20 Jul 10 9:45
There are coupling effects between stories associated with determining C.O.R. for a multi-story building such that you do not get the same C.O.R. as you would just looking at the plan by itself. See below from ETABs documentation:
Calculate Diaphragm Centers of Rigidity
The Analyze menu > Calculate Diaphragm Centers of Rigidity command is a toggle. When a check precedes this command on the menu, ETABS will calculate the diaphragm centers of rigidity during the analysis. When no check precedes this command on the menu, this calculation is not performed.
The original concept of center of rigidity dates back to manual rigidity analysis techniques associated with the lateral analysis of single-story shear wall buildings. The center of rigidity was defined as the location of the centroid of the stiffnesses of single-story lateral resisting elements (typically planar) arbitrarily located in plan. For single-story structures the definition worked well because the stiffness for each wall frame was a 1 by 1 matrix with no interstory coupling or compatibility factors to complicate the problem. The analysis technique was extrapolated for multistory lateral analysis whereby multistory buildings were analyzed as a series of single-story buildings stacked over one another with no interstory displacement compatibility. Needless to say, for complex three dimensional structures this assumption was approximate at best.
Modern computer techniques do not require the explicit evaluation of the center of rigidity. However, the center of rigidity still needs to be evaluated because some building codes refer to it as a reference point to define design eccentricity requirements in multistory buildings.
In the general three-dimensional analysis of a building, where the behavior is coupled in plan as well as through the height of the structure, the center of rigidity requires a broader definition. In this broader definition when translational lateral loads are applied at the center of rigidity of a particular floor diaphragm, with no loads applied to any of the other floor diaphragms, the displacements of that diaphragm will have only translational components with no rotations. it should be noted that the resulting displacements of the diaphragms at other levels in general will contain translational as well as rotational components.
To evaluate the center of rigidity at a particular diaphragm, the structure is analyzed for three load cases. The loads are applied at the center of mass (or any arbitrary point). Load case 1 has a unit load applied in the global X direction and results in a diaphragm rotation of Rzx. Load case 2 has a unit load applied in the global Y direction and results in a diaphragm rotation of Rzy. Load case 3 has a unit moment applied about the global Z-axis, giving a diaphragm rotation of Rzz.
The center of rigidity relative to the center of mass (or the arbitrary point) is then given in by the coordinates (X, Y), where