ENV-2E1Y Fluvial Geomorphology
2004 – 2005
A triaxial testing rig for soils
Slopes and related topics
Section 4 Shear Behaviour of Soils
N. K. Tovey
ENV-2E1Y: Fluvial Geomorphology 2004– 2005
Section 4
Slope Stability and Related Topics
4. Shear Behaviour of Soils
Fig. 4.1 Resolution of Forces
4.1 Introduction
In the previous section we have seen how a soil behaves
under a normal load. The behaviour of soils under shear
load determines whether a slope will be stable or not, and it
is thus important to understand the nature of how a soil
deforms under such loading.
The condition relating to moments we shall deal with later in
Slope Stability.
If we resolve forces parallel to P1 then:P1 = P2 cos 2 + P3 cos 3
4.2 Definitions
·
Similarly at right angles to P1
a normal load or force is one which acts parallel to the
normal (i.e. at right angles) to the surface of an object
·
a shear load or force is one which acts along the plane
of the surface of an object
·
the stress acting on a body (either normal or shear) is
the appropriate load or force divided by the area over
which it acts.
...........4.1
P2 sin 2 = P3 sin 3
...........4.2
4.3 Shear Strength Relationships
Coulomb (of electricity fame) was a French Military
Engineer and was the first person to establish a fundamental
relationship for soils. He was charged with the design of
fortifications and noted that many ramparts and trenches
failed, and in trying to understand why this was occurring
the basic ideas of Geotechnics were formulated.
In some circumstances we will be dealing with forces, in
other situations we are dealing with stresses, and it is
important to recognise the difference between them.
We shall need to consider whether an object is stationary
and in equilibrium (i.e. it has not failed).
There are three conditions that must be satisfied for
equilibrium:1) all forces parallel to one direction must be zero
2) all forces orthogonal (at right angles) to the above
direction must be zero
3) the sum of the moments of the forces must be zero
We may specify the first two conditions by resolving forces
(e.g. see Fig. 4.1)
Fig. 4.2 Box Sliding on Horizontal plane under action of
two forces
If we take a box of sand and tilt it the sand does not move
until a critical angle is reached when failure of the whole
slope occurs. Equally if we try to make a sand castle with
dry sand as we did in the introductory part of the course, we
find that failure always occurs and that there is a limit to how
steep we can have a stable slope.
In a similar way, if we have a block of wood resting on a
flat plane and we tilt the plane, the block will remain stable
until a critical angle is reached when the block will slide
down the slope. The angle at which the block first starts to
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N. K. Tovey
ENV-2E1Y: Fluvial Geomorphology 2004– 2005
Section 4
readily do the conversion by dividing by the relevant area to
get equations 4.5 and 4.6. respectively.
move is known as the angle of friction and by analogy,
the maximum angle that can be constructed in dry sand is
known as the angle of internal friction. This angle is
usually given the symbol ().
We may conduct an alternative experiment in which we have
a block on a horizontal plane and we apply a normal load
(N) to it (see Fig. 4.2). We also measure the horizontal
force (F) that is required to move the block. If we plot the
results we will find that we have a relationship such as
shown in Fig. 4.3 - this is a straight line through the origin
and the relationship between F and N is given by:-
F = N tan 
..........4.3
Fig. 4.4 Relationship[p between normal load and shear load
for a typical soil (total force terms)
i.e.  =

tan 
............4.5
and

c +  tan 
...............4.6
Note in this latter equation a lower case c is used (as this
refers to the intrinsic cohesion in stress terms) whereas the
upper case C is use to specify a cohesive force.
Fig. 4.3 Relationship between normal and shear load for a
granular medium
Equation 4.6 is the general equation specifying shear
behaviour as equation 4.5 is really a special case of equation
4.6
Now suppose there is a little glue on the base of the block.
If we try to shear the block we will initially find that the
block does not move at all, but after a certain horizontal
load the block will move. In this case, the relationship is
shown by Fig. 4.4 and is once again a straight line, but this
time there is a positive intercept representing the inherent
strength in the glue. Only after the glue bond has broken
will the block behave in the way the original block did. The
relationship is now :-
F = C + N tan 
=
There are three types of material to consider
1) granular media (e.g. sands and gravels) where c = 0 and
equation 4.5 is valid
2) wet clays which are purely cohesive and have no
frictional component where
= 0 i.e. =
c
................4.4
3) most other soils for which the governing equation is 4.6
where C is the strength of the glue bond. In soils exactly the
same thing happens. We have a stickiness associated with
clays which is called cohesion. The intercept C is called the
cohesion of the soil.
