All beam formulas
Cantilever Beam with a Point Load at the Free End
Reactions, shear, maximum moment, slope and maximum deflection for a cantilever beam carrying a single point load at its unsupported end.
A cantilever beam is fixed (rigidly restrained against both translation and rotation) at one end and completely free at the other. A point load P at the free end is the classic cantilever case — a balcony, canopy, or bracket loaded at its tip all reduce to this.
Because the fixed end can resist a moment as well as a force, the reactions here include a reaction moment, not just a vertical force — the key difference from the simply supported case.
Worked Example
- Span L = 3 m
- Point load P = 20 kN at the free end (x = L)
- Flexural rigidity EI = 50,000 kN·m² (assumed, for the deflection/slope example only)
Reactions
R = P
fixed end
Worked example: 20 kN
M_fixed = PL
fixed end (hogging)
Worked example: 60 kN·m
Shear
V(x) = P
constant along the full span
Worked example: 20 kN
Maximum moment
M(x) = −P(L − x)
linear, from −60 kN·m at the fixed end to 0 at the free end
Worked example: −60 kN·m at x = 0, 0 at x = 3 m
Slope
θ_free = PL² / 2EI
free end
Worked example: 0.0018 rad (≈ 0.103°)
Maximum deflection
δ_max = PL³ / 3EI
free end (x = L)
Worked example: 3.6 mm at x = 3 m
Assumptions & Limitations
- Assumes constant EI along the full span.
- Assumes the fixed end is genuinely rigid — a real base with any flexibility (a column it frames into, a bolted base plate) will deflect and rotate more than this idealization predicts.
- Small-deflection theory, same as the simply-supported case.
- A classical result, not a design check against any code. Open it in the Beam calculator to apply load combinations and a stress/deflection check against a real section.