To reduce the hazards posed by blast shock waves, important potential targets are usually shielded by cantilever walls. The main factor that indicates whether there is damage is the back-wall overpressure, and of concern here is predicting the back-wall overpressure behind a rigid cantilever wall. This was measured in full-scale blast experiments using 20 kg of trinitrotoluene (TNT) located on the ground, and the experimental data reveal how the cantilever wall mitigates the shock wave. A 3D numerical model was established to determine how the wall size influences the diffraction of the shock wave and the peaks and attenuation of the back-wall overpressure. Based on the numerical results and dimensional analysis, a model is proposed that provides an effective means of predicting the back-wall overpressure rapidly from the TNT equivalent, the standoff distance, and the height of the wall.

## I. INTRODUCTION

Terrorist bombing attacks can cause huge loss of life and property,^{1} so important potential targets are made explosion-proof by shielding them with cantilever walls (usually made of reinforced concrete). In the event of a shock wave, it is reflected and diffracted by the cantilever wall, thereby mitigating the overpressure behind the wall and thus protecting the target located a certain distance behind the wall.

Mitigating explosions by using different types of barriers has been the subject of continuous research. Smith reviewed the research on blast-wall structural protection and introduced the structural design cases of permanent and temporary blast walls for military and civil use.^{2} Hussein *et al.* conducted experiments on the blast resistances of a thin oriented strand board wall and the WSaW blast protection wall composed of wood and sand,^{3} and the results indicated the potential for using simple blast walls to partially mitigate blast loading at extremely low cost and with minimal installation effort.^{4,5} Xiao *et al.* conducted experimental and numerical research on the shock-wave reduction performance of a blast wall topped by an inclined canopy and found that the optimum inclination angle of the canopy was 105°–125°, and the blast wall with an inclined canopy could reduce the overpressure peaks of the shock wave by 59%–88%.^{6} Xiao *et al.* also discussed the shock-wave mitigation effect of protective barriers made of steel posts with a hollow cross section^{7} and protective barriers made of woven wire.^{8} Water walls are also effective for blast prevention.^{9–12}

Also, research has been carried out into predicting the back-wall overpressure (BWOP), which refers to the overpressure measured by a transducer located a certain distance behind the wall. Rose *et al.* carried out blast experiments at a one-tenth scale to measure the BWOP behind a rigid barrier, resulting in contour plots of the BWOP as a percentage of the pressure measured when the wall was absent.^{13} Using that experimental data, Chapman *et al.* analyzed how the mass of charge, the wall height, and the standoff distance influenced the reflected overpressure at a target point behind the cantilever wall; as a result, a prediction technique was developed for assessing the peak reflected overpressure and the reflected impulse at a point on a target structure behind a blast wall, and the relevant parameters associated with the blast wall/target structure configuration were taken into account for this technique.^{14} Zhou and Hao proposed a pseudo-analytical formula derived from best fits to numerical results, and it can be used for estimating the blast pressure on a rigid wall behind a blast barrier.^{15} Remennikov and Rose used a neural network to develop contour plots of the overpressure and impulse adjustment factors; the neural network was trained and validated using data from small-scale experiments, and the work helped to simplify the process of predicting the effectiveness of blast barriers.^{16} From small-scale Schlieren experiments, Gautier *et al.* captured the organization and evolution of the shock waves behind the obstacle, finding that its length and height affected the wave diffraction to a significant extent; accordingly, an equation for predicting the diffracted overpressure was developed based on the bypass wave, the lateral wave, and the combination wave.^{17} Xu *et al.* carried out full-scale blast-resistance experiments aimed at the reduction effect and mechanism of a gabion wall on shock waves from large equivalent explosions; the BWOP was measured during ground explosions of 1 and 10 tons of 2,4,6-trinitrotoluene (TNT), and based on the test data and a 2D numerical model, an equation for predicting the BWOP was established.^{18}

The areas behind a cantilever wall where people and structures would be affected by a shock wave can be estimated from the overpressure criterion. Previous studies were focused more on the propagation of blast waves and the blast loads on structures behind walls as obtained experimentally and numerically. However, although the aforementioned contour plots, neural networks, and empirical equations allow quick assessment of the BWOP and the reflected overpressure at target structures behind barriers, most of those studies were based on small-scale experiments and have not been verified for application to full-size structures. Also, there is a lack of published data on the BWOP generated by tens of kilograms of TNT equivalent as a reference.

