Bearing and uplift capacities of under-reamed piles in soft clay underlaid by stiff clay using lower-bound finite element limit analysis

Abstract

Ensuring a safe foundation design in soft clay is always a challenging task to engineers. In the present study, the effectiveness of under-reamed piles in soft clay underlaid by stiff clay is numerically studied using the lower-bound finite element limit analysis (LB FELA). The bearing and uplift capacities of under-reamed piles are estimated through non-dimensional factors N_cul and F_cul, respectively. These factors increased remarkably and marginally compared to N_cul and F_cul of the piles without bulbs when the bulb is placed in stiff and soft clay, respectively. For a given ratio of undrained cohesion of stiff to soft clay (c₂/c₁), the factors N_cul and F_cul moderately increased with the increase in the length-to-shaft-diameter ratio (L_u/D) and adhesion factors in soft clay (α_s1) and stiff clay (α_s2). The variation of radial stress along the pile–soil interface, distribution of axial force in the under-reamed piles, and state of plastic shear failure in the soil are also studied under axial compression and tension. The results of this study are expected to be useful for the estimation of the bearing and uplift capacities of under-reamed piles in uniform clay and soft clay underlaid by stiff clay.

Graphical abstract

Keywords

bearing capacity / uplift capacity / under-reamed pile / clay / limit analysis

Introduction

With the rapid industrialization and urbanization, the construction of structures, such as high-rise buildings, chimneys, transmission and satellite towers, heavy industrial foundations, water tanks, machine foundations, and bridges, has significantly increased. These structures produce high compressive loads on the foundations. Structures, such as transmission towers, also experience high tensile loads due to cable tension. Moreover, under high wind pressure, a significant amount of tensile and compressive loads are generated on the foundations of these structures. Due to the scarcity of lands, such structures are sometimes recommended to be constructed on soils with low bearing capacity, such as soft clays. Soft clay being underlaid by stiff clay is very common. The conventional foundation systems used in soft clay are raft foundation and pile foundation. Pile foundations are sometimes used along with raft foundations. In most cases, piles are extended up to the farm strata lying below soft clay. The bearing and uplift capacity of pile foundations can be further increased by providing bulbs. Piles with bulb-like projections are also called under-reamed piles. The bulb-shaped projections of under-reamed piles provide additional resistance to axial compression and tension. In the past, under-reamed piles were mostly used in expansive soils to resist forces caused by volume changes. Nowadays, under-reamed piles are also used in all types of soils.

Several studies have been conducted on piles without bulbs in clay under axial compression [1–8] and tension [9–14]. A few studies have also been performed to examine the ultimate capacity of under-reamed piles in clay under axial compression [15–26] and tension [18,19,24,26–28]. Essentially, among the studies on under-reamed piles under axial compression, Cooke and Whitaker [15] considered homogeneous soft clay; Mohan et al. [16] and Prakash and Sharma [18] considered pure clay (soft to stiff); Martin and DeStephen [17], Kurian and Srilakshmi [21], Watanabe et al. [22], Farokhi et al. [24], and Vali et al. [25] considered clayey soils with sand; Peter et al. [19] considered stiff clay underlaid by soft clay; Shrivastava and Bhatia [20] considered elastic soil; Kong et al. [23] considered soft clay underlaid by sand; and Kumar et al. [26] considered clay with linearly increasing undrained cohesion. Furthermore, among the studies on under-reamed piles under axial tension, Prakash and Sharma [18] considered pure clay (soft to stiff), Peter et al. [19] considered stiff clay underlaid by soft clay, Farokhi et al. [24] considered clayey soils with sand, Kumar et al. [26] and Khatri et al. [28] considered with linearly increasing undrained cohesion, and Golait et al. [27] considered homogeneous soft clay. Therefore, the available studies on under-reamed piles in soft clay focus on homogeneous soft clay, stiff clay underlaid by soft clay, and soft clay underlaid by sand. However, only a few studies [6,7] on piles without bulbs in soft clay underlaid by stiff clay are available, and the bearing and uplift capacities of under-reamed piles in soft clay underlaid by stiff clay have not yet been studied.

Therefore, in the present study, the bearing and uplift capacities of under-reamed piles in soft clay underlaid by stiff clay are estimated using the lower-bound finite element limit analysis (LB FELA). Lower- and upper-bound limit analyses give the range in which the actual collapse load may occur. The lower-bound values are more demanding than upper-bound values because they are conservative, so they are beneficial from the design point of view. Boundary value problems can be solved using the finite element method [29,30], meshfree methods [31,32], isogeometric analysis [33,34], deep neural networks [35,36], and nonlocal operator method [37]. However, limiting values (lower and upper bounds) cannot be obtained using these methods.

In the present study, the lower-bound formulations of Khatri and Kumar [6,14] are considered for the estimation of the collapse loads of under-reamed piles under axial compression and tension. The bearing and uplift capacities of under-reamed piles are presented through design charts and tables. The variation of radial stress, distribution of axial force, and state of plastic shear failure in the soil domain are also studied. The variables considered in the present analysis are the embedment length, ratio of undrained cohesion of stiff and soft clay, bulb position, and pile–soil adhesion factor. A MATLAB program is also developed for the computations.

