Mohr's Circle Calculator

Formula

• Center = (σx + σy) / 2
• Radius R = √((σx − σy)²/4 + τxy²)
• σ1,2 = center ± R
• τ_max = R
• θp = ½ atan(2τxy / (σx − σy))

About this calculator

Part of our Mechanical Calculators collection. Beams, bolts, springs, gears, bearings, and core ME calculations.

How it works

Mohr's circle center is the average of the two normal stresses. Radius comes from normal stress difference and shear stress. Principal stresses are center plus and minus radius. Use principal stresses for yield checks and fatigue screening on 2D stress states.

Enter your values in the inputs above and click CALCULATE. Results appear on the right without a page reload. No login and no server upload.

Worked example

σx = 80 MPa, σy = −40 MPa, τxy = 30 MPa.

1. Center = 20 MPa
2. R = √(60² + 30²) = 67.08 MPa
3. σ1 = 87.08 MPa, σ2 = −47.08 MPa
4. Principal angle θp = 13.28°

Result: σ1 = 87.1 MPa, σ2 = −47.1 MPa, τ_max = 67.1 MPa.

When to use

• Finding principal stresses from 2D stress components
• Maximum shear stress for yield or fatigue screening
• Orienting strain-gauge rosette results

Limitations

• Plane stress only (σz = 0, out-of-plane shear zero)
• Does not apply failure criteria (von Mises, Tresca) directly
• Sign convention: tension positive, clockwise shear positive on face

FAQ

What is principal angle θp?

θp is the angle of the principal plane from the x-face. The calculator outputs both 2θ (Mohr construction) and θp = 2θ/2.

How do I use principal stresses?

Compare σ1 and σ2 to yield strength for ductile materials, or use them in fatigue mean/stress amplitude calculations.