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The Sagulator

^{New Features!}The Sagulator helps you design shelves by calculating shelf sag (deflection) given type of shelf material, shelf load, load distribution, dimensions, and method of attachment. You can also specify an edging strip to further stiffen the shelf. See the notes below for usage tips.

Shelf Characteristics
	Shelf Material	---------- Select ---------- Afrormosia Albarco Alder, Red Andiroba Angelin Angelique Ash, Black Ash, Blue Ash, Green Ash, Oregon Ash, White Aspen, Bigtooth Aspen, Quaking Avodire Azobe Baldcypress Balsa Banak Basswood Beech, American Benge Birch, Paper Birch, Sweet Birch, Yellow Bubinga Bulletwood Butternut Cativo Cedar, Alaska Cedar, Atlantic White Cedar, Eastern Redcedar Cedar, Incense Cedar, Northern White Cedar, Port-Orford Cedar, Western Redcedar Ceiba Cherry, Black Chestnut, American Cottonwood, Balsam Poplar Cottonwood, Black Courbaril Cuangare Cypress, Mexican Dark red meranti Degame Determa Douglas-fir, Coast Douglas-fir, Interior North Douglas-fir, Interior South Douglas-fir, Interior West Ekop Elm, American Elm, Eastern Elm, Rock Elm, Slippery Fir, Balsam Fir, California Red Fir, Grand Fir, Noble Fir, Pacific silver Fir, Subalpine Fir, White Goncalo Greenheart Hackberry Hemlock, Eastern Hemlock, Mountain Hemlock, Western Hickory, Bitternut Hickory, Mockernut Hickory, Nutmeg Hickory, Pecan Hickory, Pignut Hickory, Shagbark Hickory, Shellbark Hickory, Water Honeylocust Hura Ipe Iroko Jarrah Jelutong Kaneelhart Kapur Karri Kempas Keruing Larch, western Light red meranti Limba Locust, Black Macawood Magnolia, Southern Magnolia,Cucumbertree Mahogany, African Mahogany, true Manbarklak Manni Maple, Bigleaf Maple, Black Maple, Red Maple, Silver Maple, Sugar Marishballi Merbau Mersawa Mora Oak, Black Oak, Bur Oak, Cherrybark Oak, Chestnut Oak, Laurel Oak, Live Oak, Northern Red Oak, Overcup Oak, Pin Oak, Post Oak, Scarlet Oak, Southern Red Oak, Swamp Chestnut Oak, Swamp White Oak, Water Oak, White Oak, Willow Oak Obeche Okoume Opepe Ovangkol Para-angelim Parana-pine Peroba de campos Peroba rosa Pilon Pine, Caribbean Pine, Eastern white Pine, Jack Pine, Loblolly Pine, Lodgepole Pine, Longleaf Pine, Pitch Pine, Pond Pine, Ponderosa Pine, Red Pine, Sand Pine, Shortleaf Pine, Slash Pine, Spruce Pine, Sugar Pine, Virginia Pine, Western white Pine, ocote Pine, radiata Piquia Primavera Purpleheart Ramin Redwood, Old-growth Redwood, Young-growth Robe Rosewood, Brazilian Rosewood, Indian Sande Santa Maria Sapele Sassafras Sepetir Shorea, bullau group Spanish-cedar Spruce, Black Spruce, Engelmann Spruce, Red Spruce, Sitka Spruce, White Sucupira Sweetgum Sycamore, American Tamarack Teak Tupelo, Black Tupelo, Water Wallaba Walnut, Black White meranti Willow, Black Yellow meranti Yellow-poplar llomba D–2 Particleboard D–3 Particleboard H–1 Particleboard H–2 Particleboard H–3 Particleboard M–1 Particleboard M–2 Particleboard M–3 Particleboard Melamine (see note 8) MDF - LD MDF - MD MDF - HD OSB (min stiff.) OSB (max stiff) Plywood, fir Waferboard (min stiff.) Waferboard (max stiff.) Glass
	Shelf attachment	Fixed  Floating
	Shelf load			per ft (30.5 cm)   total
	Load units	lbs    kgs			Shelf and edging strip diagram
	Load distribution	Uniform load  Center load			Build shelves with less waste
	Shelf span			in   cm   mm
	Depth (front to back)
	Thickness
[Optional]  Edging Strip
	Material	None Same as shelf Afrormosia Albarco Alder, Red Andiroba Angelin Angelique Ash, Black Ash, Blue Ash, Green Ash, Oregon Ash, White Aspen, Bigtooth Aspen, Quaking Avodire Azobe Baldcypress Balsa Banak Basswood Beech, American Benge Birch, Paper Birch, Sweet Birch, Yellow Bubinga Bulletwood Butternut Cativo Cedar, Alaska Cedar, Atlantic White Cedar, Eastern Redcedar Cedar, Incense Cedar, Northern White Cedar, Port-Orford Cedar, Western Redcedar Ceiba Cherry, Black Chestnut, American Cottonwood, Balsam Poplar Cottonwood, Black Courbaril Cuangare Cypress, Mexican Dark red meranti Degame Determa Douglas-fir, Coast Douglas-fir, Interior North Douglas-fir, Interior South Douglas-fir, Interior West Ekop Elm, American Elm, Eastern Elm, Rock Elm, Slippery Fir, Balsam Fir, California Red Fir, Grand Fir, Noble Fir, Pacific silver Fir, Subalpine Fir, White Goncalo Greenheart Hackberry Hemlock, Eastern Hemlock, Mountain Hemlock, Western Hickory, Bitternut Hickory, Mockernut Hickory, Nutmeg Hickory, Pecan Hickory, Pignut Hickory, Shagbark Hickory, Shellbark Hickory, Water Honeylocust Hura Ipe Iroko Jarrah Jelutong Kaneelhart Kapur Karri Kempas Keruing Larch, western Light red meranti Limba Locust, Black Macawood Magnolia, Southern Magnolia,Cucumbertree Mahogany, African Mahogany, true Manbarklak Manni Maple, Bigleaf Maple, Black Maple, Red Maple, Silver Maple, Sugar Marishballi Merbau Mersawa Mora Oak, Black Oak, Bur Oak, Cherrybark Oak, Chestnut Oak, Laurel Oak, Live Oak, Northern Red Oak, Overcup Oak, Pin Oak, Post Oak, Scarlet Oak, Southern Red Oak, Swamp Chestnut Oak, Swamp White Oak, Water Oak, White Oak, Willow Oak Obeche Okoume Opepe Ovangkol Para-angelim Parana-pine Peroba de campos Peroba rosa Pilon Pine, Caribbean Pine, Eastern white Pine, Jack Pine, Loblolly Pine, Lodgepole Pine, Longleaf Pine, Pitch Pine, Pond Pine, Ponderosa Pine, Red Pine, Sand Pine, Shortleaf Pine, Slash Pine, Spruce Pine, Sugar Pine, Virginia Pine, Western white Pine, ocote Pine, radiata Piquia Primavera Purpleheart Ramin Redwood, Old-growth Redwood, Young-growth Robe Rosewood, Brazilian Rosewood, Indian Sande Santa Maria Sapele Sassafras Sepetir Shorea, bullau group Spanish-cedar Spruce, Black Spruce, Engelmann Spruce, Red Spruce, Sitka Spruce, White Sucupira Sweetgum Sycamore, American Tamarack Teak Tupelo, Black Tupelo, Water Wallaba Walnut, Black White meranti Willow, Black Yellow meranti Yellow-poplar llomba D–2 Particleboard D–3 Particleboard H–1 Particleboard H–2 Particleboard H–3 Particleboard M–1 Particleboard M–2 Particleboard M–3 Particleboard Melamine (see note 8) MDF - LD MDF - MD MDF - HD OSB (min stiff.) OSB (max stiff) Plywood, fir Waferboard (min stiff.) Waferboard (max stiff.) Glass
	Width			See note 10
	Thickness
		total     per foot
		Target sag: 0.020" per foot or less

