Computational performance evaluation of sacrificial protective walls composed of lightweight concrete blocks: a parametric study of blast loads in a tunnel

Abstract Sacrificial wall systems consisting of lightweight concrete blocks have several advantages including versatility and high installation speed, making them effective tools for reducing the destructive effects of blast waves. Particularly, in confined structures, they can be placed along the path of potential explosion waves such that internal failure mechanisms in the concrete blocks cause the explosion wave energy to be dissipated. Numerous factors can affect the performance of such wall systems, including the material properties, location, and physical dimensions of the concrete blocks. This research presents an in-depth computational study on parameters influencing the performance of such sacrificial wall systems.


Introduction
The intensive pressure of blasts causes critical damage to people as well as the natural and built environment in the vicinity of the explosion.To reduce the losses of lives, assets, and resources, protecting structures against blast loads has been extensively studied by exploiting different techniques and methods.One of these techniques which is increasingly being considered in open and confined spaces is to use barriers to diffract the blast wave, leaving behind them a complex flow field that reduces the blast wave pressure and changes the load exerted on the target [1][2][3].Several approaches to shock wave attenuation such as the use of textiles [4], porous materials [5], perforated plates/walls [6][7][8], granular filters [9], solid obstacle barriers [10], rigid barrier arrays [11][12][13], water and suspended particles [14,15], and metallic grids [16] have been reported in the literature.Furthermore, various types of rigid and deformable blast wave mitigating barriers such as reinforced concrete cladding with metallic foam [17], steel angles arranged in various ways [18], deforming thick steel panel barriers [19], and glass fiber reinforced polymer blast barriers [20] have been extensively investigated.The aforementioned efforts aimed to reduce the shock wave loads on a target wall at the end of a specific tunnel.
A thorough study on the effects of rigid barrier shapes, dimensions, and inclinations on blast mitigation was presented by Berger et al. [21], who placed multiple barriers into a shock tube and measured the peak overpressure and its corresponding time.In another work, using a single barrier configuration, Berger et al. [22] investigated the effect of barrier width, opening ratio, and inclination angle on shock wave attenuation.Moreover, they concluded that for a fixed opening ratio, barriers with high inclination angles could reduce shock waves more effectively than barriers with narrower widths.In another related study, Berger et al. examined the dependence of shock wave attenuation on the double-barrier configuration with a fixed opening ratio [10].They concluded that the inclination angle of the first barrier was the dominant factor that absorbed the shock wave in the double-barrier configuration.The distance between the barriers determined the second factor in the barrier's efficiency.The geometry of dynamic barriers with changing orientations in response to loading was also studied by Berger et al. [21].In their more recent experimental studies, they concluded that the influence of different geometrical shapes on the shock-wave attenuation is negligible [10].Miura et al. [23] conducted a numerical investigation on the pressure wave mitigation effects of a rigid dike.They used 2 D and 3 D axisymmetric fluid dynamic numerical simulations to describe the interaction between a shock front and a rigid barrier.A study by Rouse and Consultants examined the effects of a rigid wall on the peak overpressure before and after it [24].Another research by Miura et al. [23] explored how an explosion affects rigid barriers.They used the finite element software ANSYS to describe the interaction between the blast wave and rectangular rigid barriers.Sugiyama et al. [25] studied the influence of rigid trapezoidal dikes of different shapes on blast wave propagation.They identified complex flow features created by the dike and discussed the relationship between the shape of a dike and its mitigation or intensification properties.Rigid rectangular barriers placed in a logarithmic spiral shape were used by Ivanov et al. to attenuate shock waves [26].Their experiments demonstrated that the three-groove barrier geometry successfully attenuated reflected shock waves' peak pressure and pressure impulse.Britan et al. studied the interaction of shock waves with barriers and grids of different shapes, sizes, and cross-sections [27].They showed that the pressure history on the target wall was strongly affected by the barrier area, not the barrier geometry.In addition, the normalized pressure measured on the target wall was reduced linearly with the distance between the barrier and the target wall.Keshavarz et al. evaluated the performances of water and concrete walls under blast loading considering physical and mechanical parameters [28,29].They reported the strength and fracture energy of concrete as the most crucial factors influencing the efficiency of the walls.Moreover, Khalilpur et al. optimized flat steel explosion-proof doors by the application of inner stiffening profiles [30].
