being given, the boundary curve corresponding to limit equilibrium can be plotted on the sphere. This closed curve obviously does not enter triangles 1, 2, 3 and 1.2, 2.3, 3.1. It cuts each zone Z o. or Z~ into two parts (Fig. 5). In zone Z12, for instance, it is an arc of small circle, intersection on (S) of the friction cone relative to/'3- The pole of this small circle is point 3, its angular distance from point 3 being ~a. In zone Z~ the boundary curve is the arc of gleat circle, which joins the arcs of small circles relative to zones Z~2 and Z13. Along this arc of great circle there is a linear relation between tan ~1 and tan 43.

The above construction is immediately obtainable by using geographical co-ordinates on a solid sphere. Thus half-meridians having a common vertical diameter are lines of equal strike and horizontal parallels are lines of equal dip (Fig. 6). In this manner, points 1, 2, 3 are directly plotted from the geological report. Points 1.2, 2.3, 3.1 are then obtained with a compass.

For convenience of language, a force E passing through O and represented by point e on the sphere (s) may be said to have a strike d e and a dip Pe, these two angles being defined as above from point e and corresponding to the plane normal to E.

Point r corresponding to the resultant -R is easily determined from the components:

-F = - (W+Q)

-U1 , -U2 , -U3 .

Force -F is generally well-known, whereas the uplift forces are not. For ease of discussion, as will be shown below, it is convenient to write

U i - - u~UiTu i

vi being the unit vector normal to plane P~ and directed towards the inside of the rock volume; Un- is the magnitude of the maximum possible uplift (for instance full

U13 U13T

U! T T

. f ,\

,. detail .,, 3.. u

"J" "5--- .i- / / | ~oi I I , I " " >" "/ "/ "/" l

98



hydrostatic pressure); u~ is a factor of between 0 and 1 giving U~ any magnitude between zero and the maximum possible value UiT.

Figure 7 shows a view in perspective of the construction in space resulting from the vectorial equality which gives the direction of -R and consequently point r.

- R = - (F + ui UiTV~ + U2 U2Tv2 + u3 U3TV3)

In this figure F and R are the ends of vectors -F and -R which issue from O. The three edges of the large parallelepiped issuing from point F can be graduated linearly in terms of u~, from u~ = 0 at F to u~ = 1 at U~T.

Let us consider (Fig. 7) the figure obtained on the sphere by conical projection from O of the parallelepipeds of the forces (U~ is projected to uiT, U~ to u~, etc.): a homologous construction on (S) is made to correspond to the previous construction in space. It gives point r and can easily be drawn with a compass. It is based on the property that any straight line in space parallel to the direction v~ is projected along a great circle passing through point i of the sphere.

FIG. 8.

99


P. Londe

Thus, the three edges FU~, of the large parallelepiped will be projected along the three well-defined great circles: (f- l) (passing through point f and point 1), (f-2) and (f-3). Each is graduated in u~ starting at f, with the help of a simple auxiliary construction.

3. Plane projection of the sphere Everything said above about the sphere can easily be represented by a plane construction, using the conventional methods of mapping the sphere.

Two geometrical projections are particularly convenient:

(a) Stereographic projection (the centre of projection being on the sphere). This has the advantage of giving a circle for all circles on the sphere, such as the boundary between zones and the curves of limit equilibrium (Fig. 8).

FIG. 9.

100



(b) Central projection from point O. The great circles, and hence the boundary between zones, are represented by straight lines. However, each point on the diagram corresponds to two points on the sphere. This drawback is not serious because, as a rule, the problem is clearly limited a priori to a few zones of the sphere (Fig. 9).

(c) In addition, it is possible to transform the construction on the sphere algebraic- ally so as to have a plane representation which gives, for instance, linear scales in ul, u2, u3. This can be done by computer (Fig. 10). Any additional force, e.g. cohesion, can be introduced in the diagram in terms of equivalent uplift components).

