Introduction

A ring foundation is commonly used to support axisymmetric structures such as silos, water towers, storage tanks, transmission towers, chimneys, and bridge piers. The use of this kind of foundation can reduce the amount of constructing material which results in a more economical design. To investigate the bearing capacity of ring foundations, Saha [1] and Saran et al. [2] performed experiments for determining the bearing capacity of ring foundations resting on a sand layer. Using the method of characteristics, Kumar and Ghosh [3] presented the plastic solutions for the bearing capacity factor of ring foundations with smooth and rough soil-structure interfaces. The finite difference method was also employed by Zhao and Wang [4] to derive the solutions of the vertical bearing capacity for ring footings on cohesionless soil. Benmebarek et al. [5] and Remadna et al. [6] used the FLAC code to numerically investigate the bearing capacity factors of smooth and rough ring foundations on soils by considering the effects of associated and non-associated flow rules. The bearing capacity factors including Nc, Nq, and Nγ for ring foundations were proposed by Kumar and Chakraborty [7] using the finite element limit analysis and Keshavarz and Kumar [8] using the method of characteristics. Tang and Phoon [9] employed the finite element limit analysis to examine the influence of the stress level effect on the bearing capacity of ring foundations on dense sand. For the cases of ring foundations on layered soils, some researchers have presented the bearing capacity of ring foundations resting on two-layered soils for the cases of two-layered clays [10], sand overlying clay [11], and weak sand overlying dense sand [12, 13]. In addition, various researchers have also investigated the bearing capacity of a ring or circular foundation resting on the surface of reinforced soil (e.g., [14,15,16,17,18]).

All aforementioned works were the studies regarding the bearing capacity of the ring foundation resting on the surface of soils in which the effect of embedded depth was not taken into account. An annular foundation with an embedded depth is more suitable and economical and can carry the load transferred from axisymmetric structures such as offshore structures. Benmebarek et al. [19] and Lee et al. [20] have proposed the solutions for the annular foundation embedded in clay with constant shear strength and linearly increasing shear strength with depth, respectively. However, the existing solutions by Benmebarek et al. [19] and Lee et al. [20] considered only the cases of annular foundations with the rough interface condition. The effects of the ratio between the internal radius and the external radius, the embedded depth ratio, and the adhesion factor at the soil-foundation interface have never been considered together in the literature.

Problem Definition

This paper aims to present new bearing capacity solutions for annular foundations embedded in clay with linearly increasing shear strength as shown in Fig. 1. The annular foundation has internal radius ri and external radius ro and is embedded in clay with the depth L. The shear strength of clay increases linearly with the depth from the ground surface, as given by Eq. (1).

$$s_{u} = s_{u0} + \rho z,$$
(1)
Fig. 1
figure 1

Problem geometry of a rigid annular foundation embedded in clay

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where su0 is the shear strength at the ground surface,

ρ is the linear strength gradient,

z is the depth measured from the ground surface.

This study considers only the case of su0 = 0, which corresponds to a normally consolidated clay in a very deep offshore area. As a result, the strength of clay at any depth can be expressed as su = ρz. Note that this assumption of increasing shear strength has been considered by several works in the past for solving the capacity of offshore foundations (e.g., [21,22,23,24,25]). It should be noted that several real cases elaborated that the shear strength represented by SPT, Vs, and CPT could increase with depth (e.g., [35, 36]).

It is well recognized that the magnitude of the bearing capacity of foundations is commonly affected by the roughness at the soil-foundation interface, which is defined by the adhesion factor (α). The definition of this factor is the ratio between the undrained shear strength at the soil-foundation interface and that of the surrounding soil, which can be expressed in Eq. (2).

$$\alpha = \frac{{s_{ui} }}{{s_{u} }},$$
(2)

where sui is the undrained shear strength at the soil-foundation interface, and su is the undrained shear strength of the surrounding soil. The complete range of the adhesion factor varies from 0 (fully smooth interface) to 1 (fully rough interface). Note that the existing solutions in the work by Lee et al. [20] are limited to the case of the rough interface (α = 1). This study considers the complete cases of α = 0, 0.25, 0.5, 0.75, and 1, which is the novel finding of this paper. It should be also noted that several previous works have proofed that the variation of the adhesion factor has a significant impact on the magnitude of the foundation capacity (e.g., [22,23,24,25,26,27,28,29]).

