Modelling the mechanical behaviour of unsaturated soils using a genetic algorithm-based neural network

Computers and Geotechnics 38 (2011) 2–13

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Modelling the mechanical behaviour of unsaturated soils using a genetic algorithm-based neural network A. Johari a, A.A. Javadi a,⇑, G. Habibagahi b a b

Computational Geomechanics Group, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF, UK Department of Civil Engineering, Shiraz University, Shiraz, Iran

a r t i c l e

i n f o

Article history: Received 28 July 2009 Received in revised form 27 August 2010 Accepted 29 August 2010

Keywords: Unsaturated soil Constitutive modelling Neural network Genetic algorithm Stress–strain relationship

a b s t r a c t Modelling the mechanical behaviour of unsaturated soils has been the subject of many research works in the past few decades. A number of constitutive models have been developed to describe the complex behaviour of unsaturated soils. Despite the significant advances in the constitutive theories for unsaturated soils, none of the existing models can completely describe the various aspects of the real behaviour of unsaturated soils. In this paper, a new unified approach is presented, based on the integration of a neural network and a genetic algorithm, for the modelling of unsaturated soils. In the proposed approach, a genetic algorithm was used to optimise the weights of the neural network. A three-layer sequential architecture was chosen for the neural network. The network had eight input neurons, five neurons in the hidden layer and three neurons in the output layer. The eight input neurons represented the initial gravimetric water content, initial dry density, degree of saturation, net mean stress with respect to pore-air pressure, axial strain, deviatoric stress, soil suction and volumetric strain, and the three neurons in the output layer represented the deviatoric stress, suction and volumetric strain at the end of each increment. The network was trained and tested using a database that included results from a comprehensive set of triaxial tests on unsaturated soils from the literature. The predictions of the proposed model were compared with the experimental results. The comparison of the results indicates that the proposed approach was accurate and robust in representing the mechanical behaviour of unsaturated soils. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction The limitations in describing the mechanical behaviour of unsaturated soils based on a single effective stress equation, similar to the one proposed by Bishop and Donald [2], has led to the development of different models to describe the observed behaviour of these soils. Consequently, the mechanical behaviour of unsaturated soils has been the topic of numerous investigations in recent years. Among recent important contributions are the works of Fredlund and Morgenstern [5], Toll [21], Alonso et al. [1], Gens and Alonso [6] and some others. Fredlund and Morgenstern [5] proposed the concept of a ‘‘state surface” defined in terms of independent state variables such as (r  ua) and (ua  uw), where r is the total stress, ua is the pore-air pressure, and uw is the pore-water pressure. Using these state variables, suitable state surfaces were defined for the stress–strain–strength behaviour of unsaturated soils. Alonso et al. [1] and Toll [21] proposed critical state frameworks by considering the effect of the net mean stress and suction separately. The

⇑ Corresponding author. Tel.: +44 1392 263640; fax: +44 1392 217965. E-mail addresses: [email protected] (A. Johari), [email protected] (A.A. Javadi), [email protected] (G. Habibagahi). 0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2010.08.011

works of Chateau and Dormieux [3], Sun et al. [19] and Khalili [13] are among some of the recent investigations. Though constituting significant steps forward in the constitutive modelling of unsaturated soils, many of these contributions are incapable of dealing with all aspects of unsaturated soil behaviour in a consistent and unified manner. Only limited attempts have been made to incorporate the pre- and post-peak nonlinear behaviour of unsaturated soils into a formal constitutive framework (e.g., see [14,15]. In addition, there currently exist no constitutive models of unsaturated soils in which a point-by-point matching of test data as observed in the laboratory can be achieved. Habibagahi and Bamdad [9] proposed a neural network-based approach to model the constitutive behaviour of unsaturated soils. They used a sequential architecture for the neural network with nine neurons in the input layer, five neurons in the hidden layer and three neurons in the output later. The database used for training and validation of the artificial neural network ANN was the same as the one used in the current work. The results indicated the good performance of the ANN for predicting the mechanical behaviour of unsaturated soils. Although neural networks have been shown to be very efficient in the constitutive modelling of materials, they also have some shortcomings. One of the drawbacks of a neural network is that there is a possibility for the search to be trapped in local minima

A. Johari et al. / Computers and Geotechnics 38 (2011) 2–13

during error minimisation, and the algorithm may fail to converge to the global minimum. In this paper, a genetic algorithm-based neural network (GABNN) is presented as a new approach to modelling of stress–strain behaviour of unsaturated soils, using basic soil properties such as initial gravimetric water content, initial dry density, and initial degree of saturation, together with net mean stress with respect to pore-air pressure, axial strain, suction and volumetric strain. A genetic algorithm (GA) was used to optimise the weights of the neural network. The following sections present an introduction to neural networks and genetic algorithms,

Fig. 1. A typical ANN (A1, . . . An = input nodes; B1, . . ., Cn = hidden nodes; b = bias nodes; D1, . . ., Dn = output nodes).

