Finite element investigation and ANNbased prediction of the bearing capacity of strip footings resting on sloping ground
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Abstract
Footings placed on or near a slope, the slope face act as a finite boundary that leads to an inappropriate and inadequate development of the plastic region of failure as the foundation approaches the limit state under the applied loading. Depending upon the position of the footing and steepness of the slope, the abovementioned phenomena might lead to considerable decrease in the bearing capacity of the foundation. A sequence of finite element analysis has been carried out using Plaxis 2D v2015.02 to inspect the ultimate bearing capacity of strip footings placed at the crest of the c–φ soil slope. The effect of various geoparameters namely internal angle of friction (φ), cohesion (c), setback distance (b), width of footing (B), slope angle (β) and embedment depth of footing (D_{f}) on the ultimate bearing capacity of the footing have been investigated and the outputs obtained are appropriately elucidated. In addition, large database of numerically simulated ultimate bearing capacity has been considered to develop and verify the ANN model to establish the relative importance of the input parameters.
Keywords
Ultimate bearing capacity Slope Strip footing Finite element Plaxis ANNIntroduction
The ultimate bearing capacity (UBC) of foundation is defined as the maximum stress that it can carry without undergoing a shear failure. Based on the shear strength parameters of the soil, Terzaghi [39] was the first to estimate the UBC of a strip footing placed on a uniform horizontal ground. The basic proposal for the bearing capacity of strip footings has undergone several modifications, primarily related to the bearing capacity factors, as well as inclusion of several new contributory factors [20, 29, 40]. Moreover, in order to accommodate different shapes of the footings (square, rectangular, circular or combined), shape factors were introduced in the bearing capacity expressions [20, 40].
Rapid growth of urbanization in hilly regions of India has resulted in myriads of residential and commercial constructions. The foundations of such constructions are mostly shallow, and are located either on the crest or on the benched face of the slopes. Apart from the urban constructions, transmission towers, mobile tower, retaining walls, water tanks, bridge abutments and even foundations for transportation links are mostly placed on the slopes. Foundation on slopes is a challenging and complex problem for the geotechnical engineers.
There exists limited researches on experimental and numerical investigations connected to the estimation of the bearing capacity of a footing located on a slope [1, 4, 5, 6, 9, 22, 23, 37, 38]. Most of the investigations have been done to estimate the bearing capacity of shallow footings, resting on dry cohesionless sandy soil slope to examine the effects of geoparameters. Very few literatures exist related to the numerical investigations for strip footings resting on a c–φ soil slope [26].
Neural networks have been initiating from the research of McCulloch and Pitts [28], who recognized the capability of interconnected neurons to estimate some logical functions. Hebb [21] investigated the importance of the synaptic connections in the learning process. Later, Rosenblatt [35] had provided the first operational model of a neural network: the ‘Perceptron’. The perceptron, built as an analogy to the visual system, was able to learn some logical functions by modifying the synaptic connections. Artificial neural networks (ANNs) usually considered when the relationship between the input and output is complex in nature or application of another available method takes a large computational time and effort is very expensive.
Many researchers in different field of the civil engineering have considered ANNs. Lee and Lee [25], Das and Basudhar [11] and Momeni et al. [31] used ANN for forecasting the pile bearing capacity. Ghaboussi et al. [18] had showed that ANNs were powerful tools for the mathematical constitutive modelling of geomechanics. Shahin et al. [36] had applied successfully back propagation and MultiLayer Perceptron’s (MLPs) to predict the settlement of shallow foundations on granular soils. Goh [19] had demonstrated that ANNs have capability to model the complex relationship between seismic soil parameters and liquefaction potential using actual field records. The prediction of swelling pressure and hydraulic conductivity characteristics of clayey soil have been investigated by several researchers [12, 13, 16, 30]. Noorzaei et al. [34], Kuo et al. [24], Behera et al. [7, 8] had used ANNs to predict the UBC of strip footing resting on horizontal ground surface loaded with centrically and eccentrically loading.
Habitats in the hilly regions mostly comprise of the houses resting either on the slope face or on the slope crest. A ‘Compilation of the catalogue of the building typologies in India’ revealed that most of the buildings located in the hilly terrains in the NorthEastern regions of India are supported by shallow footings [33]. It has been perceived from the field study [33] that in the hilly region the foundations are constructed on the slope by cutting and filling the slope face. Strip footings are commonly considered for building foundations over the slope. Hence, it is important to research and comprehend the failure mechanism of strip footings resting on the c–φ soilslope. Hence, based on 2D finite element (FE) analysis considering Plaxis 2D, this article reports the effect of various geoparameters on the normalised UBC [q_{u}/γH_{s} where, q_{u} = ultimate bearing capacity, γ = unit weight of soil and H_{s} = height of slope] of a strip footing resting on crest of c–φ soilslopes. The 2D numerical model also provides a description of the failure mechanism involved in the process of loading and failure of the footing. An artificial neural network (ANN) model has been made for the prediction of normalised ultimate bearing capacity (q_{u}/γH_{s}) of strip footing resting on crest of c–φ soilslope from the outcomes of the parametric investigation. From the ANN results, a sensitivity analysis has been performed to comprehend the importance of various geoparameters considered for assessing UBC.
Numerical analysis
Plaxis 2D v2015.02 is a finite element (FE) tool intended for twodimensional analysis of deformation, stability and ground water flow in geotechnical engineering. Plaxis 2D is equipped with several features to deal with various aspects of complex geotechnical problems incorporating advanced constitutive models for analysis of timedependent, nonlinear and anisotropic behaviour of soil or rock. It has been observed from the past researches [1, 22] that dry cohesionless sandy soilslope has been considered for experimental and numerical investigation for estimating UBC of footings resting on or near the slope. In case of practical scenario, hillslopes are made of different types of soils, ranging from fine silts, marginal soil mixtures, gravels, as well as highly weathered rock masses. As a result, finite element 2D analysis has been carried out to study the behaviour of a strip footing resting on a c–φ soil slope with an aim to represent the commonly occurring building foundations on the hill slopes of India.
Description of the modelling
To perform finite element calculation, the model geometry is discretized into smaller finite number of elements. In Plaxis 2D, the basic soil elements are the 6noded and 15noded triangular elements. In the present research, 15noded triangular elements were used (Fig. 2) as it provides more nodes and Gauss points aiding in comparatively precise determination of displacements and stresses as compared to the 6noded triangular elements. Plaxis 2D program allows for a fully automatic generation of finite element meshes using the ‘robust triangulation scheme’. In Plaxis 2D, five basic types meshing is available namely ‘very coarse’, ‘coarse’, ‘medium’, ‘fine’, and ‘very fine’, each having progressively refined mesh coarseness factor. The mesh should be sufficiently and optimally fine to obtain correct numerical results. A very coarse mesh may fail to capture the important responses of the domain while beyond optimally fine meshed, there are chances of the accumulation of numerical errors. Moreover, very fine meshing should be avoided since it will take excessive time for calculations. Any basic meshing scheme can be adopted, with further provisions of local refinements, as demanded by the merit of the problem and the location of the response points in the numerical simulation. The present numerical investigation has been done with fine meshing as shown in Fig. 2.
Properties of concrete used for strip footing
Unit weight (γ) (kN/m^{3})  Modulus of elasticity E (GPa)  Poisson’s ratio (ν) 

