COLUMN BUCKLING FORMULAS AND EFFECTIVE LENGTH CONSTANTS


Column buckling formulas and effective length constants are given in the following chart.


Column Buckling Formulas:

Step Parameter Equation
1 Radius of gyration [k] $$k=\sqrt { \frac { I }{ A } } $$
2 Eccentricity ratio [er] $$er=\frac { ec }{ { k }^{ 2 } } $$
3 Slenderness ratio [S] $$S=\frac { L }{ k } $$
4 Effective slenderness ratio [Seff] $${ S }_{ eff }=\frac { LC }{ k } $$
5 If er=0 and Seff > (2π2E/Sy)^0.5 then go to step 6
6 Force (according to Euler column formula) [Pcr] $${ P }_{ cr }=\frac { { \pi }^{ 2 }EI }{ { L }^{ 2 }{ C }^{ 2 } } $$
7 If er=0 and Seff ≤(2π2E/Sy)^0.5 then go to step 8
8 Force (according to Parabolic/J.B. Johnson formula) [Pcr] $${ P }_{ cr }=[{ S }_{ y }-({ \frac { { S }_{ y }L }{ 2\pi k } ) }^{ 2 }\frac { { C }^{ 2 } }{ E } ]A$$
9 If er≠0 and S>0.282(AE/P)^0.5 then go to step 10
10 Force (according to secant formula) [Pcr]* $${ P }_{ cr }=\frac { { S }_{ yc }A }{ 1+(\frac { ec }{ { k }^{ 2 } } )sec[(\frac { LC }{ 2k } )\sqrt { { P }_{ cr }/AE } ] } $$
11 If er≠0 and S≤0.282(AE/P)^0.5 then go to step 12
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.

List of Parameters :


Symbol Parameter
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

Effective Length Constant :

Boundary Conditions Theoretical Suggested Engineering
Free-Free 1.0 1.2
Pinned-Free 1.0 1.2
Pinned-Pinned 1.0 1.0
Guided-Free 2.0 2.1
Guided-Pinned 2.0 2.0
Guided-Guided 1.0 1.2
Fixed-Free 2.0 2.1
Fixed-Pinned 0.707 0.8
Fixed-Guided 1.0 1.2
Fixed-Fixed 0.5 0.65

Supplements:

Link Usage
Compression member design The calculator which calculates parameters of the compression member (column) for different end conditions and loading types.

Reference: