Engineering
Plane-stress von Mises and Tresca equivalent stress and yield ratios
Combine multiaxial normal stresses sigma_x, sigma_y and shear stress tau_xy into FEA Von Mises distortion-energy stress SQRT(sig_x^2 - sig_x*sig_y + sig_y^2 + 3*tau_xy^2) and Yield Factor of Safety.
Compare two idealized ductile-yield criteria
Plane stress means sigma_z, tau_xz and tau_yz are zero. The signed in-plane normal stresses and tensor shear component are entered in MPa; yield strength must use the same units. Plane strain is a different condition.
Von Mises and Tresca are idealized criteria commonly used for isotropic ductile materials. They do not prove the behavior of every metal or address fatigue, buckling, brittle fracture, anisotropy or stress concentrations omitted from the input.
The ratio of yield strength to equivalent stress is a static yield ratio, not a complete design approval. A ratio above one only places this stress state below the selected model threshold. At zero stress the ratio is unbounded and returned as a text label.
Tresca uses the largest difference among all three principal stresses. Include the out-of-plane principal stress of zero even though the inputs are two-dimensional; dropping it gives a wrong result for equal biaxial tension.
| Multiaxial Stress Case | Formula | sig_x = 120, sig_y = -40, tau = 60 MPa (Sy = 250) | Pure Shear (0, 0, tau = 100 MPa) |
|---|---|---|---|
| Von Mises Stress sigma_vM | SQRT(B2^2 - B2*B3 + B3^2 + 3*B4^2) | SQRT(31,600) = 177.76 MPa | SQRT(30,000) = 173.21 MPa |
| Yield Factor of Safety (FoS) | B5 / sigma_vM | 1.41 (model yield ratio) | 250 / 173.21 = 1.44 |
2D Plane-Stress Von Mises Equivalent Stress (MPa)
Preserves algebraic signs of sigma_x (B2) and sigma_y (B3) so biaxial tension-compression properly increases -B2 * B3.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(B5<=0,"Positive yield strength required",ROUND(SQRT(B2^2-B2*B3+B3^2+3*B4^2),2)))Ductile Yield Factor of Safety = YieldStrength / sigma_vM
Compares tensile yield strength S_y (B5) against equivalent Von Mises stress.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(B5<=0,"Positive yield strength required",IF(SQRT(B2^2-B2*B3+B3^2+3*B4^2)=0,"Unbounded at zero stress",ROUND(B5/SQRT(B2^2-B2*B3+B3^2+3*B4^2),2))))Tresca equivalent stress (MPa)
Compute the three principal stresses including zero. Use full precision before rounding the reported result.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(B5<=0,"Positive yield strength required",ROUND(LET(meanStress,(B2+B3)/2,stressRadius,SQRT(((B2-B3)/2)^2+B4^2),MAX(meanStress+stressRadius,meanStress-stressRadius,0)-MIN(meanStress+stressRadius,meanStress-stressRadius,0)),2)))Tresca static yield ratio
Compute the three principal stresses including zero. Use full precision before rounding the reported result.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(B5<=0,"Positive yield strength required",LET(equivalent,LET(meanStress,(B2+B3)/2,stressRadius,SQRT(((B2-B3)/2)^2+B4^2),MAX(meanStress+stressRadius,meanStress-stressRadius,0)-MIN(meanStress+stressRadius,meanStress-stressRadius,0)),IF(equivalent=0,"Unbounded at zero stress",ROUND(B5/equivalent,2)))))Try the example
Use the original values or change them to test your understanding.