Interaction behavior and load sharing pattern of piled raft using nonlinear regression and LM algorithm-based artificial neural network

Abstract

In the recent era, piled raft foundation (PRF) has been considered an emergent technology for offshore and onshore structures. In previous studies, there is a lack of illustration regarding the load sharing and interaction behavior which are considered the main intents in the present study. Finite element (FE) models are prepared with various design variables in a double-layer soil system, and the load sharing and interaction factors of piled rafts are estimated. The obtained results are then checked statistically with nonlinear multiple regression (NMR) and artificial neural network (ANN) modeling, and some prediction models are proposed. ANN models are prepared with Levenberg–Marquardt (LM) algorithm for load sharing and interaction factors through backpropagation technique. The factor of safety (FS) of PRF is also estimated using the proposed NMR and ANN models, which can be used for developing the design strategy of PRF.

Graphical abstract

Keywords

interaction / load sharing ratio / piled raft / nonlinear regression / artificial neural network

1 Introduction

Piled raft foundation (PRF) has earned massive attention in the recent era as a sustainable foundation technique for offshore and onshore structures. It has been applied in supporting most of the high-rise buildings and marine structures such as oil platforms, wind turbines, etc. Thus, its proper design and serviceability concern become significant aspects for its application in massive structures and offshore structures [ 1, 2]. Being the combination of traditional deep and shallow foundation strategy, it helps in reducing differential settlement of any structure and also can resist external forces of high magnitude. However, this combination of pile and raft shows a complex load transmission nature where the applied load is partially shared through the raft and the pile. In the past few years, several studies have been conducted to understand this complex load sharing model, and the researchers have idealized this load sharing pattern as a function of soil property, foundation stiffness, and the foundation’s geometry [ 3– 5]. The transmission of external load through the raft and pile is achieved through the interaction between the raft and the pile that usually arises because of the overlapped stress field and the settlement variations [ 6]. This interaction mechanism is very complex and needs due attention during the design and application of PRF. Various studies have been conducted to find the interaction factors; however, there is little research available regarding the relationship between the load sharing factors and the interaction factors. A simplified prediction model was developed to predict the load sharing factor and interaction factor for PRF, and a design strategy of the piled raft was completed using a level of safety and serviceability [ 7]. Although various design parameters were investigated in several studies that had a significant effect on load sharing and interaction behavior, soil stratum was not accounted for in any literature. Most of the studies considered a homogeneous single soil system, but it is very common to have multiple soil layers in the practical case. With the change of the layer of the soil, the resistance of soil against the external forces and the pile-soil interface also changes and thus it is required to check the influence of the soil layer on the interaction and load sharing pattern. Therefore, the change in the soil layer thickness plays an imperious part in the design of PRF, and in this study, this parameter holds great importance.

A series of numerical tests are performed in the present study based on various design parameters, and the results are analyzed using nonlinear multiple regression (NMR) and artificial neural network (ANN) modeling. NMR is a very effective tool when multiple input parameters and a nonlinear relation exist among the input and output variables. ANN is a form of artificial intelligence (AI) technique, and in the past few years, it is extensively being used for solving problems in geotechnical engineering. Several researchers have used ANN technique to model the bearing capacity, settlement, and other shallow and deep foundations parameters, as it is considered a superior alternative to other analysis methods [ 8– 17]. ANN can deal with the data’s complexity and predict a model with a suitable degree of accuracy by adopting extensive iterated computational works [ 18, 19]. Shahin et al. [ 20] have predicted the settlement of a shallow foundation with the help of ANN modeling and mentioned that the trained ANN model is a quick tool that eliminates the manual work for settlement estimation. The settlement of piles based on cone penetration data was investigated by Nejad et al. [ 21], and it was observed that the ANN model performed well up to a settlement of 138 mm. A similar study was carried out by Nejad and Jaska [ 22] using standard penetration data, and it was witnessed that the ANN model could be utilized for predicting pile settlement up to 185 mm. The load-settlement response of concrete pile was modeled with ANN technique using Levenberg–Marquardt (LM) algorithm having various inputs like pile length, applied load, axial rigidity, slenderness ratio, etc., and it was found that the correlation coefficient was near to unity with a negligible error [ 23]. The feasibility of using ANN for estimating the bearing capacity was also checked by some researchers and the studies revealed that the ANN is a reliable and efficient technique for estimating this geotechnical feature [ 24]. Although, there is no known application of ANN in the piled raft problems or particularly in load sharing and interaction behaviors, it can be assumed to have a feasible outcome based on its other successful applications in geotechnical problems. Hence, the goals of the present study are a) to check the effect of various design parameters on the load sharing and interaction aspects of PRF; b) to check the feasibility of using ANN modeling for load sharing and interaction features of PRF; and c) to make a composite load sharing and interaction model through traditional nonlinear regression and ANN model.

2 Load sharing and interaction behavior

In pile foundation and raft foundation, the load is sustained by the piles and raft, respectively, but for PRF, as it is a combination of piles and raft, the load is partially shared by raft and partially by the piles. When any load is functional on the raft, initially, it is withstood by the piles, and after the mobilization of the pile, the additional load is then withstood by the raft. To represent this distribution of applied load among the raft and piles, the researchers introduced a term called ‘load sharing ratio’. Load sharing ratio may be attained by dividing the load sustained by the piles with the total enforced load, and it is conveyed as

