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					             ENV-2E1Y Fluvial Geomorphology

                                     2004 - 2005

Multiple Landlsides at Yuen Mo Village, Kowloon East during the rain storm of 29 - 31st May 1982
when over 530 mm of rain fell. The collapse occurred in the late morning of 30 th May and most of
the huts in the village were destroyed or severely damaged. Three people were killed. At 16:15,
the site was inspected by Emergency Duty Officer, N. K. Tovey who had previously inspected 4
other landslides in neighbouring villages, each one of which involved deaths. All remaining huts
were condemed by Dr N.K. Tovey and a permanent evacuation order on all 120 inhabitants of
Yuen Mo was issued. From that time Yuen Mo Village ceased to exist.

                        Slopes and related topics

                        Section 5 Slope Stability
N. K. Tovey                ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                 Section 5

                                   Slope Stability and Related Topics

                                                5. Slope Stability
5.1. Introduction                                                 Assuming we know the geometry of the slope and the
                                                                  underlying strata, the relevant material properties, and
The stability of slopes and whether or not massive failure        we also understand the water flow, then all methods of
in the form of landslides occurs is dependent on several          analysis begin with postulating a failure mechanism.
factors as were described in the introduction to this
course. These may be summarised as:-                              It is essential that we correctly identify the most critical
                                                                  mechanism, and this usually is a matter of experience.
the geometry of the slope including the geometric                 In the past, some slopes have been analysed and given a
configuration of the varying strata - determined by               clean bill of health, but as a less than critical failure
surveying methods,                                                mechanism was identified failures have occurred on
water flow within the slope - analysed using techniques           "theoretically" stable slopes sometimes with potentially
covered in section (2) of this course,                            disastrous consequences (e.g. the Tsing Yi, Hong Kong
the material properties of the differing strata, including        failures above the PEPCO oil storage depot following the
the unit weight angle of friction and cohesion, which are         rainstorm of 29th - 31st May 1982).
in turn dependent on the previous consolidation history
of the soil,                                                      5.2. Types of failure
additional loading by man.
                                                                  Failures in slopes may:-
There are several methods by which the stability of a
slope may be analysed, many are valid only under certain                    1) be straight lines (particularly so in granular
conditions. There is also a group of more general                              media)
solutions which can be applicable in all cases, but                         2) approximate to arcs of circles
sometimes it is difficult to find a solution even with the                  3) approximate to logarithmic spirals
aid of a computer.                                                          4) be a combination of straight lines, arcs of
                                                                               circles, and/or logarithmic spirals.

                                                                  Examples are shown in Fig. 5.1

Fig. 5.1 Examples of different methods for analysing stability of slopes.

a)   example applicable for a purely cohesive soil where          c)   Infinite slope where slope is of approximately
     slip surface is a circle;                                         constant slope over a significant distance which is
                                                                       much greater than depth to bedrock. The failure
b) example applicable for soil with both cohesion and                  surface is also parallel to this slope. Water flow can
   friction, but water flow must be absent. Failure is a               be included if it is parallel to slope as can changes in
   straight line;                                                      strata;

N. K. Tovey                  ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                Section 5

d) general case for slope of general shape. Slip surface            Strain is defined as a non-dimensional ratio which is the
   may be of any form and may be composite including                displacement during shearing over the original length of
   arcs or circles and straight line portions. In the               the sample. In a triaxial test, the sample is usually 75
   example shown, there is a straight line section                  mm long and a deformation of about 1 - 2 mm is
   which is along the bedrock plane. Water flow can                 needed to achieve peak strength in a dense test
   be incorporated as can variation in strata and the               representing 1 - 3% strain. For loose samples, the
   presence of tension cracks.                                      deformation will be around 10 mm in a sample of
                                                                    comparable length.
5.3. Progressive failure.
                                                                    In a slope the soil mass is large, and it is not possible
Most slope failures occur during or immediately after               for the whole slope to deform (with the associated
periods of heavy rain when the water table is high. Thus            volume change instantaneously. Near the base of the
slope failures on the North Norfolk Coast are more                  slope, the material can expand and bulge slightly and
common in winter during times of high water table.                  allow small strains along the potential failure plane as
Equally, movement of the Mam Tor Landslide in                       shown by the shaded region in Fig. 5.2.
Derbyshire occurs during the winter months and usually
only if more than 400 mm of rain falls in the critical              The corresponding point on the stress - strain diagram
period. In Hong Kong, landslides are rare in the winter             is shown at point A on the rising part of the curve.
months from November to March, and are very common                  Further around the failure zone, the points B and C
in the summer months (May - August) and over 500                    have low amount of strain on the stress - strain plot,
landslides occurring in a single day have been reported.            and the mobilised shear strength is thus small.

Slopes may be triggered by rainfall,              and may
catastrophically fail if the rainstorm is prolonged (e.g. Po
Shan Road, Hong Kong, 1972), but not infrequently,
the failure is progressive with small amounts of
movement until eventually the failure is catastrophic in a
particular event (e.g. Aberfan, Tsing Yi). After massive
and catastrophic failure, continued movement may take
place (e.g. Mam Tor).

                                                                    Fig. 5.3 Failure is now more advanced. The most
                                                                             highly stressed region has just passed peak
                                                                             shear strength, while at B, the strength is
                                                                             approaching peak.        Region C is still
                                                                             relatively lightly stresses. Bulging at toe
                                                                             might noticeable in aerial photographs and
                                                                             might be visible to naked eye.

                                                                    Once the lower part of the slope has deformed, the next
Fig. 5.2    Region of high stress in a slope prior to               part can deform (see Fig. 5.3. Here, the lowest part of
           failure.                                                 the failure zone is at the peak shear strength, while
                                                                    moderate strengths have been mobilised further along
The stress-strain diagram indicates the approximate                 the failure arc. The stress points corresponding to
states of stress at points along the potential failure zone.        points B and C have moved further up the curve.
Slight bulging of the toe would be discernible with
accurate survey measurements.                                       Finally (Fig. 5.4), after further deformation, bulging
                                                                    should become very evident at the base of the slope
Unlike materials such as steel which show relatively                while the stress at the point A will now be less than
little deformation before failure, soils deform by a                previously as it has past the peak strength while the
considerable amount before the peak shear strength is               stress at B is now at peak and that at C is rising rapidly.
achieve in the case of dense sand or over consolidated
clays. For loose sands and normally consolidated clays,             All along the failure arc, the mobilising shear stress
the deformation before the ultimate strength is reached             will vary and it is the integrated value of the strength
is large.                                                           along the whole failure surface which will determine

N. K. Tovey                 ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                Section 5

whether or not failure will occur. Such a failure which                   failure envelope on the Mohr - Coulomb
develops in this fashion is called a progressive failure.                 envelope is a constant irrespective of normal
                                                                          stress (i.e.  = 0).
Of importance is the fact that there will be a time delay
(albeit quite short in some cases) from the start of the                  This method can be used for any slope profile,
failure to the time of catastrophic failure. Normally,                    but is more suited to simple shapes. Water flow
evidence of failure may be detected from bulging of the                   must be absent, i.e. excess water pressures,
toe (early stage) and the development of tension cracks                   although the method may be use for slopes
at the top (later stage), and finally a settlement of the                 which are entirely submerged. Only single
crest immediately prior to failure.                                       strata must be present.

