Shear Force and Bending Moment Diagram

A shear force and bending moment diagram is a pair of graphs for a loaded beam. The shear force diagram shows the internal force at each section, and the bending moment diagram shows how strongly the beam is trying to bend at each section.

For search-intent purposes, this is the key idea: external loads act on the beam from the outside, but shear force and bending moment describe the beam's internal response along its length. If you can read where the shear jumps and where the moment peaks, the diagram is already doing its job.

What a shear force diagram and bending moment diagram show

For a beam carrying transverse loads, you can imagine cutting the beam at some position and asking what internal actions are needed to keep one side in equilibrium.

• The shear force at that section is the internal force across the cut.
• The bending moment at that section is the internal turning effect across the cut.

As you move the cut from left to right, those internal values usually change. The diagrams are just graphs of those values versus position along the beam.

Why these beam diagrams matter

These diagrams answer practical questions quickly:

• Where is the shear largest?
• Where is the bending moment largest?
• Where are the zero crossings?
• Which region of the beam is most critical for design?

They are especially useful for beams, bridges, frames, and other members where bending is more important than pure axial stretching or compression.

How to read common graph shapes

A few rules explain most introductory shear force and bending moment diagrams:

• A point load causes a sudden jump in the shear force diagram.
• An applied concentrated moment causes a sudden jump in the bending moment diagram.
• In a region with no distributed load, the shear force stays constant.
• In a region where the shear force is constant, the bending moment changes linearly.
• Under a constant distributed load, the shear changes linearly and the bending moment curves rather than staying straight.

With one common sign convention, positive bending moment means sagging, where the beam curves like a shallow smile. Another convention may flip the diagram vertically, so always check the convention your course, book, or software uses.

Worked example: simply supported beam with a center load

Take a simply supported beam of span $L$ with a single downward point load $P$ at the midpoint.

By symmetry, the support reactions are equal:

$$R_A = R_B = \frac{P}{2}$$

That immediately tells you the shear force diagram. Just to the right of the left support, the shear is $+P/2$. At the midpoint, the downward load makes the shear drop by $P$, so it becomes $-P/2$. At the right support, the reaction returns it to zero.

Written as a piecewise function,

$$V(x) = \begin{cases} \frac{P}{2}, & 0 < x < \frac{L}{2} \\ -\frac{P}{2}, & \frac{L}{2} < x < L \end{cases}$$

The bending moment is zero at both simple supports and changes linearly between them because the shear is constant on each half of the beam:

$$M(x) = \begin{cases} \frac{P}{2}x, & 0 \le x \le \frac{L}{2} \\ \frac{P}{2}(L - x), & \frac{L}{2} \le x \le L \end{cases}$$

So the bending moment diagram is triangular, with its largest value at the center:

$$M_{max} = \frac{PL}{4}$$

under the usual sagging-positive convention.

What this example teaches

This one beam shows the main pattern students need first:

• Reactions and applied point loads create jumps in shear.
• Bending moment is smooth across an ordinary point load.
• The largest bending moment often occurs where the shear changes sign.

That last point needs a condition: it is reliable in the usual beam cases where the bending moment diagram stays continuous through the region and there is no applied concentrated moment at that location.

Common mistakes

Mixing up external loads and internal diagrams

The load diagram is not the shear diagram. A downward load does not mean the shear graph also slopes downward in the same visual way. The shear and moment plots are responses, not copies.

Forgetting support reactions

If the support reactions are missing or wrong, every later value in the shear and moment diagrams will also be wrong.

Making bending moment jump at a point load

A point load changes shear suddenly. A concentrated applied moment changes bending moment suddenly. Those are different effects.

Ignoring the sign convention

Two correct solutions can look vertically flipped if they use different sign conventions. Compare magnitudes, jump sizes, and zero locations only after confirming the sign rule.

Where shear force and bending moment diagrams are used

Shear force and bending moment diagrams appear in beam design, structural analysis, and mechanics courses. They are used to estimate critical sections, connect loading to internal stresses, and check whether a support and loading arrangement makes physical sense.

Even if you never design a structure, the diagrams are a clean way to see how local internal forces emerge from global equilibrium.

Try a similar case

Keep the same simply supported beam, but move the point load away from the center. Find the two support reactions first, draw the shear force diagram from left to right, and then sketch the bending moment diagram from the shear. If your work is consistent, the moment is still zero at both simple supports, but the peak shifts away from midspan.