Mohr's Circle Calculator
Calculate principal stresses, maximum shear stress, and perform 2D stress analysis
2D Stress State
Stress Units
Normal Stresses
Normal stress acting perpendicular to X-face
Normal stress acting perpendicular to Y-face
Shear Stress
Shear stress acting on X-Y plane
Units for angle results
Material Reference (Optional)
For safety factor calculations
Mohr's Circle Analysis Results
Formulas used:
Principal stresses: σ₁,₂ = (σxx + σyy)/2 ± √[((σxx - σyy)/2)² + τxy²]
Maximum shear stress: τmax = (σ₁ - σ₂)/2 = √[((σxx - σyy)/2)² + τxy²]
von Mises stress: σvM = √(σ₁² - σ₁σ₂ + σ₂²)
Angle of orientation: 2θ = arctan(2τxy/(σxx - σyy))
Example Calculation
2D Stress State Analysis
Given: σxx = 100 MPa, σyy = 40 MPa, τxy = 30 MPa
Center: (σxx + σyy)/2 = (100 + 40)/2 = 70 MPa
Radius: √[((100-40)/2)² + 30²] = √[30² + 30²] = 42.43 MPa
Principal Stresses
σ₁ = 70 + 42.43 = 112.43 MPa (Maximum)
σ₂ = 70 - 42.43 = 27.57 MPa (Minimum)
τmax = 42.43 MPa
θ = 0.5 × arctan(2×30/(100-40)) = 22.5°
von Mises: σvM = √(112.43² - 112.43×27.57 + 27.57²) = 98.7 MPa
Stress Components
Principal Stresses
Failure Criteria
Typical Yield Strengths
Analysis Tips
Principal stresses are at 45° to max shear
No shear stress on principal planes
Circle center = average normal stress
Circle radius = maximum shear stress
Use von Mises for ductile materials
Understanding Mohr's Circle
What is Mohr's Circle?
Mohr's Circle is a graphical representation of the 2D stress state at a point in a material. It provides a visual method for analyzing stress transformations and determining principal stresses, maximum shear stresses, and the orientation of critical planes. The circle is plotted with normal stress (σ) on the x-axis and shear stress (τ) on the y-axis.
Key Applications
- •Structural Design: Determining critical stress conditions in beams, shafts, and pressure vessels
- •Material Failure: Predicting failure modes using various failure criteria
- •Safety Analysis: Calculating safety factors and stress concentrations
- •Optimization: Finding optimal orientations to minimize stress
Principal Stress Formulas
Maximum & Minimum Principal Stresses
σ₁ = (σxx + σyy)/2 + √[((σxx - σyy)/2)² + τxy²]
σ₂ = (σxx + σyy)/2 - √[((σxx - σyy)/2)² + τxy²]
σ₁ = maximum, σ₂ = minimum principal stress
Maximum Shear Stress
τmax = (σ₁ - σ₂)/2
τmax = √[((σxx - σyy)/2)² + τxy²]
Maximum shear stress equals circle radius
von Mises Stress
σvM = √(σ₁² - σ₁σ₂ + σ₂²)
Equivalent stress for ductile material failure
Drawing Mohr's Circle
Step 1: Plot Points
Plot points A(σyy, τxy) and B(σxx, -τxy) on σ-τ axes
Step 2: Find Center
Center is at ((σxx + σyy)/2, 0) on σ-axis
Step 3: Draw Circle
Circle passes through points A and B with calculated center
Step 4: Read Results
Principal stresses are where circle intersects σ-axis
Failure Criteria
von Mises Criterion
Best for ductile materials (metals). Failure when σvM ≥ σyield
Tresca Criterion
Maximum shear stress theory. Failure when τmax ≥ σyield/2
Maximum Normal Stress
For brittle materials. Failure when σ₁ ≥ σult or σ₂ ≤ -σult