                                                                      2) those methods where the potential failure
                                                                         surface approximates to a straight line. It is
                                                                         valid for solids with both friction and cohesion,
                                                                         but there must be no water flow. The analysis
                                                                         is possible for irregular shaped surfaces, but it
                                                                         is more usually used for simple shapes. It is
                                                                         not really suited if there is more than one

                                                                      3) those methods for the analysis of slopes which
                                                                         are approximately infinite in extent compared
                                                                         to the depth of the soil material. The angle of
                                                                         the slope is approximately constant over a
                                                                         large distance. Differing strata may be present,
                                                                         but only parallel to the surface.

                                                                           The method can deal with water flow provided
                                                                           that it is parallel to the surface. Both frictional
Fig. 5.4.    Whole of potential failure surface is now                     and cohesive materials may be present. This
            highly stressed with region A well beyond                      method of analysis is known as the Infinite
            peak strength at the residual strength, region                 Slope Method.
            B at peak strength and region C approaching
            peak strength. Noticeable bulging at the toe              4) those methods which are applicable to the
            which should be seen by naked eye. Failure                   analysis of general slope stability. They are
            is imminent.                                                 valid for varying ground water flow
                                                                         conditions, for various modes of failure
                                                                         (straight - line, arcs of circles or various
Though the circumstances leading up to the Aberfan                       combinations), for slopes with varying strata
disaster on October 21st 1966 were contributory to the                   which may or may not be parallel to the failure
disaster. There were many signs in the months and                        surface or slope surface, and for slopes in
years before that a potential disaster that a disaster                   which tension cracks have developed from
might occur, the consequences of the disaster could                      desiccation of the surface layers.
have been avoided even at a late stage. Two people
who were working at the top of the waste tip about 30                      These methods are collectively known as the
minutes before the disaster noted tension cracks and a                     Method of Slices, and there are several
settlement at the top. In vain they attempted to raise the                 variants depending on the extent of
alarm, but vandals had removed the wires of the                            approximations made. Generally speaking all
communication telephones.                                                  assumptions are SAFE ASSUMPTIONS in
                                                                           that they underestimate the stability of the
5.4. Methods of analysis                                                   slope.

There are many methods available for studying the                 In the case of the Infinite slope method, the failure
stability of slopes, and for some there are several               surface will always be parallel to the surface, and for a
variants. In this course we shall consider 4 basic                single stratum it can be shown (see section 5.7), that the
methods:-                                                         stability is unaffected by the depth of the potential
                                                                  failure surface.
    1) those methods for relatively shallow slopes in
       normally consolidated or lightly over                      For all other methods of analysis, the method first
       consolidated materials and in which the soil               assumes a failure surface of appropriate shape and
       material may be considered to be purely                    analyses the stability to obtain a factor of safety (Fs)
       cohesive ( and undrained) situations (i.e. the
N. K. Tovey                  ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                  Section 5

                  inherentstrengthof the soil
         Fs                                                         This cohesive force will be the resisting force, while
                 strengthrequired for stability
                                                                     the weight of the slope acting through the centre of
                                                                     gravity of the potential siding segment will be the
If the computed factor of safety is less than unity, the
                                                                     mobilising force. In this case since the cohesive force
slope is clearly unstable and likely to fail. If the factor
                                                                     and the weight do not act at a point we must also
of safety is greater than unity we cannot assume that the
                                                                     consider the moments of the forces in assessments of
slope is stable as we may not have chosen the most
                                                                     equilibrium, i.e. since the centre of gravity is not below
critical mode of failure (i.e. failure surface). It is thus
                                                                     the centre of the circle of failure, the weight will act as
necessary to repeat the calculations with a different
                                                                     a pendulum and attempt to cause the segment to rotate.
failure surface until the most critical one is found.
                                                                     This tendency to move is counteracted by the cohesive
Usually, experience will give a guidance as to what
types of failure are likely to be more critical than
                                                                     The assess the moments of the two key forces we need
                                                                     to determine the distances of their respective lines of
In the case of the straight line failures (other than the
                                                                     action from the centre of the circle in a direction at right
infinite slope cases), it is possible to assess the stability
                                                                     angles to the force. The weight acts at a distance X
on a number of different failure planes each one at a
                                                                     from the point O, while the cohesion acts along a
different angle to the horizontal. A graph such as Fig.
                                                                     tangent (i.e. at the radius of the slip surface).
5.5 is now plotted with the angle of the potential failure
surface as the X - axis, and the computed Factor of
Safety as the Y - axis. The value of Fs will be high for
shallow angles, and fall as increases. After a critical
angle, the value of s will rise again as continues to
increase. In this example, the critical value of Fs is
clearly the minimum of the curve.

A similar approach may be used for the purely cohesive
failures. In this case, the it is the radius of curvature of
the failure arc which is used as the independent variable
on the X - axis.

                                                                     Fig. 5.6    Failure of a slope in a purely cohesive
                                                                                medium. This failure surface is an arc of a
                                                                                circle. It is necessary to determine the
                                                                                distances X, R and L for analysis.

                                                                                                   Restoring Moment
                                                                     the factor of safety =      -------------------------------
                                                                      is given by:-                Mobilising Moment

                                                                                          =                          ........5.1
Fig. 5.5. Variation of factor of safety with orientation             Though this is a simple method of analysis there are
          of failure zone to horizontal in a straight line           some practical problems which require some ingenuity
          failure.                                                   to solve. A question similar to this was set as an exam
                                                                     question a few years ago.
5.5     Method of Analysis - I
      - purely cohesive failures.                                    One problem is to determine the weight of the sliding
                                                                     mass:- this is equal to the area of the slice multiplied
Fig. 5.6 illustrates this type of failure. The failure               by the unit weight. The area could be computed by
surface is an arc of a circle with centre at point O.                drawing the sliding mass on graph paper and counting
There is no frictional component in the resistance to                squares, or it can be derived from geometry (rather
failure and pure cohesion is developed along the failure             more complex!). In the case in question, since this was
arc. If the cohesion is c kPa, then the total cohesive               a 40 minutes question (rather than the 60 minute
force will be c .  . 1 (the 1 comes from unit distance              question now set), the area of the wedge was given.
at right angles to the plane of the paper).
N. K. Tovey                 ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                    Section 5

This still left the question of the position of the centre                b)    use a ruler and approximately rotate it around
of gravity, the position of the centre of the sliding                            following the shape of the curve and read off
circle, and the length of the sliding arc to determine.                          the length.

