The effective lengths of a column about the x axis and y axis are 3 m and 2.75 m, respectively. If the size of the column is 400 mm × 600 mm, classify the column.

Explanation:

We know that-

\(Slenderness\, ratio\, (\lambda) ={L_{eff}\over r_{min}}\)

Where Leff = Effective length of the column

rmin = Least radius of gyration

\(r_{min}=\sqrt{I_{min}\over A}\)

Calculation:

Given data:

Effective length of the column about the X-axis (L_eff-XX) = 3 m or 3000 mm

Effective length of the column about the Y-axis (Leff-YY) = 2.75 m or 2750 mm

Cross-section of the column = 400 mm × 600 mm

Width of column (b) = 400 mm

Depth of column (d) = 600 mm

\(I_{min}={db^3\over 12}={{600× 400^3}\over 12}=32× 10^8\, mm^4\)

\(Area\,(A)=bd=400\times 600=24\times 10^4\, mm^2\)

\(r_{min}=\sqrt{32\times 10^8\over 24\times 10^4} =115.470\, mm\)

Slenderness ratio about X-axis:

\(Slenderness\, ratio\, about\, X-axis\, (\lambda) ={L_{eff-XX}\over r_{min}}\)

\(Slenderness\, ratio\, about\, X-axis\, (\lambda_{X}) ={3000\over 115.47}=25.98\)

\(\lambda_{Y}=25.98 <32\)

Slenderness ratio about Y-axis:

\(Slenderness\, ratio\, about\, Y-axis\, (\lambda_{Y}) ={L_{eff-YY}\over r_{min}}\)

\(Slenderness\, ratio\, about\, Y-axis\, (\lambda_{Y}) ={2750\over 115.47}=23.815\)

\(\lambda_{Y}=23.815 <32\)

1. Short columns have a slenderness ratio of less than 32.
2. The medium column slenderness ratio is between 32 to 120.
3. The slenderness ratio of the long columns is greater than 120.

Hence the column is a short column.