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

NoteSign 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