Mohr's Circle Calculator
Calculate principal stresses and maximum shear stress
Shear stress acting on the element
How to Use This Calculator
Enter Normal Stresses
Input the normal stress components σ_x and σ_y acting on your element. These are the stresses perpendicular to the x and y faces. Use consistent units (MPa, psi, etc.).
Enter Shear Stress
Input the shear stress τ_xy acting on the element. Shear stress acts parallel to the face. Note that τ_xy = τ_yx for equilibrium.
Calculate Principal Stresses
Click "Calculate" to determine the principal stresses (σ₁ and σ₂), maximum shear stress (τ_max), and the angles to the principal planes. These values represent the extreme stress states.
Formula
Center = (σ_x + σ_y) / 2
Radius = √[((σ_x - σ_y) / 2)² + τ_xy²]
σ₁ = Center + Radius
σ₂ = Center - Radius
Where:
- σ₁ = Maximum principal stress
- σ₂ = Minimum principal stress
- τ_max = Maximum shear stress = Radius
- θ_p = Angle to principal planes = ½ × arctan(2τ_xy / (σ_x - σ_y))
Example:
For stress state: σ_x = 100, σ_y = 50, τ_xy = 30
Center = (100 + 50) / 2 = 75
Radius = √[((100 - 50) / 2)² + 30²] = √[625 + 900] = √1525 ≈ 39.05
σ₁ = 75 + 39.05 = 114.05
σ₂ = 75 - 39.05 = 35.95
τ_max = 39.05
About Mohr's Circle Calculator
Mohr's Circle Calculator is an essential tool for stress analysis in engineering. Mohr's Circle is a graphical representation of the transformation of stress components when rotating the coordinate system. It helps engineers determine principal stresses, maximum shear stress, and the orientations at which these occur.
When to Use This Calculator
- Structural Analysis: Determine critical stress states in structural elements
- Machine Design: Analyze stress in machine components under combined loading
- Failure Analysis: Identify maximum stresses that may cause material failure
- Material Testing: Interpret results from stress-strain tests
- Engineering Education: Learn stress transformation and principal stress concepts
Why Use Our Calculator?
- ✅ Quick Analysis: Instant calculation of principal stresses and maximum shear stress
- ✅ Visual Understanding: Understand stress transformation through Mohr's Circle
- ✅ Design Optimization: Identify critical stress orientations for design improvement
- ✅ Error Prevention: Avoid manual calculation errors in critical stress analysis
- ✅ Educational Tool: Learn stress transformation principles
Key Concepts
Principal Stresses: The maximum and minimum normal stresses acting on an element. On the principal planes, there is no shear stress. Principal stresses are critical for failure analysis using failure theories (von Mises, Tresca, etc.).
Maximum Shear Stress: The largest shear stress that can occur on any plane through the element. This occurs on planes oriented 45° from the principal planes. Maximum shear stress is used in Tresca (maximum shear stress) failure theory.
Applications
- Pressure Vessels: Analyze combined stresses from internal pressure and external loads
- Shafts: Determine principal stresses in rotating shafts with combined bending and torsion
- Beams: Analyze stress at critical locations where normal and shear stresses combine
- Failure Theories: Input for von Mises and Tresca failure criteria
Frequently Asked Questions
What are principal stresses?
Principal stresses (σ₁ and σ₂) are the maximum and minimum normal stresses that occur on an element. On the principal planes (where principal stresses act), there is no shear stress. Principal stresses represent the extreme stress states and are crucial for failure analysis.
Why is Mohr's Circle useful?
Mohr's Circle provides a visual and analytical method to determine stress states at any orientation. It makes it easy to find principal stresses, maximum shear stress, and stress at any angle, which would be tedious to calculate manually for multiple orientations.
How do I determine which is σ₁ and which is σ₂?
By convention, σ₁ is the maximum (most tensile or least compressive) principal stress, and σ₂ is the minimum (least tensile or most compressive). In 2D analysis, σ₁ ≥ σ₂. In some cases, both may be compressive (negative), but σ₁ is still the algebraically larger value.
What is the relationship between principal stresses and maximum shear stress?
The maximum shear stress equals half the difference between the principal stresses: τ_max = (σ₁ - σ₂) / 2. This also equals the radius of Mohr's Circle. Maximum shear stress occurs on planes oriented 45° from the principal planes.
Can I use this for 3D stress analysis?
This calculator is for 2D (plane stress) analysis. For 3D stress states, you would need three Mohr's Circles (one for each pair of principal stresses) or use more advanced 3D stress transformation methods. However, the 2D case is sufficient for many engineering applications.