Simply Supported Beam with Uniform Load
The first beam everyone learns and the one that still covers most floor framing: a single span, simply supported, carrying a uniform load.
The equation
Vmax = R = wL/2 at each support
δmax = 5wL4/384EI at midspan
- w is the load per unit length, in kN per meter
- L is the span between supports, in meters
- E is the modulus of elasticity of the material
- I is the second moment of area of the section
- δ is the vertical deflection
Worked example
Take a 6 m span carrying 10 kN/m, with E = 200 GPa and I = 8.0 × 107 mm4, so EI = 16000 kNm2. Each reaction is 10 × 6/2 = 30 kN. The midspan moment is 10 × 62/8 = 45 kNm. Deflection is 5 × 10 × 64/(384 × 16000) = 0.0105 m, call it 10.5 mm, or span over 569. A span over 360 limit would allow 16.7 mm, so this section passes with a third in hand. The same case run through an independent numerical solve lands on the same numbers.
When to use it
Elastic behaviour, a prismatic section, small deflections, and shear deformation ignored, which is fine once the span passes about ten times the depth. Real seats are never perfectly frictionless pins, so measured deflections usually come in below the formula: treat it as a safe upper bound. Once there is more than one span or the load pattern moves, the continuous analyzer is the honest tool.
Related Calculators and Reference
- Beam Deflection Studio: run this case with your own numbers and see the diagrams.
- Continuous Beam Analyzer: when one span becomes several.
- Section Properties Calculator: get I for the section you actually have.
- Fixed Beam with Uniform Load: the same load with clamped ends, a fifth of the deflection.
- Steel Beam Design Calculator: turn the 45 kNm into a section.
- American Institute of Steel Construction: shapes and design specifications.