Shelf sag calculator

What changes sag

A shelf is a small beam. The calculator uses the standard deflection formula for a beam resting on two supports with the load spread along it: sag = 5 x W x L³ / (384 x E x I), where W is the total load, L the span, E the stiffness of the material and I depends on the board's cross-section (depth x thickness³ / 12 for a plain board). That gives a few rules worth knowing before you buy wood:

• Span matters most. With the load per foot kept the same, sag grows with the fourth power of the span. A shelf 25% longer sags about 2.4 times as much. Adding a support in the middle cuts each span in half.
• Thickness is next. Sag falls with the cube of thickness. Going from 3/4 in to 1 in cuts it to about 42% of what it was.
• Depth helps in proportion. A shelf twice as deep is twice as stiff, but it usually carries more, too.
• Species changes it by its stiffness. Hard maple is about 1.5 times as stiff as eastern white pine, so the same shelf sags about two-thirds as much.
• An edge strip on its side works like a deeper board. A 1-1/2 in strip glued under the front edge adds a lot of stiffness for very little wood. Try it in the calculator.

Wood also keeps sagging slowly under a load that stays on it, which is called creep. The USDA Wood Handbook says that for dry timber it has been customary to design the first deflection at about two-thirds of the long-term limit. The "after years" figure applies that rule, so it is about 1.5 times the first sag.

Stiffness by species

Modulus of elasticity in bending, from small clear samples at 12% moisture content. Higher means stiffer, so less sag. Real boards with knots, slope of grain or a higher moisture content can be less stiff than this.

Sheet goods are much less stiff. The handbook reports plywood sheathing at 1.01 to 1.24 million psi when the face grain runs along the span, particleboard at 0.40 to 0.58 and MDF at 0.43 to 0.64. The calculator uses the low end of each range. Hardwood-veneer cabinet plywood is not in those tables, so treat its result as a rough guide.

What it does not tell you

• It estimates bending, not whether anything will break or pull out. Shelf pins, brackets, screws, cleats and wall fixings each have their own limits set by their makers. Follow those. Our floating shelf guide covers wall fixings.
• A shelf housed in dadoes or screwed to a cabinet back sags less than this, so the number is on the safe side for fixed shelves.
• Particleboard and MDF creep more than solid wood, so for those the "after years" figure is a minimum, not an estimate.
• A load bunched in the middle sags a shelf more than the same weight spread along it.
• Weigh what will go on the shelf. Put a foot's worth of your books on a bathroom scale rather than guessing.

Use it with your plan

Once you know the board size, put the parts into the cut-list calculator, and check the build with the project readiness checklist. Gluing an edge strip on well is covered in gluing and clamping.

Sources, privacy and corrections

Stiffness values, the deflection formula (Equation 9-2 and Table 9-1) and the creep rule come from the USDA Forest Products Laboratory, Wood Handbook: Wood as an Engineering Material (FPL-GTR-282, 2021): Table 5-3b for solid wood and Tables 12-2, 12-4 and 12-6 for plywood, particleboard and MDF. The calculator uses the bending term only and the handbook's tabulated stiffness as given, which slightly overstates sag. It runs only in your browser; nothing is uploaded. Report a problem through the contact page.