Ultimate Bearing Capacity of Ring Foundations Embedded in Undrained Homogeneous Clay

Wang, An-hui; Zhang, Yan-fang; Xia, Fan; Luo, Ru-ping; Wang, Ning

doi:https://doi.org/10.1155/2022/6382799

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Volume 2022 | Article ID 6382799 | https://doi.org/10.1155/2022/6382799

Ultimate Bearing Capacity of Ring Foundations Embedded in Undrained Homogeneous Clay

An-hui Wang,¹Yan-fang Zhang,¹Fan Xia,¹Ru-ping Luo,^2,3and Ning Wang^2,3

Academic Editor: Hao Wu

Received26 Apr 2022

Revised13 Jun 2022

Accepted23 Jun 2022

Published13 Jul 2022

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 ratio 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 . When , 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.

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. Figure 1 shows the numerical model of a circular foundation in the undrained clay. The diameter of the foundation is equal to 20 m, and the size of the numerical model is (). 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.

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.

Table 1 shows the bearing capacity coefficient () of the circular foundation calculated by different methods, where is the ultimate bearing pressure and 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 calculated by the upper bound limit analysis method is relatively large, while the 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.

2.2. Numerical Models for Ring Foundations

Figure 2 shows a schematic diagram of a typical ring foundation. The inner radius of the ring foundation is , the outer radius is , the diameter is (20 m in this study), and the embedment depth is . In our analysis model, we assumed that the ground is homogeneous undrained clay and the ring foundation is rigid.

(a)

(b)

To reveal the ultimate bearing mechanism of the ring foundation, the effects of inside-to-outside radius ratio (), embedment depth ratio , and soil unit weight on the ultimate bearing capacity were investigated. The ranges from 0 to 0.9, undrained shear strength ranges from 30 kPa to 200 kPa, soil unit weight is 0 or 18 kN/m³, and 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 of 10 m. The numerical method is consistent with that in the verification case, and the specific calculations are listed in Table 2.

To characterize the interface roughness at the soil-caisson interface, the interface roughness factor () is introduced as , where is the shear strength of the interface and is the shear strength of the clay around the interface. Theoretically, the 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

Figure 3 shows the failure patterns of the ring foundation on the ground surface with different inside-to-outside radius ratios . It can be seen that the failure pattern is affected by significantly. When is small (), 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 , 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 , 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.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4 shows the variation in bearing coefficient of the ring foundation with varied . When , is basically constant. With the increase of , decreases gradually. When , the 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 versus . It can be seen from Figure 4 that the fitting result is in well agreement with the numerical calculation.

Hereinbefore, the soil-foundation interface is assumed to be completely rough and the friction coefficient . Further, the influence of friction coefficient on the bearing capacity of the ring foundation is analyzed, as shown in Figure 5. It is observed that the influence of on is dependent on the . With the decrease of , the influence of on becomes more significant. In general, when , increasing fails to effectively improve the .

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. Figure 6 shows the failure patterns of the ring foundation with varied embedment depth ratios . The undrained shear strength of soil is 50 kPa, and the soil unit weight 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 , the range of the failure surface expands continuously, and the depth and width of the failure surface gradually increase.

(a)

(b)

(c)

(d)

Figure 7 shows the variations in bearing capacity coefficient versus embedment depth ratio 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 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 8 presents the fitting curve of coefficient versus . It can be seen that using a nonlinear function favorably reveals the variation characteristics of versus 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 , 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 is larger than 1.0. Similar to the study conducted by Lee et al. [16], the fitting formula is written as

Further, taking ring foundations with and 0.8 as examples, the failure patterns of ring foundations with varied are investigated, as shown in Figures 9 and 10. The soil undrained shear strength , and the soil unit weight . It is seen that failure patterns of the ring foundation at different embedment depths are affected by . With the decrease of , 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 , the inner soil failure surface develops effectively when reaches 0.75 (Figure 9(d)). On the contrary, for the ring foundation with equal to 0.8, the ring foundation has a weak constraining effect on the inner soil. The inner soil failure surface fully develops when (Figure 10(b)).

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

Figure 11 shows the variation curves of the bearing capacity coefficient versus the embedment depth ratio in different cases. In the weightless case (), is consistent with the definition of . Except for the ring foundation (), of ring foundations with varied increases gradually with increasing . However, when reaches a critical value, tends to be stable. is significantly affected by . With the decrease of , the restriction effect of ring foundations on the inner soil is enhanced, and the increases, resulting in a greater . For the ring foundation with equal to 0, increases with the 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 increases obviously. Take the foundations with and 0.2 as examples. As it can be seen from Figure 11, when the soil unit weight () is not considered and , the bearing capacity coefficient is about 9.0 and 7.6, respectively. However, when the soil unit weight and , the bearing capacity coefficient reaches 16.2 and 9.4, respectively. The increasing rates are 80% and 23.7%, respectively.

(a)

(b)

The ultimate bearing capacity of ring foundations is affected by many factors. As shown in Figure 10, 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 equal to 0.4 and 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 is small (Figures 12(a) and 12(c)), the failure surface inside the ring foundation can fully develop. With the increase of , 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).

(a) ,

(b) ,

(c) ,

(d) ,

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 Figure 13(a). Taking the ring foundation with equal to 0.9 as an example, the influence of cutting face inclination angle 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 equal to 50 kPa and weight equal to 0. The numerical analysis model is shown in Figure 13(b).

(a) Schematic diagram

(b) Numerical model

Figure 14 depicts the variation in bearing coefficient versus , in which (i.e., is the ratio of the bearing capacity of the ring foundation with cutting face inclination angle to that of the ring foundation without an inclination angle). It is obvious that the decreases with decreasing . However, when , the tends to be stable, and the bearing capacity is reduced by approximately 30% ( for ).

In addition to the influence of cutting face inclination angle , the influence of friction coefficient on the bearing capacity is investigated further, as shown in Figure 15. Taking the ring foundations with equal to 45° and 26.6° as examples, the basically changes linearly with . The smaller the is, the more significant the influence of 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 equal to 26.6° as an example, the bearing capacity of the ring foundation with a completely smooth interface () is about 0.52 times that of the ring foundation with a completely rough interface (). Therefore, for ring foundations such as open caissons, to aid the sinking process significantly, reducing the of the structure-soil contact surface is effective.

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 of , the decreases gradually, and a hyperbolic tangent function can well describe the variation in versus (2)For the ring foundation embedded in undrained clay, when the is small, the failure surface of the inner ring foundation can develop fully. With the increase of , 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, when , 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 coefficient 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 process

It 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.

Data Availability

The data are generated from numerical simualtions and can be available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors acknowledge the funding received from the Natural Science Foundation of Jiangsu Province of China (BK20210051), National Natural Science Foundation of China (51868021, 52168047, and 52108321), Natural Science Foundation of Jiangxi Province (20202BABL204051, S2020QNJJB1234), and Science & Technology Project of the Education Department of Jiangxi Province (GJJ200637) for supporting this research.

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Copyright © 2022 An-hui Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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