Estimating the Ultimate Bearing Capacity for Strip Footing Near and within Slopes Using AI (GP, ANN, and EPR) Techniques

Numerical and computational analyses surrounding the behavior of the bearing capacity of soils near or adjacent to slopes have been of great importance in earthwork constructions around the globe due to its unique nature.*is phenomenon is encountered on pavement vertical curves, drainages, and vertical infrastructure foundations. In this work, multiple data were collected on the soil and footing interface parameters, which included width of footing, depth of foundation, distance of slope from the footing edge, soil bulk density, slope and frictional angles, and bearing capacity factors of cohesion and overburden pressure determined for the case of a foundation on or adjacent to a slope. *e genetic programming (GP), evolutionary polynomial regression (EPR), and artificial neural network (ANN) intelligent techniques were employed to predict the ultimate bearing capacity of footing on or adjacent to a slope. *e performance of the models was evaluated as well as compared their accuracy and robustness with the findings of Prandtl. *e results were observed to show the superiority of GP, EPR, and ANN techniques over the computational works of Prandtl. In addition, the ANN outclassed the other artificial intelligence methods in the exercise.


Introduction
Building substructures are often constructed on or adjacent to slopes owing to the nonavailability of level ground, especially in hilly dominant topography encountered in highway vertical curves, embankments, erosion watersheds, etc. e study of the bearing capacity of loaded slopes is vital because they are more prone to fail than other types of earth structures [1][2][3][4][5]. Shallow foundations on slopes are mainly used for small-to medium-rise buildings. In frictional soils, the bearing capacity is mainly governed by foundation failure, while in cohesive soils, the bearing capacity of the foundation is controlled by the stability of the soil structure [6][7][8][9]. Recently, methods proposed by the researchers available to find the bearing capacity of shallow foundations on or near slopes include limit equilibrium analysis [10,11], slip line analysis [12], variational calculus [13], the method of rigorous characteristics [14], improved movement optimization [15], finite element analysis [16,17], and multiblock analysis [9]. Determining the bearing capacity of a shallow substructure is a very important component of geotechnical engineering study and practice. Prandtl [18] is generally credited with some pioneer work in bearing capacity theory, having tried to establish the punching failure mechanism for thick metals based on the theory of plasticity.
is theory gave rise to Terzaghi's [19] work, where the effects of soil internal angle of friction, soil cohesion, and surcharge (overburden pressure) were taken into consideration, giving rise to the bearing capacity factors (Nc, Nc, and Nq). Meyerhof [20] improved on Terzaghi's superimposed theory to account for soil strength, footing size and shape, and slope. Vesic later considered the effects of shallow foundation shape on the ultimate bearing capacity [21]. Nevertheless, despite studies into the effects of foundation shape and depth [19], Meyerhof (1957 and1974) [20][21][22][23], there has been few research works on the bearing capacity of footings on and/or adjacent to slopes made of c′ − v′ soils. Meyerhof studied general failure mechanisms for bearing capacity on purely cohesionless or cohesive soils adjacent to slopes using an assumed failure pattern based on the empirical observation from model footing tests in the laboratory. However, actual quantifiable results were not available for model verifications. Graham et al. [24] derived an analytical model for bearing capacity and stress distribution for footings placed upon slopes for cohesionless soils that was validated by experimental results. However, his work did not expand to soils with both cohesion and internal friction. Raj and Bharathi reported that Bowles provided a set of revised bearing capacity factors for c′ − v′ soils based on an the assumed failure mechanism resembling that from Terzaghi, but limited to set of strength values. Griffiths [25] used the finite element analysis (FEA) to determine the bearing capacity of c′ − v′ soils on slopes attaining considerable results. However, his Nc factor recorded pitfall due to a convergence issue. Applying upper bound limits analysis, Kusakabe et al. [26] presented a series of design charts with dimensionless factors showing a reduction in bearing capacity compared to the conventional bearing capacity approach for footings placed on the level of the ground for soils with and without cohesion. A better analysis of the complex, complicated failure mechanism associated with the ultimate bearing capacity of strip footings on slopes requires use of tools that can account for the complex failure kinematics related to the soil-structure interaction and nonintuitive behavior of the soil collapse mechanism [27,28]; in this case, artificial intelligence (AI) looks a better tool.
Recent discoveries in the field of artificial intelligence (AI) applications have risen in the development of accurate and dependable models for solving engineering problems [29][30][31][32]. Advances in the field of AI made it possible to produce models to adapt to difficulties associated with modelling soil and foundation behavior [33][34][35][36]. AI has been considered in the field of civil engineering for more than one and a half decade [37]. ese models involve a simple black-box model to complex distributed physicsbased models. Although there are numerous AI modelling algorithms, namely, genetic algorithm (GA), ant colony (AC), differential evolution (DE), particle swarm (PS), artificial neural network (ANN), genetic programming (GP) [35], and gene expression programming [32], artificial neural network (ANN) [36], genetic programming (GP), and gene expression programming have been widely used [29,35].

