# Solution to Problem 559 | Built-up Beams

**Problem 559**

A beam is composed of 6 planks, each 100 mm wide and 20 mm thick, piled loosely on each other to an overall dimension of 100 mm wide by 120 mm high. (a) Compare the strength of such a beam with that of a solid beam of equal overall dimensions. (b) What would be the ratio if the built-up beam consisted of a 12 planks each 100 mm wide by 10 mm thick?

**Solution 559**

$I_1 = \dfrac{100(20^3)}{12}$

$I_1 = \frac{200\,000}{3} \, \text{mm}^4$

For the whole beam

$I = 6I_1 = 6(200\,000/3)$

$I = 400\,000 \, \text{mm}^4$

$\dfrac{M}{EI} = \dfrac{M_1}{E_1 I_1}$ where E_{1} = E

$\dfrac{M}{E(400\,000)} = \dfrac{M_1}{E(\frac{200\,000}{3})}$

$\dfrac{M_1}{M} = \dfrac{1}{6}$ *answer*

Part (b)

$I_1 = \dfrac{100(10^3)}{12}$

$I_1 = \frac{25\,000}{3} \, \text{mm}^4$

For the whole beam

$I = 12I_1 = 12(25\,000/3)$

$I = 100\,000 \, \text{mm}^4$

$\dfrac{M}{EI} = \dfrac{M_1}{E_1 I_1}$ where E_{1} = E

$\dfrac{M}{E(100\,000)} = \dfrac{M_1}{E(\frac{200\,000}{3})}$

$\dfrac{M_1}{M} = \dfrac{1}{12}$ ) *answer*