Bearing Capacity of Ring Foundations on Anisotropic and Heterogenous Clays: FEA, NGI-ADP, and MARS

The use of shallow circular and spudcan footings were popular in the past few decades in supporting axisymmetric structures such as transmission towers, water towers, annular platforms, silos, storage tanks, and chimneys. Nowadays, the more economical ring foundations are often being considered, with the increasing numbers of recent research in this direction.

Several experimental studies were reported in relation to the performance of ring footing in sand (e.g., Ohri et al. 1997; Hataf and Razav 2003) as well as in clay (e.g., Demir et al. 2012; Shalaby 2017). Calculation approaches have been constantly developed, especially in the area of numerical simulation. For instance, Zhao and Wang (2008) performed the bearing capacity of ring foundations in sandy soil using FLAC; whilst Benmebarek et al. (2012) adopted FLAC and Chavda and Dodagoudar (2019) to estimate N_c, N_q, N_γ considering the effects of the dilation angle, smooth and rough ring footings. Choobbasti et al. (2010) assessed the bearing capacity and settlement of ring footings on homogeneous soils using the finite element analysis. Kumar and Chakraborty (2015) adopted the finite element limit analysis (FELA), Chavda and Dodagoudar (2021) applied FEA, and Gholami and Hosseininia (2017) used method of characteristics to estimate the bearing capacity factors for ring foundations on cohesive-frictional soils. The bearing capacity solution of a ring footing rest on clay was also investigated in Lee et al. (2016), Benmebarek et al. (2017), Khatri and Kumar (2009), Yang et al. (2020), Birid and Choudhury (2021), Keawsawasvong and Lai (2021), and Papadopoulou and Gazetas (2020). Recently, the bearing capacity of a ring foundation on rock mass was studied by Yodsomjai et al. (2021a).

It was generally recognized that undrained shear strength of soils not only increases with depth but also it is directly dependent on the orientation of the major principal stress to the vertical axis or depositional direction (see e.g., Ladd and Degroot 2003; Ladd 1991; Yu and Sloan 1994). In other words, the undrained shear strength of clay exhibits somewhat anisotropic behavior in nature. Recently, some failure criteria were proposed to consider the complex inherent of anisotropic clay such as the Anisotropic Undrained Shear (AUS) model in Krabbenhoft et al. (2019). In addition, the NGI-ADP constitutive model was presented by Grimstad et al. (2012). Although the recent finite element limit analysis is considered as a powerful technique to solve various stability problems in the geotechnical engineering field (e.g., Ukritchon et al. 2019, 2020; Shiau and Al-Asadi 2020a, b, c, d; Shiau and Al-Asadi 2021; Shiau et al. 2021; Keawsawasvong et al., 2022; Keawsawasvong and Ukritchon, 2017a, 2020, 2021; Lai et al. 2021a; Yodsomjai et al. 2021b, c; Beygi et al. 2020; Ukritchon and Keawsawasvong, 2017, 2019, 2020; Tho et al. 2014; Bhattacharya and Sahoo 2017; Bhattacharya 2016, 2018), the displacement-based finite element analysis is also useful in solving the complex boundary conditions (e.g. Shiau and Yu 2000; Shiau and Smith 2006; Shiau et al. 2006a; b, 2018; Shiau and Watson 2008; Halder and Manna 2020; Pirastehfar et al. 2020; Chakraborty and Goswami, 2021; Chatterjee and Murali Krishna 2021; Keawsawasvong and Ukritchon, 2017b; Hamouma et al. 2021; Huynh et al. 2022a, b; Ukritchon et al. 2017a, b).

The application of the NGI-ADP model has been used to analyze several geotechnical problems such as in Zhang et al. (2021), Li et al. (2021), and Langford et al. (2021) for the deep excavation problem; Li and Zhang (2020) for the pile response problem; Aamodt et al. (2021) for the slope stability problem; and Zhang et al. (2020) for the tunnel stability problem. Moreover, the anisotropy effect of soils was investigated for the works related to passive failure modes of plate anchors as well as foundations by Keawsawasvong et al. (2021), Keawsawasvong and Lawongkerd (2021), Nguyen et al. (2021), and Lai et al. (2022). Nevertheless, very few studies were reported in the literature in relation to the bearing capacity of ring footings with the combined effects of clay inhomogeneity and strength anisotropy using the NGI-ADP model.

