# Calculation Method of the Blasting Throwing Energy and Its Variation Affected by the Burden
## 2 Construction of Theoretical Calculation Model of Blast Throwing Energy
### Column Charge Blast Throw Energy Calculation Model  
In actual engineering, chamber blasting, step-throw blasting shell holes, and explosives distribution show columnar characteristics. Based on the model for calculating the initial velocity of spherical cartridge blast throwing in the previous section, the columnar explosive rolls can be regarded as composed of \(n\) spherical cartridges along the axial direction of the blast hole. Similarly, the rock mass from the blast hole to the free surface in the step is divided into \(\lambda\) units, as shown in Figure 2. Based on the principle of energy conservation and assuming that the initial casting velocity and kinetic energy induced at a point \(j\) of a unit in the free surface are constituted by the joint action of \(n\) spherical cartridges, the initial  
Figure 1: The unit of throwing blast. casting velocity induced by a divided spherical cartridge \(i\) is \(v_{ij}\). The model for calculating the magnitude of the casting velocity at point \(j\) can be expressed as:  
\[v_{j}=(\sum_{i=1}^{n}v_{ij}^{2})^{0.5} \tag{12}\]  
According to Equation (12), the equation for generating the initial casting velocity \(v_{ij}\) of the \(i\)-th cartridge at unit \(j\) point can be expressed as:  
\[v_{ij}=k\frac{W^{3}}{W_{i}^{3}}(\frac{Q}{nW^{3}})^{\alpha_{1}} \tag{13}\]  
According to Equations (12) and (13), the initial casting velocity of unit \(j\) point can be calculated as:  
\[v_{j}=[\sum_{i=1}^{n}\left[k\frac{W^{3}}{W_{i}^{3}}(\frac{Q}{nW^{3}})^{\alpha_ {1}}\right]^{2}]^{0.5} \tag{14}\]  
When the unit is detached from the rock, the maximum casting velocity reached is considered as the initial casting velocity. The kinetic energy available at this point is considered as the throwing kinetic energy. Assuming that the weight of unit \(j\) point is \(m\), the equation for calculating the kinetic energy of throwing is:  
\[E_{vj}=\frac{1}{2}m(v_{j})^{2} \tag{15}\]  
From Equations (14) and (15), it follows that:  
\[E_{vj}=\frac{1}{2}m\sum_{i=1}^{n}\left[k\frac{W^{3}}{W_{i}^{3}}(\frac{Q}{nW^{ 3}})^{\alpha_{1}}\right]^{2} \tag{16}\]  
Summing up the throwing kinetic energy of all the units in the whole step, the kinetic energy of the step rock-throwing caused by the charge explosive can be obtained as:  
\[E_{v}=\frac{1}{2}m\sum_{j=1}^{\lambda}\sum_{i=1}^{n}\left[k\frac{W^{3}}{W_{i} ^{3}}(\frac{Q}{nW^{3}})^{\alpha_{1}}\right]^{2} \tag{17}\]  
The exact prerequisites used in this throwing energy calculation model assume that the rock medium is continuous and the explosive properties are unchanged. For the unknown parameters \(k\) and \(\alpha\) in the computational model, blasting model tests with multiple burdens of length \(W\) and charge quantity \(Q/n\) were carried out under the prerequisite assumptions. The experimental data were analyzed and fitted to obtain the results. Figure 2: Velocity vector diagram. Given the existence of different rock media on the step surface in actual engineering, introducing the influence variables of rock density and step height divides the step rock in the blasted area into several micro-elements. Its thickness is \(dh\), the distance from a microelement to the bottom of the step is \(h\), and the density is \(\rho_{r}\), as shown in Figure 3. Given differences in the initial casting velocity of a micro-element at different locations in step blasting, its functional relationship is supposed as follows:  
Figure 1:0  
Similarly, assuming the micro-element density as a function of step height is:  
Figure 1:1  
Then the equation for calculating the throwing kinetic energy of a micro-element with the height of \(h\) can be expressed as:  
Figure 1:2  
where \(a_{1}\) is the hole distance. The equation for calculating the kinetic energy of step blasting throwing for all micro-elements to transform the entire step height integrally is:  
Figure 1:3  
Assume that the density of each micro-element and the casting velocity do not change with the height \(h\).