Modified hypothesis for failure of Malpasset Dam

Engineering Geology, 24 (1987) 367--394

367

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands

MODIFIED HYPOTHESIS F O R F A I L U R E OF MALPASSET DAM

W. WITTKE and G.A. LEONARDS

Institut fiir Grundbau, Bodenmechanik, Felsmechanik und Verkehrswasserbau, R WTH Aachen, Mies-van-der-Rohe-Str. 1, D-51 O0 Aachen (F.R. Germany) Purdue University, School o f Civil Engineering, Grissom Hall, West Lafayette, IN 4 7907 (U.S.A.) (Accepted for publication December, 1986)

INTRODUCTION

The objective of this paper is to contribute to our understanding of the failure of the Malpasset Dam and thus to serve as a basis for improving the design and construction of future arch dams. In 1976 Bellier and Londe presented a hypothesis for the dam failure which was based on thorough studies of the design, of the dam behaviour during impounding, and of the observations made after the failure had occurred [1--3]. This hypothesis starts from the assumption that a zone of strongly reduced permeability came to exist at the left abutment where the thrust from the arch is directed parallel to the foliation. Water pressure equivalent to the full hydrostatic head consequently acted on the upstream side o f this zone and, in combination with the thrust, initiated sliding of the foundation rock on an upstream dipping fault. At the same time, parallel to this, the schistosity cracked on the upstream side of the dam, beginning at some depth where the pore pressure was significant and the arch thrust could n o t counteract the cracking, as is the case near to the surface (Fig.l). The arch thrust was evaluated from an analysis based on the trial load method. Three-dimensional FE-analyses were carried o u t taking into account the interaction between the arch dam and the foundation rock in a more realistic manner than was possible in the past. The influence of the stresses created b y seepage are also taken into account in these analyses [ 5 - - 1 0 | . As will be outlined subsequently, these analyses show that the water pressure acting on the arch dam as well as the seepage pressure and uplift acting on the rock create tensile stresses perpendicular to the schistosity on the upstream side o f the dan~. Cracking of the schistosity at the left abutment upstream of the dam could result from these tensile stresses thus leading to a redistribution of piezometric head in that area. This would then result in a hydrodynamic pressure similar in direction and magnitude to the hydrostatic pressure visualized b y Bellier and Londe in their hypothesis of failure. Further elasticviscoplastic analyses showed that this redistribution of piezometric head could have led to the instability of the left abutment of the Maipasset Dam 0013-7952/87/$03.50

© 1987 Elsevier Science Publishers B.V.

368

A

B

noturol ground Horizontol sectionat elevation ~ . ~ m v downstream fouit ~lo~in the foliotion permeobility 0.01 ko or tess

oppi~edpressure ,,fuElhydrostoticpressure Secnon A-A

Section B-B

Fig.1. Hypothesis of failure according to Bellier and Londe [3].

by sliding upwards along one of the fault zones in the manner described by Bellier and Londe. (A zone of weaker rock at the contact between the base of the arch and its left a b u t m e n t could contribute additionally to the failure mechanism described above, hence parametric studies were initiated, but not completed in time for the paper, to investigate this aspect of the problem.) If the modified hypothesis of failure resulting from these FE-calculations is appropriate, as we believe it is, then conventional drainage of the foundation rock would n o t have been the answer at Malpasset, or at least not the only answer. In the future, it will be necessary to use analyses that take into account the interaction of the dam and its foundation in a realistic way and to adapt the design accordingly. To illustrate the manner in which this can be done is the main thrust of this paper. ARCH DAM AND ITS FOUNDATION

The topography of the site and the geometry of the arch dam, as presented in a perspective view in Fig.2, are taken from the references [1--3]. The valley b o t t o m is circa 28 m wide and the average inclination of the slopes is circa 35 ° . The arch dam, with a height of about 60 m and a crest length of about 220 m, has a thickness o f 6.79 m at the valley b o t t o m and of 1.50 m at the crest. It thus represents a thin shell structure. The geometry of the upstream and downstream face is given by two functions (Fig.2). Representative elastic constants and the unit weight of the concrete presented in Fig.2 were also taken from the available papers [1--3]. The rock at the dam site is described in the references [1--3]. Further, one of the authors (Wittke) made an excursion to the site in fall of 1984 with the kind assistance of Dr. Londe, and had the opportunity to get a personal impression o f the rock conditions.

369 U~treQm

z . 00,15

[z.

l

,,-c0s o,

D~vnt~'~m

Fig.2. Geometry of the arch dam.

