The magic of sandsDavid Muir Wood
University of Bristol, England
(20th Bjerrum Lecture: Oslo: November 2005)
HKIE Geotechnical Division: 6 November 2007
David Muir Wood
1949-1950 Folkestone
1950-1970 Beaconsfield
1967-1987 Cambridge
1975 Oslo
1978 Hong Kong
1986 Boulder
1987-1996 Glasgow
1995- Bristol
2000 Minneapolis
2003 Tokyo
The magic of sands
• Introduction: context: WAC Bennett Dam
• Particle-continuum duality
• Mohr-Coulomb model
• Elastic properties
• Critical states
• Severn-Trent sand
• Grading state index
• Conclusion
The magic of sands
• Introduction: context: WAC Bennett Dam• Particle-continuum duality
• Mohr-Coulomb model
• Elastic properties
• Critical states
• Severn-Trent sand
• Grading state index
• Conclusion
WAC Bennett Dam
on Peace River, NE British Columbia
183m high, 2km long
reservoir 70x109m3; 2.73GW
completed 1968
sinkhole 1: June 1996
2.4m diameter at crest
6.7m deep hole
76m deep extremely
loose zone
114m variable zone
WAC Bennett Dam: sinkhole incident 1996
spillway flow 3000m3/s (> Canadian Niagara Falls)
fall in reservoir level: 2m in 7 weeks
MAXIMUM NORMAL RESERVOIR EL 672
675
650
625
600
575
550
525
500
475
EL
EV
AT
ION
(M
)
2.5
1 1
2
CL GROUT CURTAIN
GROUTING CULVERT
BEDROCK SURFACE
WAC Bennett Dam: cross section
zoned earthfill dam
blended 'till-like' core
well-graded transition and filter
high capacity drain
0
10
20
30
40
50
60
70
80
90
100
0.010.101.0010.00100.00Grain size (mm)
Perc
en
t P
assin
g
El 2215.0 El 2204.0El 2197.0 El 2184.5El 2181.8 El 2165.5El 2143.5 El 2125.5El 2105.5 El 2087.0El 2085.5 El 2084.0El 2082.3 El 2063.5El 2062.5 El 2043.5El 2025.5 El 2013.5El 2005.5 El 2002.0El 1985.5 El 1972.5El 1955.5 El 1952.5El 1933.5 El 1917.5El 1915.5 El 1904.5El 1894.5 El 1887.0El 1870.5 DH96-34 Avg.As-Built Avg.
#200#100#60#40#20#10#43/8"3/4"1 1/2"3"
GRAVEL
Coarse Medium Fine
SAND FINES
Silt
DH96-34 Gradations
Core material (Sinkhole 1)
as built
1996 gradings show loss of fines
2112 1995
2092 2012 1916
1965 1863 1790
2123 2067
18
00
1840
1880
1920
1960
2000
2040
2080
2120
2160
Canyon section: 2003 seepage model
silty transition material:
reduced permeability
pore pressures consistent with
permeability changes linked with fines
movement
WAC Bennett Dam: statement of problem
non-plastic material with changing granulometry and density
•what are consequences for mechanical response of dam?
•potential for future deformations?
