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Bending moment sets the stress a beam must resist; the section modulus needed equals the moment divided by the allowable bending stress. Deflection is a separate serviceability check that depends on the moment of inertia and the modulus of elasticity. The formulas below cover the two most common simply supported cases.
- L = 20 ft, w = 500 lb/ft (UDL).
- M = 500 × 20² ÷ 8 = 25,000 lb·ft.
- S = 300,000 ÷ 24,000 = 12.5 in³.
Reference tables
Simply supported beam — max values
| Load | Max moment | Max deflection |
|---|---|---|
| Uniform w | wL²/8 | 5wL⁴/(384EI) |
| Center point P | PL/4 | PL³/(48EI) |
| Two 1/3-point P | PL/3 | — |
A number the tables do not give: a 20 ft simple span at 500 lb/ft needs about a 12.5 in³ section modulus at 24 ksi allowable — roughly a W8×15.
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FAQ
Simply supported or fixed?
These formulas are for simply supported (pinned) ends. Fixed or continuous beams have lower mid-span moments and need different coefficients.
Stress or deflection — which governs?
Either can. Long, lightly loaded beams are often deflection-governed; short, heavily loaded ones are stress-governed. Check both.
What allowable stress do I use?
Fb depends on material and code (e.g., ~0.66Fy for compact steel in ASD). Use the value from AISC/NDS for your member.
Does this replace an engineer?
No — it gives first-pass demand. Real beam design covers lateral-torsional buckling, connections and load combinations under a licensed engineer.
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