# Shake Table Test of Long Span Cable-Stayed Bridge Subjected to Near-Fault Ground Motions Considering Velocity Pulse Effect and Non-Uniform Excitation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## Featured Application

## Abstract

## 1. Introduction

## 2. Prototype Bridge and the Scaled Model

#### 2.1. Prototype Bridge

#### 2.2. Scaled Model

#### 2.2.1. Shake Table System

#### 2.2.2. Similitude Ratio

#### 2.2.3. Girder and Stayed Cables

_{z}and EI

_{z}are calculated as 4.72e-06 m

^{4}and 1.50e-07 m

^{4}. The actual bending stiffness ratios of EI

_{z}and EI

_{z}are 6.91E+08 and 5.00E+08. They are about 1.86 times and 2.56 times of target similitude ratios of bending stiffness in the Table 2.

#### 2.2.4. Additional Masses

#### 2.2.5. Sensors Arrangement

## 3. Dynamic Characteristics and Test Cases

#### 3.1. Dynamic Characteristics of the Test CSB

^{nd}mode shows vertical antisymmetric vibration of girder combining longitudinal bending vibration of tower. The 2

^{nd}, 4

^{th}and 6

^{th}modes are antisymmetric around the vertical axis, while the 3

^{rd}and 5

^{th}modes are symmetric.

#### 3.2. Input Ground Motions

^{2}. According to the similitude ratio of time in Table 2, the recorder motions are scaled in time domain. The adjusted El-Centro wave and TCU052 wave are renamed as ScEL and ScTCU respectively.

#### 3.3. Shake Table Test Cases

_{app}stands for the apparent wave velocity. In this paper, different V

_{app}is studied, such as 480 m/s, 240 m/s and 120 m/s; S

_{t}stands for the similitude ratios of time, that is 0.0707 in table 2. As the main span of the CSB is 680 m, delay time dt in the test is calculated as 0.1 s, 0.2 s and 0.4 s, listed in Table 4.

## 4. Seismic Response of CSBs Under Uniform Excitations

#### 4.1. Reproduction Validation for Uniform Excitations

#### 4.2. Seismic Responses of Towers and Ppiers

#### 4.2.1. Accelerations Responses

^{2}), maximal accelerations subjected to NNF motion (ScEL) seems a little larger than the PNF motion (ScTCU). The phenomenon can be explained by the response Fourier spectrum, such as Figure 13a,b. As operation frequency limitation of the test system, the frequency range of 0~50 Hz are shown in the figures. It is obviously the energy of tower top node (T4) are mainly at 6.9 Hz, 13.8 Hz, 27 Hz, which are close to the frequencies of the 2

^{nd}, 4

^{th}, 5

^{th}modes. Among them, the 2

^{nd}mode dominate the acceleration response of tower top node. However, the response of mid cross beam (T3) has more energy than top node (T4). For f = 6.9 Hz, the peak amplitude is 0.05 m/s

^{2}for tower top node, while that is 0.15 m/s

^{2}for mid cross beam. In addition, there are an energy crest around 20 Hz, which is close to the frequency of the 5

^{th}mode. In the result, the acceleration response of tower mid cross beam is larger than top node. As shown in Figure 13b, the very similar phenomenon can be found in the tower response subjected to ScTCU.

#### 4.2.2. Displacements Responses

#### 4.2.3. Strains and Bending Moment

#### 4.3. Seismic Responses of Girder

#### 4.3.1. Longitudinal Responses

^{2}, which is larger than 0.58 m/s

^{2}when subjected to ScEL. Figure 20b shows the Fourier amplitude of the longitudinal acceleration subjected to different type of NF motions. When subjected to the PNF (ScTCU), the Fourier amplitudes of the first and second order modes are significant larger than the NNF (ScEL). In the result, when subjected to ScTCU motion, the larger acceleration response of girder are recorded, shown in Figure 20a.

#### 4.3.2. Vertical Response

^{2}, 0.88 m/s

^{2}, 2.70 m/s

^{2}separately. It is obviously that the vertical acceleration on the G2 and G4 are significant larger than node G3 on the mid-span when subjected to the ScEL. The amplitude spectrum of the accelerations time histories was analyzed by FFT method, shown in Figure 21b. The predominant frequency of the girder vertical vibration is about 13.9 Hz. which corresponds to the 4

^{th}vibration mode, shown in Figure 8d. The mode is a typical antisymmetric mode. In the result, the vibrations of the G2 and G4 are larger than the mid-span.

^{nd}, 3

^{rd}, 4

^{th}and 5

^{th}vibration mode. In the result, the peak accelerations on the G2, G3 and G4 are close.