Recently, non-rigid materials are being used in waveresistant barriers, offering a new field of study to enhance their effectiveness.It is generally presumed that the deformation of a barrier could increase its attenuation effectiveness.If a barrier is plastically deformed, it can dissipate shock wave energy.Aleyaasim et al. focused on the attenuation effects of cellular materials and concluded that the air-blast response of sacrificial barriers made from a cellular material could well attenuate the shock wave [31].Groethe et al. examined a 4-m tall, 10-m wide, reinforced concrete wall as a sacrificial barrier located at a distance of 4 m from the source of the blast [32].Sugiyama et al. examined how nonrigid solid triangular barriers made from different materials affected the propagation pattern of blast waves, observed the wave characteristics of the barrier, and analyzed the relationship between their geometric form and pressure reduction [33].Hajek et al. suggested different ways to achieve greater safety by increasing the blast mitigation properties of barriers [34].They studied the influence of various materials and shapes on the effectiveness of a barrier and proposes methods for providing increased overpressure reduction while maintaining the same overall dimensions.The results of an experimental investigation were used to determine the influence of barrier shape and material on blast wave mitigation.Lawrence et al. studied a train station platform using the finite element method (FEM) to establish the optimal configuration for blast barriers, concluding that the shape of the barriers did not influence the pressure [35].Using the mesh-free SPH method, Prasanna Kumar et al. investigated how blast waves propagate through the air and interact with barriers [36].They demonstrated that including a gap between the array of blockages and protected structures could contribute to improving blast wave mitigation.Taha et al. developed a new structural configuration for a concrete barrier wall to withstand blast loads [37].Their study found that switching the shape of a wall barrier from flat to convex has a better effect on mitigating blast waves.Convex walls with a 60 angle of curvature were also found to provide the best barrier performance.Taha et al. in another study used an ultra-high performance concrete wall that significantly outperformed an existing concrete barrier [38].Adams et al. numerically investigated the effect of retrofitting masonry walls subjected to blast with nano-particle reinforced polymer and aluminum foam [39].Ohtomo et al. experimentally and numerically studied two different barrier arrays [40].They installed a set of sharp-edged barrier plates in vertical and oblique configurations.This study showed a more effective attenuation was achievable when the barrier plates were installed obliquely rather than vertically.
As can be seen from the literature review above, many studies have been carried out aimed at exploring the potential benefits of protective structures in the attenuation of blast waves for the protection of various facilities such as power stations and chemical plants; however, there are still many aspects to the problem, including the geometric configuration and location of protective barriers, that remain barely explored.In this study, we numerically investigate the effect of different configurations of sacrificial lightweight concrete barriers, and in particular, how the geometry and location of barriers affect their blast wave attenuating properties.The main aim of this computational parametric study is to provide insights into the effect of each parameter on the performance of the wall system which can be used for designing physical experiments involving blast loadings in tunnels.
In general, underground structures are free of obstructions that allow cars and facilities to turn easily and traffic flow freely in both directions.In this regard, the barriers need to provide two conflicting goals: Firstly, a vast area of the tunnel must be covered by barriers that stop the blast waves from propagating.Secondly, these barriers should not interfere with traffic flow.Since full-scale tests require extremely complex and expensive setups and are highly affected by the test area environmental setup, the effect of lightweight concrete barriers on a shock wave is demonstrated using a finite element model implemented in ABAQUS.The presented numerical models have been validated using available experimental data from the literature.Unlike experimental methods for shock wave propagation which are usually limited to pressure measurements and photographic techniques, the validated numerical simulation produces valuable information on the physical parameters of flow fields across the entire domain.
In this paper, results have been presented as pressure-time diagrams, peak pressure bar charts, pressure reduction factors, and pressure ratios.It is worth mentioning that, once the transmitted shock wave hits the target wall, the wave will reflect toward the back face of the barrier, initiating a set of reverberating shock waves between the barrier and the target wall.This phenomenon is also investigated in this paper.

Overview of the model
To conduct numerical analyses, a tunnel model is adopted to assess the lightweight concrete wall effect on overpressure reduction.The Coupled Eulerian-Lagrangian (CEL) explicit analysis is considered an effective and accurate technique to model explosions in confined spaces that include fluid and solid materials [22].In this technique, these two environments can simultaneously be modeled; the TNT charge and air are modeled in Eulerian, whereas solid domains such as lightweight concrete blocks and tunnel linings are modeled in a Lagrangian environment.Considering the modeling assumptions and the analysis method, the finite element software ABAQUS is used in this study.

