# Relationship between the Vibration Acceleration and Stability of a Continuous Girder Bridge during Horizontal Rotation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Main Construction Process of Horizontal Rotation Bridges

## 3. Vibration Characteristics Analysis of the Rotating Structure

^{4}; the main beam is box section, the cantilever end section height of the main beam is 4.604 m, the roof thickness is 0.354 m, the web thickness is 0.6 m, the bottom plate thickness is 0.4 m, the moment of inertia of the section is 37.95 m

^{4}, the root section height of the main beam is 7.204 m, the roof thickness is 0.554 m, the web thickness is 1 m, the bottom plate thickness is 1.2 m, the moment of inertia of the section is 174.44 m

^{4}; moreover, the size of upper turntable is 16.4 m × 10.8 m × 3.7 m, the diameter of upper spherical hinge 4.1 m, the diameter of the bottom spherical hinge 3.8 m, and the centerline diameter of the slide way 4.5 m.

#### 3.1. Vibration Modal Analysis

#### 3.2. Analysis of the Vibration Tested Data Obtained from the Shang-Yuan Bridge’s Rotating Structure

## 4. Derivation of the Relationship between the Pier-Bottom Bending Moment M and the Acceleration a

_{1}corresponded to the up direction, while the positive direction of y

_{1}corresponded to the right direction. The original position of rod 2 in the local coordinate system was at the cantilever’s support. In this case, the positive direction x

_{2}was the right direction, while the positive direction y

_{2}was the down direction.

- 1.
- The displacement of the pier bottom (rod 1) should be 0 (i.e., ${Y}_{1}\left(0\right)=0$); therefore:$${C}_{1}+{C}_{3}=0$$
- 2.
- The bending moment at the free end of the cantilever (rod 2) should be 0 (i.e., ${E}_{2}{I}_{2}{\ddot{Y}}_{2}\left({L}_{2}\right)=0$); therefore:$${C}_{5}\mathrm{cosh}{\lambda}_{2}{L}_{2}+{C}_{6}\mathrm{sinh}{\lambda}_{2}{L}_{2}-{C}_{7}\mathrm{cos}{\lambda}_{2}{L}_{2}-{C}_{8}\mathrm{sin}{\lambda}_{2}{L}_{2}=0$$
- 3.
- The shear force at the free end of the cantilever (rod 2) should be 0 (i.e., ${E}_{2}{I}_{2}{\stackrel{\u20db}{Y}}_{2}\left({L}_{2}\right)=0$); therefore:$${C}_{5}\mathrm{sinh}{\lambda}_{2}{L}_{2}+{C}_{6}\mathrm{cosh}{\lambda}_{2}{L}_{2}+{C}_{7}\mathrm{sin}{\lambda}_{2}{L}_{2}-{C}_{8}\mathrm{cos}{\lambda}_{2}{L}_{2}=0$$
- 4.
- The vertical displacement at the supported end of the cantilever (rod 2) should be 0 (i.e., ${Y}_{2}\left(0\right)=0$); therefore:$${C}_{5}+{C}_{7}=0$$
- 5.
- The rotation of the pier top (rod 1) should be equal to that of the cantilever (rod 2) at the supported end (i.e., ${\dot{Y}}_{1}\left({L}_{1}\right)={\dot{Y}}_{2}\left(0\right)$); therefore:$${C}_{1}{\lambda}_{1}\mathrm{sinh}{\lambda}_{1}{L}_{1}+{C}_{2}{\lambda}_{1}\mathrm{cosh}{\lambda}_{1}{L}_{1}-{C}_{3}{\lambda}_{1}\mathrm{sin}{\lambda}_{1}{L}_{1}+{C}_{4}{\lambda}_{1}\mathrm{cos}{\lambda}_{1}{L}_{1}-{C}_{6}{\lambda}_{2}-{C}_{8}{\lambda}_{2}=0$$
- 6.
- The horizontal forces on the superstructure should balance each other.

