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Hardening Soil model with small strain stiffness
Andrzej TrutyZACE Services
25.08.2009
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 exhibitedby soils:
densificationstiffness stress dependencyplastic yieldingdilatancystrong stiffness variation with growing shear strain amplitudein the regime of small strains (γ = 10−6 to γ = 10−3)this phenomenon plays a crucial role for modeling deepexcavations and soil-structure interaction problems
NB. This model is limited to monotonic loads
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Introduction
HS model was initially formulated by Schanz, Vermeer andBonnier (1998, 1999) and then enhanced by Benz (2006)
Current implementation is slightly modified with respect tothe theory given by Benz:
simplified treatment of dilatancy for the small strain version(HS-small)modified hardening law for preconsolidation pressuremodified form of the cap yield surface (2009)
This model seems to be one of the simplest in the class ofmodels designed to handle small strain stiffness
It consists of the two plastic mechanisms, shear and volumetric
Small strain stiffness is incorporated by means of nonlinearelasticity which includes hysteretic effects
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Notion of tangent and secant stiffness moduli
Initial stiffness modulus Eo
Unloading-reloading modulus Eur
Secant stiffness modulus at 50 % of the ultimate deviatoricstress qf
0
50
100
150
200
250
0 0.05 0.1 0.15 0.2 0.25
EPS-1 [-]
q [k
pa] 1
Eo
1E50
1Eur
qf
0.5 qf
σ3=const
q50
qun
Remark: All classical soil models require specification of Eur
modulus (Cam-Clay, Cap etc..)Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Stiffness-strain relation for soils (G/Go (γ))
G - current secant shear modulus
Go - shear modulus for very small strains
Atkinson 1991
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Notion of treshold shear strain γ07
To describe the shape ofG
Go(γ) curve an additional
characteristic point is needed
It is common to specify the shear strain γ0.7 at which ratioG
Go= 0.7
0.7
γ07
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Influence of void ratio and confining stress p′
onG/Go (γ))
Cohesionless soils
Wichtmann and Triantafyllidis (after Benz)
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Influence of plasticity index PI on G/Go (γ))
Cohesive soils
(Vucetic and Dobry (after Benz (PhD thesis))
Remarks
1 Results for PI < 30 are confirmed by other researchers whilethese for PI > 30 should be used with a special care (Benz)
2 Stokoe proposed linear interpolation for γ0.7
γ0.7 = 10−4 for PI = 0 to γ0.7 = 6× 10−4 for PI = 100
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Dynamic vs static modulus
Relation between ”static” Young modulus Es , obtained fromstandard triaxial test at axial strain ε1 ≈ 10−3, and ”dynamic”Young modulus (the one at very small strains) Ed = Eo isshown in diagram published by Alpan (1970) (after Benz)
1
10
100
1000 10000 100000 1000000
cohesive soils
granular soils
Rockss
d
EE
[kPa]sE
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Shear modulus for very small strains
Go = A f (e)OCRk
(p′
pref
)m
Hardin and Black (1978)
Soil emin emax A [kPa] f (e) m Ref.
Clean sands 0.5 1.1 57(2.17− e)2
1 + e0.4 Iwasaki
Undisturbedclayey soilsand crushedsand
0.6 1.5 33(2.97− e)2
1 + e0.5 Hardin
andBlack
Undisturbedcohesive soils
0.6 1.5 16(2.97− e)2
1 + e0.5 Kim
Loess 1.4 4.0 1.4(7.32− e)2
1 + e0.6 Kim
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Poisson coefficient for very small strains
Poisson ratio varies in the range ν = 0.1..0.3 in small straindomain
Its value in further derivations will be kept constant (bydefault ν = 0.25)
Andrzej Truty ZACE Services 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 ofthe elastic domain
0
100
200
300
400
500
600
0 100 200 300 400 500
p [kPa]
q [k
Pa]
Cap surface
Graphical representation of shear mechanism and cap surfaceAndrzej Truty ZACE Services Hardening Soil model with small strain stiffness
HS model: shear and cap yield surfaces
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
0
50
100
150
200
250
0 0.01 0.02 0.03 0.04 0.05
eps-1
q [k
Pa]
1E50
qfM-C limit
1
Eur½ qf
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
HS model: shear mechanism
0
100
200
300
400
500
600
700
0 100 200 300 400 500 600
p [kPa]
q [k
Pa]
M-C
γ=0.1=const.
