# Models of Determining the Parameters of Rock Mass Oscillation Equation with Experimental and Mass Blastings

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Recording Rock Mass Oscillation Velocity

- Type and designation of instrument: Vibraloc
- Manufactured: in Sundbyberg, Sweden
- Measuring possibilities: speed, acceleration displacement
- Number of channels: three channels—speed, fourth channel—air channel
- Number of components: three—transversal, vertical, side
- Frequency range: 2–250 Hz
- Location possibility: flat floors, boards, foundations, ground
- Trigger level, V, L, T (trigger levels): 0.1–200 mm/s
- Channel trigger level A (microphone): 2–150 Pa
- Sampling: 1000, 2000 or 4000 Hz
- Recording length: 1–100 s or automatic length
- Data transfer: Vibraloc PC software
- Data analysis: UVSZ and UVSZA software (Vibraloc, Sundbyberg, Sweden)

- horizontal component v
_{t}, normal to the direction of wave spreading (transversal component) - horizontal component v
_{l}, in direction of wave spreading (longitudinal component) and - vertical component v
_{v}.

## 3. Equation of Rock Mass Oscillation

#### 3.1. Derivation of Equation of Rock Mass Oscillation

_{0}and the distance from the blast site to monitoring point r increase in the same or approximately the same ratio, the soil oscillation velocity v remains the same. Sadovskii equation, given in Equation (1) may be expressed in the form [7]:

_{v}—the coefficient conditioned by rock mass characteristics as well as blasting conditions where the explosive amount is given through radius of the explosive charge. K is being determined by terrain surveying,

_{0}—the radius of the explosive charge,

_{0}and the amount of explosive Q are related by the equation:

_{0}from the Equation (4), in the Equation (3), we obtain:

#### 3.2. The Property of the Rock Mass Oscillation Equation

#### 3.3. Models to Determine the Parameters of Equation of Rock Mass Oscillation Velocity

#### 3.3.1. Model 1—Determining the Parameters by Applying the Least Square Method

#### 3.3.2. Model 2—Determining the Parameters through the Application of Quotient of Products of the Same Number of Reduced Distances and Oscillation Velocities

_{1}, v

_{2}, …, v

_{N}be the rock mass oscillation velocity values obtained for corresponding reduced distances R

_{1}, R

_{2}, …, R

_{N}. Proceeding from the equation for the rock mass oscillation velocity, given by the Equation (1) we can assume, that for pairs (R

_{m}, v

_{m}) the following relations are valid:

_{m}, v

_{m}), m = 1, 2, …, N, into the Equation (1), we will obtain one after the other:

#### 3.3.3. Model 3—Determining the Parameters Using the Linear Connection between the Logarithms of Reduced Distances and the Logarithms of the Oscillation Velocity

_{i}, logv

_{i}) will be below logR axis. In this case, we will increase the values of the logv ordinates by a positive number, so that all points are above the abscissa axis logR [11].

_{1}, v

_{2}, ..., v

_{N}and appropriate reduced distances R

_{1}, R

_{2}, ..., R

_{N}and having in mind relations (22) and (27), we will calculate the value of S

_{T}according to Equation (29). For each pair (x

_{i}, y

_{i}+ 3) from Equation (35), we will determine the parameters A

_{i}, i = 1, 2, ..., N, and in so doing, for parameter A, we will take the arithmetic mean value of parameters A

_{i}:

#### 3.4. Defining Statistical Criteria

- 0.0 < ρ < 0.2—none or highly poor correlation,
- 0.2 < ρ < 0.4—poor correlation,
- 0.4 < ρ < 0.7—significant correlation,
- 0.7 < ρ < 1.0—strong or highly strong correlation.

## 4. Methodology and Results of Research

- Experimental research carried out in Majdanpek open pit, situated in northern part of east Serbia and,
- Investigations carried out during mass blasting in Nepričava open pit, located in central Serbia.

