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- saratdas@rediffmail.com

Prabir Kumar Basudhar

Professor, Civil Engineering Department, Indian Institute of Technology Kanpur, India.

Email: pkbd@iitk.ac.in

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:- saratdas@rediffmail.com

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