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In my engineering mechanics course, I came across the following rotation matrix, for a stress tensor, for a counter-clockwise rotation.

Rotation Matrix

However, this is completely different, actually a transpose of the matrix I would make for a counter-clockwise rotation, given here.

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Why is there is a discrepancy here?

In addition, I want to know how we can use these matrices. If we have matrix, say, [10,5;5,12], and we want the principal stresses at an angle of 30 degree, how would we do that?

Thanks in advance!

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The discrepancy you found comes from whether you are rotating the coordinate system or the object. In order to find the equivalent stress tensor at a new angle, you would have to rotate the coordinate system because the object is not actually rotating, but rather, you are looking at its stresses from a different perspective. For further explanation, here's a lesson on Coordinate Transformations.

Additionally, rotations are applied differently for 2nd rank tensors than they are for vectors, which are first rank. See equations 1 and 2 below from this lesson on Transformation Matrices

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Now you may be asking why you have to multiply 2nd rank tensors in such an odd way. This just has to do with how the transformation equations are derived. If you want more details on that, the first section in this lesson on Stress Transformations should help. The result is equation 3 below, which is the same as equation 2.

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If you want to find the stress tensor after rotating your coordinate system counter-clockwise by 30 degrees, you can simply use equation 3, which should give you the result below.

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Finding the principal stresses will require us to find the angle at which the shear stress is zero. This angle can be found using equation 4 below, which is derived in this lesson on Principal Stresses & Invariants

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You should find that the angle is -39.345 degrees, which can be used in equation 3 to find that the principal stresses are 16.1 and 5.9.

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