Principal stresses; Mohr circle at failure

Major and minor principal stresses

At any point in a stressed soil there are three mutually perpendicular planes on which the shear stress is zero. These are the principal planes, and the normal stresses on them are the principal stresses.

• Major principal stress $\sigma_1$: the largest of the three normal stresses.
• Minor principal stress $\sigma_3$: the smallest of the three normal stresses.
• The third is the intermediate principal stress $\sigma_2$.

Effect of increasing $\sigma_1$ with $\sigma_3$ constant

The deviator stress $\sigma_1 - \sigma_3$ increases, so the Mohr circle grows larger (its centre moves right and its radius increases). The maximum shear stress, $\frac{\sigma_1-\sigma_3}{2}$, also increases. When the circle becomes tangent to the Mohr-Coulomb envelope, the soil fails. $\sigma_1$ cannot be increased beyond this point, because a Mohr circle cannot cut the failure envelope: the soil has already failed.

Mohr circle at failure for a soil with only $\phi$ (cohesionless)

The failure line passes through the origin, and the circle is tangent to it:

 tau       /  tau = sigma tan(phi)
  |      T/
  |     /.-''-.
  |    /'      '.
  |  / phi  C    \
  +--+------+-----+---- sigma
     0     s3    s1

From the right-angled triangle $OTC$:

$$\sin\phi = \frac{\sigma_1-\sigma_3}{\sigma_1+\sigma_3}$$

and the failure plane is at $45^\circ+\phi/2$ to the major principal plane.