Mohr's Circle and Principal Stresses

In case of the general 3-dimensional state stress in structures is represented by 6 stress values.
For this situation principal stresses and principal stress directions are calculated.
The principal stresses σ₁, σ₂, σ₃ are the eigenvalues of the stress tensor S:

σ_x	τ_xy	τ_xz
τ_yx	σ_y	τ_yz
τ_zx	τ_zy	σ_z

The principal stresses and the Mohr circles are visualized graphically.
In the shaded areas between the circles possible pairs of stress values (σ, τ) may be found.
The 3 red points represent stress values according to the given values with repespect to a xyz-coordinate system:
(σ_x, (τ_xy²+τ_xz²)^1/2), (σ_y, (τ_yx²+τ_yz²)^1/2) , (σ_z, (τ_zx²+τ_zy²)^1/2)
The yellow points mark the principal stresses. Under the associated directions there will be no shear stresses.

A special case is the 2-dimensional stress state given by σ_x, σ_y, τ_xy whereas σ_z=τ_yz=τ_zx=0.
In this case we can determine 2 points on Mohr's circle:
(σ_x, τ_xy) and (σ_y, -τ_xy).
The center of the circle is ((σ_x + σ_y)/2, 0).

For a normalized direction vector n the normal stress σ_n and shear stress τ_n will be calculated:
σ_n = n^T S n
|τ_n| = (n^TS^T S n - σ_n²)^1/2.

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