WEEK 8 Soil Behaviour at Small Strains: Part 1 11. Strain levels and soil behaviour Soil’s shear stress-strain relationships exhibit many typologies. For example, some of them are ductile with continuing strain-hardening, and some of them are brittle with significant post-peak strain-softening. The objective of Week 8-13 is to understand what sort of soil behaviour we should expect from a given soil at a given condition, and consider what impact the observed features have on engineering problems. In doing so, it is convenient to look at soil behaviour at different strain levels. This will allow us to focus on γ τ Small strain Large strain τ Shear stiffness at small strains, yield characteristics at medium strains and strength at large strains. This view is applicable in principle for compression behaviour that we have studied last week. The only difference is that we normally do not invoke a notion of ‘strength’ in compression. 1 Medium strain p′ p′ v ε Compression
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WEEK 8
Soil Behaviour at Small Strains: Part 1
11. Strain levels and soil behaviour
Soil’s shear stress-strain relationships
exhibit many typologies. For example,
some of them are ductile with
continuing strain-hardening, and
some of them are brittle with
significant post-peak strain-softening.
The objective of Week 8-13 is to
understand what sort of soil
behaviour we should expect from
a given soil at a given condition, and
consider what impact the observed
features have on engineering problems.
In doing so, it is convenient to look at
soil behaviour at different strain levels.
This will allow us to focus on
γ
τ
Small strain
Large strainτ
Shear
This will allow us to focus on
stiffness at small strains, yield
characteristics at medium strains
and strength at large strains.
This view is applicable in principle for
compression behaviour that we have
studied last week. The only difference
is that we normally do not invoke a
notion of ‘strength’ in compression.
1
Medium strain
p′
p′
vε
Compression
12. Small-strain stiffness and non-linearity
12-1. Definitions of soil stiffness
- Tangent stiffness (Gtan, Etan , etc.)
- Secant stiffness (Gsec, Esec , etc.)
- Initial (elastic) stiffness (G0, E0 , etc.)
(Equivalent to tangent stiffness at
very small strains)
Upon unloading and reloading,
elastic stiffness is normally
observed (but not necessarily
identical to the initial stiffness).
Normally, soils’ stiffness is largest
at very small strains, exhibiting
gradual degradation as the strain
becomes larger (due to plastic
straining).
How small is small? There is no
formal definition or consensus on
γ
τ
tanG
secG0G
secG
0G
Unloading
&
reloading
formal definition or consensus on
“small strain”, but when we say
small strains, usually we talk about
strains smaller than order of 10-4
(imagine, 1 µm over 10 mm).
12-2. Some history: Background to recognition of small-strain stiffness
Importance of the stiffness non-linearity at small strains started to be recognised mainly
after the 1970s. This development had two technical factors in its background;
sophistication in laboratory tools and the advent of personal computers. New laboratory
tools allowed resolving ever smaller strains with higher accuracy. The computer allowed
non-linear numerical analyses, which provided a way to utilise the new laboratory findings
on small-strain stiffness for practical problems. Without PCs, prediction needs to be based
on analytical solutions, which normally exist for very simple, linearly elastic stress-strain
relationships. So in many senses, general recognition of the stiffness non-linearity at small
strains coincided with the turning point of soil mechanics from the classical era to the
modern.
2
γlogUp to order of 10-4
(0.01% strain)
12-3. Testing techniques for measuring small-strain stiffness
(i) Laboratory: Static tests
Triaxial apparatus with local instrumentation
is most commonly used for both research
and practice. Hollow cylinder apparatus and
plane strain apparatus are also used, but
mainly for research purposes. Here we
limit the scope to triaxial apparatus.
However, the principle itself of local
instrumentation is same in any apparatus.
Global instrumentation is erroneous due to
- Bedding errors
- Non-parallel ends
- Load cell and system compliance
Local instrumentation is capable of avoiding
these errors, providing more accurate
strain and hence stiffness measurement.
LVDTs
Suction capLoad cell
Bender elementsystem(also in other side of soil specimen)
Mid-height PWPtransducer
Soil specimen
Tie rod
Perspex wall Drainage
Ram
Porous stone
(Global)displacementtransducer
Radial belt
Ram pressure chamberfilled with oil
Bearing
To oil/air interfaceor CRS-pump
Example of triaxial apparatus withWhy not abolish all global instrumentation
and just use local one then? It is easier said
than done; local transducers are expensive
and requires expertise in handling.