25  22  0.15 
The failure mechanism generated beneath the footing resting on horizontal ground has been postulated by Terzaghi [39] on the basis of the assumption that the base of the footing is rough. In the same fashion, the base of the footing has been considered as rough base by taken into account R_{inter} = 1 to comprehend the failure mechanism of footing resting on or near the slope. The rough base indicates high friction between footing base and soil. Hence, relative horizontal movement between soil and foundation base is zero.
In Plaxis simulations, initial stresses need to be specified. Two possibilities remain for the specification of these stresses, namely ‘K_{0}procedure’ and ‘Gravity loading’. As a rule, K_{0}procedure should be used in case of horizontal surface and with any soil layer and phreatic lines parallel to the surface. For all other circumstances, Gravity loading should be considered.
In the current investigation, load controlled analysis has been considered. The vertical load has been provided over the strip footing till failure. In the present analysis, the embedded footing has been modelled by considering the practical construction sequences. Firstly, the soil has been deactivated from the foundation pit. The boundary conditions have been applied to the excavated pit by providing horizontal and vertical fixities to the vertical walls and bottom of the pit. Afterwards, the footing and the soil cluster over the footing have been activated. Then the given fixities have been removed. Finally, load has been activated over the footing till failure.
Artificial neural networks modelling
Neural networks are data processing systems consisting of a large number of simple, highly interconnected processing elements (artificial neurons) in an architecture inspired by the structure of the central cortex of the brain. They operate as black box and powerful tools to capture and learn significant structures in data. Neural networks can provide meaningful answer even when the data to be processed include errors or are incomplete and can process information extremely rapidly when applied to solve real world problems.
Stages of analysis

Validation of the numerical model.

Convergence study to determine the optimum mesh configuration.

Effect of the variation of geotechnical and geometrical parameters, specifically angle of internal friction of soil (φ), cohesion (c), slope angle (β), footing width (B), setback distance (b) and embedment depth of footing (D_{f}).

