Calculate principal stresses, max shear stress, and rotated stress components using Mohr's circle. Includes Von Mises and Tresca failure criteria.
Mohr's circle is a graphical method for analyzing 2D stress states, transforming normal and shear stresses as an element is rotated. Given σx, σy, and τxy, it determines principal stresses (σ₁, σ₂), maximum shear stress, and the orientation of the principal planes.
This calculator computes the center C = (σx + σy)/2 and radius R = √[((σx−σy)/2)² + τxy²], giving principal stresses σ₁ = C + R and σ₂ = C − R. It also calculates stresses on any rotated plane, Von Mises equivalent stress, Tresca maximum shear, and safety factors against yielding.
The tool handles classic load cases—uniaxial tension, pure shear, biaxial equal, pressure vessel, and general plane stress—through preset buttons. A rotation angle input lets you find stresses at any orientation. Engineers use Mohr's circle for mechanical design, structural analysis, geotechnical engineering, and failure prediction. The reference table compares four failure criteria with their applicability to ductile and brittle materials.
Mohr's circle is a compact way to see how combined normal and shear stresses transform when the reference plane rotates. This calculator saves time on stress transformation, principal-stress checks, and failure criteria by showing the geometry and the derived values together.
It is useful in machine design, pressure vessels, slope stability, and any other combined-loading problem where the direction of the stress plane matters.
C = (σx + σy)/2, R = √[((σx−σy)/2)² + τxy²]. σ₁ = C + R, σ₂ = C − R, τ_max = R. θp = ½·atan2(2τxy, σx − σy). Von Mises: σ_vm = √(σ₁² − σ₁σ₂ + σ₂²). Tresca: τ_max = (σ₁ − σ₂)/2.
Result: σ₁ = 140 MPa, σ₂ = -60 MPa, τ_max = 100 MPa
C = (120+(−40))/2 = 40 MPa. R = √(80² + 60²) = 100 MPa. So σ₁ = 140 MPa, σ₂ = −60 MPa. Principal angle θp ≈ 18.4°.
The circle is built from the average normal stress at the center and the combined normal/shear radius. Rotating the stress element moves a point around the circle; the horizontal intercepts give the principal stresses and the top and bottom points give the maximum shear stress.
Von Mises is usually the preferred ductile-material check because it reflects distortion energy. Tresca is stricter and often gives a lower allowable limit. If your material is brittle, the principal-stress values themselves matter more than the shear criterion.
Mohr's circle is most valuable when the loading is not already aligned with a known axis. That is common in welded joints, shafts, retaining structures, and pressure-containing parts where stress direction changes with orientation.
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It graphically represents how normal and shear stresses change as the element is rotated. Every point on the circle corresponds to a different orientation of the stress element.
Von Mises is better for ductile materials as it accounts for the distortion energy. Tresca is more conservative (gives a lower allowable stress) and is sometimes used in pressure vessel codes.
Then σx and σy are already the principal stresses and the principal angle is 0° (or 90°). The Mohr circle is centered on the σ-axis with no shear offset.
This tool handles 2D plane stress. For full 3D analysis, you need three Mohr circles (one for each pair of principal stresses). The maximum shear stress in 3D may differ from the 2D result.
The angle θp rotates the coordinate system to align with the principal directions, where shear stress is zero and normal stresses reach their extreme values.
The principal angle is fixed by the stress state. The rotation angle is user-chosen—enter any angle to see what stresses would exist on that plane.