The well-known Hardening Soil Model is used worldwide to model geotechnical engineering problems, such as deep excavations and embankment construction, because it captures several features of real soil behavior for both sandy soils and clays and silts.
This webinar briefly introduces the Hardening Soil Model theory and formulation. The creation of a SIGMA/W analysis using the Hardening Soil Model to simulate the triaxial test is included. You will be taken through a step-by-step procedure to calibrate the model and identify the model constants based on drained triaxial test results. The simulated results are compared to the laboratory data.
<v Curtis>Welcome to this GeoStudio</v>
Material Model webinar series
on the Hardening Soil Model.
My name is Curtis Kelln,
I’m the Director of Research and Development
with the GeoSlope Engineering team at Seequent.
Today’s webinar will be approximately 30 minutes long.
Attendees can ask questions using the chat feature
and I will respond to these questions
by email as quickly as possible.
A recording of the webinar will be available,
so that you can review the demonstration, at a later time.
GeoStudio was developed for geotechnical engineers
and earth scientists.
As such, it comprises a range of products,
that can be used independently
or together to solve a wide array of problems.
Today’s webinar will focus on SIGMA/W.
Those looking to learn more about the products,
including background theory, available features,
and typical modeling scenarios,
can find an extensive library of resources
on the Geoslope website.
Here, you can find a tutorial videos,
examples with detailed explanations
and reference books on heat and mass transfer modeling,
statics stress rain modeling, slope stability modeling,
and dynamics stress strain modeling with GeoStudio.
To begin this webinar, let us consider a soil specimen
subject to an experimental procedure,
and a numerical simulation
of that experimental procedure.
And in experimental approach,
there’s a real soil sample,
from which specimens can be extracted
and subjected to various tests,
such as triaxial and odometer loading.
The data collected from these tests is then interpreted
and the interpretation has to be done in a manner
that is specific to a particular constitutive of model.
The parametrization procedure
produces the models constants.
These constants are then used as inputs for the model.
The simulated results are obtained
by solving the physics subject to boundary conditions
and a constitutive law.
And finally, the simulated results are compared
to the corresponding laboratory results
to verify the parameterization.
In today’s webinar, we will review the key elements
of the hardening soil constitutive model.
Then, a step-by-step procedure will be presented
for parameterizing the model.
The procedure will then be applied to data
from numerous triaxial tests on sandy soil.
A demonstration of the steps from modeling a triaxial test
and SIGMA will be provided,
and the simulated and measured results will be compared.
Some concluding remarks will also be provided
at the end of the webinar.
The Hardening Soil Model was proposed by Schanz,
Vermeer and Bonnier in 1999.
It was formulated on the framework
of the Classical Theory of Plasticity.
The formulation comprises three stress-dependent
stiffness quantities and two yield surfaces.
One that resembles a hexagonal cone
and a second that resembles the tip of an ellipsoid.
The shear surface predominantly governs the response
to changes in deviatoric stress
while the cap predominantly governs
the response to increments in mean effective stress.
Failure is controlled by a Mohr-Coulomb failure law
hence the hexagonal shape of the yield surface.
The model implemented in Sigma
actually comprises a smoother cap
than what is depicted in this figure.
The reference book should be consulted
to understand the details of the formulation.
In the triaxial stress state,
the total stress path is a straight line
that reaches the Mohr-Coulomb envelope
out a specific failure point.
The deviatoric stress,
axial strain response of the soil
under this loading condition
is estimated by hyperbolic curve,
as will be discussed shortly.
The characteristics of this curve
are determined by three
stress-dependent stiffness quantities,
including the initial tangent stiffness,
E subscript I,
the half failure secant stiffness
E subscript 50
and an unloading reloading stiffness E subscript UR.
In order to have a closed yield surface,
a cap function is defined in the hardening soil model.
This cap has an elliptical shape
with an aspect ratio of M
or the Greek letter mu.
In the PQ stress base,
the stress state of any normally consolidated sample
is located on the cap.
That’s at the tip of this blue stress path
that intersects the ellipse.
The stress strain behavior of the sample
in an odometer test is defined
by a tension stiffness called the odometer stiffness
or E subscript OED.
Like the other stiffness quantities,
the odometer stiffness is not constant,
but it’s value depends on the stress of the specimen.
