Movement involves shear displacement along one or more surfaces, or within a relatively narrow zone, which are visible or may reasonably be inferred. Two subgroups are identified as:
A. Rotational Where movement results from forces that cause a turning moment about a point above the centre of gravity of the unit. The surface of rupture concaves upwards. |
B. Translational Where movement occurs predominantly along more or less planar or gently undulatory surfaces. Movement is frequently, structurally controlled by discontinuities and variations in shear strength between layers of bedded deposits, or by the contact between firm bedrock and overlying detritus. |
back to Slope stability
Translational or infinite
slope movement predominantly occurs along more or less planar or gently
undulatory surfaces. Displacement is frequently, structurally controlled by
discontinuities and variations in shear strength between layers of bedded deposits,
or by the contact between firm bedrock and overlying detritus.
See also the case studies:
back to Analysis of translational slip
In this case, the soil cohesion is zero, and the slope is dry or fully submerged, so no ground water seepage occurs to generate pore water pressures.
(In the general equation c' = 0 and m = 0.)
The Factor of Safety against slip reduces to:
Note that the factor of safety is independent of the mass of the soil, the length of the slip and the depth of the slip surface. The limiting condition occurs when the slope angle (b) has the same magnitude as the angle of friction (f').
back to Drained soil with zero flow
Analysis of translational slip
In this case, the soil cohesion is zero, and there is flow parallel to and coincident with the ground surface.
(In the general equation, c' = 0 and m = 1.)
The factor of safety against slip reduces to:
Note that the factor of safety is independent of the length of the slip and the depth of the slip surface. Because, in this case, the factor of safety is dependent upon the mass of the soil, the maximum slope angle (bmax) has a magnitude significantly lower than the angle of friction (f').
back to Analysis of translational slip
A translational slip analysis may be used for slip surfaces with small depth / length ratios.
This allows the end effects to be neglected.
For a potential slip surface in any soil, the factor of safety against slip is given by:
back to General equation
The factor of safety against slip is defined in terms of the ratio of this maximum shear strength to the disturbing shear stress:
Ignoring any side forces acting on the elements, the stress conditions will be identical at every point along the slip surface. Therefore the maximum shear strength is given by the Mohr-Coulomb equation:
and the stresses are determined by resolving the load due to an element.
back to Derivation of ordinary equation
Given that the soil is saturated below
the phreatic surface,
Resolving W into components parallel and perpendicular to the slip plane and converting to a stress, gives
back to Derivation of General Equation
From a consideration of the flownet,
the pore water pressure at the slip surface is
Analysis of translational slip
It is impossible to make a vertical cut in a drained soil - this is easily demonstrated by the use of dry sand. In soils which are undrained, however, a vertical cut can be made since the negative pore pressures set up by the unloading due to the excavation will generate positive effective stresses.
If there is no tension crack present, the theoretical height of the cut is given by:
H = (4 su / g)
If a tension crack is anticipated, its theoretical value, h, is (2su / g), giving a maximum height of cut as
H = (2su / g)
We should note that even if H is kept smaller than these theoretical values, local over stressing may occur near the base of the cut. As H increases towards the theoretical maximum value, the plastic zones extend, and significant deformations will take place.
g = unit weight of soil
Consider the vertical cut of height H shown in the figure. Assume that the soil is undrained and the strength can be represented by: t = su.
Consider the collapse mechanism shown, where failure occurs along the plane surface AB, inclined at q to the horizontal. BC represents a vertical tension crack of depth h. The mass of soil represented by ABCD is in equilibrium under the action of 3 forces, namely:
W = weight of ABCD,
S = shear strength along BC,
R = normal reaction on BC.
From the triangle of forces
W = R cosq + S sinq
R sinq = S cosq
Now,
W = 0.5g(H - h) (H+h) cotq
and,
S = su.AB
where,
AB = (H-h) cosecq
Eliminating R from these equations and substituting for W and S gives:
H = (4su/g) - h
If there is no tension crack, i.e. h = 0, then,
H = (4su/g)
The theoretical value of h is (2su/g)
and then H = (2su/g)
back to Analysis of translational slip
Translational slides in rock masses
are dependent upon the spatial arrangement of the discontinuities within the
mass and their relationship to the geometry of the slope. Two arrangements
are considered:
Plane failure
In which slip is controlled by a single discontinuity, although others may exist as 'release surfaces'.
Wedge failure
In which slip occurs on two discontinuities and is governed by their line of intersection.
