Stress Tensorσij = stress tensori,j = 1,2,3i = j → normal stress (σ)i ≠ j → shear stress (τ)Given that:σij = σjiThe solution of many problems in soil mechanics is facilitated because the three-dimensional stress state is simplified and converted to two-dimensional space.
Use of the Mohr’s circle in soil mechanics
If σ₁ = the maximum principal stress, σ₃ = the minimum principal stress and σ₂ = the intermediate principal stress, in soil mechanics σ₁ and σ₃ play the predominant role, while the intermediate stress σ₂, in many circumstances doesn’t affect the mechanical behavior of soils. For this reason, in soil mechanics, we commonly examine plane stress conditions (on a plane perpendicular to the direction of σ₂) using the maximum and the minimum principal stress, σ₁ and σ₃, respectively. The Mohr’s circle is a circle in the Cartesian coordinate system, the abscissa and the ordinate (σ,τ) of each point on the circle are the normal stress and the shear stress components. The center of circle has coordinates:
and the radius of the circle is:
The Mohr’s circle has the characteristic property that the abscissa and ordinate (σθ,τθ) at a point A of the circle give us the stress state on a plane where the vertical of the plane forms an angle θ with respect to the direction of the principal stress σ₁. In soil mechanics, the stresses are simulated as follows:
normal stresses: compressive stresses are taken with a positive sign.
shear stresses: they are taken with a positive sign when they tend to rotate the plane on which they act counterclockwise.
Basic equations for the Mohr's circle
Basic observations about Mohr’s circle
The maximum shear stress is
The sum of the normal stresses acting on two planes perpendicular to each other, is constant and equal to (σ₁ + σ₃).
The shear stresses acting on two planes perpendicular to each other, are equal but have opposite signs.
Plane Principle (or Pole)
It’s a point on the Mohr’s circle (which is symbolized with O🇵) for which: if we draw a line parallel with the plane of which we want to define the stress state (σ,τ), the line will cross the circle at a point with coordinates (σ,τ).
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