# Ultimate Bearing Capacity of Ring Foundations Embedded in Undrained Homogeneous Clay

Authors: An-hui Wang; Yan-fang Zhang; Fan Xia; Ru-ping Luo; Ning Wang
Journal: Geofluids
 (2022)

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## Abstract

A ring foundation is widely used in bridges, water towers, caissons, and other engineering structures, and its ultimate bearing capacity is one of the significant concerns in engineering design. This paper is aimed at exploring the ultimate bearing capacity of ring foundations embedded in undrained clay. Based on the finite element limit analysis, effects of the inside-to-outside radius ratio, embedment depth ratio, cutting face inclination angle, and face roughness on the vertical ultimate bearing capacity of ring foundations are investigated. The results show that the ultimate bearing capacity of the ring foundation increases gradually with the embedment depth ratio. When the embedment depth ratioD/B reaches a critical value, the bearing capacity tends to be stable, and the critical embedment depth ratio is affected by the inside-to-outside radius ratio of the ring foundation, varying from 0.2 to 0.4. The ultimate bearing capacity of the ring foundation decreases with cutting face inclination angle β. When β≤40°, the ultimate bearing capacity tends to be stable, and the bearing capacity is reduced by approximately 30%. The influence of the cutting face inclination angle on the bearing capacity is highly dependent on the roughness of the cutting face.


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## Body

## 1. Introduction

A ring foundation is widely used in bridges, water towers, open caissons, and other deep foundation structures. Taking the open caisson as an example, the shaft wall and ring foundation are two main bearing elements. During a sinking of an open caisson, the soil inside the open caisson is removed. The burden pressure difference inside and outside the ring foundation is formed, resulting in the bottom soil outside the ring foundation pushing into the inside of the open caisson. The ultimate bearing capacity of the ring foundation nearly determines if the open caisson can be sunk smoothly. Therefore, a deep analysis of the ultimate bearing capacity of ring foundations is of great engineering significance.In recent years, progress has been made in studying the ultimate bearing capacity of ring foundations. For example, Meyerhof [1] and Frydman and Burd [2] calculated the bearing capacity of ring foundations based on the assumption that the interface friction coefficient between the foundation and the soil increases from 0 to 1 in the contact area from the center symmetry axis to the foundation edge. Assuming that the soil moved toward the center of the ring foundation and the outermost slip line intersected at the ring center, Berezantzev [3] considered the ring open caisson as a thin-walled ring foundation and derived the ultimate bearing capacity coefficient. Kumar and Ghosh [4] used the slip line method to analyze the bearing capacity of rigid ring foundations with smooth and rough interfaces. Yan and Shi [5] established an approximate calculation model for ultimate soil resistance at the edge foundation during caisson sinking with different shapes and depths of the cutting face and derived a formula for calculating the ultimate bearing capacity coefficient of the circular open caisson.Due to the complex boundary conditions, it is difficult to use theoretical analysis to accurately reflect the ultimate bearing characteristics of ring foundations. Nabil [6] and Saran et al. [7] determined the ultimate bearing capacity of ring foundations by a model test. Hu et al. [8] studied the bearing characteristics of open caissons in saturated sandy soil under vertical loads via a centrifugal model test and clarified that the variation patterns of the ultimate bearing capacity preliminarily changed with the embedment depth and width of the foundation. Further, a theoretical expression for the ultimate bearing capacity of the caisson foundation was deduced. Zhou et al. [9] also conducted centrifuge model tests to simulate the sinking of four caissons under different embedment depths and analyzed the distribution characteristics of earth pressure at ring footing. Fattah et al. [10–13] conducted a series of model tests to study the experimental bearing and settlement behaviors of model ring footing, circular footing, and rectangular resting on reinforced sandy soil subjected to vertical static and cyclic loads.In addition to the theoretical and experimental studies, numerical analysis is also an effective means to study the bearing behavior of foundation structures. Considering the embedment depth and inclination angle of the caisson cutting face, Solov’ev [14] carried out a numerical analysis of a circular open caisson under vertical loads by using the limit equilibrium theory and finite difference method. Combined with the finite element method, Chakraborty and Kumar [15] used the limit analysis method to determine the ultimate bearing capacity coefficient of wedge-shaped foundations and proposed a series of bearing capacity coefficient curves. The calculations were verified by comparing the numerical solutions and centrifugal test results in the existing literature. Lee et al. [16] and Birid and Choudhury [17] used the finite element analysis software PLAXIS to analyze the ultimate bearing characteristics of ring foundations and validated the numerical results through the existing literature. Further, the ultimate bearing capacity of ring foundations in Gibson ground was analyzed and corresponding design charts were presented. Royston et al. [18] used the limit analysis method to study the ultimate bearing capacity of ring foundations in undrained clay and revealed the influences of the embedment depth, cutting face angle, and foundation size on the bearing capacity of ring foundations. Keawsawasvong and Lai [19] investigate the end bearing capacity of ring foundations in clay with linearly increasing shear strength via the FELA software OptumG2, and the collapse mechanisms of ring foundations are examined and discussed. Gourvenec and Randolph [20] used the finite element software ABAQUS to analyze the bearing capacity of strip and ring foundations in clay in order to investigate the interaction between the caisson and the soil and discussed the influence of soil heterogeneity on numerical results. In addition, References [21–25] have investigated the ultimate bearing capacity of ring foundations through numerical research studies as well, and a series of design charts or formulas have been proposed.In general, current research mainly focuses on the ultimate bearing capacity of shallow ring foundations, and there are few studies involving the ultimate bearing characteristics of ring foundations with different embedment depths. Shallow foundations and deep foundations will exhibit different failure patterns under vertical loads. A shallow foundation usually exhibits general shear failure, while a deep foundation is more likely to present punch or local failure, thus having a significant impact on the ultimate bearing capacity. In particular, for the caisson foundation, as the soil inside the caisson is removed, the burden pressure difference inside and outside the ring foundation is formed, and the bottom soil outside the ring foundation is pushed into the inner caisson under vertical loads, resulting in a great difference in the ultimate bearing behavior between the shallow foundation and the embedded open caisson. In view of this, based on the open caisson foundation in homogeneous undrained clay, the finite element limit analysis method is adopted to investigate the effects of the inside-to-outside radius ratio, embedment depth, cutting face inclination angle, and face roughness on the vertical ultimate bearing capacity and failure patterns of ring foundations. A calculation method considering the embedment depth correction coefficient of ring foundations is proposed, and a series of design charts of the ring foundation’s bearing capacity under different working conditions are given to provide a reference for related engineering design and construction.


