# Pressure Characteristics of Landslide-Generated Waves on Bridge Piers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

^{3}, and for this study, a density of 2.5 g/cm

^{3}was achieved using a mixture of mud and stones, as specified in Table 2 for the dimensions. Table 3 lists the mechanical properties of the test materials.

## 3. Forms and Distribution of Wave Pressure

## 4. Pressure Rules on Bridge Pier

## 5. Discussion

^{2}value of 0.78. In scenarios of engineering concern, the most dangerous situations were examined, considering that ${P}_{max}=max\left({P}_{pu},{P}_{re}\right)$, ${P}_{max}\le 3{P}_{pu}$, leading to the formulation of Equation (4).

## 6. Conclusions

- (1)
- The pulsating pressure was generated due to wave action and exhibited a strong correlation with the wave height in front of the bridge piers. It has a frequency range of 0.2–0.5 Hz, which aligns with the wave frequency. The resonance pressure was induced by the resonance of the bridge pier and was characterized by high-frequency and intense oscillations. It occurs at frequencies ranging from 300 to 900 Hz. Notably, resonance pressure was observed only under the condition of the larger water depth, specifically at h = 1.16 m. Moreover, it was limited to the bridge pier on the opposite side (across from the landslide);
- (2)
- ${P}_{pu}$ exhibits a sawtooth-like pattern along the water depth for the opposite bank bridge pier and remains relatively constant, while for the same bank bridge pier, it decreases in a sawtooth pattern along the water depth. ${P}_{pu}$ tends to increase with larger landslide volumes and steeper landslide angles. Additionally, it reaches its maximum near the still water surface;
- (3)
- The distribution pattern of ${P}_{re}/{P}_{pu}$ along the water depth can be categorized into three types: multi-peak, single-peak, and no-peak. Large landslide volumes produce single-peak or no-peak distribution for ${P}_{re}/{P}_{pu}$, while smaller landslide volumes result in a double-peak distribution. In general, the trend along the water depth direction is an increase followed by a decrease. The maximum ${P}_{re}/{P}_{pu}$ for all scenarios is consistently at $Y/h=0.3$ (one-third of the water depth below the water surface), and $Y/h=0.2$ (one-fifth of the water depth below the water surface) represents the second-largest ${P}_{re}/{P}_{pu}$. These two positions on the bridge pier are more susceptible to damage and should be prioritized for enhanced protection during design and maintenance.
- (4)
- An equation for ${P}_{pu}$ was derived through analysis and fitting. The comparison between ${P}_{re}$ and ${P}_{pu}$ indicated that $0.5<{P}_{re}/{P}_{pu}<3$, considering that ${P}_{max}\le 1.05\rho gh{\left(H/h\right)}^{0.64}=1.05\rho g{h}^{0.36}{H}^{0.64}$.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Fritz, H.M.; Hager, W.H.; Minor, H.E. Lituya Bay case: Rockslide impact and wave run–up. Sci. Tsunami Hazards
**2001**, 19, 3–22. [Google Scholar] - Noda, E. Water waves generated by landslides. J. Waterw. Harb. Coast. Eng. Div.
**1970**, 96, 835–855. [Google Scholar] [CrossRef] - Kamphuis, J.W.; Bowering, R.J. Impulse waves generated by landslides. Coast. Eng.
**1970**, 1, 575–588. [Google Scholar] - Huber, A.; Hager, W.H. Forecasting impulse waves in reservoirs. Trans. Int. Congr. Large Dams
**1997**, 1, 993–1005. [Google Scholar] - Walder, J.S.; Watts, P.; Sorensen, O.E.; Janssen, K. Tsunamism generated by subaerial mass flows. J. Geophys. Res. Solid Earth
**2003**, 108, 2236. [Google Scholar] [CrossRef] - Fritz, H.M.; Hager, W.H.; Minor, H.E. Near field characteristics of landslide generated impulse waves. J. Waterw. Port Coast. Ocean. Eng.
**2004**, 130, 287–302. [Google Scholar] [CrossRef] - Panizzo, A.; De Girolamo, P.; Petaccia, A. Forecasting impulse waves generated by subaerial landslides. J. Geophys. Res. Ocean.
**2005**, 110. [Google Scholar] [CrossRef] - Ataie-Ashtiani, B.; Nik-khah, A. Impulsive waves caused by subaerial landslides. Environ. Fluid Mech.
**2008**, 8, 263–280. [Google Scholar] [CrossRef] - Mohammed, F.; Fritz, H.M. Physical modeling of tsunamis generated by three-dimensional deformable granular landslides. J. Geophys. Res. Ocean.
**2012**, 117. [Google Scholar] [CrossRef] - Heller, V.; Spinneken, J. Improved landslide-tsunami prediction: Effects of block model parameters and slide model. J. Geophys. Res. Ocean.
**2013**, 118, 1489–1507. [Google Scholar] [CrossRef] - Heller, V.; Spinneken, J. On the effect of the water body geometry on landslide–tsunamis: Physical insight from laboratory tests and 2D to 3D wave parameter transformation. Coast. Eng.
**2015**, 104, 113–134. [Google Scholar] [CrossRef] - Lindstrøm, E.K. Waves generated by subaerial slides with various porosities. Coast. Eng.
**2016**, 116, 170–179. [Google Scholar] [CrossRef] - Sainflou, M. Essai sur les digues maritimes verticales (test on vertical sea dikes). Ann. Ponts Chaussees
**1928**, 98, 5–48. [Google Scholar] - Tadjbakhsh, I.; Keller, J. Standing surface waves of finite amplitude. J. Fluid Mech.
**1960**, 8, 442–451. [Google Scholar] [CrossRef] - Dean, R.; Dalrymple, R. Water Wave Mechanics for Engineers and Scientists. In Advanced Series on Ocean Engineering; World Scientific: Singapore, 1991; Volume 2. [Google Scholar]
- Dingemans, M.W. Water Wave Propagation over Uneven Bottoms. In Advanced Series on Ocean Engineering; World Scientific: Singapore, 1997. [Google Scholar]
- Le Méhauté, B. An Introduction to Hydrodynamics and Water Waves; Springer: New York, NY, USA, 1976. [Google Scholar]
- Heller, V.; Hager, W.H. Wave types of landslide generated impulse waves. Ocean. Eng.
**2011**, 38, 630–640. [Google Scholar] [CrossRef] - Heller, V.; Bruggemann, M.; Spinneken, J.; Rogers, B.D. Composite modelling of subaerial landslide–tsunamis in different water body geometries and novel insight into slide and wave kinematics. Coast. Eng.
**2016**, 109, 20–41. [Google Scholar] [CrossRef] - Xue, H.; Ma, Q.; Diao, M.; Jiang, L. Propagation characteristics of subaerial landslide-generated impulse waves. Environ. Fluid Mech.
**2019**, 19, 203–230. [Google Scholar] [CrossRef] - Heller, V.; Hager, W.H.; Minor, H.-E. Landslide-generated impulse waves in reservoirs: Basics and computation. In VAW-Mitteilung; Boes, R., Ed.; ETH Zurich: Zurich, Switzerland, 2009; Volume 211. [Google Scholar]
- Kobel, J.; Evers, F.M.; Hager, W.H. Impulse wave overtopping at rigid dam structures. J. Hydraul. Eng.
**2017**, 143, 04017002. [Google Scholar] [CrossRef] - Cooker, M.J.; Weidman, P.D.; Bale, D.S. Reflection of a high-amplitude solitary wave at a vertical wall. J. Fluid Mech.
**1997**, 342, 141–158. [Google Scholar] [CrossRef] - Fenton, J.D.; Rienecker, M.M. A Fourier method for solving nonlinear waterwave problems: Application to solitary-wave interactions. J. Fluid Mech.
**1982**, 118, 411–443. [Google Scholar] [CrossRef] - Lin-Feng, H. Study on the Wave Field Characteristics of Impulse Waves Generated by Three-dimensional Landslides in Curved Gorge-type Reservoirs. Ph.D. Thesis, Chongqing Jiaotong University, Chongqing, China, 2019. [Google Scholar]
- Huang, J.; Lian, J.; Zhang, T. Safety assessment on Lechangxia Dam under surge effect caused by landslide. Chin. J. Water Resour. Hydropower Eng.
**2013**, 44, 93–97. [Google Scholar] - Jian-Min, T.; Bo-Lin, H.; Yong-Bo, Z. Pressure characteristics of landslide-generated impulse waves. J. Mt. Sci.
**2019**, 16, 1774–1787. [Google Scholar]

