WEEK 11
Soil Behaviour at Large Strains: Part 1
16. Overview: ‘The’ strength, or strength’s’?
What is soil strength? In usual contexts, it means the peak strength, or the maximum stress
(or stress ratio) that soil can bear, mobilised at large strains. However, soils’ stress-strain
relationships exhibit a variety of strain-hardening/softening types, particularly at large
strains, depending very much on soil types, density, drainage conditions, deformation
modes and other factors. It is difficult in quite a few situations to define a unique strength
(‘the’ strength) without considering relevance to a particular engineering problem in interest.
τ τPeak strength
Critical State?
Strain softening
Strain
hardening
1
γ γ
Residual strength
(see Week 12)
hardening
Over-consolidated and/or cemented clays Normally consolidated clays
Strain
hardening
Sand; undrained conditions
(Verdugo, 1992)
Sand: drained conditions
(Cole, 1967; reproduced from Oda, 1975)
17. Peak shear strength
17-1. Drained strength and undrained strength
First of all, make sure that you understand the relationship between total and effective
stress paths under drained and undrained conditions, discussed in Week 3.
Consider triaxial compression of soil obeying the Mohr-Coulomb failure criterion:
q
φ
φ
′−
′
sin3
sin6
1
( Week 4)
1
3 ( Week 1)
uu cq 2= u∆
u∆
Drained path
uu cq 2=
Half the maximum value of q (qu) is
the undrained shear strength, cu(see the right diagram for why it
should be half. Recently, it is
encouraged to express it as Su).
The strength defined in terms of
effective stress (the envelope) and
the undrained shear strength are
related via the pore water pressure
development, ∆u.
2
p′
Difference between the drained and undrained
paths are due to excess pore water pressure, ∆u
0p′Undrained path:
Moderately contractive
Very contractive
σ ′
τ
1σ ′3σ ′ q
2/qcu =
φ ′
Now imagine that isotropic pressure is applied to the soil, increasing the total stress by ∆p.
Long-term strength: The strength available after any change in the total stress is converted
into a change in the effective stress via consolidation and dissipation of
pore water pressure.
q
0p′
Undrained effective path
uu cq 2=
0p
p
p
∆+
0
p
p
∆+
′0
uu cq 2=
Drained effective path
Undrained total path
p’ and p0u∆ 0u∆
Drained total path
Short-term strength: The strength available before the change in the total stress is
converted into a change in the effective stress.
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q
0p′ 0p
p
p
∆+
0
uu cq 2=
p’ and p
0u∆
0u∆
Because the effective stress does not change,
cu does not change. The cu value (only) appears
to be independent of the total stress.
Note the following:
- If you look at the short-term strength in terms of the total stress, the strength looks like
independent of the stress. That is why it is considered as ‘cohesion’ and the letter “c” was
assigned. However, it has nothing to do with the intersect of the failure envelope (called
“true cohesion”. That is why it is encouraged to use “S” for “S”trength, in place of c.
- In usual cases, the short-term strength is relevant to clays, while the long-term strength is
relevant to sands/gravels. This is simply because of the time required for consolidation and
transfer of ∆p to ∆p’. Sometimes clays are called ‘c-material’ or ‘cohesive material’, while
sands/gravels are called ‘φ-material’ or ‘frictional material’. If the behaviour is interpreted in
terms of effective stress, such a nomenclature does not make sense.
Cut slope in the London Clay: Why does it stand (Assignment 4)?
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I was working here!
(Diagram from Kovacevic et al., 2007)
17-2. Shear strength from direct shear test
A direct shear test (shear box test) has the following characteristics:
- Constant vertical stress tests or constant volume (corresponding to undrained conditions)
tests may be performed.
- The horizontal stress is not known.
Because we do not know the horizontal
vhτ
vσ
?:hσ
(Diagram from Potts & Zdravkovic, 2001)
Because we do not know the horizontal
stress, we cannot construct Mohr’s stress
circle as we did in triaxial tests. It means
that we can derive neither cu (i.e. the radius
of the circle) nor φ’ (the slope of a tangent
to the circle) in a rigorous way. It involves
some assumptions to derive these.
