Culmann's method, active, with surcharge

Culmann's method is a graphical wedge method that finds the active thrust on a wall for any backfill surface, including inclined backfill and surcharge.

Procedure (cohesionless backfill with inclined surface and line load)

1. Draw the wall $AB$ and the backfill surface to scale. Mark the heel $B$.
2. Draw the $\phi$-line $BD$ from $B$ at angle $\phi$ to the horizontal.
3. Draw the pressure line $BE$ from $B$ at angle $\psi = \theta-\delta$ above $BD$, where $\theta$ is the angle the wall back makes with the horizontal (on the fill side) and $\delta$ is wall friction.
4. Draw several trial failure planes $BC_1, BC_2, \dots$ from $B$ to points on the ground surface.
5. Compute the weight $W_1, W_2,\dots$ of each trial wedge $ABC_i$ per metre. If a line load $Q$ lies inside a wedge, add $Q$ to that wedge's weight; for a uniform surcharge $q$, add $q\times$(horizontal width of the wedge) to the weight.
6. Choose a scale of force and mark $W_i$ along $BD$: $b_1, b_2, \dots$
7. From each $b_i$ draw a line parallel to $BE$ to meet the corresponding trial plane $BC_i$ at $c_i$.
8. Join $c_1 c_2 \dots$ to get the Culmann curve. The step in the curve at the line load position shows the effect of $Q$ (jump when the plane passes beyond $Q$).
9. Draw a tangent to the curve parallel to $BD$. The ordinate from the point of tangency $c_m$ to $BD$, parallel to $BE$, scaled by the force scale, is the active thrust $P_a$ (maximum).
10. The critical failure plane is $BC_m$; $P_a$ acts at $H/3$ above the base (uniform fill) at angle $\delta$ to the normal to the wall.

        Q (line load)
   A    v      ground surface
   |\______________________ C1 C2 Cm C3
   |                    ..
   |             .. ...    Culmann
   |        ..      curve
   |   ..    /
 B +--------/---> b1 b2 bm b3 (weights on BD)
   BD at phi, BE at psi from BD

Effect of surcharge on active pressure

• A uniform surcharge $q$ is equivalent to an extra soil height $h_e = q/\gamma$. It adds a uniform (rectangular) pressure $\Delta p_a = K_a q$ over the whole height, so the total thrust increases by $K_a q H$.
• The resultant thrust moves higher than $H/3$ because the pressure distribution changes from triangular to trapezoidal, increasing the overturning moment.
• A line load adds a localised pressure on the wall (a discontinuity in the Culmann curve) and increases the thrust only when the load is within the sliding wedge; the closer to the wall the larger the effect.
• Hence surcharges (traffic, buildings) must always be included in wall design.