Keywords

1 Introduction

Suction caissons are increasingly considered to server as foundation systems for offshore wind turbines located at North Sea developments (Tjelta 2015). Suction caissons are circular steel structures, which are equipped with a lid. Suction caissons (also suction called buckets) for offshore wind turbines typically have aspect ratios between L/D = 0.5 and L/D = 1. L is the skirt length and D is the diameter. The suction caisson installation is performed in two steps: During the first phase, the caisson skirt penetrates into the soil under self-weight. The suction installation commences once a sufficient skirt penetration depth is reach and the caisson movement terminates. The water encapsulated inside the caisson is extracted by means of a pumping system. The resulting internal suction pressure provides additional downforce and yields seepage around the caisson skirt. The seepage flow reduces the skirt tip resistance. The combination of the self-weight, suction pressure and reduced tip resistance enables the caisson installation – even in dese sand (Houlsby and Byrne 2005). However, the seepage flow is expected to modify the soil plug state. particle image velocimetry (PIV) post analysis of centrifuge tests featuring a half-model caisson (Kim et al. 2016; Ragni et al. 2018; Ragni et al. 2019) visualised the effect of suction installation on the state of the sandy soil. Numerical simulations can visualise the underlying mechanisms and provide further insights. Therefore, the material point method (MPM) (Sulsky et al. 1994; Sulsky et al. 1995) is employed here to back-calculate the results obtained from the PIV post analyses results presented in Ragni et al. (2019). In order to do so, the plane strain formulation of the MPM code Anura3D 2019.1 has been extended for axisymmetric analysis as described in Fern et al. (2019).

2 Axisymmetric MPM

The axisymmetric MPM utilises a cylindrical coordinate system, where uy represents the vertical and ur is the radial displacement. The strain tensor has got four non-zero components, which are the radial strain εr, the vertical strain εy, the shear strain γ, and the circumferential strain εθ:

$$ \varepsilon_{r} = \frac{{\partial u_{r} }}{\partial r}\,\varepsilon_{y} = \frac{{\partial u_{y} }}{\partial y}\,\varepsilon_{\theta } = \frac{{u_{r} }}{r} $$
(1)
$$ \gamma_{ry} = 2\gamma = \left( {\frac{{\partial u_{r} }}{\partial y} + \frac{{\partial u_{y} }}{\partial r}} \right) $$
(2)

The distance between the location of a material point (MP) and the axis of symmetry is denoted by the radius r. The strains are linked to displacements through the following equation:

$$ \varepsilon = \varvec{B}_{\varvec{i}} \varvec{u}_{\varvec{i}} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial r}} & 0 \\ 0 & {\frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial y}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial y}} & {\frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial r}} \\ {\frac{{\partial N_{i} \left( {x_{p} } \right)}}{r}} & 0 \\ \end{array} } \\ \end{array} } \right\}\varvec{u}_{\varvec{i}} $$
(3)

The matrix Bi is formed by the gradients of the shape functions N, which contain the local coordinates for the MPs xp. In consequence, the strains in the axisymmetric formulation are given through summation across the element nodes Nn:

$$ \left( {\varepsilon_{r} } \right)_{p} = \mathop \sum \limits_{i = 1}^{{N_{n} }} u_{r,i} \frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial r} $$
(4)
$$ \left( {\varepsilon_{y} } \right)_{p} = \mathop \sum \limits_{i = 1}^{{N_{n} }} u_{y,i} \frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial y} $$
(5)
$$ \left( {\varepsilon_{\theta } } \right)_{p} = \mathop \sum \limits_{i = 1}^{{N_{n} }} u_{r,i} \frac{{N_{i} \left( {x_{p} } \right)}}{r} $$
(6)
$$ \gamma_{ry} = \mathop \sum \limits_{i = 1}^{{N_{n} }} \left( {u_{y,i} \frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial r} + u_{r,i} \frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial y}} \right) $$
(7)

The internal force f inti is calculated in a specific way in axisymmetric MPM, which is given here:

$$ \user2{f}_{\user2{i}}^{{\user2{int}}} = \mathop \sum \limits_{{p = 1}}^{{N_{p} }} \user2{B}_{\user2{i}}^{\user2{T}} \left( {x_{p} } \right)\varvec{\sigma }_{\user2{p}} \Omega _{p} $$
(8)

