# ijege-13_bs-paparo-et-alii.pdf

*Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università*

*Editrice*

*DOI: 10.4408/IJEGE.2013-06.B-56*

**THE VAJONT LANDSLIDE, 9**

**TH**

**OCTOBER 1963:**

**LIMIT EQUILIBRIUM MODEL FOR SLOPE STABILITY ANALYSIS**

**THROUGH THE MINIMUM LITHOSTATIC DEVIATION METHOD**

**INTRODUCTION**

detached from the Mt Toc slope flew into the reser-

voir at high speed, about 20 m/s (Z

*et alii*, 2103) and the water it displaced

cluding Longarone. The end result was 1917 victims

of which 1450 belonging to Longarone, 109 to Codis-

sago and Castellavazzo, 158 to Erto and Casso and

200 employees, technicians and their families who

worked for the company operating the dam.

effects generated by a landslide, owing to the large

amount of data collected during the monitoring of the

site since 1936, the year in which the Vajont gulley

was chosen for the construction of the dam. And the

quantity of data increased even more as the conditions

of the slope became critical.

in this context that the dramatic event happened. An am-

bitious project like this would have marked a decisive

unprecedented turning in the engineering field for re-

newable energy sources. But the tragedy was around the

corner, and we could say that it had already been written

during the first test of the dam. In a letter dated April 20,

1961, Dr Semenza, the project head, says, “Dopo tanti

lavori fortunati e tante costruzioni anche imponenti, mi

**ABSTRACT**

purpose of this work is to apply classical limit-equi-

librium methods as well as a the variant developed by

t

stability of the Mt Toc flank from where the slide de-

tached and to study the effect of the various factors

influencing stability.

east side of the slide, that were taken from the slid-

ing surface as reconstructed by using pre-slide and

post-slide topographic maps and by using suitable hy-

potheses from the literature on the shape of the hidden

part of the surface (that is the surface that remained

covered even after the slide occurrence).

more than the material cohesion. The analysis shows fur-

ther that the Vajont slide was close to instability. With data

used in this paper, a drop of the basin level from 710 m

a.s.l. down to 700 m a.s.l., in conditions of very high pie-

zometric level (790 m a.s.l.) as can be produced by intense

rainfall, may have the effect of drawing the safety factor

below the critical line of 1 and give rise to instability.

**K**

**ey**

**words***: Vajont landslide, slope stability, limit equi-*

*librium method*

*M.A. PAPARO, F. ZANIBONI & S. TINTI*

*International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013*

row peaks, the highest one occurring in spring 1962.

Notice also that a sequence of precipitation peaks was

recorded in fall and winter of 1960.

fied soil over an impermeable layer. This is because rain

water infiltrations penetrate into the ground until reach-

ing the impermeable layer, over which a water saturated

band tends to form with reduced mobilised strength.

Consequently the safety factor – see the formal defi-

nition in the next section – may drastically decrease

(F

*et alii*, 2004). In more general terms the increase of

with the decrease of the safety factor (K

*et alii*,

timately slope failures, and in general the attaining of

critical conditions for stability, are more probable to oc-

cur after intense and repeated periods of precipitation.

means of a system of benchmarks exhibited a remark-

able peak of downslope creep velocity (up to 5 cm/

day) after the heavy precipitations in the fall of 1960,

as shown in Fig. 2.

mensioni mi sembra sfuggire dalle nostre mani” (After

many lucky works and impressive constructions, I really

face something that for its size seems to me to escape

from our hands) (B

librium theory that was developed between 2006

and 2008 by Tinti and Manucci (t

thors (P

throughout the paper. We will also compare some of

our results with those obtained by using the classical

methods found in the literature and reformulated in a

way coherent with the MLD model.

