Principal Stress Calculator

Calculate principal stresses and maximum shear stress. Input normal stresses (σx, σy) and shear stress (τxy). Get σ1, σ2, τmax, and principal plane angle. Based on plane stress theory. Informational only—consult an engineering textbook.

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Published: October 23, 2025 | Updated: October 23, 2025 | Reviewed by: Science/Engineering Editor

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How it works

σ1,2 = (σx + σy)/2 ± √(((σx - σy)/2)² + τxy²). We use the plane stress transformation equations to find the principal stresses and the maximum shear stress.

The maximum shear stress equals the radius of Mohr's Circle, and the principal plane angle is determined from the orientation where shear stress is zero.

Inputs explained

  • Normal Stress σx: Normal stress in the x-direction
  • Normal Stress σy: Normal stress in the y-direction
  • Shear Stress τxy: Shear stress on the xy-plane
  • Stress Units: MPa (Megapascals) or ksi (kilopounds per square inch)

Example

Inputs: σx=100 MPa, σy=50 MPa, τxy=25 MPa

Calculations:

  • σavg = (100 + 50)/2 = 75 MPa
  • R = √(((100-50)/2)² + 25²) = √(625 + 625) = 35.36 MPa
  • σ1 = 75 + 35.36 = 110.36 MPa
  • σ2 = 75 - 35.36 = 39.64 MPa
  • τmax = 35.36 MPa

Tips & notes

  • Principal stresses are the maximum and minimum normal stresses at a point
  • The principal planes are oriented at 45° to the planes of maximum shear stress
  • Mohr's Circle is a graphical representation of the stress state
  • For 3D stress states, use a more advanced analysis method

FAQs

The maximum and minimum normal stresses that occur on planes where shear stress is zero.

It's the radius of Mohr's Circle, calculated from the normal and shear stresses.

The angle of the plane on which the principal stresses act, where shear stress is zero.

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Disclaimer

Informational tool based on plane stress theory. Does not account for 3D effects. Consult an engineering textbook for detailed theory and applications.

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