The engineer needs some means of graphically representing these discontinuities, if he or she is to be able to spot potential failure mechanisms. One graphical method uses stereonets to analyse the spatial arrangement of the planar discontinuities and slope surface.
back to Translational slip in rock slopes
Plane failure occurs due to sliding
along a single discontinuity. The conditions for sliding are that:
· the strikes of both the sliding plane and the slope face lie parallel (±20°) to each other.
· the failure plane "daylights" on the slope face.
· the dip of the sliding plane is greater than f'.
· the sliding mass is bound by release surfaces of negligible resistance.
Possible plane failure is suggested
by a stereonet plot, if a pole concentration lies close to the pole of the
slope surface and in the shaded area corresponding to the above rules.
back to Translational slip in rock slopes
Wedge failure occurs due to sliding
along a combination of discontinuities. The conditions for sliding require
that f is overcome, and that the intersection of
the discontinuities "daylights" on the slope surface.
On the stereonet plot these conditions
are indicated by the intersection of two discontinuity great circles within
the shaded crescent formed by the friction angle and the slope's great circle.
Note that this intersection can also be located by finding the pole P12
of the great circle which passes through the pole concentrations P1
and P2.
back to Analysis of translational slip
Chandler, R.J., 1970. "A shallow slab slide in the Lias clay near Uppingham, Rutland". Geotechnique, 20¸ 253-260.
Early, K.R. and Skempton, A.W., 1972. "The landslide at Walton's Wood, Staffordshire".Quart. J.Eng. Geol., 5, 19-41.
Esu, F. 1966. Short-term stability of slopes in unweathered jointed clays. Geotechnique, 16, 321-328.
Hutchinson, J.N. 1961. A landslide on a thin layer of quick clay at Furre, Central Norway. Geotechnique, 11, 69-94.
Hutchinson, J.N. 1967. The free degradation of London clay cliffs. Proc. Geotechnical Conf. (Oslo) 1, 113-118.
Hutchinson, J.N. 1969. "A reconsideration of the coastal landslides at Folkestone Warren, Kent" Geotechnique, 19, 6-38.
Hutchinson, J.N. and Bhandari, R.K., 1971. "Undrained loading; a fundamental mechanism of mudflows and other mass movements", Geotechnique, 21, 353-358.
Skempton, A.W., 1964. "Long-term stability of clay slopes", Fourth Rankine Lecture, Geotechnique, 14, 77-101.
Skempton, A.W., 1966. "Bedding-plane slip, residual strength and the Vaiont landslide". Geotechnique, 16, 82-84.
Skempton, A.W. and Hutchinson, J.N., 1969. "Stability of natural slopes and embankment foundations". 7th Int. Conf. Soil Mech. and Found. Engrg. (Mexico), State-of-the-Art Vol., 291-340.
Skempton, A.W. and Petley, D.J., 1967. "The shear strength along structural discontinuities in stiff clays". Proc. Geot. Conf. (Oslo), 2, 29-46.
Weeks, A.G., 1969. "The stability of natural slopes in south-east England as affected by periglacial activity", Quart. J. Eng. Geol. 2, 49-62.
back to Slope stability
The shear strength of the soil is a
function of the normal stress s :
t = c' + s' tan f'
For this reason the method of analysis must take account of the changes in overburden pressure along the length of the slip circle. The procedure requires that a series of trial circles are chosen and analysed in the quest for the circle with the minimum factor of safety. Each circle is divided into vertical strips and the factor of safety is determined by considering the forces acting on each strip.
When specifying strips, care must be exercised to avoid:
(a) A dip angle (an) of zero magnitude, since this gives an infinite factor of safety for that slice.
(b) A steep dip angle (an) such that negative normal forces are calculated. This case corresponds to the formation of a tension crack and checks should be performed to ensure this condition does not exist.
back to Analysis of rotational slip
This method is also refered to as "Fellenius'
Method" and the "Swedish Circle Method". Consider the geometry
of the trial slip circle shown in the diagram. The slipping mass is divided
into slices in the normal way and these are numbered 1, 2, 3, etc, for ease
of identification. For a potential slip circle, the factor of safety against
slip is given by:
This solution tends to give a conservative value for the factor of safety, of between 5 and 20%. This can be expensive and therefore a more rigorous approach is favoured.
back to Ordinary method of slices
The factor of safety against rotational
slip is defined as the ratio of the restraining moments to the disturbing moments:
Substituting for Sn gives,
However, the magnitude and position of the side forces (X and E) are unknown and therefore the problem is statically indeterminate. Even so, this problem can be overcome by simply ignoring the effects of the side forces, i.e.
