Free · Physics models
Plane-Stress Principal Values and Directions
Find the two in-plane principal stresses and maximum-principal direction from a symmetric Cartesian stress tensor.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
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Formula and method
Normal stresses 100 and 40 MPa with shear 30 MPa give principal stresses about 112.426 and 27.574 MPa, with maximum-principal direction 22.5°.
σ1,2 = (σx+σy)/2 ± √[((σx−σy)/2)²+τxy²]; θ1 = atan2(2τxy,σx−σy)/2.
Worked example
Enter these known values and leave the other values blank.
- Horizontal normal stress (σ==x==)
- 100 MPa
- Vertical normal stress (σ==y==)
- 40 MPa
- XY shear stress (τ==xy==)
- 30 MPa
- YX shear stress (τ==yx==)
- 30 MPa
- Maximum principal stress (σ==max==)
- 112.43 MPa
- Minimum principal stress (σ==min==)
- 27.574 MPa
- Angle of principal stress (θ)
- 22.5 deg
- Shear-plane direction θ1 + 45°
- 67.5 deg
- Orthogonal shear-plane direction θ1 − 45°
- -22.5 deg
Assumptions and limitations
- Normal stress is positive in tension. Shear components use a symmetric Cartesian tensor, so τyx = τxy. The displayed direction is measured from the x axis under this convention.
- The atan2 branch gives the maximum-principal direction modulo 180 degrees and handles reversed and equal normal stresses.
- An isotropic in-plane state with equal normal stresses and zero shear has no unique principal direction; angle output remains unavailable.
- The two shear-plane angle labels indicate the orthogonal ±45° directions relative to the principal direction. They are not out-of-plane stress extrema.
- Inverse relations can be ambiguous because principal directions repeat and square roots choose ordered principal stresses. This page does not model a general three-dimensional stress state.
- Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
- Reviewed hash-pinned model correction: Normal stress is positive in tension. Shear components use a symmetric Cartesian tensor, so τyx = τxy. The displayed direction is measured from the x axis under this convention. The atan2 branch gives the maximum-principal direction modulo 180 degrees and handles reversed and equal normal stresses. An isotropic in-plane state with equal normal stresses and zero shear has no unique principal direction; angle output remains unavailable. The two shear-plane angle labels indicate the orthogonal ±45° directions relative to the principal direction. They are not out-of-plane stress extrema. Inverse relations can be ambiguous because principal directions repeat and square roots choose ordered principal stresses. This page does not model a general three-dimensional stress state. Enter known quantities and leave the unknown blank. Editable quantities support inverse solutions and locks; inconsistent locked quantities are reported. Unit selectors convert the quantity independently of its display.
Common questions
Why use atan2 rather than an ordinary arctangent?
It preserves the quadrant of the normal-stress difference and shear, so the direction corresponds to the maximum principal value.
Why is the direction blank for equal stresses and zero shear?
Every in-plane direction is principal in that isotropic state. There is no unique direction to calculate.
How do I use the inverse calculation?
Clear the quantity you want to find and enter the remaining known values in their selected units. The model can only solve branches with enough independent information.
References
Use a symmetric Cartesian stress tensor, and atan2 for the maximum principal direction including equal-normal and reversed-normal cases. Hydrostatic states have no unique principal direction.