3
Example of triaxial apparatus with
local instrumentation (Nishimura, 2006)
Specimen
externald
0H
Load cell
internald
0H ′
0
externalrnalaxial_exte
H
d=ε
0
internalrnalaxial_inte
H
d
′=ε
From external (global) instrumentation:
rnalaxial_exte
axialexternal
ε
σ
∆
′∆=′E
From internal (local) instrumentation:
rnalaxial_inte
axialinternal
ε
σ
∆
′∆=′E
Examples of local transducers
These devices have very high resolutions in displacement measurement. Consider how
high the resolution needs to be to measure, say, Young’s modulus for strain of 10-5
(0.001%)?
Local Displacement Transducer
(LDT; Goto et al., 1991)
Axial displacement transducer using
inclinometer (Burland & Symes, 1982)
4
Linear Variable Differential Transformer
(LVDT) for axial displacement
(Cuccovillo&Coop, 1997)
LVDT for radial displacement
(Drawing provided by
Prof. Matthew Coop)
Example of measurements
Note how different the magnitudes of stiffness are when measured externally and internally.
This is a typical result; you can find numerous similar comparisons in literature for sands,
silts, soft clays, etc. However, the error involved in global measurement of strains is more
significant for stiffer soils. The same problems of bedding and system compliance are
Triaxial compression on soft mudstone (Goto et al., 1991)
significant for stiffer soils. The same problems of bedding and system compliance are
encountered in oedometer tests too.
5
Another example: Lightly over-consolidated North Sea Clay
(Jardine et al., 1984)
(ii) Laboratory: Dynamic tests
Most of the dynamic tests are based on elastic or visco-elastic wave theory. The magnitude
of strain is associated to the magnitude of oscillation amplitude. The strain levels involved
are normally very small (<10-5), in many cases small enough to regard the obtained
stiffness as the initial elastic stiffness.
One dimensional wave equation is
where G is the shear modulus, m the viscosity
and r the mass density of soil. If the viscosity
is disregarded,
Where is the shear wave velocity.
Bender element tests:
tx
u
x
uG
t
u
∂∂
∂+
∂
∂=
∂
∂2
3
2
2
2
2
ρ
µ
ρ
x
)(xu
Case of one-dimensional shear wave2
22
2
2
x
uV
t
us∂
∂=
∂
∂
ρ/GVs =
hv
Bender elements
SoilSpecimen
A bender element is made up of piezo-ceramic
semiconductors. It generates shear waves when
energised, and conversely, it sends electric signals
when receiving shear waves. So by installing
a couple of them as transmitter and receiver,
and measuring the travel time between a given
distance, Vs and then G can be calculated.
A caution is required; soil stiffness
is anisotropic (the topic of next
week), and you need to know
which shear modulus you are
measuring; Gvh Ghv or Ghh?
6
v (or z)
h (or r)
hh
-0.5 0 0.5 1 1.5 2
Time [mSec]
-100
-50
0
50
100
Am
plit
ud
e o
f sig
nals
in a
rbitra
ry u
nits
First arrivalt = 0.514 mSec
InputOutput
TE4: After consolidationf = 9 kHz, vh-direction
Beginningof signal
Example of London Clay (Nishimura, 2006)
Resonant Column test
In contrast to bender element tests, in which typically a pulse wave is transmitted to monitor
its velocity, a sample is put in steady state oscillations in resonant column tests. By
gradually changing the input frequency at a constant input force (or torque) amplitude, the
frequency at which the oscillation becomes maximum is sought (i.e. the resonance
frequency is sought). From the resonance frequency, the sample’s stiffness is obtained.
If the oscillation is compression – extension, E is obtained (E or E’?)
If the oscillation is cyclic torsional, G is obtained.
Proper consideration of stress-strain non-linearity at small strains led to significant
improvement in settlement prediction.
Note how conventional testing methods
underestimating the small-strain stiffness
led to over-estimation of the settlement.
3-D FEM mesh
14
Non-linear stiffness Settlement: Predictions and observations
Simulation cases
(iv) Tunnelling: Addenbrooke et al. (1997)
Jubilee Line Extension Project, London
Prediction of settlement troughs with non-linear numerical models
Jubilee Line (Grey-coloured)Cross-section
15
Model L4&J4: Non-linear models
fitted to locally instrumented triaxial
extension tests
Stiffness non-linearity from experiments and models
(Continued; Addenbrooke et al., 1997)
Linear elasticity is useless in predicting the settlement trough, which is deeper and
narrower than linear elasticity predicts.
However, even the non-linear stress-strain models do not do a perfect job. Research is
going on to see any other factor is being missed, such as anisotropy and the influence of
loading histories.
Settlement trough
Tunnel
excavated2D FEM mesh
16
Settlement at the ground surface due to excavation of first (west-bound) tunnel
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