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?
Multiple principal stress refers to a state of stress where an object or material is subject to more than one type of stress, such as tension, compression, and shear stress. This can occur when an object is subject to forces acting in different directions, resulting in a combination of stresses.
Multiple principal stress can be calculated using a mathematical model called the Mohr-Coulomb theory, which takes into account the magnitude and direction of different stress components acting on an object. The principal stress values can be determined by solving for the eigenvalues of the stress tensor.
Understanding multiple principal stress is crucial in engineering as it helps determine the maximum stress that a material can withstand before failure. It also allows engineers to design structures and materials that can withstand and distribute different types of stress more efficiently.
Multiple principal stress can be observed in various real-world scenarios, such as the stress distribution in a bridge, the structural stability of a building during an earthquake, or the forces acting on an airplane wing during flight. It is also a crucial factor in the design and performance of materials used in machinery and vehicles.
Engineers use various techniques and methods to account for multiple principal stress in their designs, such as stress analysis software, physical testing, and empirical formulas. They also consider factors such as material properties, loading conditions, and safety margins to ensure that their designs can withstand the stresses they will be subjected to in real-world applications.