Rock Mech. Rock Engng.

(2000) 33 (1), 31±51

Rock Mechanics
and Rock Engineering
: Springer-Verlag 2000
Printed in Austria

Dynamic and Static Stability Assessment of Rock Slopes

Against Wedge Failures
By
È . Aydan2, and R. Ulusay3
H. Kumsar1, O

1 Pamukkale University, Department of Geological Engineering, Kinikli Campus, Denizli,

Turkey
2 Tokai University, Department of Marine Civil Engineering, Orido, Shimizu, Japan
3 Hacettepe University, Department of Geological Engineering, Ankara, Turkey

Summary
The stability of slopes during and after excavation is always of great concern in the ®eld of
rock engineering. One of the structurally controlled modes of failure in jointed rock slopes
is wedge failure. The limiting equilibrium methods for slopes under various conditions
against wedge failure have been previously proposed by several investigators. However,
these methods do not involve dynamic assessments and have not yet been validated by
experimental results. In this paper, the tests performed on model wedges under static and
dynamic loading conditions are described and the existing limiting equilibrium methods are
extended to take into account dynamic e¨ects. The applicability and validity of the pre-
sented method are checked through model tests carried out under well controlled conditions
and by actual cases studied by the authors, both in Turkey and Japan.

1. Introduction
One of the fundamental problems of engineering geology and geotechnical engi-
neering is assessing and maintaining the stability of natural and man-made slopes.
Usually natural slopes are stable when they are not disturbed by any external
force, such as seismic and hydrostatic forces, blasting and surcharge loading.
When slopes are excavated for highways, power plants, construction of buildings
and open pit mining, they may become unstable. In these situations, stability
assessments become crucial for the stability of engineering work and for economy.
In competent rocks, the stability of many slopes is controlled by wedges (Fig. 1a
and 1b) or slabs of intact rock bounded by weak discontinuities, usually joints,
faults or shear zones. Attitude of discontinuities in relation to the slope face ori-
entation determines the kinematic feasibility. The geometrical intersection of the
discontinuity sets with each other and the orientation of the slope face leads to
di¨erent types of structurally controlled slope instability.
Wedge failures are analysed using kinematical approaches and also limiting
32 H. Kumsar et al.

Fig. 1. Types of wedge failure

equilibrium methods. The limiting equilibrium method for wedge failure was ®rst
presented by Wittke (1967) and elaborated for di¨erent conditions by KovaÂri and
Fritz (1975), and Hoek and Bray (1981). Although these methods consider water
pressure, dynamic assessments are not included into the factor of safety computa-
tions. It is also noted that these previous approaches have not yet been validated
by experimental results. The authors considered that re-assessment of the existing
limiting equilibrium methods by means of model tests and numerical methods
gives a con®dence to geotechnical engineers during the design of slopes. In this
paper, the authors describe the tests performed on the model wedges under static
and dynamic loading conditions, and extend the limiting equilibrium method
proposed by KovaÂri and Fritz (1975). Static tests involve dry and submerged
conditions. The experimental results are used to check the validity of the presented
limiting equilibrium method. In addition, the applicability of the presented
method was also checked on the actual cases studied by the authors both in
Turkey and in Japan. One of the cases was selected from an earthquake prone-
region in Turkey to investigate the e¨ects of the dynamic forces on the stability of
the slope by back analysis.

2. Preparation of the Wedge Models

Concrete blocks were used as model materials in this study. Six types of wedge
model moulds were set up and ordered to be prepared in a plastic factory in order
to obtain smooth surfaces on the wedge models. Since these wedge models were
designed for testing, and required to be stable at static condition before the tests,
the plunge of the intersection line of discontinuities was kept between 28 and 31 .
The horizontal projection of each intersection line is perpendicular to the slope
strike. In order to check the in¯uence of the wedge angle, as de®ned in Fig. 8, on
the stability of the slope, half wedge angles were chosen between 23 and 56 , and
the plastic wedge block models of which dimensions are given in Table 1 were
prepared.
 Dynamic and Static Stability Assessment 33

