Research Article  Open Access
Xu Wu, Qifeng Guo, Yunpeng Zhang, "A New Mathematical Model for Predicting the Surface Vibration Velocity on the Step Topography", Shock and Vibration, vol. 2018, Article ID 8356254, 8 pages, 2018. https://doi.org/10.1155/2018/8356254
A New Mathematical Model for Predicting the Surface Vibration Velocity on the Step Topography
Abstract
In order to control production blasting and optimize mining operation, it is important to study the step topography vibration amplification and attenuation effect in the openpit mine. Surface particle vibration velocity attenuation characteristics, the formation, and change rules of the terrain effect were studied by analyzing the field measured data. Results show that local amplification effect and local attenuation amplification effect of particle vibration velocity are obvious. Amplification effect associates with bench height, and the attenuation effect is closely related with the distance from the vibration source and distance from the top. With the increase of elevation, vibration magnification of the particle on the top was 1.1∼1.4. Because of the influence of the terrain effect, particle vibration velocity on the slope toe was obviously inhibitory. Based on the measured data, elevation amplification factor and clamping effect factor which influence blasting vibration velocity are put forward, and a new mathematical model considering the attenuation coefficient, the elevation amplification coefficient, and the clamping effect coefficient for predicting the blasting vibration velocity of the step topography is further improved. The regression analysis results show that the fitting coefficient of determination of the new prediction model is 0.8152 in horizontal and 0.8902 in vertical, respectively, and the prediction error is less than 20%, which is much better than other formulas. This new model provides effective reference for blasting seismic wave propagation law research of slope engineering.
1. Introduction
With the exploitation and utilization of mineral resources, a large number of open mining pits have been formed on the surface, and some mines have entered the deep mining stage. Blasting is still the main method for exploiting the resource; however, only 20%∼30% of the explosive energy is used for breaking rock, and the rest is wasted in negative areas such as ground vibration, air blast, and fly rock [1–6]. Ground vibration induced by blasting is the major hazard for the structural safety of surrounding buildings and for the stability of the high slope. Because of the complexity of blasting and the difference of propagation medium, the prediction of blasting vibration velocity is always the technical problem urgently needed to be solved [7]. To ensure the smooth operation of the slope engineering and avoid social and economic problems created by blasting, predictive and preventive measures are the essential components [8]. Thus, it is imperative to predict the vibration level and put forward the blasting parameters prior to the operations for better safety and productivity [9].
Many researchers have devoted to the study of blasting vibration propagation law and suggested various methods to forecast the ground vibration and minimize the hazard level. Ground vibration is directly related to the quantity of explosive, distance between blast hole and monitoring point, and the geological conditions. Topographic condition especially terrain elevation becomes one of the important factors affecting the vibration velocity in the slope terrain [10–14]. Peak particle velocity used as the measure index of blasting vibration effect had been estimated based on two main and commonly used parameters (maximum charge per delay and the distance between blast hole and monitoring point) [15–21]. However, a lot of field monitoring data displayed that terrain elevation has a marked impact on PPV. The amplification of vibration velocity can be 1.5∼3 times [22–25]. Elevation amplification effect is very important to study the propagation characteristics of blasting seismic wave and the slope safety protection measures. A few predictors took the elevation into consideration; it agrees that positive elevation increases the velocity, on the contrary will reduce [26–29]. There have been few researchers focusing on the clamping effect of step topography, although it exists and always appears.
Over the past few years, artificial intelligent (AI) techniques such as artificial neural networks (ANN) and support vector machine (SVM) which had put total charge, distance, max charge per delay, frequency, blasting parameters, etc, into the prediction model have been used in mining and civil engineering [30–32], but the clamping effect has been not considered. And the recent field monitoring data showed that on the step surface blasting vibration, velocity was influenced by the height of the step and the clamping effect controlled by the distance from the step top.