Input neuron

Input neuron

Input neuron updated based on output Fig. 2. Architecture of a typical sequential NN (some input neurons are updated based on the output).

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the developed hybrid genetic algorithm-based neural network and its application to the modelling of unsaturated soils, the model prediction results, and the comparison with the experimental data. 2. Neural network modelling In recent years, the use of artificial neural networks (ANNs) has been introduced as an effective alternative approach to the constitutive modelling of complex materials. An ANN is a computer-based modelling technique for computation and knowledge representation inspired by the neural architecture and operation of the human brain. Initially developed in early 1940s, ANNs have experienced a considerable resurgence of interest in recent years. The basic architecture of an ANN has been covered in many references (e.g., [18]. An ANN consists of a large number of highly interconnected processing units. Each processing unit (neuron), acting as an idealised neuron in the human brain, receives input from the units to which it is connected, computes an activation level and transmits that activation to other processing units (Fig. 1). A multi-layer perceptron neural network has an input layer, an output layer, and a number of hidden layers connected to each other. Additionally, a bias neuron lies in input and hidden layers. It is connected to all the neurons in the next layer but none in the previous layer, and it always emits 1 as illustrated in Fig. 1. Weights are assigned to the connections between these units. The presence of hidden layers allows the network to represent and compute complicated associations between input and output patterns. A multi-layer feed-forward ANN is initially trained using data to capture the underlying relationship within the data. The generalisation capabilities of the trained network are then evaluated by application to an unseen set of data. During training, weights of the network are adjusted in an iterative process. A forward pass of information through the network, including simple computations, results in the prediction of the output variables of the network. The knowledge stored in the developed network is represented by the set of connection weights and biases. The neural network is trained by appropriately modifying its connection weights through the set of ‘‘training cases” until the predicted output variables agree satisfactorily with the desired variables. The ‘‘back-propagation” term ([18]) refers to the algorithm by which the observed error in the predicted output variables is used to modify the connection weights. When the error measure of the network is reduced below a user-defined minimum, the training is stopped, and the connection weights are locked and recorded. When a trained ANN is presented with an input pattern, a feedforward network computation results in an output pattern that is the result of the generalisation and synthesis of what it has learned.

Fig. 3. Structure of a single population evolutionary GA.

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Fig. 4. Procedure for integrating the genetic algorithm and neural network.

Table 1 Range of soil properties of specimens. Properties

Range

Initial water content (%) Dry density (Mg/m3) Degree of saturation (%) Initial suction (kPa) Axial strain (%) Deviatoric stress (kPa) Excess suction (kPa) Volume strain (%) Net mean stress with respect to pore-air pressure (kPa)

17–26.3 1.442–1.716 46.5–95 2–547 0–11.52 0–930 111.8–13.3 7.5–0.35 23–347

εa (ε v ) i+1

Table 2 Initial conditions of soil specimens. Sample

MGU1 MGU2 MGU3 MGU4 MGU5 MGU6 MGU7 MGU8 MGU9 MGU10 MGU11 MGU12 MGU13 MGU14 MGU15 MGU21 MGU22 MGU23

(ε v )i

Water content (%)

Dry density (Mg/m3)

Degree of saturation (%)

Initial suction (kPa)

ua (kPa)

r3 (kPa)

Neural network status

19.6 25.5 20.8 21.4 20.7 21 17 21.1 18.9 25.1 24.9 21.9 20 26 25 26.3 24.3 25.8

1.442 1.632 1.531 1.551 1.646 1.489 1.474 1.587 1.625 1.508 1.506 1.716 1.674 1.706 1.702 1.498 1.708 1.705

51.4 84.9 61.1 64.4 70.2 58.5 46.5 66.4 62.4 71.6 70.8 81 70.2 95 90.9 74.1 89 94.2

384 4 149 22 105 256 450 186 407 11 26 161 457 5 12 2 78 54

496 247 297 227 300 445 450 304 400 300 299 300 500 299 300 300 450 299

552 302 350 300 353 500 500 352 451 350 350 350 550 350 399 350 473 324

Train Train Train Train Train Train Train Train Test Train Train Test Train Train Train Test Train Train

The use of ANNs that are constructed directly from the experimental data offers a fundamentally different approach to the modelling of the material behaviour. Because of their ability to learn and generalise interactions among many variables, ANNs have the potential to model various aspects of material behaviour. Therefore, ANNs have received considerable attention in recent years, with a wide range of applications in civil and geotechnical engineering. They are robust in modelling the nonlinear behaviour and dealing with noisy data. Some examples are the works of

Fig. 5. The proposed neural network.

