Beam Deflection Calculator

Calculates the maximum deflection, bending moment, bending stress and reactions of a simply supported, cantilever or fixed-fixed beam under a point or uniform load, and draws the load, moment and deflection diagrams.

How to use

1. Choose the "Beam type" and the "Load" and enter the "Length L" in mm.
2. Enter "Point load P" with "Load position a", or "Distributed load w" in N/mm.
3. Pick a material under "Material (typical E)" or type "Young's modulus E", then enter "Second moment of area I" and "Section modulus Z". Values sent from the section properties calculator or the materials table fill in automatically.
4. The deflection, moment, stress and reactions appear at once with the diagrams below; "Copy results" copies the numbers.

How it works

• Simply supported: δmax = PL³ ÷ 48EI (centre load), 5wL⁴ ÷ 384EI (uniform); cantilever: PL³ ÷ 3EI, wL⁴ ÷ 8EI; fixed both ends: PL³ ÷ 192EI, wL⁴ ÷ 384EI. Off-centre loads are under "Formulas".
• Bending stress σ = Mmax ÷ Z; units are mm, N and MPa. E is entered in GPa.
• Linear-elastic, prismatic beam with small deflections; shear deformation and self-weight are not included. Judge allowable values with the standard that applies.

Questions and answers

How do I calculate beam deflection?

A 1 m simply supported beam with 1000 N at mid-span, E = 205 GPa and I = 4,166,667 mm⁴ gives δ = PL³ ÷ 48EI = 0.0244 mm. Deflection grows with the cube of the span.

How do I convert kN/m to N/mm?

1 kN/m equals 1 N/mm, so for 2 kN/m enter 2.

Where do I get the second moment of area I?

A rectangle is bh³ ÷ 12, a circle πd⁴ ÷ 64. For I-beams, channels and pipes use the section properties calculator and its send button.

What does a negative bending moment mean?

A hogging moment with the top in tension, as at a fixed end. It is drawn below the axis with a minus sign.