Stress-Strain Prediction of Jointed Rocks using Artificial Neural Networks

th

The 12 International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India

Stress-Strain Prediction of Jointed Rocks using Artificial Neural Networks G. Arunakumari PhD Scholar, Dept. of Civil Engineering, Indian Institute of Science Bangalore, India

G. Madhavi Latha Asst. Professor, Dept. of Civil Engineering, Indian Institute of Science Bangalore, India Keywords: jointed rocks, joint properties, stress-strain behavior, ANN, training and testing ABSTRACT: The applicability of Artificial Neural Networks for predicting the stress-strain response of jointed rocks at varied confining pressures, strength properties and joint properties (frequency, orientation and strength of joints) has been studied in the present paper. The database is formed from the triaxial compression tests on different jointed rocks with different confining pressures and different joint properties reported by various researchers. This input data covers a wide range of rock strengths, varying from very soft to very hard. The network was trained using a 3 layered network with feed forward back propagation algorithm. About 85% of the data was used for training and remaining15% for testing the predicting capabilities of the network. Results from the analyses were very encouraging and demonstrated that the neural network approach is efficient in capturing the complex stress-strain behaviour of jointed rocks. A single neural network is demonstrated to be capable of predicting the stress-strain response of different rocks, whose intact strength vary from 11.32 MPa to 123 MPa and spacing of joints vary from 10 cm to 100 cm for confining pressures ranging from 0 to 13.8 MPa.

1 Introduction Numerical modelling of the jointed rock masses using discrete and continuum modelling approach is investigated by several researchers like Cundall (1971), Hoek and Brown (1997), Singh (1973a, 1973b), Cai and Horii (1992). Some researchers have also developed empirical relations to estimate the equivalent material properties of the jointed rock mass from the geometrical and mechanical properties of discontinuities. These equations can be incorporated in other constitutive models for effective simulation of jointed rock masses, e.g. Barton and Bandis (1990), Hoek and Brown (1997), Ramamurthy (1993). The equivalent continuum modelling approach has gained lot of popularity because of its simplicity in application. Sitharam and Madhavi Latha (2002) Madhavi Latha and Sitharam (2004), Arunakumari and Madhavi Latha (2007a, 2007b) have carried out numerical simulation of element tests on jointed rocks under triaxial compression using equivalent continuum approach suggested by Ramamurthy et al. (1993) and studied the applicability of this model for an under ground cavern. These studies revealed some deviation of the numerical predictions from the observed results especially for nonlinear stressstrain behaviour of rocks and low confining pressures. Henceforth an attempt has been made in the present paper to demonstrate the applicability of Artificial Neural Networks for the better prediction of stress-strain behaviour of jointed rocks given the intact rock properties, joint properties and confining pressure. Tedious process of implementing the complex constitutive behaviour and failure criterion into a numerical model has been overcome by using Artificial Neural Networks. The database consists of five types of jointed rocks namely Plaster of Paris, block jointed Gypsum Plaster, Jamrani Sandstone, Agra Sandstone and Granite tested under triaxial compression under a wide range of confining pressures at different joint orientations and joint frequencies. The data is trained and tested using a three layered feed forward back propagation algorithm in a visual basic based tool box WinNN. 85% of the data has been used for the training and the remaining 15% for the testing. The data has been selected at regular intervals in order to maintain the statistical consistencies (Shahin et al., 2000). The entire database consists of 845 datasets, out of which 735 datasets have been used for training and the remaining 110 datasets for testing the performance of the network. Results from the ANN analyses have been compared against the original experimental values and the applicability of Artificial Neural Networks for the efficient prediction of stress-strain response of jointed rocks is demonstrated.

