Load-settlement response of a footing over buried conduit in a sloping terrain: a numerical experiment-based artificial intelligent approach

Settlement estimation of a footing located over a buried conduit in a sloping terrain is a challenging task for practicing civil/geotechnical engineers. In the recent past, the advent of machine learning technology has made many traditional approaches antiquated. This paper investigates the viability, development, implementation, and comprehensive comparison of five artificial intelligence-based machine learning models, namely multilayer perceptron, Gaussian processes regression, lazy K-Star, decision table, and random forest (RF) to estimate the settlement of footing located over a buried conduit within a soil slope. The pertaining dataset of 3600 observations was obtained by conducting large-scale numerical simulations via the finite element modeling framework. After executing the feature selection technique that is correlation-based subset selection, the applied load, total unit weight of soil, constrained modulus of soil, slope angle ratio, hoop stiffness of conduit, bending stiffness of conduit, burial depth of conduit, and crest distance of footing were utilized as the influence parameters for estimating and forecasting the settlement. The predictive strength and accuracy of all models mentioned supra were evaluated using several well-established statistical indices such as Pearson’s correlation coefficient (r), root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), scatter index (SI), and relative percentage difference (RPD). The results showed that among all the models employed in this study, the multilayer perceptron model has shown better results with r, RMSE, NSE, SI, and RPD values of (0.977, 0.298, 0.937, 0.31, and 4.31) and (0.974, 0.323, 0.928, 0.44, and 3.75) for training and testing dataset, respectively. The sensitivity analysis revealed that all the selected parameters play an important role in determining the output value. However, the applied load, constrained modulus, unit weight, slope angle ratio, and hoop stiffness have the highest strength with the relative importance of 18.4%, 16.3%, and 15.3%, 13.8%, 11.4%, respectively. Finally, the model was translated into a functional relationship for easy implementation and can prove useful for practitioners and researchers in predicting the settlement of a footing located over a buried conduit in a sloping terrain.


