Settlement

The settlement is computed when every stratum has $E_s$ and $\mu_s$; the units, conventions and assumptions are in Method and assumptions.

Elastic settlement

The settlement is the immediate settlement at the centre of the base, by the theory of elasticity as Das & Sivakugan present it. The zone that settles runs from the base down $5B$, or to the end of the profile if it comes first; $E_s$ and $\mu_s$ are the thickness-weighted averages of the strata in it.

$$S_e = q_0 (\alpha B') \frac{1 - \mu_s^2}{E_s} I_s I_f$$

In the settlement equation, $q_0 = V/A$ spreads the vertical load over the full base, with no reduction for eccentricity and no overburden subtracted. $\alpha = 4$ with $B' = B/2$ gives the centre of a flexible, uniformly loaded base, as four $B/2 \times L/2$ rectangles that meet there; there is no reduction for a rigid footing.

Steinbrenner’s shape factor

The shape factor $I_s$ is Steinbrenner’s, with $m' = L/B$ and $n' = H/(B/2)$. His factors are for a foundation at the ground surface, so $H$ runs from the surface to the bottom of that zone, $H = \min(D_f + 5B, \text{profile})$, and Fox’s $I_f$ then accounts for the embedment; Das & Sivakugan do the same in their Example 9.2 ($H = 6$ m, $n' = 12$):

$$I_s = F_1 + \frac{1 - 2\mu_s}{1 - \mu_s}\,F_2, \qquad F_1 = \frac{A_0 + A_1}{\pi}, \qquad F_2 = \frac{n'}{2\pi}\tan^{-1}A_2$$ $$A_0 = m'\ln\frac{\left(1 + \sqrt{m'^2 + 1}\right)\sqrt{m'^2 + n'^2}}{m'\left(1 + \sqrt{m'^2 + n'^2 + 1}\right)}$$ $$A_1 = \ln\frac{\left(m' + \sqrt{m'^2 + 1}\right)\sqrt{1 + n'^2}}{m' + \sqrt{m'^2 + n'^2 + 1}}, \qquad A_2 = \frac{m'}{n'\sqrt{m'^2 + n'^2 + 1}}$$

Rischio evaluates these expressions instead of interpolating $F_1$ and $F_2$ in tables. A strip takes $m' = 10^4$, where $I_s$ has converged.

Fox’s depth factor

The depth factor $I_f$ is interpolated linearly in Fox’s table, for $L/B$ of 1, 2 and 5, by $L/B$, $D_f/B$ and $\mu_s$. The table covers $D_f/B$ from 0.5 to 1 and $\mu_s$ from 0.3 to 0.5. Beyond it, $L/B$, $\mu_s$ and a deeper $D_f/B$ take its nearest edge (a strip, the last $L/B$ row). Between $D_f = 0$, where $I_f = 1$, and the table’s first column, $D_f/B = 0.5$, $I_f$ is linear.

What isn’t included

Consolidation settlement of clays isn’t computed, although the 25 mm limit is meant for the total settlement; a warning says so with every settlement result. Punching shear, bending and reinforcement of the footing, seismic loading and sliding under the horizontal load are also outside the tool.