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.
Understanding Engineering: Shear and Moment Diagrams
A shear or moment diagram is built by making an imaginary cut through the beam. Keep one side of the cut and treat it as a separate object. The forces and turning effects exposed at the cut are the internal shear force and internal bending moment.
Their values must balance all loads on the chosen piece. This method works at any position along the beam, so engineers can map the internal action from one end to the other. A consistent sign convention is vital.
Choose the convention used by the course or textbook, then keep it for every cut. A correct calculation with mixed signs can produce a misleading diagram.
The shapes of the diagrams reveal the loading pattern before any detailed calculation. In a region with no load, shear stays constant and moment forms a straight line. Under a uniform distributed load, shear changes at a steady rate and moment becomes curved.
Under a load that grows along the beam, the curves change more quickly. These shape checks are useful because they catch arithmetic errors. A point force changes shear suddenly, but it does not create a sudden change in moment.
An applied turning couple does the opposite. At simple supports, the bending moment is normally zero because the support allows rotation. At a fixed support, a nonzero end moment can develop because rotation is restrained.
Bending moment matters because it produces tension on one face of a bent beam and compression on the other face. For a beam that sags in the middle, the bottom region is usually in tension and the top region is in compression. Many materials are weaker in tension than compression, especially concrete.
This is why reinforced concrete beams contain steel bars near the tensile face. Shear stress is often greatest near the beam's central depth, while bending stress is greatest at the outer top and bottom surfaces.
A beam can therefore need attention near a support because of high shear, even when its largest bending moment occurs farther away. Holes, notches, and sudden changes in beam depth need care because they can concentrate stress.
Students often meet these ideas in shelf brackets, floor joists, bridge spans, crane arms, and diving boards. A load placed near the middle of a simple span usually creates a large bending demand there. A load close to a support can create a large shear demand near that support.
Start each problem by finding every support reaction from whole beam equilibrium. Then move from left to right, marking each load, support, and change in loading region. Calculate values just before and just after concentrated forces or couples.
Check that the final diagram satisfies the known end condition. Use units throughout. Shear is measured as force, while bending moment is force times distance.
A diagram is not a picture of the beam's shape. It is a record of the internal actions that the beam must resist.
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.