Beam Deflection Calculator
Four classic load cases, one square-root-free arithmetic: how far the span droops, in millimetres, with the L/360 floor guide checked beside it.
Beam Deflection Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Four idealised cases — no partial spans or multiple loads
- Elastic arithmetic; no creep or cracking
In short: A 6 m simply supported span carrying 10 N/mm (ten kilonewtons per metre) on E = 200 GPa steel with I = 100×10⁶ mm⁴ droops 8.437500 mm at midspan — 5wL⁴ over 384EI. The L/360 floor guide allows 16.666667 mm on this span, so the beam passes with room to spare. The span is the tyrant in the formula: double it and the deflection multiplies by sixteen, which is why deflection — not strength — usually sizes long floors.
Formula
δ = 5wL⁴/384EI (SS+UDL) · PL³/48EI (SS+P) · PL³/3EI (cant+P) · wL⁴/8EI (cant+UDL)
Each case is a closed-form solution of the elastic curve — integrate the bending diagram twice and the constants fall out. The pattern to notice is the span exponent: point loads deflect with L³, distributed loads with L⁴. Stiffness enters downstairs as the product EI, so a section twice as deep (I up eightfold for the same area) buys eight times the rigidity — depth is the cheapest material in structural engineering.
Worked Example
- Pick the load case that matches the real support and load.
- Enter the load, span, E and I — SI throughout.
- Read the maximum deflection in millimetres.
- Check the L/360 serviceability guide for floors.
Defaults: case 1, 10 N/mm over 6,000 mm → 8.437500 mm, inside the 16.666667 mm L/360 guide. The ajax check: case 3, 1,000 N on 2,000 mm with I = 8×10⁶ → 1.666667 mm. Aluminium's 70 GPa deflects nearly three times steel's arithmetic on the same section.
Strengths & Limits Of This Model
Where this engine is strong
- All four classics on one page
- L/360 check printed, not implied
Where it stops
- No continuous-beam cases
- No camber or precamber output
Practical Use Cases
Floor design
the L/360 serviceability check
Shaft and rail spec
sag under self-weight
Teaching
why span governs, not load
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Beam Deflection Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why does deflection grow with the fourth power of span?
Because the load's leverage grows with span, the curvature accumulates over a longer run, and the slope from one end feeds the swing at the other — integrating the bending diagram twice adds two more factors of L for distributed loads. The practical translation: doubling a floor's span multiplies its droop sixteen-fold at the same load, which is why deep beams and cambers appear in long rooms long before anything breaks.
My beam passes strength but fails deflection. Is that normal?
It is the normal outcome for long lightly-loaded spans — steel is so strong that serviceability sizes the member first. Bending stress scales with the section modulus (one power of depth); deflection scales with I (depth cubed for the same width), so adding depth buys stiffness far faster than strength. When a floor bounces, the fix is deeper or shorter — almost never stronger steel.
Which I do I enter for a non-rectangular section?
The second moment of area of your actual section about its strong bending axis — from a section table for rolled shapes, or the sum of bh³/12 parts for built-ups. The page accepts I directly precisely because real sections are tables, not formulas; entering the strong-axis value for a beam loaded on its weak axis overstates stiffness by the section's aspect ratio, which is a large lie.
Is E = 200,000 MPa always right for steel?
It is the convention every steel design code carries — the modulus barely varies across grades, which is the quiet scandal of high-strength steel: stronger, but exactly as floppy. Aluminium runs about 70,000 MPa and timber 10,000 or so (species-dependent, moisture-sensitive). The deflection arithmetic is only as honest as the E you feed it.
What does the L/360 limit actually protect?
Occupant comfort and brittle finishes — plaster and tile crack long before a floor is remotely unsafe. It is a serviceability guide, not a strength verdict, and codes tune it: L/480 for brittle finishes, L/240 where some sag is tolerable. The page prints the guide's allowance beside your deflection so the comparison is explicit rather than implied.
Does self-weight count in w?
It should — the beam carries itself before it carries anything else. Add the steel's own line load (7850 N per cubic metre times the section's area in square metres) to the applied UDL. Long spans are frequently self-weight-governed, which is the polite way of saying the beam's biggest load is the beam.
Why is the midspan value the maximum?
For these four idealised cases the elastic curve is monotonic toward its low point — midspan for the simply supported cases, the free end for the cantilevers. Real load patterns (multiple point loads, partial spans) put the maximum wherever the shear diagram crosses zero, which is what beam software finds by integration. The four cases here are the standards every first check is measured against.
Can I use this for a reinforced concrete beam?
Only as a first-order estimate, and the page says so plainly: concrete cracks, creeps and its effective stiffness depends on reinforcement and load duration — the honest I is a fraction of the gross section's, and codes carry empirical curves for it. Steel and aluminium behave elastically enough for the closed forms to be genuinely predictive; concrete is a different material wearing the same formula.