Prediction of residual friction angle of...
Transcript of Prediction of residual friction angle of...
PREDICTION OF RESIDUAL FRICTION ANGLE OF CLAYS USING
ARTIFICAL NEURAL NETWORK
Sarat Kumar Das1
Assistant Professor, Civil Engineering Department, National Institute of Technology,
Rourkela, India. Email- [email protected]
Prabir Kumar Basudhar
Professor, Civil Engineering Department, Indian Institute of Technology Kanpur, India.
Email: [email protected]
ABSTRACT
The residual strength of clay is very important to evaluate long term stability of proposed
and existing slopes and for remedial measure for failure slopes. Various attempts have
been made to correlate the residual friction angle (r) with index properties of soil. This
paper presents a neural network model to predict the residual friction angles based on
clay fraction and Atterberg‟s limits. Different sensitivity analysis was made to find out
the important parameters affecting the residual friction angle. Emphasis is placed on the
construction of neural interpretation diagram, based on the weights of the developed
neural network model, to find out direct or inverse effect of soil properties on the residual
shear angle. A prediction model equation is established with the weights of the neural
network as the model parameters.
KEYWORDS: Clays; Shear strength; neural network; statistical analysis.
INTRODUCTION
1 Corresponding author, Tel: +91-9437390601. Email:- [email protected]
Residual strength of clay is very important in evaluation of long term stability of
proposed and existing slopes and for remedial measure for failure slopes. At the residual
strength, the soil is highly remoulded and has negligible cohesion. The principal shear
resistance manifests from inter particle friction, which is described in terms of residual
friction angle. Though using ring shear apparatus (Bromhead 1979) the residual shear
strength can be measured, various attempts have been made to correlate residual friction
angle (r) with index properties of soil. Skempton (1964) related the r value with the
clay fraction (CF). For a given clay, r decreases with liquid limit (LL) and for a given
LL and CF, r decreases with increase in normal effective stress (Bowles 1988). Mesri
and Cepeda-Diaz (1986) presented a correlation between r and LL. Colotta et al. (1989)
have given a correlation between r and a parameter which is a function of LL, plasticity
index (PI) and clay fraction (CF). For sedimentary soil, Stark et al. (1994) proposed
correlations of r with LL for various ranges of CF. Wesley (2003) found that for tropical
soil r can be better related with ∆PI (i.e. the deviation from the A-line in classification
chart) where,
∆PI = PI-0.73(LL-20) (1).
The data points were scattered and the correlation is valid for LL > 50. All the above
correlations are graphical. Sridharan and Rao (2004) observed that the r is related to LL,
PI and CF but better correlation was observed for clay fraction. These relationships are
developed with single variable and empirical formulae for r with multivariable soil
properties are not available. Recently, Kaya and Kwong (2007) found that amount of clay
mineral may play an important role in prediction of r based on index properties;
however, it is difficult to find out the amount of clay mineral of soil all the time.
Artificial neural network (ANN) is now being used successfully as an alternate statistical
method with high predictability. However, it is known as black box system, as it is not
possible to explain the weights/parameters of the network and this is a subject of future
research (Goh et al. 2005). There is a chance of poor generalization of the developed
model due to insufficient data points. With the above in view, in the present work a feed
forward back propagation neural network has been used to predict the residual friction
angle of clay based on index properties of soil (LL, PI, CF and ΔPI). Different methods to
interpret the important input variables are also discussed. An attempt has been made here
to find out the relationship between input variables and the output by drawing neural
interpretation diagram. A prediction model is presented based on weights/parameters of
the developed ANN model.
ANALYSIS AND DATA
In the present study data base available in Wesley (2003) have been considered. The data
base consists of index properties of soil (LL, PL, PI, CF and ΔPI). Based on the above
data Wesley (2003) observed that the residual friction angle is related to ΔPI, and the
relationship varies differently for clay in general and also that for
volcanic ash clay. In the present study, out of 54 data points, 39 were used for training
and 15 data points were used for testing. To improve the generalization of the developed
ANN model, Bayesian regularization is used. The weight values have been automatically
regularized to minimize the combined error function in case of Bayesian regularization,
the details of which are available in Mckay (1992) and Hagan et al (2004). The Bayesian
regularization technique has been implemented using MATLAB V 6.5 (MathWorks ).