In this work, the embedded annular foundation has an internal radius ri and an external radius ro and is subjected to the vertical uniform pressure q applied at the top surface of the annular foundation. By integrating the vertical uniform pressure, the total vertical force Q can be represented as the relation to internal and external radii as shown in Eq. (3):

$$Q = \pi (r_{{\text{o}}}^{2} - r_{{\text{i}}}^{2} )q.$$
(3)

The end bearing capacity factor of the annular foundation is defined as q/ρL. To avoid the effect of the skin friction or the shaft resistances of the annular foundation on the end bearing capacity factor, the condition of the smooth interface (α = 0) is strictly imposed at the surrounding surfaces inside and outside of the annular foundation while all roughness conditions (e.g., α = 0, 0.25, 0.5, 0.75 and 1) are defined only at the base of the foundation. It can be summarized that three dimensionless input parameters have significant influences on the end bearing capacity factor of the annular foundation as shown in Eq. (4):

$$\frac{q}{\rho L} \propto \,f\left( {\frac{{r_{{\text{i}}} }}{{r_{{\text{o}}} }},\frac{L}{{r_{{\text{o}}} }},\alpha } \right),$$
(4)

where ri/ro is the radius ratio, L/ro is the depth ratio, α is the adhesion factor applied at the base of the annular foundation.

It should be noted that only the Nc factor (in this study defined as q/ρL) indicating the effect of the cohesion of clay is considered in the present study, where the Nq and Nγ factors indicating the effects of the overburden pressure and the unit weight of soil are not taken into account. As a result, all numerical models in the present study are set to be the cases of weightless soils and there is no surcharge applied at the ground surface of soils.

The finite element limit analysis (FELA) is carried out to derive the new solutions of the end bearing capacity factor of this problem. The FELA utilizes the optimization techniques in conjunction with finite element discretization techniques to provide the upper bound (UB) or lower bound (LB) solutions of the end bearing factor of the annular foundation. The formulations of UB and LB FELA are based on the plastic bound theorems cooperating with governing kinematic or equilibrium equations, respectively [30,31,32]. Using FELA, the end bearing factor can be accurately acquired by bracketing the results obtained from UB and LB methods in which the impacts of the radius ratio ri/ro, the depth ratio L/ro, and the adhesion factor at the base of the foundation α can be examined. Furthermore, the collapsed mechanisms of annular foundations embedded in clay with undrained shear strength increasing linearly with depth are also illustrated, where the effect of the adhesion factor at the base of the foundation is also taken into account.

Numerical Methods

The LB and UB FELA [32] are employed to derive the end bearing capacity solutions of annular foundations embedded in clay with linear increasing undrained shear strength as shown in Fig. 1. To perform the numerical computation, the OptumG2 FELA software [33] is used to solve the solutions of this problem under axisymmetric conditions.

Note that the OptumG2 FELA software [33] has been recently used to study several bearing capacity problems (e.g., [11, 37,38,39,40]). Figure 2 depicts the simulation of an annular foundation embedded in clay modeled by OptumG2. Since the geometry of the problem is axisymmetric, only half of the model domain is used in the simulation, where the line of axial symmetry is set to be located at the left of the domain (see Fig. 2). The boundary conditions of the domain of this problem are described hereafter. The left and right boundaries of the domain are permitted to have only vertical movements whereas horizontal movements are not allowed. At the bottom boundary of the domain, it is set to be no movement in both horizontal and vertical directions. The top boundary of the domain is the free surface in which both horizontal and vertical movements are allowed to be taken place.

Fig. 2
figure 2

A numerical model of a rigid annular foundation embedded in clay in OptumG2

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To model the annular foundation, rigid volume elements are carried out to simulate the annular foundation, where the weight of all volume elements of the annular foundation is zero. Note that the annular foundation is perfectly rigid. At the top of the annular foundation, the vertical uniform pressure q is applied over the area from the inner radius to the outer radius of the foundation. The clay is also modeled using weightless volume elements, and has the strength at the ground surface being zero, and increasing linearly with depth (see Fig. 1). The failure criterion of clay is the Tresca failure criterion with an associated flow rule. Note that the reason for using weightless elements is to omit the effect of soil weight on the undrained end bearing capacity of this problem. Hence, the computed results from FELA in this study are in the scope of the stability problem under undrained conditions. At the left and right surfaces of the annular foundation, the smooth interface condition (α = 0) is applied at both sides of the inside and outside surfaces to avoid the effect of the skin friction or the shaft resistances (see Fig. 2). At the base of the annular foundation, the roughness conditions varying from α = 0 to α = 1 is set at the base (see Fig. 2). As a result, the end bearing capacity factors computed from FELA are purely affected by the adhesion factor defined at the base of the foundation.

