# Problem 1008 | Finding the width of steel plate reinforcement

**Problem 1008**

A timber beam 150 mm wide by 250 mm deep is to be reinforced at the top and bottom by steel plates 10 mm thick. How wide should the steel plates be if the beam is to resist a moment of 40 kN·m? Assume that n = 15 and the allowable stresses in the wood and steel are 10 MPa and 120 MPa, respectively.

**Solution 1008**

$I = \dfrac{15b(270^3)}{12} - \dfrac{(15b - 150)(250^3)}{12}$

$I = 24\,603\,750b - 19\,531\,250b + 195\,312\,500$

$I = 5\,072\,500b + 195\,312\,500$

$f_b = \dfrac{Mc}{I}$

Based on allowable flexural stress of steel:

$c = \frac{1}{2}(250) + 10 = 135 ~ \text{mm}$

Thus,

$\dfrac{120}{15} = \dfrac{40(1000^2)(135)}{5\,072\,500b + 195\,312\,500}$

$b = 94.57 ~ \text{mm}$

Based on allowable flexural stress of wood:

$c = \frac{1}{2}(250) = 125 ~ \text{mm}$

Thus,

$10 = \dfrac{40(1000^2)(125)}{5\,072\,500b + 195\,312\,500}$

$b = 60.01 ~ \text{mm}$

For stronger section, use $b = 94.6 ~ \text{mm}$ *answer*