Combined stresses – Minimum normal (principal) stress under biaxial normal stresses σx, σy with in-plane shear τxy\nWhen a body is subjected to direct stresses σx and σy on mutually perpendicular planes and a shear stress τxy, what is the minimum normal stress?

Introduction / Context:
Determining principal stresses under combined biaxial normal stresses with shear is fundamental for failure analysis using criteria such as maximum normal stress, maximum shear stress, or distortion energy.

Given Data / Assumptions:

• Plane stress state with σx, σy, and τxy defined positive by standard sign convention.
• Linear elastic behavior; principal planes are orthogonal.

Concept / Approach:
Principal stresses are the extrema of normal stress on rotated planes where shear becomes zero. They are obtained from Mohr’s circle or the characteristic equation of the 2D stress tensor.

Step-by-Step Solution:
Principal stresses: σ1,2 = (σx + σy)/2 ± sqrt( ((σx − σy)/2)^2 + τxy^2 ).By definition, the minimum normal stress is σmin = (σx + σy)/2 − sqrt( ((σx − σy)/2)^2 + τxy^2 ).The maximum normal stress is the corresponding + sign.

Verification / Alternative check:
Construct Mohr’s circle with center at (σx + σy)/2 on the σ-axis and radius R = sqrt( ((σx − σy)/2)^2 + τxy^2 ). The leftmost intersection gives σmin as stated.

Why Other Options Are Wrong:
Option (b) is σmax; (c) is not a correct general expression; (d) and (e) are not invariant and ignore the quadratic coupling between σx, σy, and τxy.

Common Pitfalls:
Dropping the square on τxy; mixing up σmax and σmin; using absolute values incorrectly.

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