## 2 Methodology  
### The Reticulated Shell Structures without a Roofing System  
In this paper, the single-layer spherical reticulated shell structures without a roofing system are taken as the main structure first, which is a K-8 Kewitt reticulated shell, with spans of 40 m, 60 m, and 80 m and rise-span-ratios of 1/7, 1/5, and 1/3. The material of shell members in the reticulated shell structure is Q235 steel, the elastic modulus is 206 GPa, the yield strength is 235 MPa, and the damping ratio is 0.02. The nomenclature for shell D4073-W, refers to the following: the 40-span of the shell, 7-span-to-rise ratio of the shell, 3-roof load (3 \(\times\) 60 kg/m\({}^{2}\)), and W-the shell without a roofing system. The boundary condition of the finite element models is that the boundary shell nodes are directly connected with the ground and restrict the translational displacement (\(X\), \(Y\), \(Z\)) of the support nodes in three directions. To analyse the equivalent seismic action of the nonstructural components in reticulated shell structures, the acceleration response spectra of reticulated shell nodes are analysed. At present, the existing studies show that high-order modes can greatly affect the dynamic performance of the single-layer spherical reticulated shells; therefore, the complete quadratic combination (CQC) method is used to combine the effects of modes of each order, and the effects of coupling between different modes are considered. The single-layer spherical reticulated shell is a system with multiple degrees of freedom, and the equation of motion of the shell is shown in Equation (1):  
\[\{\phi\}^{\rm T}\{M\}\{\phi\}\Big{\{}\ddot{U}\Big{\}}+\{\phi\}^{\rm T}\{C\}\{ \phi\}\Big{\{}\dot{U}\Big{\}}+\{\phi\}^{\rm T}K\{\phi\}\{U\}=-\{\phi\}^{\rm T} \{M\}\{I\}\Big{\{}\ddot{U_{\rm G}}\Big{\}} \tag{1}\]  
where \(\{M\}\) is the mass matrix of the shell, \(\{C\}\) is the damping matrix of the shell, \(\{K\}\) is the stiffness matrix of the shell, \(\{U\}\) is the relative displacement matrix of the shell node, \(\{\phi\}\) is the modal matrix of the shell, and \(\{U_{S}\}\) is the seismic displacement matrix. Based on an accurate method [36], the \(j\)-direction velocity response and displacement response (\(q^{i}_{n,j,k}\) and \(q^{i}_{n,j,k}\)) of the \(n\)-th node in the shell under the influence of the \(k\)-order mode and the \(i\)-direction earthquake are calculated, and then the \(j\)-direction absolute acceleration response and response spectra (\(\ddot{Q}^{i}_{n,j,k}\) and \(S^{i}_{n,k,j}\)) of the \(n\)-th node of the shell caused by the \(i\)-direction ground motion under the influence of the \(k\)-order mode are obtained, as shown in Equations (2) and (3) [29]:  
\[\ddot{Q}^{i}_{n,j,k}=-\varrho_{n,j,k}\lambda^{i}_{k}\Big{(}2\zeta_{k}w_{k}q^{ i}_{n,j,k}+w^{2}_{k}q^{i}_{n,j,k}\Big{)} \tag{2}\]\[S_{n,k,j}^{i}=w_{d0}\Bigg{|}\int\limits_{0}^{t}\tilde{\mathcal{Q}}_{n,j,k}^{i}e^{- \zeta u_{0}(t-\tau)}\Bigg{[}\Big{(}1-\frac{{\zeta_{0}}^{2}}{1-{\zeta_{0}}^{2}} \Big{)}\sin w_{d0}(t-\tau)+\frac{2\zeta_{0}}{\sqrt{1-{\zeta_{0}}^{2}}}\cos w_{d 0}(t-\tau)\Bigg{]}d\tau\Bigg{|}_{\max} \tag{3}\]  
where \(\varphi_{n,j,k}\) is the \(j\)-th mode vector of the \(n\)-th node of the shell under the \(k\)-th mode, \(\lambda_{k}^{i}\) is the \(i\)-direction mode participation coefficient of the shell under the \(k\)-th mode, \(w_{k}\) is the frequency of the \(k\)-th mode, and \(\zeta_{k}\) is the damping ratio of the \(k\)-th mode. Additionally, Equation (4) is outlined below [37]:  
\[\rho_{ab}=\frac{2\sqrt{\zeta_{a}\zeta_{b}}\Big{[}\big{(}w_{a}+w_{b}\big{)}^{2} (\zeta_{a}+\zeta_{b})+\big{(}{w_{a}}^{2}-{w_{b}}^{2}\big{)}(\zeta_{a}-\zeta_{ b})\Big{]}}{4\big{(}w_{a}-{w_{b}}\big{)}^{2}+\big{(}w_{a}+{w_{b}}\big{)}^{2}( \zeta_{a}+\zeta_{b})^{2}} \tag{4}\]  
where \(\rho_{ab}\) is the correlation coefficient, \(w_{a}\) is the frequency of the \(a\)-th mode, and \(\zeta_{a}\) is the damping ratio of the \(a\)-th mode.