SHANSEP NGI-ADP
Model description and
verification examples
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Authors: S. Panagoulias
R.B.J. Brinkgreve
Date: 26 januari 2017
Version: 1.0
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Table of Contents
1.
Introduction .................................................................................................................................... 2
2.
Normalised behaviour and the SHANSEP concept ................................................................ 2
3.
The SHANSEP NGI-ADP constitutive model ........................................................................... 3
3.1 The NGI-ADP model.............................................................................................................. 4
3.2 On the use of the SHANSEP NGI-ADP model .................................................................. 5
3.3 The “Switch” to the SHANSEP concept.............................................................................. 5
3.4 State variables ........................................................................................................................ 6
3.5 Model parameters .................................................................................................................. 6
3.6 SHANSEP parameters α and m .......................................................................................... 8
3.7 Minimum undrained shear strength ..................................................................................... 9
3.8 Minimum over-consolidation ratio and pre-overburden pressure ................................... 9
3.9 Interfaces tabsheet ................................................................................................................ 9
3.10 Initial tabsheet .................................................................................................................... 10
4.
Modelling the undrained behaviour .......................................................................................... 10
4.1
NGI-ADP model limitation.............................................................................................. 10
4.2
The advantage of the SHANSEP NGI-ADP model ................................................... 10
5.
The SHANSEP NGI-ADP model in combination with other constitutive models .............. 11
6.
Verification Examples................................................................................................................. 13
6.1
Triaxial compression tests ............................................................................................. 13
6.2 OCR preservation ................................................................................................................ 16
7.
Practical applications ................................................................................................................. 21
7.1 Fluctuation of the phreatic level ......................................................................................... 21
7.2 Embankment construction on soft clay ............................................................................. 25
7.3 The SHANSEP NGI-ADP and the Soft Soil Creep model ............................................. 30
8.
Conclusions ................................................................................................................................. 33
9.
Samenvatting (Nederlands) ...................................................................................................... 34
References .......................................................................................................................................... 35
Appendix: NGI-ADP model manual ................................................................................................. 36
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1. Introduction
The SHANSEP NGI-ADP model (Stress History and Normalized Soil Engineering Properties)
constitutes a soil model implemented in PLAXIS, intended for anisotropic undrained soil
strength conditions. It enables the analysis of dikes and embankments according to the new
design requirements (based upon undrained Critical State calculations) within a FEM
environment. The model is based on the NGI-ADP model (Grimstad et al., 2011), but
modified such that it is able to simulate potential changes of the undrained shear strength su,
based on the effective stress state of the soil. The model takes into account the effects of
stress history and stress path in characterizing soil strength and in predicting field behaviour.
2. Normalised behaviour and the SHANSEP concept
Laboratory tests conducted at the Imperial College using remoulded clays (Henkel (1960)
and Parry (1960)) and at the Massachusetts Institute of Technology on a wide range of clays,
give evidence that clay samples with the same over-consolidation ratio (OCR), but different
consolidation stress σ'c and therefore different pre-consolidation stress σ'pc, exhibit very
similar strength and stress-strain characteristics, when the results are normalized over the
consolidation stress σ'c.
Figure 1 illustrates idealised stress-strain curves for isotropically consolidated undrained
triaxial compression tests on normally-consolidated clay, with consolidation stresses σ'c of
200 kPa and 400 kPa. As depicted in Figure 2, the stress-strain curves are plot on top of
each other when they are normalized over the consolidation stresses.
In practice, normalised behaviour is not as perfect as shown in Figure 2. Usually, there is
discrepancy in the normalized plots caused by different consolidation stresses, soil deposit
heterogeneity or even the fact that the conditions from one soil test to another are not
identical. However, this discrepancy is reported to be quite small (Ladd & Foott, 1974) and as
a result the observed normalized soil behaviour is adopted in engineering practice. It is worth
mentioning that tests on quick clays and naturally cemented soils, which have a high degree
of structure, will not exhibit normalised behaviour because the structure is significantly
altered during the deformation process (Ladd & Foott, 1974).
The observations of normalised soil behaviour lead to the Normalised Soil Parameter (NSP)
concept. According to NSP, Figure 3 illustrates data from Ladd & Foott (1974) which show
the variation of the undrained shear strength su normalised over the current vertical effective
stress σ'v0 against the over-consolidation ratio OCR, for five cohesive soils, in
correspondence with their index properties. The data show a similar trend of increasing
su/σ'v0 with OCR.
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Figure 1 Triaxial compression test data of homogeneous clay (Ladd & Foott, 1974)
Figure 2 Normalised triaxial compression test data of homogeneous clay (Ladd & Foott,
1974)
3. The SHANSEP NGI-ADP constitutive model
Stress History and Normalised Soil Engineering Properties (SHANSEP) is the basis of the
constitutive model hereby presented. The stress history of the soil deposit can be evaluated
by assessing the OCR variation via the current and the pre-consolidation stress profiles.
Based on the NSP concept, the undrained shear strength su is estimated as:
σ′1,max m
su = ασ′1 (
σ′1
) = ασ′1 (OCR)m
[1]
in which α and m are normalised soil parameters.
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The model is implemented in PLAXIS such that the effective major principal stress σ'1 is
considered to compute the OCR and the undrained shear strength. This is thought to be a
more objective parameter in comparison with the vertical effective stress σ'v, as it is the most
compressive value, independent of the Cartesian system of axes. Assuming horizontal soil
layering, both parameters would result in the same value of OCR. However, if the vertical
effective stress σ'v was considered in case of soil slopes, the rotation of principal axes for soil
elements adjacent to the slope would result in slightly lower values of OCR and su
respectively.
Figure 3 Variation of su/σ’v0 over OCR for five different clays (Ladd & Foott, 1974)
3.1 The NGI-ADP model
The NGI-ADP model may be used for bearing capacity, deformation and soil-structure
interaction analyses, involving undrained loading of clay. The basis of the material model is:




Input parameters for (undrained) shear strength for three different stress paths/states,
i.e. Active (suA), Direct Simple Shear (suDSS) and Passive (suP).
A yield criterion based on a translated approximated Tresca criterion.
Elliptical interpolation functions for plastic failure strains and for shear strengths in
arbitrary stress paths.
Isotropic elasticity, given by the unloading/reloading shear modulus, Gur.
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The NGI-ADP model is formulated for a general stress state, matching both undrained failure
shear strengths and strains to that of selected design profiles (Andresen & Jostad (1999),
Andresen (2002), Grimstad et al. (2010)). The reader may refer to Section 12.2 of the
PLAXIS Material Models manual for further details upon the model formulation and
implementation (see Appendix).
3.2 On the use of the SHANSEP NGI-ADP model
The SHANSEP NGI-ADP model is implemented in PLAXIS as a user-defined soil model
(UDSM). Prior to using this model, the “ngiadps.dll” and the “ngiadps64.dll” should be placed
in the sub-folder “udsm” of the folder where PLAXIS 2D and PLAXIS 3D have been saved.
In order to use this model after starting a new project or opening an existing one, the Userdefined option should be selected through the Material model combo box in the General
tabsheet of the Material sets window. In the Parameters tabsheet, the 'ngiadps.dll' should be
selected as the DLL file from the drop-down menu. The “NGI-ADP-S” is used as the Model in
DLL. Subsequently, the model parameters can be specified (Section 3.4).
The reader may refer to Chapter 14 of the PLAXIS Material Models manual for further study
on the use of user-defined soil models.
3.3 The “Switch” to the SHANSEP concept
The “switch” to the SHANSEP concept is done by a certain file which is stored by the user
within the project folder, i.e. inside the *.p2dxdat folder for PLAXIS 2D and inside the *.p3dat
folder for PLAXIS 3D. This file should have the following format:
data.ngiadps.rs#
in which the special character # represents the calculation phase number at which the model
is switched from the NGI-ADP model to SHANSEP. The switch is done only for the activated
materials. Note that, depending on the user actions in the Staged construction mode, the
number mentioned in the name of the calculation phase might be different than the actual
number of this phase. For instance, this could happen if a new calculation phase is inserted
or an already existing one is deleted. Therefore, it is better to verify the phase number, by
writing the command “echo phase_#.number” in the command line, where “phase_#” stands
for the phase ID.
The SHANSEP NGI-ADP model can also be used in the Soil Test facility. This is done by
storing a file named 'data.ngiadps.rs0' within the directory where the files for the Soil Test are
generated, i.e. %temp%\VL_xxxx.
After the switch, the active undrained shear strength suA is calculated by Eq. 1 based on the
current stress state and the maximum stress in the past (according to previous calculation
phases or predefined input values). The calculated active undrained shear strength suA is
kept constant until it is re-initialised. The re-initiation of the suA is possible if another
“SHANSEP file”, which corresponds to a later calculation phase, is stored within the project
folder. The re-initiation can be done as many times as the user desires.
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Before the SHANSEP initialisation, the shear strength profile is defined by suAref and suAinc and
the corresponding ratios to suDSS and suP. The stiffness is based on a constant Gur/suA ratio
(Section 3.5). Upon re-initialisation of the shear strength, the SHANSEP formula is applied to
suA (suAinc is disregarded) whereas existing ratios to suDSS, suP as well as to Gur will remain the
same, which means that the stiffness and strength parameters of all stress points will be
updated.
3.4 State variables
The SHANSEP NGI-ADP model provides output on five State variables. These parameters
can be visualised by selecting the State parameters option from the stresses menu in the
Output program. The State variables are:





State variable 1: σ’1,max
State variable 2: suA (shansep)
State variable 3: γp
State variable 4: rκ
State variable 5: suA
(compression is positive)
(equals zero before the switch)
(plastic shear strain)
(hardening function)
(equals suA (shansep) after the switch)
Before the switch to the SHANSEP concept, state variable 2 (suA (shansep)) equals zero.
After the switch, it is calculated based on Eq. 1. State variable 5 (suA) is calculated based on
the input parameters suA and suAinc (Section 3.5). After the switch it is set equal to the state
variable 2, suA (shansep).
Also, note that during a Safety analysis the value of state variable 2 (suA (shansep)) remains
constant and equal to the value obtained after the last (re-)initialisation of the active
undrained shear strength (see Eq. 1), while the value of state variable 5 (suA) gradually
decreases in order to compute the factor of safety (see Eq. 13).
The first time that the SHANSEP NGI-ADP model is used in a sequence of calculation
phases, the state variables are initialised. The first state variable (σ’1,max) represents the preconsolidation stress defined as the maximum major principal effective stress ever reached.
This parameter is initialised based on the existing stress state. If any OCRmin>1 or POPmin>0
are used to indicate an over-consolidated stress state, these parameters are taken into
account to initialise σ’1,max. If an advanced soil model (i.e. a hardening model that includes
the pre-consolidation stress as a state parameter) was used prior to the use of the
SHANSEP NGI-ADP model, then σ’1,max will be obtained from the existing pre-consolidation
stress. In that case, any OCRmin>1 or POPmin>0 will be ignored, since the effect over overconsolidation is supposed to be already incorporated in the existing pre-consolidation stress.
3.5 Model parameters
The SHANSEP NGI-ADP model is formulated such that it initially behaves as the NGI-ADP
model until it is switched to the SHANSEP concept by the user. It should be preferably used
in combination with undrained behaviour. This drainage type can be ignored before switching
to the SHANSEP concept by using the calculations option Ignore undrained behaviour in the
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Phases window. However, the strength will still be an undrained strength as defined by the
initial shear strength profile.
The NGI-ADP is intended to be used for Undrained (B) analysis. However, for user-defined
soil models, such as SHANSEP NGI-ADP, the only available drainage type for undrained
behaviour is Undrained (A). Nevertheless, since the strength is defined as undrained shear
strength, the SHANSEP NGI-ADP model will actually behave according to the Undrained (B)
drainage type.
Since the SHANSEP NGI-ADP model is an extension of the NGI-ADP model, the model
parameters can be classified in two groups, i.e. the NGI-ADP model parameters and the
SHANSEP parameters. For the “pre-switching” behaviour only the NGI-ADP model
parameters are used.
Since there are dependencies in the NGI-ADP model parameters, there are limitations as to
which combinations of values are acceptable. The standard NGI-ADP model in PLAXIS
includes checks to evaluate whether or not a valid combination of parameters has been
specified. Such checks are not available for user-defined soil models. Therefore, it is strongly
recommended to first define a material data set using the standard NGI-ADP model and
ONLY IF the parameters are accepted as a valid combination of parameters, a data set
should be created using the SHANSEP NGI-ADP model as a user-defined soil model with
the same NGI-ADP model parameters. If an invalid combination of model parameters is used
in the SHANSEP NGI-ADP model, an error will occur during the calculation.
Table 1 gives an overview of all model parameters. Only the SHANSEP parameters will be
discussed in the next section. For the NGI-ADP model parameters the reader may refer to
Section 12.2 of the PLAXIS Material Models manual (see Appendix).
Table 1 SHANSEP NGI-ADP model parameters
Parameter
Symbol
A
G/su
NGI-ADP
model
Parameters
γf
E
γf
DSS
γf
A
su ref
vertref
A
su inc
P
A
su /su
C
A
τ0/su
DSS
A
su /su
SHANSEP
Parameters
ν
νu
alpha (α)
power (m)
su,min
OCRmin
POPmin
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Description
Ratio unloading/reloading shear modulus over (plane strain)
active shear strength
Shear strain at failure in triaxial compression
Shear strain at failure in triaxial extension
Shear strain at failure in direct simple shear
Reference (plane strain) active shear strength
Reference depth
Increase of shear strength with depth
Ratio of (plane strain) passive shear strength over (plane
strain) active shear strength
Initial mobilisation
Ratio of direct simple shear strength over (plane strain)
active shear strength
Poisson's ratio
Poisson's ratio (undrained)
Coefficient
Power
Minimum shear strength
Minimum overconsolidation ratio
Minimum pre-overburden pressure
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Unit
%
%
%
2
kN/m
m
2
kN/m /m
2
kN/m
2
kN/m
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3.6 SHANSEP parameters α and m
Based on Eq. 1, the α parameter represents the value of su/σ'1 for a normally-consolidated
soil (OCR = 1). The power m is the value to which the OCR is raised. The magnitude of m
represents the rate of strength increase with OCR.
Ladd & DeGroot (2003) indicate that for most clayey soil types, α = 0.22  0.03 and m = 0.80
 0.1. Results of SHANSEP tests performed by Seah & Lai (2003) on soft Bangkok clay,
which is a marine silty clay in the central area of Thailand, suggest the values of αc = 0.265
and mc = 0.735 for compression tests and αe = 0.245 and me = 0.890 for extension tests.
Santagata & Germaine (2002) studied the effects of sampling disturbance by conducting
single element triaxial tests on normally-consolidated resedimented Boston blue clay
(RBBC). The SHANSEP parameters obtained by undrained triaxial compression are αi = 0.33
and mi = 0.71 for the intact RBBC, and αd = 0.33 and md = 0.83 for the disturbed RBBC.
Based on the studies presented above, it can be concluded that both SHANSEP parameters
α and m are stress path dependent. Even though the range of variation of both parameters α
and m is not wide, the proper way to estimate them is via calibration of SHANSEP triaxial
test results.
Figures 4 and 5 present the influence of both parameters on the normalised shear strength
over the OCR. In Figure 4 the power m is constant and equal to 0.80, while the coefficient α
varies from 0.20 to 0.35. In Figure 5 the coefficient α is constant and equal to 0.20, while the
power m varies from 0.75 to 0.90. As expected, both parameters result in an increase of the
undrained shear strength as they grow. Variation of the coefficient α has greater influence on
the resulting su/σ'1.
Figure 4 Influence of the coefficient α on the normalised undrained shear strength
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Figure 5 Influence of the power m on the normalised undrained shear strength
3.7 Minimum undrained shear strength
To prevent zero or very small stiffness (G) and strength (su) at small depths where σ'1 is
equal to or approximately zero, a minimum value of the undrained shear strength su,min can
be selected as input parameter. PLAXIS determines the value of su by Eq. 2. A proper value
for su,min can be determined from the field and/or laboratory characterization of the soil
deposit.
su = max {ασ′1 (OCR)m ; su,min }
[2]
3.8 Minimum over-consolidation ratio and pre-overburden pressure
A minimum value of the OCR can be used as input parameter (OCRmin) in order to control the
initial stress state. Alternatively, a minimum value of the pre-overburden pressure may be
selected (POPmin). During the generation of the initial stress state, the maximum major
effective principal stress is determined by Eq. 3.
σ′1,max = max {σ′1 ; σ′1 ∙ OCR min ; σ′1 + POPmin }
[3]
3.9 Interfaces tabsheet
The Interfaces tabsheet contains the material data for interfaces, i.e. the interface oedometer
modulus, Erefoed, and the interface strength parameters c'inter, φ'inter and ψinter. Hence, the
interface shear strength is directly given in strength parameters. In addition, two parameters
are included to enable stress-dependency of the interface stiffness according to a power law
formulation:
σ′
UD−Power
ref
Eoed (σ′n ) = Eoed
(UD−Pnref )
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where UD-Power, is the rate of stress dependency of the interface stiffness, UD-Pref is the
reference stress level (usually 100 kN/m2) and σ'n is the effective normal stress in the
interface stress point.
PLAXIS will give a warning when a zero interface stiffness or strength is defined, even if no
interface elements are being used. In order to avoid this warning, a non-zero cohesion and
Eoedref should be specified.
3.10 Initial tabsheet
Based on the value of φ'inter selected in the Interfaces tabsheet, the lateral stress coefficient
at rest K0 is automatically calculated as a default value to set up the initial horizontal stress
(Eq. 6). However, note that this value is based on the interface friction angle rather than the
soil friction angle, since SHANSEP NGI-ADP does not have friction angle as model
parameter. The suggested value may be changed by the user.
K 0 = 1 − sin(φ′inter )
[5]
4. Modelling the undrained behaviour
During the initial phase(s), before switching to the SHANSEP concept, undrained behaviour
can be ignored by selecting the calculations option Ignore undrained behaviour in the Phases
window. After switching to the SHANSEP concept, the model behaves similarly to the
Undrained (B) NGI-ADP model. However, as discussed below, the SHANSEP NGI-ADP
model is advantageous in comparison to the classic Undrained (B) drainage type of the NGIADP model in the sense that the shear strength can be defined and updated based on the
(changed) effective stress level.
4.1 NGI-ADP model limitation
The undrained shear strength in the NGI-ADP model is given by Eq. 6. Such variation of the
su over depth is valid only in case that horizontal soil layers are considered. The reference
depth (yref) is a fixed value throughout the whole model. If the soil deposit is inclined (in case
of slope, embankment etc.), according to Eq. 7, as depth increases, su increases as well
along the surface. This leads to a non-realistic distribution of the undrained shear strength.
su (y) = su,ref + (yref − y)su,inc
for y < yref
[6]
4.2 The advantage of the SHANSEP NGI-ADP model
The SHANSEP NGI-ADP model gives the advantage of a realistic, empirical way of
modelling the undrained shear strength su. Figure 6 illustrates a comparison between the real
behaviour of a soft soil (a) and the SHANSEP NGI-ADP concept (b) in terms of the ESP in p'-
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q plot. During the first undrained loading the model behaves as NGI-ADP model, Undrained
(B). A consolidation phase is introduced afterwards and the effective stress increases,
reaching the re-initiation point 1 (RP1). At that point the switch to the SHANSEP model
occurs and the undrained shear strength is updated from its initial value suinitial to suupdated_1.
The same process is repeated again after the next undrained loading and the subsequent
consolidation, leading to the updated undrained shear strength suupdated_2. This behaviour
constitutes a better approach to reality in comparison with the standard NGI-ADP model, in
which no update of the undrained shear strength occurs.
It is suggested that the consolidation phase does not lead to deviatoric stress equal to the
undrained shear strength, e.g. the re-initiation point 1 (RP1) in Figure 6 stays below the
2suinitial cap. If the opposite occurs it is suggested to split the consolidation phase in more than
one phases with shorter consolidation times and re-initiate the undrained shear strength after
each one of them.
Figure 6 Undrained behaviour of real soft soil (a) and SHANSEP NGI-ADP model (b)
5. The SHANSEP NGI-ADP model in combination with other
constitutive models
Apart from using the SHANSEP NGI-ADP model merely as a NGI-ADP model and then
switching to the SHANSEP concept, it is possible to use it as an “extension” of any other
constitutive model implemented in PLAXIS. This is particularly useful for the advanced
models in which an over-consolidated stress state can be defined through the overconsolidation ratio:






Hardening Soil model (HS)
Hardening Soil model with small-strain stiffness (HSsmall)
Modified Cam-Clay model (MCC)
Soft Soil model (SS)
Soft Soil Creep model (SSC)
Sekiguchi-Ohta model (SO)
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To switch from an advanced soil model to the SHANSEP NGI-ADP model, the process
described in Section 3.3 has to be followed. However, apart from the corresponding file
needed to be stored in the project folder, the material of the soil cluster has also to be
changed from the advanced model to the SHANSEP NGI-ADP model.
To transfer the stress history from an advanced soil model to the SHANSEP NGI-ADP
model, two state parameters are used, namely the equivalent isotropic stress peq and the
isotropic pre-consolidation stress pp (refer to Section 9.3.5 of the PLAXIS Reference
manual). Depending on the soil model, the equivalent isotropic stress peq is calculated as:



HS and HSsmall:
SS, SSC and MCC:
SO:
peq = √p2 +
peq = p′ −
peq =
q
̃2
q
α2
[7]
q2
M2 (p′ −c cotφ)
̃
q
)
Mp
exp(−
[8]
[9]
The isotropic pre-consolidation stress pp represents the maximum equivalent isotropic stress
level that a stress point has experienced up to the current load step. Based on these two
state parameters, the OCR of the current calculation step is derived as:
OCR =
pp
[10]
peq
After switching to the SHANSEP NGI-ADP model, based on the calculated OCR and σ’1 of
the current step, the first SHANSEP state variable (see Section 3.4) is defined as:
σ′1,𝑚𝑎𝑥 = 𝑂𝐶𝑅 ∙ 𝜎 ′1
[11]
To calculate the fifth state variable, i.e. the SHANSEP active undrained shear strength suA
(shansep), the result of Eq. 11 is used in combination with Eq. 1.
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6. Verification Examples
6.1 Triaxial compression tests
Triaxial compression tests were performed as a Finite Element calculation starting from three
different initial stress states, i.e. σ'1 equal to 200 kPa, 300 kPa and 400 kPa. In addition, four
different over-consolidation ratios are considered, i.e. OCR equal to 1.2, 1.5, 1.8 and 2.0.
Figure 7 illustrates the model geometry in PLAXIS 2D (a) and PLAXIS 3D (b). In PLAXIS 2D
an axisymmetric model is used. A very coarse mesh is selected for both models. The
boundary conditions are set to be Normally fixed for the left, bottom and rear (only for
PLAXIS 3D) boundaries, while they are set to Free for every other boundary. Compressive
loads are applied to the boundaries that are set to be Free.
Figure 7 Model geometry in PLAXIS 2D (a) and PLAXIS 3D (b)
Intention of the present example is to verify that the undrained shear strength is calculated
correctly in PLAXIS after switching to the SHANSEP concept. The selected drainage type
does not influence the results in terms of soil strength. Thus, the selected drainage type
before switching to SHANSEP is not relevant and Drained material is used.
To obtain the desired OCR value an isotropic unloading phase follows the initial isotropic
compression of the model. After the unloading, shearing is applied via a prescribed
displacement of 0.1 m at the top of the model. The resulting axial strain εα equals 10%. The
SHANSEP NGI-ADP input parameters adopted for the simulations are listed in Table 2.
Table 3 presents a comparison between PLAXIS and the theoretically obtained results
through Eq. 1, for all loading cases. The results are in perfect agreement.
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Table 2 Adopted SHANSEP NGI-ADP model parameters
Symbol
A
G/su
C
γf
E
γf
DSS
γf
A
su ref
vertref
A
su inc
P
A
su /su
A
τ0/su
DSS
A
su /su
ν
νu
alpha (α)
power (m)
su,min
OCRmin
POPmin
Value
200.0
10.0
10.0
10.0
46.28
1.0
0.0
1.0
0.0
1.0
0.2
0.499
0.2
0.8
1.0
1.0
0.0
Unit
%
%
%
2
kN/m
m
2
kN/m /m
2
kN/m
2
kN/m
Table 3 Comparison between PLAXIS and theoretical results
σ’1,max (kPa)
240
300
360
400
360
450
540
600
480
600
720
800
σ’1 (kPa)
200
200
200
200
300
300
300
300
400
400
400
400
OCR (-)
1.2
1.5
1.8
2.0
1.2
1.5
1.8
2.0
1.2
1.5
1.8
2.0
PLAXIS
A
su (kPa)
46.28
55.33
64.01
69.64
69.42
82.99
96.02
104.50
92.56
110.70
128.00
139.30
A
su / σ’1 (-)
0.23
0.28
0.32
0.35
0.23
0.28
0.32
0.35
0.23
0.28
0.32
0.35
SHANSEP (Eq. 1)
su (kPa) su/ σ’1 (-)
46.28
0.23
55.33
0.28
64.01
0.32
69.64
0.35
69.42
0.23
82.99
0.28
96.02
0.32
104.47
0.35
92.56
0.23
110.65
0.28
128.03
0.32
139.29
0.35
Figure 8 illustrates the variation of the normalised shear strength su over the corresponding
effective major principal stress σ'1 against OCR. PLAXIS 2D and PLAXIS 3D results are in
perfect match. The fitting power line results in an equation which is in perfect agreement with
Eq. 1 for the adopted SHANSEP parameters, i.e. α = 0.2 and m = 0.8. The undrained shear
strength increases with increasing OCR.
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Figure 8 su/σ’1 over OCR obtained in PLAXIS
Figure 9 presents the deviatoric stress q normalised over the effective major principal stress
σ'1 against the axial strain, for each one of the studied loading cases (see Table 3). The
results are plot for all studied OCR values as well. There is perfect agreement between the
cases which have different pre-consolidation and current effective stresses, but the same
OCR. Thus, each one of the lines depicted in Figure 9 consists of three lines plot on top of
each other.
Figure 9 PLAXIS results for the normalised deviatoric stress q against the axial strain, for all
studied OCR values
The obtained undrained shear strength in PLAXIS is verified analytically with Eq. 12 and the
analytical results are in perfect agreement with the results presented in Figure 9.
q𝑚𝑎𝑥 = 2𝑠𝑢
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Figure 10 compares the results between PLAXIS 2D and the Soil Test facility (Section 3.3)
for the loading case with OCR equal to 1.2 and σ’1 equal to 200 kPa. Both tests result in the
same undrained shear strength.
Figure 10 Comparison between Finite Element calculation and Soil Test facility
6.2 OCR preservation
An idealized numerical model is presented in this Section to verify that OCR values are
correctly transferred when switching from other advanced soil models to the SHANSEP NGIADP model (Section 5). A plane-strain model is considered in PLAXIS 2D. A soil column 1 m
wide in x-direction and 7 m tall in y-direction is modelled, consisting of seven soil blocks at
which different material types are assigned. At the uppermost soil block a Linear Elastic
model is assigned acting as a load for every other soil material lying beneath. The rest six
soil blocks are composed of the advanced material models mentioned in Section 5.
The Medium option is selected for the Element distribution and the default boundary
conditions are selected for deformations. Figure 11 illustrates the model geometry, the
generated mesh and the material model assigned to each soil block. When switching to the
SHANSEP concept, the advanced soil models are replaced by the SHANSEP NGI-ADP
model (Figure 12).
Drained soil behaviour is considered. The water level is placed at the top model boundary
and the default boundary conditions are used for the groundwater flow. The adopted material
parameters are presented in Table 4 to 6. Important to notice is that the OCR value for every
advanced soil model is selected equal to 2. The γunsat and γsat values are selected such that a
uniform vertical effective stress σ'yy equal to 10 kN/m2 is applied below the top soil block. This
simplification offers a more straightforward way of assessing the results.
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Figure 11 Model geometry and generated mesh before switching to the SHANSEP concept
Figure 12 Model geometry and generated mesh after switching to the SHANSEP concept
Regarding the SHANSEP NGI-ADP model, Drained behaviour is also selected as the
Drainage type is not relevant to the OCR preservation check. Table 7 presents the adopted
material parameters for the SHANSEP NGI-ADP model.
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In the Initial calculation phase, the K0 procedure is used to compute the initial stresses
based on the input parameters, with the advanced soil models assigned to the soil blocks
(Figure 11). In Phase 1, all advanced soil models, apart from the Linear Elastic model
assigned to the top soil block, are changed to the SHANSEP NGI-ADP model. A file named
“data.shansep.rs1” is added to the project folder for switching to the SHANSEP concept.
Table 4 Adopted material parameters for the Hardening Soil model (HS), the Hardening Soil
model with small-strain stiffness (HSs) and the Linear Elastic model (LE)
Symbol (unit)
3
γunsat (kN/m )
3
γsat (kN/m )
2
E’ (kN/m )
ref
2
E50 (kN/m )
ref
2
Eoed (kN/m )
ref
2
Eur (kN/m )
m (-)
2
c’ref (kN/m )
φ’ (deg)
ψ (deg)
ν’ (-)
ν’ur (-)
nc
K0 (-)
K0 (-)
OCR (-)
POP (-)
HS
0.0
10.0
3
2010
3
2010
3
6010
0.5
5.0
30.0
0.0
0.2
0.5
0.75
2.0
0.0
HSs
0.0
10.0
3
2010
3
2010
3
6010
0.5
5.0
30.0
0.0
0.2
0.5
0.75
2.0
0.0
LE
10.0
20.0
3
110
0.0
1.0
-
Table 5 Adopted material parameters for the Soft Soil model (SS), the Soft Soil Creep
model (SSC) and the Sekiguchi-Ohta model (SO)
Symbol (unit)
3
γunsat (kN/m )
3
γsat (kN/m )
λ* (-)
κ* (-)
μ* (-)
2
c’ref (kN/m )
φ’ (deg)
ψ (deg)
ν’ur (-)
M (-)
nc
K0 (-)
K0 (-)
OCR (-)
POP (-)
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SS
0.0
10.0
0.2
0.05
5.0
30.0
0.0
0.15
1.563
0.5
0.8235
2.0
0.0
SSC
0.0
10.0
0.2
0.05
0.004
5.0
30.0
0.0
0.15
1.563
0.5
0.8235
2.0
0.0
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SO
0.0
10.0
0.2
0.05
0.15
1.2
0.5
0.5
2.0
0.0
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Table 6 Adopted material parameters for the Modified Cam-Clay model (MCC)
Symbol (unit)
3
γunsat (kN/m )
3
γsat (kN/m )
λ (-)
κ (-)
ν’ur (-)
M (-)
nc
K0 (-)
K0 (-)
OCR (-)
POP (-)
MCC
0.0
10.0
0.2
0.05
0.15
1.5
0.5207
0.8650
2.0
0.0
Table 7 Adopted SHANSEP NGI-ADP model parameters
Symbol
γunsat
γsat
A
G/su
C
γf
E
γf
DSS
γf
A
su ref
vertref
A
su inc
P
A
su /su
A
τ0/su
DSS
A
su /su
ν
νu
alpha (α)
power (m)
su,min
OCRmin
POPmin
Value
0.0
10.0
200.0
10.0
10.0
10.0
2.0
0.0
0.0
1.0
0.0
1.0
0.3
0.499
0.4
0.8
2.0
1.0
0.0
Unit
3
kN/m
3
kN/m
%
%
%
2
kN/m
m
2
