WEEK 10 Soil Behaviour at Medium · PDF fileRotation of principal stress directions ......
Transcript of WEEK 10 Soil Behaviour at Medium · PDF fileRotation of principal stress directions ......
WEEK 10
Soil Behaviour at Medium Strains
14. Onset of yielding
14-1. What is yielding: Remember?
We have already studied what yielding means; if you don’t remember, refer back to Week 2.
It is defined by generation of plastic strain. According to the conventional elasto-plastic
concept that we studied in Weeks 3-5, soil behaviour was assumed to be elastic within the
yield surface. In many realistic cases, however, this is a wild idealisation. If the assumption
is true, soil would always exhibit elastic responses against cyclic loadings with a constant
stress amplitude. This week we will study real soil behaviour at medium strains, which are
larger than those to which the elastic stiffness is relevant, but smaller than those at which
the large-scale yielding leads to failure of soil.
In the previous weeks, we studied that
the stress-strain non-linearity appears
against very small strain (and hence stress)
increments. Does this signify yielding?
Remember, non-linearity per se does not
necessarily mean yielding.
q
Isotropic hardening
necessarily mean yielding.
p′
Stress path
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(Addenbrook et al., 1997)
Example of
Evu – εa curves of
London Clay
Inside: Elastic?
14-2. Yielding at ‘relatively small’ to medium strains
The answer to the question seems, “yes”. See how the stress-strain relationships form a
loop even for small strain amplitudes (meaning irrecoverability of strains; hysteresis).
The τ – γ data shown below (from Iwasaki et al., 1978) were obtained with hollow cylinder
torsion shear apparatus (refer back to Week 9). A qualitatively same view is obtained for
the q – εq relationship that is obtained with triaxial apparatus, although the τ – γ and q – εqrelationships may be quantitatively different due to anisotropy.
q
p′
τor Cyclic loading with gradually
increasing stress-strain amplitude
q
qε
τor
γor
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(Iwasaki et al., 1978)Definition of damping ratio
Change of hysteresis as strain
amplitude increases
Another effect of plasticity is dependency of the stress-strain relationship on loading
histories. Shown here are the (shear) strain contours drawn for loadings from different
origins along the K0-unloading path, obtained for the Magnus Till (Jardine, 1992). Note how
the strain contours align themselves along the unloading path. A change in the loading
direction generally leads to stiffer responses.
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(Jardine, 1992)
Another example by Atkinson et al. (1990) on reconstituted and over-consolidated London
Clay
Influence of stress-path direction changes on subsequent q – εq relationshps
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(Atkinson et al., 1990)
Influence of stress-path direction changes on subsequent p’ – εp relationshps
14-3. Multiple yield surface and kinematic hardening: Concept
To interpret the yield at relatively small to medium strains, it is necessary to prepare more
than one yield surface. This concept of multi-surface plasticity is not limited for soils (for
example, Iwan, 1967). It is often combined with kinematic hardening rule.
The idea is to limit the elastic region to very small size and describe multiple distinct stages
of yielding by adopting multiple yield surfaces.
Kinematic yield
surface
q
p′
Boundary
surface
q
q
p′
Boundary
surface
(Yield surface)
q
Plasticity immediately
appears for
reloading/unloadingElasticity
5
qε qε
End of
elasticity
Onset of ‘large-scale’ yield
Critical StateEnd of
elasticity
&Onset of ‘large-scale’ yield
Critical State
qεlog
secG
0G
qεlog
secG
0GRealistic non-
linearity at small
strains may be
described
15. Behaviour under cyclic loading and principal stress axis rotations
15-1. Cyclic loading and accumulation of volumetric strain or excess pore water pressure
The cyclic and dynamic behaviour of soils will be discussed in detail in the lecture course
provided by Professor Miura, so this lecture allocates space for this topic less than it
deserves. The soil behaviour discussed under this topic is not necessarily limited to the
range of ‘medium’ strains, however you define it. For example, liquefaction eventually leads
to extremely large strains. However, the processes to reach such an ultimate state are
dominated by a sequence of medium-scale yielding, so probably it is appropriate to discuss
them here.
