Fundamental period of a layered soil profile by the Rayleigh quotient

Usage

estimateRayleighPeriod(VSm, hs, zm)

Arguments

VSm: Positive layer shear-wave velocities in m/s.
hs: Positive layer thicknesses in metres, aligned with VSm.
zm: Layer midpoint depths from the surface in metres, aligned.

Value

One estimate of the fundamental period in seconds, unrounded.

Mathematics

For a fixed-base, free-surface column of uniform density under a uniform horizontal body force, the shear strain at depth $z$ is proportional to $z / V_S(z)^2$. Integrating it from the base gives the static deflection, used as trial mode shape $X_k$ at each layer midpoint (Dobry, Oweis and Urzua 1976), and the Rayleigh quotient yields $$T_r = 2\pi \sqrt{\frac{\sum_k h_k X_k^2}{\sum_k h_k X_k}}.$$ Layers are ordered from the base internally, so the input order does not change the result. The midpoint approximation depends on the subdivision of the profile: for a uniform column of height $H$ the estimate tends to $3.974 H/V_S$ as the layers become thin, against the exact $4 H/V_S$, while one single layer gives $\pi H/V_S$. It is a conditional evaluation of the supplied profile and carries no dispersion.

Examples

estimateRayleighPeriod(VSm = c(200, 200), hs = c(10, 10), zm = c(5, 15))
#> [1] 0.3782979