Prediction of Bearing Capacity of Stone Columns

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Prediction of Bearing Capacity of Stone Columns Placed in Soft Clay Using ANN Model Article in Geotechnical and Geological Engineering · December 2017 DOI: 10.1007/s10706-017-0436-0

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Geotech Geol Eng (2018) 36:1845–1861 https://doi.org/10.1007/s10706-017-0436-0

ORIGINAL PAPER

Prediction of Bearing Capacity of Stone Columns Placed in Soft Clay Using ANN Model Manita Das . Ashim Kanti Dey

Received: 16 January 2017 / Accepted: 22 December 2017 / Published online: 29 December 2017 Springer International Publishing AG, part of Springer Nature 2017

Abstract It is known that the construction over soft clay is always a great challenge to the geotechnical engineers. The soft clay poses high compressibility and low bearing capacity. It is a common practice to construct piles in the soft clay to transfer the superimposed load to the hard strata below. Construction of stone columns is also a technique of ground improvement normally applied to the soft clay for increase in bearing capacity and reduction in compressibility. Many theories are developed to determine the bearing of a soft soil reinforced with stone columns. However, most of the theories are site specific and do not show a very good match with the field observations. Artificial neural network (ANN) is an analytical tool which can be used to predict some specific behavior of soil like bearing capacity, settlement, etc. based on some input properties like density, shear strength parameters, void ratio, etc. In the present study, 90 data were collected from the previously published literatures to build an ANN model. Five parameters, namely, un-drained cohesion of soft clay (cu), friction angle of stone column material (/), ratio of spacing to diameter of the stone columns (s/d), length of the stone column (l) and number of the stone columns (n) were taken as M. Das (&) A. K. Dey Civil Engineering Department, N.I.T. Silchar, Silchar, India e-mail: [email protected] A. K. Dey e-mail: [email protected]

input data and the bearing capacity as an output. The predicted bearing capacity was compared with the laboratory experimental data and Plaxis 2D data. The predicted values were also compared with the values obtained from the established theories. ANN predicated values showed a very good match even better than any of the established theories. Sensitivity analysis showed that the effect of the / value is maximum on determination of the bearing capacity. Keywords Artificial neural network Bearing capacity of stone columns Soft clay

1 Introduction Construction over soft clay is always a great challenge to the geotechnical engineers. High compressibility and low shear strength are the two primary reasons for the same. It is a usual practice to construct piles in this type of soil to transfer the load to the hard stratum below. For low to medium rise buildings cost of construction of piles is very high as compared to the cost of the structure. Hence, some ground improvement techniques may be adopted to solve the problem. Construction of stone columns is one of the most commonly used ground improvement techniques in the soft clay. Stone columns increase the bearing capacity (Bouassida et al. 1995) and expedite the consolidation (Balaam and Booker 1981) of the soft

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clay. However, use of stone columns is limited for residential buildings, since settlement cannot be significantly reduced. In order to minimize the settlement it is a pre-requisite to understand the factors affecting the bearing capacity and settlement of the stone columns. In the present study, artificial neural network (ANN) is used to study the different factors affecting the bearing capacity and finally to predict the bearing capacity of the stone columns. The improved bearing capacity and expected settlement of soft clay due to installation of stone columns are available in the Indian Standard code IS 15284-Part 1 (2003) and other published papers. Many researchers calculated the bearing capacity and settlement experimentally (Afshar and Ghazavi 2014b; Ambily and Gandhi 2004, 2007; Castro 2014; Mohanty and Samanta 2015). Other researchers conducted theoretical study or used numerical solution through available commercial software and observed the behavior of the stone columns (Afshar and Ghazavi 2014a; Etezad et al. 2015; Frikha et al. 2013; Goh et al. 1998; Lee and Pande 1998; Das and Deb 2016; Six et al. 2012; Fattah and Majeed 2012; Nassaji and Asakereh 2013; Niroumand et al. 2011; Borges et al. 2009; Deb et al. 2010; Tabchouche et al. 2017; Castro 2017). Etezad et al. (2015) studied the group effect of the stone columns on the bearing capacity by assuming a general shear failure. They used a limit equilibrium method for the bearing capacity determination. Castro (2014) found an analytical solution for predicting the settlement of the stone columns beneath rigid footings. Basack et al. (2017) made a numerical model based on the lateral deformation of stone columns placed in soft clay. Although the field tests are more accurate, yet importance of analytical studies cannot be ignored. Sometimes experimental studies become more costly and time consuming and show lack in repeatability when a number of parameters are involved for a certain solution. Moreover, the experimental results have some uncertainties due to some errors such as error due to weather forecast, error due to installation effects, etc. So the analytical or numerical studies are also important. Andreou and Papadopoulos (2014) discussed on the factors, which affect on the settlement of stone column reinforced soft soils. Elshazly et al. (2008a) found out the effect of spacing between the stone columns due to installation of stone columns, because spacing is the most important factor which

