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Transcript of Naples 200713
Geotechnical modelling and critical state soil mechanicsNaples, May 2007
David Muir Wood
University of Bristol
13. Designer models: addition of extra features
(GM 2, 3, EM, GeoF)
Designer models: addition of extra features
1.Kinematic yielding2. Cam clay
3. Mohr-Coulomb
shear stress
mean stress
elastic - stiff
plastic – less stiff
shear stress
shear strain
classical elastic-plastic modelling of soil
for example, Cam clay (1963, 1968)
stress
yield?
strain
typical actual response
void ratio
vertical stress(log scale)
preconsolidation pressure
classical identification of yield from stress:strain response
geometrical construction for estimation of preconsolidation pressure
-150
-100
-50
0
50
100
150
200
0 100 200 300 400
Cam clay?
p': kPa
q: kPa
Fig 3: Anisotropic yield locus for one-dimensional stress history (after Al-Tabbaa, 1984)
Cam clay providing inspiration:
search for ‘Cam clay like’ yield loci
eg kaolin (Al-Tabbaa, 1984)
yield loci for natural clays
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
p/svc
q/svc
Rang de Fleuve
Belfast
Winnipeg
St Alban
Lyndhurst
Mastemyr
collected by Graham et al (1988)
typical experimental observation: stiffness falls steadily with monotonic straining:
is there an elastic region?
shear stiffness degradation data for Quiou sand from resonant column and torsional shear tests (after LoPresti et al, 1997)
limit of elastic response??
how do we objectively identify yielding?occurrence of irrecoverable strain?
dissipation of energy in loading/unloading cycles?change in slope of stress:strain response?
stress
strain
a.
stress
strain
b.
-20
-10
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
Y1 yield locus
Y2 yield locus
Y3 yield locus
q: kPa
p': kPa
yielding of Bothkennar clay:
boundaries deduced from inspection of stress:strain response
Y1 approximately centred on in situ stress state
Y3 reflects natural structure – damaged by any irrecoverable strain - evanescent
after Smith et al (1992)
kaolin revisited: isotropic consolidation histories
-50
0
50
100
150
200
0 50 100 150 200 250 300 350 400 450
mean effective stress p': kPa
deviator stress q: kPa
a.
data from Al-Tabbaa (1987)
kaolin revisited: one-dimensional consolidation histories
data from Al-Tabbaa (1987)
-50
0
50
100
150
200
0 50 100 150 200 250 300 350 400
mean effective stress p': kPa
deviator stress q: kPa b.
q kPa
q kPa
q kPa
q kPa
p' kPa p' kPa
p' kPap' kPa
plastic strain increments: approximate normality to kinematic yield loci
kaolin: Al-Tabbaa, 1987
Deviatoric stress response envelopes
qz:kPa
qx:kPaA
sx
sy
sz
A270
C-A300
C-A330 C-A0 C-A30
C-A60
C-A90
C-A120
C-A150C-A180C-A210
C-A240
-250
-150
-50
50
150
250
-250 -150 -50 50 150 250 A
sz
sy
sx
-250
-150
-50
50
150
250
-250 -150 -50 50 150 250
•distortional stress probe rosettes•constant mean stress•cross anisotropy?•Ev > Eh
εd = 0.05, 0.2, 0.4, 0.6, 0.8, 1, 1.2%
σx σy
σz
σxσy
σz
A: isotropic compression
Hostun sand
Sadek, 2006
ABC30
qz:kPa
qx:kPaA
sx
sy
sz
ABC60
ABC90
ABC120 ABC150
ABC180
ABC210
ABC240
ABC270ABC300
ABC300
B
ABC330
C
ABC360
-250
-150
-50
50
150
250
-250 -150 -50 50 150 250
Stress paths in the octahedral plane for Rosette AB(CCA: p' = 200kPa, Hostun Sand: D r = 65%)
AB30
sz
sy
sx
Aqx:kPa
qz:kPa
AB60AB90
AB120
AB150
AB180
AB210
AB240
AB270
B
AB300AB330
AB360
-250
-150
-50
50
150
250
-250 -150 -50 50 150 250
distortional probing
constant mean stress
non-monotonic stress paths
stress probe rosettes
ABC … probe
AB … probe
σz
σx
σx
σy
σy
σz
Sadek, 2006
A
sz
sy
qx: kPa
sx
B
-250
-150
-50
50
150
250
-250 -150 -50 50 150 250A
sz
sy
sx
qx: kPa
BC
-250
-150
-50
50
150
250
-250 -150 -50 50 150 250
distortional strain
0.05%: history recalled
1%: history ‘forgotten’
radial shearing ABtwo corners
ABC
Stress response envelopes: Hostun sand: small-medium strain
εd = 0.05, 0.2, 0.4, 0.6, 0.8, 1, 1.2%
Sadek, 2006
sz
sx
sy
qx: kPaA
BC
a
bc
-150
-50
50
150
-150 -50 50 150
comparison of 0.05% strain response envelopes for histories A, AB, ABC
stress response envelopes
small/medium strain stiffness
kinematic hardening
centre as indicator of current fabric
but strain too large
Sadek, 2006
Designer models: addition of extra features 1. Kinematic yielding
2. Cam clay3. Mohr-Coulomb
Cam clay
elastic-hardening plastic model
volumetric hardening
associated flow – normality
Cam clay
response in drained triaxial compression tests with constant p'
asymptotic approach to critical state
effect of overconsolidation ratio
sharp division between elastic and plastic response
compare response of soil on nonmonotonic loading
with capability of single yield surface model
extension to simple models using kinematic hardeningand bounding surface plasticity
