Benefits of Collaboration between Centrifuge Modeling and Numerical Modeling
Xiangwu Zeng
Case Western Reserve University, Cleveland, Ohio
ABSTRACT
There is little doubt that collaboration between centrifuge modeling and numerical
modeling has mutual benefit. A quantitative analysis of the experimental data from most
earthquake centrifuge tests needs numerical procedures. Development of numerical
procedures needs experimental data for verification purpose and data from centrifuge
tests is probably the most frequently used one.
In addition to the abovementioned benefits, collaboration between centrifuge
modeling and numerical modeling can help us to understand the techniques used in
modeling and in some cases even improve the techniques. This paper uses two examples
to demonstrate how carefully planned collaboration can benefit each other. The first
example shows that numerical simulation can help us to quantify the influence of
variation of centrifugal acceleration and model container size on accuracy of centrifuge
test. The second example shows how development in centrifuge modeling technique can
determine some of the previous unknown soil parameters that have significant influence
on the results of numerical prediction.
For centrifuge modeling, the variation of centrifugal acceleration in a model and
the boundary conditions imposed by a model container have significant influence on the
accuracy of test results. It is difficult to determine the effect quantitatively by doing tests
because it is impossible to prepare identical models. On the other hand, in numerical
simulation, identical soil parameters and initial conditions are easy to achieve. Thus, the
results of numerical simulation can be used as a yardstick to determine the effect,
provided that the numerical code has been verified beforehand. For earthquake centrifuge
tests, the effect of model containers is particularly important and more research is needed.
For numerical simulation of earthquake problems, soil parameters (such as Gmax
and K0) that determine initial stress and strain conditions are important. However, in the
past, these parameters are not measured in centrifuge tests and in particular not measured
during the flight of a centrifuge model. Numerical modelers have to guess these
parameters in their prediction and, in the example presented here, a wide range of values
was used. It was not surprising that the results were not satisfactory. In addition, it is
impossible to determine how much the difference in these soil parameters contributed to
the difference in the results of numerical prediction. The recent development of new
measuring devices that can be operated during the flight of a centrifuge can make
valuable contribution to the improvement of numerical simulation by making accurate
measurement of these parameters during the flight of a model.
INTRODUCTION
There has been considerable progress made in both centrifuge modeling and
numerical modeling. For centrifuge modeling, larger and more powerful shakers have
been developed. New types of transducers and better data acquisition systems allow us to
get more and better quality data in model tests. For numerical simulation, a few effectivestress based fully-coupled numerical codes with new constitutive models have been
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developed. Together with the enormous increase in computational power provided by
new generation of computers, numerical simulation of complex problems in geotechnical
earthquake engineering is now achievable at a reasonable cost.
Both centrifuge modeling and numerical modeling have become powerful
research tools in geotechnical earthquake engineering. However, each has its own
limitations. As an experimental technique, centrifuge modeling has inherited inaccuracies
arising from factors such as boundary conditions imposed by model container, inability to
satisfy all the scaling laws in certain situations, and system limitations of equipment,
transducers, and data acquisition systems. The application of the centrifuge modeling
technique and the accuracy of testing data depend critically on how well the effects of
these problems are understood and addressed. In recent years, a significant number of
research projects have been conducted to study problems of transducer response (Kutter
et al, 1990, and Lee, 1990), boundary effects in earthquake centrifuge tests (Hushmand,
et al, 1988, and Zeng and Schofield, 1996), and the use of viscous pore fluids (Zeng et al,
1998, Ko and Dewoolkar, 1999). For numerical simulation, verification of a code is a
complex process. There are many factors, such as soil parameters, constitutive model,
and numerical methods used, which can influence the final results. It would be very
useful for the verification process if the influence of or uncertainties associated with
some of the factors can be determined or eliminated.
NUMERICAL
SIMULATION
HELPS
THE
UNDERSTANDING
OF
CENTRIFUGE MODELING
One of the major differences between a centrifuge model and the corresponding
prototype is the variation of centrifugal acceleration in both the magnitude and direction.
