Beam formulas/Fixed-fixed

Fixed-Fixed Beam with Uniform Distributed Load

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

wABL

A beam of span L is fixed at both ends (A and B) and carries a uniform load w over its full length.

Formulas

Reactions

Reactions

Shear

Shear

Bending moment

Moment

Maximum moments

Deflection

Deflection

Maximum deflection

  • The governing moment is the hogging moment at the supports, wL²/12. Points of zero moment are at 0.211L from each end.
  • The deflection is one fifth of the simply supported value (wL⁴/384EI versus 5wL⁴/384EI).

Calculator

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

Reactions

  • RA = 30 kN
  • MA = -30 kN·m
  • RB = 30 kN
  • MB = -30 kN·m

Forces: + up. Moments: − hogging.

Maximum values

  • |V|max = 30 kN at x = 0 m
  • Mmax = -30 kN·m at x = 0 m
  • δmax = 1.688 mm at x = 3 m
  • Span/δ = 3,556

Shear, moment and deflection diagrams

Shear force V (kN)
30 kN
Bending moment M (kN·m)
-30 kN·m
Deflection δ (downward) (mm)
1.688 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 uniform distributed load?

RA = RB = wL/2, with a fixed-end moment of wL²/12 at each support.

What is the maximum bending moment of a fixed-fixed beam with uniform distributed load?

wL²/12 hogging at the supports, which governs, and wL²/24 sagging at midspan.

What is the maximum deflection of a fixed-fixed beam with uniform distributed load?

δmax = wL⁴/(384EI) 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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