Keywords

1 Introduction

When a blasting vibration wave encounters a step topography during its propagation, the phenomenon of wave scattering occurs, which significantly affects the dynamic response of the near region of the step. The velocity and frequency of particle vibration of rock masses are closely related to the damage and structural instability of nearby buildings [1]. Therefore, the study of the dynamic response of the step topography under blasting load is very important.

Currently, numerous field tests and numerical simulations have been conducted to investigate the dynamic response of the step topography under blasting load. Zhu and Liu proposed a formula for blasting vibration and concluded that the dynamic response on a slope is related to the height of the slope [2]. By monitoring the blasting vibration, Wang and Lu found that the blasting vibration increase with the height of the slope, but not monotonically, and there is a maximum value [3]. Similarly, Xia declared that the top of a step topography exhibits dynamic amplification compared to a flat ground, with the amplification coefficient first increasing and then decreasing with increasing step height [4]. Hong studied the propagation of a blasting vibration wave in a step topography using a two-dimensional finite difference program, and found that the bottom of the step has a dynamic reduction effect, while the top of the step has a dynamic amplification effect [5]. Jiang and Zhou used the dynamic finite element method to simulate the blasting vibration of a slope. The results show that the height of the slope can significantly affect the vibration velocity on the slope, and the dynamic amplification effect mainly occurs in the vertical direction [6]. Tang used the UDEC software to study the blasting vibration of a step topography and found that the dynamic response of the step has an amplification effect, and that the amplification factor is related to factors such as the step height and the distance of blasting source [7].

In summary, there are many research results on the dynamic response for step topography and much valuable information has been obtained. However, most of these studies focus on specific parts of the step (e.g., the top and bottom corners) and rarely consider the entire dynamic response of the step. In this paper, the finite difference program FLAC2D is used to study the dynamic response of step topography under blasting load, with the aim of better understanding the propagation law of blasting vibration waves in step topography.

2 Numerical Model

2.1 Blasting Load

Because the FLAC2D program cannot simulate explosion process using equation of state, a triangular pulse load is applied to the blast hole wall to excite the blasting vibration wave. Since we focus mainly on the dynamic response of the step without considering the rock damage around the blast hole, the peak value of blasting load is taken as 50 MPa, which is smaller than the compressive strength of the rock of 50 to 200 MPa. Moreover, according to the theoretical calculation, the duration of the blasting pressure is generally hundreds of microseconds, so the rise time of the blasting load is taken as 100 μs, and the overpressure time is taken as 400 μs [8]. The time history curve of the blasting load is presented in Fig. 1.

Fig. 1
A graph of pressure in megapascal versus time in microseconds. It plots a triangular-shape plot, which begins at (0, 0), peaks at (100, 50), and drops to (400, 0). Values are approximate.

The time history curve of the blasting load

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2.2 FLAC2D Calculation Model

The calculation model is mainly based on the field test conducted by Xia [4]. The size of the calculation model is 100 × 50 m, the top of the model is set as a stress-free boundary to simulate step surface and flat ground, and the bottom and both sides of the model are set as absorbing boundaries to simulate infinite space. The blasting load is applied perpendicularly on the left side of the model, and the range of the loading depth is 2.5 ~ 3.0 m. Since the duration of the blasting load is only hundreds of microseconds, the frequency of the input wave can reach thousands of hertz. To ensure the accuracy of numerical simulation, the calculation model is divided into uniform square grids, and the grid size is set as 0.25 m. Besides, to facilitate the analysis of the blasting vibration distribution law of the step topography, 13 vibration monitoring points are placed on the step surface, with 8 monitoring points located at the top and bottom of the step at a distance of 1 m, and 5 monitoring points located at the side of the step at equal distances. The schematic diagram of the calculation model and its grid division are shown in Figs. 2 and 3.

Fig. 2
A schematic depicts the calculation model. It indicates the blasting load distance, step height, and 13 monitoring points. It is 100 meters wide and 50 meters high.

The schematic diagram of the calculation model

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Fig. 3
A diagram depicts the calculation model with the grid division. Top left corner of the model has a separated piece with grids. The angle between the model and the separated piece is in a foreign language.

The grid division of the calculation model

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Since the numerical simulation here does not take into account the rock damage, the rock is considered as an elastic medium, and the absorption attenuation of blasting vibration wave by the medium is simulated by setting a Rayleigh damping. The mechanical parameters of the rock refer to the field test in [4]: the rock density is 2610 kg/m3, the elastic modulus is 49GPa, and the Poisson's ratio is 0.19. Considering that the rock mass in the field is relatively complete, the damping ratio of the medium is set as 0.01.

3 Results and Discussion

3.1 Amplification Effect of Step Topography

To represent the amplification effect of the step topography, an amplification coefficient is defined as the amplitude ratio between the top corner of the step (i.e., monitoring point 9) and the flat ground at the same distance. Figure 4 compares the variation of the amplification coefficient with the step height in the numerical simulation and in the field test.

