Column buckling formulas and effective length constants are given in the following chart.
|
Parameter |
Equation |
|
Radius of gyration [k] |
$$k=\sqrt { \frac { I }{ A } } $$ |
|
Eccentricity ratio [er] |
$$er=\frac { ec }{ { k }^{ 2 } } $$ |
|
Slenderness ratio [S] |
$$S=\frac { L }{ k } $$ |
|
Effective slenderness ratio [Seff] |
$${ S }_{ eff }=\frac { LC }{ k } $$ |
|
If er=0 and Seff > (2π2E/Sy)^0.5 then go to step 6 |
|
Force (according to Euler column formula) [Pcr] |
$${ P }_{ cr }=\frac { { \pi }^{ 2 }EI }{ { L }^{ 2 }{ C }^{ 2 } } $$ |
|
If er=0 and Seff ≤(2π2E/Sy)^0.5 then go to step 8 |
|
Force (according to Parabolic/J.B. Johnson formula) [Pcr] |
$${ P }_{ cr }=[{ S }_{ y }-({ \frac { { S }_{ y }L }{ 2\pi k } ) }^{ 2 }\frac {
{ C }^{ 2 } }{ E } ]A$$ |
|
If er≠0 and S>0.282(AE/P)^0.5 then go to step 10 |
|
Force (according to secant formula) [Pcr]* |
$${ P }_{ cr }=\frac { { S }_{ yc }A }{ 1+(\frac { ec }{ { k }^{ 2 } }
)sec[(\frac { LC }{ 2k } )\sqrt { { P }_{ cr }/AE } ] } $$ |
|
If er≠0 and S≤0.282(AE/P)^0.5 then go to step 12 |
|
Force (according to stress formulas) [Pcr]* |
$${ P }_{ cr }=\frac { { S }_{ yc }A }{ 1+\frac { ec }{ { k }^{ 2 } } } $$ |
* Note: After calculation, er=0 assumption is made and step 5 and step 7 is
revisited. Pcr is
calculated
with er=0 assumption (in step 6 or step 8 according to conditions), then smaller value is
selected by the calculator.
I:
Area moment of inertia,
A:
Area of the cross-section,
L:
Length of the column,
C:
Effective length constant,
Sy: Yield strength,
Syc :Compressive yield strength,
e:
Eccentricity – Distance between central axis of column and line of action of the
force,
c:
Perpendicular distance to neutral axis