Artificial neural network ann prediction of one-dimensional consolidation

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Artificial neural network ann prediction of one-dimensional consolidation

  1. 1. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 135 ARTIFICIAL NEURAL NETWORK (ANN) PREDICTION OF ONE- DIMENSIONAL CONSOLIDATION IN HOMOGENEOUS CLAY UNDER UNIFORM APPLIED LOAD Hyeong-Joo Kim1 , Jose Leo Mission2 , Jeong-Hee Ko3 1 Ph.D. Professor, Department of Civil Engineering, Kunsan National University, Kunsan City, South Korea, 2 Ph.D, Geotechnical Engineer, SK Engineering and Construction (SK E&C), Seoul, South Korea, 3 Ph.D. Candidate, Department of Civil and Environmental Engineering, Kunsan National University, Kunsan City, South Korea. ABSTRACT The prediction of one-dimensional (1D) consolidation in homogeneous clay is typically performed by lengthy calculations using the Finite Difference Method (FDM) or manually using tables and design charts. In addition, numerical solutions by FDM are typically made in a stepwise progressing manner in which the needed computational resources in terms of processing time, memory, and storage requirements accumulates due to the large number of iterations involved. This study presents the application of Artificial Neural Network (ANN) in the prediction of 1D consolidation in homogeneous clay under uniform applied load. Aside from predicting consolidation results comparable to the FDM, ANN offers several advantages as an accurate, direct, and quick tool for prediction of 1D consolidation with less needed computational resources. Two ANN models were being developed and presented in this study: net1 for consolidation in single drainage boundary conditions, and net2 for double drainage conditions, which were further validated in a deployed environment for the prediction of excess pore water pressure and settlement in field conditions. Keywords: Artificial Neural Network (ANN), One-dimensional consolidation, Finite difference method (FDM), Homogeneous clay, Single drainage, Double drainage 1. INTRODUCTION In one-dimensional (1D) consolidation problems, the typical solution procedure for the prediction of soil settlement and excess pore water pressure profile at any specific time during the consolidation process is performed by numerical analysis of the 1D consolidation equation using finite difference method (FDM) or manually using tables and design charts. In FDM, solutions are started at the initial time and incrementally marched forward at a small time-step up to the final time INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND TECHNOLOGY (IJCIET) ISSN 0976 – 6308 (Print) ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), pp. 135-145 © IAEME: www.iaeme.com/ijciet.asp Journal Impact Factor (2013): 5.3277 (Calculated by GISI) www.jifactor.com IJCIET © IAEME
  2. 2. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 136 of interest. For longer consolidation time having a small time-step interval and a thick clay layer deposit, a large number of iterations is usually involved than can consume a lot of computer memory and having huge sizes of data file, which can slow down the computational process and may take longer computer processing time to finish (Kim et al., 1995). With this method, it is therefore not possible to directly predict the future conditions of the soil without first chronologically solving the current and sequential conditions that lead to the final conditions. Hazzard and Yacoub (2008) presented a hybrid computational scheme for the numerical solution of 1D consolidation, based on the method described by Booker and Small (1975), to speed up the required computational time by increasing the time-step gradually as the solution progresses while maintaining the required stability and accuracy. However the suggested method still suffer from the chronological or sequential type of solution in which the needed computational resources such as computer memory, output file size, and the processing time accumulates as the solution progresses. This study presents an alternative method using Artificial Neural Networks (ANN) for the direct estimation and prediction of the consolidated state of a homogeneous clay soil layer under a uniformly distributed surcharge loading. Over the last decades, ANN has been used successfully for modeling most aspects of geotechnical engineering problems as summarized by Shahin et al. (2001; 2008), but no model has yet been presented and applied for the prediction of 1D consolidation over time. With sufficient training of the ANN model and without resorting to a stepwise progressing and sequential solution procedure, the method can provide reliable and direct estimates of the excess pore pressures and settlement in the clay layer at any time during the progress of consolidation, and at a much less needed computational resources compared to FDM. Numerical examples are presented to validate the prediction performance and demonstrate the advantages of the ANN in comparison with the conventional finite difference method. 