Under uniaxial direct tensile stress σ, the planes on which the shear stress is maximum are inclined at what angles to the normal of the section?

Introduction / Context:Under a uniaxial tensile state, the material experiences not only normal stress but also shear stress on oblique planes. Identifying the orientation of planes of maximum shear is crucial for understanding yielding and failure (e.g., ductile metals forming shear bands).

Given Data / Assumptions:

• Direct tensile stress σ acts in one direction only.
• No other normal or shear components initially.
• Linear elastic continuum assumptions.

Concept / Approach:Using stress transformation, the shear stress on a plane inclined at angle θ to the normal is tau(θ). For a uniaxial state, tau(θ) reaches its maximum when θ = 45° to the normal (i.e., plane oriented at 45° to the loading direction). The complementary plane at 135° also sees the same magnitude due to symmetry.

Step-by-Step Solution:

Verification / Alternative check:On Mohr’s circle for uniaxial stress, the maximum shear equals σ/2 and occurs at 2θ = 90°, confirming θ = 45°.

Why Other Options Are Wrong:Other angles do not maximize sin(2θ); 90° and 0° give zero shear for uniaxial tension.

Common Pitfalls:Confusing plane angle θ with physical fiber angle; mixing “plane angle” versus “direction angle.”

Final Answer:

45° and 135°