# Fast Prediction of Solitary Wave Forces on Box-Girder Bridges Using Artificial Neural Networks

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## Abstract

**:**

_{s}), relative wave heights (H/h) and deck aspect ratios (W/h). With the CFD simulation results as the training datasets, an artificial neural network (ANN) is trained utilizing the back-propagation algorithm. The maximum wave forces first increase and then decrease with the C

_{s}, while they monotonically increase with H/h. For relatively large H/h and small C

_{s}values, the relationship between the maximum wave forces and W/h presents strong nonlinearities. The observed correlation coefficients between the ANN predictions and the CFD results for the vertical and horizontal wave forces are 98.6% and 98.1%, respectively. The trained ANN-based model shows good prediction accuracy and could be used as an efficient model for the tropical cyclone risk analysis of coastal bridges.

## 1. Introduction

## 2. Numerical Model and Validation

#### 2.1. Governing Equations

#### 2.2. Solitary Wave Generation and Elimination

#### 2.3. Validation of Numerical Wave Flume

#### 2.4. Validation of Numerical Wave Force

## 3. Numerical Results and Parametric Study

_{v}and horizontal force F

_{h}on the bridge deck are the functions of the wave and bridge characteristics [53]:

_{v}, and horizontal force, F

_{h}, on the bridge deck are investigated by highlighting the influence of the parameters d, H, and W on the solitary wave forces.

#### 3.1. Solitary Wave Force on Bridge Deck

_{S}is defined as the submergence depths $\left({Z}^{\ast}=d-{Z}_{ele}\right)$ divided by the deck height (h), where C

_{S}= 0 and C

_{S}= 1 respectively indicating that the SWL is equal to the bottom of the girder and the top of the deck. The simulation results suggest that the solitary wave forces consist of two components, namely a slamming force and a quasi-static force. In Figure 7, t

_{1}and t

_{2}represent the time instants of maximum slamming force and quasi-static force, respectively. The box girder is subject to a slamming force several times greater than the maximum quasi-static force due to the existence of the flat plate, and the short-duration slamming force occurs slightly before the peak of the quasi-static force. In addition to the uplift wave force, it is noted that the box girder may suffer from a significant downward wave force.

#### 3.2. Parametric Study on Wave Force

_{S}). As shown in Figure 8, both the maximum vertical and horizontal wave forces converge around C

_{S}= 0.8. While there is always a significant increase in the vertical wave forces until they reach their peak values, a similar phenomenon is not observed for the horizontal wave forces with relatively small W/h and/or H/h values (e.g., W/h = 4.074 and/or H/h = 0.556). For H/h = 0.556 and H/h = 0.741, the peak value of maximum vertical wave forces is achieved at approximately C

_{S}= 0.35. For H/h = 0.926 and H/h = 1.111, the peak value of maximum vertical wave forces is achieved at approximately C

_{S}= 0.2.

_{s}. On the other hand, the maximum wave force increases relative to the wave height following an approximately linear relationship for large C

_{s}(e.g., C

_{s}= 1).

_{s}values (e.g., H/h = 0.37 and/or C

_{s}= 1). For the relatively small values of C

_{s}, however, the relationship between the maximum wave forces and deck aspect ratios presents strong nonlinearities at relatively large values of H/h.

## 4. Artificial Neural Network for Fast Prediction of Wave Force

#### 4.1. Datasets

#### 4.2. Training Process

_{s}, H/h, and W/h refer to the submersion coefficient, relative wave height, and deck aspect ratio, respectively. The summation sign (∑) denotes${C}_{s}\cdot {W}_{1i}^{\left(I\right)}+\left(W/h\right)\cdot {W}_{2i}^{\left(I\right)}+\left(H/h\right)\cdot {W}_{3i}^{\left(I\right)}$, where ${W}_{1i}^{\left(I\right)}$, ${W}_{2i}^{\left(I\right)}$, and ${W}_{3i}^{\left(I\right)}$ are the weights from the input layer neurons to the ith neurons in the first hidden layer, and b

_{i}is the bias for the ith neurons in the first hidden layer. This summation will go through a nonlinear active function and obtain an “activated” value of ${h}_{i}^{1}$ as the input for the next hidden layer. This feedforward procedure will repeat until the Mth layer (where M is the total hidden layer number) and obtain the prediction of the output F (i.e., the maximum wave forces F

_{v}or F

_{h}).