2. FINITE DIFFERENCE SOLUTION OF THE 1D CONSOLIDATION EQUATION Terzaghi (1943) derived the one-dimensional consolidation equation for a homogenous layer of clay with thickness Hc under a uniformly distributed surcharge load q that is given as, 2 2v u u C t z ∂ ∂ = ∂ ∂ Eq. (1) where u is the excess pore water pressure, t is the consolidation time, z is the depth, and Cv is the coefficient of consolidation. Equation (1) is based on the assumption that the coefficient of consolidation Cv remain constant during the consolidation process, the effect of self-weight consolidation is neglected, the soil profile is fully saturated, and the consolidation settlements are small or infinitesimal. The finite difference form of equation (1) for numerical analysis in time (t+∆t) is written as, ( ) ( ) ( )       +− ∆ ∆ += ∆−∆+∆+ tzztztzz v tzttz uuu z tC uu ,,, 2 ,, 2 Eq. (2) In equation (2), ∆t = time-step and the depth increment∆z = Hc/n, where n is the number of sublayer elements in the finite difference grid. Equation (2) is applied using the following initial and boundary conditions with respect to the excess pore pressure at the depth and time coordinates u(z,t) in which; u (z,0) = q (initial condition) Eq. (3)
  3. 3. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 137 u (0,t) = 0 (at a permeable top surface boundary) Eq. (4) u(Hc,t) = 0 (at a permeable bottom surface boundary) Eq. (5) u z ∂ ∂ (Hc,t) = 0 (at an impermeable bottom surface boundary Eq. (6) The impermeable boundary condition defined by equation (6) means that there can be no flow in the perpendicular direction. Equation (6) is implemented numerically by creating a dummy node in the finite difference grid after the bottom surface, which can be expressed in finite difference form as; ( ) ( ) ( ) ( )0 , or 2 c c c c H z H z H z H z u u u u z +∆ −∆ +∆ −∆ − = = ∆ Eq. (7) Equation (2) implies that if the solution for u has been determined at time t, then the values at time (t+∆t) can be calculated by marching the solution downward with depth and forward in time as shown in Fig. 1. To ensure that the approximate solution of equation (2) must converge to the exact solution as ∆t and ∆z approaches zero, the following criteria should be satisfied in determining the time and depth increments, ∆t and ∆z, respectively (Forsythe and Wasow, 1960). Figure 1: Finite difference nodes in the numerical solution of the 1D consolidation equation ( ) 2 1 2 vC t z β ∆ = ≤ ∆ Eq. (8) The total settlement S can be calculated using the coefficient of compressibility mv and excess pore pressure u by numerically integrating along the depth profile as follows: ( ) ( )1 12 c nH v v v c n n o n m z S m q u dz m qH u u + = ∆ = − = − +∑∫ Eq. (9)
  4. 4. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 138 3. ARTIFICIAL NEURAL NETWORK (ANN) MODEL FOR PREDICTION OF 1D CONSOLIDATION Table 1: Range of input parameters for ANN training data Input parameter Range Thickness of clay, Hc (m) 1, 10, 20, 30 Coefficient of consolidation, Cv (m2 /year) 1, 10, 50, 100, 150 Surcharge load, q (kPa) 5, 10, 50, 100, 150, 200 Time factor, Tv 0.0 - 2.0 Drainage condition One-way (single), Two-way (double) Neural networks are composed of simple elements called neurons operating in parallel to a set of input signals whose connections largely determine the network function. The neural network is trained to perform a particular function by adjusting the values of the connections or weights between elements so that a particular input leads to a specific target or output with the minimum error (Fausett, 1994; Zurada, 1992). The number of neurons or nodes in the input and output layers are restricted by the number of model inputs and outputs. Based on Terzaghi's 1D consolidation theory (equations 1-6), it can be seen that the relevant input factors affecting the degree of consolidation are the total thickness of the clay layer (Hc), the consolidation coefficient (Cv), the magnitude of the uniformly applied surcharge load (q), and the actual consolidation time (t), while the model output consists of the distribution of the excess pore pressure isochrones u along the depth profile z. Since the type of boundary conditions are considered as extreme cases of 1D consolidation, it is presumed that there cannot be any unique ANN model for both. Therefore, separate ANN model was formulated for each case, that is, the network model net1 for single-drainage condition and net2 for double-drainage conditions. Preliminary output data for training, testing, and validation was provided from the numerical results of 1D consolidation by FDM using the combination of the range of input parameters as shown in Table 1 that are typical for 1D consolidation problems encountered in the field. To ensure that the predicted results by FDM are accurate and always stable, the time-step ∆t was determined from equation (8) using a solution criteria β = 0.25 and a depth increment ∆z determined from n = 20 elements of the total thickness Hc. Although more accurate predictions can be provided by selecting a smaller value of β and smaller depth increments by having more number of elements, the chosen values of β and n for the training model can be sufficient to produce predicted results that are almost similar to those calculated by analytical solutions (Verruijt, 2001). To improve training and performance, the original input-output dataset were preprocessed by normalizing the output excess pore pressure u profile in terms of the applied surcharge load q, that is, u/q, and normalizing the actual consolidation time t from the input in terms of the time factor Tv defined as, Tv = Cv.t/H2 Eq. (10) Where t = actual consolidation time, H = length of the longest drainage path, and in which H = Hc for single drainage and H = Hc/2 for double drainage.