According to the available information, the rock consists of a gneiss which is traversed mainly b y three families o f discontinuities. The schistosity (Sch) strikes diagonally to the valley and reveals an average dip of 40 ° (300--50 ° ) downstream (Figs.3 and 4). Discontinuities of various extent have developed parallel to the foliation. The larger ones partly contain mylonite coatings. Furthermore the rock is traversed b y faults which contain fillings of clayey breccia with a thickness of up to 80 cm. These faults strike E--W, i.e., approximately perpendicularly to the valley and dip at circa 45 ° upstream (F1 in Figs.3 and 4). Discontinuities of smaller extent and a scatter with regard to their orientation exist more or less parallel to these faults. The faults F2 strike approximately N--S and have an average dip of 70 ° 80 ° towards the left a b u t m e n t of the dam (Figs.3 and 4). Also, parallel to these faults, discontinuities of various extent have developed within the rock mass. The available information on such rock mechanical parameters as deformability and strength as well as on the in-situ stresses is limited. Using plate loading tests performed in an adit after the dam failure had occurred, average moduli of 850 MN/m 2 in the vertical direction and of 1600 MN/m 2 in the horizontal direction were measured [ 1--3]. Furthermore, mention is made in the references of the modulus of the rock mass being approximately equal to 1/10 of the modulus of the concrete, indicating a modulus of 2500 MN/m 2. This is of the same order of magnitude as the values measured in the plate loading tests. As experience shows, the deformability of schistose rocks is quite often anisotropic and in many cases can be described b y five elastic constants (El,

370

Fig.3. Photographs taken on the occasion of an excursion October 1984. E2, vl, v2, G2 in Fig.4), this being at the same t i m e t h e simplest w a y o f describing elastic a n i s o t r o p y . T h e d e f o r m a b i l i t y tests p e r f o r m e d at t h e site were, h o w e v e r , n o t o r i e n t e d w i t h regard t o t h e schistosity. T h e available i n f o r m a t i o n o n t h e shear strength o f the discontinuities is r a t h e r limited. Also n o d a t a o n t h e in-situ stresses could be f o u n d in references [ 1 - - 3 ] . F r o m t h e results o f w a t e r pressure and grouting tests it could be c o n c l u d e d

371

Fig.4. Discontinuities,

structural

model and rock mechanical

parameters.

that the permeability of the rock is very low. In laboratory tests on small samples which were performed after the dam failure, it was found that the permeability is strongly stress-dependent. The effective stresses within the samples are, however, not clearly defined and, more importantly, the stressdependency of permeability due to discontinuities of larger extent probably were not evaluated by the laboratory tests performed. STABILITY MODEL”

ANALYSES

ACCORDING

TO THE SO CALLED

“INTEGRATED

Stability analyses according to the so-called integrated model have recently been performed in a number of cases [6-l 0] . They consist of threedimensional finite element analyses of sections which contain the arch dam and the foundation rock in so far as it influences the stability of the dam. Consequently, the interaction between the dam and its foundation is taken into account. Loading of the foundation rock resulting from seepage, which can have a considerable influence on stability, can also be taken into account. Thus, this type of analysis may be considered to be more realistic than those of conventional type. An important condition for the application of such analyses, however, is an acceptable model for both the stress-strain behaviour and the permeability of jointed rock, as well as a fair knowledge of the parameters appropriate to both models. The calculation program applied in this paper starts from a structural model of the rock mass which contains three or more families of discon-

372

tinuities of arbitrary orientation as well as single, distinctly developed faults or master joints (Fig.5, [5] ). The geometry of the discontinuities can of course vary in different areas of the foundation. A homogeneous model is applied for the stress--strain behaviour (Fig.6), as is c o m m o n for soils where the size of the grains is usually small in comparison to the considered section. In the case of rock masses, mainly the spacing of the discontinuities of the various families must be small in comparison to the considered section. This condition is not always fulfilled b y faults and master joints, and thus such discontinuities had to be represented separately within the analyses. For stresses which do not approach the strength, linear elastic stress--strain behaviour can be taken into account and may be either isotropic, transversely isotropic, as presented in Fig.7, or orthotropic [5]. When the strength is approached a time
Fig.5. Structural model.

373

g~niedrock Fig.6. Homogeneous model for ~--e-behaviour of a jointed rock (after Wittke, 1984 [5 ] ).

Fig.7. Elastic-viscoplastic behaviour.

374

Fig.8. Failure criterion of a rock mass with one set of discontinuities.

Fig.9. Homogeneous model for seepage through jointed rock. in this f a m i l y o f 1 m, t h e c o e f f i c i e n t o f p e r m e a b i l i t y o f t h e r o c k mass (kT) can b e d e s c r i b e d as s h o w n in Fig.10 [ 5 ] . I t can also be seen f r o m Fig.10 t h a t t h e p e r m e a b i l i t y increases c o n s i d e r a b l y w i t h increasing w i d t h o f o p e n i n g a l o n g t h e f r a c t u r e s . T h u s , an increase in t h e w i d t h o f o p e n i n g f r o m 0.2 t o 0.4 m m increases t h e p e r m e a b i l i t y b y a p o w e r o f ten. T h e r e f o r e , it is t h e

375

20t

k/O~

kT {m/s; ]tkt(soil)

o,,o. o,.~i

~12,u.lm ~t2.v.[1.BS.(k/[kl~s].1m

(k/O.sO032; (k/~'0032}

o.zs

O/,mm~ 0,7mm ~

=m==]

1,Cmrn

i =¢QO]Z 0,25

o.z.-~ O,Z,• 11~' -~.~ .~o~

. . 0.25

0.2.10~ . . 0.t .10~

,,o.o3z

u~.to~

~ 0.032

~ ......

