Study of the Effect of Installation of Stone Columns on the Stability of Soil Slopes using Finite Element Method

There are several ways to mitigate the failure of a soil slope. The installation of stone columns for this purpose, is easy as well as economical. As their strength is greater than that of soil, they disrupt the slip surfaces. Also being more porous, they dissipate the pore pressure generated in the soil in the vicinity of the stone columns to a certain extent. This paper illustrates 2D finite element analyses carried out to simulate the behavior of stone columns in cohesive soil using the computer program PLAXIS. Parametric study was conducted to analyze the effect of number, length and diameter of the stone columns on the Factor of Safety of the soil slope. The F.O.S. of the soil slope was predicted using prediction models created using artificial neural networks (ANN) considering slope geometry like slope angle and height, material properties like cohesion and angle and internal friction of soil as well as granular material used in the columns, and the stone column parameters like the stone column’s number, length, diameter and the centre to centre (c/c) distance between two adjacent columns as input parameters. It was found that introduction of stone columns significantly improved the stability of the soil slopes. Further, it was seen that all the parameters considered in the study influenced the Factor of Safety of the soil slope.


Introduction
There are several ways to mitigate the failure of a soil slope.The installation of stone columns for this purpose, is easy as well as economical.As their strength is greater than that of soil, they disrupt the slip surfaces.Also being more porous, they dissipate the pore pressure generated in the soil in the vicinity of the stone columns to a certain extent.The main objectives served by the installation of stone columns are reduction of settlement and improvement of bearing capacity and stability of soil slopes [1][2][3][4].Various numerical methods such as Finite Element Method (FEM) and Finite Difference Method (FDM) have been used successfully to estimate the stability of slopes by calculating the Factor of Safety (F.O.S.) [4][5][6][7].The F.O.S. has been seen to increase with an increase in the diameter of the stone column [5,8].This paper describes the influence of number, length as well as diameter of the stone columns on the F.O.S. of the soil slope.The F.O.S. of the soil slope was predicted using prediction models created using artificial neural networks (ANN) considering slope geometry like slope angle and height, material properties like cohesion and angle and internal friction of soil as well as granular material used in the columns, and parameters of the stone columns like the stone column's number, length, diameter and their distance from one another at the centre, as input parameters.ANN is a nonlinear projection method and because of its propensity to overparameterize, ANN tends to be relatively insensitive to problems of multicollinearity, that is encountered often in regression analysis.When training is done with adequate data, ANN was found to be have considerably lesser predictive error than other basic machine learning tools like linear regression [9].1282 (2023) 012020 IOP Publishing doi:10.1088/1757-899X/1282/1/012020 2 A geometry model is a two-dimensional representation of a three-dimensional problem, consisting of points, lines, and clusters.The geometry is created in the global coordinate system's X-Y plane, with the z-axis pointing away from the plane.The default element chosen for the study is a 15-node triangular element.The 15-node element is more accurate and yields more realistic results.It should have a representative subsoil division into separate soil layers, building stages, and loading.The model must be large enough that boundary conditions do not affect the conclusions of the problem under investigation.The Mohr-coulomb failure criteria and plane strain condition is used for developing the model.For the soil slope, material type is selected as undrained while for the stone columns it is slected as drained.The fine element distribution is used for generation of mesh.
The unsaturated unit weight (γunsat) is taken as 19 kN/m 3 and 15 kN/m 3 for the soil slope and stone columns respectively.The saturated unit weight (γsat) is taken as 20 kN/m 3 for both soil slope and stone columns.Both kx and ky (coeffecients of permeability in x and y direction) are 1x10 -3 m/day for soil and 10.368 m/day for stone.The Young's modulus (E) is taken as 1x10 4 kN/m 2 for soil and 5x10 4 kN/m 2 for stone.The Poisson's ratios for soil and stone are taken as 0.35 and 0.3 respectively.
The parameters considered are slope angle, β (°), slope height, H (m), cohesion of soil, csoil (kN/m 2 ), cohesion of stone, cstone (kN/m 2 ), friction angle of soil, φsoil (°), friction angle of stone, φstone (°), stone column number, N, stone column length, L (m), stone column diameter, d (m) and c/c spacing between two stone columns, S (m).The methodology is divided into deterministic and probabilistic studies.In the deterministic approach, a soil slope of height 15 m and slope angle 45°is modelled in PLAXIS 2D with csoil = 35 kN/m 2 , cstone = 42 kN/m 2 , φsoil = 10°φstone = 0.1°and stone columns are installed on the slope (figure 1).  1.A trial-and-error approach is used because there is no established algorithm in the literature for finding the figure of hidden layer neurons.The figure required in a model is calculated using Jeff Heaton's approach and various neural networks are then built between the highest and lowest values [10].In this study, the network training function (TRAINLM) based on

