$${\text{P}} = \frac{{4{\pi ^2}{\text{E}}I}}{{{{\text{L}}^2}}}$$ is the equation of Euler’s crippling load if A. Both the ends are fixed B. Both the ends are hinged C. One end is fixed and other end is free D. One end is fixed and other end is hinged

Both the ends are fixed
Both the ends are hinged
One end is fixed and other end is free
One end is fixed and other end is hinged

The correct answer is: C. One end is fixed and other end is free.

Euler’s crippling load is the maximum load that a column can support without buckling. The equation for Euler’s crippling load is:

$$P = \frac{{4{\pi ^2}{\text{E}}I}}{{{{\text{L}}^2}}}$$

where:

  • $P$ is the Euler’s crippling load
  • $E$ is the Young’s modulus of the material
  • $I$ is the moment of inertia of the cross-section of the column
  • $L$ is the length of the column

For a column with one end fixed and the other end free, the moment of inertia is given by:

$$I = \frac{1}{3}bh^3$$

where:

  • $b$ is the width of the column
  • $h$ is the height of the column

Substituting this into the equation for Euler’s crippling load gives:

$$P = \frac{{4{\pi ^2}{\text{E}} \left( \frac{1}{3}bh^3 \right)}}{{{{\text{L}}^2}}}$$

$$P = \frac{{16{\pi ^2}{\text{E}}bh^2}}{{9{{\text{L}}^2}}}$$

Therefore, the equation for Euler’s crippling load for a column with one end fixed and the other end free is:

$$P = \frac{{16{\pi ^2}{\text{E}}bh^2}}{{9{{\text{L}}^2}}}$$

The other options are incorrect because they do not give the correct equation for Euler’s crippling load for a column with one end fixed and the other end free.