Evaluation of the Cable Force by Frequency Method for the Hybrid Boundary between the Ear Plate and the Anchor Plate
Abstract
:1. Introduction
2. Theoretical Aspects on Cable Forces
2.1. Test Principle of Frequency Method
2.2. Measurement Method of the Intrinsic Frequency of the Cable
2.2.1. Acquisition of the Intrinsic Frequency of the Cable
2.2.2. The Measurement of the Intrinsic Frequency of the Cable Should Pay Attention to the Problem
- (1)
- Filter Frequency: The filtering frequency is selected based on the maximum frequency of interest of the measured cables. Generally, the self-oscillation frequency is calculated based on the ultimate cable force of each cable, and a value five-times its maximum is taken as the filtering frequency;
- (2)
- Sampling frequency: To restore the original signal without distortion, the sampling frequency is at least two-times higher than the filtering frequency;
- (3)
- Sampling time: Due to the simple structure of the cable, the vibration signal is a strong signal; therefore, the sampling time does not need to be very long, and can be determined by comparing the measured frequencies obtained by different adoption times;
- (4)
- Refinement: To improve the resolution of the frequency, the refined FFT technique is used.
2.3. Exploring the Relationship between Cable Force and Frequency
3. Investigated Cables and Work Method
3.1. Engineering Background
3.2. Work Method
3.2.1. The Measurement Method of the Intrinsic Frequency of the Cable of This Project
(1) The Hardware Equipment Used in This Project to Measure the Intrinsic Frequency of the Cable
(2) Practical Selection of Equipment Parameters
3.2.2. Research on the Relationship between Cble Force and Frequency under the Hbrid Boundary between the Ear Plate and the Anchor Plate
(1) Sensitivity Analysis of Cable Parameters
- Boundary Conditions
- 2.
- Inclination Angle
- 3.
- Bending Stiffness
- 4.
- Linear Density
(2) Process for Determining the Bending Stiffness and Boundary Conditions of Cables
- (1)
- The preliminary boundary conditions were set. The upper end of the cable was connected to the upper ear plate of the arch through the fork ear, which is hinged in the longitudinal direction and solid in the transverse direction [35]. In the case of the upper ear plate connection of the cable of the curved beam and skewed arch and the installation position of the on-site sensor, the upper end is hinged, and the lower end is connected to the anchor plate of the beam through the integral anchor, which can be solid;
- (2)
- The intrinsic frequency when the cable was tensioned to 100% of the design value was selected. An acceleration sensor was used to collect the time domain signal of the response of a point on the cable under natural excitation. Then, the frequency domain spectrum was obtained by performing a frequency domain analysis of the time domain signal. Next, the frequency domain spectrum was used to select some order of the intrinsic frequency;
- (3)
- The relationship between cable force and frequency was determined for the hinged upper end and solid lower end boundary conditions. A combination of the finite element method and spline fitting technique was used to calculate the cable force versus frequency for a given cable force range;
- (4)
- The bending stiffness of the cable was determined under the boundary condition of the hinged support at the upper end and solid support at the lower end. The cable bending stiffness was selected as 0.02EImax–0.2EImax, with a step increment of 0.02EImax. By using the combination of the finite element method and spline fitting technique, the relationship between frequency and cable bending stiffness was determined by studying the cable bending stiffness values under the hinged support at the upper end and solid support at the lower end boundary conditions. By using the principle of difference, the calculated cable force value under some order of frequency was obtained. In addition, the bending stiffness of the cable was determined based on the uniqueness of the cable force;
- (5)
- The boundary conditions were adjusted. If the difference between the average value of the calculated cable force at some order of inherent frequency and the value of the hydraulic jack tension at the site of the cable pulling device was large and the dispersion of the calculated cable force at some order of inherent frequency was greater than 15, the boundary condition of the cable was changed to the solid support at both ends;
- (6)
- The relationship between cable force and frequency was determined under solid support boundary conditions at both ends. The aforementioned method was used to determine the effect of frequency on the solid cable force at both ends for a given range of the cable force;
- (7)
- The bending stiffness of the cable was determined under solid support boundary conditions at both ends. The same identification method was used to determine the bending stiffness of the cable under the boundary condition of the hinged support at the upper end and the solid support at the lower end, and the actual bending stiffness of the cable in this state was obtained;
- (8)
- The actual boundary conditions and bending stiffness of the cable were obtained by comparing and analyzing the hinged support at the upper end and the solid support at the lower end boundary conditions and the calculated cable force values under the boundary conditions of the solid support at both ends and the hydraulic jack tension values of the field cable pulling device.