kN/m /m
2
kN/m
2
kN/m
Results are presented only for the first 6 m of the soil column, excluding the top soil block.
Figure 13(a) depicts the effective major principal stress σ'1 which is uniform, equal to10
kN/m2.
The isotropic OCR is presented in Figure 13(b). Despite the fact that an input value of 2 is
assigned to the OCR of every advanced soil model, the output results deviate. This is
because the OCR value presented in output is calculated by Eq. 10. The equivalent isotropic
stress peq is calculated by the formulae presented in Section 5. The isotropic preconsolidation stress pp is calculated based on the pre-consolidation stress state (p', q), which
is a function of the vertical pre-consolidation stress σp (affected by the input OCR, which
equals 2) and the K0nc value. The reader may refer to Section 2.8 of the PLAXIS Material
Models manual for further study.
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Figure 13 Effective major principal stress σ'1 (a) and OCR (b)
Figure 14 State parameter 1 σ'1,max (a) and State parameter 2 suΑ (shansep) (b)
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Figure 14(a) illustrates the results after switching to the SHANSEP concept. State parameter
1 (σ'1,max) is calculated by Eq. 11. Thus, the depicted values can be easily verified based on
Figure 14(a) and Eq. 11. Figure 14(b) presents state parameter 2 (suA (shansep)). The
results can be verified based on Figure 14(b) and Eq. 1, considering that σ'1 equals 10
kN/m2.
It is concluded that the OCR values are correctly transferred when switching from advanced
soil models to the SHANSEP NGI-ADP model.
7. Practical applications
7.1 Fluctuation of the phreatic level
The seasonal fluctuation of the phreatic level in a submerged slope is simulated to evaluate
the effect of the varying OCR, due to the water level fluctuation, on the slope stability. A
plane-strain model is used in PLAXIS 2D with dimensions equal to 60 m in the horizontal xdirection and 25 m in the vertical y-direction. Figure 15 illustrates the model geometry. The
slope is 30 long and 13 m tall. The water level is initially set at its average level, i.e. “Level 1”,
which is 9 m above the height of the toe. The water level fluctuation equals  3 m, thus
“Level 2” equals 12 m and “Level 3” equals 6 m. The standard boundary conditions are used
for deformations and ground water flow (also for consolidation purposes). Undrained
behaviour is considered. The Very fine option is selected for the Element distribution. The
generated mesh is illustrated in Figure 15.
Figure 15 Model geometry and generated mesh in PLAXIS 2D
The soil deposit is homogenous and the material properties are listed in Table 8. The
SHANSEP parameters α and m are chosen based on the normalized behaviour of disturbed
resedimented Boston blue clay (RBBC), as reported by Santagata & Germaine (2002).
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Table 8 Adopted SHANSEP NGI-ADP model parameters
Symbol
γunsat
γsat
A
G/su
C
γf
E
γf
DSS
γf
A
su ref
vertref
A
su inc
P
A
su /su
A
τ0/su
DSS
A
su /su
ν
νu
alpha (α)
power (m)
su,min
OCRmin
POPmin
kx
ky
Value
17.0
20.0
200.0
3.0
4.0
3.7
5.0
25.0
2.2
1.0
0.3
1.0
0.3
0.498
0.33
0.83
5.0
1.0
0.0
-4
1.010
-5
5.010
Unit
3
kN/m
3
kN/m
%
%
%
2
kN/m
m
2
kN/m /m
2
kN/m
2
kN/m
m/day
m/day
The Gravity loading calculation type is used to generate the initial stress state, with the
phreatic level placed at Level 1 (average level). For the Gravity loading calculation type the
option Ignore undrained behaviour is automatically selected in the Phases window. For all
subsequent calculation phases undrained behaviour is considered. In Phase 1, an increase
of the water level up to Level 2 is simulated with a Plastic calculation. A consolidation
analysis follows (Phase 2), in which the Minimum excess pore pressure loading type is used
and a minimum value of excess pore pressures (1 kN/m2) is reached throughout the whole
model. Subsequently, the water level is set to Level 1 again and the same phases sequence
is followed (Phase 3 is Plastic calculation and Phase 4 is consolidation analysis to minimum
excess pore pressures). Afterwards, the water level is set to Level 3 and once more the
same phases sequence is adopted (Phase 5 is Plastic calculation and Phase 6 is
consolidation analysis to minimum excess pore pressures). For the last two calculation
phases the water level is placed again at Level 1 and the same phases sequence is adopted
(Phase 7 is Plastic calculation and Phase 8 is consolidation analysis to minimum excess
pore pressures). The switch to the SHANSEP model occurs in Phase 1. The shear strength
is re-initiated after every consolidation phase, i.e. at Phases 3, 5 and 7.
Table 9 presents the results of each calculation phase for the stress point in which the
maximum stress occurs (right-bottom of the model). The phases in which updating of shear
strength occurs are marked with an asterisk. Figures 16 and 17 illustrate an example of state
parameter 1 (σ'1,max) and state parameter 2 (suA (shansep)) variation throughout the whole
model after the last consolidation phase (Phase 8).
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Table 9 PLAXIS calculation results
Phase
Initial
1*
2
3*
4
5*
6
7*
8
σ’1,max (kPa)
269.1
269.1
269.1
269.1
269.1
269.1
286.0
286.0
286.0
σ’1 (kPa)
269.1
268.8
249.2
249.4
267.1
267.4
286.0
285.7
265.9
OCR (-)
1.00
1.00
1.08
1.08
1.01
1.01
1.00
1.00
1.08
A
su (kPa)
60.00
88.80
88.80
87.64
87.64
88.69
88.69
94.38
94.38
A
su / σ’1 (-)
0.22
0.33
0.36
0.35
0.33
0.33
0.31
0.33
0.35
Figure 16 State parameter 1 (σ'1,max), Phase 8 (units in kPa)
Figure 17 State parameter 2 (suA (shansep)), Phase 8 (units in kPa)
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The material is initially normally-consolidated and when the water level increases (Phase 1),
it experiences the first unloading of its stress history. The resulting higher OCR (1.08) leads
to an increase of the ratio suA/σ'1 from 0.33 in Phase 1 to 0.36 in Phase 2. For the same
phases, σ'1 reduces from 268.8 kPa to 249.2 kPa. Thus, the observed increase of the
normalised shear strength is dominated by the increase in OCR, based on Eq. 1.
In Phases 5 and 6 the water table is at the lowest level (Level 3). The effective stress state
increases and the previous σ'1,max is overcome after the consolidation in Phase 6. The reinitiation of the shear strength in the subsequent phase (Phase 7) shows an increase of suA
following the increase of σ'1, even though the material is still normally consolidated (OCR ≃
1.00).
In both Phases 4 and 8 the water table is at Level 1. However, the OCR and the undrained
shear strength suA are higher in Phase 8. This is because between those two phases the
water level decreases leading to an increase of the effective stresses throughout the soil
deposit. Both OCR and su are affected by the soil stress history.
A safety analysis is performed after each consolidation phase. Table 10 summarises the
results. The number in brackets when the water table is at Level 1 indicates the passing
sequence, i.e. [1] stands for the first passing (from Level 2 to Level 1 in Phase 3) and [2]
stands for the second passing (from Level 3 to Level 1 in Phase 7). The fluctuation of the
water level causes variation of the global Factor of Safety (FoS). The latter is calculated in
PLAXIS as:
FoS =
input
su
equilibrium
su
[13]
Table 10 Factor of Safety for various water levels
#
1
2
3
4
Preceding phase
2
4
6
8
Water table level
2
1 [1]
3
1 [2]
FoS (-)
1.60
1.23
1.10
1.29
Note that during the Safety analysis the value of suA (shansep) remains constant and equal to
the value obtained after the last (re-)initialisation of the active undrained shear strength
(given by Eq. 1), while the value of suA gradually decreases in order to compute the FoS
(given by Eq. 13).
When the water table is at the highest level (Level 2) the water load applied at the slope acts
as an extra balance force and assists equilibrium. The FoS equals approximately 1.60 in this
case. When the water level reduces to Level 1 and then Level 3, the undrained shear
strength does not change significantly (Table 9) but the balance force induced by the water
load diminishes. As a result, the FoS equals 1.23 and 1.10 correspondingly. Focusing on the
results when the water table is at Level 1, the FoS is higher in case of the second passing
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(1.29). In both cases the water load acting on the slope is the same, but in the second
passing (Phase 8), the undrained shear strength is higher (Table 9).
Assuming the same project, if the minimum value of the OCR was selected equal to 2
(OCRmin = 2), the variation of the undrained shear strength would merely be affected by the
variation of the effective major principal stress σ'1, as the water fluctuation does not lead to
an OCR higher than the predefined minimum value. Figure 18 illustrates the variation of the
normalised undrained shear strength over the project phases. The higher OCRmin leads to
higher undrained shear strength.
The effect of phreatic level fluctuation on the undrained shear strength is demonstrated with
this example, by using the SHANSEP NGI-ADP model. Considering various elevations of the
water table, safety analyses are performed to evaluate the stability of the embankment. The
results indicate that FoS is highly dependent on the imposed stress history.
Figure 18 Normalised undrained shear strength over project phases for different predefined
values of OCRmin
7.2 Embankment construction on soft clay
The construction of an embankment is simulated to compare the results between the
SHANSEP NGI-ADP material model and the NGI-ADP. A plane-strain model is used in
PLAXIS 2D. Since the geometry is considered symmetric, only half of the embankment is
modelled and the drainage is prevented only through the plane of symmetry (left model
boundary), while the standard boundary conditions for deformations are adopted. The model
length equals 35 m in the horizontal x-direction. The height of the initial soil deposit (without
the embankment) equals 7 m. The water level is set 1 m below the top soil surface.
