Beam deflection: formulas for the 6 common cases and calculator
Published November 9, 2026 — the maximum deflection and bending moment of a simply supported, cantilever or fixed-fixed beam under a point or uniform load, with two worked examples (in mm and inches) and a calculator.
In short: a beam's deflection is proportional to the load and to the cube of the span (fourth power for a uniform load), and inversely proportional to E·I, the material's Young's modulus times the section's area moment of inertia. Simply supported with a mid-span load F: δ = F·L³ / (48·E·I). Cantilever with F at the free end: δ = F·L³ / (3·E·I), 16 times more.
1. Beam deflection calculator
Pick the case and the unit system, then enter the load, the span, Young's modulus and the section's moment of inertia (get it from the area moment of inertia calculator). The section modulus W is optional: it gives the maximum stress.
Default values: a 20 × 60 mm steel flat bar on edge (I = 360,000 mm⁴, W = 12,000 mm³), simply supported over 1 m, with 2,000 N at mid-span: 0.55 mm deflection, i.e. L/1814, and 41.7 MPa. The details are in the worked examples.
2. Formulas for the 6 common cases
Straight beam of constant section, elastic material, small deflections (Euler-Bernoulli theory). F: point load; q: uniform load per unit length; L: span; E: Young's modulus; I: area moment of inertia about the bending axis.
The support reactions of a symmetric beam are half the total load (F/2 or q·L/2); a cantilever takes the whole load at the fixed end (F or q·L), plus the moment. The uniform load q includes self-weight if you want to account for it: q = mass per unit length × g.
3. Young's modulus and allowable deflection
Deflection depends on Young's modulus E, not on strength: an S355 steel deflects exactly like an S235. Usual design values:
Allowable deflection. There is no universal value: it is a serviceability criterion. In buildings, L/200 to L/300 is common depending on the member, with the values set by Eurocode 3 and its national annex; in the US, the IBC uses L/360 for floor members under live load. For a machine frame, a guide or a sensor bracket, the limit comes from the required accuracy and is often far stricter: start from the position tolerance to be held.
Deflection says nothing about strength: also check the stress σ = M / W against the material's yield strength, see S235, S275 and S355 steels.
4. Worked examples
Steel flat bar, simply supported (metric units)
- 20 × 60 mm section on edge: I = b·h³/12 = 20 × 60³ / 12 = 360,000 mm⁴; W = b·h²/6 = 12,000 mm³.
- Span L = 1,000 mm, load F = 2,000 N at mid-span, E = 210,000 MPa.
- Deflection: δ = 2,000 × 1,000³ / (48 × 210,000 × 360,000) = 0.551 mm, i.e. L/1814.
- Moment: M = F·L/4 = 500,000 N·mm; stress σ = M/W = 41.7 MPa, far below the yield strength of S235.
- Self-weight: 1,200 mm² × 7,850 kg/m³ = 9.42 kg/m, i.e. q = 0.0924 N/mm; δ = 5 × 0.0924 × 1,000⁴ / (384 × 210,000 × 360,000) = 0.016 mm. Both deflections are at mid-span and add up: 0.567 mm in total. Density of other metals: density chart.
Steel bar as a cantilever (US units)
- 1 × 2 in bar on edge: I = 1 × 2³ / 12 = 0.667 in⁴; W = 1 × 2² / 6 = 0.667 in³.
- 24 in cantilever, 200 lbf at the free end, E = 29,000,000 psi.
- Deflection: δ = 200 × 24³ / (3 × 29,000,000 × 0.667) = 0.0477 in (1.21 mm), i.e. L/503.
- Moment at the fixed end: M = F·L = 4,800 lbf·in; stress σ = M/W = 7,200 psi (49.6 MPa).
5. Reducing deflection, and the limits of the formulas
For a b × h rectangular section, deflection varies as F·L³ / (E·b·h³). The levers therefore carry very different weights:
- Shorten the span or add an intermediate support: the most powerful lever.
- Put material far from the axis: stand a flat bar on edge, choose a tube or an I-section rather than a solid bar of the same weight, see the area moment of inertia of sections.
- Fix the ends: under a mid-span load, a fixed-fixed beam deflects 4 times less than a simply supported one (5 times less under a uniform load), provided the fixing is truly rigid.
- Superposition: as long as the material stays elastic, deflections from several loads at the same point add up, like self-weight and load in the example above.
Limits. These formulas assume an elastic material (σ below the yield strength), small deflections and a slender beam: they ignore shear deformation, which is no longer negligible when the span is short compared with the depth (L/h below about 10). A real fixed end is never perfectly rigid: the true deflection lies between the fixed and the simply supported cases. For an off-centre load, several loads or a varying section, use superposition, a fuller strength-of-materials reference or a finite element analysis.
Printable version: the beam deflection formulas as a PDF (A4, no sign-up), to keep with the moment of inertia formulas.
6. FAQ
What is the deflection formula for a simply supported beam?
Under a load F at mid-span: δ = F·L³ / (48·E·I). Under a uniform load q over the whole length: δ = 5·q·L⁴ / (384·E·I). In both cases, the maximum deflection is at mid-span.
What is the deflection of a cantilever beam?
Load F at the free end: δ = F·L³ / (3·E·I). Uniform load q: δ = q·L⁴ / (8·E·I). Deflection is greatest at the free end, and the maximum moment is at the fixed end: F·L or q·L²/2.
What allowable deflection should I use?
In buildings, L/200 to L/300 is common depending on the member, with values set by Eurocode 3 and its national annex; in the US, the IBC uses L/360 for floor members under live load. A machine frame is sized from the required accuracy, which is often far stricter.
Which units should I use in deflection formulas?
Consistent units: N, mm, MPa (N/mm²) and mm⁴ give a deflection in mm; lbf, in, psi and in⁴ give a deflection in inches. The uniform load q is then in N/mm or lbf/in.
How can I reduce a beam's deflection?
Shortening the span is the most effective, since deflection varies as L³ or L⁴. Next, increase the section depth (I varies as h³), fix the ends (a fixed-fixed beam deflects 4 to 5 times less than a simply supported one) or choose a stiffer material.