In the past, the study of vibration amplification effect mainly focuses on the field of earthquake. Since the 1970s, researchers had observed the vibration amplification effect on the ridge top or cliff top in many earthquake events such as San Fernando earthquake in 1971, Central Chile earthquake in 1985, Loma Prieta earthquake in 1989, Northridge earthquake in 1994, and Wenchuan earthquake in 2008, and the amplification effect of topography had been found by experts in the analysis of seismic data [33–37]. Generally, buildings at the top of the steep slope suffered more serious damage. Experts had began a longterm theoretical and field study on the topographic amplification effect of earthquake events and had obtained some qualitative results; however, it is still insufficient in quantitative research, the topographic effects of vibration could not be predicted accurately, and efficient mathematical prediction models were not be established yet.
In recent years, Chinese researchers are beginning to pay more attention to the topographic effects of artificial blasting seismic waves. Vibration observation data had been obtained from the openpit mine. The researchers arranged the monitoring points on the different steps to obtain the influence law of elevation on vibration velocity, but the vibration velocity data on the same step were not obtained and analyzed [27–29]. The vibration velocity prediction formulas were very limited and not suitable for every position of the step topography. Therefore, previous relevant studies could not describe the attenuation law of vibration velocity of step topography and could not effectively provide the protection suggestions and guide the equipment installation.
In the paper, different empirical formulas have been used to predict PPV, and a new multifactor vibration velocity prediction model with high accuracy has been established.
2. Case Study
The vibration monitoring test was carried out at the Sijiaying Iron Mine which belongs to Hebei Iron and Steel Group Mining Co., Ltd. and locates to the east of Hebei province, in 118°45′30″ longitude and 39°38′∼39°42′ latitude and at 30 m above sea level. It has been extracting six million tons of raw iron per year. The rocks in that area comprise granulite, schist, and quartzite. Granulite is the main orebearing rock. The area belongs to the southwest edge of Yanshan folded belt, which is principally composed of Sinian and Quaternary. The main mineral of the deposit is magnetite, and the proved reserve is approximately 2.348 billion tons with an average grade of 31.09%. The mine is exploited by openpit mining in the early stage and covers an area of 2.7797 million square meters with 1630 m long from north to south and 1500 m wide from east to west. The height and slope of working benches are 12 m∼15 m and 65°, respectively. The maximum height of final slope is about 432 m.
Vibration monitoring instrumentation was TC4850, as shown in Figure 1, and it was made up of acquisition equipment and three vector speed sensor. The main technical indicators are the frequency range X/Y: 1∼300 Hz, Z: 1∼500 Hz, the sensitivity 28 V/m/s, and the harmonic distortion ≤0.1%.
(a)
(b)
The monitoring test was conducted twice in Figure 1; the total amount of explosive, respectively, was 14200 kg and 16710 kg, the maximum charge per delay was 750 kg, and the remaining parameters are listed in Table 1. Measuring points are located at the toe and the top of adjacent steps. The blasting vibration observation results are shown in Table 2.


3. Prediction by Conventional Predictors
Table 3 illustrates the various conventional vibration prediction formulas widely used in different countries or put forward by different researchers. The attenuation coefficient and site coefficient can be obtained from the data regression analysis, and the calculated values of site and attenuation coefficient in different formulas are shown in Table 4 and in Figure 2.
 
Note. V is the PPV, cm/s; Q is the maximum charge per delay, kg; R is the distance between blast hole and measuring point, m; K is the site coefficient; and α is the attenuation coefficient. 
 
Note. CoD is the fitting coefficient of determination. 
(a)
(b)
However, in the slope engineering, the above empirical formula results have large errors and low reliability. Field monitoring data show that the elevation of the terrain has obvious amplification effect on the vibration velocity. Zhu and Zhou [28, 29] had partially corrected the empirical formula, introducing the elevation variable and established the vibration velocity prediction model with terrain elevation shown in Table 5.
 
Note. H is the terrain elevation, m; β is a coefficient related to terrain elevation difference. When the location of measuring point is higher than the explosion source, beta is positive; on the contrary, it is negative. 
The formulas which add the elevation H as the variable in Table 5 are based on Sadaovsk formula, the effect of terrain elevation on vibration velocity is reflected, and the terrain effect of vibration velocity can be described more accurately.
