The discussion centers on the interpretation of multiple principal stresses in the context of three-dimensional loading with shear. Participants explore the application of principal stress formulas and the implications for using the von Mises criterion to determine yielding limits.
Participants express differing views on the interpretation of principal stresses and the relationship between eigenvalues and eigenvectors. The discussion remains unresolved regarding the implications of having multiple vectors for a single eigenvalue.
The discussion highlights limitations in applying 2D stress analysis methods to 3D scenarios and the need for clarity on the mathematical relationships involved in eigenvalue problems.
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.vin300 said: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.
The is only one 3D vector for each lambda.vin300 said: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?