#### 4.3. Prediction Accuracy

_{s}, H/h, and W/h. The observed correlation coefficient (R) between the prediction and ground truth for the vertical and horizontal wave forces are 98.6% and 98.1%, respectively. The comparisons between the ANN predictions and the CFD simulations for selected C

_{s}, H/h, and W/h values are shown in Figure 15. These comparison results suggest that the proposed ANN-based surrogate model can accurately capture both vertical and horizontal wave forces on various bridge decks.

## 5. Conclusions and Future Directions

- (1)
- Both the maximum vertical and horizontal wave forces first increased and then decreased with the submersion coefficient C
_{s}, and finally converged around C_{s}= 0.8. The peak values of maximum vertical wave forces were achieved in the range of C_{s}equal to 0.2–0.35, depending on the relative wave height (H/h) and the deck aspect ratio (W/h). For the maximum positive horizontal wave forces, the peak values were obtained in the range of C_{s}equal to 0–0.2. - (2)
- Both the maximum vertical and horizontal wave forces monotonically increased with respect to H/h. The relationship between the maximum wave forces and relative wave heights presented strong nonlinearities for small C
_{s}. On the other hand, the maximum wave force increased with H/h following an approximately linear relationship for large C_{s}. Both the maximum vertical and horizontal wave forces linearly increased with W/h for relatively small H/h and large C_{s}values. For relatively large H/h and small C_{s}values, however, the relationship between the maximum wave forces and deck aspect ratios presented strong nonlinearities. - (3)
- The ANN model efficiently and accurately predicted wave forces on bridge decks for various C
_{s}, H/h, and W/h values.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${C}_{s}$ | Submergence coefficients |

$H$ | Wave height |

$h$ | Height of bridge deck |

$W$ | Width of bridge deck |

$H/h$ | Relative wave height |

$W/h$ | Deck aspect ratio |

$\mathrm{RMSE}$ | Root mean squared error |

${F}_{V\_mean}$ | Mean values for the maximum vertical wave force |

${F}_{H\_mean}$ | Mean values for the maximum horizontal wave force |

$d$ | Still water depth |

${\overline{u}}_{i}$ | Average velocity in the i direction |

${\overline{u}}_{j}$ | Average velocity in the j direction |

${u}_{i}^{\prime}$ | Fluctuation velocity along the i direction |

${u}_{j}^{\prime}$ | Fluctuation velocity along the j direction |

${x}_{i}$ | Coordinate axis along the i direction |

${x}_{j}$ | Coordinate axis along the j direction |

$\overline{p}$ | Average pressure |

$\rho $ | Density of fluid |

$\mu $ | Viscosity of fluid |

$t$ | Time |

$k$ | Turbulence kinetic energy |

$\omega $ | Dissipation |

y+ | Nondimensional wall-coordinate |

$\eta $ | Distance from the wave surface to the still water level |

$c$ | Wave celerity |

$x$ | Axis of horizontal direction |

$z$ | Axis of vertical direction |

$g$ | Gravity acceleration |

$\delta \left(x\right)$ | Damping coefficient |

${L}_{s}$ | Length of the wave elimination zone |

${x}_{1}$ | Start coordinates of the wave elimination zone |

${x}_{2}$ | End coordinates of the wave elimination zone |

$\beta $ | Empirical parameter of the damping coefficient |

${F}_{v}$ | Maximum vertical wave force |

${F}_{h}$ | Maximum horizontal wave force |

${L}_{d}$ | Deck length |

${d}_{r}$ | Height of rail |

${Z}_{c}$ | Clearance of bridge deck |

${Z}_{ele}$ | Bottom elevation of the bridge superstructure |

$\alpha $ | Angle of incidence to the structure |

${Z}^{\ast}$ | Submergence depths |

t_{1} | Time of maximum slamming force |

t_{2} | Time of maximum quasi-static force |

t_{3} | Time of maximum downward force |

${W}_{1i}^{\left(I\right)},{W}_{2i}^{\left(I\right)},{W}_{3i}^{\left(I\right)}$ | Weights from the input layer neurons to the ith neurons in the first hidden layer |

b_{i} | Bias for the ith neurons in the first hidden layer |

${h}_{i}^{1}$ | The “activated” value of the ith neurons in the first hidden layer |

Target value | |

${O}_{k}$ | Output value |

$R$ | Correlation coefficient |

$\overline{T}$ | Mean of the target value |

$\overline{O}$ | Mean of the output value |

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**Figure 2.**Free surface profiles for solitary waves at various locations: (

