Blast mitigation effect of the layered concrete structure with an air gap: A numerical approach

Abstract

A novel layered concrete structure with an air gap was proposed to resist explosion attacks from terrorist bombings or conventional weapons. The proposed layered structure potentially benefits mitigating the blast wave induced from detonation in the predamaged concrete shelter layer by projectile penetration, because the wave impedance between air and concrete material is far different. To quantitatively evaluate the blast mitigation effects and reveal the blast-resistant mechanism, a fine numerical model of the proposed layered structure was built in LS-DYNA, in which trinitrotoluene explosive was detonated in predamaged concrete shelter at different depths. The developed model was validated by field tests of concrete slab under contact explosion. Failure mode of concrete shelter and pressure time histories on the protective structure layer were captured and compared. The distinct blast mitigation effect of the proposed layered structure was demonstrated. A parametric study on the perforation limit, height, and residual depth of concrete shelter was also performed. The calculated results revealed that trinitrotoluene explosive detonated at the impact perforation limit of concrete shelter by projectile penetration is a dangerous critical condition that may cause negative effects on blast mitigation. The blast mitigation rates can always exceed 99%, as long as the concrete shelter is not blasted to perforation. A fitting formula to obtain a higher blast mitigation rate in guiding the design of a layered concrete structure with an air gap was put forward.

Keywords

Blast wave blast mitigation layered structure air gap

Introduction

The vulnerability of human lives and structures to explosion has been demonstrated by terrorist bombing attacks, military conflicts, and the occasional burst in the industry in recent years. It is well acknowledged that different loading environments usually require the application of different types of protective structural layouts, and the selection would be typically based on a facility’s estimated effectiveness under the expected conditions. Such consideration would include the following general possibilities: above-ground, partially buried, or fully buried structures. Moreover, construction materials must be selected for specific structural applications, considering both the structure and the backfill around it. Therefore, a number of new materials and methodologies for blast mitigation, such as porous materials (Levy et al., 1993; Radulescu and Lee, 2002; Teodorczyk and Lee, 1995; Tian et al., 2016; Wu and Zhou., 2011; Zhu et al., 2010), solid particles (Britan et al., 2001; Wagner et al., 2012), waver sheet (Meekunnasombat et al., 2006), concrete wall (Hao and Hao, 2014; Li et al., 2015; Lin and Ashraf, 2017), and water barrier (Chen et al., 2016), have been proposed and studied by researchers. It is much more meaningful to propose a new structural layout with traditional material, such as steel and concrete, to mitigate the blast loads (Whitney, 1996), considering the economic and technical benefits.

FEMA 427 (Hinman, 2003) announced that floor failure is particularly common in closed-in and internal explosions. The localized failure of the floor system immediately below the weapon is one of the typical damage types. The combined blast and fragment impact may result in failure of the concrete slab (Del Linz et al., 2016). However, it is announced that failure of reinforced concrete (RC) floor slabs in the RC structure has a significant reducing effect on the blast load induced from contact explosion. As for the significant reducing effect of concrete failure on blast load and the far different wave impedance between the air and concrete, a layered concrete structure with an air gap was suggested as a matter of course, which potentially benefits mitigating the blast wave induced from detonation in the predamaged concrete shelter layer by projectile penetration.

The proposed layered concrete structure is composed of a concrete shelter layer, an air gap layer, and the protective structure layer. The sketch of the proposed structure is shown in Figure 1. Besides the energy consumption in the explosion accounting for the failure of concrete material, the proposed layered concrete structure also possesses great potential for mitigating the blast wave in propagation to the layer of protective structure via the air gap, although the concrete shelter layer is primarily applied to resist the penetration of projectile.

Figure 1.

Sketch of the layered concrete structure with an air gap: (a) isometric view and (b) sectional view.

Existing studies referring to the blast mitigation effect of the layered concrete structure against explosion are very limited in the open literature. Krauthammer (2008) concludes that the damaging effects on the buried protective structure are usually considered to be related to the transfer of detonation energy to the structure or a delayed burst following the weapon’s penetration into the overlay (or soil). Two principal types of events are always considered, as illustrated in Figure 2: CASE 1, overhead burst on, or an explosive within layers of concrete or rock rubble, loading the roof slab and CASE 2, side or slant burst, or weapon in the soil loading the walls and floor.

Figure 2.

Geometry for explosion against a buried facility (US Army, 1986).

The phenomena associated with the two conditions, including cratering, ejecta, and wave propagation, have been investigated by researchers (Brock and Achenbach, 1973; Kolsky, 1963). In CASE 1 and CASE 2, where the projectiles detonate on or in burster slabs, or in the soil near the target, a series of empirical formulae were proposed for designing the protective structure (Krauthammer, 2008). However, only the soil backfill was assumed in the existing formulae, while the backfill with free air was not considered. Actually, the blast wave propagation and transition in the layered structure with an air gap are complicated and lead to difficulty in calculation of the blast mitigation effect of the overlay and the potential load on the roof of the protective structure.

Lai et al. (2015) reported a series of blast tests in ultrahigh performance concrete (UHPC) that was predamaged by 14.5 mm bullet penetrations. The explosion damage of UHPC with trinitrotoluene (TNT) explosive embedded at different depths of penetration was measured and the explosion process was simulated by the finite element (FE) method. It is demonstrated that the mass and the placing depth of explosive have important effects on the damage of UHPC. However, the blast energy consumed in the failure of concrete and the propagation of the blast wave were not considered in the test and analysis.

The advantages of the proposed layered structure, such as easy material availability, efficient construction, and low cost, show great potential for the widespread use of protective structures. This structural layout has already been adopted in some existing concrete shelters to resist military attacks, although the blast-resistant mechanism was not quantitatively revealed. It is necessary to find out the behaviors of the layered concrete structure with an air gap under explosion, so as to reveal the blast mitigation mechanism and to develop design guidance and formulae for engineering applications.

