Effects of combustion parameters on antiexplosion performance: model tests and numerical simulations

3.2.1. Analysis of typical load conditions

DAMAGE is used as an indicator to measure structural damage. The closer the magnitude is to 1, the greater the damage; the magnitude 1 indicates complete destruction of the structure. The objects analyzed in this section are the load condition four (explosion depth 8.0 m, explosion distance 2.5 m) and working condition six (explosion depth 8.0 m, explosion distance 5.0 m). The detailed load and response process of the modelling under two of the load conditions (load condition four and load condition six) was presented below as examples.

Fig. 13 shows the damage evolution process of the RC pile under the fourth load condition, and the shown graphs are at the time of 3 ms, 15 ms, 30 ms, 45 ms, 60 ms, respectively. In the stage of shock wave, the damage of the RC pile was insignificant. With the development of the explosion, bending deformation and damage of the RC pile gradually increases. Damage of the back surface of the pile was found severer than that of the impact surface, and the concrete damage distributed in a large area. Damage at the impact surface mainly concentrated on the bottom and body of the RC pile within a certain range of the explosion source; while damage at the back surface mainly concentrated in the pile body. Finally, the body of the RC pile broke and the bottom of the pile also damaged.

Fig. 13Damage evolution diagram of reinforced concrete pile under working condition four

Fig. 14 shows the damage evolution process of the RC pile under the load condition six, and the shown graphs are at the time of 9 ms, 24 ms, 36 ms, 48 ms, 66 ms, respectively. It can be seen from the damage evolution process that in the early stage of the blasting load, damage of the RC pile was insignificant. Under the action of the explosion, the water exerted an impact force on the pile and caused deformation in bending. When the development of the bubble pulsation was sufficient, the impact of the water on the pile become severer and resulted in a large displacement in the middle section of the pile. The damage of the RC pile was shown significant particularly shown at the bottom of the pile and the back surface. In the end, the concrete damaged, and the RC pile was shown damaged in shearing.

Table 3 shows the damage analyses of the load conditions 1, 2 and 4, Table 4 shows those of the load conditions 6, 7 and 9, and Table 5 shows those of the load conditions 11, 12 and 14. From the result analysis, it can be seen that the RC pile damage effect is affected by the explosion shocking wave. The effect is small, but it is greatly affected by the bubble pulsation process.

Fig. 14Damage evolution diagram of reinforced concrete pile under working condition six

Based on above analyses and comparison of the dynamic response and damage process of the RC pile under the action of the near-field non-contact explosions, the following conclusions can be drawn: the RC pile is in a certain range of explosion distance, and the pile is mainly in local failure. With the increase of blasting distance, shear failure, shear failure and bending failure occurred successively in reinforced concrete piles. As the blasting distance increases, the concrete damage gradually decreases. The damage of concrete increases with the increase of blasting depth. As the depth of explosion increases, the failure of pile bottom will increase.

Table 3Failure analysis of the RC pile in the load conditions 1, 2 and 3

	The load condition 1	The load condition 2	The load condition 4
Final damage and morphology of reinforced concrete piles	$t=$138 ms	$t=$138 ms	$t=$144 ms
Failure analysis	Severe punching and shearing; serious damage at the bottom of the pile; damage at other parts of the pile to some extent	Severe punching and shearing; serious damage to the bottom of the pile	Bending shear failure; damage to other parts of the pile to some extent

Table 4Failure analysis of the RC pile in the load conditions 6, 7 and 9

	The load condition 6	The load condition 7	The load condition 9
Final damage and morphology of reinforced concrete piles	$t=$ 162 ms	$t=$156 ms	$t=$177 ms
Failure analysis	Severe punching and shearing; Severe shear failure at the bottom of the pile; serious damage to the concrete on the back of the pile	Bending shear damage; Pile back explosion surface concrete damage is serious; Serious shear failure at the bottom of the pile	Bending and shearing failure; concrete damage on the back of the pile back; general shear failure of the pile bottom

Table 5Failure analysis of 14.5 m reinforced concrete pile with depth of explosion

	The working condition 11	The working condition 12	The working condition 14
Final damage and morphology of reinforced concrete piles	$t=$ 150 ms	$t=$150 ms	$t=$276 ms
Failure analysis	Severe punching and shearing at the pile bottom	Severe punching and shearing at the pile bottom	Bending failure; large area concrete damage at the bottom of the pile

3.3.1. Structural damage assessment method

According to previous research results [3], the residual bearing capacity was adopted in this study to evaluate the damage level of the RC pile under blast loading. The damage index $D_f$ is defined as:

where $N_r$ is the residual axial bearing capacity of the RC pile with damage caused by the explosion; $N_u$ is the axial ultimate bearing capacity of the RC pile without damage caused by the explosion According to the $D_f$ value, the damage of RC piles can be classified to four levels, as shown in Table 6.

