Stability Analysis of Homogenous Embankment using Statistical Method

Homogeneous earth embankments are man-made or artificial structure whose construction is an important part of highway, railway and flood structure projects. Soil exhibits different properties in different places. Reinforcing the soil with Geogrid can be effectively used for improving the durability and stability of embankment, this ensures uniform settlement of embankment and result in cost effective solution. This paper aims to develop two-dimensional stability charts and correlation for slope failure (slip circle) in homogeneous soil embankment. Sensitivity analysis is also done to determine the contribution of different soil parameter. The chart and correlation developed for stability analysis can be a convenient tool for preliminary estimation of factor of safety (FOS) and possibility of failure.


Introduction
Embankment is a structure built of earth/soil that is used to support roadways or railways above the ground surface.The cross section of embankment is trapezoidal, narrow at the top and wide at the bottom.In road construction embankment is provided when the water level rise above the ground level.Slope stability is one of the most important criteria for the design of embankment.The various factors affecting the slope stability viz., geometry of the slopes, ground water, cohesion, types of equipment used during the construction, angle of internal friction and so on.Translational, rotational, wedge and compound failure are the types of failure which occurs in an embankment.Under the rotational failure, Slope failure occurs when soil above the toe is weak.Condition under which the circular failure occurs, when particles do not get properly locked due to their shapes and compaction is not done up to maximum dry density.On the basis of area affected by the failure surface, circular failures may include, toe failure and base failure.Slope failure in road embankments occur when the load is in the form of a vehicle, which acts as a surcharge, increasing the soil's lateral earth pressure.Normally, we flatten the slope to make it more stable.As a result, it requires additional breadth, leads to the increase in construction costs.
A geosynthetic material made of polymer called geogrid can be used to reduce costs and increase stability.Geogrid are made of polyester, polypropylene etc.The function of Geogrid includes holding and capturing of aggregates together.Based on manufacturing, Geogrid can be extruded, woven and bonded.Also, on the basis of stretching direction done during manufacturing, geogrids are uniaxial and biaxial.Fellenius method of slices was the first to analyse the slope stability of embankment.Furthermore, various methods like Bishop Method developed over the time.Limit Equilibrium method 1282 (2023) 012017 IOP Publishing doi:10.1088/1757-899X/1282/1/012017 2 (LEM) is the most widely used method for analysing the slope stability.
Fellenius [1] published the first methodology of research which was related to slope stability analysis.However, due to ignorance of all interslices forces the fulfilment of equilibrium of individual slices was not met.Thus, effective stress at the slice base was incorrectly computed.Bishop [2] had given a method that fulfils the vertical force equilibrium and moment equilibrium around the centre of a circular slip surface.Still, this method cannot be used for the other slip surface.Janbu [3] provided a method that balances each slice's vertical force equilibrium but the net moment equilibrium was not met.So, he gave a correction factor for this inadequacy.Morgenstern and Price [4] provided a new method that included all considerations which were the drawbacks in the above three methods and can be used for any type of slip surface.Fredlund and Krahn [5] compared various 2D slope stability method and developed the FOS equation for solving various failure surfaces and loading conditions.A new method was introduced which satisfied both force equilibrium as well as moment equilibrium [6].LEM, FEM, and FDM are also some of the different approaches used for slope stability analysis.Out of these methods, LEM is the most frequent method used to calculate the factor of safety based on the Mohr-Coulomb Criterion.Burman et al. [7] found that for the best consistent calculation of factor of safety value, the Ordinary slice method is the best.Majedi et al. [8] found that most suitable place to provide Geogrid as reinforcement is above the toe of the embankment.Also, the Horizontal displacement is reduced by the use of Geogrid.Raouf [9] after investigation founded that the ideal spacing between the Geogrid layers is 0.5m.Esmaeili et al. [10] found that with increase in tensile strength of Geogrid the bearing capacity increases.One of the initial FEM was placed forwarded by Smith and Hobbs [11] for φu=0°soil.Naseer and Evans [12] found that in FEM critical slip surface is near slope face whereas it is further away in LEM.Slope failure of an embankment is the most common failure type of an embankment.In order to provide correction for the slope failure, a good understanding of failure mechanism of embankment is required.Since then, various methods and techniques have evolved.Limit equilibrium methods are predominantly used for slope stability.Analysing stability of huge number of slopes using convectional LEM is difficult as it is very time consuming to develop slope model.Multiple regression method is better alternative which provide simple equation and statistical techniques to calculate factor of safety of slopes without using iterative procedure thereby decreasing time and complexity in slope stability evaluation [13].Khan and Wang [14] developed correlations between Factor of Safety (FOS) and soil parameters using linear regression analysis to stabilize slope and achieve high shear strength.Azmoon et al. [15] gave deep learning-based methods to evaluate slope stability.To achieve this, a multiclass classification and regression models were employed.The objectives of the present study are given in the following section.1.To determine the FOS for slopes with varying soil parameter and Geogrid spacing from the toe of embankment using LEM and to develop stability charts for calculating the FOS. 2. To carry out a regression analysis to develop a correlation for determining the FOS of the slopes with Geogrid 3. To carry out sensitivity analysis to determine the most sensitive parameter.

