*Background Literature*

Ideally, each assessment of the composite system should encompass an independent numerical analysis for the structure and foundation. For instance, the analysis of the latter would likely involve modelling the soil as continuum which is typically carried out using advanced geotechnical finite element methods. However, the limitation of the high computational cost and modelling complexities make it impractical to be utilised in preliminary design stages, yet useful in verifying the final design of the foundation. Consequently, both approaches (i.e., analytical and numerical solutions) tend to idealise the structural dynamics problem through replacing the foundation by a set of lumped springs or in the case of deep foundations distributed springs. The overall stability and foundation stiffness can be purely expressed in terms of functions that describe the force resultants and their conjugate displacements and rotations of these lumped springs. Figure 4 illustrates the breakdown of the structure-foundation problem.

**Figure 4.** Breakdown of the structure-foundation problem.

The work of this paper continues the efforts of the research group which aims at providing simplified expressions for the computation of the foundation stiffness. For instance, Shadlou and Bhattacharya [12] studied the lateral dynamic stiffness of deep foundations and proposed spring stiffness functions for both rigid and flexible monopiles. In their model, the foundation is replaced by four springs; KL (lateral spring), KR (rocking spring), KV (vertical spring) and KLR (cross-coupling spring) to capture the degrees of freedom. This methodology was later utilised by Arany et al. [13] in designing monopile foundations. Similarly, Jalbi et al. [14] obtained static stiffness functions for the lateral stiffness terms of a rigid monopod caissons. Moreover, Jalbi and Bhattacharya [15] provided closed form solutions to calculate the natural frequency of jackets supported on multiple foundations incorporating soil-structure interaction (SSI). The foundation flexibility was represented by a set of vertical springs which emphasizes the importance of predicting the vertical stiffness of foundations. Thus, it is now essential to continue the work and obtain the vertical stiffness components of the foundations.

The assessment of static and dynamic vertical spring constants has been the subject of extensive studies in the field of machine foundations and seismic analysis. Most of the elastic solutions available in the literature provide guidance regarding surface and embedded footings. Generally, most literature reports that stiffness decreases with increasing strains and increasing forcing frequencies. However, work on the elastic and non-linear stiffness of a skirted caisson is inadequate. Bell [16] presented a comprehensive review of the existing stiffness coefficients of surface footings, whereas the effect of embedment of the circular footings was extensively discussed in Gazetas [17]. On the other hand, there is less work assessing the stiffness of suction caisson foundations. These foundations are quite similar to the embedded-type foundations with the difference of the soil mass is trapped beneath the lid and within the enclosed volume. Two extreme models could be adopted to represent the caissons: one in which the lid is treated as a rigid circular foundation on the surface while ignoring the effect of the skirts, and another in which the caisson is completely rigid. The latter was analysed by Doherty et al. [18] who provided tabulated coefficients for completely rigid caisson. The analyses incorporated practical variation in soil stiffness, embedment depth and Poisson's ratios. It also provided correction factors to account for the skirt flexibility. Skau et al. [19] focused on the effect of caisson flexibility following the observations of the extensive behaviour monitoring for Borkum Riffrung 01 -Suction Bucket Jacket (BKR01-SBJ). An elastic correction to the response of a rigid foundation response was suggested to address the foundation flexibility particularly due to the lid which appeared to significantly influence the total vertical stiffness of the system. Table 1 summarises the vertical stiffness formulae found by different researchers for different foundation types including some additional guidance found for deep foundations.




**Table 1.** *Cont.*

N.B: KV is vertical stiffness of the foundation, Gs is shear modulus of the soil, *y*<sup>s</sup> is Poisson's ratio of the medium, D is diameter, L is embedment depth, H is thickness of the soil layer and Ep is modulus of elasticity of pile material.

> From the table above, it is evident that the available methodologies are limited either by the shape of the footing and the idealised soil profiles which do not reflect the actual heterogeneity in the soil. This paper aims to tackle one aspect of that where solutions are provided for rigid caissons through numerical modelling. The solutions provide the vertical stiffness Kv are for homogeneous, parabolic, and linear ground profiles.