#### 4.4. Seismic Responses of Bearing

#### 4.5. Comparison of the Seismic Response

## 5. Seismic Responses of CSBs Under Non-Uniform Excitations

#### 5.1. Non-Uniform Test Cases

#### 5.2. Seismic Responses Subjected to the Non-Uniform Excitations

^{th}, 6

^{th}modes, are more seriously excited by the non-uniform excitations. Secondly, comparing the non-uniform cases with different dt, the deformation on the G4 is more sensitive than G2. It means wave passage effect has a greater impact on the nodes located farther away, such as G4, when the excitation is propagating from G1 to G5.

## 6. Conclusions

- (1)
- The first six modes and the corresponding frequencies of the scaled CSB were identified using the SSI method. The fundamental mode shows as girder and tower longitudinal vibration with a frequency of 0.79 Hz. The 2nd mode shows as girder vertical antisymmetric vibration combing tower longitudinal bending with a frequency of 6.61 Hz. In the first six in-plane modes, the 2
^{nd}, 4^{th}and 6^{th}modes are antisymmetric, while the 1^{st}, 3^{rd}and 5^{th}modes are symmetric. - (2)
- The maximum displacement of the tower occurs on the tower top node, the maximum acceleration response of the tower occurs on the middle cross beam, and the maximum bending moment of the tower occurs on the bottom section
- (3)
- The deformation of the tower and girder subjected to uniform excitation is not always larger than that subjected to non-uniform excitation, and therefore the non-uniform case should be considered in the seismic design of CSBs.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Xu, Y.; Hu, S. Seismic design of high-rise towers for cable-stayed bridges under strong earthquakes. Front. Arch. Civ. Eng. China
**2011**, 5, 451–461. [Google Scholar] [CrossRef] - Soneji, B.; Jangid, R. Passive hybrid systems for earthquake protection of cable-stayed bridge. Eng. Struct.
**2007**, 29, 57–70. [Google Scholar] [CrossRef] - Loh, C.-H.; Liao, W.-I.; Chai, J.-F. Effect of near-fault earthquake on bridges: Lessons learned from Chi-Chi earthquake. Earthq. Eng. Struct. Vib.
**2001**, 1, 86–93. [Google Scholar] [CrossRef] - Xu, Y.; Duan, X.; Li, J. Seismic design strategy of cable stayed bridges subjected to strong ground motion. Struct. Eng. Mech.
**2012**, 51, 909–921. [Google Scholar] [CrossRef] - Yi, J.; Li, J. Effect of Seismic-Induced bearing uplift of a Cable-Stayed bridge. J. Bridge Eng.
**2019**, 24, 4018125. [Google Scholar] [CrossRef] - Fang, Z.; Zhang, C.; Chen, Y.; Zheng, Z.; Xu, L. Research on the shaking table test of three towers cable-stayed bridge based on three shaking table system. China Civ. Eng.
**2012**, 45, 25–29. (In Chinese) [Google Scholar] - Yadi, S.; Suhendro, B.; Priyosulistyo, H.; Aminullah, A. Shake table test of floating cable-stayed bridge under earthquake excitation during construction with balanced cantilever method. Int. J. Civ. Eng. Technol.
**2018**, 9, 2063–2081. [Google Scholar] - Xu, L.; Zhang, H.; Gao, J.; Zhang, C. Longitudinal seismic responses of a cable-stayed bridge based on shaking table tests of a half-bridge scale model. Adv. Struct. Eng.
**2018**, 1–13. [Google Scholar] [CrossRef] - Xie, W.; Sun, L. Experimental and numerical verification on effects of inelastic tower links on transverse seismic response of tower of bridge full model. Eng. Struct.
**2019**, 182, 344–362. [Google Scholar] [CrossRef] - Yadi, S.; Suhendro, B.; Priyosulistyo, H.; Aminullah, A. Dynamic response of long-span bridges subjected to nonuniform excitation: A state-of-the-art review. Matec. Web. Conf.
**2019**, 258, 5017. [Google Scholar] [CrossRef] [Green Version] - Shiravand, M.R.; Parvanehro, P. Spatial variation of seismic ground motion effects on nonlinear responses of cable stayed bridges considering different soil types. Soil Dyn. Earthq. Eng.
**2019**, 119, 104–117. [Google Scholar] [CrossRef] - Li, S.; Zhang, F.; Wang, J.-Q.; Alam, M.S.; Zhang, J. Seismic responses of super-span cable-stayed bridges induced by ground motions in different sites relative to fault rupture considering soil-structure interaction. Soil Dyn. Earthq. Eng.
**2017**, 101, 295–310. [Google Scholar] [CrossRef] - Liang, F.; Jia, Y.; Sun, L.; Xie, W.; Chen, H. Seismic response of pile groups supporting long-span cable-stayed bridge subjected to multi-support excitations. Soil Dyn. Earthq. Eng.
**2017**, 101, 182–203. [Google Scholar] [CrossRef] - Jia, H.; Lan, X.; Zheng, S.; Li, L.; Liu, C. Assessment on required separation length between adjacent bridge segments to avoid pounding. Soil Dyn. Earthq. Eng.
**2019**, 120, 398–407. [Google Scholar] [CrossRef] - Kojima, K.; Takewaki, I. Critical earthquake response of elastic-plastic structures under Near-Fault ground motions (Part 2: Forward-Directivity input). Front. Built Environ.
**2015**, 1, 1–13. [Google Scholar] [CrossRef] [Green Version] - Ardakani, S.; Saiidi, M.S. Simple method to estimate residual displacement in concrete bridge columns under near-fault earthquake motions. Eng. Struct.
**2018**, 176, 208–219. [Google Scholar] [CrossRef] - Adanur, S.; Altunişik, A.C.; Bayraktar, A.; Akköse, M. Comparison of near-fault and far-fault ground motion effects on geometrically nonlinear earthquake behavior of suspension bridges. Nat. Hazards
**2012**, 64, 593–614. [Google Scholar] [CrossRef] - Yang, D.; Zhao, Y.; Li, G. Influence analysis of motion characteristics of near-fault ground motions on seismic responses of long-period structures. J. Disaster Prev. Mitig. Eng.
**2007**, 27, 133–140. (In Chinese) [Google Scholar] - Zhou, L.; Wang, X.; Ye, A. Effectiveness evaluation and optimal design of nonlinear viscous dampers for inelastic structures under pulse-type ground motions. Earthq. Eng. Struct. Dyn.
**2018**, 47, 2802–2820. [Google Scholar] - Zhou, L.; Wang, X.; Ye, A. Shake table test on transverse steel damper seismic system for long span cable-stayed bridges. Eng. Struct.
**2019**, 179, 106–119. [Google Scholar] [CrossRef] - Lesgidis, N.; Kwon, O.-S.; Sextos, A. A time-domain seismic SSI analysis method for inelastic bridge structures through the use of a frequency-dependent lumped parameter model. Earthq. Eng. Struct. Dyn.
**2015**, 44, 2137–2156. [Google Scholar] [CrossRef] - Shahi, S.K.; Baker, J.W. An efficient algorithm to identify Strong-Velocity pulses in multicomponent ground motions. Bull. Seismol. Soc. Am.
**2014**, 104, 2456–2466. [Google Scholar] [CrossRef] - Ancheta, T.; Darragh, R.; Stewart, J.; Silva, W.J.; Chiou, B.S.J.; Wooddell, K.E.; Graves, R.W.; Kottke, A.R.; Boore, D.M.; Kishida, T.; et al. PEER NGA-West2 Database; Technical Report 2013/03; Pacific Earthquake Engineering Research Center: Berkeley, CA, USA, 2013. [Google Scholar]
- Okamoto, Y.; Nakamura, S. Static and seismic studies on steel/concrete hybrid towers for multi-span cable-stayed bridges. J. Constr. Steel Res.
**2011**, 67, 203–210. [Google Scholar] [CrossRef] - Martinez-Rodrigo, M.D.; Filiatrault, A. A case study on the application of passive control and seismic isolation techniques to cable-stayed bridges: A comparative investigation through non-linear dynamic analyses. Eng. Struct.
**2015**, 99, 232–252. [Google Scholar] [CrossRef]

