Beam Deflection Calculator (Simply Supported, Centre Load)
Find the maximum sag of a simply supported beam carrying a central point load from its span, load, modulus and moment of inertia.
How it works
δ = P·L³ ÷ (48·E·I) · at mid-span of a simply supported beam
A beam resting on two supports with a single load at its centre sags most in the middle by δ = PL³/(48EI). Deflection grows with the cube of the span, so doubling the length increases the sag eightfold — length dominates everything. Stiffer material (higher E) or a deeper section (higher I) reduces the sag proportionally, which is why beams are made tall rather than wide.
Worked example
A 1,000 N load sits at mid-span of a 3 m steel beam (E = 200×10⁹ Pa) with I = 8×10⁻⁶ m⁴. δ = 1000 × 3³ / (48 × 200×10⁹ × 8×10⁻⁶) = 27000 / 76,800,000 ≈ 0.000352 m = 0.35 mm.
Sources: Deflection (engineering) — Wikipedia
Frequently asked questions
Why is span cubed but everything else linear?
The bending moment grows with span, and integrating curvature twice to get deflection brings in the length again — three factors of L in total. That cubic term is why long spans deflect so dramatically.
What is the second moment of area (I)?
It describes how a cross-section resists bending. For a solid rectangle of width b and height h it is I = b·h³/12, so making a beam deeper (larger h) is far more effective than making it wider.
Does this apply to a distributed load?
No — this formula is for a single point load at the centre. A uniformly distributed load uses δ = 5wL⁴/(384EI), which gives a different maximum.
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This calculator is for educational purposes. Double-check important results before acting on them.