In the question, candidates were given a cardboard                        c)    adopt the method used by one candidate who
template of the exact shape of the sliding wedge, and a                          pulled out a strand of her hair and laid this to
drawing pin on which to balance to wedge. So the                                 follow the circle. Finally, the relevant length
solution to the question began with attempting to                                was measured on a ruler!.
balance the wedge on the drawing pin. Once the
approximate centre of gravity had been found, the                      5.6 Method of Analysis - II
template was pricked through so that the diagram on the                      - Straight Line Failures
question paper could be marked with the centre of
gravity.                                                               Fig. 5.8 shows this type of failure and the key forces to
                                                                       be considered.
First join ends of slip circle, then through middle draw
line at right angles. Point D is determined from                       To solve such a problem we need the weight of the
equation 5.2, and hence the centre of the circle can be                sliding wedge. Once again this is derived from the area
found.                                                                 on a scale diagram (i.e. weight is area multiplied by unit
                                                                       weight). In all examples, the wedge is triangular, and
                                                                       the area may be obtained either from:-

                                                                        area = 0.5 x base      x    height
                                                                                                   (i.e. 0.5 b h in this case).

                                                                          or area = 0.5 a b sin C where a and b are the
                                                                       lengths of two sides of the triangle and C is the
                                                                       included angle.

                                                                       In this example, all the forces pass through a single
                                                                       point (i.e. directly below the centre of gravity of the
                                                                       wedge and on the sliding surface). Though reference is
Fig. 5.7 Geometric construction to find centre of circle               made to the centre of gravity, we do not actually have
         accurately when using first method of analysis.               to determine its position in this method of analysis.

                                                                       The weight acting downwards will have two
The centre of the sliding circle can be estimated by trial             components, one acting down the slope, and the other
and error using a pair of compasses, or alternatively a                acting perpendicular to the slope. If the slope is just
geometric theorem can be used. The extreme ends of                     stable then the frictional force will just balance this
the slip circle are connected by a line which is then                  component of the weight acting down the slope and
bisected at right angles (Fig. 5.7), such that AY = BY.                attempting to cause failure. Equally, the component of
These two distances can be measured from the diagram                   the weight acting at right angles will balanced by the
as can the distance CY.                                                normal force N.

Finally, the geometric theorem states that

          DY x CY = AY x BY           ......................5.2

Hence it is possible to determine the distance DY and
then the diameter of the circle CD. Finally, the
distance CD is halved (i.e. OC = OD) to get the centre
of the circle

The length of arc may be determined in one of three

a)    join the lines from the extreme ends of the slip
      circle to the centre of the circle and measure the
      angle. Convert this angle to radians () and
      multiply by the radius of the circle.

        i.e.   L    = R.                                                       Fig. 5.8 Straight line failure of a slope. The
                                                                                key forces are the weight (W), the normal
N. K. Tovey                     ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                   Section 5

         force (N), and the fractional force (F) which all               5.7.1 Dry Cohesionless Slope
         act through a single point.
                                                                           Four forces act on a block of slope (Fig. 5.9):
The governing equations are thus (by resolving parallel
and perpendicular to the slope):-                                                 W = weight of block of depth Z
                                                                                  R = Reaction upwards
       F = W sin                                                                 X1 & X2 are lateral forces on block.
and N = W cos     ........................5.3
                                                                         It is readily shown that X1 and X2 have a common line
For our factor of safety we note that:-                                  of action [from Moments], and resolution parallel to
                                                                         the slope shows that they are equal and opposite. The
               inherent shear strength of soil                           effect of this is that we can effectively disregard these
 Fs =      -------------------------------------------------             in subsequent analysis.
               force which would just resist failure
                      (i..e mobilising force)                               Hence    W = R
                                                                              but    W =  . bzcos 
the mobilising force =      F = W sin                                                                  |
                                                                                                    area of block
Also the restraining force will be the inherent strength
of the soil which can be derived from the Mohr -
Coulomb envelope, i.e.

      cL      +     N tan  ............................5.4
Note we MUST use forces here, and so we are dealing
with the cohesive force (not the stress as defined by the
intrinsic property of the material).

Thus the factor of safety may be specified as:-

              c L  W cos  tan                                         Fig. 5.9 Diagram of a typical slice on an infinite slope
      Fs                                         ............5.5                 showing relevant forces which act. The
                    W sin                                                        reaction force may be further split into two
                                                                                  components one parallel and the other
 For those who are Mathematically inclined:-                                      perpendicular to the slope.

                                                                         Resolving perpendicular to the slope:-
 Note: with this geometry, it is possible to define the
 weight in terms of the slope angle  and the failure
                                                                         Rcos  = N = W cos                 = dzbcos2 
 angle , and also the length (L) of the failure
 surface in terms of  and  Equation 5.5 can thus                       N is the normal force so the normal stress
 be generalised, and it is possible to differentiate                                                       = dzcos2.
 equation 5.5 to get               and hence determine the
                          d                                             Resolving   parallel to the slope
 minimum factor of safety from all possible failure
 angles directly. If you are mathematically inclined                     Rsin  = F = Wsin            = dzcos sin .
 you may wish to check this out.
                                                                         This may be plotted on a Mohr Diagram, remembering
                                                                         that for cohesionless materials c = 0.
5.7 Method of Analysis - III
    - Infinite Slope Method                                              The angle the point subtends at the origin () is known
                                                                         as the angle of obliquity,
This method of analysis assumes that the slope is of
infinite extent, and that any water seepage is parallel to
the slope. There are several cases which can be
considered, some of which will be treated in detail

N. K. Tovey                 ENV-2E1Y: Fluvial Geomorphology 2004– 2005                               Section 5

   Fig. 5.10 Mohr - Coulomb Diagram for dry case
   stability. The point A represents the stress point on a
   plane at depth z below the surface.

                  zcos. sin                                   Fig. 5.11 Evaluation of the pore water pressure in the
i.e.   tan 1  d                1
                  z cos2    tan (tan )  
                                                                             full seepage case. The pore water pressure at
                    d                                                      A is the vertical distance AB.

i.e. in this case the angle of obliquity   () equals the          Thus pore water pressure equals   w x AB.
slope angle  .
                                                                          But AC =    z cos 
If  is increased the point moves along the dashed
line which is part of a circle which also passes through                 so.   AB = AC cos  = z cos2
the origin.
                                                                   and pore water pressure =    w z cos2.
Failure would occur if, or when      reaches the value
.                                                                 However, the pore water pressure only affects normal
                                                                   so new normal stress equals:-
Hence in this case failure occurs when      =  = ,
i.e. the slope angle equals the angle of internal friction.
                                                                        ' =  - u =  z cos2  -  wz cos2 
This latter is sometimes known as the angle of repose
and is normally in the region of 30o - 40o for granular            [Note: We must also change the unit weight from its
media. Its value is higher for denser sands and lower for                 dry value to its saturated value.]
loose sands.
                                                                   Thus  ' = ( - w)z cos2 or 'z cos2,
5.7.2. Fully Wet Cohesionless Slope - Full Seepage                 apart from the change in unit weight, the shear stress
                                                                   remains the same.
 Along any equipotential, there is no change in excess
pore water pressure. Thus the pressure at A can be
evaluated in static head terms alone if the vertical
distance AB corresponding to the equipotential AC is
used (Fig. 5.11).

