Principal stresses and planes: Through a point in a loaded soil mass, how many mutually orthogonal planes exist on which the stress is purely normal (no shear stress acts)?

Introduction / Context:
Stress at a point in a continuous medium can be represented by a second-order tensor. On certain planes passing through that point, called principal planes, shear stresses vanish and only normal stresses act. Understanding this concept is foundational for failure criteria and stress transformations in soil mechanics.

Given Data / Assumptions:

• Three-dimensional stress state in a continuum.
• Real, symmetric Cauchy stress tensor at a point.
• Materials considered are classical continua without couple stresses.

Concept / Approach:
The stress tensor has three real eigenvalues (principal stresses) with associated mutually orthogonal eigenvectors (principal directions). Planes normal to these principal directions are the principal planes. On each principal plane, shear stress is zero and only the corresponding principal normal stress acts.

Step-by-Step Solution:

Verification / Alternative check:
Mohr’s circles (3D) also reflect three principal stresses where the circles intersect the normal-stress axis; shear is zero at these points.

Why Other Options Are Wrong:
One or two planes would be incomplete; four mutually orthogonal planes cannot exist in 3D Euclidean space.

Common Pitfalls:
Confusing 2D plane stress (two principal planes) with the full 3D case, which has three.

Final Answer:
3