#### 4.1. Experimental Investigation in Majdanpek Open Pit

#### 4.1.1. General Characteristics of the Work Environment in Majdanpek Open Pit

- compressive strength, σ
_{p}= 109 MPa - bending strength, σ
_{s}= 19 MPa - tensile strength, σ
_{I}= 13 MPa - cohesion, C = 7.5 MPa
- strength coefficient, f = 11
- volume density, γ = 26 kN/m
^{3} - angle of internal friction, φ = 50°
- porosity, p = 3.6%

#### 4.1.2. Method of Blasting in Majdanpek Open Pit

- depth of borehole: 0.5 m,
- weight of explosive charge per borehole: 0.10 kg,
- number of boreholes: 10,
- distance between boreholes: 1.0 m,
- distance between measuring point and the first borehole: 5.0 m,
- time delay between initiation of boreholes: 0.5 s.

#### 4.1.3. Calculation of Parameters of Rock Masses Oscillation Equation in Majdanpek Open Pit

_{t}, v

_{v}, v

_{l}and resulting rock mass oscillation velocities v

_{rez}for a total of ten explosions, are given in Table 1.

_{r}, calculated oscillation velocities v

_{i}

_{1}, v

_{i}

_{2}, v

_{i}

_{3}as well as the difference between recorded and calculated soil oscillation velocities for models 1, 2 and 3.

- for model 1:The curved line dependency index ρ
_{1}between the reduced distance R and rock mass oscillation velocity v is:ρ_{1}= 0.9730 (there is a strong correlation between R and v, given in the Equation (44)), - for model 2:ρ
_{2}= 0.9694 (there is a strong correlation between R and v, given in the Equation (45)), - for model 3:ρ
_{3}= 0.9789 (there is a strong correlation between R and v, given in the Equation (54)).

#### 4.2. Investigation during Mass Mining in Nepričava Open Pit

#### 4.2.1. General Characteristics of Nepričava Open Pit

- Compressive strength 21.1–93.0 MPa
- Volume weight 17.1–26.3 kN/m
^{3} - Density 0.97–0.99 kN/m
^{3} - Porosity 0.0004%–0.026%
- Water suction 0.08%–0.23%

#### 4.2.2. Blasting Method

_{b}, the overall explosive amount Q

_{uk}, the maximal explosive amount by deceleration interval Q

_{i}, overall borehole depth L

_{uk}and average stemming length L

_{pc}, are presented in Table 4.

#### 4.2.3. Calculation of Rock Mass Oscillation Equation Parameters

_{t}, v

_{v}, v

_{l}and resulting oscillation velocities v

_{rez}for blasting from I to III of totally five measuring points MM are given in Table 5.

_{r}, calculated oscillation velocities v

_{i}

_{1}, v

_{i}

_{2}, v

_{i}

_{3}, as well as the difference between recorded and calculated rock mass oscillation velocities for models 1, 2 and 3.

- for model 1:The curved line dependency index ρ
_{1}between the reduced distance R and rock mass oscillation velocity v is:ρ_{1}= 0.7786 (there is a strong correlation between R and v, given in the Equation (55)). - for model 2:ρ
_{2}= 0.7831 (there is a strong correlation between R and v, given in the Equation (56)). - for model 3:ρ
_{3}= 0.6937 (there is a significant correlation between R and v, given in the Equation (65)).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Recording of oscillation velocity of the ground intended for blasting No. I—measuring point No. 3 at PK Nepričava.

**Figure 3.**Graphic representation of values logR and logv + 3 for calculating the parameters in Majdanpek open pit

**Figure 5.**Graphic representation of values logR and logv + 3 for calculating the parameters in Nepričava open pit.

No. | Blasting | r (m) | Q (kg) | R | v_{t} (cm/s) | v_{v} (cm/s) | v_{l} (cm/s) | v_{rez} (cm/s) |
---|---|---|---|---|---|---|---|---|