Reported herein are full-scale blast-resistance experiments carried out with 20 kg of TNT to test the effectiveness of a cantilever wall for shock-wave mitigation. A numerical model was also developed to analyze the blast mitigation effect of the wall. Based on the experimental and numerical results, a predictive equation for the overpressure attenuation rate is proposed and verified.

## II. EXPERIMENTAL PROGRAM

*In situ* blast experiments were conducted to evaluate the effect of a cantilever wall in explosion mitigation. The tested cantilever wall was a reinforced concrete structure that extended 1 m into the ground and was 2.5 m high, 6 m long, and 0.4 m thick above ground. The exterior of the cantilever wall is shown in Fig. 1, and its reinforcement is shown in Fig. 2.

The concrete had a strength grade of C30, a standard compressive strength of 20.1 MPa, a standard tensile strength of 2.01 MPa, and a modulus of elasticity of 30.0 GPa. The longitudinal bars had a diameter of 20 mm and a center spacing of 120 mm, the distributed bars had a diameter of 18 mm and a center spacing of 200 mm, and the tie bars had a diameter of 8 mm. The longitudinal and distributed bars had a yield strength of 335 MPa and an ultimate strength of 510 MPa, and the tie bars had a yield strength of 235 MPa and an ultimate strength of 310 MPa.

Note that in the experiments, sandbags were stacked on each side of the wall to form sandbag walls with a length of 3.5 m and a height of 2.5 m.

The layout of the overpressure transducers is shown in Fig. 3. They were fixed on the ground with their sensitive surface flush with the ground, as shown in Fig. 4. Transducers PGH-1, PGH-2, and PGH-3 were used to measure the BWOP near the ground, and the free-field overpressure transducers were in place when the wall was absent. Two experiments were carried out, and the mass of TNT in each experiment was 20 kg. The charge was located on the ground at a standoff distance of 3 m in the first experiment and 2 m in the second experiment, this being the distance from the charge center to the blast face of the wall.

## III. EXPERIMENTAL RESULTS

All the free-field overpressure signals were collected in the experiments. Some overpressure transducers behind the wall were severely disturbed, thereby distorting the collected signals, but invalid data were eliminated during data processing.

The free-field overpressure attenuation recorded when 20 kg of TNT exploded at a distance of 3 m from the wall is shown in Fig. 5. The peak BWOP collected at 4.4 m behind the wall (7.4 m from the charge center) was 46 kPa. Compared to the trend line of the overpressure attenuation, the peak BWOP is equivalent to 29% of the free-field overpressure at the same location when the wall was absent.

Figure 6 shows that when 20 kg of TNT exploded at a standoff distance of 2 m, the collected BWOP peaks were 45 and 30 kPa. The 45-kPa peak was recorded at 2.4 m behind the wall (4.4 m from the charge center), and the 30-kPa peak was recorded at 6.4 m behind the wall (8.4 m from the charge center). Compared to the trend line of the overpressure attenuation, the BWOP at the two locations is equivalent to about 5% and 19% of the free-field overpressure, respectively. After the two experiments, the wall was intact.

The experimental results show that the cantilever wall constructed of reinforced concrete had an obvious effect in mitigating the shock wave. However, because of the experimental risks and the quantitative limitations of the overpressure transducers, the subsequent analysis of more blast scenarios is based mainly on numerical simulations.

## IV. NUMERICAL ANALYSIS

Numerical simulation is an effective method for analyzing the mechanism of reflection and diffraction of shock waves. The numerical model established in this study is the fluid-structure coupled problem. AUTODYN includes a Lagrange processor for modeling structures and an Euler processor for modeling fluids. These different numerical processors can be coupled together in space and time to effectively calculate structural, fluid, or gas dynamic problems including coupled problems (e.g., fluid structure). The above processors use explicit time integration.

