Shelf sag is 5 × load × span⁴ ÷ (384 × E × I): a 36 in shelf of ¾ in grade 130 MDF holding 25 psf sags 0.35 in, about L/104, and fails the L/240 limit. Pick your material, thickness, depth, span and load above and this sagulator shows the sag, the longest span that passes and whether the board is strong enough.
How shelf sag is calculated
A shelf is a small beam. Treat it as a board resting on its two supports with the load spread evenly. Its sag depends on the load, the span, the stiffness of the material (E, the modulus of elasticity) and the stiffness of its cross section (I). Span is the killer: sag grows with the fourth power of the span, so a shelf that is twice as long sags 16 times as much.
Ends only: sag = 5 × w × L⁴ ÷ (384 × E × I). Center support (two equal spans): sag = w × L⁴ ÷ (185 × E × I). I = depth × thickness³ ÷ 12. w = load in pounds per inch of shelf.
Thickness matters even more, because it is cubed: a 1 in shelf is 2.4 times as stiff as a ¾ in one.
The limit is span divided by 240. That is the value the Composite Panel Association uses in its shelving tables: a 24 in span may sag 0.10 in, a 36 in span 0.15 in. It is chosen to keep the sag small over years of load, since particleboard and MDF creep. Where you want the shelf to look flat, use L/360.
Worked example: a 36 in MDF shelf
¾ in grade 130 MDF (E = 313,000 psi), 12 in deep, spanning 36 in between supports, carrying 25 psf. The shelf area is 1 ft × 3 ft, so the load is 75 lb, or 2.08 lb per inch.
- I = 12 × 0.75³ ÷ 12 = 0.422 in⁴.
- Sag = 5 × 2.08 × 36⁴ ÷ (384 × 313,000 × 0.422) = 0.35 in, which is L/104.
- Longest span at L/240: about 27 in. At L/360: about 24 in.
- Strength: the bending stress is 38 percent of the working stress, so the board is not close to breaking. It sags long before it fails.
Swap in a No. 2 spruce-pine-fir or Southern pine board (E = 1,400,000 psi) of the same size and the sag drops to 0.077 in, L/467, because that lumber is 4.5 times as stiff.
The Composite Panel Association example
The association's shelving bulletin works one example: ¾ in grade 130 MDF, 12 in deep, 50 psf. With a center support the longest span of each half is 29.0 in, and the same shelf with supports at the ends only can span 21.6 in. Both match this calculator.
Shelf material E and strength
Modulus of elasticity and modulus of rupture (MOR) for the board grades in the calculator, from the Composite Panel Association (ANSI A208.1 and A208.2, 2016):
| Board | E, psi | MOR, psi |
|---|---|---|
| MDF grade 115 | 180,000 | 1,800 |
| MDF grade 130 | 313,000 | 3,130 |
| MDF grade 155 | 405,000 | 4,050 |
| Particleboard PBU underlayment | 250,200 | 1,595 |
| Particleboard M-2 | 290,100 | 1,885 |
| Particleboard M-3 | 362,600 | 2,176 |
The bulletin uses 25 percent of the MOR as the working stress. Solid wood is stiffer: sugar maple 1,830,000 psi, red oak 1,820,000, white oak 1,780,000, black walnut 1,680,000, yellow-poplar 1,580,000 and black cherry 1,490,000 psi for clear wood (USDA Wood Handbook), and 1,300,000 to 1,600,000 psi for No. 2 softwood lumber (American Wood Council). Clear wood values are for straight-grained boards without knots; a knotty pine board is not as stiff. Plywood is not in the picker because no verified E was available; if you have a value, use the beam deflection calculator with your own E.
How to stop a shelf sagging
Options in order of what they buy you: add a center support (a 29 in half-span replaces a 21.6 in single span in the example above); go one thickness up (¾ to 1 in makes the board 2.4 times stiffer); move to a stiffer material; shorten the span between brackets; or glue and screw a stiffening strip to the front edge. The association also notes that a shelf up to 12 in wide fixed along its rear edge at 6 in intervals may use double the tabulated span.
Next questions
- Sag for other beams, loads and supports: the beam deflection calculator.
- Lumber for the shelves: the board foot calculator and the plywood calculator.
- Stronger horizontal members: the LVL beam calculator.
- More span tools: framing and lumber calculators.
The tables assume raw, unfinished boards, an evenly spread load and do not include the weight of the shelf itself. Concentrated loads such as a TV or a stack of heavy books in one spot need an engineer. A shelf that could hurt someone if it fell needs brackets and fasteners sized to the load: have an engineer or a qualified installer verify it.