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affects the bearing capacity and settlement of stone columns. Elshazly et al. (2008b) studied on the reliability of the settlement value for the soft soil reinforced with stone columns. Al-Homoud and Degen (2006) discussed on the dynamic properties of soil reinforced with stone column. They found a new idea to prevent the liquefaction by using marine stone columns. Artificial neural network (ANN) is a computational technique to find out an output from a given number of inputs. It has the capability of input–output mapping. In geotechnical engineering, many researchers used ANN to solve various problems (Goh et al. 1998; Kosti and Vasovic 2015). Chik et al. (2014) and Chik and Aljanabi (2014) used ANN to find out the settlement of the stone columns. Taha and Firoozi (2012) applied artificial intelligence system to predict the cohesion value of clayey soil. In the present study, ANN is used to predict the bearing capacity of the stone columns. For obtaining an ANN model, data were collected from the published literatures. To select the best ANN model, four different validation technique and four algorithms were used. The results from the best fit ANN model were compared with the results from the laboratory tests, from Plaxis 2D analyses and from well established theories. A good match was observed with the laboratory results. An equation to find the bearing capacity of stone columns was derived from the best fit ANN model. Finally a sensitivity analysis was carried out to obtain the most important input parameter affecting the bearing capacity of the stone columns.

2 Short Description of the ANN Model Back-propagation neural network (BPN) was used instead of radial basis function network (RBFN) since BPN has a superior capability in pattern classification system (Haykin 1998; Chik and Aljanabi 2014). Around 90 data were collected from the previously published literatures. Four validation techniques, namely, 90-10 validation (q90-10, ten-cross validation), 80-20 validation (q80-20), 70-30 validation (q70-30) and 50-50 validation (q50-50) were used. For every model, four algorithms namely, Bayesian Regularization (BR), Levenberg–Marquardt (LM), Gradient descent with momentum (GDM), and Broyden-Fletcher-Goldfarb-Shannon (BFGS) were used as shown in Fig. 1.

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Fig. 1 Flow chart of proposed ANN model

Thus, altogether 16 algorithms were performed to investigate the performance of ANN in determination of bearing capacity of the stone columns. Results from all these 16 models were compared to observe the accuracy of their performances. The algorithms used in this study are briefly discussed below: (a)

Bayesian Regularization (BR) Learning Algorithm: Bayesian back propagation algorithm was introduced by MacKay (1991) and Neal (1992), which is based on the Bayesian statistical approach (Box and Tiao 1973). BR is a mathematical phenomenon that transforms a nonlinear regression to a ‘‘well-posed’’ statistical approach in the similar manner of a ridge regression. This algorithm creates a well generalized network and forms simple weight functions. Like most other algorithms, it does not predict the single optimum prediction; rather it provides the probability distribution over all the predicted values which are very important. It provides characteristic error information of the prediction which arises from the uncertainties developed with interpolating the noisy data. This algorithm contains an objective function which includes sum of squared weights and residual sum of squares to reduce the errors. In the present study, BR algorithm was used to observe the performance of ANN model for the

(b)

(c)

prediction of bearing capacity of stone column and it shows the better performance than any other algorithms as it shows a very good agreement between the results (Fig. 4b). Levenberg–Marquardt (LM) learning algorithm: LM algorithm works as an iterative technique which reduces the performance function in each iteration. That is why it is the fastest method to train the feed-forward neural network of moderate size as in the present case. In case of supervised learning, this algorithm gives the most accurate result. In mathematics and computing techniques, the LM algorithm is also known as the damped least-squares (DLS) method. This algorithm can also be used in case of curve fitting problems. LM algorithm contains the minimum of a multivariate function which is expressed in terms of sum of the squares of non-linear real valued functions. Gradient descent with momentum (GDM) learning algorithm: It is a first-order iterative optimization algorithm, which is used to find local minima of a function. This algorithm is the steepest descent with momentum which ignores the small features existing in error surface. GDM algorithm is the most commonly used algorithm for the optimization in neural network. This algorithm can also be used as a black-box optimization algorithm because in

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(d)

Geotech Geol Eng (2018) 36:1845–1861

this case, practical explanations of the strengths and weaknesses are very difficult. This algorithm can be used to measure the output error, to estimate the error gradient with the adjustment of the weights in the direction of descending gradient. Broyden–Fletcher–Goldfarb–Shannon (BFGS) quasi-Newton back-propagation learning algorithm: The BFGS algorithm is an iterative optimization method which can be used to solve unconstrained nonlinear problems as in the present case. This algorithm is the computationally lowest cost algorithm which is error tolerant. This algorithm gives best results for the solution of low to moderate size problems. This algorithm utilises a local quadratic approximation of the corresponding error function and update the weights by approximating the inverse of the Hessian matrix. BFGS algorithm results good solutions through a small number of iterations (Dennis and Schnabel 1996).