Problem definition

A single under-reamed pile is embedded in soft clay underlaid by stiff clay. Two bulb positions are considered: bulb lying in stiff clay and bulb lying in soft clay. For the first position, the location of the bulb is at 0.55D_u from the base, which is the minimum bulb position required for placing the bucket of the under-reaming tool [29]. Here, D_u is the diameter of the bulb. For the second position, the location of the bulb is at 1.15D_u from the base, considering an under-ream angle (β) of 45° [21,38]. A pile without bulb with the same length and shaft diameter as the under-reamed pile is also considered. The under-reamed pile and pile without bulb are separately subjected to axial compression and tension. In axial compression, the bearing capacity of the under-reamed pile is estimated through a dimensionless factor, designated by N_cul.

(1)N_{\mathrm{c}\mathrm{u}\mathrm{l}}={\displaystyle \frac{Q_{\mathrm{c}\mathrm{l}}}{A_{\mathrm{s}}{c}_{\mathrm{1}}}},

where Q_cl is the ultimate collapse load under axial compression, A_s is the cross-sectional area of the shaft, and c₁ is the undrained cohesion of the upper layer (soft clay).

In axial tension, the uplift capacity of the under-reamed pile is estimated through a dimensionless factor, designated by F_cul.

(2)F_{\mathrm{c}\mathrm{u}\mathrm{l}}={\displaystyle \frac{Q_{\mathrm{u}\mathrm{l}}}{A_{\mathrm{s}}{c}_{\mathrm{1}}}},

where Q_ul is the ultimate collapse load under axial tension.

The assumptions considered in the present analysis are as follows: (i) the soil is rigid and perfectly plastic; (ii) the plastic deformation of soil is determined by an associated flow rule; and (iii) the failure of the soil is determined by the Mohr–Coulomb failure criteria. The LB FELA is based on the lower-bound theorem of the plasticity theory, which applies to rigid, perfectly plastic materials following an associated flow rule [39–41]. The Mohr–Coulomb failure criteria is widely used in the LB FELA [6,14,42,43]. In the case of pure clay in an undrained condition (φ = 0°), the expression of the Mohr–Coulomb failure criteria becomes the same as that of the Tresca failure criteria. It is also assumed that the self-weight of the soil is zero, following Khatri and Kumar [6,14] and Griffiths [44]. A fully bonded pile–soil interface is considered in axial compression and tension.

The variables considered in the present analysis are the embedment ratio (L_u/D), ratio of the undrained cohesion of stiff clay to soft clay (c₂/c₁), and adhesion factors for soft (α_s1) and stiff (α_s2) clay. The value of D_u/D is considered 2.5 [21,38], and L_u/D is varied from 5 to 20 [28]. The value of c₂/c₁ is varied from 1 to 100 following Chakraborty and Kumar [7]. α_s1 and α_s2 depend on c₁ and c₂. The adhesion factors in stiff and soft clays vary from 0.33–0.96 and 0.96–1, respectively [45]. Moreover, the adhesion factor in the case of the piles fully embedded in soft clay and partially embedded in stiff clay depends on the embedment length of the pile in stiff clay [46]. In this case, a single adhesion factor can be considered. Therefore, in the present study, two different values of adhesion factors are considered (α_s1 = α_s2 = 0.50 and 0.67). The value of α_b is considered 1 [7].

Details of the domain, boundary condition, and mesh

Because the present problem is axisymmetric about the centerline of the pile, a domain in the r-z plane is considered. Figures 1(a) and 1(c) show the typical domains and boundary conditions for the bulb in stiff clay and soft clay, respectively. The soft clay layer of thickness L₁ and undrained cohesion c₁ is underlaid by a stiff clay layer of thickness L₂ and undrained cohesion c₂. An under-reamed pile with length L_u, shaft diameter D, bulb diameter D_u, under-ream angle β, and bulb distance from the base h_b is fully embedded in soft clay and partially embedded in stiff clay. The horizontal extent of the domain from the pile shaft and the vertical extent of the domain below the pile base are represented by L_h and L_v, respectively. The extent of the domain in the horizontal and vertical directions is that the domain size does not affect the total collapse load and failure mechanism. In axial compression, L_h and L_v are 31.37D–78.50D and 10.23D–28.72D, respectively. In axial tension, L_h and L_v are 38.50D–89.50D and 10.23D–28.72D, respectively. The stress boundary conditions along the axis of symmetry and ground surface are given by (1) τ_rz = 0 and (2) σ_z = 0 and τ_rz = 0, respectively. The stress boundary conditions along the pile–soil inter-faces are given by (1) |τ_rz|≤α_s2c₂ for the shaft in stiff clay, (2) |τ_rz|≤α_s1c₁ for the shaft in soft clay, (3) |τ_nt|≤α_s2c₂ for the bulb in stiff clay, (4) |τ_nt|≤α_s1c₁ for the bulb in soft clay, and (5) |τ_rz|≤α_bc₂ for the pile base. α_s1 and α_s2 denote the pile–soil adhesion factor of the soft and stiff clay, respectively; α_b denotes the adhesion factor along the pile base; and τ_nt denotes the tangential stress along the bulb surface. Three nodded triangular elements are used in the finite element mesh. A mesh refinement analysis is performed to obtain the size and number of elements. Small elements are used adjacent to the pile shaft, base, bulb, and layer interface to obtain a better stress distribution. The typical finite element meshes for the bulb in stiff clay and the bulb in soft clay are shown in Figs. 1(b) and 1(d), respectively.

