How much will this beam deflect, and how hard is it working?
An IPE 200 in S235 steel, simply supported over 4 m with 10 kN/m, deflects 8.35 mm (L/479) and sees a bending stress of 105 MPa, or 45 % of its yield strength. The calculator does the same sum for IPE, HEA, HEB and UPN sections from the site’s tables, for box sections, round tubes, flat and round bar, in steel, stainless and aluminium, and for three support cases.
Short answer
A simply supported beam under a uniform load deflects δ = 5·q·L⁴ / (384·E·I); under a point load at midspan, δ = P·L³ / (48·E·I). The stress in the extreme fibre is σ = M / W. Deflection is governed by the stiffness E·I: aluminium (E = 70 000 MPa) deflects almost three times as much as steel with the same section. Stress is governed by the section modulus W; the material only changes what you compare it with. Both grow fast with span: a beam twice as long under the same load per metre deflects 16 times as much.
The result is an elastic estimate of deflection and stress for a simple beam under characteristic loads. It does not replace structural design to EN 1993 (steel) or EN 1999 (aluminium) and is not a resistance check. Left out are:
- lateral-torsional buckling, member buckling and local buckling of the walls;
- shear, bearing and web crippling under a point load;
- connections, welds and weld softening — in aluminium the heat-affected zone can halve the strength;
- load combinations and partial factors to EN 1990 (recommended 1.35 for permanent and 1.5 for variable actions);
- fatigue, vibration, long-term deflection and real supports, which are never a perfect pin or a perfectly rigid fixing.
Load-bearing members — floors, mezzanines, platforms, canopies, hoist beams — are sized and checked by a structural engineer.
Formulas for the three cases
L — span (cantilever: overhang), P — point load at midspan (cantilever: at the tip), q — uniform load per metre including self-weight. With both loads at once the results add up: in these cases the maximum moment and the maximum deflection of both loads fall at the same point.
| Case and load | Maximum moment | Maximum deflection | Support reaction |
|---|---|---|---|
| Simply supported, point load P at midspan | P·L / 4 | P·L³ / (48·E·I) | P / 2 |
| Simply supported, uniform load q | q·L² / 8 | 5·q·L⁴ / (384·E·I) | q·L / 2 |
| Cantilever, point load P at the tip | P·L | P·L³ / (3·E·I) | P |
| Cantilever, uniform load q | q·L² / 2 | q·L⁴ / (8·E·I) | q·L |
| Fixed at both ends, point load P at midspan | P·L / 8 | P·L³ / (192·E·I) | P / 2 |
| Fixed at both ends, uniform load q | q·L² / 12 | q·L⁴ / (384·E·I) | q·L / 2 |
Use consistent units: N, mm, MPa (N/mm²). I then goes in as mm⁴ (1 cm⁴ = 10 000 mm⁴), W as mm³, and a load of 1 kN/m is exactly 1 N/mm.
Worked example: IPE 200, 4 m span, 10 kN/m
The same inputs the calculator opens with: S235 steel, simply supported, self-weight included.
- From the table: IPE 200 has Iy = 1943 cm⁴, Wel,y = 194.3 cm³ and a mass of 22.4 kg/m. Steel: E = 210 000 MPa, fy = 235 MPa.
- Self-weight:
g = 22.4 · 9.81 / 1000 = 0.220 kN/m. In totalq = 10 + 0.220 = 10.22 kN/m. - Moment:
M = q·L² / 8 = 10.22 · 4² / 8 = 20.44 kN·m. - Stress:
σ = M / W = 20.44·10⁶ / (194.3·10³) = 105 MPa, i.e. 45 % of 235 MPa. - Deflection:
δ = 5·q·L⁴ / (384·E·I) = 5 · 10.22 · 4000⁴ / (384 · 210 000 · 1943·10⁴) = 8.35 mm. - Ratio:
L / δ = 4000 / 8.35 = 479. L/250 allows 16.0 mm, L/300 allows 13.3 mm: the beam meets both.
IPE beams over 5 m: deflection and stress
Simply supported, 10 kN/m plus self-weight, S235 steel. Calculated with the same function as the calculator.
| Section | Mass, kg/m | Iy, cm⁴ | Deflection, mm | L / δ | σ, MPa | L/250 | L/300 |
|---|---|---|---|---|---|---|---|
| IPE 160 | 15.8 | 869 | 45.3 | L/110 | 292 | ✗ | ✗ |
| IPE 180 | 18.8 | 1316 | 30.0 | L/166 | 218 | ✗ | ✗ |
| IPE 200 | 22.4 | 1943 | 20.4 | L/245 | 164 | ✗ | ✗ |
| IPE 220 | 26.2 | 2771 | 14.3 | L/348 | 127 | ✓ | ✓ |
| IPE 240 | 30.7 | 3891 | 10.3 | L/487 | 99 | ✓ | ✓ |
| IPE 270 | 36.1 | 5789 | 6.9 | L/721 | 75 | ✓ | ✓ |
Over this span IPE 200 gives L/245 — just short of L/250, although its stress of 164 MPa is far from yield. For beams spanning a few metres it is usually deflection, not strength, that decides.