Introduction
The tunneling and underground infrastructure is a salient feature of modern urbanization. The economic and safety benefits of the buried conduits have made them the most frequently used mode of utility conveyance. The scarcity of land to ever-increasing population growth has resulted in the construction activity over the buried infrastructure. The influence of the imposed loading on a buried conduit is always incorporated in its design and installation (Moser and Folkman 2001). The studies on the effect of the applied surface pressure on the soil-conduit interaction and the resulting stress distribution and structural response of the conduit can be found in the current literature (Dhar et al. 2004;Talesnick et al. 2011;Bryden et al. 2015;Robert et al. 2016;Wang et al. 2017;Al-Naddaf et al. 2019;. However, the research on the presence of the buried conduit on the settlement and bearing capacity of a surface footing is very limited. Srivastava et al. (2013) investigated the load-settlement response of a circular footing placed over a PVC conduit buried under the level ground. Using laboratory model tests, the loadsettlement behavior and bearing capacity of the footing were analyzed in loose-medium (relative density = 50%) Extended author information available on the last page of the article and very dense sands (relative density = 88%). The experimental results were also compared with the results obtained from finite element analysis of the same model. The results showed that in the case of the loose-medium dense sand, the induction of stiffer conduit material improved the load-settlement response of the footing. As a result, its bearing capacity increased by about 25%, whereas, for the very dense sand, the presence of the flexible conduit reduced the bearing capacity of the overlying footing by approximately 8%. Therefore, it can be concluded that under the static load conditions, the relative density of the sand surrounding the buried conduit and the resulting relative stiffness with the conduit material governs the settlement and bearing capacity of the surface footing. Similarly, Laman (2015, 2019) conducted laboratory model tests to analyze the effect of a buried PVC conduit on the load-settlement response and bearing capacity of an overlying strip footing. The study was conducted by varying the burial depth and the horizontal distance of the conduit from the footing. The settlement of the surface footing was measured by employing two deflection transduces instrumented on both sides of the surface footing. The results showed that the load-settlement behavior and bearing capacity of the footing improved significantly as the horizontal distance between the footing and the buried conduit was increased. Also, it was noted that as the buried conduit was moved away from the stress zone under the footing, the bearing capacity of the footing increased. At a burial depth of more than 4 times the conduit diameter, the buried conduit seized to impact the load-settlement behavior and bearing capacity of the surface footing. While the aforementioned studies investigated the effect of buried conduits on the load-settlement response and bearing capacity of footings located over the horizontal ground, only one study can be found in the literature that has analyzed the footing settlement in a sloping terrain. Khan and Shukla (2020) conducted laboratory model tests to investigate the settlement and bearing capacity of a strip surface footing located over a conduit buried within the soil slope. Using two linear variable displacement transducers (LVDTs) installed on both sides of the footing, the effects of un-plasticized polyvinyl chloride (PVC-U) conduits of diameters 80 mm and 160 mm were studied in detail. The shear failure mechanisms of the footing were analytically computed and illustrated to understand the resulting soil-conduit interaction. The study concluded that when the shear failure planes of the footing intersected with the buried conduit, its bearing capacity was reduced by about 40%. However, an increase in the burial depth of the conduit and the crest distance of the footing enhanced the distance between the buried conduit and failure planes of the footing, resulting in a decrease in the effect of buried conduit on the settlement and bearing capacity of the surface footing. Further, the sensitivity analysis categorized the burial depth of the conduit and the crest distance of the surface footing from the edge of the soil slope as the most influential parameters affecting the load-carrying behavior of the surface footing located over a conduit buried within a soil slope.
Summarizing, the limited number of related studies, as discussed above, has concluded that the load-settlement response and bearing capacity of footings is affected by the relative stiffness with the conduit material and the surrounding soil, burial depth of the conduit, and the crest distance of the conduit from the slope surface. However, the studies have only analyzed limited values of these influential parameters due to the experimental restraints. Furthermore, these experimental studies have been conducted on small-scale 1 g laboratory models, which hinders the veracity of such studies due to the scale effect. While the small-scale model tests may explain the relevant mechanisms/ trends, the observed measurements may not reflect the actual field values Cerato and Lutenegger 2003). Additionally, the use of only one type of conduit material significantly limits the generalized use of related studies. To the authors' best knowledge, no study exists in the current literature that can be used for the direct estimation of the settlement of a strip footing located over a buried conduit with a soil slope.
Finite element modeling (FEM) can be used to solve complex geotechnical problems and achieve more accurate results (Khan and Shukla 2021b). However, the use of expensive software for FEM analysis significantly limits their application (Kim et al. 2012). In recent times, the use of machine learning techniques has been widely used in mapping the nonlinear relationships between the input and output variables (e.g., Ahmadi et al. 2019;Yekani Motlagh et al. 2019;Aamir et al. 2020;Dorosti et al. 2020;Ghorbani et al. 2021;Kaloop et al. 2021). The novel metaheuristic algorithms are also developed for optimization purposes in big data analysis (Abualigah and Alkhrabsheh 2021;Abualigah et al. 2021a). Similarly, for geotechnical problems, the soft computing approaches are now commonly used for prediction purposes (e.g., Nguyen et al. 2019;Xiao and Zhao 2019;Bardhan et al. 2021;Kardani et al. 2021a, b;Khan et al. 2021;Raja and Shukla 2021a, b;Raja et al. 2021). The machine learning (ML) models that are based on large quantities of FEM data have also been developed to solve complex problems like soil-conduit interaction and settlement of foundations. Kim et al. (2012) employed FEM-based artificial neural network (ANN) to predict deflections of buried corrugated conduits. The data collected from three-dimensional finite element modeling were used to develop a backpropagation (BP) neural network that examined the factors affecting the structural response of different corrugated conduits buried at various depths under the level ground. Shokouhi et al. (2013) used a FEM-ANN approach to develop an ANN model that could be used to predict the bending strains developed in conduits buried within a fault zone. Kardani et al. (2020) used the FEM-based data to successfully predict the factor of safety of a soil slope using the hybrid stacking ensemble machine learning modeling technique. The aspect of footing settlement has also been studied by using the FEM-ML approach. Moayedi and Hayati (2018) used large FEM data to develop a number of soft computing models in order to predict the settlement of a strip surface footing located near a sandy soil slope. Similarly, Moayedi et al. (2020a) developed optimized neural networks such as differential evolution algorithm (DEA), adaptive neuro-fuzzy inference system (ANFIS) to predict the ultimate bearing capacity of a shallow footing on two-layered soil condition, utilizing FEM data. In this paper, an attempt has been made to predict the settlement of a surface footing located over a conduit buried within a soil slope, using various machine learning/intelligent modeling techniques, namely multilayer perceptron (MLP), Gaussian processes regression (GPR), Lazy K-Star (LKS), decision table (DT), and random forest (RF). Using the finite element modeling, largescale data were generated for the settlement of the footing located over the conduits of varying stiffness. The main objectives of this paper are as follows: (1) Development and assessment of five machine learning models such as to MLP, GPR, LKS, DT, and RF for direct estimation of the settlement of a strip footing, located over a buried conduit with a soil slope.
(2) Comprehensive analysis and comparison of these models for the same problem. (3) Feature selection to choose the most important parameters affecting the footing settlement. (4) Assess the robustness of the developed models and conduct the sensitivity analysis. 5) Develop and present a trackable functional ANNbased formula for direct estimation of the footing settlement locate over a buried conduit.