Figure 3 shows the relationship between scaled distance and PPV described by the different empirical formulas. The CoD is minimum for Japan formula, in which the scale distance is linear with PPV, and the constant term is zero. According to the formulas in Table 5, the data are analyzed with binary regression, and the fitting results are shown in Table 6.
(a)
(b)
(c)
(d)
(e)
(f)

The fitting coefficients of determination in Table 6 are higher than that in Table 4. To some extent, the models with elevation factor demonstrate the amplification effect of terrain elevation on vibration velocity, the value of β obtained by fitting also indicates that the terrain elevation has an enhancement effect on the vibration velocity. But the prediction accuracy of vibration velocity has not improved significantly, and the CoD is still low. According to the field monitoring data in Table 2, the propagation of the PPV along the step surface is not only influenced by the distance attenuation effect and elevation amplification effect. The PPV at the toe position of the same elevation attenuates rapidly. The measurement point 1 is compared with point 3, and PPV of point 3 with higher elevation is lower; therefore, it can be obtained that PPV at the toe of slope is restricted by rock mass, and amplitude of PPV drops sharply.
4. Prediction by New Model
Considering the PPV changing characteristics of the top and toe of the steps, the clamping effect factor and elevation amplification factor were introduced in the new prediction model, and it is as follows:where is the PPV, cm/s; is the maximum charge per delay, kg; is the distance between blast hole and measuring point, m; is the site coefficient; is the attenuation coefficient, ; is the elevation amplification coefficient, ; is the distance between measuring point and the top of step at the same elevation; and is the clamping effect coefficient, .
The fitting results of the new prediction model are shown in Table 7.

The PPV mathematical prediction model containing the elevation amplification factor and clamping effect factor can respond the propagation rule of the surface vibration velocity more accurately, the CoD is higher than 0.8, and predicting results are more reliable and more consistent with the monitoring data. Six more sets of vibration data have been selected for comparison with the calculation results of the new formula. The new data and comparison results are shown in Tables 8 and 9. Obviously, prediction by new model is closer to the measured velocities, the PPV errors of these common formulas are about 17%∼41% in horizontal and about 29%∼33%% in vertical, and by contrast, the errors respectively are about 9.8% and 10.34%; the comparative results are shown in Figure 4.


In conclusion, the new prediction model can fully describe the topographic elevation effect and the clamping effect and accurately predict the PPV propagating along the surface on the steps. The new formula can evaluate the vibration of the step topography at Sijiaying Iron Mine. The new formula is more suitable for prediction of blasting vibration in this area than other formulas.
5. Conclusions
The vibration velocity of the surface on the steps is related to the topographic elevation, the blasting distance, and the distance between measuring point and the top of slope. The PPV on the top has obvious amplification effect, and the magnification factor is 1.15∼1.39 times, and oppositely, the PPV which is limited on the toe of step decreases rapidly. So, vibration velocity is a complex product of multiple factors.
PPV prediction formulas are compared and analyzed based on the measured data, and the fitting results show that the new model can get the higher CoD and the minimum error.
The new prediction model is aimed at the step topography, and it is more intuitive and accurate to reflect the elevation amplification effect and the rock clamping effect on the step terrain. The new prediction model is more consistent with the change rule of the measured data, and the guiding significance of the slope blasting operation is more targeted.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The research was supported by the National Key R&D Plan (2017YFC0804101).
