Shear Force and Bending Moment Diagrams

A beam in overall equilibrium tells you nothing about any particular cross-section. Internal forces vary along the length and the design must survive the worst of them, so the diagrams exist to locate the maximum — not to be drawn for their own sake.

1. 1Sever the beam at the section of interest and discard one side.
2. 2The remaining piece was in equilibrium before, so it must be after.
3. 3Vertical forces must sum to zero — the cut face supplies the shear force.
4. 4Moments must sum to zero — the cut face supplies the bending moment.

That is the entire method. Everything else is bookkeeping about which loads fall on which side of the cut.

Load, shear and moment are linked: dV/dx = −w and dM/dx = V. So the shear diagram is the integral of the load, and the moment diagram is the running area under the shear diagram.

• The slope of the shear diagram at any point is the load intensity there.
• The slope of the moment diagram is the shear force there.
• Therefore the maximum moment occurs where the shear crosses zero — which finds the critical section without solving anything.

The step in the moment diagram caused by an applied point moment is the one most often missed. Knowing the expected shape before you compute anything is the cheapest available error check.

Where the bending moment changes sign, the curvature reverses — the beam stops sagging and starts hogging. Sagging puts the bottom fibres in tension; hogging puts the top ones there.

Concrete is weak in tension and steel is placed where the tension is, so this point decides where reinforcement moves from the bottom of a continuous beam to the top over its supports. Getting it wrong puts the steel in the compression zone, where it does very little.

• M_max goes into σ = My/I and sizes the section.
• V_max gives the shear stress, which usually governs only for short, deep beams.
• M(x) is integrated twice to get the deflection, so the moment diagram is that calculation's raw material.

The whole treatment assumes small deflections and a beam long relative to its depth. A deep beam distributes load quite differently, and beam theory should not be applied to it.