The axial ultimate bearing capacity of the RC pile without explosive damage and the residual vertical bearing capacity with various level of explosive damage were obtained by using the numerical modelling. For the pile without explosive damage, the vertical bearing capacity of the RC pile was calculated based on the same numerical model for the explosion analysis. The axial compression was applied on the top of the pile until failure. The velocity load was applied slowly.

The residual axial bearing capacity of the RC pile with explosive damage was relatively complicated, which can be obtained following the steps below:

(1) Establish a finite element model of the RC pile and apply axial pressure on the top of the pile. The axial load value equaled to 10 % of the axial ultimate bearing capacity of the pile without explosive damage. The axial compressive pre-load kept constant during the period of the following explosive loading.

(2) Simulate the near-field non-contact explosion on the RC pile. As measurement of the residual vertical bearing capacity of the RC pile after the explosion should be carried out when the pile structure reaches a state of relative static balance. Referring to the method for evaluating the damage of structures subjected to explosion in the air [3-5], the simulation calculation should be stopped when the joint speed of RC pile is lower than 0.1 m/s.

(3) Measure the remaining vertical bearing capacity of the pile on the basis of the damaged model without the water and explosion conditions. The displacement load is slowly applied to the top of the RC pile until the RC pile is damaged.

3.3.2. Damage evaluation of the reinforced concrete pile

Since the above presented modellings resulted in ignorable residual bearing capacity, five more load conditions were selected again for analysis. The specific load conditions are shown in Table 8. For all the modellings based on the verified modelling method, the charge amount of the explosive is 60 kg and the depth of the explosive is 8.0 m. The explosion distance is considered as a variable and the effect of which was investigated. The evaluation results are shown in Table 7. It can be seen that with the increase of the explosion distance, the RC pile mainly suffers from bending failure, and the degree of damage gradually decreases.

3.4.1. Effect of explosive quality

Explosive quality is related to the input of the explosion energy for the modellings, and it also affects the structural damage. Assuming the explosion depth was 8.0 m, the axial compression ratio was 0.1, and the explosive quantity was 10.0 kg, 20.0 kg, 40.0 kg, and 100.0 kg, respectively, and the charging method is still a spherical charge, and the others are the same as the aforementioned model. The safety distance and damage degree of the RC pile under the explosion with different explosive doses are shown in Table 8. It shows the recent explosion distance of reinforced concrete piles with less than 10 % damage under different doses.

The fitting relation between the charging radius and the safe distance calculated according to different loading quantities is shown in Fig. 15. The abscissa is the charging radius (the explosive mass is related to the charging radius, and the radius number is the ratio between the blasting distance and the loading radius), and the ordinate is the safety distance. According to the above results, the relation between charge radius ($R$) and safe distance ($S_d$) can be obtained, namely:

where the unit of $S_d$ is meter; $N_R^{\mathrm{\text{'}}}$ is the safety factor only related to the explosive quality, and the value of the factor is weighted to be 78.75; the unit of $R$ is meter.

Fig. 15Relationship between safety distance and explosive radius

3.4.2. Effect of explosion depth

Depth of the explosion mainly affects the impact position of the RC pile subjected to the explosion, and also results in different reflective phenomenon of the surface and the bottom of the water. Damage of the RC pile subjected to the explosion would be affected by the explosion depth. Considering the explosive quality was set as 60.0 kg, the explosion depth was set to 3.0 m, 5.0 m, 10.0 m and 12.0 m, respectively. Table 9 shows the safety distance and damage of the RC pile under different explosion depths.

The depth of explosion has a significant impact on the damage of the RC pile in the near-field non-contact explosion. ${\lambda }_h$ ($=h/H_h$) is defined as normalized depth or depth coefficient, where $h$ is the depth of the charge with the unit in m, and $H$ is the calculated depth in m. The fitted curve is shown in Fig. 16. Combined with the previous research results, the comprehensive consideration of the charge amount and the explosion depth can be obtained. The formula, is:

where $S_d$ is the safety distance with the unit in m; $N_{Rh}^{\mathrm{\text{'}}}$ is a safety factor considering the charge quality and the charge depth and the coefficient $N_{Rh}^{\mathrm{\text{'}}}$ is calculated to be 78.07; ${\lambda }_h$ is the normalized depth coefficient. $R$ is the charge radius in m.

Fig. 16Relationship between safety distance and depth coefficient of the explosion

3.4.3. Influence of steel hoop ratio

The hoop ratio is an important parameter related to strength of reinforced concrete structural members. Increase of the hoop ratio can enhance the bearing capacity, integrity and ductility of the RC structural members, and also change the deformation performance. Considering the circumstance of the explosion, increase of the hoop ratio can reduce the damage degree of the RC structural members or reduce the safety distance to a given explosion condition.

