It applies textbook elastic formulas to inputs you supply. It does not apply load factors, material safety factors, code checks, lateral-torsional buckling, web crippling, connection design, fire rating or serviceability limits beyond simple deflection. Anything that gets built must be designed and stamped by a licensed structural engineer working to the governing code. Use this to understand relationships and sanity-check a figure — never as the basis for construction.
Deflection, bending moment and shear for the five standard beam cases, with section properties solved for you and deflection checked against the usual serviceability limits. Every formula is shown, so the result can be verified rather than trusted.
Beam Deflection Calculator
LiveThe five standard cases
| Case | Max deflection | Max moment | Where |
|---|---|---|---|
| Simply supported, uniform | 5wL⁴ / 384EI | wL² / 8 | Midspan |
| Simply supported, point at centre | PL³ / 48EI | PL / 4 | Midspan |
| Cantilever, uniform | wL⁴ / 8EI | wL² / 2 | At the support |
| Cantilever, point at end | PL³ / 3EI | PL | At the support |
| Fixed both ends, uniform | wL⁴ / 384EI | wL² / 12 | At the supports |
The contrast worth internalising: a cantilever deflects 48 times more than a simply supported beam under the same uniform load, and carries four times the moment. Fixing both ends does the reverse — deflection drops to a fifth. Support conditions dominate everything else in beam behaviour.
Deflection limits, and why they exist
A beam can be perfectly safe in strength and still unacceptable in service. Deflection limits are serviceability criteria — they protect finishes, function and the occupants' confidence, not the structure itself:
| Limit | Typically applies to | What it protects against |
|---|---|---|
| L/360 | Floors with plaster or brittle finishes | Cracking in ceilings and partitions below |
| L/240 | Floors with flexible finishes, roofs with ceilings | Visible sag, doors and windows binding |
| L/180 | Roofs without ceilings, secondary members | Ponding of water, visual appearance |
| L/480 or tighter | Sensitive equipment, long spans, vibration-critical floors | Perceptible bounce, which occupants report as a defect even when it is entirely safe |
These are common values, not universal ones. Your governing code and the project specification decide, and they vary by jurisdiction, occupancy and whether the limit applies to live load alone or to the total.
Vibration is the criterion people actually notice
Long-span timber and light steel floors routinely satisfy every strength and deflection check and still feel unacceptably bouncy. Static deflection does not capture it, because bounce is a dynamic problem — it depends on natural frequency, mass and damping, none of which appear in a deflection formula.
The historical evidence is direct. AISC Design Guide 11 notes that floors which generated complaints typically had natural frequencies between 5 and 8 Hz — and that this was largely a consequence of designing to a span/360 deflection rule and nothing else. Meeting L/360 has never guaranteed a floor that feels solid.
| Frequency | Classification | What it means in practice |
|---|---|---|
| Below 3 Hz | Avoid entirely | AISC advises against these — vulnerable to rhythmic loading such as jumping, where resonance can build dangerously |
| 3–5 Hz | Low frequency | Walking excites resonance directly. Occupants will notice |
| 5–8 Hz | The complaint band | Where historically annoying floors sat. Passes L/360 routinely and still feels wrong |
| 9–10 Hz | Cut-off | AISC treats this as the point above which resonant build-up stops being significant. A separate static stiffness check applies instead — typically 1 kN/mm under a concentrated load |
| 14–15 Hz | Light-frame target | The Woeste-Dolan method recommends 14 Hz for furnished rooms and 15 Hz empty, for timber joist floors |
Different codes draw the lines differently. SCI P354 and the UK approach accept 8.4 Hz vertically as a threshold above which resonance need not be evaluated. JGJT441-2019 imposes 3 Hz as an absolute minimum and works to a peak acceleration limit of 0.05 m/s² for offices and residences. Eurocode 5 takes a different route again for timber, combining frequency, deflection under a point load, and peak velocity from footfall.
If a floor feels bouncy, tightening the deflection limit is often the wrong fix — it addresses a static criterion for a dynamic problem. Adding depth raises frequency; adding mass lowers it but increases damping. Which helps depends on where you sit in the table above, and getting it wrong can make the floor worse. This is a case for a proper vibration assessment rather than a rule of thumb.
Sources: AISC Design Guide 11 (Vibrations of Steel-Framed Structural Systems Due to Human Activity) · SCI P354 · Woeste & Dolan light-frame method · JGJT441-2019 · Eurocode 5 Annex A. Cited for the specific figures above.
Why depth matters far more than width
For a rectangular section, I = bh³/12. Width enters linearly; depth is cubed. The consequence is dramatic:
| Change | Effect on I | Effect on deflection |
|---|---|---|
| Double the width | ×2 | Halved |
| Double the depth | ×8 | Reduced to one eighth |
| Increase depth 25% | ×1.95 | Roughly halved |
| Double the span | — | ×16 — deflection scales with L⁴ |
Two practical consequences. Adding depth is the efficient fix when a beam is too flexible — a 25% deeper section roughly halves deflection for 25% more material. And span is punishing: doubling it multiplies deflection sixteenfold at the same load, which is why long spans need disproportionately deep members.
