The definition of center of rigidity (CR) is as follows:
The center of rigidity is a point at a particular story as the location of application of lateral load at that point will not produce rotation of that story.
The above definition is valid when slab is modelled as a rigid diaphragm. A Diaphragm Constraint causes all of its constrained joints to move together as a planar diaphragm that is rigid against membrane deformation. For semi-rigid diaphragm, the inplane deformation on a floor points could be different due to inplane-deformation of slab. Therefore, there is no unique solution to center of rigidity i.e. the Center of rigidity is indeterminate quantity.
The center of rigidity is just a reporting item. The inherent eccentricity due to change in mass centroid and stiffness centroid is automatically included in the analysis regardless the choice of diaphragm. Therefore, the reported value for center of rigidity for semi-rigid diaphragm has no meaning and this reported CR does not affect the analysis results.
Center of rigidity is the stiffness centroid within a floor-diaphragm plan. When the center of rigidity is subjected to lateral loading, the floor diaphragm will experience only translational displacement. Other levels are free to translate and rotate since behavior is coupled both in plan and along height. As a function of structural properties, center of rigidity is independent of loading. Certain building codes require center of rigidity for multistory-building design-eccentricity requirements.
For a given floor-diaphragm, CR is calculated through the following process in case of the floor defined as Rigid Diaphragm:-
Case-1: applies a global-X unit load to an arbitrary point such that the diaphragm rotates RXY
Case-2: applies a global-Z unit load to an arbitrary point such that the diaphragm rotates RYZ
Case-3: applies a unit moment about global-Y causing rotation RYY
If there are “N” no. of rigid diaphragms present in the model, there will be “3N” no. of load cases generated for which static analysis will be performed.
Center of Rigidity (X, Z) is then computed as:
X= RYZ / RYY
Z = - RXY / RYY
In STAAD, the unit loads are applied at CM
The global coordinates of CR at each floor is given by (CRx, CRy and CRy), where CRx = CMx + X, CRy = CMy and CRz = CMz + Z
Where, centre of mass (CM) at each floor level and find its coordinates (CMx, CMy, and CMz)
NOTE: Center of rigidity is only applicable to rigid diaphragms because in-plane slab deformation is variable across laterally loaded semi-rigid diaphragms. During computation, an arbitrary coordinate is selected and loaded, and then center of rigidity is derived, as a function of stiffness, according to the displacement at this specific point. If a diaphragm constraint is not applied, displacement at any point will also depend upon variable local membrane deformation. As a result, no unique solution is available for center of rigidity since formulation assumes that all joints translate together in planar motion.