In this paper, the NGI-ADP model is adopted to investigate the bearing capacity factor and failure mechanism of ring foundations resting on anisotropic and heterogeneous clays. The effects of the three dimensionless input parameters, namely the ratio of the inner to outer radius of the ring foundation, the increasing shear strength gradient ratio, and the anisotropic strength ratio of the cohesive clay on the performance of the ring foundation are investigated. The final outcomes are presented using design tables, charts, and an empirical equation. Due to the complex effect of each parameter on the bearing capacity factor, the multivariate adaptive regression splines (MARS) model is used to examine the sensitivity of each parameter using the output results, as well as to build a correlation equation between the multi-input parameters and output results. The tools provided in this paper would contribute to practical designs of ring footings resting on anisotropic and heterogeneous clays.

Shown in Fig. 1 is the problem definition of a rigid ring footing resting on anisotropic and heterogeneous clay. The soil anisotropy is simulated using the NGI-ADP constitutive model, whilst the heterogeneous clay is studied with the undrained shear strength linearly increasing with depth. The rigid ring footing is subjected to a uniform pressure q and has an external radius r_o and an internal radius r_i.

The main soil parameters of the NGI-ADP model are divided into 2 groups: stiffness and strength. The stiffness parameters include \(G_{ur} /s_{u}^{A}\) (ratio of unloading/reloading shear modulus over the “active” undrained shear strength), \(\upsilon_{u}\) (Poisson’s ratio), and \(\gamma_{f}^{C} ,\gamma_{f}^{E} , \gamma_{f}^{DSS}\) (shear strain at failure in triaxial compression, triaxial extension, and direct simple shear respectively). The strength parameters include \(s_{u}^{A}\) (active undrained shear strength), \(s_{u}^{P}\) (passive undrained shear strength), and \(s_{u}^{DSS}\) (direct simple undrained shear strength). Accordingly, the strength ratios required in the program input are \(s_{u}^{P} /s_{u}^{A}\), \(s_{u}^{DSS} /s_{u}^{A}\), \(_{0} /s_{u}^{A}\) (initial mobilization), \(s_{u}^{C,TX} /s_{u}^{A}\) (ratio of triaxial compressive shear strength over active shear strength), and the soil unit weight \(\gamma\). The ratio of direct simple shear strength over the active shear strength, i.e., \(s_{u}^{DSS} /s_{u}^{A} = \frac{{2s_{u}^{P} }}{{s_{u}^{A} }}/\left( {1 + s_{u}^{P} /s_{u}^{A} } \right)\), is assumed to be a harmonic mean (Krabbenhoft et al. 2019; Keawsawasvong et al. 2022). Figure 2 shows the failure criterion of the NGI–ADP model in the π—plane. Detailed descriptions of the model are stated in Grimstad et al. (2012), Ukritchon and Boonyatee (2015) and Brinkgreve and Vermeer (2019) and will not be repeated here.

Following the discussion of the anisotropic soil model, heterogeneous clay is expressed by using the three anisotropic undrained shear strengths that are linearly increasing with depth. They are presented in Eqs. (1)–(3).where \(s_{u0}^{A} , s_{u0}^{DSS} , s_{u0}^{P}\) are the anisotropic undrained shear strengths at the ground level, ρ is the linear-gradient with a unit of kPa/m (per meter depth), z indicates the depth determined from the ground surface (see Fig. 1). Note that these undrained shear strengths \(s_{u}^{A} , s_{u}^{DSS} , s_{u}^{P}\) can be obtained from three modes of shearing, including undrained tests of triaxial compression (for \(s_{u}^{A}\)), triaxial extension (for \(s_{u}^{P}\)), and direct simple shear (for \(s_{u}^{DSS}\)). For typical cohesive soils, undrained shearing strength of triaxial compression, \(s_{u}^{A}\) is the largest, followed by direct simple shear, \(s_{u}^{DSS}\), and triaxial extension, \(s_{u}^{P}\), which is the lowest.