Systematic numerical calculations were performed in this study to investigate the performance of the proposed layered concrete structure with concrete shelter and an air gap cladding on the protective structure. A fine FE model of the proposed layered structure was built in LS-DYNA, in which TNT explosive was detonated in concrete shelter at different depths. The developed FE model was validated by comparison with field tests of concrete under contact explosion. Propagation of the blast wave without concrete shelter was calculated and was compared with those with concrete shelter to evaluate the blast mitigation efficiency. Some key affecting parameters, such as perforation limit, height (H), and residual depth (H_r) of the concrete shelter, were discussed. Applicable design guidance was derived for engineering construction.

FE model of the layered concrete shelter

Schematic diagram

The layered concrete structure with an air gap, as shown in Figure 3, is proposed to resist the impact and explosion from terrorist attacks and military weapons. The concrete shelter layer is primarily used to resist high-speed penetration of projectile. However, the point of this study is not penetration in concrete, but the explosion in the predamaged concrete shield at different penetration depths, as shown in Figure 3. In fact, the local failure of concrete shelter can consume large amounts of energy. On the other hand, due to the far different wave impedance between the air and concrete material, the blast wave could be significantly mitigated in propagation from the concrete shelter to the protective structure via the air gap.

Figure 3.

Sketch of penetration and explosion of projectile in concrete shelter: (a) penetration and (b) explosion.

The schematic diagram of the computing model is given in Figure 4. The layered structure from the top to the bottom layers comprises the concrete shelter, the air gap, and the protective structure. For computing efficiency, the length, width, and height of the cuboid concrete shelter are determined as 1, 1, and 0.4 m, respectively. The dimensions of the cuboid air gap are 1 m in length, 1 m in width, and 0.3 m in depth. It was assumed that a projectile vertically penetrated into the concrete shelter and detonated at the depth it penetrated, which is a most dangerous situation. The charge of the projectile is determined as a TNT explosive of 2 kg and the diameter and height of the cylinder TNT are 0.08 and 0.244 m, respectively. The penetration depth of the projectile is predetermined as 0.1 m according to the military code. The detonation point is assumed to be at the center of the projectile charge, which is 0.722 m above the protective structure.

Figure 4.

Schematic model of the layered concrete structure with an air gap.

The FE model was developed on the LS-DYNA platform. The most commonly used material model in LS-DYNA for concrete-like materials is Mat_72^#Rel3 (LS-DYNA, 2006), which has been proven reliable for analysis of RC structures. It was employed to model the behaviors of concrete shelter under explosion. The TNT explosive was described by the Mat_8^# high explosive burn model with Jones–Wilkins–Lee (JWL) equation of state. Free air in the gap was simulated using the Mat_9^# null material model with linear polynomial equation of state. *ALE_MULTI-MATERIAL_GROUP was defined for interface reconstruction of the ALE material groupings, that is, the TNT explosive and air. The model is discretized with an 8-noded SOLID 164 element. The impact of blast wave on the concrete shelter was implemented by coupling the meshes of the ALE and Lagrange and, thus, the partial meshes of ALE materials (TNT and air) and concrete shelter were overlapping.

Material models

Mat_72^#Rel3 in LS-DYNA is a three-invariant model, which includes damage and strain-rate effects, and was originally based on the pseudotensor model. This material model consists of three shear failure surfaces: the initial yield surface, the maximum yield surface, and the residual yield surface (LS-DYNA, 2006). It could be used to simulate the behaviors of concrete subject to large strains, high strain rates, and high pressures.

The tensile dynamic increase factor (TDIF) of concrete was obtained by equations (1) and (2) as (Shi, 2009)

T D I F = \frac{f_{t d}}{f_{t s}} = {(\frac{{\overset{\cdot}{ε}}_{d}}{{\overset{\cdot}{ε}}_{t s}})}^{δ} {\overset{\cdot}{ε}}_{d} \leq 1 s^{- 1}

(1)

T D I F = \frac{f_{t d}}{f_{t s}} = β {(\frac{{\overset{\cdot}{ε}}_{d}}{{\overset{\cdot}{ε}}_{t s}})}^{1 / 3} {\overset{\cdot}{ε}}_{d} \geq 1 s^{- 1}

(2)

where f_td is the dynamic tensile strength of concrete under strain rate ${\overset{\cdot}{ε}}_{d}$ , f_ts is static tensile strength of concrete under strain rate ${\overset{\cdot}{ε}}_{t s} ({\overset{\cdot}{ε}}_{t s} {= 10}^{- 6} s^{- 1})$ , $\log β = 6 δ - 2$ , $δ = 1 / (1 + 0.8 f_{c}^{'})$ , $f_{c}^{'}$ is the static axial cylinder compressive strength of concrete, and $f_{c}^{'} = 0.79 f_{c u}$ , $f_{c u}$ is the static cubic compressive strength of concrete.

The compressive dynamic increase factor (CDIF) of concrete was calculated by equations (3) and (4) as

C D I F = \frac{f_{c d}}{f_{c s}} = {(\frac{{\overset{\cdot}{ε}}_{d}}{{\overset{\cdot}{ε}}_{c s}})}^{1.026 α} {\overset{\cdot}{ε}}_{d} \leq 30 s^{- 1}

(3)

C D I F = \frac{f_{c d}}{f_{c s}} = γ {(\frac{{\overset{\cdot}{ε}}_{d}}{{\overset{\cdot}{ε}}_{c s}})}^{1 / 3} {\overset{\cdot}{ε}}_{d} \geq 30 s^{- 1}

(4)

where f_cd is the dynamic compressive strength of concrete under strain rate ${\overset{\cdot}{ε}}_{d}$ , f_cs is the static compressive strength of concrete under strain rate ${\overset{\cdot}{ε}}_{c s} ({\overset{\cdot}{ε}}_{c s} {= 30 \times 10}^{- 6} s^{- 1})$ , and log $γ = 6.156 α - 2$ , $α = 1 / (5 + 0.9 f_{c}^{'})$ , $f_{c}^{'}$ is the static axial cylinder compressive strength of concrete.