Objective function

The objective functions for the axial compression and tension are obtained by integrating stresses along the pile–soil interface. For a fully bonded pile–soil interface, the resistances in the axial compression and tension are obtained from the base, shaft, and bulb. The total collapse load under axial compression (Q_cl) is expressed through Eqs. (3)–(6), and the total collapse load under axial tension (Q_ul) is expressed through Eqs. (7)–(10). A linear variation of stresses is assumed between two consecutive nodes along the pile–soil interface.

(3)Q_{\mathrm{c}\mathrm{l}}=Q_{\mathrm{c}\mathrm{l},\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}+Q_{\mathrm{c}\mathrm{l},\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{f}\mathrm{t}}+Q_{\mathrm{c}\mathrm{l},\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{b}},

(4)Q_{\mathrm{c}\mathrm{l},\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}={\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}}(-{\sigma }_z}\left)\mathrm{2}\mathrm{\pi }\right({\displaystyle \frac{r_i+r_{i+\mathrm{1}}}{\mathrm{2}}})\mathrm{d}{r}_i,

(5)Q_{\mathrm{c}\mathrm{l},\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{f}\mathrm{t}}={\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{s},1}}({\tau }_{rz}})\mathrm{2}\mathrm{\pi }{r}_{\mathrm{l},i}\mathrm{d}{z}_i+{\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{s},2}}({\tau }_{rz}})\mathrm{2}\mathrm{\pi }{r}_{\mathrm{l},i}\mathrm{d}{z}_i,

(6)Q_{\mathrm{c}\mathrm{l},\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{b}}={\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{b}\mathrm{s},1}}\left[\right(-{\sigma }_z\left)\mathrm{2}\mathrm{\pi }\right({\displaystyle \frac{r_{\mathrm{u},i}+r_{\mathrm{l},i}}{\mathrm{2}}})\mathrm{d}{r}_i+({\tau }_{rz}\left)\mathrm{2}\mathrm{\pi }{r}_{\mathrm{u},i}\mathrm{d}{z}_i\right]}+{\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{t}\mathrm{s},1}}\left[\right({\sigma }_z\left)\mathrm{2}\mathrm{\pi }\right({\displaystyle \frac{r_{\mathrm{u},i}+r_{\mathrm{l},i}}{\mathrm{2}}})\mathrm{d}{r}_i+({\tau }_{rz}\left)\mathrm{2}\mathrm{\pi }{r}_{\mathrm{l},i}\mathrm{d}{z}_i\right],}

(7)Q_{\mathrm{u}\mathrm{l}}=Q_{\mathrm{u}\mathrm{l},\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}+Q_{\mathrm{u}\mathrm{l},\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{f}\mathrm{t}}+Q_{\mathrm{u}\mathrm{l},\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{b}},

(8)Q_{\mathrm{u}\mathrm{l},\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}={\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}}({\sigma }_z}\left)\mathrm{2}\mathrm{\pi }\right({\displaystyle \frac{r_i+r_{i+\mathrm{1}}}{\mathrm{2}}})\mathrm{d}{r}_i,

(9)Q_{\mathrm{u}\mathrm{l},\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{f}\mathrm{t}}={\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{s},1}}(-{\tau }_{rz}})\mathrm{2}\mathrm{\pi }{r}_{\mathrm{l},i}\mathrm{d}{z}_i+{\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{s},2}}(-{\tau }_{rz}})\mathrm{2}\mathrm{\pi }{r}_{\mathrm{l},i}\mathrm{d}{z}_i,

(10)Q_{\mathrm{u}\mathrm{l},\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{b}}={\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{b}\mathrm{s},1}}\left[\right({\sigma }_z\left)\mathrm{2}\mathrm{\pi }\right({\displaystyle \frac{r_{\mathrm{u},i}+r_{\mathrm{l},i}}{\mathrm{2}}})\mathrm{d}{r}_i+(-{\tau }_{rz}\left)\mathrm{2}\mathrm{\pi }{r}_{\mathrm{u},i}\mathrm{d}{z}_i\right]}+{\displaystyle \sum _{i=\mathrm{1}}^{e_{\mathrm{t}\mathrm{s},1}}\left[\right(-{\sigma }_{\mathrm{z}}\left)\mathrm{2}\mathrm{\pi }\right({\displaystyle \frac{r_{\mathrm{u},i}+r_{\mathrm{l,}i}}{\mathrm{2}}})\mathrm{d}{r}_i+(-{\tau }_{rz}\left)\mathrm{2}\mathrm{\pi }{r}_{\mathrm{l},i}\mathrm{d}{z}_i\right],}

