Principal Stress

\[\sigma_1=\frac{\sigma_x+\sigma_y}{2}+\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau^2}\]

Variables

sigma1[σ₁]major principal stress (Pa)

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

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

tau[τ]shear stress (Pa)

Description

What is this formula?

The principal stress formula calculates the maximum normal stress acting on a material element when subjected to a combination of normal and shear stresses. Principal stresses occur on planes where the shear stress is zero.

When to use it

Use this formula when analyzing stress states in structural members, pressure vessels, machine components, welded joints, or finite element models where multiple stresses act simultaneously.

Example

A steel plate is subjected to:

σx = 120 MPa

σy = 40 MPa

τ = 30 MPa

Formula:

σ1 = (σx + σy)/2 + √(((σx - σy)/2)² + τ²)

Substitution:

σ1 = (120 + 40)/2 + √(((120 - 40)/2)² + 30²)

σ1 = 80 + √(1600 + 900)

σ1 = 80 + 50

Result:

σ1 = 130 MPa

Applications

Structural steel design, pressure vessels, fatigue analysis, finite element analysis, machine design, welded structures, and failure assessment.