γ=0.01=const.
γ=0.001=const.γ=0.0001=const.
f1 =qa
E50
q
qa − q− 2
q
Eur− γPS
qf =2 sin(φ)
1− sin(φ)(σ3 + c ccotφ)
qa =qf
Rf
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Flow rule for shear mechanism, dilatancy and hardening
g1 =σ1 − σ3
2− σ1 + σ3
2sinψm
sinψm =sinφm − sinφcs
1− sinφm sinφcs
sinφm =σ1 − σ3
σ1 + σ3 + 2c cotφ
-0.5-0.4-0.3-0.2-0.1
00.10.20.30.40.50.6
0 10 20 30 40 50 60phi_m [deg]
sin
psi_
m
Domain ofcontractancy
Domain ofdilatancy
Contractancy cut-off
Rowe’s dilatancy
dγPS = dλ1
(∂g1
∂σ1− ∂g1
∂σ2− ∂g1
∂σ3
)= dλ1
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Cap mechanism
Yield condition: f2 =q2
M2 r2(θ)+ p2 − p2
c
r(θ) is defined via van Ekelen’s formula (like in Cam-Claymodel
Plastic potential: g2 =q2
M2+ p2
Hardening law: d pc = dλ2 2H
(pc + c cotφ
σref + c cotφ
)m
p
Remarks:
1 M and H parameters can be estimated for assumed KNCo and
tangent Eoed modulus set up at a given vertical stress
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Additional strength criteria
Mohr-Coulomb yield condition
f ∗1 = σ1 −1− sinφ
1 + sinφσ3 −
2c cosφ
1 + sinφ= 0
Mohr-Coulomb plastic flow rule
g∗1 = g1
NB. Here same plastic flow rule is used as for the shearmechanism f1Rankine yield condition (tensile cut-off)
f3 = σ1 − ft = 0
where: ft is the assumed tensile strength (default is ft = 0)
Rankine plastic flow rule(associated flow rule is used)
g3 = f3
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Stiffness stress dependency
Eur = E refur
(σ∗3 + c cotφ
σref + c cotφ
)m
E50 = E ref50
(σ∗3 + c cotφ
σref + c cotφ
)m
Remarks
1 Stiffness degrades with decreasing σ3 up to σ3 = σL (bydefault we assume σL=10 kPa)
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Extension to small strain: new ingredients
To extend standard HS model to the range of small strain Benzintroduced few modifications:
1 Strain dependency is added to the stress-strain relation, forstress paths penetrating the elastic domain
2 The modified Hardin-Drnevich relationship is used to relatecurrent secant shear modulus G and equivalent monotonicshear strain γhist
3 Reversal points are detected with aid of deviatoric strainhistory second order tensor Hij ; in addition the currentequivalent shear strain γhist is computed by using this tensor
4 Hardening laws for γPS and pc are modified by introducing hi
factor; this factor for very small strains is much larger than1.0 and decreases to 1.0 once the shear strain γhist exceedscertains strain amplitude γc
5 Certain constractancy is allowed in the plastic flow rule forshear mechanism
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
How does it work ?
