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Derive the expression for Rankine's Active earth pressure for cohesive back fill.

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Consider a retaining wall of height $\mathrm{H}$ with a smooth vertical back, retaining a cohesive backfill. The relationship between the major principal stress $\sigma_{1}$ and minor principal stress $\sigma_{3}$ at failure (Plastic equilibrium) can be expressed in the form $$\sigma_{1}=\sigma_{3}\left(\frac{1+\sin \phi}{1-\sin \phi}\right)+2 c \sqrt{\frac{1+\sin \phi}{1-\sin \phi}}$$ Active State In active state, lateral stress $\sigma_{h}$ reduces to its minimum value i.e., $p_{a}$ while the vertical stress $\sigma_{v}$ remains unchanged. Since, $$\sigma_{v}\gt\sigma_{h}$$ Hence, $$\begin{array}{l} \sigma_{1}=\sigma_{v} \\ \sigma_{3}=\sigma_{h}=p_{a} \end{array}$$ Substituting the values of $\sigma_{1}$ and $\sigma_{3}$ in above eq., we get $\sigma_{v}=\sigma_{h}\left(\frac{1+\sin \phi}{1-\sin \phi}\right)+2 c \sqrt{\frac{1+\sin \phi}{1-\sin \phi}}$ $\sigma_{v}=p_{a}\left(\frac{1+\sin \phi}{1-\sin \phi}\right)+2 c \sqrt{\frac{1+\sin \phi}{1-\sin \phi}}$ $p_{a}=\left(\frac{1-\sin \phi}{1+\sin \phi}\right) \sigma_{v}-2 c \sqrt{\frac{1-\sin \phi}{1+\sin \phi}}$ $p_{a}=k_{a} \sigma_{v}-2 c \sqrt{k_{a}} \quad\left[\because k_{a}=\frac{1-\sin \phi}{1+\sin \phi}\right]$