Under the condition that one side has an important protection object, the blasting construction method of reserved rock wall is often adopted (Yang et al., 2010), that is, a rock wall of certain thickness is reserved on the side close to the protection object (Figure 1). The rock block blasting in the whole blasting area is divided into internal main rock block blasting and external rock wall demolition blasting. During blasting, the internal main stone is first exploded, and then the external rock wall is removed. The rock mass inside the rock wall is constructed by a conventional deep hole step control blasting method. When the inner rock mass of the rock wall is blasted, the rock wall, as a natural barrier, can prevent the lateral escape of the explosion pile, and at the same time, the rock wall can maintain a relative stability, avoiding the damage, stripping and rolling of slope rock block. In this way, the shielding effect of the reserved rock wall can reduce the difficulty of the main rock block blasting construction and the harm to the surrounding environment (Meng, 2015; Miao, 2020).

Meng et al. (2012) used rock wall control blasting technology in the expansion project of Chongqing–Fuzhou Railway adjacent to the existing line, pointing out that the internal main blast area of the deep hole blasting the direction of the air front parallel to the direction of the existing line. This technology can effectively control the impact of flying rocks, rolling rocks, and blasting vibration on the existing line, and then put the safety control point of the whole project on the demolition of the reserved rock wall. The construction method of the rock wall becomes the key to ensure the safety of the adjacent protection objects (Meng, 2015).

In the process of reserved rock wall removal, Tang et al. (2019) used diamond rope saw to cut the reserved rock wall, and this method is similar to the mechanical removal by rock drill, which can effectively ensure the safety of existing railway lines or surrounding buildings. However, both of them have the disadvantages of high economic cost and slow construction progress, which are not suitable for large area promotion and use. In order to achieve convenient construction and economic and reasonable purposes, the use of blasting demolition of rock walls has become possible.

Reserved rock wall blasting is a kind of step blasting, but it also has its particularity. The main characteristics are as follows: 1) it is greatly affected by groove blasting in the early stage. Due to the large amount of clamps used in trench blasting, it is necessary to increase the single consumption of explosive and the ratio of drilling hole properly, but it cannot cause damage to the rock wall. 2) There are often important facilities around the explosion area that are needed to be protected. The collapse of blasting heap and the rolling stones generated by high steep slope during blasting and rock excavation are easy to damage the environment. The flying stones and vibration of blasting must also be strictly controlled. 3) The blasting area is long and the geological conditions vary greatly, so the blasting parameters must be constantly adjusted, and the blasting medium must be fully broken or disintegrated. 4) Rock wall with the main blasting area synchronization drop, rock wall blasting adopts precision deep hole loose blasting control or shallow hole blasting; the blasting area is covered with the protective and facade multivariate stereoscopic protection system of the comprehensive protection, and the gun head machine and high-power drivers to construction are arranged, to ensure that the protected objects are safe during the rock wall blasting (Guo and xue, 2015).

The blasting resistance line of rock wall demolition deviates from the protected object, which can effectively control blasting vibration and the harm of the flying rock (Tang et al., 2019). Rock wall demolition blasting strictly belongs to parallel double-free face multi-boundary blasting. The throwing effect at this time depends on two factors: boundary conditions and charge amount (Wang et al., 2008).To tie in with the construction, the side of the ideal casting blasting effect is a fully broken rock mass, casting, protect the rock on one side of the object to the “broken and not loose” (damage and do not break), the critical instability state, and it does not allow individual slungshot, both to ensure the safety of protection objects, and can easily remove the broken part of the stone. Due to the particularity of parallel double-free face blasting, the initiation of row-by-row and hole-by-hole must be adopted (Wang, 1992). If strengthening loose blasting is used to charge, the blasting effect only depends on the width from the last row of holes to the inner side of the rock wall, which is called the critical damage width.

Meng (2015) believed that when the stress wave reached the inner side of the rock wall, its strength should not be greater than the compressive strength of the rock mass, and the critical damage width should not be less than 1.5 times the minimum resistance line. Meng et al. (2012) found that the critical damage width was greater than two times the minimum resistance line in the cutting expansion project adjacent to the existing line of Yuan–Fuzhou Railway and Yang Lin (Yang et al., 2010) in the An Tuoshan remediation project.

Most of the previous determinations of critical damage widths come from engineering experience and lack a certain theoretical basis. Based on the multi-boundary blasting theory, an equivalent surcharge-package model was established to derive the relationship between parallel double-free surface resistance lines to determine the theoretical value of the critical damage width for rock wall demolition blasting. In addition, the minimum resistance line of 2.5 times of the critical damage width was determined based on the blasting of the rock wall of the Yangtou–Yuexing station renovation project of the Yuhuai 2 line, where the blasting effect of “broken but not scattered” unexploded rock was achieved after three blasts.

The multi-boundary condition refers to the boundary condition of micro-terrain, which belongs to the shape geometry condition compared with the horizontal boundary condition. Terrain changes not only affect the blasting effect but also affect the calculating blasting role within the scope of damage index and effective utilization of explosive energy (Huang, 2006).