**THE GEOLOGY OF THE SLIDE**

the gorge along an E-W trending axis, by eroding it

along a synclinal (g

to two erosive phases: the widest part formed in the

Würmian glacialism, and the deepest and narrow part

during an intermediate or post-glacial phase (C

many geological and geotechnical investigations espe-

cially over the years immediately before and after the

slide occurrence. The slope was found to be formed by

a succession of layers of dolomitic limestone, spaced

by thin layers of clays (g

clay in the rock layers was not commonly accepted and

considered questionable (B

Jurassic rocks and that the slide took place in the Fonza-

so formation along a slipping surface corresponding to

interbeds of clay (r

of the Vajont slopes. During the four years of filling and

lowering of the basin level after the dam was built, rain-

fall was object of careful monitoring. Official monthly

and daily precipitation records taken at Erto on the Va-

*Fig. 1 - Precipitation in mm recorded at Erto from 1960*

*to 1963 (after H*

*endron*

*& P*

*atton*

*, 1985). Monthly*

*(upper panel) and daily (lower panel) values*

**THE VAJONT LANDSLIDE, 9TH OCTOBER 1963: LIMIT EQUILIBRIUM MODEL FOR SLOPE STABILITY ANALYSIS**

**THROUGH THE MINIMUM LITHOSTATIC DEVIATION METHOD**

*Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università*

*Editrice*

*φ(x)*is the angle of shearing resistance (F

*x-dx/2*and

*x+dx/2*where

*dx*is the horizontal slice width, and let’s denote the

slip surface and the top surface of the body by the

functions

*z*

*1*

*(x)*and

*z*

*2*

*(x)*respectively. If the base and

*α*and

*β*,

then the following equilibrium conditions should hold

(see t

of the forces acting on the slice, while the third one

comes from the momentum balance. In the above

equations

*P*and

*S*denote the respective normal and

shear components of the stress taken at the base of the

slice,

*E*and

*X*are internal normal and vertical forces

applied on the vertical slice walls,

*A*is related to the

momentum associated with internal forces,

*D*is a load

function applied on the slice top, ρ is the soil density

and g the gravity acceleration. In the problem, the

functions

*X(x)*,

*E(x)*and

*A(x)*have to satisfy the con-

ditions at boundaries of the slide, i.e. in the positions

*x*

*i*

*x*

*f*

*P(x), S(x), X(x), E(x)*and

*A(x)*are

number of equations which describe the problem,

there cannot exist a unique solution (t

rial of the valley is limestone, whose permeability

is higher than that of clay, if on increasing the basin

level a limestone layer is intercepted, then the basin

water may infiltrate laterally the underwater slope and

form an aquifer confined on the top by impermeable

clay layers, and this may cause a decrease of the shear

stress along the sliding surface (F

*et alii*, 2008; v

**LIMIT EQUILIBRIUM THEORY**

the limit equilibrium approach that is one of the most

commonly used in the engineering field. For the sake

of brevity and clarity, we will provide only an outline

of the method, and present the main formulas we ap-

plied in our study.

describe the equilibrium of a mass having the potential

to detach and slide along a slope. Usually and also here

it is applied to two-dimensional profiles and the mass

is divided into blocks or slices by vertical cuts (B

which takes the name of safety factor, given by the ratio

*S*

*max*

*(x)*is the mobilized shear strength at any

*S(x)*is the shear stress

at the corresponding point. This parameter can assume

different values, depending on the condition of the

slope: when it is equal to 1 the blocks are subject to the

maximum shear stress sustainable along their base.

Therefore when the ratio falls below 1 the blocks are

no longer in balance and the whole system becomes

unstable and is ready to move. On the contrary, if the

value is greater than 1, the system is stable.

Coulomb in the following way

*S*

*max*

*(x)=c(x)+[P(x)-u(x) ] tan φ (x)*

*c(x)*is the cohesion,

*P(x)*is the normal stress

along the slip surface,

*u(x)*is the pore pressure and

*Fig. 2 Rate of creeping recorded on the slope that was in-*

*volved by the Vajont slide from 1960 to 1963 (after*

*H*

*endron*

*& P*

*atton*

*, 1985)*

*M.A. PAPARO, F. ZANIBONI & S. TINTI*

*International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013*

mon classical methods. In the application of the

MLD method, the solution was searched by imposing

that

*X(x)*is a sine series truncated to the third order

(see t

tions

*E(x), X(x)*and

*A(x)*and should vanish at the be-

ginning and at the end of the trial slide as imposed by

the boundary conditions. It can be seen that the plot-

ted curves do not differ too much from one another,

but only the curves calculated through the Spencer’s

method and through the MLD method fulfil the con-

ditions in all the graphs.