Np represents the number of MPs which are located inside an element in the beginning of a time step. Ωp is the integration weight of a MP and σp is the related stress tensor. The radial and vertical components are given in the following equations:

$$ f_{r,i}^{int} = \mathop \sum \limits_{p = 1}^{{N_{p} }} \left\{ {\left( {\left( {\sigma_{r} } \right)_{p} \frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial r} + \left( {\sigma_{ry} } \right)_{p} \frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial y} + \left( {\sigma_{\theta \theta } } \right)_{p} \frac{{\partial N_{i} \left( {x_{p} } \right)}}{r}} \right)\varOmega_{p} } \right\} $$
(9)
$$ f_{y,i}^{int} = \mathop \sum \limits_{p = 1}^{{N_{p} }} \left\{ {\left( {\left( {\sigma_{y} } \right)_{p} \frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial y} + \left( {\sigma_{ry} } \right)_{p} \frac{{\partial N_{i} \left( {x_{p} } \right)}}{\partial r}} \right)\varOmega_{p} } \right\} $$
(10)

Despite the presented differences, the calculation scheme of the axisymmetric formulation is similar to the procedure for a conventional plane-strain MPM, which is also presented in detail in Fern et al. (2019).

3 Numerical Model

The MPM analyses aim to visualise the mechanisms underlying the suction caisson installation. This study features a comparison to the centrifuge test results presented in Ragni et al. (2019) in order to validate the numerical model. The numerical simulations are performed with a modified version of the MPM code Anura3D 2019.1. The modifications discussed in Sect. 2 enable the simulations of the skirt penetration with frictional soil-structure interaction while seepage flow around the skirt tip is present.

The numerical model shown in Fig. 1 features the caisson that is pre-embedded in the soil. The dimensions, shown in Fig. 1a were derived under consideration of the centrifuge tests and were also found to minimise boundary effects. The two-dimensional axisymmetric model consists of triangular elements containing three MPs each in the beginning of the analysis. The mesh shown in Fig. 1b illustrates that small elements are located close to the caisson. A coarser mesh is utilised further away from the area where the largest deformations are expected. The caisson diameter is D = 5 m, the skirt length is L = 5 m and the wall thickness is t = 5 cm in prototype scale.

Fig. 1.
figure 1

Dimensions (a) and discretization (b) of the numerical model utilised for MPM analyses of the suction caisson installation in a uniform sand profile

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Horizontal displacements are prevented at the axis of symmetry and outside vertical edge of the model geometry, while vertical displacements are permitted. The bottom features zero vertical displacements boundary conditions. These three edges are also hydraulically impermeable. The displacements at the soil surface are free, while solid and liquid pressure boundary conditions are applied on the MPs as shown in Fig. 1a in order to establish the differential pressure. The caisson is modelled as a quasi-rigid body as the horizontal displacement rate is żh = 0 and a prescribed vertical displacement rate żv is utilised to control the caisson penetration.

The (jacked) self-weight penetration in the centrifuge test was performed slowly and in a controlled fashion, as it is executed in the field. This is assumed to result in drained conditions. Hence, it was possible to model the soil as a drained one-phase continuum for the jacked penetration. The course of the suction installation is determined by the pore pressure regime. Thus, a v-w-formulation is required for the simulations of the suction installation. The state variables for the soil and the liquid phase are stored in the same MP – i.e. the one martial point, two-phase MPM is employed. The mixed integration scheme is utilised in order to minimise numerical errors due to grid crossing, which can result in artificial stress and pore pressure oscillations (Jassim et al. 2013 and Fern et al. 2019).

The elastic-plastic constitutive model featuring the Mohr-Coulomb plasticity criterion is utilised. The model parameters summarised in Table 1 are determined under consideration of laboratory test results summarised in Tran (2005), Lehane et al. (2013), and Chow et al. (2018). A reduced liquid bulk modulus is utilised in order to reduce computational costs. The value given in Table 1 is a minimum value that avoids a major influence on the results (Fern et al. 2019). The initial porosity n and the interface friction coefficient µ = 0.15 are derived from the results of the centrifuge experiments discussed in Ragni et al. (2018) and Ragni et al. (2019).

