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

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This subject was discussed extensively by Warren, Bates, Krishnan, Luth,
and Nevatia.  After reading all responses, I have to agree with Bates
based on my past experience of writing my own program for dynamic
analysis of multi-story buildings with rigid floor diaphragms.

As pointed out by Bates, the crux of the matter is the definition of
master/slave relationship.  We all have to make up our mind as to what
is master/slave relationship.  The definition given by Ghali & Neville
in "Structural Analysis" as shown below assures that the master and
slave nodes have the same rotation but different displacements.  Are we
satisfied with this definition?  Or do we want master and slave nodes to
have same rotation as well as displacement?

As a matter of fact, the definition given below by Ghali and Neville was
the same equation I used to model a rigid diaphragm more than 25 years
ago.  Thus, to avoid future confusion, may I suggest that a true
master/slave relationship be defined as the one which assures the same
displacement and rotation for all assigned master/slave degrees of
freedom.  The definition given below by Ghali and Neville should be
reserved for relationship between nodes on a rigid diaphragm.

IMHO, If one wants to model a rigid diaphragm, he or she should either
use the equation below by Ghali and Neville in writing his or her own
program, or use rigid links in general purpose programs, e.g., SAP,
RISA3D, or STAAD.

Regards,

Chang Chen, Ph.D., P.E.
Apollo Consulting, Inc.



Rudra Nevatia wrote:

> 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)
>                     [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
>
> I have verified that the master/slave method when applied in
> this fashion gives correct results for translations *and*
> in-plane rotations.
>
> Incidentally, static condensation is a process of folding
> (usually) internal DOF's into external DOF's of an element. This
> process is not applicable to master/slave DOF's.
>
> Rudra Nevatia
> Central Computing Facility
>