The formula
sag = (5 × W × L³) / (384 × E × I)
W is total load in pounds. L is span between supports in inches. E is modulus of elasticity for the species in psi (red oak: 1,820,000). I is moment of inertia for the cross-section: I = (b × h³) / 12 for a rectangular shelf b wide and h thick.
Example: 30" red oak shelf, 60 lb load
3/4" red oak shelf, 12" deep, 30" span, 60 lbs of books spread evenly. I = (12 × 0.75³) / 12 = 0.422 in⁴. Sag ≈ (5 × 60 × 30³) / (384 × 1,820,000 × 0.422) ≈ 0.028 inches — barely visible. Double the span to 60" and the cube on length pushes sag to 0.45" — clearly bowed. Span dominates the equation.
Why it matters
About 1/8" of sag in a 36-inch span is the eye’s threshold — past that, the shelf looks like it’s failing even if it isn’t. Dead On includes MOE for all 30 species, so you can trade species, thicken the stock, or shorten the span until the deflection number lands below visible.
Step-by-step
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1
Pick the species
MOE varies by species — red oak 1.82M, soft maple 1.64M, southern yellow pine 1.80M, mahogany 1.40M.
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2
Measure the span and section
Span is the unsupported distance. Section is the shelf’s width and thickness.
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3
Estimate the load
Books average 1 lb per linear inch of shelf. Stoneware and tools weigh more — estimate generously.
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4
Calculate moment of inertia
I = (width × thickness³) / 12. Thickness cubes — that’s why doubling thickness reduces sag eight-fold. -
5
Apply the formula
Plug all four into the sag equation. Anything under 1/8" over a 36" span is fine; over 1/4" is visibly bowed.
Skip the math. Build the diagram.
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Download Dead On — FreeFrequently asked questions
How much sag is too much?
About 1/8" over a 36" span starts to look bad. Building codes for floor joists allow L/360 (1/120 of the span) under live load — that’s about 1/10" for 36" — but for shelves the bar is mostly visual.
Why does span matter more than load?
Because span is cubed in the formula. Doubling the load doubles the sag; doubling the span multiplies sag by eight.
Does plywood sag the same as solid wood?
Plywood’s MOE is lower than the solid species the face veneer comes from — cross-grain plies stiffen but cross-plies also reduce average MOE. Dead On has plywood-specific values for 3/4" and 1/2" sheets.
Can I add a center support to fix sag?
Yes — halving the span quarters the sag, because the math is on individual span lengths, not total run. A single dado or shelf pin in the middle is the easiest fix on a long bookshelf.