compare response of soil on nonmonotonic loading with capability of single yield surface model
elastic-hardening plastic model expects elastic behaviour on reversal, sudden drop in stiffness at yield
soils typically show hysteretic behaviour on unload-reload cycles, steady change in incremental stiffness
kinematic hardening extension
yield locus carried around with stress state – 'bubble' – strongly influenced by recent history
stiffness falls as yield 'bubble' approaches bounding surface – controlled by distance b
when loading with 'bubble' in contact with bounding surface model is identical to Cam clay
assume relative size R of 'bubble'
assume rule for translation of 'bubble'
assume interpolation rule linking plastic stiffness with b
…otherwise identical to Cam clay
kaolin
constant p' cycles
hysteresis
build up of volumetric strainexperiment simulation
volumetric strain
distortional strain
η
η
η
η
migration of 'bubble' during constant p' unloading after one-dimensional normal compression
hardening of 'bubble' and bounding surface
q kPa
p' kPa
experiment simulation
constant q cycles after one-dimensional normal compression
add further effects in a similarly hierarchical way
cementation and structure: extension to 'bubble' model
natural soils often contain structure: bonding between particles: destroyed with mechanical or chemical damage
design model in which yield surface has increased size as a result of the bonding
with plastic straining (or chemical weathering) the yield surface gradually shrinks to the yield surface, for remoulded, structureless material
extension of 'bubble' kinematic extension of Cam clay
all features of 'bubble' model retained
ratio of sizes of structure surface and reference surface gives indication of current degree of structure
add measure of structure or bonding: single scalar parameter r
'bounding' surface now called 'structure' surface: size r times larger than a reference surface
structure lost whenever plastic strains occur
damage law:
damage plastic strain increment δεdp combines plastic volumetric
and plastic distortional strain increments:
– additional parameter to control their relative importance
structure progressively disappears:
r 1 as plastic deformation increases
pd1r
kr
logical: structureless soil is one which has been so mechanically pummelled that it has no remaining bonds between particles
particular forms of laboratory testing (triaxial testing, for example) may not be able to provide sufficient damage
evolution law and definition of damage strain may need to include some more subtle reference to the nature of the strain path
shearing with rotation of principal axes is likely to be especially damaging
feasible to introduce other evolution laws which relate change (increase or decrease) of scalar measure of structure r to chemical environment or time or temperature effects
Cam clay can be regained by setting r = 1, R = 1
hierarchical extension of 'bubble' model to include effects of structure
other evolution laws: relate change (increase or decrease) of scalar measure of structure r to chemical environment or time or temperature effects
Cam clay can be regained by setting r = 1, R = 1
Norrköping clay – calibration tests
Rouainia & Muir Wood (2000)
Norrköping clay – parametric variation
Rouainia & Muir Wood (2000)
Norrköping clay – undrained – isotropic consolidation
Rouainia & Muir Wood (2000)
Norrköping clay – undrained – anisotropic consolidation
Rouainia & Muir Wood (2000)
Norrköping clay – undrained – isotropic overconsolidation
Rouainia & Muir Wood (2000)
simulation experimentBothkennar clay
results normalised by Hvorslev equivalent consolidation pressure p'e for structureless soil
Gajo & Muir Wood, 2001
Hierarchical extensions of Cam clay
•it is relatively straightforward to add extra features to a soil model
•advantage in using well known model as basis – check implementation – acceptability
•extra features imply additional soil parameters and additional calibration tests
•seek adequate complexity in modelling – match complexity of model to availability of data and needs of application
Designer models: addition of extra features 1. Kinematic yielding
2. Cam clay
3. Mohr-Coulomb
standard elastic-perfectly plastic Mohr-Coulomb model
non-associated plastic flow
simplicity
sharp stiffness changes
tangent stiffness either elastic or zero
continuing volume change
standard elastic-perfectly plastic Mohr-Coulomb model
available in all numerical analysis programs
subjectivity in selecting values of soil parameters – stiffness, strength, dilatancy
elastic-hardening plastic Mohr-Coulomb model
non-associated flow steady fall in stiffness continuing volume change
is post-peak softening important?