A dimensionless bearing capacity factor N is defined as the ratio of the uniform pressure q over \(s_{u0}^{A}\), and it is a function of the three dimensionless design parameters, namely the ratio of the inner and outer radii \(r_{i} /r_{0}\), the strength gradient ratio \(m = \rho r_{0} /s_{u0}^{A}\), and the anisotropic ratio \(r_{e} = s_{u}^{P} /s_{u}^{A}\). This is shown in Eq. (4)

The finite element model with axial symmetry is adopted to investigate the problem, as shown in Fig. 3, by using Plaxis2D v20 (Brinkgreve and Vermeer 2019). The ring footing is modelled as rigid plates subjecting to a uniform pressure q. The soils are simulated by using 15-noded triangular elements with the NGI-ADP soil model. The model size was carefully chosen so that the overall velocity field would not interfere the boundary and the effect on the produced results can be minimized. This is shown in Fig. 3. The bottom boundary is fixed in x, y-directions, the right-hand boundary is fixed only in the x-direction, the left-hand boundary is an axisymmetric line, and the ground surface is free. The ranges of parameters chosen for the study are: (1) \(r_{i} /r_{0} =\) 0, 0.25, 0.33, 0.5, and 0.75; (2) \(m = \rho r_{0} /s_{u0}^{A}\) = 0, 1, 2.5, 5, and 15; (3) \(s_{u}^{P} /s_{u}^{A}\) = 0.4, 0.5, 0.6, 0.7, 0.8, and 1.0. It is to be noted that \(s_{u}^{P} /s_{u}^{A}\) = 1 indicates an isotropic clay. The range of parameters of \(\frac{{r_{i} }}{{r_{0} }} {\text{and}} m\) was selected based on work of Lee et al. (2016), Benmebarek et al. (2017). According to D’Ignazio et al. (2017) and Hansen and Clough (1981), typical values of \(s_{u}^{P} /s_{u}^{A}\) ratios vary from 0.3 to 0.81. Following the recommendation in the FE Plaxis code, the values of \(s_{u}^{C,TX} /s_{u}^{A} \) = 0.99, \(_{0} /s_{u}^{A}\) = 0.7, \(\upsilon_{u}\) = 0.495 are adopted for all analyses in the paper. The selected \(E_{u} /s_{u}^{A}\) ratio follows a previous stability analysis using FEA (Ukritchon et al. 2017a, b), such that there is very small to none effect of this parameter on the limit load of this stability problem. It should be noted that \(_{0}\) is initial in situ maximum shear stress and \(E_{u}\) is undrained young modulus.

To improve the confidence in later parametric analyses, numerical results from finite element analysis (FEA) must be compared with published results (Shiau et al. 2014, 2016a, b, 2017). This is done by comparing with those in Lee et al. (2016) and is presented in Fig. 4 for cases of ring footing on isotropic clay (where the anisotropic ratio \(s_{u}^{P} /s_{u}^{A}\) = 1) and heterogeneous clay (the increase shear gradient ratio m = 0, 1, 2.5, 5, and 15). The comparison study has shown that the present FEA results of bearing capacity factor N are in good agreement with those in Lee et al. (2016) and that the current model can be further be used to study the anisotropic effect with reasonable confidence.

