Appendix D
Guidance on Estimating the Vertical Stiffness of Foundations

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
images
To account for the roughness of the footing base that allow full transmission of shear stress, Spence (1968) proposed;
images
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
images
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
images images
Wolf (1988) General prismatic footing embedded in a linear elastic half space
images
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
images
Sharma and El Naggar (2015) Single pile under axial load for seismic design of highway bridges
images

N.B: KV is vertical stiffness of the foundation, images is shear modulus of the soil, ѵs is Poisson's ratio of the medium, D is diameter, L is embedment depth, H is thickness of the soil layer, and Ep 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.

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