Problem: For the W4 x 13 and the loading shown, with an additional distributed load of 20 kip/ft across the entire beam (not shown), and for L = 8 ft and M_o = 6 kip-ft, determine:

a) the slope at point A
b) the deflection at point D

Solution: From the sum of the forces, it's clear that the reactions must be equal and be 80 kips each.  Using the fourth order equation and the distributed load,

Since EIy''' is the shear, V = 80 kip at x = 0, C₁ = 80 kip. Continuing with the integration,

Since this is the moment, the external moments have to be included:

But the moment is zero at x = 0, so the constant is zero. Integrating again,

The slope is not known anywhere, so the constant cannot be determined at this point. Integrating again,

At x = 0, y = 0, so C₄ = 0. At x = 8 ft, y = 0, so

At x = 0,

At x = 4 ft,

This is huge and would probably be unsafe.  I should have chosen one of the larger structural shapes.  Incidentally, as I was hiking the Appalachian Trail in 1993, I crossed a bridge on a narrow walkway (part of the interstate).  The trucks going by caused at least an inch of vertical vibration!

¹ The beam and loading were adapted from problem 9.38 in Beer & Johnston's Mechanics of Materials, 2nd Edition.