## 1 Introduction
The analytical basis for conical and other sloped footings originates from the pioneering work of [PERSON] [11], who introduced the theory of plastic equilibrium for soil under punching shear--a concept particularly applicable to cone-shaped loading zones. Later, [PERSON] [12] formalized these ideas by deriving bearing-capacity equations for strip footings, which were subsequently adapted to cone geometries through the use of shape and depth correction factors. In the 1950s, [PERSON] extended bearing-capacity theories to account for sloped, eccentric, and inclined loads, which are directly relevant to conical foundations [13]. His corrections for shape, embedment depth, and load inclination significantly improved the realism of conical footing design [10]. [PERSON] [14] further refined these models by introducing empirical shape and load factors, making it easier to incorporate irregular geometries like conical profiles. [PERSON] [15] advanced the theoretical framework by introducing soil compressibility and elasticity into bearing-capacity calculations. Figure 2: Subsidence trough and distribution of deformations of the surface of an undermining underground working: \(\delta_{0}\), \(\delta\) and \(\psi\) are angles indicating the boundaries of the zone of influence of underground mining operations along the strike of the seam; T is the slope; K is the curvature; \(\varepsilon\) is the horizontal deformation; u is the horizontal displacement. The subsidence trough and the distribution of surface deformation: \(\delta_{0}\), \(\delta\) and \(\psi\)—the angles designating the boundaries of the zone of influence of underground mining along the strike of the formation; T—tilt; K—curvature; \(\varepsilon\)—horizontal strain; u—horizontal displacement. Conical foundations have attracted growing interest in recent decades, due to their ability to improve vertical load-bearing performance, particularly in soft or variable soils. Their geometry allows for enhanced load dispersion and the mobilization of broader passive resistance zones compared to flat or shallow foundations. These characteristics have been studied extensively, using both experimental testing and numerical modeling approaches. [PERSON] [16] addressed a significant gap in foundation engineering by developing novel plastic solutions for the bearing capacity of conical footings in anisotropic and non-homogeneous clays, using finite element limit analysis (FELA). The study introduced key dimensionless parameters--such as cone angle and strength gradient--to provide more reliable design predictions for offshore conical foundations, including spudcans. [PERSON] and [PERSON] [17] investigated conical footings in sandy soils through centrifuge testing, demonstrating that cone geometry mobilizes broader failure surfaces and offers higher normalized bearing capacities than flat bases. Similarly, [PERSON] and [PERSON] [18] presented theoretical bearing-capacity factors for circular conical footings on undrained clays, incorporating variables such as embedment depth cone angle, and increasing undrained strength with depth--findings especially relevant for gravity bases and other large-scale offshore foundations. Advancements in material innovation and foundation optimization have led to novel systems like the cone-shaped hollow flexible reinforced foundation (CHFRF), introduced by [PERSON] et al. [19]. This system reduces material use while improving lateral load performance through embedded depth and the use of laminated rubber-steel layers for energy dissipation. In a related effort, [PERSON] et al. [20] combined FELA and the MARS model to show that base roughness significantly improves the bearing capacity of conical footings. [PERSON] et al. [21] and [PERSON] et al. [22] further demonstrated, using machine learning and FEA-ANN methods, that conical footing behavior is highly sensitive to cone geometry and soil heterogeneity, yielding accurate predictive models for design.