**MINIMUM LITHOSTATIC DEVIATION**

**METHOD**

ti and Manucci (t

equilibrium in the formulation of the limit equilibrium

theory, it was noted that one can always find a solution

where

*F*is smaller than 1 or larger than 1, and there-

fore find that the slope is at the same time unstable

and stable. This inherent ambiguity was corrected by

introducing a minimization criterion and by consider-

ing

*F*not as an unknown like it is assumed in the clas-

sical methods (F

*et ali*, 2006; S

be the one minimising the lithostatic deviation, that

was defined as:

mass. A range of values of

*F*is given as input, [

*F*

*min*

*F*

*max*

*F*

*min*

*F*

*max*

*F*

providing the solution with the smallest value of δ is

taken as the safety factor of the slope.

*Fig. 3 - Curves of the horizontal inter-slice force E(x) for*

*different methods. At the boundaries the function E*

*should vanish. Notice that none of the Bishop meth-*

*ods (simplified and generalised) provide a solution*

*that goes to zero at the right boundary. Curves refer*

*to the profile 2 of the Vajont slide in case of dry soil*

*and water basin at the level of 700 m a.s.l*

*Fig. 4 - Curves of the vertical inter-slice force X(x) for*

*different methods. Notice that, like in Fig.3, the*

*generalised Bishop method provides a solution*

*that does not vanish at the right boundary. Notice*

*further that X is assumed to*

*be identically equal*

*to zero by the simplified Bishop method. The*

*horizontal x axis is the same as in Fig.3. See*

*caption of Fig. 3 for further details*

*Fig. 5 - Curves of the momentum A(x) for different methods.*

*Only for the Spencer and MLD method A vanishes*

*at both boundaries. The horizontal x axis is the*

*same as in Fig. 3.Other details in caption of Fig.3*

**THE VAJONT LANDSLIDE, 9TH OCTOBER 1963: LIMIT EQUILIBRIUM MODEL FOR SLOPE STABILITY ANALYSIS**

**THROUGH THE MINIMUM LITHOSTATIC DEVIATION METHOD**

*Italian Journal of Engineering Geology and Environment - Book Series (6) www.ijege.uniroma1.it © 2013 Sapienza Università*

*Editrice*

two sides (Z

*et alii*, 2013).

of the area that was swept by the moving slide is giv-

en. In this paper we have selected two profiles as trial

curves for the stability analysis, also plotted in the Fig.

7: profile 1 on the west and profile 2 on the east side.

pography) together with the level of water basin and

of a possible piezometric line are shown. The trial

surfaces are approximated by circumference arcs,

also plotted in Fig. 7, we used to apply classical limit

equilibrium methods

profiles 1 and 2, that are taken from the literature

(H

the assumptions that there is no water in the basin

**ANALYSIS OF STABILITY**

rupture is going to take place. In principle, the rupture

surface in 3D analyses, or the rupture curve in 2D, is

unknown. In this case a number of potential curves

are examined and one takes the one giving the small-

est value of the safety factor, under the assumption that

it is the most prone to break. This kind of approach is

appropriate to examine stability before failures, which

is the most common application. In this paper we study

a slope, namely Mt Toc flank, where a failure already

occurred, and that therefore showed to be unstable. The

purpose therefore is to find the main factors leading to

the instability over a surface that is already known.

authors to investigate the dynamics of the slide (see

Z

*et alii,*2013). The

raphies of the pre-slide (in front of the slide) and of the

post-slide (in the detachment niche) slopes and by con-

necting them in the intermediate section on the basis of

the conjecture that the west side and the east side of the

surface have distinctly different shapes: namely chair-

like downslope cross-sections on the west and parabol-

ic on the east (see S

of an exposed clay layer on the west while affected

limestone on the east and it was proven to imply that

*Fig. 6 - Topography of the sliding surface (after Z*

*aniboni*

*et alii, 2013) and boundary of the area swept by*

*the slide during its motion. The slide is subdivided*

*in two parts. For each part we selected one profile*

*Fig. 7 - Cross-sections of the profiles 1 and 2 selected for*

*the stability analysis. The best fitting circumfe-*

*rence arcs are the trial curves we used for the*

*classical stability methods. The basin level and a*

*possible piezometric level*

*are also shown*

*M.A. PAPARO, F. ZANIBONI & S. TINTI*

*International Conference Vajont 1963-2013. Thoughts and analyses after 50 years since the catastrophic landslide Padua, Italy - 8-10 October 2013*