N
N+1N-1
plot from paper by Ishihara 1986
At step N : γhistN−1= 8× 10−5 γhistN = 10−4
At step N + 1 : γhistN = 0 γhistN+1= 2× 10−5
Primary loading: γhistN+1> γmax
hist
Unloading/reloading: γhistN+1≤ γmax
hist
Hardin-Drnevich law: G =Go
1 + aγhist
γ0.7
(secant modulus)
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Shear tangent modulus cut-off
γc
G
γ
Gur
γc =γ0.7
a
(√Go
Gur− 1
)Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Modifications: Dilatancy
PHI = 40, PSI=10
-35
-30
-25
-20
-15
-10
-5
0
5
10
0 10 20 30 40
PHI_m [deg]
PSI_
m [d
eg]
Dafalias,Li(after Benz)
Rowe’s dilatancy
Scaled Rowe’s dilatancyD = 0.25
PHI = 30, PSI=5
-30
-25
-20
-15
-10
-5
0
5
10
15
20
0 10 20 30 40
PHI_m [deg]PS
I_m
[deg
]
Dafalias,Li(after Benz)
Rowe’s dilatancy
Scaled Rowe’s dilatancyD = 0.25
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Setting initial state variables: γPSo and pco
Given: σo , OCRFind: γPS
o and pco
0
100
200
300
400
500
600
0 100 200 300 400 500
p [kPa]
q [k
Pa]Cap surface
Shear mechanism
σSR
σο
Procedure:
Set effective stress state at the SR pointσSR
y = σyo OCR
σSRx = σSR
z = σy KSRo
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Setting initial state variables: γPSo and pco
0
100
200
300
400
500
600
0 100 200 300 400 500
p [kPa]
q [k
Pa]
Cap surface
Shear mechanism
σSR
σο
Procedure:
For given σSR state compute γPSo from plastic condition
f1 = 0
For given σSR state compute pco from plastic condition f2 = 0
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Setting initial state variables: γPSo and pco
Remarks
1 KSRo = KNC
o ≈ 1− sin(φ) in the standard applications(approximate Jaky’s formula)
2 KSRo = 1 for case of isotropic consolidation (used in triaxial
testing for instance)
3 For sands notion of preconsolidation pressure is not asmeaningful as for cohesive soils hence one may assumeOCR=1 and effect of density will be embedded in H and Mparameters
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Setting M and H parameters based on oedometric test
0
100
200
300
400
500
600
0 100 200 300 400 500
p [kPa]
q [k
Pa]
p*
q*
σ
εσref
1
Eoed
oed
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Setting M and H parameters based on oedometric test
Assumptions:
1 At a given σrefoed vertical stress both shear and volumetric
mechanisms are active
2 p∗ =1 + 2KNC
o
3σref
oed while q∗ = (1− KNCo )σref
oed
3 A strain driven program is applied with vertical strainamplitude ∆ε = 10−5 and resulting tangent oedometric
modulus is computed as Eoed =∆σ
∆ε4 The two conditions must be fulfiled: Ko coefficient generated
by the model must be equal to the one set by the user (usingJaky’s formula for instance Ko = 1− sinφ) and tangentoedometric modulus generated by the model must be equal tothe value given by the user
5 If we take the data from the experiment we must be sure thatthe given oedometric modulus corresponds to the primaryloading branch of σ − ε curve
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Material properties
Parameter Unit HS-standard HS-smallE ref
ur [kPa] yes yesE ref
50 [kPa] yes yesσref [kPa] yes yesm [—] yes yesνur [—] yes yesRf [—] yes yesc [kPa] yes yesφ [o ] yes yesψ [o ] yes yesemax [—] yes yesft [kPa] yes yesD [—] yes yesM [—] yes yesH [kPa] yes yesOCR/qPOP [—/kPa] yes yesE ref
o [kPa] no yesγ0.7 [—] no yes
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
User interface
Remark1 HS/HS-small model can be actived only in the � Advanced
modeAndrzej Truty ZACE Services Hardening Soil model with small strain stiffness
User interface: Elastic properties
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
User interface: Elastic properties (HS)
Remarks
1 Standard HS model is activated if � Advanced checkbox isset OFF
2 E refur is the unloading/reloading Young modulus given at the
reference stress σref
3 νur is the unloading/reloading Poisson coefficient; it variesfrom 0.15 to 0.3, hence for sands it is recommended toassume νur = 0.2..0.25 and for clays νur = 0.25..0.3
4 m is the exponent in stress dependency power law; it variesfrom m = 0.4 to m = 0..6; it is smaller for dense sands andlarger for clays
5 σL is the minimum allowed reference stress value used forevaluation of stiffness moduli
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
User interface: Elastic properties (HS-small)
Remarks
1 HS-small model is activated if � Advanced checkbox is setON
2 The HS-small model requires two additional parameters:Young modulus at very low strains E ref
o at the reference stressσref and threshold shear strain γ0.7
3 In case of lack of information on E refo one may try to estimate
E refo based on Alpan’s diagram assuming Es = Eur
4 In the current implementation γ0.7 is assumed to be constant