Methodology
the Levenberg-Marquardt back-propagation training technique is utilised, which has a faster convergence than other methods and is frequently recommended [11][12].The flow chart for the development of ANN is shown in figure 2.

Effect of stone column length
From the plot shown in figure 3, it can also be seen that for different numbers of stone columns, F.O.S. increased as the stone column length increased.

Effect of stone column diameter
Considering 6 stone column number of length 10 m, factors of safety were calculated for stone column diameters varying from 0.6 m to 0.9 m.It was observed that the F.O.S. increased with the increase in the diameter of the stone columns (figure 4).

Prediction Model
The development of prediction model is done using ANN.A hidden layer with 7 neurons is found to be the most effective for F.O.S. prediction in an ANN architecture after each network is assessed based on its R and MSE values.The adopted network architecture is shown in figure 5.The overall regression plot of the adopted network model is shown in figure 6. Very high correlation coefficient is observed for training and testing data in ANN topology.

Conclusions
The study's findings lead to the following conclusions: • Introduction of stone columns significantly improved the stability of the soil slopes.
• The F.O.S. of soil slopes increased as the stone column number increased.
• The F.O.S. of soil slopes increased as the stone column length increased.
• The F.O.S. of soil slopes increased as the stone column diameter increased.
• Predictive model can be successfully developed using ANN for the prediction of F.O.S. of soil slopes considering slope geometry like slope angle and height, material properties like cohesion and angle and internal friction of soil as well as granular material used in the columns, and parameters of the stone columns like number, length, diameter of the stone columns and c/c spacing between two adjacent stone columns, as input parameters.The RMSE and MAE are found to be 0.060 and 0.047 respectively.The forecasted results were found to be very similar to the calculated results.

Fig. 1
Fig. 1 Soil slope with stone columns modelled in PLAXIS 2D The effects of N, L, d and S on the F.O.S. of the soil slope are studied by keeping the other parameters constant.The probabilistic study includes development of prediction models for predicting the F.O.S. of the soil slope using ANN considering all the parameters as input parameters.A total of 150 datasets are considered for development of the models are another 20 datasets are used for validation of the models.The input parameters and their corresponding ranges used are shown in Table1.A trial-and-error approach is used because there is no established algorithm in the literature for finding the figure of hidden layer neurons.The figure required in a model is calculated using Jeff Heaton's approach and various neural networks are then built between the highest and lowest values[10].In this study, the network training function (TRAINLM) based on

Fig 4 .
Fig 4. Effect of diameter of stone columns on F.O.S.

Fig 6 .
Fig 6.Regression plot for adopted ANN model 20 datasets are used for validation of the models.The error analysis is carried out by calculating the RMSE and MAE.The RMSE and MAE values come out to be 0.060 and 0.047 respectively.A comparison is done between the calculated value and predicted value and it is shown in figure7.It can be seen that the forecasted results were found to be very similar to the calculated results.

Fig. 7
Fig. 7 Comparison between calculated and predicted values of F.O.S.