**Figure 7.** *Cont*.

**Figure 7.** Internal force diagram for square ring cable structure. (**a**) Internal force of struts in the square ring cable; (**b**) internal force of struts in string cables; (**c**) internal force of stay cables; (**d**) internal force of square cables.

4.2.2. Analysis of the Internal Force Response of Roof Grid Beams

The internal force response of the grid beams under No. 1 seismic array excitation is shown in Figure 8. The ring-cable structural frame system is irregular, and the internal force of the roof frame (BX1 to BX9) in the EW direction is distributed axisymmetrically with the mid-span beam BX5, which has the smallest internal force response. The X-shaped steel pipes set in the SN direction in the mid-span grid beam strengthen the local roof stiffness. As there is a long span between BX3 and BX7, their internal force response is the largest. The internal force response of the roof grid beams (BY1 to BY10) are irregularly distributed and generally greater than that in the EW direction. As the input direction of combined seismic waves is perpendicular to the roof framing elements, a larger internal force response occurs in out-of-plane vibration, maximizing at BY7 about 1440 kN.

**Figure 8.** Internal force diagram for roof grid beams. (**a**) EW direction; (**b**) SN direction.

4.2.3. Analysis of the Roof Node Deformation

The vertical displacement of roof nodes moving along the positive direction of the Z axis is represented as positive displacement, and vice versa. It can be seen from Figure 9 that the displacement and deformation of the roof grid beam nodes (JX1 to JX18) in the EW direction are basically symmetrically distributed. While the maximal displacement is reached at nodes JX7 and JX12, the increased local stiffness at the middle span greatly decreases the displacement of JX9 and JX10. The vertical displacement of roof grid beam nodes (JY1 to JY11) in the SN direction is distributed asymmetrically, with the greatest displacement appearing at node JY7, which is about 45 mm. Comparing Figure 8a with Figure 8b, the difference between the node displacement and internal force response of the roof grid beams can be seen. The reason why the overall grid beam node displacement in

the SN direction is smaller than that in the EW direction is that the long-span arch structure facilitates the antisymmetrical vibration mode under combined seismic excitation, leading to a large vertical displacement.

**Figure 9.** Vertical displacement diagram for roof grid beams nodes. (**a**) EW direction; (**b**) SN direction.

### *4.3. Dynamic Response under Excitation of No. 2 and No. 3 Seismic Arrays*

### 4.3.1. Analysis on the Internal Force Response of Roof Square Ring Cables

The structural stiffness of the gymnasium in the SN direction is relatively weak, and the structural dynamic response is larger and similar under the Excitation of No. 2 and No. 3 seismic arrays. Therefore, these two seismic arrays were considered simultaneously for better comparison and analysis.

As seen in Figure 10, the internal force response of the struts and the cables under No. 2 seismic array excitation was larger than that under the No. 3 seismic array and was about four to six times larger than that under the No. 1 seismic array. As an additional vertical acceleration is generated by the altitude difference between the north and south sides when the seismic waves move along the Y direction, the small rise span of the roof structure in this direction, coupled with less out-of-plane stiffness than in-plane stiffness, resulting in a greater structural dynamic response. In addition, the internal force response of the struts (SC1 to SC12) shows a downward trend from the inner ring to the outer ring. The additional struts (SC13 to SC16) of the outer ring have the least internal force response, followed by the struts in the string cables (SC17 to SC24), with an average value of about 80 kN. The internal force response of the middle ring stay cables (SX5 to SX8) and loop cables (SH5 to SH8) are slightly smaller than those of the inner ring. The outer ring stay cables (SX9 to SX12), loop cables and string cables (SH9 to SH14) have the largest internal force response.

### 4.3.2. Analysis of the Internal Force Response of Roof Grid Beams

As seen in Figure 11, the internal force response of the roof grid beams (BX1 to BX9) in the EW direction under No. 2 seismic array is larger than that under No. 3 seismic array, which is contrary to that of the beams from BY1 to BY10. The internal force response of the roof grid beams from BX1 to BX9 under No. 2 seismic array is symmetrically distributed, in which the grid beams BX3 and BX7 have the largest internal force response that is obviously greater than that in the SN direction, thus serving as the main stress component of the structural system. The internal force response of the grid beam in the SN direction is irregularly distributed with an average value of about 200 kN under No. 2 seismic array, and with the largest value of 1120 kN under No. 3 seismic array.

**Figure 10.** Internal force diagram of the square ring cable structure. (**a**) Internal force of struts in square ring cable; (**b**) internal force of struts in string cable; (**c**) internal force of stay cables; (**d**) internal force of square cables.

**Figure 11.** Internal force diagram of roof grid beams. (**a**) EW direction; (**b**) SN direction.

Compared with No. 1 seismic array excitation, the internal force response of the roof grid beams in the EW direction under No. 2 seismic array excitation has a significant increase, the maximum value of which is about 2.5 times that of No. 1, while a substantial decrease occurs in the internal force response of the roof grid beams in the SN direction, the maximum value of which only accounts for about 25% of that in the EW direction. The reason is that when the seismic wave moves in the Y direction, the grid beams in the EW direction become perpendicular to the moving direction of combined seismic waves and thus generate out-of-plane vibration, greatly increasing the internal force of grid beams. As the grid beams in the SN direction are parallel to the moving direction of the seismic wave, they suffer less an effect.

