## 4 Discussion
### The Reticulated Shell Structures with a Roofing System
\begin{array}{l}\frac{T_{\mathrm{a}}}{ T_{\mathrm{s1}}}(AF_{\max}S_{n,0}-S_{n,0})+S_{n,0},T_{\mathrm{a}}\leq T_{\mathrm{s1}} \\ S_{n,0}AF_{\max}\,T_{\mathrm{s1}}\leq T_{\mathrm{a}}\leq T_{\mathrm{s}}\\ \frac{S_{n,0}AF_{\max}}{1+AF_{0}(1-\frac{T_{\mathrm{a}}}{T_{\mathrm{s}}})^{2} },T_{\mathrm{s}}\leq T_{\mathrm{a}}\\ \end{array} \right. \tag{17}\]  
\[AF_{0,\mathrm{H}}=8.5\times\frac{f}{L}+1.7 \tag{18}\]  
\[AF_{0,\mathrm{V}}=A_{\mathrm{S1}}\frac{D_{\mathrm{i}}}{D_{\mathrm{max}}}+A_{ \mathrm{S2}} \tag{19}\]  
\[A_{\mathrm{S}}=A_{1}+B_{1}\exp\Bigl{[}-\exp\Bigl{(}\frac{f/L-C_{\mathrm{i}}}{ D_{1}}\Bigr{)}\Bigr{]}+E_{1}\exp\Bigl{[}-\exp\Bigl{(}\frac{L-F_{\mathrm{i}}}{C_{ \mathrm{i}}}\Bigr{)}\Bigr{]} \tag{20}\]  
\[AF_{\max}=\Bigl{(}A\frac{0.02}{\xi_{0}}\frac{D_{i}}{D_{\mathrm{max}}}+B\Bigr{)} \frac{1}{\sqrt{\xi_{0}}} \tag{21}\]  
Figure 13: The \(AF_{0}\) values of the shells with a roofing system: (**a**) horizontal (\(AF_{0,\mathrm{H}}\)); (**b**) vertical (\(AF_{0,\mathrm{V}}\)). The design values of the node acceleration response spectra of the spherical reticulated shell with a roofing system are obtained according to Equations (16)-(21), which will be compared with the results of the time history analysis. The spherical reticulated shells D6053-1500 and D8031-1500 with bolted and welded roofing systems are selected as examples for analysis, and the acceleration response spectra of the representative nodes in each ring of the spherical reticulated shells are extracted and compared with the design values in this paper. As shown in Figure 15, it is found that the calculation method proposed in this paper can better envelope the peak values of the acceleration response spectra of the spherical reticulated shell with a roofing system. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Roofing System** & **Fitting Coefficient** & \(\mathbf{A_{1}}\) & \(\mathbf{B_{1}}\) & \(\mathbf{C_{1}}\) & \(\mathbf{D_{1}}\) & \(\mathbf{E_{1}}\) & \(\mathbf{F_{1}}\) & \(\mathbf{G_{1}}\) & \(\mathbf{H_{1}}\) \\ \hline Bolted purlin roofing system & \(A_{\text{S1}}\) & \(5.03\) & \(1.32\) & \(0.16\) & \(0.016\) & \(4.99\) & \(42.26\) & \(7.25\) & \(9.31\) \\ Welded purlin roofing system & \(8.74\) & \(1.48\) & \(0.16\) & \(0.024\) & \(0.056\) & \(41.82\) & \(3.84\) & \(13.04\) \\ \hline Bolted purlin roofing system & \(A_{\text{S2}}\) & \(11.38\) & \(-8.87\) & \(0.171\) & \(0.132\) & \(16.1\) & \(89.44\) & \(68.62\) & \(-18.1\) \\ Welded purlin roofing system & \(11.98\) & \(-10.38\) & \(0.116\) & \(0.093\) & \(16.2\) & \(44.52\) & \(55.26\) & \(-14.9\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Calculation coefficients. Figure 14: The \(AF_{\text{max}}\) of each ring node of the shells with a roofing system: (**a**) D8073 horizontal; (**b**) D8053 horizontal; (**c**) D8033 horizontal; (**d**) D8073 vertical; (**e**) D8053 vertical; (**f**) D8033 vertical. Note: D8053-LSWM_002 is the damping ratio of the nonstructural component in D8053-LSWM, which is 0.02. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Coefficient** & \begin{tabular}{c} **Bolted Purlin Roofing System** \\ **System** \\ **Horizontal** \\ \end{tabular} & \begin{tabular}{c} **Welded Purlin** \\ **Roofing System** \\ **Horizontal** \\ \end{tabular} & \
\begin{tabular}{c} **Bolted Purlin Roofing** \\ **System** \\ **Vertical** \\ \end{tabular} &
\begin{tabular}{c} **Welded Purlin** \\ **Roofing System** \\ **Vertical** \\ \end{tabular} \\ \hline \(A\) & \(3.75\times\frac{f}{L}-0.25\) & \(4.0\times\frac{f}{L}+0.30\) & \(3.5\times\frac{f}{L}+0.30\) & \(2.5\times\frac{f}{L}+0.30\) \\ \(B\) & \(1+2\times\frac{f}{L}\) & \(1+1.65\times\frac{f}{L}\) & \(1.60\) & \(1.60\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Calculation coefficients (\(A\) and \(B\)) of the acceleration response spectra. representative node in each ring of the shell, and the envelope lines are in agreement with the node acceleration response spectra curve shape and peak value positions.