How it works
A shelf is a beam. Under an even load of books it bends like a loaded plank on two supports, and the middle drops. How much depends on the load, the span, and the stiffness of the section. The section stiffness is the wood's modulus of elasticity times the second moment of area, which grows with the cube of the thickness.
That is why span and thickness matter so much: sag rises with the fourth power of the span and falls with the cube of the thickness. Stretching a span from 24 to 36 inches makes the shelf sag about five times as much. Wood also keeps sagging under a constant load over the years, and the Wood Handbook says that after several years the extra creep can roughly equal the first deflection, so the tool can double it.
Worked example
A 3/4 inch thick, 10 inch deep red oak shelf spanning 3 feet on cleats, carrying 25 lb per foot of books:
- I = depth × thickness³ ÷ 12 = 10 × 0.75³ ÷ 12 = 0.3516 in⁴
- E = 2,002,000 psi; w = 25 lb/ft ÷ 12 = 2.083 lb/in; L = 3 ft × 12 = 36 in
- Sag = 5 × w L⁴ ÷ (384 E I)
- = 5 × 2.083 × 36⁴ ÷ (384 × 2,002,000 × 0.3516) = 0.065 in
- Span ÷ sag = 36 ÷ 0.065 = 556 (your limit: span ÷ 360 = 0.1 in)
| Input | Value |
|---|---|
| Span between supports | 3 ft |
| Shelf depth (front to back) | 10 in |
| Shelf thickness | 0.75 in |
| Load per foot of shelf | 25 lb/ft |
| Material | Northern red oak |
| Modulus of elasticity (only used for "Other") | 1000000 psi |
| End conditions | Resting on cleats or pins (simply supported) |
| Time under load | Initial deflection (right after loading) |
| Sag limit: span ÷ | 360 |
| Result | Value |
|---|---|
| Sag at the middle of the shelf | 0.065 in |
| Span ÷ sag | 556 : 1 |
| Allowed sag at your limit | 0.1 in |
| Longest span that meets your limit | 3.47 ft |
| Modulus of elasticity used | 2,002,000 psi |
Stiffness by species
| Species | Handbook MOE (10⁶ psi) | Used here, +10% (10⁶ psi) |
|---|---|---|
| White oak | 1.78 | 1.96 |
| Northern red oak | 1.82 | 2 |
| Bur oak | 1.03 | 1.13 |
| Sugar (hard) maple | 1.83 | 2.01 |
| Red maple | 1.64 | 1.8 |
| Silver maple | 1.14 | 1.25 |
| White ash | 1.74 | 1.91 |
| Green ash | 1.66 | 1.83 |
| Yellow birch | 2.01 | 2.21 |
| Paper birch | 1.59 | 1.75 |
| Black walnut | 1.68 | 1.85 |
| Black cherry | 1.49 | 1.64 |
| American elm | 1.34 | 1.47 |
| Hackberry | 1.19 | 1.31 |
| Shagbark hickory | 2.16 | 2.38 |
| Black locust | 2.05 | 2.26 |
| Quaking aspen | 1.18 | 1.3 |
| American basswood | 1.46 | 1.61 |
| Eastern cottonwood | 1.37 | 1.51 |
| Yellow-poplar | 1.58 | 1.74 |
| Eastern white pine | 1.24 | 1.36 |
| Red pine | 1.63 | 1.79 |
| Tamarack | 1.64 | 1.8 |
| Douglas-fir (coast) | 1.95 | 2.15 |
| Western redcedar | 1.11 | 1.22 |
| Balsam fir | 1.45 | 1.6 |
| White spruce | 1.43 | 1.57 |
Show the stiffness by species table
| Species | Handbook MOE (10⁶ psi) | Used here, +10% (10⁶ psi) |
|---|---|---|
| White oak | 1.78 | 1.96 |
| Northern red oak | 1.82 | 2 |
| Bur oak | 1.03 | 1.13 |
| Sugar (hard) maple | 1.83 | 2.01 |
| Red maple | 1.64 | 1.8 |
| Silver maple | 1.14 | 1.25 |
| White ash | 1.74 | 1.91 |
| Green ash | 1.66 | 1.83 |
| Yellow birch | 2.01 | 2.21 |
| Paper birch | 1.59 | 1.75 |
| Black walnut | 1.68 | 1.85 |
| Black cherry | 1.49 | 1.64 |
| American elm | 1.34 | 1.47 |
| Hackberry | 1.19 | 1.31 |
| Shagbark hickory | 2.16 | 2.38 |
| Black locust | 2.05 | 2.26 |
| Quaking aspen | 1.18 | 1.3 |
| American basswood | 1.46 | 1.61 |
| Eastern cottonwood | 1.37 | 1.51 |
| Yellow-poplar | 1.58 | 1.74 |
| Eastern white pine | 1.24 | 1.36 |
| Red pine | 1.63 | 1.79 |
| Tamarack | 1.64 | 1.8 |
| Douglas-fir (coast) | 1.95 | 2.15 |
| Western redcedar | 1.11 | 1.22 |
| Balsam fir | 1.45 | 1.6 |
| White spruce | 1.43 | 1.57 |
Modulus of elasticity at 12% moisture, USDA Wood Handbook Table 5-3b. The Handbook says bending stiffness is the tabulated value increased by 10%.
Assumptions and limits
- Uniform load along the whole shelf, a shelf that rests on its supports, and a solid wood section of constant size.
- Modulus of elasticity is the Wood Handbook average at 12% moisture content, increased by 10% as the Handbook advises when shear deflection is excluded. Individual boards vary a lot.
- "Fixed ends" assumes perfectly rigid joints, which glued dados only approach. Use simple supports to be safe.
- Not included: the shelf's own weight, concentrated loads, strength failure, or sagging at the supports. Plywood and MDF are entered as "Other" with the maker's modulus.
Common questions
How thick should a bookshelf be?
At 25 lb per foot on a 3 foot span, a 3/4 inch red oak shelf 10 inches deep sags about 0.065 inch at first and about 0.13 inch (span ÷ 278) after years under load. A 1 inch shelf sags about 42% as much. Length matters more: a 4 foot span sags about 3.2 times the 3 foot value.
How much can a shelf sag before it looks bad?
There is no code limit for shelves, so it is a judgment. Span ÷ 360 is the familiar floor live-load limit in the residential code and makes a conservative goal for a shelf that will carry weight for years.
Why does a shelf sag more over time?
Wood creeps under constant load. The Wood Handbook notes that after several years the added deformation may roughly equal the initial one. Temperature and humidity swings increase creep.
How do I stiffen a sagging shelf?
Add a center support, glue a hardwood edge strip on the front (a face-frame or cleat on edge multiplies stiffness), reduce the span, or use a thicker board.
Sources
- USDA Forest Products Laboratory, Wood Handbook: Wood as an Engineering Material (FPL-GTR-282, 2021): Table 5-3b (modulus of elasticity at 12% moisture content and the note on shear deflection); Chapter 5, "Creep and Relaxation".
- Beam deflection formulas for a uniformly loaded beam: 5wL⁴/384EI (simple supports) and wL⁴/384EI (fixed ends), from standard structural mechanics.
- Read them: USDA Wood Handbook, Chapter 5 (Table 5-3b and creep)
Updated 2026-09-30