The ratio of the theoretical critical buckling load for a column with fixed ends to that of another column with the same dimensions and material, but with pinned ends is equal to

Concept:

Theoretical critical buckling load is given by,

\({P_{cr}} = \dfrac{{{\pi ^2}EI}}{{L_{eff}^2}}\)

The above formula does not take into account the axial stress and the buckling load is given by this formula may be much more than the actual buckling load

Where Leff = effective length of column which depends on the end condition of the column.

Calculation:

∵ Critical buckling load, \({P_{cr}} = \dfrac{{{\pi ^2}EI}}{{L_{eff}^2}}\)

When both ends of the column are fixed then,

 \({P_{cr1}} = \dfrac{{{\pi ^2}EI}}{{{{\left( {L/2} \right)}^2}}} = \dfrac{{{\pi ^2}EI}}{{{L^2}}} × 4\)  ............................. (1)

When both ends of the column are hinged then critical buckling load,

 \({P_{cr2}} = \dfrac{{{\pi ^2}EI}}{{{L^2}}}\)  ....................... (2)

Dividing equations 1 and 2 we get

\({P_{cr1}\over P_{cr2}} = 4\)