This section reports the 3D numerical model established by using AUTODYN, as shown in Fig. 7, and its accuracy is verified by comparison with the experimental results. In Sec. V, the results of the numerical model are used to study how the relevant variables influence the mitigation of shock waves.

The 3D numerical model comprises the cantilever wall, TNT, and the surrounding air, with only half of the wall modeled because of the symmetry of the model. The height, length, and thickness of the cantilever wall model are 2500, 3000, and 400 mm, respectively. The bottom of the air domain is set to be rigid, representing the ground. The plane perpendicular to the Z axis (XOY) is the symmetry plane. The other faces of the air domain are set as flow-out boundaries. The mesh size is 50 mm.

The multi-material Euler formulation and the fluid–structure interaction algorithm were applied. By 2D simulation, the shock wave propagated to 1 m from the wall, then the results were mapped to the 3D model.

### A. Material model

*P*is the pressure,

*γ*is the adiabatic exponent,

*ρ*

_{air}is the air density, and

*e*is the internal energy. Herein, we use

*ρ*

_{air}= 1.225 kg/m

^{3},

*γ*= 1.4, and

*e*= 2.068 × 10

^{5}kJ/kg.

*ν*is the specific volume,

*E*

_{0}is the Chapman–Jouguet (CJ) energy per unit volume, and

*A*,

*R*

_{1},

*B*,

*R*

_{2}, and

*ω*are material constants. In addition, the density

*ρ*

_{tnt}, detonation velocity

*D*, and CJ pressure

*P*

_{CJ}of the TNT should be determined. The parameters used for the simulations are listed in Table I.

ρ_{tnt} (kg/m^{3})
. | D (m/s)
. | P_{CJ} (GPa)
. | A (GPa)
. | B (GPa)
. | R_{1}
. | R_{2}
. | ω
. | E_{0} (MJ/m^{3})
. |
---|---|---|---|---|---|---|---|---|

1630.0 | 6930.0 | 21.0 | 373.77 | 3.7471 | 4.15 | 0.9 | 0.35 | 6000.0 |

ρ_{tnt} (kg/m^{3})
. | D (m/s)
. | P_{CJ} (GPa)
. | A (GPa)
. | B (GPa)
. | R_{1}
. | R_{2}
. | ω
. | E_{0} (MJ/m^{3})
. |
---|---|---|---|---|---|---|---|---|

1630.0 | 6930.0 | 21.0 | 373.77 | 3.7471 | 4.15 | 0.9 | 0.35 | 6000.0 |

When the charge center is closed, the cantilever wall constructed of reinforced concrete may be deformed and damaged and the back of the wall may even collapse, with the scattered fragments causing secondary damage behind the wall. However, the aim is to establish a simplified engineering analysis model of the BWOP, so the above complex factors are not considered. Nevertheless, wall deformation and failure are very important, and the influences of the above factors on the shock-wave flow field will be studied in future work. Referring to the simulation method of Zhou and Hao,^{15} the wall is set as rigid in the present numerical model.

### B. Comparison of experimental and numerical results

Figure 8 shows the time history of free-field overpressure at 5.1 m from the charge center when 20 kg of TNT was detonated. Compared with the measured overpressure peak of 462 kPa, those simulated and given by CONWEP are 298 and 388 kPa, respectively. The overpressure calculated via the numerical model is the smallest, which is related mainly to the mesh size of the model.

Figure 9 shows that at 3.1 m from the charge center, the measured free-field overpressure peak is the largest. As the distance increases, the measured and calculated overpressures gradually approach the values given by CONWEP. The overall attenuation trend of the experimental and calculated free-field overpressure is consistent with that given by CONWEP.

The BWOP (diffracted overpressure) measured by transducer PGH-1 is plotted in Fig. 10. The transducer was 2.4 m from the blast face of the wall, and the explosion was generated by 20 kg of TNT located at a standoff distance of 2 m. The calculated BWOP peak at this location is 33 kPa, which is smaller than the measured one of 45 kPa.

