Rankine’s formula is the modification of Euler’s column theory. Euler’s column theory was related to only long column but Rankine formula can also be used for short columns.
Empirical formation of column is as following
1 / Wcr = 1/WC + 1/WE
where in the above expression Wcr is the crippling load, WC is the ultimate crushing load and WE is the crippling load taken from Euler’s formula = π2EI / L2.
If we observe the above relation, we will know that WC remains constant for any particular material and it will not change. For short columns, WE will be high which means that it’s reciprocal will be small and vice versa
By comparing, we observe that the value of crippling load is very similar to the value taken from Euler’s formula. Since the formula contains the solution of Euler’s column therefore, it can be used for both small as well as for long columns.
1 / Wcr = 1/WC + 1/WE = (WE + WC) / (WC x WE)
1 / Wcr = WC / (1 + WC / WE)
Since WE = π2EI / L2
WC = σC x A
Therefore putting the values of both types of loads in the above expression
1 / Wcr = WC / (1 + (σC x A) / (π2EI / L2) ]
1 / Wcr = (σC x A) / (1 + (σC x A) / (π2EI / L2) ]
Since I = ak2
1 / Wcr = (σC x A) / (1 + (σC x A) / (π2E.ak2 / L2) ]
1 / Wcr = (σC x A) / [ 1 + a(L / k)2 ]
where in the above expression a is the rankine constant and its value is a = σC / π2E, L is the equivalent length of column and k is the least radius of gyration.