**4. Seismic Response Analysis**

### *4.1. Contact Force between Adjacent Structures at the Expansion Joint*

A comparative analysis of several cases was carried out on the displacement, stress, contact force, and damage of the bridge structure obtained through the dynamic analysis. Compare the peak value of the normal contact force at each expansion joint of the three combined shock absorption cases with the case A. The peak value of the normal contact force of each case is shown in Figure 9. The normal contact force time history of cases A and G are listed in Figures 10 and 11.

**Figure 9.** The maximumnormal contact force of expansion joints adjacent structures of every case (unit:N).

**Figure 10.** The normal contact force time-histories of expansion joints adjacent structures of case A (time:s). (**a**) Short unit expansion joints (**b**) Middle expansion joints (**c**) Long unit expansion joints.

**Figure 11.** The normal contact force timehistories of expansion joints adjacent structures of case G (time:s). (**a**) Short unit expansion joints. (**b**) Middle expansion joints. (**c**) Long unit expansion joints.

From the analysis in Figure 9, the combined shock absorption working case provides more effective pounding mitigation. The maximum contact force of case F (the combination of VED and SCRB) is reduced by 47.6% at the maximum; case G (the combination of VFD and SCRB) is reduced by 57.9% at the maximum; and case H (the combination of LRB and SCRB) is reduced by 57.4% at the maximum. Thisindicates that the energy dissipation effect of dampers or seismic isolator plus the cushioning effect of rubber pads can greatly reduce the contact force between adjacent beams. Analyzing the anti-collision effect of

each combination device from the perspective of contact force, the energy dissipation and seismic mitigation effect of the VFD and LRB combined with SCRB are more effective.

From the comparison of Figures 10 and 11, it can be seen that in cases G and A, the contact force time history curves are similar and the moment of the maximum contact force is almost the same, but the maximum contact force is quite different. For case A, the times corresponding to the peak contact force at the three expansion joints (the peak contact force) are 23.95 s (13.22 MN), 20.16 s (5.23 MN), and 10.26 s (16.45 MN), and for case G, they are 23.98 s (5.56 MN), 20.72 s (2.25 MN), and 10.24 (6.92 MN). This shows that the maximum contact force was greatly reduced after installing the combined VFD and SCRB devices.

### *4.2. Contact Stress of the Contact Surface of the Expansion Joint*

Extracting the normal contact stress of the contact surface at the expansion joint, the anti-collision effect of the combined case was analyzed from the perspectiveof stress. The position of each collision surface is shown in Figure 4, and the maximum stress of each combined case is shown in Figure 12. The peak stress of cases A and G, where the effect of the seismic mitigation is the strongest, is shown in Figures 13 and 14.

**Figure 12.** The maximum normal contact stress of adjacent surface at expansion joints in every case (unit:N/m2).

**Figure 13.** *Cont*.

**Figure 13.** The normal contact stress of adjacent surface at expansion joints in case A(unit: N/m2). (**a**) The contact surface of the short unit expansion joint at the abutment position (t = 23.95 s). (**b**) The contact surface of the short unit expansion joint in the middle position (t = 20.16 s). (**c**) The contact surface of the long unit expansion joint at the abutment position in the middle position (t = 20.16 s). (**d**) The contact surface of the long unit expansion joint at the abutment position (t = 10.26 s).

**Figure 14.** The normal contact stress of adjacent surface at expansion joints in case G(unit: N/m2). (**a**) The contact surface of the short unit expansion joint at the abutment position (t = 23.83 s). (**b**) The contact surface of the short unit expansion joint in the middle position (t = 20.01 s). (**c**) The contact surface of the long unit expansion joint at the abutment position in the middle position (t = 20.01 s). (**d**) The contact surface of the long unit expansion joint at the abutment position (t = 10.11 s).

According toFigure 12, these mitigation and unseating prevention devices could effectively reduce the maximum contact stress of contact surfaces. Case F (the combination of VED and SCRB) is reduced by 36.2% at the maximum; case G (the combination of VFD and SCRB) is reduced by 42.3% at the maximum; and case H (the combination of LRB and SCRB) is reduced by 41.5% at the maximum. Cases G and H are more effectiveat seismic mitigation and unseating prevention.