Significance of the research
This study is useful in ensuring the stability of surface footings that are frequently located over tunnels and underground infrastructure in the current urban environment. Using extensive finite element modeling, it incorporates the effect of a large number of input parameters on the load-settlement response of a large-scale surface footing located over different types of buried conduits. The inclusion of numerous input parameters employed to define soil and the buried conduits and their complicated relationships with the output parameter results in highly complex geotechnical models. Further enhancing this complexity are the intricate correlations between the input parameters, where a change in one input parameter causes a change in another or more than one input parameters. Considering these convoluted relationships and the resulting rigorous finite element calculations, this study utilizes advanced machine learning/intelligent modeling techniques to provide accurate and straightforward solutions to the complex soil-conduit interactions. The developed MLP-based formula can be used by the practicing engineers to directly estimate footing settlement when loaded over a conduit buried within a sloping terrain. The steps involved in the model developments can be summarized as follows: (1) Data generation through finite element modeling (FEM) (2) Data preprocessing, feature validity, and data division into training and testing data (3) Development and implementation of AI-based models (4) Statistical analysis of the results and selection of best model (5) Model robustness and sensitivity analysis 2 Material and methods

Finite element modeling
The commercial PLAXIS 2D software was used for data collection by conducting the finite element analysis of a large-scale model, as presented in Fig. 1. Using the Mohr-Coulomb model, the soil was modeled as per the field properties of the most common sandy soils, presented in Table 1 (Ghazavi and Eghbali 2008;Moayedi and Hayati 2018). The soils, numbered as one to five were differentiated in terms of their strength and stiffness parameters, namely total unit weight c, elastic modulus E s , friction angle u, Poisson's ratio m s , and dilation angle w. As suggested by Brinkgreve et al. (2018), the value of cohesion c was set as 0.3 to avoid any complications during software calculations. Also, the aspect of increasing soil stiffness with an increase in depth was simulated by employing E inc ¼ 500 kN/m 3 . The stiffness parameters of the buried conduit were selected as per the properties defined by Elshimi and Moore (2013). Table 2 details the parameters of different types of conduits, presented in terms of their normal stiffness EA, flexural rigidity EI, and Poisson's ratio m c . The strength interaction parameter R inter was selected as 0.8 to simulate the realistic frictional resistance between the buried conduit and the surrounding soil (Wadi et al. 2015). The pressure q was applied on a surface footing of width B that was located centrally above the buried conduit of diameter B c . The footing was modeled as a plate, having the stiffness properties as: EA ¼ 5 9 10 6 kN/m and EI ¼ 8.5 9 10 3 kNm 2 /m (Brinkgreve et al. 2018). The standard model fixities and the default medium mesh size were used for conducting finite element simulations. In order to obtain the settlement of the footing located over a buried conduit with a soil slope, the crest distance of the footing e=B, the burial depth of the conduit z=B, and the slope angle i were varied.

