Shear and Moment Diagrams by Inspection

You do not need to integrate to sketch shear force and bending moment diagrams. Three relationships let you draw them by inspection, and they are the same relationships any calculator uses internally.

The three rules

1. The shear diagram’s slope equals the load intensity. dV/dx = −w(x). No load → horizontal shear. Uniform load → straight sloping shear. Linear load → parabolic shear.
2. The moment diagram’s slope equals the shear. dM/dx = V. Zero shear → moment is at a peak (or flat). Constant shear → straight-line moment. Linear shear → parabolic moment.
3. Concentrated effects jump. A point load jumps the shear by its magnitude; an applied moment jumps the moment by its magnitude. Nothing else jumps.

Reading a simply supported beam with a central point load

• From the left reaction, shear starts at +P/2 and stays flat (no load).
• At midspan the point load drops shear by P → −P/2.
• Shear is constant between loads, so the moment is straight-lined, peaking where shear crosses zero — under the load. Peak PL/4. ✓

Reading a full-span uniform load

• Shear starts at +wL/2 and slopes down uniformly, crossing zero at midspan.
• Moment is a parabola, maximum at midspan where shear is zero: wL²/8. ✓

Quick checks

• The moment diagram must return to zero at a simple end support (no applied end moment).
• Wherever shear crosses zero, expect a moment extremum.
• The area under the shear curve between two points equals the change in moment between them. If your peak moment doesn’t match the triangle area, one of them is wrong.

Where inspection breaks down

Statically indeterminate beams (continuous over supports) need the reactions first — see the three-moment equation. Once the reactions are known, inspection works again.

Try it yourself with the beam calculator — enter a load case, sketch the diagrams on paper first, then check against the plots.