Cantilever Beam Calculator

Enter the tip load, length and rectangular section to get deflection (δ = PL³/3EI) and bending stress at the fixed end — with a utilization check against steel yield.

Last updated: 2026-09-28

How the calculation works

The rectangular section's second moment I = w·h³/12 drives everything: depth enters cubed, width linearly.
Deflection follows δ = P·L³/(3·E·I) — the cube of length means doubling the span increases deflection 8×.
Bending stress peaks at the fixed end's outer fiber: σ = P·L·(h/2)/I, compared here against 250 MPa typical steel yield.

Formula

δ = P·L³ / (3·E·I)
σ = M·c / I   with M = P·L, c = h/2
I = w·h³ / 12   (rectangular section)

Formula variables
Symbol	Meaning	Unit
`δ`	Tip deflection	mm
`σ`	Max bending stress at the fixed end	MPa
`I`	Second moment of area	mm⁴
`E`	Young's modulus — 200 GPa (steel)	GPa

Worked example

A 1 m steel bar 50 mm deep × 20 mm wide, 500 N at the tip: I = 20 × 50³/12 = 208,333 mm⁴. Deflection = 500 × 1³ / (3 × 200e9 × 2.083e-7) = 4.0 mm. Stress = 500 × 1 × 0.025 / 2.083e-7 = 60 MPa — 24% of yield. Rotate the bar flat-wise (20 deep) and deflection jumps 15×.

Interpreting the result

The depth-cubed law is the design lever: mount the section tall, not flat. Check serviceability before strength — a shelf bracket can be far from yield yet still sag visibly; L/360 deflection is a common acceptability limit (2.8 mm on a 1 m span). Add self-weight for long beams (about half the beam mass as an equivalent tip load), and remember fatigue, not yield, governs vibrating cantilevers.

Assumptions

Linear-elastic steel, E = 200 GPa, small deflections.
Rectangular solid section; a point load at the free end.
Self-weight and dynamic amplification not included.

Limitations

Aluminum (E = 69 GPa) deflects ~2.9× more — swap E mentally or scale the result.
Short deep beams shear-govern before bending theory applies (rule of thumb: span < 5× depth).
No lateral-torsional buckling check — tall thin sections under heavy load can buckle sideways.

Frequently asked questions

What is the cantilever deflection formula?

δ = P·L³ ÷ (3·E·I) for a point load at the tip. The L³ is why spans are expensive: twice the length deflects eight times as much.

How do I make a cantilever stiffer?

Increase section depth (cubed effect), shorten the span (cubed), or add support — in that order of effectiveness. Widening the section is the weakest lever (linear).