Database collection, preprocessing, and feature validity
A database of 3600 full-scale numerical simulations was generated by conducting extensive finite element modeling. As suggested by McGrath (1998), the problem related to the buried conduits can be described in terms of the constrained modulus of soil M s , defined as, where E s and m s represent the elastic modulus and the Poisson's ratio of the soil. Similarly, the stiffness parameters of the conduit, namely normal stiffness EA and flexural rigidity EI, are usually normalized to incorporate the effect of conduit diameter B c and wall thickness t. The resulting hoop stiffness PS H (Mcgrath 1999) and bending stiffness PS B (ASTM D2412, 2002) of the conduit are defined as, where R is the radius to centroid of the conduit wall. The aspect of slope stability and the angle of repose of the granular soil are a function of its friction angle (Duncan and Wright 2005;Atkinson 2007). The graphical presentation of the complete dataset is illustrated in the form of box and whisker plots in Fig. 2. Moreover, the statistical properties of the same are tabulated in Table 3. The dataset has been normalized between -1 and 1 before feeding it to ML algorithms.
where x, x min , and x max present the observed, minimum and maximum values of the dataset, respectively. In machine learning, dealing with high-dimensional data is a challenging task for scientists and researchers. Feature reduction is an important step that effectively omits the redundant data and chooses the most optimum combination of input parameters (Jie et al. 2017;Gao et al. 2019). For this study, correlation-based feature selection, abbreviated as the CFS method, was implemented in a Waikato environment for knowledge analysis (WEKA) using the multivariate filter. Initially proposed by Hall (1999), CFS combines the correlation measure for appropriate feature subset selection and heuristic strategies for the mode of search. Therefore, it evaluates the importance/correlation of individual variables with the output and the degree of redundancy between them. The results of the feature selection depict that among the most relatively important parameters as summarized in Table 3, the applied load (q), constrained modulus of soil (M s ), unit weight of soil (c),slope angle ratio (i=u), hoop stiffness (PS H ), bending stiffness (PS B ), burial depth ratio of conduit (z/B), and crest distance of footing (e/B) have achieved the highest importance. Mathematically, the output that is settlement of footing located over a buried conduit in soil (s/B %) can be expressed as follows: Therefore, these input parameters are utilized for training the machine learning models described in the next section.

Theory of methods
The theory of the statistical concepts and the data-driven machine learning methods employed in this study to estimate the settlement of the footing located over a conduit buried within a soil slope are provided in this section. Moreover, 3060 samples were randomly earmarked in this study for training the MLP, GPR, LKS, DT, and RF models. Thereafter, the competency of each model was evaluated and validated against 540 samples. Additionally, the research scheme employed in this study is presented in Fig. 3.  Figure 4 shows a feed-forward MLP network consisting of an input layer, a single hidden layer, and an output layer. Each layer consists of a varying number of neurons. The number of independent input parameters and the output target defines the number of neurons in the input and output layers, respectively, whereas the number of neurons in the hidden layer depends upon the type and size of the problem (Ramezanian et al. 2019). The increase in the number of hidden neurons may enhance the prediction ability of the network but can also make the model computationally inconvenient and complex (Raja and Shukla 2020). As a thumb rule, the maximum number of hidden neurons should be limited to 2m þ 1, where m presents the number of input parameters (Shahin 2010). In an MLP algorithm, a number of neurons are connected by associated weights. At each neuron, the data from the input layer x i is multiplied by the associated weight w ik . Thereafter, the bias vector k k is added to the summation of the weighted inputs to obtain V k . Finally, the output of the processing neuron y k is obtained by passing V k through the sigmoidal activation function gð:Þ (see Fig. 4). More detailed information about the MLP neural network can be found in the existing literature (Gurney 1997).