The embankment is built in stages. Its slope equals 1:1.5 (height:length). After the last top
layer of the embankment is constructed, part of the initial soil deposit is excavated, together
with the top layer of the embankment. The purpose of the excavation is to cause an overconsolidated soil state. In between the various model geometry changes Consolidation
phases are added with consolidation time equal to 60 days in order to allow for excess pore
pressure dissipation. Afterwards a load equal to 8 kN/m/m with length of 3 m is applied on
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top of the embankment. A Consolidation phase to minimum excess pore pressures equal to
1 kN/m2 follows. The last calculation phase is a safety analysis.
The Very fine option is selected for the Element distribution. Figure 19 illustrates the model
geometry and the generated mesh. The soil clusters numbering from 1 to 4 is used in Table
11 to clarify the staged construction phases.
Figure 19 Model geometry and generated mesh in PLAXIS 2D
The soil deposit is homogenous and the material properties are listed in Table 12. Undrained
behaviour is considered. The SHANSEP parameters α and m are chosen based on the soft
Bangkok clay undrained triaxial compression tests as reported by Seah & Lai (2003). The
switch to the SHANSEP concept occurs in Phase 1. The shear strength is re-initiated after
every next phase from Phase 2 to Phase 11.
Table 11 Staged construction details
Phase
Initial
1
2
3
4
5
6
7
8
9
10
11
Calculation type
K0 prodecure
Consolidation
Consolidation
Consolidation
Consolidation
Consolidation
Consolidation
Consolidation
Consolidation
Consolidation
Consolidation
Safety
Time (days)
1
60
1
60
1
60
1
60
1
Unspecified
-
Details
Deactivated soil clusters 1, 2, 3
Activate soil cluster 1
Activate soil cluster 2
Activate soil cluster 3
Deactivate soil clusters 3, 4
Activate load
Min pexcess
-
Intention of the present example is to compare the SHANSEP NGI-ADP results with the
standard NGI-ADP model results. In order to obtain the latter the same project is run but
without activating the SHANSEP parameters by omitting the data.ngiadps.rs# files.
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Table 12 Adopted SHANSEP NGI-ADP model parameters
Symbol
γunsat
γsat
A
G/su
C
γf
E
γf
DSS
γf
A
su ref
vertref
A
su inc
P
A
su /su
A
τ0/su
DSS
A
su /su
ν
νu
alpha (α)
power (m)
su,min
OCRmin
POPmin
Value
13.0
15.0
150.0
3.0
4.0
3.7
8.0
7.0
0.45
1.0
0.75
1.0
0.33
0.498
0.265
0.735
8.0
1.0
0.0
Unit
3
kN/m
3
kN/m
%
%
%
2
kN/m
M
2
kN/m /m
2
kN/m
2
kN/m
Figure 20 (a) and (b) depict the variation of the active undrained shear strength suA for the
SHANSEP NGI-ADP and the NGI-ADP models respectively, at Phase 4. The update of suA
by means of the SHANSEP concept (Figure 20 (a)) results in a different variation of strength
over depth. To the contrary, in Figure 20 (b), the variation of suA corresponds to the selected
input values (Table 12).
Figure 21 (a) and (b) illustrate the variation of suA for the SHANSEP NGI-ADP and the NGIADP models accordingly, at the beginning of the Safety analysis (i.e. Phase 11). While the
undrained shear strength remains unchanged for the NGI-ADP model (Figure 21 (b)), the
load history results in a much different strength profile for the SHANSEP NGI-ADP model
(Figure 21 (a)).
The settlement at a point under the embankment is considered to compare the analyses. The
selected point is located 1 m at the right of the plane of symmetry, at height equal to 7 m.
Figure 22 presents the results. Stiffness is different between the models since it is linked to
the variation of suA via the input parameter G/suA. As a result, the NGI-ADP model has stiffer
behaviour, which is constant throughout the whole construction time, while for the SHANSEP
NGI-ADP stiffness gradually increases based on the update of suA.
The difference in stiffness is also verified by the total consolidation time needed till the end of
Phase 10 (Figure 22). The consolidation time is inversely proportional to the soil stiffness,
given that permeability is the same. The total consolidation time equals about 1370 days for
the NGI-ADP model and about 1540 days for the SHANSEP NGI-ADP model.
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Figure 20 Undrained shear strength at Phase 4 for the SHANSEP NGI-ADP (a) and the NGIADP model (b)
Figure 21 Undrained shear strength at the beginning of the Safety analysis (Phase 11) for
the SHANSEP NGI-ADP (a) and the NGI-ADP model (b)
Figure 23 presents a comparison of the computed Factors of Safety (FoS) for both models.
For the SHANSEP NGI-ADP model the FoS equals about 1.25, while for the NGI-ADP model
equals about 1.13. The explanation of such a difference between the Factors of Safety lies
on the suA updated profile of the SHANSEP NGI-ADP model (Figure 21). Once more, note
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that during the Safety analysis the value of suA (shansep) remains constant and equal to the
value obtained after the last (re-)initialisation of the active undrained shear strength (given by
Eq. 1), while the value of suA gradually decreases in order to compute the FoS (given by Eq.
13).
Figures 24 (a) and (b) illustrate the corresponding failure mechanisms, in terms of total
displacements at the end of the Safety analysis calculation. As it can be seen the failure
mechanism in case of the NGI-ADP model goes much deeper into the soil deposit. On the
contrary, the failure mechanism for the SHANSEP NGI-ADP model is much shallower. Due
to the adopted load history (especially after the deactivation of two soil clusters in Phase 7),
the undrained shear strength increases below the embankment. Because of the enhanced
suA profile the slip surface is “forced” closer to the slope, resulting in higher Factor of Safety.
Figure 22 Comparison of settlement over time obtained with NGI-ADP and SHANSEP NGIADP model
Figure 23 Comparison of Factor of Safety obtained with NGI-ADP and SHANSEP NGI-ADP
model
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Figure 24 Failure mechanism (total displacement at the end of the Safety analysis) for the
SHANSEP NGI-ADP (a) and the NGI-ADP model (b)
As shown in this example, the SHANSEP NGI-ADP model may be used to capture the
changes in undrained shear strength due to stress variation imposed by the construction of
an embankment (under constant phreatic level). Safety analyses are performed to
demonstrate the difference in results between the original NGI-ADP and the SHANSEP NGIADP models.
7.3 The SHANSEP NGI-ADP and the Soft Soil Creep model
The settlement below an embankment due to creep is simulated in order to study its effects
on the Factor of Safety (FoS). A plane-strain model is used in PLAXIS 2D. Since the
geometry is considered symmetric, only half of the embankment is modelled. The standard
boundary conditions for deformations are adopted. The model length equals 35 m in the
horizontal x-direction. The height of the initial soil deposit (without the embankment) equals 7
m. The embankment is 3 m tall with slope of 1:1.5 (height:length). The Very fine option is
selected for the Element distribution. Figure 25 illustrates the model geometry and the
generated mesh. The Soft Soil Creep model is used to simulate the soil deposit. The adopted
material properties are presented in Table 13.
Each calculation phase corresponds to different time period, i.e. 10 days, 100 days, 1,000
days, 10,000 days and 100,000 days. The calculation type is Plastic analysis. Drained
behaviour is considered. After a calculation phase with the Soft Soil Creep model is finished,
an extra calculation phase is added in order to switch from the Soft Soil Creep model to the
SHANSEP NGI-ADP model. The undrained shear strength is re-initiated and the stress
history is transferred from the Soft Soil Creep model to the SHANSEP NGI-ADP model
(Section 5). A safety analysis follows in order to calculate the FoS for the various considered
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time periods. The adopted material properties for the SHANSEP NGI-ADP model are
presented in Table 14.
Figure 26 illustrates the FoS evolution over time. As time increases, the isotropic preconsolidation stress pp increases, while the equivalent isotropic stress peq remains
unchanged. Thus, based on Eq. 10, OCR increases. As discussed in Section 5, the
undrained shear strength increases as well. After each switch to the SHANSEP NGI-ADP
model, the undrained shear strength is re-initiated, leading to higher FoS.
Once more, note that during the Safety analysis the value of suA (shansep) remains constant
and equal to the value obtained after the last (re-)initialisation of the active undrained shear
strength (given by Eq. 1), while the value of suA gradually decreases in order to compute the
FoS (given by Eq. 13).
Figure 25 Model geometry and generated mesh in PLAXIS 2D
Table 13 Adopted Soft Soil Creep model parameters
Symbol
γunsat
γsat
λ*
κ*
μ*
c’ref
φ’
ψ
ν’ur
nc
K0
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SSC
15.0
17.0
0.105
0.016
0.004
5.0
32.0
0.0
0.15
0.4701
Unit
3
kN/m
3
kN/m
2
kN/m
deg
deg
-
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Table 14 Adopted SHANSEP NGI-ADP model parameters (drained behaviour)
Symbol
γunsat
γsat
A
G/su
C
γf
E
γf
DSS
γf
A
su ref
vertref
A
su inc
P
A
su /su
A
τ0/su
DSS
A
su /su
ν
νu
alpha (α)
power (m)
su,min
OCRmin
POPmin
Value
15.0
17.0
150.0
3.0
4.0
3.8
5.0
7.0
0.5
1.0
0.7
1.0
0.3
0.498
0.33
0.83
5.0
1.0
0.0
Unit
3
kN/m
3
kN/m
%
%
%
2
kN/m
m
2
kN/m /m
2
kN/m
2
kN/m
Figure 26 Factor of Safety evolution over time
As discussed in Section 5, the SHANSEP NGI-ADP may be used as an extension of other
advanced constitutive models. In this particular example the effect of creep on the undrained
shear strength is demonstrated by using the Soft Soil Creep model and changing to the
SHANSEP concept, considering various time intervals. Safety analyses are used to study the
progressive change of FoS.
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8. Conclusions