- 7.
- According to the moment balance at the pier top (i.e., ${E}_{1}{I}_{1}{\ddot{Y}}_{1}\left({L}_{1}\right)+{h}_{k}{E}_{1}{I}_{1}{\stackrel{\u20db}{Y}}_{1}\left({L}_{1}\right)=2{E}_{2}{I}_{2}{\ddot{Y}}_{2}\left(0\right)$), then:$$\begin{array}{l}{E}_{1}{I}_{1}{\lambda}_{1}^{2}(\mathrm{cosh}{\lambda}_{1}{L}_{1}+{h}_{k}{\lambda}_{1}\mathrm{sinh}{\lambda}_{1}{L}_{1}){C}_{1}\\ +{E}_{1}{I}_{1}{\lambda}_{1}^{2}(\mathrm{sinh}{\lambda}_{1}{L}_{1}+{h}_{k}{\lambda}_{1}\mathrm{cosh}{\lambda}_{1}{L}_{1}){C}_{2}\\ +{E}_{1}{I}_{1}{\lambda}_{1}^{2}({h}_{k}{\lambda}_{1}\mathrm{sin}{\lambda}_{1}{L}_{1}-\mathrm{cos}{\lambda}_{1}{L}_{1}){C}_{3}\\ -{E}_{1}{I}_{1}{\lambda}_{1}^{2}(\mathrm{sin}{\lambda}_{1}{L}_{1}+{h}_{k}{\lambda}_{1}\mathrm{cos}{\lambda}_{1}{L}_{1}){C}_{4}\\ -2{E}_{2}{I}_{2}{\lambda}_{2}^{2}{C}_{5}+2{E}_{2}{I}_{2}{\lambda}_{2}^{2}{C}_{7}=0\end{array}$$
- 8.
- According to the moment balance at the pier bottom (i.e., ${E}_{1}{I}_{1}{\ddot{Y}}_{1}\left(0\right)\mathrm{sin}\left(\omega t+\theta \right)=-k{\dot{Y}}_{1}\left(0\right)\mathrm{sin}\left(\omega t+\theta \right)$), then:

## 5. Verification by the FEM

^{3}, and the standard compressive strength 20.1 MPa. Meanwhile, the concrete of the upper turntable and of the continuous beam had an elastic modulus of 35,500 MPa, a bulk density of 25,000 N/m

^{3}, and a standard compressive strength of 32.4 MPa. At the center of the upper turntable, the translation motion in the X, Y, and Z directions and the rotation in the X and Z directions were constrained to simulate the constraint conditions of the spherical hinge. At the edge of the upper turntable, the translational motion in the Z direction was constrained to simulate the action of the rotational spring ($k$). A time history analysis was conducted using the FEM. The forced displacement load of the sine function was applied on the position of the upper turntable bracing. The amplitude of the forced displacement load was 1 mm, and the frequency of the load corresponded to the vibration mode frequency of the rotating structure. The parameters of the materials were obtained from the design code of the Chinese railway bridge [27,28].

- 1.
- Unit length mass of the pier

- 2.
- Unit length mass of the girder

- 3.
- Additional point mass (m
_{s})

_{s}was the following:

- 4.
- Rotating stiffness of the upper turntable

- 5.
- Distance between the pier top and the gravity center of the cantilever section

- 1.
- The trends of acceleration of the pier-bottom bending moment calculated through the analytical formula were basically consistent with the results of the finite element analysis. Hence, it is acceptable to calculate the pier-bottom bending moment of the rotating structure based on the acceleration and using the analytical formula.
- 2.
- The ratio of the horizontal acceleration at the pier top (${a}_{h}$) to the pier-bottom bending moment calculated through the analytical formula was more similar to the simulated FEM results than to the ratio of the vertical acceleration at the end cantilever (${a}_{v}$). It can be seen from the analysis that, in order to simplify the calculation, the variable section beam is simplified as an equal section support rod, which makes a certain difference in the stiffness of the main beam between the analytical formula and the finite element. This difference affects the vertical acceleration response of the cantilever end and reduces the calculation accuracy of the analytical formula. As for the horizontal acceleration at the top of the pier, the main factor affecting its response is the pier stiffness rather than the girder stiffness. However, both the analytical formula and the finite element simulation show that the pier is of an equal section, so the ratio of the horizontal acceleration at the top of the pier calculated by the analytical formula to the bending moment at the bottom of the pier is closer to the finite element simulation result.
- 3.
- The horizontal acceleration at the pier top (${a}_{h}$) can be used to calculate the bending moment of the pier bottom and to evaluate the stability of the rotating structure, because the horizontal acceleration is not affected by the symmetric vibration mode of the girder.

