3D Stress Transformation

State of Stress & Stress Tensor $ ^i $

Stress components in a 3D element

$\sigma_x=$

$\sigma_y=$

$\sigma_z=$

$\tau_{xy}=$

$\tau_{xz}=$

$\tau_{yz}=$

$\sigma=$ $ \begin{bmatrix} \sigma_{x} & \tau_{x y} & \tau_{x z} \\\ \tau_{yx} & \sigma_{y} & \tau_{yz} \\\ \tau_{zx} & \tau_{zy} & \sigma_{z} \end{bmatrix}=$

Sx Txy Txz

Tyx Sy Tyz

Tzx Tzy Sz

Transformation of Stress $ ^i $

Transform stresses according to specified orientation angles in 3D space.

Direction Cosine Angles (α, β, θ) with Respect to x, y, and z Axes

$\alpha=$

$\beta=$

$\theta=$

$ cos^2(\alpha) + cos^2(\beta) + cos^2(\theta)=$

Possible angle corrections (Click to set)

$ \alpha= $ $ \beta= $ $ \theta= $ >
$ \alpha= $ $ \beta= $ $ \theta= $
$ \alpha= $ $ \beta= $ $ \theta= $

$ \begin{bmatrix} \alpha \\\ \beta \\\ \theta \end{bmatrix}=$

$n=$ $ \begin{bmatrix} cos\alpha \\\ cos\beta \\\ cos\theta \end{bmatrix}=$

Waiting for inputs... $ cos^2(\alpha) + cos^2(\beta) + cos^2(\theta)=1 $ sould be satisfied for stresses at the oriented direction

$ cos^2(\alpha) + cos^2(\beta) + cos^2(\theta)=1 $ sould be satisfied for stresses at the oriented direction

Normal and Shear Stress in the Oriented Direction

$\sigma\prime=$

$\tau\prime=$

Principal Stresses

Characteristic Equation

$$ \det(\mathbf{\sigma} - \lambda I)=\lambda^3 - I_1 \lambda^2 + I_2 \lambda - I_3 = 0 $$

Invariants

$I_{1}=$

$I_{2}=$

$I_{3}=$

Principal Normal Stresses & Principal Directions

$\sigma_{1}=$

$n_1=\begin{bmatrix} cos\alpha_1 \\\ cos\beta_1 \\\ cos\theta_1 \end{bmatrix}=$

$ \begin{bmatrix} \alpha_1 \\\ \beta_1 \\\ \theta_1 \end{bmatrix}=$

$\sigma_{2}=$

$n_2=\begin{bmatrix} cos\alpha_2 \\\ cos\beta_2 \\\ cos\theta_2 \end{bmatrix}=$

$ \begin{bmatrix} \alpha_2 \\\ \beta_2 \\\ \theta_2 \end{bmatrix}=$

$\sigma_{3}=$

$n_3=\begin{bmatrix} cos\alpha_3 \\\ cos\beta_3 \\\ cos\theta_3 \end{bmatrix}=$

$ \begin{bmatrix} \alpha_3 \\\ \beta_3 \\\ \theta_3 \end{bmatrix}=$

Max Shear Stress

$\tau_{max}=$

Mohr's Circle