General state of plastic equilibrium

A soil mass is in a state of plastic equilibrium when every point in it is on the verge of shear failure; the shear stress at each point equals the shear strength given by the Mohr-Coulomb criterion $\tau_f = c + \sigma\tan\phi$. The Mohr circle of stress at every point then just touches the failure envelope.

• Rankine's general state: this condition exists throughout the whole semi-infinite soil mass, which happens when the soil is stretched or compressed horizontally enough.
• Active state: the soil expands horizontally (e.g. the wall moves away). The vertical stress $\sigma_v$ is the major principal stress and the horizontal stress falls to the minimum, $\sigma_h = K_a\sigma_v - 2c\sqrt{K_a}$.
• Passive state: the soil is compressed horizontally. The horizontal stress becomes the major principal stress, $\sigma_h = K_p\sigma_v + 2c\sqrt{K_p}$.
• The failure planes make an angle of $45^\circ + \phi/2$ with the major principal plane. This is $45^\circ + \phi/2$ to the horizontal in the active state and $45^\circ - \phi/2$ to the horizontal in the passive state.

It is the basis of Rankine's earth pressure theory.