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2 Stress-strain prediction of jointed rocks using artificial neural networks 2.1 Artificial Neural Network Approach Artificial Neural Networks are computational models in which the mode of data processing replicates the mode of synaptic dynamics in the biological neural networks. ANNs consist of closely interconnected numerical processing units called neurons, which offer a rich structure exhibiting some properties of biological neural networks (Yegnanarayana 1999). Artificial Neural Networks have been used for a wide variety of applications in Geomechanics and Rock Engineering (Toll, 1996; Jan et al. 2002; Sitharam et al. 2003). The primary goal of training the network is to minimize the error function by searching for a set of connection strengths and threshold values that cause the ANN to produce outputs that are equal or close to targets. In order for an ANN to generate an output vector ‘Y’ that is as close as possible to the target vector ‘T’, a training process, also called learning, is employed to find optimal weight matrices ‘W’ and bias vectors ‘V’, that minimize a predetermined error function that usually has the form:

E=

∑∑ ( y P

i

− ti )

2

(1)

p

Training is a process by which the connection weights of an ANN are adapted through a continuous process of stimulation by the environment in which the network is embedded. There are primarily two types of training, supervised and unsupervised learning. A supervised training algorithm requires an external teacher to guide the training process. The inputs are cause variables of a system and the outputs are the effect variables. The training procedure involves the iterative adjustment and optimization of connection weights and threshold values for each of nodes. Most Geomechanics and rock engineering applications have utilized supervised training and the same is adapted in this study. The choice of type of particular algorithm is completely application dependent. Though there are several types of algorithms available in literature for the use of training the ANN, only Back-Propagation algorithm has been used in the present paper. Back-propagation is the most popular algorithm for training ANNs. It is essentially a gradient descent technique that minimizes the network error function. Each input pattern of the training data set is passed through the network from the input layer to the output layer. The network output is compared with the desired target output, and an error is computed based on (1). This error is propagated backward through the network to each node, and correspondingly the connection weights are adjusted based on equation

Δwij (n ) = −ε *

∂E + α * Δwij (n − 1) ∂wij

(2)

where Δ wij (n) and Δ wij (n-1) are weight increments between node i and j during the nth and (n- 1)th pass, or epoch. A similar equation is written for correction of bias values. In (2), ε and α are called learning rate and momentum, respectively. The back-propagation algorithm involves two steps. The first step is a forward pass, in which the effect of the input is passed forward through the network to reach the output layer. After the error is computed, a second step starts backward through the network. The errors at the output layer are propagated back toward the input layer with the weights being modified according to (2).

2.2 Data processing An optimal data set should be representative of the probable occurrence of an input vector and should facilitate the mapping of the underlying nonlinear process. The data from experimental results on triaxial compression tests on jointed rocks reported by various researchers in the literature forms the database in the present paper. Large amount of experimental data on jointed rock samples under triaxial compression tested under various confining pressures and at different joint inclinations and frequencies has been collected from the literature for this purpose. The jointed rock types were namely, Plaster of Paris, Gypsum Plaster, Jamrani sandstone, Agra Sandstone and Granite (experimental data after Arora 1987; Brown and Trollope 1970; Roy 1993; Yaji 1984). The intact rock properties of the five rocks and the confining pressures, joint orientations and joint frequencies under which the tests carried out are shown in Table 1. The database includes the stress-strain results on very low strength Plaster of Paris (11.32 MPa) to a very high strength Granite (123 MPa) tested at confining pressures ranging from 0 to 13.8 MPa. Performance of an ANN model is also affected by the proportion of the data chosen for the testing and training datasets. It is indispensable to consider the statistical properties, like the mean and standard deviation, of the training and testing datasets to ensure that each dataset represents the same population (Shahin et al. 2000). The statistical method of division presents a very simple yet effective mode of data division. The data was divided into training and testing datasets using sorting methods, to maintain statistical consistency. In the present paper, out of 845 data sets, 15% (110 data sets) of the database formed the testing data remaining 85% (735 datasets) of the database formed the training data for ANN. Results from the statistical analysis on all the parameters is shown in Table 2.

