Mohr’s Circle Explained: Principal Stresses & Maximum Shear

1. What is Mohr’s Circle?

Mohr’s circle is a graphical representation of the plane-stress transformation equations. Every stress state (σx, σy, τxy) maps to a circle whose geometry encodes the principal stresses, the maximum shear stress, and the orientation of the principal planes — without solving any transcendental equations.

2. Sign Conventions

• Tensile normal stress is positive (compression negative) on the σ-axis.
• Counter-clockwise shear is positive on the face, plotted positive on the τ-axis.
• Point A(σx, τxy) represents the x-face; point B(σy, −τxy) represents the y-face — the shear components are equal and opposite across perpendicular faces.

3. Construction: Center and Radius

4. Principal Stresses & Orientation

• Principal stresses: σ₁,₂ = σ_avg ± R, the two intersections of the circle with the σ-axis.
• Principal plane orientation: tan(2θp) = 2τxy / (σx − σy) — rotate the σ-axis 2θp from the radius to A.
• Maximum shear: τ_max = R, acting on planes rotated 45° from the principal planes (θp + 45°).
• Invariant: I₁ = σx + σy = σ₁ + σ₂ — a built-in arithmetic self-check.

5. Worked Example: Biaxial Tension

Take σx = 80 MPa, σy = 40 MPa, τxy = 20 MPa — the standard example in Mohr Master's suite.

6. Worked Example: Pure Shear

For a shaft in pure torsion, σx = σy = 0 with τxy = 50 MPa:

Pure shear is the classic result: equal and opposite principal stresses of magnitude τ_max at 45° — exactly why ductile shafts in torsion fail along 45° helices.