Rankine active pressure wedge

Rankine passive pressure wedge

Remarks:
Both positive and negative values of β yield the same earth pressure coefficient. Therefore, Rankine’s method applies to cases with a level or upward-sloping active side and a level or downward-sloping passive side.

The resultant active thrust P_a is inclined parallel to the backfill surface, making an angle β with the horizontal. Accordingly, the horizontal component of the thrust,P_ah, is obtained by multiplying P_a by cosβ.

Similar adjustment is also required for the passive case.

Governing Equations:

Bowles (1996) and Geoguide 1:

$$ K_a = \cos\beta \, \frac{\cos\beta - \sqrt{\cos^2\beta - \cos^2\phi}}{\cos\beta + \sqrt{\cos^2\beta - \cos^2\phi}} $$ $$ K_p = \cos\beta \, \frac{\cos\beta + \sqrt{\cos^2\beta - \cos^2\phi}}{\cos\beta - \sqrt{\cos^2\beta - \cos^2\phi}} $$

$$ \begin{align*} p_a &= \gamma z K_a \\ P_a &= \int_0^H \gamma z K_a \, dz = \frac{1}{2} \gamma H^2 K_a \end{align*} $$

$$ \begin{align*} p_p &= \gamma z K_p \\ P_p &= \int_0^H \gamma z K_p \, dz = \frac{1}{2} \gamma H^2 K_p \end{align*} $$

Horizontal force acting on the wall surface:

$$ P_{a,h} = P_a \cos\beta $$ $$ P_{p,h} = P_p \cos\beta $$