# Impact of Variable Parameters of Expansion Joints and Bearing Supports on the Vehicle-Induced Vibration of Curved Girder Bridges

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## Abstract

**:**

## 1. Introduction

## 2. Analysis Method for the Dynamics Response of Vehicle–Bridge Coupling Considering Centrifugal Forces

#### 2.1. Without Considering E-B Parameters

_{1}and rear MZ

_{2}axle transient centers. The rigid rod was then supported by the vertical springs of the front and rear suspensions. The effect of centrifugal force on the wheel loads is shown in Figure 1.

_{vr}V

^{2}/R acts on the center of mass, MC, of the vehicle body with mass M

_{vr}, where R is the radius of curvature of the vehicle’s path under uniform circular motion. This centrifugal force produces a moment M

_{vr}V

^{2}l

_{h}/R around the instantaneous axis, where l

_{h}is the distance from MC to the instantaneous center. Since the body rotates around the instantaneous axis, its MC experiences a lateral displacement l

_{h}sinχ that generates an additional moment M

_{vr}gl

_{h}sinχ ≈ M

_{vr}gl

_{h}χ. The total moment is given by Equation (1) as

_{vrm}is transmitted to the axle via the suspension spring. Given that K

_{v}

_{1}and K

_{v}

_{2}represent the roll angle stiffnesses of the front and rear axles, respectively, the moment T

_{vrm}can also be expressed by Equation (2) as

_{vrm}

_{1}and T

_{vrm}

_{2}represent the spring moments of the front and rear axles, respectively. The centrifugal force is distributed to two instantaneous centers via the position of the center of mass. In other words, the front and rear axles distribute the centrifugal force as M

_{vr}V

^{2}l

_{m}

_{2}/[R(l

_{m}

_{1}+ l

_{m}

_{2})], M

_{vr}V

^{2}l

_{m}

_{1}/[R(l

_{m}

_{1}+ l

_{m}

_{2})], respectively.

_{s}

_{1}(the distance between the left and right wheel centers of the front axle) are balanced by the sum of the three moments: (a) the spring moment T

_{vrm}

_{1}of the front axle; (b) the centrifugal force M

_{vr}V

^{2}l

_{m}

_{2}/[R(l

_{m}

_{1}+ l

_{m}

_{2})] distributed to the front axle and its corresponding moment formed by its force arm l

_{p}

_{1}(the distance from the instantaneous center of the front axle MZ

_{1}to the bridge deck); (c) The moment formed by the centrifugal force acting on the front axle $\left({M}_{sL}^{1}+{M}_{sR}^{1}\right){V}^{2}/R$ and its corresponding force arm l

_{f}

_{1}(the distance from the center of mass of the front axle MS

_{1}to the bridge deck). Therefore, the aforementioned moment–balance relationship can be expressed as follows:

#### 2.2. With E-B Parameters Taken into Account

#### 2.3. Flow Chart of the Analysis Method

#### 2.4. Method Validation

## 3. Influence of Variable Parameters of the EJ on the Vehicle-Induced Dynamic Response of the Curved Girder Bridge

#### 3.1. Height Differences

#### 3.1.1. Height Difference of the Middle Beam

#### 3.1.2. Height Difference of the Side Beam

#### 3.2. Support Stiffness of the Expansion Joints

_{0}, both the front and rear axles exhibited their maximum DAF values of 2.227 and 1.987, respectively, representing increases of 111.1% and 95.4%. Furthermore, Figure 15b illustrates that, as the stiffness decreased, there was a steady increase in the DAF at the end of the main girder. At 0.2 k

_{0}, a maximum DAF of 1.180 was reached, corresponding to an increase of 11.4%. In addition, it was important to note that other measurement points on the main girder did not exhibit sensitivity to changes in the vertical support stiffness of the EJ. The change rule of the variation rate of the 0−1# BS reaction force with the support stiffness of the EJ is shown in Figure 16.

## 4. Effects of Support Variable Parameters on the Vehicle-Induced Responses of Curved Girder Bridges

#### 4.1. Reduction in BS Stiffness

_{0}, the maximum increase rates in the DAFs at the end of the first span, the 1/4 point of the first span, the 1/2 point of the first span, the 1/2 point of the second span, and the 1/2 point of the third span were 24.18%, 7.97%, 6.40%, 7.10%, and 5.51%, respectively. This indicates that changes in BS stiffness have a significant impact on the DAFs at various measurement points on the main girder, especially the beam end of the first span.

#### 4.2. Support Voids

#### 4.2.1. Single-Support Void

#### 4.2.2. Double-Support Void

## 5. Impact of Coupled Changes in the E-B Parameters on Dynamic Responses of Curved Girder Bridges to Vehicular Loads

#### 5.1. Orthogonal Experimental Design

_{9}(3

^{4}) orthogonal table, presented in Table 9.

#### 5.2. Coupled Parameter Influence Analysis

## 6. Conclusions

- (1)
- The proposed analysis method can be applied to analyze the vehicle-induced dynamic response of a curved girder bridge, taking into account the variable parameters of expansion joints (EJs) and bearing supports (BSs). The errors between the theoretical analysis results and the field-testing results were in the range of 4.01~8.24%.
- (2)
- The DAF of the middle beam of an expansion joint (DAF-EJ) increased with the increase of its relevant height difference; the DAF of the beam end of the main girder increased with increasing or decreasing relevant height difference. There was a speed limitation which made the DAFs-EJ increase with the increasing height difference of the side beam, while the DAF of the beam end decreased with the increasing relevant height difference. The DAFs of the middle beam and the beam end both significantly increased with the decreasing support stiffness of the EJ. The height difference and support stiffness reductions to the EJ will both significantly vary the bearing reaction force of the BS, and may result in the fatigue failure of the BS.
- (3)
- The DAFs of the middle beam of the EJ and the main girder increased with the support stiffness reduction in the BS near the beam end of the main girder. The reduction in the support stiffness of the BS had little effect on the DAF-EJ, but had a significant effect on the DAF of the main girder, especially the beam end. The DAFs of the middle beam of the EJ, the beam end, the 1/4 and 1/2 points of the first span, as well as the 1/2 point of the second span, were obviously influenced by the single-support void near the traffic lane or corresponding measurement points. Moreover, the DAFs of the middle beam of the EJ and the main girder were more influenced by the double-support void, especially the beam end of the main girder.
- (4)
- The DAF-EJ was more sensitive to the height difference of the middle beam of the EJ and thevertical support stiffness variation in the BS and EJ. The DAF of the beam end of the main girder was most sensitive to the height difference of the middle beam of the EJ. The influences of the height difference of the middle beam of the EJ on the DAFs of the beam end, the 1/4 and 1/2 points of the first span, as well as the 1/2 points of the second and third spans, gradually decreased. The DAF of the beam end of the main girder was most significantly affected by the reduction in support stiffness.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Flow chart of the analysis method for vehicle–joint (bearing)–bridge coupling vibration taking into account centrifugal force.

**Figure 5.**Diagram of the EJ, including a support box (a), a side beam (b), a central beam (c), a compression support (d), a support beam (e), a waterproof sealing (f).

**Figure 6.**Section size of the main components of the EJ (cm), including a side beam (a), a central beam (b), and support beams (c and d).

**Figure 10.**Displacement curves over time of the vertical displacements of individual measuring points under different HMBs (30 km/h).

**Figure 12.**Variations in the reaction force of the BS with different HMBs (0−1#, 30 km/h). (

**a**) Displacement curves over time of the variation rate of the reaction force for the BS. (

**b**) Peak variation rate of the reaction force of the BS.

**Figure 14.**Variations in the BS reaction forces with different HSBs. (

**a**) Displacement curves over time of the variation rates of the BS reaction force. (

**b**) Peak variation rates of BS reaction force.

**Figure 16.**Variations in the BS reaction forces with different support stiffnesses of EJ. (

**a**) Displacement curves over time of the variation rate of the 0-1# BS reaction force. (

**b**) Peak variation rate of BS reaction force.

**Figure 18.**DAF curves of the bridge measuring points under different bearing support vertical stiffnesses.

Technical Parameters | Technical Conditions |
---|---|

Height difference of the EJ | <2 mm, meets the design requirement |

Stiffness of the EJ | Equivalent to the design parameter |

Bearing capacity of the EB | Equivalent to the design parameter |

Vertical stiffness of the EB | Equivalent to the design parameter |

Strength of the main girder | 53.1 MPa, meets the design requirement |

Transverse connection conditions among main girders | Effective and without any damage |

Road roughness | Close to ideal in general |

EJ (GQF-MZL160) | |||
---|---|---|---|

Parameters | Numerical Value | Parameters | Numerical Value |

Elastic modulus E/Pa | 2.05 × 10^{11} | Shear Modulus G/Pa | 8 × 10^{10} |

Material density ρ/(kg·m^{−3}) | 7800 | Poisson’s ratio v | 0.2 |

Length of support beam L_{2}/m | 0.52 | Length of middle beam L_{1}/m | 40 |

Width of side beam ls/(m) | 0.04 | Width of middle beam lc/(m) | 0.04 |

Joint gap s_{0}/(m) | 0.08 | Width of side beam hs/(m) | 0.08 |

Inertia moment of support beam I_{2}/m^{4} | 1.35 × 10^{−5} | Inertia moment of middle beam I_{1}/m^{4} | 7.7 × 10^{−6} |

Inertia moment of side beam 1 I_{2}/m^{4} | 1.82 × 10^{−6} | Inertia moment of side beam 2 I_{1}/m^{4} | 1.82 × 10^{−6} |

Support stiffness of middle beam | 4 × 10^{7} | Support stiffness of support beam | 2 × 10^{8} |

Support damping of support beam C_{2}/(N·s) m^{−1} | 5000 | Support damping of middle beam C_{1}/(N·s) m^{−1} | 5000 |

BS (GPZ2.5 DX) | |||

Parameters | Numerical value | Parameters | Numerical value |

Design bearing capacity (kN) | 2500 | Design deformation capacity(mm) | ±100 |

Ultimate bearing capacity (kN) | 2750 | Sliding friction coefficient | <0.02 |

Vertical stiffness (kN/m) | 2.4 × 10^{6} | Horizontal stiffness (kN/m) | 1.5 × 10^{5} |

Modal Order | Calculation Results (Hz) | Testing Results (Hz) | Error |
---|---|---|---|

1 | 3.803 | 3.891 | 2.26% |

2 | 4.584 | 4.893 | 6.32% |

3 | 6.902 | 7.194 | 4.06% |

Measuring Points | Testing Values (mm) | Calculating Values (mm) | Error % |
---|---|---|---|

1/4 point of the first span | 3.12 | 2.92 | 6.41 |

1/2 point of the first span | 4.35 | 4.08 | 6.21 |

1/2 point of the second span | 3.49 | 3.35 | 4.01 |

1/2 point of the third span | 4.25 | 3.90 | 8.24 |

Parameters | Variable parameters of the EJ | BS parameters | Vehicle speed | Road roughness | vehicle position | ||

Height difference (mm) | −20~20 | Support stiffness k_{1} (N/m) | 2.40 × 10^{9} | 30 km/h | Ideal | Outside | |

Support stiffness (k_{0}, N/m) | initial k_{0}, 0.8 k_{0}, 0.6 k_{0}, 0.4 k_{0}, 0.2 k_{0} | ||||||

Seam width (mm) | 80 |

Parameters | Parameters of EJs | Support parameters | Vehicle speed | Unevenness of the road surface | Lateral position of the vehicle | ||

Height difference (mm) | 0 | Vertical support stiffness of the support | Intact k_{1}, 0.8 k_{1}, 0.6 k_{1}, 0.4 k_{1}, 0.2 k_{1} (0 abutment 1#) | 30 km/h | Ideal | Outside | |

Support stiffness (k_{0}, N/m) | 8.00 × 10^{7} | ||||||

Seam width (mm) | 80 | Support void | Void condition (support stiffness is 0 and the main girder is not in contact with the support) |

Parameters of | Working Conditions of Voids | |
---|---|---|

Single-Support Void | 0-1#, 0-2#, 0-3#, 0-4#, 1-1#, 1-2#, 1-3#, 1-4# | |

Double-Support Void | 0-1, 2# | 0-1, 3# |

0-2, 3# | 0-2, 4# | |

1-1, 2# | 1-1, 3# | |

1-2, 3# | 1-2, 4# | |

0-4#, 1-4# | 0-4#, 1-4# | |

0-4#, 2-4# | 1-4#, 1-4# | |

1-4#, 2-4# | 2-4#, 1-4# |

Working Condition of Double Support Void | Category | Measurement Points | ||||||
---|---|---|---|---|---|---|---|---|

Front Axle Passing over EJ Middle Beam | Rear Axle Passing over EJ Middle Beam | End of the First Span | 1/4 Point of the First Span | 1/2 Point of the First Span | 1/2 Point of the Second Span | 1/2 Point of the Third Span | ||

The support is intact | DAF | 1.055 | 1.017 | 1.059 | 1.006 | 1.002 | 1.003 | 1.003 |

0-1, 2# | 1.082 | 1.102 | 1.548 | 1.447 | 1.131 | 1.003 | 1.003 | |

0-1, 3# | 1.074 | 1.050 | 1.403 | 1.124 | 1.039 | 1.003 | 1.003 | |

0-1, 4# | 1.075 | 1.056 | 1.431 | 1.159 | 1.053 | 1.003 | 1.003 | |

0-2, 3# | 1.059 | 1.021 | 1.239 | 1.008 | 1.002 | 1.003 | 1.003 | |

0-2, 4# | 1.059 | 1.021 | 1.229 | 1.008 | 1.003 | 1.003 | 1.003 | |

0-3, 4# | 1.056 | 1.018 | 1.132 | 1.007 | 1.004 | 1.003 | 1.003 | |

1-1, 2# | 1.055 | 1.016 | 1.024 | 1.305 | 1.534 | 1.417 | 1.011 | |

1-1, 3# | 1.055 | 1.017 | 1.065 | 1.126 | 1.227 | 1.129 | 1.004 | |

1-1, 4# | 1.055 | 1.017 | 1.066 | 1.127 | 1.230 | 1.138 | 1.004 | |

1-2, 3# | 1.055 | 1.017 | 1.067 | 1.003 | 1.004 | 1.014 | 1.006 | |

1-2, 4# | 1.055 | 1.017 | 1.064 | 1.004 | 1.003 | 1.013 | 1.005 | |

1-3, 4# | 1.055 | 1.017 | 1.051 | 1.004 | 1.006 | 1.007 | 1.004 | |

0-4, 1-4# | 1.074 | 1.050 | 1.385 | 1.264 | 1.277 | 1.130 | 1.004 | |

0-4, 2-4# | 1.074 | 1.049 | 1.377 | 1.121 | 1.040 | 1.192 | 1.187 | |

0-4, 1-4# | 1.074 | 1.050 | 1.382 | 1.122 | 1.039 | 1.003 | 1.039 | |

1-4, 2-4# | 1.055 | 1.017 | 1.057 | 1.129 | 1.233 | 1.375 | 1.189 | |

1-4, 1-4# | 1.055 | 1.017 | 1.020 | 1.127 | 1.228 | 1.129 | 1.040 | |

2-4, 1-4# | 1.055 | 1.017 | 1.021 | 1.004 | 1.002 | 1.193 | 1.235 |

Numbering of Orthogonal Tests | Factor 1 | Factor 2 | Factor 3 | Factor 4 |
---|---|---|---|---|

1 | 0 mm | k_{0} | k_{1} | No void |

2 | 0 mm | 0.6 k_{0} | 0.2 k_{1} | 0 abutment 1# |

3 | 0 mm | 0.2 k_{0} | 0.6 k_{1} | 1 span 1# |

4 | −20 mm | k_{0} | 0.2 k_{1} | 1 span 1# |

5 | −20 mm | 0.6 k_{0} | 0.6 k_{1} | No void |

6 | −20 mm | 0.2 k_{0} | k_{1} | 0 abutment 1# |

7 | 20 mm | k_{0} | 0.6 k_{1} | 0 abutment 1# |

8 | 20 mm | 0.6 k_{0} | k_{1} | 1 span 1# |

9 | 20 mm | 0.2 k_{0} | 0.2 k_{1} | No void |

_{0}is the design support stiffness of EJ; k

_{1}is the design support stiffness of BS.

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## Share and Cite

**MDPI and ACS Style**

Zheng, Y.; Lu, C.; Huang, X.; Xu, W.; Zhou, D.; Li, J.; Li, J.; Hou, L.; Wang, K.; Sun, Y.
Impact of Variable Parameters of Expansion Joints and Bearing Supports on the Vehicle-Induced Vibration of Curved Girder Bridges. *Buildings* **2024**, *14*, 293.
https://doi.org/10.3390/buildings14010293

**AMA Style**

Zheng Y, Lu C, Huang X, Xu W, Zhou D, Li J, Li J, Hou L, Wang K, Sun Y.
Impact of Variable Parameters of Expansion Joints and Bearing Supports on the Vehicle-Induced Vibration of Curved Girder Bridges. *Buildings*. 2024; 14(1):293.
https://doi.org/10.3390/buildings14010293

**Chicago/Turabian Style**

Zheng, Yu, Chunfang Lu, Xiaomin Huang, Weibing Xu, Daxing Zhou, Jin Li, Jianxiang Li, Liqun Hou, Kuan Wang, and Yulong Sun.
2024. "Impact of Variable Parameters of Expansion Joints and Bearing Supports on the Vehicle-Induced Vibration of Curved Girder Bridges" *Buildings* 14, no. 1: 293.
https://doi.org/10.3390/buildings14010293