In order to understand the influence of the hoop ratio on the RC pile and determine the corresponding safety distance, the spacing of the hoops is set to 100 mm, 150 mm, 200 mm, 250 mm and 300 mm respectively. The explosion depth was set as 8.0 m for all the modellings in this section. The rest of the settings are the same as the previous part. The safety distance and damage of the RC piles with different hoop ratios are shown in Table 10.

Fig. 17Relationship between safety distance and hoop ratio

The fitted relationship between the different hoop ratios ${\rho }_{sv}$ and the safety distance of the RC pile is shown in Fig. 17. Combined with the previous research results, the comprehensive consideration of the charge, explosion depth and hoop can be obtained. The fitted formula for the rate, namely:

where $S_d$ is the anti-explosion safety distance, the unit is meter; $N_{Rh\rho }^{\mathrm{\text{'}}}$ is the safety factor considering the charge quality, burst depth and the hoop ratio. This coefficient is calculated to be 14.50 after fitting calculation. ${\rho }_{sv}^r$ is the hoop ratio, ${\rho }_{sv}^r=A_{sv}/bs$, $A_{sv}$ is the cross-sectional area of the stirrup, $b$ is the width of the member, which is perpendicular to the shear direction, $s$ is the hoop spacing, and $r$ is the coefficient. The value $r$ is –0.228; ${\lambda }_h$ is the normalized depth coefficient; $R$ is the charge radius in meter.

3.4.4. Influence of concrete strength

Concrete strength mainly affects the bearing capacity and stiffness of RC structures. Increasing the strength of concrete can reduce the damage degree of explosion to the structure, thus improving the safety performance of the structure.

In order to understand the influence of concrete strength on the RC pile under the near-field non-contact explosion, the concrete strength grades were set to 35 MPa, 60 MPa, 80 MPa, 100 MPa, 120 MPa and 140 MPa, and the explosion depth was set as 8.0m for all the modelling in this section. The rest of the settings were the same as before. The safety distance and the damage of the RC pile of different concrete strength subjected to the explosion are shown in Table 11.

Fig. 18Relationship between safety distance and strength coefficient of concrete

The fitting relationship between the concrete strength and the safety distance of the RC pile under the explosion is shown in Fig. 18. Based on Eq. (8), a fitting formula that comprehensively considers the charge amount, explosion depth, hoop ratio and concrete strength grade can be obtained, namely:

where $S_d$ is the anti-explosion safety distance, the unit is meter. $N_{Rh\rho \mathrm{c}}^{\text{'}}$ is the safety factor considering the charge quality, burst depth, concrete strength grade and hooping ratio, and this coefficient is calculated to be 11.25.$C$ is concrete. The strength grade coefficient is defined as 0.01 times the concrete strength grade value, $B$ is the concrete strength grade coefficient index, and the value is –0.392; ${\rho }_{sv}^r$ is the hoop ratio, and $r$ is the coefficient, the value is –0.228; ${\lambda }_h$ is normalized Depth factor; $R$ is the charge radius in meter.

3.4.5. Effect of longitudinal steel bars ratio

The longitudinal steel bars ratio of reinforced concrete structures has a significant impact on the damage and damage of reinforced concrete piles under near-field blast loading. Increasing the longitudinal steel bars ratio will increase the structural integrity and improve the safety performance of the structure. In order to understand the effect of the longitudinal steel bars ratio on the damage and damage of the near-field explosion reinforced concrete pile, the longitudinal steel bars diameter is set to 20 mm, 22 mm, 28 mm and 32 mm, and the explosion depth is 8.0 m, as shown in Table 12. The rest are the same as the previous part.

In the near-field blasting load, the fitting relationship between the longitudinal steel bars ratio ${\rho }_L$ of the pile and its explosion safety distance is shown in Fig. 19. Combined with the previous calculation results, a fitting formula that comprehensively considers the charge amount, explosion depth, hoop ratio, concrete strength grade, and longitudinal steel bars ratio can be obtained, namely:

where $S_d$ is the anti-explosion safety distance, the unit is meter. $N$ is the safety factor considering the charge quality, burst depth, hooping ratio, concrete strength grade and longitudinal reinforcement ratio, and the coefficient is calculated to be 15.50 after fitting calculation. ${\rho }_L$ is the longitudinal reinforcement ratio, ${\rho }_L=A_S/A$, $A_S$ is the reinforcement area, $A$ is the area of the cross section of the test piece, and $Z$ is the coefficient, the value is –2.725. $C$ is the concrete strength grade coefficient, $B$ is concrete of strength grade coefficient index, which is –0.392. ${\rho }_{sv}^{}$ is the hoop ratio, and $r$ is the coefficient, the value is –0.228. ${\lambda }_h$ is the normalized depth coefficient. $R$ is the charge radius, and the unit is meter.

Fig. 19Relationship between safety distance and reinforcement ratio