(1) {\alpha }_{\text{PR}} = \frac{Q_{\text{P}}}{Q_{\text{PR}}} = 1 - \frac{Q_{\text{R}}}{Q_{\text{PR}}} ,

where {\alpha }_{\text{PR}} is the load sharing ratio, Q_{\text{P}} and Q_{\text{R}} are the load carried by the pile and raft, respectively and Q_{\text{PR}} is the load imposed on PRF. The interactions in a piled raft can be categorized in pile-to-pile (P-P) interaction, pile-to-raft (P-R) interaction, and raft-to-pile (R-P) interaction. P-P interaction occurs due to the change in settlement value of a particular pile in the pile group because of the superpositioning of the stress field over that single pile. R-P interaction is the change in settlement of the raft due to the attached pile group, and P-R interaction is the change of settlement in the pile due to the existence of the raft. Poulos and Davis [ 25] gave the idea of P-P interaction, considering the effect of neighboring piles on the settlement behavior of a single pile. Clancy and Randolph [ 26] have formulated P-R interaction considering the stiffness of the pile group and the raft. The same was again modified by Randolph [ 27] by incorporating the stiffness of soil. Applying finite element (FE) method, Nguyen et al. [ 28] and Kumar and Choudhury [ 29] have found out P-P, R-P, and P-R interactions. Although researchers have formulated these interaction factors, there is still a lack of illustration regarding their application in the design of PRF, which is considered an important objective in this study.

3 3-D finite element (FE) analysis

3.1 Overview of the numerical model

The load-sharing nature and the interaction factors were obtained by developing a numerical model utilizing the three-dimensional FE software ABAQUS. To examine the effect of soil stratum on the load sharing and interaction behavior, a two-layer soil profile consisted of clay and sandy soil was used in this study. To account for the nonlinear stress strain relationship of soil, modified Drucker-Prager model was used and all the piles along with the raft were considered linear elastic materials. The raft and soil were modeled as 20-nodded hexahedral brick elements, while the triangular prism element was used for modeling the pile. To model the frictional behavior between the pile and soil, the Coulomb friction model was used to relate the maximum allowable shear stress (friction) between the contacting bodies. Surface to surface contact was selected using the master-slave concept, where the harder surface was considered the master surface. The pile-raft contact surface was considered perfectly rigid without any relative motion between the nodes. Further details of the modified Drucker−Prager model and contact behavior can be found elsewhere [ 7, 30].

3.2 Boundary condition and finite element mesh

As the model was symmetric, a quarter portion was taken to reduce the analysis time. The horizontal boundary of the model (for a quarter portion) was extended up to 3 B_R, and the boundary on the vertical side was kept as (3 B_R + L/3), where B_R is the raft width, and L is the pile length. As the vertical load on the model was applied in the y-direction, the face nodes were set as XSYMM (translation in z-direction was zero) or ZSYMM (translation in x-direction was zero) the corner nodes were fixed about x and z translation. The bottom nodes were entirely restricted about any translation or rotation. The schematic depiction of the model is shown in Fig. 1. For developing the FE mesh, due care should be taken in mesh refinement as the mesh pattern, and the mesh size of the element has a significant effect on the final outcome [ 31]. Generally, finer or thicker mesh provides better results compared to the coarser or thinner mesh, but it delays the computation time and requires highly configured systems for computing. To reduce the computation time and cost, the surroundings of the loading area were provided with finer mesh, and the relatively coarse mesh was provided at the rest of the places, succeeding a smooth alteration from finer mesh to coarser mesh (Fig. 1). The final size of the mesh was taken after a sensitivity analysis with several mesh refinement.

3.3 Analysis scheme and parametric study

In this study, vertical load was applied on the raft, and the ultimate load-carrying capacity was attained at a settlement level of 10% of the raft width ( w = 0.1 B _R). The loading was simulated into different analysis steps: geostatic step, where the model was in equilibrium condition under gravity; pile was installed inside the soil in the second step and self-weight was introduced to bring the model in the equilibrium condition and in final step, the load was employed in the model. The consolidation change was not considered in the study because the vertical load was applied very quickly [ 32, 33]. The strength behavior and the effect of strain-softening of clay were also ignored, as the settlement taken in the research is very small [ 34].

For conducting the parametric analysis of the present study, various design variables were taken into account. The number of piles in the piled raft, the spacing among the piles, and the raft's width were taken as design variables. As the subsoil consisted of two layers, the effect of the soil stratum was also analyzed, and the thickness of the clay layer was altered as t/ L = 0.5, 1, and 1.5, where t is considered the thickness of clay bed, and L is the pile length. Three different types of pile groups were connected to the raft, such as 3 × 3, 4 × 4, and 5 × 5 pile groups showing the number of piles as 9, 16, and 25. The width of the raft was taken as 20 d, 25 d, and 30 d, where d is the diameter of the pile ( d = 1 m). The spacing was also changed to 3 d, 5 d, and 7 d. A similar numerical modeling on group pile and the single pile was also performed as a reference study.

3.4 Numerical model validation

To test the compatibility of the numerical model established in this study, it was validated with two other studies. At first, the model developed in this study was matched with a laboratory test, and secondly, it was compared with the numerical reference model [ 35]. The load settlement behavior achieved from a fabricated model test set-up is compared to the numerical model that exactly mimics that experimental set-up. The laboratory test comprises a testing tank attached with a loading frame for applying vertical compressive load by means of a hydraulic jack. The dimension of the testing tank was 90 cm × 90 cm square cross-section with a height of 60 cm. A 2 × 2 PRF was placed inside the tank with a 30 cm squared raft and four piles of 20 cm diameter. The load and settlement were measured through several calibrated sensors like load cells and LVDTs. Two-layer soil system with clay and sandy soil was maintained inside the tank where the clay was prepared with a bulk density conforming to S_u = 10 kPa (where, S_u is undrained shear strength). The sandy soil was prepared with a relative density equal to 70% using the ‘sand raining method’. The same experimental test is then reformed in the numerical modeling with similar material properties, and the variation in load settlement responses are shown in Fig. 2. Both the experimental and numerical models’ graphs show almost similar variation at every t/ L ratio which indicates that the developed numerical model can precisely simulate the experimental tests.