References
 M. Monjezi, M. Hasanipanah, and M. Khandelwal, “Evaluation and prediction of blastinduced ground vibration at Shur River Dam, Iran, by artificial neural network,” Neural Computing and Applications, vol. 22, no. 78, pp. 1637–1643, 2013. View at: Publisher Site  Google Scholar
 M. Khandelwal, P. K. Kankar, and S. P. Harsha, “Evaluation and prediction of blast induced ground vibration using support vector machine,” Mining Science and Technology, vol. 20, no. 1, pp. 64–70, 2010. View at: Publisher Site  Google Scholar
 M. Khandelwal, T. N. Singh, and S. Kumar, “Prediction of blast induced ground vibration in opencast mine by artificial neural network,” Mineral Engineering, vol. 9, p. 23, 2005. View at: Google Scholar
 T. N. Singh and A. K. Verma, “Sensitivity of total charge and maximum charge per delay on ground vibration,” Geomatics, Natural Hazards and Risk, vol. 1, no. 3, pp. 259–272, 2010. View at: Publisher Site  Google Scholar
 Z. Leng, W. Lu, M. Chen et al., “Improved calculation model for the size of crushed zone around blasthole,” Explosion and Shock Waves, vol. 35, no. 1, pp. 101–107, 2015. View at: Google Scholar
 H. Inanloo Arabi Shad, F. Sereshki, M. Ataei, and M. Karamoozian, “Investigation of rock blast fragmentation based on specific explosive energy and insitu block size,” International Journal of Mining and GeoEngineering, vol. 5, no. 1, pp. 1–6, 2018. View at: Publisher Site  Google Scholar
 X. Zhang, H. Yi, X. Guoshuai et al., “Influence of elevation on the blasting vibration law in the slope of an openpit mine,” Metal Mine, vol. 46, no. 7, pp. 55–59, 2017. View at: Google Scholar
 M. Khandelwal and T. N. Singh, “Prediction of blast induced air overpressure in opencast mine,” Noise & Vibration Worldwide, vol. 36, no. 2, pp. 7–16, 2005. View at: Publisher Site  Google Scholar
 M. Khandelwal and T. N. Singh, “Evaluation of blast induced ground vibration predictors,” Soil Dynamics and Earthquake Engineering, vol. 27, no. 2, pp. 116–125, 2007. View at: Publisher Site  Google Scholar
 F. Marrara and P. Suhadolc, “Site amplifications in the city of Benevento (Italy) comparison of observed and estimated ground motion from explosive sources,” Journal of Seismology, vol. 2, no. 2, pp. 125–143, 1998. View at: Publisher Site  Google Scholar
 H. B. Havenith, M. Vanini, D. Jongmans, and E. Faccioli, “Initiation of earthquakeinduced slope failure: influence of topographical and other site specific amplification effects,” Journal of Seismology, vol. 7, no. 3, pp. 397–412, 2003. View at: Publisher Site  Google Scholar
 G. Vladimir, “Lowvelocity zone and topography as a source of site amplification effect on Tarzana hill, California,” Soil Dynamics and Earthquake Engineering, vol. 29, no. 2, pp. 324–332, 2009. View at: Google Scholar
 W. U. Xu, Y. Zhang, and Q. Guo, “Amplification and attenuation effect of blasting vibration on step topography,” Explosion and Shock Waves, vol. 37, no. 6, pp. 1017–1022, 2017. View at: Google Scholar
 M. Chen, W. Lu, L. I. Peng et al., “Elevation amplification effect of blasting vibration velocity in rock slope,” Chinese Journal of Rock Mechanics and Engineering, vol. 30, no. 11, pp. 2189–2195, 2011. View at: Google Scholar
 W. I. Duvall and B. Petkof, Spherical Propagation of Explosion of Generated Strain Pulses in Rocks, USBM RI5483, Bureau of Mines, Washington, DC, USA, 1959.
 Indian Standard Institute, Criteria for Safety and Design of Structures Subjected to Underground Blast, ISI Bulletin IS6922, Indian Standard Institute, New Delhi, India, 1973.
 G. Hu, Forecast on Intensity of Vibration and Study on Effect of Ground Blast Induced Vibration of Underground Engineering in City, Chongqing University, Chongqing, China, 2005.
 U. Langefors, B. Kihlstrom, and H. Westerberg, “Ground vibrations in blasting,” Water Power, vol. 9, pp. 335–338, 1958. View at: Google Scholar
 N. R. Ambraseys and A. J. Hendron, Dynamic Behaviour of Rock Masses Rock Mechanics in Engineering, Practices, Wiley, Hoboken, NJ, USA, 1968.
