Maximum In-Plane Shear Stress under Combined σx and τxy A body is subjected to a direct tensile stress (sigma_x) in one plane and a simple shear stress (tau_xy). What is the expression for the maximum in-plane shear stress?

Introduction:
The goal is to identify the correct formula for maximum in-plane shear stress when a member has one normal stress and an in-plane shear stress.

Given Data / Assumptions:

• Plane stress with σx given and σy assumed zero unless stated.
• Non-zero in-plane shear τxy.
• Linear elasticity; Mohr’s circle relations apply.

Concept / Approach:
For plane stress, the Mohr’s circle radius equals the maximum in-plane shear stress. With σy = 0: tau_max = sqrt( ((σx - σy)/2)^2 + τxy^2 )Since σy = 0, tau_max = sqrt( (σx/2)^2 + τxy^2 )

Step-by-Step Solution:
Start from general radius: R = sqrt( ((σx - σy)/2)^2 + τxy^2 ).Set σy = 0 to get R = sqrt( (σx/2)^2 + τxy^2 ).Therefore maximum in-plane shear stress equals R, giving the required expression.

Verification / Alternative check:
Alternative form sometimes shown is: tau_max = 0.5 * sqrt( (σx - σy)^2 + 4 * τxy^2 ) Substituting σy = 0 reduces it to 0.5 * sqrt( σx^2 + 4 * τxy^2 ) which is algebraically identical to the chosen expression after simplifying under the square root.

Why Other Options Are Wrong:

• (σx + τxy)/2 and (σx − τxy): Not principal-shear relations; incorrect dimensionally.
• σx/2: Only valid when τxy = 0 and σy = 0.
• 0.5 * sqrt(σx^2 + 4 τxy^2): Equivalent form but missing the explicit σy assumption; preferred canonical answer is the radius form shown in option B.

Common Pitfalls:
Confusing principal normal stresses with principal shear and omitting the division by 2 within the radius term.

Final Answer:
tau_max = sqrt( (sigma_x / 2)^2 + tau_xy^2 )