Secondly, the numerical model established in the present study is validated with a different numerical model explained by the researchers Sinha and Hanna [ 35]. Three different sizes of square-shaped rafts such as 24 m × 24 m, 28 m × 28 m, and 32 m × 32 m having a spacing of 3 D, 7 D, and 8 D, respectively, were chosen for comparison purposes ( D = diameter of pile = 1 m). The comparison curve for both the numerical models are presented in Fig. 3, and it is evident that the developed model possesses a reliable correlation with the existing model prepared by Sinha and Hanna [ 35]. Both Figs. 2 and 3 check the legitimacy of the numerical model prepared in the current study, and it ensures that the results attained from the numerical analysis or FE analysis can be used for further analysis.

4 Results obtained from numerical analysis

4.1 Vertical load settlement of PRF

The load settlement response for PRF acquired for every case with different t/ L ratios and different numbers of the pile is presented in Fig. 4. The load was normalized via dividing the obtained load of each case by the maximum load attained among all models and the settlement was normalized by taking the ratio of the settlement to the raft width and the ultimate load of piled raft is considered at a settlement of w = 0.1 B_R. The figure illustrates that the load settlement curves are not linear, and there exists a polynomial relationship between the vertical load and the settlement of PRF. The ratio of the clay layer depth to the length of the pile (i.e., t/ L ratio) has a substantial effect on the load-carrying capacity of PRF, as the increase of t/ L ratio causes the decrease in the load sustaining capacity. This can be accredited to the properties of the soil stratum on which the piles of the piled raft are resting. In this study, it is already mentioned that two-layer soil systems are taken where clay layer is kept on the top layer underlain by a layer of sand. At t/ L = 0.5, the half part of the pile remains on clay, and the rest of the portion remains in the sand layer; at t/ L = 1, the pile rests on the clay layer, but the pile tip rests in the sand layer, and at t/ L = 1.5, all the piles entirely rest on the clay layer. When the pile is on the clay bed, the frictional resistance along the periphery of the pile is lesser than the pile resting in the sand. Hence the frictional resistance developed in the sand is higher than clay, which increases the load-carrying capacity of PRF. Considering the number of piles in the piled raft, when the number of piles under the piled raft increases, the mobilization of pile group also increases due to the interaction among the piles in the group which boosts the yielding of piles. This is why the number of the pile in the piled raft also plays a significant role in the load-carrying capacity of PRF.

The effect of spacing to diameter ratio and normalized raft width is shown in Fig. 5, and it is witnessed that the load-carrying capacity of PRF constantly increases if the spacing to diameter ratio and normalized raft width is increased. If the raft size is increased, more area under that raft becomes confined, which upsurges the pile mobilization, thereby reducing the settlement of the pile group and enhancing the yielding of the pile group. Similarly, if the spacing among piles is expanded, the overlapped stress zone is reduced, which increases the load sustaining capacity of the PRF.

4.2 Vertical load sharing response

The vertical load sharing ratio for all the models were evaluated from numerical results using Eq. (1) as mentioned in the earlier context. The load sharing ratio of the piled raft is shown in Fig. 6 for different t/ L ratios and the number of piles. The load sharing ratio also follows a nonlinear variation with the normalized settlement of the piled raft. It is noteworthy to mention that the load sharing at the initial stages, i.e., just after the load application, is much higher, and it gradually decreases as the normalized settlement increases. This phenomenon indicates that initially, the applied load is sustained by the piles under the raft, and when the piles tend to yield or being mobilized, the extra load is shared by the raft. The figure shows that the load sharing ratio gradually decreases with an increase in t/ L ratio, whereas the load sharing ratio remains increasing with the number of the pile under the raft. When the number of piles is increasing in the pile group, the yielding time of the pile gradually increases due to which more load can be sustained by the piles at the initial stages. For this reason, the load sharing ratio increases with the number of the pile in the pile group under the raft. When considering the effect of the thickness of the clay on the increase in the t/ L ratio, as the thickness of the clay layer increases, the yielding time of the pile reduces by reducing the frictional resistance. As the yielding time reduces for the piles, the fraction of load shared by the piles also decreases leading to the reduction of the load sharing ratio.

The normalized width of the raft and the s/ d ratio among the piles in the piled raft also positively influence the load sharing ratio as shown in Fig. 7 (For n = 16 and w = 0.001). As the spacing to diameter ratio between the piles and the normalized width of the raft increases, the load sharing ratio continues to increase. If the spacing among the piles in PRF increases, the overlapping stress zone in a pile due to the adjacent pile tends to reduce, increasing the mobilization of the pile group, thereby increasing the load sharing ratio. Similarly, when the normalized width of the raft increases, the confined soil below the raft improves the yielding of the pile, and as a result, the load sharing ratio of PRF increases if the normalized width of the raft increases.

