Question

The effective length of a column of length \(L\) fixed against rotation and translation at one end and free at the other end is

• A. \(0.5L\)
• B. \(0.7L\)
• C. \(1.414L\)
• D. \(2.0L\)

Hint:

Effective length factors: \[ \boxed{ \begin{array}{c|c} \text{End Condition} & K\\ \hline \text{Pinned--Pinned} & 1\\ \text{Fixed--Fixed} & 0.5\\ \text{Fixed--Pinned} & 0.7\\ \text{Fixed--Free} & 2 \end{array} } \]

Answer 1

The Correct Option is D

Step 1: Picture how the column actually bends.
A column fixed at the base and completely free at the top (a cantilever column) bends like one quarter of a full sine wave, bulging outward only near the free end.
Step 2: Match this shape to an equivalent pin ended column.
A pin ended column of length $L_e$ buckles into a full half sine wave. Since a quarter sine wave of the cantilever is the same curve as half of a sine wave twice as long, the cantilever behaves like a pin ended column of length $2L$.
Step 3: Read off the effective length.
\[ L_e = 2L. \]
\[ \boxed{2L} \]