Gaussian processes regression (GPR)
Gaussian processes regression (GPR) uses a probabilistic approach and predicts through kernel functions that evaluate on the basis of the similarity between two data points.
The GPR technique integrates a number of machine learning tasks, such as model training, parameter estimation, and uncertainty evaluation. This helps in reducing the subjectivity of the GPR results and makes them more interpretable. The GPR is based on a Gaussian process (GP) that works on the assumption of Gaussian priors for changed function values (Rasmussen 2006). A GP can be statistically presented as, where lðxÞ presents the mean and kðx; x 0 Þ presents the covariance function of g. Any finite number of random variables in a GP have a joint multivariate Gaussian distribution (Suthar 2020). Assuming g ¼ ½ĝðx i ; wÞ m i¼1 presents the model outputs in correspondence to the input dataset X, or simply, if g ¼ uw, then the prior distribution of g is Gaussian pðg X; j hÞ $ Nð0; KÞ ð 8Þ where u is the design matrix. In this study, various kernel functions, namely radial bias function (RBF), Pearson VII Universal kernel function (PUKF), and polynomial kernel function, are employed, and the best results are obtained via PUKF function.

Lazy K-star (LKS)
Lazy K-star (LKS) uses an instant base learning (IBL) classification system to generalize the training dataset. During the learning process, the learning algorithm spends most of its computation time for consultation, and learners do not operate until the system receives the query call (Webb 2011). Unlike other machine learning techniques, LKS algorithm does not predict from the instances in the training dataset but rather employs the nearest neighbor approach to provide a response from the data memory (Altman 1992). The LKS classifies a dataset by drawing a comparison with a pre-classified sample. By employing a distance function, the IBL adds up all the possible transitions of two instances and categorizes them into a simple class (Cleary and Trigg 1995). Thereafter, the generated classification function is used to provide new solutions. For example, new test data samples x are distributed to the most suitable class among the k closest information focuses y i . The corresponding LKS formulation can be given as follows (Cleary and Trigg 1995;Gao et al. 2019): where P Ã is the probability function that presents the all possible transitions from instances x to y.

Decision table (DT)
Decision table (DT) can be used to organize logic in a manner that helps in easy analysis (Nanda et al. 2017). A DT consists of different sections, namely condition and action stubs, and condition and action entries. While the ''condition stub'' presents the possible conditions or problems, the ''action stud'' illustrates potential actions or solutions. The condition and action entries are located across the corresponding stubs, in terms of rules and classes tabulated in columns and rows, respectively (Cragun and Steudel 1987). When provided with a new instance, a DT algorithm tries to find the match in the table (Kohavi 1995). It assists in testing a set of rules for conditions of completeness, redundancy, and ambiguity. The condition of completeness occurs when the rules in the table address all the possible combinations of logic. Redundancy is said to exist when more than one rule having the same actions is satisfied by the same logical Load-settlement response of a footing over buried conduit in a sloping terrain: a numerical… 6845 conditions. Ambiguity exists when two or more different rules with different actions are satisfied by the same logical conditions (Cragun and Steudel, 1987). In comparison with the hierarchical structure of the decision tree technique, the simple straightforward architecture of DT is considered to be more stable for problem solutions (Gao et al. 2019).

Random forest (RF)
Random forest (RF) uses an ensemble-learning approach that employs numerous classification trees for solving regression and classification problems (Ho 1995; Gehrke 2011). The RF creates a grove of trees whose predictive relationship alters randomly. The average output of each tree is then provided as the output. In order to generate the forest, the user has to define some parameters, such as the variable splitting the nodes and the number of trees. The accuracy of the model is determined in terms of the forest population i.e., the number of trees. The generalization error (GE) is estimated unbiasedly during the computation of the RF model. The RF evaluates the increase in the GE error to estimate the significance of the predictive variables. A variable is said to have increased significance if the value of GE increases. Also, by employing the bootstrap aggregating technique, the RF model reduces the risk of overfitting and provides a more stable solution (Breiman 2001).

Model performance and assessment
After the development and implementation phase, the next most important step is the model assessment. For datadriven modeling, the accuracy is measured in terms of the following: (i) Statistical criteria, that is, ''goodness of fit''; and (ii) Robustness and sensitivity. The former deals with model performance by evaluating it fit to the calibration data using several statistical criteria. In contrast, the latter is used to access its accuracy, reliability, and rationality according to the underlying physical behavior of the investigated system. A model can only be considered suitable if it makes accurate and realistic predictions over a wide range of data (Shahin et al. 2009). Therefore, for this study, the best ML model was selected based on these criteria.

