Beam formulas/Simply supported

Simply Supported Beam with Moment at One End

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

M₀ABL

A beam of span L on a pin and a roller has a concentrated moment M₀ applied at the left support (A). No other loads act.

Formulas

Reactions

Reactions (a couple)

Shear

Shear

Bending moment

Moment

Maximum moment

Deflection

Deflection

Maximum deflection

  • The reactions form a couple that balances M₀, so the shear is constant along the span.
  • The end slopes are θA = M₀L/3EI at the loaded end and θB = M₀L/6EI at the far end.

Calculator

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

Reactions

  • RA = -5 kN
  • RB = 5 kN

Forces: + up. Moments: − hogging.

Maximum values

  • |V|max = 5 kN at x = 0 m
  • Mmax = 30 kN·m at x = 0 m
  • δmax = 3.464 mm at x = 2.536 m
  • Span/δ = 1,732

Shear, moment and deflection diagrams

Shear force V (kN)
-5 kN
Bending moment M (kN·m)
30 kN·m
Deflection δ (downward) (mm)
3.464 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 simply supported beam cases

More beam formulas

FAQ

What are the reactions for a simply supported beam with moment at one end?

The reactions form a couple that balances the applied moment: M₀/L at each support, one up and one down.

What is the maximum bending moment of a simply supported beam with moment at one end?

Mmax = M₀ at the loaded support, falling linearly to zero at the other support.

What is the maximum deflection of a simply supported beam with moment at one end?

δmax = M₀L²/(9√3·EI) ≈ 0.0642M₀L²/EI at x ≈ 0.423L from the loaded end.

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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