A Study of Blast Vibration Propagation Law under Negative Altitude Terrains

Gao, Jing; Huang, Cong; Huang, Xiaomin; Ren, Jian-Jun; Wang, Na

doi:https://doi.org/10.1155/2022/4289057

Mathematical Problems in Engineering

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Volume 2022 | Article ID 4289057 | https://doi.org/10.1155/2022/4289057

A Study of Blast Vibration Propagation Law under Negative Altitude Terrains

Jing Gao,¹Cong Huang,²Xiaomin Huang,³Jian-Jun Ren,⁴and Na Wang²

Academic Editor: Fuqiang Ren

Received05 Jan 2022

Accepted01 Feb 2022

Published10 Mar 2022

Abstract

The propagation characteristics of blast waves and the prediction accuracy of blast vibration velocities along negative slopes are important because those can be used to guide engineering application and theoretical research. In this paper, the wave theory was applied to better understand the propagation mechanism of blast waves along negative slopes. Regression analysis was used for the Sadovsky and the CRSRI blast vibration velocity prediction models during on-site operations. The magnification of peak blast vibration velocity along a negative slope was introduced to determine the threshold altitude difference for the magnification effect to occur. Based on this parameter, the relative errors between the two prediction models were compared. The obtained results indicate that the superposition of incident and reflected blast waves on a negative slope creates the “slope effect” which locally amplifies the blast vibration velocity. The relative error of the CRSRI prediction model was as small as 17.53%, demonstrating a greater accuracy than Sadovsky’s prediction model. The magnification effect of a negative slope was observed at specific altitude differences and was more noticeable in the perpendicular direction. This paper creates a theoretical basis for studying the propagation mechanisms of blast vibration waves along negative slopes as well as predicting the blast vibration velocities.

1. Introduction

Blasting operations usually induce a topographic change within the investigated area. These modifications decrease the accuracy of the Sadovsky formula which is commonly used for predicting the blast vibration velocity. The slope magnification effect caused by the propagation of blast waves over undulating terrain has become an essential part of studying the propagation of blast vibrations during rock slope excavations [1].

Previous studies focused on the propagation mechanism of blast waves in rock slopes have laid the foundation for an accurate prediction of the blast vibration velocity. Chen et al. [2] suggested that the altitude magnification effect is caused by the “whipping effect,” as the main frequency of the blast wave load is equal to the natural vibration frequency of the bench. Guo et al. [3], Soltys et al. [4], and Man et al. [5] combined theoretical analyses with field tests and confirmed the “slope effect” for both negative and positive slopes, leading to an increase in the slope blast vibration velocity. Qiu [6] considered the rock slope as a multilayered environment and suggested that the blast waves would be superimposed by refraction and reflection after propagating to the interfaces of different layers. In consequence, this would create the “interface group effect” and lead to the altitude magnification effect.

Several studies have been carried out to accurately predict the blast vibration velocity in terrains of varying altitudes. Liu et al. [7], Roy et al. [8], Elevli and Arpaz [9], and Torres et al. [10] applied Sadovsky’s formula with modifications in the altitude difference to predict the blast vibration velocity based on a large volume of field data. The conclusions showed that the modified formulas generated safer prediction results, but their solutions were not good supported by theoretical analysis since no normalization was implemented. Zhu et al. [11], Faradonbeh et al. [12], and Himanshu et al. [13] proposed a modified formula for predicting the blast vibration velocities for sloped terrains based on field data analysis and comparison experiment. They performed dimensional analyses on the altitude difference and obtained acceptable results when they applied the modified formula to practical situations. Tang [14] and Ainalis et al. [15] used dimensional analysis and deduced a formula for predicting blast vibration velocity based on the altitude difference. After applying the results to real-life scenarios, they concluded that the prediction error of the formula was as little as 10%. Li et al. [16] and Kumar et al. [17] studied the blast vibration of a shaft. Their results indicate that the altitude magnification effect had an elevation threshold above which the blast vibration velocity would be attenuated. Wu et al. [18] and Kim and Lee [19] implemented the linear regression approach to generate a predictive model for the blast vibration velocity in benched terrains based on numerical simulations and field tests. The results suggest that the correlation coefficient of the obtained formula was higher compared to the other formulas and could better reflect the influence of altitude difference on the blast vibration velocity. Liu et al. [20] used field tests as well as theoretical analyses and concluded that their modified prediction formula performed better than Sadovsky’s formula for terrains with a negative slope when the altitude difference was taken into consideration. Lei et al. [21] and Roy et al. [22] applied an already known altitude difference correction formula and regression fitting analysis approach and concluded that the slope coefficient (β) decreased during the initial stage but afterwards rapidly increased. Lu and Huetrulid [23], Agrawal and Mishra [24], and Hudaverdi [25] analyzed the issue from different theoretical angles and proposed a modified formula that was better theoretically adjusted and more comprehensive in predicting blast vibration velocity. Zhang [26] obtained a high-precision prediction formula for bench blast vibrations by analyzing the propagation mechanism of blast waves and by associating it with the elastic wave mechanics. Yang et al. [27] and Avellan et al. [28] studied the seismic topographic effect of blasting on the follow-wave slope and derived a model for predicting the blast vibration velocity of a follow-wave slope with higher differential elevations.

