Shelf Span Calculator: Estimated Shelf Sag on Two Supports

One model only: a rectangular shelf on two supports, free to turn at each end — a shelf lying on pins or cleats near each end — with the load either spread evenly along it or concentrated at the centre. For that model, elastic beam theory gives the sag at mid-span exactly. Real shelves differ, and the calculator says what it leaves out.

Worked example: a 36 in span of northern red oak (E = 1.82 × 10⁶ lbf/in²), 10 in deep and 3/4 in thick, with 100 lb spread along it. I = 10 × 0.75³ ÷ 12 = 0.3516 in⁴, and δ = 5 × 100 × 36³ ÷ (384 × 1,820,000 × 0.3516) = 0.095 in — about 3/32 in, or L/379. With the same 100 lb at the centre, δ is 8/5 of that, 0.152 in.

• Thickness. I grows with the cube of the thickness: a shelf twice as thick is eight times as stiff in this model.
• Span. For the same load per inch, sag grows with the fourth power of the span — double the span and the sag is sixteen times as much; for the same total load, eight times.
• Depth. I grows in step with depth, so a shelf twice as deep is twice as stiff — but a deeper shelf usually carries more, so keep the load you enter realistic.
• Load. Sag is in proportion to the load: twice the load, twice the sag.
• Species. Stiffness varies from 1.24 × 10⁶ lbf/in² for eastern white pine to 2.01 × 10⁶ for yellow birch among the twelve listed — a smaller effect than thickness or span.

The span ÷ sag figure is shown for description only. Ratios such as L/240 or L/360 come from building design for floors and beams; no furniture standard is cited here that makes any ratio a pass or a fail for a shelf.

The twelve species figures are the Wood Handbook's Table 5–3b values for clear wood at 12% moisture content. The handbook measured them on centre-loaded beams at a span 14 times the depth, so they include some shear deflection; it says the figure can be raised 10% to remove it, which is not done here. Real boards contain knots, slope of grain and other variation that clear specimens do not. Plywood, MDF and other panels have no preset: their stiffness is not a solid-wood figure, so choose 'Other material' and enter the value from the panel maker's data.

• Creep. The Wood Handbook explains that wood under a sustained load keeps deforming; at typical design levels, after several years the added sag may roughly equal the initial elastic sag, and swings in temperature and humidity increase it. A loaded bookshelf is a sustained load. The calculator gives the first, short-term figure only.
• Strength. Whether the shelf, its pins, cleats, screws, joints or wall fixings hold the load is not calculated; low sag is not a safe load.
• Other supports. A cantilevered or floating shelf, and shelves whose ends are glued or fixed in dadoes, behave differently from this model.
• The shelf's own weight, knots and other defects, local crushing at the supports, and tipping of the whole bookcase.

Among BenchRoot's plans, the adjustable shelves of the built-in bookcase rest on pins near each end, which this model approximates — the pins' own strength is not modelled. The fixed shelves of the bookshelf sit in dadoes and the floating shelf is held on a hidden cleat; neither is the model here. How to plan and build case furniture is covered in furniture making.