ues of the safety factors are given in Fig. 8. Classical

methods were conceived for arc-like trial curves only

and hence the profile base has been approximated by

a circumference arc. On the other hand, we mention

that an extension of those methods was developed

by two of the authors (P

however between 1.10 and 1.4, with the generalized

Janbu’s method giving the smallest value and the gen-

also evident that approximating the bottom surface by

a circumference arc leads to a systematic increase of

the estimates for

*F*. Differences between the computed

values are small, but not negligible. For the rest of the

analysis we will use only the MLD method and drop

the unnecessary approximation of the arc-like bottom.

tors intervening in the process. The main idea is that

the slope is affected by lateral infiltration of the water

from the basin and by infiltration from the top of rain

water. Both processes determine the increase of the pi-

ezometric level with consequent increase of the inter-

stitial pore pressure. We make the further hypothesis

that varying the water level in the basin affects also

the properties of the rock, passing from unsaturated

to saturated soil conditions (see Tabs 1 and 2) (l

*et alii*, 2004). Our analy-

time needed to transform dry soil into saturated soil.

Further we assume that the soil is saturated below the

level of the water in the basin and unsaturated above,

which means that increasing the basin level increases

the portion of the bottom surface characterised by

saturated soil properties.

in Figs. 10 and 12 for profile 1.

In case 1 we use the properties of a dry soil (see Tables

1 and 2) and we consider the load of hydrostatic na-

ture exerted by the water in the reservoir on the slope,

*Tab. 1 - Geothecnical parameters for profile 1 (H*

*endron*

*&*

*P*

*atton*

*, 1985)*

*Tab. 2 - Geothecnical parameters for profile 2 (H*

*endron*

*&*

*P*

*atton*

*, 1985)*

*Fig. 8 - Values of the factor of safety computed for profile*

*2 with unsaturated soil by means of all the meth-*

*ods mentioned in the paper*

*Fig. 9 - Trend of F with varying reservoir level for cases*

*from 1 to 5 of Tab. 3 for profile 2*

*Tab. 3 - Cases analyzed in the slope stability study*

**THE VAJONT LANDSLIDE, 9TH OCTOBER 1963: LIMIT EQUILIBRIUM MODEL FOR SLOPE STABILITY ANALYSIS**

**THROUGH THE MINIMUM LITHOSTATIC DEVIATION METHOD**

*Editrice*

ezometric level is also changed exactly as in case 2.

The effects of cohesion and friction angle changes are

studied separately. It is seen that lowering cohesion

(case 3) is much less effective than lowering the fric-

tion angle (case 4). And this is also confirmed where

both effects are considered together (case 5), since

the safety factor curve of case 5 is yet lower but only

slightly than the curve of case 4.

*F*is ex-

pectedly corresponding with the highest basin level

considered in the analysis. If we compare findings

for profiles 1 and 2, we see that both profiles remain

stable for any level of the water and that profile 2 is

closer to instability conditions than profile 1. Further

we observe that curves of in Fig. 10 are more regular

than the ones of Fig. 9. Especially when the changes

of friction angle are taken into account (cases 4 and 5)

curves show a quick decrease (around level 660-690)

for profile 2, where they decrease steadily for profile

1, which is probably to be linked with the different

geometries of the two profiles.

the piezometric line is increased from 710 m to 790

m (H

water infiltration in the slide from the top surface.

So what is studied here is the effect on stability of

an increased pore pressure, all the rest remaining un-

changed. The results are shown in the plots of Figs.

11 and 12 respectively for profiles 2 and 1. For both

profiles one can observe that the safety factor further

increases, along with the level of the water in the ba-

sin. The maximum water level considered in the anal-

ysis is 710 m asl. Though the highest allowed level

according to engineering specification for the Vajont

dam was 722 m asl, the level of 710 m asl was the

highest reached during the reservoir filling tests car-

ried out by the dam engineers.

0.1 in both profiles. In addition to the stabilizing water

load, case 2 includes laterally infiltrated water and as-

sumes that the piezometric level is the same

the raise of the piezometric level entails a reduction of

the mobilized shear stress and weakening of the slope.