## 2. Numerical Analysis Model

### 2.1. Ultimate Bearing Capacity of Circular Foundations

Considering the axisymmetric characteristics, a two-dimensional finite element limit analysis (FELA) using OptumG2 was used to analyze the ultimate bearing capacity of ring foundations. The FELA of OptumG2 combines the finite element discretization technique and boundary conditions and utilizes the plastic bound theorem to obtain the rigorous ultimate load [26]. The limit analysis method can directly obtain the ultimate bearing capacity and failure pattern of ring foundations compared with the finite element method. It avoids the insufficiency of the finite element method in determining the ultimate bearing capacity through load-deformation curves and greatly improves the calculation efficiency. In addition, based on the adaptive mesh technology built in OptumG2, the mesh of a numerical model can automatically be refined in the failure area, which improves the accuracy of numerical calculations.Taking a circular foundation in the undrained clay as an example, the rationality of OptumG2 in analyzing the ultimate bearing behavior was verified. Figure1 shows the numerical model of a circular foundation in the undrained clay. The diameter B of the foundation is equal to 20 m, and the size of the numerical model is 100m×50m (length×width). In this case, boundary effects can be eliminated. In the model, displacement is not allowed at the outer boundary. To obtain a more accurate result, the 6-node mixed Gauss elements are used in this study, and a rather fine mesh with 20000 elements was adopted in the numerical model. Based on the 6-node mixed Gauss element, the calculation result is between the upper bound limit analysis solution and the lower bound limit analysis solution, which is more consistent with the actual situation. The boundary conditions of the numerical model are described hereafter. Only vertical movements are permitted to the left and right boundaries of the numerical model. For the bottom boundary of the numerical model, it is set to be no movement in both the horizontal and vertical directions. The top boundary of the numerical model is the free surface in which both the horizontal and vertical movements are allowed to be taken place.Figure 1
Numerical model of a circular foundation.In addition, automatic mesh adaptivity was used to achieve a more precise limit analysis. A vertical load multiplier was applied to the foundation to obtain the limiting bearing capacity. The program will automatically increase the load multiplier until the ultimate failure of the foundation. The ultimate bearing capacity of the foundation can be obtained via the final load multiplier.Table1 shows the bearing capacity coefficient Nc (Nc=pu/Su) of the circular foundation calculated by different methods, where pu is the ultimate bearing pressure and Su is the soil undrained shear strength. It can be seen that the bearing capacity coefficients obtained by different methods are in general consistency, which verifies the rationality of the calculation method in this paper. The Nc calculated by the upper bound limit analysis method is relatively large, while the Nc calculated by the lower bound limit analysis method is relatively small. Generally, the present bearing capacity coefficient is close to that of the lower bound limit analysis.Table 1
Bearing capacity coefficientNc of the circular foundation.
No.NcMethod of analysisPercent difference with this paperReference17.41Elastic plastic theory-18.76Terzaghi [27]26.00Plastic theory0.33Skempton [28]36.05Plastic theory and experiments-0.50Meyerhof [29]46.05Plastic theory-0.50Eason and Shield [30]56.17Elastic plastic theory-2.43Hansen [31]66.34Stress characteristics method-5.05Tant and Craig [32]75.856 (LB)6.227 (UB)FE limit analysis2.80-3.32Salgado et al. [33]86.09Small strain FE analysis-1.15Edwards et al. [34]96.166FE analysis-2.37Taiebat and Carter [35]105.945Small strain FE analysis1.26Gourvenec and Mana [36]116.08Small strain FE analysis-0.99Lee et al. [16]126.05Stress characteristics method-0.50Keshavarz and Kumar [37]136.056Finite difference method-0.59Benmebarek et al. [38]146.05Stress characteristics method-0.50Martin [39]156.01Lower bound limit analysis0.17Kumar and Khatri [40]166.16Upper bound limit analysis-2.27Kumar and Chakraborty [41]175.98Finite element method0.67Birid and Choudhury [17]186.04Finite element limit analysis0.33Keawsawasvong and Lai [19]196.02FE limit analysis—Present studyNote: LB = lower bound limit analysis; UB = upper bound limit analysis.