Then substitute for
xn = Rsinan and
un = (Wn run) / (Ln cosan)
to give the ordinary equation.
back to Derivation of ordinary method
From Coulomb's equation, the shear
strength mobilised along the failure surface for slice n is:
tn = c' + (sn - un) tanf'
or, for a unit thickness:
tn = c' + [(Nn / Ln) - un] tanf'
Therefore the shear force acting along the base of the slice is:
Sn = tn .Ln
= Ln {c' + [(Nn / Ln) - un] tanf'}
By considering the forces acting on the slices and resolving normal to the slip surface, we can show that
Nn = (Wn + Xn+1 - Xn) cos an - (En+1 - En) sin an
giving
Sn = c'Ln + (Wn .cos an - un).tanf' + [(Xn+1 - Xn).cos an - (En+1 - En).sin an].tanf'
back to Analysis of rotational slip
Bishop derived an expression which takes
account of the interslice forces and gives a more accurate solution to the
idealised geometry of circular slip. The importance of such forces can be
demonstrated by considering a slice directly below the center of rotation:
this slice has an independent safety factor of infinity since a
= 0 at that point. Bishop's solution requires that the factor of safety is
constant along the complete slip circle. For a potential slip circle, the
factor of safety against slip is given by:
or
However, the determination of the interslice forces is labourious and therefore Bishop's simplified method is prefered.
back to Bishop rigorous method
The factor of safety against rotational
slip is defined as the ratio of the restraining moments to the disturbing
moments:
Substituting for Sn and xn gives the rigorous solution.
back to Derivation of bishop rigorous method
From Coulomb's equation, the shear
strength mobilised along the failure surface for slice n is:
tn = c' + (sn - un) tanf'
therefore the shear force acting along the base of a slice of unit thickness is:
Sn = tn .Ln
= c'.Ln + N'n .tanf'
Resolving the forces on the slice vertically gives:
Wn + Xn+1 - Xn = (N'n + un Ln) cos an + Sn sin an
But this is true for Sn = c' Ln + N'n tanf' at the limiting state only. Therefore, the factor of safety is introduced into this expression for all other states (Sn / Fs) giving:
Wn + Xn+1 - Xn = (N'n + un Ln) cos an + ((c'Ln /Fs) + (N'n /Fs) tanf' ) sin an
giving,
and thus,
back to Analysis of rotational slip
Although the rigorous method is more
"accurate" it is a time consuming process and can be easily simplified by
ignoring the X terms. The errors associated with such a simplification have
been shown to be small.
The values for run vary from slice to slice but, unless zones of high pore pressure exist, an average value weighted according to area can be used throughout. Like the Ordinary Method the factor of safety calculated is conservative. The value is underestimated by about 2%, but can be as large as 7%. The nature of the equation, makes solution using a computer programme or spreadsheet desirable.
back to Bishop simplified method
Note that an may have both positive and negative values.
In order to solve this equation, which has Fs on both sides, the right hand value is first estimated using the Ordinary Method and then the left hand value is calculated. This new value is then used and the procedure repeated until the two values converge.
Create a table of calculations for each slice. Note that this calculates the factor of safety for this circle, not the slope. There may be another circle or failure mechanism with a lower factor of safety.
Solving the Bishop simplified equation
|
n |
1 |
2 |
3 |
... |
||
|
area |
(m²) |
... |
... |
... |
... |
|
|
Wn |
(kN) |
... |
... |
... |
... |
|
|
an |
(deg) |
... |
... |
... |
... |
|
|
sinan |
... |
... |
... |
... |
||
|
Wn sinan |
(kN) |
... |
... |
... |
... |
S(1) |
|
bnBISHOPSIMPLEQT_B |
(m) |
... |
... |
... |
... |
|
|
exp(1) |
(kN) |
... |
... |
... |
... |
|
|
exp(2) |
... |
... |
... |
... |
||
|
exp(1)/exp(2) |
(kN) |
... |
... |
... |
... |
S(2) |
Fs = S(2) / S(1)
an
= the dip of the slip circle at the centre of the base of the slice, most easily
obtained by drawing a line from the centre of rotation to the centre of the
slice at the slip circle and measuring the angle it makes with the vertical
(remember its sign!)
back to Bishop simplified method
This method is commonly referred to
as the "Total Stress Analysis". If we consider the case of a cohesive
soil and analyse its stability in the short term, we can substitute the total
stress soil parameters into the Bishop simplified equation. Such that,
c' Þ su
f' Þ fu = 0
and the equation reduces to
substituting for Ln = bn secan and sin an = xn / R, gives
Substitute for SLn = Rq, where q is in radians. Then because the resistance to slip is constant along the slip circle, the slices are redundant and the slip can be treated as one mass (ABCD), giving
Where W acts through the centre of gravity of the slipping mass.