From Figs. 9 and 10 it seems that the resulting safety

factor remains almost constant, varying only by less

than 0.03. This means that the destabilization effect of

pore pressure increase counteracts and slightly exceeds

the hydrostatic stabilization of the basin.

to saturated values of the cohesion and of the friction

angle. As already enunciated above, it is assumed that

at the interface between the slide and the underlying

rock, that is on the bottom surface of the slide, one

finds the saturated lower cohesion and lower friction

angle values below the basin level, while above that

level the dry values are found. It follows that increas-

ing the water in the basin implies an extension of the

area of the bottom surface where the mobilized shear

strength is lower, and hence a lower safety factor is

*Fig. 11 - Trend of F for case 6, profile 2. While keeping the*

*reservoir level constant at 710 m, we raise the piezo-*

*metric level from 710 to 790 m (H*

*endron*

*& P*

*atton*

*,*

*1985). Lowering the level basin from 710 m to 700 m,*

*instability conditions are attained (blue dot)*

*Fig. 10 - Trend of F for profile 1 for different cases from 1*

*to 5 (see Tab. 3)*

*M.A. PAPARO, F. ZANIBONI & S. TINTI*

the inner cohesion and of decreasing the angle of fric-

tion of the material forming the slide. As expected, the

pure hydrostatic load was found to increase stability

(case 1) of the slope, while increasing the piezometric

level and decreasing coherence and friction coeffi-

cient were found to favour instability (cases 2-5). One

of the findings was that after heavy rain and with the

basin level at 710 m both profiles are very close to in-

stability, but still stable. However, lowering the level

down to 700 m causes the safety factor to go below the

critical threshold of 1 (case 6), which reflects the last

sequence of the historical facts leading to the collapse

of the slope on 9

profiles) and a validation of the geotechnical param-

eters that have been used to study instability. It seems

that the angle of friction is by far more influential than

the cohesion for the stability analysis of this case, and

therefore more specific studies should be addressed

to a better determination of its values and how they

can change in the transition from dry, unsaturated and

saturated rocks. It is worth noting that the hypoth-

esis of a heterogeneous slide surface with the west

side differing from the east side (see Tabs 1 and 2)

is corroborated by the dynamical study of the Vajont

slide performed by Z

*et alii*(2013) where it

smaller on the west than on the east to obtain a good

fit between the numerical results and the observations

(namely slide deposit and peak velocity).

**ACKNOWLEDGEMENTS**

condition of failure.

m, without any further change of the other conditions.

As seen in the analysis of case 1, decreasing the level

destabilizes the slope. And indeed what is obtained is

that for both profiles the safety factors drops below 1

(blue dot points in the graphs) and both profiles reach

conditions for failure. This last step, though sounding

artificial, respects indeed what happened in the Vajont

valley on October 9

after a period of heavy rainfall.

**DISCUSSION AND CONCLUDING RE-**

**MARKS**

volved in the Vajont slide and the consequent disaster.

In this paper we have applied the limit equilibrium

theory in a variant, denoted minimum lithostatic de-

viation (MLD) method that was developed by t

method was given. It was also noted that the classical

limit equilibrium methods in their original formula-

tion require a slide bottom with an arc shape and that

a generalisation was developed by two of the authors

(Paparo and Tinti) to account for generic bottom sur-

faces. The most common classical methods and the

MLD method were applied to an exemplary case and

were shown to provide similar results though with

differences that cannot be neglected. Indeed the case

examined as an example was the profile 2 of the Va-

jont slide, and the values of

*F*we found are in the

range 1.1-1.4 with a spread that is much higher than

changes produced by physical factors (pore pressure,

cohesion, friction angle).

that were obtained by cross-cutting the sliding surface

of the Vajont slide as reconstructed by Z

*et alii*

east-side where geometries and conditions are different

following conjectures by H

*Fig. 12 - Trend of F for case 6, profile 1. See caption of*

*Fig.11 for further details*

**THE VAJONT LANDSLIDE, 9TH OCTOBER 1963: LIMIT EQUILIBRIUM MODEL FOR SLOPE STABILITY ANALYSIS**

**THROUGH THE MINIMUM LITHOSTATIC DEVIATION METHOD**

*Editrice*

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