Von Mises Equivalent Stress

\[\sigma_v=\sqrt{\sigma_x^2-\sigma_x\sigma_y+\sigma_y^2+3\tau^2}\]

Variables

sigmav[σv]Von Mises equivalent stress (Pa)

sigmax[σx]normal stress in the x direction (Pa)

sigmay[σy]normal stress in the y direction (Pa)

tau[τ]in-plane shear stress (Pa)

Description

What is this formula?

The Von Mises equivalent stress converts a combined plane stress state into a single equivalent stress value. It is widely used to evaluate yielding in ductile materials such as structural steel.

When to use it

Use this formula when normal and shear stresses act together and you need to compare the combined effect against the material yield strength.

Example

A steel plate is subjected to:

σx = 120 MPa

σy = 40 MPa

τ = 30 MPa

Formula:

σv = √(σx² - σxσy + σy² + 3τ²)

Substitution:

σv = √(120² - 120*40 + 40² + 3*30²)

Result:

σv = 118.7 MPa

Applications

Steel design, failure assessment, finite element analysis, welded joints, pressure vessels, mechanical components, and ductile material yielding checks.

Note

This formula is based on the Von Mises yield criterion for ductile materials under plane stress. It is not a universal failure criterion for brittle materials, concrete, masonry, timber, or materials whose failure is governed by cracking, crushing, or directional weakness.