Shaft resistance

Along with the toe resistance, the shaft resistance $Q_s$ makes up a pile’s ultimate capacity, $Q_u = Q_s + Q_p$. Units, symbols and assumptions are set out in Method and assumptions.

The shaft is cut at the strata boundaries. In stratum $i$ the pile has a length $\Delta L_i$ and a unit shaft resistance $f_s$, so it adds $Q_{s,i} = f_s \, p \, \Delta L_i$ to $Q_s$. How $f_s$ is found depends on the stratum.

In clay: the α method

In an undrained stratum, $f_s = \alpha \, c_u$. The adhesion factor α is interpolated linearly from $c_u/p_a$ in this table, from Das & Sivakugan (2017); beyond its ends, α keeps its end values.

$c_u/p_a$	$\alpha$	$c_u/p_a$	$\alpha$
≤ 0.1	1.00	1.2	0.42
0.2	0.92	1.4	0.40
0.3	0.82	1.6	0.38
0.4	0.74	1.8	0.36
0.6	0.62	2.0	0.35
0.8	0.54	2.4	0.34
1.0	0.48	≥ 2.8	0.34

The effective stress plays no part here, so in clay neither the water table nor the critical depth changes $f_s$.

In sand: the β method

In a drained stratum, $f_s = K \, \sigma'_v \tan\delta$, that is, $\beta \, \sigma'_v$ with $\beta = K \tan\delta$. The same rules apply to every drained stratum, sand or clay in the long term. $K$ is a fixed multiple of the coefficient of earth pressure at rest, $K_0 = 1 - \sin\phi'$, set by how the pile is installed and, for a driven pile, by its size: one with $D \ge 0.3$ m is taken as high displacement, a thinner one as low displacement.

Pile	$K$
Bored	$K_0$
Driven, $D < 0.3$ m	$1.3 \, K_0$
Driven, $D \ge 0.3$ m	$1.7 \, K_0$

The friction angle between pile and soil is $\delta = 0.8 \, \phi'$, and $\sigma'_v$ is the vertical effective stress along the stratum, limited by the critical depth.

The critical depth

In drained strata, $\sigma'_v$ stops growing at the critical depth $z_c = 15D$: from there down it is held at $\sigma'_v(z_c)$. For the stretch of pile in one stratum, from $z_t$ to $z_b$, the $\sigma'_v$ that enters $f_s$ is:

• $\sigma'_v$ at its mid-length, if it ends above $z_c$ ($z_b \le z_c$);
• $\sigma'_v(z_c)$, if it starts below $z_c$ ($z_t \ge z_c$);
• if it crosses $z_c$, the mean of both parts, weighted by their lengths:

$$\sigma'_v = \frac{\tfrac{1}{2}\left[\sigma'_v(z_t) + \sigma'_v(z_c)\right](z_c - z_t) + \sigma'_v(z_c)\,(z_b - z_c)}{z_b - z_t}$$

Below $z_c$, $f_s$ still changes from one drained stratum to the next, through $K$ and $\delta$. The limit acts on the shaft only: the toe takes the effective stress at its own depth.