1. Martin Fahey The University of Western Australia
Geomechanics 255 ( )
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Geomechanics Group
School of Civil & Resource Engineering The University of Western Australia
Part 2: Soil Strength
Professor Martin Fahey

2. Martin Fahey The University of Western Australia
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Outline
Shearing behaviour of sand (cohesionless soil)
friction
dilatancy
concept of critical state (critical void ratio)
Shearing behaviour of clays (cohesive soil)
critical state concept for clayey soils
drained and undrained shear strength in triaxial tests
relationship between pore pressure change in undrained tests, and volume change in drained tests
The aim is to show that the shearing behaviour of all soils (sands and clays) can be presented within the unified framework of Critical State Soil Mechanics. This links the volume change behaviour in drained shearing with the pore pressure changes that occur when drainage is not able to occur. For sands, undrained behaviour generally can only occur when the boundary conditions prevent – otherwise, shearing is generally slow enough to allow any pore pressures (positive or negative) that tend to occur to dissipate as the shearing progresses. (The exception may be very fast loading, as in an earthquake, or where the scale of the problem is very large, as with very large offshore gravity platforms). On the other hand, the permeability of clay soils is so low that it is very difficult to apply loads slowly enough for drained conditions to apply, and hence many problems involving applying loads to clayey soils deal with the undrained shear strength.
Lecturers:
Professor Martin Fahey, A1.08,

3. Soil Strength: Angle of Internal Friction f'
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Soil Strength: Angle of Internal Friction f' N N N f' R F F F F
f': Angle of internal friction; m: coefficient of friction
tan f' = m = F/N f'
Soil Strength
The shear strength of all uncemented soils (sands or clays) is explainable purely on the basis of friction.
The basic parameter is the coefficient of internal friction (m) or the angle of internal friction (f').
The angle of friction f' can be thought of as the angle between the resultant force R and the vertical (where R is the resultant of the normal force N and the shear force F). It is also seen directly in the two cases shown on the bottom of the diagram above.
For a frictional material, the shear strength on a potential failure plane is a function of the normal stress on the failure plane.
(In all cases, it is the effective stress that matters - that is, the total stress less the pore water pressure. See next slide. Think of a hovercraft - it moves easily because the total stress due to the weight of the hovercraft is balanced by the air pressure generated underneath, giving zero effectives stress, and hence very little resistance to horizontal movement). f'
f': Angle of plank when block slides
f': Angle of repose of sand heap

4. Principle of Effective Stress
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Principle of Effective Stress N F
Water pressure u N F F
Effective Stress
Since saturated soil is a two-phase material (soil particles and water), the normal stress that is effective in producing the frictional strength is the stress between the particles. The principle of effective stress states that the effective stress is the total stress less the pore water pressure.
The strength behaviour of soil is dominated by the behaviour of the pore pressure. Any disturbance to the soil that tends to cause the soil to reduce in volume (compress) will result in an increase in the pore water pressure. This can result in a previously-stable foundation becoming unstable.
Compression of previously stable the soil can be caused by a number of factors, such as cyclic shear stresses due to wave loading or due to earthquakes.
In some circumstances (eg for foundations for offshore structures), the foundation design is primarily governed by the behaviour in storm loading or in earthquake loading, not only because of the increased loads on the foundations due to this type of loading, but also to the reduction in shear strength brought about by pore pressure increase resulting from the cyclic nature of the loading.
The most extreme form of strength reduction involves complete liquefaction of the soil, when the shear strength reduces to zero due to the effective stress being reduced to zero.
Note: As u ® s (i.e. s' ® 0) strength (t) ® 0 (liquefaction)

5. Direct Shear Box Apparatus
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Shear Box Apparatus
The most basic form of strength test for soil is the shear box test, carried out in a shear box apparatus. This consists of a box, rectangular or circular in plan view, that is split horizontally into two halves, with the top half being able to slide over the bottom half.
The box is filled with the soil to be tested, a vertical stress is applied to the top of the sample using a dead weight N on a hanger, bearing onto a piston.
The test consists of forcing the bottom half to move horizontally using a screw-drive system (the bottom half sits on roller bearings). The top half is restrained from moving by a load cell (or proving ring), so that the shear force generated between the two halves of the box (ie in the soil) is recorded, and this gives the shear stress on the horizontal plane through the soil that coincides with the division between the two halves of the box (the plane P-P in the diagram).
Dial gauges (or displacement transducers) measure the horizontal displacement (x) of one half of the box relative to the other, and also the vertical movement (y) of the piston relative to the top half of the box.
Drainage is allowed through porous discs on the top and bottom of the sample (for tests on saturated soil).
(This diagram, and a number that follow, taken from “A Guide To Soil Mechanics”, by Malcolm Bolton. See reading list for this course).