Fig. 4
A graph of amplification coefficient versus step height in meter. It illustrates 2 plots for field test and numerical simulation. Both the plots follow an increasing to decreasing trend.

The variation of the amplification coefficient with the step height

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As illustrated in Fig. 4, the numerical results agree well with the test results. Both the numerical results and the test results show that the amplification coefficient first increases and then decreases as the step height increases, and the amplification effect is most significant at the step height is 7–8 m. However, there are still some differences between the numerical results and the field test results. When the step height is small, the numerical results are larger than the field test results, but when the step height is large, the numerical results are smaller than the field test results. This is because the numerical simulation uses a two-dimensional model, while the field test is a three-dimensional problem.

3.2 Vibration Distribution on Step Surface

As shown in Fig. 1, 13 vibration monitoring points are arranged at equal intervals on the step surface to study the overall dynamic response of the step topography. The amplitude at each monitoring point is represented by a dashed line perpendicular to the surface, while the amplitude at two corners is represented by a dashed line along the diagonal. Figure 5 shows the law of vibration distribution the step surface at step heights of 2, 4, 7 and 10 m, respectively.

Fig. 5
4 graphs, a to d, plot amplitude in centimeters per second versus distance in meters. A to D present the plots with step height of 2, 4, 7, and 10 meters, respectively. All graphs have a square to which the plots are connected with nodes.

The distribution of vibration amplitude on the steps of different heights

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As presented in Fig. 5, the step height has a significant effect on the dynamic response of the step topography. For example, when the step height is 2 m, the amplitudes at the top and bottom of the step do not differ significantly, indicating that the step topography has little effect on the propagation of the blast vibration wave in this case. However, as the step height increases, the amplitude difference between the top and bottom of the step gradually increases. For example, when the step height is 4, 7 or 10 m, the amplitude of the top corner of the step is the largest, showing a local dynamic amplification effect, while the amplitude of the rest of the top of the step is smaller than the bottom of the step, indicating a dynamic reduction effect. Moreover, the degree of dynamic reduction effect increases with increasing step height. This is mainly because as the height of the step increases, the propagation distance of the blasting vibration wave to the top of the step increases, so the degree of geometric attenuation and absorption attenuation of the blasting vibration wave increases.

Figure 5 also shows that regardless of the step height, the amplitude near the bottom corner of the step is significantly smaller than in the rest of the step, indicating a dynamic reduction effect. This is because the angle at the bottom corner of the step is 270°. When the blasting vibration wave propagates to the bottom corner, the wave front suddenly changes from an angle of 180° to 270°, resulting in a sudden increase in geometric attenuation and a significant decrease in amplitude.

3.3 Frequency Variation of Blasting Vibration Wave

When the blasting vibration wave passes through the step topography, not only the amplitude but also the frequency changes. Obviously, the change in frequency is also related to the step height. To show the influence of the step height on the frequency of the blasting vibration wave, Fig. 6 compares the amplitude spectra at the bottom and top corners of the step at different step heights.

Fig. 6
2 graphs, a and b, plot amplitude spectra versus frequency in hertz. They illustrate 4 wave forms for 2, 4, 7, and 10 meters. Both the graphs follow a fluctuating, declining trend.

The amplitude spectrum of the bottom corner and the top corner of the steps with different heights

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First, it can be seen in Fig. 6(a) that the amplitude spectrum of the bottom corner of the step contains many middle and low frequency components when the step height is small, but as the step height increases, the middle and low frequency components show a decreasing trend. The reason is that the larger the step height, the smaller the influence of the wave reflected from the top of the step on the bottom corner. Second, Fig. 6(b) indicates that as the step height increases, the low-frequency component of the top corner of the step gradually increases, while the high-frequency component gradually decreases, i.e., the energy ratio of the low-frequency component increases, which is consistent with the results of the field test in [9].

To sum up, when the blasting vibration wave passes through a step topography, the amplitude and low-frequency components of the top corner of the step are larger than those of the other parts, making buildings (structures) more susceptible to damage. Therefore, the buildings (structures) should be located as far as possible from the top corner of the step topography.

4 Conclusion

In this paper, the dynamic response of the step topography under blasting load is investigated using the finite difference program FLAC2D, and the feasibility of the numerical analysis is illustrated by comparison with field tests. Based on the numerical results, the following conclusions are drawn.

  1. (1)

    Under the same blasting load, the amplification coefficient first increases and then decreases with increasing step height, reaching a maximum value at a step height of 7–8 m.

  2. (2)

    The dynamic amplification effect of the step topography is mainly concentrated near the top corner, while there is a significant dynamic reduction effect near the bottom corner.

  3. (3)

    When a blasting vibration wave passes through the step topography, the amplitude and low-frequency components of the top corner of the step are larger than those of the other parts, thus the buildings (structures) should be located as far as possible from the top corner of the step topography.