@tomleslie

Voila. Thank you Tom. I cannot imagine how much time and patience it must take to go through someone's sheet. I gave up on mine eventually!

I am following you. I will try to give you a background of what I am doing and hopefully, it will be clearer:

Basically, I have unidirectional laminae, which are plies of composites. Composites can be different mix of heterogeneous materials. Unidirectional laminae are sheets of matrix (resin) where there are carbon fibers aligned along the same direction.

The main feature to focus on when it comes to unidirectional laminae is the fact that their stiffness is described through a tensor.

More particular is that the stiffness tensor at first is defined according to a "local axis".

The local axis is quite typical: x1' follows the fiber direction, x2' is normal to the fibers' length and x3' is normal to x1' and x2' and is always the axis poking out of the lamina.

So when I have my lamina properties (modulus according to direction 1, 2 3 and shear modulus according to 1, 2 3 and poisson according to the three directions), I can come up with the expression of the tensor in the local coordinates.

The thing is when I orient my lamina according to the global axis (at 45degress or 90 or 30), the tensor has to be transformed according to the rotation.

Stiffness of composites and heterogeneous materials needs to be recalculated in the global axis once we settle on their location ( hence the Stiffness_At_Wall= Rotation*Stiffness_LocalCoord*Transpose(Rotation)

Which is fine as long as it doesn't become too confusing.

But I happen to have a stack of laminae, let's say six. And I have to create a beam out of them in 45degrees.

This means basically means a first rotation of 45degrees.

Then rotation according to the global axis so that I can get the stiffness of every "wall".

The walls are basically a rectangle sitting in a global axis referential.

And I am basically taking every laminae, rotating it 45degrees, gluing to a wall and pondering how to match the local coords with the global coords so that I can get my stiffness at the wall.

Which can be fine if I am the only observer and spent some time making sure Im taking all my rotations in the right sense.

But when my colleagues try to use the tool, they have their own sense of rotation in their head and things get very confusing and results as well.

So, I was basically hoping that, knowing the initial configuration, and the end configuration, I could come up with something that would spare us the lengthy discussion over Ampere man and orientation of rotation and signs issue.

I hope it is clearer now. I attached a doodle I made on paint of an example of the operation. Thank you Tom, thank you.