6. Martin Fahey The University of Western Australia
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Other Versions at UWA
Pneumatic jack (computer controlled) to apply vertical load
Load cells
Direct Via Lever
Hangers for load
Other Versions
The simplest versions of the direct shear box use hangers to apply a vertical load (either directly, or via a lever system, as shown in the upper photograph above), and measurements of displacement (vertical and horizontal) and horizontal force are taken manually (reading dial gauges). The horizontal force is measured using a "proving ring" - a stiff circular ring which acts like a very stiff spring. As it is compressed, the amount of compression is measured across the diameter by a dial gauge. The load is then obtained from a calibration factor (which is effectively the spring stiffness).
Another version, which can be seen in the UWA lab (lower photograph) applies the vertical load through a computer controlled air (pneumatic) jack. The loads are measured using load cells (strain-gauged load measuring devices), with readings taken by computer. The test is computer controlled, and both monotonic (loading a single time) or cyclic loading can be applied.
Another important feature is that the vertical load can be varied by the computer according to some pre-determined criterion. It is possible to carry out tests with this device in which vertical movement can be suppressed completely, and the change in load to achieve this measured. Thus, with this device, dilation can be suppressed by increasing the load, or contraction can be prevented by reducing the load. These alternative "load paths" will be discussed later.

7. Behaviour of Sand in Direct Shear Box
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Behaviour of Sand in Direct Shear Box
Shear behaviour of dense Sand in the shear box
A Direct Shear Box allows a soil sample to be sheared through the mid-plane, with the normal stress on this plane being applied by a vertical dead load (N) on the lid of the box.
The shear force versus displacement plot (F - x) shows the shear strength rising to a peak, and then reducing (“strain softening”) back to a residual value. The lid of the shear box moves upwards due to dilation of the soil. A plot of vertical movement of the lid (y) versus horizontal movement of the box (x) has a maximum slope at the point of maximum strength, with the upward movement ceasing as the residual strength is reached.
At any point, the angle that the tangent to the displacement (y-x) curve makes with the horizontal is called the angle of dilation () of the soil.

8. Direct Shear Tests on Sand
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Direct Shear Tests on Sand
D, M, L: Dense, Medium, Loose
Direct shear behaviour of sand
The plots show the shear behaviour on three different samples of sand prepared at three different densities (dense, medium and loose) at three different normal stresses (20, 40 and 80 kPa). The shear stress (t) is plotted against the horizontal displacement (x), where:
with F being the shear force, and A being the cross-sectional (plan) area of the shear box. These show:
for any particular density, the shear strength depends on normal stress
at any particular normal stress, the peak strength depends on (relative) density
the residual strength at any particular normal stress appears to be independent of density
From these tests, we can identify a peak shear stress and a residual shear stress for each of the 9 tests.

9. Direct Shear Box: Summary of Results
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Direct Shear Box: Summary of Results
Friction angles from direct shear tests on sand
When the peak and residual shear forces in the 9 tests on the previous page are plotted as shear stress (t = F/A) versus normal stress (s' = N/A), various values of the effective friction angle (f') can be determined.
The friction angle is the angle between the horizontal and the the line through each point: f' = tan-1 (t/s').
It can be seen that:
the peak friction angle (f'peak) is greatest for the dense test at the lowest stress level (50°)
f'peak reduces with increasing stress level
f'peak reduces as the density reduces
f'peak for the loose samples is the same as the residual value f'cv
The residual value of friction is called f'cv because at this stage, the shearing is occurring without any further change in volume (f'cv at "constant volume")

