Beam Deflection: Euler–Bernoulli Formulas in Practice

What beam deflection actually is

Push down on the middle of a yardstick supported at both ends and watch it bend. The downward distance the centre of the stick travels before it stops moving is the deflection. For a real structural beam — a steel joist over a basement, a timber rafter over a porch, an aluminium balcony cantilever — the same thing happens but on a scale of millimetres rather than centimetres, and the engineer's job is to keep that movement small enough that no one upstairs feels the floor bounce, no plaster ceiling cracks, and no glass partition racks out of its frame. The beam deflection calculator evaluates the textbook closed-form solution to this problem for the four most common support and load combinations, returning the peak deflection in millimetres and the peak bending moment in kilonewton-metres for any span, load, modulus and section you give it.

Deflection matters separately from strength. A beam can be nowhere near breaking and still be useless: a floor that deflects 25 mm under a couple of people is structurally safe but feels alive underfoot, and the same floor will eventually crack any rigid finish bonded to it. Modern design codes — Eurocode, the International Building Code, AISC 360, BS 5268 for timber — set explicit serviceability limits on deflection that are usually the governing criterion for floor and roof beams in normal buildings. Sizing for strength alone produces sections that are too flexible; sizing for stiffness produces sections that are also comfortably strong.

The Euler–Bernoulli formulas

For a slender, prismatic, linearly elastic beam in small deflection, the curvature at any point along the span equals the bending moment divided by the flexural rigidity EI, where E is Young's modulus and I is the second moment of area of the cross-section. Integrate that relation twice along the span with the appropriate boundary conditions and you get a closed-form expression for the deflection at any point — including the maximum. Doing the integration once per load case gives the four formulas the beam deflection calculator uses:

Simply supported, midspan point load P:        δ_max = P · L³ / (48 · E · I)
Simply supported, uniform load w per metre:     δ_max = 5 · w · L⁴ / (384 · E · I)
Cantilever, point load P at the free end:       δ_max = P · L³ / (3 · E · I)
Cantilever, uniform load w per metre:           δ_max = w · L⁴ / (8 · E · I)

The peak bending moments that go with them are PL/4, wL²/8, PL and wL²/2 respectively — at midspan for the simply supported cases and at the fixed end (the root) for the cantilever cases. These eight identities live in Table 3-23 of the AISC Steel Construction Manual and §12.2 of Hibbeler's Mechanics of Materials, and have been the working bread and butter of structural design for more than a century.

The dependence of deflection on span is what makes long beams hard. For a simply supported beam under uniform load, halving the span cuts deflection by a factor of 16 (L⁴), while doubling the moment of inertia only cuts it by 2. Span dominates everything. Designers instinctively reach for shallower sections to keep ceiling heights generous, but doubling a beam's depth raises I by roughly 8× — bending stiffness per kilogram of steel goes up sharply with depth, which is why I-sections, hollow rectangular sections and trussed beams all push the material as far from the neutral axis as the rest of the design allows.

Worked example: a 5 m steel beam under a point load

Take the calculator's default scenario: a simply supported steel I-beam, span L = 5 m, midspan point load P = 10 kN, Young's modulus E = 200 GPa (structural steel), second moment of area I = 1,000 cm⁴. Step through the simply supported point case in SI base units:

P = 10,000 N
L = 5 m       L³ = 125 m³
E = 200 × 10⁹ Pa
I = 1,000 cm⁴ = 1 × 10⁻⁵ m⁴

δ_max = (10,000 × 125) / (48 × 200 × 10⁹ × 1 × 10⁻⁵)
      = 1.25 × 10⁶ / 9.6 × 10⁷
      = 0.01302 m
      = 13.02 mm

M_max = P · L / 4 = 10 × 5 / 4 = 12.5 kN·m

Plug the same numbers into the beam deflection calculator and it returns the same answer to the millimetre. Now check the result against the usual serviceability limits. A floor beam under live load is typically capped at L/360, which for L = 5,000 mm is 13.9 mm. We are right on the edge — passing, but with no real margin. Under L/250 (rougher, for roofs or non-brittle ceilings) the allowance is 20 mm and we pass comfortably. Under L/480 (brittle ceiling like plaster) the limit is 10.4 mm and we fail; the section would have to be upgraded to the next size up, typically a section with I around 1,400–1,500 cm⁴.

Notice that nothing in this calculation has any margin against bending failure. The bending stress is σ = M·c/I, where c is the distance from the neutral axis to the extreme fibre. For a section with c = 50 mm and I = 1,000 cm⁴, σ = 12,500 × 0.050 / 1×10⁻⁵ = 62.5 MPa — well under the 275 MPa yield stress of S275 mild steel. The beam is governed by stiffness, not strength. That pattern is overwhelmingly common in low-rise buildings, which is why deflection calculations dominate everyday structural arithmetic and why the beam deflection calculator is a far more frequently useful tool than a yield-stress check.

