Beam formulas/Simply supported

Simply Supported Beam with Point Load at Any Position

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

PABab

A beam of span L on a pin and a roller carries a point load P at distance a from the left support (b = L − a from the right).

Formulas

Reactions

Reactions

Shear

Shear

Bending moment

Moment (x ≤ a)

Maximum moment

Deflection

Deflection (x ≤ a)

Maximum deflection (a ≥ b)

Deflection under the load

  • The maximum moment is always under the load.
  • The maximum deflection is not under the load: it lies in the longer segment, between the load and midspan. If a < b, swap a and b and measure x from the right support.

Calculator

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

Reactions

  • RA = 13.33 kN
  • RB = 6.667 kN

Forces: + up. Moments: − hogging.

Maximum values

  • |V|max = 13.33 kN at x = 0 m
  • Mmax = 26.67 kN·m at x = 2 m
  • δmax = 3.871 mm at x = 2.734 m
  • Span/δ = 1,550

Shear, moment and deflection diagrams

Shear force V (kN)
13.33 kN
Bending moment M (kN·m)
26.67 kN·m
Deflection δ (downward) (mm)
3.871 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 point load at any position?

RA = Pb/L and RB = Pa/L, where a and b are the distances from the load to the left and right supports. The nearer support carries more.

What is the maximum bending moment of a simply supported beam with point load at any position?

Mmax = Pab/L, directly under the load.

What is the maximum deflection of a simply supported beam with point load at any position?

For a ≥ b, δmax = Pb(L² − b²)^(3/2) / (9√3·EIL) at x = √((L² − b²)/3) from the left support. The deflection under the load is Pa²b²/(3EIL).

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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