3 Artificial Neural Network (ANN) Formation The ANN model for the present study was formed through following steps: 3.1 Collection of Data For the neural network model, total 90 data were collected from the published papers (Ali et al. 2010; Black et al. 2007; Choobbasti et al. 2011; Deb et al. 2011; Fattah and Majeed 2009; Golait et al. 2009; Hassen et al. 2010; Malarvizhi and Ilampurthi 2007; Mohanty and Samanta 2015; Murugesan and Rajagopal 2006, 2008; Nassaji and Asakereh 2013; Poorooshasb and Meyerhof 1997; Shivashankar et al. 2010, 2011; Vekli et al. 2012; Zhang et al. 2013). Some data are from experimental study, some are from numerical study and the rest from analytical study. 40% of the total data were taken from the experimental studies. In order to reduce the variations in the values due to different types of tests, a data is presented by a scaled value given by Eq. (1).

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Scaled value ¼

ðunscaled value minimum valueÞ ðmaximum value minimum valueÞ ð1Þ

It is obvious from the above equation that the range of the scaled value is 0–1. Table 1 shows the various data collected from the literature. The statistical dispersion is measured by the coefficient of variation (CV). The CV is determined by the following equation CV ¼

SD 100 Mean

ð2Þ

where SD is the standard deviation. 3.2 Optimizing Number of Hidden Neurons The ANN problem is normally a function of optimization problem, where the best network parameters such as weights and biases can be determined by minimizing the network error. For multilayer perceptron-neural network (MLP-NN), the optimization of number of hidden neurons is the most important factor. The use of more number of hidden neurons results more accurate output. But use of too many neurons results lengthy and time consuming outputs and overfitting of data. In this study, the optimum number of hidden neurons was obtained from the minimum value of the mean squared error (MSE). Taking the number of neurons from 1 to 20, the optimum number of hidden neurons was found out as follows: • •

•

•

•

By plotting MSE versus number of hidden neuron graphs. For the 90-10 validation technique with BR algorithm, the MSE versus number of hidden neuron graph was plotted as shown in Fig. 1. From the graph, corresponding to minimum MSE value, the number of hidden neuron was taken as optimum number of hidden neuron. In Fig. 2, the optimum number of hidden neuron is noticed as four. Likewise, for all other models with all the algorithms, the MSE versus number of hidden neuron graph was plotted separately and optimum number of hidden neurons was determined. For all the models, the optimum numbers of hidden neurons are listed in Table 2.

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Table 1 Details of collected data Number of stone columns, n

Undrained cohesion, cu in kPa

s/d ratio

Friction angle, / in degree

Length, (l) in mm

Load carrying capacity (q) in kPa

Minimum

7

5

1.5

20

10.5

20

Maximum

14

32

4

48

600

905

Mean

10.7

14.7783

2.431698

36.95755

298.7594

260.0283

SD

6.8

8.713176

0.735956

11.54476

191.9605

208.4899

CV

28.33

58.95925

30.26511

31.2379

64.25253

80.1797

0.0972

0.0972

MSE

0.0971

0.097

0.097

0.0969

0

2

4

6

8

10

12

14

16

18

20

No of hidden neuron

Fig. 2 Number of neurons versus mean standard error (MSE) graph

3.3 Training and Testing of ANN Model The collected data from the previous literature on bearing capacity of stone columns was divided into training, testing and validation data. With the training data, the network was developed and with the testing and validation data, the trained model was verified. 3.3.1 90-10 Validation Technique (Tenfold Cross Validation, q90-10) In 90-10 validation model, the whole data were divided into ten equal sections. Firstly, the first nine sets of data were used for training and the remaining one set was used for testing. In the second iteration, out of these ten sets of data, other nine subsets were used for training and remaining one was used for testing. By

