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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.

Overview

Speakers

Curtis Kelln

Duration

34 min

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Video Transcript:

[00:00:00.730]
<v Curtis>Welcome to this GeoStudio</v>

[00:00:02.270]
Material Model webinar series

[00:00:04.370]
on the Hardening Soil Model.

[00:00:06.070]
My name is Curtis Kelln,

[00:00:07.610]
I’m the Director of Research and Development

[00:00:09.540]
with the GeoSlope Engineering team at Seequent.

[00:00:12.440]
Today’s webinar will be approximately 30 minutes long.

[00:00:15.770]
Attendees can ask questions using the chat feature

[00:00:18.780]
and I will respond to these questions

[00:00:20.550]
by email as quickly as possible.

[00:00:22.880]
A recording of the webinar will be available,

[00:00:25.810]
so that you can review the demonstration, at a later time.

[00:00:30.500]
GeoStudio was developed for geotechnical engineers

[00:00:33.460]
and earth scientists.

[00:00:34.930]
As such, it comprises a range of products,

[00:00:37.700]
that can be used independently

[00:00:39.640]
or together to solve a wide array of problems.

[00:00:43.010]
Today’s webinar will focus on SIGMA/W.

[00:00:46.130]
Those looking to learn more about the products,

[00:00:49.090]
including background theory, available features,

[00:00:52.300]
and typical modeling scenarios,

[00:00:54.340]
can find an extensive library of resources

[00:00:56.990]
on the Geoslope website.

[00:00:58.570]
Here, you can find a tutorial videos,

[00:01:01.130]
examples with detailed explanations

[00:01:03.630]
and reference books on heat and mass transfer modeling,

[00:01:06.930]
statics stress rain modeling, slope stability modeling,

[00:01:10.430]
and dynamics stress strain modeling with GeoStudio.

[00:01:15.810]
To begin this webinar, let us consider a soil specimen

[00:01:19.850]
subject to an experimental procedure,

[00:01:22.480]
and a numerical simulation

[00:01:24.160]
of that experimental procedure.

[00:01:26.320]
And in experimental approach,

[00:01:28.560]
there’s a real soil sample,

[00:01:30.330]
from which specimens can be extracted

[00:01:32.610]
and subjected to various tests,

[00:01:34.990]
such as triaxial and odometer loading.

[00:01:38.110]
The data collected from these tests is then interpreted

[00:01:41.880]
and the interpretation has to be done in a manner

[00:01:45.210]
that is specific to a particular constitutive of model.

[00:01:48.940]
The parametrization procedure

[00:01:51.200]
produces the models constants.

[00:01:54.550]
These constants are then used as inputs for the model.

[00:01:59.060]
The simulated results are obtained

[00:02:01.400]
by solving the physics subject to boundary conditions

[00:02:04.060]
and a constitutive law.

[00:02:05.540]
And finally, the simulated results are compared

[00:02:08.810]
to the corresponding laboratory results

[00:02:11.340]
to verify the parameterization.

[00:02:16.840]
In today’s webinar, we will review the key elements

[00:02:20.820]
of the hardening soil constitutive model.

[00:02:24.780]
Then, a step-by-step procedure will be presented

[00:02:27.750]
for parameterizing the model.

[00:02:30.160]
The procedure will then be applied to data

[00:02:32.310]
from numerous triaxial tests on sandy soil.

[00:02:36.050]
A demonstration of the steps from modeling a triaxial test

[00:02:39.027]
and SIGMA will be provided,

[00:02:40.820]
and the simulated and measured results will be compared.

[00:02:44.320]
Some concluding remarks will also be provided

[00:02:47.380]
at the end of the webinar.

[00:02:52.320]
The Hardening Soil Model was proposed by Schanz,

[00:02:55.630]
Vermeer and Bonnier in 1999.

[00:02:59.240]
It was formulated on the framework

[00:03:01.477]
of the Classical Theory of Plasticity.

[00:03:04.860]
The formulation comprises three stress-dependent

[00:03:07.930]
stiffness quantities and two yield surfaces.

[00:03:12.070]
One that resembles a hexagonal cone

[00:03:16.180]
and a second that resembles the tip of an ellipsoid.

[00:03:20.450]
The shear surface predominantly governs the response

[00:03:23.580]
to changes in deviatoric stress

[00:03:26.350]
while the cap predominantly governs

[00:03:28.200]
the response to increments in mean effective stress.

[00:03:32.080]
Failure is controlled by a Mohr-Coulomb failure law

[00:03:35.910]
hence the hexagonal shape of the yield surface.

[00:03:39.440]
The model implemented in Sigma

[00:03:41.910]
actually comprises a smoother cap

[00:03:44.440]
than what is depicted in this figure.