  5. 5. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 ISSN 0976 – 6316(Online) Volume 4, Issue 4, July Table 2: Sensitivity analysis for the relative importance of the input variables in a preliminary ANN model with four inputs ( network H net1 0.0055 net2 0.0011 In an attempt to identify which of the on the consolidation predictions, a sensitivity analysis is carried out by training a preliminary ANN model with four inputs (Hc, Cv, q, and technique proposed by Garson (1991) is u variables by examining the connection weights of the trained network. The method is also illustrated and applied by Shahin et al. (2002). The results of the sensitivity analysis model having three neurons in the single hidden layer time factor Tv has the most significant effect on the prediction of consolidation having a relative importance factor almost equal to 100% and the rest of t importance. With the normalization of the time and likewise with the normalization of the output excess pore pressure load q, the inputs Hc, Cv, and q are number of input variables may be reduced where n = 20, corresponding to the profile of the normalized excess pore pressure spaced at the normalized depth z/Hc of 1D consolidation is then shown in Fig. 2. (a) Figure 2: Typical architecture of the ANN model for 1D consolidation prediction: (a) Original ANN input-output model, and (b) normalized ANN input From the various combinations of the range of input parameters shown in Table 1, the original number of input-output dataset for ANN training consisted of a total of 384,000 samples for net1 and 96,000 samples for net2, corresponding to the total number of the final time factor Tv = 2.0 for each case. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 139 Sensitivity analysis for the relative importance of the input variables in a preliminary ANN model with four inputs (Hc, Cv, q, and Tv) and normalized outputs u/q Relative importance of input variables (%) Hc Cv q Tv 0.0055 0.0062 0.0003 99.9880 0.0011 0.0026 0.0025 99.9938 In an attempt to identify which of the original input variables has the most on the consolidation predictions, a sensitivity analysis is carried out by training a preliminary ANN , and Tv) and normalized outputs u/q. A simple and innovative technique proposed by Garson (1991) is used to interpret the relative importance of the input variables by examining the connection weights of the trained network. The method is also illustrated and applied by Shahin et al. (2002). The results of the sensitivity analysis for the preliminary ANN model having three neurons in the single hidden layer are shown in Table 2. It can be seen that the has the most significant effect on the prediction of consolidation having a relative importance factor almost equal to 100% and the rest of the input variables having negligible relative With the normalization of the time t into the time factor Tv as defined by and likewise with the normalization of the output excess pore pressure u in terms of the surcharge are indeed not required in the ANN model and thereby the original may be reduced from four to one. The number of output variables is ( = 20, corresponding to the profile of the normalized excess pore pressure u c. The typical architecture of the final ANN model for prediction of 1D consolidation is then shown in Fig. 2. (b) Typical architecture of the ANN model for 1D consolidation prediction: output model, and (b) normalized ANN input-output model From the various combinations of the range of input parameters shown in Table 1, the output dataset for ANN training consisted of a total of 384,000 samples for , corresponding to the total number of required iterations to reach = 2.0 for each case. In other words, it takes about 4 times as much the International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), Sensitivity analysis for the relative importance of the input variables in a preliminary ANN u/q 99.9880 99.9938 input variables has the most significant effect on the consolidation predictions, a sensitivity analysis is carried out by training a preliminary ANN . A simple and innovative sed to interpret the relative importance of the input variables by examining the connection weights of the trained network. The method is also illustrated for the preliminary ANN are shown