where Q_cl,base and Q_ul,base denote the base resistance from the bulb under axial compression and tension, respectively; Q_cl,shaft and Q_ul,shaft denote the shaft resistance under axial compression and tension, respectively; Q_cl,bulb and Q_ul,bulb denote the resistance from the bulb under axial compression and tension, respectively; e_base denotes the total edges at the base; r_i and r_i+1 denote the radial distance of the ith and (i+1)th nodes of the ith edge along the base measured from the pile center; e_s,1 and e_s,2 denote the total edges along the shaft below the bulb and shaft above the bulb, respectively; e_bs,1 and e_ts,1 denote the total edges along the bottom and top surfaces of the bulb, respectively; r_u,i and r_l,i denote the radial distance of the upper and lower nodes of the ith edge along the shaft, bottom, and top surfaces of the bulb; dr_i denotes the radial distance between two end nodes of the ith edge along the base and surface of the bulb; and dz_i denotes the vertical distance between two end nodes of the ith edge along the shaft and bottom and top surfaces of the bulb.

Validation of the lower-bound model

To check the accuracy of the lower-bound model, the present results are compared with the lower-bound model results of Khatri and Kumar [14], lower- and upper-bound model results of Salgado et al. [47] and Martin and Randolph [48], and finite element method results of Nguyen [49] Table 1 shows the comparison of the present Q_cl,base/(A_sc₁) results with those of Salgado et al. [47], Martin and Randolph [48], and Nguyen [49] for piles without bulb under axial compression. These values are obtained for c₂/c₁ = 1, adhesion factor at the pile base, α_b = 1, and adhesion factor along shaft, α_s = 0. The present lower-bound values are bracketed between the lower- and upper-bound values of Salgado et al. [47] and Martin and Randolph [48]. The present lower-bound values are in good agreement with the finite element method results of Nguyen [49]. For axial tension, the present Q_ul,base/(A_sc₁) values are compared with the lower-bound values of Khatri and Kumar [14] for piles without bulbs (i.e., c₂/c₁ = 1, α_b = 1, and α_s = 1), as shown in Table 2. In Table 2, the present lower-bound values are very close to the lower-bound values of Khatri and Kumar [14]. The above difference may arise due to the difference in meshes used.

Results

Variations of N_cul and F_cul

The variations of N_cul and F_cul with c₂/c₁ for different L_u/D are shown in Figs. 2 and 3, respectively. For a given L_u/D, N_cul increases with an increase in c₂/c₁ for all the cases, i.e., the pile without a bulb, the bulb in stiff clay, and the bulb in soft clay. For a given L_u/D, N_cul for the bulb in stiff clay is significantly higher than that for the bulb in soft clay, and N_cul for the bulb in soft clay is marginally higher than that for the pile without a bulb. For c₂/c₁ = 100 and L_u/D = 20, the percent increase in N_cul for the bulb in stiff clay and the bulb in soft clay with respect to the pile without bulb are 218.59% and 2.99%, respectively. For c₂/c₁ = 1, N_cul is unaffected by the change in the bulb position. Moreover, for c₂/c₁ = 1, N_cul for the pile with bulb is significantly higher than N_cul for the pile without bulb. For c₂/c₁ = 1, L_u/D = 20, and α_s1 = α_s2 = 0.67, the percent increase in N_cul with respect to the pile without bulb is 70.40%. For a given c₂/c₁, N_cul moderately increases with an increase in L_u/D.

The variations of F_cul with c₂/c₁ are similar to the variations of N_cul with c₂/c₁. However, the values of F_cul are marginally smaller than N_cul. F_cul is the highest when the bulb is located in stiff clay. For c₂/c₁ = 100 and L_u/D = 20, the percent increase in F_cul for the bulb in stiff clay and the bulb in soft clay with respect to the pile without bulb are 197.67% and 3.49%, respectively. For c₂/c₁ = 1, L_u/D = 20, and α_s1 = α_s2 = 0.67, the percent increase in F_cul with respect to the pile without bulb is 74.56%. In Figs. 2 and 3, α_s1 and α_s2 are 0.67.

Moreover, for the bulb in stiff clay, N_cul and F_cul are computed for α_s1 = α_s2 = 0.50 and compared with N_cul and F_cul for α_s1 = α_s2 = 0.67, as shown in Tables 3 and 4, respectively. The factors N_cul and F_cul for α_s1 = α_s2 = 0.50 are smaller than N_cul and F_cul for α_s1 = α_s2 = 0.67. In addition, N_cul and F_cul for α_s1 = α_s2 = 0.50 are suitable for higher c₂/c₁, and N_cul and F_cul for α_s1 = α_s2 = 0.67 are suitable for medium to lower c₂/c₁.