**a**) 0 m; (

**b**) 45 m; (

**c**) 90 m; (

**d**) 135 m.

**Figure 3.**Comparisons of wave forces by CFD model and Seiffert et al. [51]: (

**a**) vertical wave forces; (

**b**) horizontal wave forces.

**Figure 4.**Comparisons of wave forces by the CFD model and Seiffert et al. [52]: (

**a**) vertical wave forces; (

**b**) horizontal wave forces.

**Figure 7.**Vertical wave force on the bridge deck with C

_{s}= −0.185, H = 1.5 m, and W = 15 m: (

**a**) wave force time history; (

**b**) snapshot of the wave–deck interaction.

**Figure 8.**The relationship between the maximum wave force and submergence coefficient: (

**a**) vertical wave force for W/h = 4.074; (

**b**) horizontal wave force for W/h = 4.074; (

**c**) vertical wave force for W/h = 4.815; (

**d**) horizontal wave force for W/h = 4.815; (

**e**) vertical wave force for W/h = 5.556; (

**f**) horizontal wave force for W/h = 5.556.

**Figure 9.**The relationship between the maximum wave force and relative wave height: (

**a**) vertical wave force for C

_{s}= 0; (

**b**) horizontal wave force for C

_{s}= 0; (

**c**) vertical wave force for C

_{s}= 0.5; (

**d**) horizontal wave force for C

_{s}= 0.5; (

**e**) vertical wave force for C

_{s}= 1; (

**f**) horizontal wave force for C

_{s}= 1.

**Figure 10.**The relationship between the maximum wave force and deck aspect ratio: (

**a**) vertical wave force for C

_{s}= 0; (

**b**) horizontal wave force for C

_{s}= 0; (

**c**) vertical wave force for C

_{s}= 0.5; (

**d**) horizontal wave force for C

_{s}= 0.5; (

**e**) vertical wave force for C

_{s}= 1; (

**f**) horizontal wave force for C

_{s}= 1.

**Figure 11.**The maximum wave forces in datasets: (

**a**) vertical wave forces resulting from various submersion coefficients; (

**b**) horizontal wave forces resulting from various submersion coefficients; (

**c**) vertical wave forces resulting from various relative wave heights; (

**d**) horizontal wave forces resulting from various relative wave heights; (

**e**) vertical wave forces resulting from various deck aspect ratios; (

**f**) horizontal wave forces resulting from various deck aspect ratios.

**Figure 15.**The comparison between ANN predictions and CFD simulations: (

**a**) vertical wave force for W/h = 5.556 and H/h = 0.556; (

**b**) vertical wave force for W/h = 5.556 and H/h = 0.741; (

**c**) vertical wave force for W/h = 5.556 and H/h = 0.926; (

**d**) vertical wave force for W/h = 5.556 and H/h = 1.111; (

**e**) horizontal wave force for W/h = 5.556 and H/h = 0.556; (

**f**) horizontal wave force for W/h = 5.556 and H/h = 0.741; (

**g**) horizontal wave force for W/h = 5.556 and H/h = 0.926; (

**h**) horizontal wave force for W/h = 5.556 and H/h = 1.111.

Location | Numerical Wave Height | Theoretical Wave Height | Error |
---|---|---|---|

45 m | 2.528 m | 2.5 m | 1.12% |

90 m | 2.502 m | 2.5 m | 0.08% |

135 m | 2.477 m | 2.5 m | 0.92% |

Parameter | Minimum | Maximum |
---|---|---|

Submersion coefficient | −0.370 | 1.111 |

Relative wave height | 0.370 | 1.111 |

Aspect ratio | 4.074 | 6.296 |

Vertical force | 9.200 kN/m | 1643.2 kN/m |

Horizontal force | 1.720 kN/m | 654.77 kN/m |

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## Share and Cite

**MDPI and ACS Style**

Lu, M.; Li, S.; Wu, T.
Fast Prediction of Solitary Wave Forces on Box-Girder Bridges Using Artificial Neural Networks. *Water* **2023**, *15*, 1963.
https://doi.org/10.3390/w15101963

**AMA Style**

Lu M, Li S, Wu T.
Fast Prediction of Solitary Wave Forces on Box-Girder Bridges Using Artificial Neural Networks. *Water*. 2023; 15(10):1963.
https://doi.org/10.3390/w15101963

**Chicago/Turabian Style**

Lu, Minglong, Shaopeng Li, and Teng Wu.
2023. "Fast Prediction of Solitary Wave Forces on Box-Girder Bridges Using Artificial Neural Networks" *Water* 15, no. 10: 1963.
https://doi.org/10.3390/w15101963