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Support Reaction — Simple Beam, Uniform Load (R = wL/2)

Also known as simply supported beam uniform load reaction · wL over 2 · end reaction uniformly distributed load · UDL beam reaction · half the total load · maximum shear simple beam

R=wL2R = \frac{w L}{2}

Worked example: 12 kN/m over 6 m → 36 kN at each support — press Try an example to run it live, then adjust anything.

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See also
Max Bending Moment — Uniform Load — the moment that goes with these reactions, wL²/8 at midspanBeam Deflection — Simply Supported, Uniform Load — the deflection of the same beam under the same loadBeam Reaction by Moment Equilibrium — Two Point Loads — the same ΣM = 0 argument when the load arrives as discrete point loads insteadFixed-End Moment — Fixed-Fixed Beam, Uniform Load — what happens to the same load when the ends are built in rather than pinned

Learning zone

The whole load on a simply supported beam under a uniformly distributed load is wLwL. The supports are identical and the load is symmetric, so each takes half: R=wL/2R = wL/2. A 6 m span carrying 12 kN/m puts 12×6/2=3612 \times 6/2 = 36 kN into each end. In field units, 500 lbf/ft over 20 ft is 10 000 lbf total and 5000 lbf at each bearing.

It is worth being clear about what does the work, because the formula is so short that people generalise it without noticing. Equilibrium alone only tells you the two reactions sum to wLwL. It is symmetry that splits them evenly, and symmetry needs the load to run the full span at constant intensity on exactly two supports. A partial-length load, a triangular load, an overhang past a support, or a third support anywhere: all of them break it, and then you write ∑M=0\sum M = 0 about one support and solve properly.

The same number is the maximum shear in the beam, at the face of each support, which is why it is the figure that sizes the bearing length, the end connection and the web crippling check. Follow it through to the moment and you get wL2/8wL^2/8 at midspan — note that the reaction is linear in LL while the moment is quadratic, so doubling a span doubles the connection force and quadruples the bending demand. That asymmetry is the reason long spans are governed by moment and deflection while short heavily loaded ones are governed by shear.

Support Reaction — Simple Beam, Uniform Load (R = wL/2)
R=wL2R = \frac{w L}{2}
Where
  • RR= Reaction at each support (N)
  • ww= Uniform load per unit length (N/m)
  • LL= Span (m)
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