INTRODUCTION
Compacted aggregate-clay mixtures are currently successfully used as
the cores of embankment dams. These materials, called composite clays
by Jafari and Shafiee (2004), are usually broadly graded and are composed
of clay as the main body with sand, gravel, cobble or even boulders floating
in the clay matrix. Karkheh and Gotvand dams in Iran are some recent examples
of dams with cores composed of aggregate-clay mixtures.
It is also a current practice to employ the mixture of high plastic clay
with aggregates as impervious blankets in waste disposal projects (Lee
and Shackelford, 2005). It is generally assumed that the coarser fraction
of such soils imparts a relatively higher shear strength and high compacted
density while the permeability of the soil is governed by the proportion
and index properties of the finer fraction (Shafiee, 2008).
Yin (1999) experimentally investigated the behavior of Hong Kong marine
deposits with different clay contents. Test results revealed that the
friction angle of deposits decreases with an increase in plasticity index.
Young`s modulus also increases with an increase in effective confining
pressure but decreases with an increase in clay content. Vallejo and Mawby
(2000) carried out a series of direct shear tests and porosity measurements
on sand-clay mixtures. It was found that the percentage of sand in the
mixtures had a marked influence on their shear strength. It was determined
that when the concentration by weight of the sand in the mixtures was
more than 75%, the shear strength of the mixtures was governed mainly
by the frictional resistance between the sand grains. When the concentration
of sand varied between 75 and 40%, the shear strength of the mixture was
provided in part by the shear strength of the clay and in part by the
frictional resistance between the sand grains. When the sand concentration
was less than 40% by weight, the shear strength of the mixture was entirely
dictated by strength of clay.
Muir Wood and Kumar (2000) conducted drained and undrained triaxial compression
tests on isotropically normally consolidated and overconsolidated mixtures
of kaolin clay and coarse uniform sand. It was found that the deviator
stress, clay volumetric strain and pore pressure were unaffected by the
presence of the sand until the clay content fell below 40%. Jafari and
Shafiee (2004) carried out a series of strain-controlled monotonic and
cyclic triaxial tests on gravel-kaolin and sand-kaolin mixtures to investigate
the effects of aggregate on the mechanical behavior of the mixtures. Compression
monotonic test results revealed that the angle of shearing resistance
increases with aggregate content. Prakasha and Chandrasekaran (2005) conducted
an experimental study on reconstituted Indian marine soils having different
proportions of sand and clay. Test results revealed that sand grains in
clay leads to reduction in void ratio and increase in friction and pore
pressure response resulting in decrease in undrained shear strength. Similarly,
the inclusion of clay in sand leads to decrease in void ratio but imparts
metastable character to the structure of sand and marks the sand behave
in a manner consistent with a loose structure. All mixtures that fail
during cycling considered in the study exhibit flow liquefaction and the
soil with 90% sand content at low cyclic stress level exhibits limited
liquefaction due to phase transformation.
A review of the published literature in monotonic loading reveals that,
in general shear strength either increases with aggregate content or remains
constant until a limiting aggregate content, then increases as the aggregate
content increases. On the other hand, although expensive methods such
as full scale and physical model tests can be conducted for evaluating
behavior of composite clays with oversized particles, in many cases element
tests are accomplished on materials of modified gradation. This modification
in engineering projects can even be reduced to conduct tests only on pure
clay by extracting the oversized particles. Published literature in monotonic
loading clearly shows that the modification is a conservative estimation
of the mechanical behavior of the mixture. However, Jafari and Shafiee
(1998) showed that in the case of cyclic undrained loading on compacted
aggregate-clay mixtures, the assumption that adding aggregate to pure
clay improves its mechanical properties is questionable. They showed that
the mixtures containing 50% sandy gravel and 50% high plastic clay, failed
in a lower number of loading cycles and consequently exhibited less cyclic
strength (defined as the ratio of deviator stress to initial confining
stress causing 5% axial strain) than pure clays. Further comprehensive
investigations by Jafari and Shafiee (2004) confirmed their previous study.