1 0.25

l

0.3.10 -3

! sond

gr ~t

i'

Fig.10. Permeability of rock mass (after Wittke, 1984 [5 ] ).

effect of stress level on the width of the main discontinuities that controls the relation between stress and permeability of the rock mass. This relationship is not well defined in the present state of art. The permeability of rock masses is also anisotropic in m a n y cases. Only in the case of three families of discontinuities which are perpendicular to one another and which have an equal width of opening, can the permeability be considered to be isotropic

[5]. The resultant of the forces transmitted to an element of rock mass by the seepage water can be subdivided into hydrostatic and h y d r o d y n a m i c components, the so-called uplift (A) and seepage pressure (S), both being volume

Fig.11. Seepage and uplift forces.

376

forces (Fig.ll). The seepage pressure is proportional and parallel to the hydraulic gradient (I). Both forces have a considerable influence on the stability of the foundation and abutments of a dam. In the first step of the analysis, the seepage existing prior to construction o f the dam is simulated according to the results of piezometric measurements in exploratory boreholes (Figs.12 and 13). From this calculation the seepage pressure Sp and the uplift Ap are obtained. Knowledge of these is important because displacements within the rock mass resulting from this loading have occurred before measurement is started. Due to impounding, larger portions of the rock mass are submerged and subjected to uplift (As). Seepage underneath and around the dam also occurs as a consequence of impounding in a manner depending on the rock permeability, and on the grouting and drainage to be provided. This leads to a variation of seepage pressure in comparison to the state prior to construction (Ss), which is simulated in a separate step of the analysis (Figs.12 and 13). Further loads are the dead weight of the rock mass (GF), the dead weight of the dam (GM)and the water pressure (W) acting on the upstream face of the dam (Fig.12). An example of the sequence of the analysis is shown in Fig.13. In a first step, as mentioned, the seepage existing in the rock prior to construction is simulated. From the second step, in which the foundation rock is loaded b y its dead weight and the forces Ap and Sp, the so-called primary stresses (ap) and displacements (~p) are obtained. The latter have already occurred and cannot be measured. Also any tectonic stresses present can be taken into account in this step. In the third step seepage resulting from impounding is analyzed (Ss, As). Here also the effect of any grouting and drainage measures can be taken into account. The fourth step considers all loads and leads to 5Decificotion of loads 5~ dead weight of rock moss 5M dead weight of dam

Fig. 12. Acting forces.

W woterpressure on upstream face of dam ~ seepagepressureand upl ft

377

the stresses after impounding (Os). The deformations Ge have to be subtracted from the displacements obtained in this step (~s) in order to evaluate the displacements which result from the dead weight of the dam and from impounding (Fig.13). In the case of purely elastic behaviour, these displacements can also be calculated if the following loads are applied to the system (GM, W, A S = S s - - Sp, AA = As -- Ae). It is possible to calculate the separate

~cificQtion ~ !oods ~rior to construction

GF deod weight of rock moss

n.. a=,,a wetght of dora ¢l~,Jfe oN ~m foce of dom ~e pressurepnor to i ~

p~x to npou~j

Fig.13. Sequence of analyses (example).

Fig.14. Finite element mesh.

378

effects of these loads and to analyze cases in which the reservoir is only partly impounded. The dead weight of the dam can be taken into account with or without arching action; in this paper the dam was considered to act as an arch under all loads. THE FINITE ELEMENT MESH

The finite element mesh includes the complete dam and b o t h abutments (Fig.14). This is necessary because the anisotropy of the rock-mass strength, which results from the schistosity striking diagonally to the valley, is not symmetric to the valley axis. The large extent of the section b e y o n d the dam foundation is required because of the effects of seepage and uplift, as found in previous analyses [ 5--10]. The upper boundary of the analyzed section is horizontal at the elevation of the dam crest. The influence of the slope above this elevation is accounted for b y nodal forces equivalent to the dead weight of the overburden. The selected finite element mesh is relatively coarse due to the costs of the analyses. The rock mass is represented b y elements with 8 nodes and the dam b y one row of curved elements with 20 nodes (Fig.14). In order to check the accuracy o f the results, Case M1 with isotropic, elastic stress--strain behaviour of rock mass and dam and with the loads GMand W was also investigated using a finer mesh (Fig.15). As can be seen from the lines of equal displacements in a vertical section through the b o t t o m of the valley, the difference between the two calculations is not pronounced. However, in important cases and for more general loadings and rock properties, the finer mesh would be preferred. In some situations it would be advantageous to analyze the problem in two steps, i.e. to investigate the complete section with a coarser mesh and

Fig.15. Investigation of the FE-mesh.

