The Sagulator

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 ———-AfrormosiaAlbarcoAlder, RedAndirobaAngelinAngeliqueAsh, BlackAsh, BlueAsh, GreenAsh, OregonAsh, WhiteAspen, BigtoothAspen, QuakingAvodireAzobeBaldcypressBalsaBanakBasswoodBeech, AmericanBengeBirch, PaperBirch, SweetBirch, YellowBubingaBulletwoodButternutCativoCedar, AlaskaCedar, Atlantic WhiteCedar, Eastern RedcedarCedar, IncenseCedar, Northern WhiteCedar, Port-OrfordCedar, Western RedcedarCeibaCherry, BlackChestnut, AmericanCottonwood, Balsam PoplarCottonwood, BlackCourbarilCuangareCypress, MexicanDark red merantiDegameDetermaDouglas-fir, CoastDouglas-fir, Interior NorthDouglas-fir, Interior SouthDouglas-fir, Interior WestEkopElm, AmericanElm, EasternElm, RockElm, SlipperyFir, BalsamFir, California RedFir, GrandFir, NobleFir, Pacific silverFir, SubalpineFir, WhiteGoncaloGreenheartHackberryHemlock, EasternHemlock, MountainHemlock, WesternHickory, BitternutHickory, MockernutHickory, NutmegHickory, PecanHickory, PignutHickory, ShagbarkHickory, ShellbarkHickory, WaterHoneylocustHuraIpeIrokoJarrahJelutongKaneelhartKapurKarriKempasKeruingLarch, westernLight red merantiLimbaLocust, BlackMacawoodMagnolia, SouthernMagnolia,CucumbertreeMahogany, AfricanMahogany, trueManbarklakManniMaple, BigleafMaple, BlackMaple, RedMaple, SilverMaple, SugarMarishballiMerbauMersawaMoraOak, BlackOak, BurOak, CherrybarkOak, ChestnutOak, LaurelOak, LiveOak, Northern RedOak, OvercupOak, PinOak, PostOak, ScarletOak, Southern RedOak, Swamp ChestnutOak, Swamp WhiteOak, WaterOak, WhiteOak, WillowObecheOkoumeOpepeOsage OrangeOvangkolPara-angelimParana-pinePeroba de camposPeroba rosaPilonPine, CaribbeanPine, Eastern whitePine, JackPine, LoblollyPine, LodgepolePine, LongleafPine, PitchPine, PondPine, PonderosaPine, RedPine, SandPine, ShortleafPine, SlashPine, SprucePine, SugarPine, VirginiaPine, Western whitePine, ocotePine, radiataPiquiaPrimaveraPurpleheartRaminRedwood, Old-growthRedwood, Young-growthRobeRosewood, BrazilianRosewood, IndianSandeSanta MariaSapeleSassafrasSepetirShorea, bullau groupSpanish-cedarSpruce, BlackSpruce, EngelmannSpruce, RedSpruce, SitkaSpruce, WhiteSucupiraSweetgumSycamore, AmericanTamarackTeakTupelo, BlackTupelo, WaterWallabaWalnut, BlackWhite merantiWillow, BlackYellow merantiYellow-poplarllombaD–2 ParticleboardD–3 ParticleboardH–1 ParticleboardH–2 ParticleboardH–3 ParticleboardM–1 ParticleboardM–2 ParticleboardM–3 ParticleboardMelamine (see note 8)MDF – LDMDF – MDMDF – HDOSB (min stiff.)OSB (max stiff)Plywood, firWaferboard (min stiff.)Waferboard (max stiff.)Glass
	Shelf attachment	Fixed (attached to sides)  Floating (not attached)
	Shelf load			per foot (305 mm)   total
	Load units	lbs    kgs
	Load distribution	Uniform load  Center load
	Shelf span			in   cm   mm
	Depth (front to back)
	Thickness
[Optional]  Edging Strip (see note #10)
