Equations · Piles

Every equation behind Rischio's checks of piles and pile groups, with the name and unit of each term.

These are the steps of the calculation report, in the same order, each with its equation and what its terms are.

1Shaft resistance

Effective stress at mid-length, in each stratum
$\sigma'_v \text{ at mid-length}$
Adhesion factor, in each stratum
$\alpha = f(c_u / p_a)$
Unit shaft resistance, in each stratum
$f_s = \alpha \, c_u$
where $\alpha$ adhesion factor, in each stratum
Effective stress at mid-length, in each stratum
$\sigma'_v \text{ along the stratum, held at } \sigma'_v(z_c) \text{ below } z_c$
Lateral earth pressure coefficient, in each stratum
$K = K_0 \cdot f(\text{installation, } D)$
Pile–soil friction angle, in each stratum
$\delta = 0.8 \phi$
Unit shaft resistance, in each stratum
$f_s = K \sigma'_v \tan\delta$
where $K$ lateral earth pressure coefficient, in each stratum; $\sigma'_v$ effective stress at mid-length, in each stratum (kPa); $\delta$ pile–soil friction angle, in each stratum (°)
Critical depth
$z_c = 15 D$
Shaft resistance, in each stratum
$Q_s = f_s \, p \, L$
where $f_s$ unit shaft resistance, in each stratum (kPa); $p$ perimeter of the pile (m); $L$ length, in each stratum (m)
Shaft resistance
$Q_s = \Sigma Q_{s,i}$
where $Q_{s,i}$ shaft resistance, in each stratum (kN)

Meyerhof's bearing capacity factor
$N_q^* = f(\phi) \quad \text{(Meyerhof)}$
Effective stress at the toe
$q' = \sigma'_v(L)$
Unit toe resistance
$q_p = \min(q' N_q^*, \; 0.5 \, p_a N_q^* \tan\phi)$
where $q'$ effective stress at the toe (kPa); $N_q^*$ Meyerhof's bearing capacity factor
Toe resistance
$Q_p = q_p A_p$
where $q_p$ unit toe resistance (kPa); $A_p$ area of the toe (m²)
Unit toe resistance
$q_p = 9 \, c_u \quad \text{(Meyerhof)}$

Ultimate capacity of a pile
$Q_u = Q_s + Q_p$
where $Q_s$ shaft resistance (kN); $Q_p$ toe resistance (kN)

Sum of the single piles
$n \, Q_u$
where $Q_u$ ultimate capacity of a pile (kN)
Capacity of the group as a block
$Q_{block} \text{ (the group as one pile, } \alpha = 1, \delta = \phi)$
Group efficiency
$\eta = Q_{block} / (n \, Q_u)$
where $Q_{block}$ capacity of the group as a block (kN); $n \, Q_u$ sum of the single piles (kN)
Capacity of the group
$Q_g = \min(\eta, 1) \, n \, Q_u$
where $\eta$ group efficiency; $n \, Q_u$ sum of the single piles (kN)
Pile group
$FS = \frac{Q_g}{Q} \; \ge 3.0$
Capacity of the group as a block
$Q_{block} \text{ (the group as one pile, } \alpha = 1, \delta = \phi, N^*_c \text{ by Skempton)}$