Von Mises Stress Calculator: Equivalent Stress, Principal Stresses, Tresca and Safety Factor
Von Mises equivalent stress from plane stress (σx, σy, τxy), a full 3D state or principal stresses, with Tresca stress and a safety factor against yield.
How it works
2D: σv = √(σx² − σx·σy + σy² + 3τxy²) · 3D: σv = √{½[(σx − σy)² + (σy − σz)² + (σz − σx)²] + 3(τxy² + τyz² + τzx²)} · Principal: σv = √{½[(σ1 − σ2)² + (σ2 − σ3)² + (σ3 − σ1)²]} · Tresca: τmax = (σ1 − σ3)/2 · Safety factor n = Sy ÷ σv
The von Mises stress turns any combination of normal and shear stresses into one number that can be compared with the yield strength from a simple tensile test. It comes from the distortion-energy theory: a ductile metal yields when the energy that changes its shape (not its volume) reaches the value it has at yield in a tension test. Equal pressure on all sides (hydrostatic stress) does not distort the material, so it drops out of the formula; that is why σ1 = σ2 = σ3 gives a von Mises stress of 0. Pure shear τ gives σv = √3 × τ, so a part in pure shear yields at about 0.577 × Sy. Pick plane stress for thin parts or surface points where σz and the out-of-plane shears are zero, the general 3D state for results from a finite-element model or a hand analysis with all six components, or enter principal stresses directly. The calculator also finds the principal stresses (from the eigenvalues of the stress tensor), the Tresca maximum shear stress, which is slightly more conservative than von Mises by up to 15%, and a safety factor against yield for a chosen or custom material. Enter stresses in any unit; results are in MPa with psi or ksi. Unlike the Mohr’s circle calculator, which focuses on stress transformation and principal directions, this page is a yield check. The approach suits ductile metals under static load; fatigue, buckling and brittle fracture need other checks.
Worked example
A point has σx = 120 MPa, σy = −80 MPa and τxy = 60 MPa. σv = √(120² − 120 × (−80) + (−80)² + 3 × 60²) = √(14,400 + 9,600 + 6,400 + 10,800) = √41,200 = 203.0 MPa (29.44 ksi). The principal stresses are 20 ± 116.6, so σ1 = 136.6 MPa and σ2 = −96.6 MPa, and the Tresca maximum shear is (136.6 − (−96.6)) ÷ 2 = 116.6 MPa. With a 250 MPa yield strength the von Mises safety factor is 250 ÷ 203.0 = 1.23 and the Tresca one is 250 ÷ 233.2 = 1.07.
Sources: Omni Calculator: Von Mises stress (2D, 3D and principal formulas; 6.5 MPa pure-shear example) · FIRGELLI: Von Mises stress calculator, combined loading (σx = 120, σy = −80, τxy = 60 MPa example)
Frequently asked questions
What is von Mises stress?
It is an equivalent (effective) stress that combines all the normal and shear stresses at a point into a single value. If it reaches the yield strength measured in a tensile test, a ductile material is predicted to start yielding. Finite-element programs show it as the default stress plot for metals.
How do I calculate von Mises stress from σx, σy and τxy?
Use σv = √(σx² − σx·σy + σy² + 3τxy²). For σx = 100 MPa, σy = 0 and τxy = 50 MPa: √(10,000 + 7,500) = 132.3 MPa.
What is the von Mises stress for pure shear?
For pure shear τ the formula gives σv = √3 × τ ≈ 1.732τ. A shaft surface with 6.5 MPa of torsional shear therefore has a von Mises stress of 11.26 MPa, and shear yield is about 0.577 times the tensile yield strength.
What is the difference between von Mises and Tresca?
Tresca says yielding starts when the maximum shear stress (σ1 − σ3)/2 reaches half the yield strength; von Mises uses distortion energy. Tresca is more conservative by up to about 15% (in pure shear) and both agree for uniaxial stress. Test data for ductile metals usually lie between them, closer to von Mises.
Can von Mises stress be negative?
No. It is a square root of a sum of squares, so it is always zero or positive and does not say whether the point is in tension or compression. Look at the principal stresses or the mean stress for that.
What safety factor should I use?
Divide the yield strength by the von Mises stress. Typical minimums for static loads range from about 1.5 for well-known loads and materials to 3 or more for uncertain loads; design codes and company standards set the exact values.
Related calculators
For learning and preliminary checks. Safety-critical parts must be designed and verified by a qualified engineer using the applicable code and certified material properties.