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Table 1. Properties of intact rock, joint properties and confining pressures used in the numerical analysis. Property

Plaster of Paris

Gypsum Plaster

Jamrani Sandstone

Agra Sandstone

Granite

Uniaxial comp. strength of intact rock (σci)

11.32

20

55.07

110

123

Confining Pressure (σ3) MPa

2.0, 5.0

1.4, 3.4, 6.9, 13.8

2.5, 5.0, 10.0

2.5, 5.0

1.0, 2.5, 5.0

Number of joints in sample

1

2

3

7 to 8

1

3

1

2

3

1

Orientation of joints (β°)

30o,

50o

50o

30o/60o Block jointed

30o,

50o

30o, 50o , 70o

50o, 70o

50o, 70o

60o,75o, 90o

Joint frequency (Jn)

13

26

39

20

13

39

13

26

39

13

Joint factor (Jf)

520

168

252

590

306

148

289

93

140

28,19

47,23

46

69

13

50o

50o

84 Number of training datasets used

50 388

26

183

185

63

Table 2. Statistical analysis of data on all types of rocks. Parameter

Data Set

Mean

Median

Standard Deviation

Max. Value

Min. Value

σ3

Training

4.6099

5

3.04986

13.8

1

Testing

4.3786

5

2.82156

10

1

Training

49.7325

20

43.40349

123

11.32

Testing

55.99145

55.07

44.9859

123

11.32

Training

59.09091

60

19.284

90

0

Testing

53.54

60

19.3433

80

30

Training

28.1493

26

16.2257

66

13

Testing

30.8547

13

21.38371

66

13

Training

0.458175

0.46

0.2932

1

0.05

Testing

0.516154

0.63

0.342558

1

0.05

Training

0.5

0.5

0

0.5

0.5

Testing

0.5

0.5

0

0.5

0.5

Training

0.708112

0.5518

0.620677

4

0

Testing

0.852591

0.6654

0.774758

4

0

Training

25.4133

11.4381

33.013

166.85

0

Testing

28.64364

14.0648

34.2158

135.8489

0

σci

β

Jn

n

r

ε

σcj

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2.3 Input and output parameters The selection of an appropriate input vector that will allow an ANN to successfully map to the desired output vector is not a trivial task. Unlike physically-based models, the set of variables that influence the system are not known a priori. In this sense of nonlinear process identification, an ANN should not be considered as a mere black box. A firm understanding of the engineering application under consideration is an important prerequisite for successful application of ANNs. This will help in avoiding loss of information that may result if key input variables are omitted, and also prevent inclusion of spurious inputs that tend to confuse the training process. The input variables that do not have a significant effect on the performance of an ANN can be eliminated from the input vector, resulting in a more compact network. In the present paper, the input variables are confining pressure (σ3), uniaxial compressive strength of intact rock (σci), joint orientation (β), joint inclination parameter (n), joint roughness parameter (r) and axial strain (ε) which are chosen on the basis of Ramamurthy’s (1993) joint factor (Jf) model. The output parameter is a single entity i.e. deviatoric stress (σ cj).

2.4 ANN model Determination of the ANN architecture and selection of a training algorithm is an important step. A feed forward back propagation algorithm is used in the present study. The number of hidden layer neurons significantly influences the performance of a network; with too few nodes the network will approximate poorly, while with too many nodes it will over fit the training data. In the present paper the stress-strain behaviour of five different jointed rocks is predicted given the confining pressure, properties of intact rock and joint properties. Single network has been used to predict the stress-strain behaviour of all five jointed rocks put together. For this purpose, data has been taken from all the five types of rocks at regular intervals. The network was trained using the feed forward– back propagation algorithm in WinNN. The basic architecture used and the training parameters of the ANN are shown in Table 3. Figure 1 shows configuration of typical network architecture used in the present study. Table 3. ANN architecture. Parameter Number of layers

ANN 3

Number of neurons in Input layer

7

Number of neurons in Hidden layer

5

Number of neurons in output layer

1

Size of training dataset

735

Size of testing dataset

110

Number of epochs

100000

Learning rate

0.1

Momentum Target error

0.4 10%

Figure 1. Configuration of three layered network used.

