In the formulating of beam elements, is the concept of the basic coordinate system still valid when we have internal DOF

[jiadarenUA]

Hi all, I am trying to study how to develop a new displacement-based beam element in the OpenSees source code, and I got confused with the basic coordinate system.

For a general displacement-based beam element, 6 DOFs are expected in the local coordinate system. In the source code of the OpenSees, the basic coordinate system is involved containing 3 DOFs without the rigid body DOFs, and a transformation matrix will be employed.

If we have an internal rotational DOF at the middle of the beam (i.e., 7 DOFs in total). Can we still use the concept of the basic coordinate system? If so, how are we going to handle the internal DOF in the basic coordinate system?

Your help will be greatly appreciated. Thank you.

[mhscott]

You should be able to still use the basic system concept with 4 DOFs along with static condensation to get down to 3 DOFs. Or you can just formulate element in the full 7 DOF space without a basic system ... this is probably a better approach to get things working, then you can look at the basic system in "version 2.0" of your element.

[jiadarenUA]

Hi Dr Scott,

Thank you so much for the suggestions. It is so amazing that it is you who replied to my question. I am actually a big fan of your research work.

For my question, I also think I should still be able to use the basic system with 4 DOFs , and then I will use static condensation to get it down to 3 DOFs. I have a more detailed question when I want to define this basic system with 4 DOFs. My question is how to define the 4th DOF in the basic system?

Mathematically, with one internal rotational DOF in the middle of the beam, we define the nodal displacement vector in the local system as u={u_1, alpha_1, v_1, u_2, alpha_2, v_2, alpha_3}, with u indicating longitudinal displacement, alpha indicating rotational displacement and v indicating transverse displacement. alpha_3 means the internal rotational DOF. We define the displacement vector in the basic system as deta={deta_1,deta_2,deta_3,deta_4}.

Should we define deta_4=alpha_3+(v_1-v_2)/L in the relative rotation way? Or can we define deta_4=alpha_3 in the "absolute" rotation way, which is independent of the coordinate system.

Thanks so much again for your help. If my description is not clear to you, can I email you with some figures that can describe my question better? I failed to upload the figures here.

You should be able to still use the basic system concept with 4 DOFs along with static condensation to get down to 3 DOFs. Or you can just formulate element in the full 7 DOF space without a basic system ... this is probably a better approach to get things working, then you can look at the basic system in "version 2.0" of your element.

[mhscott]

I think you'll want to have 6 local DOFs and 3 basic DOFs, then have an internal DOF in the basic system that you condense out. It's been a while since I've looked at this type of formulation, but if it's displacement-based, I think this is the way to go.

[jiadarenUA]

Hi Dr Scott,

Thank you very much for your reply. I am sorry that I can not understand well that you mentioned "you'll want to have 6 local DOFs and 3 basic DOFs, then have an internal DOF".

I am trying to develop a flexure-shear coupled displacement-based beam element in OpenSees, so I want to use a shape function containing 7 DOFs in the local system to avoid the shear-locking problem, i.e., 6 external DOFs at the two ends of the beam PLUS 1 internal DOF at the middle of the beam. This is shown as the first attached picture.

The concept of the basic system is used in the OpenSees source code. In order to program this new element more easily, I want to formulate this element in the basic system, but I am confused about how should I define the basic system. Do you think it is ok to define the basic system as shown in the second attached picture? Maybe I have some misunderstanding about the concept of the basic system and the related FE formulation, your help is greatly appreciated.

I think you'll want to have 6 local DOFs and 3 basic DOFs, then have an internal DOF in the basic system that you condense out. It's been a while since I've looked at this type of formulation, but if it's displacement-based, I think this is the way to go.

[mhscott]

The rotation of the node in the middle of the beam should be an internal variable, not something you get from a regular node. The element keeps track of the state of the internal node and does whatever interpolation you need along with static condensation to get back to three basic DOFs.

[jiadarenUA]

Thank you very much for your help and patience Dr Scott. I think basically I get your meanings, but maybe I need to learn more knowledge about the FE formulation based on the basic system. Could you please recommend me some references about the displacement-based FE formulation based on the basic system? In those literature I reviewed, very little of them discussed the basic system and the formulation is based on the local/global system. It would be greatly appreciated. Thank you.

The rotation of the node in the middle of the beam should be an internal variable, not something you get from a regular node. The element keeps track of the state of the internal node and does whatever interpolation you need along with static condensation to get back to three basic DOFs.