Variation of radial stress along the pile–soil interface

The radial stress around the pile foundation plays an important role in the shaft capacity and failure of the soil mass around the pile [50,51]. The radial stress at the pile–soil interface is very significant because it affects the shaft capacity. Therefore, the normalized radial stress (σ_r/c₁) at the pile–soil interface is obtained for various pile and soil parameters, as shown in Fig. 4. In the case of a pile without bulb under axial compression, for c₂/c₁ = 1, the normalized radial stress increases with depth and reaches a maximum value at the pile base (Fig. 4(a)). In the case of an under-reamed pile under axial compression, for c₂/c₁ = 1, the normalized radial stress increases and decreases abruptly below and above the bulb, respectively (Fig. 4(a)). In the case of axial tension, for c₂/c₁ = 1, the normalized radial stress is tensile near the ground surface and base of the pile and compressive in the remaining portion in all the cases, i.e., bulb in stiff clay, bulb in soft clay, and without bulb (Fig. 4(d)). Moreover, in axial tension, the normalized radial stress increases and decreases abruptly above and below the bulb, respectively. In all the cases, for c₂/c₁>1, the normalized radial stress increases marginally in the soft clay layer and significantly in the stiff clay layer under axial compression and tension (Figs. 4(b) and 4(e)). For c₂/c₁>1, the normalized radial stress in the bulb in stiff clay is significantly higher than that in the other two cases under axial compression and tension (Figs. 4(b) and 4(e)). For a given c₂/c₁, the normalized radial stress in the bulb in stiff clay increases marginally with the increase in L_u/D under axial compression and tension (Figs. 4(c) and 4(f)). This finding implies that the stresses in weightless soil are independent of the depth of embedment. Moreover, for high c₂/c₁, a major portion of the load is carried by the shaft and bulb embedded in stiff clay. Therefore, high stresses are induced in the stiff clay layer to mobilize the high shear strength of the soil.

Distribution of the axial force

The distributions of the normalized axial force in the under-reamed pile under axial compression (Q_cl/A_sc₁) and tension (Q_ul/A_sc₁) are obtained for various parameters, as shown in Fig. 5. In the pile without bulb, for c₂/c₁ = 1, the normalized axial force increases uniformly from the base to the top of the pile under axial compression and tension (Figs. 5(a) and 5(d)). In the under-reamed pile, for c₂/c₁ = 1, the normalized axial force increases immediately at the locations of the bulbs, and the ultimate axial load at the pile top also increases marginally with the decrease in the bulb distance from the base (h_b/D_u) (Figs. 5(a) and 5(d)). In all the cases, i.e., for the pile without bulb, bulb in stiff clay, and bulb in soft clay, where c₂/c₁>1, the normalized axial force increases significantly in the stiff clay layer and marginally in the soft clay layer (Figs. 5(b) and 5(e)). This finding implies that both piles without bulb and the under-reamed pile behave as an end-bearing pile if the bottom portion of the pile is embedded in stiff clay. A significant increase in the ultimate axial load is observed for the bulb located in stiff clay compared to the bulb in soft clay and pile without bulb under axial compression and tension. For the bulb in stiff clay, the normalized axial force moderately increases with the increase in L_u/D under axial compression and tension (Figs. 5(c) and 5(f)). The load carried by the bulb and hence the total load (under axial compression and tension) carried by the under-reamed pile increases with the increase in c₂/c₁.

Failure patterns

The state of plastic shear failure in the soil domain is obtained through the plots of a/d, r/d, and z/d, where a = [(σ_r–σ_z)²+4τ_rz²] and d = 4c² are the components of the Mohr–Coulomb yield criteria [6,14]. The ratio a/d varies from 0 to 1, where a/d of 1 indicates that the plastic shear failure has occurred in the domain and a/d of less than 1 represents that the soil is in the elastic state. Figures 6 and 7 show the plastic shear failure plots for the pile without bulb and under-reamed pile under axial compression and tension, respectively. In these figures, the lightest color indicates a/d = 0, and the deepest color indicates a/d = 1. In the pile without bulb under axial compression and tension, the failure zone starts below the pile base and extends up to the ground surface. In the under-reamed pile under axial compression and tension, the failure zone starts below the bulb and base and extends up to the ground level. In all the cases, i.e., without bulb, bulb in stiff clay, and bulb in soft clay, the horizontal and vertical extents of the failure zone decrease with the increase in c₂/c₁ under axial compression (Figs. 6(a)–6(f)) and tension (Figs. 7(a)–7(f)). The horizontal extent of the failure zone for the bulb in soft clay is slightly higher than that for the bulb in stiff clay under axial compression (Figs. 6(d) and 6(f)) and tension (Figs. 7(d) and 7(f)). For a given c₂/c₁, the horizontal and vertical extents of the failure zone decrease with the decrease in L_u/D under axial compression (Figs. 6(g) and 6(h)) and tension (Figs. 7(g) and 7(h)). The horizontal extent of the failure zone under axial tension is higher than that under axial compression. From the zoomed view of the failure zone near the bulb under axial compression (Figs. 6(g) and 6(h)) and tension (Figs. 7(g) and 7(h)), the failure of the soil clearly takes place in the region near the bulb under axial compression and tension. From the radial stress and axial force distributions, stresses are significantly higher in the region near the bulbs. The high stresses in the region near the bulb caused an increase in a/d, resulting in the failure of the soil mass in that region. In the future, the failure patterns can be compared with other numerical techniques, such as the phase field model [52–57], which is based on fracture propagation.