[00:03:47.550]
The reference book should be consulted

[00:03:49.620]
to understand the details of the formulation.

[00:03:56.040]
In the triaxial stress state,

[00:03:58.130]
the total stress path is a straight line

[00:04:00.940]
that reaches the Mohr-Coulomb envelope

[00:04:04.000]
out a specific failure point.

[00:04:06.740]
The deviatoric stress,

[00:04:08.620]
axial strain response of the soil

[00:04:11.020]
under this loading condition

[00:04:12.610]
is estimated by hyperbolic curve,

[00:04:15.560]
as will be discussed shortly.

[00:04:17.600]
The characteristics of this curve

[00:04:19.560]
are determined by three

[00:04:21.080]
stress-dependent stiffness quantities,

[00:04:23.550]
including the initial tangent stiffness,

[00:04:26.750]
E subscript I,

[00:04:28.540]
the half failure secant stiffness

[00:04:31.050]
E subscript 50

[00:04:32.720]
and an unloading reloading stiffness E subscript UR.

[00:04:40.700]
In order to have a closed yield surface,

[00:04:43.540]
a cap function is defined in the hardening soil model.

[00:04:47.540]
This cap has an elliptical shape

[00:04:49.890]
with an aspect ratio of M

[00:04:52.600]
or the Greek letter mu.

[00:04:54.730]
In the PQ stress base,

[00:04:56.840]
the stress state of any normally consolidated sample

[00:05:00.750]
is located on the cap.

[00:05:03.110]
That’s at the tip of this blue stress path

[00:05:05.920]
that intersects the ellipse.

[00:05:08.200]
The stress strain behavior of the sample

[00:05:10.860]
in an odometer test is defined

[00:05:12.960]
by a tension stiffness called the odometer stiffness

[00:05:16.090]
or E subscript OED.

[00:05:19.160]
Like the other stiffness quantities,

[00:05:21.340]
the odometer stiffness is not constant,

[00:05:24.120]
but it’s value depends on the stress of the specimen.

[00:05:30.810]
To review the key ingredients of the hardening soil model,

[00:05:34.030]
Let’s consider a theoretical triaxial response.

[00:05:38.920]
The axial strain of the sample in this condition

[00:05:41.860]
is to defined by a hyperbolic function.

[00:05:46.290]
The hyperbolic curve continues

[00:05:48.750]
until the Mohr-Coulomb failure stress is reached,

[00:05:51.710]
Q subscript F.

[00:05:54.260]
The failure occurs before the curve reaches

[00:05:56.870]
a asymptotic tonic value.

[00:05:59.060]
The ratio of the failure to asymptotic deviatoric stresses

[00:06:03.680]
is R subscript F.

[00:06:06.920]
The behavior of the sample at the beginning of loading

[00:06:10.040]
is modeled by the initial stiffness

[00:06:12.580]
E subscript I.

[00:06:14.920]
Since this quantity is difficult to measure,

[00:06:18.330]
its value is presented in terms of

[00:06:20.630]
the Half failure Secant stiffness,

[00:06:22.970]
E subscript 50.

[00:06:26.020]
The half failure secant stiffness

[00:06:28.350]
is a stress dependent stiffness

[00:06:30.580]
that depends on the minimum principle stress,

[00:06:33.220]
Sigma subscript three.

[00:06:35.360]
A similar stress dependency is also considered

[00:06:38.320]
in the Hardening soil model

[00:06:39.860]
for unloading reloading stiffness.

[00:06:42.870]
The model’s constants are now highlighted in gray.

[00:06:49.350]
In an odometer test on normally consolidated soil,

[00:06:53.180]
the slope of the stress path

[00:06:54.800]
is a function of the coefficient of lateral earth pressure,

[00:06:58.550]
K nought superscript NC.

[00:07:02.100]
Moreover, the cap yield surface and PQ space is an ellipse

[00:07:06.490]
with a horizontal major axes.

[00:07:09.180]
In a non cohesive soil,

[00:07:11.230]
the elliptical yield surface is centered around the origin.

[00:07:16.270]
The odometer stiffness is also a stress dependence stiffness

[00:07:20.150]
presented in terms of the vertical stress,

[00:07:22.330]
Sigma one in a one dimensional odometer tests.

[00:07:26.070]
Notice that the dependence on Sigma one

[00:07:28.630]
is different from the other moduli

[00:07:30.760]
that were dependent on Sigma three.

[00:07:33.120]
The model’s constants are once again

[00:07:34.990]
highlighted in light gray.

[00:07:42.150]
To summarize,

[00:07:43.320]
the hardening soil model requires

[00:07:45.610]
three stiffness Constants,

[00:07:48.030]
a reference stress,

[00:07:49.970]
and an exponent M.