5 In case of lack of information on γ0.7 the diagram by Vuceticand Dobry can be used for cohesive soils and diagram byWichtmann and Triantafyllidis for cohesionless ones
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
User interface: Plastic properties (HS/HS-small)
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
User interface: Plastic properties (HS/HS-small)
Remarks1 All material properties collected in group Nonlinear are
common for HS and HS-small models2 In the advanced mode one may activate tensile and dilatancy
cut-off conditions, set up the multiplier D for Rowe’sdilatancy law in the contractant domain (for HS model thedefault value is D = 0.0 and for HS-small D = 0.25),
3 E ref50 is the secant Young modulus at 50 % of failure deviatoric
stress qf derived from the q − ε1 curve in drained triaxial test4 φ is the friction angle5 ψ is the dilatancy angle6 c ′ is the effective cohesion7 Rf is the failure ratio (default Rf = 0.9)8 ft is the tensile strength (default ft = 0)9 emax is the maximum allowed void ratio; if current void ratio
exceeds the emax dilatancy angle is switched to ψ = 0
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
User interface: Plastic properties (HS/HS-small)
Remarks
1 Cap surface parameter M and hardening parameter H arederived by using a simple calculator which simulates anoedometric test; for given tangent oedometric modulus Eoed
at a given reference vertical stress σrefoed and for assumed KNC
o
parameter (here Jaky’s formula can be used) values of H and
M are evaluated (press button Evaluate M,H ); one may
assume Eoed = E ref50
(σref
oed + c cotφ
σref + c cotφ
)m
as a default value
2 Setting the initial state variables γPSo and pco can be carried
out by means of assumed OCR or preoverburden pressureqPOP
3 To compute KNCo from Jaky formula press button
Use Jaky’s formula for KoNC
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
User interface: Plastic properties (HS/HS-small)
Remarks
1 Pairs KSRo and OCR (OCR ≥ 1.0) or KSR
o and qPOP areneeded to setup the initial position of the cap surface and theinitial value of the hardening parameter γPS
2 pminco is the minimum allowed value for the initial
preconsolidation stress
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Converting MC to HS model: general idea
Question: Having calibrated standard MC can we convert it to HSmodel ?
Stiffness modulus E refur and cap surface parameters H and M
can be estimated by running an inverse analysis of a planestrain problem of a soil layer loaded by a strip loading q
q = 0.5 qult with qult being the approximate ultimate limitload density
The template data files for MC and HS model can be found inthe CFG directory under names: template-foot-MC andtemplate-foot-HS
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Converting MC to HS model: indentation problem
10m
10m
q = 0.5 qult
1m
A
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
User interface: Converting MC to HS model
Given: γdry , K insituo ,νur , σref , σL, m, φ, ψ, c ′, OCR, KSR
o ,
E refur
E ref50
= .... andE ref
50
E refoed
= .... and Young modulus that user
would assume in the simulation with a standard MC model
Find: E refur
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Convert MC to HS model: algorithm
The estimation idea is as follows:
1 We know parameters to be used in the simulation with aid ofa standard MC model: E , γdry , K insitu
o ,νur , φ, ψ, c ′
2 Now we want to use HS/HS-small model but we do not knowon how to estimate E ref
ur , H and M parameters
3 We select a plane-strain problem of a strip loading q appliedto a uniform layer of soil as a template problem
4 We assume the additional parameters for HS model: σL, m,
OCR, KSRo and the two coefficients
E refur
E ref50
= .... (default is 3)
andE ref
50
E refoed
= .... (default is 1.0)
5 We run the optimization procedure which yields the E refur , M
and H such that the settlement at point A obtained from MCand standard (!!!) HS model are the same
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Example: triaxial test on dense Hostun sand
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0 0.02 0.04 0.06 0.08 0.1-EPS-Y [-]
SIG
-1 /
SIG
-3 [k
Pa]
HS-stdHS-small
0
20000
40000
60000
80000
100000
120000
0.00001 0.0001 0.001 0.01 0.1 1EPS-X - EPS-Y [-]
G [k
Pa]
HS-stdHS-small
(a)σ1
σ3
(ε1) (Z Soil) (b) G (γ) (Z Soil)
1
1.5
2
2.5
3
3.5
4
0 0.002 0.004 0.006 0.008 0.01-EPS-Y [-]
SIG
-1 /
SIG
-3 [k
Pa]
HS-stdHS-small
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.080 0.02 0.04 0.06 0.08 0.1
-EPS-Y [-]
-EPS
-V [-
]
HS-stdHS-small
(c)σ1
σ3
(ε1) (zoom) (Z Soil) (d) εv (ε1) (Z Soil)
(e) Solution by Benz [?]
