# Problem 1002 | Increase in moment capacity due to steel plate reinforcement

**Problem 1002**

A timber beam is reinforced with steel plates rigidly attached at the top and bottom as shown in Fig. P-1002. By what amount is moment increased by the reinforcement if n = 15 and the allowable stresses in the wood and steel are 8 MPa and 120 MPa, respectively?

**Solution 1002**

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

$M = \dfrac{f_b I}{c}$

Without the steel plate reinforcement

$c = \dfrac{300}{2} = 150 ~ \text{mm}$

$M_1 = \dfrac{8(562\,500\,000)}{150}$

$M_1 = 30\,000\,000 ~ \text{N}\cdot\text{mm} $

$M_1 = 30 ~ \text{kN}\cdot\text{m}$

With the steel plate reinforcement

I & = \dfrac{15(120)(300 + 10 + 10)^3}{12} - \dfrac{[ \, 15(120) - 250 \, ](300^3)}{12} \\

& = 1\,427\,700\,000 ~ \text{mm}^4

\end{align}$

$\begin{align}

c & = \dfrac{300 + 10 + 10}{2} \\

& = 160 ~ \text{mm}

\end{align}$

Convert steel to wood, use f_{bw} = f_{bs} / n

$M_2 = \dfrac{\dfrac{120}{15}(1\,427\,700\,000)}{160}$

$M_2 = 71\,385\,000 ~ \text{N}\cdot\text{mm}$

$M_2 = 71.4 ~ \text{kN}\cdot\text{m}$

Increase in moment capacity

$\Delta M = M_2 - M_1 = 71. 4 - 30$

$\Delta M =41.4 ~ \text{kN}\cdot\text{m}$ *answer*