The density of concrete $ρ$ is assumed as $2440 {kg / m}^{3}$ , Poisson’s ratio v is 0.19, the unit conversion factor for length RSIZE is 39.37 in/m, and the unit conversion factor for stress UCF is 145 psi/MPa.

The keyword *MAT_HIGH_EXPLOSIVE_BURN, that is, Mat_8^# in LS-DYNA, is frequently used to describe the detonated high explosive. Key variable D is the detonation velocity and PCJ is the Chapman–Jouget pressure. The Jones–Wilkins–Lee (JWL) state equation was used to model the overpressure generated by chemical energy in an explosion. It is defined as

P = A (1 - \frac{ω}{R_{1} v}) e^{- R_{1} v} + B (1 - \frac{ω}{R_{2} v}) e^{- R_{2} v} + \frac{ω E}{v}

(5)

where A, B, R₁, R₂, $ω$ are material constants, P is the hydrostatic pressure, v is the specific volume, and E is the specific internal energy.

TNT explosive was considered as the charge in this study. The mass density of high explosive $ρ$ is 1630 kg/m³, A = 373.77 GPa, B = 3.747 GPa, R₁ = 4.15, R₂ = 0.9, Chapman–Jouget pressure PCJ is 27 GPa, energy E is 2.7e⁹ J/m³, the detonation velocity D is 6930 m/s, v = 1.0, and ω = 0.35.

Air in the gap was assumed to be the ideal air that was modeled by the material *MAT_NULL, Mat_9^# in LS-DYNA, with the linear polynomial equation of state equation defined in equation (6) as

P = C_{0} + C_{1} V + C_{2} V^{2} + C_{3} V^{3} + (C_{4} + C_{5} V + C_{6} V^{2}) E

(6)

where p is pressure, E is the specific internal energy, and V is the volume. The other parameters for air in this study are C₀ = −0.1 MPa, C₁ = C₂ = C₃ = C₆ = 0, C₄ = C₅ = 0.4, and the air density ρ = 1.3 kg/m³, respectively.

Deformation and damage of the protective structure were not considered in the developed model because this study mainly focused on blast mitigation effect and distribution of blast loads on the protective structure. To reduce the computation cost, therefore, a rigid model *MAT_RIGID, Mat_20^# in LS-DYNA was employed to model the protective structure.

Erosion algorithm

The erosion algorithm is usually employed to capture the failure or crushing process of an element for specific material in the commercial FE software. A variety of criteria can be used to define the erosion algorithm, for example, the maximum principle stress, the maximum effective plastic strain, shear strain, and the principal tensile strain. Each of the criteria could be defined independently. Once any one of them is satisfied, the corresponding element would be deleted from the calculation process. However, the value of erosion algorithm is approximately defined empirically, where different criteria and failure values should be adjusted for corresponding simulations. Xu and Lu (2006) adopted the principal tensile strain criterion for concrete and the principal tensile strain at failure was set as 0.01. When simulating the penetration process in high-performance concrete, the shear strain criterion used by Unosson (2000) as the shear strain at failure was defined as 0.8 or 0.9. In this study, after a series of preparing simulations, the maximum effective plastic strain erosion algorithm with the failure value of 0.01 was determined for concrete shelter.

Mesh size

Mesh size is very important to calculate the propagation of blast wave. Inappropriate determination of the mesh size usually leads to great error or significant computational difficulty. In order to obtain a suitable mesh size to maintain reliability and efficiency during calculation, a free air blast model with different mesh sizes was built and numerically tested. The calculated results were compared with TM5-855-1 (US Army, 1986), which provides reliable blast wave data, such as overpressure, duration, and impulse, by the given charge weight and scaled distance. It is noted that the shape of charge in TM5-855-1 (US Army, 1986) is assumed as a sphere, which is different from the charge in a projectile.

Figure 5 shows the detailed schematic model of free air blast. An eighth of the symmetric surrounding air and TNT explosive was modeled in order to reduce the cost of calculation. The dimension of the eighth model is 0.5 m in radius and 2.5 m in height. The TNT charge weight is 2 kg with a spherical radius of 0.0646 m. The arbitrary Lagrange–Euler algorithm, which combines merits of the Lagrange and Euler algorithms that are often applied to eliminate numerical difficulties induced by severe distortion of solid meshes, was adopted to simulate the TNT explosive and air. The symmetric boundary conditions were applied to the three cutting planes, and the nonreflecting boundary conditions were applied to the external surfaces.

Figure 5.

Model of free air explosion.

The built model was meshed at three different sizes: 1, 0.75, and 0.5 cm. The captured incident peak overpressures $P_{s o}$ at the scaled distance from 0.40 to 1.9 m/kg^1/3 were compared with those calculated using TM5-855-1. The corresponding overpressure survey points in the model are indicated in Figure 5. They are distributed 0.5–2.5 m away from the detonation point. The propagation of blast wave in the free air is presented in Figure 6. It shows that the blast wave develops into a sphere gradually with the detonation point as the spherical center. No significant disturbance could be found on the model boundaries. The calculated curves of peak overpressure versus scaled distance are shown in Figure 7.

Figure 6.

Propagation of the blast wave: (a) 20 μs, (b) 100 μs, (c) 210 μs, and (d) 3000 μs.

