# COLUMN BUCKLING FORMULAS AND EFFECTIVE LENGTH CONSTANTS

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

### 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

### Reference:

• Budynas.R , Nisbett.K . (2014) . Shigley's Mechanical Engineering Design . 10th edition.  McGraw-Hill
• Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials, 7th Edition , 2nd edition. McGraw-Hill, Chapter 4-5-7-8-9