Materials in the calculator
Yield strength is used here only as a yardstick for the elastic stress. The strength values are the same as in the S235–S355 steel properties table.
| Material | E, MPa | fy (Rp0.2, f0), MPa | At a weld f0,haz, MPa | Source |
|---|---|---|---|---|
| S235 | 210 000 | 235 | — | EN 10025-2, thickness up to 16 mm; E — EN 1993-1-1 |
| S275 | 210 000 | 275 | — | EN 10025-2, thickness up to 16 mm; E — EN 1993-1-1 |
| S355 | 210 000 | 355 | — | EN 10025-2, thickness up to 16 mm; E — EN 1993-1-1 |
| 1.4301 (304) | 200 000 | 230 | — | EN 10088-2, cold-rolled sheet; E — EN 1993-1-4 |
| 1.4404 (316L) | 200 000 | 240 | — | EN 10088-2, cold-rolled sheet; E — EN 1993-1-4 |
| EN AW-6060 T6 | 70 000 | 140 | 60 | EN 1999-1-1, Table 3.2b, extruded profiles (6082: wall up to 5 mm) |
| EN AW-6082 T6 | 70 000 | 250 | 125 | EN 1999-1-1, Table 3.2b, extruded profiles (6082: wall up to 5 mm) |
What deflection is acceptable
EN 1993-1-1 (clause 7.2) gives no numbers: deflection limits are set by the National Annex or agreed with the client. Common reference values for beams are L/250 and L/300 — for a 5 m beam that is 20 mm and 16.7 mm. Annexes differ: the Danish one (DS/EN 1993-1-1 DK NA) gives l/400 for floors and l/200 for roofs under one variable action, and takes l as twice the overhang for a cantilever. Under partitions, brittle finishes and crane runways the annexes are stricter. The calculator compares its result with L/250 or L/300 as typical values; it does not perform a code check.
For catalogue sections the calculator also names the lightest section in the same series that meets the chosen limit and σ ≤ fy. Treat that as a starting point for the engineer, not a member selection.
Where the data comes from
Masses, dimensions and section properties come from the same tables as the beam weight calculator and the channel weight calculator. Tubes and box sections are calculated from their dimensions, as in the tube and hollow section calculator and the aluminium box section calculator; for a slotted system profile enter Iy and Wy from the maker’s data sheet. Full data for a single size, for example IPE 200.
Frequently asked questions
How do you calculate beam deflection?
For a simply supported beam under a uniform load, δ = 5·q·L⁴ / (384·E·I). A steel IPE 200 spanning 4 m with 10 kN/m (10.22 kN/m with self-weight) deflects 8.35 mm, i.e. L/479. Under a point load at midspan it is δ = P·L³ / (48·E·I); cantilevers and fixed-ended beams have their own coefficients (see the formula table above).
What size IPE beam do I need for a 5 m span?
With 10 kN/m on a simply supported S235 beam, the lightest section for L/250 is IPE 220 (deflection 14.3 mm, L/348, σ = 127 MPa) and for L/300 it is IPE 220 (L/348). IPE 200 gives L/245 and fails on deflection. For a different load, enter it in the calculator; a load-bearing member is sized by an engineer, since buckling and the bearing of the beam come into it.
How much more does an aluminium profile deflect than steel?
Almost three times with the same section: E is 70 000 MPa for aluminium and 210 000 MPa for steel. A 60×40×3 box section as a 1 m cantilever with 0.2 kN at the tip: steel 1.25 mm, 6082-T6 aluminium 3.58 mm, or 2.9 times as much (not exactly 3, because the lighter profile loads itself less). On top of that, 6082-T6 loses half its strength next to a weld: 250 MPa → 125 MPa.
Does this calculator replace structural design?
No. It works out elastic deflection and stress for three simple cases. It does not check buckling, shear, connections or welds, and applies no load combinations or partial factors from EN 1990. Load-bearing members are designed to EN 1993 (steel) or EN 1999 (aluminium).
What is the allowable deflection of a beam?
It is set by the National Annex to EN 1993-1-1 or agreed with the client; typical values for beams are L/250 and L/300. A 5 m beam may therefore deflect 20 mm or 16.7 mm. Under brittle finishes and partitions the limits are tighter.
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