Rankine and Coulomb Earth Pressure Coefficients

For retaining walls, computing these is important; but the textbooks and reference books are frequently confusing and sometimes wrong. These formulae, derived using Maple, should clear up a few things, although they’re a) not always in the format you’re used to and b) subject to the Terms and Conditions of this site.

Let’s start with a diagram and the basic Coulomb formulae, from NAVFAC DM 7.02.

It’s important for to have a nomenclature chart for any lateral earth pressure coefficient formulae.

Coulomb Active Coefficients

All angles nonzero:

$K_{{a}}=\left (\cos(\phi-\theta)\right )^{2}\left (\cos(\theta)\right)^{-2}\left (\cos(\theta+\delta)\right )^{-1}\left (1+\sqrt {{\frac {\sin(\phi+\delta)\sin(\phi-\beta)}{\cos(\theta+\delta)\cos(\theta-\beta)}}}\right )^{-2}$

Vertical Wall:

$K_{{a}}=\left(\cos(\phi)\right)^{2}\cos(\beta)\left(\cos(\delta)\cos(\beta)+2\,\cos(\delta)\sqrt{{\frac{\sin(\phi+\delta)\sin(\phi-\beta)}{\cos(\delta)\cos(\beta)}}}\cos(\beta)+\sin(\phi+\delta)\sin(\phi-\beta)\right)^{-1}$

Level Backfill:

$K_{{a}}=\left (\cos(\phi-\theta)\right )^{2}\left (\cos(\theta)\right)^{-1}\left (\cos(\theta+\delta)\cos(\theta)+2\,\cos(\theta)\cos(\theta+\delta)\sqrt {{\frac {\sin(\phi+\delta)\sin(\phi)}{\cos(\theta+\delta)\cos(\theta)}}}+\sin(\phi+\delta)\sin(\phi)\right )^{-1}$

Vertical Wall and Level Backfill:

$K_{{a}}=\left(\cos(\phi)\right)^{2}\left(\cos(\delta)+2\,\cos(\delta)\sqrt{{\frac{\sin(\phi+\delta)\sin(\phi)}{\cos(\delta)}}}+\sin(\phi+\delta)\sin(\phi)\right)^{-1}$

Coulomb Passive Coefficients

All angles nonzero:

$K_{{p}}=\left (\cos(\phi+\theta)\right )^{2}\left (\cos(\theta)\right)^{-2}\left (\cos(\theta-\delta)\right )^{-1}\left (1-\sqrt {{\frac {\sin(\phi+\delta)\sin(\phi+\beta)}{\cos(\theta-\delta)\cos(\theta-\beta)}}}\right )^{-2}$

Vertical Wall:

$K_{{p}}=\left(\cos(\phi)\right)^{2}\cos(\beta)\left(\cos(\delta)\cos(\beta)-2\,\cos(\delta)\sqrt{{\frac{\sin(\phi+\delta)\sin(\phi+\beta)}{\cos(\delta)\cos(\beta)}}}\cos(\beta)+\sin(\phi+\delta)\sin(\phi+\beta)\right)^{-1}$

Level Backfill:

$K_{{p}}=-\left (\cos(\phi+\theta)\right )^{2}\left (\cos(\theta)\right )^{-1}\left (-\cos(\theta-\delta)\cos(\theta)+2\,\cos(\theta)\cos(\theta-\delta)\sqrt {{\frac {\sin(\phi+\delta)\sin(\phi)}{\cos(\theta-\delta)\cos(\theta)}}}-\sin(\phi+\delta)\sin(\phi)\right )^{-1}$

Vertical Wall and Level Backfill:

$K_{{p}}=\left(\cos(\phi)\right)^{2}\left(\cos(\delta)-2\,\cos(\delta)\sqrt{{\frac{\sin(\phi+\delta)\sin(\phi)}{\cos(\delta)}}}+\sin(\phi+\delta)\sin(\phi)\right)^{-1}$

Rankine Active Coefficients

All angles nonzero (except obviously $\delta = 0$ ):

$K_{{a}}=\left (\cos(\phi-\theta)\right )^{2}\left (\cos(\theta)\right)^{-3}\left (1+\sqrt {{\frac {\sin(\phi)\sin(\phi-\beta)}{\cos(\theta)\cos(\theta-\beta)}}}\right )^{-2}$

Vertical Wall:

$K_{{a}}=\left (\cos(\phi)\right )^{2}\cos(\beta)\left (\cos(\beta)+2\,\sqrt {{\frac {\sin(\phi)\sin(\phi-\beta)}{\cos(\beta)}}}\cos(\beta)+\sin(\phi)\sin(\phi-\beta)\right )^{-1}$

Level Backfill:

$K_{{a}}=\left (\cos(\phi-\theta)\right )^{2}\left (\cos(\theta)\right)^{-1}\left (\left (\cos(\theta)\right )^{2}+2\,\left (\cos(\theta)\right )^{2}\sqrt {{\frac {\left (\sin(\phi)\right )^{2}}{\left (\cos(\theta)\right )^{2}}}}+1-\left (\cos(\phi)\right )^{2}\right )^{-1}$

Vertical Wall and Level Backfill:

$K_{{a}}={\frac {\left (\cos(\phi)\right )^{2}}{\left (1+\sqrt {\left (\sin(\phi)\right )^{2}}\right )^{2}}}$

Rankine Passive Coefficients

All angles nonzero (except obviously $\delta = 0$ ):

$K_{{p}}=\left (\cos(\phi+\theta)\right )^{2}\left (\cos(\theta)\right)^{-3}\left (1-\sqrt {{\frac {\sin(\phi)\sin(\phi+\beta)}{\cos(\theta)\cos(\theta-\beta)}}}\right )^{-2}$

Vertical Wall:

$K_{{p}}=-\left (\cos(\phi)\right )^{2}\cos(\beta)\left (-\cos(\beta)+2\,\sqrt {{\frac {\sin(\phi)\sin(\phi+\beta)}{\cos(\beta)}}}\cos(\beta)-\sin(\phi)\sin(\phi+\beta)\right )^{-1}$

Level Backfill:

$K_{{p}}=-\left (\cos(\phi+\theta)\right )^{2}\left (\cos(\theta)\right )^{-1}\left (-\left (\cos(\theta)\right )^{2}+2\,\left (\cos(\theta)\right )^{2}\sqrt {{\frac {\left (\sin(\phi)\right )^{2}}{\left (\cos(\theta)\right )^{2}}}}-1+\left (\cos(\phi)\right )^{2}\right )^{-1}$