Evaluate stress, stiffness and shear-wave velocity of ordered layers

Deterministic physics of a layered column whose index properties are supplied: effective unit weight under the water table, octahedral mean effective stress, overconsolidation ratio, small-strain shear modulus from the correlations of the soil group and the shear-wave velocity of each layer. The layers may come from sampleSoilLayers() or from measured or specified data with the same columns.

Usage

evaluateSoilLayers(layers, Hs, water, POP, m1 = HardinExponent)

Arguments

layers: A data.frame or data.table with one row per layer from the surface down and columns USCS, hs (m), zm (m), eo, gsat (kN/m3) and IP (percent, NA for coarse soils); emin and emax are used for the relative density when present.
Hs: One positive column height in metres.
water: Water-table height as a fraction of Hs (0 for a dry column).
POP: Preconsolidation pressure in kPa, added to the mean stress in the overconsolidation ratio.
m1: Nodes of the interpolation of $m_1$ in IP: a data.frame or data.table with numeric columns IP (percent) and m1. The default is Table 1 of Hardin and Drnevich (1972): IP 0, 20, 40, 60, 80, 100 and Inf with 0, 0.18, 0.30, 0.41, 0.48, 0.50 and 0.50.

Value

A data.table with the input columns and group (Gravels, Sands or Fines), Dr (relative density, when the void-ratio limits are supplied), gs (effective unit weight, kN/m3), pm (kPa), OCR, Gm (MPa) and VSm (m/s). Each row is a conditional evaluation of its inputs; it carries no dispersion of its own.

Mathematics

With $K_o = 0.5$ and the effective unit weight $\gamma'$ of each layer, assigned by its midpoint depth $z_m$: $\max(\gamma_{sat} - 10, 0)$ kN/m3 when $z_m$ lies below the water table $z_w = H_s (1 - w)$, else $\gamma_{sat}$; the octahedral stress increases by $\Delta p = (1 + 2K_o) \gamma' h / 3$ across each layer; the layer mean $p_m$ is the average of its top and bottom values, and $OCR = \max((p_m + POP)/p_m, 1)$ with the preconsolidation POP in kPa. For every correlation j of the layer's soil group in ShearModelParameters the modulus at the mean stress is $$G_j = A_j F_j(e_o)\, OCR^{m_1}\, 100^{n_j} (p_m/100)^{n_j},$$ with $F_j(e) = (C_e - e)^2/(1+e)$, except the linear $F(e) = C_e - e/(1+e)$ of SS75, and $m_1$ the exponent $K$ of the overconsolidation ratio of Hardin and Drnevich (1972) for fine soils and zero otherwise (hydrated copy, equation eq-design-gmax, printed page 671). $m_1$ is interpolated linearly in IP through the nodes of m1 and is NA outside them, so a fine layer whose IP lies below the first node or above the last has an NA modulus whenever $OCR > 1$. The default nodes are Table 1 of that paper (printed page 672): IP 0, 20, 40, 60 and 80 with 0, 0.18, 0.30, 0.41 and 0.48, and the row "PI $\ge$ 100: 0.50", encoded as the nodes (100, 0.50) and (Inf, 0.50) so that $m_1$ is 0.50 at and above 100. The Inf node is that encoding of the published row, not a further choice. The layer modulus is the arithmetic mean $G_m = \overline{G_j}$ over the group's correlations, in MPa, and $$V_{S} = \sqrt{9.81 \cdot 1000\, G_m / \gamma_{sat}}$$ in m/s uses the saturated unit weight. These are the relations of newmark 1.7.1 evaluated layer by layer, except the default nodes: 1.7.1 ended them at (100, 0.48), with NA above, and buildShearTable() passes those nodes to reproduce it. The spread of $G_j$ across correlations is not retained by this version.

Examples

Layers <- data.frame(USCS = c("SM", "ML"), hs = c(5, 5), zm = c(2.5, 7.5),
                     eo = c(0.7, 0.9), gsat = c(20, 18), IP = c(NA, 10))
evaluateSoilLayers(Layers, Hs = 10, water = 0, POP = 100)
#>      USCS    hs    zm    eo  gsat    IP  group    gs       pm      OCR       Gm
#>    <char> <num> <num> <num> <num> <num> <char> <num>    <num>    <num>    <num>
#> 1:     SM     5   2.5   0.7    20    NA  Sands    20 33.33333 4.000000 53.60363
#> 2:     ML     5   7.5   0.9    18    10  Fines    18 96.66667 2.034483 68.31749
#>         VSm
#>       <num>
#> 1: 162.1499
#> 2: 192.9586