379

continue the analysis using a detailed section with given displacements along the b o u n d a r y and a much finer element mesh [5]. This detailed section could, for example, be the foundation area at one abutment and/or a highly stressed segment of the arch. INVESTIGATED CASES

The nine cases investigated are compiled in Fig.16. They differ with regard to the assumed storage level (Cases M2 and M3) and with regard to the combinations o f applied loads (subscripts 1, 2 . . . ) . Because of the tensile stresses perpendicular to the schistosity along the left abutment revealed b y the analyses, it was also assumed that a crack appears in this area leading to locally increased permeability and thus to both an alteration in the distribution of equipotentiais and a change of seepage pressure in this area (Cases M6--M9 in Figs.10 and 16--18). In the Cases M1--M3 the rock mass was assumed to be isotropically elastic with a Young's modulus of E = 2500 MN/ m 2. For comparison Case M4 was analyzed with a rock mass modulus of E = 1000 MN/m 2 and in Case M5 transversely isotropic, elastic behaviour was adopted (Figs.16 and 17). Finally, Cases M7--M9 were analyzed with the assumption of elastic-viscoplastic behaviour of the rock mass. Here the tensile strength perpendicular to the foundation level and along the schistosity was assumed to be zero. In addition, the influence of a reduction of shear strength along the discontinuities was investigated in Cases M8 and M9 (Fig.16). The seepage analyses for all cases were carried out under the assumption o f isotropic rock-mass permeability. P~aneters

Case

lst~c~lng

MII'=

M5

'

e~ast I~ "

'

l oadtr4g case

vlsc-~lAstlE J

"

NG, •

Oescr)~tlon r,,,,,

"=

•

i ~ t , ~ °0 •

t ~ u ,

i =

m - o

;:::+,;: 1

•

i

wit~ ~ractlng s l o ~ Sc~ on t e ~ w ~ u m n t fo~ seeome i t n e l y t t s

Fig.16. S u m m a r y o f t h e analyzed cases (for legend see Fig.17 ).

380

Pst'aIeters

• I

I

t rar~rsely

I

i

t~l~t ro~J c

~

t$otr~tc

'~2 o.1o

I I

SDeclflr~ltl~ of I ~ GN

ae~a ~lgl~t ~f

w

watet~re=wj-eon = t r e ~

1

face of i

prior to l ~ l r ~ Ss, As seelo~ge~ ' e s ~ e ar~ uplift after I I ~ l ~ l l r ~ AS=Ss-Sp ~ ~lfferer~e of seeoage AA=As_Ao ) t~ltft: aftec ar~ prior to

Fig.17. S u m m a r y o f t h e a n a l y z e d cases and rock m e c h a n i c a l p a r a m e t e r s (legend for Fig.16).

RESULTS OF ANALYSES

Stress distribution w i t h i n the rock mass for elastic behaviour -- Cases M2, M 3 and M 5

As mentioned, Cases M2 and M3 were analyzed under the assumption of isotropic and elastic stress--strain behaviour o f the rock mass and the dam, assuming a modulus o f E = 2500 MN/m 2 for the rock mass, i.e. 1/10 of t hat o f th e dam (Figs.16 and 17). Fig.19 illustrates the increase in dam d e f o r m a t i o n at the crest, the f o o t and a horizontal section at an elevation o f 65.00 m caused by an advance in impounding f r o m 79.75 m (A) to 87.30 m (C). It can be seen t hat the evaluated displacements along the f o u n d a t i o n level as well as along the crest are practically constant. Measured d e f o r m a t i o n s at only three points on a horizontal section at a height of 65.00 m are stated in the reports available

381

i

/ i /

Fig.18. Illustration of Cases M6--M9, with regards to seepage.

r2 El ~m

O+mens~K~

'-"

(4 El 65m

[

Oof~z~mes

|le~4ep revel H]tt

moe,+ e~r,~,N IO0'IZ

HZ/ZliwT~Jndin9 to 13m tmderneothdora crest,GcW.AS.AA)

. . . . . . . . . . C

EtGSm

"

2

_ measured(~{lier/t0nde)

Fig.19. Deformation of the arch dam. to the authors (Fig.19; [3, 4] ). The displacements calculated at these points are less than the measured values; for example, 5.3 m m calculated vs. 11.3 m m measured at point H. The number of measurement points is not sufficient for an accurate estimate of the moduli for the concrete and rock mass by means of back-calculation. Therefore, with the exception of Cases 4 and 5, the analyses carried out for this paper are based on a ratio of the moduli E/E~ = 1/10, as adopted for the dam's design, with a modulus for the concrete of EB = 25,000 MN/m 2, (Fig.16). As described below, a failure hypothesis has been developed on the basis of these analyses. The failure hypothesis is based on the conclusion that tensile stresses perpendicular to the schistosity arise upstream of the dam in the left valley side in loading