Short term analysis for an undrained soil
Choosing the centre for the circle
can be daunting at first! In 1936, Fellenius proposed the following method
for locating the centre of a circle passing through the toe of the slope:
|
slope |
bº |
a1 º |
a2 º |
|
1:58 |
60 |
29 |
40 |
|
1:1 |
45 |
28 |
37 |
|
1:1.5 |
34 |
26 |
35 |
|
1:2 |
27 |
25 |
35 |
|
1:3 |
18 |
25 |
35 |
|
1:5 |
11 |
25 |
37 |
For deeper circles, the centre of rotation is generally vertically above the mid-point of the slope.
back to Analysis of rotational slip
In undrained conditions, a tension crack
may develop at the top of the slope and hence no shear strength can occur
over that length. The angle q must be reduced accordingly
(see diagram). Furthermore, water in the crack will supply an additional hydrostatic
force, acting to reduce the factor of safety. This can be incorporated into
the analysis by treating it as an additional disturbing moment,
= Fw zw.
The depth of the tension crack is given by:
In practice, limit equilibrium methods of analysis are generally adopted, in which it is considered that failure is on the point of occurring along an assumed or a known failure surface. The shear strength required to maintain a condition of limiting equilibrium is compared with the available shear strength of the soil, giving the average factor of safety along the failure surface.
The problem is normally considered in two dimensions, with the conditions of plane strain being assumed, but a rule of thumb states that the factor of safety in 3-D is 10% greater. This is usually ignored, giving an additional safety margin.
The limit equilibrium method can be used in cases of undrained or drained loading, provided that the appropriate shear strength parameters are used. It is important to note that these strengths define the ultimate collapse states. In order to design safe structures or to limit ground movements, they may be reduced.
In using the limit equilibrium method, the geometry of the assumed slip surfaces must form a mechanism that will allow collapse to occur, but since they may be of any shape, they do not necessarily meet all the conditions of compatibility. In addition, the overall conditions of equilibrium of forces on blocks within the mechanism must be satisfied although the local states of stress within the blocks are not investigated.
The steps in calculating a limit equilibrium solution are as follows:
1. Draw an arbitrary collapse mechanism of slip surfaces. This mechanism may consist of a combination of straight lines or curves.
2. Determine the static equilibrium of the mechanism by resolving forces or moments and hence calculate the strength mobilised in the soil.
3. Compare the strength mobilised with the available shear strength of the soil, and hence define an average value for the factor of safety for the mechanism considered.
4. Repeat the procedure for other mechanisms and thus find the critical mechanism which defines the minimum value for the factor of safety.
back to Analysis of rotational slip
Early, K.R. and Skempton, A.W., 1972. "The landslide at Walton's Wood, Staffordshire". Quart. J. Eng. Geol. 5, 19-41.
Hutchinson, J.N. 1969. "A reconsideration of the coastal landslides at Folkestone Warren, Kent", Geotechnique, 19, 6-38.
Hutchinson, J.N. and Bhandari, R.K., 1971. "Undrained loading; a fundamental mechanism of mudflows and other mass movements, Geotechnique 21, 353-358.
Ireland, H.O., 1954. "Stability analysis of the Congress Street open cut in Chicago". Geotechnique, 4, 163-168.
Kjaernsli, B. and Simons, N., 1962. "Stability investigations of the north bank of the Drammen River". Geotechnique, 12, 147-167.
Sevaldson, R.A., 1956. "The slide at Lodalen". Geotechnique, 6, 167-182.
Skempton, A.W., 1964. "Long-term stability of clay slopes", Fourth Rankine Lecture. Geotechnique, 14, 77-101.
Skempton, A.W. and Brown, J.D., 1961. "A landslide in boulder clay at Selset, Yorkshire", Geotechnique, 11, 280-293.
Skempton, A.W. and Golder, H.Q., 1948. "Practical examples of the f = 0 analysis of the Stability of clays". Proc. 2nd Int. Conf. Soil Mechs (Rotterdam),2, 63-70.
Skempton, A.W. and Hutchinson, J.N., 1969, "Stability of natural slopes and embankment foundations". 7th Int. Conf. Soil Mech. and Found. Engrg. (Mexico), State-of-the-Art Vol., 291-340.
Skempton, A.W. and La Rochelle, P., 1965. "The Bradwell slip, a short term failure in London Clay". Geotechnique, 15. 221-242.