1 | I | 5.0 | 0.1 | 10.7722 | 0.700 | 0.900 | 1.150 | 1.6194 |

2 | I | 6.0 | 0.1 | 12.9266 | 0.300 | 0.925 | 1.150 | 1.5060 |

3 | I | 7.0 | 0.1 | 15.0810 | 0.255 | 0.540 | 0.850 | 1.0388 |

4 | I | 8.0 | 0.1 | 17.2355 | 0.260 | 0.350 | 0.650 | 0.7827 |

5 | I | 9.0 | 0.1 | 19.3899 | 0.245 | 0.325 | 0.550 | 0.6842 |

6 | I | 10.0 | 0.1 | 21.5443 | 0.155 | 0.240 | 0.405 | 0.4956 |

7 | I | 11.0 | 0.1 | 23.6988 | 0.205 | 0.275 | 0.440 | 0.5579 |

8 | I | 12.0 | 0.1 | 25.8532 | 0.110 | 0.200 | 0.285 | 0.3651 |

9 | I | 13.0 | 0.1 | 28.0077 | 0.085 | 0.190 | 0.230 | 0.3102 |

10 | I | 14.0 | 0.1 | 30.1621 | 0.090 | 0.130 | 0.200 | 0.2550 |

**Table 2.**Representation of data for calculating the parameters in Majdanpek open pit according to Model 3.

R_{i} | x_{i} | v_{i} | y_{i} | y_{i} + 3 | h_{i} = x_{i+1} − x_{i} | (y_{i} + 3 + y_{i+1} + 3)h_{i} |
---|---|---|---|---|---|---|

10.7722 | 2.376969 | 1.6194 | 0.482056 | 3.482056 | 0.182318 | 1.256447 |

12.9266 | 2.559287 | 1.5060 | 0.409457 | 3.409457 | 0.154149 | 0.993879 |

15.0810 | 2.713436 | 1.0388 | 0.038066 | 3.038066 | 0.133535 | 0.773576 |

17.2355 | 2.846971 | 0.7827 | −0.245006 | 2.754994 | 0.117781 | 0.633130 |

19.3899 | 2.946752 | 0.6842 | −0.379505 | 2.620495 | 0.105359 | 0.518209 |

21.5443 | 3.070111 | 0.4956 | −0.701986 | 2.298014 | 0.095313 | 0.449347 |

23.6988 | 3.165424 | 0.5579 | −0.583576 | 2.416424 | 0.087010 | 0.388613 |

25.8532 | 3.252434 | 0.3651 | −0.007584 | 1.992416 | 0.080045 | 0.305922 |

28.0077 | 3.332479 | 0.3102 | −0.170538 | 1.829462 | 0.074107 | 0.256630 |

30.1621 | 3.406586 | 0.2550 | −0.366492 | 1.633508 | - | - |

No. | R | v_{r} (cm/s) | v_{i}_{1} (cm/s) | v_{i}_{2} (cm/s) | v_{i}_{3} (cm/s) | v_{r} − v_{i}_{1} | v_{r} − v_{i}_{2} | v_{r} − v_{i}_{3} |
---|---|---|---|---|---|---|---|---|

1 | 10.7722 | 1.6194 | 1.8821 | 1.9200 | 1.8043 | −0.2627 | −0.3006 | −0.1849 |

2 | 12.9266 | 1.5060 | 1,3475 | 1.3686 | 1.3200 | 0.1585 | 0.1374 | 0.1860 |

3 | 15.0810 | 1.0388 | 1.0159 | 1.0279 | 1.0135 | 0.0229 | 0.0109 | 0.0253 |

4 | 17.2355 | 0.7827 | 0.7954 | 0.8022 | 0.8061 | −0.0127 | −0.0195 | −0.0234 |

5 | 19.3899 | 0.6842 | 0.6410 | 0.6446 | 0.6588 | 0.0432 | 0.0396 | 0.0254 |

6 | 21.5443 | 0.4956 | 0.5284 | 0.5301 | 0.5499 | −0.0328 | −0.0345 | −0.0543 |

7 | 23.6988 | 0.5579 | 0.4437 | 0.4441 | 0.4670 | 0.1142 | 0.1138 | 0.0909 |

8 | 25.8532 | 0.3651 | 0.3783 | 0.3778 | 0.4023 | −0.0132 | −0.0127 | −0.0372 |

9 | 28.0077 | 0.3102 | 0.3267 | 0.3257 | 0.3507 | −0.0165 | −0.0155 | −0.0405 |

10 | 30.1621 | 0.2550 | 0.2852 | 0.2838 | 0.3089 | −0.0302 | −0.0288 | −0.0539 |

Blasting | N_{b} | Q_{uk} (kg) | Q_{i} (kg) | L_{uk} (m) | L_{pč} (m) |
---|---|---|---|---|---|

