Combined plane stress: A body has tensile stresses of 1200 MPa and 600 MPa on mutually perpendicular planes, with a shear stress of 400 MPa on the same planes. What is the maximum in-plane shear stress?

Introduction / Context:In a general plane stress state, the maximum in-plane shear governs yielding for Tresca and influences fatigue assessments. It equals the radius of Mohr's circle constructed from the normal and shear components.

Given Data / Assumptions:

• σx = 1200 MPa (tension), σy = 600 MPa (tension).
• τxy = 400 MPa (shear on the same planes).
• Plane stress; linear elasticity.

Concept / Approach:Maximum in-plane shear stress is given by:tau_max = √[ ((σx − σy)/2)^2 + τxy^2 ]Geometrically, this is the radius of Mohr's circle.

Step-by-Step Solution:

Verification / Alternative check:Plot Mohr’s circle with points (σx, τxy) and (σy, −τxy). The radius measured horizontally equals 500 MPa, consistent with the computation.

Why Other Options Are Wrong:400 MPa: ignores the normal stress difference.900 MPa and 1400 MPa: overestimates inconsistent with the circle radius.300 MPa: equals (σx − σy)/2 only, not including shear.

Common Pitfalls:Using τmax = |σx − σy| / 2 (only valid when τxy = 0); arithmetic errors when squaring and adding.

Final Answer:

500 MPa