What is the expression for the crippling load of a column with both ends hinged?
Try it first: pick an option, then read the explanation.
Correct answer: C. P = π²EI/l²
Asked in SSC JE 23/09/2019 (Morning)
Detailed Explanation
The pin-ended column is Euler's fundamental case. Its buckled shape is a half sine wave between the two hinges.
\[ EI\frac{d^2y}{dx^2} = -Py \;\Rightarrow\; y = C\sin\!\left(x\sqrt{P/EI}\right),\quad y(l)=0 \Rightarrow l\sqrt{P/EI} = \pi \;\Rightarrow\; P = \frac{\pi^2EI}{l^2} \]
Why “P = π²EI/l²” Is Correct
With le = l, the crippling load is P = π²EI/l².
Why the Other Options Are Wrong
- A. P = π²EI/(6l²): No standard end condition gives π²EI/6l².
- B. P = 3π²EI/(6l²): 3π²EI/6l² = π²EI/2l² — not a standard case.
- D. P = 5π²EI/l²: 5π²EI/l² does not match any standard case (fixed–fixed gives 4π²EI/l²).
Key Point
Pin-ended column: P = π²EI/l² — every other case is this formula with le in place of l.
Key Concepts for Students
- Use the least I of the section (buckling occurs about the weaker axis).
- Euler's load is independent of the yield strength.
- Fixed–fixed gives 4×, fixed–hinged 2×, fixed–free 1/4× this load.
Practice more: this is question 7 in Columns and Struts MCQs – Part 1. Try the whole set and check every answer instantly.