$5\mathrm{.}6-23$ A retaining wall $6ft$ high is constructed al

horizontal wood planks $2\mathrm{.}5$ in$.$ thick $($actual dimen$-$

sion$)$ that are supported by vertical wood piles of a

$12$ in$.$ diameter $($actual dimension$),$ as shown in the

figure. The lateral earth pressure is $p_1=125l\frac{b}{f}{t}^2$ at

the top of the wall and $p_2=425l\frac{b}{f}{t}^2$ at the bottom.

$($a$)$ Assuming that the allowable stress in the wood

is $1175,$ calculate the maximum permissible

spacing $s$ of the piles.

$($b$)$ Find the required diameter of the wood piles

so that piles and planks $(t=2\mathrm{.}5in.)$ reach the

allowable stress at the same time.

Hint: Observe that the spacing of the piles may be

governed by the load$-$carrying capacity of either the

planks or the piles. Consider the piles to act as canti$-$

lever beams subjected to a trapezoidal distribution al

load, and consider the planks to act as simple beamt

between the piles. To be on the safe side, assume that

the pressure on the bottom plank is uniform and

equal to the maximum pressure.