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Risk-targeted ground motion for one hazard curve (ASCE 7-22 Sec. 21.2.1)

solveRTGM.Rd

Solves for the spectral ordinate whose lognormal collapse fragility, convolved with the site hazard curve, yields the target collapse probability over the target time horizon. With the defaults this is ASCE/SEI 7-22 Sec. 21.2.1: a fragility with a 10 percent conditional probability of collapse at the ordinate and a logarithmic standard deviation of 0.60, targeting 1 percent probability of collapse in 50 years.

Usage

solveRTGM(
  sa,
  aep,
  returnPeriod = 2474.915822,
  collapseProb = 0.1,
  beta = 0.6,
  targetProb = 0.01,
  timeHorizon = 50,
  scaleMin = 0.1,
  scaleMax = 10,
  step = log(1.05),
  tolerance = 0.01,
  maxIter = 20L
)

Arguments

sa

numeric, ground motion amplitudes (g), strictly positive and unique.

aep

numeric, annual frequencies of exceedance corresponding to sa. These are rates, not probabilities, and must be strictly positive and non-increasing in sa. poeToAep() produces them from a probability of exceedance and an investigation time.

returnPeriod

numeric, return period (years) of the uniform-hazard ordinate that anchors the first guess and the resampling bounds. Default 2474.915822, the return period whose 50-year exceedance probability is exactly 2 percent.

collapseProb

numeric, conditional probability of collapse at the risk-targeted ordinate. Default 0.10.

beta

numeric, logarithmic standard deviation of the collapse capacity. Default 0.60.

targetProb

numeric, target probability of collapse over timeHorizon. Default 0.01.

timeHorizon

numeric, time horizon in years for the target collapse probability. Default 50.

scaleMin

numeric, lower resampling bound as a multiple of the uniform-hazard ordinate. Default 0.10.

scaleMax

numeric, upper resampling bound as a multiple of the uniform-hazard ordinate. Default 10.

step

numeric, resampling increment in log space. Default log(1.05).

tolerance

numeric, relative tolerance on the annual collapse rate. Default 0.01, the reference implementation's own criterion.

maxIter

integer, iteration cap. Default 20.

Value

A list with rtgm (the risk-targeted ordinate, g), uhgm (the uniform-hazard ordinate at returnPeriod, g), riskCoeff (their ratio, the risk coefficient), colRisk (the achieved annual collapse rate), iterations, and converged.

Details

The procedure follows the USGS Risk-Targeted Ground Motion Calculator (Makdisi 2024, doi:10.5066/P9A2F2B8 ), which ASCE 7-22 Commentary C21.2.1 traces to Luco et al. (2007). The hazard curve is resampled and log-log extrapolated onto [scaleMin, scaleMax] times the uniform-hazard ordinate before integrating; restricting the integral to the tabulated curve domain instead is what makes low-hazard, long-period cases fail to bracket.

This is the probabilistic branch only. The Sec. 21.2.2 deterministic motions, the Sec. 21.2.3 80-percent floor, and the Sec. 21.2 maximum-direction scaling are outside its scope and are not applied.

The result is an ordinate, not a spectrum shape: ASCE 7-22 Sec. 21.3 defines the design value as two thirds of it at every period. Do not pass this output to designUHS(), which expects a uniform-hazard spectrum and performs the unrelated Sec. 21.4 parameterisation.

Every tunable is a defaulted argument rather than a package constant, so that a different target set (for example the AASHTO guide specification: 1033-year anchor, 5 percent conditional collapse, 1.5 percent in 75 years) can be selected without editing the source.

Examples

sa <- exp(seq(log(0.005), log(3), length.out = 20))
aep <- 0.002 * (sa / 0.1)^-2.5
OUT <- solveRTGM(sa = sa, aep = aep)
OUT$rtgm
#> [1] 0.182258
OUT$riskCoeff
#> [1] 0.9612812