Maximum deflection, bending moment, shear and bending stress for simply supported, cantilever and fixed-end beams under point or uniformly distributed load. Includes ten rolled sections plus rectangular, circular, I-beam and hollow custom shapes.
All cases use Euler–Bernoulli theory, in which deflection is governed by the flexural rigidity EI and bending stress by the section modulus Z. Span enters deflection to the third or fourth power, which is why long beams are almost always governed by stiffness rather than strength.
δ maximum deflection ·
w distributed load (N/m) ·
P point load (N) ·
L span (m) ·
E Young's modulus (GPa) ·
I moment of inertia (mm⁴) ·
Z section modulus (mm³) ·
Fy yield strength (MPa)
An IPE300 steel beam spans 5 m, simply supported, carrying a uniformly distributed load of 20 kN/m.
Both checks pass with reasonable margin. Deflection uses 70 % of its allowance while stress uses only 45 % of yield — the usual pattern, and a reminder that stiffness normally governs before strength on spans of this order.
| Quantity | Symbol | Unit | Accepted range |
|---|---|---|---|
| Span | L | m, mm, ft or in | > 0 |
| Distributed load | w | kN/m or N/m | > 0 |
| Point load | P | kN, N, kip or lbf | > 0 |
| Young's modulus | E | GPa | > 0 |
| Yield strength | Fy | MPa | > 0 |
| Maximum deflection | δ | mm | output |
| Bending moment | M | kN·m | output |
| Bending stress | σ | MPa | output |
| Moment of inertia | I | mm⁴ | output |
Imperial inputs are converted before calculation: 1 ft = 304.8 mm, 1 in = 25.4 mm, 1 kip = 4 448.22 N and 1 lbf = 4.44822 N.
| Section | I (mm⁴) | Z (mm³) | A (mm²) |
|---|---|---|---|
| IPE160 | 8.693 × 10⁶ | 123 000 | 2 009 |
| IPE200 | 1.943 × 10⁷ | 194 200 | 2 848 |
| IPE240 | 3.892 × 10⁷ | 324 300 | 3 912 |
| IPE300 | 8.356 × 10⁷ | 557 400 | 5 381 |
| HEA100 | 3.490 × 10⁶ | 72 760 | 2 124 |
| HEA140 | 1.033 × 10⁷ | 173 500 | 3 142 |
| HEA180 | 2.790 × 10⁷ | 324 000 | 4 525 |
| HEA200 | 3.692 × 10⁷ | 388 800 | 5 383 |
| UB 203×133×30 | 2.850 × 10⁷ | 279 000 | 3 820 |
| UB 305×165×54 | 1.170 × 10⁸ | 765 000 | 6 860 |
| Material | E (GPa) | Fy (MPa) | Density (kg/m³) |
|---|---|---|---|
| Structural steel | 200 | 250 | 7 850 |
| Aluminium | 69 | 276 | 2 700 |
| Timber | 12 | 30 | 500 |
| Concrete | 30 | 25 | 2 400 |
Deflection depends on the load, the span raised to the third or fourth power, and the flexural rigidity EI. For a simply supported beam under uniform load the maximum deflection is 5wL⁴/384EI. For a central point load it is PL³/48EI. Because span appears to the fourth power for distributed load, doubling the span increases deflection sixteen-fold at the same load intensity.
This calculator checks against L/360, the common limit for floors supporting brittle finishes such as plaster or tile. Other limits are used in practice: L/250 for general floor beams, L/200 for roof members, and L/180 for purlins and secondary members. Cantilevers are usually checked on twice the projection. Confirm the limit required by your governing code.
Moment of inertia I governs stiffness and therefore deflection, with units of mm⁴. Section modulus Z governs strength and therefore bending stress, with units of mm³. Z equals I divided by the distance from the neutral axis to the extreme fibre. A beam can be strong but flexible, or stiff but weak, which is why both checks are needed.
Strength scales with section modulus while stiffness scales with moment of inertia, and deflection grows much faster with span than stress does. On long spans deflection almost always governs before yield. Increasing the depth of the section is far more effective than increasing width, because moment of inertia grows with the cube of depth while area grows only linearly.
Simply supported assumes both ends are free to rotate, the safe default for beams on bearings or simple connections. Fixed-fixed assumes both ends fully restrained against rotation, which reduces mid-span moment and deflection substantially but requires genuinely rigid connections. Cantilever assumes one end fully fixed and the other free. Real connections usually fall between simply supported and fixed, so simply supported is the conservative choice unless the restraint is verified.