FORMULAS FOR COLUMNS Machinery's Handbook, 31st Edition Rankine and Euler Formulas for Columns
283
Symbol
Quantity
English Unit
Metric SI Units
Ultimate unit load Total ultimate load
Lbs/in 2 Pounds Lbs/in 2 Inches Inches Inches 4 Inches 2 Lbs/in 2 Inches Inches
Newtons/mm 2 Newtons Newtons/mm 2 Millimeters Millimeters Millimeters 4 Millimeters 2 Newtons/mm 2
p P S
Ultimate compressive strength of material Length of column or strut Least radius of gyration Least moment of inertia Moment of inertia/area of section Modulus of elasticity of material Distance from neutral axis of cross section to side under compression Distance from axis of load to axis coinciding with center of gravity of cross section
l r I
r 2 E c
Millimeters Millimeters
z
Rankine Formulas
Both Ends of Column Fixed S 1 25 000 2 2 l + l S 1 5000 2 2 + l S 1 35 000 2 2 , r r +
One End Fixed and One End Rounded
Material
Both Ends Rounded
, l S 1 12 500 2 2 + l S 1 2500 2 2 r + , l S 1 17 500 2 2 + l S 1 1500 2 2 r r +
r l S 1 6250 2 2 + l S 1 1250 2 2 + r l S 1 8750 2 2 + l S 1 750 2 2 r +
p
p
p
=
=
=
Steel
p
p
p
=
=
=
Cast Iron
p
p
p
=
=
=
Wrought Iron
,
r
l S 1 3000 2 2 +
p
p
p
=
=
=
Timber
r
r
r
Formulas Modified for Eccentrically Loaded Columns
Both Ends of Column Fixed S 1 25 000 2 2 l +
One End Fixed and One End Rounded
Material
Both Ends Rounded
l S 1 12 500 2 2 +
l S 1 6250 2 2 +
p
p
p
=
=
=
r cz 2
r cz 2
r cz 2
Steel
+
+
+
,
,
r
r
r
For materials other than steel, such as cast iron, use the Rankine formulas given in the upper table and add to the denominator the quantity cz / r 2 Euler Formulas for Slender Columns Both Ends of Column Fixed One End Fixed and One End Rounded Both Ends Rounded One End Fixed and One End Free
P l IE 4 2 2 = π
P l IE 2 2 = π
P l IE 4 2 2 = π
P l IE 2 2 2 = π
Allowable Working Loads for Columns: To find the total allowable working load for a given sec tion, divide the total ultimate load P (or p 3 area), as found by the appropriate formula above, by a suitable factor of safety.
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