Shelf sag calculator

How far a shelf will sag under books or anything else spread along it: when you first load it, and after years under the load. Pick a wood species or a panel, give the span, depth and thickness, and see the answer against the sag most people notice.

How the sag is worked out

A shelf resting on two supports with its load spread along it is a simply supported beam under a uniform load. The USDA Wood Handbook gives its deflection at midspan as 5wL⁴ ÷ (384EI) (Eq. 9-2 with Table 9-1): w is the load per inch, L the span, E the material's stiffness (modulus of elasticity) and I the moment of inertia of the shelf's cross section, depth × thickness³ ÷ 12. Thickness is cubed, so a shelf twice as thick sags an eighth as much; span is to the fourth power, so a shelf a quarter longer sags about two and a half times as much.

Wood species use the Handbook's stiffness at 12% moisture content (Tables 5-3b and 5-5b). Those figures already include a little shear deflection, measured at a span 14 times the depth, so for long, thin shelves the answer leans slightly high. Panels use their makers' own data sheets: the median of three makers for MDF, of four for particleboard, the average of two for multi-ply plywood, and one maker's panel for veneer-core plywood, each shown with its range.

Sag after years under load

Wood and wood panels keep sagging slowly under a load that stays. The Handbook says that at ordinary loads and conditions, after several years the added creep may approximately equal the first sag (p. 5-39), so the long-term answer here is twice the first. It notes that designers have customarily allowed about one and a half times for dry wood (p. 9-4), and that heat and humidity raise creep. Unloading the shelf brings back the first sag at once and about half the creep over time.

How much sag shows

There is no standard for the sag people notice. Two sources put it at about 1/32 in per foot of span: Harris Hyman, P.E., writing in the Journal of Light Construction in 1995, took about 1/32 in per foot as the acceptable limit for shelves, and WoodBin says the eye notices 1/32 in per running foot. The result says which side of that line your shelf falls on, at first and after years, so you can judge.

The books preset is 30.5 lb per foot: WoodBin weighed hardcover books at 20 to 32 lb per foot, and Hyman weighed engineering textbooks at 35. Paperbacks weigh far less (WoodBin measured 6 lb per foot); weigh a foot of your own if the number matters.

A Sagulator alternative

WoodBin's Sagulator is the best-known shelf sag calculator. By its own description it computes the initial sag, suggests a target of 0.02 in per foot or less to allow for wood sagging about 50% more over time, and uses average stiffness of clear, straight-grained wood, mainly from the Forest Products Laboratory. This calculator works from the same beam equation; it shows the sag after years as well as at first, names the source behind every material's stiffness on the result, and keeps your inputs in the page address and your browser so a reload or a shared link opens with the same numbers. Coming from it? See where each of its inputs is here.

More on the woods themselves: the types of wood and each species page, and the Janka hardness chart.