3 Results and discussions This section of the paper presents discussion of the results obtained from ANN analysis in detail. ANNs are used to predict the stress-strain behaviour of five different types of jointed rocks namely Plaster of Paris, Gypsum Plaster, Jamrani Sandstone, Agra Sandstone and Granite under triaxial compression. The uniaxial compressive

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strengths of the respective jointed rocks are 11.32, 20, 55.07, 110 and 123 MPa respectively. The analyses have been carried out under a wide range of confining pressures of 0 MPa to 13.8 MPa and also for wide range of joint orientations of 30o, 50o, 70o, 75o and 90o. The joint frequency varied from 13 to 39. The efficiency of ANN model in predicting the nonlinear and complex stress-strain behaviour of jointed rocks in triaxial compression is shown in Figures 2 to 5.

3.1 Effect of confining pressure (σ 3) The capability of ANN model to predict the behaviour under different confining pressures is shown in Figures 2 (a), (b) and (c) for the case of Agra Sandstone, block jointed Gypsum Plaster and Jamrani Sandstone respectively. From Figure 2 it can be identified that ANN is capturing the effect of confining pressure on the overall behaviour of the jointed rock very well for the case of Agra Sandstone and Jamrani Sandstone. Little deviation from the experimental values is observed in case of Gypsum Plaster under lower confining pressures i.e. 3.4 MPa. However the deviation is relatively small and can be acceptable.

σ3=10 MPa

140

100 80

40

3

σ3=6.9 MPa

20

σ3=0

40

60

30

σ3=2.5 MPa

60

σ3=13.8 MPa

50

σ3=7.5 MPa σ3=5 MPa

50

120

Experiment ANN

60

Agra Sandstone

Deviatoric Stress (MPa)


160

ANN Experimental

20

σ3=3.4 MPa

10

0

0

0

0.5

1 Axial Strain (%)

1.5

2

0.01

0

(a) Agra Sandstone

0.04

0.05

(b) Gypsum Plaster

80

Jamrani-1Joint

σ3=10 MPa

70 Deviatoric Stress (MPa)

0.02 0.03 Axial Strain

30o

60

ANN

50

Experimental

40

σ3=5 MPa

30 20 σ3=2.5 MPa

10 0 0

0.005

0.01


0.025

0.03

0.035

(c) Jamrani Sandstone Figure 2. Prediction of stress-strain response of rocks at different confining pressures.

3.2 Effect of joint orientation (β) Effect of joint orientation on the stress-strain behaviour of jointed rocks is shown in Figures 3a and 3b respectively for the case of Agra Sandstone and Granite for different values of β. The most vital parameter influencing the behaviour of jointed rocks is the joint orientation. Figure 3 depicts the efficiency of ANN model to capture the response of jointed rocks for different joint orientations ranging from most critical orientation of 30o to o the horizontal joint 90 . It can be observed that the values predicted by ANN are closely following the experimental stress-strain response at all strain levels for both the rock types considered.

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160

Agra Sandstone 1Jointed σ3=5 MPa β=70

140

30o

50o

70o

120 100 80

β=30o

60 40 ANN Experimental

20 0.005 0.01 Axial Strain

σ3=5 MPa

90o

120 100

ANN Experimental

80 60 75

40 20

0 0

Granite-1Joint

140

β=50o Deviatoric Stress (MPa)


160

o

0

0.015

0

(a) Agra Sandstone


0.015

(b) Granite

Figure 3. Prediction of stress-strain response of rocks at different joint orientations.

