## Principal Stresses — Mohr's Circle

### Fundamental :

The state of stress at a point is typically represented by a cube with axes in the x, y, and z directions. The stress vector on each face is decomposed into a normal stress (e.g., in the x-direction) and two shear stresses (e.g., in the y and z directions).

For equilibrium, shear stresses on perpendicular planes must be equal ( τ xy = τ yx τ_{xy}=τ_{yx} ).

### Definition :

- σ xx %sigma _{xx} : Normal stress on the plane perpendicular to the x-direction.

- τ xy %tau _{xy} : Shear stress on the plane perpendicular to the x-direction and acting in the y-direction.

- τ xz %tau _{xz} : Shear stress on the plane perpendicular to the x-direction and acting in the z-direction.

Stresses on an elemental cube

### Note : Sign Convention

Compressive stresses are considered positive, as they are most common in soil mechanics. The sign convention for shear stresses is shown in the figure below .

Sign convention for positive stresses

For two perpendicular planes, the normal and shear stresses on any other plane can be related to the stresses on the perpendicular planes as follows:

Wedge in equilibrium under normal and shear stresses

From equilibrium conditions, the following equations are derived:

σ y A cos α + τ xy A sin α + τ A sin α − σ A cos α = 0 %sigma _{y}A cos %alpha +%tau _{xy}A sin %alpha +%tau A sin %alpha -%sigma A cos %alpha =0

σ x A sin α − τ yx A cos α − τ A cos α − σ A sin α = 0 %sigma _{x}A sin %alpha -%tau _{yx}A cos %alpha -%tau A cos %alpha -%sigma A sin %alpha =0

Where A in the surface of the inclined face.

σ = σ y + σ x 2 + σ y − σ x 2 cos 2 α − τ xy sin 2 α %sigma = {%sigma_{y}+%sigma_{x}} over {2} +{%sigma_{y}-%sigma_{x}} over {2} cos 2 %alpha-%tau_{ xy} sin 2 %alpha

τ = σ y − σ x 2 sin 2 α − τ xy cos 2 α %tau = {%sigma_{y}-%sigma_{x}} over {2} sin 2 %alpha-%tau_{ xy} cos 2 %alpha

If the planes perpendicular to the x and y directions are principal planes (zero shear stress), the equations simplify to:

σ = σ 1 + σ 3 2 + σ 1 − σ 3 2 cos 2 α %sigma = {%sigma_{1}+%sigma_{3}} over {2} + {%sigma_{1}-%sigma_{3}} over {2}cos2%alpha

τ = σ 1 − σ 3 2 sin 2 α %tau = {%sigma_{1}-%sigma_{3}} over {2}sin2%alpha

Where σ 1 %sigma_{1} and σ 3 %sigma_{3} are the major and minor principal stresses, respectively.

These equations yield the relationship:

[ σ − σ 1 + σ 3 2 ] 2 + τ 2 = ( σ 1 − σ 3 2 ) 2 [%sigma-{%sigma_{1}+%sigma_{3}} over {2}] ^2+%tau^2=( {%sigma_{1}-%sigma_{3}} over {2} )^2

This is the equation of a circle (Mohr's Circle[1]) with center at ( σ 1 + σ 3 2 {%sigma_{1}+%sigma_{3}} over {2} ,0) and radius σ 1 − σ 3 2 {%sigma_{1}-%sigma_{3}} over {2}

An example of Mohr's Circle for a triaxial test at failure is shown in the figure below

Mohr's Circle for a triaxial test

## Notes

1. Mohr's Circle

   A graphical representation of the state of stress at a point.