10. Relative Density – Density Index (ID)
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Relative Density – Density Index (ID)
Absolute value of soil density not so important – what matters is how dense is the soil relative to its maximum possible value and its minimum possible value ID
Densest possible state (emin, or rdmax)
(obtained by vibration under load)
1 or 100%
Density index ID (relative density) –
where density lies in the range min. to max. -
or rather where void ratio lies between loosest (emax) and densest (emin) state
RELATIVE DENSITY
Many aspects of soil behaviour depend on the density of packing. However, for any particular soil, the absolute value of the density does not tell us that much about how it will behave. What matters is where is the current density relative to the maximum possible value that soil can have, and the minimum possible value it can have.
There are standard tests to determine the values of maximum dry density (rdmax = emin) and minimum dry density (rdmin = emax). The former involves vibration in a standard manner under a surcharge load equivalent to 5 kPa. The latter involves very loose pouring, using a funnel.
The relative density is simply a measure of where the density lies between these limits (or rather, where the void ratio lies between the void ratio limits). High values (close to 1, or to 100%) indicate very dense state; close to 0 indicate very loose state.
Relative density could be defined as where the density lies between max. and min. values. However, it is actually defined as where the void ratio lies between max. and min. values. Example: rs = 2.65 t/m3, rdmax = 1.6 t/m3 (emin = 0.65), rdmin = 1.2 t/m3 (emax = 1.21), actual dry density rd = 1.4 t/m3 (e = 0.89), gives:
Loosest (stable) state (emax, or rdmin) (obtained by pouring with funnel)
As e approaches emin, ID approaches 100%.
Note: In this example, the dry density is ‘half-way’ between max. and min. values, but ID is not 50%.

11. Apparent Cohesion in Sand
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Apparent Cohesion in Sand
Mohr-Coulomb Failure Criterion: tf = c' + s' tan f'
Failure surface is actually curved
Straight line through tests results at s' of 40, 60 and 80 kPa implies a cohesion intercept (c') of 10 kPa
This implies a strength at zero effective stress: NOT CORRECT
Mohr-Coulomb failure criterion
The Mohr-Coulomb failure criterion is written as:
tf = c' + s' tan f'
where c' is the "apparent cohesion"
The cohesion intercept (c') implies that the material has a shear strength when the normal effective stress is zero. However, this is not the case. In reality, the Mohr-Coulomb failure surface is curved at low stresses, so that for uncemented materials it passes through the origin (no true cohesion).
(This is an example where extrapolation of test data outside the range of the data can give a misleading picture).
As will be shown later, the "cohesive strength" exhibited by clayey soils can be explained on the basis of friction and effective stress.
Note: Sands can have ‘true cohesion’ – if there is cementation for example. So, the weak limestones that we see along King’s Park on Mount’s Bay road would have some true cohesion, as well as friction.
“Apparent cohesion” c'

12. "Saw-tooth" Model of Dilation
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
"Saw-tooth" Model of Dilation
Dilation has effect of increasing the apparent friction angle on interface above the true value (f'cv)
Apparent friction angle from sawtooth model: f'peak = f'cv + n
Dilation angle = n
Observed relationship: f'peak » f'cv n (Bolton)
Collapsing material (negative dilation) shows friction angle less than f'cv
Saw-tooth model for dilation
Dilation occurs because, in the dense state, soil grains must "ride up over each other" to allow a shearing plane to develop.
The saw-tooth model provides a good analog for dilation. In the model, the fundamental material friction angle can be called f'cv. However, when the two halves of the block are sheared, with a vertical force N, the shearing resistance is greater than if the interface was horizontal. The extra resistance depends on the angle of the saw-tooth planes with the horizontal (n).
According to this model, the relationship between apparent peak friction angle f'peak and the fundamental friction angle f'cv is obtained simply from the geometry of the sliding plane and the fundamental friction f'cv :
f'peak = f'cv + n
This is also not a bad representation for sands, though Bolton found from experiment that it slightly overestimates the apparent friction angle, and proposed an alternative model:
f'peak = f'cv n
(Malcolm Bolton is a Professor of Soil Mechanics at Cambridge University, UK – and a frequent visitor to UWA).

13. Stress-Ratio Dilation Relationship (Taylor)
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Stress-Ratio Dilation Relationship (Taylor)
Peak stress ratio (tan f'peak)
DENSE
Stress ratio (t/s'n)
"Constant volume" stress ratio (tan f'cv)
LOOSE
dy/dx = 0
DENSE dx dy
Vertical displacement y (vol. strain)
Point of max. slope (nmax)
Friction-dilation relationship derived using work principles
Taylor examined the relationship between work input in shearing, work output due to dilation (vertical movement against the normal force) and work dissipated in friction, and showed that :
i.e. the relationship between the current or mobilised friction angle f ' and the fundamental friction angle f'cv is given by the current gradient of the dilation curve.
This also applies at the point of maximum shear stress, such that:
For typical values, this is not much different from Bolton's relationship.
Note that according to this relationship, the dilation rate is maximum at the point of maximum strength, is zero at stress ratio = tan f'cv (both in initial loading and at ultimate state, and is negative while f' is less than f'cv.
dy/dx = 0
LOOSE
dy/dx negative, increasing towards zero