Figure 7.

Comparison of incident peak overpressure at different scaled distances.

As shown in Figure 7, compared to TM5-855-1, error of the calculated incident overpressure basically decreased with the scaled distance. It is obvious that the developed model lumped by solid elements with an average dimension size of 0.5 cm gives more acceptable predictions. The tendency of curves calculated with 1 and 0.75 cm meshes are similar to that given by TM5-855-1, however, the deviation is much bigger. A size of 0.5 cm is small enough to obtain reliable results.

A further quarter model of a rigid cylinder plate was added 0.722 m below the explosive (scaled distance 0.573 m/kg^1/3); it was built to check the reflected pressure. The distance 0.722 m was determined in accordance with the schematic model described in Figure 4. As shown in Figure 8, the section length and width of the model are 0.5 m and the height is 0.882 m. The depth of the rigid plate is 0.01 m. The rigid plate is fixed at the bottom. Figure 9 shows the propagation of spherical blast wave in the free air, as well as the reflected blast wave against the rigid plate.

Figure 8.

Model for reflected pressure.

Figure 9.

Propagation of the blast wave: (a) 50 μs, (b) 100 μs, (c) 200 μs, and (d) 350 μs.

Distribution of the peak reflected pressure P_r₀ (the subscript r represents reflected pressure and 0 represents height of concrete shelter as 0 m) on the rigid plate in the range 0 ≤ r ≤ 0.5 m (r is distance to the center point A, as shown in Figure 8) is compared in Figure 10. It is obvious that the 0.5 cm mesh size provides a much better prediction with an error within 5%, compared with those calculated by TM5-855-1. However, further decreasing the element size has insignificant influence on the numerical results but may lead to the risk of computer memory overflow and substantially increase the computing time (Zhang et al., 2016). Therefore, due to the limitation of workstation, the mesh size of 0.5 cm is determined for the following simulation in this study.

Figure 10.

Comparison of peak reflected pressure at different scaled distances.

Contact explosion test and validation of the FE model

It is well acknowledged that the accuracy of numerical predictions mainly depends on the model parameters, such as material parameters, erosion algorithm, boundary parameters, and mesh size. As described in the schematic model, the concrete shelter is sandwiched between the explosive charge and the air gap. Failure characteristics of the concrete shelter under explosion, such as crater, scabbing and spallation, are significantly important to determine the distribution of blast load on the protective structure. In other words, the calculated failure characteristics of concrete shelter are very important indices to validate the reliability of the developed FE model. Therefore, a field test was performed on the high-strength concrete shelter under contact explosion. The corresponding FE model was built to benchmark the determined material parameters, erosion algorithm, boundary parameters, and mesh size.

Test of concrete shelter under contact explosion

A concrete shelter specimen was cast and constrained in a circle steel culvert with a thickness of 0.5 cm, as shown in Figure 11. The diameter and height of the shelter are 1 and 0.4 m, respectively. A 2-kg cubic TNT charge was set at the center of the upper surface of the concrete shelter. There are two symmetric hooks welded on the steel culvert. The concrete shelter specimen was hung up by the two hooks and laid on three concrete units to keep the bottom surface free. The uniaxial compressive strength of the concrete material is 67 MPa. The strength and Young’s modulus of the steel culvert are 310 MPa and 210 GPa, respectively.

Figure 11.

Test setup of the contact explosion.

The damage characteristics of the concrete shelter under contact explosion are shown in Figure 12. It is obvious that a quasi-circular crater forms on the upper surface of the shelter. The average diameter and depth of the crater are 0.43 and 0.096 m, respectively. Moreover, there are many radial microcracks surrounding the crater on the upper surface. The dark areas on the upper surface of concrete shelter exhibit the splashing direction of the fireball and concrete debris, which can clearly indicate the typical shape of blast wave generated by the cubic TNT. At the center of the bottom surface, many large cracks have developed and are connected through, which shows the tendency of spallation of concrete fragments. The concentrated cracking area on the bottom surface also indicates severe damage and cracks inside the concrete shelter.

Figure 12.

Damage characteristics of concrete shelter under contact explosion: (a) upper surface and (b) bottom surface.

Validation of the FE model

A quarter of the concrete shelter under contact explosion was built along the FE model, as shown in Figure 13. Dimensions of the cubic TNT charge, concrete shelter, and steel culvert are in accordance with those in the test. The element size is 0.5 cm and the total element number is 2,741,204. *ALE_MULTI-MATERIAL_GROUP was defined for interface reconstruction of the ALE material groupings, that is, TNT explosive and air. A symmetric boundary condition was applied to the cutting planes. Nonreflecting boundary conditions were applied to the surrounding surfaces of the air domain. The material parameters for concrete, explosive, and air are in accordance with those described in section “FE model of the layered concrete shelter.” The static compressive strength of the concrete is 67 MPa. The erosion value of the maximum effective plastic strain is 0.01 for concrete. The steel culvert was simulated by *MAT_PLASTIC_KINEMATIC, Mat_3#. The mass density of steel is 7800 kg/m³, Young’s modulus is 210 GPa, Poisson’s ratio is 0.3, and yield strength is 310 MPa.

Figure 13.

FE model of concrete shelter under contact blast.

The calculated failure characteristics of the concrete shelter are shown in Figure 14. Figure 14(a) shows damage of the upper surface with the average diameter and depth of quasi-circular crater at 0.398 and 0.095 m, respectively. They are in good agreement with those captured in the test, where the average diameter is 0.43 m and the depth is 0.096 m. It is obvious that a lot of large cracks have been formed and connected through at the bottom of the specimen, as shown in Figure 14(b). The average diameter of the cracking area is 0.342 m, although the scabbing crater has not been generated on the bottom surface, which is also similar to the testing scenario. Figure 14(c) shows the calculated vertical cutting section of the concrete shelter. A seriously damaged zone can be observed above the bottom surface, which is induced by the reflected tensile shock wave from the free surface of concrete specimen. It is validated that the used arbitrary Lagrange–Euler algorithm, material models, erosion algorithm, and mesh size in the FE model are precise enough to simulate the failure characteristics of concrete shelter under contact explosion. The following simulation was conducted on the basis of these validated parameters.

