Beam deflection formula: the standard cases, worked, with real numbers.
A beam can pass its stress check and still fail in service because it sags too much. Deflection is a stiffness problem, not a strength problem, and it is governed by a different set of formulas. Here are the standard cases, what the EI term really does, and a worked example you can copy onto your own section.
When a guard rail bounces, a conveyor frame droops, or a machine base flexes under a spindle load, the issue is almost never that the steel yielded. It is that the section was not stiff enough. Deflection scales with the cube of span and inversely with stiffness, so small changes in geometry produce large changes in sag. Understanding the deflection formula is what stops you from over-designing on strength while quietly failing on serviceability.
Strength vs stiffness
Two checks decide whether a beam works. The strength check asks whether bending stress stays below the allowable — that protects against fracture and yield. The stiffness check asks whether deflection stays below a serviceability limit — that protects against sag, vibration, cracked finishes and the simple perception that something is too bendy. A beam can be enormously strong yet unacceptably flexible. Deflection is where the second check lives.
The EI term, explained
Every deflection formula contains the product EI, called flexural rigidity:
- E is the modulus of elasticity — a material property. Steel is about 200 GPa, aluminium about 69 GPa. So an identical aluminium section deflects roughly three times as much as steel.
- I is the second moment of area — a geometry property. For a rectangle, I = b·h³/12, so depth dominates: doubling the height multiplies stiffness by eight.
Because deflection is inversely proportional to EI, the cheapest way to halve sag is almost always to add depth, not material area.
Standard deflection formulas
These are the maximum-deflection formulas for the cases you meet most often. P is a point load, w is a uniformly distributed load per unit length, L is the span, and EI is flexural rigidity.
| Case | Loading | Max deflection |
|---|---|---|
| Simply supported | Central point load P | P·L³ / (48·EI) |
| Simply supported | UDL w | 5·w·L⁴ / (384·EI) |
| Cantilever | Point load P at free end | P·L³ / (3·EI) |
| Cantilever | UDL w | w·L⁴ / (8·EI) |
| Fixed-fixed | Central point load P | P·L³ / (192·EI) |
Notice the denominators. A cantilever with an end load (divide by 3) sags sixteen times more than a simply supported beam of the same span and load (divide by 48). Fixing both ends (divide by 192) cuts deflection to a quarter of the simply supported value. The support condition matters as much as the section.
A worked example
Take a simply supported steel beam, span L = 3 m, carrying a central point load P = 10 kN. Section is a 150 × 75 mm rectangular bar, so I = b·h³/12 = 75 × 150³ / 12 = 21.1 × 10⁶ mm⁴. Steel E = 200,000 N/mm².
- EI = 200,000 × 21.1 × 10⁶ = 4.22 × 10¹² N·mm².
- P = 10,000 N, L = 3000 mm, so P·L³ = 10,000 × 2.7 × 10¹⁰ = 2.7 × 10¹⁴.
- Deflection = 2.7 × 10¹⁴ / (48 × 4.22 × 10¹²) = 0.133 mm.
Against a span/360 limit of 3000/360 = 8.3 mm, this beam is comfortably stiff. Now rotate the same bar so the 75 mm dimension is vertical: I collapses to 9 × 150 × 75³... in fact I = 150 × 75³/12 = 5.3 × 10⁶ mm⁴, one quarter of before, and deflection quadruples to 0.53 mm. Same bar, same load, orientation alone changed the result — a reminder to always orient sections for depth in the load plane.
Allowable deflection limits
Limits are expressed as a fraction of span. The right number depends on what the deflection would damage or annoy.
| Application | Typical limit |
|---|---|
| Floor beam, live load | Span / 360 |
| Floor beam, total load | Span / 240 |
| Roof beam, no plaster | Span / 180 |
| Crane girders | Span / 600 to / 1000 |
| Machine tool structures | Span / 1000 or tighter |
These are serviceability values, not safety values, but they are contractual on most projects. A floor that meets strength but violates span/360 will feel bouncy and may crack brittle finishes — and the client will reject it.
How to make a beam stiffer
- Add depth. I rises with the cube of height, so this is the most efficient lever by far.
- Shorten the span. Deflection falls with L cubed (point load) or L to the fourth (UDL); an extra support is dramatic.
- Improve end fixity. Moving from simply supported to fixed-fixed cuts central deflection to one quarter.
- Change material only as a last resort. Going from aluminium to steel triples E but adds weight and cost; geometry is usually cheaper.
Common mistakes
- Mixing units. Keep N and mm throughout, or convert cleanly; an E in GPa with an I in mm⁴ gives nonsense.
- Wrong axis for I. Use the second moment of area about the bending axis, not the larger value.
- Ignoring support conditions. Assuming simply supported when the real connection is partially fixed under-predicts or over-predicts sag.
- Checking strength only. The single most common omission — a beam that passes stress can still fail serviceability.
Frequently asked questions
Maximum deflection at mid-span is P·L³ divided by 48·EI, where P is the point load, L the span, E the modulus and I the second moment of area.
For a point load at the free end, deflection is P·L³ divided by 3·EI — sixteen times more than the same load on a simply supported span, because the support offers no help at the tip.
Span/360 for floor live load and span/240 for total load are common; machine and crane structures often require span/1000 or tighter. Check the governing code for your application.
Because the second moment of area I = b·h³/12 grows with the cube of height. Doubling depth multiplies stiffness by eight; doubling width only doubles it.
When the section is finalised and the drawing goes out for inspection, ballooning every dimension by hand is the slow step. CadNexa's auto-ballooning tool (Smart Detect plus Box+Balloon OCR) reads the drawing and numbers every characteristic, so the inspection sheet for your fabricated beam is ready in minutes.