The previously mentioned studies focused on the efficiency of the blast waves and the prediction of blast vibration velocities, but only some of them considered the negative slopes. Therefore, there is a need for further research. In this study, we investigated the propagation mechanism of blast waves along negative slopes using the elastic wave theory. Afterwards, we applied the regression analysis approach to two prediction models based on the altitude difference, and it was concluded that the prediction results of the CRSRI model were more accurate than those of the Sadovsky model. Finally, we introduced the magnification of peak vibration, so that the threshold altitude difference of the magnification effect of a negative slope could be quantified.

2. Analysis of the Mechanism of the Negative Slope Effect

2.1. Generation Mechanism of Seismic Waves

When explosives are detonated in rocks, a large amount of energy is instantaneously released. The generated energy is dissipated by shock wave overpressure, compressive stress waves, as well as seismic waves. This causes the formation of a crushing zone, fracture zone, and vibration zone spreading outward from the explosion center (Figure 1). Within the vibration zone, seismic waves will cause elastic vibrations of the rock matrix, resulting in damage or even destruction of the matrix structure. Therefore, it is important to investigate the propagation mechanisms of blast waves in the vibration zone [29].

2.2. Basic Classification of Seismic Waves

The propagation of seismic waves in various media is affected by many factors, and the variety of seismic waves is extremely complex. Seismic waves propagating within a medium are called body waves, while those propagating near the surface are known as surface waves. The body waves can either be longitudinal or transverse. The longitudinal waves, whose propagation direction is aligned with the vibration direction of the medium, cause tensile and compressive damage to the medium, while the transverse waves, which propagate perpendicularly to the medium vibration direction, cause shear damage. Surface waves, on the other hand, are generally considered secondary waves which are generated by multiple reflections of the body waves from stratigraphic interfaces. The surface waves include Love waves and Rayleigh waves [30] (Figure 2).

2.3. Propagation Mechanism of Seismic Waves along a Negative Slope

According to the Fermat principle, the blast-generated seismic waves propagating in the negative slope direction will travel along the shortest path between the blast center and a measurement point, instead of a straight line [31] (Figure 3).

When the seismic wave propagates along a negative slope, the incident wave will be completely reflected from the slope, thus forming a reflective stretching area. Assuming that the slope is made of a homogeneous matrix, the propagation velocity of the incident wave can be expressed accordingly [32] as follows:

The propagation velocity of the reflected wave can be expressed as follows:

By integrating the vector sum of the propagation velocities of the incident and the reflected wave, the displacement of slope particles can be calculated as follows:

In equations (1)–(3), A₁ is the amplitude of the incident wave vibration velocity, is the propagation velocity of the incident wave, and is the propagation velocity of the reflected wave. It can be observed from equation (3) that a superposition of the incident and the reflected wave occurs when they travel along the negative slope direction, resulting in a local increase in vibration intensity.

3. Predictive Models for Negative Slopes and Their Coefficients

Sadovsky’s and CRSRI formulas are widely used in practical activities to predict blast vibration velocities. In this study, regression analysis is applied to these two common models to obtain the relative errors between the two prediction models.