It was concluded that when aggregate content is raised, excess pore pressure
is also increased and consequently cyclic shear strength would decrease.
This shows the need for understanding of different features of aggregate-clay
mixtures behavior under cyclic undrained loading.
This research is composed of two parts: in the first part the cyclic
deformation properties (i.e., shear modulus, damping ratio and pore pressure)
of aggregate-clay mixtures tested previously by Jafari and Shafiee (2004)
are thoroughly investigated and in the second part Artificial Neural Networks
(ANNs) are used to examine the possibility of modeling the cyclic deformation
properties of the heterogeneous materials.
MATERIALS AND METHODS
Materials tested: Pure clay with six mixtures of gravel-clay and
sand-clay were used in this study. The physical properties of the materials
were measured just prior to the beginning of the shear tests. Commercial
kaolin clay was selected as the cohesive part. The kaolin had a specific
gravity of 2.74 and its plasticity index was 38%. The X-ray diffraction
analysis revealed that the clay was mainly composed of montmorillonite.
The X-ray diffraction analysis was performed in the Iranian Geological
Survey. The grain-size distribution curve for the clay is shown in Fig.
1. The aggregates that were mixed with the clay were retrieved from
riverbed and composed of subrounded particles with a specific gravity
of 2.66. Figure 1 shows the grain-size distribution
curve of the parent granular material from which two types of aggregates,
i.e., coarse sand and fine gravel were sieved. The aggregates passing
through the 2.0 mm sieve and retained on the 1.68 mm sieve, with a minimum
and maximum void ratio of 0.655 and 0.901, respectively, were used as
the sand. The aggregates passing through the 6.35 mm sieve and retained
on the 4.7 mm sieve with a minimum and maximum void ratio of 0.639 and
0.896, respectively were used as the gravel. The gap graded gradation
was considered for the aggregates in order to focus on the impact of aggregate
size on the mechanical behavior while minimizing the effect of particle
size distribution.
Kaolin was mixed with respective amounts of sand and gravel to obtain
various mixtures. The seven samples of aggregate-kaolin were mixed in
volumetric proportion and named as K100, K80S, K80G, K60S, K60G, K40S
and K40G, where the first letter is an abbreviation of Kaolin, the second
number is the volumetric clay percent in the mixture and the third letter
indicates the type of aggregate in the mixture (S stands for Sand and
G for Gravel). A minimum of 40% clay content was considered since this
is a limit value for materials used as cores in embankment dams.
|
| Fig. 1: |
Grain-size distribution for the soils used in the study |
Specimen preparation: The specimen preparation technique was chosen
in a manner to model as precisely as possible the in situ conditions
of core materials of embankment dams. All the specimens, typically 50
mm in diameter and 100 mm in height were prepared, with a dry density
of 95% of the maximum dry density obtained from the standard compaction
test method and water content of 2% wet of optimum. Table 1 shows the
specimens initial dry density and water content.
The appropriate amounts of kaolin and aggregate for each layer were first
thoroughly mixed. Each layer was then mixed with water at least 24 h before
use and sealed. The material was poured in 6 layers into a cylindrical
mold and compacted. To achieve a greater uniformity of specimens, a procedure
similar to undercompaction technique (Ladd, 1978) was used. For each layer,
the compactive effort was increased towards the top by increasing the
number of blows per layer. Each layer was then scored after it was compacted
for better bonding with the next layer.
To reduce the effect of cap friction during the triaxial test, two thin
rubber sheets coated with silicone grease were placed between the lower
and upper porous stones and the specimen. Further, the sheets in contact
with the specimen were divided into four sectors. This was done to let
the specimen deform more easily in lateral directions. Five drainage holes
with a diameter of about 5 mm were also provided in the rubber sheets
to facilitate the saturation and consolidation process. The specimen preparation
technique was verified when repeated testing of similar specimens yielded
consistent results.