Note that the time history curves of the simulated overpressure are smooth because of the large mesh size used in the numerical model. Using a smaller size mesh would have improved the calculation accuracy,^{18} but doing so would have required more computing resources than were affordable, so analysis with a smaller mesh size was not performed.

### C. Shock-wave diffraction

The BWOP distribution is closely related to the diffraction of the shock wave. Based on Schlieren experimental results, Gautier *et al.* noted that the length and height of the obstacle govern the evolution of the diffracted wave.^{17} Obstacles are split into three categories according to the length of the wall *l*. For the first and second categories (*h* ≤ *l <* l.5*h* and 1.5*h* ≤ *l <* 3*h*), the shock wave is only reduced in the range of approximately one wall height (*h*) behind the wall, and this range is called the shadow region. For the third category (*l* ≥ 3*h*), the shock wave is effectively mitigated in a certain range behind the obstacle. The geometric parameters of the obstacle are shown in Fig. 11.

Based on the above conclusions, the shock-wave evolution was analyzed numerically, and Fig. 12 shows the overpressure contours of the diffracted waves for two walls with different length-to-height ratios. The wall height *h* is 2.0 m, and the wall length *l* is 2.0 m (*l* = *h*) or 6.0 m (*l* = 3*h*). The explosion is generated by 20 kg of TNT located at a standoff distance of 4.5 m.

In Ref. 17, the incident wave that diffracts over the obstacle and reflects on the ground is called the bypass wave, and the shock wave that diffracts around each lateral side of the obstacle is called the lateral wave. The bypass wave and the lateral wave depend on the wall height and the wall length, respectively. When the bypass and lateral waves arrive at the same time, they interact and create a combination wave.

Figure 12(a) shows that for *l* = *h*, the lateral wave arrives at the back of the wall before the bypass wave. After 2 ms, the bypass wave reaches the wall back and interacts with the lateral wave, thus the combination wave is created [Fig. 12(b)]. Meanwhile, for *l* = 3*h*, only the bypass wave arrives at the back of the wall and reflects on the ground [Figs. 12(c) and 12(d)], and no obvious combination wave is observed from the side of this model.

The evolution of the shock wave is also illustrated from the bottom view in Fig. 13. For the case in which the length and height of the wall are both 2.0 m (*l* = *h*), the two lateral waves diffract around the wall [Figs. 13(a) and 13(b)] and interact [Fig. 13(c)]. The combination wave is created by the interaction of the two lateral waves. The bypass wave joins in but is not dominant because the encounter of the two lateral waves makes the combination wave stronger.

For the case in which the wall length is 6.0 m and the wall height is 2.0 m (*l* = 3*h*), a large area of reflection is first generated on the blast surface of the wall [Fig. 13(d)], then the bypass wave and the two lateral waves arrive at the wall back [Fig. 13(e)]. The bypass wave and the two lateral waves gradually interact in the propagation, thus generating the combination wave [Fig. 13(f)]. Note that Mach reflection may occur at a certain distance behind the wall,^{17,18} although no obvious Mach waves are observed in Fig. 12. The above analysis indicates that the length and height of the wall significantly influence the organization and evolution of the shock-wave diffraction, and the numerical results basically agree with the findings in Ref. 17.

## V. PREDICTIVE MODEL OF BACK-WALL OVERPRESSURE

### A. Parameter analysis of overpressure attenuation rate

*η*, which is defined as

*P*

_{W}is the overpressure near the ground behind the cantilever wall, and Δ

*P*is the overpressure in the free field. The overpressure attenuation is mainly related to the blast equivalent

*W*, the stand-off distance

*d*, the length of the wall

*l*, the height of the wall

*h*, and the horizontal distance

*r*between the back-wall transducer and the blast face of the wall, which are shown in Fig. 11.

