Modal period of the inhomogeneous truncated shear beam

Usage

evaluateBeamPeriod(Hs, lo, mo, VSo, an)

Arguments

Hs: Slope heights from crest to base, in metres.
lo: Truncation ratios $\lambda_o$ in $[0, 1)$, aligned with Hs; see buildGeometry().
mo: Inhomogeneity exponents, below 2.
VSo: Basal shear-wave velocities in m/s.
an: Characteristic roots of the requested mode at mo and lo.

Value

Periods in seconds, aligned with the inputs, unrounded.

Mathematics

Dakoulas and Gazetas (1985) measure the depth $z$ from the apex where the extended slopes meet: the crest lies at $z = h$, the base at $z = H$, the modulus is $G = G_b (z/H)^m$ and the truncation ratio is $\lambda = h/H$ (hydrated copy, Fig. 1 on printed page 167 and equations eq-modulus-depth-power and eq-free-truncation-ratio). The slope of buildGeometry() has height $H_s = H - h$ from crest to base, and its truncation ratio $\lambda_o = b/b_{max}$ is that $h/H$, because the extended slopes meet $h = b/(2s)$ above the crest; hence $H_s = H (1 - \lambda_o)$. The period of the mode whose characteristic root is $a_n(m_o, \lambda_o)$, with basal velocity $V_{So}$, is (eq-natural-frequency-period) $$T_n = \frac{4 \pi H}{a_n (2 - m_o) V_{So}}, \qquad H = \frac{H_s}{1 - \lambda_o}.$$ With exact roots it reproduces the shear-beam periods of the five dams of Table 8 of the paper (printed page 175) within 0.1 percent from their printed height, truncation ratio, exponent and average velocity.

The exponent $m_o$ and velocity $V_{So}$ that this package estimates come from the fit of characterizeProfile(), whose origin is the crest ($G = G_o (z/H_s)^{m_o}$, $z$ below the crest), while the roots belong to the apex-origin profile of the paper. Evaluating the period of a fitted profile this way is an approximation. The theory base measures it on a continuous profile at 1.4, 3.7 and 6.0 percent for $m = 0.5$ and $\lambda$ = 0.075, 0.20 and 0.35; for a layered realization the measure is the difference that propagateBeamPeriod() reports against the numerical period, with its divergence flag.

The result is a conditional evaluation of its five inputs; evaluating it on summary statistics of $m_o$ and $V_{So}$ does not give the statistics of the period, which propagateBeamPeriod() obtains realization by realization.

Examples

# Dam A of Table 8 of Dakoulas and Gazetas (1985): 0.534 s, with the basal
# velocity derived from its average velocity of 200 m/s and the exact root.
evaluateBeamPeriod(Hs = 40, lo = 0.043, mo = 0.57, VSo = 228.25, an = 3.0123)
#> [1] 0.5342111