[00:07:52.680]
The strength constants and failure ratio

[00:07:56.930]
are subscript F,

[00:07:59.280]
the coefficient of lateral earth pressure.

[00:08:02.130]
Some other parameters will also be needed

[00:08:04.480]
to describe the initial condition of a specific soil sample.

[00:08:08.730]
These include the initial void ratio,

[00:08:11.650]
the unit weight of the soil,

[00:08:13.720]
and an isotropic over consolidation ratio,

[00:08:16.840]
which we’ll touch on briefly in the subsequent slides.

[00:08:24.000]
We can now discuss a step-by-step parametrization procedure

[00:08:28.000]
for determining the Hardening soil model Constants

[00:08:30.520]
from a number of drain triaxial tests.

[00:08:35.130]
Step one involves determination

[00:08:37.600]
of the effective friction angle

[00:08:39.370]
and the effect of cohesion C.

[00:08:42.290]
The failure stress from numerous triaxial tests

[00:08:45.810]
at different confining pressures

[00:08:47.820]
are theoretically located

[00:08:49.800]
on the Mohr-Coulomb failure line in PQ space.

[00:08:53.670]
As such the slope of the line,

[00:08:56.350]
which is 10 beta,

[00:08:58.170]
as well as the Y intercept alpha,

[00:09:01.890]
is calculated using the method of least squares

[00:09:05.130]
where N is the number of data points.

[00:09:09.800]
Alternatively, the trendline functionality

[00:09:12.890]
and programs like Excel

[00:09:14.570]
can be used instead of the least squares formulas.

[00:09:18.800]
The effective friction angle and effective cohesion

[00:09:21.710]
of the soil can be found by changing the coordinate system

[00:09:25.380]
to a shear normal stress base.

[00:09:28.250]
The equations corresponding to that transformation

[00:09:31.430]
are shown here.

[00:09:36.880]
The second step of the procedure

[00:09:38.830]
is to consider a reference stress

[00:09:41.480]
and to estimate reference stiffnesses.

[00:09:44.680]
One of the available triaxial tests

[00:09:47.340]
is considered as a reference test.

[00:09:50.170]
The confining pressure of this test

[00:09:52.070]
is then called the reference stress.

[00:09:54.750]
The failure of deviatoric stress for this particular test

[00:09:58.060]
is determined based on Mohr-Coulomb strength constants

[00:10:01.920]
determined in the previous step.

[00:10:04.250]
The half failure referenced stiffness

[00:10:06.530]
is the slope of a straight line that passes through

[00:10:09.570]
the point carrying 50% of the failure deviatoric stress.

[00:10:14.640]
The least squared methods is applied again

[00:10:17.620]
on the unloading reloading branch

[00:10:19.610]
of the stress strain curve

[00:10:21.140]
to calculate E subscript UR.

[00:10:28.540]
In the third step of the procedure,

[00:10:30.900]
the stress dependency exponent M is estimated.

[00:10:35.100]
In stress strain curves of drain triaxial tests

[00:10:37.880]
with different confining pressures,

[00:10:39.910]
half failure stresses are first calculated

[00:10:43.110]
using the method described in step two.

[00:10:47.200]
A new natural log space can then be defined

[00:10:50.320]
that represents the half failure stiffnesses of each test

[00:10:55.340]
in the form of a single point.

[00:10:59.060]
The point corresponding to the reference test

[00:11:01.430]
is located on the origin.

[00:11:04.750]
The stress dependency exponent M

[00:11:07.220]
is the slope of a trend line

[00:11:09.280]
that passes through these points.

[00:11:11.790]
The similar steps can be taken for unloading reloading

[00:11:15.420]
in order to calculate

[00:11:16.620]
the stress dependency exponent twice.

[00:11:20.050]
In the hardening soil model however,

[00:11:22.290]
it is assumed that the constant M

[00:11:24.860]
is the same for both half failure

[00:11:27.140]
and unloading reloading stiffnesses.

[00:11:33.060]
The fourth step of the parameterization procedure

[00:11:35.960]
is to calculate the failure ratio, RF.

[00:11:39.410]
This constant has a value between 0.5 and one,

[00:11:43.320]
and is usually assumed to be 0.9.

[00:11:46.180]
A more accurate value for this constant can be calculated

[00:11:49.750]
based on the results of the triaxial tests.

[00:11:52.820]
First, the stress strain hyperbolic function

[00:11:55.960]
is rewritten in terms of the failure ratio,

[00:11:59.610]
R subscript F.

[00:12:03.600]
This expression introduces a new coordinate system

[00:12:07.270]
in which each hyperbolic curve

[00:12:09.440]
is projected to a straight line.

[00:12:12.540]
The failure ratio is the slope of these parallel lines.