## 6. Recommend Formula for Calculating the Allowable Acceleration

## 7. Conclusions

- The vibration of the rotating structure in the facade is mainly influenced by the first three order vibration modes; meanwhile, the pier-bottom section bending moment (which is directly related to the rotating structure stability) is only affected by the first two order asymmetric vibration modes.
- The analytic formula, which considers the cantilever beam and the pier as an infinite-degree-of-freedom rod, can describe the relationship between the vibration acceleration and the pier-bottom bending moment.
- The allowed maximum acceleration at the pier top (a
_{h}) should correspond to the minimum value calculated to ensure safety during the rotation process. - The simplified form of the main beam reduces the accuracy of calculating the vertical acceleration of the cantilever end by the analytical formula, and a more optimized simplified form needs further study. Furthermore, the ratio of the pier-bottom bending moment in the first asymmetric mode to that in the second asymmetric mode is not sufficient for accurate evaluations and need further research.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The special construction process of horizontal rotation construction method. (

**a**) Rotating system construction, (

**b**) Unbalanced weighting, (

**c**) Balance counterweight, (

**d**) Trial rotation, (

**e**) Formal rotation, (

**f**) Monitoring.

**Figure 4.**Vibration modes of the Shang-Yuan bridge rotating structure. (

**a**) First-order vibration mode (First-order asymmetric vibration mode), (

**b**) Second-order vibration mode (First-order symmetric vibration mode), (

**c**) Third-order vibration mode (Second-order asymmetric vibration mode).

**Figure 5.**Vibration displacement time and frequency domain distributions for the Shang-Yuan bridge rotating structure. (

**a**) Displacement time history of vertical vibration, (

**b**) Discrete Fourier spectrum.

**Figure 7.**Acceleration and pier-bottom bending moment ratios of the rotating structure vibrating in the first asymmetric vibration mode. (

**a**) Span: 40 m + 64 m + 40 m, (

**b**) Span: 48 m + 80 m + 48 m, (

**c**) Span: 60 m + 100 m + 60 m, (

**d**) Span: 70 m + 125 m + 70 m.

**Figure 8.**Acceleration and pier-bottom bending moment ratios of the rotating structure vibrating in the second asymmetric vibration mode. (

**a**) Span: 40 m + 64 m + 40 m, (

**b**) Span: 48 m + 80 m + 48 m, (

**c**) Span: 60 m + 100 m + 60 m, (

**d**) Span: 70 m + 125 m + 70 m.

Vibration Mode | Mode Shape | Vibration Frequency (Hz) | Sum of the Modal Participating Mass Ratios (%) | Mode Directional Factor (%) | ||||
---|---|---|---|---|---|---|---|---|

DX | RY | DZ | DX | RY | DZ | |||

1 | Vibration of the main beam along the bridge | 0.7278 | 27.31 | 68.44 | 0.00 | 28.52 | 71.48 | 0.00 |

2 | First-order symmetric bending of the main beam | 2.4994 | 27.31 | 68.44 | 33.32 | 0.00 | 0.00 | 100.00 |

3 | First-order antisymmetric bending of the main beam | 2.9827 | 81.53 | 85.36 | 33.32 | 76.21 | 23.79 | 0.00 |

4 | Second-order antisymmetric bending of the main beam | 8.3363 | 84.00 | 86.70 | 33.32 | 64.84 | 35.08 | 0.08 |

Span Combination (m) | 40 + 64 + 40 | 48 + 80 + 48 | 60 + 100 + 60 | 70 + 125 + 70 | |
---|---|---|---|---|---|

Pier | Cross-section width (cm) | 900 | 960 | 900 | 1000 |

Cross-section height (cm) | 360 | 360 | 400 | 400 | |

Cross-section area (m^{2}) | 29.619 | 31.779 | 32.566 | 36.566 | |

Section inertia moment (m^{4}) | 29.240 | 31.573 | 39.233 | 44.566 | |

Upper turntable | Radius of the brace slideway (m) | 4.0 | 4.5 | 4.5 | 5.0 |

Thickness of the upper turntable (m) | 2.0 | 2.5 | 2.5 | 3.0 | |

Thickness of the down turntable (m) | 8.0 | 9.0 | 9.0 | 10.0 |

Span Combination (m) | 40 + 64 + 40 | 48 + 80 + 48 | 60 + 100 + 60 | 70 + 125 + 70 |
---|---|---|---|---|

${\overline{m}}_{1}$—Unit length mass of the pier (kg/m) | 7.5512 × 10^{+4} | 8.1019 × 10^{+4} | 8.3027 × 10^{+4} | 9.3224 × 10^{+4} |

${E}_{1}$—Elastic modulus of the pier concrete (N/m^{2}) | 3.30 × 10^{+10} | 3.30 × 10^{+10} | 3.30 × 10^{+10} | 3.30 × 10^{+10} |

${I}_{1}$—Moment inertia of the pier (m^{4}) | 29.240 | 31.573 | 39.233 | 44.566 |

${\overline{m}}_{2}$—Equivalent unit length mass of the girder (kg/m) | 3.0127 × 10^{+4} | 3.3027 × 10^{+4} | 4.0094 × 10^{+4} | 4.1386 × 10^{+4} |