It is often assumed that the shear stress,
τvh, is at the crown of the Mohr’s stress
circle (assuming σv = σh). Hence,
Or, it is also often assumed that the
maximum stress ratio is mobilised along
the horizontal plane. Hence,
These two assumptions are not compatible
(think why), but both are often invoked.
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σ ′
τ
1σ3σ
),( vhv τσ ′
)/(tan 1
vvh στφ ′=′ −
),( vhh τσ ′
φ’ from direct shear test
φ’ from triaxial test
=
?
=
?
=
?
cu from
triaxial test
cu from
direct shear
test
max)( vhuc τ=
17-3. Peak shear strength of soils: Drained conditions
Peak shear strength of soils are closely related to dilatancy characteristics. It is observed
better under drained conditions.
Experimental findings:
Higher density and lower confining
stress
Higher peak stress ratio and
more dilative behaviour
Below is an interpretation of how
dilatancy adds apparent frictional
strength.
τ
σ
δγ
δε−
6
Drained simple shear of Leighton Buzzard Sand
(Cole, 1967; reproduced from Oda, 1975)
Dilatancy of granular material
(Diagrams taken from Wood, 1990)
)( δεστδγδ −−=W
δγτδ fE =
The work done by external forces:
The energy dissipated internally
due to friction:
where is the internal friction stress.
From ,EW δδ =
fτ
τδγ
δεστ =−+ )(f
DilatancyObserved
in test
True
friction
stress
Do you remember the data from Assignment 2? They were from drained triaxial tests on
normally consolidated Soma Silica Sand. Here indeed you see an ‘apparent’ cohesion for
the denser condition, caused by dlatancy.
0 50 100 150 2000
100
200
300
400
Dr
50% 90%
p0' = 20 kPa
p0' = 40 kPa
p0' = 80 kPa
q [kP
a]
p' [kPa]
0 2 4 6 8 10 12 14 16 18 200
100
200
300
400
500 Dr
50% 90%
p0' = 20 kPa
p0' = 40 kPa
p0' = 80 kPa
q [kP
a]
εq [%]
στ ′−
στ ′−f
Corrected for
From drained
test, uncorrected
0
2
4
6
8 Dr
50% 90%
p0' = 20 kPa
p0' = 40 kPa
p0' = 80 kPa
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σ ′
Corrected for
dilatancy, or from
undrained test
Peak strength envelope for
uncemented (no true cohesion) soils0 2 4 6 8 10 12 14 16 18 20
-12
-10
-8
-6
-4
-2
ε p [%
]
εq [%]
Bjerrum & Simons (1960) showed that
corrected φ’ from drained triaxial tests
and φ’ from undrained triaxial tests are
broadly identical (diagram reproduced
from Ishihara, 1988).
17-4. Peak shear strength under undrained conditions: Clays
Under undrained conditions, we do not have the problem of dilatancy’s influence on the
apparent friction, because the constant-volume condition prohibits dilatancy. However, the
suppressed dilatancy leads to changes in the effective stress.
So under undrained conditions, knowing strength comes down to knowing which direction
the undrained effective stress path goes, as well as knowing the location of the peak
strength envelope.
sp ′′,
tq,
00 , sp ′′
Contractive
Neither dilative
or contractive
Dilative
We have already studied a model (the Cam Clay Model) to interpret these effective stress
path behaviour in relation to overconsolidation. For heavily over-consolidated soils,
however, it is difficult to observe effective stress paths converging to the Critical State, as
the strain-softening leads to bifurcation (i.e. shear banding) phenomena in most testing
methods.
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00 , sp ′′
Effective stress paths of reconstituted Todi Clay (Burland et al., 1996)
17-5. Peak shear strength under undrained conditions: Sands
Shown in the diagrams are undrained triaxial compression data of clean, Toyoura sand by
Verdugo (1992; reproduced from Yoshimine & Ishihara, 1998).
Note the following:
- Dense sand can exhibit extremely high
shear strength, while the opposite is
true for loose sand (static liquefaction, or
flow failure).