Habibagahi [8] and Johari et al. [12]. Ellis et al. [4] modelled the stress–strain relation of sands using an ANN and showed good agreement between laboratory data and modelling results. A series of undrained triaxial tests on mortar sand was used to develop the models. Two different types of architecture were used to evaluate the ability of the ANN for modelling sand behaviour: the conventional neural network without feedback and the sequential neural network with feedback. In a sequential network (Fig. 2), a pattern is input to the plan units at the initial phase of the training. A feed-forward process occurs as in the standard back-propagation algorithm, producing the first output pattern. This output is then copied back to the current state units for the next feed-forward process. The sequential neural network has the potential to incorporate the path dependency of mechanical behaviour into the model. Another neural networkbased constitutive relationship was presented by Penumadu and Zhao [16] to model the stress–strain and volume change behaviour of sand and gravel under drained triaxial compression test conditions. Despite the considerable amount of research work on constitutive modelling of saturated soils, the application of ANNs in the

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A. Johari et al. / Computers and Geotechnics 38 (2011) 2–13 Table 3 Connection weights and biases for the proposed network. Weights between the input and hidden neurons Input neuron Hidden neuron

1

2

3

4

5

6

7

8

Bias

1 2 3 4 5

0.284 0.784 0.518 0.049 0.521

0.204 0.118 0.277 0.038 0.084

0.315 0.184 0.561 0.115 0.192

0.363 0.895 0.416 0.724 1.369

0.822 0.175 0.196 0.089 0.285

0.046 0.862 0.218 0.697 0.142

0.718 0.050 0.078 1.020 0.196

0.864 0.564 0.823 0.002 0.070

1.599 1.212 0.179 0.042 0.862

Weights between the hidden and output neurons Output neuron 1

2

3

0.098 0.340 0.305 0.010 0.953 0.208

0.056 0.019 0.082 1.111 0.102 0.104

0.342 0.206 1.201 0.008 0.197 0.767

modelling of unsaturated soil has been very rare and limited to a single paper by Habibagahi and Bamdad [9], which could be attributed to the complexities involved in the modelling of unsaturated soils and the lack of comprehensive experimental data. One of the drawbacks of a back-propagation neural network (BPNN) is that there is a possibility for the search to be trapped in local minima during error minimisation, and the algorithm may fail to find the global minimum of the error function. To overcome this deficiency, a genetic algorithm was employed in this study to minimise the error function in the neural network. This method was effective in finding the global or near global minimum.

Table 4 Normalised mean sum of squared error. Neuron

MGU1 MGU2 MGU3 MGU4 MGU5 MGU6 MGU7 MGU8 MGU9 MGU10 MGU11 MGU12 MGU13 MGU14 MGU15 MGU21 MGU22 MGU23

Mean

Deviatoric stress

Suction

Volumetric strain

0.00134 0.00627 0.00366 0.00360 0.00186 0.00359 0.00038 0.00695 0.00552 0.00220 0.00419 0.00201 0.00825 0.00683 0.00426 0.00162 0.00266 0.00855

0.00177 0.00001 0.00036 0.00205 0.00168 0.00168 0.00079 0.00068 0.00748 0.00031 0.00036 0.00148 0.00012 0.00021 0.00022 0.00209 0.00088 0.00062

0.00209 0.00055 0.00251 0.00239 0.00225 0.00179 0.00707 0.00100 0.00601 0.00340 0.00147 0.00417 0.00163 0.00152 0.00196 0.01029 0.00431 0.00223

0.00173 0.00228 0.00218 0.00268 0.00193 0.00235 0.00275 0.00288 0.00634 0.00197 0.00200 0.00255 0.00333 0.00285 0.00215 0.00467 0.00262 0.00379

Table 5 Correlation coefficient values (R2).

A genetic algorithm (GA) is a statistical method for optimisation and searching. The characteristics of this algorithm prevent it from being considered as a pure random search method. The idea of this method, which was inspired by the theory of natural evolution, was first proposed by Holland [10]. One important feature of a GA is its durability and adaptability because it provides a flexible balance between effectiveness and necessary characteristics for survival in different environments and conditions. If the adaptability of a system increases, it will be able to function longer and more effectively [7]. Searching in a GA is usually done with a group or population of binary or real strings, which are included in the form of decision variables. Each string is analogous to a chromosome, and each binary or real bit is analogous to a gene on that chromosome. The procedure of finding the best string can be described as follows. The objective (fitness) function is evaluated for each string, and then three operations, namely selection, crossover and mutation, act on some individuals in the population (Fig. 3). Using these operations, fitter members of the population are created over time, and the population evolves to optimal or near optimal solutions.