To review the key ingredients of the hardening soil model,
Let’s consider a theoretical triaxial response.
The axial strain of the sample in this condition
is to defined by a hyperbolic function.
The hyperbolic curve continues
until the Mohr-Coulomb failure stress is reached,
Q subscript F.
The failure occurs before the curve reaches
a asymptotic tonic value.
The ratio of the failure to asymptotic deviatoric stresses
is R subscript F.
The behavior of the sample at the beginning of loading
is modeled by the initial stiffness
E subscript I.
Since this quantity is difficult to measure,
its value is presented in terms of
the Half failure Secant stiffness,
E subscript 50.
The half failure secant stiffness
is a stress dependent stiffness
that depends on the minimum principle stress,
Sigma subscript three.
A similar stress dependency is also considered
in the Hardening soil model
for unloading reloading stiffness.
The model’s constants are now highlighted in gray.
In an odometer test on normally consolidated soil,
the slope of the stress path
is a function of the coefficient of lateral earth pressure,
K nought superscript NC.
Moreover, the cap yield surface and PQ space is an ellipse
with a horizontal major axes.
In a non cohesive soil,
the elliptical yield surface is centered around the origin.
The odometer stiffness is also a stress dependence stiffness
presented in terms of the vertical stress,
Sigma one in a one dimensional odometer tests.
Notice that the dependence on Sigma one
is different from the other moduli
that were dependent on Sigma three.
The model’s constants are once again
highlighted in light gray.
the hardening soil model requires
three stiffness Constants,
a reference stress,
and an exponent M.
The strength constants and failure ratio
are subscript F,
the coefficient of lateral earth pressure.
Some other parameters will also be needed
to describe the initial condition of a specific soil sample.
These include the initial void ratio,
the unit weight of the soil,
and an isotropic over consolidation ratio,
which we’ll touch on briefly in the subsequent slides.
We can now discuss a step-by-step parametrization procedure
for determining the Hardening soil model Constants
from a number of drain triaxial tests.
Step one involves determination
of the effective friction angle
and the effect of cohesion C.
The failure stress from numerous triaxial tests
at different confining pressures
are theoretically located
on the Mohr-Coulomb failure line in PQ space.
As such the slope of the line,
which is 10 beta,
as well as the Y intercept alpha,
is calculated using the method of least squares
where N is the number of data points.
Alternatively, the trendline functionality
and programs like Excel
can be used instead of the least squares formulas.
The effective friction angle and effective cohesion
of the soil can be found by changing the coordinate system
to a shear normal stress base.
The equations corresponding to that transformation
are shown here.
The second step of the procedure
is to consider a reference stress
and to estimate reference stiffnesses.
One of the available triaxial tests
is considered as a reference test.
The confining pressure of this test
is then called the reference stress.
The failure of deviatoric stress for this particular test
is determined based on Mohr-Coulomb strength constants
determined in the previous step.
The half failure referenced stiffness
is the slope of a straight line that passes through
the point carrying 50% of the failure deviatoric stress.
The least squared methods is applied again
on the unloading reloading branch
of the stress strain curve
to calculate E subscript UR.
In the third step of the procedure,
the stress dependency exponent M is estimated.
In stress strain curves of drain triaxial tests
with different confining pressures,
half failure stresses are first calculated
using the method described in step two.
A new natural log space can then be defined
that represents the half failure stiffnesses of each test
in the form of a single point.
The point corresponding to the reference test
is located on the origin.
The stress dependency exponent M
is the slope of a trend line
that passes through these points.
The similar steps can be taken for unloading reloading
in order to calculate
the stress dependency exponent twice.
In the hardening soil model however,
it is assumed that the constant M
is the same for both half failure
and unloading reloading stiffnesses.
The fourth step of the parameterization procedure
is to calculate the failure ratio, RF.
This constant has a value between 0.5 and one,
and is usually assumed to be 0.9.
A more accurate value for this constant can be calculated
based on the results of the triaxial tests.
First, the stress strain hyperbolic function
is rewritten in terms of the failure ratio,
R subscript F.
This expression introduces a new coordinate system
in which each hyperbolic curve
is projected to a straight line.
The failure ratio is the slope of these parallel lines.