### 2.2. Numerical Models for Ring Foundations

Figure2 shows a schematic diagram of a typical ring foundation. The inner radius of the ring foundation is Ri, the outer radius is R0, the diameter is B (20 m in this study), and the embedment depth is D. In our analysis model, we assumed that the ground is homogeneous undrained clay and the ring foundation is rigid.Figure 2
Schematic diagram of the ring foundation.
(a)(b)To reveal the ultimate bearing mechanism of the ring foundation, the effects of inside-to-outside radius ratioη (η=Ri/R0), embedment depth ratio D/B, and soil unit weight γ on the ultimate bearing capacity were investigated. The η ranges from 0 to 0.9, undrained shear strength Su ranges from 30 kPa to 200 kPa, soil unit weight γ is 0 or 18 kN/m3, and D/B varies from 0 to 1.0, which covers most problems of practical interest. In addition, the influences of the soil unit weight and undrained shear strength were analyzed. The soil was assumed as homogeneous undrained clay, and the Tresca constitutive model was adopted. The ring foundation is rigid with an external radius R0 of 10 m. The numerical method is consistent with that in the verification case, and the specific calculations are listed in Table 2.Table 2
Numerical analysis cases.
ηD/BSu (kPa)γ (kN/m3)0 (circular foundation), 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 (ring foundation)0 (footing on the surface)50180, 0.2, 0.4, 0.6, 0.8 (ring foundation)0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1.0 (embedded footing)5000.4, 0.8 (ring foundation)0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1.0 (embedded footing)50, 30, 20018To characterize the interface roughness at the soil-caisson interface, the interface roughness factor (α) is introduced as α=Su,i/Su, where Su,i is the shear strength of the interface and Su is the shear strength of the clay around the interface. Theoretically, the α factor ranges from zero to one, corresponding to the fully smooth and fully rough interfaces, respectively. In this study, the interface of the caisson side wall is assumed to be fully smooth, the bottom interface is assumed to be fully rough, and the ultimate bearing capacity of the caisson is only provided by end resistance, as shown in Figure 2(b).


## 2.1. Ultimate Bearing Capacity of Circular Foundations

Considering the axisymmetric characteristics, a two-dimensional finite element limit analysis (FELA) using OptumG2 was used to analyze the ultimate bearing capacity of ring foundations. The FELA of OptumG2 combines the finite element discretization technique and boundary conditions and utilizes the plastic bound theorem to obtain the rigorous ultimate load [26]. The limit analysis method can directly obtain the ultimate bearing capacity and failure pattern of ring foundations compared with the finite element method. It avoids the insufficiency of the finite element method in determining the ultimate bearing capacity through load-deformation curves and greatly improves the calculation efficiency. In addition, based on the adaptive mesh technology built in OptumG2, the mesh of a numerical model can automatically be refined in the failure area, which improves the accuracy of numerical calculations.Taking a circular foundation in the undrained clay as an example, the rationality of OptumG2 in analyzing the ultimate bearing behavior was verified. Figure1 shows the numerical model of a circular foundation in the undrained clay. The diameter B of the foundation is equal to 20 m, and the size of the numerical model is 100m×50m (length×width). In this case, boundary effects can be eliminated. In the model, displacement is not allowed at the outer boundary. To obtain a more accurate result, the 6-node mixed Gauss elements are used in this study, and a rather fine mesh with 20000 elements was adopted in the numerical model. Based on the 6-node mixed Gauss element, the calculation result is between the upper bound limit analysis solution and the lower bound limit analysis solution, which is more consistent with the actual situation. The boundary conditions of the numerical model are described hereafter. Only vertical movements are permitted to the left and right boundaries of the numerical model. For the bottom boundary of the numerical model, it is set to be no movement in both the horizontal and vertical directions. The top boundary of the numerical model is the free surface in which both the horizontal and vertical movements are allowed to be taken place.Figure 1
Numerical model of a circular foundation.In addition, automatic mesh adaptivity was used to achieve a more precise limit analysis. A vertical load multiplier was applied to the foundation to obtain the limiting bearing capacity. The program will automatically increase the load multiplier until the ultimate failure of the foundation. The ultimate bearing capacity of the foundation can be obtained via the final load multiplier.Table1 shows the bearing capacity coefficient Nc (Nc=pu/Su) of the circular foundation calculated by different methods, where pu is the ultimate bearing pressure and Su is the soil undrained shear strength. It can be seen that the bearing capacity coefficients obtained by different methods are in general consistency, which verifies the rationality of the calculation method in this paper. The Nc calculated by the upper bound limit analysis method is relatively large, while the Nc calculated by the lower bound limit analysis method is relatively small. Generally, the present bearing capacity coefficient is close to that of the lower bound limit analysis.Table 1
Bearing capacity coefficientNc of the circular foundation.
No.NcMethod of analysisPercent difference with this paperReference17.41Elastic plastic theory-18.76Terzaghi [27]26.00Plastic theory0.33Skempton [28]36.05Plastic theory and experiments-0.50Meyerhof [29]46.05Plastic theory-0.50Eason and Shield [30]56.17Elastic plastic theory-2.43Hansen [31]66.34Stress characteristics method-5.05Tant and Craig [32]75.856 (LB)6.227 (UB)FE limit analysis2.80-3.32Salgado et al. [33]86.09Small strain FE analysis-1.15Edwards et al. [34]96.166FE analysis-2.37Taiebat and Carter [35]105.945Small strain FE analysis1.26Gourvenec and Mana [36]116.08Small strain FE analysis-0.99Lee et al. [16]126.05Stress characteristics method-0.50Keshavarz and Kumar [37]136.056Finite difference method-0.59Benmebarek et al. [38]146.05Stress characteristics method-0.50Martin [39]156.01Lower bound limit analysis0.17Kumar and Khatri [40]166.16Upper bound limit analysis-2.27Kumar and Chakraborty [41]175.98Finite element method0.67Birid and Choudhury [17]186.04Finite element limit analysis0.33Keawsawasvong and Lai [19]196.02FE limit analysis—Present studyNote: LB = lower bound limit analysis; UB = upper bound limit analysis.


