Beam formulas/Propped cantilever

Propped Cantilever with Center Point Load

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

PABL

A beam of span L is fixed at the left end (A) and supported on a roller at the right end (B). A point load P acts at midspan.

Formulas

Reactions

Reactions

Shear

Shear

Bending moment

Moment (x ≥ L/2)

Maximum moments

Deflection

Deflection under the load

Maximum deflection

  • x is measured from the fixed end. The fixed-end moment (3PL/16) governs over the moment under the load (5PL/32).
  • The maximum deflection is 0.447L from the roller, a little past the load toward the roller.

Calculator

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

Reactions

  • RA = 13.75 kN
  • MA = -22.5 kN·m
  • RB = 6.25 kN

Forces: + up. Moments: − hogging.

Maximum values

  • |V|max = 13.75 kN at x = 0 m
  • Mmax = -22.5 kN·m at x = 0 m
  • δmax = 2.012 mm at x = 3.317 m
  • Span/δ = 2,981

Shear, moment and deflection diagrams

Shear force V (kN)
13.75 kN
Bending moment M (kN·m)
-22.5 kN·m
Deflection δ (downward) (mm)
2.012 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 propped cantilever beam cases

More beam formulas

FAQ

What are the reactions for a propped cantilever with center point load?

RA = 11P/16 at the fixed end, RB = 5P/16 at the roller, and a fixed-end moment of 3PL/16.

What is the maximum bending moment of a propped cantilever with center point load?

3PL/16 at the fixed end, which governs, and 5PL/32 under the load.

What is the maximum deflection of a propped cantilever with center point load?

δmax = PL³/(48√5·EI) ≈ 0.00932PL³/EI at 0.447L from the roller.

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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