382

case "impounding complete". Bearing in mind the above-mentioned deformations, the deformability of the rock mass was probably assumed to be t o o low. Analyses were therefore carried o u t under the assumption of a smaller Young modulus of the rock mass for the purposes of comparison (Cases 4 and 5). It was found that tensile stresses perpendicular to the schistosity also arise at the left valley side when a higher deformability of the bedrock is assumed. Thus a cause for the failure of the structure in accordance with the failure hypothesis described in this paper has been demonstrated to exist. For completeness, the corresponding calculation results for full storage are also presented in Fig.19 (Case M3/2). This result probably does n o t coincide with reality. However, as described in subsequent paragraphs, the points which are critical with regard to the stability of the dam can still be identified from the results of these elastic calculations. In Fig.20 the evaluated distributions of tensile stresses (Oz) and compressive stresses (OD) along the foundation are shown. It is interesting to note that for storage to 13 m underneath the dam crest and for the load GM + W only very small tensile stresses occur on the upstream side of the foundation (Case M2/1 in Fig.20). An increase of circa 0.7 MN/m 2 results however from the effect of seepage due to impounding (Case M2/2 in Fig.20), which is normally n o t taken into account in stability analyses of arch dams. Significant tensile stresses of circa 2.0 MN/m 2 result when full storage and the loads GM + W are taken into account (Case M3/1 in Fig.20). The effect of seepage again leads to an increase in the tensile stresses of circa 0.9 MN/m 2 (Case M3/2 in Fig.20). Thus, when all loads are considered, tensile stresses of nearly 3.0 MN/m 2 occur at foundation level. It is also interesting to note that tensile stresses perpendicular to the schistosity occur along the upstream side of the foundation in the case of az lerlslle s|ress a~ compressivestress

~J ~m "~6M r~lO" ,O~M~lr#

~

__

~ me~

,~03~Vdm ~

M3n(G~.W) Fig.20. Vertical stresses b e t w e e n dam and f o u n d a t i o n rock.

~IJm

383

I

slice O

MZ13IC~.G~4V.Ss.As)

I

i

1 element I

X3/3IG.C~.W.~Asl

.~

tension

Fig.21. Normal stresses on schistosity Sch (Cases M2/3 and M3/3).

full storage and loading b y G F + GM + W % S s + As (Case M3/3 in Fig.21). Because the schistosity strikes diagonally to the valley these stresses only occur at the left a b u t m e n t and at the valley b o t t o m . In the perspective view o f the dam in Fig.21, the average stresses evaluated for the corresponding elements are presented (0.12, 0.32 and 0.39 MN/m 2 in Fig.21). In addition, for element I, the tensile stresses evaluated for four of the eight Gaussian points are presented in Fig.21. It can be seen that the maximum stress amounts to 0.92 MN/m 2 and is 2 to 3 times larger than the average. A finer FE-mesh would probably lead to even higher tensile stresses along the schistosity. A further very interesting point is that much smaller and less extensive tensile stresses occur for impounding to 13 m below the dam crest (Case M2/3 in Fig.21). The distribution of tensile stresses perpendicular to the schistosity along a vertical section through the valley b o t t o m is presented in Fig.22. It may be seen that the maximum tensile stress occurs at the ground surface and that there is a decrease to circa 0, or at least to very small stresses, at a depth of 2 to 3 times the width of the dam, if only loads due to the dam's own weight and impounding are taken into account (Case M3/2 in Fig.22). If the dead weight of the rock mass is also taken into account, then the decrease occurs at a smaller depth and in deeper zones compressive stresses result perpendicular to the schistosity (Case M3/3 in Fig.22). Another interesting feature is the shear stresses along the F1 discontinuities on the downstream side of the dam. In perspective views the resultant shear

384

Q7OMN/mz

/]

O.?:JMN/mz

051MN/rnz

R~r

MG

rt',Ox,impounding with crQcking (]long Sch for

tension Fig.22. Normal stresses on schistosity Sch (Cases M3/2, M3/3 and M6).

