## 1 Introduction  
Sustainable energy sources such as wind, hydro and solar are critical for the planet and are currently underutilized. Global warming, increasing CO\({}_{2}\) in the atmosphere and fears of increasing frequency and severity of natural disasters such as extreme weather, flooding and tsunamis have driven government bodies, research institutes and industry to move at an increased pace towards cultivating more energy from these sources, in a bid to reduce dependency on fossil fuels. Specifically, exponential growth of wind energy has been achieved from onshore and offshore wind farms (OWF), with individual offshore wind turbines (OMTs) with a capacity of 14 MW reaching the market [1]. Figure 1 shows the energy output from offshore wind farms in Europe between 2000-2020. This increase in capacity has been realized using heavier and taller turbines, often leading to very large foundation systems, which can constitute up to 40% of the total cost [2; 3; 4]. Different foundation systems are employed to support wind turbines, such as fixed gravity base foundations (GBF), tripod structures, jackets, suction caissons, monopiles (MP) and buoyant fixed structures [6; 7; 8; 9]. Figure 2 presents some of the foundation systems currently used to support OWT. The foundation-design process is iterative in nature and begins with an estimate of the lateral applied loads from the wind, waves and currents, in addition to the aerodynamic forces. Since the geometry of the foundation is unknown, the size is usually assumed initially and then the foundation response is evaluated. If the assumed foundation geometry is insufficient to meet the ultimate or serviceability limit states, a new geometry is assumed. This loop repeats itself until a suitable foundation geometry is found [3]. Extensive records of the performance of monopiles from other offshore structures, such as oil platforms, have aided engineers with the design of early forms of OWFs. For these piles, the design is based on the famous p-y curve approach [10; 11; 12; 13; 14]. In this approach, the soil layers are replaced with independent springs, each having a different load-displacement curve, known as the p-y curve. This approach was first proposed by [12] and validated by [15], for the case of a slender pile in soft soil, and then by [16] who proposed the p-y formulae for stiff soils. However, a few issues have been raised  
Figure 1: Capacity of wind farms installed in Europe [5]. Figure 2: Available foundation options for offshore wind turbines. (**a**) GBF, (**b**) monopile, (**c**) suction caisson, (**d**) tripod/tetrapod piles, (**e**) tripod/tetrapod suction caissons, (**f**) multiple foundation options, and (**g**) guys with anchors [10]. when applying this method to offshore wind turbines. Firstly, the p-y approach was validated [15] by testing slender piles in soft soil with a diameter of only 0.32 m and a length of 12.8 m, giving an L/D ratio of 39.5. However, a typical offshore wind turbine has an order of magnitude of a larger diameter monopile with a lower L/D ratio. Numerous researchers have pointed out that diameter effects were not included in the original method (e.g., [10; 17; 18; 19; 20; 21; 22]. Secondly, other deficiencies arise from the fact that the tested pile only experienced 20 cycles of loading, whereas the expected number of load cycles experienced by an OWT in its lifetime can be of the order of \(10^{7}\). Thirdly, the ratio of vertical load to lateral load for oil and gas platforms is much higher than that for wind turbines, making direct application of current American Petroleum Institute (API) recommendations for monopile design questionable. Finally, the effect of cyclic loading is treated uniformly along the pile depth with a reduction factor A of 0.9, irrespective of load level [23]. Improved p-y formulae have since focused on closing these gaps in the literature, such as the recent PISA project.