14. Critical State Concept
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Critical State Concept
When sheared, state of soil tends to migrate to a unique line in t - s' - e space. This is called the critical state line (CSL).
CSL has same gradient as NC line (l)
Critical State Concept
The basis of the Critical State family of soil models is that when soil is sheared, the state tends to migrate towards ultimate states in t - s' - e space which lie along a single line, the Critical State Line (CSL). The projection of this line on the t - s' is a straight line with gradient f'cv. The projection on the e - s' plane is a curved line.
If plotted as e versus loge s', the CSL has a gradient of l.
If samples lie on the "dense" side of the CSL on the e - s' plane, then dilation will occur during shearing if the normal stress s' is kept constant (as in the direct shear box test).
If samples lie on the "loose" side of the CSL on the e - s' plane, then contraction will occur during shearing if the normal stress s' is kept constant (as in the direct shear box test).

15. Dilation depends on density and stress level
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Dilation depends on density and stress level
Critical State Line (CSL)
At low stress, even loose samples may dilate
LOOSE
Void ratio e
At high stress, even dense samples may contract
DENSE
Dilation is density and stress dependent
The relative density alone is not enough to determine whether a sample will dilate or contract when sheared at constant s'.
Effectively, for a dense sample, if the normal effective stress is sufficiently high, the tendency for dilation to occur will be suppressed. (Shearing under these circumstances is accompanied by some grain breakage - the serrations on the "saw" are broken off to allow the failure plane to develop; grains break rather than forcing other grains to ride up over them).
Similarly, for a loose sample, if the normal effective stress is sufficiently low, dilation can occur.
Thus, relative density (or density index ID) on its own is not a good measure of how soil will behave. For this reason, Bolton proposed a new "Relative Density Index" IR, that depends on the confining stress (p') where:
Normal effective stress s'n (or mean effective stress p')

16. Relative Density Corrected for Stress Level
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Relative Density Corrected for Stress Level
For plane strain, Bolton found that:
nmax (º) = 6 IR for plane strain
f´max – f´cv = 0.8 nmax
f´max – f´cv  5 Irº
For triaxial conditions
Must define 'dilatancy' in general as
where ev is volumetric strain = e1 + e2 + e3.
e1 is the major principal strain (generally ea in triaxial tests)
(negative sign, because expansion - I.e. dilation - is negative by normal sign convention, but want 'dilatancy' to be positive)
p' (kPa) n 10
100
1,000
10,000
0.2 (20%)
3.2º
0.5º
-2.3 (?)
0.5 (50%)
17.1º
10.2º
3.3º I D
0.8 (80%)
(30.9º ?)
19.9º
8.8º
Dilation (and hence peak friction angle) is density and stress dependent
nmax (º) = 6 IR for plane strain; f´max – f´cv = 0.8 nmax;  f´max – f´cv  5 Irº
shows that peak friction angle depends on IR, and hence it depends on both relative density ID and stress level.
at very low stress, can have very high dilation angle and hence very high peak friction angle
at high stress, can suppress dilation altogether - peak friction angle reduced to f´cv
-2.2º (?)
p' (kPa) n 10
100
1,000
10,000
0.2
3.2º
0.5º
-2.3º ?
0.5
17.1º
10.2º
3.28º I D
0.8
(30.9º?)
19.9º
8.8º
-2.2º ?

17. Drained & Undrained Shear Strength
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Drained & Undrained Shear Strength
Shear stress t
Drained strength sd
f'cv
Undrained strength su
Undrained strength su
Drained strength sd
s'n
Void ratio e
Suction increases effective stress
Positive pore pressure reduces effective stress
Dilation
Undrained test Þ no volume change allowed
Loose states eo
Dense states
Contraction
"Undrained" shearing behaviour of saturated soil (sand)
The critical state concept also applies if a sample is sheared with no volume change allowed. This occurs naturally for saturated samples if no water is allowed to leave or enter the sample during shearing (particularly if the "background" water pressure is high, as would be the case for sub-seabed sediments). The sample state will still tend to move towards the critical state, but with no change in volume occurring (void ratio e constant).
Undrained conditions also occur where the rate of shearing is fast enough for the time for water to escape (the consolidation time) is much greater than the time for the load to be applied (e.g. earthquake or wave loading, particularly for large structures, where the scale term "d" in the consolidation equations is large).
From the discussion in the previous slide, the confining stress for "dense" samples will have to increase to prevent dilation in an undrained test, whereas for "loose" samples, the confining stress will have to reduce, as shown above.
Where the zero volume change condition is being achieved by preventing drainage, the "confining stress" referred to in the previous paragraph is the effective confining stress (applied stress minus the pore water pressure u). Thus, for "dense" samples, the pore water pressure will have to reduce (suction) to keep zero volume change conditions, whereas for "loose" samples it will increase.
CSL
Normal effective stress s'n (or mean effective stress p')