0.050 GABNN Model

0.045

ANN Model

0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 0.000

Neuron

Training Testing

3. Genetic algorithm

Normalized mean sum squre error

1 2 3 4 5 Bias

0

Deviatoric stress

Suction

Volumetric strain

0.986 0.965

0.998 0.996

0.992 0.988

250

500

750

1000

1250

1500

1750

Generation / Epock Fig. 6. Performance and comparison of the GABNN and ANN models during training.

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The first step is selection, which introduces a pressure towards fitter solutions by giving them a higher probability of being selected for the next iteration relative to less fit individuals. After selection, crossover takes place. This operation works on one pair of chromosomes. In general, this operation does not operate on all chromosomes of the population, as it may otherwise lead to an increased probability of the population losing certain characteristics. For this reason, using a user-specified probability (Pc), the crossover operator creates new solutions by exchanging genetic information between selected individuals. After crossover, the mutation operator randomly switches a bit or bits in the chromosomes with the user-specified probability, Pm. This specified probability is usually very small (less than 0.05). In binary coding, for example, this mutation operator will change a bit from zero to one and vice versa. Fig. 3 shows the structure of a simple GA. 4. Application of a GA in training neural networks In this paper, a new hybrid genetic algorithm-based neural network is presented that overcomes the shortcomings of the standard neural network by increasing the probability of locating the global optimum (minimum) for the error function. This procedure is shown in the flow chart in Fig. 4. In the proposed method, a GA was used to determine the optimal weights and structure of the neural network. In conventional ANNs, the ‘‘back propagation law” is used as a device for minimising the error function with respect to the connection weights of the layers. By using the GA, the search for the optimum set of weights and hidden layers was started from different points (equal to the number of individuals in the popula-

tion), and hence, reaching the global minimum was more probable. In this method, the error function (E) is defined as: P  2 1X Op  T p P p¼1

E¼

ð1Þ

where TP = target value for the pattern (P), OP = output value of network, given by:

Op ¼

Nh X

wjk

j¼1

Ni X

wij Ipi

ð2Þ

i¼1

wjk = weights between hidden and output layers, wij = weights between input and hidden layers, Ipi = input variable i for pattern (p), Ni = number of input neurons, and Nh = number of hidden neurons. The problem was formulated as an optimisation problem to find a set of weights for the ANN that minimised the difference between the ANN predictions and the target values in the training set of data. Different sets of weights were generated randomly and evolved using the GA, and the accuracy of the ANN model predictions was analysed to evaluate the fitness of each set of weights in competition with other generated sets. This methodology was very robust in finding optimum set of weights for the ANN model. 5. Database Results from a set of triaxial tests on Lateritic gravel reported by Toll [20] were adopted for the analysis. In the tests, the samples were sheared in compression under constant water content conditions, though the samples could change in volume due to the flow 450

400

400

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ua-uw (kPa)

300

Experment

200

GBNN Habibagahi and Bamdad (2003)

150

300 250 Experment

200

GBNN Habibagahi and Bamdad (2003)

150 MGU1 W=19.6 Dry Density = 1.442 Sr=51.4

100 50 0 0.0

100 50 0

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q (kPa)

250

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Axial Strain (%) Fig. 7. Best simulation results among tests used for training GABNN and ANN models. Normalised mean sum of squared error = 0.00173.

8.0

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A. Johari et al. / Computers and Geotechnics 38 (2011) 2–13

of air [21]. Table 1 indicates the range of basic soil properties. Table 2 shows the initial condition of soil specimens adopted for this study and also indicates whether the results of a particular test were used for training or testing the GABNN. This database consists of the results from 23 unsaturated specimens prepared using static or dynamic compression. However, for the sake of consistency, only 18 specimens prepared by static compression were considered in this investigation. The other five test specimens were prepared by a different procedure (dynamic compaction), and hence, they were not included in the database. The experimental results (graphs) presented in Toll [20] were digitised. Digitisation resulted in a database with a total of 6433 patterns that were used for training and testing the neural network. Before training, each component of the data set was normalised to lie in the interval of [1, 1] using a max–min approach.