The SHANSEP NGI-ADP model is an advanced constitutive model developed to enable the
analysis of dikes and embankments according to the new design requirements (based upon
undrained Critical State calculations) within a FEM environment. It overcomes limitations of
the traditional NGI-ADP model with respect to the varying undrained shear strength of soils
due to load history. Five additional input parameters are needed for this model, namely α, m,
su,min, OCRmin and POPmin. The model is based on the SHANSEP concept, which describes
the undrained shear strength as a function of the effective stress history. The two SHANSEP
parameters α and m influence the value of the undrained shear strength. The effect of these
parameters was investigated and the results show that the effect of α is more dominant.
However, the range of variation of these two parameters is short.
Comparison between numerical results and analytical solutions for triaxial compression tests,
starting from different initial stress states and OCR, verifies that the model is well
implemented in PLAXIS. OCR preservation when switching from other advanced constitutive
models to the SHANSEP NGI-ADP model is checked as well. PLAXIS results are compared
with analytical formulations and they are found to be in perfect agreement.
Three practical applications were performed to illustrate how the model features influence the
soil behaviour. In the first practical case, the fluctuation of the water level and its influence on
the undrained shear strength is studied. In the second application, it is demonstrated that the
stress history during an embankment construction affects the variation of the undrained
shear strength and the corresponding Factors of Safety. In the third case, a way of using the
SHANSEP NGI-ADP model in combination with another advanced soil model (Soft Soil
Creep model) is described. All three cases show a proper functioning of the model.
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9. Samenvatting (Nederlands)
Het SHANSEP NGI-ADP model is een geavanceerd constitutief model, ontwikkeld om de
berekening van dijken en waterkeringen volgens de nieuwe ontwerpregels (gebaseerd op
ongedraineerd Critical State rekenen) mogelijk te maken in een EEM omgeving. Het
overstijgt de beperkingen van het traditionele NGI-ADP model met betrekking tot de variatie
van ongedraineerde schuifsterkte van grond als gevolg van de belastinggeschiedenis. Vijf
invoerparameters zijn nodig voor dit model. Het model is gebaseerd op het SHANSEP
concept, die de ongedraineerde schuifsterkte beschrijft als functie van de effectieve
spanningsgeschiedening. De twee SHANSEP parameters α en m beïnvloeden de waarde
van de ongedraineerde schuifsterkte. De invloed van beide parameters is onderzocht en de
resultaten laten zien dat het effect van α het grootst is. Echter, de range van variatie van
beide parameters is beperkt.
Vergelijking tussen numerieke resultaten en analytische oplossingen voor triaxiaal
compressie proeven, startend vanaf verschillende initiële spanningen en OCR-waarden,
laten zien dat het model correct is geïmplementeerd in PLAXIS. Het behouden van de OCR
wanneer vanaf een ander geavanceerd model wordt ‘geswitcht’ naar SHANSEP NGI-ADP is
ook geverifiëerd. PLAXIS resultaten zijn vergeleken met analytische formules en blijken
hiermee in volledige overeenstemming.
Drie praktische toepassingen zijn uitgevoerd om te laten zien hoe de modeleigenschappen
het grondgedrag beïnvloeden. In de eerste praktische toepassing zijn de fluctuatie van het
waterniveau en de invloed op de ongedraineerde schuifsterkte onderzocht. In de tweede
toepassing is getoond hoe de spanningsgeschiedenis tijdens het aanleggen van een
aardebaan de variatie van de ongedraineerde schuifsterkte en de bijbehorende
stabiliteitsfactor beïnvloedt. In de derde situatie is beschreven hoe het SHANSEP NGI-ADP
model in combinatie met een ander geavanceerd grondmodel (Soft Soil Creep model) kan
worden gebruikt. Alle drie de toepassingen tonen een correcte werking van het model aan.
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References
Andresen, L. (2002). Capacity analysis of anisotropic and strain-softening clay. PhD thesis,
University of Oslo, Institute of Geology, Norway.
Andresen, L. and Jostad, H.P. (1999). Application of an anisotropic hardening model for
undrained response of saturated clay. In Proc. NUMOG VII, Graz, Austria, 581–585.
Grimstad, G. and Andresen, L. and Jostad, H.P. (2012). NGI‐ADP: Anisotropic shear
strength model for clay. International Journal for Numerical and Analytical Methods in
Geomechanics, 36(4):483-497.
Henkel, D.J. (1960). The shear strength of saturated remolded clays. In Proc. of the ASCE
Specialty Conference on Shear Strength of Cohesive Soils, University of Colorado,
Boulder, 533–554.
Ladd, C.C. and DeGroot, D.J. (2003). Recommended practice for soft ground site
characterization: Arthur casagrande lecture. In Proc. of the 12th Panamerican Conference
on Soil Mechanics and Geotechnical Engineering, Massachusetts Institute of Technology,
Cambridge, MA, USA.
Ladd, C.C. and Foott, R. (1974). New design procedure for stability of soft clays. Journal of
the Geotechnical Engineering Division, 100(7), 763–786.
Santagata, M.C. and Germaine, J.T. (2002). Sampling disturbance effects in normally
consolidated clays. Journal of Geotechnical and Geoenvironmental Engineering, 128(12),
997–1006.
Seah, T.H. and Lai, K.C. (2003). Strength and deformation behavior of soft bangkok clay.
Geotechnical Testing Journal, 26(4), 421–431.
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Appendix: NGI-ADP model manual
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THE NGI-ADP MODEL (ANISOTROPIC UNDRAINED SHEAR STRENGTH)
12
THE NGI-ADP MODEL (ANISOTROPIC UNDRAINED SHEAR STRENGTH)
The NGI-ADP model may be used for capacity, deformation and soil-structure interaction
analyses involving undrained loading of clay. The basis of the material model is:
•
Input parameters for (undrained) shear strength for three different stress paths/
states (Active, Direct Simple Shear, Passive).
•
A yield criterion based on a translated approximated Tresca criterion.
•
Elliptical interpolation functions for plastic failure strains and for shear strengths in
arbitrary stress paths.
•
Isotropic elasticity, given by the unloading/reloading shear modulus, Gur .
12.1
FORMULATION OF THE NGI-ADP MODEL
The NGI-ADP model is formulated for a general stress state, matching both undrained
failure shear strengths and strains to that of selected design profiles (Andresen & Jostad
(1999), Andresen (2002), Grimstad, Andresen & Jostad (2010)). The model formulation
is presented in steps, starting with 1D anisotropy in triaxial test condition. In Section
12.1.2 a simplified expression for plane strain is presented. Thereafter the formulation is
extended to full 3D stress state. In this formulation compressive stresses are positive.
In the NGI-ADP model the Tresca approximation after Billington (1988) together with a
modified von Mises plastic potential function (von Mises (1913)) is used to circumvent the
possible corner problems. The yield and plastic potential function are independent of the
mean stress hence zero plastic volume strain develops.
12.1.1
1D MODEL PRESENTATION
Under triaxial tests condition two undrained shear strengths can be determined, i.e. suC
and suE . The test measures the response in vertical stress σ 'v and horizontal stress σ 'h
for applied shear strain γ . The Tresca yield criteria can be modified, Eq. (12.1), to
account for the difference in undrained shear strength in compression and extension:
sC + suE
suC − suE =0
f = τ − (1 − κ)τ0 − κ
− κ u
2
2
(12.1)
where τ = 0.5(σ 'v − σ 'h ) and the initial in situ maximum shear stress τ0 is then defined as
τ0 = 0.5(σ 'v 0 − σ 'h0 ) = 0.5σ 'v 0 (1 − K0 ).
To account for difference in failure shear strain a stress path dependent hardening
parameter is introduced. The stress path dependent hardening is made possible by
p
different plastic failure shear strain γf in compression and extension. The hardening
function is given by:
κ=2
q
γ p /γfp
1 + γ p /γfp
p
when γ p < γf
else κ = 1
(12.2)
p
where γ p and γf are the plastic shear strain and the failure (peak) plastic shear strain
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respectively.
stress path
stress-strain
τmax = 1(σ1 − σ3 )
2
φcs − line
TXC
suC
τ0
K0 − line
γfC
γfE
p'
γ = ε1 − ε3
suE
TXE
φcs − line
Figure 12.1 Typical stress paths and stress strain curves for triaxial compression and triaxial
extension
12.1.2
THE NGI-ADP MODEL IN PLANE STRAIN
The yield criterion for the NGI-ADP model in plane strain is defined by:
v
!
!
u
u σyy − σxx
suA − suP 2
suA + suP 2
suA + suP
t
− (1 − κ)τ0 − κ
+ τxy
=0
f =
−
κ
2
2
2
2suDSS
(12.3)
Restriction to clays with horizontal surfaces are made to simplify the presentation.
Further y is taken as the vertical (depositional) direction. For isotropy in hardening (i.e. κ
independent of stress orientation) Eq. (12.3) plots as an elliptical shaped curve in a plane
strain deviatoric stress plot. When κ equals 1.0, the criterion in Eq. (12.3) reduces to the
formulation given by Davis & Christian (1971). While hardening the yield curves are
characterized by slightly distorted elliptical shapes. The shape is dependent on the
interpolation function used and values of failure strain. The NGI-ADP model uses
elliptical interpolation between failure strain in passive stress state, direct simple shear
and active stress state. In the implementation of the NGI-ADP model the yield surface is
ensured to remain convex by restricting the input.
12.1.3
THE NGI-ADP MODEL IN 3D STRESS SPACE
This section describes the actual implementation of the NGI-ADP model in PLAXIS,
whereas the previous sections should be regarded as an introduction using simplified
formulation. For the general stress condition a modified deviatoric stress vector is defined
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THE NGI-ADP MODEL (ANISOTROPIC UNDRAINED SHEAR STRENGTH)
Figure 12.2 Typical deviatoric plane strain plot of equal shear strain contours for the NGI-ADP model
as:

