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Re: SAP, RISA3D, STAAD - Master/Slave Relationships

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> From: BRBATES <BRBATES(--nospam--at)aol.com>
> To: seaoc(--nospam--at)seaoc.org
> Subject: Re: SAP, RISA3D, STAAD - Master/Slave Relationships
> Date: Wednesday, December 31, 1997 1:35 PM
> 
> 
> 

> Here's an example. Imagine a simple table top model, 4 columns with the
tops
> of the columns linked together via a diaphragm. If we model this top
diaphragm
> using in-plane master/slave links, this means the four top nodes will
share
> the same in-plane DOF's. We'll say this is the X- Y plane, so they share
the
> X,Y and Z Rotation DOF's.
> 
> Now, for the solution, a stiffness method program calculates one
displacement
> per DOF, so the four top nodes will all have the same X,Y and Z rotation
> displacements because they all share the same DOF's. Is this right? Nope.
If
> this diaphragm rotates, the X,Y displacements for each of the nodes will
be
> different. Only the in-plane rotation is always the same for all the
nodes on
> the diaphragm.
> 
> Another point is the loading. If we apply a force to one of the top slave
> nodes, a master/slave link will move that force to the master node
because,
> again, there is only one DOF and the load vector is formed on a DOF
basis. So
> if we have a master node in the center and we load one of the corners of
the
> tabletop, we won't get any rotation (as we should) because the load is
moved
> to the center of the diaphragm!

The load is not just moved to the center of the diaphragm. But a moment is
introduced
as well, as correctly indicated in Rudra Nevatia's response (follows):

>>>>Rudra's response:
I agree with Warren's point that master/slave, diaphragm and
rigid link are all synonymous mathematically. The displacements,
actions and stiffness matrices corresponding to slave DOF's are
transferred to master DOF's by the following relations :

(See section 16-7 of "Structural Analysis" by Ghali & Neville)

                    {q} =  [C].{D}
                    {F} =  transpose [C] . {Q)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Clearly, the above transformation indicates that if a load is imposed at
the slave degree
of freedom, then the C matrix containing the "distance between the slave &
the master
nodes" not only translates the load from the slave node to the master node
but also 
introduces a moment (about the z axis) to account for the lever arm between
the master
and slave nodes.


                    [Sm] = transpose [C] . [Ss] . [C]
where

{q}  = displacements at slave DOF's
{D}  = displacements at master DOF's
{Q}  = actions at slave DOF's
{F}  = equivalent actions at master DOF's
[Ss] = stiffness matrix at slave DOF's
[Sm] = stiffness matrix at master DOF's

         | 1  0 -y |
[C]  = | 0  1  x |
         | 0  0  1 |

x,y  =  co-ordinates of slave node with origin at master node
>>>>>



> 
> The master/slave method will give correct results for a diaphragm so long
as
> there is no in-plane rotation of the diaphragm (perhaps the case when you
were
> running SAP?), but it falls apart as soon as the first rotation appears.
> 

Nope. The resulting z rotation at the master node because of the moment
introduced is
the "diaphragm rotation". Thus the diaphragm will have resultant x & y
displacements
as well as a z rotation. In ETABS, if you want to see the "torsional" drift
in the columns
due to this diaphragm z-rotation, you need to introduce dummy frames
(basically a 
column line) at the four corners of the building.

So, in short, I too agree that rigid diaphragms can be modeled by using the
master-slave 
node in the right fashion.

Bruce, a quick question. Does this mean that the master-slave option in
RISA does not
use the above transformation before eliminating the slave-dofs, but instead
just replaces
the slave-dofs with the master-dofs? If that's the case, then clearly,
there are two 
separate definitions for "master-slave option" prevalent and this needs to
be unified and
agreed upon. I have always thought that master-slave option includes that
key 
transformation before the reduction in the no.of dofs in the model.

-Swaminathan Krishnan

> >> 
>  The upshot of all this is that the software writers must fully 
>  document what assumptions they are using. 
> <<
> 
> Our documentation is crystal clear as to what assumptions we use.
> 
> 
> Bruce Bates
> RISA Technologies