# Parametric Analysis of Rotational Effects in Seismic Design of Tall Structures
## 5 Detailed FEM Analyses  
The procedure for determining the response parameters, which describe the increase in the rotational seismic components to the overall seismic response according to the previous section, is only an assessment because of its simplicity. Nevertheless, it provides an appropriate quick overview for the designers when a more complex analysis is necessary. This coefficient with a somewhat limited precision can be also used for the estimation of the error of the simplified solution. We decided to verify it on more complex structures. Four different structural types have been assumed (Figure 2) and analyzed using FEM ANSYS software: (a) frame + wall (\(H=47\ m\)), (b) R/C core building with perimeter columns (\(H=74\ m\)), (c) tower (\(H=280\ m\)) and finally a special structure, (d) chimney. The last one is with two different heights, \(H=65\ m\) and \(H=133\ m\). The well-known formula Eq. (19) has been assumed with a structural coefficient = 0.03; 0.043; 0.085; 0.045 for the (a) frame + wall, (b) R/C core with perimeter columns; (c) tower and (d) chimney, respectively, according to [26,27], cl.4.3.3.2.2(3), respectively. Due to the diversity of structural systems, it was necessary to choose a suitable comparative quantity. As a comparative quantity expressing the seismic effects in a simplified way, a fictitious bending moment at the fixed end calculated from the amplitudes of inertial forces according to Equation (35) was used. This variable was an appropriate benchmark for a mutual comparison of results in accordance with Sections 3 and 4. The procedure was as follows:  
1. Work has started with a complex 3D FEM model (Figure 2), taking into account all relevant constructional details, including different types of foundation. In case (a), it was piles; in case (b), a foundation slab and two underground floors; in case (c), diaphragm walls; in case (d), a circular foundation slab. For these structures, the natural frequencies and the mode shapes have been calculated, from which the first two bending mode shapes in a given direction were selected (see the arrows of Figure 2), which have to match the calculations of steps (b), (c). 2. For each structure, we have prepared a simplified alternate dynamic model with six degrees of freedom representing the horizontal displacements in the considered direction of the vibrations (Figure 2). The flexibility matrix of the simplified model  
Figure 2: Influence of rotational component of seismic effects—response parameter: (**a**) frame + wall building with weak ground story; (**b**) R/C core + perimeter columns building; (**c**) tower building; (**d**) chimney. was calculated using unit forces acting in places of the six degrees of freedom on the complex model  
\[\delta=\left[ \begin{array}{ccc}\delta_{11}&\dots&\delta_{16}\\ \dots&\dots&\\ \delta_{16}&&\delta_{66}\end{array} \right] \tag{34}\]  
where \(\delta_{ij}\) are the displacements of the selected six degrees of freedom, evenly distributed along the height of the building (as in Figure 1a) in the direction \(i\) from the unit force applied in the direction \(j\). From the flexibility matrix, the stiffness matrix \(\mathbf{K}=\delta^{-1}\) was calculated. The mass matrix was calculated as a diagonal matrix with concentrated masses (lumped mass matrix) attributable to the six degrees of freedom. 3. The first two bending eigenfrequencies and mode shapes were calculated on the simplified model according to (b), and these were compared with the calculated frequencies from the complex model of (a). Since this is a linear calculation, the first two bending eigenfrequencies and mode shapes matched well.