Cantilever Beam Calculator (Moment, Shear & Deflection)
Calculate the fixed-end moment, support reaction, tip deflection and rotation of a cantilever beam with an end point load and a uniform load, from span, material E and moment of inertia I.
How it works
Reaction R = P + wL · fixed-end moment M = PL + wL²/2 · tip deflection δ = PL³/(3EI) + wL⁴/(8EI) · tip rotation θ = PL²/(2EI) + wL³/(6EI) · deflection at x from the support: Px²(3L − x)/(6EI) + wx²(6L² − 4Lx + x²)/(24EI).
A cantilever is fixed at one end and free at the other, so the support must resist both the vertical load and a bending moment. Shear and moment are largest at the fixed support and fall to zero at the free end; deflection is the opposite, largest at the tip. The deflection formulas come from Euler–Bernoulli beam theory as tabulated in the AISC Steel Construction Manual and the American Wood Council Design Aid 6. A common serviceability limit for cantilevers is span/180 (twice the L/360 used for simple spans, because the cantilever length is half an equivalent span).
Worked example
A 3 m steel cantilever (E = 200 GPa, I = 8,000 cm⁴) carries a 10 kN load at its tip and a 4 kN/m uniform load. The reaction is 10 + 4 × 3 = 22 kN and the fixed-end moment is 10 × 3 + 4 × 3²/2 = 30 + 18 = 48.00 kN·m. EI = 200×10⁹ × 8×10⁻⁵ = 16×10⁶ N·m², so the tip deflection is 10,000 × 27/(3 × 16×10⁶) = 5.62 mm from the point load plus 4,000 × 81/(8 × 16×10⁶) = 2.53 mm from the uniform load, 8.16 mm in total — L/368, within the L/180 limit. The tip rotates 0.226°.
Sources: American Wood Council — Design Aid No. 6: Beam Design Formulas (cases 22–24) · Deflection (engineering) — cantilever formulas
Frequently asked questions
Where is the maximum moment in a cantilever beam?
At the fixed support. For an end point load it is P × L and for a uniform load wL²/2; with both, add them. The moment is hogging — the top of the beam is in tension — which matters for where reinforcement goes in a concrete cantilever.
What deflection limit should a cantilever meet?
Building codes usually check spans against L/240 to L/360 for simple spans; for a cantilever, IBC Table 1604.3 footnote h and most designers use twice the cantilever length, which works out to about L/180 of the cantilever itself (L/120 for some roofs).
Why is cantilever deflection so much larger than a simply supported beam?
A tip load on a cantilever deflects PL³/3EI, 16 times more than the same load at the centre of a simple beam of the same length (PL³/48EI). Cantilevers are governed by stiffness far more often than by strength.
What moment of inertia should I enter?
The second moment of area about the axis of bending, from a steel table (IPE, HEA, W-shapes) or b·h³/12 for a solid rectangle. For timber use the actual (dressed) size, and for cracked concrete use the effective moment of inertia, roughly 35–50% of the gross value.
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