Preamble, Data Collection, and Statistical Study.
Prantdl [18] proposed a mathematical formula for the ultimate bearing capacity of strip footing in a semi-infinity soil model with a horizontal ground surface based on an experimentally observed failure shape below footing as shown in equation (1). For strip footing near or within the slope, Prantdl suggested modified parameters N′c � Nc (L1/L0) and N′q � Nq (A1/A0), where L0, L1, A0, and A1 are illustrated in Figure 1 and could be calculated from the geometry. Nc is still the same because the triangular wedges below the footings are the same in all cases. e aim of this research is to develop mathematical formulas for the modified N′c and N′q based on slope geometry and soil proprieties using different artificial intelligence (AI) techniques.
where qult: ultimate bearing capacity of strip footing, C: soil cohesion, q: overburden pressure at foundation depth, c: bulk density of soil below the footing, B: strip footing width, Nc, Nq, and Nc are parameters based on the soil internal friction angle ϕ, Nq: e π tan ϕ. tan 2 (45 + ϕ/2), Nc: (Nq − 1) cot ϕ (Nc � 5 for ϕ � 0), and Nc: (Nq − 1) tan ϕ. From the foregoing, the methodology was to generate a database of 300 records; each record contains slope angle (β), angle of internal friction of soil (ϕ), ratio between the distance to the slope edge and footing width (b/B), and the ratio between foundation depth and footing width (Df/B) besides the calculated values for both N′c and N′q. Table 1 includes the complete dataset, while Tables 2 and 3 summarize their statistical characteristics and the Pearson correlation matrix. Finally, Figure 2 shows the histograms for both inputs and outputs.

Research Program.
ree different artificial intelligent (AI) techniques were used to predict the values of both N′c and N′q. ese techniques are genetic programming (GP), artificial neural network (ANN), and polynomial linear regression optimized using genetic algorithm which is known as evolutionary polynomial regression (EPR). All the three developed models were based on the generated database . Each model of the three developed models was based on different approaches (evolutionary approach for GP, mimicking biological neurons for ANN, and an optimized mathematical regression technique for EPR). However, for all developed models, prediction accuracy was evaluated in terms of sum of squared errors (SSE). e following section discusses the results of each model. e accuracies of the developed models were evaluated by comparing the SSE between the predicted and calculated N′c and N′q values.

Model (1)-Using the GP Technique.
e developed GP model started with the one level of complexity and settled at five levels of complexity. e population size, survivor size, and number of generations were 10000, 30000, 50000, 75000, and 100000, respectively. Equations (2) and (3)   Journal of Engineering

Model (2)-Using the ANN Technique.
A back-propagation ANN with one hidden layer and Hyper Tan activation function was used to predict the values of N'c and N'q. e used network layout is illustrated in Figure 5, and its connation weights are listed in Table 4. Since the used ANN has a nonlinear activation function, the equivalent equation is very complicated to be presented mathematically. e average errors in % of this network are 6.9% and 13.0% for N′c and N′q, respectively, while the corresponding R 2 values are 0.995 and 0.991. e relation between the calculated and predicted values is shown in Figures 3(b) and 4(b). e summation of connection weights of each input parameter is a good indication for its importance; accordingly, it was found that ϕ was the most important parameter with 78.6% of the total weights. β came in the second place with 10.1% and then b/B and D f /B with 7.7% and 3.6%, respectively.

Model (3)-Using the EPR Technique.
Finally, the developed EPR models were limited to the pentagonal level, for 4 inputs; there are 226 possible terms (56 + 35+20 + 10+4 + 1 � 126) as follows: e GA technique was applied on these 126 terms to select the most effective 6 terms to predict the values of N′c and N′q. e output is illustrated in equations (3) (5) and (6) show that ϕ existed in each term and it got the highest power, which indicated that it is the most effective parameter, and other parameters almost had the same importance. Meanwhile, the results of all the developed models are summarized in Tables 5 and 6.

Conclusions
is research presents three models using three AI techniques, namely, GP, ANN, and EPR, to predict the modified bearing capacity parameters N′c and N′q using slope angle (β), angle of internal friction of soil (ϕ), ratio between the distance to the slope edge and footing width (b/B), and the ratio between foundation depth and footing width (D f /B). e results of comparing the accuracies of the developed models could be concluded in the following points: (i) Although five levels of complexity (128 genes in the chromosome) were used in the GP model, the achieved accuracy was still low (79.3% and 68.4% for N′c and N′q, respectively) (ii) e prediction accuracies of ANN and EPR models are close, 93.1% and 92.8% for N′c and 87.0% and 82.7% for N′q, which gives an advantage to the EPR model because its output is a simple equation and could be applied either manually or implemented in software unlike the complicated output of the ANN which cannot be applied manually (iii) e importance analysis of both ANN and EPR models indicated that (N′c and N′q) values were mainly governed by ϕ, while other parameters are minors (iv) All developed models showed that N′c and N′q values increase with increasing ϕ, b/B, and D f /B values and with decreasing β value (v) GA technique successfully reduced the 126 terms of the conventional PLR quadratic formula to only 6 terms without a significant impact on its accuracy (vi) Like any other regression technique, the generated formulas are valid within the considered range of parameter values; beyond this range, the prediction accuracy should be verified Abbreviations AI: Artificial intelligence C: Cohesion q: Overburden pressure at foundation depth c: Bulk density of soil below footing B: Strip footing width

L0:
Length of the failure wedge of normal footing L1: Length of the failure wedge of slopy footing A0: Area of a rectangular mass on normal footing A1: Area of a triangular mass on slopy footing b: Distance of the slope from the footing edge β: Slope angle ϕ: Friction Nc: Bearing capacity coefficient of cohesion Nq: Bearing capacity coefficient of overburden pressure Nc: Bearing capacity coefficient of bulk density N′c: Predicted bearing capacity coefficient of cohesion on the slope N′q: Predicted bearing capacity coefficient of overburden pressure on the slope ANN: Artificial neural network GP: Genetic programming' GA: Genetic algorithm EPR: Evolutionary polynomial regression PLR: Polynomial linear regression SSE: Sum of squared errors R 2 : Coefficient of determination.

Data Availability
e underlying data supporting the results of this research are reported in the manuscript.