4.3 Interaction behavior of piled raft

4.3.1 Pile-pile (P-P) interaction

P-P interaction occurs due to the positioning one pile adjacent to another pile by the superposition of the stress field. This overlapped stress field alters the settlement of one single pile adjacent to another pile, and this phenomenon is known as P-P interaction. The general expression for P-P interaction is

(2) {\beta }_{\text{p-p}} = \frac{Q_{\text{GP}}}{n\times Q_{\text{SP}}} .

where {\beta }_{\text{p-p}} is the pile-pile interaction factor, Q_{\text{GP}} is the load carrying capacity of group pile, Q_{\text{SP}} is the load carrying capacity of single pile, and n is the number of pile. The load-carrying capacity for piles in group and for a single pile are evaluated separately for each configuration and the variation of {\beta }_{\text{p-p}} is presented in Table 1 for the settlement of w = 0.1 B_R. It is noticed that {\beta }_{\text{p-p}} is slightly decreasing with an increase in t/ L ratio. For t/ L ratio of 0.5, when the pile is resting in the sand, the pile group is withstanding slightly higher value than the sum of the single individual pile; however, at t/ L = 1.5, the total load carried by the pile group is relatively lesser than the sum of the single pile. When the pile group rests on the sand layer, the resistive force develops due to the frictional resistance of sand, and the periphery of the pile becomes more than the single pile, and thus the pile group can sustain more load compared to the sum of single individual pile. On the other hand, when the pile group rests entirely in the clay layer, the sum of the loads carried by the single pile is slightly higher than the group pile, giving the value of {\beta }_{\text{p-p}} less than 1. At t/ L ratio of 1, the ultimate load sustained by the pile group is almost similar to the sum of the load carried by single piles, showing {\beta }_{\text{p-p}} as unity. Considering the number of piles in the pile group and the spacing to diameter ratio among the piles, {\beta }_{\text{p-p}} increases with the number of piles as well as the spacing among the piles, although the variation in {\beta }_{\text{p-p}} is very less.

4.3.2 P-R interaction

The settlement of PRF is entirely different from the pile group because of the existence of the raft. When the raft is attached to the pile head, the settlement behavior of the pile group changes, which is termed the P-R interaction. It is expressed as

(3) {\beta }_{\text{p-r}} = \frac{Q_{\text{PR-GP}}}{Q_{\text{GP}}} ,

where {\beta }_{\text{p-r}} is the pile-raft interaction factor and Q_{\text{PR-GP}} is the load carrying capacity of pile in a piled raft. The P-R interaction has both positive and negative impacts on PRF because of the confinement effect of soil mass during the presence of the raft. The change of {\beta }_{\text{p-r}} with varying normalized raft width is shown in Fig. 8 (for n = 9 and t/ L = 1). It is observed that {\beta }_{\text{p-r}} increases with the normalized settlement irrespective of the normalized width of the raft, and at a settlement of 0.1 B_R, the value of {\beta }_{\text{p-r}} reaches up to 70%. The value of {\beta }_{\text{p-r}} is also gradually decreasing with the reduction in the normalized width of the raft. The lower raft size reduces the mobilization of pile group under the raft due to which {\beta }_{\text{p-r}} also decreases. The effect of the t/ L ratio, number of piles, and the spacing between the piles are also shown in Fig. 9, and it is observed that {\beta }_{\text{p-r}} mostly depends on the number of piles rather than the t/ L ratio and the spacing between the piles. When the number of piles is changed from 9 to 16, there is almost 9.3% increase in {\beta }_{\text{p-r}} and it is increased by about 19.7% when the number of piles is changed from 9 to 25. However, there is significantly less effect of spacing on the P-R interaction, as the percentage change in {\beta }_{\text{p-r}} lies in the range of 2%–4%, when the s/ d ratio is changed from 3 to 7. Similarly for the t/ L ratio, {\beta }_{\text{p-r}} is continuously decreasing with the increasing t/ L ratio and the percentage change in {\beta }_{\text{p-r}} lies in the range of 8%–15%.

4.3.3 Raft-to-pile (R-P) interaction

R-P interaction is opposite to P-R interaction, where the settlement of the raft is affected by the presence of piles under the raft. It is defined as the load sustained by the raft present in PRF to the load sustained by the unpiled raft at a certain settlement. It can be found as

(4) {\beta }_{\text{r-p}} = \frac{Q_{\text{PR-R}}}{Q_{\text{UR}}} .

where {\beta }_{\text{r-p}} is the raft-pile interaction, Q_{\text{PR-R}} is the load carrying capacity of raft in a piled raft and Q_{\mathrm{U}\mathrm{R}} is the load carrying capacity of unpiled raft. When piles are attached to the raft, the load distribution mechanism completely changes compared to the unpiled raft foundation, as the load is divided among the piles and the raft, and due to this, the settlement behavior of the raft changes. Deb and Pal [ 7] have introduced another equation for the R-P interaction considering the load sharing ratio and P-R interaction factor, which is given as

(5) {\beta }_{\text{r-p}} = \frac{1}{1-{\alpha }_{\text{PR}}} - \frac{{\beta }_{\text{p-r}}}{\phi } ,

where \phi is the load-carrying capacity ratio \left(\phi ={\displaystyle \frac{Q_{\text{UR}}}{Q_{\text{GP}}}}\right). It can be clearly observed that the R-P interaction factor is inherently dependent upon the P-R interaction factor and the load sharing ratio, and once {\alpha }_{\text{PR}} and {\beta }_{\text{p-r}} can be adequately evaluated, {\beta }_{\text{r-p}} can also be obtained from Eq. (5). It is noteworthy to mention that the influence of different design variables in the R-P interaction factor is already considered while evaluating the {\alpha }_{\text{PR}} and {\beta }_{\text{p-r}} and it is not at all necessary to check their effects on {\beta }_{\text{r-p}}.

