$M_{live-load} = 18\,000(9) + \frac{1}{2}(9)(18\,000)$
$M_{live-load} = 243,000 \, \text{lb}\cdot\text{ft}$
$S_{required} \ge \dfrac{M_{live-load}}{(\,f_b\,)_{max}} \ge \dfrac{243\,000(12)}{20\,000}$
$S_{required} \ge 145.8 \, \text{in}3$
From Appendix B, Table B-7 Properties of Wide-Flange Sections (W Shapes): US Customary Units, of text book:
| Designation |
Section Modulus |
| W12 × 120 |
163 in3 |
| W14 × 99 |
157 in3 |
| W16 × 89 |
155 in3 |
| W18 × 76 |
146 in3 |
| W21 × 73 |
151 in3 |
| W24 × 68 |
154 in3 |
Use W24 × 68 with S = 154 in3. answer
Checking:
$S_{resisting} \ge S_{live-load} + S_{dead-load}$
Where
$S_{live-load} = 145.8 \, \text{in}^3$
$S_{dead-load} = \dfrac{\frac{1}{8}(68)(36^2)(12)}{20\,000}$
$S_{dead-load} = 6.61 \, \text{in}^3$
$S_{live-load} + S_{dead-load} = 145.8 + 6.61$
$S_{live-load} + S_{dead-load} = 152.41 \, \text{in}^3$
Thus,
$(S_{resisting} = 154 \, \text{in}^3) > 152.41 \, \text{in}^3$ (okay!)
Actual bending moment:
$M = M_{live-load} + M_{dead-load}$
$M = 243\,000 + \frac{1}{8}(68)(36^2)$
$M = 254\,016 \, \text{lb}\cdot\text{ft}$
Actual stress:
$(\,f_b\,)_{max} = \dfrac{M}{S} = \dfrac{254\,016(12)}{154}$
$(\,f_b\,)_{max} = 19\,793.45 \, \text{psi}$
$(\,f_b\,)_{max} = 19.79 \, \text{ ksi}$ answer