mens, and the other five neurons, namely net mean stress with respect to pore-air pressure, axial strain, deviatoric stress, suction and volumetric strain, were updated incrementally during the training and testing of the network based on the outputs from the previous increment (Fig. 5). In the incremental modelling process, after an increment of axial strain from (ea)i to (ea)i+1, the parameters, qi ðua  uw Þii and (ev), were updated for the following increment directly from the output of the GABNN models:

qi ¼ qiþ1

ð3Þ

ðua  uw Þi ¼ ðua  uw Þiþ1 ðev Þi ¼ ðev Þiþ1

ð4Þ ð5Þ

The other parameter, p  ua, was updated using the following expression:

6. Network architecture A computer program, coded in Visual Basic, was developed for training the proposed genetic algorithm-based neural network (GABNN). In the proposed approach, a GA was used to optimise the weights of the neural network, and the topology of the neural network was determined by trial and error. Due to the incremental nature of soil stress–strain modelling in practical applications, it was necessary to use a type of network called a sequential neural network. The network had three layers with eight input, five hidden and three output neurons. In the input layer, the first three neurons, namely gravimetric water content, dry density and degree of saturation, represented the initial conditions of soil speci-

ðp  ua Þi ¼ ðp  ua Þi þ

qiþ1  qi 3

ð6Þ

All updating was done in response to an increment of axial strain given by the following expression:

ðea Þi ¼ ðea Þiþ1

ð7Þ

The output layer had three neurons, namely deviatoric stress, suction and volumetric strain, corresponding to the following incremental step. To find the optimum number of hidden neurons, the number of hidden neurons was decreased from a maximum of 10 neurons while checking the error measure of the network. This

450

300

400

Experiment

250

350

Habibagahi and Bamdad (2003)

ua-uw (kPa)

300 250 MGU6 W = 21.0 Dry Density = 1.489 Sr= 58.5

200 150

Experiment

100

GBNN Habibagahi and Bamdad (2003)

200 150 100 50

50 0

0.0

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Axial Strain (%)

3.0

4.0

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0.5 0.0 0.0 -0.5

Volumtric Strain (%)

q (kPa)

GBNN

1.0

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Axial Strain (%) Fig. 8. Near average simulation results among tests used for training GABNN and ANN models. Normalised mean sum of squared error = 0.0023.

8.0

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procedure resulted in a total of five neurons for the hidden layer. Fig. 5 shows the proposed network configuration. Other characteristics of the optimum GABNN model are described below. The recommended value of the crossover probability (Pc) for single-point crossover is generally between 0.7 and 0.95. In this research, different values of Pc within this range were examined by trial and error, and a value of 0.9 was found to be appropriate in terms of the diversity of the design variables in the search space and convergence of the algorithm. The mutation probability of chromosome i, Pmi , was selected by adopting a rule with the following formula (Pham and Karaboga [17]):

Pmi ¼ ð1  ð1=SSEp Þ  ððPmmax  Pmmin Þ þ Pmmin Þ

7. Results and discussion From the 18 triaxial tests used in this study, 15 tests were used to train the network, and the remaining three tests were used to test the generalisation capability of the trained network. Generally, in pattern recognition procedures (e.g., neural network, fuzzy logic or genetic programming), it is common that the model construction is based on adaptive learning over a number of cases, and the performance of the constructed model is then evaluated using an independent validation data set. In data mining techniques, the way in which the data are divided into training and validation sets has a significant effect on the results [11]. In this study, the entire database was divided into several random combinations of training and validation sets until a robust representation of the whole population was achieved for both training and validation sets. After digitisation of the test results, 5533 patterns from a total of 6433 patterns generated were used for model development, and the remaining 899 patterns were allocated for testing. To find the optimum network, the sum of squared differences between the predicted three output parameters and the actual values were monitored. Genetic algorithm generations continued until this error measure did not decrease appreciably. The optimal weights thus obtained are presented in Table 3. The mean sum of squared error for each output neuron and the mean of these errors were calculated for each sample individually, and the results are presented in Table 4. Also, the correlation coefficient value (R2) for each output neuron was calculated, and the results are presented in Table 5. Fig. 6 shows the variation of the

ð8Þ

where SSEp is the sum of squared error for member p of the population, P mmin is the minimum probability of mutation, selected as 0.001, and Pmmax is the maximum probability of mutation, selected as 0.01. The population size was 50 members, and the maximum number of generations was 2000. The transfer function in the hidden neurons was selected as the tangent sigmoidal function, given by the following equation:

  y ¼ 1 þ 2= 1 þ eð2xÞ

ð9Þ

The transfer function in the output neurons was a linear function.

350

60

300

50

ua-uw (kPa)

200 150

MGU23 W = 25.8 Dry Density = 1.705 Sr= 94.2

100

40 30

10 0

0 0

1

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50

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Axial Strain (%) Fig. 9. Worst simulation results among tests used for training GABNN and ANN models. Normalised mean sum of squared error = 0.00379.

