General 2D Stress Transformation Calculator

General 2D Stress Transformation Equation

$$\sigma_{N} = \frac{\sigma_x + \sigma_y}{2} + \left(\frac{\sigma_x - \sigma_y}{2}\right)\cos\left(\theta \right) + \tau_{xy}\sin \left(2\theta \right) $$ $$\tau_s = \left(\frac{\sigma_x - \sigma_y}{2}\right)\sin(2\theta) - \tau_{xy}\cos(2\theta) $$ where:

σ_N = Normal Stress (Pa, N/m², kPa, MPa)

τ_s = Shear Stress (Pa, N/m², kPa, MPa)

σ_x = Normal Stress in x direction (Pa, N/m², kPa, MPa)

σ_y = Normal Stress in y direction (Pa, N/m², kPa, MPa)

τ_xy = Shear Stress in xy direction (Pa, N/m², kPa, MPa)

θ = Angle (° or rad)

Calculator

Property to Calculate:

Normal Stress in x direction (σ_x):

Normal Stress in y direction (σ_y):

Shear Stress in xy direction (τ_xy):

Angle (θ):

Equation Assumptions

The material is homogeneous and continuous.
The material follows Hookes 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.