How much of an effect does the longitudinal grade of a beam have on bending moments? Typically this is something we'd ignore, as road alignments aren't noticeably steep (at least road alignments which have bridges).
But it can make an impact to the vertical loads.
The 2nd free-body diagram in the image above is the *correct* way to model the behaviour of the beam, but nobody wants to have supports at different heights in their analysis models. We'd much prefer to make a few simplifications and call it a day.
So how much of a difference does this really make?
To figure that out, we have to look at our beam in reality. This vertical load can be split into two components: one perpendicular to the beam and one acting parallel along the member.
If the vertical load is w, we can determine the perpendicular components easily with trigonometry, taking the angle from horizontal as θ.
So the relationship is just based on cosine. As the grade of the road increases, the difference between analysing the span as a flat element vs. its true geometry just follows the cosine curve.
TL;DR
This doesn't make any practical difference for even very steep road geometries. Our engineering intuition is correct. Good to prove it, though.