Find the shortest length L for a steel column with pinned ends having a cross-sectional area of 60 mm × 100 mm, for which the elastic Euler formula applies. Let E = 200 Gpa and assume the proportional limit to be 250 Mpa. Choose the correct answer from the following options.

Explanation:

Euler's formula to calculate crippling load in the column

\({P_{cr}} = \frac{{{\pi ^2}EI_{min}}}{{L_e^2}} \)

Where P = Crippling load, E = young's modulus, Imin = Minimum area moment of inertia of column, Le = Effective length of a column

Crippling stress is given by:

\({σ _C} = \frac{{{\pi ^2}E}}{{{{\left( {\frac{{{L_{e}}}}{k}} \right)}^2}}}= \frac{{{\pi ^2}E}}{{{{\left( S \right)}^2}}}\)

Where S = Slenderness ratio &

The slenderness ratio is defined as the ratio of effective length and radius of gyration. i.e. \(S=\frac{L_e}{k}\)

Calculation:

Given data

A = 60 mm × 100 mm

E = 200 Gpa 

Stress = 250 Mpa

I_xx = \(60 \times 100^3\over 12\) = 5000000 mm⁴

I_yy = \(100 \times 60^3\over 12\) = 1800000 mm⁴

Euler's formula to calculate crippling load in the column

\({P_{cr}} = \frac{{{\pi ^2}EI_{min}}}{{L_e^2}} \)

\({L_e^2} = \frac{{{\pi ^2}EI_{min}}}{{P_{cr}}} \)

\({L_e^2} = \frac{{{\pi ^2}\times 200000\times 1800000}}{{250 \times 60 \times 100}} \)

L_e = 1539.05

Hence, the shortest length L for a steel column with pinned ends is 1540 mm.