week 8 soil behaviour at small strains: part 1 · soil behaviour at small strains: part 1 11. ......
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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.
The resonant column apparatus is
normally purpose-built, unlike
auxiliary tools such as bender elements.
This poses some inconveniences.
However, it has a big advantage; by
changing the input force, the oscillation
amplitude (hence strain amplitude) can
be changed. This is a useful feature for
estabilishing G – γ curves over a wider
strain range.
Various types of resonant column
Active Active
Active
Passive
F F
F
Ka
Ca
Ca
(a) Fixed-free (b) Fixed-base-spring –top (c) Free-free
Ka
7
Shear modulus measured in crag and Tertiary soils
(LC: London Clay, TC: Thanet Sand; Hight et al., 1997)
(iii) Field
Shear wave velocity measurement: Cross-hole and down-hole methods
The principle of these field methods is same as that of bender element tests. A receiver
(and transmitter in down-hole methods) is placed inside a borehole, or if the soil is soft, it
may be installed in a penetration cone (seismic cone penetration test; SCPT).
These method measures shear wave
velocity, which is a body wave. There
are also techniques which use
surface wave (Reighley wave).
Making waves above a seismic cone
8
Cross-hole measurement
(Hight et al., 1997)
Down-hole measurement
(Hight et al., 1997)
Example of comparison between different method:
30
20
10
0
De
pth
be
low
GL
[m
]0 100 200
Gvh [MPa]
-10
0
10
20
Ele
va
tio
n [
m O
D]
Down-hole (BH407, North)*
Down-hole (BH407, East)*
Resonant column (rot. core)
Bender element
Resonant column(range for blocks)
*Shear wave was transmittedfrom two sides of borehole
Biii
Bii
Bi
B1
C
FinallyR
In old days, the stiffness moduli measured in dynamic and static tests used to be
considered two fundamentally different things due to the strain-rate effects, because the
dynamic moduli were always far larger than the static ones. After it was found that the
static moduli had been underestimated by global measurement, the agreement of the
moduli between dynamic and static tests has been seen (Tatsuoka & Shibuya, 1991).
One problem solved?
9
Shear modulus Gvh of natural London Clay measured
by different laboratory and field methods (Nishimura, 2006)
50
40
-20
A3Lithological unit:
12-4. Importance of small-strain stiffness non-linearity: Case studies
(i) Excavation: Simpson et al. (1979)
One of the early examples of geotechnical non-linear finite element analysis is on
construction of an underground car park in front of the Palace of Westminster in the 1970s.
To avoid affecting the historic building,
the ground deformation caused by
the excavation needed to be predicted
with high accuracy.
A Class A prediction had been given
by elastic analysis by Ward and
Burland (1973). The problem was
revisited by Simpson et al. (1979)
by non-linear analysis.
Palace of Westminster with Big Ben Clock Tower
10
Cross-section
(Continued; Simpson et al., 1979)
The non-linear analysis was capable of
simulating the observed ground movements
with good accuracy.
An interesting episode is that the linear elastic
and non-linear analyses predicted the tower’s
leaning towards opposite directions.
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Modelling of stress-strain relationships
Predicted ground movements
(ii) Shallow foundation: Jardine et al. (1995)
Experiments at Bothkennar site, Scotland
Loading on a 2.4m x 2.4m footing on soft silty clay.
Analysis with a non-linear model predicted better the observed settlement than with linear
elasticity. The elastic analysis predicts that the influence of the footing settlements reaches
very far. In reality, it does not, as the non-linear analysis indicates.
Testing pad
12
Testing pad
Predicting and observed settlements
D
r
rδcδ
(iii) Shallow - deep foundation: Izumi et al. (1997)
Rainbow Bridge, Tokyo
(Construction work: 1987-1993)
140,000 tf anchorages built on Tertiary
Mudstone
(©Google 2011)
13
Cross-sections
(Continued: Izumi et al., 1997)
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
References
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stiffness on the numerical analysis of tunnel construction,” Geotechnique 47(3) 693-712.
Burland, J.B. and Hancock, R.J.R. (1977) “Underground car park at the House of
Commons, London: Geotechnical aspects” The Structural Engineer,” The Journal of The
Institution of Structural Engineers 87-100.
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triaxial apparatus,” Geotechnique 32(1) 62-65.
Cuccovillo, T. and Coop, M.R. (1997) “The measurement of local axial strains in triaxial
tests using LVDTs,” Geotechnique 47(1) 167-171.
Goto, S., Tatuoka, F., Shibuya, S. Kim, Y.-S. and Sato, T. (1991) “A simple gauge for local
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