Fundamental period of a layered soil column: Rayleigh approximations and numerical solution

The fundamental period $T_r$ of the soil profile, or site period, describes its free, undamped horizontal shear vibration on rigid rock. The deposit is represented as a linear elastic shear column of thickness $H$, extending indefinitely in the horizontal directions. Its mass density $\rho$, shear-wave velocity $V$ and shear modulus $G=\rho V^2$ vary with the vertical coordinate. With $z$ measured upward from the base, the modal displacement is zero at $z=0$, and the shear stress is zero at the free surface $z=H$. The fundamental mode has the smallest positive angular frequency $\omega$, and its period is

$$T_r = \frac{2\pi}{\omega}.$$

For the fundamental modal shape $X(z)$, equality of the maximum kinetic and elastic energies gives the Rayleigh quotient:

$$\omega^2 = \frac{\displaystyle\int_0^H \rho(z)V^2(z)\left(\frac{dX}{dz}\right)^2 dz}{\displaystyle\int_0^H \rho(z)X^2(z)\,dz}.$$

The numerator weights the squared shear deformation by the shear modulus; the denominator weights the squared displacement by the mass distribution. Substituting an estimated shape gives an approximation to the fundamental frequency and period. The first Rayleigh iteration uses the static deflection generated by a horizontal force per unit volume proportional to the density. With $S(z)$ denoting the mass per unit area above level $z$, the mass distribution is described by

$$S(z) = \int_z^H \rho(z_b)\,dz_b.$$

The corresponding first-iteration shape is

$$X^{(2)}(z) = \int_0^z \frac{S(z_a)}{G(z_a)}\,dz_a,$$

where $z_a$ and $z_b$ are integration coordinates. This shape satisfies the fixed-base and free-surface conditions. The superscript identifies the shape obtained from the initial constant shape $X^{(1)}=1$; the complete Rayleigh procedure continues with successive modal iterations, whereas the library’s approximation functions evaluate this first iteration in layered form.

For a layer $i$ with constant properties, $H_i$ is its thickness, $\rho_i$ its density and $G_i$ its shear modulus. The mass above its midpoint includes the overlying layers and half its own mass. Integrating $S/G_i$ across the layer gives the change in the estimated shape between its boundaries. The layered approximation replaces the shape by the average of these boundary values, $\bar{X}_i$, and the slope by its midpoint value. For the soil column with $n$ layers, the approximation to its fundamental period is

$$T_r \approx 2\pi\sqrt{\frac{\displaystyle\sum_{i=1}^{n}\rho_iH_i\bar{X}_i^2}{\displaystyle\sum_{i=1}^{n}\rho_iH_i\bar{X}_i}}.$$

estimateWeightedRayleighPeriod(VSm, hs, zm, gsat) evaluates this density-dependent approximation. The arguments are layer shear-wave velocities in $\mathrm{m/s}$, thicknesses in $\mathrm{m}$, midpoint depths below the surface in $\mathrm{m}$, and unit weights in $\mathrm{kN/m^3}$, respectively. Unit weight $\gamma$ replaces density through $\rho=\gamma/g$, where $g$ is gravitational acceleration; the common factor cancels. The midpoint depths order the layers, while their thicknesses and unit weights determine the mass above each layer midpoint.

estimateRayleighPeriod(VSm, hs, zm) evaluates the constant-density form. For uniform unit weight and midpoint depths consistent with the layer thicknesses, the weighted expression reduces to that form. Both functions return one unrounded period in seconds. Their layered approximation depends on subdivision: refining the layers changes the evaluation of the first-iteration shape and quotient; it does not perform further modal iterations.

solveColumnPeriod(layers, elements) approximates the same fundamental period through a finite element eigenvalue problem with linear elements, consistent mass, a fixed base and a free surface. The table supplies hs, zm, gsat and VSm; Gm, the shear modulus in $\mathrm{MPa}$, is accepted when VSm is absent. The declared mesh has elements elements per layer, and the returned Tr uses twice that number. TrMeshChange reports $1-T_e/T_{2e}$, where $T_e$ and $T_{2e}$ are the periods on those meshes. The result also includes TrRayleighWeighted and its relative difference from Tr. These fields describe numerical approximation for the supplied profile; they do not quantify physical variability among soil profiles.