Understanding the Failure Plane Angle

This question relates to the orientation of the failure plane in soil or rock mechanics, specifically within the context of the Mohr-Coulomb failure criterion. We need to determine the angle between the failure plane and the major principal stress plane.

Mohr-Coulomb Failure Criterion Explained

The Mohr-Coulomb criterion is a widely used model to predict the shear strength of soils and rocks. It defines the condition under which a material fails when subjected to shear stress. The criterion depends on the material's effective angle of internal friction ($ \phi' $) and cohesion ($ c' $).

When analyzing stresses on a failure plane using Mohr's circle, the angle of this plane relative to the principal stress axes is important. The major principal stress is denoted by $ \sigma_1 $, and the minor principal stress is denoted by $ \sigma_3 $.

Determining the Failure Plane Orientation

In geotechnical engineering, the angle ($ \alpha $) of the failure plane, measured from the major principal plane (the plane subjected to $ \sigma_1 $) towards the direction of shear failure, is given by a specific formula derived from the Mohr-Coulomb theory.

The relationship is established using the geometry of the Mohr's circle at the point of failure. The standard formula for this angle is:

$ \alpha = 45^\circ + \frac{\phi'}{2} $

In this formula:

• $ \alpha $ represents the angle of the failure plane with respect to the major principal plane ($ \sigma_1 $).
• $ \phi' $ is the effective angle of internal friction of the soil or rock.

This angle indicates the orientation at which shear failure is most likely to occur under the given principal stresses.

Evaluating the Options Provided

Let's compare the standard formula with the options given:

• Option 1: \( 45^\circ + \phi' \). This option adds the full angle of internal friction, which is not the correct formula for the failure plane angle relative to $ \sigma_1 $.
• Option 2: \( \(45^\circ+\frac{\phi^{'}}{2}\) \). This option correctly represents the angle of the failure plane with the major principal plane according to the Mohr-Coulomb criterion.
• Option 3: \( \(45^\circ-\frac{\phi^{'}}{2}\) \). This angle ($ 45^\circ - \frac{\phi'}{2} $) typically represents the angle of the failure plane measured from the minor principal plane ($ \sigma_3 $) or may be used in different contexts or conventions.
• Option 4: \( 45^\circ - \phi' \). Similar to option 1, this does not represent the standard angle derived from the theory.

Conclusion on Failure Plane Angle

Based on the established principles of soil mechanics and the Mohr-Coulomb failure criterion, the angle of the failure plane with the major principal plane ($ \sigma_1 $) is $ 45^\circ + \frac{\phi'}{2} $. Therefore, option 2 accurately provides this relationship.