Given:

$b = 150 ~ \text{mm}$

$d = 300 ~ \text{mm}$

**Part 1: ***P* = 30 kN

$V = \frac{1}{2}P = \frac{1}{2}(30)$
$V = 15 ~ \text{kN}$

$f_v = \dfrac{3V}{2bd'}\left( \dfrac{d}{d'} \right)^2$

$1.0 = \dfrac{3(15\,000)}{2(150d')}\left( \dfrac{300}{d'} \right)^2$

$d' = 238 ~ \text{mm}$

depth of notch = 300 - 238 = 62 mm *answer*

**Part 2: Depth of notches = 100 mm**

$d' = d - 100 = 300 - 100$
$d' = 200 ~ \text{mm}$

$f_v = \dfrac{3V}{2bd'}\left( \dfrac{d}{d'} \right)^2$

$1.0 = \dfrac{3V(1000)}{2(150)(200)}\left( \dfrac{300}{200} \right)^2$

$V = 8.89 ~ \text{kN}$

Safe value of *P* = 2*V* = 17.78 kN *answer*

**Part 3: ***P* = 25 kN and depth of notches = 150 mm

$d' = d - 150 = 300 - 150 = 150 ~ \text{mm}$
$V = \frac{1}{2}P = \frac{1}{2}(25) = 12.5 ~ \text{kN}$

$f_v = \dfrac{3V}{2bd'}\left( \dfrac{d}{d'} \right)^2$

$f_v = \dfrac{3(12\,500}{2(150)(150)}\left( \dfrac{300}{150} \right)^2$

$f_v = 3.33 ~ \text{MPa}$ *answer*