I redraw the load to make the calculations easier:
First, I need the reactions at the supports:
Taking the usual very short free body diagrams on either end, I get a shear of - 18 kips at B and a moment of - 135 kip-ft at B. The moment and shear at A are zero. These will serve as checks on the values obtained using the area relations:
This checks at x = 0 and at x = 18 ft. Equating this to zero to find the place where the shear is zero and therefore the position of the maximum moment yields nothing useful as you can see if you do it. The moment is
Where is this a maximum? As I already said, equating the derivative to zero is not much use. But since both terms are negative it is clear that the absolute value of the moment grows monotonically until x = 18 ft. Therefore, the maximum of the absolute value of the moment is 135 kip-ft.
Now I compute the section modulus S:
From Appendix C,
| designation | Sx (in3) |
| W18 x 50 | 88.9 |
| W16 x 57 | 92.2 |
| W14 x 53 | 77.8 |
| W12 x 72 | 97.4 |
| W10 x 68 | 75.7 |