1 Introduction

Cylindrical caissons, also known as skirted or bucket foundations, have been proven to be an effective foundation system for bridges, piers, floating platforms and other offshore foundations in deep water (Deng et al. 2020; Mello et al. 2021; Sales et al. 2021). Caisson is generally made from a large steel thin-walled cylindrical structure that is open at the bottom and closed at the top. A comprehensive review of the offshore foundation was presented in Randolph and Gourvenec (2011), who reported that the depth ratio between the diameter and the depth of caissons is in the range of 1–6.

Early studies on the behaviour of caissons in cohesive soils were conducted by using field experiments and centrifuge model tests (Andersen et al. 1993; Dyvik et al. 1993; Clukey and Morrison 1993; Cauble 1996). Numerical methods such as using the displacement-based finite element and the finite element limit analysis were also reported by numerous researchers (see e.g., Bransby and Yun 2009; Geer 1996; Gourvenec 2008; Gourvenec and Barnett 2011; Mana et al. 2013; Ukritchon et al. 2018; Keawsawasvong et al. 2021; Keawsawasvong and Lawongkerd 2021; Keawsawasvong and Ukritchon 2016; Ukritchon and Keawsawasvong 2016; Yun and Bransby 2007). Most of the research work done was for the bearing and pull-out capacity of caissons in undrained clays. Very few works were reported in relation to soils in drained condition. In addition, the studies on the behaviours of open caissons were also considered by Lai et al. (2020, 2021) and O’Dwyer et al. (2018, 2020).

The bearing capacity of strip foundations with structural skirts was presented by Al-Aghbari and Mohamedzein (2004) who performed a series of model tests on these footings embedded in dense sand. Later, Al-Aghbari and Dutta (2008) and Eid et al. (2009) conducted several experiments to investigate the behaviours of square skirted footings in sand. A numerical finite element analysis was employed by Eid (2013) to evaluate the bearing capacity and settlement of skirted shallow foundations on sand. The effect of different types of sands (e.g., medium dense and dense sands) on the bearing capacity of circular skirted footings in sand was examined by Wakil (2013). Khatri et al. (2017) later performed a series of small-scale model test to study the behaviour of rectangular and square skirted footings on sand. Using the finite element limit analysis (FELA), Khatri and Kumar (2019) proposed the depth factor for soil weight Fγd for circular and strip skirted footings with small embedment depth ratios (L/D) varying from 0 to 2.

Very recently, Shiau and Al-Asadi (2020a, b) adopted a stability factor approach, that is analogous to the traditional bearing capacity problem, to study the drained stability solutions of underground tunnelling. The method has been proven to be both efficient and effective. So far, there is no comprehensive results of the depth factors (Fcd, Fqd and Fγd) for the caisson problems in the literature. Following the stability factor approach in Shiau and Al-Asadi (2020c), the focus of this study is to assess the ultimate bearing pressures on cylindrical caissons in cohesive-frictional soils using the advanced upper and lower bound limit analysis. Comprehensive bearing capacity factors (Nc0, Nq0 and Nγ0) and depth factors (Fcd, Fqd and Fγd) are presented in tables and figures to assist designers and practising engineers in calculating the critical pressure that can apply to the caissons in their preliminary stage of design. Note that the bearing capacity factors (Nc0, Nq0 and Nγ0) are used for the cases of circular footings resting on the surface of soils. To consider the circular skirted footings with an embedment depth (L/D > 0), the depth factors (Fcd, Fqd and Fγd) are then adopted. This will be demonstrated in the example section of the paper.

2 Stability Factor Approach

The problem definition of a cylindrical caisson in axisymmetry (AX) for a cohesive-frictional soil is shown in Fig. 1. The caisson has a diameter D and an embedment depth L. The ground surface is subject to a vertical surcharge (q), while a bearing pressure at the top of the caisson is (qu). The soil medium obeys the Mohr–Coulomb failure criterion with three parameters including cohesion (c), unit weight (γ), and friction angle (ϕ).

