Mohr–Coulomb Failure – Orientation of Failure Plane Relative to Major Principal Plane According to Mohr–Coulomb theory, the angle between the failure plane and the major principal plane is equal to:

Introduction / Context:
Failure planes in soils orient themselves relative to principal stress directions in a way governed by Mohr–Coulomb strength. Recognizing these angles aids in interpreting triaxial tests, shear bands in the field, and active/passive earth pressure mechanisms.

Given Data / Assumptions:

• Material follows Mohr–Coulomb with angle of shearing resistance φ.
• Principal stresses are well-defined at failure.

Concept / Approach:

From Mohr’s circle at failure, the plane of maximum shear stress makes an angle of 45° with the major principal plane. Due to frictional strength (φ), the actual failure plane is inclined by an additional φ/2 from the maximum shear direction, yielding an angle of 45° + φ/2 between the failure plane and the major principal plane.

Step-by-Step Solution:

Verification / Alternative check:

Direct derivation from Mohr’s circle geometry shows the failure plane forms an angle of 45° + φ/2 with σ1 and 45° − φ/2 with σ3, consistent with classic earth pressure theory as well.

Why Other Options Are Wrong:

Options adding 90° are geometrically incorrect; 45° − φ/2 corresponds to the angle with the minor principal plane; coincidence (zero angle) is impossible except in trivial cases.

Common Pitfalls:

Confusing angles measured from principal planes vs principal directions; misapplying the ±φ/2 shifts.

Final Answer:

45° + half of the angle of shearing resistance