Comparisons

The present results are compared with the results in the available literature on piles without bulbs and under-reamed piles, as shown in Tables 5–7. Table 5 shows the comparison of the normalized bearing capacity (Q_cl/A_sc₂) with those in the study of Chakraborty and Kumar [7] for piles without bulbs, where α_s1 = α_s2 = 0.67 and α_b = 1. From Table 5, it is conclusive that the present values are very close to the values of Chakraborty and Kumar [7]. For the under-reamed pile under axial compression, the computed bearing capacities are compared with those in the studies of Cooke and Whitaker [15] and Mohan et al. [16], as shown in Table 6. The present values are computed considering the fully bonded interface and assuming the immediate breakaway at the top surface of the bulb. The present values for the immediate breakaway condition match well with those in the study of Cooke and Whitaker [15] and slightly lower than those in the study of Mohan et al. [16]. Particularly, Cooke and Whitaker [15] and Mohan et al. [16] neglected the resistance from the top surface of the bulb for the estimation of the bearing capacity.

For the under-reamed pile under axial tension, the computed uplift capacities are compared with those in the studies of Golait et al. [27] and Khatri et al. [28], as shown in Table 7. The present values are obtained for the fully bonded interface and immediate breakaway at the base and bottom surface of the bulb. The present values for the immediate breakaway condition are in good agreement with those in the study of Golait et al. [27]. The uplift capacities in the study of Khatri et al. [28] are higher than the present values for the fully bonded and immediate breakaway conditions; however, the difference is less for the fully bonded condition. Of note, Khatri et al. [28] considered the weight of the pile. However, in the present analysis, the weight of the pile is not considered. Moreover, the collapse load was not well defined in the study of Khatri et al. [28], obtained from the load–deformation response when the increase in the load is not significant to the increase in the deformation. For D = 0.3 m, L_u/D = 15, h_b/D_u = 0.55, c₁ = c₂ = 15 kPa, α_s1 = α_s2 = 1, and unit weight, γ = 16 kN/m³, the ultimate collapse load from the present analysis and that of Kumar et al. [26] are equal to 123.66 and 134.53 kN, respectively. The difference between the present analysis and that of Kumar et al. [26] may be attributed to the approximate estimation of the collapse load, assumption of the elastic pile and elastic–plastic soil, and consideration of the self-weight of the pile in the study of Kumar et al. [26].

Conclusions

In the present study, the non-dimensional bearing (N_cul) and uplift (F_cul) capacity factors are estimated using the LB FELA. The variation of normalized radial stress along the pile–soil interface, distribution of normalized axial force in the under-reamed pile, and proximity of the plastic shear failure in the soil domain are also obtained. The major conclusions of the present study are listed below.

1) The non-dimensional bearing capacity factor N_cul and uplift capacity factor F_cul increase significantly when the bulb is placed in the stiff clay. In the uniform clay (c₂/c₁ = 1), the factors N_cul and F_cul for the under-reamed piles are significantly higher than N_cul and F_cul for those without bulb. For a given c₂/c₁, the factors N_cul and F_cul increase moderately with increases in L_u/D, α_s1, and α_s2.

2) In axial compression, the radial stress increases and decreases significantly below and above the bulb, respectively. In axial tension, the radial stress increases and decreases significantly above and below the bulb, respectively. The radial stress at the pile–soil interface increases significantly in the stiff clay layer for the pile with and without bulb. The radial stress is the highest for the bulb in stiff clay under axial compression and tension.

3) In the axial force distributions, the bulb carries a major portion of the ultimate load under axial compression and tension. The load carried by the bulb and the ultimate load carried by the under-reamed pile (under axial compression and tension) increase significantly when the bulb is placed in stiff clay.

4) The failure zone originates below the base in the case of pile without bulb and below the base and bulb in the case of under-reamed pile and extends up to the ground surface under axial compression and tension. The extent of the failure zone under axial tension is higher than that under axial compression. The vertical and horizontal extents of the failure zone decrease with the increase in c₂/c₁ under axial compression and tension.

Nonetheless, the present study provides only the lower-bound (safe) estimate of the bearing and uplift capacities of under-reamed piles. In the future, further studies may be conducted to determine the upper-bound bearing and uplift capacities of under-reamed piles. The lower- and upper-bound solutions can be considered the bracketed values of the actual collapse load.

