Beam Deflection Calculator
Deflection, bending moment and bending stress for a beam, computed from the elastic modulus already stored for each material and from the dressed size of a real board. Four support and load cases, and the arithmetic printed underneath the answer.
9.22 mm (0.363 in)
span / 397 · maximum at 1829 mm from the support · moment 3.34 kN·m · bending stress 9.54 MPa
Where that number comes from
recompute it before you build on it| Second moment of area I | b·h³/12 = 38.1 × 234.9³ / 12 = 41,178,458 mm⁴ |
|---|---|
| Section modulus S = I/c | 41,178,458 ÷ 117.5 = 350.5 cm³ |
| Modulus of elasticity E | 12,273 MPa — from the material record |
| Span L | 3658 mm |
| Load | 2.00 kN/m |
| Deflection | δ = 5wL⁴ / 384EI = 5 × 2.00 × 3657.6⁴ / (384 × 12,273 × 41,178,458) |
| Bending stress | σ = M / S = 3.34 kN·m ÷ 350.5 cm³ = 9.54 MPa |
The formula shown is the closed form for the selected support and load case; the coefficients are 5/384, 1/48, 1/8 and 1/3 for the four cases. Downward deflection is positive.
The same beam against four deflection limits
which limit applies is set by the code, not by this page| Limit | Allowed deflection | This beam |
|---|---|---|
| span / 180 | 20.3 mm (0.800 in) | within — 0.454× the limit |
| span / 240 | 15.2 mm (0.600 in) | within — 0.605× the limit |
| span / 360 | 10.2 mm (0.400 in) | within — 0.908× the limit |
| span / 480 | 7.6 mm (0.300 in) | over — 1.21× the limit |
span/360 is the figure most often quoted for floors carrying plaster or brittle finishes, and span/240 for roofs; the number that actually applies comes from the governing code and from what the beam carries. Passing one of these lines is not an approval, and failing one is not a failure of the member.
What this page is not. These are linear-elastic, short-term, dry-condition figures. Timber creeps under sustained load, and its stiffness is adjusted for moisture content, load duration, size and lateral stability by the applicable design code; steel has its own factors and the calculation ignores lateral-torsional buckling entirely. Neither the deflection nor the stress here is a design check.
Why the numbers move the way they do
what to change, and what it costs you| Change | Deflection | Effect on the default beam |
|---|---|---|
| Span doubled | × 16 | 148 mm |
| Depth doubled | ÷ 8 | 1.15 mm |
| Width doubled | ÷ 2 | 4.61 mm |
| Load doubled | × 2 | 18.44 mm |
| Turned flat | × 38.0 | 350.7 mm |
This is the whole reason a deeper board beats a wider one: depth enters cubed, width only linearly. It is also why a 2 × 10 lying flat behaves worse than a 2 × 4 on edge.
Frequently asked questionsdeflection, stiffness and the limits of this page
How much does a 2 × 10 timber beam deflect over 12 ft?
A oak 2 × 10 spanning 3.66 m with a uniform load of 2 kN/m deflects about 9.22 mm (0.363 in) at midspan, which is span/397. Change the species and it barely moves — E runs from about 3 GPa for balsa to 22 GPa for hickory, a factor of seven. Change the depth and it moves by the cube.
Why does deflection depend on the fourth power of span?
Because deflection is the double integral of curvature, and curvature is M/EI. The bending moment itself grows with the square of the span (wL²/8 for a uniform load), and integrating twice more brings in another L². Double the span and a beam deflects sixteen times as much under the same load per metre — which is why long spans are dominated by stiffness, not strength.
Is this a structural design tool?
No, and it should not be used as one. The figures are linear-elastic, short-term and at one moisture condition. A timber member in service also gets adjusted for load duration, moisture content, size and lateral stability, and those adjustments live in the design code, not here. Steel brings lateral-torsional buckling, which this page does not consider at all. Use it to understand a beam, not to approve one.
What deflection limit should a floor beam meet?
That is set by the governing code and by what the floor carries. The figure repeated most often in timber guidance is span/360 for members supporting plaster or brittle finishes and span/240 for roof members, and those are the two lines shown in the table. Passing one is not an approval; failing one is not necessarily a failure of the member — it is a conversation with whoever is responsible for the design.
Why do I enter 38 mm when the board is called 2 × 10?
Because the nominal size is the rough size before planing, and stiffness follows the dressed size. A 2 × 10 board in the database measures 38.1 × 234.95 mm. Using the nominal 50 × 250 mm would overstate the second moment of area by about 68%, which is not a rounding difference.
Does a 2 × 10 work flat instead of on edge?
Badly. Turned flat the same board deflects 38.0 times as much, because the second moment of area scales with the cube of the depth and the depth drops from 234.95 mm to 38.1 mm. This is the single largest mistake made with sawn timber in amateur construction.
What happens if the same section is steel instead of timber?
Deflection falls from 9.22 mm to 0.57 mm — the ratio of the two elastic moduli, about 16.3×. In practice a steel beam of that section would never be used: the calculation also leaves out lateral-torsional buckling, which is what actually governs a slender steel member.
Where do the elastic moduli come from?
Timber values are the USDA Forest Products Laboratory clear-wood figures (Table 5-3b) already carried by each species page — small clear specimens, not design values. The metals use the modulus stated on their own material record, from the ASTM or EN condition listed there. Nothing on this page invents an E: the dropdown reads the material records directly.