Mohr Circle Calculator
Free calculate principal stresses and maximum shear stress using mohr's circle method. Get instant, accurate results with our easy-to-use calculator.
Input Parameters
Results
Enter stress components to calculate
What is Mohr's Circle?
Mohr's Circle is a graphical method for analyzing stress transformation in materials under plane stress conditions. It provides a visual representation of how normal and shear stresses change as the orientation of the plane changes.
The circle's center is at ((σₓ + σᵧ)/2, 0) and its radius is √[((σₓ - σᵧ)/2)² + τₓᵧ²]. The intersections with the horizontal axis give the principal stresses (σ₁ and σ₂), where shear stress is zero. The maximum shear stress equals the circle's radius.
Mohr's circle is essential in materials science and structural engineering for predicting failure, understanding stress states, and designing safe structures. It's particularly useful for analyzing brittle materials and determining maximum stress concentrations.
Mohr Circle Formulas
Where:
- • σ₁, σ₂ = Principal stresses (maximum and minimum normal stresses)
- • τ_max = Maximum shear stress
- • σₓ, σᵧ = Normal stresses in x and y directions
- • τₓᵧ = Shear stress on xy-plane
Note: σ₁ is the maximum principal stress, σ₂ is the minimum. Maximum shear occurs at 45° from principal planes.
How to Calculate
-
1
Calculate center of circle
Center = (σₓ + σᵧ)/2. This is the average of the two normal stresses.
-
2
Calculate radius
R = √[((σₓ - σᵧ)/2)² + τₓᵧ²]. This is the distance from center to edge of circle.
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3
Calculate principal stresses
σ₁ = center + R, σ₂ = center - R. These are the intersections of the circle with the horizontal axis.
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4
Calculate maximum shear stress
τ_max = R. The maximum shear stress equals the circle's radius and occurs at 45° from principal planes.
Practical Examples
Example 1: Plane Stress
σₓ = 100 MPa, σᵧ = 50 MPa, τₓᵧ = 30 MPa.
Solution:
Center = (100 + 50)/2 = 75 MPa
R = √[((100-50)/2)² + 30²] = √[625 + 900] = 39.05 MPa
σ₁ = 75 + 39.05 = 114.05 MPa
σ₂ = 75 - 39.05 = 35.95 MPa, τ_max = 39.05 MPa
Applications
Structural Engineering
Analyzing stress in beams, columns, and structural members. Determining failure modes and designing safe structures.
Mechanical Design
Designing machine components, analyzing stress concentrations, and predicting material failure in mechanical systems.
Materials Science
Understanding material behavior under stress, analyzing failure criteria, and studying stress-strain relationships.
Education
Teaching stress analysis, understanding Mohr's circle construction, and demonstrating stress transformation principles.
Frequently Asked Questions
What are principal stresses?
Principal stresses (σ₁, σ₂) are the maximum and minimum normal stresses at a point. They occur on planes where shear stress is zero. These are the orientations where normal stress is extreme.
Why is maximum shear at 45°?
Maximum shear stress occurs on planes oriented 45° from the principal planes. This is because shear stress is maximum when the plane is halfway between principal stress directions.
What if σ₁ = σ₂?
If principal stresses are equal (hydrostatic stress), Mohr's circle becomes a point. All planes have the same normal stress, and there's no preferred orientation. This is isotropic stress state.
How do I find stress at an angle?
Rotate around Mohr's circle by 2θ (where θ is the physical rotation angle). The coordinates on the circle give normal and shear stress on that plane. σ = center + R×cos(2θ), τ = R×sin(2θ).
What about 3D stress?
For 3D stress, you get three Mohr circles. The largest circle (between σ₁ and σ₃) gives the absolute maximum shear stress. Plane stress (2D) is a special case where one principal stress is zero.