Yield coefficient for an admissible seismic displacement

The yield coefficient $k_y=a_y/g$ expresses the acceleration $a_y$ required to initiate sliding relative to gravitational acceleration $g$. In a pseudostatic stability analysis, the factor of safety is calculated by limit equilibrium with a static horizontal inertial force equal to the product of a seismic coefficient and the weight of the sliding mass (Bray and Travasarou 2009); the yield coefficient is the value of that coefficient at which the factor of safety reaches unity. A displacement-based criterion relates this coefficient to the permanent seismic displacement $D_n(k_y)$ and an admissible displacement $D_a$, both expressed in $\mathrm{cm}$. For a target exceedance probability $p_e$, with $D_a>0$ and $0<p_e<1$, the criterion on a non-increasing random displacement curve $\bar D(k_y)$ defines the coefficient by

$$k_{\max}(p_e\mid D_a)=\inf\left\{k_y>0:P[\bar D(k_y)>D_a]\le p_e\right\}.$$

If every $k_y>0$ satisfies the condition, the infimum is zero; if the set is empty, the infimum is $+\infty$. The regression $D_n(k_y)$ of each model and realization is projected onto $\bar D(k_y)$; the two are not interchangeable when the regression has an increasing segment. In the flexible regressions, the concave quadratic of $\ln D_n$ in $\ln k_y$ tends to $-\infty$ as $k_y\to0$, so the unprojected displacement tends to zero outside the calibration range. The global infimum of the criterion, applied to the unprojected regression in place of $\bar D$, can then be zero although a positive crossing of $D_a$ exists on its decreasing branch.

The probability concerns the represented distribution of displacement curves. With realizations from evaluateDisplacements(), the flexible models contribute their positive lognormal branches, with the probability of negligible displacement omitted. When the demand varies, those branches retain the original demand distribution; their mixture is not the full population conditioned on nonzero displacement.

For a model realization $s$, let $\bar D_s(k_y)$ denote the curve $\bar D(k_y)$ of that realization, continuous and non-increasing for $k_y>0$. Its threshold coefficient is

$$K_s=\inf\left\{k_y>0:\bar D_s(k_y)\le D_a\right\}.$$

Under these continuity and monotonicity conditions, $\bar D_s(k_y)>D_a$ is equivalent to $K_s>k_y$. If $K$ denotes the random threshold coefficient and $F_K$ its cumulative distribution function, the displacement criterion therefore becomes

$$k_{\max}(p_e\mid D_a)=F_K^{-1}(1-p_e).$$

Here the inverse is the generalized inverse of the distribution function (Kroese et al. 2011), and the corresponding quantile level is $q=1-p_e$. The inversion acts on each realization before the roots from different models are combined with their weights. The arithmetic mean of those roots describes their center without representing a target exceedance probability.

Numerical curves are stored at increasing coefficients $k_{y,1}<\cdots<k_{y,J}$, where $J$ is the number of grid points. For the displacement $D_i$ at grid index $i$, the upper non-increasing envelope is

$$\bar D_j=\max_{i\ge j}D_i.$$

At a grid point $j$, this envelope retains the largest displacement at that coefficient or any larger stored coefficient. Between distinct positive envelope values, the inverse interpolates linearly in $(\ln D_n,\ln k_y)$. The calculated coefficient thus depends on the supplied grid as well as on the displacement model.

Outside that positive support, numerical assignments have a different meaning from an exact root. Except for the separate constant-AM88 inversion, a target above the stored envelope maximum receives a zero coefficient; this records the finite grid’s ceiling, not the absence of displacement demand over every positive $k_y$. Displacements below $10^{-4}\ \mathrm{cm}$ are set to zero before ordinary inversion. When the target lies below the positive part of a curve with a zero tail, the assigned coefficient is the first grid point of that tail. For a target below a curve that remains positive on the grid, continuation uses three positive, strictly decreasing displacements at its largest coefficients. For AM88 and SR08 continuation, the search is limited to $k_y/\mathrm{PGA}\le0.9999$, where PGA is the realization’s peak ground acceleration expressed in $g$; saturation at that limit need not satisfy $D_n=D_a$.

solveKmax() performs the inversion for one model: Dn contains one realization per row and one coefficient from ky per column, while Da supplies the admissible displacements. In its result, realization identifies the row number in Dn; kmax and beyond are reported for each target. The beyond flag identifies continuation or inversion of a constant AM88 curve; it excludes zero assignments and assignments to the grid floor. It describes numerical support separately from physical dispersion.

summarizeKmax() combines roots with the positive model weights normalized to sum to one, requiring shared realization coverage at each Da. Zero roots retain their weights. The reported quantiles are interpolated estimates of the population quantiles: they are not the generalized inverse of the finite sample’s empirical distribution, so the fraction of sampled roots above a reported quantile can exceed $p_e$. Its beyond output is the weighted fraction of flagged roots.

For project tables, buildKmaxTable() applies inversion and summaries by demand, geometry and material, using admissible displacements derived from the slope heights. When pooled roots contain ties, this route computes quantiles from values represented with 15 significant digits; its mean uses unrounded roots. The general summary uses full-precision roots. The product reports Kh as the dimensionless ratio of kmax to the site’s amplified PGA for that demand, geometry and material. assembleKmaxTables() checks and combines the site tables into the global coefficient table.

References

Bray, Jonathan D., and Thaleia Travasarou. 2009. “Pseudostatic Coefficient for Use in Simplified Seismic Slope Stability Evaluation.” Journal of Geotechnical and Geoenvironmental Engineering 135 (9): 1336–40. https://doi.org/10.1061/(ASCE)GT.1943-5606.0000012.

Kroese, Dirk P., Thomas Taimre, and Zdravko I. Botev. 2011. Handbook of Monte Carlo Methods. Wiley Series in Probability and Statistics. John Wiley & Sons. https://doi.org/10.1002/9781118014967.