18. Martin Fahey The University of Western Australia
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Triaxial Test
The triaxial test enables a variety of stress or strain controlled tests to be carried out on cylindrical soil specimens. Fa
scell
Area, A u
The triaxial test
This test is the most widely used shear strength test. It is suitable for all types of soil allowing a number of different stress paths to be carried out .
A cylindrical soil sample is placed on the pedestal and enclosed between rigid end caps inside a thin rubber membrane. Rubber “O-rings” are fitted over the membrane to provide a seal. The cell is closed and the specimen is subjected to a stress by filling the cell with water and increasing the pressure to a prescribed value.
With the cell pressure held constant, the axial load is then increased until the sample shears or the ultimate stress is reached. The variation in sample height (Dh), axial load (F), cell pressure (scell) and pore pressure (u) are monitored during loading.
The following parameters can then be deduced:
Deviator stress: q = sa – scell = sv – sh = Fa/A. In this definition, q can be positive or negative – i.e. can have positive (compressive) or negative (tensile) F)
Total axial stress: sa = scell + Fa/A
Mean (average) stress: p = (s1 + s2 + s3)/3 = (sa + 2.scell)/3 = (sv + 2.sh)/3
Pore pressure variation: Du = u - uo ( uo is the initial pore pressure, or back pressure)
Axial strain: ea = Dh/ho (ho is the initial height of the sample)
Volumetric strain : ev = DV/Vo (Vo is the initial volume of the sample)

19. One of the UWA Triaxial Systems
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
One of the UWA Triaxial Systems
Axial motor drive system
Cell cover lowered once sample in place
Sample goes here
UWA Triaxial Systems
In addition to “standard” triaxial systems (one of which is used in the laboratory experiment in this unit), there are a number of more sophisticated triaxial setups at UWA for research purposes (all built at UWA). These are used for applying complicated stress paths, and for cyclic loading testing of offshore samples (for research and for commercial clients), as well as more general testing.
The equipment is fully computer controlled, enabling long-duration tests to be conducted un- attended. The computer controls the test and logs the data.
Additional instrumentation includes:
Internal displacement transducers, attached over the central part of the sample, to enable strains to be measured away from the disturbed ends of the sample.
An internal load cell, to measure the axial load applied to the sample more accurately than an external load cell.
A method of sending shear waves through the sample, to enable shear wave velocity in the sample to be measured (the value of shear wave velocity measurement will be discussed in the "site investigation" part of this unit).
Cell pressure controller
Sample, enclosed in rubber membrane, with axial strain measuring devices attached
Control and data logging system

20. Triaxial Test: Background
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Triaxial Test: Background
Direct shear test useful, but limited
Know only 1 normal stress (s'n), don't know horizontal normal stresses
Failure plane pre-defined - must coincide with the shear box
Triaxial test still limited:
vertical and horizontal directions still principal directions
horizontal stress equal in all directions
“true triaxial” test would allow different s'1, s'2, s'3 on three faces of cubical sample
even more general - allow shear stresses to be applied to the three faces
s'v
s'v (=s'1)
s'1
thv
The direct shear test is very simple, but the complete stress state in the soil is unknown (the horizontal stresses are not controlled, or measured).
In the triaxial test, a more general stress state is applied, but this is limited. Thus, the horizontal stresses are "axisymmetric" - the same in all directions, whereas a general state of stress could have different horizontal stresses in different directions. Also, in the triaxial test, the vertical and horizontal directions are and remain principal directions (I.e. there are no shear stresses on vertical and horizontal planes, whereas in reality, shear stresses could be applied to these planes in a real problem. For example, when horizontal forces are applied to a foundation, the resulting shear stresses on horizontal planes means that this can no longer be a principal plane). This rotation of principal directions can result in strain or failure, but this cannot be investigated in a triaxial test.
A “true triaxial” test would allow the three principal stresses to be varied independently, but these would still be principal planes (no shear stresses on these planes).
A “simple shear test” is like a direct shear test, except that horizontal stresses are applied. In the UWA simple shear apparatus, the sample is in a pressurised cell, with a membrane, like a triaxial sample. The sample height is much less than the diameter. It can be loaded by increasing the vertical stress, or the cell pressure (or both) and by applying a horizontal shear stress to the top face. It is used extensively for offshore testing (both “monotonic” and “cyclic” loading).
s'h
s'h
s'h (=s'2)
s'2
s'h (=s'3)
s'3
“True triaxial” (s'1s'2 s'3)
“Simple shear”
Triaxial

