Hardening Soil model with small strain stiffness - Zace Services Ltd.

Hardening Soil model with small strain stiffness
Andrzej Truty
ZACE Services
1.09.2008
Andrzej Truty ZACE Services
Hardening Soil model with small strain stiffness

Introduction
Hardening Soil (HS) and Hardening Soil-small (HS-small)
models are designed to reproduce basic phenomena exhibited
by soils:
densification
stiffness stress dependency
plastic yielding
dilatancy
strong stiffness variation with growing shear strain amplitude
in the regime of small strains (γ = 10 −6 to γ = 10 −3 )
this phenomenon plays a crucial role for modeling deep
excavations and soil-structure interaction problems
NB. This model is limited to monotonic loads
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Hardening Soil model with small strain stiffness

Introduction
HS model was initially formulated by Schanz, Vermeer and
Bonnier (1998, 1999) and then enhanced by Benz (2006)
Current implementation is slightly modified with respect to
the theory given by Benz:
simplified treatment of dilatancy for the small strain version
(HS-small)
modified hardening law for preconsolidation pressure
This model seems to be one of the simplest in the class of
models designed to handle small strain stiffness
It consists of the two plastic mechanisms, shear and volumetric
Small strain stiffness is incorporated by means of nonlinear
elasticity which includes hysteretic effects
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Hardening Soil model with small strain stiffness

Triaxial test: illustration
Undrained triaxial test:
Drained triaxial test :
video
video
Annimations by P.Baran (University of Agriculture, Kraków,
Poland)
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Hardening Soil model with small strain stiffness

Notion of tangent and secant stiffness moduli
Initial stiffness modulus E o
Unloading-reloading modulus E ur
Secant stiffness modulus at 50 % of the ultimate deviatoric
stress q f
0
250
q [kpa]
200
150
100
50
1
E o
1 E 50
q 50
E ur
1
q f
0.5 q f
σ 3 =const
q un
0 0.05 0.1 0.15 0.2 0.25
EPS-1 [-]
Remark: All classical soil models require specification of E ur
modulus (Cam-Clay, Cap etc..)
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Hardening Soil model with small strain stiffness

Stiffness-strain relation for soils (G/G o (γ))
G - current secant shear modulus
G o - shear modulus for very small strains
Atkinson 1991
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Hardening Soil model with small strain stiffness

Notion of treshold shear strain γ 07
G
To describe the shape of (γ) curve an additional
G o
characteristic point is needed
It is common to specify the shear strain γ 0.7 at which ratio
G
G o
= 0.7
0.7
γ 07
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Hardening Soil model with small strain stiffness

Dynamic vs static modulus
Relation between ”static” Young modulus E s , obtained from
standard triaxial test at axial strain ε 1 ≈ 10 −3 , and ”dynamic”
Young modulus (the one at very small strains) E d = E o is
shown in diagram published by Alpan (1970) (after Benz)
100
E
E
d
s
Rocks
10
cohesive soils
granular soils
E s
[kPa]
1
1000 10000 100000 1000000
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Hardening Soil model with small strain stiffness

HS model: general concept
Double hardening elasto-plastic model (Schanz, Vermeer,
Benz)
Nonlinear elasticity for stress paths penetrating the interior of
the elastic domain
600
q [kPa]
500
400
300
200
Cap surface
100
0
0 100 200 300 400 500
p [kPa]
Graphical representation of shear mechanism and cap surface
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness

HS model: shear mechanism
Duncan-Chang model as the origin for shear mechanism
250
q f
q [kPa]
200
150
100
50
0
E 50
½ q f
1
1
M-C limit
0 0.01 0.02 0.03 0.04 0.05
eps-1
E ur
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Hardening Soil model with small strain stiffness

Stiffness stress dependency
Remarks
E ur = E ref
ur
E 50 = E ref
50
( σ
∗
3 + c cotφ
σ ref + c cotφ
( σ
∗
3 + c cotφ
σ ref + c cotφ
) m
) m
1 Stiffness degrades with decreasing σ 3 up to σ 3 = σ L (by
default we assume σ L =10 kPa)
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Hardening Soil model with small strain stiffness

Extension to small strain: new ingredients
To extend standard HS model to the range of small strain Benz
introduced few modifications:
1 Strain dependency is added to the stress-strain relation, for
stress paths penetrating the elastic domain
2 The modified Hardin-Drnevich relationship is used to relate
current secant shear modulus G and equivalent monotonic
shear strain γ hist
3 Reversal points are detected with aid of deviatoric strain
history second order tensor H ij ; in addition the current
equivalent shear strain γ hist is computed by using this tensor
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Hardening Soil model with small strain stiffness