repeating this process, ten iterations were completed until all the subsets were used for training. Figure 3a–d show the Mean Standard Error (MSE) versus epoch graph of 90-10 (tenfold cross validation) model for the algorithms BR, LM, GDM, and BFGS algorithms respectively. From the figure, it can be seen that in all the cases as the number of epoch increases, MSE value decreases which confirms that there is no chance of over-fitting. Figure 4a–d show the BPN network output versus training of 90-10 (tenfold cross validation) model for the algorithms BR, LM, GDM, and BFGS respectively. From the figure, it is seen that the BR algorithm shows the best prediction with experimental value among all other algorithms. In case of 90-10 validation model, the BR algorithm shows the best result for four numbers of hidden neurons. It is observed from the Fig. 4, that the four algorithms (BR, LM, GDM, and BFGS) show different results for the same ANN model (90-10 i.e. tenfold cross validation model). In case of BR algorithm, the correlation value, R, defined as the coefficient of determination is obtained as R = 0.99781; for LM algorithm, R = 0.99617 and for GDM and BFGS algorithms these values are R = 0.98134 and R = 0.97926 respectively. Standard error (SE) is defined as the minimum distance between the data points and the fitted line. The Standard Error values for these algorithms are SE = 1.6124, 1.612, 2.305 and 2.321 respectively. All the prediction values are within the 95% prediction interval. From R and SE values it can be seen that the BR and LM learning algorithms show almost similar results but BR shows the best performance.

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1850 Table 2 Optimum number of hidden neurons

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ANN models

Algorithms

R

SE

q90-10

BR

0.99781

1.613

4

LM

0.99617

1.612

11

GDM

0.98134

2.305

18

BFGS

0.97926

2.321

16

BR

0.98865

1.914

4

LM

0.9841

2.342

11

GDM

0.96038

2.505

9

BFGS BR

0.96862 0.979

2.621 2.114

7 4

LM

0.973

2.53

7

GDM

0.959

2.625

13

BFGS

0.958

2.632

14

BR

0.972

2.414

11

LM

0.964

2.73

9

GDM

0.95335

2.825

16

BFGS

0.94347

3.032

15

q80-20

q70-30

q50-50

3.3.2 80-20 Validation (Fivefold Cross Validation, q80-20) In 80-20 validation model, the whole data were divided into five equal sections. Firstly, the first four sets were used for training and remaining one set was used for testing and validation. Iteratively the remaining sets were used for training, testing and validation. In this technique, four algorithms were used to know the performance such as BR, LM, GDM and BFGS. 3.3.3 70-30 Validation (Non-cross Validation, q70-30) In 70-30 validation model, 70% of the total collected data were used for training and the remaining 30% of the data were used for testing and validation of the ANN model. In this case also, previously defined four algorithms were used. The details of all the algorithms can be seen in Table 2. 3.3.4 50-50 Validation (Two-Cross Validation, q50-50) In 50-50 validation model, in the first iteration, first 50% of the total collected data were used for training and the remaining 50% of the data were used for testing and validation of the ANN model. In second iteration, second 50% of the total collected data were used for training and the first 50% were used for

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No of hidden neurons

testing and validation. To know the best performance of this technique, four algorithms were also used. The correlation value (R), SE and optimum number of hidden neurons (n) for all the 16 models are listed in Table 2. From the table, it is seen that among all the 16 models, the BR algorithm of 90-10 model shows the best performance as its R value is the largest and the SE value is the lowest of all the values from other models.

4 Validation of ANN Model with Experimental and Plaxis-2D Results 4.1 Experimental Investigation An experiment is an investigating method by which a hypothesis can be scientifically tested. In an experiment, by assuming various independent variables, the effect of dependent variable can be measured. Normally the independent variables are assumed as the cause and the dependent variables as the effect. Moreover, the experimental methods allow the control of variables. In this study, total ten plate load tests were conducted and the bearing capacity for each test was determined. All the tests were conducted in a square tank of size 1 m 9 1 m 9 1 m. Locally available soft clayey soil and stones of size 2–6 mm were used for the investigation. To construct the model, first of all, the tank was wrapped with a polythene sheet and

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Best Validation Performance is 0.095966 at epoch 1000

Best Validation Performance is 0.082873 at epoch 2 10

10

10

Train Validation Test Best

-1

Mean Squared Error (mse)


10

0


0

10

10

-2

0

200

400

600

800

-1

-2

0

1000

0.5

(a)

10


-1

10

0

10

10

-2

0

5

10

2

Best Validation Performance is 0.1021 at epoch 2

0



10

1.5

(b)

Best Validation Performance is 0.064272 at epoch 13 10

1

2 Epochs

1000 Epochs

15

19 Epochs

(c)


-1

-2

0

0.5

1

1.5

2

2 Epochs

(d)

