Mu$$strame c$$mo resolver $10\mathrm{.}2$ The initial stresses on a soil are ${}_1^{\text{'}}={}_z^{\text{'}}$ and ${}_3^{\text{'}}=K_o{}_z^{\text{'}},$

where $K_o$ is the lateral earth pressure coefficient at rest

$($see Chapter $7).$ The soil was then brought to failure

$($critical state$)$ by reducing ${}_3^{\text{'}}$ while keeping ${}_1^{\text{'}}$ con$-$

stant. At failure, ${}_3^{\text{'}}=K{}_1^{\text{'}},$ where $K$ is a lateral earth

pressure coefficient. Show that

$\frac{{}_{cs}}{{}_{zx}}=\frac{1-K}{1-K_o}cos{}_{cs}^{\text{'}}$

where ${}_{zx}$ is the maximum shear stress under the initial

stresses and ${}_{cs}$ is the failure $($critical$)$ shear stress.