Shelf span and sag calculator

Shelf span and sag calculator#

Enter the clear span, the board and the load. The calculator returns the sag on the day it is loaded and the sag years later once the panel has crept, and checks each against its own limit.

Inches Centimetres

Clear span between supports in

Shelf depth in

MaterialMelamine-faced chipboardMDFBirch plywoodSoftwood, pineHardwood: oak, ash, maple

Thickness5/8 in (16 mm)3/4 in (18 mm)1 in (25 mm)1 1/4 in (32 mm)1 1/2 in (38 mm)

What goes on itPaperbacks, 15 lb/ftMixed hardbacks, 22 lb/ftArt and reference books, 32 lb/ftVinyl records, 50 lb/ftFolded linen, 10 lb/ftCrockery, 32 lb/ft

Front edgePlain edgeFront stiffener, 1.5 to 2 in deep

Enter your measurements above.

Deflection is calculated for a simply supported beam under a uniform load, and is checked twice: the day it is loaded against a 1/180 limit, and years later against a 1/120 limit once the panel has crept. Creep factors are 2.0 for chipboard and MDF, 1.5 for plywood and 1.3 for solid timber.

Nothing you type leaves your browser: the maths runs on this page, and no values are stored or sent anywhere.

The six inputs, and the one people enter wrong#

Span is the clear distance between the inner faces of the supports, measured face to face. It is not the length of the board. A 36 in (91 cm) shelf sitting in a case with 3/4 in (18 mm) sides spans 34.5 in (88 cm), and that 1.5 in matters more than it looks, because the answer follows the fourth power of this number.

A support is anything the shelf cannot deflect past: the sides of a carcass, a vertical divider, a bracket. Shelf pins count, even though the shelf only rests on them. A cleat behind the shelf does not, unless the shelf is fixed to it along its whole length, and neither does a back panel.

The depth, material, thickness, load band and front edge complete the picture. All six run through one formula in your browser as you type. Nothing you enter is sent anywhere.

Why span is raised to the fourth and thickness to the third#

A loaded shelf is a simply supported beam under a uniform load, so

sag = 5 x w x L^4 / (384 x E x I)

where w is the load per inch of run, L the clear span, E the modulus of elasticity of the material and I the second moment of area, which for a rectangular board is b x d^3 / 12 with b the depth and d the thickness.

Two exponents carry the whole result. Span is raised to the fourth power and thickness to the third. Load, depth and stiffness are all linear. The consequences of those exponents are worked through at length in shelf span, thickness and sag; what the calculator adds is the second test that comes after them.

1/180 on day one, 1/120 after creep#

The panel checks the same shelf twice.

1. The day it is loaded, against a limit of 1/180 of the span. That is the point at which the eye stops reading a shelf as level: 0.20 in (5 mm) over 36 in (91 cm).
2. Years later, against a limit of 1/120 of the span, with a creep factor applied. Wood panels keep deforming under sustained load, and the factors are in the table below.

It reports whichever test is closer to failing. A shelf that is comfortable on day one and bowed in five years is the normal failure mode, and the second test is the one that catches it.

The plywood value assumes the face grain runs along the span. Turned the other way it loses roughly half its stiffness, and site-cut shelves often are, because it wastes less sheet.

A 36 in chipboard shelf, worked by hand#

A 36 in span of 3/4 in melamine chipboard, 11 in (28 cm) deep, carrying mixed hardbacks.

1. Second moment of area: 11 x 0.75 x 0.75 x 0.75 divided by 12 = 0.387 in4.
2. Load: 22 lb per foot divided by 12 = 1.83 lb per inch of run.
3. Span to the fourth: 36 x 36 x 36 x 36 = 1,679,616.
4. Top of the fraction: 5 x 1.83 x 1,679,616 = 15,396,000.
5. Bottom: 384 x 350,000 x 0.387 = 51,975,000.
6. Sag: 0.30 in (7.5 mm), against a 1/180 limit of 0.20 in (5 mm). It fails on day one.
7. Long term: 0.30 x 2.0 = 0.59 in (15 mm), against a 1/120 limit of 0.30 in (7.6 mm). It fails again, by a factor of two.

Swap the board for birch plywood and only line five changes: 384 x 1,300,000 x 0.387 = 193,000,000, and the sag falls to 0.08 in (2 mm), or 0.12 in (3 mm) after creep. Same shelf, same load, same span, and the difference between an obvious fault and one nobody notices.

What it tells you to do when the shelf fails#

When the ratio goes over 1, the panel stops describing and starts prescribing. It gives the span the shelf would need to pass, which on the chipboard example is about 28.5 in (72 cm), and notes that one support at mid span cuts the sag to roughly a sixteenth. If the front edge is plain, it shows what a 2 in (51 mm) stiffener would bring the long-term figure down to, which here is 0.12 in (3 mm). And if the load is vinyl it says so directly, because records at 50 lb per foot (74 kg per metre) are the one domestic case where ordinary shelving practice is not conservative enough. The formats behind that figure are in book, vinyl and media storage dimensions.

What it will not tell you: strength, fixings, tipping#

It predicts deflection under a uniform load. It is a planning tool, not a structural check.

• It does not test strength. A shelf can be inside a deflection limit and still break under a point load, and glass is governed by breaking strength rather than sag, which is why it is not in the list.
• It assumes an even load. One heavy object at mid span deflects a shelf about 1.6 times as much as the same weight spread along it.
• It treats every shelf as simply supported. Housing a shelf into a dado gains a little end restraint, and brackets set in from the ends behave differently again.
• It says nothing about fixings or tipping. Anchoring and carcass proportions are in bookcase dimensions and built-in cabinetry dimensions.
• It has no view on whether the shelf is at a useful height, set out in reach zones and useful storage heights, or on how a run of shelving is proportioned, which is in bookshelf and shelving dimensions.

For the rest of the set, see the calculators index.