Example $13\mathrm{.}14$: A retaining wall $9$ m high retains granular fill weighing $18k\frac{N}{m^3}$ with level

surface. The active thrust on the wall is $180$ kN per metre length of the wall. The height of the

wall is to be increased and to keep the force on the wall within allowable limits$,$ the backfill in

the top$-$half of the depth is removed and replaced by cinders. If cinders are used as backfill

even in the additional height, what additional height may be allowed if the thrust on the wall

is to be limited to its initial value? The unit weight of the cinders is $9k\frac{N}{m^3}.$ Assume the

friction angle for cinders the same as that for the soil.