## 2 Theoretical Aspects
### Centre of Stiffness  
CS, corresponding to a particular floor, is defined as the application of the lateral load to the point, and it does not produce a rotation of that floor. This definition is valid when the slab is modelled with the infinite in-plane floor behaviour. The infinite in-plane floor behaviour causes all of its constrained joints to move together as a planar diaphragm that is rigid against membrane deformation. Firstly, it has to determine the values of the stiffness (\(K_{x},\ K_{y},\ K_{xy}\)), and one must consider and analyse every structural element corresponding to one storey. The number of the columns between the floors is denoted with \(n\). For this purpose, one chooses an arbitrary orthogonal reference system denoted with \(xOy\) relative to the floor, and another local reference system \(x_{k}Oy_{k}\ (k\ =\ \overline{1,n})\) corresponds to the cross-section of the column \(k\) of the storey (located between two consecutive floors). The stiffness factor depends on the following: moment of inertia, the Young's modulus of the material and the distance between two floors. The distance between floors and the material property are given due to the chosen structural system, so we are required to determine the moment of inertia corresponding to the floor. In the general case corresponding to an irregular shape of the floor's section, one uses section meshing in order to obtain simple and regular geometric shapes. Steiner's relations are used to determine the moment of inertia \(I_{xk},I_{yk},I_{xyk}\) corresponding to each element of the storey. Then, one may compute the principal directions of inertia denoted with \(\alpha_{k1}\) and \(\alpha_{k2}\) corresponding to the section \(k\) of the element of the storey. The corresponding principal moments of inertia denoted with \(I_{1k}\) and \(I_{2k}\), respectively, are also computed. Then, the principal rigidity factors of the cross section of the column \(k\) of the floor are obtained, denoted with \(K_{1k}\) and \(K_{2k}\), respectively. At this moment, the stiffness factors \(K_{1k}\) and \(K_{2k}\) are computed with respect to the local coordinate system \(x_{k}Oy_{k}\). To define the rigidity factors corresponding to the section \(k\ =\ \overline{1,n}\) with respect to the global coordinate system \(xOy\), it is used the following Equations (3)-(5) [15] whose physical significance is shown in Figure 1a,b:  
\[K_{xk}\ =\ K_{1k}\cos^{2}\alpha_{k}+K_{2k}\sin^{2}\alpha_{k}; \tag{3}\]  
\[K_{yk}\ =\ K_{1k}\sin^{2}\alpha_{k}+K_{2k}\cos^{2}\alpha_{k}; \tag{4}\]  
\[K_{xyk}\ =\ (K_{1k}-K_{2k})\sin\alpha_{k}\cos\alpha_{k}. \tag{5}\]  
Figure 1: The rigidity factor corresponding to a unitary displacement applied on: (**a**) x axis, and (**b**) y axis. To compute the stiffness factors (\(K_{x}\), \(K_{y}\), \(K_{xy}\)) corresponding to the entire floor with respect to the global coordinate system \(xOy\), the contribution of all structural elements of the storey is considered, shown in the following Equations (6)-(8) [15]:  
\[K_{x} = \sum_{k}K_{xk}; \tag{6}\] \[K_{y} = \sum_{k}K_{yk};\] (7) \[K_{xy} = \sum_{k}K_{xyk}. \tag{8}\]  
This is possible in the case of rigid diaphragm and spatial connection of the structural elements. As a function of structural properties, CS is an elastic characteristic of the storey and it is independent of loading [3]. It is already known that the rotation does not take place if CM coincides with CS. In a mathematical approach of the problem, the CS is the point defined by the intersection of two lines presented in Equation (9) [15]:  
\[\mathrm{CS}\ =\ (\mathrm{D}_{1})\cap(\mathrm{D}_{2}) \tag{9}\]  
In Equation (9), the mechanical significance of the lines \((\mathrm{D}_{1})\) and \((\mathrm{D}_{2})\) is that if the lateral load is applied along these lines, then the storey performs only translational movements (\(\Delta_{x}\), \(\Delta_{y}\)). Stiffness means the force caused by the unit displacement.