## 6 The Spectrum Methods for Incoherent Ground Displacement Excitations
### The Simplified Power Spectrum Method
\begin{table}
\begin{tabular}{c c c c c} \hline \(\omega\) & \(p_{d}\) & \(q_{d}\) & \(\omega\) & \(p_{d}\) & \(q_{d}\) \\ \hline \(\omega_{1}=4.81\) & 2.71 & 0.51 & \(\omega_{6}=8.97\) & 2.81 & 0.49 \\ \(\omega_{2}=5.61\) & 2.75 & 0.50 & \(\omega_{7}=18.71\) & 2.65 & 0.53 \\ \(\omega_{3}=6.11\) & 2.77 & 0.50 & \(\omega_{8}=18.88\) & 2.65 & 0.53 \\ \(\omega_{4}=6.92\) & 2.79 & 0.49 & \(\omega_{9}=20.93\) & 2.62 & 0.53 \\ \(\omega_{5}=7.98\) & 2.81 & 0.49 & \(\omega_{10}=23.03\) & 2.60 & 0.54 \\ \hline \end{tabular}
\end{table} 
Table 4: The peak factors of generalized single-degree-of-freedom (GSDOF) systems. \begin{table}
\begin{tabular}{c c c} \hline \(\zeta\) & \(p_{z}\) & \(q_{z}\) \\ \hline \(0.05\) & 2.72 & 0.51 \\ \(0.05\) & 2.72 & 0.52 \\ \(0.04\) & 2.73 & 0.51 \\ \(0.03\) & 2.74 & 0.51 \\ \(0.02\) & 2.75 & 0.51 \\ \(0.01\) & 2.76 & 0.51 \\ \hline \end{tabular}
\end{table} 
Table 3: The peak response factor of beam bending moment at Point 9. Now, \(\mu_{p_{d}}=2.750\) and \(\mu_{q_{d}}=0.509\) can be obtained by calculating the average value of the peak response factors for the first three vibration modes. Clearly, they are very close to the mean values of the structural peak response factors, \(\mu_{p_{z}}=2.752\) and \(\mu_{q_{z}}=0.508\). The greatest difference between \(\mu_{p_{d}}\) and \(p_{z}\), or \(\mu_{q_{d}}\) and \(q_{z}\), is 2.8% or 3.3%; if conducting similar calculation is conducted for the first six modes, \(\mu_{p_{d}}=2.779\) and \(\mu_{q_{d}}=0.502\). The greatest difference between \(\mu_{p_{d}}\) and \(p_{z}\), or \(\mu_{q_{d}}\) and \(q_{z}\), is 3.1% or 3.8%. Hence, structural peak response factors can be replaced using the mean peak response factor for the first \(k\) GSDOF systems, that is,  
\[\overline{p}_{z}=\frac{1}{k}{\sum_{j=1}^{k}{p_{d,j}}}, \tag{62}\]  
\[\overline{p}_{z}=\frac{1}{k}{\sum_{j=1}^{k}{p_{d,j}}}, \tag{63}\]  
where \(p_{d,j}\) and \(q_{d,j}\) are calculated by Equations (59), (60), (48), (49), (50), and (51); \(k\) is the order number, which has the biggest contribution to the structural response and is taken as 3\(\sim\)6 for the example in Figure 4. \(\overline{p}_{z}\) and \(\overline{q}_{z}\) are the collective peak response factors based on the incoherent ground displacement input. Correspondingly, the response spectrum method using \(\overline{p}_{z}\) and \(\overline{q}_{z}\) is called as the collective response spectrum method. Now, the results above can be summarized as a few points:  
1. \(\overline{p}_{z}\) and \(\overline{q}_{z}\) in Equations (62) and (63) simplify the peak response factors, \(p_{z}\) and \(q_{z}\), in Equations (48) and (49). \(\overline{p}_{z}\) and \(\overline{q}_{z}\) can calculate the peak response and the variance of the peak response in the structural linear-elastic stage under incoherent ground motion. 2. \(\overline{p}_{z}\) and \(\overline{q}_{z}\) have three merits when compared with the peak response factors, \(p_{z}\) and \(q_{z}\): less computation, less process extent in simplifying and approximating, and without involving in the response spectrum. 3. When compared with the existing power spectrum method introduced above, the collective response spectrum method reduces the computational effort by about 50%.