Correct Answer - Option 3 : Area of cross-section
Explanation:
Column:
- If a beam element is under a compressive load and its length is an order of magnitude larger than either of its other dimensions such a beam is called a column.
- Due to its size, its axial displacement is going to be very small compared to its lateral deflection called buckling.
Euler's Buckling Load:
\( {P_{cr}} = \frac{{{{\rm{\pi }}^2}E{I_{\min }}}}{{l_e^2}}\)
where, Pcr = crtical load for buckling; E = Young's Modulus (GPa); Imin = Area moment of inertia, le = effective length; kmin = minimum raduis of gyration
Slenderness Ratio (S):
The ratio between the length and least radius of gyration.
\(S = \frac{{{l_e}}}{{{k_{\min }}}}\)
Hence, buckling load does not depend on the area of the cross-section.
Buckling load for various end conditions is given in the table below.
End conditions
|
Le
|
Buckling load |
Both ends hinged
|
Le = L
|
\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{L^2}}\) |
Both ends fixed
|
\({L_e} = \frac{L}{2}\)
|
\({P_b} = \frac{{{4\pi ^2}E{I_{}}}}{{L^2}}\) |
One end fixed and another end is free
|
Le = 2L
|
\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{4L^2}}\) |
One end fixed and another end is hinged
|
\({L_e} = \frac{L}{{\sqrt 2 }}\)
|
\({P_b} = \frac{{{\pi ^2}E{I_{}}}}{{2L^2}}\) |