Model performance based on statistical indices
For ''goodness of fit,'' five statistical indices, namely Pearson's correlation coefficient (r), root mean square error (RMSE), Nash-Sutcliffe efficiency (NSE), scatter index (SI), and relative percentage difference (RPD), were used to access the accuracy of the developed models. Moreover, based on these criteria, a ranking system was developed by assigning the scores to the models in training and testing dataset. It may be noted that this ranking system was successfully applied in many previous (Gao et al. 2019;Moayedi et al. 2020a;Zhang et al. 2020). The mathematical forms of all the indices, namely r, RMSE, NSC, SI, and RPD, are given in Eqs. (10-14) where s=B iobs , s=B ipre , s=B obs , s=B pre , and n represent the ith observed value of settlement, ith predicted value of settlement, mean value of observed settlement, mean value of predicted settlement, and number of data samples, respectively. It is to be noted that the model is considered to be accurate if it has high r, NSE, and RPD values and low RMSE and SI values.  (0.964, 0.349, 0.916, 0.52, and 3.46), respectively, for MLP, GPR, LKS, DT, and RF (Table 5). After reviewing both the training and the testing performance, MLP technique can be introduced as the most accurate model in determining the settlement of footing located over a conduit buried within a soil slope. Moreover, the performance of MLP is followed by RF and GPR models and thus can be considered as the second and third best models in the hierarchy. Also, the LKS and DT have shown rather poor predictive performance in comparison with their counterparts.
The combined performance of all the models in training and testing datasets is computed in Table 6. In this regard, a total rank is obtained by summing the partial scores given to the model based on the statistical performance indicators, that are r, RMSE, NSE, SI, and RPD values (Tables 4  and 5). From the results, the supremacy of the MLP model can be established with the highest total ranking score of 48. The second best performance is obtained by RF with a total score of 42. The total ranking scores for GPR, LKS, and DT were 28, 12, and 20, respectively. Furthermore, Fig. 5a-e depicts the regression correlation coefficient between the observed and predicted values for all the present models in the testing dataset. It can be observed from the regression chart that the MLP model has achieved the highest R 2 , that is, 0.948, in comparison with 0.931, 0.820, 0.770, and 0.756, respectively, for RF, GPR, LKS, and DT. This also proves that the developed MLP model has outperformed all other ML models applied in the context of predicting the settlement of footing located over a conduit buried within a soil slope.
The predictive performance of all the models was also accessed via Taylor's diagram in Fig. 6. Taylor's diagram is a useful graphical tool to illustrate the accuracy of the developed data-driven models on a single platform (Taylor 2001). The strength between the predicted and simulated field is evaluated on the basis of the combine effect of three statistical parameters that are centered RMSE, correlation coefficient and standard deviation (SD). In the given figure, the solid black lines depict the correlation coefficient, solid radial lines represent the standard deviation, and dotted radial lines show the centered RMSE values in the simulated field. The reference model is shown by a black dot with the correlation coefficient of unity, measured SD of 1.21, and zero centered RMSE. It can be seen from the figure that the best performance is obtained by MLP model with the correlation coefficient of 0.974, SD of 1.33, and RMSE of 0.3168. The RF model has also shown a good correlation strength with coefficient of 0.965, RMSE of 0.347, but the spatial variability is low with an SD of 1.02 in reference to observed value. The other models such as GPR, LKS and DT have correlation coefficient (0.905, 0.876, and 0.870), centered RMSE (0.517, 0.673, and 0.613), and SD (1.04, 0.73, and 0.92), respectively. This depicts that these models are associated with high bias and have poor prediction strength compared to the MLP model. Therefore, to this point, it is admissible that the developed MLP model predicts the settlement of footing located over a conduit buried within a soil slope in a reliable and intelligent way. Additionally, the time consumed by each approach is shown in Fig. 7. It can be observed that apparently, the MLP and GPR approaches had consumed less time in comparison with other models. Total ranking score 24 14 6 10 21 Load-settlement response of a footing over buried conduit in a sloping terrain: a numerical…