**Figure 5.**Assembling of the components (

**a**) cable anchor; (

**b**) fixed tower foundation platform; (

**c**) sliding bearing.

**Figure 8.**Identified modal shapes and frequencies (

**a**) the 1

^{st}mode, f

_{v1}= 0.79 Hz; (

**b**) the 2

^{nd}mode, f

_{v2}= 6.61 Hz; (

**c**) the 3

^{rd}mode, f

_{v3}= 9.34 Hz; (

**d**) the 4

^{th}mode, f

_{v4}= 14.19 Hz; (

**e**) the 5

^{th}mode, f

_{v5}= 20.97 Hz; (

**f**) the 6

^{th}mode, f

_{v6}= 27.16 Hz.

**Figure 9.**Input ground motions (

**a**) Acceleration time histories; (

**b**) Acceleration Fourier Amplitude; (

**c**) Velocity time histories; (

**d**) Displacement time histories;

**Figure 11.**Measured accelerations of two tables subjected to uniform ScEL (Case1) (

**a**) Time history (

**b**) Fourier Amplitude.

**Figure 13.**Accelerations Fourier amplitude of 3# tower (

**a**) subjected to ScEL; (

**b**) subjected to ScTCU.

**Figure 20.**Longitudinal acceleration of girder. (

**a**) Acceleration time histories; (

**b**) Acceleration amplitude.

**Figure 24.**Longitudinal displacement of tower and girder on bearing connection (

**a**) Bearing relative displacement (

**b**) Fourier Amplitude.

**Figure 28.**The girder vertical deformation time history subjected to ScTCU with (

**a**) uniform; (

**b**) dt = 0.1 s; (

**c**) dt = 0.2 s; (

**d**) dt = 0.4 s.

Table NO. | Table Size | Self-Weight | Payload | Degree of Freedom | Stroke Length | Velocity | Acceleration | Operation Frequency |
---|---|---|---|---|---|---|---|---|