## 2.2. Numerical Models for Ring Foundations

Figure2 shows a schematic diagram of a typical ring foundation. The inner radius of the ring foundation is Ri, the outer radius is R0, the diameter is B (20 m in this study), and the embedment depth is D. In our analysis model, we assumed that the ground is homogeneous undrained clay and the ring foundation is rigid.Figure 2
Schematic diagram of the ring foundation.
(a)(b)To reveal the ultimate bearing mechanism of the ring foundation, the effects of inside-to-outside radius ratioη (η=Ri/R0), embedment depth ratio D/B, and soil unit weight γ on the ultimate bearing capacity were investigated. The η ranges from 0 to 0.9, undrained shear strength Su ranges from 30 kPa to 200 kPa, soil unit weight γ is 0 or 18 kN/m3, and D/B varies from 0 to 1.0, which covers most problems of practical interest. In addition, the influences of the soil unit weight and undrained shear strength were analyzed. The soil was assumed as homogeneous undrained clay, and the Tresca constitutive model was adopted. The ring foundation is rigid with an external radius R0 of 10 m. The numerical method is consistent with that in the verification case, and the specific calculations are listed in Table 2.Table 2
Numerical analysis cases.
ηD/BSu (kPa)γ (kN/m3)0 (circular foundation), 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 (ring foundation)0 (footing on the surface)50180, 0.2, 0.4, 0.6, 0.8 (ring foundation)0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1.0 (embedded footing)5000.4, 0.8 (ring foundation)0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.75, 1.0 (embedded footing)50, 30, 20018To characterize the interface roughness at the soil-caisson interface, the interface roughness factor (α) is introduced as α=Su,i/Su, where Su,i is the shear strength of the interface and Su is the shear strength of the clay around the interface. Theoretically, the α factor ranges from zero to one, corresponding to the fully smooth and fully rough interfaces, respectively. In this study, the interface of the caisson side wall is assumed to be fully smooth, the bottom interface is assumed to be fully rough, and the ultimate bearing capacity of the caisson is only provided by end resistance, as shown in Figure 2(b).


## 3. Analysis of Numerical Results

### 3.1. Ultimate Bearing Behavior of Ring Foundations

Figure3 shows the failure patterns of the ring foundation on the ground surface with different inside-to-outside radius ratios η. It can be seen that the failure pattern is affected by η significantly. When η is small (η≤0.1), the soil failure surface mainly occurs at the outside of the ring foundation. As soil inside the ring foundation is constrained by the boundary, the soil failure surface is difficult to develop. With the increase of η, the constraint effect of the ring foundation on the inner soil gradually weakens, and the soil failure surface inside the ring foundation gradually develops. When η≥0.4, the soil failure surface inside the ring foundation is completely connected, and the failure surface distribution pattern is consistent with that of the strip foundation, with a symmetrical distribution of the soil failure surface inside and outside the ring foundation.Figure 3
Failure patterns of the ring foundation with different inside-to-outside radius ratios.
(a)
η=0(b)
η=0.1(c)
η=0.2(d)
η=0.4(e)
η=0.5(f)
η=0.6Figure4 shows the variation in bearing coefficient Nc of the ring foundation with varied η. When η≤0.1, Nc is basically constant. With the increase of η, Nc decreases gradually. When η=0.9, the Nc is 5.23, about 87% of the bearing capacity coefficient of the ring foundation. As shown in Equation (1), a hyperbolic tangent function is used to fit the variation in Nc versus η. It can be seen from Figure 4 that the fitting result is in well agreement with the numerical calculation.
(1)Nc=4.73+1.29×tanh0.38η,η≠0.Figure 4
Variation curve ofNc versus η.Hereinbefore, the soil-foundation interface is assumed to be completely rough and the friction coefficientμ=1.0. Further, the influence of friction coefficient μ on the bearing capacity of the ring foundation is analyzed, as shown in Figure 5. It is observed that the influence of μ on Nc is dependent on the η. With the decrease of η, the influence of μ on Nc becomes more significant. In general, when μ≥0.5, increasing μ fails to effectively improve the Nc.Figure 5
Variation curve ofNc versus μ.


### 3.2. Influence of the Embedment Depth

Firstly, the influence of the embedment depth on the ultimate bearing capacity of the circular foundation is investigated. Figure6 shows the failure patterns of the ring foundation with varied embedment depth ratios D/B. The undrained shear strength Su of soil is 50 kPa, and the soil unit weight γ is 0 kN/m3 (i.e., the influence of the soil unit weight on the ultimate bearing capacity is ignored). As shown in Figure 6, the soil failure surface presents an arc-shaped distribution pattern under different embedment depth ratios. With the increase of D/B, the range of the failure surface expands continuously, and the depth and width of the failure surface gradually increase.Figure 6
Failure patterns of the ring foundation with different embedment depth ratios.
(a)
D/B=0.1(b)
D/B=0.2(c)
D/B=0.5(d)
D/B=1.0Figure7 shows the variations in bearing capacity coefficient Nc versus embedment depth ratio D/B of the ring foundation in the literature. The soil unit weight is assumed as 0, and the friction angle is 0 for undrained clay. It can be seen that a discrepancy occurs in Nc calculated by different methods. The results from this work are generally distributed in the middle distribution range of those from the existing literature and are close to those calculated by Lee et al. [16] and Gourvenec and Mana [36]. This finding verifies the rationality of the numerical model in this paper.Figure 7
Variation curve ofNc versus D/B.Figure8 presents the fitting curve of coefficient Nc versus D/B. It can be seen that using a nonlinear function favorably reveals the variation characteristics of Nc versus D/B of the ring foundation, and the fitting results are in rather well agreement with the numerical results. Besides, as it can be seen from Figure 8, when D/B=1, Nc is close to 9.0, which is the theoretical value of the end bearing capacity of the pile foundation. Based on the calculation results, it can be concluded that the circular foundation can be approximately regarded as the pile foundation when the embedded ratio D/B is larger than 1.0. Similar to the study conducted by Lee et al. [16], the fitting formula is written as
(2)Nc=6.02+2DB+DB.Figure 8
Fitting curve ofNc versus D/B for the circular foundation.Further, taking ring foundations withη=0.4 and 0.8 as examples, the failure patterns of ring foundations with varied D/B are investigated, as shown in Figures 9 and 10. The soil undrained shear strength Su=50kPa, and the soil unit weight γ=0kN/m3. It is seen that failure patterns of the ring foundation at different embedment depths are affected by η. With the decrease of η, the restriction effect of the ring foundation on the inner soil is enhanced, and the failure surface inside the soil is more difficult to develop. For example, when η=0.4, the inner soil failure surface develops effectively when D/B reaches 0.75 (Figure 9(d)). On the contrary, for the ring foundation with η equal to 0.8, the ring foundation has a weak constraining effect on the inner soil. The inner soil failure surface fully develops when D/B=0.1 (Figure 10(b)).Figure 9
Failure patterns of the ring foundation at different embedment depths (η=0.4).
(a)
D/B=0.2(b)
D/B=0.3(c)
D/B=0.5(d)
D/B=0.75Figure 10
Failure patterns of the ring foundation with different embedment depths (η=0.8).
(a)
D/B=0.05(b)
D/B=0.1(c)
D/B=0.2(d)
D/B=0.3Figure11 shows the variation curves of the bearing capacity coefficient pu/Su versus the embedment depth ratio D/B in different cases. In the weightless case (γ=0), pu/Su is consistent with the definition of Nc. Except for the ring foundation (η=0), pu/Su of ring foundations with varied η increases gradually with increasing D/B. However, when D/B reaches a critical value, pu/Su tends to be stable. pu/Su is significantly affected by η. With the decrease of η, the restriction effect of ring foundations on the inner soil is enhanced, and the D/B increases, resulting in a greater pu/Su. For the ring foundation with η equal to 0, pu/Su increases with the D/B continuously. In addition, the soil unit weight has a significant influence on the bearing capacity. With increasing soil unit weight, the overburden pressure to the foundation increased, the resistance to be overcome for the ground soil to reach the failure state enhanced, and the bearing capacity coefficient pu/Su increases obviously. Take the foundations with Ri/R0=0 and 0.2 as examples. As it can be seen from Figure 11, when the soil unit weight (γ=0) is not considered and D/B=1.0, the bearing capacity coefficient pu/Su is about 9.0 and 7.6, respectively. However, when the soil unit weight γ=18kN/m3 and D/B=1.0, the bearing capacity coefficient pu/Su reaches 16.2 and 9.4, respectively. The increasing rates are 80% and 23.7%, respectively.Figure 11
Variation curves of the bearing coefficient versusD/B under different soil unit weights.
(a)
γ=0kN/m3(b)
γ=18kN/m3The ultimate bearing capacity of ring foundations is affected by many factors. As shown in Figure10, due to the earth pressure induced by the burden soil outside the ring foundation, the bottom soil outside the ring foundation is pushed into the inner ring foundation. To reveal the failure mechanism of the ring foundation at different embedment depths, failure patterns of the ring foundation with different soil shear strengths and weights are analyzed. Taking the ring foundation with η equal to 0.4 and D/B equal to 0.3 as an example, the soil failure patterns of the ring foundation with different soil shear strengths and weights are illustrated in Figure 12. It can be seen that when the Su/γ is small (Figures 12(a) and 12(c)), the failure surface inside the ring foundation can fully develop. With the increase of Su/γ, the extrusion resistance of the inner soil increases markedly due to the restriction of the ring foundation, and the failure surfaces tend to develop toward the outside soil, as shown in Figures 12(b) and 12(d).Figure 12
Failure patterns of the ring foundation under different working conditions.
(a)
Su=30kPa, γ=18kN/m3(b)
Su=50kPa, γ=0kN/m3(c)
Su=50kPa, γ=18kN/m3(d)
Su=200kPa, γ=18kN/m3