21. Triaxial Test: Conduct of Test
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Triaxial Test: Conduct of Test
Almost always use saturated samples (using high backpressure uo to achieve full saturation)
Almost always consolidate the sample to some stress state (in situ stresses often) before carrying out the strength test
isotropic consolidation: vertical and horizontal stresses equal (increase cell pressure only, allowing drainage against constant back pressure)
s'h = s'3 = sc - uo, and s'1 = s'v = s'h = s'3 in this stage
anisotropic consolidation: generally vertical stress greather than horizontal stress: increase cell pressure and apply additional vertical load
s'h = s'3 = sc - uo, and s'1 = s'v > s'h = s'3 in this stage
“Shearing” phase (in the simplest test): increase the vertical load (stress) until the sample fails
other “stress paths” also possible - see later
Example: Anisotropic consolidation to in situ stress state:
Sample,10 m depth, water table 2 m below ground surface. Assume total unit weight above water table g = 18 kN/m3, and g = 20 kN/m3 below the water table.
In situ stresses on sample:
Vertical total stress sv = kPa
Pore water pressure (hydrostatic): uo = kPa
 Vertical effective stress s'v = sv - uo = kPa
Horizontal stress? Not usually known!!
Assume normal consolidation, with Ko = 0.6
Definition: Ko = s'h/s'v  s'h = Ko s'v
 s'h = 0.6 s'v = kPa
Pore pressure same in all directions: uo = kPa
 sh = s'h + uo = kPa
Therefore, consolidate with cell pressure sh = kPa; Back pressure uo = kPa
Vertical stress sv = kPa
But vertical stress = cell pressure + "extra vertical stress" (q)  q = sv - sh = kPa
(Note: It is not important that the back pressure be equal to the in situ pore pressure. What is important is that the effective stresses are correct - which can be achieved by higher total stresses and correspondingly higher back pressure. In our tests for offshore industry, we use very high back pressure - 1 MPa - to ensure saturation).
10 m
2 m
g = 18 kN/m3
g = 20 kN/m3
sv =
uo =
s'v =
s'h =
sh =

22. Stress Paths in Triaxial Tests
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Stress Paths in Triaxial Tests
Different stress paths in “shearing” phase:
keep cell pressure constant (Dsh = 0) and increase vertical stress (Dsv +)
keep vertical stress constant (Dsv = 0) and reduce cell pressure (Dsh -)
keep vertical stress constant (Dsv = 0) and increase cell pressure (Dsh +)
keep cell pressure constant (Dsh = 0) and reduce vertical stress (Dsv -)
vary both cell pressure and vertical stress in some predetermined way, to produce any type of stress path
Stress path in q-p space: Dq = Dsv - Dsh Dp = (Dsv + 2Dsh)/3
Dsh = 0 and Dsv = +  Dq = +Dsv and Dp = +Dsv/3  Dq/Dp =3
Shearing phase
Shearing phase q q
Stress path: a plot showing how the stresses vary during a test.
In this case, this is a Total Stress Path (TSP).
In this case, shearing starts from an isotropic stress state, following isotropic consolidation.
Total stress paths for 2, 3 and 4 above:
2. Dsv = 0, Dsh = –  Dq = 0 – (– Dsh) = + Dsh and Dp = +2 (– Dsh)/3 = – 2/3Dsh
 Dq/Dp = 1/(–2/3) = –3/2. Negative slope, with Dq positive from above, and Dp negative
3. Dsv = 0, Dsh = +  Dq = Dp =
4. Dsv = –, Dsh = 0  Dq = Dp =
In this case, shearing starts from an anisotropic stress state, following anisotropic consolidation. 3 q 3 1 1 +3
Anisotropic consolidation phase p p
Anisotropic consolidation phase –2 p