Model robustness and sensitivity
In this section, a sensitivity analysis was carried out to investigate the reliability and robustness of the developed MLP model. For this, incremental stepwise sensitivity analysis, also known as one-at-time analysis, was conducted to examine the robustness of the model. In this method, each variable is increased in a stepwise manner while other variables remain constant at their mean value. However, this is practically not possible if the variables have both independent and correlated effect, that is, the change in one variable causes an inherent change in another variable such as unit weight and constrained modulus of soil, and hoop and bending stiffness of conduit, for this study. Therefore, the combined effect is calculated in the sensitivity analysis for these variables, as illustrated in Fig. 8. The results show that an increase in the footing settlement increases significantly with an increase in the applied loading. This is a very common effect that relates to the movement of the underlying soil particles that slide along the shear failure planes due to the downward motion of the loaded footing, hence providing more settlement space for the surface footing (Terzaghi 1943). The constrained modulus and unit weight of the soil are interrelated variables that correspond to the density and the resulting stiffness of the soil. The increased soil stiffness reflects an increase in the shear strength of soil that provides resistance to the footing settlement (Berardi and Lancellotta 1991;Mayne and Poulos 1999). The slope angle ratio symbolizes the aspect of slope stability of an unconfined granular material in terms of the state of failure of a soil at which the angle of repose equals its friction angle (Al-Hashemi and Al-Amoudi 2018; Miura et al. 1997). An increase in the slope angle ratio increases the slope instability and hence reduces the support available to the footing on the slope side, explained in terms of the asymmetric footing failure mechanism (Hovan 1985;Graham et al. 1988). Due to the increase in asymmetric footing failure with an increase in slope angle ratio, the soil along the unstable slope yields under load application, resulting in increased footing settlement (Keskin and Laman 2013;Dey et al. 2021). The hoop and bending stiffness of the conduit are the design parameters that are employed to classify a conduit as either rigid or flexible (Mcgrath 1999). When buried under a loaded footing, the relative stiffness of the conduit to the adjacent soil determines the footing settlement. An increase in relative stiffness increases the stiffness of the soil located between the buried conduit and the overlying footing, thereby reducing the footing settlement (Srivastava et al. 2013). The effect of the burial depth of the conduit on the footing settlement is related to the aspect of the intersection of the shearing failure plane of the loaded footing with the buried conduit. As the burial depth increases and the conduit is located below the shear failure places of the footing, it serves as a support and reduces the settlement of the overlying footing (Khan and Shukla 2020). The increase in the crest distance relates to the support available to the surface footing on the slope side, as discussed above. An increase in the crest distance reduces the asymmetric nature of the failure mechanism, allowing more support to the slope side of the footing and causing a decrease in the footing settlement (Dey et al. 2021). The trends illustrated in Fig. 8 comprehensively prove that the developed MLP model network correctly predicts the underlying physical behavior of the investigated system according to the known knowledge pertaining to geotechnical engineering and thus can be considered reliable and robust.
In order to find the importance of each variable affecting the settlement of footing located over a conduit buried within a soil slope, a sensitivity analysis was also conducted using the Garson's algorithm (Garson 1991). In the case of a single hidden layered network, this technique involves the deconstruction of the model weight connections. The algorithm is explained in Appendix section for the MLP network with eight inputs, six hidden layer nodes, and one output node. From the results illustrated in Fig. 9, it can be observed that the most important parameter for estimating the settlement is applied load with the relative importance of 18.4%, followed by the unit weight of soil and constrained modulus of soil with relative importance of 16.3% and 15.3%, respectively. The relative importance of other parameters such as slope angle ratio, hoop stiffness, bending stiffness, burial depth, and footing crest distance is 13.8%, 11.4%, 9.9%, 7.7%, and 6.9%, respectively.

MLP model formulation
In this section, the developed optimal MLP model was translated into a trackable equation for hand or spreadsheet calculations. The mathematical form of MLP is given as follows (Ghorbani et al. 2020): where g ho is the applied transfer between hidden-output layer, k o is the bias at output layer node, w ko is the synaptic weight between node k of hidden layer and single output node, g ih is the applied transfer function between inputhidden layer, k hk is the bias value for node k of hidden layer (k = 1,h),w ik is the synaptic weight between input i and node k of hidden layer, and x i is the ith input node (variable). The weights and biases of the network are summarized in Table 7.
In order to predict the settlement of footing located over a buried conduit with eight inputs (q; M s ; c; i=u; PS H ; PS B ; z=B; e=B), the optimal MLP model can be formulated as follows: where ðs=BÞ np , ðs=BÞ max , and ðs=BÞ min are the normalized settlement value, maximum values of the settlement, and minimum value of the settlement, respectively. The normalized settlement value can be estimated as follows: where the subscript n denotes the normalized values of the corresponding input parameters. The mathematical form of sigmoid activation function is given in Eq. (19).
For easy comprehension, the design numerical example is also presented below.

