Author Contributions
Conceptualization, T.Z.; Software, X.C. and X.Q.; Validation, X.C. and X.Q.; Formal analysis, W.G.; Resources, W.G.; Writing—original draft preparation, T.Z.; Writing—review and editing, Y.Z. All authors have read and agreed to the published version of the manuscript.
Figure 1.
Three-component force on the girder section in the body axis coordinate system.
Figure 1.
Three-component force on the girder section in the body axis coordinate system.
Figure 2.
Aerodynamic force coefficient of the main girder of Xiangshan Harbor Bridge in body axis coordinate system.
Figure 2.
Aerodynamic force coefficient of the main girder of Xiangshan Harbor Bridge in body axis coordinate system.
Figure 3.
Calculation flow chart of checkpoint method for reliability analysis.
Figure 3.
Calculation flow chart of checkpoint method for reliability analysis.
Figure 4.
Layout of main bridge (m).
Figure 4.
Layout of main bridge (m).
Figure 5.
Cross-section layout of steel box girder (mm).
Figure 5.
Cross-section layout of steel box girder (mm).
Figure 6.
Finite element model of Xiangshan Harbor Bridge.
Figure 6.
Finite element model of Xiangshan Harbor Bridge.
Figure 7.
Flowchart of the static wind instability solution program.
Figure 7.
Flowchart of the static wind instability solution program.
Figure 8.
ANSYS solution process diagram. (a) User Toolbar; (b) Parameter input dialog box.
Figure 8.
ANSYS solution process diagram. (a) User Toolbar; (b) Parameter input dialog box.
Figure 9.
Variation curves of the maximum displacement of the main beam with wind speed. (a) Transverse displacement; (b) Vertical displacement; (c) Torsion angle.
Figure 9.
Variation curves of the maximum displacement of the main beam with wind speed. (a) Transverse displacement; (b) Vertical displacement; (c) Torsion angle.
Figure 10.
Maximum displacement curve of the main beam with wind speed under different initial wind attack angles; (a) Lateral displacement (b) Vertical displacement; (c) Torsion angle.
Figure 10.
Maximum displacement curve of the main beam with wind speed under different initial wind attack angles; (a) Lateral displacement (b) Vertical displacement; (c) Torsion angle.
Figure 11.
Variation curve of critical instability wind speed with the initial wind angle of attack.
Figure 11.
Variation curve of critical instability wind speed with the initial wind angle of attack.
Figure 12.
Maximum displacement curve of main beam with wind speed under different wire breakage rates; (a) Lateral displacement (b) Vertical displacement; (c) Torsion angle.
Figure 12.
Maximum displacement curve of main beam with wind speed under different wire breakage rates; (a) Lateral displacement (b) Vertical displacement; (c) Torsion angle.
Figure 13.
Critical aerostatic instability wind speed with the steel wire breakage rate.
Figure 13.
Critical aerostatic instability wind speed with the steel wire breakage rate.
Figure 14.
Maximum displacement curve of main beam with wind speed under different three-component force coefficients; (a) Lateral displacement; (b) Vertical displacement; (c) Torsion angle.
Figure 14.
Maximum displacement curve of main beam with wind speed under different three-component force coefficients; (a) Lateral displacement; (b) Vertical displacement; (c) Torsion angle.
Figure 15.
Stay cable broken position (m).
Figure 15.
Stay cable broken position (m).
Figure 16.
Maximum displacement curve of main beam with wind speed under different broken positions of the stay cable; (a) Lateral displacement; (b) Vertical displacement; (c) Torsion angle.
Figure 16.
Maximum displacement curve of main beam with wind speed under different broken positions of the stay cable; (a) Lateral displacement; (b) Vertical displacement; (c) Torsion angle.
Table 1.
Modal analysis results of Xiangshan Harbor Bridge.
Table 1.
Modal analysis results of Xiangshan Harbor Bridge.
Number | Calculated Frequency/Hz | Measured Frequency/Hz | Error/% | Mode Shape |
---|
1 | 0.129 | —— | —— | Longitudinal drift of main beam |
2 | 0.206 | 0.213 | 2.82 | First-order symmetrical lateral bending of main beam |
3 | 0.249 | 0.250 | 0.40 | First-order symmetrical vertical bending of main beam |
4 | 0.316 | 0.300 | 5.33 | First-order antisymmetric vertical bending of main beam |
5 | 0.453 | 0.438 | 3.20 | First-order antisymmetric lateral bending of bridge tower |
6 | 0.457 | —— | —— | First-order symmetrical lateral bending of bridge tower |
7 | 0.482 | —— | —— | Second-order symmetrical vertical bending of main beam |
8 | 0.570 | 0.525 | 9.52 | First-order antisymmetric lateral bending of main beam |
9 | 0.578 | 0.600 | 3.17 | Second-order antisymmetric vertical bending of main beam |
10 | 0.628 | —— | —— | Third-order symmetrical vertical bending of main beam |
Table 2.
Static instability wind speeds under different three-component force coefficients.
Table 2.
Static instability wind speeds under different three-component force coefficients.
Item | Normal | 1.2 Times Moment Coefficient | 1.2 Times Lift Coefficient | 1.2 Times Drag Coefficient |
---|
Static wind instability speed (m/s) | 298 | 275 | 295 | 285 |
Table 3.
Aerostatic instability wind speeds under different broken positions of the stay cable.
Table 3.
Aerostatic instability wind speeds under different broken positions of the stay cable.
Cable Broken Position | Normal | Condition 1 | Condition 2 | Condition 3 |
---|
Static wind instability speed (m/s) | 298 | 260 | 290 | 295 |
Table 4.
Parameter distribution and statistical characteristic values.
Table 4.
Parameter distribution and statistical characteristic values.
Random Parameters | Distribution Type | Average Value | Variation Coefficient |
---|
Moment coefficient | Normal distribution | 1.00 | 0.15 |
Cross-sectional area of stay cable | Normal distribution | 1.00 | 0.05 |
Stay cable broken position | Normal distribution | 1.00 | 0.50 |
Table 5.
Results of LHS for 30 times for different parameters.
Table 5.
Results of LHS for 30 times for different parameters.
Sample No. | Moment Coefficient | Steel Wire Breakage Rate | Stay Cable Broken Position |
---|
1 | 0.920728 | 0.949115 | 1.082463 |
2 | 0.974271 | 1.041689 | 0.981851 |
3 | 0.771126 | 1.061711 | 0.746454 |
4 | 1.287531 | 1.095844 | 0.908971 |
5 | 1.254599 | 0.966521 | 2.168303 |
6 | 0.961617 | 1.005954 | 0.929463 |
7 | 0.934908 | 0.999166 | 1.598472 |
8 | 1.079174 | 0.979184 | 1.478054 |
9 | 0.997497 | 0.973576 | 1.338008 |
10 | 0.980966 | 1.024584 | 0.281436 |
11 | 1.017861 | 0.959554 | 1.110001 |
12 | 1.048353 | 1.000593 | 0.204652 |
13 | 0.847346 | 1.067997 | 1.128450 |
14 | 1.106737 | 0.987206 | 1.405852 |
15 | 0.856827 | 0.993655 | 1.243036 |
16 | 1.036913 | 1.021351 | 0.374567 |
17 | 1.001778 | 1.084866 | 0.545285 |
18 | 1.125068 | 0.923709 | 0.866323 |
19 | 1.064053 | 0.991424 | 0.596021 |
20 | 0.779631 | 1.046271 | 1.492150 |
21 | 1.161102 | 0.943786 | 0.514781 |
22 | 1.185132 | 1.053701 | 1.733024 |
23 | 0.831357 | 1.012304 | 1.883946 |
24 | 1.138812 | 0.903784 | 0.659194 |
25 | 1.203990 | 1.026391 | 1.181329 |
26 | 0.878663 | 1.035579 | -0.089507 |
27 | 1.073751 | 0.959554 | 0.820229 |
28 | 0.711353 | 0.952276 | 1.285940 |
29 | 0.899563 | 0.926544 | 1.011250 |
30 | 0.937552 | 1.016118 | 1.046224 |
Table 6.