Factors that drive how much a beam deflects

Span length

Span is the single biggest lever. Deflection scales with L³ for point loads and L⁴ for uniform loads, so a small change in span has an enormous effect on movement. Doubling a timber joist's span from 3 m to 6 m increases deflection under self-weight by a factor of 16. Halving the span by introducing an intermediate support cuts deflection by the same factor. Where ceiling heights and floor plans allow, intermediate columns, beams or load- bearing partitions are the most efficient single change you can make.

Cross-section: the second moment of area I

I depends on how the cross-sectional material is distributed about the neutral axis. For a solid rectangle of breadth b and depth d, I = b·d³/12 — depth cubed, breadth only linear. That is why structural sections look the way they do: I-beams concentrate material in the flanges at the top and bottom and use a thin web to keep the flanges apart, channels and Z-sections do the same trick in a different geometry, and box sections close the loop for torsional stiffness. For standard rolled steel, look I up in the AISC Steel Construction Manual, the British Constructional Steelwork Association blue book, or any mill catalogue; for timber and concrete, derive it from the dimensions. The concrete calculator and square footage calculator are useful companions when you are working out how much section depth you can afford in a given floor build-up.

Material — Young's modulus E

Steel sits at E ≈ 200 GPa, aluminium ≈ 69 GPa, structural timber 7–13 GPa along the grain, concrete 20–35 GPa. Higher E for the same I means proportionally less deflection. Going from aluminium to steel for the same section cuts deflection by a factor of three; switching softwood timber for hardwood improves matters by perhaps a third. Material choice tends to be locked in by other constraints (cost, fire rating, weight, corrosion, appearance), so the designer treats E as fixed and adjusts I and L instead. For a deeper primer on what E really means and how to measure it, see the Young's modulus calculator.

Load magnitude and distribution

Doubling the load doubles the deflection — a linear effect, because the formulas are linear in P and w. Distribution matters too: a single point load P at midspan and a total uniform load wL = P deposited along the same simply supported beam give very different deflections. The uniform case has δ_max = 5wL⁴/(384EI) = 5PL³/(384EI), versus PL³/(48EI) for the point case. The ratio is 5/8: a uniform load that totals the same as the point load produces only about 5/8 of the deflection. Loads spread over a span are kinder to a beam than equivalent loads dropped on a single spot.

Support conditions

Cantilevers deflect dramatically more than simply supported beams of the same span and load. For an equal point load, the cantilever formula has 3 in the denominator versus 48 for the simply supported case — a factor of 16 worse before you change anything else. Worse still, the bending moment at a cantilever's fixed root is PL, four times the PL/4 peak moment in the simply supported case. A balcony beam built into a wall does roughly 16× the work of a floor beam of the same span carrying the same load. That is why cantilever balconies have very deep sections, often with concealed back-spans that turn them into propped cantilevers for stiffness.

How to reduce deflection in a beam design

• Shorten the span. Add an intermediate support — a column, an inner load- bearing wall, a flush beam under a steel header. Because deflection scales as L³ or L⁴, this is by far the biggest lever.
• Choose a deeper section. Doubling depth multiplies I by about 8 for a rectangular section and by a similar factor for an I-section, while doubling weight at most. Deeper is far more efficient than wider when the geometry allows.
• Move to a stiffer material. Swap aluminium for steel, softwood for hardwood, plain concrete for steel-reinforced concrete. The E ratio cuts straight into deflection.
• Use continuity. A two-span continuous beam over the same total length deflects much less than two separate simply supported spans. The negative moment over the intermediate support pulls the central deflection back up.
• Pre-camber the beam. Order the steel beam with an upward camber equal to the calculated dead-load deflection, so that when the floor is built and the dead load applies, the beam ends up flat. The live load then deflects from flat, halving perceived movement.
• Combine more than one of the above. Doubling depth and halving span together is a 128-fold improvement in deflection — well within the range of routine design adjustments and much cheaper than upgrading to exotic materials.

Common mistakes

• Mixing units before the formula. The Euler–Bernoulli formula is unforgiving on units: use newtons, metres, pascals and metres-to-the-fourth, or convert everything consistently. Plugging GPa straight into a denominator with cm⁴ in the numerator produces an answer wrong by 10¹³. The beam deflection calculator takes the structural-drawing conventions (m, kN, GPa, cm⁴) and does the SI conversions internally to avoid this whole class of error.
• Treating a cantilever like a simply supported beam. The PL³/(3EI) and PL³/(48EI) formulas differ by a factor of 16. Use the wrong case and the beam is either 16× too flexible or 16× too stiff. Confirm the support condition at the start of every deflection check.
• Ignoring serviceability deflection limits. A beam can pass a bending stress check by a wide margin and still deflect well past the L/250 or L/360 service limit. Designers used to working in steel are particularly prone to this because steel's high strength means strength rarely controls. Always check deflection against the relevant code limit; it is almost always the governing criterion for floor beams.
• Forgetting self-weight. The dead load of the beam itself, plus the slab or floor build-up sitting on it, is usually a major part of the uniform load. Calculate it from the section weight per metre and the floor area tributary to the beam, then add it to the live load before computing deflection.
• Picking I about the wrong axis. Sections have two principal moments of inertia, I_x and I_y. Bending about the strong axis uses I_x (the larger value); about the weak axis uses I_y. Use the strong-axis value only if the beam is oriented to bend that way — and verify by sketching the cross-section with the loading direction marked on it.
• Applying linear superposition past yield. Adding deflections from a point load and a uniform load case is correct only while both keep the beam elastic. Once any fibre yields the formula no longer applies and the beam needs a plastic analysis.

