# Experimental study on the applicability of Westergaard’s formula for calculating earthquake-induced hydrodynamic pressure in small lake

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## Abstract

Moraine-dammed lake outbursts usually threaten highways, railways, and key facilities in alpine regions. The varying amplitudes and distribution of hydrodynamic pressures significantly affect the stability of the dam. We utilize a shaking table to investigate the development of hydrodynamic pressure caused by different sinusoidal waves and seismic Wolong wave. A series of shaking table tests indicate that the hydrodynamic pressure variation significantly follows seismic acceleration wave motion. The maximum hydrodynamic pressures calculated by Westergaard’s equation are compared with the experimental values under different waves. It is shown that the Westergaard’s values are lower than the experimental ones under the sinusoidal waves. However, the Westergaard’s method is able to predict the earthquake-induced hydrodynamic pressure caused by Wolong wave in small lake with desirable accuracy.

## Keywords

Moraine dam Hydrodynamic pressure Shaking table Westergaard Small lake## 1 Introduction

Sichuan–Tibet railway has been included in China western development strategy. Sichuan–Tibet railway is expected to across the Hengduan mountains from the east of Tibet plateau, and this area has the geomorphic features of high altitude and large elevation difference and distributes a large amount of glacial lakes. In the Parlung Zangbo river basin, for example, there are 461 glaciers and 131 glacial lakes of different sizes [1, 2]. In nearly 40 years, there had eight large-scale debris flow from glacial lake outburst floods (GLOF), blocking and damaging roads from 20 to 270 days. For example, in July 1988, debris flow resulting from GLOF in Midui destroyed nearly 30-km Sichuan–Tibet roads and blocked the route for half a year [3]. The lake outbursts pose a catastrophe risk to the Sichuan–Tibet railway coming across the Parlung Zangbo river basin, which is a challenge in China’s railway history [4]. Moreover, Parlung Zangbo river basin is located in strong earthquake regions where more than 40,000 earthquakes have been recorded since 1970, and more than 3000 of these earthquakes were greater than Ms3.0, posing a potential threat to the stability of moraine dams [5]. In recent years, with the global warming and glacial retreating, moraine-dammed lake disaster has a tendency to increase. Therefore, the moraine-dammed lake outbursts caused by earthquakes have become a special environmental geological hazard to railway route [6].

*y*denotes the water heights,

*P*is the maximum hydrodynamic pressures at height

*y*,

*h*is the reservoir depth, \(\rho\) is the unit density of water,

*k*

_{ h }is the seismic acceleration coefficients in the horizontal direction, and

*g*is the acceleration of gravity (9.8 m/s

^{2}). The approximate formula provides a convenient way for solving the response of the dam–water interactions. In 1953, Zangar developed an experimental solution for the same problem using an electrical analogue and reported extensive results for a variety of non-vertical upstream faces [11]. Based on momentum method and two-dimensional potential flow theory, Chwang solved the hydrodynamic pressure problem with a more general configuration of a dam [12, 13]. Wang et al. [14] modified Westergaard’s equation considering the influences of the dam height, elasticity, and reservoir bottom condition. Saleh and Madabhushi [15] investigated the effect of dam–foundation interaction on the hydrodynamic pressure response on the dam face under different earthquake waves using the dynamic centrifuge modeling technique. The results generated by these works were more or less equivalent to the results of Westergaard, and thus, the approximate formula has been widely used to calculate the hydrodynamic pressure loads.

Therefore, it is now widely believed that the earthquake-induced hydrodynamic pressure is mainly affected by parameters such as the peak ground acceleration (PGA) and the initial water depth. In this paper, the conditions of rigid vertical upstream face and flat bottom were simulated to obtain the hydrodynamic pressure. However, in many cases, the moraine-dammed lake is small, where the boundary reflections may be not negligible. Therefore, it is necessary to study whether Westergaard’s formula can be applied to calculate the hydrodynamic pressure in small reservoirs or lakes.

In addition, the moraine dams are formed by an unconsolidated, poorly sorted rock debris or highly heterogeneous mixture of particles [16, 17, 18]. The pressure loads acting on this type of dams may lead to failure of the dams and become a critical factor causing the dam to break in the earthquake areas [19]. In this paper, experiments were conducted to study the characteristics of hydrodynamic pressures caused by sinusoidal waves and Wolong wave on a shaking table, and the results were compared with those by Westergaard’s formula.

## 2 Simulation experiment

### 2.1 Experimental setup

_{1}–P

_{6}. A detailed illustration of sensor distribution is shown in Fig. 2.

### 2.2 Experimental design

In each test, the input seismic excitation wave was created by the computer control system. When the vibrator receives the signal, the shaking table moves with the triggering wave. In addition, the hydrodynamic pressures are recorded by the water pressure acquisition system. Even after the seismic wave passes by, the acquisition system keeps running until the entire water body becomes stable. The measured hydrodynamic pressures during the earthquakes were calculated by deducting the initial hydrostatic pressure from the total water pressure recorded by the sensors at the corresponding heights. Moreover, each test was repeated twice to reduce the equipment error or human error.

Seismic wave types and corresponding PGAs

Seismic wave types | PGA (m/s | |||||
---|---|---|---|---|---|---|

Wolong wave | 0.05 | 0.1 | 0.15 | 0.2 | 0.25 | 0.3 |

2–6 Hz sinusoidal wave | 0.1 | 0.2 | 0.3 | – | – | – |

White noise | 0.1 | – | – | – | – | – |

## 3 Experimental results and discussion

### 3.1 Fluctuation characteristics of hydrodynamic pressure

_{1}pulsating water pressure sensor. Then, FFT was used to derive the pressure spectrum, and the water fundamental frequency is 0.3325 Hz (Fig. 7). It is found that the input seismic excitation frequency is far away from the fundamental frequency, indicating that the resonance did not appear.

