3D Stress Transform

You can find guidance below to help you use the 3D Stress Transformation tool effectively.

State of Stress & Stress Tensor

• Enter the six independent stress components: \(\sigma_x\), \(\sigma_y\), \(\sigma_z\), \(\tau_{xy}\), \(\tau_{xz}\), and \(\tau_{yz}\). Due to symmetry, components like \(\tau_{yx}\), \(\tau_{zx}\), and \(\tau_{zy}\) are not needed—they are automatically taken as equal to their counterparts.
• Immediately below the input area, the stress tensor is displayed in matrix form. Any changes made to the input values are instantly reflected in the tensor.



• An info icon (i) next to the title opens a 3D visual of the stress element, showing the directions and placement of each stress component on the element.

Transformation of Stress

• Enter the direction cosine angles \(\alpha\), \(\beta\), and \(\theta\), which define the orientation of the plane where stresses are to be evaluated. These angles represent the direction cosines between the normal to the desired plane and the x, y, and z axes.
• Angles are entered directly into the input boxes. The app automatically checks whether the condition \(\cos^2\alpha + \cos^2\beta + \cos^2\theta = 1\) is satisfied, which ensures the orientation is valid.



• If the condition is not satisfied, the app provides up to three angle corrections that do meet the requirement. Clicking one of these suggestions auto-fills the input fields and updates the results accordingly.



• The following results are displayed for the selected oriented plane:
  • Stress components in the x (\(\sigma_{n,x}\)), y (\(\sigma_{n,y}\)), and z (\(\sigma_{n,z}\)) directions,
  • Magnitude of the normal stress (\(\sigma_n\)), magnitude of the shear stress (\(\tau\)), and the total stress intensity on the plane.



• Click the i icon next to the section title to view two helpful illustrations:
  • The first shows a 3D representation of the oriented plane and the direction of the normal stress \(\sigma_n\).
  • The second illustrates the meaning and orientation of the angles \(\alpha\), \(\beta\), and \(\theta\) with respect to the coordinate axes.

Principal Stresses

• The stress invariants of the characteristic equation are calculated and shown at the top of this section.



• The three principal stresses \(\sigma_1\), \(\sigma_2\), and \(\sigma_3\) are listed in descending order.
• For each principal stress \(\sigma_i\), the direction cosines of the corresponding normal vector \(n_i\) are displayed both as:
  • Cosine components: \(\cos\alpha_i\), \(\cos\beta_i\), and \(\cos\theta_i\)
  • Angle form: \(\alpha_i\), \(\beta_i\), and \(\theta_i\)



• The maximum shear stress for the current state of stress is also provided.

Mohr's Circle

• The full state of stress is illustrated on a Mohr’s circle representation. While this is a 2D visualization, it effectively reflects the 3D stress state.
• The diagram highlights:
  • The principal stresses (\(\sigma_1\), \(\sigma_2\), \(\sigma_3\))
  • Maximum shear stress (\(\tau_{\text{max}}\))
  • The points corresponding to the X, Y, and Z directions
  • The point corresponding to the oriented plane

  

Clear Inputs or Recalculate

• You can modify the input stresses and direction angles at any time to explore different stress conditions and their implications.