Figures 8, 9 and l0 correspond to the same problem. The choice between these various ways of 'mapping' the sphere is a matter of personal

preference. In some cases it is advisable to use more than one map so as to select the best one after trials. At this point, the engineer has at his disposal an efficient tool for

60

45" 40!,

30"

20".

FOR THE POINT, r ~ ' , ~;t

F3) / !

50- ~o- 4~- i s - " : /~

t13

ul, I

O~ \

FIG. I O.

I01

~t, POLE OF LINES i "",. X3 : CONSTANT I %.%

ZONE _}_ :

/, i

"---SCALE OF U .-..-,


P. Londe

discussing the conditions of equilibrium and for assessing the weight of the various parameters involved in the problem of stability he is investigating.

4. Weight of the various parameters In rock masses, the parameters concerned are almost always imprecisely estimated, particularly friction angles and uplift forces. Hence it is vital to assess the possible consequences of any error in their evaluation and to examine whether a small variation Apt of the parameter p~, can bring point r onto the boundary curve. If so, parameter p is critical; in other wbrds, it has great 'weight'. Attempts must therefore be made to measure its value more accurately (by laboratory or in situ tests for instance) or to impose certain artificial limits upon it (limitation of uplift by drainage for instance).

Weight of the various parameters in the case of the geometrical constructions (Figs. 8 and 9). The proposed method makes it easy to assess the weight of each kind of parameter in turn, since they are only introduced in successive stages. What follows is valid either on the sphere (S) or in its plane representations.

(a) Geometrical parameters (directions of geological discontinuities). The determina- tion of the eight zones Zo, Z1, etc. depends exclusively on the geometrical parameters. The group of point 1, 2 and 3 corresponding to the in situ geological measurements can be plotted. This gives a direct view of the scattering pattern. In order to be sufficiently conservative, a spherical triangle (1, 2, 3) is drawn small enough for most of the points to fall outside it. This reduces the zone Zo of perfect stability.

Obviously, the more the contact on plane P~ is vital to stability and the closer r is to the corresponding portion of the boundary curve, the more critical the direction of this plane will be. Its weight is therefore closely related to that of the parameter ffi.

(b) Friction coefficients. These coefficients determine the curve of limit equilibrium, but each coefficient only influences part of the curve. If r is in zone Z~j, only the parameter ffk has any weight. In a zone Z~, only the two parameters 'kt and ~ have any weight. The respective weights of each of these two parameters is then seen directly on the sphere. They are in inverse proportion to the increments A~bj or Affk which make the boundary curve pass through r (Fig. 11).

(c) Uplift. Uplift is only brought into play at the last stage of the plotting of the diagram. The construction of r shows the influence of the Au~ increments.

Several rock volumes limited by three sets of joints or faults belonging to the same three families will often have to be studied on the same site. The result is that the constructions above can be drawn once and for all. Only r will have to be studied for each rock volume.

(d) Global visualization of the margins of error. Any curve of limiting equilibrium can be replaced by a strip covering the possible real location of the actual curve. Its width will point to the imprecision of the geometrical and strength parameters.

In the same way, given that the various uplift forces are situated between limits u~ and u~ +Aug, r is found to be in an area limited by a hexagon. The

102



result is the same if account is taken of the margins of error on the components of F.

It can then be seen at once whether point r has any chance of reaching the boundary curve, and what is the most unfavourable combination of the large number of parameters involved.

Boundary_ curve corresponcling_ __~j~k ' ~,~k_

Point j.k .z ............. -"--- ...... \Rnund. vcurvel~fa__~_j~__corres.pondmg_

FIG. 11.

WEIGHTING CHART

lO0 % -~.'~T_~

.

\ \ . UI ~ O\ t~

I

I

/ so-/ . \

THE PARAMETERS OF EQUILIBRIUM

OF LIMIT EQUILIBRIUM { F , I . ~

11 t I

\'~ \ l \ \ \ \ i o 50% IOO%

UPLIFT U I U z U 3

FRICTION 1~11 (~) (~) 3

EXAMPLE : VARIATIONS OF (~Z FOR AU I , 6U z AU s ( U p - t

FiG. 12.

HAS GREATER WEIGHT)

103


P. Londe

Point

1

Weight of the various parameters in the case of computer programme (Fig. 10). In this case, the graph is entirely determined by the computer and the construction does not enter the various parameters in turn. The discussion of the weight of the parameters is, however, basically the same. For uplift forces the evaluation is straightforward by a mere trial variation Au~ of each term u~ (Fig. 12). The same applies to the friction angles. If the directions of geological planes are considered together with the concept of strip of errors, it is necessary to run the computer several times. This however is not a serious drawback as each computation takes less than 10 seconds of execute time (IBM 360-40), and less than half an hour of plotting. The use of the programme is a routine work easily per- formed by a draughtsman.

Remark on earthquake effects. It is often assumed that an earthquake with an acceleration of ~g is equivalent to the application of a supplementary force to the rock mass: IGI = ~lw[. It can have any direction. If R' is the resultant of the other directly applied forces, then R = R' + G and the point R can be located at any point of a sphere centred at R' and having the radius ~[W[ (Fig. 13).

The most adverse directions of -R are obviously in the cone of revolution having its centre at O, circumscribed to this sphere and having a half-angle co:

~lwl ~lwl sin co - whence co ~

IR'I IR'I

If G is small as compared with R', as is frequently the case, it can be seen that the most adverse directions of G are almost normal to R', i.e. they are not necessarily horizontal.

The cone cuts the sphere (S) along a small circle, which is the locus of r. It is centred at a point r' (constructed like r above). The angular diameter 2oJ of this latter circle is known from the weight W of the rock mass and from the three components of the force - R = - (F + U1 + I52 + U3) in a rectangular system of axes.

These considerations of course imply increasing the area, locus of r, mentioned above. Figure 9 is an example of the earthquake computation, assuming horizontal acceleration

only.

Safety factor. It is widespread practice to measure the margin of safety of a structure by a factor of safety. The margin of safety is a physical concept. In contrast, the factor of safety is a conventional figure with a value depending on its definition, which is always

\ FIG. 13.

104



arbitrary. In rock mechanics problems, such as the one presented here, too many parameters are involved for it to be possible to define a factor of safety. A single ratio for the three tan r and for the three u~ would obviously be meaningless as some of them have more weight with respect to stability than others. Moreover, their magnitudes are not known with the same degree of accuracy.

This would suggest that a safety factor is not only meaningless but could, at times, be dangerously misleading and should be used with caution, if at all.

The method proposed here makes it possible to discuss the influence of each of the parameters on the conditions of equilibrium. The engineer thus has a sound basis on which to judge the determining factor and to act accordingly, if necessary. For instance, it is possible to single out the surface of separation whose shear strength is vital for equilibrium and measure it accurately. In the same way, knowledge of the uplift which it is essential to control may promote rational drainage design.

This, of course, could also be done by comparing the safety factors computed for all possible values of the parameters, but the amount of computation would be a burden and there would still be a risk that some critical combinations had been overlooked.

The diagrams used in the proposed method cover all possible cases of equilibrium and point clearly to the weight of the parameters, as they have been made precisely for that purpose.

o

. . . . -- ~~ fk aUz J I

I t AUI I

- - - - x

/ ~ PIU1)

(a)

f

~ 0 il~ Aft, fk

5. Probability concept Not all the values for a given parameter have equal probability of occurrence. In coming closer to reality, an attempt has to be made to ascribe a certain probability to each value for each parameter. As a result, the diagram on which the weight of the parameters is discussed, can be divided into zones of differing probability. In other words, any set of parameters shown on the diagram gives conditions of limit equilibrium but one set has a higher probability than the others, i.e. not all the points of the diagram have the same probability. If curves of equal maximum probability are then plotted, meaningful conclusions can be made as to the weight of the different parameters.

To take a practical case, the meaning of the probability of occurrence of a state of equilibrium represented by a

(b) p(fk} (c) FIG. 14.

105


P. Londe

point r on the diagram will be examined (Fig. 14a). Let this probability be called P(r) and assume first that r is in a zone Z~j where sliding can occur on plane Pk only.

If tan fie = fk, then P(fk) = P(fk) " A(fk)

where P(fk) is the probability of the value fk,

P(fk) is the function of distribution of P,

A(fk) is the selected interval (Fig. 14b).

We have necessarily: f=oo

P(fk) A(fk) = 1 f=0

The probability of occurrence of a given set (u 1, u2, ua)n is"

Pn(u) = Pn(ul) " Pn(u2) " Pn(u3)

and the probability of all the sets corresponding to point r is"

e(u) = r , e . (u~) . e . (u~) .e,~(u3) n

i.e.

P(u) = ~ Pn(Ul) Aul x Pn(U2) AU2 x pn(u3) AU3 n

The probability of the equilibrium at point r is thus"

e( r ) = e( fk) " e(u) .

If r is in a zone Z~, where sliding can occur on both planes Pj and Pk simultaneously, the relation is the same, but P(f) has a slightly different meaning. As is known, there is a linear relation between tan ~i and tan ~k, that is betweenfj, andfk (see Fig. 14c).

To each point of line D correspond the values Anf j, Anfk, Pn(fj), P,~(J}~). Then:

e( f ) = ~ P r~(fs)Pr~(fk) Anfj Anfk

In order to have comparable values of P( f ) in all zonesZ of the diagram, Afhas to be the same everywhere. Plane (fj, fk) is thus covered by elementary squares with their sides equal to Af For each square, there is one set (p(fj), P(fk)) and the entire plane covers all the possible sets (p(fj), P(fk)). The sets giving the limit equilibrium are defined by the squares touching line D (Fig. 14c).

Figure 15 (corresponding to Fig. I0) gives an example of this probabilistic approach. The main difficulty is to ascribe a probability function to each parameter. It is impossible to do so rigorously, but very crude functions are adequate for the purpose. The functions shown in Figure 15 for ul, u2, u3, ~1, ~2 are made up of portions of straight lines.

The contour lines of constant probability of occurrence for point r show a marked peak. At this peak the conditions are those most likely to occur. The peak points to the region of the diagram that is most representative of the probable natural conditions and where the study of the weight of the parameters is most useful.

106



6. Malpasset Dam It was the Malpasset disaster which threw doubt on a technique which, until then, had been considered as being thoroughly proved in practice and brought home to dam builders throughout the world the absolute necessity of being able to understand and forecast the behaviour and strength of the rock foundations of dams better than before. Laboratory research and in situ tests were actively pursued. The science of rock mechanics, then in its infancy, was appearing just at the right time.

The failure of the abutment at Malpasset Dam (Fig. 16) could not be satisfactorily or completely explained until after five years of research, during which time eminent experts were called to make a rigorous critical study of the facts and hitherto unknown properties of rock were discovered in the laboratory. And yet from the first visit to the site after the accident (Figs. 17 and 18), there was the feeling that hydraulic uplift pressures in the heart of the rock had played a capital part in the process of destruction, but with no idea as to the mechanism and even less as to the intensity of these uplift pressures. If a method of stability analysis were to have a chance of representing actual conditions, it had to be a three-dimensional one, as a plane section could evidently not hope to reproduce the spatial complexity of the planes of weakness existing in the rock (Fig. 19) and the forces acting on them.

These two considerations, the difficulty of assessing the internal hydrostatic forces and the need for a three-dimensional type of analysis, were the basis for the method

Uz f"x~

0 0 it~

P(Ut) I~l

!~ 0.5 l,lO Ul

/(([iJlll O ~ 15~ 30 a 45~

FIG. 15.

107

P(r) j__-Q

I

.~.

f,L-


P. Londe

subsequently developed by Coyne & Bellier, which has been regularly used for a number of years now to analyze the stability of rock slopes.

Figure 20 shows the rock volume on the left bank used for the three-dimensional analysis. Its shape is as observed after the accident (Fig. 18).

FIG. 16. Dam before failure. Thrust block in foreground.

FIG. 17. Right bank after failure

FIG. 18. Left bank after failure. Thrust block at top.

