Why do the 10 elastic links in the attached model need different beta angles to point perpendicular to the two column pier, despite having the same connectivity and local x-axis pointing down? I was expecting the beta to be the same for all links.

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Answer:

Yes, that is correct.  If the links have the same orientation, then the same beta angle has to be applicable.  However, in this case, the local x isn't exactly vertical for few of the links.  Kindly have a look at the image below.

As it could be noted from the image above, the elastic link connecting nodes 4764 and 4786 isn't exactly vertical, while the link connecting nodes 4766 and 4783 is exactly vertical.

This discrepancy has been removed in the attached modified model for all links and corresponding nodes above the link.  Now, the same beta angle works well for all links.