5 Prediction of load sharing and interaction factors using multiple nonlinear regression model

As all the variables influence the load sharing ratio and the interaction factor, a generalized equation is developed with these input variables through NMR analysis. The analysis of variance (ANOVA) technique was applied to check the statistical significance of the data with a 95% confidence level. The input variables are taken as normalized width of the raft ( B_R/ d), spacing to diameter ratio ( s/ d), t/ L ratio, and normalized settlement ( w/ B_R). The output variables are load sharing ratio ( {\alpha }_{\text{PR}}), P-P interaction factor ( {\beta }_{\text{p-p}}) and P-R interaction factor ( {\beta }_{\text{p-r}}). The R-P interaction factor ( {\beta }_{\text{r-p}}) is not considered for statistical analysis because it is intrinsically dependent upon the load sharing ratio ( {\alpha }_{\text{PR}}) and P-R interaction factor ( {\beta }_{\text{p-r}}). As {\beta }_{\text{r-p}} is highly correlated with the other two parameters, it is excluded from the NMR analysis. All the results are first checked for an outlier to determine whether the data set is homogeneous. Chauvenet’s outlier criterion was used to check the outlier, and it was observed that there was no such outlier in the obtained results for each case. However, the regression analysis for P-P interaction factor ( {\beta }_{\text{p-p}}) was not possible, as there was no significant effect of the input variables on {\beta }_{\text{p-p}}. The one-way ANOVA table for each variable and its effect on {\beta }_{\text{p-p}} is shown in Table 2. In the case of the number of piles ( n) and s/ d ratio, the obtained p-value is greater than 0.05, and hence the null hypothesis could not be rejected and therefore, the development of regression equation for P-P interaction factor ( {\beta }_{\text{p-p}}) is not possible. But for t/ L ratio, the obtained p-value is less than 0.05, i.e., t/ L ratio has a significant effect on the P-P interaction. It has obtained that the P-P interaction factor i.e., {\beta }_{\text{p-p}} varies linearly with a negative gradient when it is plotted against t/ L ratio (Fig. 10).

The linear regression equation for {\beta }_{\text{p-p}} is shown in Eq. (6). For {\alpha }_{\text{PR}} and {\beta }_{\text{p-r}}, the NMR analysis is carried out, and the equations for {\alpha }_{\text{PR}} and {\beta }_{\text{p-r}} are represented in Eqs. (7) and (8).

(6) {\beta }_{\text{p-p}} = - 0.383 t / L + 1.44 ,

(7) \begin{array}{rl}{\alpha }_{\text{PR}}=& 0.71+0.002n+0.002B_{\text{R}}/d-0.56t/L+0.03s/d\\ & -21.2w/B_{\text{R}}+0.25(t/L{)}^2+290.4{\left(w/B_{\text{R}}\right)}^2\\ & -0.02(t/L)(s/d)+4.78\left(w/B_{\text{R}}\right)(t/L)-1487{\left(w/B_{\text{R}}\right)}^3\\ & -1.920w(t/L{)}^2,\end{array}

(8) \begin{array}{rl}{\beta }_{\text{p-r}}=& 0.039+0.045B_{\text{R}}/d+0.08t/L+10.38w/B_{\text{R}}\\ & -0.05(t/L{)}^2-184.3{\left(w/B_{\text{R}}\right)}^2+(0.14n+0.23B_{\text{R}}/d\\ & -1.37t/L+0.41s/d)w/B_{\text{R}}+1132.4{\left(w/B_{\text{R}}\right)}^3\\ & -(0.88n-1.49B_{\text{R}}){\left(w/B_{\text{R}}\right)}^2-0.04\left(w/B_{\text{R}}\right)(s/d{)}^2.\end{array}

The statistical estimates for both the regression models are summarized in Tables 3 and 4, respectively. From the tables, it can be noticed that the input variables have a significant influence on the output variables, as the p-values are less than 0.05. Moreover, the interaction among the input variables also has a significant impact on load sharing and interaction factors. There is a linear relationship between {\alpha }_{\text{PR}} and the number of piles, normalized width of the raft, and s/ d ratio. A quadratic relationship exists between the t/ L ratio and load sharing ratio, whereas normalized settlement is best suited for cubic relationship. On the other hand, the normalized width of the raft and number of piles have linear relationship with P-R interaction factor, whereas, the t/ L and s/ d ratios show a quadratic relation, and normalized settlement shows a cubic relationship with the P-R interaction. Once the equations are obtained, it is essential to examine the validity of the developed equation, i.e., how much deviation there is in the actual data with the data obtained using the prediction equations. The variation in the data for predicted and calculated equations is shown in Figs. 11 and 12. As the correlation coefficients are ranging between 0.91 and 0.96, it can be said that the developed equation performs well to predict the load sharing ratio and P-R interaction factor.

6 Prediction of load sharing and interaction factors using artificial neural network

6.1 Modeling of neural network

ANN is a soft computing tool based on AI inspired by the functioning of the human nervous system where information is transferred from one neuron/node to another in the form of a signal. In a neural network, the multilayer perceptions (MLP) and processing elements are organized in different layers such as a) input layer, b) output layer, and c) hidden layers. The input layer contains the input variables, i.e., the factors on which the output parameter depends, and the output layer contains the required output variables, whereas the hidden layers are the intermediate layers that cannot be accessed from the outside of the network. Each layer of the network contains a number of nodes having a certain weightage, and the signals are transformed from layer to layer using any activation function. A simple neural network is shown in Fig. 13, and the relation of input and output variables are presented in Eqs. (9) and (10). As the complete description of the neural network is beyond the scope of the paper, only the details related to the modeling of the present network are discussed here, and the comprehensive descriptions of neural networks can be found in the previous studies [ 8]. The steps involved in developing the ANN models are divided into various parts: selection of inputs and outputs, division and pre-processing of data, determination of model architecture, training of the model, stopping criteria, and validation of the model, which are further illustrated in the following contexts. In this paper, the ANN modeling was carried out using MATLAB software.

