## 1 Introduction  
As the demand for renewable energy continues to grow globally, construction of larger wind turbines (WTs) is foreseen. To improve the long-term safety and performance of the WTs and to reduce their costs, research on the design and dynamic behavior of their foundations has been the focus of many studies in recent years. This paper addresses a behavior type of shallow footings termed \"rocking\", which refers to repeated tilting rotations to an extent that may cause repeated partial uplifts of its base from the ground. Rocking of objects on the ground has been studied for various applications, as summarized below, and has recently been investigated for its effects on WT foundations [1, 2, 3]. Rocking of rigid blocks and flexible structures has been investigated for its implications for safety of equipment [4], valuable objects [5], masonry elements of buildings [6], and structural systems [7]. [PERSON] et al. [8] showed that a rocking foundation can reduce the structural damage due to the base excitation of a shear wall by acting as a method of base isolation. [PERSON] [9] and [PERSON] and [PERSON] [10] showed that rocking of a multi-storey building on its base alters its fundamental period of movements and thus its dynamic behavior considerably. [PERSON] and [PERSON] [11] underscored the base isolation effect of foundation rocking for bridge piers. [PERSON] and [PERSON] [12] and [PERSON] and [PERSON] [13] used non-linear Wrinkler springs to model rocking of rigid footings. [PERSON] et al. [14] suggested that structures can be protected from damage by deliberately under-designing their foundation and thus allowing for bearing failure in the soil. Finite element (FE) analyses of rigid footings on clay illustrated that while rocking of light structures was dominated by uplift, heavier structures experienced more soil failure and settlements [15]. Some studies suggested that factors of safety less than 1.0 can be used in the seismic geotechnical design of foundations [16]. However, they warned that residual rotations and settlements may exceed the serviceability limits. [PERSON] et al. [17] investigated the effects of rounding soil under rocking foundations. As the foundation uplift increases, the contact area between the footing base and soil converges to a critical value, below which the contact area cannot decrease. The critical area as well as the horizontal pressure of side soil determine the ultimate moment capacity of the footing. Rocking foundations on cohesive soil demonstrated good recentering ability in field tests of [18]. [PERSON] et al. [19] analyzed the results of rocking foundation tests and defined a rocking coefficient defined as the ultimate moment capacity of the footing normalized by the product of the structure's weight and effective height. They demonstrated that the rocking coefficient is a determining factor for permanent settlements and energy dissipation of the foundation and that the maximum rotation is dependent on the peak ground acceleration and the ratio of the effective height to the footing width. [PERSON] et al. [20] conducted shaking table tests on rocking foundations on sand and reported that settlements and rotations increased by decreasing the frequency of the shaking. Additionally, rocking foundation tests on sand showed that the shape of the foundation and its length to width ratio affects its recentering capability and hysteresis shape [21]. The design of shallow foundations under eccentric loads commonly adopts the concept of effective area for calculation of bearing capacity [22]. Although the effective area shape for circular footings is geometrically similar to the footing base area remaining in contact with soil when uplift occurs in rocking, it is an imaginary area to limit the region where the block of bearing pressures is assumed to exist and is not intended to accurately model the uplift phenomenon.