Academic Level: 2. YearAdvanced 15m Read
Mohr's Circle Stress Analysis (Principal Stresses)
In multi-axial loading, the stress at a point depends on the orientation of the plane you are looking at. Mohr's Circle is a powerful graphical method (and mathematical model) to find the maximum and minimum stresses a material will ever see.
Governing Formula
σ1,2 = (σx + σy)/2 ± √[((σx - σy)/2)² + τxy²]σ1,2Principal Stresses (MPa)
σx, σyNormal stresses in X and Y (MPa)
τxyShear stress (MPa)
Principal Stress Simulator
Physics Lab v2.1
Mohr's Circle Interactive Lab
Real-time Stress Transformation Engine
Live Simulation
Max Principal
Min Principal
Normal Stress σx100 MPa
Normal Stress σy40 MPa
Shear Stress τxy40 MPa
Principal σ1
120.0MPa
Principal σ2
20.0MPa
Max Shear τmax
50.0MPa
Princ. Angle θp
26.6°
Insight: The radius of the circle is equal to the maximum shear stress. Use this to determine if the material will fail under Tresca or Von Mises criteria.
Step-by-Step Calculation
- 1Determine the initial stress state (σx, σy, τxy).
- 2Plot the points (σx, τxy) and (σy, -τxy) on the stress axes.
- 3Draw the circle passing through these two points.
- 4Locate the center and calculate the radius (R).
- 5Identify σ1 (max) and σ2 (min) as the intersections with the horizontal axis.
Worked Example
Input Parameters
- σx = 100 MPa
- σy = 40 MPa
- τxy = 40 MPa
Calculation
Center = 70. Radius = √[(30)² + 40²] = 50. → σ1 = 120 MPa, σ2 = 20 MPa.
Why This Matters
- Ductile materials typically fail due to maximum shear stress (τmax).
- Brittle materials fail due to maximum principal stress (σ1).
- Mohr's Circle allows designers to predict the angle of failure.
Common Mistakes
- Confusing the 2θ angle in Mohr's Circle with the θ angle in the physical world.
- Incorrectly plotting the sign of the shear stress.
- Forgetting to check the 3D stress state (Triaxial stress).
Reference Material & Handbooks
Mohr Dairesi Analizi Sunumu
Technical Q&A
What is a principal stress?
It is a normal stress that acts on a plane where the shear stress is exactly zero.
Live Simulation Engine
Open the Analysis Lab to visualize complex stress states