3.3 Effect of joint frequency (Jn) Figures 4a and 4b show the predictions of ANN in capturing the effect of joint frequency on the stress-strain response of jointed rocks for the case of Agra Sandstone and Plaster of Paris respectively. The stress-strain behaviour predicted by ANN for Agra Sandstone (70o joint inclination) and Plaster of Paris (50o joint inclination) are compared with the experimental results. In case of Agra Sandstone the results are matching very well with the experimental values. But in case of Plaster of Paris ANN is predicting slightly stiffer behaviour than experimental values for single jointed specimen. For two and three joint specimens, the ANN predictions are very close to experimental observations. Overall, the agreement between predicted and measured response is good and hence the ANN model is effective in predicting the variation in stress-strain response with varying number of joints per unit depth. 160

Agra Sandstone-σ3=5 MPa

70o

DeviatoricStress (MPa)

120

70o


1-Joint 2-Joints

140 70o

3-Joints

100 80 60 40

Experimental

20

ANN

0 0


0.015

Plaster of Paris σ3=5 MPa

25

50

50

50

20 15 Experimental-1-Joint

10

ANN-1-Joint Experimental-2Joints ANN-2-Joints

5 0 0

a) Agra Sandstone


0.015

(b) Plaster of Paris

Figure 4. Stress-strain comparisons for Agra Sandstone and Plaster of Paris at different joint frequencies.

3.4 Effect of uniaxial compressive strength of intact rock (σ ci) Figure 5 shows the prediction of stress-strain response for four types of jointed rocks, whose intact rock strength vary from 11.32 MPa (Plaster of Paris) to 110 MPa (Agra Sandstone) using a single network. The joint orientation o is 30 for all cases, which is most critical angle. It is observed from the figure that the ANN model is very efficient in capturing the stress-strain behaviour for all the cases.

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140

Agra Sandstone σ3=5 MPa

σci Varied Experimental


120

30o

ANN

100 80 60 Jamrani Sandstone σ3=5 MPa 40 Plaster of Paris σ3=5 MPa

20

Gypsum Plaster σ3=3.4 MPa

0 0

0.005

0.01

0.015


0.03

0.035

0.04

0.045

Figure 5. Stress-strain comparisons for different types of rocks for joint inclination of 30o. As understood from the analyses, the effect of various joint parameters on the stress-strain behaviour of different types of jointed rocks is predicted very well by the ANN model. The reason for this is the highly nonlinear structure and rich pattern recognition inherent in neural networks, which allows the predicted stress-strain response to follow experimental curve more closely unlike the conventional trendsetting methods. Also the use of ANN avoids the tedious mathematical representations involved in modelling the constitutive behaviour and failure mechanisms. This study successfully demonstrates the applicability of ANN model for predicting the stress-strain response of jointed rocks. Further refinement of the model and extension of database including more varieties of rock types could still improve the capabilities of the model.

4 Conclusions A neural network model to predict the stress-strain response of different jointed rocks is developed and its applicability to several rock types is tested. It is observed from the study that the ANN model is capable of predicting the complex stress-strain response of the jointed rocks very well. The efficiency of the model to capture the stress-strain behaviour for wide range of rock types at various confining pressures, different joint orientations and different joint frequencies has been brought out in this paper. The study demonstrated that a single network can be used for predicting the stress-strain response of jointed rocks given the intact rock strength, joint properties and confining pressure as input. Tedious process of either incorporating all joint sets and properties in the analyses as done in case of discrete modelling or implementing the complex constitutive behaviour and failure criterion into a numerical model as done in case of equivalent continuum modelling has been overcome by using Artificial Neural Networks without compromising in the accuracy of results. Neural network approach could be employed to rock mechanics applications like underground excavations and slopes in calculating the reduced strength of jointed rock mass due to the inherent presence of discontinuities. Hence the approach can replace the complicated programs and subroutines to be written to incorporate the complex behaviour of jointed rocks by a simple neural network code.

