**1. Introduction**

The European Union's (EU) strategy to fight climate change and air pollution issues has accelerated the investments into sustainable energy sources. This is essentially to meet the targets of 55% reduction in the greenhouse emissions by 2030 as well as paving the way for climate neutrality by 2050 [1]. Offshore wind in Europe in particular has witnessed a substantial growth, with the UK leading the market (10,428 MW cumulative capacity) and is expected to add 15 GW capacity in the next 5 years. Countries with large offshore wind developments also include Germany (7689 MW capacity), Belgium (2261 MW capacity), and the Netherlands (2611 MW capacity) [2]. Other global leaders include mainland China (approximately 10 GW capacity) [3]. There are new entries to the market including Taiwan (through the Formosa 1 and 2 offshore wind farms) and in the final planning stages for the East Coast of the United States.

This growing demand for renewable energy is responsible for the rapid pace of the technological developments emerging in the industry. Such developments mainly target the turbine size and installations in deeper waters. Wind Europe [2] has stated that the rated capacity of OWT has been enhanced by 102% in the last 20 years. At present, the newlyinstalled turbines have an average rated capacity of 7.8 MW, though most new installments have turbine capacities exceeding 10 MW. Monopiles support 81% of all installed OWT in Europe [2]. Yet, for some locations, their design poses several engineering challenges and environmental issues to satisfy the requirements needed to support larger wind turbines in deeper waters. This explains the continuous efforts to innovate in this field. Recently, the trend in the construction of new wind farms has considered OWTs supported on jackets as an attractive alternative to conventional monopiles in deeper waters.

**Citation:** Salem, A.; Jalbi, S.; Bhattacharya, S. Vertical Stiffness Functions of Rigid Skirted Caissons Supporting Offshore Wind Turbines. *J. Mar. Sci. Eng.* **2021**, *9*, 573. https:// doi.org/10.3390/jmse9060573

Academic Editor: Eugen Rusu

Received: 27 April 2021 Accepted: 18 May 2021 Published: 26 May 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Jackets supported on piles or caissons, illustrated in Figure 1, are suitable for water depths of 30–60 m which allow them to be potentially used for future rounds of wind energy developments [4]. Aberdeen and Borkum Riffgrund 2 offshore wind farms are recent developments utilising caisson jackets foundations to support 8 MW turbines [5]. Other on-going projects include Seagreen and Zhuanghe 2.

**Figure 1.** Schematic of a 3-legged jacket on suction caissons reproduced from [6], with permission from Ørsted, 2019.

There is an inherent difference in the way monopiles and jackets resist overturning moments due to lateral loads. As illustrated in Figure 2, single foundations (typically monopiles) transfer the loads via overturning moments to the surrounding soil. On the other hand, multiple foundations such as jacket on piles/caissons mainly transfer the loads through axial push-pull interaction. This obvious difference will later drive the simple mechanical modelling of the system where the foundation is replaced by equivalent springs for analysis purposes. Hence whilst the lateral stiffness plays a major role in the dynamic performance of monopile supported offshore wind turbines, the vertical stiffness plays a more detrimental role in jackets due to the propensity of rocking type of vibrations. It has shown by Jalbi and Bhattacharya that low-frequency rocking modes of jacket vibration must be avoided as it may coincide with the low frequency 1P rotor frequency and more importantly the peak wave frequency [7].

**Figure 2.** Load transfer in different foundation systems.

Current design aims to place the natural frequency of the bottom-fixed structures within the soft-stiff band, see Bhattacharya [4] for fundamentals of design. It is well established in literature that the natural frequency of the system is reliant on the support condition (i.e. foundation stiffness) which in turn is a function of the properties of the foundation and subsoil. Considering the dynamic sensitivity of the OWTs, modes of vibration are considered a key element in the design procedure. Similar to the load transfer process discussed above, the modes of vibration of an OWT system are primarily dependent on the foundation and superstructure stiffness [8].

Studies carried out by Bhattacharya et al. [9] showed that the first eigenfrequency of vibration for OWTs supported on multiple shallow foundations (such as jackets on three or four suction caissons) correspond to low frequency rocking modes of vibration about the principle axes. The work is based on scaled model tests on three types of foundations: monopiles, tetrapods (4-legged jacket on caissons), and asymmetric tripods (3-legged seabed frame on caissons). Rocking modes of vibration are also reported in offshore structures such as the Brent B Condeep platform, see [4].

As mentioned before, rocking modes of vibration tend to have a lower frequency and may interfere with the 1P (rotor) frequency range and wave frequency, see Figure 3 for schematics. This is particularly challenging for large turbines where the soft-stiff target frequency is shifting towards the wave frequency. For example, a typical 8 MW turbine will have a target of 0.22 Hz and a 12 MW turbine will have a target frequency in the range of 0.15 Hz. Furthermore, wave loads will have a higher energy of excitation and may impose serious fatigue damage on the structure if rocking modes are allowed. It is therefore advisable to avoid rocking modes for jackets supported on shallow foundations. In addition, for asymmetric arrangements, scaled model tests showed that they have experienced two closely-spaced natural frequencies associated with the rocking modes of vibration [10]. This corresponds to the variability of the ground reflected in the vertical stiffness of the foundation. Not only does it widen the range of frequencies that can be excited by the loading conditions but also may introduce an additional design problem such as the beating phenomenon and both can have an impact in the fatigue limit state. Moreover, through analytical methods, Jalbi et al. [11] and Jalbi and Bhattacharya [7] showed that that a jacket may be engineered towards a no-rocking solution by optimising two parameters: (a) ratio of vertical stiffness of the foundation stiffness to lateral superstructure stiffness; and (b) aspect ratio of the jacket-tower geometry. A low value of vertical foundation stiffness values together with a low aspect ratio will promote a rocking mode of vibration.

**Figure 3.** Vibration modes in different foundation systems.

From the discussions above, it is essential to have a method to calculate the vertical stiffness of foundations early on in the design stages of a project. Hence, given the importance of SSI on the dynamic performance, the objectives of the paper are as follows:


It should be noted 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 scarce. As the design progresses from conceptual to detailed design, a higher computational complexity of the analysis is required to further optimise the foundations. This includes using refined soil constitutive models incorporated in 3D finite element analysis (FEA) packages. In addition, this would also require more input such as site-specific ground investigations and geotechnical laboratory testing.
