Cantilever Deflection — End Load

\delta = \frac{P L^{3}}{3 E I}

Worked example: 2 kN on a 2 m cantilever → 6.4 mm tip sag — press Try an example to run it live, then adjust anything.

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Cantilever Deflection — End Load explained

The cantilever is the shape Galileo drew in 1638 and the one every diving board, balcony and pipe-support bracket still copies. With the load at the tip, δ = PL³/3EI — sixteen times the sag a simply supported beam of the same span, load and section would show, because the fixed end must resist the full moment PL rather than sharing it with a second support. A 2 kN load on the end of a 2 m steel cantilever with I = 4.167 × 10⁻⁶ m⁴ droops δ = 2000 × 8 ÷ (3 × 200 × 10⁹ × 4.167 × 10⁻⁶) = 0.0064 m, 6.4 mm.

Everything hinges on the word fixed. The formula assumes the built-in end has zero rotation, and in real life that is the hardest thing to achieve — a bracket bolted to a flexible column, a beam pocketed into masonry, or a weld that is not full-strength will all rotate, and the tip deflection then comes out well above prediction. Measure the true rotation, or design conservatively. If instead the load is spread uniformly along the cantilever, the coefficient becomes wL⁴/8EI; and enter I as a plain number in m⁴, since this engine has no m⁴ unit.

Cantilever Deflection — End Load formula

\delta = \frac{P L^{3}}{3 E I}

Where

$\delta$ = Tip deflection (m)
$P$ = End load (N)
$L$ = Cantilever length (m)
$E$ = Young's modulus (kPa)
$I$ = Area moment of inertia (mm⁴)

Missing one of these? Work it out first, then come back

Tip deflection — Cantilever Deflection — Uniform Load
End load — Pulley System Effort Force, Max Bending Moment — Centre Point Load
Cantilever length — Cantilever Deflection — Uniform Load, Normal Strain (ε = δ/L)
Young's modulus — Young's Modulus (E = σ/ε), Axial Deformation (δ = PL/AE)
Area moment of inertia — Bending Stress (σ = Mc/I), Beam Deflection — Simply Supported, Centre Load

Part of this formula set

Beam bending and deflection

9 formulasMaximum moment, bending stress by Mc/I or M/S, and the deflection of simply supported and cantilever beams under point and uniform loads.