## 6 The Spectrum Methods for Incoherent Ground Displacement Excitations
### The Simplified Power Spectrum Method  
Here, a simplification of the peak factor, \(p_{z}\) or \(q_{z}\), is made through the analysis of the structural response in the example above. The site condition of the structure is the Category IV and the seismic intensity IX. Structural damping ratio is supposed as 0.02. Structural response mean values and variance peak factors are calculated by Equations (44), (45), (48), and (49), as listed in Table 2. In Table 2, the mean values are \(\mu_{p_{z}}=2.752\) and \(\mu_{q_{z}}=0.508\); the standard deviations are \(\sigma_{p_{z}}=0.0400\) and \(\sigma_{q_{z}}=0.00884\); the variation coefficients are \(\delta_{p_{z}}=0.015\) and \(\delta_{q_{z}}=0.017\); the largest difference between \(p_{z}\) and \(\mu_{p_{z}}\) is 2.75%; the largest difference between \(q_{z}\) and \(\mu_{q_{z}}\) is 3.22%. The calculating result indicates that the mean values and variance peak factors change to a small extent. Therefore, for a simplification, the peak factors to structural responses can be approximately calculated according to the seismic and structural features. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Position**} & \multicolumn{2}{c}{**Peak Factor**} & \multirow{2}{*}{**Position**} & \multicolumn{2}{c}{**Peak Factor**} \\ \cline{2-2} \cline{4-5}  & \(p_{z}\) & \(q_{z}\) & & \(P_{z}\) & \(q_{z}\) \\ \hline VD 1 at Point 2 & 2.79 & 0.50 & BBM at the left of Point 13 & 2.72 & 0.52 \\ VD at Point 3 & 2.78 & 0. The mean value, \(p_{g}\) and the peak variance factor, \(q_{g}\) of ground motion displacement can be calculated by substituting Equations (59) and (60) into Equations (48), (49), (50), and (51)  
\[\lambda_{m}=\int_{-\infty}^{\infty}\omega^{m}S(\omega)d\omega, \tag{59}\]  
\[\overline{\omega}=\frac{\sqrt{\lambda_{2}}}{\sqrt{\lambda_{0}}}. \tag{60}\]  
The calculated result for Point 9 is \(p_{g}=2.533\) and \(q_{g}=0.563\), respectively, near to the corresponding values of the structural peak response factors, \(p\) and \(q\), as listed in Table 2. But, the difference still exists between the ground displacement factors and structural peak response factors. This difference seems to be because \(p_{g}\) and \(q_{g}\) do not include structural damping and natural frequency characteristics, which are considered in \(p_{z}\) and \(q_{z}\). However, since \(p_{z}\) and \(q_{z}\) do not obviously change with the damping ratios, as listed in Table 3, the difference is not caused by the damping ratio, \(\zeta\). Hence, it is not a reasonable choice to replace the peak response factor using those deduced in the case of ground displacement exercitation. The analytical results above imply a different approach for describing structural peak response factors. This approach could be easily obtained through the generalized single-degree-of-freedom (GSDOF) system with lumped mass under seismic displacement action. The peak response factors of GSDOF systems corresponding to natural frequencies of the structure, \(p_{d}\) and \(q_{d}\), reflect the collective effect of the ground displacement, structural frequency, and damping characteristics. Therefore, \(p_{d}\) and \(q_{d}\) should be closer to the structural peak response factors. Equation (61) (reference) is adopted for the \(m\)th order spectral moment of the GSDOF system corresponding to the \(j\)th mode  
\[\lambda_{m,j}=\int_{-\infty}^{\infty}\omega^{m}H_{j}(i\omega)H_{j}(-i\omega)S (\omega)d\omega, \tag{61}\]  
where \(S(\omega)\) is the power spectrum density function of ground displacement. For the structural model in Figure 4, the mean value and peak variance factors of GSDOF systems, \(p_{d}\) and \(q_{d}\) can be calculated by Equations (60), (61), (48), (49), (50), and (51), and the result is shown in Table 4.