in Table 2. It can be seen that the has the most significant effect on the prediction of consolidation having a relative he input variables having negligible relative as defined by equation (10), in terms of the surcharge not required in the ANN model and thereby the original The number of output variables is (n+1), u/q that is equally ANN model for prediction Typical architecture of the ANN model for 1D consolidation prediction: output model From the various combinations of the range of input parameters shown in Table 1, the output dataset for ANN training consisted of a total of 384,000 samples for required iterations to reach In other words, it takes about 4 times as much the
  6. 6. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 140 required time for single drainage conditions compared to that of double drainage conditions to attain the same degree of consolidation as can be seen in equation (10). In order to speed up the network training and learning process, only one sample was arbitrarily chosen for every 20th time-step or iteration from the database results in double drainage, and only one sample for every 80th time-step was arbitrarily taken for training from the database results in single drainage. The total number of reduced samples used for training in each case is 4,800 and it had been justified that this number was sufficient based on the observed prediction performance. Table 3: Optimization of the final ANN model architecture Trial Number of hidden layers Number of hidden layer nodes Coefficient of correlation, R net1 net2 1 1 3 0.9995 0.999 2 1 4 0.999 .0999 3 2 [4 2] 0.9995 0.9995 4 2 [6 3] 0.999 0.999 The training process of the network was based on supervised learning (Masters, 1993), where the network was presented with the historical set of model inputs and their corresponding targets or outputs as shown in Fig. 2. In addition, back-propagation learning algorithm (Rumelhart et al., 1986) was used in which the connection between the processing elements through their weights were adjusted both in the forward and backward directions until the error between predicted and measured outputs are minimized based on performance criteria. Using cross-validation (Stone, 1974) as a stopping criterion, the database is randomly divided into three sets and proportions as follows: training (70%), testing (15%), and validation (15%). Although a single hidden layer may be sufficient to model any solution of practical interest (Cybenko, 1989; Hornik et al., 1989; Hecht- Nielsen, 1990), there is no unified approach for determination of an optimal ANN architecture, which is generally achieved by fixing the number of hidden layers and choosing the number of nodes in each. Keeping the number of hidden layers and hidden nodes to a minimum is preferable for reduced training time, better generalization performance, prevents over fitting, and allows the network to be easily analyzed (Shahin, 2008). The best approach by Nawari et al. (1999) was to start with a small number of hidden layer and nodes and to slightly increase the number until no significant improvement in model performance is achieved. For this purpose several trials were made as shown in Table 3, where the main criteria that is used to evaluate the prediction performance of the optimal ANN model is the coefficient of correlation (R), which is a measure that is used to determine the correlation and the goodness-of-fit between the predicted and observed data. A performance goal of 10-6 was chosen and based on the trial with the highest coefficient of correlation (R) and minimum number of layers and nodes for both cases as presented in Table 3, the final size of the selected network architecture for the model shown in Fig. 2 has then one input layer, one hidden layer with four neurons, and 21 neurons in the output layer. This is also the optimum network size which produces the minimum number of epochs and iteration during the training process and thus producing the least required learning time and consequently the fastest processing time in terms of producing the estimated and predicted output. For both cases, the networks net1 and net2 were modeled and trained using the Matlab function 'newff'' that is used to create a feed-forward back- propagation network and the 'train' function that is used to train the neural network (Demuth et al., 2009). Figure 3 shows the regression plot of the training data, validation, and testing of the respective ANN model in Matlab. The high correlation coefficient (R = 0.999) for each case proves
  7. 7. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 141 the satisfactory prediction performance of the model and network architecture. The general performance of the trained networks, net1 and net2, were then further validated individually in the deployed environment and field conditions as described in the following sections. (a) (b) Figure 3: Regression plot of the ANN model showing the prediction performance and correlation coefficient (R) of the training data, validation, and testing in Matlab: (a) net1 - single drainage condition, and (b) net2 - double drainage condition 4. NUMERICAL VALIDATION CASE EXAMPLES Table 4: Case 1 - properties of marine clay profile under fill Thickness of clay, Hc (m) 25.0 Coefficient of consolidation, Cv (m2 /year) 70.0 Coefficient of compressibility, mv (m2 /kN) 0.00025 Surcharge load, q (kPa) 112.70 Drainage boundary condition single (one-way) drainage 4.1. Validation Case 1: 25 m thick marine clay under reclamation fill (Bjerrum et al., 1969) Bjerrum et al. (1969) reported results of a consolidation test on a land that was reclaimed from the sea by placing about 8 m of fill over the sea bed at the Heroya site, Norway. The consolidating soil consists of about 25 m of marine clay. Table 4 shows the chosen properties for the clay profile in which the consolidation parameters and magnitude of the applied surcharge load were deduced from the published data (Poulos and Davis, 1980; Alonso et al., 1984; Fellenius, 2006). Piezometer measurements indicated that one-way drainage conditions were present at the site. The total thickness was subdivided into 20 elements for consolidation analysis by FDM using a solution criteria β = 0.25 for the time and depth increment as described in Section 2. ANN prediction was performed using the network model (net1) that was trained, tested, and validated for single drainage
  8. 8. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 142 condition. Figure 4(a) shows the predicted excess pore pressure isochrones at different time factor Tv in which the ANN predictions compare well with FDM results. Figure 4(b) shows the profile of the excess pore pressure in May 1967 corresponding to a time factor Tv of about 0.38, in which the predicted results compares fairly with measured data. (a) (b) Figure 4: Example 1: (a) Comparison of predicted excess pore pressure isochrones, and (b) Excess pore pressure (u) profile in May 1967 (Tv = 0.38) and calculated effective stress (σ') 4.2. Validation Case 2: 15.5 m thick stiff clay under embankment fill (Walker et al., 1973) Walker et al. (1973) reported results of a consolidation test of a highly stratified soil consisting of a 2 m of recent fill, 7 m of sand overlying a stiff silty clay 15.5 m thick, and underlain by sandy silt/dense sand layers. Surcharge load was applied by constructing a test embankment 100 m x 200 m x 3 m high and in which monitoring was made for about 8 months. The clay layer was confined by cohesionless soil layers above and below such that two-way drainage was assumed. Table 5 shows the chosen properties for the clay profile that were deduced from the published data (Wong and Teh, 1995; Fellenius, 2006; Kim and Mission, 2009). Numerical analysis of 1D consolidation was performed using 20 elements and the time-step was determined using a solution criteria β = 0.25. ANN prediction was performed using the network model (net2) that was trained, tested, and validated for double drainage condition. Figure 5(a) shows the predicted excess pore pressure isochrones at different time factor Tv in which the ANN predictions compare well with numerical results. Figure 5(b) shows the comparison between the measured and predicted total settlement in which good agreement is observed. Table 5: Case 2 - properties of stiff clay profile under embankment fill Thickness of clay, Hc (m) 15.5 Coefficient of consolidation, Cv (m2 /year) 135.0 Coefficient of compressibility, mv (m2 /kN) 0.000042 Surcharge load, q (kPa) 51.0 Drainage boundary condition double (two-way) drainage