Figure 14.

Damage characteristics of concrete shelter under contact explosion: (a) upper surface, (b) bottom surface, and (c) vertical section.

Blast mitigation effect of the concrete shelter and the air gap

Distribution of blast loads on the protective structure without concrete shelter

First, distribution of the pressure on the protective structure without concrete shelter was calculated for comparison. As shown in Figure 15, a quarter FE model without concrete shelter was built according to that indicated in Figure 4. The protective structure is modeled by the rigid element similar to that in Figure 8. A cylinder TNT explosive of 2-kg weight was considered in the simulation account for the charge of projectile. The radius and height are 0.04 and 0.244 m, respectively, as presented in Figure 4. The detonation point is set at the center of the TNT, and it is hung at 0.722 m above the protective structure. The mesh size is 0.5 cm and a total of 1,789,294 elements are lumped in the model. Figure 16(a) to (e) presents the process of wave propagation. The reflection of blast wave against the rigid protective structure is shown in Figure 16(f) to (h).

Figure 15.

Model of projectile explosion above the structure without concrete shelter.

Figure 16.

Propagation of the blast wave: (a) 0 μs, (b) 5 μs, (c) 50 μs, (d) 100 μs, (e) 150 μs, (f) 190 μs, (g) 250 μs, and (h) 400 μs.

As shown in Figure 16, the oval shape of the generated blast wave is different from a spherical shape generated by the spherical TNT charge. This is because greater amount of blast energy was concentrated on the axial direction than in the radial direction due to the cylindrical shape of TNT explosive. The peak reflected pressure P_r₀ distributed on the rigid protective structure is shown in Figure 17.

Figure 17.

Peak reflected pressure distribution on the protective structure.

As shown in Figure 17, it is obvious that the peak reflected pressure P_r₀ decreases very rapidly as r increases in the range 0 ≤ r ≤ 0.5 m. This phenomenon is mainly attributed to the oval shape of blast wave, as well as variation of the scaled distance caused by increase in the blast incident angle. P_r₀ induced by the same spherical charge calculated by TM5-855-1 is also plotted in Figure 17, and it is much lower than the P_r₀ induced by the cylinder TNT when r is smaller than 0.2 m. However, P_r₀ of the spherical TNT decreases much slower than P_r₀ of the cylinder TNT as r increases. It surpasses P_r₀ of the cylinder TNT in the range r > 0.2 m. The significant difference in the simulation results compared to the calculated results of TM5-855-1 is mainly attributed to the explosive shape. Only the spherical shape of explosive was considered in TM5-855-1; however, the shape effect of explosive cannot be neglected in the near field, because the height of air gap is very limited due to the architectural and economical restrictions.

P_r₀ of the cylinder TNT at r = 0 designated as Point A is 142.01 MPa. Correspondingly, P_r₀ at the distance r = 0.3 m is 6.62 MPa, which is less than 4.66% of the pressure at Point A. Thus, it is proved that the blast pressure is mainly concentrated in the area range of r < 0.3 m. The average diameter of this area is 83% of the hanging height of TNT charge. Figure 18 shows the reflected pressure time histories at different positions on the protective structure. By integrating the pressure time histories, the corresponding curve for the impulse distribution is obtained and is shown in Figure 19.

Figure 18.

Reflected pressure time histories on the protective structure without concrete shelter.

Figure 19.

Impulse distribution on the protective structure.

As shown in Figure 19, similar to the distribution of the peak reflected pressure, it is found that the tendency of the impulse curve against r also shows the distinct influence of the TNT geometry. Moreover, the area of r < 0.02 m, which is exactly beneath the TNT, shows a nearly uniform distribution of the impulse.

To quantitatively evaluate the blast mitigation effect of the layered concrete shelter with an air gap, a series of FE models were further built according to the schematic model presented in Figure 4.

Blast mitigation effect at the impact perforation limit of the concrete shelter

It is well known that the penetration depth x and the height of rear scabbing crater H_c were determined by the impact velocity of projectile in each concrete shelter. The reflected tensile wave from the free surface results in a rear scabbing crater in concrete shelter. If the impact velocity was very low, H_c and x were relatively small and independent. As the velocity increased, a critical impact perforation limit was achieved as the penetration depth x extended through to the rear scabbing crater, that is, H_c + x = H. Charge of the projectile is directly exposed to the air gap and there is no other barrier on the path of propagation of the blast wave. Hence, it is very necessary to consider this critical and dangerous limit for the engineering application.

A quarter FE model of the projectile in perforated concrete shelter was developed, as shown in Figure 20. The dimensions of projectile, the penetration depth, and the height of air gap are all the same as the schematic model in Figure 4. A corresponding rear scabbing crater was modeled in the concrete shelter. The height of the concrete shelter is H = 0.215 m and $α$ is the cone slope angle, that is, the angle between the moving direction of the projectile and the conical edge of the ejected plug (Chen et al., 2008). Dancygier (1998) suggested the cone slope angle $α$ be determined as $66 °$ 66º and 76º $76 °$ for normal strength concrete and high-strength concrete, respectively. $α = 76 °$ is determined for high-strength concrete with a uniaxial compressive strength of 67 MPa. Mesh of the developed FE model is also shown in Figure 20, where the total element number is 2,076,312.

Figure 20.

Model of the concrete shelter in perforation limit.