### 4.3.3. Analysis of the Roof Nodes Deformation

The displacement response of roof grid beam nodes under No. 2 and No. 3 seismic array excitation is displayed in Figure 12. The displacement response in the EW direction under No. 2 seismic array excitation is weaker than that under No. 3 seismic array excitation, in addition to nodes JX9 and JX10, while the displacement response in the SN direction is different. The displacement response is basically symmetrically arranged in the EW direction (JX1 to JX18), with the displacement maximizing at the midspan nodes of JX9 and JX10 under No. 2 seismic array, and at the nodes of JX7 and JX13 under No. 3 seismic array excitation. The displacement peak in the SN direction appears at JY4, and the displacement of the north nodes (JY2 to JY5) under seismic excitation exceeds that of the south nodes (JY7 to JY10), because of weak lateral rigidity caused by the longer steel pipe columns on the north side. Compared with No. 1 seismic array, the displacement of the roof grid beam nodes under No. 2 seismic array excitation experiences a decrease in the EW direction, while an increase is seen in the SN direction, with the maximum value almost doubled.

### **5. Effect Law of Different Types of Ground Motions on the Seismic Response of Roof Structures**

Seismic action comes with strong randomness and uncertainty. Located at the junction of the Pacific plate and the Asia-Europe plate, Fuzhou has witnessed the occurrence of different types of earthquakes from time to time. Therefore, further research on the dynamic response of the gymnasium structure under different seismic fields and seismic impulses is of grea<sup>t</sup> significance for seismic monitoring and emergency response tasks related to the gymnasium.

### *5.1. Input of Different Types of Seismic Waves*

Imperial Valley-06 far-field seismic wave files from the PEERC were selected, including two groups of pulse near field(PNF) seismic waves, two groups of nonpulse near field (NNF) seismic waves and two groups of pulse far field(PFF) seismic waves. The spectrum curve of seismic wave acceleration response (ARS) is displayed in Figure 13. In consideration of rarely occurring earthquakes, the PGA was adjusted to 0.22 g and the horizontal direction (X-direction) and vertical direction (Z-direction) of the six sets of seismic waves were chosen as the earthquake inputs according to the partial coefficients. The seismic parameter information is detailed in Table 2, where the nonlinear time history analysis method based on "self-weight" is used. The damping ratio was set as 0.05 and the average value of the response under various types of seismic array conditions was selected as the structural dynamic response.

**Figure 13.** Seismic wave response spectrum curves under load case. (**a**) EL06-ARS; (**b**) EL07-ARS; (**c**) BCR-ARS; (**d**) CHI-ARS; (**e**) EL11-ARS; (**f**) DLT-ARS.



### *5.2. Analysis of the Internal Force Response of Roof Square Ring Cables*

The internal force responses of the structural components of the multiple square ring cables under the excitation of the three types of seismic waves are shown in Figure 14. It can be seen from Figure 14a,b that the internal force response law of the struts (SC1 to SC24) remains basically the same with comparable values. Except for some struts (such as SC1, SC11, SC21, etc.), the internal force response under the excitation of the PNF combined seismic wave is the smallest, indicating that, in general, seismic impulses weaken the internal force response of the struts, while the near and far seismic fields equip the struts with greater internal force response.

**Figure 14.** Internal force diagram of the square cable structure. (**a**) Internal force of struts in square ring cable; (**b**) internal force of struts in string cable; (**c**) internal force of stay cables; (**d**) internal force of square cables.

The internal force response of stay cables and ring cables are shown in Figure 14c,d. In general, the internal force response under the PFF combined seismic wave is the biggest, while that under the PNF combined seismic wave is the smallest. The outer ring stay cables (SX9 to SX12) and the ring cables (SH9 to SH14) undergo the largest increase in the internal force response under the NNF and PFF seismic excitation, and the smallest under the PFF seismic excitation. SH13, under the PFF seismic excitation, has the largest internal force response with an axial force value of 503 kN, while SH14, under the NNF seismic excitation, has the largest internal force response with an axial force value of 535 kN and an increase of 0.203. That proves that the seismic impulse fails to cause larger internal force responses of stay cables and ring cables. Instead, it weakens their internal force response. Other than SH14, the seismic excitation in the far field generates a greater internal force response than the near field.

### *5.3. Analysis of the Internal Force Response of Roof Grid Beams*

Figure 15 shows the internal force response of the roof grid beams under the three types of seismic excitation. It can be seen that the change law of the structure system under different seismic excitation remains basically the same. The PNF seismic array has the biggest excitation, followed by NNF and then PFF, which indicates that the seismic impulse generates a larger structural dynamic response to the roof grid frame beams, the near seismic field produces a greater internal force response than that of the far seismic field under the same impulse, and the influence of the near and far seismic fields exceed that of the impulse. In addition, as the frame beam BX5 of the roof vault is rigid enough and the axial forces under different seismic conditions are equivalent, with the smallest value, the aforementioned two factors exert little effect on this cross-grid beam.

**Figure 15.** Internal force diagram for the midspan frame beam. (**a**) EW direction; (**b**) SN direction.

### *5.4. Analysis of the Roof Nodes Deformation*

The vertical displacement response of the roof grid beam nodes under three types of seismic excitation is shown in Figure 16, in which the overall distribution law is consistent with that of the grid beam internal force response. PNF combined seismic excitation has the largest vertical displacement, followed by NNF and then PFF. In addition, the positive deformation of nodes JX11–JX17 and the negative deformation of JX2–JX8 are distributed alternately under the excitation of NNF and PFF, with little difference. This indicates that the seismic impulse excitation further increases the displacement response of the roof grid beam nodes, and the seismic excitation of the near field produces a greater nodal displacement response than the far field. Overall, the near field factor exceeds the impulse factor.

**Figure 16.** Vertical displacement diagram of roof span nodes. (**a**) EW direction; (**b**) SN direction.