  9. 9. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 143 (a) (b) Figure 5: Example 2: (a) Comparison of predicted excess pore pressure isochrones, and (b) Comparison of measured and predicted total settlement Having validated the accuracy of the ANN for prediction of 1D consolidation whose results are comparable with the FDM, the advantages in terms of the solution speed and efficiency of the ANN are then compared with the FDM as shown in Table 6. The FDM and ANN predictions were implemented using the Matlab program in a 2.83 GHz computer with 6 Gb memory and quad-core processor. The central processing unit (CPU) time in seconds (s) as well as the size of the output file in kilobytes (kb) were compared for the range of time factor Tv shown in Table 6. Due to the sequential nature or time-marching solution process of the FDM, the CPU time and output file sizes or memory requirements are thus increased especially when consolidation results are needed at longer consolidation times. In contrast, equivalent and accurate predictions are still being provided by ANN in which direct and quick results can be made at any consolidation time of interest. As shown in Table 6, prediction method of 1D consolidation can therefore be reliably made by ANN that can be more efficient by about 6 % to 278 % compared to the FDM and thus minimizing the needed computational resources. Table 6: Comparison of CPU times and output file sizes for analyses Case 1 and 2 Tv Case CPU Time (s) Output file size (kb) (FDM-ANN) ANN (%) FDM ANN FDM ANN CPU time Output file size 0.10 Case 1 0.196 0.007 28 1 27 27 Case 2 0.082 0.007 7 1 11 6 0.50 Case 1 0.734 0.007 140 1 104 139 Case 2 0.254 0.007 35 1 35 34 1.00 Case 1 1.381 0.007 279 1 196 278 Case 2 0.482 0.007 70 1 68 69
  10. 10. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 144 5. CONCLUSIONS This paper presents the novel application of Artificial Neural Network (ANN) in the prediction of one-dimensional consolidation of a homogeneous clay layer under uniform applied load. ANN can provide accurate and direct estimates of the excess pore pressures and settlement at any time during consolidation without resorting to the stepwise progressing solution procedure by the Finite Difference Method (FDM). The prediction performance of the ANN has been validated by the equivalent results with FDM. Prediction of consolidation by ANN has the advantage that it can be used as a direct, accurate, and quick tool for estimating excess pore pressures and settlement at any time without a need to perform any lengthy manual calculations or approximately using tables and charts. Two ANN models were being developed and presented in this study: net1 for consolidation in single drainage boundary conditions, and net2 for double drainage conditions, which were further validated in a deployed environment for prediction of excess pore pressures and settlement in field conditions. Compared to FDM, ANN offers several advantages with regards to efficiency, speed, and economy by having lesser computational resources needed and faster time for the calculations and thus makes it a powerful and practical tool for the prediction of 1D consolidation in homogeneous clays. ACKNOWLEDGEMENTS This paper was supported by research funds of Kunsan National University. REFERENCES 1. Kim, H.J., S. Hirokane, H. Yoshikuni,. T. Moriwaki, and O.Kusakabe. (1995). “Consolidation behavior of dredged clay ground improved by horizontal drain method.” Compression and Consolidation Clay Soils,Vol.1, pp.99 -104. 2. Hazzard, J. and Yacoub, T. (2008). “Consolidation in multi-layered soils: a hybrid computational scheme.” GeoEdmonton '08 :61st Canadian Geotechnical Conference and 9th Joint CGS/IAH-CNC Groundwater Conference, pp. 182-189. 3. Booker, J. R. and Small, J. C. (1975). “An investigation of the stability of numerical solutions of Biot's equations of consolidation.” International Journal of Solids and Structures, Vol. 11, pp. 907-917. 4. 22. Shahin, M. A., Jaksa, M. B., and Maier, H. R. (2001). “Artificial neural network applications in geotechnical engineering.” Australian Geomechanics, Vol. 36, No. 1, pp. 49- 62. 