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error measure during training by the GABNN and ANN (artificial neural network) models. We can see that the mean sum square error of GABNN model decreased significantly during the initial generations and remained almost constant after about 750 generations. Figs. 7–9 compare the predictions of the GABNN model, the ANN model of Habibagahi and Bamdad [9] and the experimental data for the three specimens used in training of the model. From these figures, it may be concluded that the proposed GABNN had a good potential for predicting stress–strain curves with a good accuracy. Similarly, Figs. 10–12 present the prediction of stress– strain curves for the three typical test specimens that were not used in the model construction process. The results were compared with those of the ANN model and the experimental results. From these figures, it may be concluded that the proposed method was also capable of generalising the training to simulate new test results, and it outperformed the ANN model. 8. Sensitivity analysis To evaluate the model’s response to changes in input parameters, a sensitivity analysis was carried out. For this purpose, a typical specimen with basic soil properties given in Table 6 was selected. Three input parameters, the dry density, initial net mean stress with respect to pore-air pressure (p  ua) and initial degree of saturation, were considered in the sensitivity analysis. To evaluate the influence of changes in the dry density, this parameter was changed within a range of ±5% while keeping the other parameters constant. The results are shown in Figs. 13–15. Fig. 13 shows the influence of the dry density on the stress–strain behaviour while other parameters are kept constant. As expected, an increase in dry density resulted in the stress–strain curve shifting upwards,

indicating that a sample with higher density had a higher rupture point and larger elastic zone. Fig. 14 shows the influence of the dry density on the soil suction. For a soil sample, increasing the dry density decreased the void ratio and decreased the soil suction, as correctly predicted by the model. Fig. 15 shows that an increase in dry density did not change the contraction behaviour appreciably. However, the rate of volumetric strain, especially in the main dilation part of the curve (related to the angle of dilation), increased considerably. Fig. 16 indicates the influence of p  ua on the stress–strain behaviour, while the other parameters are kept constant. An increase in p  ua shifted the stress–strain curve upwards, indicating that a sample with higher net mean stress became stiffer and stronger. Fig. 17 indicates the influence of p  ua on the soil suction. As expected, as p  ua decreased, the void ratio increased, and the soil suction increased. Fig. 18 shows that an increase in p  ua, increased contraction and decreased dilation in the soil sample, although the changes were small. To evaluate the influence of changes in the degree of saturation, this parameter was changed within a range of ±10% while keeping the other parameters constant. The results are shown in Figs. 19–21. Fig. 19 shows the influence of the degree of saturation on the stress–strain behaviour, while other parameters are kept constant. With an increase in the initial degree of saturation, the stress–strain curve shifted downwards, which shows that a sample with higher degree of saturation had a lower rupture point and smaller elastic modulus. Fig. 20 shows the influence of the degree of saturation on the soil suction. For a soil sample, increasing the degree of saturation decreased the soil suction, as correctly predicted by the model. Fig. 21 shows that, as expected, an increase in degree of saturation led to increased contraction in the soil sample.

800

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ua-uw (kPa)

150

MGU12 W=21.9 Dry Density = 1.716 Sr=81.0

400 300

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100

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200

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100

0 0.0

0 0.0

0.5

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Axial Strain (%)

Axial Strain (%) 1

Volumtric Strain (%)

q (kPa)

500

0 0.0 -1

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Experiment GABNN Habibagahi and Bamdad (2003)

-2 -3 -4 -5

Axial Strain (%) Fig. 10. Best simulation results among tests used for testing GABNN and ANN models. Normalised mean sum of squared error = 0.00255.

5.0

10

A. Johari et al. / Computers and Geotechnics 38 (2011) 2–13

500

50 MGU21 W=26.3 Dry Density = 1.498 Sr=74.1

30 20

Experiment GABNN Habibagahi and Bamdad (2003)

300

ua -uw (kPa)

q (kPa)

400

40

200

10 0 0.0 -10

2.0

4.0

6.0

-20 100

8.0

10.0

12.0

Experiment GABNN

-30

Habibagahi and Bamdad (2003)

-40 0 0.0

2.0

4.0

6.0

8.0

10.0

-50

12.0

Axial Strain (%)

Axial Strain (%) 4 3

Experiment

Volumtric Strain (%)

GABNN Habibagahi and Bamdad (2003)

2 1 0 0.0 -1

2.0

4.0

6.0

8.0

10.0

12.0

-2 -3 -4

Axial Strain (%) Fig. 11. Near average simulation results among tests used for testing GABNN and ANN models.