4. Results and Discussions
4.1. Determination of the Bending Stiffness of the Cable under the Hinged Support at the Upper End and the Solid Support at the Lower End of the Boundary
4.1.1. Relationship between Cable Force and Frequency
4.1.2. Determining the Bending Stiffness
4.2. Determination of the Bending Stiffness of the Cable under the Solid Support Boundary at Both Ends
4.2.1. Relationship between Cable Force and Frequency
4.2.2. Determination of Bending Stiffness
4.3. Boundary Conditions and Bending Stiffness Results of the Cable
5. Conclusions
- (1)
- Boundary conditions, bending stiffness, and linear density affect the cable force considerably, whereas the inclination angle has little influence on the cable force;
- (2)
- During the measurement of the cable force of this bridge, the results of the bending stiffness of the cable in the tensioning process was 0.12EImax under the boundary conditions of the hinged support at the upper end, the solid support at the lower end, and the solid support at both ends;
- (3)
- The actual bending stiffness of the cable under the boundary conditions of the two ends of the solid support which calculated the cable force value and the actual tension value of the hydraulic jack of the site cable pulling device showed good agreement; the maximum relative error was within 3.5%;
- (4)
- When the cable belongs to the ultra-short cable (within 5 m), the influence of the bending stiffness and boundary conditions of the cable on the cable force test will become larger, and the values of the bending stiffness and boundary conditions identified in this paper may not be applicable; however, the method of identifying the bending stiffness and boundary conditions proposed in this paper is still available;
- (5)
- When the cable belongs to the extra-long cable, the bending stiffness and boundary conditions of the cable have less influence on the cable force test, and the values of the bending stiffness and boundary conditions have less influence on the cable force test results; moreover, even without considering the influence of the bending stiffness and boundary conditions, the results of the cable force test do not differ much.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Parameters | 12th Cable | 19th Cable | 21st Cable | 23rd Cable |
---|---|---|---|---|
Length (m) | 39.037 | 29.368 | 22.142 | 13.188 |
Linear density (kg/m) | 25.684 | 25.684 | 25.684 | 25.684 |
Inclination angle (°) | 47.324 | 49.090 | 50.64 | 53.030 |
Diameter of the full cross-section of the cable (m) | 0.108 | 0.108 | 0.108 | 0.108 |
Number of steel strands | 19 | 19 | 19 | 19 |
Maximum bending stiffness (kN·m2) | 1301.6 | 1301.6 | 1301.6 | 1301.6 |
Subsystem Number | Subsystem Name | Performance Indicators |
---|---|---|
1 | Acceleration Sensor |
|
2 | Collector |
|
3 | Transmitter |
|
4 | Power supply |
|
Subsystem Number | Subsystem Name | Actual Setting Value |
---|---|---|
1 | Filter frequency | Low-pass filtering, filtering frequency of 50 Hz |
2 | Sampling frequency | 128 Hz |
3 | Sampling time | 1~2 min |
4 | Refinement | FFT technique |
Cable Number | Calculated Cable Force for Solid Support at Both Ends (kN) | Calculated Cable Force for Hinged Support at Both Ends (kN) | Relative Error (%) |
---|---|---|---|
12 | 383 | 411 | 7.31 |
19 | 845 | 898 | 6.27 |
21 | 670 | 735 | 9.70 |
23 | 469 | 571 | 21.75 |
Cable Number | Calculated Cable Force at 0° (kN) | Calculated Cable Force at 90° (kN) | Relative Error (%) |
---|---|---|---|
12 | 380 | 385 | 1.3 |
19 | 842 | 846 | 0.4 |
21 | 667 | 670 | 0.4 |
23 | 467 | 469 | 0.4 |
Cable Number | Calculated Cable Force at Zero Bending stiffness (kN) | Calculated Cable Force at Maximum Bending Stiffness (kN) | Relative Error (%) |
---|---|---|---|
12 | 429 | 345 | 19.58 |
19 | 915 | 750 | 18.03 |
21 | 768 | 567 | 26.17 |
23 | 612 | 227 | 62.91 |
Cable Number | Calculated Cable Force at Design Linear Density (kN) | Deviation from Design Linear Density +5% | Deviation from Design Linear Density +10% | ||
---|---|---|---|---|---|
Calculate Cable Force (kN) | Relative Error (%) | Calculate Cable Force (kN) | Relative Error (%) | ||
12 | 383 | 404 | 5.5 | 425 | 11.0 |
19 | 845 | 890 | 5.3 | 934 | 10.5 |
21 | 670 | 706 | 5.4 | 744 | 11.0 |
23 | 469 | 497 | 6.0 | 526 | 12.2 |
Cable Number | Certain Orders of Measured Frequency (Hz) | |||||
---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | |
12 | — | — | 4.901 | 6.561 | 8.344 | 10.125 |
19 | 3.182 | 6.375 | 9.625 | 12.938 | 16.313 | — |
21 | 3.813 | 7.688 | 11.688 | 15.813 | 20.123 | — |
23 | 5.668 | 11.688 | 18.125 | — | — | — |
Status | Cable Force Value (kN) | |||
---|---|---|---|---|
12th Cable | 19th Cable | 21st Cable | 23rd Cable | |
The pulling force value of the hydraulic jack | 389 | 842 | 667 | 453 |
Hinge support at the upper end and solid support at the lower end | 397 | 872 | 703 | 522 |
Relative error (%) | 2.1 | 3.6 | 5.4 | 15.2 |
Status | Cable Force Value (kN) | |||
---|---|---|---|---|
12th Cable | 19th Cable | 21st Cable | 23rd Cable | |
The pulling force value of the hydraulic jack | 389 | 842 | 667 | 453 |
Solid support at both ends | 383 | 845 | 670 | 469 |
Relative error (%) | 1.5 | 0.4 | 0.4 | 3.5 |
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Xu, Y.; Xie, Y.; Chen, S.; Zhu, M. Evaluation of the Cable Force by Frequency Method for the Hybrid Boundary between the Ear Plate and the Anchor Plate. Buildings 2022, 12, 1853. https://doi.org/10.3390/buildings12111853
Xu Y, Xie Y, Chen S, Zhu M. Evaluation of the Cable Force by Frequency Method for the Hybrid Boundary between the Ear Plate and the Anchor Plate. Buildings. 2022; 12(11):1853. https://doi.org/10.3390/buildings12111853
Chicago/Turabian StyleXu, Yufeng, Yunfei Xie, Si Chen, and Mengyang Zhu. 2022. "Evaluation of the Cable Force by Frequency Method for the Hybrid Boundary between the Ear Plate and the Anchor Plate" Buildings 12, no. 11: 1853. https://doi.org/10.3390/buildings12111853