23. Total and Effective Stress Paths (TSP, ESP)
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Total and Effective Stress Paths (TSP, ESP) q
TSP: Total stress path (imposed by apparatus)
ESP: Effective stress path (soil response) 3 1
Du (+)
(ESP) B
q = q'
(b') B'
"Standard" stress path: sh constant sv increased to failure
sv increasing
sh constant
Dq = Dsv
Dp = D sv/3
Dq/ Dp = 3
(Du may be negative)
p, p'
Total and Effective Stress Paths
The “stress path” is simply a plot of how the state of stress changes during a test (or during the application of a foundation load in the field).
In a triaxial test, the most common test is to:
consolidate isotropically (q = 0) to some initial effective stress state - point A)
carry out an undrained loading tests in which the cell pressure is kept constant and the axial stress is increased to failure (e.g. to point B)
The total stress path (TSP) is therefore predetermined (A-B). It has a gradient of 3:1 in q-p space.
The soil response (or more precisely the pore pressure response) determines the horizontal offset of the effective stress path (ESP) from the TSP. If the sample tends to reduce in volume (but is prevented because the test is undrained), the pore pressure increases, and the ESP will lie to the left of the TSP (e.g.. A-B' above). Conversely, if the sample tends to dilate, suction will be generated, and the ESP will lie above the TSP (A-b' above).
Note that the effective stress path is generally curved, as indicated above, but in the following discussion, it will be shown as straight, for simplicity. Note also that the pore pressure generated depends on the soil response (the ESP) and on the applied total stress path (TSP), which might be different from that shown above. p' A p
p' = p - Du

24. Drained & Undrained Strength (Clays)
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Drained & Undrained Strength (Clays)
CSL
Deviator stress q Bd
TSP
Drained strength sd
Au , Bu
Du -
Undrained strength su
Undrained strength su
Du + 3 Ad
Undrained strength depends on p'o and OCR
Drained strength sd
ESP 1 A B
Void ratio e
mean effective stress p' Ad
NC line
Dilation
Undrained test Þ no volume change allowed B
"Wet of critical" eo A
Contraction
Critical State concept applied to clays
The same Critical State concept can be applied also to clayey soils (including calcareous muddy and silty materials).
The initial state of the clayey soil is achieved by some consolidation history, as explained previously. Thus, the soil may be on the normal consolidation line (NCL) if it has never been stresses to a higher level than it is at present. It may also be overconsolidated, meaning that it has been stresses to a high level, and then the stress has been reduced allowing swelling to occur. In this case, the state lies on an overconsolidation line.
For NC soils and "lightly" overconsolidated soils, the soil "state" will lie to the left of the critical state line (CSL) in the e - p' plane. These samples will tend to compress in a drained triaxial test, as shown above. Samples with initial state to the left of the CSL on the e-p' plane will tend to dilate.
In undrained tests, the pore pressure generated during shearing will be either positive or negative, depending on whether the state lies to the right or the left of the CSL.
In the examples shown, the two samples (A and B) have the same undrained shear strength (su) even though the initial p' for the OC sample A is much lower than for the NC sample B. However, the drained shear strengths (sd) are different. For A, dilation occurs in the drained test (A - Ad), whereas for B compression occurs (B - Bd).
OC line Bd
"Dry of critical"
CSL
Mean effective stress p'

25. Initial and Final Undrained Strength
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Initial and Final Undrained Strength q
su for NC soil increases after consolidation
Tank or GBS ® Dsv su
su after consolidation
CSL
suo
NC soil
In situ su 1
Depth (m) e p' k
In situ eo
su after consolidation
e after consolidation
NCL
In situ su
su = k.z (k = 1 to 2 kPa/m)
(or su = suo + k.z)
CSL
Strength increase due to applied load
For many soil deposits (some onshore, many offshore), the initial state is normally consolidated (on the NC line). For some onshore clay layers that are normally consolidated, there is often an overconsolidated surface "crust". The shear strength versus depth profile will therefore depend on the effective overburden pressure (or the mean effective stress p'), the gradient of the CSL on the q - p' plane, and the "spacing" of the NCL and CSL on the e - p' plane. (These are parameters of the Critical State models). For fine grained calcareous soils, the su-z relationship may be linear, with zero strength at the surface:
su = k.z
where su is in kPa, z is the depth in metres, and k is the gradient of the line (kPa/m).
There may also be some finite shear strength at the surface (suo) as indicated by the dashed line above.
If an embankment, oil tank, or large gravity base structure is placed on the surface, the total stress increases, and this will eventually lead to an increase in the shear strength. (This is illustrated in a very simplistic way in the diagrams above on the left).
However, it may take a very long time for this strength increase to occur, especially for very fine grained materials (low cv) or large structures (large "d").
Therefore, the stability of a structure increases with time after construction.. p'
How long for strength increase to occur ???
GBS ® Dsv ® p'

