# Coulomb and Rankine Earth Pressure Coefficients for Vertical Walls

Virtually all sheet pile walls are vertical; thus, it is not necessary to worry about the wall angle when considering lateral earth pressure coefficients for these walls. This means that equations for vertical walls and vertical walls with level backfills are all that is left.

First, a complete chart, showing the nomenclature, from NAVFAC DM 7.02:

For sheet pile walls, $\theta = 0$. The lateral earth pressure equations for Coulomb earth pressure theory with sloping backfill are as follows:

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

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

And for Rankine, same conditions:

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

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

Now let us reduce $\beta = 0$ for level backfill. The Coulomb coefficients for vertical walls and level backfill are as follows:

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

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

And for Rankine theory:

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

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

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