${E}_{2}$— Elastic modulus of the main girder (N/m^{2}) | 3.55 × 10^{+10} | 3.55 × 10^{+10} | 3.55 × 10^{+10} | 3.60 × 10^{+10} |

${I}_{2}$—Moment inertia of the girder (m^{4}) | 28.438 | 42.669 | 79.469 | 108.016 |

${L}_{2}$—Cantilever length of the girder (m) | 31.0 | 39.0 | 49.0 | 61.5 |

${m}_{s}$— Additional mass (kg) | 3.7373 × 10^{+5} | 5.7178 × 10^{+5} | 8.8476 × 10^{+5} | 1.5798 × 10^{+6} |

${h}_{0}$—Section height at the cantilever supported end (m) | 6.05 | 6.65 | 7.85 | 9.20 |

${h}_{z}$—Section height at the midpoint of the cantilever (m) | 3.79 | 4.46 | 5.58 | 6.37 |

$k$—Rotating stiffness of the upper turntable (N·m) | 1.4200 × 10^{+11} | 2.7734 × 10^{+11} | 2.7734 × 10^{+11} | 4.7925 × 10^{+11} |

${\mathit{\mu}}_{1}={\mathit{a}}_{\mathit{h}1}/\mathit{M}$ 10 ^{−6}(m/s^{2})/(kN·m) | Span Combination (m) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

40 + 64 + 40 | 48 + 80 + 48 | ||||||||||||

Rotating stiffness of the upper turntable (N·m) | 1.00 × 10^{+11} | 2.00 × 10^{+11} | 5.00 × 10^{+11} | 1.00 × 10^{+11} | 2.00 × 10^{+11} | 5.00 × 10^{+11} | |||||||

Moment inertia of the pier (m^{4}) | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | |

Height of the pier (m) | 14 | 9.33 | 13.55 | 8.90 | 12.76 | 8.50 | 11.54 | 4.39 | 7.07 | 4.16 | 6.56 | 3.95 | 5.85 |

18 | 9.29 | 12.67 | 8.96 | 12.21 | 8.66 | 11.48 | 4.59 | 7.04 | 4.38 | 6.66 | 4.20 | 6.12 | |

22 | 8.97 | 11.53 | 8.74 | 11.28 | 8.53 | 10.89 | 4.66 | 6.79 | 4.48 | 6.52 | 4.34 | 6.14 | |

26 | 8.49 | 10.38 | 8.33 | 10.25 | 8.20 | 10.06 | 4.63 | 6.41 | 4.49 | 6.23 | 4.37 | 5.97 | |

30 | 7.93 | 9.30 | 7.84 | 9.25 | 7.76 | 9.17 | 4.53 | 5.98 | 4.42 | 5.86 | 4.33 | 5.68 | |

34 | 7.35 | 8.34 | 7.30 | 8.33 | 7.26 | 8.32 | 4.38 | 5.54 | 4.30 | 5.47 | 4.23 | 5.37 | |

${\mathit{\mu}}_{1}={\mathit{a}}_{\mathit{h}1}/\mathit{M}$10^{−6}(m/s^{2})/(kN·m) | Span Combination (m) | ||||||||||||

60 + 100 + 60 | 70 + 125 + 70 | ||||||||||||

Rotating stiffness of the upper turntable (N·m) | 1.00 × 10^{+11} | 2.00 × 10^{+11} | 5.00 × 10^{+11} | 1.00 × 10^{+11} | 2.00 × 10^{+11} | 5.00 × 10^{+11} | |||||||

Moment inertia of the pier (m^{4}) | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | |

Height of the pier (m) | 14 | 1.76 | 3.11 | 1.67 | 2.90 | 1.59 | 2.61 | 0.81 | 1.49 | 0.76 | 1.39 | 0.73 | 1.26 |

18 | 1.90 | 3.25 | 1.80 | 3.05 | 1.73 | 2.80 | 0.88 | 1.60 | 0.83 | 1.49 | 0.80 | 1.36 | |

22 | 1.99 | 3.29 | 1.90 | 3.12 | 1.83 | 2.90 | 0.93 | 1.66 | 0.89 | 1.56 | 0.86 | 1.44 | |

26 | 2.06 | 3.26 | 1.97 | 3.12 | 1.91 | 2.94 | 0.98 | 1.70 | 0.94 | 1.61 | 0.91 | 1.50 | |

30 | 2.09 | 3.18 | 2.02 | 3.06 | 1.96 | 2.92 | 1.02 | 1.72 | 0.98 | 1.63 | 0.95 | 1.53 | |