1000 490

Experiment GABNN Habibagahi and Bamdad (2003)

470 450

600

MGU9 W=18.9 Dry Density = 1.625 Sr=62.4

500 400

Experiment GABNN Habibagahi and Bamdad (2003)

300

ua-uw (kPa)

800 700

430 410 390

200 370

100 0 0.0

2.0

4.0

6.0

8.0

10.0

350 0.0

12.0

2.0

4.0

6.0

8.0

10.0

Axial Strain (%)

Axial Strain (%) 1 0

0.0

Volumtric Strain (%)

q (kPa)

900

-1

2.0

4.0

6.0

8.0

10.0

12.0

Experiment GABNN

-2

Habibagahi and Bamdad (2003)

-3 -4 -5 -6

Axial Strain (%) Fig. 12. Worst simulation results among tests used for testing GABNN and ANN models. Normalised mean sum of squared error = 0.00634.

12.0

11

A. Johari et al. / Computers and Geotechnics 38 (2011) 2–13 Table 6 Basic soil properties of the specimen used for sensitivity analysis.

700 Value

Initial water content (%) Dry density (Mg/m3) Degree of saturation (%) Initial suction (kPa) Mean effective stress with respect to pore-air pressure (kPa)

20.7 1.646 70.2 105 52.448

600 500

q (kPa)

Properties

400 300

700 200

600

q (kPa)

MGU5 W=20.7 Dry Density = 1.646 Sr=70.2

500

100

400

0

Real Prediction P-ua - 50kPa P-ua + 50kPa

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Axial Strain (%)

300 MGU5 W=20.7 Dry Density = 1.646 Sr=70.2 Real Prediction Dry Density - 5% Dry Density + 5%

200 100

Fig. 16. Influence of p  ua on the stress–strain curve.

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6 150.0

Axial Strain (%) Fig. 13. Influence of the dry density on stress–strain curve.

100.0

Ua-Uw (kPa)

150.0

Ua-Uw (kPa)

100.0

50.0

Real Prediction

50.0

Real Prediction P-ua - 50kPa P-ua + 50kPa

0.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

Dry Density - 5% Dry Density + 5%

0.0 0.0

1.0

2.0

3.0

4.0

5.0

-50.0 6.0 -100.0

-50.0

Axial Strain (%) Fig. 17. Influence of p  ua on the suction–strain curve.

-100.0

Axial Strain (%)

1.0

1.0

0.5

0.5

0.0 0.0 -0.5

0.0 1.0

2.0

3.0

4.0

5.0

6.0

Real Prediction Dry Density - 5% Dry Density + 5%

-1.0 -1.5 -2.0

Volumtric Strain (%)

Volumtric Strain (%)

Fig. 14. Influence of the dry density on the suction–strain curve.

0.0

1.0

2.0

3.0

4.0

-0.5

5.0 Real Prediction P-ua - 50kPa

-1.0

P-ua + 50kPa

-1.5 -2.0

-2.5

-2.5

-3.0

-3.0 -3.5

-3.5

Axial Strain (%) Fig. 15. Influence of the dry density on the volumetric strain–strain curve.

Axial Strain (%) Fig. 18. Influence of p  ua on the volumetric strain–strain curve.

6.0

12

A. Johari et al. / Computers and Geotechnics 38 (2011) 2–13

600

500

q (kPa)

400

300 MGU5 W=20.7 Dry Density = 1.646 Sr=70.2

200

100

Real Prediction Degree of Saturation - 10% Degree of Saturationy + 10%

0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Axial Strain (%) Fig. 19. Influence of the degree of saturation on the stress–strain curve.

150.0

Ua-Uw (kPa)

100.0

50.0

0.0 0.0

1.0

2.0

3.0

4.0

5.0

6.0

Real Prediction

-50.0

Degree of Saturationy + 10% Degree of Saturation - 10%

-100.0

Axial Strain (%) Fig. 20. Influence of the initial degree of saturation on the suction–strain curve.

