## 2 Methodology
### The Reticulated Shell Structures with a Roofing System  
The roofing system is an essential part of the single-layer spherical reticulated shell structure. Scholars have generally ignored the influence of roofing systems in previous designs and analyses [39]. However, according to existing studies, the roofing system notably affects the static and dynamic performance of the reticulated shell structure [40; 41; 42]. Therefore, the influence of the roofing system should be considered in the analysis of the node acceleration response spectra of the shell. As shown in Figure 3, the finite element model (FEM) of a shell with a roofing system consists of an equivalent panel, a weakened area, a purlin hanger, a purlin, a pillar, and the shell. The FEM considers the semirigid character of the purlin joints, the orthotropic characteristics of the roofing panel, and the strength and stiffness of the self-tapping screw connections between the purlin and roof panel. The applicability of the FEM has been studied by experiments and finite element analysis in Ref. [41]. The nomenclature for the shells D6053-LSWM and D6053-HHWM refers to the following: the 60-span of the shell, 5-span-to-rise ratio of the shell, 3-roof load (3 \(\times\) 60 kg/m\({}^{2}\)), LSWM-bolted purlin roofing system, and HHWM-welded purlin roofing system. The elastic modulus and the yield strength of the purlin members and pillars are 206 GPa and 235 MPa, respectively; the elastic modulus and the yield strength of the roof panel are 146 GPa and 460 MPa, respectively; the elastic modulus and the yield strength of weakened area are 160 MPa and 40 MPa, respectively. The boundary condition of the FEMis that the boundary shell nodes are directly connected with the ground. The degrees of freedom in three directions (\(X\), \(Y\), \(Z\)) of the support nodes are restricted, and the ground motions are input in three directions in the time history analysis.  
In the modal analysis of the shell with a roofing system, the components such as the roof panel and purlin have obvious local vibrations in the high-order modes while the relative displacements of all nodes in the shell are very small. Thus, even if more modes are adopted in Equations (1)-(7), the sum of the contribution coefficients cannot approach 1.0, resulting in a large difference between the results of the calculations with the formulas and the simulation values. Therefore, the acceleration response spectra of nodes in the shell with a roofing system under seismic action are analysed by the time history analysis method. Seven Class II site ground motions [43] are used to analyse the acceleration response of nodes in the shell with a roofing system. The shell node in each ring with the maximum peak value of combined acceleration in three directions is defined as the representative node in each ring of the shell, as shown in Equation (8). The acceleration response spectra of the representative node of each ring are analysed in the shell with a roofing system. The information on the ground motions is shown in Table 2.  
\[\begin{array}{c}A_{n}=\max\{A_{n,1},A_{n,1},A_{n,1}\ldots A_{n,m}\}\\ A_{n,m}(t)=\sqrt{A^{2}{}_{n,m,x}(t)+A^{2}{}_{n,m,y}(t)+A^{2}{}_{n,m,z}(t)}\end{array} \tag{8}\]  
where \(A_{n}\) is the maximum peak value of the combined acceleration for the representative node in the \(n\)-th ring, \(A_{n,m}\) is the combined acceleration of the \(m\)-th node of the \(n\)-th ring in the shell, and \(A_{n,m,x}\) is the acceleration of the \(m\)-th node of the \(n\)-th ring in the \(X\)-direction.