Fig. 3 Mean Standard Error (MSE) versus epoch graph of 90-10 (tenfold cross validation) model for the algorithms a BR, b LM, c GDM, and d BFGS algorithms respectively

then 0.9 m depth of the tank was filled with the soft soil in layers with uniform compaction to each layer. The tank was covered with a thick polythene sheet and kept idle for about a month for self consolidation of the soil. Table 3 shows the properties of the soft clay which was used for all the experiments. To construct a stone column, a hole was made with the help of an auger of size 50 mm, the stones were inserted and compacted with uniform compaction in layers. As per IS 15284-Part 1 (2003), seven stone columns in a triangular patter as shown in Fig. 5 are to be installed to determine the behavior of a single column placed at

the centre. For experimental set-up, one hydraulic jack of capacity 5 t was used for the loading, two dial gauges were used to measure the settlement and one proving ring was used to measure the applied load as shown in Fig. 6. For all the tests, one circular footing of diameter 10 cm was placed centrally over the arrangement of stone columns, i.e. over the central stone column. It was tried to keep the water content constant during all the experiments. The experiments were conducted by varying the input parameters such as the compactness of stone i.e. friction angle of stones (u), spacing to diameter ratio (s/d) and, length of the

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Fig. 4 BPN network output versus training of 90-10 (tenfold cross validation) model for the algorithms a BR, b LM, c GDM, and d BFGS algorithms respectively

stone column (l). The data from the experiments are listed in Table 4. 4.2 Plaxis 2D PLAXIS 2D is a FEM package which is being used for a two dimensional analysis for the deformation and stability of different types of structures in geotechnical

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engineering. With this user-friendly interface, the models can be efficiently created with a logical geotechnical workflow. The finite element mesh can be immediately created with the automatic meshing procedure. In this study, for determination of the bearing capacity of stone columns five parameters such as the number of stone columns (n), undrained cohesion of

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Table 3 Properties of clay Specific gravity

2.51

Bulk unit weight

1.7–1.75 gm/cc

Liquid limit

61%

Plastic limit

34.7%

Unified soil classification system

CH

mesh which was constructed in Plaxis-2D, is presented in Fig. 7. The ultimate bearing capacity of the stone column was obtained from the double tangent method applied to the load settlement curve. The results from the Plaxis-2D analysis are presented in Table 5. 4.3 Comparison of Bearing Capacity Values

Fig. 5 Plan view of seven stone column

Hydraulic jack

Proving ring

Since from the ANN study BR algorithm of q90-10 model shows the best performance, hence, all the experimental and Plaxis-2D data are validated with this model. The comparisons of all the bearing capacity values are shown in Table 6. The first column of Table 6 shows the experimental and Plaxis 2D results and the second column shows the results of ANN model. A comparison with the established theories on bearing capacity obtained from IS 15284-Part 1 (2003), Etezad et al. (2015) and Afshar and Ghazavi (2014a) is also shown in Table 6 against column number 3, 4 and 5, respectively. From the Table 6, it is observed that the ANN model shows a better match with the experimental and Plaxis 2D results than any other established method. The comparison shows that ANN model is reliable in predicting the bearing capacity of the stone columns. It also shows that whereas the established theories are site specific, the ANN model is soil specific.

So clay Dial gauge Stone column 1mx1m tank

Fig. 6 Experimental set up

soft soil, (cu), the friction angle of stones (u), spacing to diameter ratio (s/d) and, length of the stone column (l) were varied. The triangular arrangement of the stone columns was considered for the analysis. The plate size was kept constant as 10 cm for all the analyses. In the finite element mesh, a 15-noded triangular element and Mohr–Coulomb failure criteria were considered. One of the figures of the deformed

4.4 Performance Evaluation of the Proposed Models The performance of all the prediction models are evaluated by different statistical parameters such as Mean Absolute Percentage Error (MAPE), Variance Absolute Relative Error (VARE), Median Absolute Error (MEDAE) as mentioned in Eqs. (3), (4), and (5). Table 7 shows the performance of all the prediction models with their corresponding algorithms. " # n X 1 t x i i 100 MAPE ¼ ð3Þ n ti i¼1 " !# n X t i xi ti xi 2 1 VARE ¼ t mean t n i i i¼1 100 ð4Þ MEDAE ¼ medianðti xi Þ

ð5Þ

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Table 4 Experimental data / (degree)

Number of stone columns, n

cu (kPa)

s/d ratio

7

22

1.5

40

400

206

7

21.5

2

40

400

185

7 7

21.3 21.5

2.5 3

40 40

400 400

165 148

7

21.2

3.5

37.27

400

120

7

21.5

1.5

37.27

500

145

7

21.5

2

37.27

500

134

7

21.5

2.5

35

500

128

7

21.5

3

35

500

118

7

21.5

3.5

35

500

106

L (mm)

qu (kPa) measured value

5 MVR Model Development To validate the best ANN model presented in this study, an MVR analysis (multi-variable regression analysis) was also performed with the same dataset for the prediction of bearing capacity of the stone columns. In the present study, to conduct the MVR model, n, cu, s/d, / and l were considered as the independent variables and the bearing capacity qu was considered as the dependent variable. The relation between the dependent and independent variables is expressed in form of a standard equation as in Eq. (6) Y ¼ c þ B1 X1 þ B2 X2 þ B3 X3 þ . . .Bn Xn e

Fig. 7 Deformed mesh of Plaxis-2D model

where ti is the measured value of bearing capacity of the stone columns, and xi is the predicted value of the bearing capacity of stone column. From Table 7, it can be seen that for the model q9010 (ten-cross validation), the algorithm BR has the lowest values of MAPE, VARE, and MEDAE. For the model q80-20 (five-cross validation), the algorithm BR has the lowest values of all the statistical parameters (MAPE, VARE, and MEDAE). For the other two models (i.e. q70-30 and q50-50) also, the algorithm BR has the lowest values. Kayri (2016) also obtained a better performance with BR algorithm than with LM algorithm.

