Beam deflection, explained

The elastic curve

Euler–Bernoulli beam theory assumes plane sections stay plane, the material is linear elastic and the deflections are small. Then the curvature of the beam equals M/EI. Write the bending moment as a function of x, integrate once for the slope and again for the deflection, and fix the two constants with the supports: zero deflection at a pin, zero deflection and zero slope at a fixed end.

For loads that combine, use superposition: the deflection under two loads is the sum of the deflections under each one. That is why deflection tables are the first tool an engineer reaches for.

Stiffness is a design limit

A beam can be strong enough and still unusable. Floors that bounce, shafts that misalign their bearings and machine frames that lose accuracy all fail on deflection, not stress. Building codes typically limit floor deflection under live load to around span/360; precision machines need far less.

Deflection scales with L³ or L⁴ and with 1/I. Halving a span cuts the deflection under a point load by eight; doubling a rectangular section’s depth cuts it by eight as well. Depth is the cheapest stiffness there is.

A worked number

A 1 m steel cantilever, 20 mm wide and 40 mm deep, carries 500 N at its tip. I = 20 × 40³/12 ≈ 1.07 × 10⁵ mm⁴, which is 1.07 × 10⁻⁷ m⁴. The tip deflection is 500 × 1³/(3 × 200 × 10⁹ × 1.07 × 10⁻⁷) ≈ 7.8 mm.

Lay the same bar flat and I drops by a factor of four, so the deflection rises to about 31 mm.

Common mistakes

• Mixing millimetres and metres inside EI, which puts the answer off by powers of a thousand.
• Using the wrong boundary conditions, for example forgetting that the slope is zero at a fixed end.
• Leaving out self-weight on a long beam, where it can be most of the load.