[00:12:21.950]
In the second last step of the parametrization procedure,

[00:12:26.470]
the odometer reference stiffness

[00:12:29.260]
and the coefficient of lateral earth pressure are estimated.

[00:12:33.170]
The odometer test result is required for the step.

[00:12:36.420]
However, these values can be approximated

[00:12:39.350]
in the case where the appropriate laboratory result

[00:12:41.860]
is not available.

[00:12:43.330]
As mentioned earlier, the odometer reference stiffness

[00:12:46.460]
is defined in a one-dimensional compression test.

[00:12:49.470]
By definition, it is the tangent stiffness

[00:12:52.440]
at the point where the vertical stress

[00:12:54.230]
is equal to the reference stress.

[00:12:56.610]
In addition, the coefficient of lateral earth pressure

[00:12:59.610]
for a normally consolidated soil, K nought NC,

[00:13:03.120]
is the ratio of lateral stress, Sigma three,

[00:13:06.770]
to vertical stress, Sigma one, in the odometer test.

[00:13:11.640]
These two constants must be approximated

[00:13:14.520]
in the absence of laboratory data.

[00:13:17.010]
For instance, the value of the odometer reference stiffness

[00:13:20.390]
is usually greater than 50%

[00:13:23.130]
of E 50 rough and less than 80% of E 50 rough.

[00:13:29.490]
And also the coefficient of lateral earth pressure

[00:13:32.450]
can be estimated using the well-known Jackie’s formula,

[00:13:35.770]
which relies on the friction angle of the soil.

[00:13:41.960]
One additional step in the privatization procedure

[00:13:45.140]
is required if the soil is over consolidated,

[00:13:48.270]
and that is the determination of the isotropic over

[00:13:51.820]
consolidation ratio,

[00:13:53.510]
which is a ratio equivalent isotropic stresses.

[00:13:59.150]
The denominator of this expression

[00:14:02.000]
is simply the confining pressure of the triaxial test.

[00:14:06.440]
In contrast, the numerator, P subscript C,

[00:14:11.020]
defines the size of the yield locus

[00:14:14.180]
passing through the yields stresses QY and PY

[00:14:20.850]
As such, P subscript C

[00:14:23.110]
must be calculated from the equation

[00:14:25.250]
of the yield function.

[00:14:27.090]
The yields stresses are generally determined

[00:14:29.690]
from plots of deviatoric stress versus axial strain

[00:14:33.400]
and void ratio versus natural log mean effect of stress.

[00:14:37.130]
And in both of these plots, the yield point,

[00:14:40.320]
that is that transition from

[00:14:41.790]
elastic to elastic plastic behavior

[00:14:46.700]
it deduce by an inflection point

[00:14:49.480]
in the measured laboratory data.

[00:14:52.160]
That brings us to the constant mu,

[00:14:55.330]
which is in the equation for the cap yield function.

[00:15:01.020]
It is calculated by the material model in the solver

[00:15:04.640]
from this expression here.

[00:15:06.610]
We elaborate on this expression in the reference book,

[00:15:10.410]
but for the purpose of this webinar,

[00:15:13.610]
it suffices to say that theta is

[00:15:18.070]
Q/P stress ratio

[00:15:20.700]
that can be calculated from the user input,

[00:15:23.400]
K nought NC,

[00:15:24.470]
which we discussed on the previous slide

[00:15:26.930]
and lambda/Kappa is calculated from

[00:15:30.627]
failure’s ratio

[00:15:32.510]
and

[00:15:33.343]
this ratio of reference stiffnesses.

[00:15:38.020]
Now the entire procedure for determining OCR

[00:15:41.370]
can be simplified if the stress path is instead

[00:15:45.550]
that of isotropic compression, because in that case,

[00:15:48.860]
the stress path tracks along

[00:15:50.450]
the mean effect of stress axes.

[00:15:52.690]
Meaning that the deviatoric stress is zero

[00:15:56.010]
and PC is equal to PY.

[00:16:01.900]
To demonstrate the parametrization procedure

[00:16:04.850]
we are going to consider 12 drain triaxial compression tests

[00:16:08.660]
on Ottawa sand.

[00:16:10.920]
These tests were performed at different confining pressures

[00:16:13.810]
ranging from 50 to 600 kilopascals.

[00:16:23.680]
The effective friction angle and the cohesion of the soil

[00:16:26.520]
are calculated from the first step

[00:16:28.610]
of the parametrization procedure.

[00:16:31.650]
The deviatoric compression stress in 12 triaxial tests

[00:16:35.530]
continued until failure was achieved.

[00:16:38.360]
The failure state of each task can be represented

[00:16:41.070]
by a point in PQ space.