When to seek professional advice

The four cases in the beam deflection calculator are adequate for teaching, for sizing a garden-room lintel, for sanity-checking a finite-element model, or for choosing a candidate section before formal design. For any load-bearing member in an occupied building, a bridge, lifting equipment, or anything that fails dangerously, the answer needs to be signed off by a chartered or licensed structural engineer who can apply the full code and account for what the textbook formula leaves out: shear deflection in deep short beams, long-term creep in timber and concrete, the partial fixity of real connections, multi-span continuity, lateral-torsional buckling, and the appropriate factor of safety for the consequence of failure.

Frequently asked questions

What is the difference between deflection and bending stress?

Deflection is how far the beam moves when loaded — a serviceability concern, measured in millimetres, governed by EI and the span. Bending stress is the internal stress in the extreme fibres of the cross-section — a strength concern, measured in MPa, governed by the bending moment and the section modulus. A beam can fail either by deflecting too much (cracked plaster, bouncy floors, jammed doors) or by yielding/breaking in bending. The two checks are independent; both have to pass.

What deflection is acceptable?

It depends on the code and the application. Common serviceability limits under live load only are L/360 for general floor beams, L/240 for roof beams, L/480 for beams carrying brittle finishes like plaster, and L/600 for floors supporting vibration-sensitive equipment. Under total load (dead plus live) the limits are usually about L/200 to L/300. Eurocode (BS EN 1990 plus its National Annex) and the IBC both reference these ranges. The calculator gives you the deflection number; check it against whichever limit governs your project.

Does this work for steel, timber, concrete and aluminium beams?

Yes — the formula only assumes linear elastic behaviour, small deflection and a prismatic cross-section. Steel and aluminium fit this naturally up to yield. Timber fits in the short term but you should de-rate E by the long-term creep factor (around 0.5 for typical service classes per Eurocode 5) before computing deflection at the end of the design life. Reinforced concrete is more involved because cracked-section behaviour reduces effective I; in that case use the cracked moment of inertia I_cr, not the gross-section I_g.

Can I handle a uniform load and a point load on the same beam?

Yes, in the elastic range by linear superposition. Run the beam deflection calculator once for the uniform load case and once for the point load case, then add the two midspan deflections. This is exactly how Table 3-23 in the AISC manual is intended to be used and gives the correct answer for any combination of standard cases provided the beam stays elastic.

What is Young's modulus and where do I get it?

Young's modulus E is the slope of the linear part of the material's stress–strain curve, in pascals. Standard values are 200 GPa for structural steel, 69 GPa for aluminium, 7–13 GPa for structural timber along the grain, 20–35 GPa for concrete depending on the mix. The Young's modulus calculator shows how E is derived from a tensile test, lists typical values for every common material, and explains the temperature and anisotropy caveats that matter for beam design.

Where do I find I for a standard steel section?

For UK and European sections, the BS EN 10365 dimension tables published by the British Constructional Steelwork Association ("the blue book"). For US sections, Part 1 of the AISC Steel Construction Manual. Both list I about the strong axis (I_x) and the weak axis (I_y); convert in⁴ to cm⁴ by multiplying by 41.62. For timber, calculate I = bd³/12 from the dressed dimensions.

Does this account for shear deformation?

No — the formulas are pure Euler–Bernoulli (flexural deflection only) and are exact for slender beams with span-to-depth ratio above about 10. For deep, short transfer beams (ratio below 5) shear deflection becomes significant and Timoshenko beam theory is more accurate. For everyday joists, rafters, lintels and floor beams, the error from neglecting shear is under 2 %.

Related calculators

Solve any of the four cases directly with the beam deflection calculator. To check the material modulus E that feeds into the formula, use the Young's modulus calculator. For the loading side, the force calculator evaluates F = m·a when the load on the beam is an inertial reaction. For the geometry of the floor or roof being supported, the square footage calculator gives tributary area and the roof pitch calculator sizes rafters and ridge beams. When the beam is concrete rather than steel or timber, the concrete calculator works out the pour volume; the brick calculator covers masonry that beams routinely sit on; the stair calculator handles the rise-and-run geometry that often controls floor-to-floor beam depths; and the boiler size calculator sizes the heating equipment that beams sometimes have to be designed around in mechanical-services voids.