### 3.3. Influence of the Cutting Face Inclination Angle

To reduce the end bearing capacity and aid the sinking process, the caisson walls typically feature a tapered base, referring to the cutting face, as shown in Figure13(a). Taking the ring foundation with η equal to 0.9 as an example, the influence of cutting face inclination angle β on the bearing capacity of the ring foundation is analyzed. Similar to the above analysis, the Tresca model is adopted for soil, with undrained shear strength Su equal to 50 kPa and weight γ equal to 0. The numerical analysis model is shown in Figure 13(b).Figure 13
Analytical model of the ring foundation with the cutting face.
(a)
Schematic diagram(b)
Numerical modelFigure14 depicts the variation in bearing coefficient k versus β, in which k=Nc,β/Nc,90° (i.e., k is the ratio of the bearing capacity of the ring foundation with cutting face inclination angle β to that of the ring foundation without an inclination angle). It is obvious that the k decreases with decreasing β. However, when β≤40°, the k tends to be stable, and the bearing capacity is reduced by approximately 30% (k=0.717 for β=26.6°).Figure 14
Variation in the bearing coefficient of the ring foundation with different inclination angles.In addition to the influence of cutting face inclination angleβ, the influence of friction coefficient μ on the bearing capacity is investigated further, as shown in Figure 15. Taking the ring foundations with β equal to 45° and 26.6° as examples, the k basically changes linearly with μ. The smaller the β is, the more significant the influence of μ on the bearing capacity of the ring foundation is. This result is consistent with the conclusion from Royston et al. [18]. Taking the ring foundation with β equal to 26.6° as an example, the bearing capacity of the ring foundation with a completely smooth interface (μ=0) is about 0.52 times that of the ring foundation with a completely rough interface (μ=1.0). Therefore, for ring foundations such as open caissons, to aid the sinking process significantly, reducing the μ of the structure-soil contact surface is effective.Figure 15
Variation in the bearing coefficient of the ring foundation with different friction coefficients.


## 3.1. Ultimate Bearing Behavior of Ring Foundations

Figure3 shows the failure patterns of the ring foundation on the ground surface with different inside-to-outside radius ratios η. It can be seen that the failure pattern is affected by η significantly. When η is small (η≤0.1), the soil failure surface mainly occurs at the outside of the ring foundation. As soil inside the ring foundation is constrained by the boundary, the soil failure surface is difficult to develop. With the increase of η, the constraint effect of the ring foundation on the inner soil gradually weakens, and the soil failure surface inside the ring foundation gradually develops. When η≥0.4, the soil failure surface inside the ring foundation is completely connected, and the failure surface distribution pattern is consistent with that of the strip foundation, with a symmetrical distribution of the soil failure surface inside and outside the ring foundation.Figure 3
Failure patterns of the ring foundation with different inside-to-outside radius ratios.
(a)
η=0(b)
η=0.1(c)
η=0.2(d)
η=0.4(e)
η=0.5(f)
η=0.6Figure4 shows the variation in bearing coefficient Nc of the ring foundation with varied η. When η≤0.1, Nc is basically constant. With the increase of η, Nc decreases gradually. When η=0.9, the Nc is 5.23, about 87% of the bearing capacity coefficient of the ring foundation. As shown in Equation (1), a hyperbolic tangent function is used to fit the variation in Nc versus η. It can be seen from Figure 4 that the fitting result is in well agreement with the numerical calculation.
(1)Nc=4.73+1.29×tanh0.38η,η≠0.Figure 4
Variation curve ofNc versus η.Hereinbefore, the soil-foundation interface is assumed to be completely rough and the friction coefficientμ=1.0. Further, the influence of friction coefficient μ on the bearing capacity of the ring foundation is analyzed, as shown in Figure 5. It is observed that the influence of μ on Nc is dependent on the η. With the decrease of η, the influence of μ on Nc becomes more significant. In general, when μ≥0.5, increasing μ fails to effectively improve the Nc.Figure 5
Variation curve ofNc versus μ.