In Figures 13 and 14, it is shown that the collision betweenadjacent components does not occur on the entire contact surface, but locally. The location of the maximum contact stress at different expansion joints is different. For the position of the short unit expansion joint, the collision occurs between the junction of the top plate of the box girder and the outer web and the bridge abutment in cases A and G. For the position of the middle expansion joint, the collision occurs between the outer side of the box girder top plate of the adjacent girder. For the position of the long unit expansion joint, the collision occurs between the junction of the top plate of the box girder and the inner web and the bridge abutment. After installing the pounding mitigation and unseating prevention devices, because of the installation of the rubber pad at the beam end on the side of the expansion joint, the contact stress only appears in the area with the rubber pad. Thisshows that the adjacent girders did not directly collide, but indirect collisions were caused by the force transmission of rubber pad.

### *4.3. Damage to Adjacent Structures at Expansion Joints*

We extracted the tensile damage and compression damage at the last moment of the ground motion of each case. The maximum cumulative damage of each combined case is shown in Figure 15. The compression damage of the girder at the expansion joints of cases A and G are shown in Figures 16 and 17, respectively. The values in the figure represent the cumulative degree of plastic damage; 0 means the material is intact, 1 means complete loss of strength.

**Figure 15.** The cumulative damage of expansion joints adjacent surface of every case.

**Figure 16.** Damage of girder in expansion joints of case A. (**a**) Compression damage at abutment position of short unit expansion joint. (**b**) Compression damage in the middle of the short unit expansion joint. (**c**) Compression damage in the middle of long unit expansion joint. (**d**) Compression damage at abutment position of the long unit expansion joint.

**Figure 17.** Damage of girder in expansion joints of case G. (**a**) Compression damage at abutment position of short unit expansion joint. (**b**) Compression damage in the middle of the short unit expansion joint. (**c**) Compression damage in the middle of long unit expansion joint. (**d**) Compression damage at abutment position of the long unit expansion joint.

Figure 15 shows that the tensile damage toconcrete is more serious than the compression damage. After installing the seismic mitigation and unseating prevention device, both tension and compression damage are reduced to a certain extent, but the reduction ratio of the tensile damage is smaller and the ratio of compression damage reduction is larger. For tensile damage, the maximum reduction percentage in each combined case is about 24%. For compressive damage, case F (the combination of VED and SCRB) is reduced by 32.2% at the maximum; case G (the combination of NFVD and SCRB) is reduced by 36.6%

at the maximum; and case H (the combination of LRB and SCRB) is reduced by 36.3% at the maximum.

From the comparison of Figures 13 and 16, it can be found that the location where the maximum contact stress is generated is also the place where the compression damage is the most serious. This shows that the huge contact pressure generated by collision damages and destroysthe bridge structure.

### *4.4. The Number of Collisions at the Edge of Expansion Joints*

The timehistory contact stress of nodes at the edges of different expansion joints in each case was extracted, and the number of collisions of each node was obtained by statistics. Figure 18 shows the number of collisions at the edge of the expansion joints in each case. Refer to Figure 5 for the location of each node.

**Figure 18.** The contact number of node at the edge of expansion joints of every case.

Figure 18 shows that each combined case can effectively reduce the number of collisions of each edge node, and cases G and H exerta better effect on reducing the number of collisions. The collision responses between the inner node and the outer node of the same expansion joint arequite different. At the short unit expansion joint, the number of collisions on the outer side (node 3) is less than on the inner side (node 4).At the middle expansion joint, the number of collisions on the outer side (node5) is much higherthan on the inner side (node 6), and the outer contact stress is greater.At the long unit expansion joint, the number of collisions on the outside (node 9) is far lower than on the inside (node 10), and the contact stress on the inside is greater. This phenomenon shows that when the seismic wave is input along the connecting direction of piers 2 and 3 , the short unit mainly twists horizontally, counterclockwise, around the Z axis, while the long link mainly twists horizontally, clockwise, around the Z axis; the length of the long unit bridge is longer, and the torsion effect is obvious, which makes the numberof collisions and the collision stress between the inner and outer nodes obviously different.
