Question 1

How do you find principal stresses from Mohr's circle?

Accepted Answer

Principal stresses are σ₁,₂ = (σx+σy)/2 ± R, where R = √{[(σx−σy)/2]² + τxy²} is the radius of Mohr's circle. Construction steps: (1) Plot point A=(σx, τxy) and point B=(σy, −τxy) on the σ-τ plane; (2) the midpoint C = ((σx+σy)/2, 0) is the circle centre; (3) radius |CA| = R; (4) the circle intersects the σ-axis at σ₁ (right) and σ₂ (left). For example, σx=80, σy=−40, τxy=60 MPa: C=20 MPa, R≈84.9 MPa, σ₁≈104.9 MPa, σ₂≈−64.9 MPa.

Question 2

Where does maximum shear stress appear on Mohr's circle?

Accepted Answer

The maximum shear stress τ_max = R = (σ₁−σ₂)/2 corresponds to the top and bottom points of Mohr's circle (at σ = C). The planes of maximum shear are rotated 45° from the principal planes. On these planes the normal stress equals the average stress C = (σ₁+σ₂)/2. Von Mises yield: σ_eq = √(σ₁²−σ₁σ₂+σ₂²); Tresca yield: τ_max = σ_y/2.

Question 3

How is the principal stress angle θ_p calculated?

Accepted Answer

The principal stress angle is θ_p = (1/2) arctan(2τxy / (σx−σy)). This is the angle from the reference x-axis to the first principal stress direction. On Mohr's circle, the arc from point A=(σx, τxy) to σ₁ sweeps an angle of 2θ_p (angles on the circle are twice real-space angles). On principal planes the shear stress is exactly zero.

Question 4

Can Mohr's circle be extended to 3D stress states?

Accepted Answer

For a 3D stress state with three principal stresses σ₁ ≥ σ₂ ≥ σ₃, three Mohr's circles can be drawn (for the σ₁-σ₂, σ₂-σ₃, and σ₁-σ₃ planes). Any stress state on an arbitrary plane plots inside the outermost circle. The absolute maximum shear stress is τ_abs_max = (σ₁−σ₃)/2. For plane stress (σ₃=0), if σ₁ and σ₂ have the same sign, τ_abs_max = max(|σ₁|, |σ₂|)/2, which is larger than the in-plane τ_max.

Mohr's Circle Tool

Principal Stresses

Maximum Shear Stress

Principal Stress Angle

Steps to Draw Mohr's Circle