In the second last step of the parametrization procedure,
the odometer reference stiffness
and the coefficient of lateral earth pressure are estimated.
The odometer test result is required for the step.
However, these values can be approximated
in the case where the appropriate laboratory result
is not available.
As mentioned earlier, the odometer reference stiffness
is defined in a one-dimensional compression test.
By definition, it is the tangent stiffness
at the point where the vertical stress
is equal to the reference stress.
In addition, the coefficient of lateral earth pressure
for a normally consolidated soil, K nought NC,
is the ratio of lateral stress, Sigma three,
to vertical stress, Sigma one, in the odometer test.
These two constants must be approximated
in the absence of laboratory data.
For instance, the value of the odometer reference stiffness
is usually greater than 50%
of E 50 rough and less than 80% of E 50 rough.
And also the coefficient of lateral earth pressure
can be estimated using the well-known Jackie’s formula,
which relies on the friction angle of the soil.
One additional step in the privatization procedure
is required if the soil is over consolidated,
and that is the determination of the isotropic over
which is a ratio equivalent isotropic stresses.
The denominator of this expression
is simply the confining pressure of the triaxial test.
In contrast, the numerator, P subscript C,
defines the size of the yield locus
passing through the yields stresses QY and PY
As such, P subscript C
must be calculated from the equation
of the yield function.
The yields stresses are generally determined
from plots of deviatoric stress versus axial strain
and void ratio versus natural log mean effect of stress.
And in both of these plots, the yield point,
that is that transition from
elastic to elastic plastic behavior
it deduce by an inflection point
in the measured laboratory data.
That brings us to the constant mu,
which is in the equation for the cap yield function.
It is calculated by the material model in the solver
from this expression here.
We elaborate on this expression in the reference book,
but for the purpose of this webinar,
it suffices to say that theta is
Q/P stress ratio
that can be calculated from the user input,
K nought NC,
which we discussed on the previous slide
and lambda/Kappa is calculated from
this ratio of reference stiffnesses.
Now the entire procedure for determining OCR
can be simplified if the stress path is instead
that of isotropic compression, because in that case,
the stress path tracks along
the mean effect of stress axes.
Meaning that the deviatoric stress is zero
and PC is equal to PY.
To demonstrate the parametrization procedure
we are going to consider 12 drain triaxial compression tests
on Ottawa sand.
These tests were performed at different confining pressures
ranging from 50 to 600 kilopascals.
The effective friction angle and the cohesion of the soil
are calculated from the first step
of the parametrization procedure.
The deviatoric compression stress in 12 triaxial tests
continued until failure was achieved.
The failure state of each task can be represented
by a point in PQ space.
The failure line that passes through these failure points
has a slope of 1.18
and a Y intercept of zero.
Therefore the effect of friction angle
as calculated as 29.6 degrees
while the cohesion is calculated as zero,
since alpha is zero.
At the second step of the procedure,
the test with the confining stress of 100 kPA
is considered as the reference test.
The half failure deviatoric stress for this reference test
is calculated to be 97.6 kilopascals.
According to the measured stress strain curve,
the axial strain corresponded into this half failure
stress is equal to 0.0055.
This means that the half failure referenced stiffness
for the soil is 17,745 kilopascals.
Again, that is simply the slope of this line.
Finally, the unloading reloading branch
of the measured stress drain curve
gives us the value of the unloading reloading
And the slope of that line
is estimated to be 45,000 kilopascals.
The third step of the parametrization procedure
estimates the stress dependency exponent M.
As shown here this constant can be estimated
from either the unloading reloading
or the half failure stiffnesses in natural log,
natural log space.
The slope of these two lines is almost the same
and their average is 0.680.
And this is what we consider to be the constant M.
As can be seen here,
the proposed expressions can adequately estimate
the unloading reloading and half failure stiffnesses
in terms of their corresponding confining stress
The proposed curve, however,
is more consistent with the measured results
for the unloading reloading stiffnesses
than for the half failure stiffnesses.
This is commonly the case.
The failure ratio, R subscript F
is calculated at the fourth step.
The linear projections of the hyperbolic curves
are shown in the suggested space in the left graph.
The near vertical branches of these curves
represent unloading reloading phases,
where Q approaches zero
causing the ratio of QF/Q
to approach infinity.
