# Form-Finding Analysis of the Rail Cable Shifting System of Long-Span Suspension Bridges

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mechanical Characteristics of the RCS System under the Non-Loading Condition

#### 2.1. Brief Introduction to RCS Technology

#### 2.2. Mechanical Model of the RCS System under Non-Loading Condition

- (1)
- The flexible cable can only be tensed but not be pressed or bent.
- (2)
- The stress-strain of the flexible cable is consistent with Hooke’s theorem.
- (3)
- The cross-sectional area before deformation is used for the calculation of the tensile stiffness of the main cable, rail cable, and sling before and after stress.
- (4)
- The friction between the rail cable and the saddle is ignored, and the sling does not tilt after the rail cable tension is completed.

#### 2.2.1. Force Analysis of the Main Cable

_{zi}-A

_{zi}

_{+1}and A

_{zi}

_{+1}-A

_{zi}

_{+2}under the non-loading condition. In addition, the corresponding segmental catenary element equations [34] can be established and solved to obtain l

_{zi}, h

_{zi}, l

_{zi+1}, h

_{zi+1}, S

_{zi}:

_{1}and A

_{1}are elastic modulus and cross-sectional area of the main cable, respectively; q

_{1}is the self weight per unit length of the main cable; S

_{zi}is the segmental cable length after the sling installed to the unloaded cable; and i = 1~n. Other parameters are shown in Figure 2.

_{zi}node and the sling, two additional equations can be established

_{zi}node.

#### 2.2.2. Force Analysis of the Rail Cable

_{2}and A

_{2}are respectively the elastic modulus and cross-sectional area of the rail cable; q

_{2}is the self-weight per unit length of the rail cable; S

_{gi}is the cable length of the rail cable segment; P

_{gi}is the sling force increment; and i = 1~m. Other parameters are shown in Figure 4.

#### 2.2.3. Force Analysis of the Sling

_{3}and A

_{3}are the elastic modulus and cross-sectional area of the sling, respectively; q

_{3}is the self-weight per unit length of the sling; and S

_{di}is the unstressed cable length of the sling. Other parameters are shown in Figure 2.

#### 2.3. Solution Analysis of the System

_{g}of the two anchorage points, the pretension force T

_{g}

_{0}and the unstressed cable length of the sling are known. After the installation of the sling, the cable force, line shape, cable tower distance l

_{z}, and the height difference h

_{z}can be computed according to actual conditions. On this basis, other unknown variables can be solved by an iterative method. The iterative scheme is shown below:

- (1)
- Assume the horizontal force H
_{g}_{0}and the elevation h_{g}_{0}at the left support of the rail cable; the horizontal force H_{z}_{0}at the left support of the main cable, and the increment of the cable force of the sling P_{g}_{1}; - (2)
- Obtain H
_{gi}, H_{zi}from Equations (4) and (11); - (3)
- Substitute them into Equations (10), (13), (7), (8), and (9); then V
_{g}_{0}, V_{g}_{1}, l_{g}_{1}, h_{g}_{1}, and T_{g}_{1}are obtained; - (4)
- From the coordination relation of deformation, Y
_{g}_{1}, Y_{z}_{1}and h_{z}_{1}are obtained by combing with Equations (15), (17), and (4), respectively; - (5)
- Substitute Y
_{g}_{1}, Y_{z}_{1}and h_{z}_{1}into Equations (1) and (2); V_{z}_{0}and l_{z}_{1}can then be obtained; - (6)
- Substitute V
_{z}_{0}and l_{z}_{1}into Equations (16), (6), and (13), P_{z}_{1}; V_{z}_{1}and V_{g}_{2}can then be obtained; - (7)
- Substitute P
_{z}_{1}; V_{z}_{1}and V_{g}_{2}the results of step 6 into Equations (2), (1), (10), (7), and (8); h_{z}_{2}, l_{z}_{2}, T_{g}_{2}, l_{g}_{2}, and h_{g}_{2}can be acquired, respectively; - (8)
- Substitute them into Equations (4), (15), and (17); Y
_{z}_{2}, Y_{g}_{2}, and P_{g}_{2}are then obtained; - (9)
- Following the similar iterative manner, all values of l
_{gi}, l_{zi}, h_{gi}and h_{zi}are obtained; - (10)
- Check convergence conditions Δ
_{z}_{1}= $\left|{\sum}_{i=1}^{n}{l}_{zi}\right|-{l}_{z}\le \epsilon $ ($\epsilon $ = 1.0 × 10^{−6}is the given error limit), Δ_{z}_{2}= $\left|{\sum}_{i=1}^{n}{h}_{zi}\right|-{h}_{z}\le \epsilon $, Δ_{g}_{1}= $\left|{\sum}_{i=1}^{m}{l}_{gi}\right|-{l}_{g}\le \epsilon $, and Δ_{g}_{2}= $\left|{\sum}_{i=1}^{m}{h}_{gi}\right|-{h}_{g0}\le \epsilon $. If they are satisfied, go to the next step. If not, let H_{z}_{0}= H_{z}_{0}+ Δ_{z}_{1}/DH_{z}, P_{g}_{1}= P_{g}_{1}+ Δ_{z}_{2}/DP_{g}, H_{g}_{0}= H_{g}_{0}+ Δ_{g}_{1}/DH_{g}and h_{g}_{0}= h_{g}_{0}+ Δ_{g}_{2}/Dh_{g}, and return to step 1, iteratively recalculating, where DH_{z}, DP_{g}, DH_{g}, and Dh_{g}are respectively the first derivatives of H_{z}_{0}, P_{g}_{1}, H_{g}_{0}, and h_{g}_{0}[38].