Test procedure: The specimens were saturated with a Skempton B-value
in excess of 95%. To facilitate saturation process Carbon Dioxide (CO2)
was first percolated through the specimens (this was more effective for
saturation of the low clay content specimens), then de-aired water flushed
into the specimens.
| Table 1: |
Specimens properties of initial dry density our water content |
 |
| Table 2: |
Summary of testing program on each specimen |
 |
Finally a back pressure of 150 kPa was incrementally
applied to accelerate the saturation rate. Then specimens were isotropically consolidated under three different effective
confining stresses of 100, 300 and 500 kPa. Following consolidation, undrained
cyclic triaxial tests were carried out under strain-controlled condition.
Strain-controlled approach were also preferred over stress-controlled for
cyclic loading tests, since previous researches strongly suggest that shear
strain is a more fundamental parameter for studying pore pressure build-up
than shear stress (Matasovic and Vucetic, 1992). The cyclic tests were continued
until 50 cycles of loading as well. The loading rate shown in Table 2, was
chosen so that pore pressure equalization through the specimen was ensured.
The system used for conducting cyclic tests, was also an advanced automated
triaxial testing apparatus.
STRAIN DEPENDENT CYCLIC DEFORMATION PROPERTIES
Shear modulus and damping ratio are important soil properties that are
used extensively in dynamic analyses. When cyclic triaxial tests are performed,
a hysteresis loop will be formed by plotting the deviator stress,σd,
versus axial strain, ε. The slope of the secant line connecting the
extreme points on the hysteresis loop is the Young modulus, E, (Fig.
2) where:
As the axial strain amplitude increases, the Young modulus decreases,
as shown in Fig. 2. Having obtained the values of E
and ε, the shear modulus, G and shear strain, γ can be found
through the following equations:
and
Where:
| V |
= |
Poisson`s ratio and may be estimated at 0.5 for saturated,
undrained specimens |
The damping ratio, D, is a measure of dissipated energy, WD,
versus elastic strain energy, WS and may be computed with the
Eq:
Referring to Fig. 2, the area inside the hysteresis
loop is WD and the shaded area of the triangle is WS.
|
| Fig. 2: |
Definition of strain dependent shear modulus and damping ratio in
cyclic loading |
|
| Fig. 3: |
Variation of (a) shear modulus and (b) damping ratio
against shear strain amplitude in gravel-kaolin mixtures,  and
N = 2 |
The results are presented for the tests performed under an initial confining
stress, 
of
100 kPa and at 2nd loading cycle, N (the trend for other confining stresses and cycles is the same and not shown herein).
|
| Fig. 4: |
Variation of (a) shear modulus and (b) damping ratio against shear
strain amplitude in sand-kaolin mixtures, and
N = 2 |
As can be seen in Fig. 3 and 4, for all mixtures, when shear strain amplitude increases, the shear modulus decreases and damping ration increases. In addition, when aggregate content is raised both shear modulus and
damping ratio increase. However, the effect of aggregate content on shear
modulus diminishes at high shear strain amplitudes of order 1.5%. Thus,
in general, K40G and K40S specimens exhibit the highest
values of shear modulus and damping ratio, whilst the pure clay (i.e.,
K100) specimens exhibit the lowest ones.
DEPENDENCE OF SHEAR MODULUS ON AGGREGATE CONTENT AND LOADING CYCLES
Herein, the variations in shear modulus are presented for different shear strain
amplitudes, γcunder an initial confining stress of 100 kPa (the
trend under other confining stresses is the same and not shown herein).
|
|
|
| Fig. 5: |
Dependence of shear modulus on loading cycles in gravel-kaolin
mixtures under
(a) γc = 0.15% (b) γc = 0.75% and
(c) γc = 1.5% |
As shown
in Fig. 5a and 6a, under a low shear strain
amplitude of 0.15%, in all mixtures shear modulus is independent of loading
cycles and increases when aggregate content is raised. When the shear strain
amplitude is increased to 0.75 or 1.5%, the behavior is quite different.
