Shear and moment diagrams show how internal forces change along a beam, which is essential for designing safe structures. Engineers use them to find where a beam is most likely to crack, bend too much, or fail. A free body diagram gives the external loads and support reactions, while the shear force diagram and bending moment diagram translate those loads into internal effects.
These diagrams connect the real structure to the equations used for stress, deflection, and material selection.
The key idea is that distributed load, shear, and moment are linked by slope relationships. The rate of change of shear equals the negative of the load intensity, and the rate of change of moment equals the shear force. Point loads create jumps in the shear diagram, while applied couples create jumps in the moment diagram.
Maximum bending moment usually occurs where the shear crosses zero or changes sign.
Key Facts
- For vertical equilibrium of a beam: ΣFy = 0.
- For moment equilibrium of a beam: ΣM = 0.
- Load, shear, and moment are related by dV/dx = -w(x).
- Shear and moment are related by dM/dx = V(x).
- The change in shear over an interval equals the negative area under the load diagram: ΔV = -∫w(x) dx.
- The change in moment over an interval equals the area under the shear diagram: ΔM = ∫V(x) dx.
Vocabulary
- Shear force
- The internal transverse force in a beam that resists sliding of one cross section past another.
- Bending moment
- The internal moment in a beam that resists bending caused by external loads.
- Simply supported beam
- A beam supported by a pin at one end and a roller at the other, allowing rotation but preventing vertical translation.
- Distributed load
- A load spread over a length of a beam, measured in force per unit length such as N/m or lb/ft.
- Sign convention
- A consistent rule for assigning positive and negative directions to loads, shear forces, and bending moments.
Common Mistakes to Avoid
- Skipping support reactions, which is wrong because shear and moment diagrams must start from a correct free body diagram in equilibrium.
- Treating a point load as a sloped line on the shear diagram, which is wrong because a point load causes an immediate vertical jump in shear.
- Placing the maximum moment at the largest load automatically, which is wrong because maximum moment occurs where shear is zero or where the moment reaches an endpoint extreme.
- Mixing sign conventions within one problem, which is wrong because inconsistent signs make the shear and moment areas give incorrect changes.
Practice Questions
- 1 A simply supported beam is 6 m long with a 12 kN point load at midspan. Find the support reactions, the maximum shear magnitude, and the maximum bending moment.
- 2 A simply supported beam is 8 m long and carries a uniform distributed load of 3 kN/m over the entire span. Find the support reactions and the maximum bending moment.
- 3 A beam has a shear force diagram that starts positive, decreases linearly, crosses zero, and then becomes negative before the right support. Explain where the bending moment is maximum and why.