Shear strength
Near any geotechnical
construction (e.g. slopes, excavations, tunnels and foundations) there
will be both mean and normal stresses and shear stresses. The mean or normal
stresses cause volume change due to compression or consolidation.
The shear stresses prevent collapse and help to support the geotechnical
structure. Shear stress may cause volume change.
Failure will occur when the shear stress exceeds the limiting shear stress (strength).
In practice, the state of stress in the ground will be complex.
There are simple theories for two special cases.
 Triaxial (axial symmetry)
 Parameters used for analysis:
 · deviator stress
 · shear strain
 · normal stress
 · volumetric strain
 · specific volume


 Direct or simple shear
 Parameters used for analysis:
 · shear stress
 · shear strain
 · normal stress
 · volumetric (normal) strain
 · void ratio


It is not possible to draw a Mohr circle for a shear test unless stresses on vertical planes are measured.
In very simple terms, the strength of soil is the maximum shear stress
(t_{f})
it can sustain, or the shear stress acting on a shear slip surface along
which it is failing. There are three distinct strengths: peak, critical
(or ultimate) and residual. Shearing may be simple or direct.
Drained direct (ring) shear
 Drained simple shear

We explore the relationship between the maximum shear stress and the effective normal stress
(s') by ploting a graph of t_{f}
against s'.
Some aspects of the behaviour show up more clearly if we normalise the data by plotting
t_{f} / s' against
s' / s'
The peak strength is the maximum value of the shear stress or the maximum value of the
ratio of shear stress to effective mean or normal stress. For drained tests these will
occur simultaneously, for undrained tests they may occur at different points and the
definition used here is the maximum stress ratio.
 Peak strengths can only occur at shear stresses above the critical state line and at water contents below the CSL.
 Peak states can occur anywhere in the regions above and below the CSL.
 Peak states at the same water content fall on unique smoth envelopes.
 The peak states can be represented on a graph in 3 dimensions.
 All peak states fall on a surface in this graph.
The circle represent the results of a set of shear tests on samples at the same moisture content but different normal stresses.
The squares represent the results of a second set of tests at a different moisture content.
We normalise the data by plotting
t_{f} / s' against s' / s'
The basic peak states, before normalisation, fall on different curves each
for a particular water content or void ratio.
After normalisation all the peak states fall on a single unique envelope.
Equations
At a given water content or void ratio, all the peak states fall on a single smooth envelope.
This may be represented in one of two ways:
As a power law
t'_{p} = Bs' ª
As a linear (MohrCoulomb) envelope
if the curvature is relatively small over a given range.
t'_{p} = c'_{p} + s'
tan f'_{p}
The parameters a, B and c'_{p}, f'_{p}
depend on the water content or void ratio.
Even at a given water content or voids ratio, the parameters c'_{p}
and f'_{p} depend on the range of stress
for linear approximation.
 