This is also why an I-section exists at all. It puts material where the depth term rewards it — in flanges far from the neutral axis — and removes it from the web where it contributes little to bending. An I-beam achieves the stiffness of a solid rectangle at a fraction of the weight.
What this calculator does not do
| Not included | Why it matters |
|---|---|
| Load factors | Codes require factored combinations — 1.2D + 1.6L and similar. The loads here are unfactored, so the results are service-level, not design-level |
| Lateral-torsional buckling | An unrestrained steel beam can fail sideways well below its bending capacity. This frequently governs, and is not checked here |
| Shear and web checks | Short deep beams and heavily loaded webs can fail in shear or web crippling before bending governs |
| Material capacity | The stress figure is elastic and compared against nothing. Whether it is acceptable depends on grade, partial factors and the code |
| Connections | Beams fail at their connections more often than in the span. Nothing here addresses them |
| Long-term effects | Concrete creeps and timber sags under sustained load — long-term deflection can be two to three times the immediate value |
Span is the variable that punishes you
Deflection under a distributed load follows the fourth power of the span. Nothing else in the equation grows that fast:
| Change | Effect on deflection | Why |
|---|---|---|
| Double the span | 16 times worse | L⁴ — the dominant term by a wide margin |
| Double the load | 2 times worse | Linear |
| Double the width | Half | b appears once in bd³/12 |
| Double the depth | One eighth | d is cubed |
| Switch timber to steel | About one twentieth | Young’s modulus, roughly 10 GPa against 200 |
A 3-metre joist that deflects 5 mm will deflect 84 mm at 6 metres — the same timber, the same load per metre, and a floor that is now unusable. Adding an intermediate support is almost always cheaper than deepening a beam, because halving the span divides the deflection by sixteen.
Deflection limits are about damage, not strength
A beam can be nowhere near failure and still be unacceptable. The limits exist to protect what is attached to it:
| Limit | Applies to | What it protects |
|---|---|---|
| L/360 | Floors with plaster or tiled finishes | Cracking. Rigid finishes fail long before the structure does |
| L/240 | Floors with flexible finishes | Doors binding, visible sag, uneven furniture |
| L/180 | Roofs without ceilings below | Ponding — water collecting in a sag, which deepens the sag |
| L/120 | Secondary and temporary members | Appearance and serviceability only |
On a 4-metre span, L/360 is 11 mm and L/240 is 17 mm. Neither is anywhere near breaking — a timber joist typically fails at many times that. The limit is set by the plasterboard ceiling underneath, not by the timber.
Common mistakes
Frequently asked questions
Can I use this to design a beam for my house?
No. Use it to understand how span, depth and support conditions interact, and to sanity-check a figure someone has given you. Structural design requires factored load combinations, code checks, buckling and connection design, and in most jurisdictions a licensed engineer's stamp — for good reason. A beam that fails does not fail gradually.
What deflection limit should I use?
L/360 for floors with brittle finishes, L/240 for general floors and roofs with ceilings, L/180 for roofs without. These are common values rather than universal ones — your code and project specification govern, and they differ on whether the limit applies to live load alone or to the total including dead load. Check which before comparing anything.
Why does doubling the depth help so much more than doubling the width?
Because I = bh³/12 — width is linear, depth is cubed. Doubling width doubles stiffness; doubling depth multiplies it by eight. This is why adding depth is the efficient way to stiffen a beam, and why I-sections put material in flanges far from the neutral axis rather than spreading it evenly.
My beam passes deflection but the floor feels bouncy. Why?
Because bounce is a vibration problem, not a deflection one. Static deflection under load says nothing about natural frequency, and long-span timber or light steel floors routinely satisfy every strength and deflection check while feeling unacceptable. Residential floors are usually kept above roughly 8 Hz. Fixing it generally means more depth or more mass, not a tighter deflection limit.
Should I use gross or cracked properties for concrete?
Cracked, for any realistic deflection estimate. A reinforced concrete beam cracks in tension under service load, and the effective moment of inertia falls to roughly 35–50% of the gross value. Creep under sustained load can double the long-term deflection again. Using gross properties gives an answer that is optimistic by a factor of three or more.
Does this include the beam's own weight?
No — add it to the uniform load yourself. For steel, multiply the section area by 78.5 kN/m³; for concrete use 24 kN/m³ and for softwood about 5. On long spans self-weight is a substantial share of the total, and omitting it understates deflection, moment and shear together.
Why is a cantilever so much worse than a simply supported beam?
Because it has support at one end only, so the entire load must be resisted by bending at that single point. Under the same uniform load a cantilever deflects 48 times more and carries four times the moment of an equivalent simply supported beam. Cantilevers also fail without warning at the support, which is why they are detailed conservatively.
Is my input stored?
No. Everything runs in your browser with no server request, and works offline once the page has loaded.
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