Malpasset Dam.

108



FIG. 19. The main geological features of the Malpasset Dam site.

FIG. 20. Malpasset Dam. Rock volume used for three-dimensional analysis.

109


P. Londe

Figure 21 shows the results obtained from the computer analysis, giving all possible combinations of uplift pressures and friction coefficients for limit equilibrium. The graphical constructions in Figs. 8 and 9 represent the same relationships. It can be seen from these that with full uplift on the upstream plane P1 and a more or less triangular distribution on planes/)2 and/)3 (ul = l, u2 -- 0.5, u 3 = 0.5), limit equilibrium requires that the friction angle r on the downstream fault (plane/)2) must be greater than 30 ~ and laboratory tests on the clayey breccia filling this fault gave a value of 30 ~ Hence, this uplift assumption explains the failure.

On the other hand, it is clear that for full uplift on the upstream plane but zero uplift on the downstream and base planes (ul = 1, u2 = 0, u3 = 0) failure is impossible as the limit friction angle r on plane/)2 is only 15 ~ This means that the dam could not slide on the downstream fault without uplift pressures existing downstream of the dam. This diagram also shows that the assumption as to uplift u3 has little effect on the result, but the assumption as to u2 and especially ua is on the contrary a determining factor for equilibrium.

Over the years during which the reservoir was slowly filling, the uplift pressures on the three faces rose steadily. The diagram shows that the rock volume tended to slide by moving upwards on the downstream fault, since in all cases there is a tendency for the

U2 (%) \

tO 0

r (~on ~'l)

U~(%) \

4~

'S 70 U3(%) 80

100

~00

6c ............... ~ - - -7 - . . . . .

FOR POINT r

4c i 7 J - - - - / /

3(:

/ / i /

20" ~

0 - - [J5 20~ 30~ 40~ 45~

(a) (b) FIG. 21. Malpasset Dam. Results of computer analysis giving all possible combinations of

uplift pressures and friction coefficients for limit equilibrium.

~'2 (Jan 0'2} v

110



FIG. 22. Vouglans arch dam on the river Ain during construction.

FIo. 23. Horizontal massive Jurassic limestones interbedded with clays at the site of Vouglans dam.

111


P. Londe

base plane P3 to open; this tendency results from the only two types of failure possible, Z3 and Zal.

After a probing study of the Malpasset gneiss, special mechanisms were revealed connected with the nature and structure of the site leading to the progressive appearance of a high uplift value ul on the upstream plane P1. The movement was slow but, aided by the pressure on plane P~, finally produced a crack in the zone that was to become plane /3- The crack broke out into the reservoir and gave free passage for water to flow into the deepest part of plane P~, which suddenly had to support the full hydrostatic load of the reservoir. In the final stage, the uplift pressures were probably u~ = 1, u2 > 0.3, u 3 > 0.5, shown in Fig. 21 by point r'. In this case, failure must occur if the friction angle is not greater than 30 ~ .

J

400-

, .L__ 300"

q5

200 F-=---=-:---:

FIG. 24.

Z/It

F~G. 25.

\ \

112



It is interesting to note that the shear strength of the upstream surface plays no part as regards stability. It is sufficient to suppose that it has not tensile strength, which is in fact quite reasonable for a family of fine shear cracks containing films of mylonite.

This analytical method has since been used at the design stage to study the abutments of several dams of which a few examples are given in later sections. They are all arch dams but the method has also been used recently to analyze the rock slopes downstream of a large earth dam.

7. Vouglans Dam Electricit6 de France completed Vouglans arch dam on the river Ain in 1968 (Fig. 22). The dam is 130 m high. The rock is a massive Jurassic limestone lying in horizontal beds with thin but continuous interstratified layers of clay, especially at level 340, i.e. near the foot of the valley flanks (Figs. 23 & 24). In addition to these horizontal planes of weakness, there is a double system of vertical tectonic fractures running at 45 ~ across the valley (Fig. 25). The clayey horizontal joint at elevation 340 (covering the whole area of the site) and the vertical fractures meant that the dam abuts on both banks on large rock volumes which could slide under the combined effect of dam thrust and pore pressures.