need model of soil behaviour which can incorporate changes in density and grading of the soil
The magic of sands
• Introduction: context: WAC Bennett Dam
• Particle-continuum duality• Mohr-Coulomb model
• Elastic properties
• Critical states
• Severn-Trent sand
• Grading state index
• Conclusion
paradox of sand:
blows in the wind
flows like a liquid
but supports loads like a solid
Particle-continuum duality
for analysis we need to treat soil as a continuum (stress, strain)
… but its properties emerge from its particulate nature
Leighton Buzzard sand (picture width 37 mm)
Particle-continuum duality
The magic of sands
• Introduction: context: WAC Bennett Dam
• Particle continuum duality
• Mohr-Coulomb model• Elastic properties
• Critical states
• Severn-Trent sand
• Grading state index
• Conclusion
model = appropriate simplification of reality
modelling for working loads:
•serviceability limit states
•deformations of geotechnical systems
usually requires numerical analysis:
•for example, finite element, finite difference
Mohr-Coulomb model
equilibrium compatibility of deformations
stresses strains
stress:strain relationship
constitutive model
eg Mohr-Coulomb
constitutive model: central element of numerical modelling
linking change in stress with change in strain
generalised evolving incremental stiffness
Mohr-Coulomb model
Mohr-Coulomb familiar as strength/failure model
limiting stress ratio characterised by frictional strength φ' (ignore cohesion)
drives laboratory and in situ investigation of soil properties
Mohr-Coulomb model: failure
0
100
200
300
400
500
0 0.05 0.1 0.15 0.2
0
0.01
0.02
0.03
0.04
0 0.05 0.1 0.15 0.2
deviator
stress
shear strain
shear strain
volumetric
strain
Mohr-Coulomb model: parameter selection
loose Hostun sand
Benahmed, 2001
choice of stiffness, strength, dilatancy
pedagogic exercise
… as many answers as people!
compression +ve
The magic of sands
• Introduction: context: WAC Bennett Dam
• Particle continuum duality
• Mohr-Coulomb model
• Elastic properties• Critical states
• Severn-Trent sand
• Grading state index
• Conclusion
Elastic properties
stiffness does not imply elasticity (recoverable deformations)
fall of stiffness ⇒ plasticity (permanent changes on unloading)
… hence need for constitutive models
Quiou sand (LoPresti et al., 1997)
shear strain
shear stresslimit of elastic
response??
unloading
loading
perceptions of laboratory soil stiffness have changed as instrumentation for
measurement of deformations has improved
limit of elastic
response?then … paradox of low laboratory
stiffness – high in-
situ stiffness
now … consistency
of dynamic and static measurements
secant
stiffness
then
now
Elastic properties
bender element (s-waves)
extender element (p-waves)
+
–
+
+
–
+
laboratory geophysics: piezoceramic elements
v
h belt
h 90
V s(hv)
V s(hh)
V s(vh)b
V s(vh)
bender elements
on triaxial sample
deduction of elastic stiffness
elastic stiffness deduced from shear (or compression) wave velocity G = ρ vs
2
velocity = distance/time
arrival time? complex (attenuated) received signals –reflections, dispersion
elastic properties: numerical simulation
FLAC-3D grid
reflection at boundaries
conversion of s-wave to p-wave energy
bender element
non-absorbing boundaries
experimental observation: note p-wave
arrival in s-wave trace
interpretation of numerical simulations
numerical simulation: objective
deduction of arrival time??
input signal: single sine pulse!!
compare theoretical p-wave and s-
wave arrival times
interpretation of numerical simulations
typical choices for arrival time from
inspection of received signal
interpretation of numerical simulations
no clear trends?
p-wave dominating with distance
cross-correlation
frequency domain
non-absorbing boundaries
distance from bender tip
vp/vs = 1.5
normalised shear wave
velocity
1.0
The magic of sands
• Introduction: context: Bennett Dam
• Particle continuum duality
• Mohr-Coulomb model
• Elastic properties
• Critical states
• Severn-Trent sand• Grading state index
• Conclusion
Bristol
River Severn
Severn-Trent sand
Severn – river adjacent to Bristol
Trent – Alessandro Gajo, Italy
Trento
Hostun sand (Benahmed)
Severn-Trent sand
?
?
?
Mohr-Coulomb model: defects?
•incremental stiffness elastic or zero
•indefinite dilation/compression at failure
•strength chosen as soil constant
volumetric strain
constant strength
what is peak strength?
•property of the soil which
changes with stress level, density
•dependent quantity
Severn-Trent sand: strength
Been & Jefferies
state parameter ψ = volume distance
from critical state line
function of density and stress level
more useful than void ratio alone –indicating effect of density and stress'dense'
'loose'
ψ
critical state line
mean stress
specific volume
Severn-Trent sand: strength
what is peak strength?