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ð6Þ

where c is the constant on Y intercept. X1 , X2 , X3 and Xn are the independent variables and B1 , B2 , B3 and Bn are the slopes associated with the above mentioned independent variables; Y is the dependent variable and e is the approximated error. 5.1 Comparison of ANN Model with the Statistical Approach of MVR Model The BR algorithm with ten-fold cross validation technique is also validated with the MVR model with the same set of data. The comparisons of all the results are presented in Table 8. The statistical presentation from Table 8 shows the superior capability of ANN model than MVR model for the prediction of bearing capacity of stone columns. The regression analysis of data was performed with F-test and t test at 95% confidence interval. The statistical results of MVR

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Table 5 Plaxis 2D data / (degree)


cu (kPa)

s/d ratio

7

25

1.5

40

400

232

7

25

2

40

400

210

7 7

25 25

2.5 3

40 40

500 500

194 175

7

25

3.5

37.27

500

150

7

22.5

1.5

37.27

600

190

7

21.5

2

37.27

600

165

7

21.5

2.5

35

600

142

7

23

3

35

600

125

12

25

2.5

37.27

400

227

12

23

2.5

37.27

400

199

12

21.5

2.5

37.27

400

175

19

25

2.5

37.27

400

250

19

23

2.5

37.27

400

227

19

21.5

2.5

37.27

400

200

model are presented in Table 9. The t-stat value, p value and upper and lower limits associated with the individual coefficients are listed in Table 9. From Table 9, it is observed that p values associated with the parameters n and l are very high i.e. the confidence level is very low [(1 - P) \ 95%] which represents an insignificant results for the MVR model. Again, for the parameters cu, s/d, and /, the P value is very low, which results a significant coefficient in MVR model. Therefore, it is concluded that the MVR model rejects the significance of the parameters n and l in bearing capacity prediction and hence, it fails to predict precisely the bearing capacity of stone columns.

6 ANN Model Equation to Determine the Bearing Capacity of Stone Columns (q) Based on Trained Neural Network Goh et al. (1998) expressed the mathematical equation based on the weights of input and output for ANN model as shown in Eq. (7). ( " ! #) h m X X Y ¼ fsig b0 þ wk fsig bhk þ wik Xi ð7Þ k¼1

i¼1

where fsig is the sigmoid transfer function (in this case logistic); b0 is the bias at the output layer; wk is the connecting weight between kth neuron of the hidden layer and the single output neuron; bhk is the bias at

L (mm)

qu (kPa) measured value

neuron k of the hidden layer; wik is the connecting weight between input variable i and neuron k of the hidden layer; Xi is the input variable i; and Y is the output parameter. In this study, by using this equation, the normalized Y value (in this case 0–1) is found out. In this case an ANN model is performed to find out the bearing capacity of stone column. The three layered neural network structure of this ANN model is presented in Fig. 8. The middle layer is called hidden layer which receives inputs from the input layer and sends output to the output layer. The connection weights from input to hidden and hidden to output are shown in this figure which was determined from the matlab R2013a programming. The values of weights and biases of the trained ANN model of BR algorithm with four numbers of hidden neurons are presented in Table 10. To obtain the correlation of Y, the following expressions are written: s A1 ¼ 1:235 þ 0:2345n 0:1915cu þ 0:4325 d þ 0:3125/ þ 0:1190l ð8Þ A2 ¼ 0:2420 þ 0:3122n þ 0:2321cu þ 0:3128

s d

0:1280/ þ 0:2235l ð9Þ

123

1856

Geotech Geol Eng (2018) 36:1845–1861

Table 6 Comparison of qu (kPa) values obtained from experiments and Plaxis, ANN model and other established theories qu (kPa) measured value from experiment and Plaxis 2D

qu (kPa) predicted value from ANN

qu (kPa) from IS CODE (2003)

qu (kPa) from Etezad et al. (2015)

qu (kPa) from Afshar and Ghazavi (2014a)