Fig. 1
figure 1

The axisymmetric problem of a caisson

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Terzaghi’s bearing capacity equation, as shown in Eq. (1), can be used to calculate the ultimate uniform pressure qu applied at the top cap of a 2D plane strain caisson when L = 0.

$$q_{u} = cN_{c}^{{}} + qN_{q}^{{}} + 0.5\gamma DN_{\gamma }^{{}}$$
(1)

where c is the soil cohesion; q is the surcharge loading; γ is the soil unit weight; Nc is the 2D plane strain bearing capacity factor for cohesion; Nq is the 2D plane strain bearing capacity factor for surcharge loading; Nγ is the 2D plane strain bearing capacity factor for soil weight.

Equation (1) is further modified to Eq. (2) to include the axisymmetric effects in 3D using (Nc0, Nq0 and Nγ0) as well as the depth factor (L/D) effects using (Fcd, Fqd and Fγd).

$$q_{u} = cN_{c0}^{{}} F_{cd}^{{}} + qN_{q0}^{{}} F_{qd}^{{}} + 0.5\gamma DN_{\gamma 0}^{{}} F_{\gamma d}^{{}}$$
(2)

where (Nc0, Nq0, and Nq0) are the axisymmetric bearing capacity factors for cohesion, surcharge, and soil weight respectively of a surface circular footing (L/D = 0). The three bearing capacity factors (Nc0, Nq0 and Nγ0) are a function of only soil internal friction angle (ϕ). On the other note, the depth factors (Fcd, Fqd and Fγd) of cylindrical caissons are a function of both the soil internal friction angle (ϕ) and the depth ratio (L/D) of the caissons. Interestingly, the three depth factors are unity when L/D = 0, and Eq. (2) reduces to Eq. (3) for the problem of surface circular footings.

$$q_{u} = cN_{c0}^{{}} + qN_{q0}^{{}} + 0.5\gamma DN_{\gamma 0}^{{}}$$
(3)

As stated in Shiau and Al-Asadi (2020b), to obtain the respective factors such as (Nc0, Nq0, and Nγ0) and (Fcd, Fqd, and Fγd), it is necessary to impose zero value to c (cohesion) or q (surcharge) or γ (soil unit weight). This procedure works well to obtain individual pressure qu for cohesion, surcharge, and unit weight. They are then used to calculate the respective factor of a given depth ratio (L/D). The factor Nc0 can be obtained by assuming no surface pressure (q = 0) and weightless soil (γ = 0) whereas the factor Nq0 be acquired by assuming no cohesion (c = 0) and weightless soil (γ = 0). In addition, the factor Nγ0 can also be obtained by assuming no cohesion (c = 0) and surface pressure (q = 0). The principal of superposition has been discussed and validated by Shiau and Al-Asadi (2020a), and in this paper again, has been successfully applied to the examples in a later section.

3 FELA Model

Recent advances in Finite Element Limit Analysis (FELA) with both upper bound (UB) and lower bound (LB) estimates are powerful as they can provide an error indicator towards the true collapse load (Sloan 2013). The development began with linear programming in Sloan (1988, 1989). Nonlinear programming formulations provide better solution accuracy with shorter cpu solution time (Lyamin and Sloan 2002a, b; Krabbenhoft et al. 2007). Recently, the FELA (Optum CE 2021) has been widely used to solve a variety of drained and undrained stability problems in geotechnical engineering (Shiau and Smith 2006, Shiau et al. 2016a, b, c, 2021a, b; Shiau and Al-Asadi 2020c, d, 2021; Keawsawasvong and Ukritchon 2017a, b, 2020, 2021; Ukritchon and Keawsawasvong 2017a, b, 2019a, b, 2020a, b; Ukritchon et al. 2019, 2020; Yodsomjai et al. 2021) and it was chosen in this study to compute the bearing capacity factors (Nc0, Nq0 and Nγ0) and the depth factors (Fcd, Fqd and Fγd) of the cylindrical caissons in axisymmetric condition.

In the LB method, three-node triangular elements are used in the analysis. Each triangular element has the nodal stresses of σr, σz, σθ, and τrz for an axisymmetric problem. The statically admissible stress discontinuities are allowed for producing the continuity of normal and shear stresses along with the interfaces of all the elements. The conditions of stress equilibrium, stress boundary condition, and the Mohr–Coulomb failure criterion are the constraints in a typical LB analysis, in which the objective function is to maximize the collapse load of problems. On the other hand, the upper bound theorem requires a kinematically admissible velocity field where the external work is greater or equal to the plastic shear dissipation. In the UB method, three-node triangular elements are used in the formulation. At each node of the element, there are the horizontal (u) and vertical velocities (v) defined as the basic unknown variables. The setting of kinematically admissible velocity discontinuities is applied at the interfaces of all the elements. The material is set to obey the associated flow rule. These two theorems are perfectly fitted to the nonlinear programming optimization problems using the second-order cone programming (SOCP). The constraints involved in this procedure are nonlinear and non-smooth but remain convex and amenable to analysis.