•property of the soil which changes
with stress level, density
data confirm link between strength
and state parameter ψ
Mohr-Coulomb model with current strength dependent on current stress level and density
Been & Jefferies
peak strength
Severn-Trent sand: strength
'dense'
'loose'
ψ
mean stress
specific volume
state parameter ψ
Severn-Trent sand: dilatancy
Benahmed
volume strain
shear strain
dilatancy: volume change during
shearing
'dense' sand expands – negative state
parameter
'loose' sand contracts – positive state
parameter
dilatancy depends on density
dilatancy varies during test
link with state parameter (stress level
and density)
Severn-Trent sand: stiffness evolution
mobilised
strength φ'mob
shear strain
currently mobilised strength
currently available strength monotonic relationship
1ratio
available strength φ': varies with ψ
model complete
•monotonic travel towards current strength
•current strength depends on current density (state parameter)
•shearing leads to change in density (dilatancy): soil seeking
critical state
•change in density leads to change in strength
etc
predicted softening for dense sand as emergent property
Severn-Trent sand
drained triaxial compression tests
different initial density (state
parameter)
model automatically homes in on
critical state
peak strength is moving target
reached at infinite distortional strain –critical state
current peak
strength
Severn-Trent sand: simulations
increasing density
increasing density
calibrated against triaxial test data
for Hostun sand
undrained triaxial compression
effect of different initial density
automatically described
Gajo & Muir Wood, 1999
Severn-Trent sand: calibration
Mohr-Coulomb model with extra features
building on familiar foundations
•central role of state parameter
•feedback through dilatancy
•rich patterns of response simulated
mathematically elegant
economical in demand for soil parameters
Severn-Trent sand
The magic of sands
• Introduction: context: WAC Bennett Dam
• Particle continuum duality
• Mohr-Coulomb model
• Elastic properties
• Critical states
• Severn-Trent sand
• Grading state index• Conclusion
occurrence of crushing
change in grading
irreversible
Chattahoochee River sand
Vesic & Clough, 1968
Grading state index
particle size distribution
1: before testing
2: after compression to 6.21MPa
3: after triaxial compression
21
3
0
20
40
60
80
100
100 1000Size, log d (µm)
% p
assin
g b
y v
olu
me Original PSD
20 MPa before shear
20 MPa axial strain = 0.3
20 MPa axial strain = 0.5
20 MPa axial strain = 0.6
20 MPa axial strain = 0.7
Influence of axial strain:
(Cheng, 2004)numerical simulations – compression and shearing of assembly of agglomerates
gradings tend to self similar 'fractal' grading
continuous 'fractal' grading: every void space filled with progressively smaller particles
389 agglomerates
6.66 mm
Grading state index
Cheng
crushing
Maeda (2005)
crushing?
coordination number (number
of contacts) larger for larger particles
smaller particles tend to crush
Grading state index: crushing
Brazil cylinder test: tensile strength of
concrete
Grading state index IG: definition
definition of IG?
IG = area ABC/area ABD
0 < IG < 1
single size AB: IG = 0
fractal limit AD: IG = 1
0
0.2
0.4
0.6
0.8
1
0.0001 0.001 0.01 0.1 1
d/dmax: particle sizelogarithmic scale
fraction finer
d = dmax
current
grading
A
BC
D
(fractal) limiting grading
0
10
20
30
40
50
60
70
80
90
100
0.010.101.0010.00100.00Grain size (mm)
Perc
en
t P
as
sin
g
El 2215.0 El 2204.0El 2197.0 El 2184.5El 2181.8 El 2165.5El 2143.5 El 2125.5El 2105.5 El 2087.0El 2085.5 El 2084.0El 2082.3 El 2063.5El 2062.5 El 2043.5El 2025.5 El 2013.5El 2005.5 El 2002.0El 1985.5 El 1972.5El 1955.5 El 1952.5El 1933.5 El 1917.5El 1915.5 El 1904.5El 1894.5 El 1887.0El 1870.5 DH96-34 Avg.As-Built Avg.
