Deformable sliding-mass displacement

Evaluates the lognormal distribution of the nonzero seismic displacement of a deformable sliding mass with yield coefficient \(k_y\) and initial fundamental period \(T_s\) under an earthquake of moment magnitude \(M_w\), with one of the three flexible models of the displacement ensemble, named by its author-year identifier model. The spectral acceleration is read at the degraded period of the model: \(1.5 T_s\) for BT07 and BM17, \(1.3 T_s\) for BM19.

Usage

evaluateFlexibleDisplacement(model, ky, Ts, Sa, Mw, PGV, NFC, pulse)

Arguments

model

One flexible model identifier: "BT07", "BM17" or "BM19".

ky

Yield coefficients in g, finite and positive.

Ts

The initial fundamental period of the sliding mass in seconds, one finite non-negative value.

Sa

Spectral accelerations in g at the degraded period of the model, finite and positive.

Mw

One finite positive moment magnitude.

PGV

Peak ground velocities in cm/s, finite and positive, aligned with Sa, for example estimatePGV() of each PGA; only the pulse equations of BM19 read them.

NFC

Near-fault component of the BM19 pulse equations, "D100" or "D50".

pulse

One logical: TRUE for a near-fault pulse motion of BM19, FALSE for ordinary motions.

Value

A data.table with IDn (the model), muLnD and sigmaLnD, the mean and standard deviation of \(\ln D_n\) (\(D_n\) in cm) given the inputs and a nonzero displacement: \(e^{\mu_{\ln D}}\) is the median nonzero displacement and \(\sigma_{\ln D}\) its dispersion in natural logarithms. One row per element of the longer of ky and Sa.

Details

Each published model treats the displacement as a mixed random variable: a probability of negligible ("zero") displacement, with a threshold of 1 cm in BT07 and 0.5 cm in BM17 and BM19, and a lognormal distribution of the nonzero displacement. This function evaluates only the nonzero branch, equation 14.5 of Bray (2007, p. 344), equation 4 of Bray, Macedo and Travasarou (2018, p. 5) and equations 6, 8 and 10 of Bray and Macedo (2023, pp. 6-7), and omits the probability of zero displacement of equation 14.4 (p. 343), equations 2 and 3 (p. 5) and equations 5, 7 and 9 (pp. 6-7) of the same sources. Its result is the distribution of \(D_n\) conditional on the inputs and on a nonzero displacement.

ky and Sa are evaluated element by element: one of length one is repeated for every element of the other, otherwise both must have the same length, and PGV has the length of Sa; the function stops and names the lengths otherwise.

Mathematics

With \(D_n\) in cm, \(S_a\) in g, \(T_s\) in s and \(x = \ln k_y\), \(y = \ln S_a\):

BT07, Bray and Travasarou (2007), equation 5, with \(a_0 = -1.10\), replaced by \(-0.22\) for nearly rigid masses (\(T_s < 0.05\) s): $$\ln D_n = a_0 - 2.83 x - 0.333 x^2 + 0.566 x y + 3.04 y - 0.244 y^2 + 1.50 T_s + 0.278 (M_w - 7), \qquad \sigma = 0.66$$

BM17, Bray, Macedo and Travasarou (2018), equation 4, subduction interface earthquakes, with \((c_1, c_2, c_3) = (-5.864, -9.421, 0)\) for \(T_s < 0.1\) s and \((-6.896, 3.081, -0.803)\) otherwise: $$\ln D_n = c_1 + 0.550 M_w + c_2 T_s + c_3 T_s^2 - 3.353 x - 0.390 x^2 + (3.060 + 0.538 x) y - 0.225 y^2, \qquad \sigma = 0.73$$

BM19, Bray and Macedo (2019) as corrected in 2023, shallow crustal earthquakes. Ordinary motions (pulse = FALSE, equation 10): $$\ln D_n = a_0 + (2.649 + 0.344 x) y - 0.090 y^2, \qquad \sigma = 0.72$$ with \(a_0 = -4.684 + 0.603 M_w - 9.471 T_s - 2.482 x - 0.244 x^2\) for \(T_s < 0.1\) s and \(a_0 = -5.981 + 0.603 M_w + 3.223 T_s - 0.945 T_s^2 - 2.482 x - 0.244 x^2\) otherwise. Near-fault forward-directivity pulse motions (pulse = TRUE), a property of the scenario declared by the analyst: $$\ln D_n = c_1 - b_1 x - b_2 x^2 + b_3 x y + b_4 y - b_5 y^2 + c_2 T_s + c_3 T_s^2 + b_m M_w + c_4 \ln PGV$$ with PGV in cm/s and the coefficients of equation 6 for the maximum component (NFC = "D100": \(b = (2.632, 0.278, 0.527, 1.978, 0.233)\), \(b_m = 0.01\), \(\sigma = 0.56\)) or equation 8 for the median component ("D50": \(b = (2.931, 0.319, 0.584, 2.261, 0.241)\), \(b_m = 0.05\), \(\sigma = 0.54\)). \(c_1\) and \(c_4\) change at PGV = 150 cm/s, \(c_1, c_2, c_3\) at \(T_s = 0.1\) s: D100 \(c_1 = -6.724 / 1.991\) and \((c_2, c_3) = (-2.744, 0)\) below 0.1 s, \(c_1 = -6.951 / 1.764\) and \((1.069, -0.498)\) above, \(c_4 = 1.547 / -0.097\); D50 \(c_1 = -7.497 / 2.480\) and \((-2.731, 0)\), \(c_1 = -7.718 / -0.369\) and \((1.031, -0.480)\), \(c_4 = 1.458 / 0.025\) (PGV up to / above 150 cm/s). For D50 above 150 cm/s and \(T_s < 0.1\) s the paper's \(c_1 = 2.480\) differs from the authors' spreadsheet (-0.148); the paper is followed.

References

Bray, J. D. and Travasarou, T. (2007). Simplified procedure for estimating earthquake-induced deviatoric slope displacements. Journal of Geotechnical and Geoenvironmental Engineering, 133(4), 381-392. doi:10.1061/(ASCE)1090-0241(2007)133:4(381)

Bray, J. D. (2007). Simplified seismic slope displacement procedures. In K. D. Pitilakis (ed.), Earthquake Geotechnical Engineering, 327-353. Springer.

Bray, J. D., Macedo, J. and Travasarou, T. (2018). Simplified procedure for estimating seismic slope displacements for subduction zone earthquakes. Journal of Geotechnical and Geoenvironmental Engineering, 144(3), 04017124. doi:10.1061/(ASCE)GT.1943-5606.0001833

Bray, J. D. and Macedo, J. (2019). Procedure for estimating shear-induced seismic slope displacement for shallow crustal earthquakes. Journal of Geotechnical and Geoenvironmental Engineering, 145(12), 04019106.

Bray, J. D. and Macedo, J. (2023). Performance-based seismic assessment of slope systems. Soil Dynamics and Earthquake Engineering, 168, 107835. doi:10.1016/j.soildyn.2023.107835 The BM19 coefficients implemented here are those printed in its equations 6, 8 and 10.

Examples

evaluateFlexibleDisplacement(model = "BM19", ky = 0.1, Ts = 0.3, Sa = 0.8, Mw = 7,
                             PGV = estimatePGV(0.5), NFC = "D100", pulse = FALSE)
#>       IDn    muLnD sigmaLnD
#>    <char>    <num>    <num>
#> 1:   BM19 3.124364     0.72