3.1. Sadovsky’s Predictive Model Coefficients

where is the particle peak vibration velocity, m/s; Q represents the amount of explosive, kg; R is the distance from the observation point to the explosion center, m; K is a coefficient related to topographic and geological conditions; α is the attenuation coefficient.

By applying the logarithm for both sides of equation (4), the following formula is obtained:

If we consider that , a = lgK, b = α, and x = lg , (5) can be transformed into a single-variable linear function as follows:

The least squares method can be used to calculate the coefficients a and b as follows:

3.2. CRSRI Predictive Model Coefficients

where H is the bench height; β is the influence coefficient of elevation difference; the other parameters have the same definition as in the previous equations.

By applying the logarithm on both sides of equation (8), the following formula is obtained:

If we consider that z = lg, a = lgK, b = α, x = lg , c = β, and y = lg(), equation (9) can be transformed into a two-variable linear function as follows:

The least squares method can then be used to obtain the following equations:

When A ≠ 0, the only valid solution for M can be obtained by solving the system of linear equations as follows:

4. Case Study

4.1. Geographical Location

The test site is in Guiyang City, Guizhou Province, China. The surroundings are complex, mainly comprising the low Zhongshan erosion residual hills and depressions. Overall, the terrain is relatively flat, and the ground elevation varies between 1330 m and 1350 m, that is, the maximum height difference is 20 m.

Based on information from drilling logs and field observations, the site profile consists of soil, red clay, and plain fill. The thickness of the upper layer is 0.5–2 m, with an average thickness of 1.0 m. The bedrock is the first and the second section of Tsz1 + 2. The layers are consisted of argillaceous dolomite, claystones, tuffaceous claystones. The rocks are usually fractured. The site is located 60 m away from the Guizhou e-commerce industrial park to the east, 65 m away from a small-sized pig shed to the south, 72 m away from a single residential house, and 113 m away from the Binyang Road to the southeast, in proximity to residential areas to the west, and 331 m away from residential houses to the northwest. There are industrial parks, roads, high-intensity residential housing areas, and other similar elements located around the blasting area (Figure 4) [33].

4.2. Blasting Procedures

According to the on-site geological conditions and blasting requirements, this test adopts a multirow deep-hole blasting method, a hole-by-hole initiation technology, and the relevant [34, 35]. Rock powdery emulsion explosives and bulk ammonium nitrate explosives are used in the field, and the specific blasting parameters are shown in Tables 1 and 2.

This design points out that it is allowed to use the above parameters in the indicated area only if the maximum single dose of explosives is less than 21 kg. The specific delay design is shown in Table 3.

In multirow deep-hole blasting, the triangle hole arrangement is selected on the step flat plate, as shown in Figure 5.

The initiation sequence is illustrated in Figure 5. The numbers in the figure represent the delay time. Based on the delay network, the movement and the throwing direction of the explosive pile are indicated in Figure 6.

The network has a parallel connection. It registers the electronic detonator, as well as the number of the main line of the network. The electronic detonator clip of each hole is connected to the main line and is firmly attached to it. After the registration is completed, the detonation network extension is pressed. The time difference can be set manually, and a special detonator is used for charging and detonation. The schematic diagram of the connection between the electronic detonators is illustrated in Figure 7.

4.3. Field Monitoring

TC-4850 and NUBOX-8016 blast vibration meters were used during on-site operations. Both instruments can collect data from tangential, vertical, and radial channels (Figure 8).

Two rounds of data collection were carried out when the test was performed. Five measurement points were selected for each round, and ten field data records were obtained, as shown in Table 4.

4.4. Analysis of Blast Vibration Monitoring Results

Group 1 experimental data were selected from Table 4 to plot the vertical, tangential, and radial peak velocities according to the distance from the blast center (Figure 9).

It can be observed from Figure 9 that for the studied terrain with a negative slope, the peak velocities of blast vibrations in each of the three directions generally decreased with the increase in the distance from the blast center. However, when the distance from the blast center was equal to 150 m and 80 m, the peak velocities of blast vibration showed local magnification in the vertical and tangential direction, respectively. The magnification effect in the vertical direction was bigger than that in the tangential direction.