                                                                   Fig. 5.12 Mohr - Coulomb Diagram for full seepage
                                                                             case. The stress point A has moved to point
                                                                             A' as a result of the excess pore water
                                                                             pressure. This stress point is closer to the
N. K. Tovey                    ENV-2E1Y: Fluvial Geomorphology 2004– 2005                             Section 5

           failure line and the slope is thus less stable.
           The locus of the point A' is now an ellipse.            the imposed shear stress will be:-
                                                                                                (d + z) cos.sin 
The effect of the positive pore water pressure is to bring                     (  z  d)cos. sin          z  d
the stress point closer to the failure line, and thus the         and tan                           
                                                                               ( z  d   z)cos2    ( z  d   z) tan
slope is less stable.                                                                       w                         w

      Once again failure occurs when
                                                 
              zcos. sin                                     For typical values of z and d = 1m and the same
and tan                    
             (    )zcos   (    ) tan
                                                                  values for the unit weights as before:-
                    w                w
                                                                                    20 x 1  20 x 1              40
                                                                   tan                                 tan      tan 
Typical values: - for a sand          = 20 kNm-3                            ( 20 x 1  20 x 1  10 x 1)         30

                            and       w = 10 kNm-3.              Failure once again occurs when  =      
                    Hence   tan         = 2 tan                 In this case  is approximately 0.75, in
                       or                 2 .                  other words the surface dry layer has increased the
                                                                  stability again
Thus failure occurs when  is approximately  /2
and the slope approximately half as stable as in the dry
case.                                                             5.7.4 Case with damp soil - i.e. negative pore water
                                                                  pressures throughout.
5.7.3. Case with dry top layer and full seepage below             If the whole section is damp with negative pore water
                                                                  pressures throughout, then the stress point in the Mohr
The next case to consider is the one where there is full          - Coulomb Diagram (Fig. 5.10) will move towards the
seepage below a given depth d and material is dry                 right making the slope more stable (i.e. the reverse
above. We shall neglect the effect of capillary action at         direction to that with full seepage..
the interface (this has the effect of improving the
stability so by neglecting it will be a SAFE
                                                                  5.7.5 Case with soil with cohesion.

                                                                   If the material forming the slope has cohesion as well
                                                                  as frictional properties, then some interesting things
                                                                  happen. If we return to the first case, we note that the
                                                                  slope is table provided that the stress point lies below
                                                                  the Mohr - Coulomb line. If cohesion is present, then
                                                                  the failure line is displaced upwards, and the slope
                                                                  becomes more stable. Indeed it is quite possible for a
                                                                  slope to plot as a stress point shown as B (Fig. 5.14 )
                                                                  and be stable.

                                                                  However, the question of stability must be qualified
                                                                  since it was noted that failure on an infinite slope was
Fig. 5.13 case with full seepage with a dry layer of soil         independent of depth: i.e. at greater depth both the
          above. The dry layer increases the normal               normal and frictional components increased at a
          stress faster than the shear stress and hence           constant proportion so that the angle of obliquity
          the slope becomes more stable.                          remained constant, and hence the factor of safety.

                                                                  In the case of material with cohesion and the stress
In this example we would consider a potential failure in          point lying above the purely friction envelope, the
the full seepage region (which is the more critical               above statement is no longer true. Thus as the depth
region).                                                          increases, the stress point will move along the line of
                                                                  constant obliquity and will eventually cross the failure
 the total normal stress will now be:-                            envelope when failure will occur. The conclusion that
                               ( d + z) cos2                   can be drawn from this is that infinite slopes with
                                                                  material having cohesion are stable but only up to a
 the water pressure will be:               w cos2               given normal stress.

N. K. Tovey                 ENV-2E1Y: Fluvial Geomorphology 2004– 2005                               Section 5

What does this mean in practice?                                  when dealing with Mohr's Circles. However, the result
                                                                  is simple, the maximum depth that will remain stable is
                                                                  given by:-

                                                                     depth of stable vertical slope =              .......5.6

                                                                  Since c is approximately 20 - 40 kPa for many soils,
                                                                  and is approximately 20 kN m-3, the maximum depth
                                                                  for a stable vertical slope will be 1 - 2 m.

                                                                  5.7.6     An Infinite Slope which is completely

                                                                  An infinite slope which is completely submerged will
                                                                  not have any water seepage (except in extreme
                                                                  conditions), and analysis will show that the stability is
Fig. 5.14 Mohr - Coulomb Diagram for case with soil
                                                                  identical to that in the dry case. It is suggested that you
          having cohesion. Slope is more stable and
                                                                  check this out for yourself. Remember in this case you
          slope angles steeper than the angle of friction
                                                                  should use the submerged unit weight to work out both
          are possible but only for a limited slope
                                                                  the normal stress and shear stress and that excess pore
                                                                  water pressures are now zero.

                                                                  5.7.7 Depth of Tension Cracks
The normal stress is dependent only on the angle of the
slope, the depth to the failure surface, and the unit
                                                                  Tension cracks will appear in cohesive materials when
weight of the material. Since for a given slope it will
                                                                  desiccated. This will appear on flat surfaces, the most
only be the depth that can vary, it is clear that any
                                                                  classic examples are the cracking that occurs on
cohesion present will allow slopes of an angle steeper
                                                                  reservoir floors when the water level falls in summer.
than the angle of friction only for a limited depth. We
                                                                  From the above, we can predict the maximum depth of
may estimate this critical depth by drawing the diagram
                                                                  such cracks. If the cracks deepen, then minor failures
as shown in Fig,. 5.14 and reading off the normal stress
                                                                  will occur and debris will fill the base of the crack such
() on the X - axis                                               that the maximum depth will never be much greater
                                                                  than that predicted for long.
Hence the critical depth is given by:-
                                                                  The development of tension cracks can also be of
                          z                                      importance in the stability of finite slopes as will be
                              cos2                              considered in section 5.8. The maximum depth of such
                                                                  cracks can be predicted from the relationship in equation
                                                                  5 6 and used in more detailed analysis of the stability of
It is also possible to work out what z is directly without
plotting the graph by trigonometry.

Thus the vertical distance on the stress plot of the
                                                                  5.7.8 The factor of safety for Infinite Slope Analysis.
critical points is given by two relationships - one from
the Mohr - Coulomb line and the other from the shear
stress i.e.:-

from Mohr Coulomb....... c   tan 
                ............  z cos  sin 
from shear stress
                          (c   tan  )
              i. e. z 
                           cos  sin 

A consequence of the above observation is to note that
we can readily dig vertical trenches in clay when it is
not possible to do so in dry sand. A question that
immediately arises is what is the maximum depth for               Fig. 5.15 Determination of factor of safety in the dry
which a vertical slope in a cohesive material will be             cohesionless case.
stable? This analysis is beyond the scope of this course
but will be covered indirectly in the Seismology Course
N. K. Tovey                    ENV-2E1Y: Fluvial Geomorphology 2004– 2005                              Section 5

The factor of safety may be readily determined when
using the infinite slope method of analysis.   It is
obtained directly from the Mohr - Coulomb Diagram.
Thus Fig. 5.15 is a revised form of Fig. 5.10.

The point A represents the stress point for a slope of
angle  while the point C represents the point on the
failure envelope directly above the stress point.

Thus the factor of safety is   AB / BC. i.e.