(i) Sands
Let us start with sands, with which liquefaction under cyclic loadings is always a great
concern (the significance of liquefaction phenomena will be discussed in detail in the other
e3 post-graduate course by given the lecturer, “Disaster Mitigation Geotechnology”).
The data shown here were obtained
for the Toyoura Sand in drained
cyclic simple shear, performed
in hollow cylinder apparatus.
See how volumetric strain
accumulates over number of
loading cycles.loading cycles.
How can we interpret, or how
can we not interpret this
behaviour from what we have
learnt previously?
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Shahnazari & Towhata (2002)
Simple shear in hollow cylinder
apparatus
In undrained cyclic simple shear,
the volumetric strain is forced to be
zero, but this time the pore water
pressure cannot be controlled. As a
result, the pore water pressure
increases and p’ decreases.
The ultimate state can be
liquefaction.
How can we interpret the behaviour
under drained and undrained conditions
in a unified way?
Actually, you already know the (or, ‘an’)
answer; remember what you studied
in Week 3?
Undrained conditions:
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Effective stress-paths and shear
stress-strain curves of loose and
dense Fuji River Sand (Ishihara, 1985;
reproduced after Iai et al., 1991)
e
p′
Drained conditions:
∆e due to plastic straining
Undrained conditions:
Because of plastic straining due
to the cyclic loading, e wants to
decrease, but it cannot (∆e must
be zero). So p’ is forced to
decrease instead.
(ii) Clays
In Clays, cyclic loading also leads
to excess pore water generation
and hence reduction in p’.
It is not common, however, for p’
to reach zero and attain a state of
liquefaction. Literature does not
cite liquefaction in clays.
Towhata (2008) notes some
similarity between behaviour of
dense sands and clays.Pore water pressure changes during
cyclic simple shear of Kaoline
(Ohara & Matsuda,1988)
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Effective stress-path and shear stress-strain relationships of Eastern Osaka Clay
(Adachi et al., 1995)
15-2. Soil behaviour under rotation of principal stress axes
First of all, get familiar with rotations of principal stress axes in 2D.
σ
τ( )
xyx τσ ,
( )xyy τσ ,
1σ3σy
x
yσ
xσxyτ
PD (Pole with regard
to direction)1σ
1σ
σ
τ
( )xyx τσ ,
σσy
yσ
xσxyτ
1σ
If the stress state (i.e. σx, σy and τxy) is changed in such a way that the Mohr’s stress circle’s
centre (= (σx + σy )/2) and radius (= [[(σx - σy )/2]2+[τxy]2]0.5) are not changed, a pure rotation of the principal stress axes occurs. It means that only the directions of σ1 and σ3 change, but not their magnitudes.
If σ1 and σ3 do not change, q, t, p, s, etc.do not change either (ignore σ2 for themoment). So, according to the models
based on these ‘invariant’ quantities
would not predict any change in state.
You stay where you are. Is this realistic?
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σ
( )xyy τσ ,
1σ3σy
x
1σ
q or t
p or s
In what situations do rotations of the principal stress axes matter particularly?
In quite many a situation, actually.
Examples:
(Bjerrum,1973)
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Rotation of principal stress directions
due to embankment construction
(Jardine & Smith., 1991)
Cyclic rotation of principal stress
directions in seabed due to wave loading
(Ishihara & Towhata., 1983)
The rotation of principal stress
directions is accompanied by
increases in p’ and q.
The rotation of principal stress
directions occurs almost with constant
p’ and q.
15-2. Soil behaviour under rotation of principal stress axes
If we assume elastic behaviour,
(see p.11, Week 2)
or,
A strain increment is parallel to
the stress path.
