Principal Stresses and Strains MCQ with answer and solution

Concept

Shear stress in hydrostatic state of stress

Formula

τ = 0 (Mohrs circle radius R = 0)

Step-by-step solution

1. Under a state of hydrostatic pressure p = 30 MPa, equal normal compressive stresses act in all directions: σ_x = σ_y = σ_z = -30 MPa. 2. On any plane oriented at arbitrary angles, the normal stress is identically -30 MPa. 3. The radius of Mohr's circle is R = (σ₁ - σ₂) / 2 = (-30 - (-30)) / 2 = 0 MPa. 4. Therefore, shear stress on any plane or face is exactly 0 MPa. Answer: B

Explanation

Under a state of hydrostatic pressure (hydrostatic stress), equal compressive normal stresses act on all mutually perpendicular faces (σ_x = σ_y = σ_z = -p = -30 MPa) and all shear stresses on orthogonal planes are zero. The corresponding 3D Mohr's circle degenerates into a single point on the normal stress axis.

Why the other options are wrong

A) 30 MPa is the normal hydrostatic pressure, not shear stress. C) 15 MPa is half pressure, which would only apply in uniaxial tension/compression. D) 60 MPa is double the pressure.

Shortcut

Hydrostatic pressure = Spherical stress tensor => Shear stress on all planes is zero.

Common mistake

Assuming shear stress equals the hydrostatic pressure (30 MPa) or half of it (15 MPa).