Beam Deflection Formula
Calculate how much a beam bends under a load.
Essential for structural engineering and construction design.
The Formula
δ = PL³ / (48EI) (center load, simply supported beam)
Beam deflection tells you how much a beam bends under an applied load. The formula varies based on support conditions and load type. This is the most common case.
Variables
| Symbol | Meaning |
|---|---|
| δ | Maximum deflection at the center (meters) |
| P | Applied load at the center (Newtons) |
| L | Length of the beam (meters) |
| E | Young's modulus of the beam material (Pa) |
| I | Second moment of area (moment of inertia) of the cross section (m⁴) |
Example 1
A 4 m steel beam (E = 200 GPa, I = 8.33 × 10⁻⁶ m⁴) supports 10 kN at the center
δ = (10,000 × 4³) / (48 × 200 × 10⁹ × 8.33 × 10⁻⁶)
δ = 640,000 / 79,968,000
δ ≈ 0.008 m = 8 mm
Example 2
Same beam but 6 m long instead of 4 m. How much more deflection?
δ = (10,000 × 6³) / (48 × 200 × 10⁹ × 8.33 × 10⁻⁶)
δ = 2,160,000 / 79,968,000
δ ≈ 0.027 m = 27 mm
3.375 times more deflection (deflection scales with L³)
When to Use It
Use the beam deflection formula when:
- Checking if a beam meets deflection limits in building codes
- Sizing beams for floors, bridges, and platforms
- Comparing the stiffness of different beam materials
- Preventing excessive bending that could damage finishes or cause vibrations