206

215

182.76

262

350

185

182

170

238.05

335

165

161

155.8

212

330

148

148

131.77

179

327

120

116

118

152

194.07

145

152

164.43

213

257

134

133

142

203

245.89

128

131

153.3

196

182.45

118

118

161

190

170.11

106

109

152

182

159.32

232

250

230

306.8

380.44

210

217

218

296

380.06

194

189

211.5

262

356

175

173

199

228.5

329.7

150 190

150 185

185 189.072

190 233

228.55 206.038

165

160

168.5

206

195.89

142

145

148

182

171.45

125

121

134

205.45

179.5

227

232

199

218.16

259

199

205

188

205

240.8

175

172

149

194

217.2

250

247

211

242.5

298.55

227

223

205

232

260.6

200

203

197

223

226.79

A3 ¼ 0:790 þ 0:0353n þ 0:5323cu 0:2972

s d

B3 ¼

1:214 ½1 þ eA3

ð14Þ

B4 ¼

0:2573 ½1 þ eA4

ð15Þ

þ 0:0491/ 0:3232l ð10Þ A4 ¼ 0:2143 þ 0:0544n þ 0:6520cu þ 0:1781

s d

C1 ¼ 0:3125 þ B1 þ B2 þ B3 þ B4

þ 0:6563/ 0:3724l ð11Þ

Y¼

The Y value in Eq. (17) is the normalized value which is to be denormalized to obtain the actual bearing capacity of the stone column (q).

B1 ¼

0:2212 ½1 þ eA1

ð12Þ

B2 ¼

0:3250 ½1 þ eA2

ð13Þ

123

1 ½1 þ eC1

ð16Þ

q ¼ Ymin þ ðYmax Ymin ÞY

ð17Þ

ð18Þ

Geotech Geol Eng (2018) 36:1845–1861

1857

For cu = 30 kPa, applying Skempton’s formula for square footing on the ground surface, the net ultimate bearing capacity of soil is given by Df B qnu ¼ 5cu 1 þ 0:2 1 þ 0:2 L B ¼ 5 30 ð1 þ 0:2 0Þð1 þ 0:2Þ ¼ 180 kN=m2

Table 7 Statistical errors for all the ANN models for different algorithms ANN models

Algorithms

MAPE

VARE

MEDAE

q90-10

BR

2.54166

2.537

1.075

LM

2.5421

2.539

0.995

GDM

3.1612

3.142

1.743

BFGS

3.0625

3.062

1.401

BR

2.342

2.343

1.213

LM

2.35012

2.322

1.003

GDM

3.1111

3.107

1.821

BFGS

3.0272

3.018

1.522

BR

2.67056

2.648

1.402

LM

2.646

2.607

1.009

GDM

3.062

3.041

1.915

BFGS

2.9711

2.970

1.523

BR

2.630244

2.641

1.431

LM

2.53025

2.510

1.011

GDM BFGS

3.1121 3.0642

3.107 3.047

1.942 1.550

q80-20

q70-30

q50-50

Considering around two and half times improvement in qnu due to installation of stone columns, the improved net ultimate bearing capacity qnu ¼ 2:5 180 ¼ 450 kN/m2 : Assuming a suitable factor of safety, the net safe bearing capacity, qns ffi 250 kN/m2 (say). Therefore, size of the footing can be assumed as follows: rffiffiffiffiffiffiffiffi 800 B¼ ¼ 1:78 ffi 1:8 m: 250 Again, for stone column, qns ffi 250 kN/m2 , minimum and maximum values of Y are considered as Ymin = 200 kPa and Ymax = 260 kPa respectively. Now, to design the group of stone columns, the following steps are used:

Table 8 Comparision of ANN and MVR model Model

R

SE

MAPE

VARE

MEDAE

ANN

0.99781

1.613

2.54166

2.537

1.075

MVR

0.99584

1.825

2.98441

2.512

1.135

• • •

where Ymax and Ymin are maximum and minimum values of Y respectively which are taken from the dataset.

From Eq. (18), Y value is determined. C1 value is found out from Eq. (17). Then by using Eqs. (8)–(16), with the trial and error method, all the design parameters (n, s/d and l) are estimated as presented in Table 11.

7 Mechanistic Analysis

8 Sensitivity Analysis of the Input Parameters of Experimental and Plaxis-2D Data

Let us consider a column taking an axial load of 800 kN resting on a group of stone columns installed in a soft clayey soil with cu = 30 kPa, angle of friction of stone materials, / = 42. The stone column group is to be designed.