Methodology
Three different slope angles of 30°, 45°and 60°for homogeneous earth embankment were considered.For each slope angle, three different embankment heights of 10m, 20m, and 30m were taken into consideration.For each slope and three different heights, random cohesion values were taken ranging from 10-40 kPa with random values of internal friction angle ranging from 10°-40°in reference with Maharashtra PWD Handbook on Earth and Composite Dam [16].The Geogrid layers were placed at different spacing from toe of embankment for different embankment height and slope angle.Using 1282 (2023) 012017 IOP Publishing doi:10.1088/1757-899X/1282/1/0120173 LEM, FOS were obtained for 2100 cases of embankment design for Fellenius method and Bishop's simplified method.

Development of Stability Charts
The FOS values were used to develop two stability charts based on data forecasting method for Ordinary and Bishop's method.The development of the new charts was prompted by the introduction of a legitimate tool for the quick evaluation of the slopes' safety, based on the stringent LEM.For a range of inclination angles, Bell [17] suggested that (F / tan φd) can be used as a function of (cd / γ H tan φd).He used the modified stability number, N*, denoted by (cd / γ H tan φd).Utilizing such a representation has the benefit of making parameter N* independent on safety factor F. Therefore, no iterative processes are needed for the estimate of the safety factor from charts that are provided as functions of N*.In order to take into account the spacing of the geogrid layer, (F m / tan φ) was used as a function of (c m / γ H tan φ) where (m = Gs / H).A typical example of embankment with geogrid is shown in Fig 1 .

Development of prediction model
The data used to develop the stability charts were also used to develop the prediction model.For the development of prediction model, regression analysis was used in MS Excel.The independent or input parameters include, slope angle (β), slope height (H), cohesion (c), angle of internal friction (φ) and spacing of geogrid layer (Gs).FOS is used as a dependent or output parameter.

Sensitivity Analysis
Sensitivity analysis was carried out on the pre-trained regression model to examine the impact of each predictor variables on the prediction of dependent variables.An approach based on data perturbation was employed for the sensitivity analysis [18].Each input parameter was successively raised and lowered while all other input variables remained constant.Here, each input parameter was modified by ± 10%, and the percentage change in output was then calculated.The sensitivity analysis was done by taking the average value of all independent parameters.The dependent parameter is calculated by using the regression formula given in equations (1) or (2).If ΔX is the percentage increment or decrement and ΔY is the difference between the FOS considering 10% increment or decrement and the predicted FOS, then relative sensitivity (RS) is expressed as RS = (X .ΔY) / (Y' .ΔX).
The trained MLR models' prediction performance is assessed by determining the correlation percentage.Furthermore, the error analysis is carried out by calculating the root mean square error (RMSE) and mean absolute error (MAE).

