Beam deflection analysis is fundamental to structural and mechanical design. Whether you're sizing a machine frame, designing a lifting beam, or checking a shelf bracket, understanding how loads create deflection and stress prevents failures and over-engineering.
Key Beam Properties
Three properties determine beam stiffness:
- E (Modulus of Elasticity): Material stiffness. Steel ≈ 29×10⁶ psi, aluminum ≈ 10×10⁶ psi
- I (Moment of Inertia): Cross-section's resistance to bending. Larger = stiffer
- L (Length): Span between supports. Deflection increases dramatically with length
Common Beam Cases
Simply Supported, Center Load
δ_max = PL³ / 48EI
Where P is the concentrated load, L is span, E is modulus, I is moment of inertia.
Simply Supported, Uniform Load
δ_max = 5wL⁴ / 384EI
Where w is load per unit length.
Cantilever, End Load
δ_max = PL³ / 3EI
Cantilevers deflect much more than simply supported beams of the same span.
Calculate Beam Deflection Instantly
Enter load, span, and beam properties to get deflection and stress for common beam configurations.
Moment of Inertia for Common Shapes
| Shape | I Formula |
|---|---|
| Rectangle (bh) | bh³/12 |
| Circle (diameter D) | πD⁴/64 |
| Tube (OD, ID) | π(OD⁴-ID⁴)/64 |
| I-beam | Use tables or CAD |
Orientation matters: a 2×4 on edge has 8× the stiffness of a 2×4 laid flat.
Bending Stress
Maximum bending stress occurs at the outer fibers:
σ = Mc/I
Where M is maximum bending moment, c is distance from neutral axis to outer fiber, I is moment of inertia.
For a simply supported beam with center load:
M_max = PL/4
Design Limits
Deflection Limits
- L/360: Floor beams (prevents plaster cracking)
- L/240: General structural
- L/180: Industrial, roof members
- Machine frames: Often 0.001-0.005" depending on precision
Stress Limits
Keep bending stress below the yield strength divided by safety factor:
- Steel: σ_allow ≈ 0.6 × Fy (36 ksi for A36 → 22 ksi allowable)
- Aluminum: σ_allow ≈ 0.5 × Fy (varies by alloy)
Practical Example
A 4' shelf using 1" × 2" steel tubing (0.065" wall) with 100 lb center load:
- I ≈ 0.13 in⁴ (from tube tables)
- E = 29×10⁶ psi
- L = 48"
- δ = (100 × 48³) / (48 × 29×10⁶ × 0.13) = 0.061"
L/360 = 48/360 = 0.133", so this passes the deflection check with margin.
Increasing Stiffness
When deflection is excessive, you have options:
- Increase depth: I increases with h³—most effective
- Reduce span: Add supports; deflection varies with L³ or L⁴
- Change material: Steel is ~3× stiffer than aluminum
- Add bracing: Tube or box sections are stiffer than open sections
Common Mistakes
- Wrong axis: Using I about the wrong axis for the loading direction
- Unit confusion: Mixing inches and feet, or lb and kips
- Support assumptions: Real connections aren't perfectly pinned or fixed
- Ignoring shear: Short, deep beams may be shear-critical
Beam calculations become intuitive with practice. For critical applications, verify hand calculations with structural software or engineering review, especially when safety is involved.