ŝ
 xx

 ŝyy


 ŝzz


 ŝxy


 ŝxz

ŝyz

σ 'xx − σ 'xx0 (1 − κ) + κ 1 (suA − suP ) − p̂


3

 
2
P
A

  σ 'yy − σ 'yy0 (1 − κ) − κ (su − su ) − p̂ 

3
 

 
  σ 'zz − σ 'zz0 (1 − κ) + κ 1 (suA − suP ) − p̂ 

 
3

=
A
P

s +s
 

τxy u DSSu
 

 
2su

 

 

τxz
 



suA + suP
τyz
DSS
2su


(12.4)
where σ 'xx0 , σ 'yy 0 and σ 'xx0 are the initial stresses and p̂ is the modified mean stress.
The modified mean stress is defined as:
(σ 'xx − σ 'xx0 (1 − κ)) + (σ 'yy − σ 'yy0 (1 − κ)) + (σ 'zz − σ 'zzx0 (1 − κ))
3
= p' − (1 − κ)p'0
p̂ =
(12.5)
where p' is the mean stress. Modified second and third deviatoric invariants are defined
accordingly in Eqs. (12.6) and (12.7).
2
2
2
Ĵ2 = −ŝxx ŝyy − ŝxx ŝzz − ŝyy ŝzz + ŝxz
+ ŝxz
+ ŝyz
(12.6)
2
2
2
Ĵ3 = ŝxx ŝyy ŝzz + 2ŝxy ŝyz ŝxz − ŝxx ŝyz
− ŝyy ŝxz
− ŝzz ŝxy
(12.7)
The yield criterion is expressed as:
f =
q
H(ω)Ĵ2 − κ
suA + suP
=0
2
(12.8)
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where, to approximate the Tresca criterion, the term H(ω) is defined as:
H(ω) = cos
2
1
arccos(1 − 2a1 ω)
6
with ω =
27 Ĵ32
4 Ĵ23
(12.9)
By letting the value of a1 go to 1.0 an exact Tresca criterion is obtained. The parameter
a1 can be directly linked to the rounding ratio suC /suA . This ratio takes typically a value just
below 1.0 and a value of 0.99 is chosen as an appropriate default. Figure 12.3 shows the
failure criterion of the NGI-ADP model in the π -plane (for Cartesian stresses) with default
rounding ratio. This criterion is continuous and differentiable and it is described by a
single function.
Figure 12.3 Failure criterion of the NGI-ADP model in the π -plane
The combinations of strength ratios are limited by lower limit for combinations of suC /suA
and suP /suA .
p
The value of γf is given by elliptical interpolation:
γfp (θ̂) =
q
R̂B R̂D (R̂D2 − R̂C2 )cos2 (2θ̂) + R̂C2 − R̂D2 R̂A cos(2θ̂)
R̂B2 − (R̂B2 − R̂D2 )cos2 (2θ̂)
(12.10)
where
R̂A =
γfp,E − γfp,C
2
+ γfp,C
γp
R̂B = f ,E
q 2
R̂C = γfp,E γfp,C
R̂D =
p
p
γfp,DSS R̂B
R̂C
p
and γf ,C , γf ,DSS and γf ,E are the failure plastic maximum shear strain in triaxial
132 Material Models Manual | PLAXIS 2016
(12.11)
(12.12)
(12.13)
(12.14)
THE NGI-ADP MODEL (ANISOTROPIC UNDRAINED SHEAR STRENGTH)
compression, direct simple shear and triaxial extension respectively. Note that θ̂ is not the
Lode angle, but is defined as:
cos(2θ̂) =
√
3 ŝyy
q
2
Ĵ2
(12.15)
A non-associated flow rule is used such that the derivative of the plastic potential g is:
s
1
∂g
∂p ∂ ŝ T ∂ Ĵ2 1
=
Î +
∂ ŝ Ĵ2
∂σ ' 2
∂σ ' ∂ p̂
(12.16)
where Î is a modified unit vector:








Î = 








1
1
1
suA + suP
2suDSS













1

suA + suP 
2suDSS
(12.17)
The increment in plastic shear strain is defined as:
2
dεpxx − dεpyy 2 + dεpxx − dεpzz 2 + dεpyy − dεpzz 2 +
∂γ = H(ω)
3
1/2
p 2
p 2
p 2
(12.18)
+ dγyz
+ dγxz
dγxy
p
p
where the plastic strains are defined as:
dεp = dλ
∂g
∂σ '
(12.19)
with dλ being the plastic multiplier. Hence, Eq. (12.18) gives the relation between the
hardening parameter γ p and the plastic multiplier dλ.
12.1.4
THE NGI-ADP MODEL TRACTION CRITERION FOR INTERFACES
For plane strain conditions a traction criterion, corresponding to the plane strain failure
criterion is formulated. This criterion is intended to be used on interface elements in finite
element calculations. The interface strength is controlled by a lower and upper limit,
which are dependent on the direction of the interface, β .
Let a plane being oriented by the direction β to the horizontal. The plane has a tangential
direction t and a normal direction n and the adjacent continuum defines the stresses σnn ,
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σtt and τn by:



