Simply supported centre load
Given: P = 10 kN, L = 4 m, E = 200 GPa, I = 8×10⁻⁶ m⁴
Working: δ = 10000 × 4³ / (48 × 200×10⁹ × 8×10⁻⁶)
Maximum deflection ≈ 8.33 mm.
Structural Engineering
Beam Deflection Formula
Learn common beam deflection formulas for simply supported and cantilever beams, including point-load and UDL cases, with worked examples and a live calculator.
Governing equation
Simply supported, centre point load: δmax = P × L³ / (48 × E × I)
Beam deflection is the displacement of a beam from its unloaded position. Classical elastic formulas depend strongly on span length, loading, Young's modulus and the second moment of area.
Other load/support cases use different coefficients. For example, a cantilever with an end point load uses δmax = P × L³ / (3EI), while a simply supported beam under full-span UDL uses 5wL⁴ / (384EI).
Live calculator
Centre point-load deflection calculator
Simply supported beam with a point load at midspan.
Result
8.333mm
Simply supported beam, centre point load
Formula variables and units
| Symbol | Meaning | Typical SI unit |
|---|---|---|
| δmax | Maximum elastic deflection | m or mm |
| P | Point load | N |
| L | Beam span | m |
| E | Young's modulus | Pa |
| I | Second moment of area | m⁴ |
How to calculate it
- 1Identify the support and loading case.
- 2Keep E and I in units compatible with load and span.
- 3Use the correct deflection coefficient for that case.
- 4Calculate the elastic deflection.
- 5Compare the result with the applicable serviceability limit for the project.
Common mistakes
- Using the wrong support/load formula.
- Mixing mm⁴ and m⁴ for section inertia.
- Entering E in GPa without converting when the equation uses Pa.
- Treating a serviceability calculation as a complete structural design check.
Worked examples
Why span matters
Given: Same beam and load, but span doubles
Working: Point-load deflection varies with L³
Doubling span increases this elastic deflection by about 8 times.
Go deeper
Use the dedicated OneCalcApp tool
The quick calculator above demonstrates the governing equation. Open the related engineering tool for broader inputs, outputs and calculation context where available.
Frequently asked questions
What is the beam deflection formula for a centre point load?
For a simply supported beam with a point load at midspan, δmax = PL³/(48EI).
What is the cantilever beam deflection formula?
For a cantilever with a point load at the free end, δmax = PL³/(3EI).
Why does beam deflection increase so quickly with span?
Many beam deflection equations contain L³ or L⁴, so span length has a strong nonlinear effect.
Does a lower deflection always mean the beam is safe?
No. Strength, stability, local buckling, connections and code-specific checks must also be satisfied.
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