According to the blasting efficiency and geological characteristics, the micro-terrain boundary conditions can be divided into horizontal boundary conditions, tilt boundary conditions, boundary conditions of convex multi surface free terrain and topographic boundary conditions of the concave pass, as shown in Figure 2. 1) The characteristic quantity of the tilt boundary condition is that the slope angle α ranges from 0 to 90°, and the general flat terrain is a special case of the slope boundary condition α = 0. 2) For boundary conditions of convex multi surface free terrain, general characteristics are on either side of the ridge slopes α1 and α2, (rock wall demolition blasting is the convex face more empty terrain, α1 and α2 approximate to 90°), through the study of hill facing more empty blasting, with energy distribution coefficient control medicine package location arrangement, usable slow lateral ground slope angle of the characteristics of representative face more empty terrain, and the multi-faced empty terrain can be converted into inclined boundary condition. 3) For topographic boundary conditions of the concave pass, the reciprocal of convex boundary condition can be approximated. In this way, the slope angle in the boundary conditions of multi-boundary blasting can be used as the basis of model calculation by the ground slope angle α, a characteristic quantity of slope boundary conditions.

a) Horizontal boundary condition,

In multi-boundary blasting, the blasting effect is measured by a throwing rate (E), which refers to the volume of rock thrown out as a percentage of the overall volume of rock being blasted, expressed in Eq. 1: E = V t h r o w V b l a s t × 100 % . (1)

When the throw rate E = 27%, the throw rate is defined as the standard state. Hongqu Wang and Wenxue Gao (Gao and Liu, 2007) derived the calculation formula of multi-boundary blasting charge based on the principle of conservation of mechanical energy and functional balance, as shown in Eq. 2: Q = K W 3 F ϕ ( E , a ) . (2)

In the equation, K is the charge amount per cubic meter when forming a standard throwing funnel; W is the minimum resistance line length; and F _ϕ (E,α) is the theoretical explosive package property index. Wang and Gao derived the theoretical explosive package property index as: F ϕ ( E , α ) = 10 0.0129 E ( 0.05 α + 1 ) [ 1.11 − 86.133 E lg f ( α ) ] . (3)

In the aforementioned equation, E is the throw rate; α is the natural ground slope; and f(α) is the throw factor, which is consistent with the meaning of the terrain coefficient; f _ϕ (α,E) is the terrain factor, which means the terrain is favorable, the effective utilization rate of explosive energy increases or the projectile volume increases, and the charge should be reduced, also known as the charge attenuation coefficient (Gao et al., 2010). f ( α ) = { 1 − α 2 7000   o r   cos ⁡ α , α < 30 °, 26 α , α ≥ 30 ° . (4)

When explosive conversion coefficient e and plugging coefficient d are added, the theoretical calculation formula of multi-boundary blasting charge is as follows (Liu, 1999): Q = e d K W 3 F ϕ ( E , α ) = e d K W 3 10 0.0129 E ( 0.05 α + 1 ) [ 1.11 − 86.33 E lg f ( α ) ] . (5)

The aforementioned formula describes the inner relationship among the throwing effect, boundary condition, and charge amount. Q is the charge quantity, and the throwing effect is reflected by the throwing rate E, α can be used to describe the inclined boundary conditions of blasting rock and soil mass (horizontal boundary conditions, convex and concave boundary conditions can all be used as models for subsequent calculation).

The explosive amount of a multi-directional charge is decomposed into sub-charge directed by each surface, according to the proportion of the space resistance line of each surface, so that its blasting action and blasting parameters are equivalent to one-way charge, which is called equivalent sub-charge.

According to the equivalent sub-charge blasting energy distribution model and multi-direction group charge blasting energy distribution model, the following model is established for rock wall demolition blasting (double resistance line and double face blasting).Suppose a small cylindrical cartridge of equal diameter is set on the axis of a homogeneous, unequal radius, and infinite length thick-walled circle, and its volume is negligible compared with that of the thick-walled circle. Take a unit length from it, as shown in Figure 3. There is no problem of energy dissipation in the axis direction of the intercepted cylindrical charge of unit length, and the explosive energy can only spread evenly along the radial direction (resistance line direction) of the thick-walled circle, so that it is broken and generates speed. At this point, the momentum obtained by monomers on both sides (the throwing funnel of single concentrated charge blasting, the joint action area of adjacent charge blasting of single row unidirectional group charge blasting, and the action area of single unidirectional prolonged charge blasting, except for the end being thrown out of the body, are collectively called monomers) is equal to the impulse of explosive gas acting on the hole air cavity, as shown in Eq. 6. m 1 v 1 = ∫ 0 t P a θ d t = m 2 v 2 , (6) m = 1 2 θ W 2 ρ . (7)

In the equation, v ₁ and v ₂ are the velocity of monomers on both sides, m ₁ and m ₂ are the mass of monomers on both sides, a is the radius of the gun hole, P is the pressure of the explosion gas, θ is the angle of the gun aperture toward the center of the circle, t is the action time, W is the minimum resistance line, and ρ is the density of rock mass. By combining Eq. 6 with Eq. 7, it can be obtained that the square of resistance line of monomer velocity on both sides is inversely proportional, as shown in Eq. 8. v 1 W 1 2 = v 2 W 2 2 . (8)

Under the condition of double air surface (bilateral resistance line), the conventional strip charge was decomposed into two equivalent sub-charge packets, with the equivalent charge quantity of Q ₁ and Q ₂ , respectively.

