How it’s calculated
For a prismatic, simply supported beam (pin at one end, roller at the other) the classic elastic beam formulas give the mid-span deflection, the maximum moment and the reactions. Assumptions: linear-elastic material, small deflections, constant E and I along the span, load in the plane of bending, no lateral-torsional buckling and negligible shear deformation.
Example: a steel beam (E = 200 GPa) spanning 6 m with I = 8,000 cm⁴ (8.0 × 10⁻⁵ m⁴) carries 10 kN/m. δ = 5 × 10,000 × 6⁴ ÷ (384 × 200 × 10⁹ × 8.0 × 10⁻⁵) = 0.01055 m ≈ 10.5 mm, which is L/569, so it passes an L/360 limit (16.7 mm). Mmax = 10 × 6² ÷ 8 = 45 kN·m and each reaction is 30 kN.
Deflection limits such as L/360 (floors under live load) and L/240 (total load) come from building codes (e.g. IBC Table 1604.3). Deflection is only a serviceability check: the beam must also be checked for bending and shear strength, bearing and stability by a qualified engineer.
Frequently asked questions
What does L/360 mean?
The allowable deflection is the span divided by 360. A 12 ft (144 in) floor joist limited to L/360 may deflect at most 144 ÷ 360 = 0.4 in under live load.
Which E should I use for lumber?
Use the design modulus of elasticity for the species and grade from your lumber design tables. Clear-wood values in the USDA Wood Handbook are averages for defect-free specimens and are higher than graded lumber values.
Does this include the beam’s self-weight?
Only if you add it. Include self-weight in w for the uniform case. For the point-load case, superpose the two results (deflections add linearly in elastic beams).
Why is deflection so sensitive to span?
Deflection grows with L⁴ for a uniform load and L³ for a point load, so doubling the span makes a uniformly loaded beam deflect 16 times as much.
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Sources
- Beam Design Formulas with Shear and Moment Diagrams (Design Aid No. 6) — American Wood Council
- Mechanics of Materials (Roylance), 4.3 Beam Displacements — Engineering LibreTexts
- University Physics Vol. 1, 12.3 Stress, Strain, and Elastic Modulus (Table 12.1) — OpenStax
- Wood Handbook, Chapter 5: Mechanical Properties of Wood (Table 5-3a) — USDA Forest Service, Forest Products Laboratory
- IBC 2021, Section 1604.3 Serviceability (Table 1604.3 deflection limits) — International Code Council
Formulas are taken from the free public references above. Results are provided “as is” for informational and educational purposes only. See our disclaimer.
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