Figure 1.
Dimensional details of bridge specimen: (a) section view of main girder, (b) section view of column, (c) pier (unit: cm).
Figure 1.
Dimensional details of bridge specimen: (a) section view of main girder, (b) section view of column, (c) pier (unit: cm).
Figure 2.
Layout diagram of reinforcement: (a) elevation view, (b) side view, (c) cross section (unit: cm).
Figure 2.
Layout diagram of reinforcement: (a) elevation view, (b) side view, (c) cross section (unit: cm).
Figure 3.
Picture of bridge model.
Figure 3.
Picture of bridge model.
Figure 4.
Constitutive relationship: (a) the concrete, (b) the steel bar.
Figure 4.
Constitutive relationship: (a) the concrete, (b) the steel bar.
Figure 5.
Procedure of the modeling.
Figure 5.
Procedure of the modeling.
Figure 6.
Comparison of relative longitudinal displacement between the pier and the beam: (a) El Centro wave, (b) Chi−chi wave.
Figure 6.
Comparison of relative longitudinal displacement between the pier and the beam: (a) El Centro wave, (b) Chi−chi wave.
Figure 7.
Longitudinal test seismic wave: (a) El Centro wave, (b) Chi−chi wave.
Figure 7.
Longitudinal test seismic wave: (a) El Centro wave, (b) Chi−chi wave.
Figure 8.
Longitudinal collapse mode of bridge model under El Centro wave.
Figure 8.
Longitudinal collapse mode of bridge model under El Centro wave.
Figure 9.
Longitudinal collapse mode of bridge model under Chi-chi wave.
Figure 9.
Longitudinal collapse mode of bridge model under Chi-chi wave.
Figure 10.
Time history curve of vertical seismic input of El Centro wave.
Figure 10.
Time history curve of vertical seismic input of El Centro wave.
Figure 11.
Time history curve of the relative displacement between the side pier and the girder under El Centro wave.
Figure 11.
Time history curve of the relative displacement between the side pier and the girder under El Centro wave.
Figure 12.
The time history curve of acceleration response of middle pier cap under El Centro wave.
Figure 12.
The time history curve of acceleration response of middle pier cap under El Centro wave.
Figure 13.
The time history curve of principle compressive strain of middle pier at the bottom under El Centro wave.
Figure 13.
The time history curve of principle compressive strain of middle pier at the bottom under El Centro wave.
Figure 14.
Time history curve of vertical seismic input of Chi−chi wave.
Figure 14.
Time history curve of vertical seismic input of Chi−chi wave.
Figure 15.
Time-history curve of the relative displacement between the side pier and the girder under Chi−chi wave.
Figure 15.
Time-history curve of the relative displacement between the side pier and the girder under Chi−chi wave.
Figure 16.
The time history curve of acceleration response of middle pier cap under Chi−chi wave.
Figure 16.
The time history curve of acceleration response of middle pier cap under Chi−chi wave.
Figure 17.
The time history curve of principle compressive strain of middle pier at the bottom under Chi−chi wave.
Figure 17.
The time history curve of principle compressive strain of middle pier at the bottom under Chi−chi wave.
Figure 18.
Collapse process under bidirectional Chi−chi wave input.
Figure 18.
Collapse process under bidirectional Chi−chi wave input.
Figure 19.
Collapse mode of bridge model under Chi−chi wave input: (a) column length of 1.8 m; (b) column length of 2.4 m; (c) column length of 3 m; (d) column length of 3.6 m.
Figure 19.
Collapse mode of bridge model under Chi−chi wave input: (a) column length of 1.8 m; (b) column length of 2.4 m; (c) column length of 3 m; (d) column length of 3.6 m.
Figure 20.
Layout diagram of fluid viscous damper.
Figure 20.
Layout diagram of fluid viscous damper.
Figure 21.
The change curve of damping force peak.
Figure 21.
The change curve of damping force peak.
Figure 22.
The change curve of damper stroke peak.
Figure 22.
The change curve of damper stroke peak.
Figure 23.
Peak relative displacement between the side pier and the beam.
Figure 23.
Peak relative displacement between the side pier and the beam.
Figure 24.
The change curve of internal force peak at the bottom of the middle pier: (a) shear force; (b) moment.
Figure 24.
The change curve of internal force peak at the bottom of the middle pier: (a) shear force; (b) moment.
Figure 25.
Collapse process of shock mitigation system under El Centro seismic wave.
Figure 25.
Collapse process of shock mitigation system under El Centro seismic wave.
Figure 26.
Suggested anti-collapse ductility design flow.
Figure 26.
Suggested anti-collapse ductility design flow.
Table 1.
Similar relation of bridge specimen.
Table 1.
Similar relation of bridge specimen.
| Physical Quantity | Dimension | Relationship | Coefficient of Similarity |
---|
Geometric properties | Length | [L] | Sl | 1/3 |
Area | [L] | Sl2 | 1/9 |
Material properties | Mass | [M] | Sl2·Sσ/Sa | 1/18 |
Density | [ML−3] | Sσ/(Sl·Sa) | 1.5 |
Stress | [ML−1T−2] | Sσ | 1 |
Strain | [1] | Sσ/SE | 1 |
Elastic Modulus | [ML−1T−2] | Sσ | 1 |
Load performance | Force | [MLT−2] | Sl2·Sσ | 1/9 |
Moment | [ML2T−2] | Sl3·Sσ | 1/27 |
Dynamic properties | Time | [T] | Sl0.5·Sa−0.5 | 0.408 |
Frequency | [T−1] | Sl−0.5·Sa0.5 | 2.449 |
Acceleration | [LT−2] | Sa | 2.0 |
Gravity | [LT−2] | 1 | 1 |
Damping Ratio | [MT−1] | Sl1.5·Sσ·Sa−0.5 | 1/9 |
Table 2.