Since the problem of an cylindrical caisson can be modelled under an axisymmetric (AX) condition, only half of the domain is employed in the simulation, as shown in Fig. 2. The line of axial symmetry is set to be located at the left of the domain. Noting that the underlying bound theorems assume a rigid-perfectly plastic material with associated plasticity, the soil mass is discretised as triangular elements and modelled as Mohr–Coulomb material. The caisson is modelled by using rigid plate elements. The interface condition at the contact surface between the caisson and the soil is set to be fully rough. The feature “Fan Mesh” in the program is activated at the tip of the caisson to improve the solution accuracy (Krabbenhoft et al. 2015). The bottom boundary of the model is fixed in both horizontal and vertical directions, while the left and the right boundary can move in the vertical direction. The domain size was chosen to be large enough so that the solution is not affected by the development of the overall velocity field.

Fig. 2
figure 2

Numerical model under an axisymmetric condition in OptumG2

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An automatically adaptive mesh refinement was employed in both the UB and LB simulations to compute the tight UB and LB solutions of the ultimate pressure qu. Figure 3 presents a typical example of the mesh refinement technique for a depth ratio of (L/D) = 2. The initial FELA mesh with approximately 5,000 elements is shown in Fig. 3a, whilst the final mesh with 10,000 elements is shown in Fig. 3b. It should be also noted that all presented numerical results hereafter are the average solutions from LB and UB FELA after the five adaptive mesh refinement steps with approximately 10,000 elements.

Fig. 3
figure 3

A typical mesh for Fcd solution (L/D = 2). a Initial setting with 5,000 elements; b final setting with 10,000 elements

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4 Discussing the Axisymmetric Factors N c0, N q0, and N γ0

Numerical results of the axisymmetric (AX) cohesion factor Nc0 (L/D = 0) with soil frictional angles ϕ varying from 5° to 35° are presented in Fig. 4. Noting that Nc0 increases nonlinearly with the increasing ϕ, they are in excellent agreement with the slip line solutions in Martin (2004) and Kumar and Khatri (2011). As discussed in Sloan (2013), the slip line solutions by Martin (2004) can be considered as nearly exact solutions. The published Nc0 results of Terzaghi (1943), Meyerhof (1963), and the displacement-based finite element of Chavda and Dodagoudar (2019) are consistently lower than those in the present study and they are considered as conservative solutions.

Fig. 4
figure 4

Variation and comparison of Nc0 with ϕ for surface cylindrical caissons (L/D = 0)

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Figure 5 presents numerical results of the axisymmetric (AX) surcharge factor Nq0 (L/D = 0) with ϕ varying from 5° to 35°. Similar trends as in the cohesion factor Nc0, the nonlinearly increasing results have compared well with published ones, except those conservative solutions of Terzaghi (1943), Meyerhof (1963) and Chavda and Dodagoudar (2019). For the axisymmetric (AX) soil unit weight factor Nγ0 (L/D = 0), again, numerical results in Fig. 6 have suggested great confidence in using the present averaged results of rigorous upper and lower bounds. Interestingly, the finite difference results reported by Zhao and Wang (2008) are unsafe (or unconservative) as they are consistently greater than our solutions. The values of Nc0, Nq0, and Nγ0 are explicitly shown in Table 1. It should be noted that the all presented results are in good accordance with the research of Kumar and Khatri (2011) since this study and Kumar and Khatri (2011) employed the same FELA method to solve the solutions of circular foundation problems. However, Kumar and Khatri (2011) only proposed the bearing capacity factors for surface circular foundations. The solutions of caissons with a given depth ratio (L/D) have never been presented in the past. To the best of our knowledge, this study is the first work to consider the depth factors Fcd, Fqd, and Fγd for the caisson problem.

