Simply Supported Beam with Uniformly Distributed Load

A simply supported beam rests on a pin at one end and a roller at the other, so it can’t resist a moment at either support — only vertical reactions develop. A uniformly distributed load (UDL) applies a constant force per unit length, w, along the full span L. This is the single most common load case in beam design: a floor or roof slab framing uniformly onto a beam looks like this to first approximation.

The results below follow directly from classical elastic (Euler–Bernoulli) beam theory: static equilibrium gives the reactions and shear diagram, and integrating the moment diagram twice (with the two support conditions as boundary values) gives the slope and deflection.

Worked Example

• Span L = 6 m
• Uniformly distributed load w = 10 kN/m
• Flexural rigidity EI = 50,000 kN·m² (assumed, for the deflection/slope example only — reactions and moment don’t depend on EI)

Reactions

Shear

Maximum moment

Slope

Maximum deflection

Assumptions & Limitations

• Assumes constant EI along the full span — a beam with a stepped or tapered section needs a more general method (e.g. virtual work / unit-load, or numerical integration).
• Assumes the load is genuinely uniform over the full span. A UDL over only part of the span, or a linearly-varying (trapezoidal) load, uses different formulas — the Beam calculator handles both directly.
• Small-deflection theory: valid while deflections stay small relative to the span (the normal case for a properly-sized beam) — not valid for a highly flexible member undergoing large deflection.
• This is a hand-verifiable classical result, not a design check against any code — it doesn’t include load factors, material safety factors, or a capacity check. Open it in the Beam calculator to apply load combinations and a stress/deflection check against a real section.