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Calculate beam deflection and slope for various beam types and loading conditions. Determine if your beam meets deflection limits (L/360, L/240, L/180) and get detailed analysis.
I = (b × h³) / 12 for rectangular sections
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Beam deflection is the vertical displacement of a beam under load. Even if a beam is strong enough not to break (adequate bending stress), it can still sag or bend excessively, causing various problems.
Deflection is measured in inches or millimeters and is typically limited to a fraction of the beam span (L/360, L/240, etc.) to ensure:
| Case | Formula | Max Deflection |
|---|---|---|
| Simply Supported - Point Load (Center) | δ = (P × L³) / (48 × E × I) | At center |
| Simply Supported - UDL | δ = (5 × w × L⁴) / (384 × E × I) | At center |
| Cantilever - Point Load (End) | δ = (P × L³) / (3 × E × I) | At free end |
| Cantilever - UDL | δ = (w × L⁴) / (8 × E × I) | At free end |
| Fixed-Fixed - Point Load (Center) | δ = (P × L³) / (192 × E × I) | At center |
| Fixed-Fixed - UDL | δ = (w × L⁴) / (384 × E × I) | At center |
Where: P = Point Load (lbs), w = Distributed Load (lbs/ft), L = Beam Span (in), E = Elastic Modulus (psi), I = Moment of Inertia (in⁴)
Standard deflection limits for various applications and building codes:
| Application Type | Deflection Limit | Description |
|---|---|---|
| Residential Floors | L/360 | Standard residential floor beams |
| Roof Beams | L/240 | Roof and attic trusses |
| Industrial Floors | L/480 | Industrial and machinery areas |
| Cantilever Beams | L/240 | Cantilever and overhang sections |
| Stiff Beams | L/180 | Very rigid structures requiring minimal deflection |
L/360 example: For a 12-foot span, L/360 = 144 inches ÷ 360 = 0.4 inches maximum deflection allowed
Deflection formula: δ = (Coefficient × Load × L⁴) / (E × I)
L/360 means the maximum deflection should not exceed the beam span divided by 360. For example, a 12-foot (144-inch) span can deflect no more than 144÷360 = 0.4 inches. This is the standard for residential floors.
Even if a beam is structurally strong, excessive deflection can cause cracked drywall, stuck doors and windows, sloped floors, cracked tile, and other cosmetic damage. Deflection limits ensure user comfort and prevent secondary damage.
Standard limits are L/360 for floors (0.33 inches per foot of span) and L/240 for roofs (0.5 inches per foot). Some sensitive applications may require L/480 or L/600. Always check local building codes for your specific application.
Yes, most building codes require deflection checks for all beams. Both bending stress and deflection must be within acceptable limits. A beam that's strong enough (stress-wise) might still be too flexible (deflection-wise).
Deflection is the vertical displacement (how much the beam sags), measured in inches. Slope is the angle at which the beam deflects, measured in radians or degrees. Both should be limited for proper design.
Deflection increases with the fourth power of span (L⁴). Doubling the span increases deflection by 16 times! This is why longer spans require much larger beams and why intermediate supports are so important.
No, they are independent considerations. A beam might be strong enough for the load (passes stress check) but too flexible (fails deflection check). You need to pass both checks - adequate strength AND adequate stiffness.
For rectangular sections: I = (b × h³) / 12, where b is width and h is height. For standard lumber and beams, use published tables. Many manufacturers provide I-values in their product specs. Use the axis about which bending occurs (usually vertical).
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