Three-dimensional stability analyses were made for a number of these rock volumes and one of these will be used as an example. Plane P1 is vertical and is an upstream fracture; plane P2, also vertical, is a fracture downstream and plane/'3, which is horizontal, is the clayey joint at elevation 340. Figure 26, which shows the result obtained with a computer, is particularly eloquent. Point r corresponds to the set of uplift pressures ul, u2, u3, which would probably exist if there were no drainage at all (full uplift upstream and triangular uplift diagram on the other two faces). The type of failure possible is then simultaneous sliding on planes P2 and P3 (zone Z1). If uplift is reduced, the same type of failure can occur but only for very low values of the friction coe~fficient. Point r' represents the case where conventional grouting and drainage are usec~ ~ that is, where there is still

,~,.)., u1.0 9

FIG. 26.

113


P. Londe

appreciable pressure on the upstream plane. Point r" represents a drainage system which, in addition, also reduces pressure on the upstream plane. A glance at the relative positions of the lines representing cases r, r' and r" on the graph of tan 4,ff versus tan 4,2' in Fig. 27 shows that stability depends basically on good drainage. The graph also shows that, in all cases, the friction angle 4,3' on the horizontal clayey joint has more 'weight' than angle 4,2' on the vertical joint.

It was therefore decided to (a) adopt a system of grouting and drainage which would reduce uplift on the upstream plane, and (b) make in situ and laboratory measurements of the shear strength of the clayey joint. The residual friction angle was found by the various tests to be close to 25 ~ (Fig. 28). It was at this time that the large shear machine, described in the first paper, was developed by the SElL laboratory.

As far as drainage and grouting were concerned, it was decided to incline the curtains upstream, as shown in Fig. 29, so as to protect planes such as P1 against uplift. The dam is now in operation and is behaving satisfactorily. The piezometric measurements give no detrimental pressures and the leakage is nominal.

FIG. 27.

FIG. 28. FIG. 29.

114



8. Rape l Dam

Rapel arch dam in Chile, built for the Electric Power Company ENDESA, is 110 m high (Fig. 30). The geological formation is a massive granite and the most salient feature of the site is the presence of faults of very large extension. A regional fault, called the La

FIG. 30. Rapel arch dam, Chile.

/ I , ! ~v (r /&'/ . . a ~ \

I (~"-7 )

e o oo

J

.T

FIG. 31.

115


FIG. 32. Rapel arch dam, Chile. Fault in granite on the left bank.

e PLA.E z ~ FZ

~I ~ ANE3 9W}" r

/,, / 7 ton ~' 3 ~ . ,

Y

/ - / " ,,,

F IG . 33.

Lit I 0.0

/ / / /

/ j /

116

{ \

\ \

,,\

[ U~: I F Uz : 0.~ U3 = 0 .3



Piscina fault, is 20 m thick and crosses the gorge just upstream of the dam. The other faults, although of large extension too, are very thin (Fig. 31). Some, such as the Sablazo fault, are nothing more than thin fissures containing a millimetre of clayey minerals.

The geometry of these faults and the existence of a sub-horizontal jointing system raised the question as to the stability of a number of large rock volumes once the arch- thrust and pore-water pressures were applied to them.

On the left bank, a remarkable fault which the geologists called F2 (Fig. 32), lies in a particularly unfavourable position as it joins the major La Piscina fault in depth, which itself lies in the reservoir. This fault F2 could therefore become a slide surface on which high water pressures would be acting. It was represented in the stability analysis by an approximately plane surface, although it is not in actual fact plane.

The massive concrete spillway shores up the bank downstream of the dana abutment and the rock volume studied is the monolithic block thus created contained inside the downstream and base boundary planes. The base plane, which follows the sub-horizontal joints, does not emerge at the surface, except on the highly pessimistic assumption of the large mass of rock at the toe of the spillway being destroyed.