|
| Fig. 6: |
Dependence of shear modulus on loading cycles in sand-kaolin mixtures
under
(a) γc = 0.15% (b) γc = 0.75% and
(c) γc = 1.5% |
As shown in Fig. 5b, c and 6c,
under high shear strain amplitudes, shear modulus depends on loading cycles
and decreases when loading proceeds. As can be seen, prior to 10th cycle of loading, shear modulus generally
increases with aggregate content.
|
| Fig. 7: |
Effect of loading cycles and shear strain amplitude on degradation
index in specimen K60G, |
|
| Fig. 8: |
Variation of degradation index in gravel-kaolin mixtures under
(a) γc = 0.75% and (b) γc = 1.5% |
After the cycle 10, shear modulus is
nearly identical for all mixtures, because of severe decrease in shear
modulus of specimens with high aggregate content. As will be explained
later, the decrease in shear modulus is attributed to the pore pressure
build-up when loading proceeds.
|
| Fig. 9: |
Variation of degradation index in gravel-kaolin mixtures under
(a) γc = 0.75% and (b) γc = 1.5% |
To quantify the shear modulus dependence on cycles of loading and have
more insight on the behavior, it is prudent to verify the variation of
degradation index, δ (Idriss et al., 1978) in different mixtures.
δ is defined as follows:
where, G1and GN denote the secant shear modulus
in the first and Nth cycles of loading respectively. Figure
7 presents typically the variation of δ in terms of N for the
specimens containing 40% gravel (i.e., K60G) for different shear strain
amplitudes. The variation of δ in other mixtures obeys the similar
trend shown for K60G specimens and hence it is not shown herein. As shown
in Fig. 7, the value of δ depends on both cycles
of loading and shear strain amplitude in such a manner that it decreases
when these parameters increase.
The variation of δ with N, in gravel-kaolin and sand-kaolin mixtures,
is shown in Fig. 8 and 9, respectively
for shear strain amplitudes of 0.75 and 1.5%, under an initial confining
stress of 100 kPa. As can be seen, in general, when aggregate content
is raised, δ decreases. Hence, K100, K80G and K80S
specimens have the highest values of δ, while K40G and
K40S specimens have the lowest ones. The increase in degradation of
shear modulus of high aggregate content mixtures may be attributed to
the appreciable pore pressure build-up, as will be discussed later.
DEPENDENCE OF DAMPING RATIO ON AGGREGATE CONTENT AND LOADING CYCLES
As can be shown in Fig. 10 and 11,
under a typical initial confining stress of 100 kPa, damping ratio increases
with aggregate content. If enhancement in aggregate content is interpreted
as decrease in plasticity, the observed trend is consistent with the curves
of Vucetic and Dobry (1991) on the effect of soil plasticity on damping
ratio. In addition, as shown in Fig. 10 and 11,
damping ratio is almost independent of loading cycles. The reason can
be readily explained. Figure 12, for example, presents
the hysteresis loop formed in 2nd and 30th cycles of loading on specimen
K80G under a shear strain amplitude of 0.75% and an initial confining
stress of 500 kPa. As can be seen, the area of hysteresis loop (i.e.,
the dissipated energy) in the 30th cycle is less than that of 2nd cycle.
However, the secant shear modulus in 30th cycle is less than 2nd cycle.
In other words, in the 30th cycle the specimen absorbs less energy than
2nd cycle. It can be shown when loading proceeds the dissipated energy
decreases with nearly the same order as absorbed energy decreases and
hence according to Eq. 4, damping ratio does not change
remarkably with loading cycles.
EFFECT OF AGGREGATE CONTENT ON PORE PRESSURE BUILD-UP
As indicated previously, the appreciable decrease in shear modulus of
aggregate-clay mixtures may be attributed to the pore pressure build-up
during cyclic loading. Figure 13 and 14
display the variation of
(which is pore pressure normalized to initial confining stress and computed
where shear strain is zero) in terms of number of loading cycles, in gravel-kaolin
and sand-kaolin mixtures respectively under shear strain amplitudes of
0.15, 0.75 and 1.5% and a typical confining stress of 100 kPa (the behavior
under other confining stresses is the same and not shown herein). As seen,
in general, pore pressure increases when aggregate content is raised.
