21.20.2. Case 2: Soil Mechanics

Try these first

1. 1.

   What is a stress tensor?

   Hint

   Think about how forces are distributed at a point.

2. 2.

   Define principal stresses.

   Hint

   These are critical for structural safety.

3. 3.

   What does a stress tensor represent?

   • External forces
   • Internal forces at a point
   • Tensile strength

   Hint

   Focus on what happens inside the soil.

4. 4.

   True or False: Principal stresses are always positive.

   • True
   • False

   Hint

   Consider the nature of stresses in soil.

5. 5.

   Given a stress tensor as $\sigma = \begin{bmatrix} 10 & 5 \\ 5 & 15 \end{bmatrix}$, find the principal stresses and directions.

   Hint

   Don't forget to set up the determinant for the characteristic polynomial!

6. 6.

   Consider a soil mass subjected to varying loads resulting in a stress tensor of $\sigma = \begin{bmatrix} 20 & -10 \\ -10 & 30 \end{bmatrix}$. Analyze how the principal stresses affect the design of an embankment.

   Hint

   Think about what happens if stress surpasses soil yield levels!