Notes

2. The eye will notice a deflection of 1/32" (0.03") per running foot, or 3/32" (0.09") for a 3' wide bookshelf.  The Sagulator computes initial sag only. As an engineering rule of thumb, wood beams/shelves will sag an additional 50% over time beyond the initial deflection induced by the load. Thus, a suggested target for allowable sag is 0.02" per foot or less.

3. Once you have a maximum allowable sag figure in mind, you can design your shelf by tweaking the material types and dimensions. Many bookshelves have a depth of 8" to 12", a width of 24" to 36", and a thickness of 3/4" to 1".

4. A fully loaded bookshelf weighs 20-25 pounds (9-11 kg) per running foot, or 60-75 pounds (27-34 kg) for a 3' wide shelf. 

5. You can enter fractional dimensions as decimals (8.75), or as conventional fractions (8 3/4). If you use fractions, just make sure to leave a space between any leading whole number and the fraction.

6. The shelf thickness value is actual thickness, not nominal. If you're using common construction lumber, a 1 inch board is actually 3/4" and a 2 inch board is 1-1/2" thick.

7. This calculator can also be used to measure beam deflection.  Because beams are typically positioned on edge, use "thickness" to represent beam depth and "depth" to represent the thickness of the beam. The shelf span parameter represents the beam span.

8. The Melamine material choice assumes M-1 grade particleboard with a melamine resin/paper facing. If your melamine sheet has a different particleboard grade (M-2, M-3, etc), select that grade of particleboard in the pull-down menu. The melamine facing has a negligible effect on the stiffness of the shelving material.

9. If using a hardwood ply with a composite core - veneer center plies, with relatively thick MDF outer layers under the face veneers, select MDF for the shelf material.

10. An optional edging strip can be specified to stiffen the shelf. The "thickness" of the edge strip will be the same direction as "depth" of the shelf, and the "width" of the strip will be the same direction as "thickness" of the shelf. For the purpose of computing sag, it doesn't matter if the strip is attached to the front of the shelf or its underside.

Tips for Managing Shelf Sag

• If shelf span is reduced by one-fifth, stiffness is roughly doubled (deflection is halved).
• If shelf span is increased by one-fourth, deflection doubles.
• If shelf span is doubled, deflection is eight times greater.
• If shelf thickness is doubled, deflection is reduced to one-eighth.
• If shelf depth is doubled, deflection is cut in half.

Credits and References

The Sagulator employs established engineering formulas for calculating beam deflection. Some references that I found useful include:

• Roark's Formulas for Stress and Strain by Warren Young
• Understanding Wood: A Craftsman's Guide to Wood Technology by Bruce Hoadley
• Engineers Edge structural beam bending equations
• Beam Design Formulas with Shear and Moment Diagrams by American Forest and Paper Association

Special thanks to Steve Stephenson for providing initial formulas and Jeff Bratt for his very helpful input (including the spiffy shelf diagram), and derivation of formulae for handling an edging strip.