The influence of the wall length on the BWOP is shown in Fig. 14. The analyzed case is 20 kg of TNT detonated at a standoff distance of 4.5 m. The height of the wall is 2.0 m and the length of the wall varies. The BWOP at 2 m behind the wall is collected for the comparison. As shown in Fig. 14(a), the peaks of the BWOP generally decrease with the length of the wall. For *l* = *h* and *l* = 1.5*h*, the first peak of the BWOP is generated by the lateral wave because it travels less far than the bypass wave. For *l* ≥ 2*h*, the first peak of the BWOP is produced by the bypass wave because it travels less far than the lateral wave. For *l* = 1.5*h* and *l* = 2.0*h*, the bypass wave and the lateral wave overlay at a similar time in the positive phase of the shock wave, thus influencing the overpressure peaks. For *l* ≥ 2*h*, the first peaks of the BWOP calculated by different models are almost the same. According to the research of Gautier *et al.*^{17} and the arrival-time consistency of the BWOPs, the first peaks are deemed to be generated by the bypass waves for *l* ≥ 2*h*. Also, the overpressure peak generated by the lateral wave decreases with the length of the wall. For *l* = 6*h*, there is only one main peak generated by the bypass wave because the lateral wave is much weaker after traveling farther.

Figure 14(b) shows that the ratio of the wall length and height significantly affects the BWOP attenuation. Note that the BWOP is the lowest for *l* ≥ 3*h*. The bypass wave is dominant behind the wall for these cases because the wall is long enough to delay the lateral wave. Overall, the phenomenon and the results mentioned above are consistent with the experimental results in Ref. 17, so the numerical model is deemed feasible.

In fact, protected targets are usually enclosed by walls. For these cases, the lateral waves are delayed so much by the closed walls that they are unlikely to disturb the bypass wave. In the subsequent numerical analysis, the length of the wall model was set to be larger than the width of the air domain, so the wall can be regarded as being infinitely long. Also, more cases were calculated to analyze the influence of the parameters *W*, *d*, and *h* on the overpressure attenuation rate behind the wall. In the analysis, the model of the wall is 40 mm thick. The overpressure transducers in the numerical model were located at distances of 2–20 m behind the wall and at heights of 0.2, 1, 2, and 3 m. The maximum overpressure corresponding to each distance was selected for further regression analysis.

Figure 15(a) shows that as the mass of TNT *W* increases from 20 to 60 kg, the overpressure attenuation rate increases accordingly. The corresponding initial conditions are *d* = 4 m and *h* = 2.5 m. Note that in the area of effectiveness behind the wall, the overpressure attenuation rate should be less than 1.0. Hence, a target is protected if it is located at a scaled distance of less than 3.2 m/kg^{1/3} (4.8 *h*) behind the wall for *W* = 60 kg and *h* = 2.5 m. In the contour plot proposed by Rose *et al.*,^{13} the area with an overpressure attenuation rate less than 1.0 is within 5*h*–6*h* behind a wall, and that result agrees with the present findings.

As the standoff distance *d* changes from 2 to 6 m, the largest overpressure attenuation rate corresponds to the standoff distance of 2 m, while the smallest corresponds to 4 m [Fig. 15(b)]. The corresponding initial conditions are *W* = 20 kg and *h* = 3.5 m. There seems to be no obvious positive or negative correlation between the standoff distance and the overpressure attenuation rate.

As the height of the wall *h* increases from 2.5 to 4.5 m, the overpressure attenuation rate decreases accordingly [Fig. 15(c)]. The analysis conditions are that 60 kg of TNT exploded at a standoff distance of 4 m. There is a clear negative correlation between the wall height and the overpressure attenuation rate.

Therefore, the mass of charge, the standoff distance, and the height of the wall all influence the shock-wave mitigation by the cantilever wall. The effective area for overpressure attenuation is where the overpressure attenuation rate is less than 1.0. Beyond this effective range, the overpressure attenuation rate is greater than or equal to 1.0, and the wall loses its mitigation effect on shock waves.