[00:16:43.940]
The failure line that passes through these failure points

[00:16:46.670]
has a slope of 1.18

[00:16:49.481]
and a Y intercept of zero.

[00:16:52.200]
Therefore the effect of friction angle

[00:16:55.448]
as calculated as 29.6 degrees

[00:16:59.860]
while the cohesion is calculated as zero,

[00:17:03.000]
since alpha is zero.

[00:17:09.000]
At the second step of the procedure,

[00:17:11.220]
the test with the confining stress of 100 kPA

[00:17:14.480]
is considered as the reference test.

[00:17:17.900]
The half failure deviatoric stress for this reference test

[00:17:21.680]
is calculated to be 97.6 kilopascals.

[00:17:27.220]
According to the measured stress strain curve,

[00:17:29.780]
the axial strain corresponded into this half failure

[00:17:32.800]
stress is equal to 0.0055.

[00:17:37.770]
This means that the half failure referenced stiffness

[00:17:41.190]
for the soil is 17,745 kilopascals.

[00:17:46.880]
Again, that is simply the slope of this line.

[00:17:51.060]
Finally, the unloading reloading branch

[00:17:54.740]
of the measured stress drain curve

[00:17:57.260]
gives us the value of the unloading reloading

[00:18:00.200]
reference stiffness.

[00:18:01.560]
And the slope of that line

[00:18:03.200]
is estimated to be 45,000 kilopascals.

[00:18:08.670]
The third step of the parametrization procedure

[00:18:11.700]
estimates the stress dependency exponent M.

[00:18:15.010]
As shown here this constant can be estimated

[00:18:18.420]
from either the unloading reloading

[00:18:20.580]
or the half failure stiffnesses in natural log,

[00:18:23.500]
natural log space.

[00:18:25.230]
The slope of these two lines is almost the same

[00:18:28.550]
and their average is 0.680.

[00:18:32.780]
And this is what we consider to be the constant M.

[00:18:42.180]
As can be seen here,

[00:18:43.990]
the proposed expressions can adequately estimate

[00:18:46.910]
the unloading reloading and half failure stiffnesses

[00:18:50.450]
in terms of their corresponding confining stress

[00:18:53.240]
Sigma three.

[00:18:54.620]
The proposed curve, however,

[00:18:56.530]
is more consistent with the measured results

[00:18:59.170]
for the unloading reloading stiffnesses

[00:19:01.570]
than for the half failure stiffnesses.

[00:19:04.020]
This is commonly the case.

[00:19:07.860]
The failure ratio, R subscript F

[00:19:10.540]
is calculated at the fourth step.

[00:19:13.120]
The linear projections of the hyperbolic curves

[00:19:15.840]
are shown in the suggested space in the left graph.

[00:19:19.980]
The near vertical branches of these curves

[00:19:22.480]
represent unloading reloading phases,

[00:19:25.110]
where Q approaches zero

[00:19:27.250]
causing the ratio of QF/Q

[00:19:30.230]
to approach infinity.

[00:19:32.120]
These near vertical branches are not included

[00:19:35.000]
in the calculation of RF,

[00:19:37.130]
rather it is the sloping portions

[00:19:39.500]
of the curves that are described

[00:19:41.180]
by the equation of a line

[00:19:42.950]
where the slope of each line is our subscript F.

[00:19:47.330]
The graph on the right shows the slopes

[00:19:49.840]
for all 12 triaxial tests.

[00:19:52.390]
The average of 0.941 is then estimated

[00:19:56.730]
for the failure ratio of the soil.

[00:20:00.170]
For the odometer related constant in step five,

[00:20:03.870]
we apply the approximate method

[00:20:05.810]
of calculating the odometer reference stiffness.

[00:20:09.040]
According to the value

[00:20:10.090]
of the half failure reference stiffness,

[00:20:12.380]
which was calculated at the second step,

[00:20:14.720]
we estimate the odometer reference stiffness to be 115,

[00:20:18.960]
excuse me, 11,500 kilopascals.

[00:20:23.220]
The Jackie’s formula is also used to calculate the

[00:20:26.090]
coefficient of lateral earth pressure

[00:20:27.970]
in terms of the effect of friction angle

[00:20:30.340]
and from that relationship,

[00:20:31.770]
we obtained 0.506.

[00:20:36.750]
The results the parametrization procedure

[00:20:39.010]
for the Ottawa sand can be summarized as follows.

[00:20:42.280]
Our two strength constants gave us a friction angle

[00:20:45.330]
of 29.6 and a cohesion of zero.

[00:20:48.400]
The reference dress was 100 kilopascals,

[00:20:50.980]
and the reference stiffnesses were calculated accordingly.