With the success of model validation, the next task is to study the effects of the ratio of inner and outer radius (\(r_{i} /r_{0}\)), the anisotropic ratio (\(s_{u}^{P} /s_{u}^{A}\)), and the shear strength gradient ratio (m) on the bearing capacity factor (N). Figure 5 shows the linear relationships of the anisotropic ratio (\(s_{u}^{P} /s_{u}^{A}\)) on the bearing capacity factor N for the various cases of m and r_i/r_o. Noting that (\(s_{u}^{P} /s_{u}^{A}\) = 1) indicates an isotropic clay, a decrease in anisotropic ratio (\(s_{u}^{P} /s_{u}^{A}\)) means an increased effect of soil anisotropy. For the various m considered in Fig. 5 (m = 0, 1, 2.5, 15), numerical results have shown that a decrease in (\(s_{u}^{P} /s_{u}^{A}\)) results in a decrease in the bearing capacity factor N. They have also shown that the bearing capacity factor N decreases with the increasing \(r_{i} /r_{0}\). Noting that the larger the \(r_{i} /r_{0}\), the smaller the footing contact area, it is therefore not surprised to see the decreased bearing capacity owing to the reduced footing area (increasing \(r_{i} /r_{0}\)). Besides, it was found that the rate of decrease is more pronounced for larger values of the shear strength gradient ratio m.

Using the same results, Fig. 6 shows a nonlinear relationship of the shear strength gradient ratio (m) on the bearing capacity factor N_. The greater the value of m, the larger the N is. The rate of increase (the gradient) in N decreases as the ratio of inner and outer radius r_i/r_o increases (less footing area).

The potential failure mechanisms of ring footings in anisotropic and heterogeneous clay are investigated in Figs. 7, 8 and 9. The influence of the anisotropic ratio (\(s_{u}^{P} /s_{u}^{A}\)) on the potential failure mechanisms of ring footings is presented in Fig. 7 for the case of (m = 5 and r_i/r_o = 0.5). The results of the failure zone and shear band have indicated an unchanged mechanism despite the increase of the \((s_{u}^{P} /s_{u}^{A}\)). It means that the anisotropic ratio (\(s_{u}^{P} /s_{u}^{A}\)) has a little effect on the failure mechanism. More specifically speaking, it is true for the undrained clay only where the soil frictional angle is zero. The same observation was made by the recent studies of undrained anisotropic clay in Nguyen et al. (2021) and Keawsawasvong (2021). Despite this, it is not yet to be concluded for drained soils with non-zero frictional angles owing to the lack of published literatures.

A further failure mechanism study on the influence of shear gradient ratio (m) is shown in Fig. 8 for the case of (\(s_{u}^{P} /s_{u}^{A}\) = 0.5 and r_i/r_o = 0.5). The plots of failure mechanisms have shown that the overall failure zone is reduced in both horizontal and vertical directions as the value of shear strength gradient ratio (m) increases. The interference effect (overlapping) of the failure zone diminishes as m increases and the smallest failure zone occurs at the largest value of m = 15. For the more interesting study, the effects of inner and outer radius ratio (\(r_{i} /r_{0}\)) on the failure mechanism of ring footing are presented in Fig. 9 for the case (\(s_{u}^{P} /s_{u}^{A}\) = 0.5 and m = 5). The failure mechanism for a solid circular footing is firstly presented for \(r_{i} /r_{0} =\) 0. A ring foundation forms as \(r_{i} /r_{0}\) > 0, and the overlapping of failure zone diminishes as the value of \(r_{i} /r_{0} \) increases, resulting in a Prandtl type of general failure.

The multivariate adaptive regression splines (MARS) model is an automated regression modelling tool. Using several piecewise linear segments (splines) with differing gradients, the relationship between the input variable and output results can be established in multi-dimensions. Recently, the use of the MARS model as the machine learning method in analyzing geotechnical data has become more and more common. Lai et al. (2021a, b) adopted the MARS model to assess the impacts of input design parameters on the output ground movements due to the effects of installing twin caisson foundations. Zhang et al. (2017) proposed an empirical equation to determine horizontal wall displacement of braced excavation in cohesive soil. With five input parameters, the normalized maximum wall deflection was determined using the empirical equation generated by the MARS model. Other applications of the MARS model in geotechnical engineering are seen in the works of Zhou et al. (2021), Zhang et al. (2018), Zheng et al. (2019, 2020). Below is a brief description of the MARS model. More details can be found in Zhang (2019).

