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I know how to get the stiffness matrix of a beam with any type of end releases (hinges and rollers) by applying Guyan reduction (static condensation) to the beam stiffness matrix in local axes. However, by doing this, the rollers sliding direction is parallel to the beam local axes.

First question: Would I get the rollers parallel to the global axes if I apply the Guyan reduction to the beam stiffness matrix expressed in global (rather than local) axes?

Second question: What if I wish to get a roller in a direction which is not parallel to neither local nor global axes? Is it just a matter of transforming the matrix in two steps? I mean: first step, apply a transformation matrix from local axes to roller axes. Apply Guyan reduction there. Then apply a second transformation matrix from roller axes to global axes, and you have the beam stiffness matrix with the desired end release, ready for assembly because it's in global axes.

Would it be as easy as that? Is there any book covering this? (I found end releases explained in several books, but only in local axes).

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  • $\begingroup$ "Would it be as easy as that?" - Yes. But sometimes you don't want to do the "second transformation" back to global axes - it might be conceptually easier to specify the loads, constraints, etc in the local axis system. $\endgroup$ – alephzero Jul 16 '17 at 0:37
  • $\begingroup$ @alephzero: I'm talking about when you want to add end releases not parallel to the beam local axes. Imagine a roller/slider which is not parallel to the beam local axes. How to you implement it as an end release? Have you seen any book discussing this? $\endgroup$ – cesss Jul 16 '17 at 11:56
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If I remember correctly, first, you do the static condensation and then you rotate the degrees of the roller appling the transformation only on those. So it's like going from your roller local axis to the local axis of you element.

And yes, getting rid of degrees in the global system will be the same with doing it locally and then transforming it.

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