References

[1] Meyerhof G G. Some recent research on the bearing capacity of foundations. Canadian Geotechnical Journal, 1963, 1(1): 16–26

[2] Coyle H M, Reese L C. Load transfer for axially loaded piles in clay. Journal of the Soil Mechanics and Foundations Division, 1966, 92(2): 1–26

[3] Vesic A S. A Study of Bearing Capacity of Deep Foundations. Final Report, Project B-189. Atlanta: Georgia Institute of Technology, 1967

[4] Poulos H G. The influence of shaft length on pile load capacity in clays. Geotechnique, 1982, 32(2): 145–148

[5] Zhou H, Chen Z. Analysis of effect of different construction methods of piles on the end effect on skin friction of piles. Frontiers of Architecture and Civil Engineering in China, 2007, 1(4): 458–463

[6] Khatri V N, Kumar J. Bearing capacity factor N_c under φ = 0 condition for piles in clays. International Journal for Numerical and Analytical Methods in Geomechanics, 2009, 33(9): 1203–1225

[7] Chakraborty D, Kumar J. Bearing capacity of piles in soft clay underlaid by cohesive frictional soil. International Journal of Geomechanics, 2013, 13(3): 311–317

[8] Ismail A. ANN-based empirical modelling of pile behaviour under static compressive loading. Frontiers of Structural and Civil Engineering, 2018, 12(4): 594–608

[9] Meyerhof G G, Adams J I. The ultimate uplift capacity of foundations. Canadian Geotechnical Journal, 1968, 5(4): 225–244

[10] Das B M. A procedure for estimation of ultimate uplift capacity of foundations in clay. Soil and Foundation, 1980, 20(1): 77–82

[11] Das B M, Seeley G R. Uplift capacity of pipe piles in saturated clay. Soil and Foundation, 1982, 22(1): 91–94

[12] Shin E C, Das B M, Puri V K, Yen S C, Cook E E. Ultimate uplift capacity of model rigid metal piles in clay. Geotechnical and Geological Engineering, 1993, 11(3): 203–215

[13] Veeresh C, Rao S N. Vertical pullout capacity of model batter anchor piles in marine clays. Marine Georesources and Geotechnology, 1996, 14(3): 205–215

[14] Khatri V N, Kumar J. Uplift capacity of axially loaded piles in clays. International Journal of Geomechanics, 2011, 11(1): 23–28

[15] Cooke R W, Whitaker T. Experiments on model piles with enlarged bases. Geotechnique, 1961, 11(1): 1–13

[16] Mohan D, Murthy V N S, Jain G S. Design and construction of multi-under-reamed piles. In: Proceedings of the 7th International Conference on Soil Mechanics and Foundation Engineering. Mexico, 1969, 183–186

[17] Martin R E, DeStephen R. Large diameter double under-reamed drilled shafts. Journal of Geotechnical Engineering, 1983, 109(8): 1082–1098

[18] Prakash S, Sharma H D. Pile Foundations in Engineering Practice. New York: Wiley-Interscience, 1990

[19] Peter J A, Lakshmanan N, Devadas Manoharan P. Investigations on the static behavior of self-compacting concrete under-reamed piles. Journal of Materials in Civil Engineering, 2006, 18(3): 408–414

[20] Shrivastava N, Bhatia N. Ultimate bearing capacity of under-reamed pile-finite element approach. In: The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG). Goa, India, 2008, 1–6

[21] Kurian N P, Srilakshmi G. Studies on the geometrical features of under-reamed piles by the finite element method. Journal of Karunya University, 2010, 2(1): 1–14

[22] Watanabe K, Sei H, Nishiyama T, Ishii Y. Static axial reciprocal load test of cast-in-place nodular concrete pile and nodular diaphragm wall. Geotechnical Engineering Journal of the SEAGS & AGSSEA, 2011, 42(2): 11–19

[23] Kong G Q, Yang Q, Liu H L, Liang R Y. Numerical study of a new belled wedge pile type under different loading modes. European Journal of Environmental and Civil Engineering, 2013, 17(sup1): s65–82

[24] Farokhi A S, Alielahi H, Mardani Z. Optimizing the performance of under-reamed piles in clay using numerical method. Electronic Journal of Geotechnical Engineering, 2014, 19: 1507–1520

[25] Vali R, Mehrinejad Khotbehsara E, Saberian M, Li J, Mehrinejad M, Jahandari S. A three-dimensional numerical comparison of bearing capacity and settlement of tapered and under-reamed piles. International Journal of Geotechnical Engineering, 2019, 13(3): 236–248

[26] Kumar A, Khatri V N, Gupta S K. Effect of linearly increasing cohesion on the compression and uplift capacity of the under-reamed pile in clay. SN Applied Sciences, 2020, 2(2): 315

[27] Golait Y S, Padade A H, Cherian T. Prediction of quantitative response of under-reamed anchor piles in soft clay using laboratory model study. Journal of Testing and Evaluation, 2017, 46(2): 507–522

[28] Khatri V N, Kumar A, Gupta S K, Dutta R K, Gnananandarao T. Numerical study on the uplift capacity of under-reamed piles in clay with linearly increasing cohesion. International Journal of Geotechnical Engineering, 2019 (in press)

[29] Zienkiewicz O C, Taylor R L, Nithiarasu P, Zhu J Z. The Finite Element Method. London: McGraw-Hill, 1977

[30] Hughes T J R. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Mineola, NY: Dover Publications, 2000

