Question

What will be position of principal plane if atrrese in two direction are same?

Answer 1

When you set up a solid mechanics problem the first step is usually the choice of a suitable coordinate system. This is typically some sort of right handed cartesian coordinate system aligned with some convenient part geometry. You then use this coordinate system without having to think about it again to calculate shear and normal stresses. The problem with this though is that you tend to forget that your choice of coordinate system in step one was completely arbitrary. Your calculation of these stresses is however dependent on this coordinate systems. These axes are more than likely however not aligned with the loading of the problem and as such there probably exists another choice of coordinate system where stresses are higher than in the system you chose.

Naively you could then go about trial and error choosing coordinate systems until the stresses where maximum but that isn't the greatest idea. Instead because we know that the stress is a tensor we can simply find its eigenvalues and eigenvector. This is then the maximum values for the stresses and the direction of the coordinate system in which this is the case respectively. If your matrix manipulation isn't so good you can instead choose to draw the stress tensor on a mohr’s circle and rotate it geometrically instead. This works well for 2D, but gets clumsy but though doable in 3D. Doesn't really matter what method you use to get there or the dimensionality of the problem all you are essentially doing is rotating an arbitrarily chosen coordinate system to ensure that it is aligned with the applied stresses. This then maximizes the individual components and allows you to check whether the material properties are adequate in the direction of the maximum stresses.

I hope this will help you

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