Stresses acting on an infinitesimal soil element are shown in the figure (with \( \sigma_z>\sigma_x \)). The major and minor principal stresses are \( \sigma_1 \) and \( \sigma_3 \), respectively. Considering the compressive stresses as positive, which one of the following expressions correctly represents the angle between the major principal stress plane and the horizontal plane?
Show Hint
To calculate the angle between the principal stress plane and the horizontal, use the relationship derived from Mohr's circle: \( \tan \theta = \frac{\tau_{zx}}{\sigma_1 - \sigma_x} \).
The angle between the major principal stress plane and the horizontal plane can be determined using Mohr's circle for stress. From Mohr's circle, we can derive the following expression for the angle:
\[
\tan \theta = \frac{\tau_{zx}}{\sigma_1 - \sigma_x}
\]
Where:
- \( \theta \) is the angle between the major principal stress plane and the horizontal plane,
- \( \tau_{zx} \) is the shear stress,
- \( \sigma_1 \) is the major principal stress,
- \( \sigma_x \) is the normal stress in the x-direction.
To find the angle \( \theta \), we take the inverse tangent (arctan) of the ratio of shear stress to the difference between the major principal stress and the normal stress in the x-direction:
\[
\theta = \tan^{-1} \left( \frac{\tau_{zx}}{\sigma_1 - \sigma_x} \right)
\]
Thus, the correct answer is Option (A).
