Plane stress with in-plane shear: Given normal stresses σx and σy on perpendicular planes and shear τxy, what is the maximum normal (principal) stress?

Introduction / Context:
Transformations of plane stress determine principal stresses and directions. The maximum normal stress controls yielding under maximum normal stress criteria and is widely used to assess safety margins in flat plates and machine elements.

Given Data / Assumptions:

• Plane stress state: σx, σy, τxy known.
• Homogeneous, isotropic, linear elastic material.
• Standard sign conventions.

Concept / Approach:
From Mohr's circle or transformation equations, principal stresses are
σ1,2 = (σx + σy)/2 ± sqrt( ((σx − σy)/2)^2 + τxy^2 )
The maximum normal stress is σ1, obtained using the plus sign.

Step-by-Step Solution:

Verification / Alternative check:
Special cases: if τxy = 0, σ1 = max(σx, σy); if σx = σy, σ1 = σx + |τxy|, matching Mohr's circle for pure shear superposed on hydrostatic stress.

Why Other Options Are Wrong:

• (σx + σy)/2 is the center value, not the maximum principal stress.
• |σx − σy|/2 is valid only when τxy = 0 and then equals the radius, not σ1 directly.
• sqrt( σx^2 + σy^2 + τxy^2 ) is not a principal-stress formula.
• (σx − σy)/2 + τxy is not invariant and lacks proper vector combination.

Common Pitfalls:
Dropping the 1/2 inside the square root; mixing up radius and diameter on Mohr's circle; sign mistakes for τxy.

Final Answer:
(σx + σy) / 2 + sqrt( ((σx − σy) / 2)^2 + τxy^2 )