Table 1. Geometric parameters of the prepared model wedges

Wedge no. Intersection Half wedge

angle ± ia …0 † angle oa …0 †

TB1 29 56
TB2 29 51
TB3 31 45
TB4 27 36
TB5 30 30
TB6 30 23

Fig. 2. Concrete wedge models used in the study

Base and wedge models were formed by using mortar, whose geomechanical
parameters are similar to those of rocks. The composition of the mortar used for
the preparation of the models includes 17.5 kN of ®ne sand, 3.5 kN of cement and
1.75 kN of water.
In order to model the wedge base and wedge blocks, wooden boxes with a
length of 28 cm, a width of 14 cm and a depth of 12 cm were used. The prepara-
tion of the concrete models was carried out in two stages consisting of base and
wedge model preparations. Base model preparation requires forming two discon-
tinuity surfaces, that make a wedge failure surface, and the space for the wedge
block volume. This was done by attaching specially prepared plastic wedge blocks
having smooth surfaces and sharp edges at the sides of the boxes.
Plastic moulds were extracted from the base models after 8 hours, in order to
minimize the damage to them during the extraction, so that discontinuity surfaces
for wedges were obtained. By ®lling the rest of the space in each box with the
mortar prepared earlier, the concrete wedge base blocks were modelled. After 24
hours, when the base and wedge models had become dry enough, the concrete
wedge models were removed from the moulds. For each wedge type, three wedge
models were prepared with the same geometry and material composition (Fig. 2).
The base and wedge models then were cured for seven days to gain high strength.
34 H. Kumsar et al.

Fig. 3. Perspective view of the tilting test on a wedge model

3. Model Tests
A model testing program was undertaken to check the validity of the limiting
equilibrium method to be presented in the next section. The concrete wedge mod-
els were tested under dynamic and static conditions. The static tests were carried
out by considering dry and submerged conditions. The tests were brie¯y described
in the following sections.

3.1 Static Tests

The friction angle between the concrete base block and the wedge models was
determined from a tilting test, using a portable tilting device. Two concrete blocks,
each having the same composition with the base and wedge models, were put on
each other and the tilting angle at which the upper block slides on the lower block
is assumed as the friction angle (f ˆ 35 ), mobilised along the discontinuities.

3.1.1 Dry Tests

A portable tilting test device (Aydan et al., 1995) was used for the stability tests on
the model wedges. After ®xing the base blocks on the tilting test device, the wedge
block models were assembled (Fig. 3). Then the base platform was gradually tilted
by keeping the tilting speed as slow as possible. The inclination angle of each
wedge block during sliding was recorded by means of an inclinometer. Each
wedge block was tested three times to obtain the average inclination angle
recorded during the sliding. For each wedge type, three wedge models were tested.
The total number of static-dry wedge tests was 54.

3.1.2 Submerged Tests

The stability assessment of the wedge blocks was also carried out under submerged
condition. The wedge base and wedge blocks were put in a water tank, which was
 Dynamic and Static Stability Assessment 35

Fig. 4. Front view of the submerged stability test of a wedge model

®lled with water up to a level at which all the model blocks were fully submerged
(Fig. 4). The tilting test device was again used to measure the inclination angle of
the base of the wedge models under submerged condition. Each wedge model was
tested three times. For each wedge type, three concrete wedge models were tested
and the average of the inclination angles at failure was assumed as a critical failure
inclination angle for each wedge model.

