Woodworking guide
Shelf Span and Sag: How Thick a Shelf Needs to Be for Your Load
How much a loaded shelf sags: the beam formula, span and thickness cubed, a chart of 13 materials and the span ÷ 360 limit. Typical values only.
Quick answer
A 3/4 in by 10 in white pine shelf spanning 36 in, carrying 40 lb spread evenly on simple supports, sags about 0.056 in (1.4 mm), well inside a span ÷ 360 limit of 0.1 in. Sag grows with the cube of the span and falls with the cube of the thickness: at 48 in the same shelf sags 0.132 in, eight times as much as at 24 in and only just inside its limit. A 1/2 in pine shelf sags 0.188 in at 36 in and fails, and typical MDF at the same size sags 0.154 in and fails too.
Formula
sag = k × load × span³ ÷ (E × I) | I = depth × thickness³ ÷ 12 | k = 5/384 (simple ends, even load) | limit = span ÷ 360
Accuracy
Reference table
Search by name or sort by any column. Each row links to the matching calculator.
13 of 13 rows · sorted by material
| Calculator | |||||||
|---|---|---|---|---|---|---|---|
| Birch plywood | 1.2 | 0.058 | 1.46 | 625 | Yes | 0.62 | Shelf sag calculator |
| Black walnut | 1.68 | 0.041 | 1.04 | 875 | Yes | 0.56 | Shelf sag calculator |
| Cherry | 1.49 | 0.046 | 1.18 | 776 | Yes | 0.58 | Shelf sag calculator |
| Douglas-fir | 1.95 | 0.035 | 0.90 | 1016 | Yes | 0.53 | Shelf sag calculator |
| Eastern white pine | 1.24 | 0.056 | 1.42 | 646 | Yes | 0.62 | Shelf sag calculator |
| Hard maple | 1.83 | 0.038 | 0.96 | 953 | Yes | 0.54 | Shelf sag calculator |
| MDF | 0.45 | 0.154 | 3.90 | 234 | No | 0.87 | Shelf sag calculator |
| Particleboard | 0.35 | 0.198 | 5.02 | 182 | No | 0.94 | Shelf sag calculator |
| Poplar | 1.58 | 0.044 | 1.11 | 823 | Yes | 0.57 | Shelf sag calculator |
| Red oak | 1.82 | 0.038 | 0.96 | 948 | Yes | 0.54 | Shelf sag calculator |
| Southern yellow pine | 1.6 | 0.043 | 1.10 | 833 | Yes | 0.57 | Shelf sag calculator |
| White oak | 1.78 | 0.039 | 0.99 | 927 | Yes | 0.55 | Shelf sag calculator |
| Yellow birch | 2.01 | 0.034 | 0.87 | 1047 | Yes | 0.53 | Shelf sag calculator |
How it works
The beam formula behind shelf sag
A shelf resting on two supports is a beam, and its sag in the middle is k × load × span³ ÷ (E × I). E is the stiffness of the material (modulus of elasticity), and I = depth × thickness³ ÷ 12 describes the shape of the cross-section. For the default 10 in deep, 3/4 in thick shelf, I = 10 × 0.75³ ÷ 12 = 0.3516 in⁴. White pine is typically about 1.24 million psi, and for simple supports with an even load k = 5/384. So sag = 5/384 × 40 lb × 36³ ÷ (1,240,000 × 0.3516) = 0.0557 in. Two cubes do the work: halve the thickness and the sag rises eightfold, and double the span and it does the same. That is why a small increase in thickness or a middle support fixes a bowing shelf faster than a stronger-sounding wood.
Span limits, supports and how the load sits
Designers compare sag with the span divided by a number: 240 is a visible-but-tolerable limit, 360 is a common guide for shelves, and 480 is stricter. On a 36 in span, ÷ 360 allows 0.1 in. How the shelf is held and loaded changes k. A shelf glued or tightly dadoed at both ends is nearer to fixed ends and sags a fifth as much as the same shelf on simple supports (k = 1/384 against 5/384), but loose fits, pins or screws behave more like simple supports, so assume simple unless the joint really is rigid. A single heavy item in the centre (k = 1/48) sags 1.6 times as much as the same weight spread evenly. Books spread along the shelf are closer to an even load.
Choosing a material and thickness
The chart on this page runs 13 materials through the default shelf. Solid woods in the typical table fall between 1.24 and 2.01 million psi and all pass. Birch plywood at a typical 1.2 million psi sags 0.058 in and also passes. Typical MDF (0.45) and particleboard (0.35) are far less stiff: MDF sags 0.154 in and needs about 0.87 in of thickness to pass, or a span of no more than about 29 in, against 48 in for pine. To stiffen a shelf without changing the board, shorten the span with a middle support, use a thicker board, or glue a deeper strip to the front edge. Real boards vary, so treat the numbers as a first check and leave a margin if the shelf will carry a steady load for years.
Work it out
Change the inputs to work out your own numbers. Each step of the working is shown with the answer.
Your measurements
Enter the shelf and the load. Everything updates as you type.
The shelf
Modulus of elasticity (stiffness) values are typical and rounded; real boards vary by 20% or more. Enter your own if you know it. For plywood, the face grain should run along the span.
