Combined loading — maximum normal stress for a member under direct tensile stress σx and in-plane shear τxy: Find the expression for the maximum principal (normal) stress.
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Aσ_max = σx/2 + sqrt((σx/2)^2 + τxy^2)
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Bσ_max = sqrt(σx^2 + 4τxy^2)
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Cσ_max = σx + τxy
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Dσ_max = σx/2 + τxy
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Eσ_max = σx
Answer
Correct Answer: σ_max = σx/2 + sqrt((σx/2)^2 + τxy^2)
Explanation
Introduction:Plane stress transformation is fundamental for components under combined normal and shear actions. Determining the maximum principal stress ensures safe design against tensile failure modes.Given Data / Assumptions:
- Plane stress with σy = 0 (uniaxial σx) and in-plane shear τxy.
- Linear elasticity; small strains.
Concept / Approach:The principal stresses are the eigenvalues of the 2D stress tensor. For σy = 0, the closed-form expression yields symmetric shift about σx/2 with a radius depending on τxy.Step-by-Step Solution:
General principal stress: σ_{1,2} = (σx + σy)/2 ± sqrt(((σx − σy)/2)^2 + τxy^2)With σy = 0: σ_{1,2} = σx/2 ± sqrt((σx/2)^2 + τxy^2)Therefore, maximum normal stress: σ_max = σx/2 + sqrt((σx/2)^2 + τxy^2)Verification / Alternative check:Mohr's circle: center at σx/2, radius sqrt((σx/2)^2 + τxy^2); the rightmost point is σ_max as stated.Why Other Options Are Wrong:
- sqrt(σx^2 + 4τxy^2): incorrect combination; ignores the mid-value shift.
- σx + τxy or σx/2 + τxy: dimensional mixing and incorrect transformation.
- σx: neglects shear completely.
Common Pitfalls:Forgetting σy term structure; sign conventions for τxy; misreading Mohr's circle center and radius.Final Answer:
σ_max = σx/2 + sqrt((σx/2)^2 + τxy^2)