Beam Deflection Calculator
Calculate the maximum deflection, bending moment, shear force and reactions of a beam. Choose simply supported, cantilever or fixed-fixed supports with a uniform or point load, enter the span, E and I in metric or US units, and see the deflected shape.
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Empty = mid-span (simply supported) or free end (cantilever)
Empty = mid-span (simply supported) or free end (cantilever)
Steel 200, aluminum 69, softwood ≈ 11, concrete ≈ 25–30
Steel 29,000, aluminum 10,000, lumber ≈ 1,200–1,900
2×10 lumber (1.5 × 9.25 in) is 98.9 in⁴
Adds bending stress
Adds bending stress
How to use the Beam Deflection Calculator
- Choose the units, the support type and the load type.
- Enter the span and the load (w for a uniform load, P and its position for a point load).
- Enter E and I for the beam material and section (and S for stress if you know it).
- Read the deflection, moment and shear with the formula used.
- Check the L/360 limit and look at the deflected shape.
Beam deflection formulas
| Case | Max deflection | Max moment | Max shear |
|---|---|---|---|
| Simply supported, uniform load w | 5wL⁴ ÷ 384EI | wL² ÷ 8 | wL ÷ 2 |
| Simply supported, point load P at centre | PL³ ÷ 48EI | PL ÷ 4 | P ÷ 2 |
| Simply supported, point load at a (b = L − a, b ≤ a) | Pb(L² − b²)^1.5 ÷ (9√3·EI·L) | Pab ÷ L | max(Pb/L, Pa/L) |
| Cantilever, point load P at free end | PL³ ÷ 3EI | PL | P |
| Cantilever, uniform load w | wL⁴ ÷ 8EI | wL² ÷ 2 | wL |
| Fixed both ends, uniform load w | wL⁴ ÷ 384EI | wL² ÷ 12 | wL ÷ 2 |
| Fixed both ends, point load P at centre | PL³ ÷ 192EI | PL ÷ 8 | P ÷ 2 |
Use consistent units: the calculator converts metric inputs to newtons, metres and pascals (I in cm⁴ × 10⁻⁸ = m⁴) and US inputs to pounds, inches and psi before applying the formulas.
Typical modulus of elasticity
| Material | E (GPa) | E (ksi) |
|---|---|---|
| Structural steel | 200 | 29,000 |
| Stainless steel | 193 | 28,000 |
| Aluminum alloy | 69 | 10,000 |
| Titanium alloy | 114 | 16,500 |
| Concrete (normal weight) | 25–35 | 3,600–5,000 |
| Softwood lumber (along grain) | 8–14 | 1,200–2,000 |
| Glulam | 12–14 | 1,800–2,000 |
Frequently asked questions
What is the deflection formula for a simply supported beam?
With a uniform load w, the maximum mid-span deflection is δ = 5wL⁴ ÷ (384EI). With a single point load P at mid-span it is δ = PL³ ÷ (48EI). L is the span, E the modulus of elasticity and I the second moment of area.
What is the deflection of a cantilever beam?
A cantilever with a point load P at the free end deflects δ = PL³ ÷ (3EI) at the tip; with a uniform load w it deflects δ = wL⁴ ÷ (8EI). The maximum moment occurs at the fixed end: PL or wL² ÷ 2.
What is L/360?
It is a serviceability limit: the deflection should not exceed the span divided by 360. Building codes often use L/360 for floors under live load, L/240 for total load and L/180 for some roofs. A 12 ft (144 in) span at L/360 allows 0.4 in.
Where do I find E and I?
E depends on the material (steel about 200 GPa or 29,000 ksi, aluminum about 69 GPa or 10,000 ksi, structural softwood roughly 8–14 GPa). I depends on the cross-section; for a rectangle I = b·h³ ÷ 12, and steel section tables list I for each profile.
Does this include the beam's own weight?
No. Add the self-weight per unit length to the distributed load w if you want it included. For a point-load case you can run the uniform-load case separately and add the deflections, since linear elastic results superpose.
Is this a structural design tool?
It is an educational and preliminary calculator using standard elastic beam formulas. Real design must also check strength, lateral-torsional buckling, bearing, connections, load combinations and the applicable building code, ideally by a qualified engineer.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.