Table 1. Model parameters for MPM simulations featuring fine silica sand.
Full size table

The simulations are conducted according to the following procedure: First, the effective stresses and, if necessary the hydrostatic pressure, are applied during the K0 procedure. The caisson is penetrated displacement controlled in a second step.

The caisson penetration rate żv and the internal suction pressure s are estimated under consideration of the fast caisson penetration of Test 3 (Ragni et al. 2019). The penetration rate in the numerical model is assumed to decrease linearly with ongoing caisson depth from żv = 0.2 m/s to żv = 0.08 m/s. The differential pressure is applied simultaneously, if required. This differential (or suction) pressure is calculated following the procedure presented in Houlsby and Byrne (2005) and using the parameters given in Ragni et al. (2019). A ratio of internal to external effective permeabilities ki/ko = 1 has previously been found to agree well with measurements from centrifuge tests (Bienen et al. 2018). The development of the normalised suction pressure from the prediction of Test 3 (Ragni et al. 2019) is considered: The applied normalised suction pressure starts at s/(γ’D) = 0 and increases preeningly linear towards s/(γ’D) = 1 at a relative penetration depth of ∆z/L = 1. The suction pressure is normalised by the effective specific density of the sand γ’ = 10.5 kN/m3 and the caisson diameter D = 5 m.

4 Results and Discussion

The simulated suction caisson installation is compared with experimental results obtained from PIV analyses to ensure that the MPM model captures the relevant mechanisms. The presented displacements are normalised by the considered vertical caisson displacement ∆z as referred in Ragni et al. (2018). Figure 2 compares the calculated results of a jacked caisson installation to the corresponding experimental data pretend in Ragni et al. (2019). The calculated results on the right-hand side (RHS) of Fig. 2 show a separated mechanism at shallow penetration depths, which is similar to the experiments shown on the left-hand side (LHS) of Fig. 2. A combined mechanism indicating plug heave has formed at a penetration depth of z/L = 0.3 in the experimental and the numerical results. Furthermore, symmetrically soil displacement at the skirt tip is evident in the experiment and in the numerical simulation. Hence, the numerical model is considered to be suitable for the simulation of the jacked caisson installation.

Fig. 2.
figure 2

Comparison of normalised resultant displacements during the self-weight penetration in dense sand obtained from PIV analyses (LHS, Ragni et al. 2019) and axisymmetric MPM analyses (RHS) at increasing normalised penetration depths z/L = 0.15 (a, b); z/L = 0.30 (c, d).

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The calculated results of the suction installation phase presented in Fig. 3 feature the same inverted V-shaped mechanism that was observed by Ragni et al. (2019). The resultant displacements show minor downward movement of the soil outside the caisson skirt due to the frictional interaction. Upward displacement is observed inside the caisson. The largest heave occurs in the centre of the caisson. The internal frictional interaction reduces the heave close to the skirt. This behaviour is similar to the experimental results, albeit Fig. 3d shows only minor uplift.

Fig. 3.
figure 3

Comparison of normalised resultant displacements during the suction installation in dense sand obtained from PIV analysis (LHS, Ragni et al. 2019) and axisymmetric MPM analyses (RHS) at increasing normalised penetration depths z/L = 0.28 (a, b); z/L = 0.51 (c, d).

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The simulations of the suction installation had to be re-started at a certain penetration depth, which is analogue to the PIV post analyses that were also conducted stepwise (Ragni et al. 2019). Hence, the accumulation of the plug heave and therefore the upward movement of the internal soil surface differ, but the resultant displacement contours are comparable. Consequently, it is demonstrated that the presented numerical model is suitable for the simulation of the suction caisson installation in saturated sand.

5 Concluding Remarks

An MPM model capable of the simulation of the self-weight and suction installation of a caisson foundation into sand is established. The numerical model, validated through the comparison with centrifuge test results, visualises the underlying mechanisms. The presented numerical model is expected to facilitate further investigation of the suction caisson installation and the response to monotonic and repetitive loading.