Fig. 4.4 shows the inherent relationship at failure between
normal stress and shear stress. Any point which plots below
the failure line (Fig. 4.5) e.g. point A is stable, but a sample
with a state of stress indicated by point B will be stable, but
if as a result of weathering the failure line F - F changes to
line G - G, then failure can then occur. This in part indicates
Both equations 4.3 and 4.4 are given in terms of forces. It is
normally more convenient to talk in terms of stress. We can
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N. K. Tovey
ENV-2E1Y: Fluvial Geomorphology 2004– 2005
how a slope which may initially be stable will fail in the
longer term. In other cases, the strength of the soil may
increase with time and the soil will become more stable.
Diagrams such as Fig. 4.5 are called Mohr - Coulomb
Diagrams. Those of you doing the Seismology Course will
no doubt have come across Mohr's Circles, and this refers to
the same person. In fact we can analyse soil behaviour using
Mohr's Circles and this does give us a greater insight into
what is happening, but this is beyond the scope of the
present course..
Section 4
Load N remains constant while the shear load is
progressively increase. On the Mohr - Coulomb Diagram
the stress path moves from the initial point A (where there is
no shear load) to point B which is one the failure line when
failure will occur.
Fig. 4.6 Shear Box Test and associated stress path
Fig. 4.5 Mohr - Coulomb Diagram
4.4.2 Stress - strain relationships from the Standard Shear
Box.
Fig. 4.4 shows the inherent relationship at failure between
normal stress and shear stress. Any point which plots below
the failure line (Fig. 4.5) e.g. point A is stable, but a sample
with a state of stress indicated by point B will be stable, but
if as a result of weathering the failure line F - F changes to
line G - G, then failure can then occur. This in part indicates
how a slope which may initially be stable will fail in the
longer term. In other cases, the strength of the soil may
increase with time and the soil will become more stable.
Diagrams such as Fig. 4.5 are called Mohr - Coulomb
Diagrams. Those of you doing the Seismology Course will
no doubt have come across Mohr's Circles, and this refers to
the same person. In fact we can analyse soil behaviour using
Mohr's Circles and this does give us a greater insight into
what is happening, but this is beyond the scope of the
present course..
If we undertake two tests at the same normal load but at
different initial unit weights we will find that in the case of
the dense sample there is a definite peak which occurs after a
strain of 2 - 3% (Fig.. 4.7), whereas in a fully loose sample
there will be no peak at all - the shear strength will
eventually reach an steady value.
Provided that we allow both test to progress sufficiently we
should find that for the same normal load all tests
irrespective of how dense these were initially will end up at
the same shear strength.
If we undertake dense tests on the same material at different
normal loads we will find that we have a series of similarly
shaped curves (Fig. 4.8).
4.4. Measurement of the Shear Strength of a
Soil
There are several methods by which the shear strength of a
soil may be measured. There are both field methods (e.g.
the shear vane and unconfined compression methods used in
the practical), and laboratory based methods including the
Shear Box and the Triaxial Test.
4.4.1 The Standard Shear Box
A typical shear box is shown in Fig. 4.6. The sample is
contained within two halves of a box which can move
relative to one another causing the sample to shear along an
approximately horizontal plane. During the test the Normal
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N. K. Tovey
ENV-2E1Y: Fluvial Geomorphology 2004– 2005
Section 4
Fig. 4.7 Standard Shear Box Tests at the same normal load
on a dense and loose sample
behaviour of two types of test together with the stress-strain
curves from Fig. 4.7.
Fig. 4.8 Standard Shear Box Tests on dense samples at
different normal loads
Fig. 4.9. Unified Curves for all stress levels using nondimensional stress parameter.
As with previous sections of this course, we can unify the
results by plotting the Y - axis as the non-dimensional stress
parameter t / s when all tests on materials at the same unit
weight will fall on an unique curve (Fig. 4.9).
Normally we would undertake several tests each one at a
different normal load as with a single test we can only obtain
a single point on the failure envelope. In the practical class,
each group used a single normal load, and we shall be
pooling the class data so that a true failure envelope may be
drawn.
4.4.3 Types of Test in the Standard SHear Box
There are two different types of test that can be conducted
on a dry sand:1) stress - controlled
2) strain-controlled
Fig. 4.10
In a stress - controlled test, the horizontal load is applied via
a pulley. We can plot the shear load as it increases to
failure, but as soon as failure is reached there will be a
catastrophic failure and we will not be able to get any further
readings. Fig. 4.10 shows the differences in the two tests as
far as the extent of readings that can be taken.
.
In a strain controlled test, we drive one half of the box
relative to the other at a constant rate and measure the load
with a load measuring device such as a proving ring. This
allows us to obtain reading post failure and this is vital if we
are to understand how slopes fail.
Differences between stress-controlled and
strain -controlled tests.
In the dense test there is sometimes a small compression as
the particles pack closer to one another, but soon there is a
substantial expansion with the rate of expansion being
greatest when the peak strength is reached. Thereafter the
volume expansion levels off and become constant as the
shearing continues past peak.