However, the drained hollow
cylinder test results by Gutierrez
et al. (1991) indicate significant
plastic strains for any ‘rotational’
stress path. Towhata and Ishihara
(1985) demonstrated that even
liquefaction can be triggered by Increase of q, with p’ and principal stress
directions fixed
∆−∆
=
∆
∆−∆
xy
yx
xy
yx
G
G
τσσ
γεε 2/)(
/10
0/1
2/)( θσσ −= zX
2/θτ zY =z
θ
∆−∆
=
∆
∆−∆
xy
yx
xy
yx
G τσσ
γεε 2/)(1
liquefaction can be triggered by
pure rotation of the principal stresses.
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directions fixed
Rotation of principal stress directions, with p’
and q fixed
(Gutierrez et al.,1993)
References
Adachi, T., Oka, F., Hirata, T., Hashimoto, T., Nagaya, J., Mimura, M. and Pradhan, T.B.S.
(1995) “Stress-strain behavior and yielding characteristics of Eastern Osaka Clay,” Soils
and Foundations, 35(3) 1-13.
Addenbrooke, T.I., Potts, D.M. and Puzrin, A.M. (1997) “The influence of pre-failure soil
stiffness on the numerical analysis of tunnel construction,” Geotechnique 47(3) 693-712.
Atkinson, J.H., Richardson, D. and Stallebrass, S.E. (1990): “Effect of recent stress history
on the stiffness of overconsolidated soil,” Geotechnique 40(4) 531-540.
Bjerrum, L. (1973) “Problems of soil mechanics and construction on soft clays and
structurally unstable soils (collapsible, expansive and others),” Proceedings of 8th
International Conference on Soil Mechanics and Foundation Engineering, Moscow, Vol.3,
111-159.
Gutierrez, M., Ishihara, K. and Towhata, I. (1991): ”Flow theory for sand during rotation of
principal stress direction,” Soils and Foundations 31(4) 121-132.
Iai, S., Matsunaga, Y. and Kameoka, T. (1992): “Strain space plasticity model for cyclic
mobility,” Soils and Foundations 32(2) 1-15.
Ishihara, K. (1985) “Stability of natural deposits during earthquakes,” Proceedings of 11th
International Conference on Soil Mechanics and Foundation Engineering, San Francisco,
Vol.1, 327-376.
Ishihara, K. and Towhata, I. (1983) “Sand response to cyclic rotation of principal stress
directions as induced by wave loads,” Soils and Foundations 23(4) 11-26.
Iwan, W.D. (1967): “On a class of models for the yielding behavior of continuous and
composite systems,” Journal of Applied Mechanics 34(E3) 612-617.composite systems,” Journal of Applied Mechanics 34(E3) 612-617.
Iwasaki, T., Tatsuoka, F. and Takagi, Y. (1978): “Shear modulus of sands under cyclic
torsional shear loading,” Soils and Foundations 18(1) 39-56.
Jardine, R.J. (1992): “Some observations on the kinematic nature of soil stiffness,” Soils
and Foundations 32(2) 111-124.
Jardine, R.J. and Smith, P.R. (1991) “Evaluating design parameters for multi-stage
construction,” Proceedings of the International Conference on Geotechnical Engineering
for Coastal Development, Geo-coast ‘91, Vol.1, 197-202.
Ohara, S. and Matsuda, H. (1988) “Study on the settlement of saturated clay layer induced
by cyclic shear,” Soils and Foundations, 28(3) 103-113.
Shahnazari, H. and Towhata, I. (2002) “Torsion shear tests on cyclic stress-dilatancy
relationship of sand,” Soils and Foundations 42(1) 105-119.
Towhata, I. (2008) “Geotechnical earthquake engineering,” Springer-Verlag Berlin
Heidelberg.
Towhata, I. and Ishihara, K. (1985) “Undrained strength of sand undergoing cyclic rotation
of principal stress axes,” Soils and Foundations 25(2) 135-147.
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