To avoid the insufficient boundary effect, the sizes of the domains are chosen to be sufficiently large for all numerical models of annular foundation problems. Thus, the intersection of the plastic shear zone to the right and bottom boundaries of the domain is not permitted to occur. However, the intersection of the plastic shear zone to the left boundary (at the line of axial symmetry) and the top boundary (free surface) can take place. To increase the accuracy of the computed numerical results from UB and LB FELA, the automatic mesh adaptivity, which is a feature in OptumG2, is employed in the computation. This feature can improve the accuracy of the computed UB and LB bearing capacity solutions. During the process of the automatic mesh adaptivity, the number of elements will extremely increase in the zone that has high plastic shearing strain from the first step to the final step of mesh adaptivity iterations. More information regarding the automatic mesh adaptivity in OptumG2 can be found in Ciria et al. [34]. In this study, five adaptive steps are chosen, where the setup of the first adaptivity iteration is the use of 5000 elements while that of the fifth adaptivity iteration (or final adaptivity iteration) is approximately 10,000 elements.

Numerical Results and Discussion

The results of the end bearing capacity factor are first verified with the existing solutions by Lee et al. [20] for the cases of the rough interface condition (α = 1) defined at the base of foundations. Table 1 shows a comparison between the present FELA results and those FEM results from Lee et al. [20] for the cases of ring foundations resting on the surface of clays (L/ro = 0) by varying the shear strength at the ground surface su0. In Table 1, the normalized strength gradient is defined as ρro/su0 and the end bearing capacity factor is defined as q/su0. The average (Avg) solutions shown in Table 1 are the average values from the UB and LB FELA solutions. It can be found from Table 1 that the FEM results by Lee et al. [20] are slightly larger than the Avg FELA solutions about 0.2–0.9%.

Table 1 A comparison of obtained q/su0 values with the existing solutions by Lee et al. [20] for the cases of L/ro = 0
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Figure 3 presents a comparison of the present study and the existing solutions by Lee et al. [20] for the cases of annular foundations embedded in clay. In Fig. 3, the existing solutions by Lee et al. [20] for the cases of ρro/su0 = 15 are employed for the verification. Note that all solutions presented in Fig. 3 and hereafter in this paper are the average (Avg) solutions of the end bearing capacity factor defined as q/ρL. It can be seen from Fig. 3 that the present solutions agree very well with those of Lee et al. [20]. It should be noted that the results by Lee et al. [20] are limited to the cases of the rough interface condition, which are not the practical value of the roughness at the soil-foundation interface. The value of the inherent adhesion factor is in between rough (α = 1) to smooth (α = 0) conditions.

Fig. 3
figure 3

Comparison of end bearing capacity factors for rough annular foundations embedded in clay (α = 1)

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Table 2 shows the percent of the difference between the UB and LB solutions of the end bearing capacity factor defined as q/ρL. The %Diff in Table 2 can be calculated as shown in Eq. (5):

$$\% {\text{Diff}} = \frac{{\left[ {\left( {\frac{q}{{\rho L_{{({\text{UB}})}} }}} \right) - \left( {\frac{q}{{\rho L_{{({\text{LB}})}} }}} \right)} \right] \times 100\% }}{{\left[ {\left( {\frac{q}{{\rho L_{{({\text{UB}})}} }}} \right) + \left( {\frac{q}{{\rho L_{{({\text{LB}})}} }}} \right)} \right]/2}}.$$
(5)
Table 2 The percent of the difference between the UB and LB solutions
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It can be found from Table 2 that the %Diff is less than 1.3% for all cases expressed in Table 2.

The end bearing capacity factors of annular foundations embedded in clay with linearly increasing strength by considering the adhesion factor are presented in Fig. 4a–e for the different values of α = 1, 0.75, 0.5, 0.25, and 0, respectively. These figures show the relationship between the end bearing capacity factors q/ρL and the depth ratio L/ro. The contour lines in Fig. 4 represent the values of ri/ro varying from 0, 0.25, and 0.5, respectively. It can be seen from Fig. 4 that the relationship between q/ρL and L/ro is highly non-linear. When the value of ri/ro is equal to 0.5, the variation of q/ρL decreases as L/ro increases. However, for the cases of ri/ro = 0 and 0.25, the value of q/ρL reduces when the value of L/ro increases, and reaches the lowest point at L/ro is about 1 or 1.5. After the lowest point, the value of q/ρL turns to increase when L/ro increases. It is also found from Fig. 4 that an increase in ri/ro results in a decrease in q/ρL. The effect of the adhesion factor on the end bearing capacity factor is demonstrated in Fig. 5a–d for the cases of L/ro = 0.25, 0.5, 2, and 3, respectively. The relationship between q/ρL and α is non-linear. The influence of α is prominent when L/ro = 0.25 and 0.5 (shallow foundations) as can be seen in Fig. 5a, b. The impact of α on the end bearing capacity factor becomes less when the depth of a foundation is large (see Fig. 5c, d).