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Example: triaxial test on dense Hostun sand
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0 0.02 0.04 0.06 0.08 0.1EPS-1 [-]
SIG
-1 /
SIG
-3 [k
Pa]
HS-stdHS-small
0
20000
40000
60000
80000
100000
120000
140000
160000
180000
200000
0.00001 0.0001 0.001 0.01 0.1 1EPS-1 - EPS-3 [-]
G [k
Pa]
HS-stdHS-small
(a)σ1
σ3
(ε1) (Z Soil) (b) G (γ) (Z Soil)
1
1.5
2
2.5
3
3.5
4
0 0.002 0.004 0.006 0.008 0.01EPS-1 [-]
SIG
-1 /
SIG
-3 [k
Pa]
HS-stdHS-small
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.080 0.02 0.04 0.06 0.08 0.1
EPS-1 [-]
EPS-
V [-]
HS-stdHS-small
(c)σ1
σ3
(ε1) (zoom) (Z Soil) (d) εv (ε1) (Z Soil)
(e) Solution by Benz [?]
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Example: triaxial test on dense Hostun sand
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
0 0.02 0.04 0.06 0.08 0.1
EPS-1 [-]SI
G-1
/ SI
G-3
[kPa
]
HS-stdHS-small
0
50000
100000
150000
200000
250000
300000
0.00001 0.0001 0.001 0.01 0.1 1EPS-1-EPS-3 [-]
G [k
Pa]
HS-stdHS-small
(a)σ1
σ3
(ε1) (Z Soil) (b) G (γ) (Z Soil)
1
1.5
2
2.5
3
3.5
4
0 0.002 0.004 0.006 0.008 0.01
EPS-1 [-]
SIG
-1 /
SIG
-3 [k
Pa]
HS-stdHS-small
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.080 0.02 0.04 0.06 0.08 0.1
EPS-1 [-]
EPS-
V [-]
HS-stdHS-small
(c)σ1
σ3
(ε1) (zoom) (Z Soil) (d) εv (ε1) (Z Soil)
(e) Solution by Benz [?]Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: input data
Given 3 drained triaxial test results for 3 confining pressures:σ3 = 100 kPaσ3 = 300 kPaσ3 = 600 kPa
Shear characteristics q − ε1
Dilatancy characteristics εv − ε1
Stress paths in p − q planeMeasurements of small strain stiffness moduli Eo (σ3) for theassumed confining pressures (through direct measurement ofshear wave velocity in the sample)
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: stress paths in p-qplane
Estimation of friction angle φ = φcs and cohesion c
p
q
φφ
sin3cos6*−
=cc
1
φφ
sin3sin6*−
=MResidual M-C envelope
If we know M∗ and c∗ then we can compute φ and c :
φ = arcsin3 M∗
6 + M∗c = c∗
3− sinφ
6 cosφ
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: stress paths in p-qplane
Estimation of friction angle φ = φcs and cohesion c
0
500
1000
1500
2000
2500
3000
0 300 600 900 1200 1500 1800
p [kPa]
q [k
Pa]
1386
2358 12358/1386=1.7
Here: φ = arcsin3 ∗ 1.7
6 + 1.7≈ 42o c = 0
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: dilatancy angle
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0 0.02 0.04 0.06 0.08 0.1
EPS-1 = - EPS-3 [-]
EPS-
V [-]
1
d Dilatancy cut-off
ψ = arcsin
(d
2 + d
)
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: dilatancy angle
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
1
d=0.75Vε
1ε
ψ = arcsin
(0.75
2 + 0.75
)≈ 16o
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: E refo and m
Analytical formula: Eo = E refo
(σ∗3 + c cotφ
σref + c cotφ
)m
Measured: shear wave velocity vs at ε1 = 10−6 and at givenconfining stress σ3
Compute : shear modulus Go = ρv2s
Compute : Young modulus Eo = 2 (1 + ν) Go
σ3 [kPa] Eo [kPa]
100 250000
300 460000
600 675000
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: E refo and m
