Rigid sliding-block displacement

Evaluates the conditional lognormal distribution of the Newmark displacement of a rigid block with yield acceleration \(k_y\) under a ground motion of peak acceleration PGA and Arias intensity \(I_a\), with one of the three rigid models of the displacement ensemble, named by its author-year identifier model.

Usage

evaluateRigidDisplacement(model, PGA, ky, AI)

Arguments

model

One rigid model identifier: "AM88", "JB07" or "SR08".

PGA

Peak ground accelerations in g, finite and positive.

ky

Yield accelerations in g, finite and positive.

AI

Arias intensities in m/s, finite and positive, for example estimateAI() of each PGA; AM88 does not read them.

Value

A data.table with IDn (the model), muLnD and sigmaLnD, the mean and standard deviation of \(\ln D_n\) (\(D_n\) in cm), one row per element of the longest vector the model reads.

Mathematics

With \(r = k_y / PGA\), \(D_n\) in cm, PGA in g and \(I_a\) in m/s:

AM88, Ambraseys and Menu (1988), with the ratio capped at 0.9999 so that \(1 - r > 0\) when the yield acceleration reaches the peak: $$\log_{10} D_n = 0.90 + \log_{10}\left[(1 - r)^{2.53} r^{-1.09}\right], \qquad \sigma_{\log_{10} D_n} = 0.30$$

JB07, Jibson (2007), equation 10: $$\log_{10} D_n = 0.561 \log_{10} I_a - 3.833 \log_{10} r - 1.474, \qquad \sigma_{\log_{10} D_n} = 0.616$$

SR08, Saygili and Rathje (2008), equation 6 with the (PGA, \(I_a\)) coefficients of tables 1 and 2 (the authors recommend the (PGA, PGV) vector; this published variant keeps the rigid models on Arias intensity): $$\ln D_n = 2.39 - 5.24 r - 18.78 r^2 + 42.01 r^3 - 29.15 r^4 - 1.56 \ln PGA + 1.38 \ln I_a, \qquad \sigma_{\ln D_n} = 0.46 + 0.56 r$$

The base-10 results are returned in natural logarithms (\(\times \ln 10\)).

The vectors a model reads, PGA and ky and, for JB07 and SR08, AI, are evaluated element by element: a vector of length one is repeated for every element of the others, and the vectors longer than one must have one common length; otherwise the function stops and names the lengths.

References

Ambraseys, N. N. and Menu, J. M. (1988). Earthquake-induced ground displacements. Earthquake Engineering & Structural Dynamics, 16(7), 985-1006. doi:10.1002/eqe.4290160704

Jibson, R. W. (2007). Regression models for estimating coseismic landslide displacement. Engineering Geology, 91(2-4), 209-218. doi:10.1016/j.enggeo.2007.01.013

Saygili, G. and Rathje, E. M. (2008). Empirical predictive models for earthquake-induced sliding displacements of slopes. Journal of Geotechnical and Geoenvironmental Engineering, 134(6), 790-803. doi:10.1061/(ASCE)1090-0241(2008)134:6(790)

Examples

evaluateRigidDisplacement(model = "SR08", PGA = 0.5, ky = c(0.05, 0.1), AI = estimateAI(0.5))
#>       IDn    muLnD sigmaLnD
#>    <char>    <num>    <num>
#> 1:   SR08 4.562406    0.516
#> 2:   SR08 3.725351    0.572