Beam deflection calculator
This beam deflection calculator covers simply supported beams and cantilevers with a point or a uniform load: from length, load, cross-section and material it gives deflection, bending moment and bending stress.
Source: Wikipedia – Biegelinie (Durchbiegung von Balken, Formelsammlung). Updated: .
How it is calculated
A simply supported beam with a point load F at midspan deflects by f = F·L³ / (48·E·I). A 100 × 200 mm steel beam over 4 m carrying 10 kN at midspan sags by about 0.95 mm.
How it is calculated
Deflection grows with the load and with the third or fourth power of the length, and falls with the modulus of elasticity E and the second moment of area I of the section. Depth counts to the third power: a beam twice as deep deflects only one eighth as much. The bending stress follows from σ = M / W.
The four load cases
- Simply supported, point load at midspan: f = F·L³/(48·E·I), M = F·L/4
- Simply supported, uniform load: f = 5·q·L⁴/(384·E·I), M = q·L²/8
- Cantilever, point load at the tip: f = F·L³/(3·E·I), M = F·L
- Cantilever, uniform load: f = q·L⁴/(8·E·I), M = q·L²/2
As a guide, floor and roof beams are often limited to a deflection of L/300. This calculator does not replace a structural calculation to the Eurocodes.
Frequently asked questions
How do you calculate the deflection of a beam?
Use the formula for the load case, for example f = F·L³/(48·E·I) for a simply supported beam with a point load at midspan. E is the modulus of elasticity of the material and I the second moment of area of the section.
What deflection is acceptable?
A common guide for floor and roof beams is L/300, stricter for sensitive finishes. The limits in the relevant standard and the structural design are what count.
How much does a deeper section help?
A lot: the second moment of area grows with the cube of the depth. A beam twice as deep deflects only one eighth as much.
Sources and legal basis
- Wikipedia – Biegelinie (Durchbiegung von Balken, Formelsammlung)
- Wikipedia – Deflection (engineering): beam deflection formulas
- Wikipedia – Flächenträgheitsmoment (Rechteck, Kreis, Rohr)
- Wikipedia – Elastizitätsmodul (Richtwerte Stahl, Aluminium, Holz, Beton)
As of: · Editorial policy
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