Comparison between calculated and experimental values of vibration mode frequency of finite element model.
Table 3.
Original shaking wave details.
Table 3.
Original shaking wave details.
Wave | Observation Station | Year | Duration/s | PGA/m·s−2 | Predominant Period/s |
---|
El Centro | Imperial Valley, USGS | 1940 | 53.460 | 3.4 | 0.19 |
Chi−Chi | TCU117 | 1999 | 89.995 | 1.2 | 0.57 |
Table 4.
Comparison of seismic response and collapse condition.
Table 4.
Comparison of seismic response and collapse condition.
Wave condition | Case of 0.4g PGA | Collapse Case |
---|
Peak Longitudinal Relative Displacement/mm | Peak Acceleration Response of Middle Pier Cap/m·s−2 | Peak Principle Compressive Strain of Middle Pier at the Bottom/με | PGA of the Wave |
---|
El Centro wave | X | 12.14 | 6.75 | 519 | 1.2 g |
X + 0.6Z | 12.45 | 7.68 | 531 | 1.2 g |
Chi−chi wave | X | 15.53 | 6.59 | 611 | 1.2 g |
X + 0.8Z | 15.47 | 7.52 | 602 | 1.2 g |
Table 5.
Longitudinal collapse information of El Centro wave input condition.
Table 5.
Longitudinal collapse information of El Centro wave input condition.
Span/m | PGA/g | Failure Time of Middle Pier/s | Collapse Mode |
---|
3.3 | 1.4 | 1.94 | Structural collapse caused by failure of middle pier |
5.3 | 1.2 | 0.93 |
6.7 | 1.2 | 0.9 |
8.3 | 1.0 | 0.91 |
Table 6.
Longitudinal collapse information of Chi−chi wave input condition.
Table 6.
Longitudinal collapse information of Chi−chi wave input condition.
Span/m | PGA/g | Cracking Time of Upper Edge of Beam/s | Collapse Mode |
---|
3.3 | 1.6 | 17.09 | Excessive relative displacement between pier and beam led to falling beam |
5.3 | 1.4 | 17.3 |
6.7 | 1.2 | 17.48 |
8.3 | 1.2 | 24.26 |
Table 7.
Longitudinal collapse information of El Centro wave input condition.
Table 7.
Longitudinal collapse information of El Centro wave input condition.
Pier Height/m | Natural Period of Vibration/s | PGA/g | Failure Time of Middle Pier/s | Collapse Mode |
---|
1.2 | 0.283 | 1.2 | 1.05 | Structural collapse caused by failure of middle pier |
1.8 | 0.338 | 1.2 | 1.07 |
2.4 | 0.409 | 1.4 | 1.04 |
3 | 0.495 | 1.4 | 1.37 |
3.6 | 0.594 | 2.2 | 1.97 |
Table 8.
Longitudinal collapse information of Chi−chi wave input condition.
Table 8.
Longitudinal collapse information of Chi−chi wave input condition.
Pier Height/m | Natural Period of Vibration/s | PGA/g | Cracking Time of Upper Edge of Beam/s | Collapse Mode |
---|
1.2 | 0.283 | 1.2 | 17.45 | Excessive relative displacement between pier and beam led to falling beam |
1.8 | 0.338 | 1.0 | 25.28 |
2.4 | 0.409 | 1.0 | 25.34 |
3 | 0.495 | 0.8 | 25.22 |
3.6 | 0.594 | 0.8 | 25.13 |
Table 9.
Damper parameter optimization scheme.
Table 9.
Damper parameter optimization scheme.
Scheme Number | α | C/kN·(s/m)α | Scheme Number | α | C/kN·(s/m)α |
---|
1 | 0.2 | 1 | 22 | 0.6 | 1 |
2 | 0.2 | 2 | 23 | 0.6 | 2 |
3 | 0.2 | 4 | 24 | 0.6 | 4 |
4 | 0.2 | 6 | 25 | 0.6 | 6 |
5 | 0.2 | 8 | 26 | 0.6 | 8 |
6 | 0.2 | 10 | 27 | 0.6 | 10 |
7 | 0.2 | 12 | 28 | 0.6 | 12 |
8 | 0.3 | 1 | 29 | 0.8 | 1 |
9 | 0.3 | 2 | 30 | 0.8 | 2 |
10 | 0.3 | 4 | 31 | 0.8 | 4 |
11 | 0.3 | 6 | 32 | 0.8 | 6 |
12 | 0.3 | 8 | 33 | 0.8 | 8 |
13 | 0.3 | 10 | 34 | 0.8 | 10 |
14 | 0.3 | 12 | 35 | 0.8 | 12 |
15 | 0.4 | 1 | 36 | 1 | 1 |
16 | 0.4 | 2 | 37 | 1 | 2 |
17 | 0.4 | 4 | 38 | 1 | 4 |
18 | 0.4 | 6 | 39 | 1 | 6 |
19 | 0.4 | 8 | 40 | 1 | 8 |
20 | 0.4 | 10 | 41 | 1 | 10 |
21 | 0.4 | 12 | 42 | 1 | 12 |
Table 10.
Damper parameter alternative scheme.
Table 10.
Damper parameter alternative scheme.
Scheme Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|
C/kN·(s/m)α | 6 | 6 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 |
α | 0.2 | 0.3 | 0.2 | 0.3 | 0.4 | 0.2 | 0.3 | 0.4 | 0.2 | 0.3 | 0.4 | 0.6 |