Simply Supported Beam Calculator
Enter the center load, span and rectangular section to get midspan deflection (δ = PL³/48EI), max stress and support reactions — with a utilization check.
Last updated: 2026-09-28
How the calculation works
- Both supports carry half the load; the peak moment sits at midspan where the load applies.
- Deflection uses δ = PL³/48EI — the simply-supported companion of the cantilever's PL³/3EI.
- Stress is checked against 250 MPa typical steel yield for a utilization read.
Formula
δ = P·L³ / (48·E·I) M_max = P·L / 4 σ = M·c/I I = w·h³ / 12
| Symbol | Meaning | Unit |
|---|---|---|
δ | Midspan deflection | mm |
M | Max bending moment at midspan | N·m |
E | Young's modulus — 200 GPa (steel) | GPa |
Worked example
Interpreting the result
Support conditions dominate stiffness — both-ends-supported is 16× stiffer than cantilevered, which is why floors span between walls instead of cantilevering from one. The L³ law still rules: double the span and deflection multiplies 8×, the reason long unsupported spans need engineered sections. As with the cantilever, serviceability (L/360) usually binds before stress does.
Assumptions
- Linear-elastic steel, rectangular section, point load at exact midspan.
- Self-weight excluded; lateral-torsional buckling not checked.
Limitations
- Distributed loads and off-center loads use different equations (δ = 5wL⁴/384EI for uniform).
- No shear deflection on short deep spans; no buckling check on tall thin sections.
Frequently asked questions
What is the deflection formula for a simply supported beam?
For a center point load: δ = P·L³ ÷ (48·E·I). For a uniform load: δ = 5wL⁴ ÷ (384·E·I). Both are linear-elastic midspan values.
Why is a supported beam stiffer than a cantilever?
Geometry of bending: the cantilever's PL³/3EI versus the supported beam's PL³/48EI — 16× less deflection for identical load and section. Both ends held means both ends resist.