Slenderness ratio of a square column A vertical column has a square cross-section of side 10 cm and an overall (effective) length of 500 cm with pinned ends. Compute its slenderness ratio λ = L_e / r.
Mechanical Engineering
Strength of Materials
Difficulty: Easy
Choose an option
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A117.2
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B17.3
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C173.2
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D137.2
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E13.72
Answer
Correct Answer: 173.2
Explanation
Introduction / Context:Slenderness ratio indicates the susceptibility of a column to buckling. For a given effective length L_e and radius of gyration r, λ = L_e / r classifies a column as short, intermediate, or long.
Given Data / Assumptions:
- Square cross-section: side a = 10 cm.
- Overall (effective) length L_e = 500 cm (pinned–pinned).
- Radius of gyration r = sqrt(I/A).
Concept / Approach:For a square of side a: I = a^4 / 12 and A = a^2, so r = sqrt(I/A) = a / √12. Then λ = L_e / r.
Step-by-Step Solution:
Compute r = a / √12 = 10 / 3.464 = 2.8868 cm (approx.).Compute λ = L_e / r = 500 / 2.8868 ≈ 173.2.Match to options ⇒ 173.2.Verification / Alternative check:Using r^2 = I/A = (a^4/12) / a^2 = a^2/12 yields the same r value.
Why Other Options Are Wrong:
- 117.2 and 137.2 arise from incorrect r or L assumptions.
- 17.3 and 13.72 are off by an order of magnitude, likely mixing cm and mm.
Common Pitfalls:Confusing effective length with clear length, and mixing units; here all dimensions are in centimetres.
Final Answer:173.2