 H. Yu and S. Yu, “Study on forecasting intensity of ground motion induced by engineering explosion,” Earthquake Engineering and Engineering Vibration, vol. 24, no. 1, pp. 122–129, 2004. View at: Google Scholar
 K. Ali, “Analysis of parameters of ground vibration produced from bench blasting at a limestone quarry,” Soil Dynamics and Earthquake Engineering, vol. 24, no. 11, pp. 887–892, 2004. View at: Publisher Site  Google Scholar
 Y. Zhang, X. Yang, L. I. Yan et al., “Numerical simulation analysis for the influence of slope angles on elevation blasting amplification,” Mining R&D, vol. 37, no. 7, pp. 23–25, 2017. View at: Google Scholar
 G. Liu, J. Zhou, and H. Yu, “Study on amplification effects of blasting vibration under positive elevation,” Mining R&D, vol. 35, no. 12, pp. 84–87, 2015. View at: Google Scholar
 B. Fu, Y. Hu, W. Lu et al., “Local amplification effect of blasting vibration in high rock slope,” Blasting, vol. 31, no. 2, pp. 1–7, 2014. View at: Google Scholar
 M. Chen, W. Lu, and L. Wu, “Safety threshold of blasting vibration velocity to high rock slope of Xiaowan hydropower station,” Chinese Journal of Rock Mechanics and Engineering, vol. 26, no. 1, pp. 51–56, 2007. View at: Google Scholar
 J. E. Field, S. M. Walley, W. G. Proud, H. T. Goldrein, and C. R. Siviour, “Review of experimental techniques for high rate deformation and shock studies,” International Journal of Impact Engineering, vol. 30, no. 7, pp. 725–775, 2004. View at: Publisher Site  Google Scholar
 H. Tang and H. Li, “Study of blasting vibration formula of reflecting amplification effect on elevation,” Rock and Soil Mechanics, vol. 32, no. 3, pp. 820–824, 2011. View at: Google Scholar
 C. Zhu and H. Liu, “Selection of formula on propagation of the parameters of explosive seismic wave along slope,” Blasting, vol. 5, no. 2, pp. 30–34, 1988. View at: Google Scholar
 T.L. Zhou and L. I. YuShou, “Application of blasting vibration formula related to highness,” Jiangsu Coal, vol. 22, no. 4, pp. 2122, 1997. View at: Google Scholar
 M. Khandelwal and T. N. Singh, “Prediction of blast induced ground vibration using artificial neural network,” International Journal of Rock Mechanics and Mining Sciences, vol. 46, no. 7, pp. 1214–1222, 2009. View at: Publisher Site  Google Scholar
 M. Monjezi and H. Dehghan, “Evaluation of effect of blasting pattern parameters on back break using neural networks,” International Journal of Rock Mechanics and Mining Sciences, vol. 45, no. 8, pp. 1446–1453, 2008. View at: Publisher Site  Google Scholar
 M. Monjezi, M. Ahmadi, and M. Sheikhan, “Predicting blastinduced ground vibration using various types of neural networks,” Soil Dynamics and Earthquake Engineering, vol. 30, no. 11, pp. 1233–1236, 2010. View at: Publisher Site  Google Scholar
 D. M. Boore, “A note on the effect of simple topography on seismic SH waves,” Bulletin of the Seismological Society of America, vol. 62, no. 1, pp. 275–284, 1972. View at: Google Scholar
 M. Celebi, “Topographic and geological amplification determined from strongmotion and aftershock records of 3 March 1985 Chile earthquake,” Bulletin of the Seismological Society of America, vol. 77, no. 4, pp. 1147–1167, 1987. View at: Google Scholar
 S. H. Hartzell, D. L. Carver, and K. W. King, “Initial investigation of site and topographic effects at Robinwood Ridge, California,” Bulletin of the Seismological Society of America, vol. 84, no. 5, pp. 1136–1149, 1994. View at: Google Scholar
 S. A. Ashford and N. Sitar, “Topographic amplification in the 1994 Northridge earthquake analysis and observations,” Earthquake Engineering, vol. 34, no. 1, pp. 210–214, 1997. View at: Google Scholar
 Y. Luo and Y. Wang, “Mountain slope ground motion topography amplification effect induced by Wenchuan earthquake,” Journal of Mountain Science, vol. 31, no. 2, pp. 200–210, 2013. View at: Google Scholar
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Copyright © 2018 Xu Wu 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.