Cantilever Beam with End Point Load

Reference

The balcony, the bracket, the sign post arm: one end gripped, the other end free, and a load at the tip.

The equation

Mwall = PL at the fixed end
V = P along the whole length
δtip = PL3/3EI
slope at the tip = PL2/2EI
  • P is the load at the free end, in kN
  • L is the cantilever length, in meters
  • E and I are the modulus and second moment of area
  • δ is the tip deflection
P = 10 kNL = 3 mBending moment, sagging plotted upMwall = PL = 30 kNm hoggingcalculatgeek.com
The whole beam hogs: tension on top from wall to tip, which is why cantilever reinforcement goes in the top face and why cutting it short at the root is the classic failure.

Worked example

A 3 m cantilever with 10 kN at the tip and EI = 16000 kNm2. The wall moment is 10 × 3 = 30 kNm and the shear is a constant 10 kN all the way back. Tip deflection is 10 × 33/(3 × 16000) = 0.0056 m, or 5.6 mm, comfortably inside the usual cantilever limit of span over 180. The same beam simply supported over 3 m would deflect 16 times less under the same load, which is the cleanest way to feel what losing a support costs.

When to use it

The formula assumes the wall grips the beam completely. A cantilever is only as stiff as its anchorage: any rotation at the root adds that angle times the length straight onto the tip, which is why site measured cantilevers routinely deflect more than the arithmetic says. Check the back span or hold down that provides the fixity before trusting PL cubed over 3EI.

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