508 Minimum height of beam with given maximum flexural stress

Problem 508
Determine the minimum height h of the beam shown in Fig. P-508 if the flexural stress is not to exceed 20 MPa.

Solurion 508
$\Sigma M_{R2} = 0$
$3R_1 = 2(5) + 2(2.5)(4)$
$R_1 = 10 \text{ kN}$

$\Sigma M_{R1} = 0$
$3R_2 = 1(5) + 1(2.5)(4)$
$R_2 = 5 \text{ kN}$

$f_b = \dfrac{Mc}{I}$
Where:
$f_b = 20 \text{ MPa}$
$M = 5 \text{ kN} \cdot \text{m} = 5(1000^2) \text{ N} \cdot \text{m}$
$c = \frac{1}{2} h$
$I = \dfrac{bh^3}{12} = \dfrac{80h^3}{12} = \frac{20}{3} h^3$

Thus,
$20 = \dfrac{5(1000^2)(\frac{1}{2} h)}{\frac{20}{3} h^3}$
$h^2 = 18 \, 750$
$h = 137 \text{ mm}$