Both the Rankine as well as the Euler formula are used for buckling calculation, however, the accuracy of the buckling calculation results depends upon the proper application of the Rankine and Euler formulas for metric steel columns or strut.
What is a Column or Strut?
Any machine member, subjected to the axial compressive loading is called a strut and the vertical strut is called column
The columns are generally categorized in two types: short columns and long columns. The one with length less than eight times the diameter (or approximate diameter) is called short column and the one with length more than thirty times the diameter (or approximate diameter) is called long column.
Euler’s Buckling Formula
To get the correct results, this formula should only be applied for the long columns. The buckling load calculated by the Euler formula is given by:
Fbe = (C*Π2*E*I)/L2…………..Eqn.1.1
Fbe = Buckling force calculated by the Euler’s formula.
C = End fixity coefficient to be selected based on the end conditions of the column.
E = Modulus of elasticity
I = area moment of inertia of the cross section of the column
L = length of the column
The end fixity constant (C) used in the above Eqn.1.1 to be selected as below:
- For the column with both end hinged – 1
- For the column with both end fixed – 4
- For the column with one end fixed and other end hinged – 2
- For the column with one end fixed and other end free – 0.25
Please note that out of the two area moment of inertia Ixx and Iyy, the least one has to be considered for the above equation. The things will be clearer, once you will go through the next example.
Example: Find out the buckling stress of a 4 meter long both ends fixed column, which has rectangular cross section of size 200mm X 100 mm. Assume the modulus of elasticity as 100 kN/mm2.
Ixx = (100*2003)/12 = 66666666.67 mm3
Iyy = (200*1003)/12 = 16666666.67mm3
Fbe = (4* 3.142*100*16666666.67)/40002 = 4108.167 kN
Rankine’s Buckling Formula for Columns
The Euler’s formula is ideal only for the long columns. To eliminate this limitation of the Euler’s formula, Prof. Rankine came out with the following empirical formula:
1/Fbr = 1/Fc + 1/Fbe ……………Eqn.1.2
Fbr = buckling load calculated using Rankine’s formula
Fc = ultimate crushing load on the column = σuc * A
σuc = ultimate compressive stress for the column material
A = area of the cross section of the column
Fbe = buckling load calculated using Euler’s formula
The buckling calculation is done using the Rankine and Euler Formulas for Metric Steel Columns or strut. The Euler formula is ideal for long column. The Rankine formula is a more general formula and can be used for both the long as well as the short column.