1# | 4 m × 4 m | 9.65 t | 22 t | 3 | ±250 mm | 75 cm/s | 1.5 g(Ux)/1.2 g(Uy) | 0.1–50 Hz |

2# (3#) | 2.5 × 2.5 m | 3.0 t | 10 t | 3 | ±250 mm | 150 cm/s | 1.5 g(Ux)/1.2 g(Uy) | 0.1–50 Hz |

Physical Properties | 1/Ratio | Material Properties | 1/Ratio | Dynamic Parameter | 1/Ratio |
---|---|---|---|---|---|

Length | 1/102 | Elastic Modulus | 1/12.81 | time | 0.0707 |

Area | 1/104 | bending Stiffness | 1/1.281 × 109 | Frequency | 1/0.0707 |

Moment of inertia | 1/108 | Equivalent density | 1/0.2562 | Velocity | 1/7.07 |

Strain | 1 | Mass | 1/256,200 | Acceleration | 2 |

Component | Material | Young’s Modulus of Elastic E (MPa) | Density (kg/m ^{3}) | Poisson’s Ratio | Yield Strength (MPa) |
---|---|---|---|---|---|

tower/pier | PMMA | 2.69 × 10^{3} | 1180 | 0.391 | 126 |

girder | Aluminum | 7.53 × 10^{4} | 2700 | 0.326 | 187 |

stayed cable | Steel | 1.95 × 10^{5} | 7850 | 0.3 | 1330 |

Test cases | Uniform Excitation | Non-Uniform Excitation | ||||||
---|---|---|---|---|---|---|---|---|

Case1 | Case2 | Case3 | Case4 | Case5 | Case6 | Case7 | Case8 | |

Exctation | ScEL | ScTCU | ScEL | ScEL | ScEL | ScTCU | ScTCU | ScTCU |

delay time (dt) | 0 s | 0 s | 0.1 s | 0.2 s | 0.4 s | 0.1 s | 0.2 s | 0.4 s |

Components | 0#Pier | 1# Pier | 2# Pier | 5# Pier | 6# Pier | 7# Pier |
---|---|---|---|---|---|---|

under ScEL | 10.23 | 10.01 | 9.98 | 10.14 | 10.23 | 10.23 |

under ScTCU | 13.84 | 14.32 | 14.87 | 13.8 | 13.18 | 13.84 |

Excitation | Longitudinal Acceleration (m/s^{2}) | Longitudinal Displacement (mm) | ||||
---|---|---|---|---|---|---|

T2-3# | T3-3# | T4-3# | T2-3# | T3-3# | T4-3# | |

ScEL | 3.68 | 4.21 | 1.49 | 1.53 | 1.00 | 1.15 |

ScTCU | 3.83 | 3.02 | 1.52 | 22.17 | 25.63 | 32.30 |

Excitation | Vertical Acceleration (m/s^{2}) | Vertical Displacement (mm) | ||||
---|---|---|---|---|---|---|

G2 | G3 | G4 | G2 | G3 | G4 | |

ScEL | 2.82 | 0.89 | 2.69 | 0.38 | 0.06 | 0.55 |

ScTCU | 1.69 | 1.06 | 1.54 | 2.07 | 0.56 | 3.76 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, C.; Fu, G.; Lai, Z.; Du, X.; Wang, P.; Dong, H.; Jia, H.
Shake Table Test of Long Span Cable-Stayed Bridge Subjected to Near-Fault Ground Motions Considering Velocity Pulse Effect and Non-Uniform Excitation. *Appl. Sci.* **2020**, *10*, 6969.
https://doi.org/10.3390/app10196969

**AMA Style**

Zhang C, Fu G, Lai Z, Du X, Wang P, Dong H, Jia H.
Shake Table Test of Long Span Cable-Stayed Bridge Subjected to Near-Fault Ground Motions Considering Velocity Pulse Effect and Non-Uniform Excitation. *Applied Sciences*. 2020; 10(19):6969.
https://doi.org/10.3390/app10196969

**Chicago/Turabian Style**

Zhang, Chao, Guanghui Fu, Zhichao Lai, Xiuli Du, Piguang Wang, Huihui Dong, and Hongyu Jia.
2020. "Shake Table Test of Long Span Cable-Stayed Bridge Subjected to Near-Fault Ground Motions Considering Velocity Pulse Effect and Non-Uniform Excitation" *Applied Sciences* 10, no. 19: 6969.
https://doi.org/10.3390/app10196969