How to Read Shear Force and Bending Moment Diagrams

Why these diagrams are needed

When you apply a load to a beam, the effect of that load differs at every point along the beam — some cross-sections are stressed more in "shear", others more in "bending". The shear force diagram (SFD) and the bending moment diagram (BMD) visualize how these internal forces change along the beam — the designer reads from these diagrams where the beam is at the greatest risk of failure (usually at the point of maximum moment).

On the SFD, a vertical "jump" occurs at the location of a point load — the shear force changes abruptly by the magnitude of that load.

On the BMD, the moment is the integral of the area under the SFD curve — so at the point where the shear force is zero, the moment usually reaches its maximum or minimum value (this is the practical way to find where the maximum stress will occur).

Under a distributed load, the SFD draws an inclined line/curve rather than a flat line, and the BMD becomes a parabolic curve.

On a simply supported beam, the moment is zero at the supports (since it rotates freely, it cannot carry a moment); on a cantilever (fixed) beam, the maximum moment occurs at the fixed support.

First, the static equilibrium equations are solved:

ΣF = 0 (the sum of vertical forces is zero). ΣM = 0 (the sum of moments about a point is zero). These two equations are sufficient to find the two unknown reaction forces on a simply supported beam.

Example: on a 4-metre-long beam, simply supported at both ends, with a 1000 N point load applied exactly at the midpoint, both supports carry 500 N each by symmetry, the maximum moment occurs at the center of the beam, and its value is M = (500 × 2) = 1000 N·m.

Source: Standard Euler-Bernoulli beam theory; see R.C. Hibbeler, Mechanics of Materials (a standard topic in engineering statics / strength of materials textbooks).