## 4 Discussion
### The Reticulated Shell Structures with a Roofing System  
The acceleration response spectra of the representative node of each ring in the shell under the action of seven ground motions are calculated. Figure 11 shows that the acceleration response spectra of the representative node usually have two peaks corresponding to the basic period of the shell and the high-order sensitive period of the representative node, which are recorded as \(T_{\rm s}\) and \(T_{\rm s1}\), respectively. By calculating the acceleration response spectra of the representative node under the action of seven ground motions, the ratios of the acceleration response spectra values of representative nodes at \(T_{\rm a}=0\) s (\(S_{n}(T_{\rm a}=0,\zeta_{0})\)) to the ground motion input amplitude (\(S_{\rm g}(T=0,\zeta)\)) are obtained and recorded as \(AF_{0}\), as shown in Equation (13). The ratios of the peak values of the representative node acceleration response spectra (MAX(\(S_{n}(T_{\rm a},\zeta_{0})\))) to \(S_{n}(T_{\rm a}=0,\zeta_{0})\) are recorded as \(AF_{\rm max}\), as shown in Equation (14). According to Figure 12, the combined distance between a node in the \(i\)-th ring and the nearest support node is defined as the effective distance \(D_{i}\) of the nodes in the \(i\)-th ring, and the distance between the central node of the shell and the nearest support node is defined as the maximum effective distance of the shell, which is recorded as \(D_{\rm max}\), as shown in Equation (15). \[AF_{0}=\frac{S_{n}(T_{\rm a}=0,\zeta_{0})}{S_{\rm g}(T=0,\zeta)} \tag{13}\]  
\[AF_{\rm max}=\frac{\rm MAX(S_{n}(T_{\rm a},\zeta_{0}))}{S_{n}(T_{\rm a}=0, \zeta_{0})} \tag{14}\]  
\[\left\{ \begin{array}{c}D_{i}=\sqrt{D_{i,x}^{2}+D_{i,y}^{2}+D_{i,z}^{2}}\\ D_{\rm max}=\sqrt{D_{\rm top,x}^{2}+D_{\rm top,y}^{2}+D_{\rm top,z}^{2}}\end{array} \right. \tag{15}\]  
Figure 11: The average acceleration response spectra of representative nodes in the D4073 shell with a roofing system: (**a**) D4073-LSWM (\(\zeta_{0}=0.02\)) horizontal; (**b**) D4073-LSWM (\(\zeta_{0}=0.02\)) vertical. Note: R0 is the central node of the shell, and R2 is the second ring near the central node. Figure 12: Effective distance between the node and support of a spherical reticulated shell. Based on the statistics of \(AF_{0}\) and \(AF_{\max}\) in the shell with the roofing system under different ground motions, both \(AF_{0}\) and \(AF_{\max}\) are described by a normal distribution. The \(AF_{0}\) and \(AF_{\max}\) values are obtained with 95% confidence, and their relationships with \(D_{i}/D_{\max}\) are shown in Figures 13 and 14. The horizontal \(AF_{0}\) (\(AF_{0,\mathrm{H}}\)) has no obvious increasing trend with the increasing \(D_{i}\), and the \(AF_{0,\mathrm{H}}\) of the shells, with three kinds of spans, increases with an increasing rise-to-span ratio. The overall trends of the vertical \(AF_{0}\) (\(AF_{0,\mathrm{V}}\)) and \(AF_{\max}\) increase with the increasing \(D_{i}\). According to the above analysis, a calculation method for the equivalent seismic action of a nonstructural component in a shell with a roofing system is proposed. The acceleration response spectra curves of the shell nodes are divided into high-frequency response sections, peak sections, and low-frequency response sections. There are generally two peak values in the acceleration response spectra of the representative nodes, and the high-order sensitive period of the shell can be set as \(T_{\mathrm{s1}}=0.5T_{\mathrm{s}}\). The design values of the acceleration response spectra in the shell can be calculated according to Equations (16)-(21). The calculation coefficients \(A_{\mathrm{S1}}\) and \(A_{\mathrm{S2}}\) are shown in Equation (20) and Table 4, and the horizontal and vertical \(AF_{\max}\) values can be calculated by equation (21), according to Table 5.  
\[S_{n,0}=AF_{0}S_{\mathrm{g,0}} \tag{16}\]  
\[S_{\mathrm{a}}(T_{\mathrm{a}})=\left\{