(9) y = \sum _{i=0}^N w_j^{(N)} Z_j \pm b_j^{(N)} ,

(10) Z_j = \frac{1}{1+e^{\left(\sum _{i=0}^N\pm w_{ij}^{(1)}{X}_i\pm b_j^{(1)}\right)}} ,

where y is the output layer, w_j^{(N)} are the weights of input variables, Z _j is the intermediate bias parameter, X _i is the input layer, N is the total number of input layer or hidden layer and b_j^{(N)} are the biases of hidden layers.

6.1.1 Selection of input and outputs

Input and output parameter selection is an essential step as it has the most significant effect on the ANN model's performance. In the present study, the load sharing ratio ( {\alpha }_{\text{PR}}), and P-R interaction factor ( {\beta }_{\text{p-r}}) are taken as the output variables. As it is already mentioned during nonlinear regression analysis that the design variables are not having any notable influence on the P-P interaction factor ( {\beta }_{\text{p-p}}) and hence these parameters are not considered in ANN modeling. Similarly, for raft-pile interaction factor ( {\beta }_{\text{r-p}}), as it is highly correlated with {\alpha }_{\text{PR}} and {\beta }_{\text{p-r}}, this parameter is not included either in the neural network modeling. The input parameters are a) normalized width of the raft ( B _R/ d), b) spacing to diameter ratio ( s/ d), c) the number of piles in the piled raft ( n), d) thickness of clay to the length of the pile ( t/ L), and e) normalized settlement ( w/ B _R). Two separate ANN models were developed and trained with the LM algorithm for two different output parameters.

6.1.2 Division and pre-processing of data

Previous literature on neural networks suggested dividing the data set into various subsets like training set, testing set, and validation set [ 21, 36, 37]. The training set is mainly used for adjusting the weights, the testing set is used for ensuring the functioning of the model in every step, whereas the validation set is used for confirming the performance of the developed network. In the present study, first, the raw data was divided into two groups- training data (90% of the total data) and validation data (10% of the total data). The training data was further subdivided into a training set (70% of training data) and a testing set (30% of the training data). As the testing and validation are completely dependent upon the training set and these represent the same population of data, it is very much essential to have similar statistical properties among these data sets, i.e., the mean, standard deviation, minimum value, maximum value, and range should be similar for all the three data sets. Before applying the data sets in the ANN model, it is also required to pre-process all data by scaling them. As sigmoid transformation is used in this study, the data s scaled between 0 and 1.

6.1.3 Model architecture

Development of model architecture involves collecting optimal hidden layers and the number of nodes in each layer in the network. However, there is no theory available for selecting an optimum model architecture, and it depends on the number of parameters to be analyzed [ 20]. The nodes in the input layer and output layer are pre-defined based on input variables and output variables. In this study, five nodes were taken for the input layer, and a single node was taken for two different sets of output layer (as the analysis was done separately). For selecting the number of hidden layers, a series of tests were carried out considering 1, 2, 3, 4, 5, 7, 9, and 11 hidden layers. Caudill [ 38] reported that the upper limit of hidden layers for five inputs should be restricted to 11.

6.1.4 Training of the model

Training of the model is done for the optimization of the connection weights. To find the optimum weight transfer, feed-forward and feed-back methods are generally used where feed-forward indicates the processing of connection weights in the forward direction and feed-back indicates the processing in both forward and backward direction. In this study, feed-forward method is chosen with the backpropagation algorithm (BP), which is mainly based on the 1st order gradient descent. Several researchers use this feed forward network with BP algorithm for modeling the geotechnical problems. For finding the optimum training parameters, the model was run several times with different learning rates and momentum terms, and based on this process, the learning rate and momentum term are selected as 0.1 and 0.6, respectively.

6.1.5 Stopping criteria

Stopping criteria decides when to stop the learning/training of the network, and it controls whether the model is optimally trained or not. Previous literature mentioned that cross-validation condition is the primary technique to control the stopping criteria [ 39], and this process was used in this study. This process does not allow the overfitting of the data and the training of the network stops automatically when the errors in the testing set tend to increase.

6.1.6 Validation of the model

Model validation is a part of model building, and it is only used for validating the developed model. The main purpose of validation is to confirm that the model is capable of generalizing its performance robustly within the set limits. For validation, the coefficient of correlation ( r), root mean square error ( RMSE), and mean absolute error ( MAE) are generally evaluated, which can be calculated as

(11) r = \frac{{\displaystyle \sum _{i=0}^N\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}}{\sqrt{{\displaystyle \sum _{i=0}^N{\left(x_i-\overline{x}\right)}^2}}\sqrt{{\displaystyle \sum _{i=0}^N{\left(y_i-\overline{y}\right)}^2}}} ,

(12) R M S E = \sqrt{\frac{1}{N}\sum _{i=0}^N{\left(x_i-y_i\right)}^2} ,

(13) M A E = \frac{1}{N} \sum _{i=0}^N x_i-y_i ,

where x_i is the predicted (model) output, y_i is the actual (observed) output and N is the total number of observations. In the present study, at the starting of the data processing and ANN modeling, the testing and testing data set was changed several times, and the cross-validation was also carried out 6 times. The values of statistical parameters are also evaluated at each cross-validation, and then their average values are checked for further processing.

6.2 Results obtained from the ANN model

6.2.1 Performance of LM model

The mean square error ( MSE) for the ANN model with different layers is shown in Figs. 14 and 15. It is already discussed that the ANN model was made with multiple hidden layers from 1 to 11 to obtain the best model architecture. For both the cases, the least prediction error was obtained when the number of the hidden layer was 7; however, the number of hidden layer in this study was chosen as 2 because the difference in MSE for 7 hidden layer and 2 hidden layers is very less (0.4%). The network performance for both the cases (for load sharing ratio and P-R interaction factor) is shown in Figs. 16(a) and 16(b), indicating a continuous fall in MSE values with the number of iterations or epochs. The optimum model performance was attained with comparatively lesser MSE values like 0.0019 at an epoch of 69 for load sharing ratio and 0.0011 at an epoch of 52 for the P-R interaction factor. It can be conjectured that the learning process stopped to avoid overfitting and can be considered as early stopping criteria. The error gradient, Marquardt adjustment parameter ( m_u) and the validation for both models are shown in Figs. 17(a) and 17(b).