3.2 Dynamic Tests

Rock slope failures may also occur under the in¯uence of tectonic events in a
region. Heavily jointed rock slopes may indicate that the region was subjected to
active tectonic events in the past. If the tectonic events still continue from time to
time, the slopes will be subjected to seismic forces. Even an engineering structure
may not fail because of an earthquake shock; it may collapse as a result of a large
scale wedge failure which is caused by the same earthquake shock. Dynamic forces
can also result from blasting, tra½c and machinery work near slopes. The in¯u-
ence of the seismic forces decreases the slope stability. Therefore, it is also an im-
portant task to investigate the dynamic stability of slopes.
Dynamic testing of the wedge models were performed in the laboratory by
means of a one-dimensional shaking table, which moves along horizontal plane.
The wave forms of the shaking table are sinusoidal, saw tooth, rectangular, tra-
pezoidal and triangle. The shaking table has a square shape with 1m side length.
The frequency of the wave to be applicable to the shaking table can range between
1 Hz and 50 Hz. The table has a maximum stroke of 100 mm, a maximum accel-
eration of 6 m/s 2 and a maximum load of 980.7 N.
Each wedge base block was ®xed on the shaking table to receive same shaking
with the shaking table during the dynamic test. The accelerations acting on the
36 H. Kumsar et al.

Fig. 5. A close up view from the shaking table during the experiment

shaking table, the base and wedge blocks were recorded during the experiment,
and saved on a data ®le as digital data (Fig. 5). The reason for recording accele-
rations at three di¨erent locations during the experiments is to determine the ac-
celeration at the moment of failure as well as any ampli®cations from the base to
the top of the block. In fact, when the amplitude of input acceleration wave is
increased, there must be a sudden decrease on the wedge block acceleration
records during the wedge failure, while the others should be increasing.
The accelerographs may not completely record the acceleration data due to
their insu½cient sampling time of a record. Therefore, incomplete records may
lead the investigator to misinterpret the acceleration at the time of failure. In Fig.
6 the acceleration records of shaking table, base and wedge blocks are shown for
the dynamic stability test on the model numbered as TB1, as mentioned in Table
1. The wedge block failed just after 9th second, as indicated in Fig. 6. The accel-
eration peaks of the shaking table and base blocks were incomplete at the moment
of failure due to insu½cient sampling time of the record. Finally, the failure
moment is detected and the acceleration during failure is obtained by comparing
the acceleration record graphs drawn for the shaking table, the base and the wedge
blocks.

4. Limit Equilibrium Method

The approach proposed by KovaÂri and Fritz (1975) for the analysis of wedge
failures is used and modi®ed to take into account the possible loading conditions
considered herein. The forces acting on a wedge block can be illustrated as shown
in Fig. 7. If the force equilibrium in the directions of s, n and t is considered, the
following equations are obtained.
 Dynamic and Static Stability Assessment 37

Fig. 6a±c. Example records of acceleration of the dynamic stability test on the wedge block numbered
TB1C: a records on the shaking table; b records on the base block; c records on the wedge block

X
Fs ˆ W sin ia ‡ E cos…ia ‡ b† ÿ Us cos ia ‡ Ut sin ia ÿ S ˆ 0; …1†
X
Fn ˆ W cos ia ÿ E sin…ia ‡ b† ‡ Us sin ia ‡ Ut cos ia ÿ N ˆ 0; …2†
X
Ft ˆ ÿ…N1 ‡ aUb1 † cos o1 ‡ …N2 ‡ aUb2 † cos o2 : …3†
Where
N ˆ …N1 ‡ aUb1 † sin o1 ‡ …N2 ‡ aUb2 † sin o2 …4†
and ia is the plunge of the intersection line and a is Biot's coe½cient. The weight of
the wedge block can be written as
W ˆ …1 ÿ n†Wbr ‡ nWbw ; …5†
where n is porosity, Wbr is the weight of solid phase of the wedge block without
regarding porosity, and Wbw is the weight of water contained in wedge block, S is
shear force, N is the normal force acting perpendicular to the line of intersection in
a plane, and E is the dynamic force, b and ia are the inclinations of the dynamic
force E and the line of intersection in turn. Us and Ut are the water forces acting
on the face and the upper part of the slope, respectively. x1 and x2 are the incli-
nations of the normal forces N1 and N2 acting normal to the surfaces A and B,
38 H. Kumsar et al.