#200#100#60#40#20#10#43/8"3/4"1 1/2"3"
DH96-34 Gradations
Chattahoochee River sand
grain crushing: IG increasing
WAC Bennett Dam core
fines removal: IG falling
Grading state index IG
•soil grading change: erosion/transport or crushing
•material changing (irreversibly) while being studied
•effect on mechanical behaviour?
0
50
100
0.001 0.01 0.1 1 10 100
Grading state index
particle size: logarithmic scale
% finer
fractal limiting gradings
residual granitic soil
glacial till
natural soils:
discovering fractal limiting gradings?
•material changing (irreversibly) while being studied
•modelling requirements:
�characterisation of evolving grading – additional grading state index
�evolution law for grading state index (mass conservation, transport, crushing criteria, etc)
�influence of grading state index on constitutive properties (for example, critical states)
•research in progress: add IG influence to existing model
Modelling grading change
Grading state index IG: influence
influence of grading state index on constitutive properties
•elastic properties – unchanged (first order)?
•friction/strength – unchanged (first order)?
•critical state line – expected to change!
evidence?
smaller particles tending to fill gaps
maximum and minimum void ratios ↓ as IG ↑
Grading state index IG: critical states
critical state line?
deduced from tests with
increasing stress levels
mean stress: log scale
specific
volume
IG increasing (irreversible)
changing material
onset of crushing
original material
crushing complete
critical state line –
before crushing: IG = 0
mean stress: log scale
reinterpretation:
critical state surface
limit of critical state lines –during crushing: 0 < IG < 1
critical state line –
crushing exhausted: IG = 1IG
specific volume
IG increasing: evolving
critical state line
Grading state index IG: critical states
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
1 10 100
Pressure, p' (MPa)
Vo
ids
rati
o, e
Dense
Loose
Loose, over-compressed
Final states of p' -constant tests:
Loose
Dense
Loose, over-compressed
simulations for assemblies of agglomerates
critical state line changes with crushing
fresh samples – pre-compressed samples
Cheng, 2005
389 agglomerates
6.66 mm
pre-compressed samples
fresh
samples
Grading state index IG: critical states
0.25
0.27
0.29
0.31
0.33
0.35
0.37
0.39
0.41
0.43
10 100 1000 10000
mean effective stress: kPa
vo
id r
ati
o
F = 34 %
F = 26 %
F = 19 %
csl_interpolation.xls
WAC Bennett Dam: interpretation of effect of fines content on location of critical state line (triaxial tests, artificial mixtures)
non-monotonic…!?
34% fines
19% fines26% fines
Grading state index IG: critical states
e
Grading state index
WAC Bennett Dam
Severn-Trent sand
transport of fines from core
void ratio ↑
grading state index ↓
critical state line ↓??
state parameter ↑
soil feels looser �
v
p’
e
Grading state index
WAC Bennett Dam
Severn-Trent sand
transport of fines from core
void ratio ↑
grading state index ↓
critical state line ↑??
state parameter ↓
soil feels denser ☺
v
p’
WAC Bennett Dam??
benefit of simple model that systematically incorporates changes in stress level and density and grading (making up
state of soil)
model has to be honed – subtle data requirements for
calibration
most testing has used artificially prepared mixtures
Grading state index
The magic of sands
• Introduction: context: WAC Bennett Dam
• Particle continuum duality
• Mohr-Coulomb model
• Elastic properties
• Critical states
• Severn-Trent sand
• Grading state index
• Conclusion
WAC Bennett Dam: Need for modelling able to include effects
of density, stress and evolving grading.
Mohr-Coulomb model: Use as basis for development of more
comprehensive models.
Severn-Trent sand: Mohr-Coulomb model with strength dependent on state parameter – softening as emergent
property.
Grading state index: How does the changing grading of a soil
affect its mechanical behaviour?
Conclusion
Nous nous sommes nourris de la magie des sables.
We are sustained by the magic of sands.
Antoine de St Exupéry: Terre des hommes (1939)