Numerical example
The 1-m wide footing is located over a conduit buried within a soil slope at 1.5 m depth below the base of footing. The crest distance of the footing is 1.75 m. Other parameters, including the constrained modulus of soil, unit weight of soil, slope angle ratio, hoop stiffness of pipe, and bending stiffness of pipe are 35,000 kPa, 19.9 kN/m 3 , 4071.2 kPa, and 8.55 kPa, respectively. Estimate the settlement of the footing under the application of load (q) of 50 kPa, 100 kPa, and 150 kPa.

Solution
Input parameters xðiÞ ¼ f q M s c i=u PS H PS B z=B e=Bg ¼ f 50 35000 19:9 0:454071:28:551:51:75g Step 1 Normalize the values using Eq. (4). The maximum and minimum values of all the parameters are mentioned in Table 3.

Conclusions
Settlement estimation of the footing located over a buried conduit in a sloping terrain is a challenging task for civil engineers. A novel approach is presented in this study to predict this settlement. It involves generating the pertaining database using extensive large-scale numerical simulations. Thereafter, five machine learning models (MLP, GPR, LKS, DT, and RF) were developed and implemented to evaluate the feasibility of the investigated system. The following general conclusions can be drawn from the above discussion. (total score = 48). The next best performance is achieved by RF model (total score = 42) followed by GPR (total score = 28). Therefore, RF and GPR can be introduced as second and third best models in estimating the settlement of footings located over buried pipes in sloping terrain. 3. DT and LKS showed subpar performance in predicting the settlement with the total score of 20 and 12, respectively. 4. Sensitivity analysis was conducted using Garson's algorithm to assess the strength of input variables in estimating the output (i.e., settlement). The results showed that the applied load ranked first with the relative importance of 18.4%, followed by unit weight of soil and constrained modulus of soil with relative importance of 16.3% and 15.3%, respectively. The relative importance of other parameters such as slope angle ratio, hoop stiffness, bending stiffness, burial depth, and footing crest distance is 13.8%, 11.4%, 9.9%, 7.7%, and 6.9%, respectively. 5. The combined predictive performance of all the model were assessed via Taylor's diagram. Based on the results, the standard deviation (SD) (1.33, 1.02, 1.04, 0.73, and 0.92), RMSE (0.3168, 0.374, 0.517, 0.673, and 0.613), and correlation coefficient (r) (0.974, 0.965, 0.905, 0.876, and 0.870), respectively, estimated MLP, RF, GPR, LKS, and DT, confirm the predictive strength of the developed MLP model. 6. Robustness analysis and generalization ability check showed that the settlement of footing over buried conduit in a sloping terrain increases with the increase in applied load and slope angle ratio, whereas the increase in the hoop stiffness, bending stiffness, burial depth, and footing crest distance causes the decrease in the footing's settlement.
Most importantly, the developed MLP model network has been translated into a functional relationship for easy hand or spreadsheet calculations. It can prove useful in saving the computational cost associated with intensive numerical simulations.

Limitations and future works
Although a wide range of data is utilized to train and validate the developed models, the models can be further improved by incorporating more data in the future. Moreover, the future research will also be dedicated in exploiting the deep learning techniques and hybrid ensemble learning approach to further increase the reliability of artificial intelligence-based modeling techniques in predicting the load-settlement behavior of the footing resting on buried conduit within a sloping ground.
1. Calculate G ik by multiplying the absolute values of hidden-output weight with the absolute value of inputhidden weight of each input variable j, that is, w ko Â w ik j j , e.g., from Table 7 (G 11 = -0.9407 9 -0.5494 = 0.5168).