A body under uniaxial direct tensile stress σ experiences a maximum shear stress equal to what fraction of the maximum normal stress?

Introduction / Context:
Relating normal and shear stresses helps predict yielding and failure using criteria such as Tresca and von Mises. In a uniaxial stress state, maximum shear stress occurs on planes at 45° to the load axis.

Given Data / Assumptions:

• Uniaxial tension: principal stresses are σ1 = σ and σ2 = 0 (plane stress).
• Small deformation linear elasticity.
• Shear stress sign convention standard.

Concept / Approach:
Maximum shear stress in plane stress is half the difference of principal stresses:
tau_max = (σ1 − σ2) / 2 = (σ − 0) / 2 = σ/2Maximum normal stress equals σ in the loaded direction. Therefore, the ratio is 1/2.

Step-by-Step Solution:

Verification / Alternative check:
Mohr’s circle has radius R = (σ1 − σ2)/2 = σ/2, which is the maximum shear; the horizontal intercept equals σ, confirming the ratio.

Why Other Options Are Wrong:
Equal/Twice: Overestimate shear; Two-thirds has no basis for uniaxial stress; Zero is only for hydrostatic stress (σ1 = σ2).

Common Pitfalls:
Confusing maximum shear for 3D principal differences; mixing engineering and true stress at large strains (not relevant here).

Final Answer:

One-half of σ