However, the trend is manifested when shear strain amplitude is increased.
|
| Fig. 10: |
Damping ratio in terms of loading cycles in gravel-kaolin mixtures
under
(a) γc = 0.15% (b) γc = 0.75% and
(c) γc = 1.5% |
Thus is highest for K40G and K40S specimens and lowest
for K100 ones. As indicated by Jafari and Shafiee (2004), since the compressibility
of clayey matrix is much more than individual grains, all of the specimen
deformations take place in the clay. Hence, during a strain-controlled
loading, the clayey matrix of specimens containing more aggregate experiences
more deformation for the same strain level, directly leading to more pore
pressure build-up.
|
| Fig. 11: |
Damping ratio in terms of loading cycles in sand-kaolin mixtures
under
(a) γc = 0.15% (b) γc = 0.75% and
(c) γc = 1.5% |
Jafari and Shafiee (2004) also reported that the heterogeneous
field of effective stress and deformation developed in clayey matrix can
be another reason for more pore pressure build-up in specimens containing
more aggregates.
The trend of pore pressure variation shown in Fig. 13
and 14 is wholly consistent with the trend of degradation
index variation (Fig. 8 and 9).
 |
| Fig. 12: |
Hysteresis loops for specimen K80G, γc = 0.75% and
 |
On
the basis of the data presented in Fig. 13 and 14,
it can be inferred that an increase in aggregate content would lead to
decrease in effective confining stress when loading proceeds. Consequently,
shear modulus would decrease more with loading cycles when aggregate content
is raised.
EFFECT OF INITIAL CONFINING STRESS ON CYCLIC PROPERTIES
To explore the impact of confining stress on cyclic deformation properties,
the tests were performed under various confining stresses of 100, 300
and 500 kPa. Figure 15a and b,
for instance, presents the effect of initial confining stress on shear
modulus in cycle 10, G10 under a typical shear strain amplitude
of 0.75% in gravel-kaolin and sand-kaolin mixtures respectively. It is
interesting to note that an increase in initial confining stress would
increase shear modulus linearly. Figure 16a and
b also present the variation of damping ratio in cycle 10, D10
under a shear strain amplitude of 0.75% in gravel-kaolin and sand-kaolin
mixtures respectively. As can be seen, damping ratio is nearly unaffected
by the initial confining stress.
EFFECT OF AGGREGATE SIZE ON CYCLIC PROPERTIES
One of the issues of concern in the experimental study was to assess
the effect of aggregate size on the cyclic deformation properties. Hence,
the tests were conducted on gravel-kaolin and sand-kaolin mixtures. The
gravel and the sand had mean grain diameter of 5.55 and 1.84 mm, respectively.
Figure 17, for instance, compares the impact of aggregate
size on cyclic properties under an initial confining stress of 100 kPa
and a shear strain amplitude of 0.75%.
|
| Fig. 13: |
Pore pressure build-up in gravel-kaolin mixtures under
(a) γc = 0.15% (b) γc = 0.75% and
(c) γc = 1.5% |
As can be seen, aggregate size
does not affect cyclic deformation properties markedly when aggregate
content is low (i.e., 20%). In high aggregate content mixtures, particularly
in mixtures containing 60% aggregates, shear modulus and pore pressure
slightly decrease and damping ratio increases when aggregate size is increased.
|
| Fig. 14: |
Pore pressure build-up in sand-kaolin mixtures under
(a) γc = 0.15% (b) γc = 0.75% and
(c) γc = 1.5% |
ANNS MODELING OF SHEAR MODULUS, DAMPING RATIO AND PORE PRESSURE
Although accurate, it is often tedious and expensive to conduct laboratory
tests on composite materials. One alternative to the lengthy laboratory
determination of cyclic deformation properties can be the utilization
of automated predictors that require only the input of easily measurable
parameters.