How does it work
N-1
N
N+1
plot from paper by Ishihara 1986
At step N : γ histN−1 = 8 × 10 −5 γ histN = 10 −4
At step N + 1 : γ histN = 0 γ histN+1 = 2 × 10 −5
Primary loading: γ histN+1 > γhist
max
Unloading/reloading: γ histN+1 ≤ γ max
Hardin-Drnevich law: G =
hist
G o
1 + a γ hist
γ 0.7
(secant modulus)
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Hardening Soil model with small strain stiffness

Shear tangent modulus cut-off
G
G ur
γ c
γ c = γ 0.7
a
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(√ )
Go
− 1
G ur
Hardening Soil model with small strain stiffness
γ

Setting initial state variables: γ PS
o
Given: σ o , OCR
Find: γ PS
and p co
o and p co
0
600
q [kPa]
500
400
300
200
100
Shear mechanism
Cap surface
σ ο
σ SR
0 100 200 300 400 500
p [kPa]
Procedure:
Set effective stress state at the SR point
σy
SR = σ yo OCR
σx
SR = σz
SR = σ y Ko
SR
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Hardening Soil model with small strain stiffness

Setting initial state variables: γ PS
o
and p co
600
q [kPa]
500
400
300
200
100
Shear mechanism
Cap surface
σ ο
σ SR
0
0 100 200 300 400 500
p [kPa]
Procedure:
For given σ SR state compute γo
PS
f 1 = 0
from plastic condition
For given σ SR state compute p co from plastic condition f 2 = 0
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Hardening Soil model with small strain stiffness

Setting initial state variables: γ PS
o
and p co
Remarks
= Ko
NC
(approximate Jaky’s formula)
1 K SR
o
2 K SR
o
≈ 1 − sin(φ) in the standard applications
= 1 for case of isotropic consolidation (used in triaxial
testing for instance)
3 For sands notion of preconsolidation pressure is not as
meaningful as for cohesive soils hence one may assume
OCR=1 and effect of density will be embedded in H and M
parameters
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Hardening Soil model with small strain stiffness

Setting M and H parameters based on oedometric test
600
500
σ
q [kPa]
400
q*
300
200
100
p*
0
0 100 200 300 400 500
p [kPa]
σ ref 1
oed
E oed
ε
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Hardening Soil model with small strain stiffness

Material properties
Parameter Unit HS-standard HS-small
Eur ref [kPa] yes yes
E50 ref [kPa] yes yes
σ ref [kPa] yes yes
m [—] yes yes
ν ur [—] yes yes
R f [—] yes yes
c [kPa] yes yes
φ [ o ] yes yes
ψ [ o ] yes yes
e max [—] yes yes
f t [kPa] yes yes
D [—] yes yes
M [—] yes yes
H [kPa] yes yes
OCR/q POP [—/kPa] yes yes
Eo ref [kPa] no yes
γ 0.7 [—] no yes
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Hardening Soil model with small strain stiffness

Converting MC to HS model: indentation problem
Assumption: q = 0.5 q ult
Given: E for MC model and E ur
E 50
= ...,
E 50
E oed
= ...
1m
A
q = 0.5 q ult
10m
10m
Find: E ref
ur , M and H for standard HS model
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Hardening Soil model with small strain stiffness

Example: triaxial test on dense Hostun sand
6
5.5
5
120000
SIG-1 / SIG-3 [kPa]
4.5
4
3.5
3
2.5
HS-std
HS-small
G [kPa]
100000
80000
60000
40000
HS-std
HS-small
2
1.5
20000
1
0
0 0.02 0.04 0.06 0.08 0.1
0.00001 0.0001 0.001 0.01 0.1 1
-EPS-Y [-]
EPS-X - EPS-Y [-]
(a) σ 1
σ 3
(ε 1 ) (Z Soil)
(b) G(γ) (Z Soil)
4
3.5
0 0.02 0.04 0.06 0.08 0.1
0.08
SIG-1 / SIG-3 [kPa]
3
2.5
2
HS-std
HS-small
-EPS-V [-]
0.07
0.06
0.05
0.04
0.03
0.02
HS-std
HS-small
1.5
0.01
0
1
-0.01
0 0.002 0.004 0.006 0.008 0.01
-EPS-Y [-]
-0.02
-EPS-Y [-]
(c) σ 1
σ 3
(ε 1 ) (zoom) (Z Soil)
(d) ε v (ε 1 ) (Z Soil)
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Hardening Soil model with small strain stiffness