Analytical formula: Eo = E refo
(σ∗3 + c cotφ
σref + c cotφ
)m
Measured: shear wave velocity vs at ε1 = 10−6 and at givenconfining stress σ3
Compute : shear modulus Go = ρv2s
Compute : Young modulus Eo = 2 (1 + ν) Go
σ3 [kPa] Eo [kPa]
100 250000
300 460000
600 675000
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: E refo and m
Plot Eo vs σ3
0
100000200000
300000
400000
500000600000
700000
800000
0 100 200 300 400 500 600 700
E
o[k
Pa]
3σ [kPa]
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: E refo and m
Reanalyze Eo vs σ3 in logarithmic scales
Averaged slope yields m; here m =13.1− 12.55
1.0= 0.55
Find intersection of the line with axis ln Eo at
ln
(σ∗3 + c cotφ
σref + c cotφ
)= 0
Here the intersection is at 12.43 henceE ref
o = e12.43 ≈ 2.71812.43 = 250000 kPa
12.2
12.4
12.6
12.8
13
13.2
13.4
13.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
⎟⎟⎠
⎞⎜⎜⎝
⎛++
φσφσ
cotcotln 3
cc
ref
oEln
1
m
12.43
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of E refo from CPT testing
To estimate small strain modulus Go at a certain depth onemay use empirical formula by Mayne and Rix:
Go = 49.4q0.695t
e1.13[MPa]
qt is a corrected tip resistance expressed in MPa
e is the void ratio
Note: this is very rough estimation
Best solution: Perform triaxial testing and project on CPTprofile to adjust empirical coefficient (49.4) for a given site
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: E ref50
Lets us find E50 for each confining stress
0
500
1000
1500
2000
2500
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
1)100( 350 kPaE =σ
)300( 350 kPaE =σ1
)600( 350 kPaE =σ1
)100( 350 =σfq
)100( 350 =σfq
)100( 350 =σfq
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: E ref50
Reanalyze E50 vs σ3 in logarithmic scalesHere we can fix m to the one obtained for small strain moduliFind intersection of the line with axis ln E50 at
ln
(σ∗3 + c cotφ
σref + c cotφ
)= 0
Here the intersection is at ≈ 10.30 henceE ref
50 ≈ e10.30 ≈ 2.71810.30 ≈ 30000 kPa
10.2
10.4
10.6
10.8
11
11.2
11.4
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
50ln E
⎟⎟⎠
⎞⎜⎜⎝
⎛++
φσφσ
cotcotln 3
cc
ref10.30
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Estimation of material properties: E refur
The unloading reloading modulus as well as oedometricmoduli are usually not accessible
We can use Alpans diagram to deduce E refur once we know
E refo (default is
E refur
E refo
= 3); for cohesive soils like tertiary clays
this value is larger
For oedometric modulus at the reference stress σref = 100kPa we can assume E ref
oed = E ref50
γ0.7 = 0.0001...0.0002 for sands and γ0.7 = 0.00005...0.0001for clays
Smaller γ0.7 values yield softer soil behavior
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness
Conclusions
Model properly reproduces strong stiffness variation with shearstrain
It can be used in simulations of soil-structure interactionproblems
Implementation is ”rather heavy”
It should properly predict deformations near the excavations
Model reduces excessive heavings at the bottom of theexcavation
Andrzej Truty ZACE Services Hardening Soil model with small strain stiffness