Evaluating the Reactions of Bridge Foundations to Combined Wave–Flow Dynamics
Abstract
:1. Introduction
2. Stokes’s Wave Theory and Its Wave–Flow Coupling Action Theory
2.1. Control Equations and Boundary Conditions for Fluid Motion
2.2. Stokes’s Wave Theory
2.3. Theory of Wave–Current Coupling Action
- (1)
- The surface of the pile column is smooth.
- (2)
- The hydrodynamic coefficient is constant along the water depth when the wave and current coexist.
- (3)
- In the wave flow field, Stokes’s wave theory is used to calculate the velocity and acceleration at the water quality point.
- (4)
- When the wave and flow coexist, the velocity and acceleration of the water quality point for the wave and the water flow are generated by the respective velocity and acceleration vectors.
3. Numerical Modeling of Bridge Foundations under Wave–Flow Coupling
3.1. Sink Simulation of Wave Flow
3.2. Numerical Simulation of Wave Flow under Different Wave Flow Parameters
3.3. Bridge Foundation Grid Size
- (1)
- Discussion of the transitional water range on both sides of the perpendicular to the direction of wave propagation (Y-direction) mesh delineation:
- (2)
- Discussion of the transitional water range on both sides of the wave propagation direction (X-direction) grid division:
3.4. Simulation Analysis Model
4. Dynamic Response of a Bridge Foundation under Wave–Current Coupling
5. Conclusions
- When the length of the water in the direction of wave propagation reaches four times the wavelength (4 L), the width of the water body on both sides of the pier is taken as more than 14 times its diameter, and the dynamic response of the pier column fluctuates less in the highest value and basically tends to be stable. Considering calculation accuracy and efficiency and the numerical analysis of the pier–wave coupling, the wave propagation direction of the water length is taken to be four times the wavelength (4 L), and the width of the water on both sides of the pier column is taken to be 14 times its diameter.
- The displacement responses in Case 2 and Case 3 basically overlap, while the displacement response in Case 1 is significantly larger, and its vertical displacement maximum value is 22.2% and 22.4% larger than the values for Case 2 and Case 3, respectively. The maximum values of the transverse displacement in Cases 1, 2 and 3 are 13.7%, 7.4% and 39.4% higher than those in cases 4, 5 and 6, respectively, while the displacement responses of Cases 2 and 5 are basically the same when the wave is the main load, which indicates that the wave has an obvious influence on the transverse displacement of the main beam at this location.
- In the combined action of a uniform flow and waves with different flow velocities, the maximum dynamic response value of a single pile increases with the increase in the uniform flow velocity, and the maximum dynamic response value of single pile is obtained at the peak and trough of the wave, indicating that the wave load plays a major role in the combined action. In the combined action of a non-uniform flow and waves with different flow velocities, the maximum dynamic response of a single pile gradually increases with an increase in the non-uniform flow velocity, and the maximum dynamic response of a single pile is also obtained at the positions of the wave’s crest and trough. Therefore, wave load plays a major role in the combined action.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Periodicity (m) | Pogo (s) | Water Depth (m) | Theory | Wavelength (m) | Frequency (rad/s) | Wave Number |
---|---|---|---|---|---|---|
6 | 3 | 20 | Stokes2 | 55.1 | 1.043 | 0.115 |
6.5 | 3 | 20 | Stokes2 | 63.4 | 0.964 | 0.097 |
7 | 3 | 20 | Stokes5 | 72.5 | 0.897 | 0.084 |
7.5 | 3 | 20 | Stokes5 | 81.2 | 0.834 | 0.085 |
8 | 3 | 20 | Stokes5 | 89.7 | 0.781 | 0.071 |
8.5 | 3 | 20 | Stokes5 | 97.8 | 0.735 | 0.064 |
9 | 3 | 20 | Stokes5 | 106.5 | 0.691 | 0.591 |
Periodicity (m) | Pogo (s) | Water Depth (m) | Theory | Wavelength (m) | Frequency (rad/s) | Wave Number |
---|---|---|---|---|---|---|
1 | 8 | 50 | Stokes2 | 99.5 | 0.786 | 0.062 |
2 | 8 | 50 | Stokes2 | 99.4 | 0.786 | 0.061 |
3 | 8 | 50 | Stokes3 | 100.3 | 0.786 | 0.062 |
4 | 8 | 50 | Stokes3 | 101.4 | 0.786 | 0.062 |
Pier height | 30 |
Radius | 3 |
Materials | C40 |
Modulus of elasticity | 3.25 × 1010 |
Poisson’s ratio | 0.2 |
Density | 2500 |
Quality | 1,200,000 |
Water depth | 5, 10, 15, 20 |
Horizontal flow rate | 1, 2, 3, 4, 5, 6 |
Water Density | 1025 |
Bulk modulus | 1 × 1020 |
Dynamic viscosity coefficient | 1.05 × 10−3 |
Work Conditions | V10 (m/s) | Vb (m/s) | Hs (m) | vu (m/s) | vt (m/s) | v (m/s) |
---|---|---|---|---|---|---|
1 | 42.91 | 53.61 | 7.84 | 0.76 | 1.16 | 1.84 |
2 | 30.14 | 37.14 | 11.63 | 0.76 | 1.16 | 1.84 |
3 | 30.14 | 37.14 | 7.84 | 1.13 | 1.16 | 2.34 |
4 | 42.91 | 53.61 | 6.59 | 0.64 | 1.16 | 1.76 |
5 | 25.69 | 33.69 | 11.63 | 0.64 | 1.16 | 1.76 |
6 | 25.69 | 33.69 | 6.59 | 1.13 | 1.16 | 2.34 |
Flow Rate | Displacement (mm) | Shear Force (MN) | Bending Moment (MN·m) | ||||
---|---|---|---|---|---|---|---|
Waves, Uniform Flow | Waves, Non-Uniform Flow | Waves, Uniform Flow | Waves, Non-Uniform Flow | Waves, Uniform Flow | Waves, Non-Uniform Flow | ||
Wave crest | 1 | 0.28 | 0.27 | 0.49 | 0.11 | 2.84 | 2.91 |
2 | 0.31 | 0.32 | 0.54 | 0.32 | 3.17 | 3.01 | |
3 | 0.38 | 0.33 | 0.64 | 0.74 | 3.71 | 3.11 | |
4 | 0.52 | 0.38 | 0.87 | 1.13 | 4.93 | 3.51 | |
Wave Valley | 1 | −0.25 | −0.24 | −0.39 | −0.43 | −2.37 | −2.34 |
2 | −0.22 | −0.28 | −0.34 | −0.41 | −2.16 | −2.51 | |
3 | −0.15 | −0.27 | −0.22 | −0.37 | −1.36 | −2.36 | |
4 | −0.08 | −0.23 | −0.06 | −0.35 | −0.58 | −1.96 |
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Xiao, X.; Nie, J. Evaluating the Reactions of Bridge Foundations to Combined Wave–Flow Dynamics. Buildings 2023, 13, 2030. https://doi.org/10.3390/buildings13082030
Xiao X, Nie J. Evaluating the Reactions of Bridge Foundations to Combined Wave–Flow Dynamics. Buildings. 2023; 13(8):2030. https://doi.org/10.3390/buildings13082030
Chicago/Turabian StyleXiao, Xian, and Jianwei Nie. 2023. "Evaluating the Reactions of Bridge Foundations to Combined Wave–Flow Dynamics" Buildings 13, no. 8: 2030. https://doi.org/10.3390/buildings13082030