|
|
| Fig. 15: |
Effect of initial confining stress on shear modulus in cycle 10
for (a) gravel-kaolin and (b) sand-kaolin mixtures under γc
= 0.75% and |
Theses predictive methods can be utilized effectively for
feasibility studies, early decision in the field, parametric analytical
studies and such. Herein, Artificial Neural Networks (ANNs) are utilized
to model the complex behavior of aggregate-clay mixtures.
Since a large body of ANN knowledge has been introduced in the literature,
this issue is not addressed in detail herein. However, the salient features
of this computational paradigm are explained. A neural network is a massively
parallel distributed processor made up of simple processing units, which
has a natural propensity for storing experiential knowledge and making
it available for use. The procedure used to perform the learning process
is called a learning algorithm, the function of which is to modify the
synaptic weights of the network in an orderly fashion to attain a desired
design objective (Haykin, 1999).
ANNs have been satisfactorily applied in the prediction of diverse problems
in civil engineering.
|
|
| Fig. 16: |
Effect of initial confining stress on damping ratio in cycle 10
for (a) gravel-kaolin and (b) sand-kaolin mixtures under γc
= 0.75% and |
The advantage of ANNs is its capacity to learn complex
functions between variables without the necessity of applying suppositions
or restrictions a priori to the data. In recent years, the use of ANNs
has increased in many geotechnical earthquake engineering issues and has
demonstrated some degree of success. A review of the literature reveals
that ANNs have been used successfully in evaluation of seismic site effects
(Paolucci et al., 2000), prediction of maximum shear modulus and
minimum damping ratio (Akbulut et al., 2004), assessment of earthquake-induced
landslide (Lee and Envangelista, 2006) and assessment of liquefaction
potential (Baziar and Jafarian, 2007).
In this study, the multilayer perceptron (MLP) was utilized to model
the cyclic deformation properties. It is the most popular class of networks
which has been applied to solve diverse civil engineering problems. A
multilayer perceptron is composed of an input layer, an output layer and
one or more hidden layers, however, it has been demonstrated that for
the majority of problems only one hidden layer is enough. The architecture
of a typical perceptron formed by an input layer, a hidden layer and an
output layer is shown in Fig. 18.
|
| Fig. 17: |
Effect of aggregate size on (a) shear modulus (b) damping ratio
and (C) pore pressure under γc = 0.75% and |
Training, testing and validation datasets: The dataset used for
training the networks is based on fifty six tests carried out on seven
specimens namely K100, K80S, K80G, K60S, K60G, K40S and K40G (Table 2).
It should be noted that by increasing the number of connection weights
in networks, the networks can overfit the training data, particularly
if the training data are noisy.
|
| Fig. 18: |
The architecture of a typical perceptron |
In order to avoid overtraining, various
rules are proposed to restrict the ratio of the number of connection weights
to the number of data samples in the training set. Alternatively, the
cross-validation technique can be used, in which the available data sets
are partitioned into training, testing and validation subsets (Haykin,
1999).
The training subset is used to adjust the connection weights, whereas the
validation subset is used to check the performance of the network at various
stages of learning. The test subset is also used for comparing the generalization
of networks and selecting the best of them. In this study, data sets were
randomly partitioned. Ten percent of the available data were used for testing,
ten percent for validation and eighty percent of the remaining data were
used for training. Subsequently, seven, seven and forty two tests were allotted
to the test, validation and training subset, respectively.
The network architectures and input variables: The objective of
training the neural networks was to predict the deformation properties
of aggregate-clay mixtures under cyclic loading conditions. Two MLP models
with different architectures were utilized.
|
|
| Fig. 19: |
The architecture of the proposed model for predicting the cyclic
properties (a) M-1 and (b) M-2 |
First model consists of three
layers: an input layer, a single hidden layer and an output layer (Fig.
