In cohesive soil, if active earth pressure is zero, n depth (z) is given by-

Explanation:

Rankine’s Active earth Pressure theory in cohesive Soil:

${\mathrm{P}}_{\mathrm{a}}={\mathrm{K}}_{\mathrm{a}}\gamma \mathrm{Z}-2{\mathrm{c}}^{\prime }\sqrt{{\mathrm{K}}_{\mathrm{a}}}$

So, at Z =0,

${\mathrm{P}}_{\mathrm{a}}=-2{\mathrm{c}}^{\prime }\sqrt{{\mathrm{K}}_{\mathrm{a}}}$

Negative sign shows that the pressure is negative.

For the stress to reverse its nature, a point will reach where the earth pressure value will be zero,

Therefore, Pa = 0,

${\mathrm{K}}_{\mathrm{a}}\gamma \mathrm{Z}-2{\mathrm{c}}^{\prime }\sqrt{{\mathrm{K}}_{\mathrm{a}}}=0$

${\mathrm{Z}}_{\mathrm{c}}=\frac{2{\mathrm{c}}^{\prime }\sqrt{{\mathrm{K}}_{\mathrm{a}}}}{{\mathrm{K}}_{\mathrm{a}}\gamma }=\frac{2{\mathrm{c}}^{\prime }}{\gamma \sqrt{{\mathrm{K}}_{\mathrm{a}}}}$ = (2C'/γ) tanα  where α = 45+?/2  

The above depth is known as the depth of tensile crack (Zc).

Additional Information

The critical depth of vertical cut is 2 × Zc = $\frac{4\mathrm{c}}{\gamma \sqrt{\mathrm{k}\mathrm{a}}}$= H

We know, $k_a=\frac{1-Sin?}{1+Sin?}=Co{t}^2(45+?/2)$

Putting the value of Ka in the above equation, we get

$H=\frac{4\mathrm{C}}{\gamma }\tan \left(45+\frac{\mathrm{\varnothing }}{2}\right)$