5 References Arora V.K. 1987. Strength and deformation behaviour of jointed rocks. Ph.D. Thesis, Indian Institute of Technology, Delhi, India Arunakumari G., Madhavi Latha G. 2007a. Effect of joint parameters on the stress-strain response of rocks. Proc. 11th Cong. of the Int. Society for Rock Mechanics, Lisbon (Portugal), Taylor & Francis group, London, 243-246. Arunakumari G., Madhavi Latha G. 2007b. Parametric studies of an underground excavation in jointed rock mass, Numerical Models in Geomechanics - NUMOG X, Pande G.N. & Pietruszczak S. (eds.),Taylor and Francis Group, London, 399-404. Barton N., Bandis S. C. 1983. Effects of block size on the shear behavior of jointed rock, In Proc. 23rd U.S. Symp. Rock Mech., Berkeley. CA., Goodman R.E. and Heuze F.E. (eds.), American Soc. Min. Eng., New York, 739-760. Brown E.T., Trollope D.H. 1970. Strength of model of jointed rock, Journal Soil Mechanics and Foundation Division, ASCE, Vol. 96 (SM2), 685-704. Cai M., Horii H. 1992. A constitutive model of highly jointed rock masses, Mechanics of Materials, Vol. 13, 217-246. Cundall P.A. 1971. A Computer Model for Simulating Progressive Large Scale Movements of Blocky Rock Systems. Proc. ASCE, Symp. Int. Soc. Rock. Mech., Nancy (France), Vol. 1, 132-150.

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Duncan J.M., Chang C.Y. 1970. Non-linear analysis of stress and strain in soils, Journal of Soil Mechanics and Foundation Engineering, ASCE, Vol. 5, 1629-1652 Jan J. C., Shin-Lin Hung M., Chi S. Y., Chern J. C. 2002. Neural network forecast model in deep excavation. Journal of Computing in Civil Engineering, ASCE, Vol. 16 (1), 59-65. Hoek E., Brown E.T. 1997. Practical estimates of rock mass strength, Int. J. Rock Mech. Min. Sciences. Vol. 34 (8), 11651186. Madhavi Latha G., Sitharam T.G. 2004. Comparison of failure criterion for jointed rock masses. Int. J. Rock Mechanics and Mining Sciences, Vol. 41 (3), CDROM. Ramamurthy T.1993. Strength and modulus responses of anisotropic rocks, Comprehensive Rock Engineering, Hudson J. A. (ed.), Vol. 1 (13), 313-329. Roy N., 1993. Engineering behaviour of rock masses through study of jointed models, Ph. D. Thesis, Indian Institute of Technology, Delhi, India. Shahin M.A., Jaksa M.B., Maier H.R. 2000. Predicting the settlement of shallow foundations on cohesionless soils using backpropagation neural networks. Research report No. R 167, Dept. of Civil and Environmental Engg., University of Adelaide, Australia.. Singh B. 1973a. Continuum characterization of jointed rock masses Part I-The constitutive equations. Int. Journal of Rock Mechanics and Mining Sciences- Abstracts. Vol. 10, 311-335. Singh B. 1973b. Continuum characterization of jointed rock masses Part II-Significance of low shear modulus. Int. Journal of Rock Mechanics and Mining Sciences- Abstracts., Vol. 10, 337-349. Sitharam T.G., Madhavi Latha G. 2002. Simulation of excavations in jointed rock masses using a practical equivalent continuum approach. Int. Journal of Rock Mechanics and Mining Sciences, Vol. 39 (4), 517-525. Sitharam T.G., Shailendra K., Madhavi Latha G., Madhusudan N. 2003. Intelligent Prediction of Elastic Properties of Jointed Rocks. Journal of Rock Mech. and Tunneling Tech., Vol. 9 (1), 11-34. Toll D. 1996. Artificial intelligence applications in geotechnical engineering. Electronic Journal of Geotechnical Engg., Vol. 1, Available online at http://www.ejge.com/1996/Ppr9608/Ppr9608.html Yegnanarayana B. 1999. Artificial Neural Networks. Prentice-Hall of India private Limited, New Delhi.. Yaji R.K. 1984. Shear strength and deformation of jointed rocks. Ph.D. Thesis, Indian Institute of Technology, Delhi, India.

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