I | 8 | 185.0 | 26 | 80.0 | 3.0–3.2 |

II | 7 | 135.0 | 20.0 | 70.0 | 3.0–3.2 |

III | 7 | 105.0 | 19.0 | 61.0 | 3.0–3.2 |

No. | Blas. | MM | r (m) | Q (kg) | R | v_{t} (cm/s) | v_{v} (cm/s) | v_{l} (cm/s) | v_{rez} (cm/s) |
---|---|---|---|---|---|---|---|---|---|

1 | I | MM–1 | 256.90 | 185.0 | 45.0858 | 0.117 | 0.136 | 0.110 | 0.2104 |

2 | I | MM–2 | 220.81 | 185.0 | 38.7521 | 0.199 | 0.296 | 0.227 | 0.4228 |

3 | I | MM–3 | 224.58 | 185.0 | 39.4137 | 0.174 | 0.313 | 0.160 | 0,3922 |

4 | I | MM–4 | 161.78 | 185.0 | 28.3923 | 0.267 | 0.417 | 0.237 | 0.5490 |

5 | I | MM–5 | 179.10 | 185.0 | 31.4320 | 0.217 | 0.211 | 0.124 | 0.3271 |

6 | II | MM–1 | 184.41 | 135.0 | 35.9479 | 0.195 | 0.171 | 0.110 | 0.2817 |

7 | II | MM–2 | 151.52 | 135.0 | 29.5365 | 0.341 | 0.276 | 0.253 | 0.5064 |

8 | II | MM–3 | 152.79 | 135.0 | 29.7840 | 0.372 | 0.466 | 0.221 | 0.6359 |

9 | II | MM–4 | 87.57 | 135.0 | 17.0704 | 1.170 | 0.821 | 0.615 | 1.5560 |

10 | II | MM–5 | 99.91 | 135.0 | 19.4759 | 0.370 | 0.480 | 0.304 | 0.6780 |

11 | III | MM–1 | 137.43 | 105.0 | 29.1308 | 0.285 | 0.254 | 0.180 | 0.4221 |

12 | III | MM–2 | 137.33 | 105.0 | 29.1096 | 0.300 | 0.376 | 0.246 | 0.5403 |

13 | III | MM–3 | 72.09 | 105.0 | 15.2808 | 1.030 | 0.565 | 0.339 | 1.2227 |

14 | III | MM–4 | 77.99 | 105.0 | 16.5314 | 0.337 | 0.256 | 0.315 | 0.5276 |

**Table 6.**Representation of data for calculating the parameters in Nepričava open pit according to Model 3.

R_{i} | x_{i} | v_{i} | y_{i} | y_{i} + 3 | h_{i} = x_{i+1} − x_{i} | (y_{i} + 3 + y_{i+1} + 3)h_{i} |
---|---|---|---|---|---|---|