26. Staged Loading (Undrained)
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Staged Loading (Undrained) q
CSL
Fully drained sd 6
su after two increments 5 4
In situ su
Dq due to total load > in situ su
® failure if applied in 1 increment 2 3
TSP 1 p' e 2 1
ESP in undrained loading
Consolidation between increments
In situ eo 4 3
NCL
Staged construction
In the previous slide, the point was made that placing a structure on normally consolidated soil increases the mean effective stress (p') after consolidation has occurred. However, it also increases the deviator stress (shear stress) immediately. This increase may be greater than the in situ su value, and failure will occur, as shown graphically above.
However, if the load is placed in stages, with time being allowed for consolidation between stages, a greater total load can be applied without causing failure.
The case illustrated involves placing the total load in two stages. The steps are labeled 1, 2 etc. Step 1-2 is the effective stress path resulting from application of the first load. This load is less than that required to cause failure. Then consolidation is allowed (2-3). The second load increment (3-4) takes the state close to failure. Subsequent consolidation (4-5) results in a further increase in shear strength, leaving a reasonable margin of safety under the eventual total load. (Undrained shearing at this stage would cause failure at point 6).
The idea with staged loading is to try to carry out drained loading (follow the TSP as closely as possible). Staged loading is very common with embankment construction on soft ground - e.g.. the construction of the freeway embankments on Burswood Peninsula. In principle, it could also be carried out offshore.
e after two increments 5 6
CSL p'

27. Drained Tx Tests: Silica & Calc. Sands
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Drained Tx Tests: Silica & Calc. Sands
Calc. sand (Dog's Bay)
Silica sand
Dilation
Dilation
Drained triaxial tests on silica and calcareous sands
The plots show the results of drained triaxial compression tests on a silica sand on the left and a calcareous sand on the right. (The calcareous sand is from Dog's Bay, a beach on the west coast of Ireland. This is a very angular sand, with in situ void ratios of about 2).
The silica sand tends to dilate at all stress levels, though the stress level does affect the rate of dilation.
However, the Dog's Bay sand shows a very strong dilation behaviour at low stress, and a very strong compression behaviour at high stress.
This difference in behaviour is due to:
the much more angular nature of the calcareous sand compared to the silica sand; and
the much greater compressibility of the calcareous sand, as previously shown.
The greater compressibility is also a result of the angularity, and the weaker nature of the sand grains (partly because calcite is softer than quartz, but also because the asperities on the angular grains break off easily).

28. Drained & Undrained Tx Tests, Calc. Sand
Geomechanics 255
Soil Strength
Martin Fahey
The University of Western Australia
Drained & Undrained Tx Tests, Calc. Sand
Dog's Bay
Dog's Bay
TSP
Drained
Undrained
Drained and undrained shearing of calcareous sand
The two plots on the left show the stress-strain (q-e) and volume strain behaviour of the calcareous sand in drained tests.
The plots on the right show the undrained behaviour. The upper one shows the effective stress paths (ESP) on the q-p' plane (one TSP is also shown for comparison). The lower plot shows the pore pressure changes versus strain. This shows that at low initial confining stress, the pore pressure reduces from the initial value (indicating a dilational tendency), but as the initial confining stress level increases, this tendency is suppressed, and the pore pressures generated are positive.
Positive pore pressure causes the TSP to "lean over to the left". Note that having reached a maximum value early in the test, the pore pressure starts to reduce again. From then, the stress path tends to "move to the right" again. This is not covered in the basic critical state model discussed earlier.
The point farthest to the left on each ESP is called the "phase transformation" point or the "characteristic state" point. It corresponds to the change from a compression tendency to a dilational tendency. This point is very important for cyclic loading.
Important point: Pore pressure change in undrained tests is directly related to volume change behaviour in drained test.