Fig. 5
figure 5

Variation and comparison of Nq0 with ϕ for surface cylindrical caissons (L/D = 0)

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Fig. 6
figure 6

Variation and comparison of Nγ0 with ϕ for surface cylindrical caissons (L/D = 0)

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Table 1 Axisymmetric bearing capacity factors Nc0, Nq0 and Nγ0 (L/D = 0)
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5 Discussing the Depth Factors F cd, F qd, and F γd

Numerical solutions of the axisymmetric (AX) cohesion depth factor Fcd with soil frictional angles ϕ varying from 5° to 35° are presented in Fig. 7 for (L/D = 0 to 6). The value of Fcd starts from one at L/D = 0, and it increases linearly with the increase in L/D for all considered fiction angles ϕ = 5° to 35°. The larger the ϕ, the greater the Fcd. The effect of ϕ is insignificant when ϕ > 30° since both lines of ϕ = 30° and 35° overlap perfectly.

Fig. 7
figure 7

The depth factor for cohesion Fcd with various L/D and ϕ

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Figure 8 shows that, for all friction angles, the surcharge depth factor Fqd increases as L/D increases. This increase is approximately linear. The deeper the caisson is, the larger the Fqd (surcharge effect) is. The greater the ϕ, the larger the Fqd. For the soil unit weight depth factor Fγd, see Fig. 9, a highly non-linear relationship between Fγd and L/D is observed. An increase in L/D yields an increase in Fγd. Noting that Fγd is the smallest when ϕ = 35°, whilst the largest when ϕ = 5°. This is dissimilar to the other factors such as Fcd and Fqd. A possible explanation for this might be due to the local punching failure mechanism near the end bearing point of the deep foundation. The complete values of (Fcd, Fqd, and Fγd) are explicitly presented in Tables 2, 3 and 4. Using the bearing capacity factors (Nc0, Nq0, and Nγ0) and the depth factors (Fcd, Fqd, and Fγd), Eq. (2) can be used by practical engineers to evaluate the ultimate capacity qu of cylindrical caissons in cohesive-frictional soils.

Fig. 8
figure 8

The depth factor for surcharge Fqd with various L/D and ϕ

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Fig. 9
figure 9

The depth factor for unit weight Fγd with various L/D and ϕ

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Table 2 The depth factor for cohesion Fcd
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Table 3 The depth factor for surcharge Fqd
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Table 4 The depth factor for unit weight Fγd
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Figure 10 compares the unit weight depth factor Fγd between the present study and the footing embedment study by Lyamin et al. (2007), as well as the study of skirt foundation on the sand by Khatri and Kumar (2019). Note that the work by Khatri and Kumar (2019) was for cylindrical skirt footings with the soils inside the skirt, whereas Lyamin et al. (2007) studied rigid embedded footings without the soils inside. The comparison shows a good agreement between the present results with those in Khatri and Kumar (2019). Though not entirely the same as our current study, the solutions provided by Lyamin et al. (2007) are conservative as they predict lower Fγd factors than those in Khatri and Kumar (2019) and in this study.

Fig. 10
figure 10

Comparison of Fγd

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6 Examples

Several examples are presented in this section to demonstrate how to use the produced results to evaluate the uniform bearing capacity of cylindrical caissons by using the formulation shown in Eq. (2). The principal of superposition using the stability factor approach is also validated with the following examples.

6.1 Example 1: Cohesionless Soil Without Surcharge Loading

A cylindrical caisson has the length L = 12 m and the diameter D = 3 m. The design parameters are given as: the unit weight γ = 18 kPa and the soil internal friction angle ϕ = 35°. The soil cohesion c is zero since the soil is cohesionless. The surcharge loading q is also zero in this example. Given ϕ = 35°, the value of Nγ0 = 41.298 is obtained from Table 1 and Fγd = 73.189 is also obtained from Table 4 using L/D = 4. Note that the values of Nc0, Nq0, Fcd and Fqd are not required since c = 0 kPa and q = 0 kPa. Using Eq. 2, the ultimate uniform pressure can be then calculated as: qu = (0.5 × 18 × 3 × 41.298 × 73.189) = 81,609.10 kPa.

An actual computer analysis of the problem gives qu = 81,672.279 kPa, which is very close to the solution using stability factors and the principal of superposition.

