*Methodology Verification and Comparison of Results*

Normalized values for KV are plotted against L/D for 0.2 < L/D < 2 considering the three different ground profiles. The results are then compared to the solutions provided by Wolf and Deeks [25] and DNVGL [23] which are in turn based on the work of Gazetas [22]. Figure 7 shows the stiffness coefficients plotted for homogeneous profile at two Poisson's ratios (0.2 and 0.499). All the simulated cases are summarised in Appendix A.

**Figure 7.** Vertical static stiffness functions for rigid caissons in homogeneous ground profiles at two Poisson's ratios (**A**) υ<sup>s</sup> = 0.2, and (**B**) υ<sup>s</sup> = 0.499.

The numerical model compares well with the formulations provided in the literature which justifies the method of extraction, the mesh used, the extent of the boundary conditions, and the rigid body assumption applied in the finite element model. Any discrepancy can be justified in terms of the foundation geometry implemented; where both Wolf and Deeks [25] and DNVGL [23] used a solid embedded foundation rather than a skirted caisson with soil mass enclosed within it. Another reason could be the effect of the Poisson ratio.

The impact of Poisson's ratio on the computed stiffness coefficients was evident from the FE analysis. Therefore, normalised values for the vertical foundation stiffness coefficients in a homogeneous stratum were plotted against υs, as shown in Figure 8. The results were normalised against their respective values at υ<sup>s</sup> = 0.1. The aspect ratio (L/D) also appears to influence the effect of Poisson's ratio as also shown in Figure 8. For all L/D cases, the stiffness decreases with increasing υ<sup>s</sup> until υ<sup>s</sup> = 0.4 and then slightly increases. Both Gazetas [22] and Wolf and Deeks [25] considered the effect of the soil's Poisson's ratio on the stiffness coefficients and incorporated it in their proposed impedance functions. Their proposed functions show similarity with the trend predicted by this study, yet, the impact of the Poisson's ratio is observed to be slightly lower than the abovementioned literature. Accordingly, a correction factor f(υs), function of both υ<sup>s</sup> and L/D, will be developed in the subsequent section.

**Figure 8.** Variation of vertical stiffness (Kv) with Poisson's ratio (υs) and aspect ratio (L/D).

#### **3. Development of the Static Stiffness Functions and Correction Factors**

Functions for the vertical stiffness were developed using non-linear regression analysis on the normalised set of stiffness coefficients. A diameter of 5m was adopted and the soil stiffness at 1m depth was set as 100 (MPa), see Figure 5. The study then plots the change of the vertical stiffness with increasing L (L values include 2.5, 5, 7.5 and 10 m) and υ<sup>s</sup> (0.1 to 0.49).

First, a new correction factor due to Poisson's ratio, f(υs), will be presented. The reason for this correction is to clarify how the Poisson's ratio modifies the vertical stiffness for different ranges of caisson dimensions. Thus, Figure 8 shows that f(υs) is not only a function the Poisson's ratio itself but also a function of L/D. This has not been extensively discussed in literature where the correction factor was only a function of the Poisson's ratio itself. Consequently, revised f(υs) functions dependent on both L/D and υ<sup>s</sup> are suggested herein.

From Figure 8, the best fit curve for the Poisson's ratio correction was a cubic function in the form of f(υs)= a0υ<sup>3</sup> s−a1υ<sup>2</sup> <sup>s</sup>+a2υs+a3. The values of the coefficients a0, a1, a2, and a3 were recorded for L/D = 0.5, 0.75, 1,1.5, 2 normalized at L/D = 0.5. The coefficients for L/D = 0.5 are a0 = 10.028, a1 = -5.8814, a2 = 0.9092, and a3 = 0.96. Figure 9 shows the normalized values for a0, a1, a2, and a3, thus simulating the dependency of f(υs) on L/D. From the figure, it is evident that a0 and a1 follow a similar trend and where given the same logarithmic function whilst a2 followed a different trend and was given another logarithmic function. Finally, a constant value of 1 was given for a3.

**Figure 9.** Variation of the polynomial coefficients with L/D.

As a result, from the analysis above, the Poisson's ratio correction may be summarized using Equation (1):

$$\mathbf{f}(\boldsymbol{\upsilon}\_{s}) = \left[ \left( 10 \mathbf{v}\_{s}^{3} - 5.88 \mathbf{v}\_{s}^{2} \right) \left( -0.34 \ln \frac{\mathbf{L}}{\mathbf{D}} + 0.77 \right) \right] + 0.91 \boldsymbol{\upsilon}\_{s} \left( -0.57 \ln \frac{\mathbf{L}}{\mathbf{D}} + 0.6 \right) + 1 \tag{1}$$

It may be noted that the same methodology was repeated in parabolic and linear inhomogeneous ground profiles where the f(υs) obtained was in close proximity to the one shown in Equation (1). Similarly, a few trials on higher aspect ratios (L/D > 2) showed no noticeable change in the formulations:

Subsequently, the normalized values of KV DESOf(υs) were then computed and plotted against L/D for all ground profiles. Therefore, a best fit curve was applied in the form of a power function. Even though previous literature did not specifically use power functions for static vertical stiffness functions of shallow caissons, it still performs accurately for the rigid caissons as the R2 values shown show a good correlation as shown in Figure 10 and the solutions are summarized in Table 2.

$$\mathbf{f}(\boldsymbol{\upsilon\_s}) = \left[ \left( 10 \boldsymbol{\upsilon\_s^3} - 5.88 \boldsymbol{\upsilon\_s^2} \right) \left( -0.34 \ln \frac{\mathbf{L}}{\mathbf{D}} + 0.77 \right) \right] + 0.91 \boldsymbol{\upsilon\_s} \left( -0.57 \ln \frac{\mathbf{L}}{\mathbf{D}} + 0.6 \right) + 11$$

**Table 2.** Vertical stiffness for shallow skirted foundations exhibiting rigid behaviour.


**Figure 10.** Best fit curves for the vertical stiffness functions.

## **4. Discussion and Validation of the Results**

The functions provided in Table 2 were used to calculate KV for the simulated cases and the highest recorded percentage was found to be below 10% which is considered of low practical significance. The results are also checked against the coefficients provided by [18] and summarized in Table 3. As shown in the table, the results generally show a good match with the obtained results while being applicable for a wide range of Poisson's ratio. Slight discrepancies exist at the higher Poisson ratio, and this is explained by the difference in the Poisson's ratio correction between the proposed method and existing literature.


