What this calculator does
A bending moment is the internal turning force a beam resists at any section along its length, caused by loads trying to bend it. This bending moment calculator finds the maximum bending moment for the four standard textbook cases: a simply-supported beam with either a point load at midspan or a uniform load along its length, and a cantilever beam with either a point load at the free end or a uniform load along its length.
Bending moment is distinct from deflection. Deflection describes how far the beam physically bends under load; bending moment describes the internal stress-causing force at a cross-section, independent of how stiff the material is. The two use different formulas and answer different questions, though both matter when sizing a beam.
The formula
For a simply-supported beam with a point load P at midspan, the maximum moment is M = PL/4. For the same beam with a uniform load W spread along its length, M = WL/8. For a cantilever with a point load P at the free end, M = PL. For a cantilever with a uniform load W along its length, M = WL/2. In every case, L is the full span or beam length and the result is the maximum moment, which occurs at midspan for the simply-supported cases and at the fixed support for the cantilever cases.
| Term | Meaning |
|---|---|
| M | Maximum bending moment, in newton-metres (N·m). |
| P | A point load applied at a single location, in newtons. |
| W | A uniform load, entered as its total value over the whole span, in newtons. |
| L | The span (simply-supported) or length (cantilever) of the beam, in metres. |
The inputs explained
| Field | What to enter |
|---|---|
| Beam and load type | Pick the support condition and how the load is applied. This decides which of the four formulas is used. |
| Load (point load P, or total uniform load W) (N) | For a point-load case, enter the single load P. For a uniform-load case, enter the total load W spread across the whole span, not the load per metre. |
| Span or beam length (L) (m) | The full span between supports, or the full length of the cantilever, in metres. |
When to use it
Sizing a floor or deck joist
A joist spanning between two supports with people or furniture on it behaves as a simply-supported beam under a roughly uniform load, so the WL/8 case gives the design moment to check against the timber or steel section capacity.
Checking a bracket or balcony
A bracket fixed at one end and loaded at the other, such as a shelf bracket or balcony beam, is a cantilever. The PL or WL/2 cases apply, and cantilevers generally see a larger moment for the same load and length than a simply-supported beam.
Comparing load cases before ordering material
Running the same length and load through both the simply-supported and cantilever cases shows how much extra moment a cantilever arrangement has to resist, which often changes the section size needed.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does bending moment change with span, for a fixed point load?
A fixed 1,000 N load at midspan, across a range of spans.
| Span | Maximum bending moment |
|---|---|
| 1 m | 250.00 N·m |
| 2 m | 500.00 N·m |
| 3 m | 750.00 N·m |
| 4 m | 1,000.00 N·m |
| 5 m | 1,250.00 N·m |
| 6 m | 1,500.00 N·m |
How does the load case change the moment, for the same load and length?
The same 1,000 N load and 3 m length, run through each of the four support and load combinations.
| Case | Maximum bending moment |
|---|---|
| Simply supported, point load | 750.00 N·m |
| Simply supported, uniform load | 375.00 N·m |
| Cantilever, point load | 3,000.00 N·m |
| Cantilever, uniform load | 1,500.00 N·m |
Questions
What is the difference between bending moment and bending stress?
Bending moment is the internal turning force at a section, in newton-metres. Bending stress converts that moment into a stress value, in pascals, using the beam cross-section shape via its second moment of area. You need the moment first before you can find the stress.
Why is the cantilever moment so much larger than the simply-supported case?
A simply-supported beam has two supports sharing the load, and the moment builds up and then comes back down across the span, peaking at PL/4 or WL/8 in the middle. A cantilever has only one fixed support carrying the whole turning effect of the load over the full length, so its formulas do not have that /4 or /8 sharing factor.
Should I enter the total uniform load or the load per metre?
Enter the total load W over the whole span. If you only know the load per metre, multiply it by the span length first to get the total before entering it here.
Does this handle beams with more than one point load or a mix of loads?
No, this covers the four single-load standard cases only. A beam with multiple point loads, or a combination of point and uniform loads, needs the individual moment contributions calculated separately and added together at the section of interest.
For how far a beam actually deflects under load rather than the internal moment it resists, see the beam deflection calculator. To convert a known bending moment into a stress value, see the bending stress calculator.