19a). Henceforth this model is named M 1. The input parameters of M 1 are as
follows:
| • |
Aggregate content, Ac (%); the range of which is between 0 to 60. |
| • |
Shear strain amplitude, γc (%); which varies from
0.15 to 1.5. |
| • |
Initial confining stress, 
(kPa); the range of which is between 100 to 500. |
| • |
Mean aggregate size, Mas (mm); in the range of 0 to 5.55. |
| • |
Loading cycles, N; the range of which is between 0 to 50. |
The output layer consists of a single neuron which predicts shear modulus
(MPa), damping ratio (%) and pore pressure (kPa). Consequently, three
networks M1-D, M1-G and M1-U predicting shear modulus, damping ratio and
pore pressure, respectively, were trained.
Since the current state of stress affect the next state in the soils,
a second model was constructed. It is similar to the first one in the
sense that both of them have three layers. The main difference between
the second model, namely M-2 and the first one (i.e., M-1) is that the
model M-2, takes into consideration the current state of the stress by
incorporating pore pressure (Ui) in Nth cycle of loading as
an extra input which its value is equal to the output from network`s forward
computation of the last iteration. The model M-2 has two outputs: one
is pore pressure in (N+1)th cycle of loading and the second is alternatively
damping ratio or shear modulus (Fig. 19b). This type
of network is also known as sequential network (McCelland and Rumelhart,
1988). As shown in Fig. 19b the predicted value of
pore pressure in each step acts as the input value for the next step.
Eventually, three networks namely M2-(D,U), M2-(G,U) and M2-U were trained,
in the second model.
Training the network: Once the network weights and biases are
initialized, the network is ready for training. During training the weights
and biases of the network were iteratively adjusted to minimize the network
performance function. The performance function used in the study was Mean
Square Error (MSE), which is the average square error between the network
outputs and the target outputs. The Levenberg-Marquardt was used as the
training algorithm, since it has the fastest convergence (Haykin, 1999).
The cross-validation technique was also utilized as the stopping criterion
for training the networks. In this technique, as soon as MSE on the validation
subset increases the training process stops. Whenever the iteration is
stopped, MSE would be calculated for each of the training, testing and
verification phases and compared.
Each of the 6 networks was trained using different number of hidden neurons
and different activation functions. Next, the number of hidden neurons
and the type of activation functions which produced minimum MSE and maximum
correlation coefficient (R) were selected. Subsequently, the appropriate
structure of each network was determined. Finally, the networks which
best predicted the target values were selected. Table 3
presents the characteristics of the networks. On the basis of the values
of MSE and R shown in Table 3, the first model predicts
damping ratio better than the second one, while the second model performs
better than the first model in predicting shear modulus and pore pressure.
In other words, the current state of stress, which was considered by a
feedback loop in the second model (Fig. 19b), plays
an important role in determination of shear modulus and pore pressure
while it is not an important parameter in predicting damping ratio. As
previously indicated in Fig. 10 and 11,
damping ratio is not affected by loading cycles, which is an indication
of current state of stress.
|
| Fig. 20: |
Comparison of predicted cyclic properties and laboratory data (a)
shear modulus (b) damping ratio and (c) pore pressure |
The predictions using ANNs against laboratory data is plotted in Fig. 20. As can be seen, in spite of large values of shear modulus and low
values of damping ratio, the ANNs appropriately predict the cyclic deformation
properties.