In the MARS model, the different splines are connected using a knot representing the end of one spline and the beginning of another. The fitted basic functions (BFs) have better flexibility to the studied model where the bends, thresholds, and other derivations from linear functions are allowed (Zhang, 2019). The basic function can be written as:where x is an input variable, and t is a threshold value.

MARS model produces BFs by searching in a stepwise process, of which the knot locations are automatically determined using the adaptive regression algorithm. MARS model is presented by a two-step procedure. The first (or called “forward”) step is to provide BFs as well as to find their potential knots to optimize the model performance and fitting accuracy. The second (or “backward”) step uses a pruning algorithm to delete the least effective terms, resulting in the generation of an optimal model used for the problem prediction. To build the correlation equation between the input and output variables, the MARS model combines all linear basic functions (BFs) which are described in Eq. (6), where a₀ is the constant, N is the number of BFs, g_n is the nth BF, a_n is the coefficient of g_n. Note that increasing the number of basic functions can increase the accuracy of the MARS model.

The current study uses the MARS model to investigate sensitivity analyses of each input variables (i.e., \(r_{i} /r_{0}\), \(s_{u}^{P} /s_{u}^{A}\) and m). The aim is to build a mathematical equation for predicting the N value, considering all effects of input variables. A training data set in the MARS model is selected from all FEA results of bearing capacity factors N. The corresponding 150 design combinations and input parameters are shown in Table 1.

To achieve the best accuracy, the chosen number of basic functions is varied to check the performance of MARS using two criteria of statistical analyses named Mean Squared Error (MSE) and the coefficient of determination (R² value). MSE represents the mean square error between the predicted output variables and the real output results. The lower MSE value, the better model can be obtained. The closer the value of R² is to 1, the better the linear regression fits the data. R² value equals 0.0 means that the model fails to predict real value, whilst 1.0 means that the forecast model is highly reliable.

Shown in Fig. 10, is the variation of MSE and R² due to the changes in the number of basic functions. Noting that the MSE decreases sharply as the number of BFs increases from 0 to 35, and it becomes constant for BFs > 35, though small. In contrast, the R² increases dramatically when the number of BFs increases from 0 to 30. R² becomes a constant (close to unity) for BFs > 30. It was decided that the number of BFs be chosen as 40 to perform sensitivity analyses as well as to generate the equation.

The sensitivity of each input variable is described by the relative important index (RII), as shown in Fig. 11. The value of RII indicates the weight of impaction. RII of 100% means that the respective input variable has the most important influence on the output results. This is shown for the increasing shear strength gradient ratio (m) in this study. The increasing shear gradient ratio has the most significant influence on the bearing capacity factor N. This is followed by the anisotropic ratio (\(s_{u}^{P} /s_{u}^{A}\)), and the ratio of inner and outer radius (\(r_{i} /r_{0}\)) with RII of 50.30% and 50.13% respectively. This finding has suggested that all the three investigated parameters play an important role in the design of ring foundations considering the anisotropic and heterogeneous behavior of cohesive soil.

Table 2 shows the basic functions and the mathematical equation generated by the MARS model. They can be written as Eq. (7).

To verify the proposed Eq. (7), a comparison between the bearing capacity factors N from FEA results and those from equation prediction is presented in Fig. 12. Numerical results have shown that the predicted N has an excellent fit to those from FEA—with the high value of R² = 99.99%. It can be concluded that Eq. (7) is an effective tool to estimate the bearing capacity of ring footings rest on anisotropic and heterogeneous clay.

Axisymmetric solutions for the bearing capacity of ring foundation resting on anisotropic and heterogenous clays have been successfully investigated in this paper using Plaxis finite element analysis and the NGI-ADP soil model—a widely used anisotropic soil model. The influences of inner and outer radius ratio (\(r_{i} /r_{0}\)), anisotropic ratio (\(s_{u}^{P} /s_{u}^{A}\)) and the shear strength gradient ratio (m) on the bearing capacity factor (N) and the failure mechanism of ring footings resting on anisotropic and heterogeneous clay are determined. The following conclusions are drawn based on the study results.