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

• To: seaoc(--nospam--at)seaoc.org
• Subject: Re: SAP, RISA3D, STAAD - Master/Slave Relationships
• From: Rudra Nevatia <nanditan(--nospam--at)giasbma.vsnl.net.in>
• Date: Thu, 2 Apr 1998 16:26:29 -0500 (GMT)

```Aleksander Urbanski wrote:

>in dynamics the presence of "rigid link " options inevitably
>spoils the diagonal structure of the mass matrix when lumped
>masses are used. In such case consistent mass matrix should be
>used

I beg to differ. One can take advantage of the fact that lumped
masses are associated with the rigidly linked DOFs. The other
DOF's can therefore be condensed out, leaving a very small
matrix for dynamic analysis. If a building has n floors, the
size of matrix reduces to 3n x 3n, irrespective of the total
number of nodes.

Consider the following example taken from "Dynamics of
Structures" by Clough & Penzien:

A rigid square slab of sides L, with a total mass m = 0.5
kips.sec2/ft is supported by three columns of height L. The
column stiffness k = 12*E*I/L3  about each axis is 0.5 kips/ft.
The torsional rigidity of each individual column is negligible.
It is required to find the frequencies of the system.

/-------------------/
/                   / |
/                   /   |
/                   /     |
/                   /       |
/-------------------/         |
|                   |         |
|                   |
|                   |         C
|                   |
|                   |
|                   |

A                   B

(Note that the definition of k implies that the columns are
fixed at both ends: The slab is infinitely rigid out of its
plane as well)

The complete stiffness and mass matrices for the system are:

12EI |  4  -2   2 |            m |  4  -1   3 |
K = ---- | -2   3  -2 |        M = - | -1   4  -3 |
L3  |  2  -2   3 |            6 |  3  -3   6 |

The resulting freqencies are { 5.036, 5.477, 9.464 } rad/sec.

This is an excellent bench-mark problem for testing dynamic
analysis features of commercial software (I was unable to get
correct results on STAAD version 19). It will be interesting to
get feedback from users of other software.

Rudra Nevatia
Central Computing Facility

```