σ
σtt
 xx





 σnn  = A σyy



τxy
τtn





(12.20)
where A is the transformation matrix.
In a local coordinate system three strains εnn , εtt and γtn are defined in the plane strain
condition. Due to the requirement of no volume change for a perfect plastic mechanism
with shearing in tangential direction, will give that εnn = εtt = 0, resulting in:
∂f
=0
∂(σnn − σtt )
(12.21)
The plain strain formulation for the NGI-ADP model is defined as follows:
s
σnn − σtt
cos(2β) − τtn sin(2β) − RA
f =
2
−RB = 0
2
RB
+
RD
σnn − σtt
sin(2β) + τtn cos(2β) 2
2
(12.22)
where
suA − suP
2
suA + suP
RB =
2
RD =suDSS
(12.23)
RA =
12.2
(12.24)
(12.25)
PARAMETERS OF THE NGI-ADP MODEL
Stiffness parameters:
Gur /suA
: Ratio unloading/reloading shear modulus over (plane
strain) active shear strength
[-]
γfC
: Shear strain at failure in triaxial compression
[%]
γfE
: Shear strain at failure in triaxial extension
[%]
γfDSS
: Shear strain at failure in direct simple shear
[%]
Hint: Note that the input value of shear strains is in percent
Strength parameters:
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THE NGI-ADP MODEL (ANISOTROPIC UNDRAINED SHEAR STRENGTH)
A
su,ref
: Reference (plane strain) active shear strength
[kN/m2 /m]
suC,TX /suA
: Ratio triaxial compressive shear strength over (plane
strain) active shear strength (default = 0.99)
[-]
yref
: Reference depth
[m]
A
su,inc
: Increase of shear strength with depth
[kN/m2 /m]
suP /suA
: Ratio of (plane strain) passive shear strength over
(plane strain) active shear strength
[-]
τ0 /suA
: Initial mobilization (default = 0.7)
[-]
suDSS /suA
: Ratio of direct simple shear strength over (plain strain)
active shear strength
[-]
Advanced parameter:
ν'
: Poisson's ratio
[-]
Ratio unloading / reloading shear modulus over plane strain active shear
strength (Gur /suA )
Ratio unloading / reloading shear stiffness as a ratio of the plane strain active shear
strength. If the shear strength is increasing with depth the constant ratio for Gur /suA gives
a shear stiffness increasing linearly with depth.
Shear strain at failure in triaxial compression (γfC )
This parameter γfC (%) defines the shear strain at which failure is obtained in undrained
triaxial compression mode of loading, i.e. γfC = 3 εC
1 from triaxial testing.
2
Shear strain at failure in triaxial extension (γfE )
This parameter γfE (%) defines the shear strain at which failure is obtained in undrained
triaxial extension mode of loading, i.e. γfE = 3 εE1 from triaxial testing.
2
Shear strain at failure in direct simple shear (γfDSS )
This parameter γfDSS (%) defines the shear strain at which failure is obtained in undrained
direct simple shear mode of loading (DSS device).
For near normally consolidated clays, the failure strain in compression loading γfC is
generally the lowest value and the failure strain in extension loading γfE is the highest
value. The failure strain from direct simple shear loading takes an intermediate value, i.e.
γfC < γfDSS < γfE . From laboratory test results reported in literature one find typically γfE
in the range 3-8 %, γfDSS in the range 2-8 % and γfC in the range 0.5 - 4 %.
If stress-strain curves from undrained triaxial and/or DSS laboratory tests are available it
is recommended to choose the elastic shear modulus and failure strains such that a good
fit to the curves are obtained. This is in particular important for deformation and SLS
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assessments. However, for pure capacity and stability (e.g. factor of safety) analyses the
values for shear strains at failure is not important and one may set all three values equal
to e.g. 5 % for simplicity.
Note that it is the failure strains from triaxial loading that is input because they are the
most readily available. When the NGI-ADP model is used for plane strain conditions the
failure strains will automatically be slightly adjusted for that loading condition. See
Grimstad, Andresen & Jostad (2010) for more details.
A
Reference active shear strength(su,ref
)
The reference active shear strength is the shear strength obtained in (plane strain)
undrained active stress paths for the reference depth yref , expressed in the unit of stress.
Ratio triaxial compressive shear strength over active shear
strength(suC,TX /suA )
This ratio suC,TX /suA defines the shear strength in undrained triaxial compression mode of
loading in relation to the shear strength in plane strain undrained active mode of loading.
The value cannot be changed by the user and is predefined at 0.99 giving practically the
same strengths in triaxial and plane strain conditions.
Reference depth (yref )
A
This is the reference depth yref at which the reference active shear strength su,ref
is
defined. Below this depth the shear strength and stiffness may increase linearly with
A
increasing depth. Above the reference depth the shear strength is equal to su,ref
.
A
Increase of shear strength with depth (su,inc
)
A
This parameter su,inc
defines the increase (positive) or decrease (negative) of the
undrained active shear strength with depth, expressed in the unit of stress per unit of
A
depth. Above the reference depth the shear strength is equal to su,ref
, below the
reference depth the shear strength is defined as:
A
A
suA (y ) = su,ref
+ (yref − y )su,inc
(12.26)
Ratio of passive shear strength over active shear strength (suP /suA )
This ratio suP /suA defines the undrained shear strength for (plane strain) passive mode of
loading.
Ratio of direct simple shear strength over active shear strength (suDSS /suA )
This ratio suDSS /suA defines the undrained shear strength for direct simple shear mode of
loading. Please note that active / passive strength input is defined for plane strain
conditions. However, it is generally acceptable and only slightly conservative to use the
strength obtained from a triaxial compression test as input for the active plane strain
condition (i.e. suA = suC,TX ) and the strength obtained from a triaxial extension test as input
for the passive plane strain condition (i.e. suP = suE,TX ). More control over the strength
difference between triaxial and plane strain loading conditions can be obtained by using
the advanced parameter suC,TX /suA .
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THE NGI-ADP MODEL (ANISOTROPIC UNDRAINED SHEAR STRENGTH)
For near normally consolidated clays, the passive strength suP is generally the lowest
strength value, while the direct simple shear strength takes an intermediate value, i.e.
suP < suDSS < suA . From laboratory results reported in literature one find typically suP /suA in
the range 0.2 - 0.5 and suDSS /suA in the range 0.3 - 0.8. If direct simple shear strengths
are not available suDSS can be estimated from: suDSS /suA = (1 + suP /suA )/2.
Initial mobilization (τ0 /suA )
The initial mobilization τ0 /suA is clearly defined for nearly horizontally deposited normally
consolidated or lightly overconsolidated clay layers where the vertical stress is the major
principle stress σ '1 . As defined in Figure 12.4, the initial mobilization can be calculated
from the earth pressure coefficient at rest K0 by the following equation:
τ0 /suA = −0.5(1 − K0 )σ 'yy0 /suA , where σ 'yy0 is the initial (in situ) vertical effective stress
(compression negative). A default value 0.7 of τ0 /suA is given which represent a typical
value for a near normally consolidated clay deposit (e.g. K0 = 0.55 and
−σ 'yy0 /suA = 3.11, or K0 = 0.6 and −σ 'yy0 /suA = 3.5).
Figure 12.4 Definition of initial mobilized maximum shear stress τ0 = 1/2 |σ 'yy0 − σ 'xx0 | for a soil
element in a horizontal deposited layer.
A more detailed evaluation of the initial (in situ) mobilization can be done by assessing
the in situ K0 value and use the relationship: τ0 /suA = −0.5(1 − K0 )σ 'v 0 /suA . Changing
the default value for the initial mobilization should be considered in particular for
over-consolidated materials where K0 generally is higher than 0.6, however the NGI-ADP
model is not intended used for heavily overconsolidated clays and should be used with
care for K0 > 1.0 (i.e. negative τ0 /suA ).
For non-horizontal layering (e.g. sloping ground) a K0 procedure is normally not
recommended. In such cases it is recommended to establish the initial stress condition
by gravity loading using a material model suited for such a purpose (e.g. drained
behavior with the Mohr-Coulomb model or the Hardening Soil model). After the
equilibrium initial stresses are established for gravity loading in the first phase, one
should switch to the NGI-ADP model in the relevant clusters for the next phase and run a
NIL step (i.e. without changing the external loads). The hardening parameter of the
NGI-ADP model will then be adjusted such that equilibrium is obtained (f=0). Then in the
third phase the external loading can be applied.
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Poisson's ratio (ν ')
Similar as in the Mohr-Coulomb model, Poisson's ratio is generally between 0.3 and 0.4
for loading conditions considering effective stress analysis. For unloading conditions a
lower value is more appropriate. See the linear elastic perfectly plastic model
(Mohr-Coulomb model) for more details.
When the Drainage type = Undrained (B) option is used an effective Poisson's ratio
should be entered that is less than 0.35. Excess pore pressures are generated using the
bulk modulus of water as described in Section 2.4. In this case, Undrained (B), the model
will calculate excess pore pressures due to mean stress changes according to (2.37).
When the Undrained (C) drainage option is used a pure total stress analysis is carried out
where no distinction between effective stresses and pore pressures is made and all
stress changes should be considered as changes in total stress. A Poisson's ratio close
to 0.5 should be entered.ν = 0.495 is given as default.
12.3
STATE PARAMETERS IN THE NGI-ADP MODEL
In addition to the output of standard stress and strain quantities, the NGI-ADP
modelprovides output (when being used) on state variables such as plastic shear strain
γp and the hardening function rκ . These parameters can be visualised by selecting the
State parameters option from the Stresses menu. An overview of available state
parameters is given below:
γp
: Plastic shear strain
[-]
γp =
rκ
: Hardening function
rκ = 2
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[-]
q
γ p /γfp
1 + γ p /γfp