Cantilever Beam: Tip Load and Deflection

1. Shear at the support: V = P = 10 kN, constant along the length.
2. Fixed-end moment: M = P × L = 10 × 2.0 = 20 kN·m, hogging (tension on the top).
3. Bending stress: σ = M ÷ S = 20 × 10⁶ N·mm ÷ 1.94 × 10⁵ mm³ = 103 MPa.
4. Tip deflection: δ = P L³ ÷ (3 E I) = 10,000 × 2,000³ ÷ (3 × 200,000 × 1.94 × 10⁷) = 6.87 mm.
5. Tip rotation: θ = P L² ÷ (2 E I) = 10,000 × 2,000² ÷ (2 × 200,000 × 1.94 × 10⁷) = 0.00515 rad (0.30°).
6. Limit: L ÷ 180 = 2,000 ÷ 180 = 11.1 mm. The deflection of 6.87 mm is within it (62% of the limit).

Answer

What it shows

• The moment is largest at the support, and so is the stress. The deflection is largest at the tip.
• Deflection grows with L³, so a cantilever that is 20% longer deflects 73% more under the same load (1.2³ = 1.73).
• Stiffness depends on E × I. A deeper section increases I much faster than it increases weight.

Common mistakes

• Using the simply supported formula (5wL⁴ ÷ 384EI or PL³ ÷ 48EI) for a cantilever.
• Mixing units: N, mm and MPa together, or kN and m with mm⁴ for I.
• Checking stress but not deflection, or the reverse. Both limits apply, and either can control.
• Forgetting that the supports of a real cantilever rotate a little, which adds to the tip deflection.