Stress concentration around an elliptical hole in a plate under tension Plate with a central elliptical hole: semi-major axis a is perpendicular to loading; semi-minor axis b is along the loading direction. Let σ be the nominal (no-hole) stress. Find the maximum stress at the hole edge.
Mechanical Engineering
Machine Design
Difficulty: Medium
Choose an option
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Aσ (1 + a/b)
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Bσ (1 + 2a/b)
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Cσ (1 + b/a)
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Dσ (1 + 2b/a)
Answer
Correct Answer: σ (1 + 2a/b)
Explanation
Given• Plate in tension with an elliptical hole (semi-major axis a ⟂ load, semi-minor axis b ∥ load).• Nominal stress (no hole) = σ.
ConceptThe stress concentration factor at the ends of the minor axis for an elliptical hole isK = 1 + 2(a/b)⇒ σmax = K · σ = σ (1 + 2a/b)
Step-by-step1) Identify orientation: a ⟂ load → the high-stress points lie on the ends of b.2) Use the classic elasticity result for an ellipse in an infinite plate.3) Multiply nominal stress by K to obtain σmax.
Common pitfallSwapping a and b reverses the factor; ensure correct axes relative to the loading direction.
Final Answerσ (1 + 2a/b)