In some cases, it is necessary to determine the center of rigidity of a building. The concept of “center of rigidity” is used, for example, in seismic analysis. If the center of rigidity does not coincide with the center of mass, torsional oscillation modes appear. Some building frame design guides recommend arranging vertical diaphragms so that the center of rigidity is close to the common center of mass.
A center of rigidity of a storey can be considered in buildings where space-planning design varies in height. The center of rigidity of a storey is defined as a point that has the following property: if the resultant of all horizontal forces passes through it, then only the translational displacement of the floor disk occurs and there is no torsion.
In accordance with the recommendations of “Instructions for Determining the Design Seismic Load for Buildings and Structures” (Moscow, Gosstroyizdat, 1962) coordinates of the center of rigidity are determined by the formulas of the following type
\[x_0= \frac{\sum\limits_{i=1}^{n}G_{xi}x_i}{\sum\limits_{i=1}^{n}G_{xi}},\]
\[y_0= \frac{\sum\limits_{i=1}^{n}G_{yi}y_i}{\sum\limits_{i=1}^{n}G_{yi}},\]
where \(G_{xi}, G_{ii}\) are rigidities of vertical elements of a storey (walls, columns, pylons etc.) in the direction of the X and Y axes, respectively, аnd \(x_i, y_i\) are coordinates of the elements’ centers.