Beam Deflection Formula
Reference for beam deflection formulas for simply supported, cantilever, and fixed beams.
Covers δ = FL³/48EI for point and distributed loads with examples.
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
Key Notes
- Simply supported beam, center load: δ = FL³ / (48EI): F is applied force, L is beam span, E is Young's modulus, and I is the second moment of area of the cross-section. Deflection scales as L³ — doubling the span increases deflection eightfold.
- Cantilever beam, end load: δ = FL³ / (3EI): A cantilever deflects 16× more than a simply supported beam of the same span and load (48 vs 3 in the denominator). This is why cantilever structures require much stiffer members.
- Second moment of area I governs stiffness: For a rectangular section, I = bh³/12. Doubling the height triples deflection reduction. I-beams concentrate material far from the neutral axis to maximize I for minimum weight — the same mass arranged as an I-beam is far stiffer than a solid rectangle.
- E × I = flexural rigidity: The product EI is the beam's total resistance to bending. A steel beam (E ≈ 200 GPa) with the same I as an aluminum beam (E ≈ 69 GPa) is approximately 3× stiffer. Both E and I must be considered together.
- Building code deflection limits: Codes typically limit floor beam deflection to L/360 under live load (prevents cracking finishes), and L/240 under total load. A 6 m floor beam may deflect no more than 6000/360 ≈ 16.7 mm under occupancy loads.