## 3.2. Influence of the Embedment Depth

Firstly, the influence of the embedment depth on the ultimate bearing capacity of the circular foundation is investigated. Figure6 shows the failure patterns of the ring foundation with varied embedment depth ratios D/B. The undrained shear strength Su of soil is 50 kPa, and the soil unit weight γ is 0 kN/m3 (i.e., the influence of the soil unit weight on the ultimate bearing capacity is ignored). As shown in Figure 6, the soil failure surface presents an arc-shaped distribution pattern under different embedment depth ratios. With the increase of D/B, the range of the failure surface expands continuously, and the depth and width of the failure surface gradually increase.Figure 6
Failure patterns of the ring foundation with different embedment depth ratios.
(a)
D/B=0.1(b)
D/B=0.2(c)
D/B=0.5(d)
D/B=1.0Figure7 shows the variations in bearing capacity coefficient Nc versus embedment depth ratio D/B of the ring foundation in the literature. The soil unit weight is assumed as 0, and the friction angle is 0 for undrained clay. It can be seen that a discrepancy occurs in Nc calculated by different methods. The results from this work are generally distributed in the middle distribution range of those from the existing literature and are close to those calculated by Lee et al. [16] and Gourvenec and Mana [36]. This finding verifies the rationality of the numerical model in this paper.Figure 7
Variation curve ofNc versus D/B.Figure8 presents the fitting curve of coefficient Nc versus D/B. It can be seen that using a nonlinear function favorably reveals the variation characteristics of Nc versus D/B of the ring foundation, and the fitting results are in rather well agreement with the numerical results. Besides, as it can be seen from Figure 8, when D/B=1, Nc is close to 9.0, which is the theoretical value of the end bearing capacity of the pile foundation. Based on the calculation results, it can be concluded that the circular foundation can be approximately regarded as the pile foundation when the embedded ratio D/B is larger than 1.0. Similar to the study conducted by Lee et al. [16], the fitting formula is written as
(2)Nc=6.02+2DB+DB.Figure 8
Fitting curve ofNc versus D/B for the circular foundation.Further, taking ring foundations withη=0.4 and 0.8 as examples, the failure patterns of ring foundations with varied D/B are investigated, as shown in Figures 9 and 10. The soil undrained shear strength Su=50kPa, and the soil unit weight γ=0kN/m3. It is seen that failure patterns of the ring foundation at different embedment depths are affected by η. With the decrease of η, the restriction effect of the ring foundation on the inner soil is enhanced, and the failure surface inside the soil is more difficult to develop. For example, when η=0.4, the inner soil failure surface develops effectively when D/B reaches 0.75 (Figure 9(d)). On the contrary, for the ring foundation with η equal to 0.8, the ring foundation has a weak constraining effect on the inner soil. The inner soil failure surface fully develops when D/B=0.1 (Figure 10(b)).Figure 9
Failure patterns of the ring foundation at different embedment depths (η=0.4).
(a)
D/B=0.2(b)
D/B=0.3(c)
D/B=0.5(d)
D/B=0.75Figure 10
Failure patterns of the ring foundation with different embedment depths (η=0.8).
(a)
D/B=0.05(b)
D/B=0.1(c)
D/B=0.2(d)
D/B=0.3Figure11 shows the variation curves of the bearing capacity coefficient pu/Su versus the embedment depth ratio D/B in different cases. In the weightless case (γ=0), pu/Su is consistent with the definition of Nc. Except for the ring foundation (η=0), pu/Su of ring foundations with varied η increases gradually with increasing D/B. However, when D/B reaches a critical value, pu/Su tends to be stable. pu/Su is significantly affected by η. With the decrease of η, the restriction effect of ring foundations on the inner soil is enhanced, and the D/B increases, resulting in a greater pu/Su. For the ring foundation with η equal to 0, pu/Su increases with the D/B continuously. In addition, the soil unit weight has a significant influence on the bearing capacity. With increasing soil unit weight, the overburden pressure to the foundation increased, the resistance to be overcome for the ground soil to reach the failure state enhanced, and the bearing capacity coefficient pu/Su increases obviously. Take the foundations with Ri/R0=0 and 0.2 as examples. As it can be seen from Figure 11, when the soil unit weight (γ=0) is not considered and D/B=1.0, the bearing capacity coefficient pu/Su is about 9.0 and 7.6, respectively. However, when the soil unit weight γ=18kN/m3 and D/B=1.0, the bearing capacity coefficient pu/Su reaches 16.2 and 9.4, respectively. The increasing rates are 80% and 23.7%, respectively.Figure 11
Variation curves of the bearing coefficient versusD/B under different soil unit weights.
(a)
γ=0kN/m3(b)
γ=18kN/m3The ultimate bearing capacity of ring foundations is affected by many factors. As shown in Figure10, due to the earth pressure induced by the burden soil outside the ring foundation, the bottom soil outside the ring foundation is pushed into the inner ring foundation. To reveal the failure mechanism of the ring foundation at different embedment depths, failure patterns of the ring foundation with different soil shear strengths and weights are analyzed. Taking the ring foundation with η equal to 0.4 and D/B equal to 0.3 as an example, the soil failure patterns of the ring foundation with different soil shear strengths and weights are illustrated in Figure 12. It can be seen that when the Su/γ is small (Figures 12(a) and 12(c)), the failure surface inside the ring foundation can fully develop. With the increase of Su/γ, the extrusion resistance of the inner soil increases markedly due to the restriction of the ring foundation, and the failure surfaces tend to develop toward the outside soil, as shown in Figures 12(b) and 12(d).Figure 12
Failure patterns of the ring foundation under different working conditions.
(a)
Su=30kPa, γ=18kN/m3(b)
Su=50kPa, γ=0kN/m3(c)
Su=50kPa, γ=18kN/m3(d)
Su=200kPa, γ=18kN/m3