These near vertical branches are not included
in the calculation of RF,
rather it is the sloping portions
of the curves that are described
by the equation of a line
where the slope of each line is our subscript F.
The graph on the right shows the slopes
for all 12 triaxial tests.
The average of 0.941 is then estimated
for the failure ratio of the soil.
For the odometer related constant in step five,
we apply the approximate method
of calculating the odometer reference stiffness.
According to the value
of the half failure reference stiffness,
which was calculated at the second step,
we estimate the odometer reference stiffness to be 115,
excuse me, 11,500 kilopascals.
The Jackie’s formula is also used to calculate the
coefficient of lateral earth pressure
in terms of the effect of friction angle
and from that relationship,
we obtained 0.506.
The results the parametrization procedure
for the Ottawa sand can be summarized as follows.
Our two strength constants gave us a friction angle
of 29.6 and a cohesion of zero.
The reference dress was 100 kilopascals,
and the reference stiffnesses were calculated accordingly.
The stress dependency exponent and the failure ratio
were estimated as 0.68
and 0.941 respectively,
and the coefficient of lateral earth pressure
was obtained from Jackie’s formula
to be 0.506.
Now that we have reviewed the step-by-step parametrization
procedure for the Hardening soil model,
I will describe how to model a Triaxial test in Sigma,
and compare the numerical results
with the previously mentioned lab data.
The numerical model configuration
is based on the geometry and boundary conditions
of the triaxial test conducted in the laboratory.
A typical triaxial sample has a cylindrical shape
with a diameter of five centimeters
and a height of 10 centimeters.
Due to axis symmetric shape of this geometry
with respect to the central vertical axis,
we use a 2D axis symmetric geometry and Sigma.
The geometry is also symmetric
with respect to the horizontal axis.
So we can model the top one quarter of the specimen.
An eight nodes single element model
is considered for this resulting rectangle.
The material definition is of course the key subject
of this webinar.
This step is performed in the software
using the defined materials dialog box.
The material model is set to the Hardening soil model,
the sample parameters
including the initial void ratio
and over consolidation ratio are entered.
It should be noted that the effect of the soil self weight
can be removed by setting the unit weight to zero.
And lastly, the constitutive model constants
are entered based on what was previously obtained
during the parametrization procedure
for the Ottawa sand.
The triaxial tests are simulated
by firstly considering the isotropic confining pressure.
And secondly, the deviatoric pressure.
To do this, the project file is set up
with an analysis tree structure for each test.
In each branch,
we see the confining pressure phase
acting as a parent,
which provides initial data
to the subsequent displacement control deviatoric phase.
In test A one as an example,
a constant normal stress of 50 kPA
is applied on both the right and top sides of the model
using a normal tangential stress boundary condition.
And in the deviatoric phase of this test,
a displacement of two centimeters
that is equivalent to 40% strain,
is applied on the top side of the model.
The displacement spline data function
contains a small plip
that just before
a time and a lapse time
and this is the unloading reloading phase.
Let’s just take a few minutes to explore
some of the finer details of these ostensibly simple
First, we’ll start at the top of the project definition.
And I will point out that you can right click on here
and add multiple geometries to the same project file.
In this analysis, of course we have only one single geometry
and it is of the type 2D Axis symmetric.
The next noteworthy thing under our geometry sits
our confining stress analysis
for all 12 triaxial tests
and then unlike what is shown
in the PowerPoint presentation,
we actually have two cases
for each deviatoric loading condition.
One that is monotonic,
meaning strained without unload reload,
and the other with the LUR or Load Unload Reload branch.
All three of the analysis in each branch
are low deformation.
Now the first analysis
is the one in which we apply the confining pressure
of 50 all the way through to 600 kilopascals.
And it is an analysis of the type load deformation.
We’re not doing an in-situ or gravity activation analysis
dimensions of the specimen
are so negligible
that the body load or the gravity load
doesn’t contribute to the stresses in any meaningful way.
The other thing that’s worth noting is that
our deviatoric phases
are also low deformation,
and the initial stresses and poor pressures
come from the parent analysis.
We also have the ability to set
the final pour water pressure conditions.
And you can see in this case,
it’s the same as the initial conditions,
meaning that there’s no increment or decrement
in the pour water pressure.
