Beam formulas/Cantilever

Cantilever Beam with Moment at Free End

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

M₀AL

A cantilever of length L is fixed at the left end (A) and has a concentrated moment M₀ applied at the free end, bending the tip downward.

Formulas

Reactions

Fixed-end reactions

Shear

Shear

Bending moment

Moment

Deflection

Deflection

Maximum deflection

Slope at the free end

  • This is pure bending: the moment is the same at every section, so the beam bends into a circular arc (small deflection theory gives the parabola above).

Calculator

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

Reactions

  • RA = 0 kN
  • MA = -15 kN·m

Forces: + up. Moments: − hogging.

Maximum values

  • |V|max = 0 kN at x = 0 m
  • Mmax = -15 kN·m at x = 0 m
  • δmax = 3.375 mm at x = 3 m
  • L/δ = 888.9

Shear, moment and deflection diagrams

Shear force V (kN)
0 kN
Bending moment M (kN·m)
-15 kN·m
Deflection δ (downward) (mm)
3.375 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 moment at free end?

There is no vertical reaction. The wall resists a moment equal to M₀.

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

The bending moment is M₀ at every section (pure bending).

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

δmax = M₀L²/(2EI) at the free end. The slope there is M₀L/EI.

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.

Related tools and topics