For a soil layer in the prototype and its corresponding centrifuge model shown in Fig.1,
there are obvious differences in the stress field. For a typical geotechnical problem in the
field, the gravitational acceleration everywhere can be considered heading vertically
down and has a constant magnitude. On the other hand, in a centrifuge model, the
magnitude of centrifugal acceleration increases with radius and the direction is always in
the radial direction. Therefore, centrifugal acceleration varies from point to point. In
addition, artificial boundaries (usually rigid end walls) are imposed by model containers,
which would further affect the stress distribution in the model. The end walls can be
smooth or rough and in each case, it will cause stress condition different from that in the
field. The effect of the variation of centrifugal acceleration on vertical stress at the center
of a model was analyzed by Schofield (1980) using a one-dimensional approach. It was
found that the stress in the upper part of a centrifuge model would typically be lower than
that in the corresponding prototype while at the bottom of a model, the stress would be
higher than its counterpart in the field. It was shown that if the depth of a model is onetenth of the radius of a centrifuge, the magnitude of vertical stress error is under ± 2%
(+2% at bottom, -2% near the top). However, in reality the problem is at least twodimensional and the boundary conditions imposed by the model container are a further
source for inaccuracy. Apart from that, accuracy in the simulation of horizontal stress and
shear stress is also very important. Thus a more comprehensive analysis which can take
into account these differences is needed. Here the results of a numerical simulation of this
problem using a finite element program is presented.
2
soil
σv '
σh '
gravitational acceleration g
a) Gravitational acceleration and the resulting stresses in a prototype
model container
centrifugal acceleration
soil
b) Centrifugal acceleration in a model
Fig. 1 A soil layer in prototype and its corresponding centrifuge model
Methodology
Ideally, the best way to investigate the influence of the non-uniform centrifugal
acceleration on the accuracy of a centrifuge test is to compare directly the stresses and
other properties measured in the field to those measured in the corresponding centrifuge
model. However, in reality, a number of conditions make such a comparison impossible.
First, soils in the field are most likely to be anisotropic and inhomogeneous. These field
conditions are very difficult, if not impossible, to be replicated in a small-scale model.
Second, there will be some discrepancies between the properties of soil in the model and
those in the field, which would contribute to some of the differences in the results. Last
but not least, there will be experimental inaccuracies arising from instrumentation and
operation of devices used both in the field and centrifuge tests. Therefore, it is impossible
to identify how much of the difference is due to each of the contributing factors.
An alternative approach is to apply an analytical method to this problem assuming
that all the soil properties are identical in the prototype and the model. Thus it can
exclude the influence of other factors so as to investigate the inaccuracy caused
exclusively by the variation of centrifugal acceleration and the artificial boundary
conditions imposed by model containers. To simplify the analysis, soils in both the
prototype and the model are assumed to be isotropic and homogeneous. For the stress
field in the prototype, assuming that the uniform soil layer is of infinite lateral extent,
vertical and horizontal effective stresses can be determined by hand calculation. On the
other hand, the stress field in the model is a complex two-dimensional problem, which
can be solved by using a finite element code. Of course, the finite element code used has
to be one that is well established. For this project, the finite element code used is the
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SIGMA/W developed by Geo-Slope International (1995). Details of this study are
reported by Zeng and Lim (2002) and a summary is presented here.
The accuracy of a finite element simulation is affected by the number of elements
and the type of elements used. In this study, 8 noded quadrilateral elements with
integration order of 9 were used. A simple problem of a soil layer of 10 m depth with an
infinite lateral extent was simulated using a varying number of elements. It was found
that when 50 (10×5) elements were used, the difference in stresses between the analytical
solution and the numerical simulation was less than 1%. Therefore, in the following
study, a finite element mesh with 50 elements was adopted.