5. Analysis of the Negative Slope Effect on Blast Vibrations

The negative slope magnification effect on blast vibration velocity was analyzed in detail by considering that the magnification effect in the vertical direction was more pronounced. The first mid-vertical monitoring data were selected for the linear regression analysis of the Sadovsky and the CRSRI formulas.

5.1. Regression Analysis of Sadovsky’s Predictive Model

The results of the linear regression approach for Sadovsky’s formula are illustrated in Figure 10.

By including the regression parameters into equations (5) and (6), equation (4) can be expressed as

5.2. Regression Analysis of the CRSRI Predictive Model

To obtain a more accurate blast vibration prediction model, the linear regression approach was applied to the CRSRI formula, and the resulting model coefficients are illustrated in Table 5.

By including the parameters from Table 4 into equations (9) and (10), equation (8) can be expressed as

The experimental data from Table 4 are included in equations (13) and (16) to obtain predicted values by using the Sadovsky and the CRSRI formulas. The relative errors of the two formulas were calculated asand are illustrated in Table 6.

According to Figure 10 and Table 6, the R² value for Sadovsky’s prediction model is 0.526, with the average relative error of the predicted velocity equal to 28.47%. This indicates a poor correlation and large errors for this prediction model. In contrast, according to Tables 3 and 4, the R² value for the CRSRI prediction model is 0.918, and the average relative error is equal to 17.53%, which means that the CRSRI prediction model of blast vibration is more accurate for negative slopes.

5.3. Analysis of Peak Velocity Magnification for the Negative Slope

The peak velocity magnification parameter (ε) was used to calculate the threshold slope difference for the negative slope magnification effect. Since the values of and are always positive, it is observed from equation (16) that when > , then values where ε > 0 result in the slope magnification effect. The formula for ε is expressed as [36]where is the peak velocity calculated by the CRSRI prediction model, and is the peak velocity calculated by the Sadovsky’s prediction model. The peak blast vibration velocities predicted using the Sadovsky and the CRSRI models listed in Table 4 were introduced into equation (16) to calculate the peak velocity magnification at the measurement points (Table 7). This allowed establishing the relationship between the peak vertical velocity magnification and the altitude difference (Figure 11). For visualization purposes, all negative altitude differences were plotted as absolute values.

As seen in Figure 11, except for the edge point (where the altitude difference was −4 m, and the peak velocity magnification was 0.3970 > 0), only when the altitude difference was −8.5 m, the peak velocity magnification was 0.4573 > 0, and the peak velocity magnification corresponding to the other altitude differences was smaller than 0 in all cases. This means that the altitude magnification in the vertical direction occurred solely when the altitude difference reached −8.5 m, that is, the altitude threshold for the altitude magnification effect was equal to −8.5 m. According to Figure 9, the magnification effect of the negative slope was produced only for certain altitude differences, and it was more pronounced in the vertical direction.

6. Conclusions

This study applied the wave theory to analyze the propagation mechanism of blast waves along negative slopes. The linear regression approach was used for on-site data for the Sadovsky and the CRSRI blast vibration velocity prediction models. To compare their relative errors, the magnification of peak blast vibration velocity along a negative slope was introduced to determine the threshold altitude difference for the magnification effect to occur. The conclusions are as follows:(1)In terrains with a negative slope, the superposition of incident and reflected blast vibration waves on the slope results in the local amplification of blast vibration velocity.(2)The influence of the negative slope on the peak blast vibration velocity should be acknowledged. The predictions of the peak blast vibration velocity using the CRSRI formula are more accurate when compared to the other formulas.(3)The magnification effect of the negative slope appears only for certain altitude differences above a threshold and is more obvious for the vertical vibration velocity.

Data Availability

The data used to support the findings of the study are available within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was financially supported by the Youth Science and Technology Talent Growth Project of Guizhou Provincial Department of Education (Grant No. ky[2018]414) and Chinese National Natural Science Foundation (51664007). The authors would like to express their gratitude to EditSprings (https://www.editsprings.cn/) for the expert linguistic services provided.

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Copyright © 2022 Jing Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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