CIVL 272 Soil Mechanics
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Topic V: A Critical State Model to Interpret Soil
Behavior (continued)
5. Critical state model (for clay)


Critical state (ultimate failure)
It is a stress sate reached in a soil when continuous
shearing occurs at constant shear stress and constant
volume.
Example for critical state model
CIVL 272 Soil Mechanics

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Equations for critical state line, normal consolidation
line, and swelling recompression line (unloadingreloading line).
In the p`-q-e space
Critical state line (CSL) in the p’-q space
q = Mc p’
6 sin 'cs
Mc 
3  sin 'cs
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CSL, NCL, and SRL (URL) on the e-lnp’ space
v = 1+e (specific volume)
Critical state line (CSL)
CSL:
v     ln p' or e    1   ln p'
Normal consolidation line (isotropic loading, NCL)
NCL:
v  N   ln p' or e  N  1   ln p'
Swelling-reloading line (SRL) or Unloading-reloading line
(URL)
SRL:
v  v k   ln p' or e  v k  1   ln p'
(elastic deformation on the SRL)
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where M, , N, ,  are all material constants and can be
determined from the triaxial test. The vk is not a material
constant. It is a state variable that characterizes the yield
locus for current e, p’ state (the projection of the yield
surface on the e-p’ space is the SRL). The vk can be
derived through known e and p’.

One-dimensional compression line (1-D consolidation,
K0-consolidation)
On the p’-q space
q = 0p’
On the e-lnp’ space
v  N 0   ln p' or e  N 0  1   ln p'
CIVL 272 Soil Mechanics
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Wet and dry side
It is convenient to distinguish between samples which
lie above and to the right of the critical state line in vlnp’ space and those which lie below and to the left. The
first group of samples will be termed wet of critical, for
each sample has a moisture content higher than that of a
sample on the critical state line at the same value of p’,
and the second group as dry of critical. This
classification is useful in that it groups together samples
with similar pore pressure response and volume change
behavior.
Wet side: Normally or lightly
overconsolidated clay; loose
sand (compression)
Dry side: Heavily overconsolidated
clay; dense sand (expansion)
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The soils at the wet side of critical tend to contract while
shearing, so the volume will decrease for a drained test
and positive pore water pressure can be generated for an
undrained test. The ultimate strength is the critical state
shear strength.
The soils at the dry side of critical tend to dilate while
shearing, so the volume will increase for a drained test
and negative pore water pressure can be generated for an
undrained test. The ultimate (maximum) strength is the
peak shear strength which is higher than critical state
shear strength.

The complete state boundary for clay (the complete
yield surface)
The complete state boundary (yield surface) is
assembled from all the effective stress paths for
normally consolidated clay, lightly overconsolidated
clay, and heavily overconsolidated clay at different
initial stress states and different test conditions (drained
and undrained).
p’-q-e space
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p’-q space
On the wet side:
The yield locus is some sort of ellipse
On the dry side:
The Hvorslev surface represents the envelope of peak
strengths, which expand as the preconsolidated mean
effective stress, pc’ increases.
The maximum value of q/p’ would be when 1’ was
large and 3’ was small. If the soil could not withstand
tensile effective stresses, the highest value of q/p’ that
could be observed would correspond to 3’ = 0 (3’
cannot be negative). Then, for a triaxial compression test,
1
q  1' ; p'  1' ; q / p'  3
3
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The yield surface for the Modified Cam clay model is
Pc’/2
CIVL 272 Soil Mechanics
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Prediction of soil behavior
Normally consolidated clay
(1) Isotropic compression
(2) Drained compression (CD test)
(3) Undrained compression (CU test)
CIVL 272 Soil Mechanics
Heavily overconsolidated clay
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CIVL 272 Soil Mechanics

Prediction of soil behavior (example in the textbook)
Lightly overconsolidated clay (R02), drained test
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CIVL 272 Soil Mechanics
Lightly overconsolidated clay (R02), undrained test
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CIVL 272 Soil Mechanics
Heavily overconsolidated clay (R0>2), drained test
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CIVL 272 Soil Mechanics
Heavily overconsolidated clay (R0>2), undrained test
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CIVL 272 Soil Mechanics
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Undrained shear strength su for normally consolidated
clay
1+es
We have two equations:
q s  Mp s'
e s    1   ln p s'
From Equation (2), we can get
  1  es
p s'  exp

(1)
(2)
CIVL 272 Soil Mechanics
From Equation (1),
q s  Mp s'  M  exp
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  1  es

For the undrained condition, es=e0=e (volume is kept
constant, so the void ratio is kept the same as the initial
void ratio e0) and therefore, the undrained strength is
q s' M
 1 e
su 
 exp
(3)
2
2

Equation (3) indicates that the undrained shear strength
for a given soil is uniquely determined by the soil initial
void ratio, initial water content, or the initial
confinement. As the initial void ratio of a given soil is
determined before shearing, its undrained strength is
also determined and independent of the total p.
For natural 1-D consolidated soils (1-D consolidation),
e  N 0  1   ln p' (equation for 1-D consolidation line)
Substituting the above equation into Equation 3, we can
get
  1  ( N 0  1   ln p' ) M
M
   N0

s u  exp
 exp
 ln p' 
2

2
 

  N0 
M
  exp
 p'
 
2
This equation indicates that su linearly increase with p’,
which in turn increase with depth or the effective
overburden pressure (z’).
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Recall the equation in Table 5.5
-------------------------------------------------------------------

Example:
The following parameters are known for a saturated
normally consolidated clay: N = 2.48,  = 0.12, =2.41
and M=1.35. Estimate the values of principle stress
differences and void ratio at failure in undrained and
drained triaxial tests on specimens of the clay
consolidated under an isotropic pressure of 300 kN/m2.
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Undrained test
After
normal
consolidation to 300
kN/m2 (pc’), the
specific volume (vc)
is:
v c  N   ln p 'c
 2.48  0.12 ln 300
 1.80
In an undrained test
on a saturated clay
the volume change
is
zero,
and
therefore
the
specific volume at
failure (vf) will be
1.80, i.e. the void ratio at failure (ef) will be 0.8.
Assuming failure to take place on the critical state line
(CSL),
q f  Mp f '
and the value of pf’ can be obtained from the equation of
CSL , v f     ln p 'f , therefore,
  vf
q f  M  exp(
)

2.41  1.80
 1.35  exp(
)  218 kN / m 2  (1   3 ) f
0.12
CIVL 272 Soil Mechanics
Drained test
For a drained test,
the slope of the
effective stress path
on the p’-q space is 3.

q f  3 p 'f  p 'c

q

 3 f  p c ' 
M

Therefore,
3  M  Pc'
qf 
3 M
3 1.35  300

3  1.35
 736 kN / m 2
 1   3 f
Then,
q ' 736
pf '  f 
 545 kN / m 2
M 1.35
v f     ln p f '  2.41  0.12 ln 545  1.65
ef  vf  1  1.65  1  0.65
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