Beam formulas/Simply supported

Simply Supported Beam with Center Point Load

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

PABL

A beam of span L on a pin and a roller carries a single point load P at midspan.

Formulas

Reactions

Reactions

Shear

Shear

Bending moment

Moment (x ≤ L/2)

Maximum moment

Deflection

Deflection (x ≤ L/2)

Maximum deflection

  • Shear is constant at P/2 and changes sign under the load, which is where the moment peaks.
  • The beam is symmetric, so the maximum deflection is at midspan.

Calculator

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

Reactions

  • RA = 10 kN
  • RB = 10 kN

Forces: + up. Moments: − hogging.

Maximum values

  • |V|max = 10 kN at x = 0 m
  • Mmax = 30 kN·m at x = 3 m
  • δmax = 4.5 mm at x = 3 m
  • Span/δ = 1,333

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 simply supported beam cases

More beam formulas

FAQ

What are the reactions for a simply supported beam with center point load?

Each support carries half the load: RA = RB = P/2.

What is the maximum bending moment of a simply supported beam with center point load?

Mmax = PL/4, under the load at midspan.

What is the maximum deflection of a simply supported beam with center point load?

δmax = PL³/(48EI) at midspan.

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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