Beam formulas/Fixed-fixed

Fixed-Fixed Beam with Center Point Load

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

PABL

A beam of span L is fixed at both ends (A and B) and carries a point load P at midspan. It is statically indeterminate to the second degree.

Formulas

Reactions

Reactions

Shear

Shear

Bending moment

Moment (x ≤ L/2)

Maximum moments

Deflection

Deflection (x ≤ L/2)

Maximum deflection

  • The hogging moment at each support equals the sagging moment at midspan (PL/8), half the simply supported value.
  • Points of zero moment are at L/4 from each end. The deflection is one quarter of the simply supported value.

Calculator

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

Reactions

  • RA = 10 kN
  • MA = -15 kN·m
  • RB = 10 kN
  • MB = -15 kN·m

Forces: + up. Moments: − hogging.

Maximum values

  • |V|max = 10 kN at x = 0 m
  • Mmax = -15 kN·m at x = 0 m
  • δmax = 1.125 mm at x = 3 m
  • Span/δ = 5,333

Shear, moment and deflection diagrams

Shear force V (kN)
10 kN
Bending moment M (kN·m)
-15 kN·m
Deflection δ (downward) (mm)
1.125 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 fixed-fixed beam cases

More beam formulas

FAQ

What are the reactions for a fixed-fixed beam with center point load?

RA = RB = P/2, with a fixed-end moment of PL/8 at each support.

What is the maximum bending moment of a fixed-fixed beam with center point load?

PL/8 hogging at both supports and PL/8 sagging under the load.

What is the maximum deflection of a fixed-fixed beam with center point load?

δmax = PL³/(192EI) 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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