The Rapel analysis was done by the late R. Vormeringer. Earthquake effects were included. It is of course well known that a static force proportional to gravity is not a correct representation of the dynamic effect of an earth tremor, but there is as yet no other way of studying the problem. It was therefore assumed that an earthquake would produce an additional horizontal force in an initially unknown direction but that its intensity would be proportional to the weight of the rock. The graphical construction shown in Fig. 33 illustrates how this force can be taken into consideration by representing

N ~- 1

tan. gl~

10" 20 ~ 30 ~ 40 ~ 50 ~

J ~ \ I , ......

o.,s -U -~- -T - T ,E - \~- -T .... 5

-~ ~ k ; ~ -1 30* .~

. . . . . . 1_ - - '~ ; ' 20*

i ! jj \.,!, o 0.5 i' tan ~'2

to.. ~

0.75

0.5

0 0.5 0.7

N e 2

t on . ~lt2

U l . : 1 U 2= 0,3 No EARTHQUAKE

U3= 0 ,3 O,12g EARTHQUAKE

F IG . 34 .

U 1 =1

U2=O

U~=O

117

No EARTHQUAKE

O,12g EARTHQUAKE

F IG . 35 .


P. Londe

it as the circular locus of point r. The friction coefficients needed to ensure limit equilibrium can be immediately derived. Figures 34 and 35 show at a glance the incidence of an earthquake on stability.

It can be seen that in the unfavourable conditions assumed, the stability of the abutment would be in danger if high uplift pressures u2 exist in fault F2 (here plane P2), because it can be seen from the graphical construction in Fig. 34 that the friction coefficients on plane P2 and base plane P3 would have to be impossibly high. An earthquake aggravates this situation still further. The bank must therefore be actively drained since grouting alone cannot guarantee that high uplift pressures will not occur downstream of the curtain. If Figs. 34 and 35 are compared, the influence of reduced uplift downstream of plane P1 on the limit friction angles will be seen.

Very thorough drainage was obviously needed. This was done by making a sort of cage of drain holes drilled from galleries (Fig. 36). In fact, the general outline of the drainage system had been designed before the stability analyses were made, but the calculations did serve to justify the arrangements planned.

9. Oymapinar Dam The Oymapinar arch dam in Turkey is now at the final design stage. It will be of the double curvature type with a height of 180 m (Fig. 37).

The site is a deep short canyon, cut in Paleozoic limestone (Fig. 38). The bedding is vertical. The left bank is particularly steep: a narrow spur of rock has to be used for the dam abutment; it forms a vertical cliff on the upstream side, more than 100 m high (Fig. 39). The salient feature of the site, not to speak of its karstic nature, is the fact

~00

90

I10

70

~0

~0

40

30

20

lO -

O - -

SECTION X_X OF FIG. 31

i Z:S~.~::.~__._2_:._?.~]:~.~_.~.__2__Zz__~2~__2L_L_. . . . .

W

/

i / -W -- I % J. I

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FAULT

. . . . . DRAIN

N~fl~ll' l ' l ' l ' l ' l 'tl i '[ ' l ' l ' lq~ G ROUTING

FIG. 36.

118


Stability o frock slopes

that it is divided by a great number of joints. They are of varying importance to the designer depending on their extension, continuity, density and spacing, filling material and their position in relation to the arch dam planned.

A total of 19 250 joints were surveyed in adits and at the surface. They have been classified as:

thin joints bedding joints tectonic tight joints

jo ints with calcite filling open joints with clay and sand filling slickensides

major discontinuities cemented brecciated zones shear zones faults

Four sub-areas (A, B, C and D) have been the subject of separate statistical studies as shown in Fig. 40. Although fracturing is complex and involves a wide range of directions and extensions, there is a marked trend for major joints to be sub-vertical and perpendicular to the river. Conversely, extensive sub-horizontal joints are rare in the left bank and hardly noticeable in the right bank.

