# Parametric Analysis of Rotational Effects in Seismic Design of Tall Structures
## 4 Influence of Soil Stiffness--Simplified Analysis  
In order to assess possible changes of the soil stiffness, simple modification can be carried out. Total displacement of the system can be assumed as the sum of both flexion of the structure and rocking displacement due to foundation rotation (Figure 1a)  
\[\delta=\delta_{\text{struc}}+\delta_{\text{soil}}. \tag{25}\]  
where  
\(\delta_{\text{struc}}\) is the displacement caused due to flexion of the structure;  
\(\delta_{\text{soil}}\) is the displacement due to foundation rotation in the soil. \begin{table}
\begin{tabular}{c c c c} \hline
**Soil** & \(\rho\) [**kg/m\({}^{3}\)**] & \(\
u\) [\(-\)] & \(v_{s}\) [**m/s**] \\ \hline A & 2500 & 0.15 & 800 \\ B & 2000 & 0.2 & 580 \\ C & 1900 & 0.3 & 270 \\ \hline \end{tabular}
\end{table} 
Table 1: Soil parameters. Figure 1: (a) Simplified dynamic model; (b) coefficient \(\beta_{\varphi}\). Equation (18) has then been modified. The fundamental period after accounting of soil stiffness is  
\[\mathrm{T_{1S}}=2\pi\sqrt{\left(\frac{1}{K}+\frac{H^{2}}{K_{\varphi}}\right)} \mathrm{m_{SDOF}}, \tag{26}\]  
where  
\(\mathrm{m_{SDOF}}\) is the assessment of the mass for a SDOF model;  
\(\mathrm{K_{\varphi}}\) is the rotational stiffness of the soil (Figure 1a);  
\(\mathrm{K}\) is the stiffness of the SDOF model in the case of rigid support only. Rotational stiffness of the soil has been assumed according to [29]  
\[\mathrm{K_{\varphi}}=\frac{G}{1-\
u}\beta_{\varphi}\mathrm{BL}^{2} \tag{27}\]  
where  
\(G,\
u\) are shear modulus and Poisson ratio of the soil taken according Table 1; \(G=\rho v_{s}^{2}\);  
\(L,\ B\) are coordinates of the foundation in-plan view;  
\(\beta_{\varphi}\) is the coefficient that depends on the ratio \(L/B\) (Figure 1b). It is easy to express the stiffness of an SDOF system  
\[\mathrm{K}=\mathrm{m_{SDOF}}\frac{4\pi^{2}}{T_{1,S}^{2}}. \tag{28}\]  
The only difficulty is to assess the mass for the SDOF model. For simplicity reasons, it would be appropriate if this value also depends only on one parameter, namely, the total height of the building \(H\). Assuming mainly buildings, the plan-view ground-level dimension \(L\) can be assessed as a function of the total height  
\[\mathrm{L}=\mathrm{c_{L}H}, \tag{29}\]  
where the coefficient \(\mathrm{c_{L}}\) starts with \(1/2\) for low buildings and reaches the value \(1/8\) for tall building with the height of about \(300\) m,  
\[\mathrm{c_{L}}=4.166\cdot 10^{-6}\mathrm{H}^{2}-0.0025\mathrm{H}+0.5. \tag{30}\]  
The coefficients of the quadratic Formula (30) as well as the coefficients of the story height (31) were derived by comparing different types of structures already built with different heights. We realize that this is not a general rule and that this is only an estimate. For our purpose, it was helpful because we needed only a rough estimation. The story height \(h_{st}\) [m] depends on the total height of the building and is assessed as  
\[\mathrm{h_{st}}=3+(5-3)\frac{H}{300}. \tag{31}\]  
Then, the total mass of the building as a function of one parameter--the height is  
\[\mathrm{m_{tot}}=\frac{H}{h_{st}}\mathrm{L}^{2}\mathrm{m_{1m^{2}}}, \tag{32}\]  
where \(m_{1m^{2}}\) is the mass of \(1\) m\({}^{2}\) of the floor area (including dead weight), assumed by a value \(700\)-\(1100\) kg/m\({}^{2}\). Finally, the mass for the SDOF model is assumed approximately as \(m_{MDOF}=m_{tot}/2\) for regular buildings without changes of plan and \(m_{MDOF}=m_{tot}/3\) for towers and chimneys.