Methods to estimate the vertical stiffness of the foundation in literature:

	Source	Formulae and their applications
Surface foundation	Poulos and Davis (1974) Spence (1968)	Circular rigid footing on surface of homogenous elastic half‐space To account for the roughness of the footing base that allow full transmission of shear stress, Spence (1968) proposed; The results from this analytical solution showed up to 10% increase in stiffness values at low ѵs values.
	Gazetas (1983) DNV (2014)	Circular footing on stratum over half space
Shallow embedded foundation	Gazetas (1991) DNV (2014)	Circular rigid footing embedded in homogenous stratum over bedrock. Developed for machine‐type inertial loading. Range of validity: L/D < 1
	Wolf (1988)	General prismatic footing embedded in a linear elastic half space Where 2l, 2b are the base dimensions of circumscribed rectangle and e is the embedment depth.
Deep foundation	Fleming et al. (1992)	Embedded piles considering shaft friction only
	Sharma and El Naggar (2015)	Single pile under axial load for seismic design of highway bridges

N.B: K_V is vertical stiffness of the foundation, is shear modulus of the soil, ѵ_s is Poisson's ratio of the medium, D is diameter, L is embedment depth, H is thickness of the soil layer, and E_p is modulus of elasticity of pile material.

In practice, t‐z type of analysis or calibrated FEA (finite element analysis) can be carried out to obtain the axial stiffness of the piles.