Had these been different due to, for example,
specifying the pour pressures
from another geostudio analysis like Sigma CAP
or the use of a water table or
special function to find the pressure heads,
then not change in pour water pressure
would be reflected
in the deformation response,
the simulated deformation response of the specimen.
I should also note while I’m here,
that there are these two options to reset
displacements and strains
and reset state variables.
These are not on automatically
when a low deformation analysis is added
to another low deformation analysis.
And so they were not set in this case.
Now that means that any of the deformations accumulated
during the isotropic compression
will carry through
into the deviatoric loading phase.
And so in actuality,
it would have been better modeling practice
to toggle the reset displacements and strains,
and these are accumulated values to toggle that on.
If you’re using a material model that has state
like the hardening soil model in the confining phase,
then you can also reset the state variables.
Now, in this case,
what we’ve done is we’ve used
an isotropic elastic material model
in the compression phase,
and we’ve given it a very high stiffness quantity
of 15 million kilopascals,
meaning that the simulated deformations
from this phase are negligible.
And therefore we need not reset
the accumulated displacements and strains.
The other point I’ll make is that
when you deal with large project files
containing multiple analysis,
there are right click options in the solve manager
to uncheck all,
to check children, which picks a single branch,
that’s the zero one branch.
And then there are a couple other options.
Similarly, if you go up into the solve manager,
it’s often more convenient
if these branches are quite elaborate,
in this case there are only two analysis
that are parenting to that,
the confining stress analysis.
But we can imagine a scenario
where they’re eight, 12, and so on analysis underneath
the parent analysis.
In which case right clicking up here
and queuing an analysis branch for solving
is more convenient.
You can also
you can clear, and you can also
queue one particular analysis.
Now, in this case,
I’m going to right click and
and then solve.
And once the solution is obtained for any one of these,
we can switch to the results view.
And it is here where we can
go to draw graph
the response through various graphs,
such as deviatoric stress versus axial strain,
deviatoric stress versus mean effective stress
and void ratio versus P.
You can see that there’s a unload branch in there.
There’s the monotonic loading case.
You can also multi-select these graphs,
which of course gives a rather strange view
since the axes are not all the same,
but I just wanted to point out
that you can copy this graph data
and then go to a spreadsheet program like Excel,
where you can then
and paste the data into the spreadsheet.
This will then make for easier comparison
with the measured laboratory data
If you turn out to the PowerPoint presentation,
the simulation results of six triaxial trust tests
with confining pressures ranging from 50 to 300 kilopascals
are compared here with laboratory results.
Colored curves are laboratory results.
And the black lines are the results
from the Sigma simulation
and analytical solution is also shown on these plots
as gray dash lines.
The full compatibility of the analytical
and simulated solutions verifies the implementation
of the hardening soil model in SIGMA/W.
the acceptable consistency
between the lab data and the simulator response
confirms that the hardening soil model
with carefully parameterized material constants
can capture the stress strain response of the Ottawa sand.
The same trends can be observed for the next six tests
that had confining pressures ranging from
350 kilopascals to 600 kilopascals.
To conclude, the parametrization procedure
of the hardening soil model
was provided in five straightforward steps
and a sixth optional step for over consolidated soil.
It was shown that the results from
multiple drain triaxial tests
and at least one odometer tests
are required to parameterize
the hardening soil model.
In the absence of odometer data,
approximations can be used to obtain
the reference odometer stiffness
and the K nought NC
or normally compressed earth pressure coefficient inputs.
The material constants were then used
in numerical simulations of the lab tests
and the numerical results were shown
to compare favorably with both the laboratory data
and analytical solutions.
During the presentation,
I referenced these top two
Shanz, Vermeer and Bannier,
that’s the original model formulation.
I also want to remind you that you can consult
our stress strain modeling
with GeoStudio 2021 reference book,
to understand the details of the formulation.
And then this is the paper that comprises
the laboratory tests on the Ottawa sand.
I’ll also remind you that
the website comprises a number of resources,
including example files.
One noteworthy example is that of
the tieback wall in Berlin,
it is completed using both the Mohr-Coulomb
and hardening soil models
and those results are compared.
With that, we’ve reached the end of the webinar.
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