- Higher confining stress can work as if
equivalent to looser conditions.
The right diagram is an interpretation of
these observations. A “phase transformation”
(Ishihara et al., 1975) from contractive to
dilative behaviour is characteristic of granular
soils.
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p′log
e
High p’Low p’
High e
Low e
Ultimate state line
(Critical State Line)
17-6. Sampling, disturbance and laboratory strength
Knowing the effective stress path characteristics under undrained conditions helps
understand the disturbing mechanisms involved in sampling. Let us consider normally
consolidated clay as an example, in which shear (i.e. changes in q) causes a reduction in p’.
p′
q
“Perfect sampling”
p’ reduces due to q reduction
for normally consolidated soils
Various actions during handling
A: In-situ stress
Start of unconfined
compression test
Not-perfect sampling in reality:
sampler intrusion, sample extrusion,
etc. causes shear
In-situ qu
Lab qu
B
qu∆
10
samples (see the diagram next page).
p’ may increase or decrease
0u
0vσ
0hσ
3/)2( 000 hvp σσ +=
qupuu ∆+−= 00
0
0
0=p
000 upp −=′quupp ∆−−=′ 00
A: In-situ stress B: After perfect sampling
During sampling (which is considered to be an undrained process for clayey soils), the
release of p is compensated by a reduction of u. Any change in p’ is therefore ideally
due to plastic deformation by shearing.
The change in p’ is recoverable by re-consolidation in a laboratory triaxial (CU) test, while
this is not possible in unconfined compression or unconsolidated triaxial (UU) test.
However, sample disturbance may cause permanent changes in the failure envelope in
‘sensitive’ soils. This loss of structure is generally not recoverable, as we see next week.
p′
q
In-situ stress
Start of unconfined
compression or UU
triaxial test
In-situ qu
Lab quRe-consolidation in CU test
(anisotropic)
Re-consolidation in CU test
(isotropic)
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triaxial test
Factors affecting p’ during sampling
(Hight et al., 1992)
References
Bjerrum, L. and Simons, N.E. (1960) “Comparison of shear strength characteristics of
normally consolidated clays,” Proceedings of Research Conference on Shear Strength of
Cohesive Soils, ASCE, 711-726.
Burland, J.B., Rampello, V.N., Georgiannou, V.N. and Calabresi, G. (1996) “A laboratory
study of the strength of four stiff clays,” Geotechnique 46(3) 491-514.
Cole, E.R.L. (1967) “The behaviour of soils in the simple shear apparatus,” PhD Thesis,
University of Cambridge.
Hight, D. W., Bike, R., Butcher, A. P., Clayton, C. R. 1. and Smith, P. R. (1992)
“Disturbance of the Bothkennar clay prior to laboratory testing,” Geotechnique 42(2) 199-
217.
Ishihara, K. and Tatsuoka, F. and Yasuda, S. (1975): “Undrained deformation and
liquefaction of sand under cyclic stresses,” Soils and Foundations 15(1) 29-44.
Ishihara, K. (1988) “土質力学 (Soil mechanics),” Maruzen (in Japanese).
Kovacevic, N., Hight, D. W. and Potts, D. M. (2007) “Predicting the stand-up time of
temporary London Clay slopes at Terminal 5, Heathrow Airport,” Geotechnique 57(1) 63–
74.
Oda, M. (1975) “On stress-dilatancy relation of sand in simple shear test,” Soils and
Foundations 15(2) 17-29.
Potts, D.M. and Zdravkovic, L. (2001) “Finite element analysis in geotechnical engineering:
Application,” Thomas Telford, London.
Verdugo, R. (1992) “Characterization of sandy soil behavior under large deformation,” PhD
Thesis, University of Tokyo.Thesis, University of Tokyo.
Wood, D.M. (1990) “Soil Behaviour and Critical State Soil Mechanics,” Cambridge
University Press.
Yoshimine, M. and Ishihara, K. (1998) “Flow potential of sand during liquefaction,” Soils
and Foundations 38(3) 189-198.
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