Shear and bending moment diagrams are the two plots that decide what size a beam needs to be. They take ten minutes to learn and people get them wrong for years, almost always because of a sign convention rather than the arithmetic.
What the two diagrams actually are
Cut the beam at some position x and look at the left-hand piece. The shear force V(x) is the vertical force that piece needs to stay in equilibrium; the bending moment M(x) is the couple it needs. Slide the cut along the beam and you have both diagrams.
The sign convention that trips everyone up
Shear is positive when the left segment pushes up. Bending moment is positive when the beam sags — tension on the bottom fibre. That is why a cantilever shows a negative moment at the wall: it hogs, putting the top in tension. If you remember one thing, remember that the sign of M tells you which face is in tension, and that is the face that cracks first in concrete and fails first in a weld.
Draw them in four steps
1. Find the reactions. Sum moments about one support to get the other, then sum vertical forces. 2. Start at the left end at V = 0 and walk right. 3. Every point load is a vertical jump in V equal to the load; every distributed load is a slope equal to −w. 4. The moment diagram is the area under the shear diagram — literally, because V = dM/dx.
The check that catches almost every mistake
The bending moment peaks exactly where the shear crosses zero. That follows from V = dM/dx: where the slope of M is zero, M is at a maximum. Find the zero crossing on your shear plot and the moment peak must line up with it. If it does not, a load or a reaction has the wrong sign. A second free check: the moment must be zero at any free end and at any simple support that has nothing beyond it.
Overhangs, and the reaction that goes negative
Move a support inboard so part of the beam cantilevers past it and something important happens: the far support can go into uplift. It has to be held down, not just supported. A beam resting on a bearing pad will simply lift off, and this catches people out regularly when a cantilevered section is added late in a design. A negative reaction is not a sign error — it is a requirement for anchor bolts.
What to do with the numbers
Take Mmax and get the bending stress from σ = M/Z, or the section modulus you need from Z = M/σallow. Then check deflection separately, because deflection governs more beams than stress does — a beam can pass comfortably on stress and still bounce unacceptably. Build any of this on the Shear & Moment Diagram tool, get Z from Section Properties, and run the code-style capacity check on Steel Beam Check.