[00:20:55.280]
The stress dependency exponent and the failure ratio

[00:20:59.200]
were estimated as 0.68

[00:21:01.110]
and 0.941 respectively,

[00:21:04.150]
and the coefficient of lateral earth pressure

[00:21:06.440]
was obtained from Jackie’s formula

[00:21:07.910]
to be 0.506.

[00:21:12.490]
Now that we have reviewed the step-by-step parametrization

[00:21:15.470]
procedure for the Hardening soil model,

[00:21:17.730]
I will describe how to model a Triaxial test in Sigma,

[00:21:21.000]
and compare the numerical results

[00:21:22.940]
with the previously mentioned lab data.

[00:21:25.600]
The numerical model configuration

[00:21:27.820]
is based on the geometry and boundary conditions

[00:21:30.440]
of the triaxial test conducted in the laboratory.

[00:21:34.170]
A typical triaxial sample has a cylindrical shape

[00:21:37.890]
with a diameter of five centimeters

[00:21:40.050]
and a height of 10 centimeters.

[00:21:42.260]
Due to axis symmetric shape of this geometry

[00:21:45.100]
with respect to the central vertical axis,

[00:21:47.810]
we use a 2D axis symmetric geometry and Sigma.

[00:21:51.840]
The geometry is also symmetric

[00:21:53.750]
with respect to the horizontal axis.

[00:21:56.570]
So we can model the top one quarter of the specimen.

[00:22:01.350]
An eight nodes single element model

[00:22:03.780]
is considered for this resulting rectangle.

[00:22:10.690]
The material definition is of course the key subject

[00:22:14.780]
of this webinar.

[00:22:16.790]
This step is performed in the software

[00:22:20.160]
using the defined materials dialog box.

[00:22:24.630]
The material model is set to the Hardening soil model,

[00:22:28.780]
the sample parameters

[00:22:30.140]
including the initial void ratio

[00:22:31.960]
and over consolidation ratio are entered.

[00:22:35.040]
It should be noted that the effect of the soil self weight

[00:22:38.830]
can be removed by setting the unit weight to zero.

[00:22:42.270]
And lastly, the constitutive model constants

[00:22:45.150]
are entered based on what was previously obtained

[00:22:48.250]
during the parametrization procedure

[00:22:50.220]
for the Ottawa sand.

[00:22:54.090]
The triaxial tests are simulated

[00:22:57.070]
by firstly considering the isotropic confining pressure.

[00:23:03.200]
And secondly, the deviatoric pressure.

[00:23:06.550]
To do this, the project file is set up

[00:23:09.850]
with an analysis tree structure for each test.

[00:23:15.670]
In each branch,

[00:23:18.360]
we see the confining pressure phase

[00:23:21.100]
acting as a parent,

[00:23:23.210]
which provides initial data

[00:23:25.200]
to the subsequent displacement control deviatoric phase.

[00:23:30.610]
In test A one as an example,

[00:23:33.620]
a constant normal stress of 50 kPA

[00:23:36.910]
is applied on both the right and top sides of the model

[00:23:40.630]
using a normal tangential stress boundary condition.

[00:23:44.320]
And in the deviatoric phase of this test,

[00:23:47.010]
a displacement of two centimeters

[00:23:49.510]
that is equivalent to 40% strain,

[00:23:51.970]
is applied on the top side of the model.

[00:23:55.270]
The displacement spline data function

[00:23:58.020]
contains a small plip

[00:24:00.770]
that just before

[00:24:03.210]
a time and a lapse time

[00:24:05.560]
of 1.2,

[00:24:08.480]
and this is the unloading reloading phase.

[00:24:12.130]
Let’s just take a few minutes to explore

[00:24:14.150]
some of the finer details of these ostensibly simple

[00:24:18.440]
numerical analysis.

[00:24:20.380]
First, we’ll start at the top of the project definition.

[00:24:24.430]
And I will point out that you can right click on here

[00:24:27.560]
and add multiple geometries to the same project file.

[00:24:31.920]
In this analysis, of course we have only one single geometry

[00:24:35.850]
and it is of the type 2D Axis symmetric.

[00:24:39.950]
The next noteworthy thing under our geometry sits

[00:24:43.560]
our confining stress analysis

[00:24:46.070]
for all 12 triaxial tests

[00:24:49.420]
and then unlike what is shown

[00:24:51.060]
in the PowerPoint presentation,

[00:24:52.950]
we actually have two cases

[00:24:55.560]
for each deviatoric loading condition.

[00:24:58.820]
One that is monotonic,

[00:25:00.710]
meaning strained without unload reload,

[00:25:03.270]
and the other with the LUR or Load Unload Reload branch.

[00:25:08.910]
All three of the analysis in each branch

[00:25:13.070]
are low deformation.

[00:25:15.010]
Now the first analysis

[00:25:17.170]
is the one in which we apply the confining pressure

[00:25:20.460]
of 50 all the way through to 600 kilopascals.