Aerostatic instability wind speeds corresponding to 30 samples for different parameters (m/s).
Table 6.
Aerostatic instability wind speeds corresponding to 30 samples for different parameters (m/s).
Sample No. | Moment Coefficient | Steel Wire Breakage Rate | Stay Cable Broken Position |
---|
1 | 304 | 257 | 278 |
2 | 301 | 263 | 260 |
3 | 316 | 265 | 282 |
4 | 277 | 267 | 280 |
5 | 282 | 259 | 298 |
6 | 301 | 261 | 269 |
7 | 303 | 261 | 290 |
8 | 292 | 260 | 290 |
9 | 298 | 259 | 286 |
10 | 299 | 262 | 295 |
11 | 297 | 259 | 279 |
12 | 295 | 261 | 293 |
13 | 309 | 265 | 280 |
14 | 291 | 260 | 290 |
15 | 308 | 260 | 281 |
16 | 296 | 262 | 290 |
17 | 298 | 266 | 290 |
18 | 289 | 255 | 280 |
19 | 294 | 260 | 287 |
20 | 313 | 264 | 290 |
21 | 287 | 257 | 290 |
22 | 286 | 264 | 292 |
23 | 311 | 262 | 293 |
24 | 288 | 254 | 285 |
25 | 282 | 263 | 281 |
26 | 307 | 263 | 298 |
27 | 293 | 258 | 281 |
28 | 318 | 258 | 283 |
29 | 306 | 256 | 260 |
30 | 303 | 262 | 264 |
Table 7.
Statistic characteristic values of wind speed for static wind instability.
Table 7.
Statistic characteristic values of wind speed for static wind instability.
Static Wind Instability Speed (m/s) | Moment Coefficient | Steel Wire Breakage Rate | Stay Cable Broken Position |
---|
Mean value | 298 | 260 | 284 |
Standard deviation | 10.18 | 3.16 | 9.67 |
Table 8.
Calculation results of static wind reliability.
Table 8.
Calculation results of static wind reliability.
Item | Moment Coefficient | Steel Wire Breakage Rate | Stay Cable Broken Position |
---|
Reliability index | 6.3375 | 5.9757 | 6.1908 |
| 0.7500 | 0.7753 | 0.7613 |
| 291.5469 | 260.4271 | 277.5833 |
| 1.6144 | 1.6003 | 1.6083 |
| 135.4469 | 126.1683 | 131.4031 |
Table 9.
Static wind instability probability corresponding to different steel wire breakage rates.
Table 9.
Static wind instability probability corresponding to different steel wire breakage rates.
Steel Wire Breakage Rate | 0% | 5% | 10% | 15% | 20% |
---|
Reliability index | 6.3645 | 6.3344 | 6.2836 | 6.1050 | 5.9757 |
Failure probability | 9.795 × 10−11 | 1.199 × 10−10 | 1.654 × 10−10 | 5.133 × 10−10 | 1.145 × 10−9 |
Table 10.
Static wind instability probability corresponding to different broken cable positions.
Table 10.
Static wind instability probability corresponding to different broken cable positions.
Stay Cable Broken Position | Random Location | Near to Bridge Tower | 1/4 of the Main Span | 1/2 of the Main Span |
---|
Reliability index | 6.1908 | 6.3063 | 6.2559 | 5.9371 |
Failure probability | 2.993 × 10−10 | 1.429 × 10−10 | 1.976 × 10−10 | 1.450 × 10−9 |