For the loose sand the behaviour is very dependant on
exactly how loose the sample is to start with. If it is very
loose, there will be a steady reduction in volume as the test
proceeds. In other cases there may be a small reduction in
volume followed by a small expansion. For a medium dense
sample the will be a small increase in volume.
4.4.4 Volume changes during shearing
It is normal practice to measure the change in volume during
shearing. In the case of the standard shear box, this may be
done by measuring any changes in the vertical displacement
of the piston as the shearing progresses. Fig. 4.11 shows the
While this behaviour seems unrelated, we can attempt to
examine what is happening by plotting the voids ratio
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N. K. Tovey
ENV-2E1Y: Fluvial Geomorphology 2004– 2005
against the displacement rather than the change in volume.
If we do that then a set of curves similar to those in Fig. 4.12
is obtained.
Section 4
total stress = effective stress + pore water pressure
and this gives us a clue how to modify the original failure
criterion (as was done by Terzaghi in the early part of the
last century).
Here all tests appear to reach a common voids ratio
irrespective of the initial starting condition. This
voids ratio is known as the critical voids ratio for the
given normal stress. This critical voids ratio will vary
depending on the normal load and is closely related to the
consolidation line.
Apart from small amounts (as will be discussed in the
section on rivers), water cannot take shear, and so the Mohr
- Coulomb equation must be modified to replace the total
normal stress by the effective normal stress.
i.e.  =
c + (  - u ) tan  ................4.7
where u is the pore water pressure.
Frequently equation 4.7 is abbreviated as

=
c +
'
tan 
..................4.8
where ' is the effective normal stress
Before proceeding it is worth considering whether equation
4.7 bears any resemblance to reality and in particular the
'floppy' membrane demonstration given earlier.
If we have no pore water pressure, then
equation 4.7 is the same as equation 4.6.
u = 0 and
If the pore water pressure is greater than zero, the term in
brackets in equation 4.7 will be less and so the shear
resistance will be less. If the pore water pressure is negative
(i.e. a suction), the term in brackets will be larger than in the
dry case, and the strength will be increased. This is exactly
what was seen in the 'floppy' membrane demonstration as so
the equation seems reasonable.
Fig. 4.11 Volumetric relationships during shearing
4.6. Consequences of water pressure on Mohr Coulomb Envelope
Fig. 4.12 Voids ratio relationships during shearing
4.5 Effect of water pressure
We saw in the initial demonstrations that water pressure has
a profound effect on the stability of soils, and yet the Mohr Coulomb failure criterion says nothing about water pressure.
Indeed as it stands the Mohr - Coulomb failure envelope
only considers TOTAL stresses when we should be dealing
with EFFECTIVE stresses.
Fig. 4.13
Remember in our discussions on consolidation we noted
that:-
Fig. 4.13 shows the point A on a Mohr - Coulomb diagram
representing the state of stress of a sample in a dry condition
(or zero excess pore water pressure). If the water pressure
increases, then the stress point A will move parallel to the X
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Effect of water pressure on Mohr - Coulomb
Diagram
N. K. Tovey
ENV-2E1Y: Fluvial Geomorphology 2004– 2005
- axis towards the left and get nearer the failure line (hence
the sample is more likely to fail). Conversely if there is a
negative pore water pressure the stress point A will move
horizontally away from the failure line, and the sample will
be more stable. Several slopes in the UK, particularly those
in south - east Essex (near Hadleigh) have soils with a
negative pore pressure and are therefore artificially
steepened as a result. If there is heavy rain or change in
climate, these slopes may well become unstable.
Section 4
water extruded from or sucked into the sample
in such tests.
2) an undrained test in which no drainage is
allowed. However, we measure the pore water
pressures during the test.
4.7. The Triaxial Test
While the nature of the standard shear box may be simple,
and gives approximate estimates of the shear strength of a
soil it does induce complex stresses on the sample which are
difficult to analyse accurately. It is possible to saturate the
sample and test samples in water but we can only do this if
we conduct the test sufficiently slowly to allow full drainage.
Fig. 4.15 Pore water pressure relationships during shearing
The apparatus is shown in simplified form in Fig. 4.14. In
the configuration shown, it is set up for a drained test. For
an undrained test, the burette is replaced by a pore pressure
transducer which measures changes in the pore water
pressure. Initially the test begins by increasing the water
pressure in the cell and allowing the sample to consolidate
under that pressure. (In the case of coarse sands, this
consolidation is small, and in any case occurs almost
instantaneously because of the high permeability of the
sand). Thereafter the sample is loaded in a strain-controlled
test and the load on the top cap, the displacement, and the
volume of water change are measured. In the undrained test,
the pore water pressure is measured instead of the volume.
The shape of the stress-strain curves are similar to those
shown in Fig. 4.7, while the volume change in the drained
test follows that shown in Fig. 4.11. For the undrained test,
the pore water pressure follows a curve similar to that shown
in Fig. 4.15.
