On the basis of results of experiments performed by Rankine, he established an empirical formula which is applicable to all columns whether they are short or long.
The empirical formula given by Rankine is known as Rankine’s formula, which is given as
For a given column material the crushing stress σc is a constant. Hence the crushing load Pc (which is equal to σc x A) will also be constant for a given cross-sectional area of the column. In equation (i), Pc is constant and hence value of P depends upon the value of PE. But for a given column material and given cross-sectional area, the value of PE depends upon the effective length of the column.
(i) If the column is a short, which means the value of Le is small, then the value of PE will be large. Hence the value of 1/PE will be small enough and is negligible as compared to the value of 1/PC . Neglecting the value of 1/PE in equation (i), we get,
Hence the crippling load by Rankine's formula for a short column is approximately equal to crushing load. Also we have seen that short columns fail due to crushing.
(ii) If the column is long, which means the value of Le is large. Then the value of PE will be small and the value of 1/PE will be large enough compared with 1/PC.Hence the value of 1/PC may be neglected in equation (i).
(iii) Hence the crippling load by Rankine's formula for long columns is approximately equal to crippling load given by Euler's formula.