Material properties, boundary condition, and pressure pattern
In this research, the air is modeled as an ideal gas with a linear polynomial equation of state (EoS) based on the gamma-law rule as follows: where c is the adiabatic coefficient, E is the specific internal energy per unit mass, and q is the initial density of air.The values of these parameters used in this model are listed in Table 1.
The Jones-Wilkins-Lee (JWL) EoS is used to calculate the blast wave's resultant pressure to examine the radiated energy from the blast.This equation is obtained based on chemical processes occurring in blasts, the general form of which is considered as Eq. ( 1) [41], leading to the following equation: where parameters A, B, R 1 , R 2 , and x are TNT's constant coefficients.Parameters A and B represent the magnitude of pressure, q is the density of the explosive charge in the solid state, and e int is the specific internal energy of TNT at atmospheric pressure.The exponential terms in Eq. ( 2) demonstrate the high pressure generated during the explosion.The parameters in this equation are summarized in Table 2.
Concrete blocks are made of lightweight concrete with a density of 500 kg/m 3 .The mechanical properties of this type of concrete are given in Table 3.
Concrete materials without steel reinforcement have brittle and fragile behavior, expected to linearly be compressed until a collapse in the compressive phase and nonlinearly cracked and detached during the tensile phase.Contrary to seismic models that consider the concrete crack growth and its plastic behavior as an indicator of brittle concrete behavior, the failure mode and destruction pattern were considered in blast models.In order to simulate these destructions, several material constitutive models such as Johnson-Cook, concrete damaged plasticity, and brittle cracking have been developed.In this study, the brittle cracking model is utilized in which it is assumed that the material is perfectly linear in the compressive phase and does not manifest any deterioration or plastic behavior.Figure 1a shows the stressstrain diagram for this type of concrete obtained from a series of compressive tests on standard cubic specimens.Based on values for stress and strain, ABAQUS removes elements that are of excessive strain.Therefore, to establish a correct model, the option "Element Deletion" should be activated in the field output module.
The tunnel's periphery is made of rock, which is considered a blast-wave reflective boundary condition.As shown in Figure 1b, the velocity of surface boundaries in the three directions of x, y, and z are fixed except for the opening at which the fluid is free to flow into or out of the surface.
With regard to the modeling of the tunnel lining and the role of its interior boundary against the blast wave, the problem can be compared with fluid flow in tubes, in accordance with Figure 1c.In particular, by examining the fluid behavior in pressure pipes and comparing it to the propagation of blast waves in tunnel pathways, Lagrangian walls can be removed and replaced with equivalent Eulerian boundary conditions.As a result, as can be seen from Figure 1c, the Eulerian boundary condition should be zero velocity, while the Lagrangian Neuman and Dirichlet boundary conditions are ignored.Therefore, since there is no need to model the concrete lining in the tunnel, the overall computational time is drastically reduced.The boundary condition, the concrete lining, and the inlet boundary conditions are schematically presented in Figure 1d.
The pressure value at each tunnel section is denoted by the average pressure across the selected elements.ABAQUS measures the pressure value at the center of each element, so the weighted mean value of a set of elements should be considered as the average.The dimensions of the FEM mesh in this study were 10 cm, uniformly scattered across the domain.Therefore, both the simple and weighted averaging methods were equally appropriate.Since the averaging system is based on selecting only one element per row, more elements selected at a point may result in more contributions to the mean value from the elements at that point.Consequently, a predetermined element set was defined in the software to increase the accuracy of the mean value.The standard deviation of pressure values was calculated to evaluate the dispersion of pressure values from the mean value.The following relationship determines the standard pressure deviation at the peak pressure moment on the pressure study surface: where P i is the pressure value of element i, P is the mean pressure, and n is the number of selected elements.A standard pressure deviation of 1.209 indicated a low dispersion of elemental pressure values.To further investigate pressure deviations, Figure 1e illustrates the mean, minimum, and maximum pressure values on the tunnel door as functions of time.Also, the pressure distribution pattern at peak pressure on the door is depicted as an inset at the corner of this diagram.It is observed that the explosion wave encounters the left of the end door at the earliest, and expectedly, the maximum pressure occurred at this point.

Validation of the numerical model
In order to verify the numerical model, we simulated a tunnel explosion scenario that had been previously investigated

Parametric study
In this section, we report the results of a parametric study in which eight variables are considered as follows.Four primary parameters are TNT weight, concrete block size, wall height, and the cross-sectional coverage of the walls, and four secondary parameters are the location, number, distance, and mixture of the walls in the tunnel.For the sake of simplicity, the concrete blocks are assumed to be perfect cubes with a side length of 50 cm, and they cover the entire width of the rectangular part of the tunnel cross-section.
Based on more than 100 simulations, here we present an indepth explanation of how each parameter affects the performance of the wall system.The geometry of the tunnel and the arrangement of the explosive and concrete wall are depicted in Figure 3.The total length of the tunnel is 55 m which includes a 45 deviation of 15 m at a distance of 40 m from the tunnel's opening.The maximum height and weight of the tunnel are 6.15 and 6.80 meters, respectively.The detonation charge is placed at the opening of the tunnel.The surface chosen for this study is at the end of the deviation, which is named the pressure study surface in this paper.For each simulation, the volume of the explosive is calculated based on the density of TNT.

Effect of explosive charge mass
In this section, the effects of changes in the explosive charge mass are examined.To this end, first, the parameters involved in reducing the pressure caused by the explosion wave must be studied.Some of these parameters are as follows: I. The friction between the air and the tunnel wall that reduces the pressure along the tunnel (even without a blast protection wall).II.Interactions and disturbances created by the barriers' layout, curves, or indirect tunnel paths in the wave transmission process.III.The damage level of concrete blocks; the more damage a concrete block has, the less effective it is.
Concerning the latter case, the amount of explosive charge and its energy level determine the amount of damage in concrete blocks.In the absence of barriers, a relatively linear relationship between the explosive charge and the resultant pressure on the door is expected.In this regard, TNT charge masses of 400, 600, 800, 1000, 1500, and 2000 kg are considered.Simulations are carried out with and without concrete walls.The concrete walls are consisted of nine rows of blocks and are located at a depth of 10 m. Figure 4a and b shows the pressure-time diagrams for all the models; as can be seen from these diagrams, reducing the amount of TNT delays the blast wave arrival time at the tunnel end.The peak pressure at the end of the tunnel is displayed as a bar chart in Figure 4c to facilitate further investigations of the results.
According to Figure 4c, as expected, the peak pressure increases with increasing explosive mass.To determine the rate of this increase, the peak pressure subtracted and normalized with the corresponding TNT mass is illustrated as a bar chart in Figure 5a.As can be observed from this figure, there is a linear relationship between TNT mass and peak pressure for the base models, whereas it is nonlinear for the models with a wall.To illustrate how effective the walls are in each case, the percentage of pressure reduction for wall-protected models compared to their respective base models is also depicted in Figure 5a.
Figure 5b presents wave propagation results, i.e. pressure versus time diagram, in a tunnel without a wall subjected to a TNT charge of 1000 kg.As can be seen from this diagram, the maximum pressure is approximately 0.784 MPa, which is considered the base pressure, and the wave arrival time is around 61 ms.This case is chosen as the benchmark base model in the following section.

Wall height effect
This section examines the impact of the number of block rows arranged at the tunnel cross-section.In this regard, 12 different models were considered, including walls with 1 to 12 rows, as shown in Figure 6a.For a set of five selected models, pressuretime diagrams and pressure reduction percentage bars are respectively presented in parts b and c of Figure 6.As can be seen from these results, the efficiency of the wall increases as the number of block rows increases.However, having more than 9 rows increases the efficiency just by 0.5%.Therefore, it can be concluded that the optimal number of block rows for the given tunnel is nine which forms a 4.5-m tall wall.