Simulation of Stresses in a Soil Layer Induced by Self-Weight
In the first attempt to investigate the influence of the variation of centrifugal
acceleration on the accuracy of centrifuge modeling, the stresses induced by self-weight
of the soil in a horizontal soil layer of infinite lateral extent is calculated and compared to
the stresses in the corresponding centrifuge model. The soil layer in the prototype is 10
meters thick and has a saturated and buoyant unit weight of 19.81 kN/m3 and 10 kN/m3,
respectively. The water table is at the surface of the layer. The test is assumed to be
conducted at a centrifugal acceleration of 50g at the center of the model and hence the
thickness of the soil layer in the model is 20 cm. The study will be concentrated on the
influence of the radius of the centrifuge R (defined as the distance from the rotating
center of the centrifuge to the center of the model) and the width of the model container
B. To illustrate the variation of centrifugal acceleration in a centrifuge model, the models
of the 10 m soil layer when tested on centrifuges with radii of 1m and 4m, respectively,
and a model container 0.6m wide are shown in Fig. 2. Clearly, the variation in centrifugal
acceleration in the model is quite significant for the case of a small centrifuge.
Two types of constitutive models were used in the finite element simulation:
linearly elastic and the original Cam-Clay models. Two parameters (Young’s modulus E
and Poisson’s ratio ν) are required in a linearly elastic model and the values used in this
study are E = 30 MPa and ν = 0.333 (which would result in a K0 = 0.5). For the original
Cam-Clay model, soil parameters used are: λ = 0.193, κ = 0.047, Μ = 1.2, G’ = 2.4 MPa,
and ν = 0.333. When the linearly elastic model is used, the stress calculation is achieved
in one load increment. On the other hand, for a non-linear model such as the Cam-Clay, a
small load increment needs to be used in order to achieve accurate results. Also the first
initial stress state has to be given or calculated by other procedures. In this study, stress
calculation when using the Cam-Clay model was finished in 50 load steps. For the stress
calculation in the field, the first step uses a linearly elastic model and a unit weight of
10/50 = 0.2 kN/m3. Then the next 49 steps use the Cam-Clay model and each step has the
same unit weight increase of 0.2 kN/m3 with the stresses from the previous step used as
the initial stresses. For the stress calculation in the centrifuge model, the initial stresses
due to 1g gravity are calculated by the linearly elastic model. Using this as the initial
stress state, the stresses at 2g are calculated using Cam-Clay model with the body forces
increased by an amount corresponding to an increase of centrifuge acceleration of 1g at
the center of the model. This procedure is repeated until the final body force is increased
to that corresponding to a centrifugal acceleration of 50g at the center of the model.
4
1.0 m
18.4
o
0.3m
0.3m
45g
47.4g
50g
15.2
0.2m
o
55g
57g
a) Centrifuge model with R = 1m, B = 0.6m
4.0 m
4.4
o
48.9g
0.3m
48.8g
50g
4.2o
0.3m
0.2m
51.3g
51.4g
b) Centrifuge model with R = 4m, B = 0.6m
Fig. 2 Variation of centrifuge acceleration in models of a 10 m soil layer
Simulation of Centrifugal Forces
The effect of the centrifugal acceleration is simulated by applying body forces on
to each element. The body forces have both the X and Y components as shown in Fig. 3.
Supposing the center of element i has a coordinate of Xi and Yi, the centrifugal
acceleration at this point would be
ai = ω2 √(Yi2 + Xi2)
where ω = angular velocity of the centrifuge. The resulting body forces per unit volume
of this element in x and y directions are
fxi = ρω2Xi
fyi = ρω2Yi
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where ρ = submerged mass density of soil. Obviously, variation in body forces can be
quite significant between elements near the center and elements at the four corners.
Rotation Center
X
Y
Yi
R
element i
f xi
f yi
Xi
B
Fig. 3 Body forces induced by centrifugal acceleration
Results from Linearly Elastic Model
For a soil layer that is 10 meter thick in the prototype and has a saturated and
buoyant unit weight of 19.81 kN/m3 and 10 kN/m3, respectively, the vertical and
horizontal normal stress distributions are linearly increase with depth for the linearly
elastic soil. There is no shear stress on the horizontal plane or vertical plane. When the
same soil layer is simulated in a model container 0.6m wide with rigid and smooth end
walls on a centrifuge with a radius of 1m, the stress distributions are shown in Fig. 4. The
vertical stress is well simulated throughout the model but there are some differences in
the horizontal stress in the model in comparison to the stress in the prototype, especially
near the end walls. In addition, there is a small shear stress in the horizontal and vertical
planes in the model.