Line of intersection

Failure
direction

Fig. 7. Forces acting on a wedge block

respectively, o1 and o2 are the angles between the surface A and vertical and
surface B and vertical in turn. The shear force components acting normal to the
line of intersection on the surfaces of A and B are ignored in the Eqs. (2) and (3).
This assumption is made by considering the fact that the movement on the normal
direction to the line of intersection can be ignored. The only accelerations with the
trend of the line of intersections are regarded. The sum of normal forces on plane 1
and plane 2 can be obtained from Eqs. (2) and (3) as

N1 ‡ N2 ˆ ‰W cos ia ÿ E sin…ia ‡ b† ‡ Us sin ia ‡ Ut cos ia Šl ÿ aUb ; …6†

where
Ub ˆ Ub1 ‡ Ub2 …7†
and l is called the wedge factor by KovaÂri and Fritz (1975) as
cos o1 ‡ cos o2
lˆ : …8†
sin…o1 ‡ o2 †
 Dynamic and Static Stability Assessment 39

Let assume that the failure planes obey Mohr-Coulomb yield criterion as given
below:

T ˆ …N1 ‡ N2 † tan f ‡ c…A1 ‡ A2 †; …9†

where c is cohesion, f is friction angle, A1 and A2 are the area of plane 1 and plane
2, respectively.
The relationship between the dynamic force E, weight of the wedge block W,
and the seismic coe½cient h can be written as follow:

E ˆ hW …10†

and the factor of safety (SF) is given by the following expression,

T
SF ˆ …11†
S
Using Eqs. (1) to (4), the following expression for the factor of safety (SF) is
obtained

‰l‰W …cos ia ÿ h sin…ia ‡ b†† ‡ Us sin ia ‡ Ut cos ia Š ÿ aUb Štan f ‡ c…A1 ‡ A2 †

SF ˆ :
W …sin ia ‡ h cos…ia ‡ b†† ÿ Us cos ia ‡ Ut sin ia
…12†

5. Special Conditions
Wedge slopes fail under the in¯uence of various disturbing e¨ects, such as seismic
forces, surface loads, increasing pore pressure, water pressure in totally and par-
tially submerged states, excavation at the toe in static dry case, etc. In the follow-
ing sections, limiting equilibrium equations for wedge blocks in dry and sub-
merged conditions without seismic loading, and dry case with seismic loading
conditions are given.

5.1 Dry Condition with no Seismic Loading

Assuming that h ˆ 0, Us , Ut , Ub1 , Ub2 ˆ 0, and c ˆ 0, the following expression for
the safety factor is obtained:

cos ia tan f
SF ˆ l : …13†
sin ia

Choosing SF ˆ 1, the apparent friction angle f  due to the geometric con®gura-

tion of the wedge is as follows:

f  ˆ tanÿ1 …l tan f†: …14†

The maximum wedging e¨ect is obtained when o1 ˆ o2 ˆ o.

40 H. Kumsar et al.

5.2 Submerged Condition with no Seismic Loading

The e¨ect of water has a vital importance on the behaviour of soils and rock
masses on which engineering structures are built or excavated. Terzaghi (1925)
introduced a concept of e¨ective stress in order to express the mechanical e¨ect of
water on the behaviour of soils. This concept was also applied to rocks after that.
Biot (1942), on the other hand, included the volumetric porosity n, and the ratio of
the sti¨ness of the solid Ks and that of the bulk modulus K in his concept for
general e¨ective stress law.
Karaca et al. (1995) carried out a series of tests to clarify the applicability of
the e¨ective stress law for rock discontinuities. These experiments showed that
Terzaghi-type e¨ective stress law is generally applicable to throughgoing dis-
continuities. The factor of safety of a wedge block subjected to water forces in a
submerged case can be written as:

‰W cos ia ‡ Us sin ia ‡ Ut cos ia ÿ a…Ub1 sin o1 ‡ Ub2 sin o2 †Štan f

SF ˆ l: : …15†
W sin ia ÿ Us cos ia ‡ Ut sin ia

If a is chosen as 1, which corresponds to Terzaghi-type e¨ective stress law, c ˆ 0,

the resulting equation by taking into account the geometry of the block will be
obtained as
f  ˆ tanÿ1 …l tan f†: …16†

This implies that the apparent friction angle of the block should be the same as
both under dry and submerged conditions unless there is a chemical reaction
between rock and water along sliding planes.

