Discussion
Mathematically, crippling load by Rankine's formula is :
Where - σc = ultimate crushing stress, α = Rankine constant, A = area of cross section, K = least radius of gyration, Le = effective length
Answer: B
Rankine's formula is \(P = \frac{\sigma_c A}{1 + \alpha\left(\frac{L_e}{K}\right)^{2}}\).
Rankine combined the crushing load of a short column, \(P_c = \sigma_c A\), with Euler's buckling load of a long column, \(P_E\), through \(\frac{1}{P} = \frac{1}{P_c} + \frac{1}{P_E}\):
\[ \begin{aligned} P &= \frac{\sigma_c A}{1 + \frac{\sigma_c}{\pi^2 E}\left(\frac{L_e}{K}\right)^2} \\ \alpha &= \frac{\sigma_c}{\pi^2 E} \end{aligned} \]
The slenderness ratio \(L_e/K\) appears squared, which rules out the first option. Squaring \(\sigma_c\) or A, as the other options do, gives a quantity that is not a force.
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