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.

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:

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: