Shelf Sag (deflection) Calculator

Understanding the shelf sag calculator — concept and principle

This calculator predicts how much a shelf bends in the middle when loaded evenly, such as a row of books. Enter the span between supports, the shelf depth and thickness, the board material and the load; it returns the sag in millimetres, the sag per metre of span and a simple verdict. Use it to choose board thickness, material or bracket spacing before you build.

30 kg on an 800 mm chipboard span sags 4.6 mm — more than the 1 mm per metre guide.

How it is calculated

The shelf is treated as a simply supported beam with a uniform load: δ = 5wL⁴ ÷ (384EI).

w = load in kg × 9.81 ÷ span, the load per millimetre in N/mm.
L = span in mm.
E = stiffness (modulus of elasticity) of the material in N/mm² (MPa); 3,500 MPa is 3.5 GPa.
I = depth × thickness³ ÷ 12, the second moment of area in mm⁴.

The verdict is "acceptable" when sag ÷ span in metres is 1 mm/m or less.

The principle

A loaded board bends until its internal stresses balance the load. How far it bends depends on stiffness E (a material property) and on I (a shape property). Two powers dominate: sag grows with the fourth power of span and falls with the cube of thickness. Doubling the span makes sag 16 times larger, while doubling the thickness cuts it to one eighth. Depth only enters linearly, so a deeper shelf helps less than a thicker one. Wood and wood-based panels also creep under long-term load, so sag can grow by 1.5–2 times over months.

Worked example

Inputs: span 800 mm, depth 250 mm, thickness 18 mm, chipboard E = 3,500 MPa, load 30 kg.
I = 250 × 18³ ÷ 12 = 121,500 mm⁴.
w = 30 × 9.81 ÷ 800 ≈ 0.368 N/mm.
δ = 5 × 0.368 × 800⁴ ÷ (384 × 3,500 × 121,500) ≈ 4.61 mm.
Per metre: 4.61 ÷ 0.8 ≈ 5.77 mm/m, above 1 mm/m, so the verdict is visible sag — add support or thicker board.

Each point is the calculator re-run with only “Thickness (mm)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Thickness (mm)	Sag (mm)
4.5	295.28
9	36.91
13.5	10.94
18	4.61
22.5	2.36
27	1.37
36	0.577

How to read the result

Where does your result fall?

Sag above about 1 mm per metre is often noticeable along a row of books. In the example, a 25 mm board cuts sag to about 1.7 mm, and adding a middle support (two 400 mm spans with half the load each) cuts it to roughly one sixteenth. Plywood or solid timber, with two to three times the stiffness of chipboard, helps a lot as well.

Common mistakes

Using the total shelf length instead of the clear span between supports.
Ignoring creep: a shelf that looks fine on day one can sag more after months.
Assuming the formula covers point loads or cantilevers; those use different formulas.

Keep in mind: This is an elastic beam estimate with typical material values; real boards vary with moisture and grade, and the calculation does not check bracket or wall-fixing strength. For heavy or safety-critical loads, consult a qualified professional.

Frequently asked questions

Does a front edge strip reduce sag?

Yes. A solid timber lip glued on edge adds a lot of I because its height is cubed.

Is MDF stiffer than chipboard?

Slightly; typical values are about 4 GPa versus 3.5 GPa, but both creep more than plywood.

What load should I assume for books?

Weigh a sample length; densely packed books can exceed 20 kg per metre.

Sources: J. M. Gere and B. J. Goodno, Mechanics of Materials (beam deflection tables) · USDA Forest Products Laboratory, Wood Handbook (FPL-GTR-190)

Checked 8 Nov 2026. Typical coverage rates and standard formulas. Always check the product label and local building rules; buy 5–10% extra for waste.

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