Thin spherical shell under internal pressure p: is the maximum in-plane shear stress in the shell wall equal to zero?

Introduction / Context:
Thin spherical pressure vessels (e.g., gas spheres) have equal biaxial tensile stresses on their walls under internal pressure. Determining the maximum shear stress helps in applying yield criteria and comparing with cylindrical shells.

Given Data / Assumptions:

• Thin spherical shell, thickness small relative to diameter.
• Internal pressure p; negligible bending and radial stress.
• Uniform biaxial membrane stress state.

Concept / Approach:
For a thin sphere, principal stresses are equal in all in-surface directions: sigma_1 = sigma_2 = p d / (4 t). Maximum in-plane shear stress equals (sigma_1 − sigma_2) / 2. If sigma_1 = sigma_2, this difference is zero, so the maximum in-plane shear stress vanishes.

Step-by-Step Solution:

Verification / Alternative check:
Mohr’s circle reduces to a single point at sigma = p d / (4 t) when principal stresses are equal—its radius (shear) is zero.

Why Other Options Are Wrong:
Non-zero shear would require unequal principal stresses (e.g., cylinders where sigma_h ≠ sigma_l).

Common Pitfalls:
Applying cylindrical shell results to spheres; forgetting that equal principal stresses imply zero in-plane shear.

Final Answer:

Yes