Only a short summary of the stability analysis will be given here. Plane P1 is taken in the system of joints marked )(3, sub-area A (Fig. 40). It is also in the vicinity of the system of tight joints J4. Whatever the case, plane P1 is not made of a continuous surface of separation and most probably has high strength. None of the major joints could be found to be a critical direction on the left bank. Plane P2 is taken in a sub-horizontal system, not fre- quent but well defined and extensive. Plane/'3, in the stratification joints, is vertical, along the upstream toe of the dam.

The rock volumes considered are six in number: two directions for joint P~, and three levels (elevations 0, 70 and 130) for joint/'2- Figure 41 shows the position of the dam in

~.r . - -~

FIG. 37. Site plan of the Oymapinar dam, Turkey. 1. Arch dam; 2. Spillway intakes; 3. Intakes; 4. Underground power station; 5. Diversion tunnel.

119


P. Londe

o

,a2

~,x : t

t::l

~o

0

"6 ~ .,...~

>

120

. ,,,,,~

0

0

N e~ 0

9

o,~


O~

0

CL

CO CO

D.

u~ a~

0

O-

L~J (.C}

v Z

F - l,a_ LId .,_1

~ l i l l l ~ ! 1 1!1 ~ -; I 1111 i- ~ . i

i=;.2_ :

I - f J i - l l J~ i i l - r , . ! W I -~

. . . . ' - l . . . . . t . . . . ] . . . . . . . i

i r . . . . . . . ~ . . . . . j r "

ii i i ! ~ I

8

- -+- i

i ! s - , i ! i -~t - -~ i , i - i J i j I

p,,, ~ , ~ ', .... 4 - -~k _ , - -d - -d -~

-TiZ~ ...... L_L%

o o

\

~ ~ ~ o o o O o o o o o

, . . ; ~ :

o~o ~ ~o + o~o o

6


P. Londe

relation to the six rock tetrahedrons. Results of the analysis, by computer, are given in Fig. 42 for three of them.

Point R in each diagram corresponds to ul = 25 per cent, u2 = 25 per cent, u3 = 100 per cent. The latter value corresponds to direct loading of plane/>3 by the water pressure of the reservoir. The values of 25 per cent on the other two planes mean, however, that a thorough system of grouting and drainage has to be carried out. In view of the design adopted (concrete lining of the rock upstream of the dam, and drainage by many adits and boreholes), it is likely that the uplifts will be substantially less than 25 per cent of full hydrostatic pressure. Points R show that, in these conditions, stability is ensured, and that there is an ample margin of safety for uplift and friction angle variations. The other combinations of joints which have been tried are not less safe than the ones presented here.

The conclusions to be drawn from this analysis are as follows:

(a) The extensive grouting and drainage system is vital in preventing seeping water from developing detrimental pore pressures in the rock mass.

(b) The margin of safety of the abutment is satisfactory considering the actual geological structure, depth of excavation and thrust of the dam.

(c) The margin of safety increases from bottom to top of the abutments.

(d) More precise analysis of the safety requires tests for a better knowledge of the shear strength of the joints playing an important part in the stability (here joints X3 and J4). Laboratory tests on large size core samples will be adequate.

(e) Careful observation of the seepage conditions in the abutment by means of drain discharge and piezometric measurements will increase safety during operation of the scheme by giving advance warning of any development of abnormal conditions.

10. Conclusion The method proposed for analyzing the stability of a rock slope cut by weak geological surfaces of separation enables the enginer to assess the factors which are vital to the safety of the rock mass. No attempt is made to define a factor of safety owing to the number of parameters involved. The approach is typical of problems where the answer is not well-defined either because the data are not sufficiently well-known or because too many variables are introduced, a condition which prevails in rock mechanics. Although several simplifying assumptions have to be made, this method has already been used with success for practical cases. Studies are under way with a view to improving the approach and to taking account of other mechanisms of failure such as sliding with rotation.

11. Reference LOND~, P., VIGmR, G. & VOrtMEmNGER, R. Stability of slopes, Graphical Methods. J. Soit

Mech. Found. Div., ASCE, Vol. 96, No. SM4, 1970, pp. 1411-34.

124


Documents

Analysis of the Stability of Rock Slopes