5.3 Dry Condition With Seismic Loading

Assuming that Us , Ut , Ub1 , Ub2 ˆ 0, and c ˆ 0, the equation of the safety factor
given below is obtained:
…cos ia ÿ h sin…ia ‡ b†† tan f
SF ˆ l : …17†
sin ia ‡ h cos…ia ‡ b†

Choosing SF ˆ 1, the seismic coe½cient h, at which the block slides, is as follows:

l cos ia tan f ÿ sin ia
hˆ : …18†
cos…ia ‡ b† ‡ l sin…ia ‡ b† tan f

6. Comparison and Discussion of the Experimental and Theoretical Results

The test results for assessing the stability of the wedge models under dynamic
loading, dry static and submerged static states were compared with the theoretical
predictions by using the presented limiting equilibrium method. For each wedge
geometry, nine experimental results are used.
 Dynamic and Static Stability Assessment 41

Fig. 8. Comparison of the dry-static model test results with theoretical solutions

In the case of dry static test, the plunge of the line of intersection during the
failure of each wedge model is measured. This angle has two components: the
plunge of the model (ia ) before tilting test, and the tilting angle of the model wedge
surface at failure during tilting test, respectively. The sum of these two angles is
called the plunge of the wedge model during the sliding in dry static test. Fig. 8
compares the theoretical predictions with the experimental results. From this
comparison it can be stated that the limiting equilibrium method is generally valid
when the strength of a discontinuity is purely frictional (c ˆ 0).
The comparison of the submerged static test results of the wedge blocks and
the theoretical estimation of the plunge during the failure of the wedge block for
di¨erent values of the friction angle (f) and zero cohesion (c), indicate that there is
not a considerable di¨erence among the failure intersection angles under static
loading conditions (Fig. 9). This con®rms that water pressure does not have an
important in¯uence on the stability of wedge models in submerged and c ˆ 0
conditions, and Terzaghi-type e¨ective stress law is applicable to rock discon-
tinuities. However, if there is a chemical deterioration (such as softening of rock)
on discontinuity surfaces due to the presence of water, the stability of a wedge
block below water table may be in¯uenced.
The results of the dynamic stability tests obtained from the wedge models were
compared with the theoretical estimations (Fig. 10). The theoretical estimations
were done by using Eq. (18) for the plunge of the line of intersection of each type
of wedge model. The friction angle of the discontinuity surfaces, which was
obtained from the tilting test, is taken 35 . The acceleration, that is required to
make the wedge block unstable, is assessed for various half wedge angles. As seen
from Fig. 10, the results of the dynamic stability tests on the wedge blocks are in a
good agreement with those obtained from the presented limiting equilibrium
method. The scattering in Fig. 10 may be attributed to the possible experimental
errors in sample preparation and to the smoothness of the surfaces. It is suggested
42 H. Kumsar et al.

Fig. 9. Comparison of the submerged-static model test results with theoretical solutions

Fig. 10. Comparison of the dynamic model test results with theoretical solutions

that this method can be employed for the stability assessment of the slopes located
in earthquake-prone regions.

7. Case Studies
A series of stability assessments of failed and stable wedge cases from Turkey and
Japan were analysed by using the presented limiting equilibrium method. The
cases from Turkey are, wedge failures in a very high bench at a strip coal mine, in
an open museum in the Central Anatolia, in a quarry in Ankara Castle, and in an
earthquake prone-region in western Turkey. A wedge failure in an active volcanic
 Dynamic and Static Stability Assessment 43

mountain in Japan called Mayuyama was also assessed to check the validity of the
presented method in this study.

