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.

Difficulty: Medium

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)

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