The monomer velocity formula is as follows (Gao et al., 2010): v = K v ( Q V W 3 ) 2 3 . (9)

In the equation, K _v is the velocity coefficient and V is the volume of monomer. Combining Eq. 8 with Eq. 9, W 1 W 2 = ( Q 2 Q 1 ) 2 3 . (10)

According to Eq. 10, Q 1 Q 2 tends to decrease with the increase of W 1 W 2 , in line with the principle of minimum resistance line. As the explosive energy propagates along the cylinder, the particle is displaced. Since there is no barrier of outer medium on each surface, the medium on the surface produces outward radial motion, which promotes the fracture of rock mass near the surface. At the same time, the reflected stretching wave is generated in each plane, and the layer-by-layer propagates to the center of the charge package from the plane, so that the medium also moves in the direction of the plane, causing damage. The motion of explosion wave and reflected wave overlapped and interfered with each other, resulting in the increase of the motion velocity of the medium along the resistance line direction, and the symmetry of motion was destroyed due to the difference of resistance line. Because the explosion energy obtained along the air surface of the minimum resistance line is the most, and the resistance is the least, the wave travel route is the shortest, the reflected wave appears the earliest, the intensity is the largest, the expansion effect of the explosion product is the largest, and along the direction of the medium movement speed is the largest, the medium throws the largest degree.

In the theoretical calculation formula of multi-boundary blasting charge, let a = e d 10 0.0129 E ( 0.05 α + 1 ) [ 1.11 − 86.133 E lg f ( a ) ] . (11)

Then the formula for calculating the amount of charge on multiple surfaces can be simplified as (Liu and Gao, 2006): Q = a K W 3 . (12)

The rock wall demolition blasting belongs to convex double-sided aerial terrain, and α ₁ and α ₂ are approximately 90°. After determining the energy distribution of both sides of the sub-charge through the equivalent sub-charge package model, Eqs 10, 12 are jointly solved as follows: W 1 W 2 = Q 1 a 1 K 3 Q 2 a 2 K 3 = Q 1 a 2 Q 2 a 1 3 = ( Q 1 Q 2 ) 1 3 ( a 2 a 1 ) 1 3 = W 2 W 1 ( a 2 a 1 ) 1 3 , (13) W 1 W 2 = ( a 2 a 1 ) 2 9 . (14)

It can be seen from the analysis that the inner relationship among the throwing effect, boundary conditions, and charge amount of throwing blasting can be quantitatively described by Eq. 12 for the calculation of charge amount of multi-boundary blasting. Eq. 14 can be used to further quantitatively determine the relationship between the resistance lines on both sides of double-free surface blasting. It can be seen from Eq. 14 that the ratio of double resistance lines is related to explosive conversion coefficient, plugging coefficient, and theoretical charge property index (drop rate and boundary ground slope angle).

The ideal throwing effect of rock wall demolition blasting is that one side is thrown, one side is loose, and the throwing rate of the throwing side is in accordance with the standard throwing rate of blasting under inclined boundary conditions, that is, E ₁ = 27%, α ₁ = 90°; On the loose side, E ₁ = 0, α ₁ = 90°, and the throw rate is zero. According to the theoretical explosive package property index table (Wang, 1994; Di Xinning Yuan, 2001), f _ϕ1 (E,α) and f _ϕ2 (E,α), so a 1 a 2 = 25.4. (15)

The ratio of bilateral resistance lines is W 2 W 1 = ( a 1 a 2 ) 2 9 = 2.05. (16)

In the equation, W ₁ is the minimum burden length; W ₂ is the critical instability width in the critical instability state; and a ₁ and a ₂ are theoretical explosive package property indexes.

According to the analysis results of Eq. 16, the theoretical value of the critical width of dynamic self-stability in rock wall demolition blasting is 2.05 W.

In this article, the critical damage width of parallel double-free surface blasting is studied and analyzed by combining theoretical analysis and the field test. The conclusions are as follows:

1) Multi-boundary blasting charge calculation formula Q = aKW ³ , which can quantitatively describe the inner relationship among the throwing effect, boundary conditions, and charge of the throwing blasting; The equivalent sub-charge package model was established, and the relationship between the resistance lines of the double-free surfaces was deduced as W 1 W 2 = ( a 2 a 1 ) 2 9 , and the theoretical value of the critical width of dynamic self-stability was determined as 2.05 W.