6.2.2 Model robustness

To check the robustness for both the models, the predicted values obtained from the LM trained networks are compared with the numerical results. The comparison for predicted and numerical results of load sharing ratio ( {\alpha }_{\text{PR}}), and P-R interaction factor ( {\beta }_{\text{p-r}}) for a piled raft system are shown in Fig. 18(a) and 18(b). Based on this graphical comparison, it can be inferred that the LM trained networks are capable enough to fit the target values and can meet the robustness of the analysis data. The regression calibration chart for both the models is shown in Figs. 19(a) and 19(b), including training, testing, and validation curves, and in all the cases, the correlation coefficient was greater than 0.97, which indicates that the application of LM algorithm in ANN modeling to predict {\alpha }_{\text{PR}} and {\beta }_{\text{p-r}} can be an effective tool.

6.2.3 Sensitivity analysis

The sensitivity analysis for the relative significance of the input variables in both cases is shown in Tables 5 and 6. It can be observed that the normalized settlement has the maximum effect on the interaction and load sharing pattern at several trials of the neural network. In the case of other remaining input variables, the normalized width of the raft and t/ L ratio have the second-highest effect on interaction behavior and load sharing behavior, respectively. The synaptic weights for the input parameters, no hidden layers and output layer for both the ANN model are presented in Tables 7 and 8 and the neural network for both models is shown in Figs. 20(a) and 20(b).

7 Comparison of developed ANN model with support vector regression (SVR)

To compare the developed ANN model, support vector regression (SVR) is used in the present study. SVR mainly uses the concept from support vector machine (SVM) and mainly involves analyzing data for classification and regression. In this study, the performance of the ANN model is checked by preparing an SVR model using the same data set. The same input variables and the output variables used during the neural networking are also used for SVR model preparation. The input parameters are a) normalized width of the raft ( B_R/ d), b) spacing to diameter ratio ( s/ d), c) the number of piles in the piled raft ( n), d) thickness of clay to the length of the pile ( t/ L) and e) normalized settlement ( w/ B_R). The output variables are taken as load sharing ratio ( {\alpha }_{\text{PR}}), and P-R interaction factor ( {\beta }_{\text{p-r}}). Here, 80% of the data are used to train the SVR model, and the rest of the 20% data are used to validate the developed SVR model. Two Kernal functions, such as radial basis kernel function (RBF) and exponential radial basis kernel function (ERBF), are used in this study. The constant parameters and the kernel parameters are chosen based on trial and error following the method used by other researchers [ 40, 41]. Here, the number of support vectors is chosen as 170, and the width of RBF and ERBF functions are chosen as 1. The values of constant C (penalty or regularization parameter) and ε (optimum deviation of the error margin) are taken as 10 and 0.00001, respectively. The same data are used for both SVR-RBF and SVR-ERBF. In this study, the SVR model is performed in MATLAB using the SVM code prepared by Gunn [ 42]. For comparison of ANN with SVR, some statistical parameters are used, such as r, RMSE, MAE, mean absolute percentage error ( MAPE), and median absolute error ( MEDAE). The comparison of these statistical parameters are obtained for both the output parameters ( {\alpha }_{\text{PR}} and {\beta }_{\text{p-r}}) using both SVR and ANN model and the details are mentioned in Table 9. From the table, it is clear that all the models, i.e., SVR-RBF, SVR-ERBF, and ANN models, perform almost the same in all the cases. This indicates that the performance of the developed ANN model to predict the numerical data are acceptable. This comparison also shows that the slow convergence or stagnancy in local minima problems that generally arise during ANN modeling are also minimized.

8 Comparison of developed ANN model with unseen data set

Developing the ANN model is also validated with an unknown data set obtained from the experimental test carried out by Debnath and Dey [ 41]. In that study, the bearing capacity of geo-grid supported stone column is obtained using different design variable. It is noteworthy to mention that the ANN model prepared in this study is case-specific and can only be used to predict the load sharing ratio and the interaction factor of the PRFunder the action of vertical loading. However, it is important to note that there is no literature available where the load sharing ratio and pile to raft interaction factor are evaluated experimentally or numerically using the same design variables used in the present study. Hence, the data available in the literature mentioned above are used as an unseen data set.

For comparison purpose, only the process of developing an ANN model which is used in the present study is validated. In that study, the settlement to the diameter of the footing ratio, thickness of unreinforced sand bed and the geogrid-reinforced sand bed to the diameter of the footing ratio, the diameter of the geogrid-reinforced layer to diameter of the footing ratio, and length to the diameter of the stone column ratio are used as input parameters. The bearing capacity of the geo-grid supported stone column is considered as the output parameter. The researchers then use these experimental data for developing an SVR model and an ANN model. The data of these two models are used for comparison purposes. In the present study, an ANN model is developed considering the same process that is used in the present study. All the steps for preparing the ANN model that is mentioned in the previous steps are repeated here. The process or method used for testing and training data selection, model architecture preparation, number of hidden layer generation, training of model, etc., are kept the same as the present study. The variation of different statistical parameters is compared with that reference study, as shown in Table 10. It is confirmed from the table that the values of all the parameters obtained using different methods are almost similar with a maximum variation of ±10%. Hence, it can be confirmed that the process used in the present study for developing the ANN model is acceptable and this process does not suffer from the slow convergence or stagnancy in local minima problem.