Results and discussions
After carrying out the various analyses, the results are plotted in the following sections.

Stability charts
The computational results are used to develop two stability charts, firstly, for the

Prediction model
The equation of multiple linear regression (MLR) for Bishop's method for prediction of FOS is given in the following equation: FOS=1.832-0.025x β-0.042 x H+0.022 x c+0.037 x φ-0.001 x Gs 1 (R2 = 0.900; MSE = 0.062) The correlation matrix for FOS with other independent parameters for Bishop's method is shown in Table 1.This helps in identifying the presence or absence of a relationship between two variables clearly

Sensitivity analysis
The data perturbation method, which involved altering the input values by ± 10% one at a time while keeping other parameters constant, was used to do sensitivity analysis on the created models.The percentage change in output due to input perturbation was determined in order to determine the parameters that were the most sensitive, and the findings are displayed in Tables 3 and Table 4 The well-trained MLR models are validated by considering 20 different data sets (shown in Table 5) not used during the model building process.In Fig. 6, model simulation results arepresented and statistical metrics, including correlation %, RMSE, and MAE are compared.

Discussion
In this study, a distinctive method was employed to develop design charts and prediction models using 2100 artificial slope cases.The capacity of the created models and design charts to predict the FOS was implied by successful model validation with 20 different data sets not used during the model building process.The data perturbation method of sensitivity analysis shows that slope angle is the most influencing factor.Prediction of slope stability has been investigated previously over a 5-to-15-year period.Current study highlights the importance of using geogrid in the model.In developing countries like India, construction of houses, roads, tunnels, widening of roads, etc. are 1282 (2023) 012017 IOP Publishing doi:10.1088/1757-899X/1282/1/01201710 going on at a very fast rate.Hence, many slope cuttings are done which results in the increase in the stability issues.Geogrid can be an effective tool for enhancing the stability of the slopes.The presented methodology provides the site engineers a rapid and an effective way to arrive at a stable slope geometry.

Conclusion
For the slope stability analyses, stability charts and regression analysis were conducted.This design chart is convenient to use and it doesn't require any iterative process to determine the FOS.However, with zero value of angle of internal friction, the charts produced are not intended to be used.The following conclusions can be drawn from the present study.1. Bishop's method gives higher FOS comparison to Fellenius method.Higher FOS is obtained when Geogrid touches slip circle.2. Regression statistics for Bishop's method provides better FOS than Fellenius method.On comparing the MLR and stability charts, the FOS predicted using stability charts have higher correlation %, lower RMSE and lower MAE.Thus, the accuracy of stability charts over MLR is found to be high.
3. According to the sensitivity analysis tornado chart, slope angle is the most influencing parameter for the slope stability.

Fig 1 .
Fig 1. Embankment with Geogrid Fellenius method and secondly, for the Bishop's method for different slope inclination having different values of geogrid spacing.The use of these charts is very straight forward.If the slope geometry, soil parameters and geogrid spacing are known, one can easily calculate the FOS thereby, eliminating the rigorous method of iteration.The stability charts for Fellenius method and Bishop's Method are shown in Fig 2 and Fig 3. Of course, these stability charts developed in terms of N* cannot be used for slopes with zero internal friction angle.

Fig 2 .
Fig 2. Stability chart for Fellenius method of slope stability .The tornado diagram [19] showing the most sensitive parameter for Bishop's method and Fellenius method is shown in Fig 4 and Fig 5.From Fig 4 and Fig 5, it can be seen that for both the slope stability methods, the slope inclination is found to be the most sensitive factor.

Table 1 .
Correlation matrix for Bishop's method of slope stability

Table 2 .
Correlation matrix for Fellenius method of slope stability

Table 3 .
Sensitivity analysis for Fellenius method of slope stability.

Table 5 .
Data set for model validation