## 3.3. Influence of the Cutting Face Inclination Angle

To reduce the end bearing capacity and aid the sinking process, the caisson walls typically feature a tapered base, referring to the cutting face, as shown in Figure13(a). Taking the ring foundation with η equal to 0.9 as an example, the influence of cutting face inclination angle β on the bearing capacity of the ring foundation is analyzed. Similar to the above analysis, the Tresca model is adopted for soil, with undrained shear strength Su equal to 50 kPa and weight γ equal to 0. The numerical analysis model is shown in Figure 13(b).Figure 13
Analytical model of the ring foundation with the cutting face.
(a)
Schematic diagram(b)
Numerical modelFigure14 depicts the variation in bearing coefficient k versus β, in which k=Nc,β/Nc,90° (i.e., k is the ratio of the bearing capacity of the ring foundation with cutting face inclination angle β to that of the ring foundation without an inclination angle). It is obvious that the k decreases with decreasing β. However, when β≤40°, the k tends to be stable, and the bearing capacity is reduced by approximately 30% (k=0.717 for β=26.6°).Figure 14
Variation in the bearing coefficient of the ring foundation with different inclination angles.In addition to the influence of cutting face inclination angleβ, the influence of friction coefficient μ on the bearing capacity is investigated further, as shown in Figure 15. Taking the ring foundations with β equal to 45° and 26.6° as examples, the k basically changes linearly with μ. The smaller the β is, the more significant the influence of μ on the bearing capacity of the ring foundation is. This result is consistent with the conclusion from Royston et al. [18]. Taking the ring foundation with β equal to 26.6° as an example, the bearing capacity of the ring foundation with a completely smooth interface (μ=0) is about 0.52 times that of the ring foundation with a completely rough interface (μ=1.0). Therefore, for ring foundations such as open caissons, to aid the sinking process significantly, reducing the μ of the structure-soil contact surface is effective.Figure 15
Variation in the bearing coefficient of the ring foundation with different friction coefficients.


## 4. Conclusions

The ultimate bearing capacity of a ring foundation embedded in undrained clay was investigated. By means of finite element limit analysis OptumG2, the undrained vertical bearing capacity factors of foundations with a wide range of parameters were calculated. The influences of the inside-to-outside radius ratio, embedded ratio, and inclination angle on the ultimate bearing capacity of the ring foundation were analyzed, and related calculation methods and design charts were proposed. The following conclusions can be summarized:(1)
With the increase ofη, the Nc decreases gradually, and a hyperbolic tangent function can well describe the variation in Nc versus η(2)
For the ring foundation embedded in undrained clay, when theSu/γ is small, the failure surface of the inner ring foundation can develop fully. With the increase of Su/γ, the failure surfaces tend to develop toward the outside soil(3)
When the embedment depth ratio reaches a critical value, the bearing capacity coefficients tend to be stable. For the ring foundation in practical engineering, whenD/B≥0.5, the bearing coefficient is basically stable(4)
For the open caisson with an inclination angle of less than 40°, the bearing coefficient tends to be stable, and the bearing capacity is decreased by approximately 30% compared with that of the open caisson without an inclination angle(5)
The bearing coefficientk changes linearly with the interface friction coefficient. The smaller the inclination angle is, the more significant the influence of the friction coefficient on the bearing capacity is. Reducing the friction coefficient is an effective means to aid the sinking processIt should be noted that the foundation soil in this study is assumed as homogeneous undrained clay. When the foundation soil is Gibson ground, layered ground, or other sandy soil layers, the applicability of the above conclusions needs to be verified further.



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*Source: 6382799-2022-07-13.xml*