| Table 4: |
Relative importance (in percent) of input neurons |
 |
|
| Fig. 21: |
Relative importance of input parameters |
Relative importance of input variables: Learning algorithms such
as the back propagation neural network do not give information on the
impact of each input parameter or influencing variable upon the predicted
output variable. In other words, it is not possible to find out immediately
how the weights of the network or the activation values of the hidden
neurons are related to the set of data being handled. Instead, ANNs have
been presented to the user as a kind of black box whose extremely complex
work transforms inputs into predetermined outputs (Montano and Palmer,
2003). To deal with this problem, different interpretative methods for
analyzing the effect or importance of input variables on the output of
a feedforward neural network have been proposed. These interpretative
methods can be divided in two types of methodologies: analysis based on
the magnitude of weights and sensitivity analysis (Montano and Palmer,
2003). Analysis based on the magnitude of weights groups together those
procedures that are based exclusively on the values stored in the static
matrix of weights to determine the relative influence of each input variable
on each one of the network outputs (Table 4). On the basis of the magnitudes
of weights, the Garson`s method (Garson, 1991) was used to determine the
relative importance of the input parameters.
As can be seen in Fig. 21, shear modulus is mostly affected by current
state of stress (i.e., Ui), aggregate content and with some
lower degree by the shear strain amplitude. On the other hand, aggregate
size does not have a remarkable effect on the shear modulus. In addition,
since pore pressure is closely related to the loading cycles, it can be
assumed that the effect of loading cycles on shear modulus has been integrated
somehow into the current state of stress.
As shown in Fig. 21, damping ratio is mostly affected
by the aggregate content and shear strain amplitude and less by the other
parameters. This is in accordance with the results of laboratory tests,
showing damping ratio is not affected considerably by loading cycles (Fig.
10, 11), initial confining stress (Fig.
16) and aggregate size (Fig. 17b). It is also interesting
to note that pore pressure in aggregate-clay mixtures is mostly affected
by aggregate content and initial confining stress and less by the current
state of stress and shear strain amplitude. In consistent with the results
shown in Fig. 17c, Fig. 21 also
reveals that the Mean Aggregate Size (Mas) has a negligible impact on
the pore pressure.
CONCLUSIONS
An experimental study was performed on the compacted mixtures of gravel-kaolin
and sand-kaolin under different initial confining stresses and shear strain
amplitudes to investigate the effects of aggregates on the shear modulus,
damping ratio and pore pressure during strain-controlled cyclic triaxial
loading. Artificial neural networks (ANNs) were also implemented to model
shear modulus, damping ratio and pore pressure in the mixtures. The following
conclusions may be drawn based on the study:
| • |
Under low shear strain amplitudes, shear modulus increases
with aggregate content, however, it is independent of loading cycles.
Under high shear strain amplitudes, shear modulus depends on loading
cycles and it decreases with loading cycles. In this case, shear modulus
increases with aggregate content prior to 10th cycle of loading, however,
it is nearly identical in all mixtures passing the cycle 10. In addition,
in all mixtures shear modulus increases linearly with initial confining
stress |
| • |
In all mixtures, the rate of decrease in shear modulus with loading
cycles (i.e., degradation index) decreases with aggregate content.
This can be attributed to the increase in pore pressure with aggregate
content |
| • |
An increase in aggregate content would lead to increase in damping
ratio. However, in all mixtures, damping ratio is nearly independent
of loading cycles and initial confining stress |
| • |
Aggregate size does not affect cyclic deformation properties when
aggregate content is low. In high aggregate content mixtures, shear
modulus and pore pressure slightly decrease and damping ratio increases
when aggregate size is increased |
| • |
The observations indicate the potential of well-trained neural networks
in developing a relatively general model that can predict shear modulus,
damping ratio and pore pressure build-up in the aggregate-clay mixtures.
Utilizing current state of state as an input parameter assist in better
prediction of the cyclic properties, particularly shear modulus and
pore pressure |
| • |
Sensitivity analysis of ANNs shows that aggregate content is the
most important parameter that affects the dynamic deformation properties.
In addition, shear modulus, damping ratio and pore pressure are highly
affected by the current state of stress, shear strain amplitude and
initial confining stress respectively. Aggregate size has negligible
effect on the cyclic properties. The results of the sensitivity analysis
fairly support the laboratory observations. |
ACKNOWLEDGMENTS
The support of the International Institute of Earthquake Engineering
and Seismology (IIEES) which made this work possible is gratefully acknowledged.
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