            BC  z cos 2  . tan    tan 
     Fs                          
            AB    z cos  . sin    tan 
                                                                    Fig. 5.16 General slope with slices drawn. Note that one
Thus the factor of safety may be determined either                            slice boundary coincides with the intersection
graphically using the Mohr - Coulomb Diagram or                               of the water table with the slip circle. Both
entirely numerically using the above equation.                                the actual slip circle and the surface are
                                                                              approximated to a series of straight lines. For
The factor of safety in all other cases may be similarly                      analysis purposes, it is convenient to number
estimated.                                                                    the slices separately.

5.8 General Solution to stability of Slopes
5.8.1 Introduction

The general method for analysing the stability of slopes
was developed by Fellenius and is usually referred to by
the method of slices. There are several variants on
the method with fewer or more simplifying assumptions.
The Swedish Slice Method has the most such
assumptions, but all are SAFE ASSUMPTIONS, i.e.
they underestimate the factor of safety.

All the methods begin by dividing the slope into a                  Fig. 5.17 Details of forces acting on a single slice. The
number of vertical slices and analysing the stability of                        effect of tension cracks at the crest can be
each. The overall stability is then given by a summation                        included as can the effects of varying shear
of the stability effects on all the slices.                                     strength properties.
                                                                    Once the factor of safety has been estimated along a
As with many problems in Geotechnics, there is a trade              single slip circle, further potential slip circles are
off between accuracy and time as to the number of slices            analysed to find the most critical one (or group of critical
chosen. Typically, the number chosen should not be less             slip circles.
than 5, and numbers more than about 8 get tedious to
compute. There are some computer methods to analyse
the stability of slopes, but these require careful data             5.8.2 Method of Analysis
input as the whole of the slope geometry must be
specified, and there can be problems with convergence if            A typical slope is shown in Fig. 5.16 with the various
simple rules are not followed.                                      slices included. In addition the approximate straight
                                                                    lines to the various curved sections of both the slip circle
The method begins by drawing a realistic slip circle and            and the surface as shown.
then dividing the area of the slope above this circle into
several slices. The slices may be of varying width, but             In Fig. 5.17 the details of the forces acting on each slice
the general rule is that the height no slice should be more         are shown. These are (in the case without water):-
than about 2 - 2.5 its width, nor should it be less than 0.4
times the width. Further, it simplifies calculations, if            1) the weight acting vertically downwards (W)
slice boundaries are arranged to coincide where different           2) the normal force acting orthogonal to the local slip
strata cut the slip circle and also where the water table              surface (N)
intersects the slip surface.                                        3) the frictional force acting parallel to the local slip
                                                                       circle. (F)

N. K. Tovey                ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                Section 5

4) vertical inter-slice forces arising from friction               1)       the area of each slice (so that we can work out
   between slices (X1 and X2).                                              the weight,
5) horizontal inter-slice reaction forces between                  2)       the orientation of the local angle of the slip
   adjacent slices (E1 and E2).                                             circle to the horizontal ().

The inter-slice forces were encountered previously in the
                                                                   The analysis is most conveniently conducted in tabular
Infinite Slope Method of Analysis, but in that case it was
                                                                   form such as that shown on the next page.
easy to show that such forces on either side of each slice
were equal and opposite and would thus cancel each
                                                                   a)   We begin by measuring the local angle of the slip
other out. In the general method, this is not the case as
                                                                        circle and enter this in column 1 of the table.
the slices are of different sizes. However, overall there
must be no net horizontal force or vertical force arising
                                                                        NOTE: some of the local angles of the slip circle
from these inter-slice forces and various approximations
                                                                        will be negative in deep seated failures such as that
are possible.
                                                                        shown in Fig. 5.16 (e.g. 1).
A)   The first approximation is to assume that the
     vertical forces on each slice balance (this is more           b)   We now need to estimate the weight of each slice
     likely than the horizontal ones). Overall they must                and we do this as in other methods for slope
     do so, but not necessarily for a particularly slice.               stability by determining the area of the slice. If
     By making this simplifying assumption to lead to an                there is a single stratum, the we determine the area
     underestimate of the factor of safety and thus it is a             of the whole slice whereas if there are several strata
     SAFE ASSUMPTION.                                                   we must sub-divide the slice so that the area of each
                                                                        stratum in each slice can be determined. Since the
     The computer method which you will use in the                      unit - weight of most materials is approximately the
     practical uses two methods, a rigorous method                      same, this refinement is not always necessary. The
     which does not make the approximation and a less                   estimated values of area are entered into column
     rigorous one which does. In execution using the                    [2]. The shape of slices may be approximated to
     rigorous method, it is necessary to first guess what               either:-
     the imbalance of the two forces on the sides are.
                                                                   triangles [ area = 0.5 x base x height. It is usually
      The calculation is then completed at which time a                       convenient to use the vertical side of the slice
     revised estimated can be determined from the                             as the base of the triangle and the distance
     calculations. If the revised estimate and the initial                    from this slice edge to the opposite apex as the
     guess are close then the computed factor of safety is                    height in these estimates].
     accepted. If the two are not close, then a second             trapezia [ area is mean height of parallel sides multiplied
     trial is made using the computed values from the                         by distance between them].
     first set of calculations. The processes is repeated
     through several iterations until convergence is               c)   The weight of the slice is then the area multiplied
     achieved. Unfortunately, if the slices are ill-                    by the unit weight. In the case of multiple strata,
     conditioned (i.e. they do not conform to the rules                 the each separate stratum can be approximated to a
     laid out above), convergence will not occur, and                   trapezium, and the resultant weight is the
     the solution can become unstable.                                  summation of all areas multiplied by the respective
                                                                        unit weights.
B)   The second approximation is to assume that both
     the horizontal and vertical forces on the sides of the        d)   We have three basic forces controlling the
     slices cancel out.       This is a more severe                     equilibrium of each slice [if we use approximation
     approximation, but is a SAFE APPROXIMATION                         B], and as with previous examples, the mobilising
     once again.                                                        force will be W sin  while the normal force will
                                                                        be W cos  We compute this values from the
5.8.3 Analysis of Slopes without water seepage                          data in columns [1] and [3] and enter the results in
                                                                        columns [5] and [4] respectively.
In most of the discussion regarding the Method of Slices
we shall assume that both the horizontal and vertical              e)   The cumulative effects of all slices may then be
inter-slice forces cancel. We will return briefly to deal               determined by summing the values in both columns
with the more accurate solution using approximation A                   [4] and [5].
above later.

Basically the analysis follows exactly the same form as            NOTE:      some of the values in column [5] will be
that for a straight line failure. (We may incorporate the                    negative!!!
effects of water pressure later). We treat each slice
separately and from each we will need to determine:-      f)       Finally, we require (for the dry case), information on the
                                                                   cohesive force. To obtain this which measure the length
N. K. Tovey                  ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                 Section 5

of the segment on the slip circle in each slice (column [6]          column [8] from which the pore water force U may be
), and multiply these values by the cohesion to give the             determined (column [9]).
values in column [7]. Finally we sum all values in
column [7].