The real thickness, not the nominal. A 3/4 in board or sheet is often a little under.
The clear distance from one support to the next, not the length of the board.
The load and the supports
The whole weight on the shelf between the supports, not the weight per foot.
Resting on brackets, pins or cleats: the ends can turn. This is the safe choice for most shelves.
A rule of thumb for what the eye notices, not a code. Span ÷ 360 is a common choice. Use 480 for a shelf that must look flat.
Results
Sag at the middle of the shelf
in
= 1.416 mm
Eastern white pine, 0.75 in thick, 36 in span, 40 lb spread evenly: the sag is within the span ÷ 360 limit of 0.1 in.
- Against the limit
- Within the limit
- span ÷ 645.8; limit span ÷ 360
- Thinnest that meets the limit
- 0.6172in
- = 15.68 mm
- round up to a thickness you can buy
- Longest span at the limit
- 48.22in
- = 1224.79 mm
- same thickness, depth and load
- Modulus used
- 1.24million psi
- 8.55 GPa
Before you rely on this (2)
- The modulus is a typical, rounded value. Real boards vary by 20% or more. Enter your own if you know it.
- The shelf's own weight, shear, and the brackets, pins and fixings are not included, and wood sags more over months under a steady load. This is a guide, not an engineering check for heavy loads.
How this was calculated
Stiffness, sag and the limit, with your numbers.
How this was calculated
Stiffness, sag and the limit, with your numbers.
Step 1: Stiffness of the shelf section
I = depth × thickness³ ÷ 12
= 10 × 0.75³ ÷ 12 = 10 × 0.421875 ÷ 12
= I = 0.3515625 in⁴I is how much the cross-section resists bending. Thickness is cubed, so a little extra thickness helps a lot, and depth only counts once.
Step 2: Stiffness of the material
E = modulus of elasticity
= Eastern white pine: 1.24 million psi
= E = 1,240,000 psi (8.55 GPa)A typical, rounded value. Real boards vary by 20% or more, and plywood and MDF vary by maker. 1 million psi is 6.894757 GPa.
Step 3: Sag at the middle
sag = 5 × W × L³ ÷ (384 × E × I)
= 5 × 40 × 36³ ÷ (384 × 1,240,000 × 0.3515625)
= 0.0557 in (1.416 mm)W is the total load in pounds and L is the span in inches. The sag grows with the cube of the span, so doubling the span with the same total load makes the shelf sag eight times as much.
Step 4: Compare with the limit
limit = span ÷ 360
= 36 ÷ 360
= 0.1 in (2.54 mm): within the limitThe sag is span ÷ 645.8. Span ÷ 360 is a common rule of thumb for what the eye notices on a shelf; it is not a code limit.
Step 5: Thinnest shelf that meets the limit
h_min = h × (sag ÷ limit)^(1/3)
= 0.75 × (0.0557 ÷ 0.1)^(1/3)
= 0.6172 inThe sag scales with 1 ÷ thickness³, so the thickness that brings the sag to the limit is the cube root of the ratio times the thickness you have. The span, depth, load and material stay the same.
Troubleshooting
What usually goes wrong, how to check for it, and how to prevent it.
A shelf that checked out on paper bows after a few months
- Cause
- Wood creeps under a steady load, and the sag limit was only just met
- How to check
- Compare the calculated sag with span ÷ 360 and check how much margin is left
- Prevention
- Aim for a clear margin or a stricter limit such as span ÷ 480 for books and heavy items, and shorten the span or add thickness
An MDF or particleboard shelf sags far more than a pine one
- Cause
- These boards are much less stiff than solid wood or plywood of the same thickness
- How to check
- Typical stiffness is about 0.45 million psi for MDF and 0.35 for particleboard, against about 1.24 for pine
- Prevention
- Use a thicker board or a shorter span, switch to plywood or solid wood, or stiffen the front edge
The shelf is stiff enough but the unit pulls away from the wall
- Cause
- The brackets, pins or wall fixings are the weak link, which this calculation does not cover
- How to check
- Check how the shelf is fixed, how many fixings there are and what they are rated for
- Prevention
- Fix into studs or use anchors rated for the load, and check the supports for heavy shelves
Common questions
How far can a 3/4 inch shelf span?
For a 10 in deep shelf carrying 40 lb spread evenly on simple supports with a span ÷ 360 limit, typical values give about 48 in for white pine, 47 in for birch plywood and 29 in for MDF. Change the depth, load or limit and the answer changes with them.
How much does a shelf sag under load?
Sag = k × load × span³ ÷ (E × I), with I = depth × thickness³ ÷ 12. For the default pine shelf that is 0.0557 in. The span and the thickness each appear as a cube, so they matter far more than the load.
How much sag is acceptable?
There is no single rule for shelves. Span ÷ 240 is visible but tolerable, span ÷ 360 is a common guide and span ÷ 480 is stricter, which suits heavy or highly visible shelves. Pick one and check it, and leave margin for sag over time.
Does doubling the thickness help?
Yes, a great deal. Stiffness depends on thickness cubed, so a shelf twice as thick sags about one eighth as much. A 1 1/2 in pine shelf on the default span sags 0.007 in, against 0.056 in at 3/4 in.