Results and Discussion
Different ANN models were tried using different combinations of the above input
variables and developed models are compared in terms of correlation coefficient (R) and
coefficient of efficiency (R2). The R
2 is defined as
1
212
E
EER
(2a)
where
2
12
2
11
N
tmrpr
N
tavemrmr
E
E
)(
(2b)
and r m, r m ave , r p are the measured, average of the measured and predicted residual
friction angle.
Some of the models and the corresponding R and R2
values are presented in Table 1. It
can be seen from Table 1 that Model 1 does not show good correlation as reflected by
poor R and R2 values during both training and testing. Model 2 and 3 show good
correlation during training but show poor correlation during testing phase which is
specially signified in R2 values 0.652 and 0.761 respectively, even though values of R
(0.883 and 0.885) show better correlation. Considering both R and R2, Model 4 is the
“best model” of the above. So the Model with CF and ∆PI has the best correlation with r
values. The weights and biases of the final network are presented in Table 2. The weights
and biases can be utilized for interpretation of relationship of inputs and output,
sensitivity analysis and framing an ANN model in equation form.
Neural interpretation diagram (NID)
Ożesmi and Ożesmi (1999) proposed neural interpretation diagram (NID) for visual
interpretation of the connection weight among the neurons. For the present study with the
weights as obtained and shown in Table 2 an NID is presented in Figure 1. The lines
joining the input-hidden and hidden-output neurons represent the weights. The positive
weights are represented by solid lines and negative weights by dashed lines and the
thickness of the lines is proportional to their magnitude. The relationship between the
input and output is determined in two steps. Direct proportionality of the input variables
is depicted by positive input-hidden and positive hidden-output weights, or negative
input-hidden and negative hidden-output weights. The positive input-hidden and negative
hidden-output and negative input-hidden and positive hidden-output weight indicates the
inverse proportionality of the input variables. The input directly related to the output is
represented with a grey circles and that having inverse effect with blank circle.
It can be seen from Figure 1 that both the inputs ΔPI and CF inversely related to the r
values. Thus it is inferred that r values decrease with increase in ΔPI and CF values.
Wesley (2003) observed that r decreases with increase in ΔPI value and Sridharan and
Rao (2004) observed that r decreases with increase in CF. So it can be seen that
developed ANN model is not a „black box‟ and could explain the physical effect of inputs
on the output.
Sensitivity analysis
Sensitivity analysis is of utmost concern for selection of important input variables.
Different approaches have been suggested to select the important input variables. The
Pearson correlation coefficient is defined as one of the variable ranking criteria in
selecting proper inputs for the ANN (Guyon and Elisseeff 2003 ; Wilby et al. 2003).
Goh (1994) and Shahin et al.(2002) have used Garson‟s algorithm (Garson 1991) in
which the input-hidden and hidden-output weights of trained ANN model are partitioned
and the absolute values of the weights are taken to select the important input variables,
the details with example has been presented in Goh (1994). So it does not provide
information on the effect of input variables in terms of direct or inverse relation to the
output. Olden et al. (2004) proposed a connection weights approach based on the NID, in
which the actual values of input-hidden and hidden-output weights are taken. It sums the
products across all the hidden neurons, which is defined as Si. The relative inputs are
corresponding to absolute Si values; the most important input corresponds to highest Si
value. The detail of above approach is presented in Olden et al. (2004). The results of
sensitivity analysis using above approaches are presented here as follows.
Table 3 shows the cross correlation of inputs with the r value. From the table it
can be seen that r is highly correlated to ΔPI as signified by the cross correlation values
of 0.62, followed by CF, PI and LL. The sensitivity analysis for the model as per
Garson‟s method to find out important input parameters are presented in Table 4. The
ΔPI is found to be the most important input parameter with the relative importance value
being 66.33% in comparison to 33.66% for the clay fraction. The relative importance of
the input variables as calculated following connection weight approach (Olden et al.
2004) is also presented in Table 4. It can be seen that, here also ΔPI is the most important
input parameter (Si value -9.65) followed by CF (Si value -8.55). The Si values being
negative imply that both the ΔPI and CF are indirectly related to r. From the above
results it can be seen that ΔPI is found to be the most important parameter in predicting r
value based on all the sensitivity analysis methods.
ANN model Equation for the r value based on trained neural network.
Another criticism regarding the application ANN is framing a model equation. However,
as in the present study, there are only two parameters (ΔPI and CF) a model equation can
be established with the weights as the model parameters (Goh 2005). The mathematical
equation relating the input variables and the output can be written as,
h
k
m
i
iikhksigksignr Xwbfwbf
1 1
0 (3)
Where r n is the normalized (in the range -1 to 1 in this case) r value,
b0 = bias at the out put layer;
wk = connection weight between kth
neuron of hidden layer and the single output neuron;
bhk = bias at the kth
neuron of hidden layer;
h = number of neurons in the hidden layer;
wik = connection weight between ith
input variable and kth
neuron of hidden layer;
Xi = normalized input variable i in the range [1,1] and
f sig = sigmoid transfer function.