In this study five input parameters were considered to obtain the bearing capacity of the stone columns. All the input parameters affect the bearing capacity little more or less. In order to obtain the parameter which

Table 9 Statistical parameters of MVR model

Coefficients

SE

t-Stat

p value

Lower 95%

Upper 95%

Intercept

- 6.86

3.52

- 5.61539

2.48E-07

- 30.07

-13.603

n

- 0.53

0.32

- 0.43

0.253

- 0.62

0.26

cu

1.7

0.003

- 1.342

0.000

- 0.35

0.32

s/d

3.1

0.07

14.8

0.000

0.44

0.98

/

1.33

0.022

11.5

1.01E-28

0.14

0.44

l

0.05

0.14

0.293

0.03

0.06

8.402

123

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Geotech Geol Eng (2018) 36:1845–1861

Fig. 8 Structure of proposed ANN model

Table 10 Connection weights and biases for the trained ANN model for BR algorithm of 90-10 model Neuron

Weights

Biases

Input 1

Input 2

Input 3

Input 4

Input 5

Output

Hidden neuron 1 (i = 1)

0.2345

- 0.1915

0.4325

0.3125

0.1190

0.2212


0.3122

0.2321

0.3128

- 0.1280

0.2235

- 0.3250

Hidden layer

Output layer

1.235

0.3125

0.2420


0.0353

0.5323

- 0.2972

0.0491

- 0.3232

1.214

0.790


0.0544

0.6520

0.1781

0.6563

- 0.3724

0.2573

0.2143

Table 11 Design of group of stone columns arranged in a triangular pattern to carry a vertical load of 800 kN


Spacing to diameter ratio, s/d

Length of stone columns, l in m

7

2

2.8

7

2.5

2.9

7

3

3.0

12

3

2.9

12

3.5

3.02

12

4

3.14

mostly affects the bearing capacity of the stone columns, a sensitivity analysis of the input parameters was performed. The sensitivity of an input parameter is defined as the change in the output expressed as percentage for 1% increase in the input parameter. For an example, 1% increase in u value of the soil changes the value of bearing capacity by 5.88% hence, the sensitivity of u is 5.88%. This sensitivity analysis was

123

performed in matlab R2013a. The outcome of the analysis is shown in Fig. 9. It is observed that the fourth parameter i.e. friction angle, u of the stone column material affects the bearing capacity the most among all other parameters. The second parameter affecting the bearing capacity is spacing to diameter ratio (s/d) of a stone column and the parameter which

Geotech Geol Eng (2018) 36:1845–1861

1859

(b)

6

Change in output in %

5

(c) 4

(d)

3

2

(e) 1

(f) 0

del n

del cu

del s/d

del phi

del l

Input parameter

(g) Fig. 9 Sensitivity analysis of each input data

has the least effect on bearing capacity is the length of the stone column.

(h)

(i) 9 Conclusions In this study, total 16 ANN models were performed with four different combinations and four different learning algorithms to predict the bearing capacity of stone columns. For this purpose, 90 data were collected from the previous technical literatures with different input values. The bearing capacity of stone column mainly depends on the friction angle (u) of the stone column materials, diameter (d), length (l), spacing between the stone columns (s), numbers of stone column (n), and undrained cohesion (cu) of the surrounding soft soil. So all these parameters were considered as inputs and bearing capacity of stone column was taken as output. To conduct the ANN model, a three-layer feed-forward back propagation neural network was used. By optimizing the number of hidden neurons for every models separately, different numbers of hidden neurons were used to get the best results. Four types of algorithms i.e. BR, LM, GDM, and BFGS were used in this case. From the analysis, the following conclusions are drawn: (a)

In case of 90-10 model, the algorithm BR with 4 numbers of hidden neurons gives the best result with correlation, R = 0.99781 and SE = 1.612.

(j)

(k)

(l)

In case of 80-20 model, the algorithm BR with 7 numbers of hidden neurons performs the best with correlation, R = 0.98865 and SE = 1.914. For 70-30 model, the algorithm BR shows the best solution with 19 number of hidden neurons with correlation, R = 0.979 and SE = 2.114. For 50-50 model also, the algorithm BR also shows the best solution with nine hidden neurons with correlation, R = 0.972 and SE = 2.414. Overall performance of the BR algorithm is the best as compared to other algorithms. ANN model has the superior capability in predicting the bearing capacity of stone columns over MVR model. From the MVR model, it is reported that the parameters cu, s/d, and / are significant in prediction of bearing capacity of stone columns. ANN model shows a better match with the experimental and Plaxis 2D results than any other established method. The established theories are site specific; the ANN model is soil specific. For all the models, the algorithm BR has the lowest value of all the statistical parameters such as Mean Absolute Percentage Error (MAPE), Variance Absolute Relative Error (VARE), and Median Absolute Error (MEDAE). From the sensitivity analysis, the bearing capacity of stone column is mostly affected by the friction angle of the stone column material and is least affected by the length (l) of the stone columns. A prediction model equation based on the weights of the trained ANN model is presented to predict the bearing capacity of stone column.

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