Peak strength in triaxial tests
The basic peak states, before normalisation, fall on different curves each
for a particular water content or specific volume.
After normalisation all the peak states fall on a single unique envelope.
equations
At a given water
content or specific volume all the peak states fall on a single smooth
envelope. This may be represented in one of two ways:
As a power law
As a linear envelope
if the curvature is relatively small over a given range.
q'_{p} = G_{p} + H_{p} p'
The parameters a, b
and G_{p}, H_{p} depend on the water content or voids ratio.
Even at a given water content or voids ratio, the parameters G_{p}
and H_{p} depend on the range of stress for linear approximation.
Peak strength and dilatancy
Stresses and displacements
in a shear sample are analagous to the forces and movements of a friction
block on an inclined plane.
At critical
state
y = 0 and t'
= s' tanf'_{c}
The additional stress ratio (above the critical state) is due to the
rate of dilation
tan y = dv/dh
Peak state and initial state
The peak stress ratio depends on the initial state given by the initial
overconsolidation ratio. The maximum rate of dilation increases with overconsolidation
ratio. For the same initial overconsolidation ratio (i.e. A and A') the
peak stress ratio is the same.
Critical state strength
At its critical state
soil continues to distort at constant effective stress and at constant
volume.
This applies for turbulent flow of the particles: if the flow becomes
laminar, as in clays at large strain, the strength falls to the residual.
When soil is at its critical state there is a unique relationship between
shear stress, effective normal stress and water content (or specific volume
or void ratio). Critical states are unique and do not depend on initial
state or stress path.
Critical states correspond to shear strains typically 10% to 40%.
Critical shear stress (critical state strength) increases with increasing
effective normal stress and with decreasing water content.
The critical state line can be represented as a graph in 3 dimensions.
For isotropic compression, shear stresses are zero and the isotropic normal
compression line can also be represented.
Critical state strength in shear tests
The graphs show the critical state line. If you know either s'
or e at the critical state you can calculate the critical state strength
t'.
f'_{c} c_{c} and e_{G}
are soil parameters.
The onedimensional normal compression line (NCL) for zero shear stress
is also shown.
The critical state line can be normalised with respect to the critical
pressure s'_{c} or the equivalent void
ratio e_{l}. The critical state line and the isotropic normal compression
line both reduce to single points.
Critical state strength in triaxial tests
The graphs show the critical state line. If you know either p' or v at
the critical state you can calculate the critical state deviation stress
q'.
M l and G are
soil parameters.
We should really use subscripts c and e for compression and extension
as the values are slightly different.
The isotropic normal compression line corresponds to zero deviator
stress
The critical state line can be normalised with respect to the critical
pressure f'_{c} or the equivalent specific
volume v_{l}. The critical state line and the isotropic normal
compression line both reduce to single points.
Typical values of critical state strength
parameters
The critical state parameters are basic soil parameters and they depend
principally on the nature of the soil grains. For fine grained soils the
CS parameters are related to the Atterberg limits; for coarsegrained soils
they are related to the mineralogy and shape of the grains.
Typical values 
l  G  M  f' 
high plasticity clay 
0.16  2.45  0.89  23º 
low plasticity clay 
0.10  1.80  1.18  29º 
quartz sand 
0.16  3.00  1.28  32º 
carbonate sand 
0.34  4.35  1.65  40º 
For finegrained soils the gradient l or
C_{c} of the critical state line is related to the Atterberg limits
by
C_{c} = (I_{p} x G_{s}) / 200
l = (I_{p} x G_{s}) / 460
For many soils
the critical state lines all pass through a single point called the W
(omega) point.
v(_{W}) = 1.25 p'(_{W}) = 10MPa
e(_{W}) = 0.25 s'(_{W})
= 15MPa.
Undrained strength
The critical state
strength is uniquely related to the water content.
If the soil is sheared without change of water content (i.e. undrained)
its strength remains the same. This is called the undrained strength s_{u}.
But if the soil is not undrained and the water content changes the strength
will also change.
The undrained strength is directly related to the liquidity index I_{L}.
Some authors give slightly different values for s_{u} but
s_{u} at PL (i.e. I_{L}=0) is always 100 times
s_{u} at LL (i.e. I_{L}=1)
Residual strength
This is the very
lowest strength which occurs after very large displacements. For sands
the residual strength is the same as the critical state strength. For clays
the residual is about ½ the critical state strength. For clays the
flat clay particles become aligned parallel to the direction of shear.
The residual strength occurs after very large (>1m) movements and is
not usually relevant for geotechnical engineering where generally ground
movements must be small. However, on old landslides there may have already
been very large movements and in such cases the strength may already be
at the residual before construction starts.
Residual strength: equations
Residual strength
applies to clays after very large shear displacements when clay particles
have become aligned in welldefined shear zones or slip planes.
 Drained case
 t = s' tan f'_{r}
 f'_{r} = residual friction angle
 In clays f'_{r} can be less than ½f'_{c}.
 For London Clay,
 f'_{c} »22º and f'_{r}»10º.
 In mixed soils f'_{r} depends on the quantity of clay present.
 Undrained case
 t = s_{ur}
 s_{ur} = undrained residual strength
 (depends on water content)