Formula and Method for Principal Stress
At any point inside a loaded structural member, the state of stress on a small 2D element is described by two normal stresses, σx and σy, and one shear stress, τxy. As you rotate that element, the normal and shear stresses on its faces change — but there is always one orientation where the shear stress drops to exactly zero. The normal stresses on that orientation are the principal stresses, σ1 (maximum) and σ2 (minimum). This calculator derives σ1, σ2, the maximum in-plane shear stress τmax, and the principal plane angle θp directly from σx, σy, and τxy.
Deriving the principal stress formula
Starting from the plane-stress transformation equations and setting the rotated shear stress τx'y' to zero gives tan(2θp) = 2τxy / (σx − σy), which locates the principal planes. Substituting that angle back into the transformation equation for normal stress yields the standard closed-form result: σ1,2 = (σx + σy)/2 ± √(((σx − σy)/2)² + τxy²). The term under the square root, R = √(((σx − σy)/2)² + τxy²), is the radius of Mohr's circle; the average stress (σx + σy)/2 is its center. The maximum in-plane shear stress is simply that radius, τmax = R = (σ1 − σ2)/2, and it acts on planes rotated 45° from the principal planes. The principal angle itself is θp = ½·atan2(2τxy, σx − σy), which locates the plane carrying σ1; the plane carrying σ2 is 90° away.
Sign convention and units
- Tensile normal stresses (σx, σy) are positive; compressive stresses are negative.
- τxy is positive using the standard engineering convention: it tends to rotate the element counter-clockwise on the face whose outward normal points in the +x direction.
- σx, σy, and τxy must all be entered in the same stress unit (e.g., all in MPa or all in psi) — the calculator does not convert between them.
Where principal stress matters
Principal stress analysis underlies most structural failure checks: pressure vessel walls, shafts carrying combined bending and torsion, welded and bolted joints, and any component under biaxial loading. Because material yielding and fracture are governed by the largest normal or shear stress at a point — not by σx and σy individually — design codes convert a raw stress state into principal stresses (or into a failure criterion such as Tresca's maximum-shear-stress theory or the von Mises equivalent stress) before comparing it against a material's allowable strength.