15.2808 | 2.726597 | 1.2227 | 0.201062 | 3.201062 | 0.078665 | 0.437507 |

16.5314 | 2.805262 | 0.5276 | −0.639417 | 2.360583 | 0.032084 | 0.186174 |

17.0704 | 2.837346 | 1.5560 | 0.442118 | 3.442118 | 0.131832 | 0.798046 |

19.4759 | 2.969178 | 0.6780 | −0.388608 | 2.611392 | 0.376940 | 1.889123 |

28.3923 | 3.346118 | 0.5490 | −0.599657 | 2.400343 | 0.024950 | 0.119379 |

29.1096 | 3.371068 | 0.5403 | −0.615631 | 2.384369 | 0.000728 | 0.003292 |

29.1308 | 3.371796 | 0.4221 | −0.862513 | 2.137487 | 0.013831 | 0.061646 |

29.5365 | 3.385627 | 0.5064 | −0.680428 | 2.319572 | 0.008346 | 0.040609 |

29.7840 | 3.393971 | 0.6359 | −0.452714 | 2.547286 | 0.053855 | 0.238567 |

31.4320 | 3.447826 | 0.3271 | −1.117489 | 1.882511 | 0.134245 | 0.485376 |

35.9479 | 3.582071 | 0.2817 | −1.266915 | 1.733087 | 0.075114 | 0.290859 |

38.7521 | 3.657185 | 0.4228 | −0.860856 | 2.139144 | 0.016928 | 0.071151 |

39.4137 | 3.674113 | 0.3922 | −0.935983 | 2.064017 | 0.134454 | 0.471298 |

45.0858 | 3.808567 | 0.2104 | −1.558745 | 1.441255 | - | - |

No. | R | v_{r} (cm/s) | v_{i}_{1} (cm/s) | v_{i}_{2} (cm/s) | v_{i}_{3} (cm/s) | v_{r} − v_{i}_{1} | v_{r} − v_{i}_{2} | v_{r} − v_{i}_{3} |
---|---|---|---|---|---|---|---|---|

1 | 15.2808 | 1.2227 | 1.0784 | 1.1934 | 0.8477 | 0.1443 | 0.0293 | 0.3750 |

2 | 16.5314 | 0.5276 | 0.9762 | 1.0712 | 0.7904 | −0.4486 | −0.5436 | −0.2628 |

3 | 17.0704 | 1.5560 | 0.9373 | 1.0251 | 0.7682 | 0.6187 | 0.5309 | 0.7878 |

4 | 19.4759 | 0.6780 | 0.7932 | 0.8553 | 0.6832 | −0.1152 | −0.1773 | −0.0052 |

5 | 28.3923 | 0.5490 | 0.4921 | 0.5098 | 0.4887 | 0.0569 | 0.0392 | 0.0603 |

6 | 29.1096 | 0.5403 | 0.4768 | 0.4926 | 0.4779 | 0.0635 | 0.0477 | 0.0624 |

7 | 29.1308 | 0.4221 | 0.4764 | 0.4921 | 0.4776 | −0.0543 | −0.0700 | −0.0555 |

8 | 29.5365 | 0.5064 | 0.4681 | 0.4829 | 0.4718 | 0.0383 | 0.0235 | 0.0346 |

9 | 29.7840 | 0.6359 | 0.4632 | 0.4774 | 0.4683 | 0.1727 | 0.1585 | 0.1676 |

10 | 31.4320 | 0.3271 | 0.4327 | 0.4433 | 0.4464 | −0.1056 | −0.1162 | −0.1193 |

11 | 35.9479 | 0.2817 | 0.3650 | 0.3687 | 0.3962 | −0.0833 | −0.0870 | −0.1145 |

12 | 38.7521 | 0.4228 | 0.3319 | 0.3326 | 0.3706 | 0.0909 | 0.0902 | 0.0522 |

13 | 39.4137 | 0.3922 | 0.3249 | 0.3249 | 0.3651 | 0.0673 | 0.0673 | 0.0271 |

14 | 45.0858 | 0.2104 | 0.2740 | 0.2702 | 0.3239 | −0.0636 | −0.0598 | −0.1135 |

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## Share and Cite

**MDPI and ACS Style**

Lutovac, S.; Gluščević, B.; Tokalić, R.; Majstorović, J.; Beljić, Č.
Models of Determining the Parameters of Rock Mass Oscillation Equation with Experimental and Mass Blastings. *Minerals* **2018**, *8*, 70.
https://doi.org/10.3390/min8020070

**AMA Style**

Lutovac S, Gluščević B, Tokalić R, Majstorović J, Beljić Č.
Models of Determining the Parameters of Rock Mass Oscillation Equation with Experimental and Mass Blastings. *Minerals*. 2018; 8(2):70.
https://doi.org/10.3390/min8020070

**Chicago/Turabian Style**

Lutovac, Suzana, Branko Gluščević, Rade Tokalić, Jelena Majstorović, and Čedomir Beljić.
2018. "Models of Determining the Parameters of Rock Mass Oscillation Equation with Experimental and Mass Blastings" *Minerals* 8, no. 2: 70.
https://doi.org/10.3390/min8020070