_{1}sensors attached to the wall under the actions of 3 and 6 Hz sinusoidal waves, together with the 0.2 g PGA. The hydrodynamic pressure amplitudes were not increasing with time, but in synchronization with input seismic excitation waves. It is observed that the changing process of hydrodynamic pressures can be divided into two stages: (a) For 0 <

*t*< 20 s, the hydrodynamic pressures obviously follow seismic acceleration wave motion, and (b) for 20 s <

*t*after the seismic wave passes by, the hydrodynamic pressure returns simultaneously to zero. The hydrodynamic pressures recorded at other sensors have similar fluctuations. Further, the maximum hydrodynamic pressures under action of different waves were analyzed.

### 3.2 Sinusoidal wave action

_{1}sensor under the actions of 3 and 6 Hz sinusoidal waves are shown in Fig. 11.

*g*, 0.2

*g*, 0.3

*g*was analyzed, as shown in Fig. 12.

When PGA is 0.1*g*, the maximum hydrodynamic pressure along the water depth has the same distribution rules. Under the PGAs of 0.2*g* and 0.3*g*, the distribution of hydrodynamic pressures with different frequencies along the water depth was basically the same, but slightly different due to the equipment error or human error. It is shown that the amplitudes of hydrodynamic pressure are independent of the seismic frequency frequencies when the input seismic excitation frequencies are much larger than water fundamental frequency.

Comparisons between the experimental hydrodynamic pressures and Westergaard’s values indicate that the experimental values are slightly larger than Westergaard’s equation. For instance, the experimental hydrodynamic pressures recorded at P_{1} sensor under the actions of 2–6 Hz sinusoidal waves were larger than those calculated theoretically by 10%–20%. The difference is attributed to the following reason. The real water tank has the boundary reflection, while the Westergaard’s formula does not include the boundary effect. As a result, the water waves reflected at the side walls will enlarge the hydrodynamic pressures. Therefore, Westergaard’s formula can accurately calculate the hydrodynamic pressure caused by sinusoidal wave in small lakes.

### 3.3 Wolong wave action

_{1}sensor attached to the wall during the Wolong earthquake, together with the 0.2

*g*PGA. Compared with the sinusoidal waves, the pressure caused by Wolong wave displays similar changes, which is always in synchronization with the response of Wolong wave. For further analysis, the maximum hydrodynamic pressures generated at different heights are shown in Fig. 14. The hydrodynamic pressures are normalized with the total hydrostatic pressure, and the sensor depths are normalized with reservoir depth. In Fig. 14,

*y*denotes the sensor heights,

*P*

_{ max }(

*y*) denotes the maximum hydrodynamic pressures at heights

*y*,

*h*is the reservoir depth,

*ρ*is the unit density of water (1.0 t/m

^{3}), and

*g*is the acceleration of gravity (9.8 m/s

^{2}).

As observed in Fig. 14, the maximum hydrodynamic pressures along the water depth are increased from top to the bottom, and it is also found that the ratio of maximum hydrodynamic pressure to total hydrostatic pressure approximately grows from 5% to 25%, increasing monotonously with seismic acceleration.

Comparison of the measured hydrodynamic pressures and Westergaard’s formula is given in Fig. 14. It indicates that generally, the measured hydrodynamic pressures mostly match well with those calculated by the Westergaard’s formula. As discussed earlier, there exists boundary effect, and hence, we can see that the experimental hydrodynamic pressures are higher than those induced in a semi-infinite reservoir. However, with the same maximum excitation acceleration, the input energy generated by the sinusoidal wave is greater than the seismic wave induced. Therefore, the hydrodynamic pressures induced by sinusoidal wave are greater than the values caused by seismic wave. As a result, the test data for hydrodynamic pressures caused by Wolong wave are mostly in good agreement with Westergaard’s formula. Small deviation may be attributed to equipment error or human error in measurement.

The Westergaard’s method is able to predict the earthquake-induced hydrodynamic pressure caused by Wolong wave in the small lake with reasonable accuracy. In the experiment, the most disadvantage condition of hydrodynamic pressure is considered, i.e., the side walls are vertical and rigid, and the hydrodynamic pressure at the side may be larger compared to the normal case. As reference to “Specifications for Seismic Design of Hydraulic Structure” (DL5073-2000), if an inclined upstream face is with a horizontal angle of \(\theta\), the hydrodynamic pressure values should be multiplied by \(\theta\)/90.

## 4 Conclusions

- 1.
When the input seismic excitation frequency is far from the fundamental frequency, the hydrodynamic pressure variation significantly followed the seismic acceleration wave motion. And the maximum hydrodynamic pressures were independent of the input seismic excitation frequencies, but have a positive correlation with PGAs.

- 2.
The comparison of the theoretical values and experimental data shows that Westergaard’s equation was slightly lower than the experimental values caused by sinusoidal waves, but in good agreement with the Wolong earthquake-induced hydrodynamic pressure in this small lake. It is indicated that Westergaard’s formula to calculate hydrodynamic pressures caused by real seismic waves in the small lake is not a conservative approach.

- 3.
In current risk analysis, the surge wave caused by earthquake or ice avalanches is the main potential threat to the glacier lake outburst, while the destructive earthquake-induced hydrodynamic pressures directly affect the stability of the moraine dam and are disastrous to human lives and properties along the shore.

## Notes

### Acknowledgements

This work is financially supported by the Natural Science Foundation of China under contract No. 41571004; National Key Research and Development Program (Grant No. 2016YFC0802206); Research and Development Program of Science and Technology of China Railway Corporation (Grant No. 2015G002-N).

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