Beam formulas/Cantilever

Cantilever Beam with Point Load at Free End

Reactions, shear, bending moment and deflection formulas, with a live calculator.

PAL

A cantilever of length L is fixed at the left end (A) and carries a point load P at the free end.

Formulas

Reactions

Fixed-end reactions

Shear

Shear

Bending moment

Moment

Maximum moment

Deflection

Deflection

Maximum deflection

Slope at the free end

  • x is measured from the fixed support. The moment is hogging (negative) along the whole length.

Calculator

Loads act downward as drawn. Steel: E = 200 GPa. I in millions of mm⁴ (100 means 100 × 10⁶ mm⁴).

Reactions

  • RA = 10 kN
  • MA = -30 kN·m

Forces: + up. Moments: − hogging.

Maximum values

  • |V|max = 10 kN at x = 0 m
  • Mmax = -30 kN·m at x = 0 m
  • δmax = 4.5 mm at x = 3 m
  • L/δ = 666.7

Shear, moment and deflection diagrams

Shear force V (kN)
10 kN
Bending moment M (kN·m)
-30 kN·m
Deflection δ (downward) (mm)
4.5 mm

x is measured from the left end. Positive moment (sagging) plots above the axis. Need more loads or supports? Solve any beam with the full beam calculator.

How these formulas were checked

Every formula on this page is evaluated by the calculator above and was cross-checked numerically against PolyCalc's stiffness-method beam solver: reactions, shear and moment at dozens of points, and deflection by integrating the solver's moment diagram twice with the support conditions.

Other cantilever beam cases

More beam formulas

FAQ

What are the reactions for a cantilever beam with point load at free end?

The wall provides a vertical reaction R = P and a fixed-end moment M = PL.

What is the maximum bending moment of a cantilever beam with point load at free end?

Mmax = PL at the fixed support.

What is the maximum deflection of a cantilever beam with point load at free end?

δmax = PL³/(3EI) at the free end. The slope there is PL²/(2EI).

What sign convention do these formulas use?

x is measured from the left end (from the fixed end for cantilevers and propped cantilevers). Loads act downward, reactions are positive upward, sagging moment is positive and deflection is positive downward. E is the elastic modulus and I the second moment of area of the section.

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