Example: triaxial test on dense Hostun sand
SIG-1 / SIG-3 [kPa]
6
5.5
5
4.5
4
HS-std
3.5
HS-small
3
2.5
2
1.5
1
0 0.02 0.04 0.06 0.08 0.1
EPS-1 [-]
G [kPa]
200000
180000
160000
140000
120000
100000
80000
60000
40000
20000
0
0.00001 0.0001 0.001 0.01 0.1 1
EPS-1 - EPS-3 [-]
HS-std
HS-small
(a) σ 1
σ 3
(ε 1 ) (Z Soil)
(b) G(γ) (Z Soil)
4
3.5
0 0.02 0.04 0.06 0.08 0.1
0.08
SIG-1 / SIG-3 [kPa]
3
2.5
2
HS-std
HS-small
EPS-V [-]
0.07
0.06
0.05
0.04
0.03
0.02
HS-std
HS-small
1.5
0.01
0
1
-0.01
0 0.002 0.004 0.006 0.008 0.01
-0.02
EPS-1 [-]
EPS-1 [-]
(c) σ 1
σ 3
(ε 1 ) (zoom) (Z Soil)
(d) ε v (ε 1 ) (Z Soil)
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Hardening Soil model with small strain stiffness

Example: triaxial test on dense Hostun sand
6
5.5
300000
5
250000
SIG-1 / SIG-3 [kPa]
4.5
4
3.5
3
2.5
HS-std
HS-small
G [kPa]
200000
150000
100000
HS-std
HS-small
2
1.5
50000
1
0
0 0.02 0.04 0.06 0.08 0.1
0.00001 0.0001 0.001 0.01 0.1 1
EPS-1 [-]
EPS-1-EPS-3 [-]
(a) σ 1
σ 3
(ε 1 ) (Z Soil)
(b) G(γ) (Z Soil)
4
0 0.02 0.04 0.06 0.08 0.1
3.5
0.08
SIG-1 / SIG-3 [kPa]
3
2.5
2
HS-std
HS-small
EPS-V [-]
0.07
0.06
0.05
0.04
0.03
0.02
HS-std
HS-small
1.5
0.01
0
1
-0.01
0 0.002 0.004 0.006 0.008 0.01
-0.02
EPS-1 [-]
EPS-1 [-]
(c) σ 1
σ 3
(ε 1 ) (zoom) (Z Soil)
(d) ε v (ε 1 ) (Z Soil)
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Hardening Soil model with small strain stiffness

Estimation of material properties: input data
Given 3 drained triaxial test results for 3 confining pressures:
σ 3 = 100 kPa
σ 3 = 300 kPa
σ 3 = 600 kPa
Shear characteristics q − ε 1
Dilatancy characteristics ε v − ε 1
Stress paths in p − q plane
Measurements of small strain stiffness moduli E o (σ 3 ) for the
assumed confining pressures (through direct measurement of
shear wave velocity in the sample)
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Hardening Soil model with small strain stiffness

Estimation of material properties: stress paths in p-q plane
Estimation of friction angle φ = φ cs and cohesion c
q
Residual M-C envelope
1
6sinφ
M*
=
3 − sinφ
6 cosφ
c*
= c
3 − sinφ
p
If we know M ∗ and c ∗ then we can compute φ and c:
φ = arcsin 3 M∗
6 + M ∗ c = c ∗ 3 − sin φ
6 cos φ
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Hardening Soil model with small strain stiffness

Estimation of material properties: stress paths in p-q plane
Estimation of friction angle φ = φ cs and cohesion c
3000
q [kPa]
2500
2000
1500
1000
2358 1
2358/1386=1.7
500
1386
0
0 300 600 900 1200 1500 1800
p [kPa]
Here: φ = arcsin 3 ∗ 1.7
6 + 1.7 ≈ 42o c = 0
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Hardening Soil model with small strain stiffness

Estimation of material properties: dilatancy angle
0.06
0.05
EPS-V [-]
0.04
0.03
0.02
0.01
d
1
Dilatancy cut-off
0
-0.01
-0.02
0 0.02 0.04 0.06 0.08 0.1
ψ = arcsin
EPS-1 = - EPS-3 [-]
( d
2 + d
)
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Hardening Soil model with small strain stiffness

Estimation of material properties: dilatancy angle
0.06
0.05
0.04
ε 0.03 V
0.02
0.01
0
-0.01
d=0.75
-0.02
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
ε 1
( ) 0.75
ψ = arcsin
≈ 16 o
2 + 0.75
1
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Hardening Soil model with small strain stiffness

Estimation of material properties: E ref
o
and m
Analytical formula: E o = E ref
o
( σ
∗
3 + c cotφ
σ ref + c cotφ
) m
Measured: shear wave velocity v s at ε 1 = 10 −6 and at given
confining stress σ 3
Compute : shear modulus G o = ρv 2 s
Compute : Young modulus E o = 2 (1 + ν) G o
σ 3 [kPa] E o [kPa]
100 250000
300 460000
600 675000
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Hardening Soil model with small strain stiffness

Estimation of material properties: E ref
o
and m
Analytical formula: E o = E ref
o
( σ
∗
3 + c cotφ
σ ref + c cotφ
) m
Measured: shear wave velocity v s at ε 1 = 10 −6 and at given
confining stress σ 3
Compute : shear modulus G o = ρv 2 s
Compute : Young modulus E o = 2 (1 + ν) G o
σ 3 [kPa] E o [kPa]
100 250000
300 460000
600 675000
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Hardening Soil model with small strain stiffness