	Material	NoneSame as shelfAfrormosiaAlbarcoAlder, RedAndirobaAngelinAngeliqueAsh, BlackAsh, BlueAsh, GreenAsh, OregonAsh, WhiteAspen, BigtoothAspen, QuakingAvodireAzobeBaldcypressBalsaBanakBasswoodBeech, AmericanBengeBirch, PaperBirch, SweetBirch, YellowBubingaBulletwoodButternutCativoCedar, AlaskaCedar, Atlantic WhiteCedar, Eastern RedcedarCedar, IncenseCedar, Northern WhiteCedar, Port-OrfordCedar, Western RedcedarCeibaCherry, BlackChestnut, AmericanCottonwood, Balsam PoplarCottonwood, BlackCourbarilCuangareCypress, MexicanDark red merantiDegameDetermaDouglas-fir, CoastDouglas-fir, Interior NorthDouglas-fir, Interior SouthDouglas-fir, Interior WestEkopElm, AmericanElm, EasternElm, RockElm, SlipperyFir, BalsamFir, California RedFir, GrandFir, NobleFir, Pacific silverFir, SubalpineFir, WhiteGoncaloGreenheartHackberryHemlock, EasternHemlock, MountainHemlock, WesternHickory, BitternutHickory, MockernutHickory, NutmegHickory, PecanHickory, PignutHickory, ShagbarkHickory, ShellbarkHickory, WaterHoneylocustHuraIpeIrokoJarrahJelutongKaneelhartKapurKarriKempasKeruingLarch, westernLight red merantiLimbaLocust, BlackMacawoodMagnolia, SouthernMagnolia,CucumbertreeMahogany, AfricanMahogany, trueManbarklakManniMaple, BigleafMaple, BlackMaple, RedMaple, SilverMaple, SugarMarishballiMerbauMersawaMoraOak, BlackOak, BurOak, CherrybarkOak, ChestnutOak, LaurelOak, LiveOak, Northern RedOak, OvercupOak, PinOak, PostOak, ScarletOak, Southern RedOak, Swamp ChestnutOak, Swamp WhiteOak, WaterOak, WhiteOak, WillowOakObecheOkoumeOpepeOvangkolPara-angelimParana-pinePeroba de camposPeroba rosaPilonPine, CaribbeanPine, Eastern whitePine, JackPine, LoblollyPine, LodgepolePine, LongleafPine, PitchPine, PondPine, PonderosaPine, RedPine, SandPine, ShortleafPine, SlashPine, SprucePine, SugarPine, VirginiaPine, Western whitePine, ocotePine, radiataPiquiaPrimaveraPurpleheartRaminRedwood, Old-growthRedwood, Young-growthRobeRosewood, BrazilianRosewood, IndianSandeSanta MariaSapeleSassafrasSepetirShorea, bullau groupSpanish-cedarSpruce, BlackSpruce, EngelmannSpruce, RedSpruce, SitkaSpruce, WhiteSucupiraSweetgumSycamore, AmericanTamarackTeakTupelo, BlackTupelo, WaterWallabaWalnut, BlackWhite merantiWillow, BlackYellow merantiYellow-poplarllombaD–2 ParticleboardD–3 ParticleboardH–1 ParticleboardH–2 ParticleboardH–3 ParticleboardM–1 ParticleboardM–2 ParticleboardM–3 ParticleboardMelamine (see note 8)MDF – LDMDF – MDMDF – HDOSB (min stiff.)OSB (max stiff)Plywood, firWaferboard (min stiff.)Waferboard (max stiff.)Glass
	Width (vertical plane)
	Thickness
		Target sag: 0.02 in per foot (1.7 mm per m) or less

Notes

1. The deflection calculations use average wood stiffness properties of clear, straight-grained samples measured in controlled laboratory conditions (mainly from the U.S. Forest Products Lab). Expect some real-life variation from the calculated results.

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 20-40 pounds (9-18 kg) per running foot, or 60-120 pounds (27-54 kg) for a 3′ wide shelf.  A value of 35 pounds per running foot is used by some for library shelving. Fine Woodworking magazines can weigh up to 40 pounds per foot.

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.

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 helpful input and derivation of formulae for handling an edging strip.