Equation 5.5 which was used for determining the
stability of a single wedge is then modified to
incorporate all slices by a summation on each part of the

i.e. Fs   
               c    W cos  tan                 ....5.7
                      W sin
This factor of safety will give the situation in the dry case
( i.e. without water seepage).
                                                                     Fig. 5.18 Flowlines and equipotentials in ground water
                                                                               flow in a slope
5.8.4 Analysis of Slopes with water seepage
                                                                     NOTE: we must work in terms of forces not stresses
The analysis of slope stability in the wet case proceeds in          (i.e. we multiply the pore water pressure by the area of
the same fashion as the dry case, and once again , it is             the base of the slice).
most convenient to use a tabular form for analysing the
slope. Columns [1] - [7] are completed as in the dry

We need to assess the effects of water seepage. There is
a more exact method involving the construction of a
flow net and an approximate method. The approximate
method will over estimate the water pressure and hence
once again it will be a SAFE APPROXIMATION.

We first need to sketch the water table (Fig. 5.18), then
remembering that with the phreatic surface (water table),            Fig. 5.19 Estimation of true water pressure        (h) and
the positions of the intersections of the equipotential                        approximate pressure (h').
drops with the water table will be spaced at points to
given equal vertical head drops. This gives us the start
of the flownet which we can then complete by sketching               The approximation will always overestimate true value
curvilinear squares in the normal way.                               and will thus be a safe approximation. In the example
                                                                     shown the water table is steeply curved and this
Once we have done this we can then estimate the water                accentuates the difference between the two values.
pressure at any point on the slip circle by following the            Normally the water table is less steep and the two values
relevant equipotential line from the point on the slip               usually differ by less than 10%.
circle up to the water table and then evaluating this
vertical head drop. (Fig. 5.19). This distance (h) will              Finally, we modify the Normal force to account for the
be less than the direct vertical distance (h') and so will           pore water pressure and enter the value in column [10].
over-estimate pore water pressures. The water table is               Column [10] is effectively column [4] - column [9]
shown steeply curved in the diagram for clarity, but
normally the water table is flatter and the difference               The factor of safety in the wet case is then given by:-
between the two estimates is much less [typically being
around 10% or less.
                                                                     Fs 
                                                                             c    (W cos   U )tan              ........5.8
The values for the head of water (hw) , whether                                       W sin
accurate or approximate, are entered for each slice in
Space for additional notes

N. K. Tovey                   ENV-2E1Y: Fluvial Geomorphology 2004– 2005                     Section 5

                                               TABLE 5.1: SLOPE STABILITY - METHOD OF SLICES
Cohesion (c) =          kPa             Angle of Friction (  ) =              Unit Weight ( )      =         kN m-3
 Slice No                      Area of      Weight of     W cos    W sin                         c                 hw       U            N=             Slice
                                 Slice        Slice                                                                           =hw  gw     W cos -U          No

                  [1]             [2]           [3]           [4]      [5]             [6]               [7]            [8]      [9]           [10]

    1                                                                                                                                                         1
    2                                                                                                                                                         2
    3                                                                                                                                                         3
    4                                                                                                                                                         4
    5                                                                                                                                                         5
    6                                                                                                                                                         6
    7                                                                                                                                                         7
    8                                                                                                                                                         8
    9                                                                                                                                                         9
   10                                                                                                                                                        10
                                                                                                                                      

            Fs( wet)      c   ( W cos  U )tan                                     Fs( dry )      c   W cos )tan            =
                                    W sin                                                                       W sin
                                                                      ==========                                                                  ========

N. K. Tovey                 ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                  Section 5

5.8.5. Modification to allow for inter-slice horizontal             and impose additional lateral stresses on the slope which
        forces - Approximation A.                                   will make the slope less stable (see Fig. 5.20).

Equations 5.6 and 5.7 give the factors of safety in both
the dry and the wet case for the situation where the inter-
slices forces are assumed to cancel out.

Bishop proposed methods to avoid this approximation
although the full analysis with no approximations is
difficult to execute in practice, needs a computer
program, and then can be beset with convergence
problems if the correct procedure is not rigidly followed.
However, the simplified method of Bishop does allow us
to obtain a more accurate estimate by removing the
restriction that the horizontal forces on each slice must
                                                                    Fig. 5.20     Slope with a tension crack arising from
                                                                                   desiccation. The presence of the crack
The full derivation of the formula is beyond the scope of
                                                                                   reduces the area of the sliding surface and
this course, but it is sufficient to note that if we do know
                                                                                   hence stability is reduced. Water filling
what the factor of safety is it is possible to estimate what
                                                                                   crack will impose lateral stress on the sides
the horizontal inter-slice forces are and include them in
                                                                                   of the crack causing a further reduction in
our analysis.
This begs a question though. How do we know what the
factor of safety is until we have estimated it using an             Analysis of the slope in the dry case will follow the
appropriate formula?                                                same procedure as described above except that the
                                                                    failure surface terminates at the base of the crack. When
The formula to use is:-                                             water is present the analysis is more involved an beyond
                                                                    the scope of this course

        Fs 
                (cb W tan ).M
                      n                                             5.8.7 Effect of water filled tension crack on a straight
                    W sin                   ...........5.9
                                                                           line failure

                   tan .tan                                     Fig . 5.21 shows a potential straight line failure surface
   where M  cos 1
                              
                                                                   with a tension crack both dry and filled. It is relatively
                      Fs                                          straightforward to analyse the reduction in strength in
                                                                    these two cases as shown below. Analysis of slope
  and bn is the width of the nth slice (not the length              stability in the general case with dry cracks is easily
parallel to the slip circle)                                        accommodated in the analysis, but the effect of water
                                                                    filled cracks is more complex and beyond the scope of
The term M involves Fs so the way we proceed is to                  this course except in general descriptive terms.
first make a guess that the factor of safety is unity (i.e.
the slope is just stable), we work through the analysis             Equation 5.5 for the stability of a straight line failure is:-
and obtain an estimate for Fs which is general will be
                                                                                   c L  W cos  tan 
different from our original guess. We now replace the                       Fs                        .....................5.5
computed value of        Fs as the denominator to the                                    W sin
expression for M and repeat. We will find that the
difference in the two values on the second interation is
much less and we continue until both the guess and the
computed value are closely the same (usually agreement
to 1% is sufficient), and this is then the correct value of
the factor of safety.

5.8.6 Effect of Tension Cracks - general description.

Tension cracks may, in dry weather, open to a depth of
2c/ (see section 5.7.7) and will affect the stability of a
slope in two ways.