[00:25:28.250]
And it is an analysis of the type load deformation.

[00:25:32.170]
We’re not doing an in-situ or gravity activation analysis

[00:25:36.010]
because

[00:25:37.160]
the

[00:25:38.460]
dimensions of the specimen

[00:25:41.140]
are so negligible

[00:25:43.340]
that the body load or the gravity load

[00:25:46.480]
doesn’t contribute to the stresses in any meaningful way.

[00:25:51.750]
The other thing that’s worth noting is that

[00:25:54.570]
our deviatoric phases

[00:25:58.180]
are also low deformation,

[00:26:00.320]
and the initial stresses and poor pressures

[00:26:02.130]
come from the parent analysis.

[00:26:04.780]
We also have the ability to set

[00:26:06.400]
the final pour water pressure conditions.

[00:26:08.420]
And you can see in this case,

[00:26:10.040]
it’s the same as the initial conditions,

[00:26:12.830]
meaning that there’s no increment or decrement

[00:26:15.230]
in the pour water pressure.

[00:26:16.860]
Had these been different due to, for example,

[00:26:20.000]
specifying the pour pressures

[00:26:21.450]
from another geostudio analysis like Sigma CAP

[00:26:24.670]
or the use of a water table or

[00:26:26.250]
special function to find the pressure heads,

[00:26:28.840]
then not change in pour water pressure

[00:26:30.620]
would be reflected

[00:26:32.230]
in the deformation response,

[00:26:34.550]
the simulated deformation response of the specimen.

[00:26:39.040]
I should also note while I’m here,

[00:26:40.950]
that there are these two options to reset

[00:26:43.110]
displacements and strains

[00:26:44.810]
and reset state variables.

[00:26:46.830]
These are not on automatically

[00:26:49.550]
when a low deformation analysis is added

[00:26:52.800]
to another low deformation analysis.

[00:26:59.560]
And so they were not set in this case.

[00:27:02.370]
Now that means that any of the deformations accumulated

[00:27:05.890]
during the isotropic compression

[00:27:08.400]
will carry through

[00:27:10.540]
into the deviatoric loading phase.

[00:27:14.430]
And so in actuality,

[00:27:16.470]
it would have been better modeling practice

[00:27:18.730]
to toggle the reset displacements and strains,

[00:27:22.280]
and these are accumulated values to toggle that on.

[00:27:25.720]
If you’re using a material model that has state

[00:27:29.020]
like the hardening soil model in the confining phase,

[00:27:31.970]
then you can also reset the state variables.

[00:27:34.800]
Now, in this case,

[00:27:35.950]
what we’ve done is we’ve used

[00:27:37.820]
an isotropic elastic material model

[00:27:42.720]
in the compression phase,

[00:27:44.080]
and we’ve given it a very high stiffness quantity

[00:27:47.210]
of 15 million kilopascals,

[00:27:49.620]
meaning that the simulated deformations

[00:27:52.500]
from this phase are negligible.

[00:27:55.730]
And therefore we need not reset

[00:27:58.780]
the accumulated displacements and strains.

[00:28:02.670]
The other point I’ll make is that

[00:28:05.210]
when you deal with large project files

[00:28:08.920]
containing multiple analysis,

[00:28:11.210]
there are right click options in the solve manager

[00:28:14.720]
to uncheck all,

[00:28:17.150]
to check children, which picks a single branch,

[00:28:21.480]
that’s the zero one branch.

[00:28:24.000]
And then there are a couple other options.

[00:28:26.070]
Similarly, if you go up into the solve manager,

[00:28:29.820]
it’s often more convenient

[00:28:31.610]
if these branches are quite elaborate,

[00:28:34.650]
in this case there are only two analysis

[00:28:38.940]
that are parenting to that,

[00:28:40.110]
the confining stress analysis.

[00:28:42.060]
But we can imagine a scenario

[00:28:43.570]
where they’re eight, 12, and so on analysis underneath

[00:28:47.712]
the parent analysis.

[00:28:50.050]
In which case right clicking up here

[00:28:52.440]
and queuing an analysis branch for solving

[00:28:56.820]
is more convenient.

[00:28:58.170]
You can also

[00:28:59.880]
end queuing,

[00:29:00.760]
you can clear, and you can also

[00:29:04.210]
queue one particular analysis.

[00:29:06.870]
Now, in this case,

[00:29:07.703]
I’m going to right click and

[00:29:10.670]
check

[00:29:12.890]
all analysis

[00:29:14.830]
and then solve.

[00:29:16.820]
And once the solution is obtained for any one of these,

[00:29:20.840]
we can switch to the results view.