34 | 2.10 | 3.07 | 2.04 | 2.98 | 1.99 | 2.87 | 1.05 | 1.71 | 1.01 | 1.63 | 0.98 | 1.55 |

${\mathit{\mu}}_{2}={\mathit{a}}_{\mathit{h}2}/\mathit{M}$ /10 ^{−6}(m/s^{2})/(kN·m) | Span Combination (m) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

40 + 64 + 40 | 48 + 80 + 48 | ||||||||||||

Rotating stiffness of the upper turntable (N·m) | 1.00 × 10^{+11} | 2.00 × 10^{+11} | 5.00 × 10^{+11} | 1.00 × 10^{+11} | 2.00 × 10^{+11} | 5.00 × 10^{+11} | |||||||

Moment inertia of the pier (m^{4}) | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | |

Height of the pier (m) | 14 | 80.80 | 112.41 | 65.96 | 74.62 | 57.55 | 52.99 | 59.77 | 82.85 | 49.03 | 55.83 | 42.88 | 40.24 |

18 | 56.66 | 89.35 | 47.51 | 60.29 | 42.35 | 43.57 | 42.13 | 64.05 | 35.50 | 43.94 | 31.71 | 32.38 | |

22 | 41.86 | 70.83 | 35.88 | 48.88 | 32.52 | 36.20 | 31.72 | 50.80 | 27.27 | 35.49 | 24.72 | 26.69 | |

26 | 31.80 | 55.11 | 27.81 | 39.15 | 25.59 | 29.92 | 24.90 | 40.78 | 21.74 | 29.05 | 19.95 | 22.31 | |

30 | 24.44 | 41.53 | 21.82 | 30.62 | 20.37 | 24.33 | 20.08 | 32.81 | 17.77 | 23.89 | 16.47 | 18.76 | |

34 | 18.75 | 29.81 | 17.12 | 23.10 | 16.24 | 19.28 | 16.47 | 26.25 | 14.76 | 19.59 | 13.80 | 15.78 | |

${\mathit{\mu}}_{2}={\mathit{a}}_{\mathit{h}2}/\mathit{M}$/10^{−6}(m/s^{2})/(kN·m) | Span Combination (m) | ||||||||||||

60 + 100 + 60 | 70 + 125 + 70 | ||||||||||||

Rotating stiffness of the upper turntable (N·m) | 1.00 × 10^{+11} | 2.00 × 10^{+11} | 5.00 × 10^{+11} | 1.00 × 10^{+11} | 2.00 × 10^{+11} | 5.00 × 10^{+11} | |||||||

Moment inertia of the pier (m^{4}) | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | 25 | 90 | |

Height of the pier (m) | 14 | 41.56 | 63.17 | 34.10 | 42.56 | 29.79 | 30.50 | 30.65 | 47.34 | 25.18 | 31.96 | 22.01 | 22.90 |

18 | 29.24 | 46.37 | 24.69 | 32.07 | 22.06 | 23.76 | 21.56 | 34.16 | 18.25 | 23.79 | 16.32 | 17.73 | |

22 | 22.15 | 35.66 | 19.09 | 25.20 | 17.33 | 19.15 | 16.37 | 25.97 | 14.15 | 18.53 | 12.86 | 14.20 | |

26 | 17.61 | 28.36 | 15.42 | 20.42 | 14.15 | 15.84 | 13.04 | 20.54 | 11.46 | 14.95 | 10.54 | 11.71 | |

30 | 14.46 | 23.09 | 12.82 | 16.91 | 11.87 | 13.35 | 10.76 | 16.74 | 9.57 | 12.39 | 8.89 | 9.88 | |

34 | 12.15 | 19.08 | 10.88 | 14.21 | 10.16 | 11.41 | 9.09 | 13.95 | 8.17 | 10.48 | 7.64 | 8.49 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, W.; Liang, K.; Chen, Y.
Relationship between the Vibration Acceleration and Stability of a Continuous Girder Bridge during Horizontal Rotation. *Sustainability* **2022**, *14*, 5853.
https://doi.org/10.3390/su14105853

**AMA Style**

Zhang W, Liang K, Chen Y.
Relationship between the Vibration Acceleration and Stability of a Continuous Girder Bridge during Horizontal Rotation. *Sustainability*. 2022; 14(10):5853.
https://doi.org/10.3390/su14105853

**Chicago/Turabian Style**

Zhang, Wenxue, Kun Liang, and Ying Chen.
2022. "Relationship between the Vibration Acceleration and Stability of a Continuous Girder Bridge during Horizontal Rotation" *Sustainability* 14, no. 10: 5853.
https://doi.org/10.3390/su14105853