Firstly, there will be a reduction in the length of the slip
surface which has inherent strength, and secondly
                                                                    Fig. 5.21 Straight Line failure with water filled tension
immediately after a drought, water can fill such a crack
N. K. Tovey                 ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                   Section 5
                                                                                             n is . 'W (   ) soc . 2dw  .
                                                                     Incorporating these modifications changes the factor of
                                                                     safety to:-
                                                                           c L'  ( W' cos   .  w d2 . sin )tan  ….5.11
                                                                     Fs                      2
                                                                                 ( W' sin  .  w d2 . cos  )
                                                                     5.8.8. Effect of Man Made Loading

                                                                     Man-made (or natural loading) on a slope can affect its
                                                                     stability. It can be incorporated by adding the load to the
                                                                     appropriate slices before analysis. The positioning of the
                                                                     load also affects the stability. In some cases it will
                                                                     increase the stability, in other cases it will decrease.
                                                                     Generally speaking, loads near the base of the slope will
Fig. 5.22 The water pressure distribution with depth for
                                                                     tend to improve stability and those near the crest will
          the tension crack in Fig. 5.21
                                                                     reduce the stability, but this is not always the case as it
                                                                     also depends on the position of the water table.
With a dry tension crack, the equation is essentially the
same except the length L' is now the relevant length, and
the weight W' will be reduced to omit the small wedge to             An example of a slope with a crack at the
the right of the tension crack.                                      top
The formula now becomes:-                                            Straight Line failure with crack at top which can become
                                                                     water filled
         c L'  W' cos tan 
 Fs                                        ..........   5.10        [based on Exam question in January 1996]
               W' sin
                                                                     The geometry is shown in Fig. 5.23
In the case of the crack filled with water, the water
pressure will be hydrostatic and thus the total water force          Height of slope = 10 m, angle of slope = 60o, c = 15
from the crack acting horizontally will be:                          kPa;  = 19o;  = 16 kN m-3
                                        wd2
                                     2                               Critical failure surface will be about (  +  ) /2, i.e.
                                                                     about 40o
Note that because the pressure is hydrostatic it has a
triangular distraction and increases linearly wt. depth,             We need the length of the failure surface (L), and also
and the above expression is effectively the pressure at              the weight of the sliding mass.
mean depth multiplied by the depth.
                                                                                 L   =     H /     sin 
This force acts horizontally and thus perpendicular to
the slope it will have a component of:-                              This is computed in column [2] of Table 5.2

               1                                                        Column [3] then computes the area of the wedge and
                 .  w d 2 . sin              acting in a               column [4] the total weight using the unit weight of
               2                                                         16 kN m-3.
direction outwards from the slope and will thus reduce
the effective normal force from W' . cos to                            the cohesive resistance is computed in colum [5]
       ( W' . cos            .  w d 2 . sin )                       column [6] computes the Normal Force
                             2                                          W cos  tan  = 0.3444 W cos 
At the same time there will also be a component of the                  the Shear Force (W sin ) is evaluated in column [7]
water force acting parallel to and downwards along the
slope equal to:-                                                        the factor of safety is calculated as ([5] + [6]) / [7]
                                  1                                  
                                    .  w d 2 . cos                     the procedure is repeated for other failure angles
                                  2                                      around 40o.
So the mobilising force increases from      W' .sin
                                                                     The factor of safety is now plotted against failure surface
                                                                     angle (Fig.5.24), or from the Data Sheets

N. K. Tovey                ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                 Section 5

        c . L  W . cos  . tan 
Fs 
                W . sin

Fig. 5.23. Geometry of the Slope showing tension crack. This question appeared in the 1996 Exam Paper.

              failure    failure     area of    weight of        cohesion x    Normal       Shear      Factor
              surface    surface     wedge       wedge               L          Force       Force        of
               angle      length                                                                       Safety
                [1]         [2]         [3]        [4]              [5]           [6]           [7]     [8]
                35         17.4        42.5        680              261           198           390    1.177
                40         15.6        30.7        491              234           129           316    1.148
                45         14.1        21.0        336              212            82           238    1.235
                38         16.2        35.1        562              243           152           346    1.142

Table 5.2 Analysis of factor of Safety at various Failure Angles without tension crack.

                                                                  Fig. 5.24 Plot of Factor of safety against failure surface

                                                                  Having obtained values for 35o, 400, and 450, it is clear
                                                                  that the minimum factor of safety representing the critical
                                                                  failure occurs between 35o and 40o, so a further angle of
                                                                  38o is used and indeed has a slightly lower Fs. If time
                                                                  permitted, a further iteration would be done, this time
                                                                  between 38o and 40o. However, the error in using the
                                                                  value at 38o will be very small.

                                                                                                2. c 2 x 15
                                                                   Tension crack depth      =               1. 9 m
                                                                                                     16

N. K. Tovey                  ENV-2E1Y: Fluvial Geomorphology 2004– 2005                                  Section 5

        failure        failure     area of     weight of cohesion x     Normal       Shear        Factor of
        surface        surface     wedge        wedge        L           Force       Force         Safety
         angle          length
           [1]           [2]         [3]         [4]       [5]      [6]          [7]           [8]
           38           13.1                    539        197      146          332          1.033
so the failure surface length (at 38o) is reduced by 1.9 /
sin 38 = 3.1 m to 13.1 m and the weight lost beyond            The slope is still stable (see Table 5.3), but is now
the crack will be 0.5 x1.9 x 1.9.tan38 x unit weight =         becoming somewhat critical as Fs is only just over
22.6.                                                          1.000.

Table 5.3 Revised Factor of Safety Calculations incorporating DRY tension crack.

It is now necessary to out the effects of the crack being filled with water after heavy rain. This is done in the manner
shown in section 5.8.7.

i.e. the pore pressure in the crack is   U  0. 5  w hc
                                                                     = 0.5 x 10 x 1.92 = 18 kPa

        where hc is the depth of the tension crack.

The normal force is now W cos  tan  - U sin  tan  and the Shear Force is ( W sin  + U cos ) and the final, and
most critical value of the factor of safety is now recalculated. Note that in this case it is only columns [5] to [8] that
have to be recalculated as the values in the other columns remain the same.

               failure        failure        area of   weight of cohesion        Normal      Shear        Factor of
               surface        surface        wedge      wedge      xL             Force      Force         Safety
                angle          length
                 [1]             [2]          [3]        [4]          [5]          [6]         [7]              [8]
                 ----             as         above       ----         197          142         346             0.979

Table 5.4 Final calculation of Factor of Safety at critical angle – water now fills the crack.

The factor of safety is now less than unity and failure is likely.

5.10      General Statement on Factors of                            As a guidance, the factor of safety required in Hong
         Safety.                                                     Kong is:-

While all analyses compute a factor of safety and this                      1.4 if the slope threatens residential buildings
forms the basis of whether a slope is stable, a further
discussion is important.                                                    1.2 if it threatens lines of communication

A factor of safety of unity indicates that a slope is just                  1.1 if it threatens other man-made resources
stable, while one which is above unity should be stable
and one which is less than unity is critical. The choice of                 1.0 ( or less) for other regions
what factor of safety to use in a risk analysis is a matter
of judgement, but there is a growing awareness that this             Remember, that all approximations tend to under
should take account of the consequences of a potential               estimate the factor of safety and that is why there are
slope failure.                                                       several slopes which have factors of safety around 0.9
                                                                     and yet appear stable. What we can say is that their
Normally, we estimate the factor of safety under the                 stability is critical and in adverse weather conditions
worst conceivable conditions (e.g. a 1000 year storm).               failure is likely.
We would estimate the likely ground water level at this
stage and carry out the analysis by one of the methods               A further reason why the factor of safety is
outline previously. We would then test alternative modes             underestimated is that when we conduct shear strength
of failure until we found the most critical one. If this was         tests on the soil we will always draw an envelope to the
above a given threshold then we should be satisfied                  weakest strengths measured rather than take a mean. In
regarding its stability.                                             this way we always work on the safe side, but this has
                                                                     the effect of underestimating the factor of safety further.
N. K. Tovey                ENV-2E1Y: Fluvial Geomorphology 2004– 2005                               Section 5