### B. Regression equation for back-wall overpressure

Some models have been proposed for predicting the BWOP. Rose *et al.* developed contour plots from a series of measurements of the blast environment behind the barrier.^{13} In the contour plots, the pressures and impulses behind the wall are shown in terms of the percentage of the quantity measured when the wall is absent. Based on the above experimental results, the effectiveness of blast barriers was predicted using a neural network.^{16}

Chapman *et al.* analyzed the variation of the blast load at a fixed point on a target structure behind a cantilever wall using the mass of charge and geometric parameters.^{14} The analysis showed that the effect of the wall is equivalent to increasing the standoff distance between the charge and the target so that the overpressure is attenuated. As the height of the wall increases, so does the protection afforded to the target point by the wall. As the heights of the charge and the target point behind the wall increase, the protection afforded to the target point is reduced. The equation for calculating the reflected overpressure at the target point behind the wall was fitted to data from multiple small-scale experiments.

*et al.*, Xu

*et al.*proposed the following equation for predicting the BWOP behind a gabion wall:

^{18}

*H*

_{1}is the height of the wall,

*H*

_{2}is the height of the measuring point behind the wall, and $Z\u0304$ is the scaled distance between the charge center and the measuring point behind the wall, with $Z\u0304=(d+r)/W1/3$.

Gautier *et al.* observed the bypass, lateral, and combination waves at the rear of the barrier using high-speed photography in small-scale explosion experiments.^{17} A calculation model for the maximum overpressure behind the obstacle was established depending on the obstacle length. In the model, the overpressures produced by the bypass, lateral, and combination waves are considered separately.

Physical quantity . | Parameter . | Units . |
---|---|---|

Explosive density | ρ_{e} | kg·m^{−1/3} |

Explosive heat | Q | m^{2}·s^{−2} |

Detonation velocity | D | m·s^{−1} |

Mass of charge | W | kg |

Initial density of air | ρ_{0} | kg·m^{−1/3} |

Stand-off distance | d | m |

Height of wall | h | m |

Distance behind wall | r | m |

Physical quantity . | Parameter . | Units . |
---|---|---|

Explosive density | ρ_{e} | kg·m^{−1/3} |

Explosive heat | Q | m^{2}·s^{−2} |

Detonation velocity | D | m·s^{−1} |

Mass of charge | W | kg |

Initial density of air | ρ_{0} | kg·m^{−1/3} |

Stand-off distance | d | m |

Height of wall | h | m |

Distance behind wall | r | m |

*D*, the explosive heat

*Q*, the explosive density

*ρ*

_{e}, and the initial density of the air

*ρ*

_{0}are constants. The variables that affect the overpressure attenuation rate are the explosion equivalent

*W*and the geometric parameters

*d*,

*h*, and

*r*. The above five π terms can be simplified further, and the overpressure attenuation rate expressed in the form of a power function is given as

*W*is the TNT equivalent,

*K*is the undetermined coefficient, and

*α*,

*β*, and

*γ*are undetermined indices. The other parameters are shown in Fig. 11.

*R*

^{2}(square of the correlation coefficient) of the best-fitted equation is 0.90, which indicates a good correlation between the numerically derived overpressure attenuation rate and that predicted by Eq. (7). After parameters

*W*,

*d*,

*h*, and

*r*are confirmed, the diffracted overpressure attenuation rate can be calculated via Eq. (7).

*W*and

*R*, the incident overpressure Δ

*P*can be confirmed via the following empirical relationships given by Henrych:

^{19}

*P*is the free-field overpressure produced by the explosive charge in the atmosphere, and $R\u0304$ is the scaled distance between the charge center and the measuring point, with $R\u0304=(d+r)/W1/3$.

*W*in Eq. (8). The BWOP Δ

*P*

_{w}can be obtained via the following equation:

To verify Eqs. (4) and (7), the measured BWOP behind gabion walls in large TNT equivalent experiments^{18} is used, as shown in Figs. 16–19. In each experiment, the TNT equivalent *W* was 1000 kg and the length of the wall *l* was 5.3 m. In addition, the height *h*_{t} of the overpressure transducers behind the wall was set to half the wall height *h*.