The calculated propagation of the blast wave is shown in Figure 21. It is interesting that only a bundle of the blast wave directly propagates to the protective structure through the rear scabbing crater. The shape of the generated blast wave is far different from that without the concrete shelter. Therefore, much detonation energy is assumed to be consumed in the failure of the concrete shelter.

Figure 21.

Propagation of blast wave: (a) 0 μs, (b) 5 μs, (c) 20 μs, (d) 30 μs, (e) 100 μs, (f) 200 μs, (g) 500 μs, and (h) 1000 μs.

Figure 22 shows the failure characteristics of the concrete shelter at perforation limit under explosion. It is found that the concrete shelter is severely damaged locally by the detonation energy. The original perforation hole is dramatically extended to a much larger perforated crater with an average diameter of 0.748 m. It is about 10 times that of the original perforation hole, with a diameter of 0.08 m.

Figure 22.

Failure characteristics of the concrete shelter: (a) upper surface and (b) vertical cutting section.

Distribution of the peak reflected pressure on the protective structure is designated as P_rp, where subscript r represents reflected pressure and p represents perforation limit. The pressure obtained from calculation in the perforation model is compared with P_r₀ without the concrete shelter in Figure 23. It is found that in the range of r < 0.08 m, that is, the diameter of the cylindrical explosive, the peak reflected pressure with the shelter is slightly smaller than that without the shelter. However, when r ≥ 0.08, P_rp with the shelter is larger than P_r₀ without the concrete shelter. The same phenomenon is found in Figure 24, which shows a comparison of the reflected pressure time histories on the protective structure. Distribution of the impulse on the protective structure is shown in Figure 25, by integrating the reflected pressure time histories.

Figure 23.

Comparison of peak reflected pressures on the protective structure.

Figure 24.

Comparison of reflected pressure time histories on the protective structure.

Figure 25.

Comparison of impulses on the protective structure.

However, an inverse phenomenon is found in Figure 25. The impulse on the protective structure with concrete shelter under the perforation limit condition is about 0.7 kPa s larger than that without the concrete shelter in the range of r < 0.5 m. This is mainly attributed to the reflected blast wave from the incline surface of rear scabbing crater. It overlaps the original blast wave to amplify the blast loads on the protective structure. This phenomenon also indicates a negative effect of the perforation concrete shelter on blast mitigation. It could be concluded that the perforation limit for concrete shelter is a dangerous limit on blast mitigation, and should be carefully avoided in the engineering design.

Blast mitigation rate of the layered concrete structure

As aforementioned, the case of critical impact perforation limit is a disadvantage for the layered concrete shelter with air gap. The impact velocity of the projectile, at which a given concrete shelter was penetrated to the critical impact perforation limit, is called the ballistic limit (Li and Tong, 2003). There would be a residual depth left in the concrete shelter when the velocity is lower than the ballistic limit, which is designated as H_r. H_r increases when the impact velocity of the projectile decreases. To mitigate the blast wave efficiently, there is an ideal limit where no scabbing occurs on the rear surface of the concrete shelter; the concrete shelter remains complete except for the penetration depth x. In contrast to the impact perforation limit, the ideal limit could be achieved by increasing the depth of shelter, or reinforcing with more steel bars. Similar to the schematic program described in Figure 4, a quarter FE model is built with the ideal limit, as shown in Figure 26. The dimensions of TNT, the height of concrete shelter, and the height of air gap are all similar to those suggested in Figure 4. The residual depth of the concrete shelter is H_r = H, that is, x = 0.3 m. The other parameters remain the same as those in the developed model at impact perforation limit. The total element number meshed in the model is 2,592,045. The calculated propagation of the blast wave is shown in Figure 27.

Figure 26.

Model of the concrete shelter without scabbing.

Figure 27.

Propagation of the blast wave: (a) 0 μs, (b) 10 μs, (c) 30 μs, (d) 50 μs, (e) 90 μs, (f) 200 μs, (g) 500 μs, and (h) 1000 μs.

As shown in Figure 27(a) to (d), a blast wave front was first generated in the concrete, whose shape is similar to the compressed crater of concrete under explosion. The blast wave generated and propagated in the concrete material was then mainly reflected back from the free surface of the shelter, as indicated in Figure 27(e) to (h). No obvious blast wave can be seen propagating in the air gap below the concrete shelter. This demonstrates the significant contribution of the far different wave impedance between the air and concrete. Therefore, most of the blast energy is consumed in the propagation and failure in the concrete shelter. Figure 28 shows the failure characteristic of the predamaged concrete shelter under explosion.

Figure 28.

Failure characteristics of concrete shelter: (a) upper surface, (b) bottom surface, and (c) vertical section.

As shown in Figure 28, the concrete shelter is seriously damaged by the explosion. Figure 28(a) and (c) shows that a quasi-circular explosion crater is generated on the upper surface of the shelter, whose average diameter and depth are 0.444 and 0.181 m, respectively. As for the bottom surface, as shown in Figure 28(b) and (c), a rear scabbing crater is also generated by concrete debris scabbing out, with average diameter and depth of the scabbing crater as 0.360 and 0.167 m, respectively.

The pressure mitigation rate R_mP*, which is an index of the mitigation effect of peak reflected pressure, is defined as the ratio of P_r* over P_r₀ in this study, as

R_{m P *} = \frac{P_{r *}}{P_{r 0}}

(7)

Similarly, the impulse mitigation rate R_mI*, which is an index of the mitigation effect of reflected impulse on the protective structure, was defined as the ratio of I_r* over I_r₀ as

R_{m I *} = \frac{I_{r *}}{I_{r 0}}

(8)

where * denotes the height of the concrete shelter.