Note. in the dense tests the pore water
pressure goes negative after a small positive pore pressure.
Fig. 4.14 The Triaxial Apparatus
A more fundamental piece of apparatus is the Triaxial
apparatus in which the sample is in the form of a cylinder
and enclosed in a rubber membrane. The sample is
surrounded by water to which is applied a steady hydrostatic
pressure, and the sample is deformed by loading the top via
a piston. By varying the cell pressure we can carry out tests
at different normal stress levels and obtain a failure envelope
similar to that obtained using the standard shear box.
4.8 Failure modes in the Triaxial Test.
As the sample is loaded in the triaxial test, its length will
shorten as the strain increases and there will be some bulging
towards the end. In over consolidated samples (and dense
sands), there is usually a very definite failure plane which
develops as soon as the peak strength is reached. In
normally consolidated clays and loose sands, such a failure
zone is not visible as there are usually numerous micro
failure zones criss-crossing the bulging region.
A further advantage of this piece of apparatus is the ability
to control drainage. Thus we can conduct two types of test:1) a drained test in which we allow complete
dissipation of the pore water pressure. The
speed of the test must be such that it allows for
the permeability of the material. For clays the
time to conduct a drained test is usually at least
a week. It is normal to measure the volume of
In an undrained test the orientation of the failure zone will
be at 45o to the horizontal, while in a drained test the
orientation will be at (45 + /2), although it is often not as
well defined as in the undrained test.
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N. K. Tovey
ENV-2E1Y: Fluvial Geomorphology 2004– 2005
Section 4
to see no volume change in a drained test and no change in
pore water pressure in an undrained one.
4.9 Unifying remarks on the behaviour of soils
under shear.
A diagram such as Fig. 4.16 gives us an insight into why
some slopes appear to fail soon after they have formed,
while in other cases they are initially stable, but fail much
later.
We have seen that in some circumstances a soil expands on
shearing, and in other cases it contracts and that this
depends on the initial compaction and / or consolidation of
the sample. Equally, we have seen that in some cases a
negative pore water pressure develops on shearing and in
other cases it is a positive pressure. Fig. 4.16 shows a
typical consolidation curve in e - log s space.
In a normally consolidated clay (or lightly over consolidated
clay, or loose sand), a sudden imposition of load such a the
construction of a slope will be fast relative to the normal
drainage time, and so positive pore water pressures will
develop and the slope may fail as the effective stress, and
consequently the shear strength as give by equation 4.8 will
be reduced. As time progresses, the excess water pressure
will dissipate, and the effective normal stress will increase
and at the same time the slope will be able to resist shear
more readily. In such cases, the critical time is during
formation of the slope.
Let us imagine that we have a sample initially on the virgin
consolidation line at A and we conduct and undrained test on
the sample. Such a sample might be a recent wet clay or a
loose sand. Such a test also implies NO VOLUME
CHANGE (as under the stresses concerned, water is
incompressible). As there is no movement of water we will
see that the pore water pressure will increase and as it does
so, the effective normal stress will reduce and so the point A
will move to the left until failure eventually occurs.
In the case of over consolidated clays (OCR > 1.7 or dense
sands), the initial rapid imposition of load is once again to
fast for equilibrium drainage, and negative water pressures
will develop thereby increasing the effective stress and the
slope will be relatively stable. However, as time progresses,
water will be sucked in and as it does so the negative water
pressure will equilibrate and the effective stress will fall
causing a reduction in the shear strength. In such cases, the
soil will weaken with time. It will be more stable
immediately after formation but may collapse later. Several
railway cuttings in London Clay failed about 50 - 100 years
after formation for precisely this reason.
If we do a drained test on a sample starting at exactly the
same point on the virgin consolidation line, then we see that
water is extruded and the point A will move vertically
downwards until failure occurs.
If we plot the end points of all tests, whether undrained or
drained, we will find that they all lie on a straight line (on
the logarithmic plot) which is parallel to the virgin
consolidation line. This line is called the Critical Voids
Ratio Line (or more usually, the Critical State Line).
What happens if the sample is not on the virgin
consolidation line, and it is over consolidated and lies at a
point such as B?
We will find that in an undrained test (constant volume), a
negative pressure develops, and the effective stress will
increase as the point B moves to the right and eventually
failure will occur when we reach the Critical State Line.
If on the other hand we do a drained test, then water will be
sucked in as the volume increases and we will move upwards
towards the Critical State Line.
Thus the Critical State Line represents the end point of
shearing from wherever the sample began.
Fig. 4.16 Critical State Line
It should be noted that if the over consolidation ratio (OCR)
is about 1.7 at the start of the test the stress point lies exactly
on the Critical State Line, and in this case we would expect
space reserved for additional notes.
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