[00:29:24.750]
And it is here where we can

[00:29:27.740]
go to draw graph

[00:29:30.320]
and

[00:29:31.500]
consider

[00:29:32.990]
the response through various graphs,

[00:29:36.710]
such as deviatoric stress versus axial strain,

[00:29:40.710]
deviatoric stress versus mean effective stress

[00:29:43.500]
and void ratio versus P.

[00:29:46.620]
You can see that there’s a unload branch in there.

[00:29:48.840]
There’s the monotonic loading case.

[00:29:51.490]
You can also multi-select these graphs,

[00:29:55.290]
which of course gives a rather strange view

[00:29:58.610]
since the axes are not all the same,

[00:30:01.280]
but I just wanted to point out

[00:30:02.580]
that you can copy this graph data

[00:30:04.870]
and then go to a spreadsheet program like Excel,

[00:30:08.890]
where you can then

[00:30:11.250]
right-click

[00:30:15.950]
and paste the data into the spreadsheet.

[00:30:18.840]
This will then make for easier comparison

[00:30:22.240]
with the measured laboratory data

[00:30:27.030]
If you turn out to the PowerPoint presentation,

[00:30:32.430]
the simulation results of six triaxial trust tests

[00:30:35.940]
with confining pressures ranging from 50 to 300 kilopascals

[00:30:40.090]
are compared here with laboratory results.

[00:30:43.440]
Colored curves are laboratory results.

[00:30:46.040]
And the black lines are the results

[00:30:47.610]
from the Sigma simulation

[00:30:49.280]
and analytical solution is also shown on these plots

[00:30:52.770]
as gray dash lines.

[00:30:54.770]
The full compatibility of the analytical

[00:30:57.560]
and simulated solutions verifies the implementation

[00:31:00.950]
of the hardening soil model in SIGMA/W.

[00:31:03.570]
In addition,

[00:31:04.540]
the acceptable consistency

[00:31:06.420]
between the lab data and the simulator response

[00:31:09.460]
confirms that the hardening soil model

[00:31:11.590]
with carefully parameterized material constants

[00:31:14.400]
can capture the stress strain response of the Ottawa sand.

[00:31:22.350]
The same trends can be observed for the next six tests

[00:31:25.670]
that had confining pressures ranging from

[00:31:27.660]
350 kilopascals to 600 kilopascals.

[00:31:37.610]
To conclude, the parametrization procedure

[00:31:41.140]
of the hardening soil model

[00:31:42.690]
was provided in five straightforward steps

[00:31:46.050]
and a sixth optional step for over consolidated soil.

[00:31:50.480]
It was shown that the results from

[00:31:52.110]
multiple drain triaxial tests

[00:31:54.550]
and at least one odometer tests

[00:31:57.020]
are required to parameterize

[00:31:58.680]
the hardening soil model.

[00:32:00.500]
In the absence of odometer data,

[00:32:02.730]
approximations can be used to obtain

[00:32:06.080]
the reference odometer stiffness

[00:32:09.090]
and the K nought NC

[00:32:12.250]
or normally compressed earth pressure coefficient inputs.

[00:32:19.630]
The material constants were then used

[00:32:21.950]
in numerical simulations of the lab tests

[00:32:24.400]
and the numerical results were shown

[00:32:26.970]
to compare favorably with both the laboratory data

[00:32:30.040]
and analytical solutions.

[00:32:33.290]
During the presentation,

[00:32:34.580]
I referenced these top two

[00:32:39.970]
articles.

[00:32:41.300]
Shanz, Vermeer and Bannier,

[00:32:42.730]
that’s the original model formulation.

[00:32:45.220]
I also want to remind you that you can consult

[00:32:47.850]
our stress strain modeling

[00:32:49.330]
with GeoStudio 2021 reference book,

[00:32:52.730]
to understand the details of the formulation.

[00:32:56.000]
And then this is the paper that comprises

[00:32:58.230]
the laboratory tests on the Ottawa sand.

[00:33:01.720]
I’ll also remind you that

[00:33:04.100]
the website comprises a number of resources,

[00:33:07.885]
including example files.

[00:33:10.950]
One noteworthy example is that of

[00:33:14.520]
the tieback wall in Berlin,

[00:33:18.050]
it is completed using both the Mohr-Coulomb

[00:33:22.090]
and hardening soil models

[00:33:23.650]
and those results are compared.

[00:33:27.290]
With that, we’ve reached the end of the webinar.

[00:33:29.620]
Our recording will be available to view online.

[00:33:32.470]
Please take time to complete the short survey

[00:33:34.690]
that appears on your screen

[00:33:36.320]
so that we know the types of webinars

[00:33:38.730]
that are of interest to you.

[00:33:40.470]
Thank you very much for joining us.

The video transcript gets copy and pasted here