## 4 Discussion  
### The Reticulated Shell Structures without a Roofing System  
The shell of D4053-W is taken as an example to analyse the acceleration response of each ring node in the shell under the action of three-dimensional ground motions. The calculation model is shown in Figure 8. First, modal analysis of the spherical reticulated shell is carried out to calculate the frequency, participation coefficient, and mode vector of the shell. The acceleration response spectra of each ring node of the shell are calculated according to Equations (1)-(7). Additionally, a time history analysis of the shell under the earthquake action is carried out, and the finite element analysis results and the calculation results, determined with Equations (1)-(7), for the acceleration response spectra are compared. Figure 9a-f shows that the peak position and size of the node response spectra of the shell can be accurately predicted by the formulas. Figure 9g-o shows that the calculation errors of the horizontal acceleration response spectra of Node 3 in the \(Y\)-direction and Node 5 in the \(X\)-direction on the first ring near the support are larger than the values from the time history analysis because the main rib directions of these two points coincide with the two orthogonal horizontal seismic input directions. As shown in Figure 8, taking Node 3 as an example, the difference in membrane stiffness in the two directions of the node has an obvious influence on the acceleration response. In the seismic time history analysis of the shells, the influences of the seismic input direction on the acceleration response of nodes near the support caused by the difference in membrane stiffness in different directions are  
Figure 8: Calculation of the node position in D4053-W.  
Figure 7: Vertical acceleration response spectra of the nodes under 2 m/s\({}^{2}\) earthquake acceleration with Eurocode 8-1: (**a**) D6053-W; (**b**) D6053-LSWM; (**c**) D6053-HHWM. considered. However, these differences cannot be accurately reflected in the modal analysis of the shells. As shown in Equations (11) and (12), the influence of the membrane stiffness difference can be introduced into the calculation of the acceleration response spectra of nodes in the first ring near the support. As shown in Figure 10, the results of the modified calculation of the response spectra of the node in the first ring near the support are in agreement with the results of the time history analysis. Based on the above analysis, this approach can be applied in the analysis of the node acceleration response spectra of a single-layer spherical reticulated shell structure without a roofing system under the earthquake action. \[B_{1}=\frac{EA_{1}}{S_{1}}+\frac{EA_{\rm c}}{S_{\rm c}}\sin^{4}\alpha,\ B_{2}= \frac{EA_{2}}{S_{2}}+\frac{EA_{\rm c}}{S_{\rm c}}\cos^{4}\alpha,\ R_{0}=\frac{B_ {1}}{B_{2}} \tag{11}\]  
where \(B_{1}\) and \(B_{2}\) are the membrane stiffnesses in the \(X\)-direction and \(Y\)-direction, respectively [39]. \[\left\{ \begin{array}{l}S_{2n,x}(T_{0},\zeta_{0})=\frac{R_{0}(1+\cos\theta)} {2}\sqrt{\left(S_{n,x}^{1}\right)^{2}+\left(S_{n,x}^{2}\right)^{2}+\left(S_{n,x}^{3}\right)^{2}}\\ S_{2n,y}(T_{0},\zeta_{0})=\frac{R_{0}(1+\sin\theta)}{2}\sqrt{\left(S_{n,y}^{1 }\right)^{2}+\left(S_{n,y}^{2}\right)^{2}+\left(S_{n,y}^{3}\right)^{2}}\\ S_{2n,z}(T_{0},\zeta_{0})=\sqrt{\left(S_{n,z}^{1}\right)^{2}+\left(S_{n,z}^{2 }\right)^{2}+\left(S_{n,z}^{3}\right)^{2}}\end{array} \right. \tag{12}\]  
Figure 9: _Cont._  
Figure 10: Comparisons of the modified and simulated values of the acceleration response spectra: (**a**) Node 3 \(Y\)-direction; (**b**) Node 5 \(X\)-direction; (**c**) Node 4 \(X\)-direction; (**d**) Node 4 \(Y\)-direction.