[31] Rabczuk T, Belytschko T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. International Journal for Numerical Methods in Engineering, 2004, 61(13): 2316–2343

[32] Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H. A simple and robust three-dimensional cracking-particle method without enrichment. Computer Methods in Applied Mechanics and Engineering, 2010, 199(37-40): 2437–2455

[33] Nguyen V P, Anitescu C, Bordas S P, Rabczuk T. Isogeometric analysis: An overview and computer implementation aspects. Mathematics and Computers in Simulation, 2015, 117: 89–116

[34] Hughes T J, Sangalli G, Tani M. Isogeometric Analysis: Mathematical and Implementational Aspects, with Applications. Insplines and PDEs: From Approximation Theory to Numerical Linear Algebra. Cham: Springer, 2018, 237–315

[35] Anitescu C, Atroshchenko E, Alajlan N, Rabczuk T. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials & Continua, 2019, 59(1): 345–359

[36] Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh V M, Guo H, Hamdia K, Zhuang X, Rabczuk T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790

[37] Rabczuk T, Ren H, Zhuang X. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Computers. Materials and Continua, 2019, 59(1): 31–55

[38] IS 2911 Part III. Indian Standard Code of Practice for Design and Construction of Pile Foundations (Part–III): Under-Reamed Piles. 1st ed. New Delhi: Bureau of Indian Standards, 1980

[39] Chen W F, Liu X L. Limit Analysis in Soil Mechanics. Amsterdam: Elsevier Science, 1990

[40] Makrodimopoulos A, Martin C M. Lower bound limit analysis of cohesive-frictional materials using second-order cone programming. International Journal for Numerical Methods in Engineering, 2006, 66(4): 604–634

[41] Keawsawasvong S, Ukritchon B. Undrained stability of an active planar trapdoor in non-homogeneous clays with a linear increase of strength with depth. Computers and Geotechnics, 2017, 81: 284–293

[42] Bottero A, Negre R, Pastor J, Turgeman S. Finite element method and limit analysis theory for soil mechanics problems. Computer Methods in Applied Mechanics and Engineering, 1980, 22(1): 131–149

[43] Sloan S W. Lower bound limit analysis using finite elements and linear programming. International Journal for Numerical and Analytical Methods in Geomechanics, 1988, 12(1): 61–77

[44] Griffiths D V. Elasto-plastic analyses of deep foundations in cohesive soil. International Journal for Numerical and Analytical Methods in Geomechanics, 1982, 6(2): 211–218

[45] NAVFAC DM (Naval Facilities Engineering Command Design Manual) 7.2. Foundation and Earth Structures. Alexandria: U.S. Department of the Navy, 1984

[46] Tomlinson M J. Pile Design and Construction Practice. 4th ed. London: E and F N Spon, 1994

[47] Salgado R, Lyamin A V, Sloan S W, Yu H S. Two-and three-dimensional bearing capacity of foundations in clay. Geotechnique, 2004, 54(5): 297–306

[48] Martin C M, Randolph M F. Applications of the lower and upper bound theorems of plasticity to collapse of circular foundations. In: Proceedings of the 10th International Conference on Computer Methods and Advances in Geomechanics. Abingdon: Taylor and Francis, 2001, 2: 1417–1428

[49] Nguyen V Q. Numerical modelling of the undrained vertical bearing capacity of shallow foundations. Thesis for the Master’s Degree. Queensland: University of Southern Queensland, 2008

[50] Clark J I, Meyerhof G G. The behavior of piles driven in clay. An investigation of soil stress and pore water pressure as related to soil properties. Canadian Geotechnical Journal, 1972, 9(4): 351–373

[51] Randolph M F, Carter J P, Wroth C P. Driven piles in clay-the effects of installation and subsequent consolidation. Geotechnique, 1979, 29(4): 361–393

[52] Zhou S, Zhuang X, Rabczuk T. A phase-field modeling approach of fracture propagation in poroelastic media. Engineering Geology, 2018, 240: 189–203

[53] Zhou S, Rabczuk T, Zhuang X. Phase field modeling of quasi-static and dynamic crack propagation: COMSOL implementation and case studies. Advances in Engineering Software, 2018, 122: 31–49

[54] Zhou S, Zhuang X, Zhu H, Rabczuk T. Phase field modelling of crack propagation, branching and coalescence in rocks. Theoretical and Applied Fracture Mechanics, 2018, 96: 174–192

[55] Zhou S, Zhuang X, Rabczuk T. Phase-field modeling of fluid-driven dynamic cracking in porous media. Computer Methods in Applied Mechanics and Engineering, 2019, 350: 169–198

[56] Zhou S, Zhuang X, Rabczuk T. Phase field modeling of brittle compressive-shear fractures in rock-like materials: A new driving force and a hybrid formulation. Computer Methods in Applied Mechanics and Engineering, 2019, 355: 729–752

[57] Zhuang X, Zhou S, Sheng M, Li G. On the hydraulic fracturing in naturally-layered porous media using the phase field method. Engineering Geology, 2020, 266: 105306

RIGHTS & PERMISSIONS

Higher Education Press