**Figure 2.**Experimental layout diagram, (

**a**) is a schematic diagram of the tank cross-section, (

**b**) is a photograph of the tank. (unit: m).

**Figure 6.**The time variation of pressure at measurement points on the opposite bank bridge pier in Case 9.

**Figure 7.**The wave pressure distribution on the same bank and opposite bank bridge piers at water depths of 0.74 m, 0.88 m, and 1.16 m, as well as the wave height distribution in front of the bridge piers.

Reference | Model Dimensional | Water Tank Length (m) | Water Tank Weith (m) | Water Tank Height (m) | Bed Slope (°) | Slide Mass Specifications |
---|---|---|---|---|---|---|

Noda [2] | 2D | - | - | - | - | Solid rectangular box |

Kamphuis and Bowering [3] | 2D | 45.00 | 1.00 | 023–0.46 | 45 | Steel box |

Huber and Hager [4] | 3D | 30.00 | 0.50 | 0.50 | 28–60 | Granular material |

Walder et al. [5] | 2D | 3.00 | 0.28 | 1.00 | 10–20 | Hollow rectangular |

Friz et al. [6] | 2D | 11.00 | 0.50 | 1.00 | 45 | Granular material |

Panizzo et al. [7] | 3D | 11.50 | 6.00 | 080 | 16–36 | Solid rectangular box |

Ataic-Ashiani and Nik-Khah [8] | 2D | 3.60 | 2.50 | 0.80–0.50 | 15–60 | Solid and granular |

Mohammed and Fritz [9] | 3D | 48.80 | 2.65 | 0.30–1.20 | 27 | Granular |

Heller and Spinneken [10] | 2D | 24.50 | 0.60 | 0.30–0.60 | 45 | Solid |

Heller and Spinneken [11] | 2D, 3D | 21,20 | 0.60, 7.40 | 0.24, 0.48 | 45 | Solid |

Lindstrom [12] | 2D | 7.30 | 0.20 | 0.10 | 35 | Solid and granular |

No. | ${\mathit{l}}_{\mathit{m}}$ (m) | ${\mathit{b}}_{\mathit{m}}$ (m) | ${\mathit{s}}_{\mathit{m}}$ (m) |
---|---|---|---|

A1 | 0.18 | 0.12 | 0.06 |

A2 | 0.15 | 0.10 | 0.05 |

A3 | 0.12 | 0.08 | 0.04 |

A4 | 0.09 | 0.06 | 0.03 |

A5 | 0.06 | 0.04 | 0.02 |

Material | Density (g/cm^{3}) | Composition | Cohesion (kPa) | Internal Friction Angle |
---|---|---|---|---|

Sandstone | 2.20~2.70 | Purple-red sandstone, siltstone, carbonate rock | Cf = 27~29 kPa | φf = 21°~24° |

Mudstone | 2.45~2.65 | Montmorillonite and illite | Cr = 10~25 kPa | φr = 5.6°~11.9° |

Test stone | 2.50 | Subclay with crushed rock fragments | C = 25 kPa | φ = 20° |

Location | A (m) | B (°) | C (°) | D (m) | E (m) | F (m) |
---|---|---|---|---|---|---|

Straight | 1.16 | 33 | 20 | 1.79 | 2.94 | 3.27 |

Curve 1# | 1.16 | 28 | 16 | 2.2 | 1.68 | 4.12 |

Curve 2# | 1.16 | 32 | 14 | 1.84 | 1.41 | 4.75 |

Curve 3# | 1.16 | 26 | 17 | 2.36 | 1.84 | 3.8 |

Curve 4# | 1.16 | 41 | 18 | 1.35 | 2.96 | 3.69 |

Curve 5# | 1.16 | 25 | 19 | 2.47 | 2.17 | 3.36 |

Curve 6# | 1.16 | 28 | 19 | 2.22 | 2.43 | 3.35 |

Curve 7# | 1.16 | 29 | 20 | 2.11 | 2.69 | 3.2 |

Curve 8# | 1.16 | 29 | 18 | 2.14 | 2.27 | 3.59 |

Curve 9# | 1.16 | 28 | 20 | 2.23 | 2.53 | 3.24 |

Case | H (m) | l (m) | b (m) | s (m) | V (m^{3}) | A (°) |
---|---|---|---|---|---|---|

C1 | 0.74 | 1.0 | 0.5 | 0.6 | 0.3 | 60 |

C2 | 0.74 | 1.0 | 1.0 | 0.4 | 0.4 | 40 |

C3 | 0.74 | 1.0 | 1.5 | 0.2 | 0.3 | 60 |

C4 | 0.74 | 1.0 | 1.5 | 0.6 | 0.9 | 60 |

C5 | 0.88 | 1.0 | 0.5 | 0.6 | 0.3 | 60 |

C6 | 0.88 | 1.0 | 1.0 | 0.4 | 0.4 | 40 |

C7 | 0.88 | 1.0 | 1.5 | 0.2 | 0.3 | 60 |

C8 | 0.88 | 1.0 | 1.5 | 0.6 | 0.9 | 60 |

C9 | 1.16 | 1.0 | 0.5 | 0.6 | 0.3 | 60 |

C10 | 1.16 | 1.0 | 1.0 | 0.4 | 0.4 | 40 |

C11 | 1.16 | 1.0 | 1.5 | 0.6 | 0.9 | 60 |

C12 | 1.16 | 1.0 | 1.5 | 0.6 | 0.9 | 40 |

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**MDPI and ACS Style**

Tian, Y.; Wang, P.; Wang, M.
Pressure Characteristics of Landslide-Generated Waves on Bridge Piers. *Water* **2023**, *15*, 3623.
https://doi.org/10.3390/w15203623

**AMA Style**

Tian Y, Wang P, Wang M.
Pressure Characteristics of Landslide-Generated Waves on Bridge Piers. *Water*. 2023; 15(20):3623.
https://doi.org/10.3390/w15203623

**Chicago/Turabian Style**

Tian, Ye, Pingyi Wang, and Meili Wang.
2023. "Pressure Characteristics of Landslide-Generated Waves on Bridge Piers" *Water* 15, no. 20: 3623.
https://doi.org/10.3390/w15203623