stresses, as well as the components across and parallel to the valley axe shown in Fig.23. For impounding to 13 m underneath the dam crest the components parallel to the valley are oriented downwards, whereas for full storage along the surface all shear stresses have components which are oriented upwards, and thus reveal a tendency of the rock mass to slide upwards, and thus reveal a tendency o f the rock mass to slide upwards, i.e. towards the downstream side (Case M3/3 in Fig.23). In order to separate the influence of the different loads in Fig.24, the shear stress along the F1 discontinuities are presented separately for the dead weight of the dam and the water pressure on the upstream face of the dam (Case M3/1) and for the effect of seepage (Case M3/4). As can be seen, the water pressure on the upstream face of the dam leads to upward-oriented components of the shear stresses on F1 close to the surface and to large inward-oriented shear stress components at the abutments resulting from the thrust (Case M3/1 in Fig.24). Only small shear stresses across the valley result from seepage, whereas the upward oriented shear stress components on F1 due to the seepage pressure reach far downwards and into the abutments (Case M3/4 in Fig. 24). Finally, the distribution of the normal stresses perpendicular to the F1 discontinuity along three vertical straight lines on the downstream side of the left a b u t m e n t is presented in Fig.25 (Case M3/1 and M3/2). It may be seen that the maximum stress occurs near the surface and mainly close to the valley. It may also be seen that the stresses decrease rather rapidly with depth and that t h e y practically result solely from the influence of the dead weight o f the dam (GM) and the water pressure on the upstream face of the dam (W). For comparison, Case M5 with the assumption of transversely isotropic

385

Fig.23.

Shear stresses on Fl

discontinuities

(Cases M2/3

and M3/3).

Fig.24.

Shear stresses on Fl

discontinuities

(Cases M3/1

and M3/4).

stress-strain behaviour of the rock mass was analyzed. The moduli perpendicular (E,) and parallel (E,) to the schistosity were assumed to be equal to the values measured in the plate loading tests for horizontal and vertical directions (Fig.16 and 17; [l] ). Assumed values were adopted for Poisson’s ratio and shear modulus. The stresses perpendicular to the Fl discontinuity resulting in this case (Case M5 in Fig.25) are equal to those evaluated for isotropic

386

It,---M312 IGM.' .w.As.

I--- H3/2 (G..W.AS.AA)]

Ai

Fig.25. N o r m a l stresses o n F1 d i s c o n t i n u i t i e s .

k[cmls] t

" ~ment

flow flow IS= ~ u

1

0 1

5

10

Fig.26. Decrease of permeability according to Bellier and Londe.

387

behaviour. Thus, there is no evidence of a difference in stress distribution underneath the left a b u t m e n t due to elastic anisotropy, though a modification of the elastic constants and consideration of any viscoplastic displacements along the schistosity might lead to different results, such as those postulated b y BeUier and Londe [3]. From the laboratory tests reported b y Bellier and Londe (Fig.26; [3] ), it followed that an increase in stress of circa 1.0 MN/m 2, which is the maximum stress evaluated (Fig.25), would decrease the permeability b y a power of ten. Apart from the fact that the stress p, shown in Fig.26, is the difference of the hydrostatic stresses inside and outside the sample, and n o t the effective stress, the size of the samples was t o o small to reflect the influence of large discontinuities.

Assumption o f a crack along the schistosity at the left abutment and its influence on the distribution of equipotentials If it is assumed that the tensile stresses evaluated perpendicular to the schistosity along the upstream side of the dam's left abutment (Figs.21 and 22) lead to cracking on the schistosity, then even small cracks lead to a considerable localized increase in the permeability of the rock mass. In order to study the effect o f such an increase in permeability on the distribution of piezometric head in the foundation rock, a crack as shown in Figs.18 and 27 was assumed at the left abutment in Cases M6--M9. The distribution of equlpotentials which results from this assumption is shown in Fig.27. For Section II--II at the left abutment, the equipotentials underneath the dam are oriented parallel to the schistosity, which is inclined downstream. Furthermore, a concentration of equipotentials results in this area. As a consequence of these results, the seepage pressures (oriented perpendicular to the equipotentials) is high, because it is proportional to the gradient in this area. In

s ~ t ~ !* |: equipotent~s without CnX'k~:j

st~[m n-if: t ~ t i 0 t s

~th cmck~

Fig.27. Alteration of equipotentials due to cracking along schistosity.

388

fact, the seepage pressure in this zone is practically equal to the hydrostatic pressure resulting from the water level in the reservoir. In the right abutment (Section I--I, Fig.27) a homogeneous distribution of equipotentials results from the analysis and, as expected, influences of cracking are not evident.

Influence of the change in the distribution of equipotentials due to cracking The influence of the change in the distribution of equipotentials due to cracking was investigated firstly for elastic behaviour of the rock mass (Case M6 in Figs.22, 28 and 29). An increase in tensile stresses perpendicular to the foundation and the schistosity relative to the corresponding Cases M3/2 (Fig.20) and M3/3 (Fig.21) results from this influence (Fig.28). Furthermore, it becomes clear that tension reaches further downwards into the rock (compare with Cases M3/2 and M6 in Fig.22) and shear stresses on the F1 discontinuities have larger components acting towards the downstream side and horizontally towards the valley (see Case M3/3 in Fig.23 and Case M6 in Fig.29).

~

slress~s be'me~ dam ~

[aun~bon rock

~ r ~ t s|resses on s c ~ t y

~h

F i g . 2 8 . M 6 (G F + G M + W + S S + AS).