Wall coverage and location effects
Another scenario of a seamless wall of blocks with various cross-sectional area coverages is examined in this section.Given that the explosion wave does not cause serious damage to the blocks and has no chance of going around them, it is expected to be reflected from the wall surface.The wall is placed at five different distances of 10, 20, 30, 43, and 48.5 m from the detonation point.Figure 7a presents the simulation results in the form of pressure-time diagrams for each of the abovementioned distances (the inset at the top left corner of Figure 7a illustrates how the blocks are laid out).
Parts b and c of Figure 7 illustrate the pressure contours in the tunnel with the wall at distances of 10 and 30 m, respectively.As can be seen from these figures, it is evident that at a distance of 10 meters, the concrete blocks were severely damaged and deformed, while at a 30-m distance, they were not seriously affected.In this regard, as can be demonstrated from Figure 7a, the performance of the wall improves as the wall moves further away from the tunnel opening.In fact, due to their preserved integrity, concrete blocks at a distance far from the detonation point are more efficient than walls in the vicinity of the point.
The first and second peak pressure values and reduction percentages for various configurations are respectively represented in parts a and b of Figure 8. Furthermore, the ratio of the second peak pressure, P max,2 , to the first peak pressure, P max,1 , is plotted in the inset of Figure 8a.The mechanism of the formation of the second peak is that after the wave reaches the branching point, a part of the wave enters the branch and the remainder enters the direct path.Since the ends of both paths are closed, the wave hits the end wall and is reflected.The reflected waves are redistributed at the branching point and collide with each other at a specific point.Since the geometric specifications and the length of the branching paths are identical, the reflected waves collide with each other with almost the same intensity in the center of the branching point.Therefore, it can be observed that the waves neutralize each other to a great extent.Again, a part of these waves is distributed between the branches at the branching point and hits the end of each branch.Given that the waves have largely neutralized each other at the branching point, it is expected that the pressure caused by the reflected waves is lower than the pressure caused by the initial wave.
It turned out the most efficient configurations can reduce the first peak pressure by approximately 93%.In all the cases, the second peak was smaller than the first peak, except for the blocks inside the split.As the waves of each split did not return simultaneously, they were not able to neutralize each other and the second peak did not increase.In general, considering the reflected waves, placing the wall at a depth of 20 or 30 m will be more effective.
To further analyze the effect of area coverage on the behavior of the blast wave, the pressure-time diagrams for three walls at distances of 48.5, 30, and 10 m from the tunnel opening are illustrated in Figure 9a-c, respectively.For the seamless wall at the beginning of the split, the pressure contours associated with the first and second blast waves are depicted in Figure 9d and e, respectively.As can be seen from these sub-figures, only a small part of the explosion wave goes into the split tunnel; in other words, the major wave propagation occurs inside the tunnel and goes through the direct pathway.However, because at this stage the first wave has already displaced the concrete blocks, most of the returned wave went through the split tunnel, and thereby, the second pressure peak (as indicated in the pressure-time diagram) is higher than the first pressure peak.
Here, we thoroughly study the effect of the location of the wall on the blast wave.In this regard, the single 4.5-m lightweight wall (containing nine rows of concrete blocks)  with a width of 6.5 meters (including 13 vertical columns of concrete blocks) is located at various distances from the detonation point, including a range of distances from 41.7 to 55 m in the branch tunnel, as illustrated in Figure 10.
The pressure-time diagrams for different wall locations are presented in Figure 11.The depth of 53.8 m (that is 1.2 m from the tunnel branch end) has the worst performance and has even created more pressure than the base model at the pressure study surface.The reason is that the blast wave energy is converted to the wall momentum, causing it to move.The disintegrating wall compresses the small space between the wall and tunnel end, and because the space is too small, the air does not have enough time to escape.The air compression gradually increases the pressure, which is even more than the pressure caused by the explosion wave in the base model.Furthermore, the closer the walls are to the detonation point, the sooner their effect will be felt and the longer the wave delay will be.For all wall positions, the pressure second peak is very low and insignificant.However, when the alls are placed 30 to 38 m away from the tunnel opening, the returned wave pressure increases (though it is still less than the first peak).It is because, in the straight pathway, the reflected wave collides with the wall, and since the wall is nearby the path split, a greater amount of wave energy enters the branch and increases the second peak pressure.Therefore, it is not reasonable to place the wall near the path split unless there is a thoughtful solution for the wave reflected at the tunnel end.
Concerning the first peaks, all the cases appear to have rather similar range and peak pressure values.However, to investigate the results more closely, zoomed-in views of pressure-time diagrams around the first and second peaks are illustrated in Figure 12a and b, respectively, for various wall locations before the split point.This figure shows that, even though the peak pressure value increased slightly by increasing the distance from the tunnel opening, the first peak values decreased between the distances of about 10 m to 35 m.When the wall is close to the opening, the pressure is too high so that the nearby walls suffer from severe damage, resulting in a drastic reduction in the effectiveness of the wall.Nevertheless, after approximately 10 m, the amount of pressure inside the tunnel became more uniform, resulting in the wall's integrity being maintained; this can be seen from Figure 12c in which   the first and second peak pressure values for each wall location are presented.This chart shows that at a distance of 1 to 38 m, the optimal mode is the depth of 38 m; the lowest first peak pressure is found in this position, despite having a higher secondary peak, as both the first and second peaks are almost equal, so it can be concluded that this is the most appropriate position for the wall.