**Table 3.** Comparison of vertical stiffness at υ<sup>s</sup> = 0.2 and υ<sup>s</sup> = 0.499. Data from Doherty et al. (2005) [18].

#### **5. Application of the Methodology**

In order to demonstrate the application of the proposed vertical stiffness functions, a solved example in the context of the prediction of the natural frequency of an OWT system is presented in this section. A 5 MW turbine supported on a symmetrical four-legged jacket was considered for design in deep waters, as shown in Figure 11. Details about the turbine specification and an approximate jacket dimensions were found in Jonkman et al. [39] and Alati et al. [40], respectively, and summarised in Table 4.

**Figure 11.** Example problem.


**Table 4.** Main input parameters of the example (Data from Jonkman et al. [39]; Alati et al. [40]).

A homogeneous stratum over bedrock is assumed with all the foundation dimensions and soil properties summarised in Table 5.

**Table 5.** Foundation details for the example problem.


Using the equations provided in Table 2, a preliminary estimate of the vertical stiffness for a rigid caisson foundation (with L/D = 1) is obtained as shown below:

$$\begin{array}{rcl} \mathbf{f}(\mathbf{v}\_{\sf s}) &=& \left[ \left( 10 \mathbf{v}\_{\sf s}^{3} - 5.88 \mathbf{v}\_{\sf s}^{2} \right) \left( -0.34 \ln \frac{1}{\sf D} + 0.77 \right) \right] + 0.91 \mathbf{v}\_{\sf s} \left( -0.57 \ln \frac{1}{\sf D} + 0.6 \right) + 1 \\ &=& \left[ \left( 10(0.28)^{3} - 5.88(0.28)^{2} \right) \left( -0.34 \ln(1) + 0.77 \right) \right] + 0.91 (0.28) \left( -0.57 \ln(1) + 0.6 \right) + 1 = 0.97 \\ & \mathbf{K}\_{\sf V} = 2.31 \left( \frac{1}{\sf D} \right)^{0.52} \text{DE} \mathbf{g} \mathbf{f}(\mathbf{v}\_{\sf s}) = 2.31 (1)^{0.52} (4) (40) (0.97) = 0.36 \, \frac{\rm C \rm N}{\rm m} \end{array}$$

The target natural frequency of the system for a soft-stiff design should ultimately be between 0.2 and 0.35 Hz to avoid 1P/3P frequencies. Following the methodology suggested by Jalbi and Bhattacharya [15], a closed-form solution of the first natural frequency of the system which considers the soil-structure interaction can be obtained as follows:

$$\mathbf{f}\_{\mathbf{0}} = \mathbf{C}\_{\mathbf{J}} \times \mathbf{f}\_{\mathbf{fb}}$$

where ffb is the fixed base natural frequency and CJ is the foundation flexibility parameter that is dependent on the vertical stiffness of the springs (KV).

For the purpose of this explanatory example, the computed fixed base natural frequency is 0.303 Hz and the CJ is equal to 0.77. Hence, the first natural frequency of the system (with the SSI effect) is calculated to be f0 = CJ × ffb = 0.77 × 0.303 = 0.23 Hz, refer to Appendix A for the detailed calculation of the example. Although, the estimated value of the natural frequency falls within the targeted range, it is still in close proximity to the 1P frequency.

For further detailed analysis group effects can also be incorporated and readers are referred to Bordón et al. [26] which suggests formulations on how to incorporate foundation group effects correction factors which are dependent on the caisson aspect ratio, diameter and spacing. These formulations are also presented in the Appendix A. This will further lower the natural frequency and the design may have to be refined in order to allow for the additional 10% safety margin as per the DNVGL [23] recommendations. Subsequently, the FLS and ULS criteria should also be checked.

#### **6. Conclusions**

Offshore wind turbines supported on multiple shallow foundations can exhibit undesirable rocking modes of vibration strongly dependent on the vertical stiffness of the foundation. In this paper, numerical analysis is carried out to explore the axial behaviour of skirted caisson foundations for three idealised ground profiles (homogeneous, linear inhomogeneous and parabolic inhomogeneous) where the soil stiffness varies with depth. A set of static vertical stiffness functions and correction factors due to Poisson's ratio are derived. The results are validated against numerical and analytical solutions found in the literature. These formulations can be utilised for optimising the caisson's dimensions for feasibility studies purposes and preliminary design stages of a project. An explanatory example is provided to illustrate the usage of the derived formulation whereby natural frequency of a typical jacket supported OWT structure is calculated.

**Author Contributions:** Conceptualization, A.S., S.J. and S.B.; methodology, A.S. and S.J.; software, A.S.; validation, A.S. and S.J.; formal analysis, A.S. and S.J.; investigation, A.S. and S.J.; resources, S.B.; data curation, A.S.; writing—original draft preparation, A.S. and S.J.; writing—review and editing, S.B.; visualization, A.S.; supervision, S.B.; project administration, S.B.; funding acquisition, A.S. and S.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