389 For a better understanding of this result the angle of friction (~FZ,req.) required to ensure equilibrium on the F1 discontinuities along the downstream side of the dam was evaluated (Fig.29). The tangent of this angle is derived from the effective normal stress and the resulting shear stress on F1 assuming zero cohesion in the manner shown in Fig.29. One can see that the required angle of friction at the valley b o t t o m and left abutment is much higher than at the right abutment.

Viscoplastic analysis Finally, three cases assuming the occurrence of viscoplastic behaviour were analyzed (Cases M7 through M9 in Fig.16). Cases M8 and M9 provide the most interesting results. Here, besides zero tensile strength perpendicular to the schistosity and the foundation, angles of friction along the F1 discontinuities were assumed as ~FZ = 20° (Case M8) and ~ z = 150 (Case M9). In Case M8, especially at the left abutment, extensive zones arise in which the tensile strength along the schistosity and the shear strength along the F1

:0

e~enr s|Jce~

005 [~l~/mz I

sheQr stresses on disconfi~es~_)on downstream side of dam

Ti~" ~SULt$~ ~heflf

stressflF1

~:~ -riot-real stress ,L F1

mmm ~

of m ¢ ~

~

~

to ¢ r 0 ~

required angle .of friction ~ on discontinuities ( ~ on aownstregm side of dam Fig.29. M6 (G F + GM + W + Ss + As).

390

discontinuities are exceeded. The viscoplastic displacements resulting from the analysis are also larger at the left a b u t m e n t (point B) than at the right (point A, see Fig.30). The displacements do however converge after 20--30 steps in the iterative calculation to simulate the viscoplastic behaviour. Assuming an angle of friction of ¢F1 = 15o , however, the viscoplastic displacements at the left abutment do not converge because the shear stress on the F1 discontinuities cannot be carried (Fig.30). Thus the analysis of this case leads to failure of the rock at the left a b u t m e n t due to sliding upwards on the F1 discontinuities and opening of the schistosity due to tension, as has been observed. On the other hand, the viscoplastic displacements at the right a b u t m e n t do converge, which indicates that, according to the analysis, the stability of the righthand side is not endangered. MODIFIED HYPOTHESIS FOR FAILURE

A hypothesis for the failure of the Maipasset Dam based on the preceding considerations will n o w be presented. The hypothesis may be regarded as a modification of that proposed b y Bellier and Londe [3]. It has been shown that a large reduction in stress (perhaps sufficient to induce cracking) occurs perpendicular to the schistosity on the upstream side along the left abutment of the dam due to the water pressure on the upstream face of the dam and to the forces resulting from the seepage under and around the dam (Fig.31). As a consequence the permeability parallel to the schistosity increases in this area resulting in a redistribution of the equipotentials; this, in turn, leads to a concentration of seepage pressure in an unfavourable, upward direction. This seepage pressure is practically equal to the full hydrostatic load from the reservoir, as assumed b y Bellier and Londe, although for a different reason (compare Figs.1 and 31). As a consequence of this phenomenon, an increase occurs in the shear stresses along the F1

f

60

=20'

50 ¢0 30 2.0

5

10 15 20 25 30

Fig.30. D e f o r m a t i o n s at f o o t of d a m o n d o w n s t r e a m side.

391

Rw.'S ~ / Fig.31. Modified hypothesis of failure of the Malpasset Dam.

discontinuities underneath the dam and on the downstream side of the left abutment. The wedge b o u n d e d b y the schistosity and a discontinuity of the F1 family tends to slide upwards due to this loading and leads to the failure o f the left abutment. To simulate this failure elastic-viscoplastic analyses required angles o f friction (¢) ranging from 15 ° to 20 ° on the F1 discontinuities. These angles seem to be low compared to the statement of Bellier and Londe [3] that an angle of friction of OF1 ~ 30 ° exists on the F1 faults. Apart from the fact that not much information is available on the rock mechanical parameters of the site, if, for example, the horizontal components of the in-situ stresses close to the surface are smaller than assumed in the calculations, the crack along the schistosity would reach a far greater depth and the loading of the left a b u t m e n t b y seepage pressure would be larger than that taken into account in the preceding analysis. Consequently, failure could then also be simulated with higher angles of friction along the F1 discontinuities. To summarize, the mechanism of failure is the same in the hypothesis presented here as in that presented b y BeUier and Londe, although the rock mechanics reasoning is different. This difference can be very important in the use of analyses as a basis for design of future arch dams. SUMMARY AND CONCLUSIONS

After a brief description of the hypothesis of failure for the Malpasset Dam given b y BeUier and Londe in 1976 [3], the geometry of the dam as well as the geological and rock mechanics conditions are described in so far as details on these aspects are available from previous reports. There follows a discussion o f the method of analysis according to the so-called integrated