As the radius of the centrifuge increases, the difference in the horizontal stress
drops quickly and the difference in the vertical stress become even less. For example, if
the radius of the centrifuge is increased to 4m with the size of the model container
remaining at 0.6m, the difference between the horizontal stress in the model and
prototype becomes much smaller and may be neglected for most purposes, as shown in
Fig. 5.
Results from Original Cam-Clay Model
When the same layer is simulated on a centrifuge with a radius of 1m and a model
container 0.6m wide using the Cam-clay model, the stress distributions are shown in Fig.
6. Again, the vertical stress is well simulated but there is noticeable difference in
horizontal stress even near the centerline.
6
Fig. 4. Effective stresses for a soil layer in a centrifuge model (R = 1m, B = 0.6m)
Fig. 5 Horizontal effective stress in the centrifuge model (R = 4m, B = 0.6m)
7
Fig. 6 Effective stresses in the centrifuge model (R = 1m, B = 0.6m)
The difference in horizontal stress distribution still exists even when the radius of
the centrifuge is increased to 4m. On the other hand, if the width of the model container
is increased to 0.8m, the difference in horizontal stress distribution is significantly
reduced, suggesting that boundary effect of the end walls plays a more important role for
this type of situation.
An Example Problem
In order to evaluate the effect of difference in stress field between the model and
the prototype on test results, the settlement of a 6 m wide flexible strip footing subject to
100 kPa surface loading is analyzed. The footing is founded on a 10 m thick normally
consolidated (by self weight) and soft clay layer and the original Cam-Clay model is used
in the finite element calculation. The load is applied in 50 steps. Assuming the same
structure is simulated in centrifuge tests with different centrifuge radii and model
8
container width, the settlement and stress conditions in each model are also calculated
using the approaches mentioned earlier in this paper.
For the problem in the prototype, the settlement of the footing and the foundation
soil is shown in Fig. 7. The maximum settlement of the footing is 1.288 m. If the model
test is conducted on a centrifuge with a radius of 1m and a model container 0.6m wide,
the maximum settlement is increased to 1.448 m, resulting in a 12.4% difference. The
magnitude of the difference may not be significant from an engineering standpoint but it
is a concern for a numerical simulation. The stress distribution in the foundation looks
similar but there are some differences in the magnitude of stresses. However, when the
radius of the centrifuge is increased to 4m, the difference between the results of the
prototype and the centrifuge model becomes negligible. The maximum settlement of the
footing is now 1.256m, only a 2.5% difference compared to the prototype. The stress
distributions and their magnitudes in the foundation in the model are similar to that in the
prototype. Several other simulations were conducted and the results are shown in Table 1.
For this particular problem, it seems that a centrifuge radius of 2m will produce
satisfactory accuracy.
Fig. 7 Settlement of a foundation in prototype (maximum settlement 1.288m)
It is important to point out that the influence of the centrifugal acceleration, the
radius of the centrifuge, and the size of model container vary from problem to problem
and from parameter to parameter measured. For the example footing problem, if a
centrifuge has a radius of 2m, the settlement of the footing would be simulated with
satisfactory accuracy. For other types of problems, the requirement may be different.
Also for the examples here, it is assumed that the tests are conducted at 50g. For tests
conducted at other centrifugal accelerations, the results may be slightly different for
analyses using the Cam-Clay model. Therefore, it is highly recommended that analyses
similar to what are reported in this paper be conducted to determine a suitable size of the
centrifuge and model container or at least determine the range of accuracy of the
experimental data due to the imperfect simulation of gravitational acceleration on a
centrifuge and the boundary conditions imposed by a model container in a centrifuge test.