9 Estimation of factor of safety of PRF

The ultimate load-carrying capacity of a piled raft system can be obtained from the equation provided by de Sanctis and Mandolini [ 5], which is given as

(14) Q_{\text{PR,ult}} = {\xi }_{\text{PR}} \left(Q_{\text{UR,ult}}+Q_{\text{GP,ult}}\right) ,

where Q _PR,ult is the ultimate load carrying capacity of piled raft, Q _UR,ult is the ultimate load carrying capacity of unpiled raft, Q _PR,ult is the ultimate load carrying capacity of unpiled raft and {\xi }_{\text{PR}} is the load distribution coefficient. By combining, Eqs. (3), (4), and (5), the load distribution coefficient can be expressed as

(15) {\xi }_{\text{PR}} = \frac{\phi {\beta }_{\text{r-p}}+{\beta }_{\text{p-r}}}{1+\phi } .

Similarly, the factor of safety of PRF can be evaluated as

(16) F S_{\text{PR}} = {\xi }_{\text{PR}} \left(F{S}_{\text{UR}}+F{S}_{\text{GP}}\right) = \frac{\phi {\beta }_{\text{r-p}}+{\beta }_{\text{p-r}}}{1+\phi } \left(F{S}_{\text{UR}}+F{S}_{\text{GP}}\right) ,

where FS _PR is the factor of safety of piled raft, FS _UR is the factor of safety of unpiled raft, FS _GP is the factor of safety of group pile, and \varphi is the load capacity ratio. Considering hyperbolic expression, the Eq. (16) can be represented as

(17) F S_{\text{PR}} = \frac{\phi {\beta }_{\text{r-p}}+{\beta }_{\text{p-r}}}{1+\phi } \left(\frac{a_{\text{r}}+b_{\text{r}}\cdot \left(w/B_{\text{R}}\right)}{\left(w/B_{\text{R}}\right)}+\frac{a_{\text{p}}+b_{\text{p}}\cdot (w/d)}{(w/d)}\right) ,

where a_r, b_r, a_p, and b_p are the hyperbolic coefficients and their values can be obtained from the database created by Akbas and Kulhawy [ 43] and Dithinde et al. [ 44]. For the settlement corresponding to 0.1 B_R, Lee et al. [ 45] modified the values of these coefficients, and the values for a_r, b_r, and a_p, b_p can be adopted as 0.02, 0.8, and 0.01, 0.09, respectively. Once, {\beta }_{\text{r-p}} and {\beta }_{\text{p-r}} are evaluated properly, the overall factor of safety for PRF can also be obtained using Eq. (17). In this study, {\beta }_{\text{r-p}} and {\beta }_{\text{p-r}} are obtained in three ways: finite element analysis, predicted equations using nonlinear regression model (NMR) and using ANN model, and accordingly the factor of safety (FS) is also evaluated. Figure 21(a) and 21(b) is depicting the variation of FS with the normalized settlement for the piled raft with 9 and 25 numbers of pile, respectively, having B_R/ d of 30, s/ d of 7 and t/ L of 1. Both the figures indicate that the projected models are highly complementing with the FE model. The correlation coefficient, RMSE and MAE for regression and ANN models are also calculated for both the cases and mentioned along with the figures. Both the prediction models are showing very good correlation with the FE model and the RMSE for both the models are also very less. Therefore, it is anticipated that the predicted equations can be further used for the idealisation of FS for PRF. Based on this widespread analysis of piled raft through NMR and ANN modeling, a simple flow chart is mentioned in Fig. 22. This flow chart proposes a simplified process/methodology for evaluating the factor of safety for PRF on the basis of load sharing ratio and interaction behavior. This proposed methodology for the design of PRF would be helpful for the modification/upgradation of the existing design strategy of the PRF.

10 Conclusions

PRF has earned massive attention as a settlement reducer in current construction practices for high rise and heavy structures where the conventional pile foundation and raft foundation are not capable of withstanding the external load. However, the combined effect of piles and raft makes the load sharing more complex than only group pile or only raft foundation. Moreover, some interaction factors also exist among the piles in the group and between the pile and the raft. In some previous studies, researchers have tried to identify the load sharing pattern and interaction behavior, however, those studies have not included some essential factors like the effect of soil stratum or the effect of foundation elements. The resistance of soil against the external forces and the pile-soil interface changes with the change of the soil layer, and therefore, it is required to examine the influence of soil layer on the load sharing pattern and the interaction behavior. This study aimed to inspect this interaction and load sharing pattern through FE modeling using multiple regression methods and ANNs.

A series of numerical models were prepared and analyzed using different pile configurations, and the results obtained from the numerical analysis were used to formulate the prediction models. Load sharing pattern is developed for each configuration, and the results are analyzed to develop statistical prediction models through NMR and ANN modeling. In interaction behavior, the P-P interaction factor is very slightly affected by most of the input variables, and hence this was not included in statistical or ANN modeling. P-R interaction factor was greatly affected by the input variables, and hence the results from numerical analysis were utilized to develop prediction models. For the R-P interaction factor, as it is entirely dependent upon the load sharing ratio and P-R interaction factor, the results were evaluated using the predicted equations of load sharing and P-R interaction factor. Finally, the factor of safety for PRF is also evaluated using those developed expressions. It is observed that the proposed equations are highly correlated with the numerical results as the results obtained from the proposed equations and numerical analysis are almost the same. Thus, the factor of safety of PRF can be evaluated at any settlement level if the capacities of group pile and unpiled raft are known. Finally, it can be concluded that the proposed methods provide a significant enrichment in the current design practices of PRF subjected to vertical loading.

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