6.2 Example 2: Cohesionless Soil with Surcharge Loading

Same as in Example 1, now the surcharge loading q = 20 kPa. From Table 1, given ϕ = 35°, the values of Nq0 = 79.893 and Nγ0 = 41.298. The values of Fqd = 7.746 and Fγd = 73.189 can also be obtained from Tables 3 and 4 respectively for ϕ = 35° and L/D = 4. Note that the values of Nc0 and Fcd are not required since c = 0 kPa in this example. Using Eq. 2, the ultimate uniform pressure can be then calculated as: qu = [(20 × 79.893 × 7.746) + (0.5 × 18 × 3 × 41.298 × 73.189)] = 93,986.13 kPa.

An actual analysis in the program using the real parameters gives qu = 95,164.657 kPa, which is 1.2% greater than the solution using stability factors and the principal of superposition.

6.3 Example 3: Cohesive-Frictional Soil Without Surcharge Loading

In this example, the cylindrical caisson has a diameter D = 4 m (i.e. L/D = 3). The design parameters are given as the unit weight γ = 16 kPa, the soil internal friction angle ϕ = 10°, and the soil cohesion c = 25 kPa. There is no surcharge loading at the ground surface so that q = 0 kPa. From Table 1, given ϕ = 10°, the values of Nc0 = 11.053 and Nγ0 = 0.379 are obtained. The values of Fcd = 4.852 and Fγd = 99.456 are also obtained from Tables 2 and 4 respectively for ϕ = 10° and L/D = 3. Note that the values of Nq0 and Fqd are not required since q = 0 kPa. Using Eq. 2, the ultimate uniform pressure can be then calculated as: qu = [(25 × 11.053 × 4.852) + (0.5 × 16 × 4 × 0.379 × 99.456)] = 2,546.93 kPa. An actual computer analysis of the problem gives qu = 2,567.307, which is very close to the solution using stability factors and the principal of superposition.

The above examples have proven that the principal of superposition with the stability factors and the depth factors can be used to evaluate the unltimate bearing pressures of caissons.

7 Conclusions

This paper has successfully produced lower and upper bound solutions of the bearing capacity factors (Nc0, Nq0, and Nγ0) and the depth factors (Fcd, Fqd, and Fγd) for cylindrical caissons in cohesive-frictional soil. The following conclusions are drawn based on this study:

  1. 1.

    The axisymmetric bearing capacity factors (Nc0, Nq0, and Nγ0) for a surface footing are a function of only the soil friction angle (ϕ). An increase in the soil friction angle (ϕ) results in an increase in all three bearing capacity factors (Nc0, Nq0, and Nγ0).

  2. 2.

    The depth factors (Fcd, Fqd, and Fγd) are functions of both the soil friction angle (ϕ) and the depth ratio (L/D). An increase in the depth ratio (L/D) causes an increase in all depth factors (Fcd, Fqd, and Fγd).

  3. 3.

    For the unit weight depth factor, Fγd, an increase in the soil friction angle (ϕ) results in a decrease in Fγd. This is different from Fcd and Fqd, where they increase with the increasing soil friction angle (ϕ). A possible explanation for this might be due to the local punching failure near the end bearing point of the foundation.

  4. 4.

    The present solutions of bearing capacity factors (Nc0, Nq0, and Nγ0) are in good agreement with the previous solutions using the method of characteristic. In addition, the present solutions of the depth factor Fγd are also in good agreement with those published ones.

  5. 5.

    The illustrated examples using the principal of superposition have proven that the use of the bearing capacity factors (Nc0, Nq0, and Nγ0) and the depth factors (Fcd, Fqd, and Fγd) is both convenient and accurate.

  6. 6.

    The proposed rigorous solutions of the bearing capacity factors (Nc0, Nq0, and Nγ0) and the depth factors (Fcd, Fqd, and Fγd) are useful and they can be used by pratising engineers to evaluate bearing capacity requirements of caissons in cohesive-frictional soils with great confidence.

The results in this study are applicable to skirted circular footing subjected to drained loading. For the open caissons which are generally sunken within the ground and have an opening at the bottom and top (during sinking), the current drained solutions may not be suitable. In addition, the interface between caissons and soils is only set to be fully rough, where the adhesion factor is set to be one. The influence of the adhesion factor on the bearing capacity factors may be significant, and future works should consider the full range of the adhesion factor (0 to 1) for more realistic simulations.