#### **2. Numerical Modelling**

Finite element method using Plaxis 3D (continuum approach) was utilised to model the soil-structure interaction. The size of the soil contour is specified such that any stress increase on the boundary is absorbed without rebounding and disturbing the model results. Suryasentana et al. [29] presented a mesh domain of 80D (D is the diameter of the suction caisson) for both diameter and depth to analyse vertically loaded foundations, while Latini et al. [30] used 100D and 30D for the diameter and depth respectively. Moreover, Sloan [31] adopted 5D for the mesh dimensions when analysing vertically loaded rigid circular footing. This clearly shows the wide range of possibilities to eliminate the boundary effects. Considering the scope of this analysis, all the models have been set up with an extent of the soil domain 10D and depth of 15D; as shown in Figure 5 (which has been obtained through trial and error). Despite the symmetry of the problem, a full model was adopted as to avoid a rotation of the caisson if the point load was placed at the centre.

**Figure 5.** Mesh dimensions and shape of caissons.

Generally, the stiffness of the foundation dictating the dynamic stability of the system, is characterised by a non-linear nature. It is dependent on the strain levels; generated from the load cycles due to the soil-structure interaction, as well as the forcing frequency (expressed in terms of static and dynamic stiffness) [32]. Since the natural frequency is associated with relatively small amplitude of vibrations (linear range), the initial foundation stiffness would be sufficient for this purpose [13]. Similarly, OWTs are considered a very low frequency application compared to seismic actions based on design charts provided in [12,20,33–36]. Hence, the effect of the forcing frequency on the stiffness values can be ignored and the static stiffness value can effectively be adopted. Based on the above justifications, the soil is modelled as a linear elastic material. This model is based on Hooke's law of isotropic elasticity and requires the identification of two basic elastic parameters; Young's modulus (E) and Poisson's ratio (vs) which can be determined using conventional site investigation techniques [37].

To encompass the realistic variation of the shear modulus of the soil with depth, three idealisations of the ground profile were adopted:


Figure 6 illustrates the three ground profiles described above where the values of Young's modulus for the different ground profile intersect at one diameter depth. For all the cases, Poisson' ratio was assumed to be uniform within each model and the soil density was set to a constant value of 18 kN/m3.

The structure forming the foundation, which consists of the skirt and the lid, has been treated under the assumption of being rigid for all the models. In other words, the response of the foundation system due to the applied loads is solely due to the deformations in the soil (no structural deformations of the caisson lid and skirt). This also includes the soil enclosed within the skirt. This assumption is considered valid considering the low aspect ratio of the caissons modelled and the high flexural and shear stiffness of the steel compared to those of the soil. Doherty et al. [18] investigated the effect of the skirt flexibility on the response of a caisson foundation. Results showed that the vertical stiffness values of both cases are almost similar for low aspect ratios of the caisson which reinforces and validates the previous assumption. In addition, the bucket lid can in-reality deflect and alter the foundation stiffness where the recent in-situ observations by Shonberg et al. [38] confirmed that the suction bucket's structural elements stiffness has an effect on the performance and can be idealised as pair of vertical springs in series, such that <sup>1</sup> ktotal <sup>=</sup> <sup>1</sup> klid <sup>+</sup> <sup>1</sup> ksoil−skirt . However, for simplicity, a lumped vertical spring compiling both elements is adopted for this study rather than treating them separately.

**Figure 6.** Variation in stiffness with depth.

Furthermore, the push-pull nature for OWT supported on jackets requires the estimation of the stiffness in both tension and compression, yet this study assumes that the vertical stiffness is the same for both tension and compression as the intention is to use these values for low amplitude vibrations. In reality, the computed compressive stiffness should be higher due to the additional contribution of the bearing below the lid. Other factors should further be investigated involving the impact of the grouted connections and its imperfections, interaction between the adjacent jacketed caissons and imperfect contact at the interface of the soil and foundation. All these factors form a strong basis for the continuation of the work produced in this study. Nevertheless, it may be reminded that the solutions provided in this paper are intended for the concept design stage and for initial sizing of the foundation when information about the structure and the ground profile is limited.