The peak reflected pressure P_r_0.4 distributed on the protective structure captured in the calculation is compared with that without concrete shelter in Figure 29. It is found that the blast mitigation effect of the proposed layered structure is very significant. The calculated R_mP at all points on the protective structure exceed 99.9%. In particular, at the center point A of the protective structure, the peak overpressure P_r_0.4 is 2.77 kPa, which is only 0.002% compared to 142.01 MPa of P_r₀, that is, 99.998% of the reflected peak pressure is eliminated by the proposed layered concrete structure with an air gap. Similarly, a significant mitigation effect on the reflected impulse of the layered concrete structure can also be seen in Figure 30. It shows that $R_{m P}$ at all points on the protective structure also exceeds 99.9%, although $R_{m I}$ first decreases and then increases as r increases. The minimum value of $R_{m I}$ at r = 0.26 achieves 99.947%, strongly demonstrating the good performance of concrete shelter on blast wave mitigation.

Figure 29.

Comparison of the peak reflected pressure on the protective structure: (a) P_r and (b) R_mP_0.4.

Figure 30.

Comparison of the reflected impulses on the protective structure: (a) I_r and (b) R_mI_0.4.

Parametric study

Height and residual depth of concrete shelter

According to the analytical results above, it is found that the total height H and residual depth after penetration H_r of the concrete shelter affect the blast mitigation effects significantly. To find the contributions of the two parameters H and H_r to the blast mitigation effect, three more FE models were developed on the basis of the model built in section “Blast mitigation rate of the layered concrete structure” by revising the height of concrete shelter H to 0.2, 0.3, and 0.5 m, while the other parameters remained unchanged. Also, the scabbing phenomenon in the penetration process was not considered in the developed model; the corresponding residual depths of the concrete shelter H_r are 0.1, 0.2, and 0.4 m, respectively. The calculated failure characteristics of the concrete shelter with four different heights, 0.2, 0.3, 0.4, and 0.5 m, are compared in Figure 31.

Figure 31.

Failure characteristics of concrete shelter with different heights and residual depths: (a) concrete shelter with H = 0.2 m and H_r = 0.1 m, (b) concrete shelter with H = 0.3 m and H_r = 0.2 m, (c) concrete shelter with H = 0.4 m and H_r = 0.3 m, and (d) concrete shelter with H = 0.5 m and H_r = 0.4 m.

All of the concrete shelters were seriously damaged under explosion, with quasi-circular explosion craters generated on the upper surface, and rear scabbing craters generated by scabbing out. As shown in Figure 31(a) and (b), the two concrete shelters are both blasted to perforation, where the explosion crater on the upper surface and the scabbing crater on the bottom surface are connected. The blast wave could directly propagate to the protective structure via the connected crater. The average diameter of the upper and rear crater in the 0.2-m concrete shelter are 0.542 and 0.58 m, respectively, and the average diameter of the upper and rear crater in the 0.3-m concrete shelter are 0.442 and 0.452 m, respectively, providing a decrease of 18.5% and 22.5%.

Benefiting from the large height and residual depth, concrete shelter with height of 0.4 and 0.5 m avoid being blasted to perforation, as shown in Figure 31(c) and (d). A residual thickness of concrete T_r_* survives between the upper and rear craters, where subscript r represents residual thickness and * represents the height of shelter. In Figure 31(c), T_r_0.4 is 0.052 m, where the depth and average diameter of the upper crater are 0.181 and 0.444 m, respectively, and the depth and average diameter of the rear crater are 0.167 and 0.360 m, respectively. In Figure 31(d), T_r_0.5 increases to 0.105 m, where the depth and average diameter of the upper crater decrease to 0.150 and 0.390 m, respectively; the depth and average diameter of the rear crater are 0.245 and 0.434 m, respectively. It can also be concluded that the local damage level in the concrete shelter under explosion decreases with the total height H and residual depth H_r, approximately.

Figure 32 shows a comparison of the peak reflected pressure P_r and the blast pressure mitigation rate R_mP distribution on the protective structure with different heights of concrete shelter.

Figure 32.

Comparison of P_r and R_mP with different shelter heights: (a) peak reflected pressure P_r and (b) reflect overpressure mitigation rate R_mP.

As shown in Figure 32(a), it can be seen that the magnitude of P_r_0.4 and P_r_0.5 is of the order of kPa, while that of P_r_0.2, P_r_0.3 is of the order of MPa, which is 1000 times of kPa. A similar phenomenon is seen while comparing the pressure mitigation rates R_mP, as shown in Figure 32(b), although the pressure mitigation effects of the four calculated layered concrete structures are commonly very significant. R_mP_0.4 and R_mP_0.5 are always larger than 99.96%. In contrast, R_mP_0.2 and R_mP_0.3 are much smaller; generally, R_mP_0.2 varies from 27.02% to 96.57%, while R_mP_0.3 varies from 66.02% to 98.15% as r changes. The interesting appearance is mainly attributed to the residual thickness of concrete under explosion T_r*, as described above and as shown in Figure 32. The thinner concrete shelters with total height of 0.2 and 0.3 m are both blasted to perforation under the explosion, which allows the air gap to be directly exposed to the blast wave and induces a drastic increase in P_r. On the contrary, the residual thickness of concrete in the shelter T_r_0.4 and T_r_0.5 isolated the air gap away from the upper crater. Great mitigation effect on the magnitude of blast wave is achieved, benefiting from the far difference on wave impedance between concrete and air.

Besides, as shown in Figure 32(a), P_r first decreases, then increases, and finally decreases as r increases, which shows a second peak value in the P_r–r curves. This second peak is probably caused by superposition of the direct incident blast wave and the reflected blast wave, which reflected first from protective structure and then reflected back toward the protective structure by the shelter. The delay in the second peak value of P_r_0.2 compared with P_r_0.3 can also support this argument. Mitigation effect on the reflected impulse of the layered concrete structure is demonstrated in Figure 33.