9
Table 1 Maximum Settlement of a Footing by Different Tests
Test Case
Prototype
R = 1m, B = 0.6m
R = 1m, B = 0.8m
R = 2m, B = 0.6m
R = 4m, B = 0.6m
Settlement (m)
1.288
1.448
1.497
1.326
1.256
Diffenernce (%)
-12.4
16.2
2.9
2.5
DEVELOPMENT OF CENTRIFUGE MODELING TECHNIQUES ENHANCES
NUMERICAL SIMULATION
A numerical simulation is critically dependent on the input parameters used. If the
parameters used in a constitutive model can be accurately measured during the flight of a
centrifuge model, it can enhance a numerical simulation through the elimination of
uncertainties associated with input parameters. Some latest development in
instrumentation for earthquake centrifuge modeling can provide valuable information
about soil properties during the flight of a centrifuge and thus, improve the usefulness of
centrifuge test data, which in return, benefit the development of numerical simulation as
data from centrifuge tests are the ones that most frequently used for the verification of
numerical codes.
For example, the initial stress and strain state of soil is very important in the
analysis of soil liquefaction. Therefore, accurate measurement of parameters such as Gmax
and K0 can play a critical role in the numerical simulation. In the past, these parameters
are not measured in a centrifuge model during the flight of a centrifuge. Numerical
predictors have to assume the values of these parameters based on empirical formula or
experience, creating uncertainties in the numerical prediction because it is unknown how
much these estimations will contribute to the difference between numerical simulation
and experimental data. For example, during the VELACS project, numerical predictors
used a wide range of values for Gmax and K0 for Nevada sand as shown in Table 2. It
means that the initial stress state was quite different between different predictors. For
instance, the corresponding initial stress state when the vertical effective stress is 100 kPa
while K0 are 0.70 or 0.36 (the maximum and minimum value used by the predictors) is
shown in Fig. 8. Using an analogy to triaxial tests, if we had two soil samples made of the
same soil but with such different initial stress states, we would expect quite different
response of these samples to external loading applied. Therefore, it was not surprising
that there were large discrepancies between the results of numerical predictions.
Moreover, it is not possible to identify the cause of these discrepancies because of the
differences in the initial stress and strain state.
This particular problem was not caused by either numerical predictors (since they
did not have these values provided by centrifuge modelers) or centrifuge modelers (since
they were not asked to provide such parameters, or even if asked, could have not been
able to do so because technology of measuring these parameters during the flight of a
centrifuge model was not available at that time). It was the consequence of insufficient
communication between numerical modelers and centrifuge modelers in a joint research
project.
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A new technique that can carry out the measurement of Gmax and K0 of soils in a
centrifuge model during the flight of a centrifuge is being developed at the Case Western
Reserve University using bender element technique. The project is sponsored by the
National Science Foundation and is expected to be finished in about a year. With accurate
measurement of Gmax and K0, numerical predictors can get the initial stress and strain
state correct before carrying out dynamic simulation, thus excluding one of the
uncertainties in the prediction.
Table 2 Parameters of Nevada Sand used by predictors in VELACS Project
n in Gmax = Kσn
1.0
Predictor
Anandarajah
Aubry
Bardet
Been
Chan
Iai
Ishihara
Kimura
Lacy
Li
Prevost
Roth
Shiomi
Siddharthan
K0
0.55
0.5
0.45
0.7
0.4
0.36
0.5
0.5
0.5
0.52
0.5
0.5
0.5
0.7
0.5
0.5
0.51
0.6
0.40
σv’ = 100 kPa
σv’ = 100 kPa
σh’ = 70 kPa
K0 = 0.7
σh’ = 36 kPa
K0 = 0.36
Fig. 8 Difference in initial stress state with different K0 value
CONCLUSIONS
In conclusion, it is obvious that collaboration between numerical modeling and
centrifuge modeling can bring mutual benefits. For earthquake centrifuge modeling, there
are a few problems related to modeling technique that can benefit from numerical
simulation such as the influence of boundary conditions of different types of model
11
container, the effect of using a viscous fluid, and the effect of smooth or frictional end
walls. For numerical modeling, development of in-flight measuring device and better
instrumentation can provide more and better data for verification purpose. In order to
maximize the benefits, the collaboration needs to start from the planning stage.
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