7.1 Case 1: Wedge Failure at a Strip Coal Mine

One of the failures, which a¨ected the operations at Eskihisar strip coal mine, SW
Turkey, occurred in the form of a wedge failure. The wedge resulted from the in-
tersection of a fault plane and a highly continuous joint (Fig. 11a). Tension cracks
of large extent behind the crest after this failure suggested that the movement into
the void was still continuing and a monitoring study was initiated. Then a second
failure occurred.
On the basis of kinematical analysis of the slope (Fig. 11b) and the back
analysis of the failure using the laboratory determined residual strength parame-
ters of both discontinuities, it was concluded that the mode of failure of the pre-
vious major instability was wedge type, involving the fault and joint set numbered
4 (Ulusay, 1991). In this study, the geometrical parameters from the stereonet
projection, and the weight of this wedge were calculated (Fig. 11a). The shear
strength parameters of the fault plane were accepted to be mobilised and dropped
to residual values during the failure (Fig. 11b). The back analysis yielded a safety
factor of 0.93, which validates the limiting equilibrium method presented in this
study. Monitoring station movement directions plotted (Fig. 11b) generally cor-
respond to the trend of the line of intersection of the fault and the joint 1. This
situation indicated that the wedge formed by I^f , j 1 has been activated towards the
void created by the previous failure.

7.2 Case 2: A Wedge Failure in an Open Museum

A wedge failure in a thick and soft tu¨ stratum was observed by the authors in
Zelve Open Museum in Cappadocia Region in Central Anatolia. Two joints,
intersecting each other, had slightly rough surfaces and formed a tetrahedral
wedge (Fig. 12a).
Determination of the internal friction angle of these slightly rough joint sur-
faces was carried out by employing the criterion proposed by Barton and Choubey
(1977). For this purpose, Schmidt Hammer rebound number and JRC value of the
surfaces were determined, and the laboratory test results by Ulusay et al. (1997)
carried out on these tu¨s were also considered. Calculations yielded an internal
friction angle of 30 . The kinematical analysis of the wedge with its geometrical
parameters is given in Fig. 12b. The analysis of the failure using the presented
method resulted in a factor of safety less than unity, which emphasises that the
wedge block has already failed.

7.3 Case 3: Wedge Failure near Ankara Castle

A very old wedge failure in a jointed andesite rock mass near Ankara Castle in
Bent Deresi region of Ankara City was investigated by the authors. Two discon-
44 H. Kumsar et al.

Fig. 11a,b. Case 1: a general view of the wedge failure and the geometrical characteristics of the wedge;
b kinematical analysis of possible lines of intersection and plots of monitoring data for Case 1 and shear
strength parameters of the discontinuities involved by the wedge (after Ulusay, 1991)

tinuity surfaces dipping to each other formed a wedge surface within andesite
(Figs. 13a and b). On the top of this wedge failure there is a house. The line of
intersection, which was obtained from the kinematic analysis of the wedge failure
shown in Fig. 13c, is greater than the friction angle and smaller than the slope
angle.
 Dynamic and Static Stability Assessment 45

(a) (b)

Fig. 12a,b. Case 2: a a wedge failure in Zelve at Cappadocia; b its kinematical analysis and geometrical
characteristics

Under these conditions the wedge block is expected to be unstable. In the

analysis, a friction angle of 30 determined from laboratory tests on joint samples
by Ercanoglu (1997) was used. The stability assessment of the wedge block under
dry-static condition was carried out by using the suggested method in this paper.
The safety factor of the slope was found as 0.73, which clari®es that the slope has
already failed.

7.4 Case 4: Wedge Failure at an Earthquake-prone Area

The town of Dinar is located at an earthquake-prone area in the western Turkey.
An earthquake with a magnitude of 6.0 occurred in Dinar and its vicinity on
October 1, 1995. During this disaster 90 people lost their lives and many houses
collapsed and were damaged.
After the earthquake, ®eld studies were carried out and a wedge failure was
observed northeast of Kizilli Village near Dinar. This failure is located at the fault
zone of Dinar (Fig. 14a). A kinematical analysis was done to determine the nec-
essary wedge parameters for the stability assessment of the wedge block (Fig.
14b). The internal friction angle of the joint surfaces was determined by perform-
ing a simple tilting test in the ®eld and found as 41 .
A back analysis of the wedge failure, using the proposed method, was carried
out to check the in¯uence of the seismic force due to Dinar earthquake on the
46 H. Kumsar et al.

