Shelf Sag Calculator
Sag of a loaded shelf, its stiffness ratio, the load for a 1:250 limit and the span that would halve the sag.
Results
What this tool does
Sag goes with the fourth power of the span and the cube of the thickness: adding one more bracket beats any amount of extra timber, and the last result shows the span that would cut the sag in half. Typical stiffness values are about 9 GPa for softwood, 12 for hardwood, 3.5 for chipboard and 4 for MDF.
Formula
deflection = 5 w L⁴ ÷ (384 E I) , I = b h³ ÷ 12
Variables
| Symbol | Meaning | Unit |
|---|---|---|
lg | Span between supports | mm |
wd | Shelf depth | mm |
th | Shelf thickness | mm |
ld | Load | kg |
em | Young's modulus | GPa |
SG | Deflection | mm |
II | Second moment of area | cm⁴ |
RT | Ratio | — |
LM | Load | kg |
L2 | Span between supports | mm |
Worked example
- Span between supports800 mm
- Shelf depth250 mm
- Shelf thickness18 mm
- Load20 kg
- Young's modulus9 GPa
- Deflection1.1958 mm
- Second moment of area12.15000 cm⁴
- Ratio669
- Load53.523 kg
- Span between supports475.7 mm
Limitations
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Density changes with temperature, moisture and composition. Use a value from the supplier's data sheet whenever precision matters.
Frequently asked questions
How much sag is acceptable?
The eye starts to notice a curve at about one part in 250 of the span, which is the ratio shown as the third result — bigger is better, and anything above 360 looks dead flat. But shelves have a second problem wood alone cannot solve: under a constant load, timber and especially chipboard keep creeping for years, ending up two or three times the initial sag. Design for a third of what you would accept, or stiffen the front edge with a lipping.