Question

Which one of the following vertical columns, of circular cross-section, sustains the highest load without buckling?

• A. Cantilever column with a length L and cross-section diameter d
• B. Column with hinge at one end and roller at the other end with a length 2L and cross-section diameter d
• C. Cantilever column with a length L and cross-section diameter 2d
• D. Column with hinge at one end and roller at the other end with a length L and cross-section diameter d

Hint:

Euler load depends heavily on diameter (\(d^4\) effect) — doubling the diameter increases load capacity 16 times. Geometry dominates over length factor.

Answer 1

The Correct Option is C

Step 1: Euler’s buckling load formula.
\[ P_{cr} = \frac{\pi^2 EI}{(K L)^2} \] where - \(E\) = Young’s modulus, - \(I\) = area moment of inertia, - \(L\) = actual column length, - \(K\) = effective length factor (depends on end conditions).

Step 2: Effective length for each case.
- Cantilever (fixed–free): \(K = 2 \Rightarrow L_{eff} = 2L\). - Hinged–roller (same as pinned–pinned): \(K = 1 \Rightarrow L_{eff} = L\). So: - Option A: \(L_{eff} = 2L\), \(d\). - Option B: \(L_{eff} = 2L\), \(d\). - Option C: \(L_{eff} = 2L\), diameter \(2d\). - Option D: \(L_{eff} = L\), \(d\).

Step 3: Compare inertia effect.
For a circular cross-section: \[ I = \frac{\pi d^4}{64} \] If diameter is doubled (\(2d\)): \[ I \propto (2d)^4 = 16 d^4 \] So inertia increases 16 times.

Step 4: Critical load ratios.
- (A) Cantilever, \(I \propto d^4\), denominator \((2L)^2 = 4L^2 \Rightarrow P \propto \frac{d^4}{4L^2}\). - (B) Length \(2L\), effective length = \(2L \Rightarrow\) denominator \( (2L)^2 = 4L^2 \Rightarrow P \propto \frac{d^4}{4L^2}\). - (C) Cantilever, \(I \propto 16 d^4\), denominator \(4L^2 \Rightarrow P \propto \frac{16 d^4}{4L^2} = \frac{4 d^4}{L^2}\). - (D) Hinged–roller, effective length \(L\), denominator \(L^2 \Rightarrow P \propto \frac{d^4}{L^2}\). Clearly, (C) is 4 times stronger than (D) and much higher than (A) or (B). Final Answer:
\[ \boxed{\text{(C) Cantilever column with length L and diameter 2d}} \]