5.11 Predicting the form and position of
potential failures.                                              In early October 1996 the largest glacier in Europe in
                                                                 Iceland became the centre of attention when a volcano
In the analysis we will always strive to find the failure        erupted beneath the glacier causing a large amount of
mechanism which gives the minimum factor of safety.              the ice to melt with the threat of extensive and perhaps
More often than not, we will find a band of possible slip        catastrophic flooding. This section has been written at a
surfaces which have approximately the same stability.            time when the outcome is not known but we can consider
Defining this critical band will indicate the position of        possible modes of failure that would lead to flooding.
the slip surface zone should failure occur.
                                                                 Fig. 5.24 shows a simplified possible cross section of the
                                                                 toe of the glacier some 150m long and 100 m high.
5.12 Summary of Slope Stability                                  Behind this is a crevasse in which water from the melted
                                                                 glacier behind is accumulating and starting to rise in the
      Methods for analysing the stability of slopes all         crevice.
       require the identification of a potential failure
       mechanism. Several such mechanisms should be              The first question to address is what possible modes of
       tested to determine the most critical one.                flooding are likely.
      Analysis can be done on current water level               Four situations could lead to extensive flooding
       conditions, to determine the current stability, or
       more usually, a particularly severe rainfall event           The water level in the crevasse may reach such a
       will be used as the basis for analysis (e.g. 1000             height that the flow of water beneath the toe glacier
       year storm).                                                  become critical causing a quicksand at the down
                                                                     steam end and excessive erosion and undermining of
      Many approximations can be made to simplify the               the toe of the glacier. As the erosion takes place, the
       analysis and in all cases, the approximations                 resistance to flow will decrease causing an
       reduce the factor of safety, and hence are all                increasing flow of water until the level of the lake
       SAFE ASSUMPTIONS.                                             eventually dissipates.
      The choice of what factor of safety to use should            The level of water continues to rise as more ice
       be determined on the basis of the consequences of             melts (up valley of the toe] causing the toe glacier to
       failure following a model such as that used in                be over-topped and flow of water over the surface.
       Hong Kong. Rarely is this done.                               This is unlikely to be catastrophic as the flow will
                                                                     eventually equal the rate of ice melting. Erosion of
5.13 Some comments about the stability of                            the surface of the toe glacier is likely causing further
the Glacier in Iceland.                                              release of water, but this should be controllable.

Fig. 5.25 Schematic representation of toe of glacier with build of water in crevasse behind.

N. K. Tovey                 ENV-2E1Y: Fluvial Geomorphology: 2002 - 2003                              Section 5

   Iceland is volcanic, and if the underlying part of the        distribution as was should in section 2 in the lecture
    glacier is volcanic ash rather than rock, it is               notes for flow beneath a weir.
    possible that the glacier will have carved out deep
    valleys. The same would be true in rock (e.g. U-              To investigate whether the toe glacier is stable we need
    shaped valleys). If as a result these valleys are             to work out the weight of the glacier. This we can do by
    steep, then exposed to a water level rise could               dividing it into say 5 slices (as in the general slope
    trigger landslides which could suddenly displace a            stability method), or by counting squares.
    large quantity of water causing a massive over-
    topping of the toe glacier and catastrophic flooding          Let us assume that we have worked this out (remember
    (as happened to the dam at Vaiont in Italy in 1962).              the unit weight of ice is 9.00 kN m-3 (see data
    Note, however, that since the slope is solely
    submerged, then that stability is the same as in air-         Let this value be W
    i.e. it is not a seepage situation. If the underlying
    material is rock, then rock slides could also occur,          The total resisting horizontal force provided by the
    depending on the dip of the strata - fissures may                surface of the sand/silt would be
    have opened up while the ice was present, but with
    water filling the gaps, the effective normal force                       (W  U).tan()
    would be reduced.
                                                                  and the factor of safety against sliding would be:-
   Uplift on the base of the toe glacier because of
    seepage pressures could reduce the effective normal                                    ( W  U ). tan(  )
                                                                                    Fs 
    stress and hence the ability of the toe glacier to                                            H
    resist sliding following the build up of horizontal
    pressure on the side of the crevasse. This would be           As previously, the situation is critical if Fs < 1. We can
    the most catastrophic failure mode.                           easily investigate how stability would be subsequently
                                                                  affected by changing the height of the water (h) and
The horizontal force (H) from the hydrostatic pressure is         reworking U and hence Fs. As in many previous
proportional to depth below the surface of the water, so          examples we would plot Fs against h. However, this
the mean force will be                                            time the value of Fs would continue to decrease as h
                                                                  increased. What we would be looking for is the value of
                            1                                     h where Fs becomes 1.0.
                       H      h2
                                                                  The total area of the toe glacier (which was determined
Since the length of the toe glacier is large compared             by slicing the section) = 9650 m2, and since the unit
with the depth of the sediment, and since the base is             weight of ice = 9.00 kN m-2 - from data tables, the total
nearly horizontal in this case, the hydrostatic uplift on         weight of the toe ice is 86850 kN per metre width of the
the base of the toe glacier will decline linearly from the        glacier.
above value at the crevasse to zero at the toe.
                                                                  Remember we have to work this out only once in the
Thus the total uplift (U) on the base of the glacier will         calculations (unlike the case of the slope stability as the
be                                                                failure will occur at the base of the glacier, if any where,
                                                                  and the weight above remains constant.
                  U      hd
                       2                                          As a first guess try a value of h = 50m. This gives a
                                                                  high factor of safety. Then try say 60, 70 m until Fs fall
where d is the length of the toe glacier (150 m in this           below 1,0. Finally perhaps, and if time permits add one
   case)                                                          intermediate point.

If the base of the toe glacier was not straight or the            Now plot the factor of safety against height of water
depth of the sandy/silt was large compared to the length          and read of value when Fs = 1.0. In this case it is 62.9
d, then it would be more accurate to plot the seepage             m.
flownet beneath the glacier to work out the pressure

          Height of water       Horizontal         Uplift force            W-U                  Fs =
              (h) (m)            force H                 U                                 ( W  U ) tan(  )
                                (kN m-1)            (kN m-1)                                     H
                 50               12500               37500                49350                1.974
                 60               18000               45000                41850                1.163
                 70               24500               52500                34350                0.902
                 65               21125               48750                38100                0.902
N. K. Tovey                ENV-2E1Y: Fluvial Geomorphology 2004– 2005                             Section 5

Table 5.5 Calculation of Factor of Safety against failure

Fig. 5.26 Factor of safety against height of water

We can thus predict that failure of the ice block will occur when the water in the crevass reaches 62.9m.

 space for additional notes.

N. K. Tovey   ENV-2E1Y: Fluvial Geomorphology 2004– 2005   Section 5