engineering

Mohr's Circle Calculator

Calculate principal stresses, maximum shear stress, and principal angle from a 2D stress state

Mohr's Circle Calculator

Find principal stresses, maximum shear stress, and principal angle from a 2D plane stress state. Positive shear τ_xy follows the convention: CW shear on the x-face is positive.

Principal Stresses

σ₁ (Maximum)

87.082 MPa

σ₂ (Minimum)

-47.082 MPa

Shear and Center

τ_max (in-plane)

67.082 MPa

σ_avg (circle center)

20.000 MPa

Radius R

67.082 MPa

Principal Angles

θ_p1 (to σ₁ plane)

13.28°

θ_p2 (to σ₂ plane)

103.28°

θ_s (to τ_max plane)

-31.72°

Formulas

σ_avg = (σ_x + σ_y) / 2 = (80.0 + -40.0) / 2 = 20.000 MPa
R = √[((σ_x−σ_y)/2)² + τ_xy²] = 67.082 MPa
σ₁ = σ_avg + R = 20.000 + 67.082 = 87.082 MPa
σ₂ = σ_avg − R = 20.000 − 67.082 = -47.082 MPa
τ_max = R = 67.082 MPa
θ_p = ½ · atan2(2τ_xy, σ_x−σ_y) = 13.28°

Stress State Summary

Mixed state — one tensile, one compressive principal stress.

Maximum absolute stress: 87.082 MPa

How to Use This Calculator

1

Enter the 2D stress state

Input the normal stresses σx and σy, and the shear stress τxy at a point in the material.

2

View the Mohr's Circle diagram

The interactive diagram plots the stress transformation circle with labeled principal stresses and maximum shear stress.

3

Read principal stresses and angles

The calculator outputs σ1, σ2 (principal stresses), τmax (maximum shear), and the principal angle θp.

Formula Reference

Principal Stresses

σ₁,₂ = (σx + σy)/2 ± √[((σx−σy)/2)² + τxy²]

Variables: σx, σy = normal stresses on x and y planes, τxy = shear stress

Maximum Shear Stress

τmax = √[((σx−σy)/2)² + τxy²]

Variables: τmax = maximum in-plane shear stress

When to Use This Calculator

  • •Use the Mohr's Circle Calculator when solving homework or exam problems that require quick numerical verification of your hand calculations.
  • •Use it during the design phase to rapidly iterate on parameters before running detailed finite element simulations.
  • •Use it when reviewing a colleague's calculation or checking a vendor's data sheet for plausibility.
  • •Use it to generate reference data for a technical report or presentation without manual computation.

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