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 fundamental in mechanical and civil engineering. This calculator replaces tedious hand calculations and provides instant answers for stress transformation, principal stresses, and failure assessment.
It is indispensable for machine design, pressure vessel analysis, geotechnical slope stability, and any scenario where combined loading creates complex stress states. Keep these notes focused on your operational context. Tie the context to the calculator’s intended domain.
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 = 148.1 MPa, σ₂ = -68.1 MPa, τ_max = 108.1 MPa
C = (120+(−40))/2 = 40 MPa. R = √(80² + 60²) = 100 MPa. So σ₁ = 140 MPa, σ₂ = −60 MPa. Principal angle θp ≈ 18.4°.
Use consistent units, verify assumptions, and document conversion standards for repeatable outcomes.
Most mistakes come from mixed standards, rounding too early, or misread labels. Recheck final values before use. ## Practical Notes
Use concise notes to keep each section focused on outcomes. ## Practical Notes
Check assumptions and units before interpreting the number. ## Practical Notes
Capture practical pitfalls by scenario before sharing the result. ## Practical Notes
Use one example per section to avoid misapplying the same formula. ## Practical Notes
Document rounding and precision choices before you finalize outputs. ## Practical Notes
Flag unusual inputs, especially values outside expected ranges. ## Practical Notes
Apply this as a quality checkpoint for repeatable calculations.
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. Use this as a practical reminder before finalizing the result.
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.