Beam Deflection Calculator
Beam deflection is how far a loaded beam bends from its unloaded position. For the classic case of a simply supported beam carrying a single point load at the centre of its span, Euler-Bernoulli beam theory gives an exact closed-form result. This calculator applies that formula to your inputs: the point load, the span length, the material modulus of elasticity, and the cross section's second moment of area. It returns the maximum midspan deflection in both metres and millimetres so you can check it against your serviceability limit. Use consistent SI units throughout.
Beam deflection formula
Max deflection = (P * L^3) / (48 * E * I)
Flexural rigidity = E * I
Deflection (mm) = deflection (m) * 1,000
Span / deflection ratio = L / deflection (m)
This is the standard Euler-Bernoulli result for a simply supported beam with a central point load. The cube of the span dominates the deflection, which is why even small increases in span have a large effect. The span-to-deflection ratio is a common serviceability check; larger ratios mean a stiffer, less bouncy beam.
Engineering context
- The formula assumes small deflections, a linear-elastic homogeneous material, and a constant cross section along the span.
- Modulus of elasticity is a material property: enter the published value for your steel, aluminium, or timber from its data sheet.
- The second moment of area depends on cross-section shape; for a rectangle it is b times h cubed over 12.
- Serviceability codes often limit deflection to a fraction of the span, such as L/360 for floors carrying plaster ceilings.
- This tool is an educational aid; a licensed structural engineer must sign off any load-bearing design.
Beam deflection: frequently asked questions
What beam case does this calculator solve?
It solves a simply supported beam of length L carrying a single point load P at the centre of the span. This is one of the standard cases in Euler-Bernoulli beam theory. The maximum deflection occurs at midspan and is given by the closed-form expression P times L cubed divided by 48 times E times I.
What is E and I in the deflection formula?
E is the modulus of elasticity (Young's modulus) of the beam material, a measure of its stiffness in pascals. I is the second moment of area (area moment of inertia) of the cross section about the bending axis, in metres to the fourth power. Their product, E times I, is the flexural rigidity that resists bending.
Why must I enter the modulus of elasticity myself?
The modulus depends entirely on the material you choose, from steel to aluminium to timber. Rather than assume a value, the calculator asks you to enter it so the result matches your actual beam. Look up the modulus for your material from the manufacturer's data sheet or an engineering materials standard.
Does this account for the beam's own weight?
No. This formula covers a single concentrated point load at midspan only. Self-weight is a distributed load and produces a different deflection formula (5 w L to the fourth over 384 E I). For combined loads, superpose the deflections, or consult a structural engineer for a design that must carry life-safety loads.
What units does the calculator use?
Enter the load in newtons, the span in metres, the modulus in pascals, and the second moment of area in metres to the fourth power. The deflection result is in metres, and also shown in millimetres for convenience. Keep all inputs in consistent SI units so the result is correct.
Official sources
- NASA Glenn Research Center: Beam bending fundamentals.
- NIST: SI units reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.