Principal Stress and Principal Angle Calculator

Principal Stresses and Principal Angles Calculator

Property to Calculate:

Principal Stress 1 (σ₁):

Principal Stress 2 (σ₂):

Principal Angle 1 (θ_p1):

Principal Angle 2 (θ_p2):

Normal Stress in x direction (σ_x):

Normal Stress in y direction (σ_y):

Shear Stress in xy direction (τ_xy):

How This Is Calculated

$$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + {\tau_{xy}}^2} $$ $$\tan(2\theta_p) = \frac{2 \tau_{xy}}{\sigma_x - \sigma_y} $$ where:

σ_1,2 = Principal Stresses (N/m² or Pa)

θ_p = Principal Angles (°)

σ_x = Normal Stress in x direction (N/m² or Pa)

σ_y = Normal Stress in y direction (N/m² or Pa)

τ_xy = Shear Stress in xy direction (N/m² or Pa)

Assumptions

The material is homogeneous, isotropic, and continuous.
The material follows Hooke's Law, i.e., the stress-strain relationship is linear.
The stress in the thickness direction (out-of-plane stress) is assumed to be zero.
Deformations are insufficient such that the original geometry is not altered.

Related Calculators

General 2D Stress Transformations

Von Mises Stress