Estimation of material properties: E ref
o
and m
Reanalyze E o vs σ 3 in logarithmic scales
13.1 − 12.55
Averaged slope yields m; here m =
1.0
Find
(
intersection of the line with axis ln E o at
σ
∗ )
3 + c cotφ
ln
σ ref = 0
+ c cotφ
Here the intersection is at 12.43 hence
= e 12.43 ≈ 2.718 12.43 = 250000 kPa
E ref
o
= 0.55
13.6
ln E o
13.4
13.2
13
m
12.8
12.6
⎛ σ
3
+ c cotφ
⎞
1
ln⎜
⎟
12.4 12.43
ref
⎝ σ + c cotφ
⎠
12.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
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Hardening Soil model with small strain stiffness

Estimation of E ref
o
from CPT testing
To estimate small strain modulus G o at a certain depth one
may use empirical formula by Mayne and Rix:
G o = 49.4 q0.695 t
e 1.13
[MPa]
q t is a corrected tip resistance expressed in MPa
e is the void ratio
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Hardening Soil model with small strain stiffness

Estimation of material properties: E ref
50
Lets us find E 50 for each confining stress
f
50
( σ
3
=
q
2500
2000
1500
100)
1000
1
E
50( σ
3
= 300kPa)
1
E
50( σ
3
= 600kPa)
E
50( σ
3
= 100kPa)
1
q
f
50
( σ
3
=
q
f
50
( σ
3
=
100)
500
100)
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
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Hardening Soil model with small strain stiffness

Estimation of material properties: E ref
50
Reanalyze E 50 vs σ 3 in logarithmic scales
Here we can fix m to the one obtained for small strain moduli
Find
(
intersection of the line with axis ln E 50 at
σ
∗ )
3 + c cotφ
ln
σ ref = 0
+ c cotφ
Here the intersection is at ≈ 10.30 hence
E50 ref ≈ e10.30 ≈ 2.718 10.30 ≈ 30000 kPa
ln E 50
11.4
11.2
11
10.8
10.6
⎛ σ
3
+ c cotφ
⎞
ln⎜
⎟
σ + c cotφ
ref
10.4
10.30
⎝
⎠
10.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
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Hardening Soil model with small strain stiffness

Estimation of material properties: E ref
ur
The unloading reloading modulus as well as oedometric
moduli are usually not accessible
We can use Alpans diagram to deduce E ref
ur
E ref
o
(default is E ur
ref
Eo
ref
this value is larger
once we know
= 3); for cohesive soils like tertiary clays
For oedometric modulus at the reference stress σ ref = 100
kPa we can assume Eoed ref = E 50
ref
γ 0.7 = 0.0001...0.0002 for sands and γ 0.7 = 0.00005...0.0001
for clays
Smaller γ 0.7 values yield softer soil behavior
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Hardening Soil model with small strain stiffness

Excavation in Berlin Sand: engineering draft
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Hardening Soil model with small strain stiffness

Excavation in Berlin Sand: FE discretization
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Hardening Soil model with small strain stiffness

Excavation in Berlin Sand: Bending moments
-500 -400 -300 -200 -100 0 100 200 300 400 500
0
-5
-10
Y [m[]
-15
-20
-25
-30
HS
HS-small
MC
M [kNm/m]
-35
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Hardening Soil model with small strain stiffness

Excavation in Berlin Sand: Wall deflections
-0.015 -0.01 -0.005 0 0.005 0.01
0
-5
-10
Y [m]
-15
-20
-25
-30
HS-small
HS
MC
Ux [m]
-35
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Hardening Soil model with small strain stiffness

Excavation in Berlin Sand: Soil deformation in cross
section x =20m
Y [m]
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
0 0.01 0.02 0.03 0.04
0
Uy [m]
HS
HS-small
MC
Vertical heaving of subsoil at last stage of excavation, relative to
the step when dewatering was finished (t = 2)
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Hardening Soil model with small strain stiffness

Excavation in Berlin Sand: Soil deformation in cross
section x =20m
Y [m]
-0.004 -0.003 -0.002 -0.001 0
0
Ux [m]
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
HS
HS-small
MC
Horizontal movement in cross section x=20m at last stage of
excavation, relative to the step when dewatering was finished
(t = 2)
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Hardening Soil model with small strain stiffness

Conclusions
Model properly reproduces strong stiffness variation with shear
strain
It can be used in simulations of soil-structure interaction
problems
Implementation is ”rather heavy”
It should properly predict deformations near the excavations
Model reduces excessive heavings at the bottom of the
excavation
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Hardening Soil model with small strain stiffness