(a) (b)

(c)

Fig. 13a±c. Case 3: a view of the wedge failure near Ankara Castle; b close up view from the wedge
near Ankara Castle; c the kinematical analysis and the geometrical characteristics of the wedge

stability of the wedge block by considering static conditions. A factor of safety of

2.02 was obtained, which indicates that the wedge block is stable under static
conditions. Therefore, a second analysis which takes dynamic conditions into
account was performed by using the amax value in EW direction (Aydan and
Kumsar, 1997), given in Fig. 14. The value of the safety factor for this direction
was 0.99. This result revealed that the slope instability was initiated when the
earthquake occurred on October 1, 1995.

7.5 Case 5: Wedge Failure of Mt. Mayuyama in 1792

Mt. Mayuyama in Japan failed following an earthquake about 8 pm and resulted
in the loss of 15000 people during a volcanic activity of Unzen volcanos in 1792. A
slope failure also occurred during this disaster. The authors carried out a series of
back analyses considering di¨erent conditions to discuss the mechanism and the
model of this wedge type slope failure.
 Dynamic and Static Stability Assessment 47

(a)

(b)

Fig. 14a,b. Case 4: a the wedge failure occurred northeast of Kõzõllõ Village (Dinar) after the 1995
earthquake; b the kinematical analysis and the geometrical characteristics of the wedge

There are three di¨erent explanations of the failure of Mt. Mayuyama

(Misawa et al., 1993):
1. Failure due to earthquake,
2. Failure due to gas pressure of heated ground water, and
3. Failure due to liquefaction of an underlying sandy layer.
Four cross-sections of the failed body are given in Fig. 15. From these sections
it was concluded that there are three distinct failure planes, namely J1 , J2 and J3 ,
and their projections on a lower hemisphere Schmidt net and wedge parameters
for the stability assessment of the wedge block are given in Fig. 16.
48 H. Kumsar et al.

Fig. 15. Present topography and cross-sections of the close vicinity of Mt. Mayuyama (after Misawa
et al., 1993)

Fig. 16. Kinematical analyses of the wedge blocks at Mt. Mayuyama and the geometrical character-
istics of the wedge

The planes J1 and J2 are the planes which should be developed long time
(Ohta, 1987). However, the plane J3 was considered to have developed during the
volcanic activity and associated earthquake in 1792, since there is thin vegetation
 Dynamic and Static Stability Assessment 49

Fig. 17. Comparison of di¨erent case results for the wedge failure at Mt. Mayuyama

on this surface (Misawa et al., 1993). There are two possible wedge failures as
shown in Fig. 16. Since the dip of the intersection of the planes J1 and J2 is greater
than the slope angle, the wedge failure along these discontinuity planes is not
kinematically possible. On the other hand the wedge failure on planes J1 and J3 is
the most likely as the plunge of the intersection line daylights on the slope face.
The geometrical parameters of the wedge were determined and given in Fig. 16.
A series of stability calculations were carried out by considering four di¨erent
conditions (Fig. 17):
Condition 1: Rock mass is dry and earthquake force is only applied by increasing
h from 0.0 to 0.5 with c ˆ 0; Us ˆ 0; Ut ˆ 0; Ub ˆ 0; a ˆ 1; b ˆ 0.
Condition 2: Rock mass is assumed to be subjected to only uplift ¯uid pressure
acting on the discontinuity planes with c ˆ 0; Us ˆ 0; Ut ˆ 0; a ˆ 1;
h ˆ 0. The ¯uid pressure was assumed to consist of two parts: static
¯uid pressure (Ubs ˆ gs W ) and excess ¯uid pressure (Ube ˆ ge W ).
The static ¯uid pressure coe½cient (gs ) is assumed to correspond to a
fully saturated situation with a value of 0.4. The value of the excess
¯uid pressure coe½cient (ge ) was increased from 0.0 to 0.5.
Condition 3: Rock mass is assumed to be subjected to static ¯uid pressure and the
earthquake force with c ˆ 0; Us ˆ 0; Ut ˆ 0; a ˆ 1. The ¯uid pres-
sure was kept constant during the increase of the earthquake force.
Condition 4: Rock mass is assumed to be subjected to both the ¯uid pressure and
earthquake force c ˆ 0; Us ˆ 0; Ut ˆ 0; a ˆ 1. The initial value of
¯uid pressure is equal to its static value. Then, excess ¯uid pressure
and earthquake force increased by the same amount.
The results of the calculations for Condition 1 indicate that the seismic intensity
(h) should be 0.36 to initiate the failure. Considering the magnitude of the earth-
50 H. Kumsar et al.