Development of ANN model for prediction of normalised ultimate bearing capacity (q_{u}/γH_{s}) and recognition of the relative importance of the various geotechnical and geometrical parameters used for evaluating the UBC (q_{u}).
Results and discussion
Validation study
In validation study, the load–displacement pattern obtained from numerical analysis is abrupt in nature for footing resting on crest of slope with zero setback distance (b/B = 0) as shown in Fig. 7a. Footing resting on crest of slope with zero setback distance (b/B = 0) is the most critical situation where the passive resistance is minimum. Hence, in case of zero setback distance, the UBC is least and load–displacement pattern is brittle in nature. The stability of foundation has been improved with increasing the setback distance as shown in Fig. 7b. Hence the UBC and passive resistance have been increased with increasing setback distance. Moreover, the load–displacement pattern (Fig. 7b) is not abrupt and the UBC can easily be obtained from it.
Parametric study
For footing resting on a sloping ground, the setback distance is perceived as one of the most important governing parameter in the assessment of bearing and deformation characteristics of the footing. The lesser the setback distance, higher is the possibility of failure of the footing exhibiting conditions of distress due to the deformation of the slope face. Hence, in order to highlight the effect of various parameters, a detailed parametric study has been conducted keeping the setback distance as one of the contributing parameters of the simulation. For a footing resting on a sloping ground, ten different setback ratios were considered in the analysis, namely b/B = 0–10, and the same is represented in Fig. 3.
Variation of cohesion (c)
Slope angle (β)
Width of footing (B)
Angle of internal friction (φ)
Embedment depth ratio of footing (D _{f}/B)
Study of failure mechanism
This phenomenon results in a considerable decrease of the confinement pressure, and hence, attenuation of the bearing capacity. As the setback ratio increases, the influencing effect of the slope face on the development of the passive mechanism gradually diminishes, as can be observed from the Fig. 14. It is noted that beyond a critical setback ratio (b/B) _{critical} of 6, the footing behaves as if resting on horizontal ground, wherein the developed displacement contours for the passive zone remains unaffected from the influence of the slope face.
ANN results
Number of hidden neurons in hidden layer
Neural network training, testing and validation
A neural computing system can adjust its behaviour in response to its environment. When sets of inputs are shown to the network, it will selfadjust to produce reliable responses through a process called training. Training is the process of altering the weights methodically in order to attain some desired results for a given set of inputs. The aim of training is to find a set of connection weights that will reduce the MSE predicting error in the shortest possible training time [14]. In total, 80% of the data are used for training and 20% are used for validation. The training data are further divided into 70% for the training set and 30% for the testing set [18].
Sensitivity analysis
 a.For each hidden neuron h, divide the absolute value of the inputhidden layer connection weight by the sum of the absolute value of the inputhidden layer connection weight of all input neurons, i.e. for h = 1 to nh, and i = 1 to ni:$$A_{ih} = \frac{{\left {W_{ih} } \right}}{{\sum\nolimits_{i = 1}^{ni} {\left {W_{ih} } \right} }}$$(4)
 b.For each input neuron i, divide the sum of the A_{ih} for each hidden neuron by the sum for each hidden neuron of the sum for each input neuron of A_{ih}, multiply by 100. The relative importance of all output weights attributable to the given input variable is then obtained. For i = 1 to ni:$$RI(\% ) = \frac{{\sum\nolimits_{n = 1}^{nh} {A_{ih} } }}{{\sum\nolimits_{n = 1}^{nh} {\sum\nolimits_{i = 1}^{ni} {A_{ih} } } }} \times 100$$(5)
Relative importance of inputparameters from Garson’s algorithm
Input  Relative importance  Relative importance (%)  Rank 

c  1.68  18.72  2 
φ  3.17  35.27  1 
B  1.47  16.39  3 
b/B  1.02  11.37  5 
β  0.60  6.64  6 
D _{ f} /B  1.05  11.61  4 
Conclusions

Mesh convergence study aided to define a nondimensional optimal mesh size for the Plaxis 2D models so as to obtain accurate solutions from the numerical simulation.

Ultimate bearing capacity increases with the increase in the angle of internal friction and cohesion for footing resting on sloping ground owing to the fact that increase in the shear strength of foundation soil.

Ultimate bearing capacity increases with an increase of embedment depth of the footing owing to increase in the degree of confinement restricting the movement of the soil towards the sloping face.

Ultimate bearing capacity gets significantly increased with the increase in the footing width.

Ultimate bearing capacity with the increase of slope angle, which is associated with the increased soil movement towards the slope.

Ultimate bearing capacity increases with the increasing setback distance. Beyond a setback ratio (b/B) _{critical} = 6, the footing behaves similar to that on horizontal ground.

The ANN model with c, φ, B, b/B, β and D_{f}/B as input parameters is the ‘best’ model, based on coefficient of efficiency, for training, testing and validation data set.

Based on sensitivity analyses; it has been perceived from Garson’s algorithm that angle of internal friction, φ and, c are the most important input parameters for estimating the ultimate bearing capacity of strip footing resting on crest of sloping ground.
Notes
Authors’ contributions
The author read and approved the final manuscript.
Acknowledgements
The author expresses his gratitude to Dr. Arindam Dey, Associate Professor, Civil Engineering, IIT Guwahati for his valuable support and guidance in comprehending the topics of strip footings on or near the slope which immensely assisted in evaluating the ultimate bearing capacity and failure mechanism of strip footing resting on crest of slope.
Competing interests
The author declares that he has no competing interest.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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