30.14. Principal Axes and Eigenvectors

Try these first

1. 1.

   What is an eigenvector?

   Hint

   Think of vectors in a direction that does not change.

2. 2.

   What does an eigenvalue indicate?

   Hint

   It is a scalar associated with an eigenvector.

3. 3.

   What is the primary purpose of eigenvectors in materials?

   • To measure temperature
   • To determine principal stresses and directions
   • To calculate the volume of materials

   Hint

   Consider how forces behave in different directions within structures.

4. 4.

   True or False: Principal stresses are eigenvalues derived from stress tensors.

   • True
   • False

   Hint

   Reflect on how stress tensors relate to these concepts.

5. 5.

   Given a 2x2 stress tensor, compute the principal stresses and directions analytically.

   Hint

   Start by calculating the determinant for the characteristic polynomial.

6. 6.

   Explain how you would approach designing a beam based on its principal stresses.

   Hint

   Consider real-world applications in construction materials.