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Principal Stress and Maximum Shear Calculator — result sheet
Resolve a two-dimensional stress state into principal stresses, maximum in-plane shear stress, and principal-angle orientation.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: σ₁,₂ = (σx+σy)/2 ± √[((σx−σy)/2)²+τxy²]; τmax,in-plane = (σ₁−σ₂)/2; θp = ½ atan2(2τxy,σx−σy).
The transformation rewrites the same plane-stress state in axes where normal stress is extreme and shear stress is zero. The angle is measured from the entered x-axis using the signed shear convention.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Normal stress σx — MPa; minimum -1000000000; maximum 1000000000
- Normal stress σy — MPa; minimum -1000000000; maximum 1000000000
- Shear stress τxy — MPa; minimum -1000000000; maximum 1000000000
Worked example
| Input | Value |
|---|---|
| Normal stress σx | 80 |
| Normal stress σy | 20 |
| Shear stress τxy | 15 |
Principal stresses ≈ 83.54 MPa and 16.46 MPa; maximum in-plane shear ≈ 33.54 MPa; θp ≈ 13.28°.
Assumptions and limits
- The inputs describe a two-dimensional plane-stress tensor with a consistent sign convention.
- The principal-angle result is one mathematically equivalent orientation; the perpendicular direction is the second principal axis.
- Out-of-plane stress, plasticity, failure criteria, residual stress, and material strength are not modeled.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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