Discussion and conclusions
This study aimed to investigate the effect of lightweight concrete walls composed of identical blocks on blast wave pressure reduction in a tunnel.To this end, the blast wave was numerically simulated in ABAQUS based on the JWL model.To simulate the fracture and brittle behavior of the lightweight concrete blocks, the brittle cracking equation of state (EoS) was utilized.The air was also modeled based on the ideal gas EoS.In all the simulations, the average mesh size of the finite element models was approximately 15 cm.
After verifying the accuracy of the utilized numerical model, a 50-m tunnel with a 15-m branch at an angle of 45 and a depth of 35 m, was studied parametrically.A total of 100 simulations were performed and analyzed in ABAQUS.The parameters examined in this study include two general categories.The first group involves investigating the effect of explosive charge weight, wall height, and the coverage percentage of the tunnel surface.In the following, the supplementary parameters' effects, including the location and height of the wall, are discussed.Here is the summary of the obtained results: i. Concerning the mean pressure values at each cross-section of the tunnel, due to the mesh similarity, it was concluded that the standard averaging method is sufficient, and there is no need for weighted averaging techniques.Additionally, since the pressure values for different elements differ in each cross-section of the tunnel, the deviation from the mean pressure value was calculated and observed to be about 1.208, which is acceptable.ii.According to Figures 4 and 5, as expected, the peak pressure increases with increasing explosive charge weight.More specifically, there is a linear relationship between the charge weight and peak pressure for the base models, while it is nonlinear for the models with sacrificial walls.It has been concluded that the efficiency of lightweight concrete wall systems closely depends on the amount of explosive charge.In fact, the more explosive charge causes the more corresponding pressure on the wall interface, resulting in the wall elements being eliminated in the FEM model.It also observed that the pressure reduction percentage has almost reached a fixed value for a lower than 1000 kg of TNT.iii.As shown in Figure 6, the efficiency of the blocks increases as the number of block rows increases, but from nine rows to the tunnel top, the efficiency only increases by 0.5% despite the additional 20 blocks.Therefore, it can be concluded that the optimal number of block rows is nine (4.5-m wall height).iv.Considering the gap between the concrete blocks and tunnel walls, it can be concluded that the more distance from the tunnel's beginning, the better the performance of gapless walls.This is because the blocks kept their integrity at distances away from the detonation point.The walls inside the split or at the beginning of the split are the most efficient ones and make about 93% pressure reduction in the first peak.In all cases, the second peak is less than the first peak, except for the blocks inside the split, in which the waves of each split were not returned simultaneously; the waves did not neutralize each other and increase the second peak.v.In accordance with Figure 12a, it has been shown that if the location of the walls is close to the detonation point, the walls encounter severe damage and cause performance reduction.Ultimately, it has been concluded that the best position of the wall is on the main tunnel path and within 10 to 30 m from the tunnel's beginning.vi.As can be seen in Figure 12c, the depth of 53.8 m (1.2 m from the tunnel branch end) has the lowest performance and has even created more pressure than the base model on the tunnel end as the blast wave energy is converted to the wall momentum, causing it to move.The gapless moving wall compresses the small space between the wall and tunnel end, and because the space is so small, the air does not have enough time to escape.The air compression gradually increases the pressure, which is even more than the pressure caused by the explosion wave in the base model.vii.Concerning the first peaks, even though the peak pressure values increased slightly by increasing the distance from the tunnel opening, they decreased between the distances of about 10 to 35 m.Importantly, for the walls close to the tunnel opening, the pressure is so great that the nearby walls suffer severe damage, resulting in a dramatic performance impairment of the wall.From these points forward, as shown in Figure 12c, the performance of the wall system is improved.viii.The bar chart of Figure 12c shows that at a distance of 1 to 38 m, the most optimal mode is the depth of 38 m.The lowest first peak pressure is found in this position, despite having a higher secondary peak, as both the first and second peaks are almost equal, so it can be concluded that this is the ideal position for the wall.