design a model to include softening
Mohr-Coulomb family
post-peak softening to critical state
three regimes of response
adaptation of hardening Mohr-Coulomb model
for η < ηp response is elastic: η < ηp, ηy = ηp δεqp = 0
after peak, linear fall in yield stress ratio with strain
0 < εqp < b: (distortional 'hardening' law)
eventual perfectly plastic critical state: εqp b, ηy = M
non-associated flow rule as before
bM
pq
p
yp
conventional triaxial drained compression test
elastic
post-peak softening
critical state
triaxial undrained compression test
p
ppp
p
i
M12
M211
bK
'p'p
effective stress path
elastic (isotropic: δp' = 0)
post-peak softening
critical state
limited model
concentration on single aspect of response
Mohr-Coulomb model with strength dependent on state variable
Severn-Trent sand
influence of density
softening
dilatancy
simplicity
build on Mohr-Coulomb model
describe journey from initial elastic response to ultimate critical state
include nonlinearity, peak strength and softening
simplicity?
adequate complexity?
Severn-Trent sand
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?
data confirm link between strength and state parameter ψ
Mohr-Coulomb model with current strength dependent on current state parameter
Been & Jefferies
peak strength
Severn-Trent sand: strength
'dense'
'loose'
ψ
mean stress
critical state line
specific volume
state parameter ψ
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 state parameter
Been & Jefferies
peak strength
Severn-Trent sand: strength
'dense'
'loose'
ψ
mean stress
specific volume
state parameter ψ
Severn-Trent sand: dilatancy
Benahmed
dilatancy: volume change during shearing
'dense' sand expands
'loose' sand contracts
dilatancy depends on density
dilatancy varies during test
what do we mean by 'dense' and 'loose'?
volume strain
shear strain
data confirm link between dilatancy and state parameter ψ
if soil is not at critical state when it is being sheared (ψ 0):
then volume changes occur towards the critical state: dilatancy
'loose': ψ > 0: contraction
'dense': ψ < 0: dilation
'loose''dense'
state parameter ψ
Been & Jefferies
dilatancy
'dense'
'loose'
ψ
critical state line
mean stress
specific volume
Severn-Trent sand: dilatancy
mobilised strength 'mob
shear strain
currently mobilised strengthcurrently available strength
monotonic relationship
1ratio
available strength ':varies with state parameter
distortional hardening
monotonic increase of ratio of mobilised to available strength (η/ηp) with distortional strain εq
p
hyperbolic hardening law: simple
Mohr-Coulomb model with strength dependent on state variable
volume change accompanies shearing
hence change in state variable
hence change in available strength
model automatically homes in on critical state
softening emerges without being described mathematically
peak strength is moving target reached at infinite distortional strain – then identical with critical state strength
conventional drained triaxial compression tests
different initial density (state variable)
current peak strength
characterisation of variation of tangent stiffness
soil response perfectly plastic model
nonlinearity and reversed plasticity observed when direction of loading is reversed
elastic-hardening plastic model: behaviour purely elastic for stress ratios lower than the previous maximum stress ratio
Severn-Trent sand
add kinematic hardening:
elastic region of high stiffness carried round with recent stress history
boundary of elastic region is the yield surface
use bounding surface plasticity:
plastic hardening stiffness depends on separation of the yield surface and bounding surface
kinematic hardening Mohr-Coulomb: strength dependent on state variable: hierarchical development
Severn-Trent sand
calibrated against triaxial test data for Hostun sand
effect of different density/stress level automatically described
ignore practical problem of maintaining homogeneity within softening sample
Gajo & Muir Wood, 1999
use model to simulate cyclic undrained loading leading to eventual liquefaction
model fails after 25 cycles
actual soil (Hostun sand) fails after 89 cycles
number of cycles to liquefaction is not a particularly reliable parameter to use for model calibration
obvious significant difference between samples which liquefy in one or two cycles and those which survive for many cycles
character of cyclic pore pressure build-up reproduced in model
messages:
•possible to develop elegant models which reproduce desirable mechanical characteristics
•especially effects of density and stress level
•mathematical complexity not essential
•build up from well known model – Mohr-Coulomb