Culmann's method for passive pressure

Culmann's passive method finds the minimum passive resistance by trial wedges in front of the wall.

Procedure

1. Draw the wall $AB$, the backfill surface in front of the wall and the heel/toe $B$ to scale.
2. Draw the $\phi$-line $BD$ from $B$ at angle $\phi$ to the horizontal, but on the opposite side from the active case (below the horizontal, going into the soil), because the resultant of the failure plane tilts the other way in the passive state.
3. Draw the pressure line $BE$ from $B$ at angle $\psi = \theta+\delta$ to $BD$, where $\theta$ is the angle of the wall back with the horizontal and $\delta$ the wall friction (the thrust acts on the other side of the normal in the passive state).
4. Draw trial failure planes $BC_1, BC_2, BC_3,\dots$ from $B$ to the ground surface, in front of the wall.
5. Calculate the weight $W_i$ of each trial wedge $ABC_i$ (per metre), adding any surcharge on it.
6. Mark $W_i$ to a scale on $BD$ to get points $b_1, b_2, \dots$
7. Through each $b_i$ draw a line parallel to $BE$ to cut the trial plane $BC_i$ at $c_i$.
8. Join $c_1 c_2 c_3 \dots$ to obtain the Culmann curve.
9. Draw a tangent to the curve parallel to $BD$ on the side closest to $BD$. The minimum distance from the curve to $BD$, measured parallel to $BE$ and multiplied by the force scale, gives the passive resistance $P_p$. (Passive failure occurs along the plane giving the least resistance, so the minimum ordinate is taken, whereas it is the maximum for active.)
10. The critical plane is $BC_m$; $P_p$ acts at one-third of the height above the base.

   A  ____ ground surface ___ C3 C2 Cm C1
   |  \
   |   \_ trial planes
 B +----+-----> b1 b2 bm b3  (W on BD, below horizontal)
   BE at psi = theta + delta from BD
   P_p = minimum ordinate of Culmann curve