Data availability statement
Some or all data, models, or code that support the findings of this study are available from the corresponding author upon request.
experimentally and numerically by Li et al.[43] and Liu et al.[41], respectively.In this scenario, a 0.6-kg TNT charge explodes at the beginning of a straight tunnel with a horseshoe cross-section and a length of 10 m, as illustrated in Figure2a.Liu's finite element model was implemented in LS-DYNA based on the equations described in the previous section.The pressure-time diagrams at the distances of 2.25 and 6.25 m from the detonation point are shown in parts b and c of Figure 2, respectively.Some key outputs of Liu's simulations and those of the current study are presented in

Figure 1 .
Figure 1.(a) Standard compressive test results (stress-strain) for lightweight concrete with a density of 500 kg/m 3 [29].(b) Peripheral boundary condition of the tunnel.(c) Fluid flow in pressure pipes with zero velocity at the boundaries (adapted from Brode [42]).(d) Zero-velocity boundary condition on Eulerian parts.(e) Minimum, maximum, and mean pressure-time diagrams.

Figure 2 .
Figure 2. (a) Cross-sectional and profile views of the tunnel model.(b) Pressure-time diagram at a distance of 2.25 m from the detonation point: (i) Li's experimental results[43], (ii) Liu's numerical results[41], and (iii) the numerical results of the present study.(c) Pressure-time diagram at a distance of 6.25 m from the detonation point: (i) Li's experimental results[43], (ii) Liu's numerical results[41], and (iii) the numerical results of the present study.

Table 4 .
Maximum pressure and wave arrival time at different distances from the detonation point.Research Distance from the detonation point 6.25 (m) 2.25 (m) Peak pressure (MPa) Arrival time (ms) Peak pressure (MPa) Arrival time (ms) Numerical results of Liu et al.

Figure 3 .
Figure 3. Geometry of the tunnel and arrangement of the explosive, concrete wall, and pressure study surface.

Figure 4 .
Figure 4. (a-b) Pressure-time diagram for various explosive charges in models (a) with a wall, and (b) without a wall.(c) Peak pressure bar chart for various explosive charges.

Figure 5 .
Figure 5. (a) Normalized peak pressure and peak pressure reduction of the base and wall-protected models for various explosive charges.(b) Pressure-time diagram of the base model for a charge mass of 1000 kg (benchmark base model).

Figure 6 .
Figure 6.(a) Protective walls with various heights.(b) Pressure-time diagram for various numbers of block rows.(c) Pressure reduction percentage bar chart for various numbers of block rows.

Figure 7 .
Figure 7. (a) Pressure-time diagrams for various for different distances of the wall from the detonation point (inset on top-left corner shows the concrete wall layout).(b &c) Pressure contours in the tunnel with the wall at distances of 10 and 30 m, respectively.

Figure 8 .
Figure 8. Comparative representation of pressure values for different locations of the protective wall.(a) Values of the first and second peak pressures and their respective ratios (inset).(b) Reduction percentages of the first and second peak pressures.

Figure 9 .
Figure 9. (a-c) Pressure-time diagrams for different area coverages at various distances.(d-e) Pressure contours across the tunnel for a seamless wall at the beginning of the split.

Figure 10 .
Figure 10.Different locations of the concrete wall in the main tunnel (top) and the branch tunnel (bottom).

Figure 11 .
Figure 11.Pressure-time diagram for various wall locations: (a) selected wall locations in the main and branch tunnels; (b) selected wall locations in the main tunnel; and (c) selected wall locations in the branch tunnel.

Figure 12 .
Figure 12.Pressure-time diagrams around (a) the first, and (b) the second peak pressure.(c) The first and second peak pressure values for various locations.

Table 4
. As can be seen from this table, the maximum value of pressure and the arrival time of the wave are in good agreement.