## 3. Global Model Test of the Moving Girder by Rail Cable

## 4. Testing Results of Model Test

## 5. Conclusions

- (1)
- The global mechanical analytical model for the main cable, slings, and rail cables of the RCS system under the non-loading condition is established through the theoretical derivation. The equilibrium equation established from the position after structural deformation can be used to perform the geometrical nonlinear analysis of the structural large deformation. The simplified calculation and analysis process of the RCS system achieve sufficient accuracy, which thus can be applied in engineering.
- (2)
- Under the condition that the initial geometric state of the RCS system is unknown, the true internal force and geometry of the rail cable can be iteratively computed by assuming the initial force or configuration of the rail cable.
- (3)
- The designed rail cable tensioning and testing device meets the requirements of the model test. The measured results of the main cable line shape and main cable force in the model test are in good agreement with the theoretical results.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Glossary

A_{1} | cross-sectional area of the main cable |

A_{2} | cross-sectional area of the rail cable |

A_{3} | cross-sectional area of the sling |

A_{zi} | node locations of main cable segment |

E_{1} | elastic modulus of the main cable |

E_{2} | elastic modulus of the rail cable |

E_{3} | elastic modulus of the sling |

H_{g}_{0} | horizontal force pretension force of the rail cable at the left support |

h_{g}_{0} | elevation at the left support of the rail cable |

h_{g}_{1} | vertical height between left anchorage points of rail cable and node A_{g}_{2} |

h_{i} | height difference of cable tower |

H_{z}_{0} | horizontal force at the left support of the main cable |

h_{z}_{1} | height difference between the top of left cable tower and node A_{z}_{2} |

h_{zi} | vertical length of main cable segment |

l_{g} | longitudinal position of the two anchorage points |

l_{g}_{1} | horizontal length between left anchorage points of rail cable and node A_{g}_{2} |

l_{z} | cable tower distance |

l_{zi} | horizontal length of main cable segment |

P_{g}_{1} | increment of the cable force of the sling |

P_{gi} | sling force increment |

P_{zi} | force of the sling at A_{zi} node |

q_{1} | self-weight per unit length of the main cable |

q_{2} | self-weight per unit length of the rail cable |

q_{3} | self-weight per unit length of the sling |

S_{di} | unstressed cable length of the sling |

S_{gi} | cable length of the rail cable segment |

S_{zi} | segmental cable length after sling installed to unloaded cable |

T_{g}_{0} | pretension force of the rail cable at the left support |

T_{g}_{1} | vertical force at the left support of the rail cable |

V_{g0} | vertical force pretension force of the rail cable at the left support |

V_{g}_{1} | vertical force between left anchorage points of rail cable and node A_{g}_{2} |

W | constant weight of the saddle |

Y_{g}_{1} | vertical displacement between left anchorage points of rail cable and node A_{g}_{2} |

Y_{z}_{1} | vertical displacement between the top of left cable tower and node A_{z}_{2} |

Δ_{g}_{1} | convergence condition |

Δ_{g}_{2} | convergence condition |

Δ_{z}_{1} | convergence condition |

Δ_{z}_{2} | convergence condition |

ε | given error limit |

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**Figure 2.**Mechanical analytical model of the rail cable shifting (RCS) system under the non-loading condition.

Items | Main Cable Displacement at Mid-Span | Main Cable Displacement at 1/4 Span on the Jishou Side | Main Cable Displacement at 1/4 Span on the Chadong Side | |||
---|---|---|---|---|---|---|

X Direction/mm | Y Direction/mm | X Direction/mm | Y Direction/mm | X Direction/mm | Y Direction/mm | |

Finite element solution | 1.4 | 28.6 | −0.6 | 9.0 | 3.6 | 7.7 |

Model test measured value | 1.5 | 27.5 | −0.5 | 8.9 | 3.5 | 7.5 |

Analytical calculation results | 1.6 | 26.3 | −0.6 | 8.7 | 3.7 | 7.2 |

Deviation of the finite element solution from the measured value (%) | 6.67 | 4.00 | 20.00 | 1.12 | 2.86 | 2.67 |

Deviation of the analytical method from the measured value (%) | 6.67 | 4.36 | 20.00 | 2.25 | 5.71 | 4.00 |

Main Cable Force at the Anchor Point | Finite Element Solution/kN | Model Test Measured Value/kN | Analytical Results/kN | Deviation of the Finite Element Solution from the Measured Value/% | Deviation of the Analytical Method from the Measured Value/% |
---|---|---|---|---|---|

Jishou side | 61.7 | 59.9 | 61.3 | 3.01 | 2.34 |

Chadong side | 58.5 | 57.1 | 58.2 | 2.45 | 1.93 |

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## Share and Cite

**MDPI and ACS Style**

Pan, Q.; Yan, D.; Yi, Z. Form-Finding Analysis of the Rail Cable Shifting System of Long-Span Suspension Bridges. *Appl. Sci.* **2018**, *8*, 2033.
https://doi.org/10.3390/app8112033

**AMA Style**

Pan Q, Yan D, Yi Z. Form-Finding Analysis of the Rail Cable Shifting System of Long-Span Suspension Bridges. *Applied Sciences*. 2018; 8(11):2033.
https://doi.org/10.3390/app8112033

**Chicago/Turabian Style**

Pan, Quan, Donghuang Yan, and Zhuangpeng Yi. 2018. "Form-Finding Analysis of the Rail Cable Shifting System of Long-Span Suspension Bridges" *Applied Sciences* 8, no. 11: 2033.
https://doi.org/10.3390/app8112033