Principal Stress Calculator
Calculate maximum and minimum principal stresses and orientations
Stress Analysis Configuration
2D Stress State: Plane stress analysis with normal and shear components
Stress Components
Output Units
Principal Stress Results
Material Properties
Name: Mild Steel
Category: Metals
Description: Common structural steel with good ductility
Yield Strength: 250 MPa
Tensile Strength: 400 MPa
Compressive Strength: 400 MPa
Principal Stress Formulas:
σmax = (σx + σy)/2 + √[((σx - σy)/2)² + τxy²]
σmin = (σx + σy)/2 - √[((σx - σy)/2)² + τxy²]
θ = ½ arctan(2τxy/(σx - σy))
τmax = √[((σx - σy)/2)² + τxy²]
Example: Biaxial Stress State
Given Values
Horizontal Normal Stress (σx): 10 MPa
Vertical Normal Stress (σy): 10 MPa
Shear Stress (τxy): 8 MPa
Calculation Steps
Center stress = (σx + σy)/2 = (10 + 10)/2 = 10 MPa
Radius = √[((σx - σy)/2)² + τxy²] = √[0² + 8²] = 8 MPa
σmax = 10 + 8 = 18 MPa
σmin = 10 - 8 = 2 MPa
θ = ½ arctan(2×8/(10-10)) = 45°
Results
The maximum principal stress is 18 MPa and minimum is 2 MPa, occurring at 45° to the horizontal plane.
Analysis Types
Typical Yield Strengths
Failure Criteria
σy = yield strength
Physics Tips
Principal stresses occur on planes with zero shear
Maximum shear stress = (σmax - σmin)/2
Principal planes are 90° apart
von Mises criterion works well for ductile materials
Brittle materials fail along principal planes
Understanding Principal Stresses
What are Principal Stresses?
Principal stresses are the normal stresses that occur on planes where shear stress is zero. They represent the maximum and minimum normal stresses at a point and are fundamental for understanding material failure. The principal plane orientations are critical for determining where cracks might initiate.
Key Formulas
σmax = (σx + σy)/2 + √[((σx - σy)/2)² + τxy²]
σmin = (σx + σy)/2 - √[((σx - σy)/2)² + τxy²]
θ = ½ arctan(2τxy/(σx - σy))
Where σ = normal stress, τ = shear stress, θ = principal angle
Engineering Applications
Failure Analysis
Predict crack initiation and propagation direction
Material Selection
Choose materials based on stress capacity
Design Optimization
Orient components to minimize critical stresses
Failure Criteria
von Mises Criterion
Best for ductile materials like metals
Tresca Criterion
Maximum shear stress theory, conservative
Maximum Principal Stress
Suitable for brittle materials like ceramics
Mohr's Circle Relationships
Center Point
(σx + σy)/2
Circle Radius
√[((σx - σy)/2)² + τxy²]
Maximum Shear
Equals the circle radius