# Parametric Analysis of Rotational Effects in Seismic Design of Tall Structures
## 3 General Simplified Procedure  
The following simplified assumptions are used, which are mostly fulfilled for slender structures such as towers, masts and chimneys:  
* The contribution of the first vibration mode is most important;
* The mass and stiffness are evenly distributed along the height, or the changes of it are gradual. Based on these assumptions, we can express the seismic effects from translational and rotational components of ground motion by inertial forces in the particular levels \(i\) (storeys). For translational components of ground motion, it follows  
\[F_{i,u}=m_{i}\bar{u}_{i,u,max}=\omega_{j}^{2}m_{i}u_{i,u,\ max}, \tag{3}\]  
where  
\(u_{i,u,max}\) is the maximal deflection of the \(i\)-th story under vibration with frequency \(\omega_{j}\) from the effects of translational components;  
\(m_{i}\) is mass concentrated in level \(i\). Using decomposition into eigen modes, we can express \(u_{i,u,max}\) as  
\[u_{i,u,max}=\[PERSON]_{i,j}\frac{\mathbf{\[PERSON]}_{j}^{T}\mathbf{m}\mathbf{I}}{\mathbf{ \[PERSON]}_{j}^{T}\mathbf{m}\mathbf{\[PERSON]}_{j}}S_{d}\left(T_{j}\right) \tag{4}\]  
where  
\(\Phi_{i,j}\) is the coordinate of the \(j\)-th vibration mode shape at location \(i\);  
\(\mathbf{\[PERSON]}_{j}\) is the vector of the \(j\)-th vibration mode;  
\(\mathbf{I}=\left\{1\ \ldots\ 1\right\}^{T}\) is a unit vector;\(S_{d}\big{(}T_{j}\big{)}=\frac{S_{e}\big{(}T_{j}\big{)}}{\omega_{j}^{2}}\) spectral displacement coordinate; \(S_{e}\big{(}T_{j}\big{)}\) spectral acceleration coordinate.  
After substitution of Equation (4) into Equation (3)  
\[F_{i,u}=m_{i}\Phi_{i,j}a_{j,u}S_{e}\big{(}T_{j}\big{)}, \tag{5}\]  
where  
\[a_{j,u}=\frac{\mathbf{\Phi}_{j}^{T}\mathbf{\mathrm{m}}\mathbf{\mathrm{l}}}{\mathbf{\mathrm{ \Phi}}_{j}^{T}\mathbf{\mathrm{m}}\mathbf{\mathrm{\Phi}}_{j}} \tag{6}\]  
is the participation factor of mode shape j under translational excitation. Similarly, for the rotational components of ground motion  
\[F_{i,rot}=m_{i}\bar{u}_{i,rot,max}=\omega_{j}^{2}m_{i}u_{i,rot,\ max}, \tag{7}\]  
where  
\[u_{i,rot,\ max}=\varphi_{j,max}h_{i} \tag{8}\]  
is the maximal deflection of the \(i\) -th story under vibration with frequency \(\omega_{j}\) from the effects of rotational components of ground motion;  
\[\varphi_{j,max}=\frac{R^{\Theta}\big{(}T_{j}\big{)}}{\omega_{j}^{2}} \tag{9}\]  
is the maximal rotation of foundation about the horizontal axis;  
\[R^{\Theta}\big{(}T_{j}\big{)}=\frac{1.7\pi}{v_{s}T_{j}}S_{e}\big{(}T_{j}\big{)} \tag{10}\]  
is the spectrum of rotational acceleration response about the horizontal axis \([\mathrm{rad}/\mathrm{s}^{2}]\) according to EN 1998-6 (2005), cl. A(5); with shear modulus \(G\) and soil density \(\rho\); \(v_{s}\) is the shear wave velocity (11). \[v_{s}=\sqrt{\frac{G}{\rho}} \tag{11}\]  
Inserting Equations (8)-(10) into Equation (7) we can write  
\[F_{i,rot}=\omega_{j}^{2}m_{i}\Phi_{i,j}\frac{\mathbf{\mathrm{\Phi}}_{j}^{T}\mathbf{ \mathrm{m}}\varphi_{j,max}\mathbf{\mathrm{h}}}{\mathbf{\mathrm{\Phi}}_{j}^{T}\mathbf{ \mathrm{m}}\mathbf{\mathrm{\Phi}}_{j}}, \tag{12}\]  
where \(\mathbf{\mathrm{h}}=\{h_{1},\ h_{2},\dots,h_{i},\dots\}^{T}\) is a vector of the heights of individual stories. After arrangement  
\[u_{i,u,max}=\[PERSON]_{i,j}\frac{\mathbf{\[PERSON]}_{j}^{T}\mathbf{m}\mathbf{I}}{\mathbf{ \[PERSON]}_{j}^{T}\mathbf{m}\mathbf{\[PERSON]}_{j}}S_{d}\left(T_{j}\right) \tag{4}\]0  
where  
\[u_{i,u,max}=\[PERSON]_{i,j}\frac{\mathbf{\[PERSON]}_{j}^{T}\mathbf{m}\mathbf{I}}{\mathbf{ \[PERSON]}_{j}^{T}\mathbf{m}\mathbf{\[PERSON]}_{j}}S_{d}\left(T_{j}\right) \tag{4}\]1  
is the participation factor of shape j under rotational excitation.