Inputs
Normal stresses σx and σy plus shear stress τxy fully describe the state; an optional element angle θ additionally gives the stress on an arbitrarily rotated cutting plane.
Statics & strength · Multiaxial stress
Plane stress transformation and Mohr's circle
Calculate the principal stresses σ1 and σ2, the maximum in-plane shear stress τmax, the corresponding principal-axis angle, and optionally the stresses at any chosen rotation angle θ, from a full plane stress state σx, σy, τxy -- the classical stress transformation behind Mohr's circle.
This covers a pure plane (2D) stress state with linear-elastic material. A third principal stress acting perpendicular to the plane, plasticity, fatigue and stability are not considered.
σ1/σ2τmax
02
Principal stresses and Mohr's circle
Enter the stress state and transform it.
Inputs and method
Plane stress transformation and Mohr's circle
Transform a full plane stress state (σx, σy, τxy) into principal stresses, maximum shear stress and the stress at any rotation angle.
Calculation
From σx, σy, τxy follow the average stress σavg=(σx+σy)/2 and the circle radius R=√(((σx−σy)/2)²+τxy²). The principal stresses are σ1=σavg+R and σ2=σavg−R, the maximum in-plane shear stress is τmax=R. For any θ the transformation equations give σx', σy' and τx'y'.
Example
σx=−20 MPa, σy=90 MPa, τxy=60 MPa give σavg=35 MPa, R≈81.4 MPa, so σ1≈116.4 MPa, σ2≈−46.4 MPa and τmax≈81.4 MPa at φp≈66.3°.
Sources and limits: Classical mechanics of materials (stress transformation, Mohr's circle); plane stress state only, no third principal stress, not a normative check.
Plane stress transformation and Mohr's circle in detail
Transform a full plane stress state (σx, σy, τxy) into principal stresses, maximum shear stress and the stress at any rotation angle.
What is a plane stress state?
A plane stress state exists when the stress perpendicular to the plane being considered (e.g. at a free component surface) is negligible. It is fully described by three quantities: the two normal stresses σx, σy and the shear stress τxy on an arbitrarily but fixed-oriented x/y element. Rotating that element changes all three values -- only certain combinations, such as the sum σx+σy or Mohr's circle as a whole, stay invariant.
Formula and variables
σ1,2 = (σx+σy)/2 ± √(((σx−σy)/2)² + τxy²)
τmax = √(((σx−σy)/2)² + τxy²)tan(2φp) = 2τxy/(σx−σy)σx' = (σx+σy)/2 + (σx−σy)/2·cos(2θ) + τxy·sin(2θ)
| Symbol / input | Meaning |
|---|---|
| σx, σy | Normal stresses on the x- and y-faces of the original element, in MPa. |
| τxy | Shear stress on those same faces, in MPa, using the usual mechanics-of-materials sign convention. |
| σ1, σ2 | Principal stresses (σ1 ≥ σ2), on the plane where the shear stress vanishes. |
| τmax | Maximum in-plane shear stress, equal to the radius of Mohr's circle. |
| θ, φp | θ is the freely chosen evaluation angle, φp the angle to the σ1 principal axis; both counter-clockwise from the x-axis, in degrees. |
Choose the inputs correctly
Normal stresses σx and σy plus shear stress τxy fully describe the state; an optional element angle θ additionally gives the stress on an arbitrarily rotated cutting plane.
How to use the calculator
Enter the element's two normal stresses σx, σy and the shear stress τxy. An element angle θ is optional and additionally gives the stresses on an arbitrarily rotated cutting plane; it is not needed for the principal stresses or τmax.
Worked example
σx=−20 MPa, σy=90 MPa, τxy=60 MPa give σavg=35 MPa, R≈81.4 MPa, so σ1≈116.4 MPa, σ2≈−46.4 MPa and τmax≈81.4 MPa at φp≈66.3°.
How should principal stresses and τmax be interpreted?
σ1 and σ2 are the extreme values the normal stress can take over all possible cutting planes; the shear stress vanishes on the corresponding planes. τmax is the radius of Mohr's circle and occurs on a plane rotated 45° from that, accompanied by the normal stress σavg. For a strength assessment, usually σ1 (the governing tensile or compressive stress) and τmax (governing for shear or equivalent-stress criteria) matter, not the originally entered σx/σy/τxy.
All stresses in MPa (or Pa, kPa, psi, ksi); the angle θ in degrees, positive counter-clockwise from the x-axis.
Typical applications
Evaluating a multiaxial stress state measured or computed at a component point, for example from superimposed tension, bending and torsion, to find the actual governing principal stress and shear stress rather than looking only at the entered components.
Assumptions, limits and common mistakes
Classical mechanics of materials (stress transformation, Mohr's circle); plane stress state only, no third principal stress, not a normative check.
Common mistake: Don't confuse τxy with the maximum shear stress τmax computed later – τxy is an input for the original element orientation, τmax is the result of the transformation. Likewise, don't confuse σ1/σ2 with σx/σy: the principal stresses act on a different, rotated cutting plane where the shear stress is zero by definition.
Frequently asked questions
What is the difference between this calculator and the GEH equivalent-stress calculator?
The GEH (von Mises) calculator combines already-known resultant normal and shear stresses into a single scalar equivalent stress and only answers 'is the strength sufficient'. This calculator instead transforms the full plane stress state itself: it returns principal stresses, their direction (principal-axis angle) and the stresses on arbitrarily rotated cutting planes -- information the GEH calculator does not output.
Why are there two candidate angles for the principal-axis angle?
The equation tan(2φ)=2τxy/(σx−σy) has two solutions 90° apart -- one belongs to σ1, the other to σ2. Which is which is checked by substituting back into the transformation equation, rather than assuming a fixed sign rule.
How are the principal-axis angle and the maximum-shear angle related?
The plane of maximum in-plane shear stress always sits exactly 45° from a principal-stress plane. The normal stress on that plane is not zero, but equal to the average stress σavg.
What does a negative σ2 mean?
σ1 and σ2 are signed normal stresses; a negative σ2 means compression on that principal plane, while σ1 can simultaneously be tensile. This is common under combined bending and torsion.
Does the calculator account for the full 3D stress state?
No, it handles only the plane (2D) stress state. At a free surface this is usually valid, since no stress acts perpendicular to the surface there. Note, however, that τmax is the maximum shear stress within the plane considered. If σ1 and σ2 have the same sign, the third principal stress σ3 = 0 lies outside the interval [σ2, σ1], and the absolute maximum shear stress is then max(|σ1|,|σ2|)/2 -- larger than the in-plane value reported here.