5. Shahin, M. A., Jaksa, M. B., and Maier, H. R. (2008). “State of the art of Artificial Neural Networks in Geotechnical Engineering.” Electronic Journal of Geotechnical Engineering (EJGE), Vol. 8, pp. 1- 26, http://www.ejge.com. 6. Terzaghi, K. (1943). Theoretical Soil Mechanics. New York: John Wiley and Sons. 7. Forsythe, G. E. and Wasow, W. R. (1960). Finite Difference Methods for Partial Differential Equations. New York: Wiley. 8. Fausett, L. V. (1994). Fundamentals of neural networks: Architecture, algorithms, and applications. New Jersey: Prentic-Hall. 9. Zurada, J. M. (1992). Introduction to artificial neural systems. Minnesota: West Publishing Company. 10. Verruijt, A. (2001). Soil Mechanics. Delft University of Technology, The Netherlands. 11. Garson, G. D. (1991). “Interpreting neural network connection weights.” AI Expert, Vol. 6, No. 7, pp. 47-51
  11. 11. International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308 (Print), ISSN 0976 – 6316(Online) Volume 4, Issue 4, July-August (2013), © IAEME 145 12. Shahin, M. A., Maier, H. R., and Jaksa, M. B. (2002). “Predicting settlement of shallow foundations using neural networks.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 128, No. 9, pp. 785-793. 13. Masters, T. (1993). Practical neural network recipes in C++. California: Academic Press. 14. Rumelhart, D. E., Hinton, G. E., and Williams, R. J. (1986). “Learning internal representation by error propagation.” Parallel distributed processing. Vol. 1, Chapter 8, D. E. Rumelhart and J. L. McClelland, eds., MIT Press, Cambridge, Mass. 15. Stone, M. (1974). “Cross-valedictory choice and assessment of statistical predictions.” Journal of the Royal Statistical Society, Vol. 36, No. 2, pp. 111-147. 16. Cybenko, G. (1989). “Approximation by superpositions of a sigmoidal function.” Mathematics of Control, Signals, and Systems, Vol. 3, pp. 303-314. 17. Hornik, K., Stinchcombe, M., and White, H. (1989). “Multilayer feedforward networks are universal approximators.” Neural Networks, Vol. 2, pp. 359-366. 18. Hecht-Nielsen, R. (1990). Neurocomputing. Massachusetts: Addison-Wesly. 19. Nawari, N. O., Liang, R., and Nusairat, J. (1999). “Artificial intelligence techniques for the design and analysis of deep foundations.” Electronic Journal of Geotechnical Engineering, Vol. 4, http://www.ejge.com. 20. Demuth, H., Beale, M., and Hagan, M. (2009). Neural Networks ToolBox User Guide. The Math Works Inc., Natick, Massachusetts. 21. Bjerrum, L., Johannessen, I. J., and Eide, O. (1969). “Reduction of negative skin friction on steel piles to rock.” Proceedings of the 7th International Conference on Soil Mechanics and Foundation Engineering, Mexico City, August 25-29, Vol. 2, pp. 27-34. 22. Poulos, H.G. and Davis, E.H. (1980), Pile Foundation Analysis and Design, New York: John Wiley Sons, Inc. 23. Alonso, E.E., Josa, A., and Ledesma, A. (1984). “Negative skin friction on piles: a simplified analysis and prediction procedure.” Geotechnique Vol. 34, No. 3, pp. 341-357. 24. Fellenius, B.H. (2006). “Results from long-term measurement in piles of drag load and downdrag.” Canadian Geotechnical Journal, Vol. 43, No. 4, pp. 409-430. 25. Walker, L.K., Le, P., and Darvall, L. (1973). “Dragdown on coated and uncoated piles.” Proceedings of the 8t International Conference on Soil Mechanics and Foundation Engineering, August, Vol. 2, Paper 3/41, pp. 257-262. 26. Wong, K.S., and Teh, C.I. (1995). “Negative Skin Friction on Piles in Layered Soil Deposit.” Journal of Geotechnical and Geoenvironmental Engineering, Vol. 121, No. 6, pp. 457-465. 27. Kim, H. J. and Mission, J. L. (2009). “Negative skin friction on piles based on finite strain consolidation theory and the nonlinear load transfer method.” Korean Society of Civil Engineers (KSCE) Journal of Civil Engineering , Vol. 13, No. 2, pp. 107-115. 28. Balamuruga Mohan Raj.G and V. Sugumaran, “Prediction of Work Piece Hardness using Artificial Neural Network”, International Journal of Design and Manufacturing Technology (IJDMT), Volume 1, Issue 1, 2010, pp. 29 - 44, ISSN Print: 0976 – 6995, ISSN Online: 0976 – 7002. 29. V. Sugumaran, V. Muralidharan, Bharath Kumar Hegde and Ravi Teja C, “Intelligent Process Selection for Ntm - A Neural Network Approach”, International Journal of Industrial Engineering Research and Development (IJIERD), Volume 1, Issue 1, 2010, pp. 84 - 93, ISSN Online: 0976 - 6979, ISSN Print: 0976 – 6987,

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