1.0

References

0.5

[1] Alonso EE, Gens A, Josa A. A constitutive model for partially saturated soils. Geotechnique 1990;40(3):405–30. [2] Bishop AW, Donald IB. The experimental study of partly saturated soil in triaxial apparatus. In: Proc 5th international conference on soil mechanic and foundation engineering, Paris, vol. 1; 1961. p. 13–21. [3] Chateau X, Dormieux L. The behavior of unsaturated porous media in the light of micromechanical approach. In: Rahardjo H, Toll DG, Leong EC, Balkema AA, editors. Proceedings, unsaturated soils for Asia, Singapore, 18–19 May, Rotterdam; 2000. p. 95–100. [4] Ellis GW, Yao C, Zhao R, Penumadu D. Stress–strain modeling of sands using artificial neural networks. J Geotech Eng, ASCE 1995;121(5):429–35. [5] Fredlund DG, Morgenstern NR. Stress–strain variables for unsaturated soils. J Geotech Eng, ASCE 1977;103(GT5):447–66. [6] Gens A, Alonso EE. A framework for the behavior of unsaturated expansive clays. Can Geotech J 1992;29:1013–32. [7] Goldberg D. Genetic algorithms in machine learning optimization and search. Addison-Wesley; 1988. [8] Habibagahi G. Reservoir induced earthquakes analyzed viaradial basis function networks. Soil Dynamics Earthquake Eng 1998;17(1):53–6. [9] Habibagahi G, Bamdad AA. Neural network framework for mechanical behavior of unsaturated soils. Can Geotech J 2003;40:684–93. [10] Holland JH. Adaptation in natural and artificial system. Ann Arbr (MIT): University of Michigan Press; 1975. [11] Javadi AA, Rezania M, Nezhad MM. Evaluation of liquefaction induced lateral displacements using genetic programming. Comput Geotech 2006;33(4–5): 222–33. [12] Johari A, Habibagahi G, Ghahramani A. Prediction of soil–water characteristic curve using genetic programming. J Geotech Eng, ASCE 2006;132(5):661–5. [13] Khalili N. Application of effective stress principle to volume change in unsaturated soils. In: Rahardjo H, Toll DG, Leong EC, editors. Proceedings, unsaturated soils for Asia, Singapore, 18–19 May. Rotterdam: A.A. Balkema; 2000. p. 119–24.

0.0

Volumtric Strain (%)

starts from different starting points (equal to the number of individuals in the population), and therefore, obtaining the global or near global minimum is more probable. Indeed, the method is generally considered to give the global extremum. Hence, in this paper, a GA was used to minimise the neural network error function. A database containing the results from triaxial tests on samples of compacted unsaturated granular soils was employed to develop the model. The model was trained using the results from 15 triaxial tests and tested using three additional test results that had not been exposed to the network during the training phase. The model predictions indicated a good accuracy for the results used in both the training and testing phases. The patterns used for testing were complete curves corresponding to a number of tests. In these tests, the entire stress paths were predicted incrementally, point-by-point, for the entire test. In this way, all the points on a curve were predicted incrementally by the model, and the errors of predictions were accumulated pointby-point. Despite this accumulation of errors at subsequent points, the fact that the predicted curves were very close to the experimental results is in itself a testament to the robustness of the proposed models. A sensitivity analysis of the model indicated that the developed model predicted correctly the influence of various parameters, such as the dry density, initial net mean stress and degree of saturation. The results indicated that although the model was trained directly from data, it was able to capture the expected physical relationships between the parameters correctly. The computational time for the GABNN method was about 15–20 times greater than that for an ANN run. However, the main advantage of the proposed method is in providing greater accuracy and better convergence of the ANN model. Considering that for the database used in this study, the typical computational time for training of a GABNN model was only about 15 min on a Pentium 4 PC with 3 GHz of RAM, and also considering the availability of very fast computers and rapid developments in computational hardware and software, the relative increase in computational time is not an issue. Furthermore, the computational time can be significantly reduced if parallel computation is adopted.

0.0 -0.5

1.0

2.0

3.0

4.0

5.0

6.0

-1.0 -1.5 -2.0 -2.5 -3.0 -3.5

Real Prediction Degree of Saturationy - 10% Degree of Saturationy + 10%

-4.0

Axial Strain (%) Fig. 21. Influence of the degree of saturation on the volumetric strain–strain curve.

9. Conclusion In conventional neural networks, ‘‘back propagation of error” is used to determine the optimum weights connecting the layers. Despite the various measures available to prevent trapping in local minima, experience has shown that this is not always successful. However, using a GA as the training rule, function minimisation

A. Johari et al. / Computers and Geotechnics 38 (2011) 2–13 [14] Khalili N, Habte M, Zargargashi S. A fully coupled flow deformation model for cyclic analysis of unsaturated soils including hydraulic and mechanical hystereses. Comput Geotech 2008;35(6):872–89. [15] Masin D, Khalili N. A hypoplastic model for mechanical response of unsaturated soils. Int J Numer Anal Meth Geomech 2008;32(15):1903–26. [16] Penumadu D, Zhao R. Triaxial compression behavior of sand and gravel using artificial neural networks (ANN). Comput Geotech 1999;24:207–30. [17] Pham DT, Karaboga D. Intelligent optimization techniques. London: SpringerVerlag; 2000.

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