In all the tests, the measured BWOP increases with the distance, as shown in Figs. 16(a), 17(a), 18(a), and 19(a). The overpressures calculated by Eq. (4) all decrease with the distance behind the wall, which is different from the experimental results. For test 1, the BWOP calculated by Eq. (7) decreases with the distance behind the wall, which is inconsistent with the experimental results [Fig. 16(a)]. For tests 2–4, the BWOPs calculated via Eq. (7) all increase with the distance behind the wall, which agrees with the experimental results [Figs. 17(a), 18(a), and 19(a)].

The relative deviations of Eqs. (4) and (7) are shown in Figs. 16(b), 17(b), 18(b), and 19(b). Equation (4) gives reasonable predictions for tests 1 and 2, as shown in Figs. 16(b) and 17(b). However, for tests 3 and 4, the prediction deviations of Eq. (4) seem somewhat unreasonable [Figs. 18(b) and 19(b)]. As can be seen, most of the overpressures predicted via Eq. (7) are lower than the experimental results. For tests 1 and 2, the maximum relative deviations of Eq. (7) are −12% and −9%, respectively [Figs. 16(b) and 17(b)]. For tests 3 and 4, the maximum relative deviations of the predictions are −25% and −33%, respectively [Figs. 18(b) and 19(b)].

For predicting the BWOP, the contour-plot^{13} and neural-network^{16} approaches have been shown to be effective, but the functional relationship among the TNT equivalent *W*, the stand-off distance *d*, the wall height *h*, and the BWOP Δ*P*_{w} is unclear. Meanwhile, the model in Ref. 17 is too complicated for calculating the effects behind the obstacle because the model distinguishes among the bypass, lateral, and combination waves. The predictive equation proposed by Xu *et al.*^{18} includes the main blast and geometry variables and the calculation is relatively simple, but its calculated values agree only partially with the experimental results.

The significance of this paper is as follows: (1) In the tests, the explosion equivalent of TNT was 20 kg and the wall size was 6 m (length) × 0.4 m (thickness) × 2.5 m (height), so the tests were close to engineering scale. (2) Compared with the predictive model reported in Ref. 18, the prediction results of the model proposed herein are more reliable. (3) The proposed model is relatively easy to use.

## VI. CONCLUSIONS

A cantilever wall constructed of reinforced concrete resisted an explosion generated by 20 kg of TNT located at a standoff distance of 2 m from the wall, and the wall was intact after the experiments. The measured BWOPs were reduced to 5%–29% of the overpressures in the free field.

Numerical analysis showed that the length and height of the wall significantly influence the organization and evolution of the shock-wave diffraction. When the wall length

*l*exceeds three times the wall height*h*, the maximum BWOP is produced by the bypass wave and the mitigation of the shock wave is the most effective. The overpressure attenuation rate is positively correlated with the mass of charge but negatively correlated with the wall height.A predictive model of the BWOP was proposed based on dimensional analysis and numerical simulation. The parameters of the model include the explosion equivalent

*W*, the standoff distance*d*between the explosion center and the blast face of the wall, the wall height*h*, and the standoff distance*r*between the measuring point behind the wall and the blast face of the wall. The predicted trend of the BWOP changing with the distance behind the wall is basically consistent with the test results, with relative deviations of −9% to −33% for predicting the peaks of the BWOP.

## ACKNOWLEDGMENTS

This work was supported by the 2023 Wuhan Knowledge Innovation Special Basic Research Project (Grant No. 2023020201010147), the National Natural Science Foundation of China (Grant No. 52378399), the Key R&D Projects in Hubei Province (Grant No. 2020BCA084), and the Innovative Group Project of Hubei Natural Science Foundation (Grant No. 2020CFA043).

Dr. Xinzhe Nian acknowledged financial support from the 2022 Scientific Research Starting Foundation for Doctors of Hubei (Wuhan) Institute of Explosion and Blasting Technology (Grant No. PBSKL-2022-QD-07).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Quanmin XIE**: Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Xinzhe NIAN**: Conceptualization (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (lead); Supervision (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Hui ZHOU**: Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). **Jianyuan WU**: Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*The Dynamics of Explosion and its Use*