Fig. 4
figure 4

End bearing capacity of annular foundations embedded in clay, where a α =  1; b α = 0.75; c α = 0.5; and d α = 0.25; and e α = 0

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Fig. 5
figure 5

End bearing capacity of annular foundations embedded in clay, where a L/ro = 0.25; b L/ro = 0.5; c L/ro = 2; and d L/ro = 3

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The impacts of L/ro, ri/ro, and α on the collapsed mechanisms of annular foundations embedded in clay with linearly increasing strength are demonstrated hereafter. Figure 6 shows the influence of L/ro on the final adaptive meshes of annular foundations embedded in clay for the cases of constant values of α = 1 and ri/ro = 0.5. Note that many elements extremely increase in the zones that contain very high plastic shear strains due to the feature of the automatic mesh adaptivity by OptumG2 [33, 34]. This can imply the correctness of the numerical results from the FELA simulations certainly reaches the very accurate values of the end bearing capacity factor of annular foundations embedded in clay. Figure 7 also presents the effect of L/ro on the incremental shear strain contours of annular foundations for the cases of α = 1 and ri/ro = 0.5. It can be seen from Figs. 6 and 7 that the final adaptive meshes and the incremental shear strain contours can be used to demonstrate the collapsed mechanisms of the annular foundation problem since the plastic shear zones take place at the same locations. From Figs. 6 and 7, the shape of the plastic shear zones is log-spiral that extends from the bases of annular foundations to the free surface of the clay. An increase in L/ro results in a decrease in the size of the plastic shear zones. The deepness of the plastic shear zones for the cases of L/ro = 0.25, 0.75, and 2 are approximately 1L, 0.5L, and 0.3L, respectively. Figure 8 demonstrates the effect of ri/ro on the incremental shear strain contours of annular foundations for the cases of α = 1 and L/ro = 0.75. The case of ri/ro = 0 reveals only one log-spiral plastic shear zone extending from the right corner to the free surface. However, when ri/ro > 0 (e.g., ri/ro = 0.25 and 0.5), two log-spiral plastic shear zones extending from both left and right corners appear. In addition, the size of the plastic shear zone also depends on the magnitude of ri/ro, where an increase in ri/ro yields a decrease in the size of the plastic shear zone. Figure 9 presents the examples of the base pressure distributions (or normal pressures) below the footing bases of annular foundations for the cases of α = 1 and L/ro = 0.75. It is found that the base pressures are not uniform, where the shapes of the base pressures significantly depend on the values of ri/ro as shown in Fig. 9. The effect of α on the incremental shear strain contours of annular foundations embedded in clay is demonstrated in Fig. 10 for the cases of ri/ro = 0.25 and L/ro = 1. It can be found that the size of the plastic shear zone for the foundation with the smooth interface condition (α = 0) is smallest, and vertically expanded when the value of α increases. The biggest size of the plastic shear zone in Fig. 10 is the case with the rough interface condition (α = 1).

Fig. 6
figure 6

Effect of L/ro on the final adaptive meshes of annular foundations embedded in clay with α = 1 and ri/ro = 0.5

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Fig. 7
figure 7

Effect of L/ro on the incremental shear strain contours of annular foundations embedded in clay with α = 1 and ri/ro = 0.5

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Fig. 8
figure 8

Effect of ri/ro on the incremental shear strain contours of annular foundations embedded in clay with α = 1 and L/ro = 0.75

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Fig. 9
figure 9

Effect of ri/ro on the base pressure distributions below the footing bases of annular foundations embedded in clay with α = 1 and L/ro = 0.75

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Fig. 10
figure 10

Effect of α on the incremental shear strain contours of annular foundations embedded in clay with ri/ro = 0.25 and L/ro = 1

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Conclusions

In the paper, new plastic solutions for the end bearing capacity of annular foundations in clay with linearly increasing shear strength are presented, where the adhesion factor at the base of foundations are considered. Using the lower and upper bound finite element limit analysis with the automatic mesh adaptivity feature, the solutions of the end bearing capacity factor of annular foundations in clay can be obtained. The main results of the present study can be summarized as follows.

  • In the verification, the present FELA solutions are larger than the existing FEM solutions by Lee et al. [20] about 0.2–0.9% for the cases of L = 0. For the cases of L > 0, the present FELA solutions are in good agreement with those from Lee et al. [20].

  • The end bearing capacity factor is significantly influenced by three dimensionless parameters including the ratio between the internal radius and the external radius, the embedded depth ratio, and the adhesion factor.

  • For the collapse mechanisms of this problem, it is found that the size of the plastic shear zone becomes smaller when the depth of foundations becomes larger.

  • The shapes of the plastic shear zones between the case of ri/ro = 0 and that of ri/ro > 0 are different, where one log-spiral shear zone appears in the cases of ri/ro = 0 whereas the case of ri/ro > 0 has two log-spiral shear zone taking place at the left and the right of the base of the annular foundation.

  • For the effect of the adhesion factor at the base of foundations on the collapse mechanisms, the size of the plastic shear zone becomes bigger when the value of α becomes larger.