What is the definition of Center of Rigidity (CR) in ETABS
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. Figure 41-1 illustrates the three load cases. The center of rigidity relative to the center of mass (or the arbitrary point) is then given by the coordinates (X, Y), where X = -Rzy / Rzz and Y = Rzx / Rzz. This point is a function of the structural properties and is independent of any loading.
As described above, the determination of the center of rigidity can be numerically intensive as it involves a static analysis of the whole structure for three independent load cases for each floor diaphragm. This procedure is implemented in ETABS so that the centers of rigidity for every diaphragm are automatically calculated as part of the solution process.
You can use Display > Set Building View Options and check the box for Building Output and click OK to find out the Center of Mass and Center of Rigidity. The diaphragm assignment must be present in model to compute the center of rigidity.