392

model applied in this paper and the main assumptions with regard to the stress--strain behaviour and the permeability of the rock mass. After explaining the selected calculative section and the finite element mesh which contains the complete dam and large parts o f the foundation, the results o f the analysis are presented. These analyses show that, for full storage, as a consequence of the water pressure on the upstream face of the dam as well as of seepage pressure exerted on the foundation, tensile stresses perpendicular to the schistosity develop on the upstream side of the left abutment, probably leading to cracking on the schistosity in this area. As a consequence, the permeability of the cracked zone is larger than that of the undisturbed rock and b o t h a redistribution of equipotentials and an unfavourable loading of the left a b u t m e n t b y seepage pressure arise. The results of the calculations can be used to explain the failure of the Malpasset Dam. In conclusion it m a y be stated that the mechanisms of failure in both hypotheses are the same, although the rock mechanics reasons provided differ. This can lead to different conclusions with regard to the present and future design of dams, e.g., the location of grout curtains, the use of upstream concrete slabs, and the t y p e and location of drainage to be provided. In addition, it will be necessary to improve our understanding of the interaction between an arch dam and its foundation, including the effect of loading the foundation b y seepage water. The method of analysis presented herein is considered to be capable of providing a better solution to the problem than has been possible in the past. It will, however, be necessary to evaluate rock conditions sufficiently carefully so that mechanical and hydraulic parameters

tensile stresses /.L foundotion , sliding upwards

Fig.32. Interaction between arch darn and foundation.

393

y

.

i

/~

.

.

.

te~s~0ndue to

ten~on±~ot greater distQnceff~n dQm

Fig.33. Different measures to increase d a m safety.

are established m u c h more reliably than was the case for the Malpasset D a m * i. The water pressure on the upstream face of a d a m frequently leads to tensile stresses on the upstream side of the foundation. These m a y however be avoided by the selection of an appropriate d a m shape and width of foundation (Fig.32). Furthermore, when a verticalgrout curtain and a downstream drainage screen are adopted, seepage pressure can cause the danger of both cracking along discontinuitiesdipping downstream from the upstream side of the d a m and upward slidingover a discontinuity with an adverse dip (Fig.32). Inclination of the grout curtain and drainage screen towards the upstream side decreases this danger, but at the same time leads to an increase in the verticaltensile stresseson the upstream side of the d a m foundation (Fig.33). A solution similarto that shown in the lower sketches in Fig.33 was therefore recommended for a project which has been designed recently [10]. ACKNOWLEDGEMENT

The authors would like to acknowledge the valuable contribution made by Dipl.-Ing. C. Erichsen in carrying out the analyses for this paper. REFERENCES [ 1 ] Final Report of the Investigating Committee of the Malpasset Dam. Ministry of Agriculture, Paris, 1960. *1The importance of the type of analysis recommended above has recently been proven [ 5--10 ]. Some of the principal results will now be discussed.

394 [2 ] Lessons from Dam Incidents. International Commission on Large Dams, Paris, 1974. [3] Bellier, J. and Londe, P., 1976. The Malpasset Dam. Engineering Foundation Conference Proceedings, "The Evaluation of Dam Safety". Pacific Grove, Calif. [4] Leonards, G. A., 1982, 1984. Investigation of Failures, 16th Terzaghi Lecture. Proc. ASCE, GT2, February 1982 (and corresponding discussions in Vol. 110, No. 1, Jan. 1984, pp.95--96 and 105--107). [5 ] Wittke, W., 1984. Felsmechanik -- Grundlagen fiir wirtschaftliches Bauen im Fels. Springer Verlag, Berlin, Heidelberg, New York, Tokyo. [6] Wittke, W., 1982. Contribution to Question 53. International Commission on Large Dams, Rio de Janeiro. [7 ] Gell, K., 1984. Der Einflu~ der SickerstrSmung im Untergrund auf die Berechnung der Spannungen und Verformungen von Bogenstaumauern. VerSffentlichungen des Institutes fiir Grundbau, Bodenmechanik, Felsmechanik und Verkehrswasserbau der RWTH Aachen, Heft 11, Aachen. [8 ] Wittke, W. and Gell, K., 1983. Wechselwirkung zwischen Staumauer und Untergrund. Wasserwirtschaft, 74: 137--141. [9 ] Wittke, W., Erichsen, C. and Kleinschnittger, M., 1985. Influence of seepage and uplift forces on stresses in the rock foundation of arch dams. Proc. 5th Int. Conf. Numerical Methods in Geomechanics, Vol. 4, Nagoya (in print). [10] Ernstbachtalsperre -- Berichte zu den Standsicherheitsnachweisen fiir das integrierte Modell aus Bogenmauer und Untergrund -- Phase E (Report on Ernstbach Dam). Prof. Dr.-Ing. W. Wittke, Beratende Ingenieure fiir Grundbau und Felsbau GmbH, Aachen, Juni 1984 and M~irz 1985 (unpublished).