Figure 33.

Comparison of I_r and R_mI with different shelter heights: (a) reflected impulse I_r and (b) reflected impulse mitigation rate R_mI.

The trend of the reflected impulse I_r distributed on the protective structure is found similar to that of P_r in Figure 33(a); however, the magnitude of I_r is quite limited compared to that without concrete shelter. As shown in Figure 33(b), R_mI_0.4 and R_mI_0.5 exceed 99.9% at all of the survey points on the protective structure, which also indicates a good effect of T_r* on blast wave mitigation. In contrast, R_mI_0.2 and R_mI_0.3 are much lower, and even negative effects, R_mI < 0, can be found. The minimum values of R_mI_0.2 and R_mI_0.3 are −79.9% and −85.4%, respectively. This means that the existing concrete shelter enlarges the reflected impulse on the protective structure, because the concrete shelter might then reflect the reflected blast wave from the protective structure. This would help to constrain the propagated blast energy in the air gap when the shelter was blasted to perforation, which is really a dangerous situation in the engineering application. In addition, the second peak in the I_r_0.2–r and I_r_0.3–r curves in Figure 33(a) also supports this standpoint.

It is concluded that the residual thickness T_r of concrete shelter under explosion is a key parameter for blast mitigation in the proposed layered concrete structure. It increases with the total height H and residual depth H_r of the shelter in the case of a given projectile and concrete strength. In order to achieve the highest blast mitigation rate, the designed height of the concrete shelter H should be determined larger than the safety height H_s, that is, the critical height protecting the concrete shelter from being impacted and subsequently blasted to perforation.

Formula of the pressure mitigation rate

An important aspect in the analysis and design of the protective structure is the comparison of effects from various weapons detonated at various distances. Such a comparison can be made by employing acceptable scaling laws (Krauthammer, 2008). One approach is introducing the scaled distance $λ$ to describe the overpressure induced by specific charge W of explosive at distance R away from the detonation point

λ = \frac{R}{W^{1 / 3}}

(9)

In this study, the scaled distance $λ$ is also introduced to describe the variation of pressure mitigation rate R_mP by the layered concrete structure with an air gap, where R denotes the vertical distance from the protective structure to the detonation point. The residual depth H_r of concrete shelter in penetration is a key designing parameter for the proposed layered concrete structure. It also depends on the total height H of the concrete shelter in the case of a given projectile and concrete strength. Therefore, a nondimensional residual depth H_N was also employed to describe R_mP as

H_{N} = \frac{H_{r}}{H}

(10)

A number of simulations at different scale distance $λ$ and nondimensional residual thickness H_N were performed. All of the simulated data and the obtained R_mP are plotted in Figure 34.

Figure 34.

Fitting data on the pressure mitigation rate R_mP.

Fitting these data with a two-dimensional (2D) parabola method provides a useful formula for the pressure mitigation rate

R_{m P} = 105.9 + 76.1 H_{N} + 15.6 H_{N}^{2} - 154.8 λ + 71.4 λ^{2}

(11)

The fitting results are also plotted in Figure 34, which agree well with the simulated data. It demonstrates a good validation of the fitting formula. This fitted formula could be applied to calculate the layout of the layered concrete structure to achieve the best blast mitigation effect.

Conclusion

A novel layered concrete structure with an air gap was proposed in this study to resist explosion attacks from terrorist bombings or conventional weapons. The blast mitigation effect was investigated using the LS-DYNA program. A series of elaborate FE models were developed, which were validated by field tests of contact explosion. The entire process of explosion, including failure of concrete shelter, propagation of blast wave, and its interaction with protective structure, was investigated. It was demonstrated that the proposed layered concrete structure was able to resist impact and subsequent explosion of the projectile, and could mitigate the blast loads on the roof of protective structure, significantly. The following conclusions are drawn:

Owing to the far difference wave impedance between concrete and air, the concrete shelter that was not blasted to perforation by projectile could provide good performances on mitigating blast loads on the roof of the protective structure. Over 99.9% of the peak reflected pressure and over 99.9% of the reflected impulse on the roof of the protective structure could be mitigated, compared with that without concrete shelter.

The critical impact perforation limit of concrete shelter in penetration is a dangerous condition that can cause negative effects on blast mitigation, which should be avoided in the engineering design.

Parametric study reveals that the larger the residual depth of concrete H_r that remains after penetration, the better the blast mitigation effect found on the protective structure. A residual thickness of concrete T_r* that survives between the upper and rear craters after explosion was found to be a critical parameter to determine the effectiveness of layered structure on blast mitigation. Low mitigation effect on the peak reflected pressure and even negative effect on reflected impulse was found, while no residual thickness of concrete survived under explosion. R_mP_0.2 varies from 27.02% to 96.57%, while R_mP_0.3 varies from 66.02% to 98.15% as r changes; R_mI_0.2 and R_mI_0.3 even provide minus values of −79.9% and −85.4%, respectively.

To achieve the highest blast mitigation rate of R_mP and R_mI, the designed height H of the concrete shelter should be determined larger than the safety height H_s, which is the critical height protecting the concrete shelter from being impacted and subsequently blasted to perforation.

A 2D parabola formula on the reflected pressure mitigation rate considering the scaled distance $λ$ and nondimensional residual thickness H_N was fitted and put forward. It was very helpful for designing the proposed layered concrete structure to achieve the best blast mitigation effect.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors acknowledge the financial support from the National Basic Research Program of China (Grant Nos 2016YFC0305200 and 2015CB058003), the National Natural Science Foundation of China (Grant Nos 51622812 and 51427807), and China Postdoctoral Science Foundation (Grant No. 2017M613379).

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