Shelf & beam span
"Will this shelf sag?" is a question most woodworkers answer with gut feel. This calculator answers it with beam-deflection math: span, load, species MOE, shelf thickness. You'll see the deflection in inches and a yes/no on whether it's within the typical L/360 limit (≈1/4″ deflection on a 6 ft shelf is usually the threshold of perception).
Calculator
Why deflection, not strength, matters
A shelf "fails" in two ways: it breaks (strength failure) or it sags (stiffness failure). For most residential wood shelves, stiffness failure happens first by a wide margin. A shelf that's strong enough to hold a hundred books without snapping can still annoy you by sagging 3/8″ over six months.
The standard rule is L/360 — the shelf can deflect 1/360th of its span without looking broken. For a 36″ shelf that's 0.1″; for a 6 ft shelf that's 0.2″. L/480 (the calculator's default) is a premium threshold where you genuinely cannot perceive the sag without a straightedge.
What this calculator is doing
It's using Euler-Bernoulli beam theory for a simply-supported beam under uniform load: δ = 5 w L⁴ / (384 E I), where:
- w = load per inch (= lb/ft ÷ 12)
- L = span in inches
- E = modulus of elasticity for the species (in Mpsi × 10⁶)
- I = moment of inertia = b × h³ / 12 (where b = shelf depth, h = thickness)
The thickness enters the formula as h³ — doubling thickness cuts deflection by 8×. That's why a slightly thicker shelf makes a dramatic difference.
Limitations
- Assumes simply-supported (a shelf resting on two side supports). Cantilever or fixed-end conditions change the math.
- Assumes the load is uniformly distributed and static. A heavy amplifier on one spot is much worse than the same weight spread evenly.
- MOE values are averages; specific boards vary ±15%. Deflection is a rule-of-thumb, not a guarantee.
- Doesn't check strength — for typical shelf loads, that's fine. For spans over ~6 ft, verify bending stress too.
- Doesn't model creep — wood under sustained load will sag more over years. If you're building a library, design to L/720 if you can.
Frequently asked questions
Why is the verdict "exceeds" when I have thicker stock?
Probably because span or load is high. Doubling thickness cuts deflection 8×; doubling span increases it 16×. For long spans, even thick shelves will sag — the answer is a centre support, not more wood.
What about plywood shelves?
Plywood is stiffer per unit weight than solid wood in the long-grain direction. The E value is similar (~1.5 Mpsi for construction plywood) but it doesn't crack or check with humidity the way solid wood does. For utility shelves, 3/4″ plywood is hard to argue with.
What about a steel angle or centre support?
Cutting span in half cuts deflection by 16×. A small steel bracket or a centre divider is the cheapest way to fix a sagging shelf. Use this calculator to find the worst-case span, then design around it.