Equations · Retaining walls

Every step of Rischio's retaining wall calculation, with its equation, in the same order as the calculation report.

These are the same steps, in the same order, as the calculation report; a report leaves out what its model doesn’t have, such as uplift without water.

1Geometry and weights

Weight of each part
$W = A \cdot \gamma_c$
where $A$ area of each part (m²)
Weight of the soil over the heel
$W_s = \Sigma A_i \gamma_i \quad (\gamma_{sat} \text{ below the water table})$

2Earth pressure

Height of the virtual back
$H' \text{ (height of the virtual back at the heel)}$
Surcharge
$q = \Sigma q_i$
Thrust of the surcharge
$\Delta P_{a,h} \approx K_a \, q \, H' \quad \text{(weight over the heel not counted)}$
where $q$ surcharge (kPa); $H'$ height of the virtual back (m)
Active thrust, horizontal
$P_{a,h} = \int_0^{H'} \left[ (\sigma'_v + q) K_a - 2c\sqrt{K_a} \right]^+ dz \cdot \cos\beta$
where $q$ surcharge (kPa); $H'$ height of the virtual back (m)
Active thrust, vertical
$P_{a,v} = \int_0^{H'} \sigma_a \, dz \cdot \sin\beta$
Water thrust
$P_w = \tfrac{1}{2} \gamma_w h_w^2$
Depth of the tension crack
$z_c: \sigma_a(z_c) = 0$
Thrust of the water in the crack
$P_{wc} = \tfrac{1}{2} \gamma_w z_c^2$
where $z_c$ depth of the tension crack (m)
Active thrust before the crack (textbook)
$P_{a,h}^{\text{no crack}} = \int_0^{H'} \left[ (\sigma'_v + q) K_a - 2c\sqrt{K_a} \right] dz \cdot \cos\beta$
where $q$ surcharge (kPa); $H'$ height of the virtual back (m)
Passive resistance
$P_p = \int_0^{D_f} \sigma_p \, dz$

3Uplift

Uplift pressure at the heel
$u_{heel} = \gamma_w h_{heel}$
Uplift pressure at the toe
$u_{toe} = \gamma_w h_{toe}$
Uplift
$U = \tfrac{1}{2} (u_{heel} + u_{toe}) B$
where $u_{heel}$ uplift pressure at the heel (kPa); $u_{toe}$ uplift pressure at the toe (kPa); $B$ base width (m)
Lever arm of the uplift
$x_U = \frac{B (u_{toe} + 2 u_{heel})}{3 (u_{toe} + u_{heel})}$
where $u_{heel}$ uplift pressure at the heel (kPa); $u_{toe}$ uplift pressure at the toe (kPa)

4Overturning

Resisting moment
$\Sigma M_R = \Sigma W_i x_i + W_s x_s + P_{a,v} B$
where $W_{stem}$ weight of each part (kN/m); $x_{stem}$ lever arm of each part (m); $W_s$ weight of the soil over the heel (kN/m); $x_s$ lever arm of each stratum (m); $P_{a,v}$ active thrust, vertical (kN/m); $B$ base width (m)
Overturning moment
$\Sigma M_O = P_{a,h} \, y_a + P_w \, y_w + P_{wc} \, y_{wc} + U x_U$
where $P_{a,h}$ active thrust, horizontal (kN/m); $y_a$ active thrust arm (m); $P_w$ water thrust (kN/m); $y_w$ water thrust arm (m); $P_{wc}$ thrust of the water in the crack (kN/m); $y_{wc}$ crack water thrust arm (m); $U$ uplift (kN/m); $x_U$ lever arm of the uplift (m)
Overturning
$FS = \frac{\Sigma M_R}{\Sigma M_O} \; \ge 2.0$

5Sliding

Total vertical force
$\Sigma V = \Sigma W_i + W_s + P_{a,v} - U$
where $W_{stem}$ weight of each part (kN/m); $W_s$ weight of the soil over the heel (kN/m); $P_{a,v}$ active thrust, vertical (kN/m); $U$ uplift (kN/m)
Total horizontal force
$\Sigma H = P_{a,h} + P_w + P_{wc}$
where $P_{a,h}$ active thrust, horizontal (kN/m); $P_w$ water thrust (kN/m); $P_{wc}$ thrust of the water in the crack (kN/m)
Friction angle at the base
$\delta = \tfrac{2}{3} \phi$
Adhesion at the base
$c_a = \tfrac{2}{3} c$
Resistance to sliding
$R = \Sigma V \tan\delta + B \, c_a + P_p$
where $\Sigma V$ total vertical force (kN/m); $\delta$ friction angle at the base (°); $B$ base width (m); $c_a$ adhesion at the base (kPa); $P_p$ passive resistance (kN/m)
Sliding
$FS = \frac{R}{\Sigma H} \; \ge 1.5$

6Bearing capacity

Eccentricity of the resultant
$e = \left| \frac{B}{2} - \frac{\Sigma M_R - \Sigma M_O}{\Sigma V} \right|$
where $B$ base width (m); $\Sigma M_R$ resisting moment (kN·m/m); $\Sigma M_O$ overturning moment (kN·m/m); $\Sigma V$ total vertical force (kN/m)
Maximum contact pressure
$q_{max} = \frac{\Sigma V}{B} \left(1 + \frac{6e}{B}\right)$
where $\Sigma V$ total vertical force (kN/m); $B$ base width (m); $e$ eccentricity of the resultant (m)
Minimum contact pressure
$q_{min} = \frac{\Sigma V}{B} \left(1 - \frac{6e}{B}\right)$
where $\Sigma V$ total vertical force (kN/m); $B$ base width (m); $e$ eccentricity of the resultant (m)
Effective width
$B' = B - 2e$
where $B$ base width (m); $e$ eccentricity of the resultant (m)
Overburden at the base
$q = \sigma'_v(D_f)$
Ultimate bearing capacity
$q_u = c N_c F_{cs} F_{cd} F_{ci} + q N_q F_{qs} F_{qd} F_{qi} + \tfrac{1}{2} \gamma' B' N_\gamma F_{\gamma s} F_{\gamma d} F_{\gamma i}$
where $q$ overburden at the base (kPa); $\gamma'$ effective unit weight under the base (kN/m³); $B'$ effective width (m); $N_c$ bearing capacity factor; $N_q$ bearing capacity factor; $N_\gamma$ bearing capacity factor; $F_{cs}$ fcs; $F_{qs}$ fqs; $F_{\gamma s}$ fgs; $F_{cd}$ fcd; $F_{qd}$ fqd; $F_{\gamma d}$ fgd; $F_{ci}$ fci; $F_{qi}$ fqi; $F_{\gamma i}$ fgi
Bearing capacity
$FS = \frac{q_u}{q_{max}} \; \ge 3.0$

7Eccentricity

Eccentricity
$e \le \frac{B}{6}$

8Gabion levels

Overturning of each level on the one below
$FS \; \ge 2.0$
Sliding of each level on the one below
$FS \; \ge 1.5$