Using the values of the weights and biases presented in Table 2 the following expression
can be written to finally arrive at a correlation of r with the input parameters.
A1 = -0.1129 – 1.6191 CF – 4.4933ΔPI (4)
A2 = -0.5814 - 0.1202 CF – 1.3113ΔPI (5)
A3 = 0.9155 - 8.1472 CF – 12.4994ΔPI (6)
A4 = 1.7969 + 2.8241CF + 1.8583ΔPI (7)
11
11
1 1772.2AA
AA
ee
eeB
(8)
22
22
2 8174.3AA
AA
ee
eeB
(9)
33
33
3 9498.0AA
AA
ee
eeB
(10)
33
33
4 3732.1AA
AA
ee
eeB
(11)
C1 = 2.233+ B1 + B2 + B3 + B4 (12)
11
11
CC
CC
nr
ee
ee
(13)
The r value as obtained from Eq. 13 is in the range [-1, 1] and this needs to be
denormalized as
r = 0.5 (r n +1) (r max- r min) + r min (14)
Where, r max and r min are the maximum and minimum values of r respectively in the
data set.
Conclusions
The following conclusions can be drawn from the above studies:
(1) The ANN model with CF and ΔPI as input parameters is the „best‟ model, based on
statistical parameters, correlation coefficient and coefficient of efficiency, for training
and testing data set.
(2) The developed ANN model could explain the physical effect of inputs on the output,
as depicted in NID. It was observed and inferred that r values decrease with increase in
ΔPI and CF values, conclusions similar to that drawn earlier by Stark and Eid (1994),
Wesley (2003) and Sridharan and Rao (2004).
(3) Based on sensitivity analyses; Pearson correlation coefficient, Garson‟s algorithm and
connection weight approaches, it was observed that ΔPI is the most important parameter.
(4) A model equation is presented based on the trained weights of the ANN.
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LIST OF TABLES
Table 1 Different ANN models and their statistical performances
Table 2 Connection weights and biases
Table 3 Cross-correlation of the input and outputs for the residual friction angle
Table 4 Relative Importance of different inputs as per Garson‟s algorithm and
connection weight approach
Table 1 Different ANN models and their statistical performances
Models Inputs Coefficient of Correlation
(R)
Coefficient of efficiency
(R2)
Training Testing Training Testing
Model 1 LL, CF 0.851 0.829 0.741 0.543
Model 2 LL, PI, CF 0.902 0.883 0.813 0.652
Model 3 LL, PI, CF,
ΔPI
0.926 0.885 0.807 0.761
Model 4 CF, ΔPI 0.906 0.942 0.826 0.868
Table 2 Connection weights and biases
Neuron Weights (w ik) Biases
Input 1 Input 2 Output bhk b0
Hidden
neuron 1
(k=1)
-1.6191
-4.4933 -2.1772
-0.1129
2.233
Hidden
neuron 2
(k=2)
-0.1202 -1.3113 3.8174
0.5814
-
Hidden
neuron 3
(k=3)
-8.1472 -12.4994 0.9498 0.9155 -
Hidden
neuron4
(k=4)
2.8241 1.8583 -1.3732 1.7969 -
Table 3 Cross-correlation of the input and outputs for the residual friction angle
LL PI CF ΔPI r
LL 1.00 0.87 0.60 0.01 -0.08
PI 1.00 0.57 0.50 -0.37
CF 1.00 0.11 -0.51
ΔPI 1.00 -0.62
r 1.00
Table 4 Relative Importance of different inputs as per Garson‟s algorithm and
connection weight approach
Parameters Garson‟s algorithm Connection weight
approach
Relative
importance
(%)
Ranking of
inputs as
per
relative
importance
Si value as
per
connection
weight
approach
Ranking of
inputs as
per
relative
importance
CF 33.6% 2 -8.55 2
ΔPI 66.33% 1 -9.65 1
LIST OF FIGURES
Figure 1 The NID showing lines representing connection weights and effects of inputs
on r.
Figure 1 The NID showing lines representing connection weights and effects of inputs
on r.
1
2
B
C
D
O
A
1. CF
2. ΔP
r