quake at the time of failure, it is not possible that the mountain can fail without
the e¨ect of ¯uid pressure.
The mountain should be stable under the fully saturated condition according
to the calculation results for Condition 2. However, if the excess pressure is to
occur, the mountain failure could be initiated when the excess ¯uid pressure is 0.12
times the dead weight of the wedge body. If the excess ¯uid pressure is due to the
heating of groundwater by the magma, the mountain failure is possible.
If the rock mass is fully saturated without generation of excess ¯uid pressure
(Condition 3), the increase of seismic intensity h can also cause the failure of the
mountain for a value of 0.08. If the Condition 4 is considered, the failure of the
mountain can occur for a value of 0.05.
At the time of failure, an earthquake did happen and the hot water did spout
from the ground. Therefore, it is most likely that the forces due to the earthquake
and the ¯uid pressure caused by the gravity, shaking and heating of the ground-
water could be the main reasons for the failure of the mountain. Nevertheless, it
should be kept in mind that the observation of the hot water spewing out of the
ground near the coast may also be caused by the energy dissipation released
during the frictional sliding of the failing mountain.

8. Conclusions
The stability assessments of the wedge models were performed under dynamic and
static loading conditions. While dynamic tests were carried out by using one di-
mensional shaking table under dry condition; the static tests were performed under
two di¨erent conditions; namely dry and submerged. A portable tilting machine
was used for the static tests. In submerged tests the base and wedge blocks were
put in a water tank, which was ®lled with water. The experimental results were
compared with the theoretical estimations. In static tests, the calculated results are
in good agreement with the experimental results for static and dynamic states
when the shearing forces between the discontinuity surface were controlled by
friction only. There is not a great deal of in¯uence of water pressure in submerged
condition under static loading conditions. This proves that Terzaghi-type of ef-
fective stress law is applicable to rock discontinuities. The estimated accelerations
for F ˆ 1 condition by using the presented limit equilibrium method are also close
to the accelerations obtained from the dynamic tests. Therefore, the presented
method can also be applied for the dynamic stability assessment of wedge failures.
The stability assessment of the wedge cases was carried out under static-dry
condition, and the stability conditions of the wedges were determined by using the
presented limiting equilibrium method. A back analysis of a wedge failure at an
earthquake-prone area showed that the wedge block was stable under static state.
When a seismic force, resulting from an earthquake shock occurred in Dinar in
Turkey in October 1995, was introduced, the slope lost its stability. Four di¨erent
conditions were considered in a case study of a wedge failure in an active volcanic
mountain. The main reasons for the instability of the wedge were estimated as the
earthquake and the ¯uid pressure caused by the gravity, shaking and heating of
 Dynamic and Static Stability Assessment 51

the groundwater. These case studies also show the validity of the limiting equilib-
rium approach.

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Authors' address: Dr. Halil Kumsar, Pamukkale University, Department of Geological

Engineering, Kinikli Kampusu, TR-20017 Denizli, Turkey.