# Multiple principal stress

1. Jan 21, 2016

### vin300

For 3 D loading with shear, if I use the principal stress formula, say for x-y direction, two principal stresses are obtained. If the same is applied to y-z, two more principal are obtained, with one supposed to be common, but not. Thus I obtain six principal values, which cannot be used with the von mises criterion to find yielding limit.

2. Jan 21, 2016

### Staff: Mentor

You can't use the 2D equations for a 3D situation. To determine the principal directions and stresses in 3D, you need to solve the following eigenvalue problem: $\vec{\sigma} \centerdot \vec{n}=\lambda \vec{n}$, where sigma is the stress tensor, n is a unit vector in one of the three principal directions, and lambda is the corresponding principal stress. This equation leads to 3 homogeneous linear algebraic equations in the components of n.

Last edited: Jan 21, 2016
3. Jan 22, 2016

### vin300

After finding the eigen values, do the eigen vectors represent their direction? If they do, what do I make of more than one vector for one lambda?

4. Jan 22, 2016

### Staff: Mentor

The is only one 3D vector for each lambda.