$M_{live-load} = 45\,000(9) + \frac{1}{2}(9)(45\,000)$
$M_{live-load} = 607\,500 \, \text{lb}\cdot\text{ft}$
$S_{required} \ge \dfrac{M_{live-load}}{(\,f_b\,)_{max}} \ge \dfrac{607\,500(12)}{20\,000}$
$S_{required} \ge 364.5 \, \text{in}^3$
From Appendix B, Table B-7 Properties of Wide-Flange Sections (W Shapes): US Customary Units, of text book:
| Designation |
Section Modulus |
| W12 × 279 |
393 in3 |
| W14 × 233 |
375 in3 |
| W24 × 146 |
371 in3 |
| W27 × 146 |
411 in3 |
| W30 × 132 |
380 in3 |
| W33 × 130 |
406 in3 |
Use W33 × 130 with S = 406 in3. answer
Checking:
$S_{resisting} \ge S_{live-load} + S_{dead-load}$
$S_{live-load} = 364.5 \, \text{in}^3$
$S_{dead-load} = \dfrac{\frac{1}{8}(130)(36^2)(12)}{20\,000} = 12.636 \, \text{in}^3$
$S_{live-load} + S_{dead-load} = 364.5 + 12.636$
$S_{live-load} + S_{dead-load} = 377.136 \, \text{in}^3$
$(S_{resisting} = 406 \, \text{in}^3) \gt 377.136 \, \text{in}^3$ (okay!)
Actual bending moment:
$M = M_{live-load} + M_{dead-load}$
$M = 607\,500 + \frac{1}{8}(130)(36^2)$
$M = 628\,560 \, \text{lb}\cdot\text{ft}$
Actual stress:
$(\,f_b\,)_{max} = \dfrac{Mc}{I} = \dfrac{628\,560(12)}{406}$
$(\,f_b\,)_{max} = 18\,578.13 \, \text{psi}$
$(\,f_b\,)_{max} = 18.58 \, \text{ksi}$ answer
