[MBDyn-users] Equilibrium equation for end reference points of a three-node beam

Pierangelo Masarati pierangelo.masarati at polimi.it
Tue Oct 18 21:08:19 CEST 2016


On 10/07/2016 01:59 PM, Yuanchuan Liu wrote:
> Dear all,
>
> I am trying to understand the finite volume beam element model in MBDyn but have had this question in my mind for several days. Suppose I create a model consisting of four three-node beam elements, numbering from #1 to #4 and with #2 located between beams #1 and #3. I understand, from Pierangelo Masarati’s PhD thesis on page 56, that for every beam, equilibrium equations for the three beam portions, divided by two evaluation points (I and II) and centred on three nodes (a, b, c), will be established.
>
> For the middle beam portion, integration boundaries are well defined at evaluation points and everything is clear. What about those end portions, for example, 2a and 2c for beam #2? As nodes 2a and 1c are basically the exactly same node, will there be two individual equilibrium equations for 2a (or 1c) corresponding to the two adjacent beams? Or the two equations will be later merged into one single equation, with an expanded boundary from evaluation point II of beam 1 to evaluation point I of beam 2?
Equations are generated by nodes.  Thus portions 1c and 2a of beams 1 
and 2 will contribute to the equations generated by the same node, 
assuming you use the same node label in the specification of the two 
beams.  For example:

structural: 0, ...
structural: 1, ...
structural: 2, ...
structural: 3, ...
structural: 4, ...

beam3: 1,
         0, position, null, # 1a
         1, position, null, # 1b
         2, position, null, # 1c, same as 2a
         ...
beam3: 2,
         2, position, null, # 2a, same as 1c
         3, position, null, # 2b
         4, position, null, # 2c
         ...

and so on

>
> A similar question would be how adjacent beam elements are connected. Is it legitimate to consider a model with multiple three-node beam elements to be one beam element with multiple nodes?
Somewhat, from a topological point of view.  From the point of view of 
displacement and rotation discretization, the interpolation is parabolic 
every three nodes, allowing for slope discontinuities at the boundaries.

Sincerely, p.
>
> Any help is very much appreciated. Cheers.
>
> Best wishes,
>
> Yuanchuan Liu
>
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-- 
Pierangelo Masarati
Professor
Dipartimento di Scienze e Tecnologie Aerospaziali
Politecnico di Milano



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