How to Read a Bending Moment and Shear Force Diagram

A bending moment diagram looks like a shape drawn under a beam, and it's tempting to just read the peak number off it and move on. But the shape itself is telling you something specific about the loading, and knowing how to read it is what lets you sanity-check a result — by hand or from software — in about five seconds.

Shear force diagram: it's just a running total

At any point along a beam, the shear force is simply the sum of every vertical force to one side of that point. Walk from the left support rightward, add each upward reaction as you pass it, subtract each downward load — that running total is the SFD. This is why:

• Under a point load, the SFD jumps instantly, straight down by the load's magnitude. A vertical line on an SFD always means a point load sits exactly there.
• Under a UDL, the SFD changes gradually — a straight sloped line, because you're adding up a little more load with every step you take.
• Between loads, with nothing acting, the SFD stays flat. No load, no change in the running total.

Bending moment diagram: the area under the SFD

This is the relationship worth actually internalising: the bending moment at any point is the running area under the shear force diagram up to that point. It isn't a coincidence that these two diagrams always pair the way they do — one is the mathematical integral of the other. That single fact explains every shape you'll ever see:

• Constant shear → straight-line moment. A cantilever with just a tip load has constant shear along its whole length, so the moment grows in a straight line, peaking at the fixed end.
• Linearly varying shear → parabolic moment. A UDL gives a sloped SFD, so the BMD curves — and the peak sits exactly where the SFD crosses zero, every time, because that's the point where you stop adding area and start subtracting it.
• A jump in shear → a kink in the moment diagram. The BMD doesn't jump at a point load (moment itself stays continuous), but its slope changes abruptly there — that's the sharp corner you see pointing to a point load's exact location.

Sign convention: sagging is positive

A beam that sags — curves so the bottom fibre stretches — is carrying positive bending moment, by the overwhelmingly standard convention. For a simply supported beam under a downward load, that's the entire diagram: positive throughout, peaking wherever the shear crosses zero. A cantilever under a downward tip load does the opposite — it hogs, curving so the top fibre stretches — and that's negative moment by the same convention, largest at the fixed support.

The fastest sanity check there is

For a simply supported beam, the moment at both end supports must be exactly zero — there's nothing there to resist a moment, so none can exist. If your diagram doesn't return to zero at a simple support, something upstream is wrong, before you even look at the peak value. It's the single fastest check available, and it costs nothing to run every time.