SIGMA/W Material Model Series: Hardening Soil Model

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

Video Transcript:

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

[00:00:02.270]
Material Model webinar series

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on the Hardening Soil Model.

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My name is Curtis Kelln,

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I’m the Director of Research and Development

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with the GeoSlope Engineering team at Seequent.

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Today’s webinar will be approximately 30 minutes long.

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Attendees can ask questions using the chat feature

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and I will respond to these questions

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by email as quickly as possible.

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A recording of the webinar will be available,

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so that you can review the demonstration, at a later time.

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GeoStudio was developed for geotechnical engineers

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and earth scientists.

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As such, it comprises a range of products,

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that can be used independently

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or together to solve a wide array of problems.

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Today’s webinar will focus on SIGMA/W.

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Those looking to learn more about the products,

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including background theory, available features,

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and typical modeling scenarios,

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can find an extensive library of resources

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on the Geoslope website.

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Here, you can find a tutorial videos,

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examples with detailed explanations

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and reference books on heat and mass transfer modeling,

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statics stress rain modeling, slope stability modeling,

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and dynamics stress strain modeling with GeoStudio.

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To begin this webinar, let us consider a soil specimen

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subject to an experimental procedure,

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and a numerical simulation

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of that experimental procedure.

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And in experimental approach,

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there’s a real soil sample,

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from which specimens can be extracted

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and subjected to various tests,

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such as triaxial and odometer loading.

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The data collected from these tests is then interpreted

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and the interpretation has to be done in a manner

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that is specific to a particular constitutive of model.

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The parametrization procedure

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produces the models constants.

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These constants are then used as inputs for the model.

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The simulated results are obtained

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by solving the physics subject to boundary conditions

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and a constitutive law.

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And finally, the simulated results are compared

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to the corresponding laboratory results

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to verify the parameterization.

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In today’s webinar, we will review the key elements

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of the hardening soil constitutive model.

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Then, a step-by-step procedure will be presented

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for parameterizing the model.

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The procedure will then be applied to data

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from numerous triaxial tests on sandy soil.

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A demonstration of the steps from modeling a triaxial test

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and SIGMA will be provided,

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and the simulated and measured results will be compared.

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Some concluding remarks will also be provided

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at the end of the webinar.

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The Hardening Soil Model was proposed by Schanz,

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Vermeer and Bonnier in 1999.

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It was formulated on the framework

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of the Classical Theory of Plasticity.

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The formulation comprises three stress-dependent

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stiffness quantities and two yield surfaces.

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One that resembles a hexagonal cone

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and a second that resembles the tip of an ellipsoid.

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The shear surface predominantly governs the response

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to changes in deviatoric stress

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while the cap predominantly governs

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the response to increments in mean effective stress.

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Failure is controlled by a Mohr-Coulomb failure law

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hence the hexagonal shape of the yield surface.

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The model implemented in Sigma

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actually comprises a smoother cap

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than what is depicted in this figure.

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The reference book should be consulted

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to understand the details of the formulation.

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In the triaxial stress state,

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the total stress path is a straight line

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that reaches the Mohr-Coulomb envelope

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out a specific failure point.

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The deviatoric stress,

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axial strain response of the soil

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under this loading condition

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is estimated by hyperbolic curve,

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as will be discussed shortly.

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The characteristics of this curve

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are determined by three

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stress-dependent stiffness quantities,

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including the initial tangent stiffness,

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E subscript I,

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the half failure secant stiffness

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E subscript 50

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and an unloading reloading stiffness E subscript UR.

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In order to have a closed yield surface,

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a cap function is defined in the hardening soil model.

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This cap has an elliptical shape

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with an aspect ratio of M

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or the Greek letter mu.

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In the PQ stress base,

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the stress state of any normally consolidated sample

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is located on the cap.

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That’s at the tip of this blue stress path

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

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The stress strain behavior of the sample

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

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by a tension stiffness called the odometer stiffness

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or E subscript OED.

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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.

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To review the key ingredients of the hardening soil model,

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Let’s consider a theoretical triaxial response.

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The axial strain of the sample in this condition

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is to defined by a hyperbolic function.

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The hyperbolic curve continues

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until the Mohr-Coulomb failure stress is reached,

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Q subscript F.

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The failure occurs before the curve reaches

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

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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

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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.

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The half failure secant stiffness

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

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that depends on the minimum principle stress,

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Sigma subscript three.

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A similar stress dependency is also considered

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

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for unloading reloading stiffness.

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The model’s constants are now highlighted in gray.

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In an odometer test on normally consolidated soil,

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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.

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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,

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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.

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The model’s constants are once again

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

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To summarize,

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the hardening soil model requires

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three stiffness Constants,

[00:07:48.030]
a reference stress,

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

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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

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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.

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We can now discuss a step-by-step parametrization procedure

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for determining the Hardening soil model Constants

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

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Step one involves determination

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

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

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The failure stress from numerous triaxial tests

[00:08:45.810]
at different confining pressures

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are theoretically located

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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.

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The equations corresponding to that transformation

[00:09:31.430]
are shown here.

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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