Blast resistance of air-backed RC slab against underwater contact explosion

Gung-dong Yng , Yong Fn ,*, Go-hui Wng , Xin-ze Cui , Zhen-dong Leng ,Wen-bo Lu

a Hubei Key Laboratory of Construction and Management in Hydropower Engineering, China Three Gorges University, Yichang, 443002, China

b College of Hydraulic & Environmental Engineering, China Three Gorges University, Yichang, 443002, China

c School of Water Resources and Hydropower Engineering, Wuhan University, Wuhan, 430072, China

Keywords:Air-backed RC slab Underwater contact explosion Dynamic behavior Parametric study Numerical simulation

ABSTRACT Reinforced concrete(RC)structures are common in engineering,and usually exposed to air or water,may be subjected to various blast scenarios.This paper aims to investigate the blast resistance of an airbacked RC slab against underwater contact explosions (UWCEs).A detailed numerical model based on CLE method considering explosive, water, air, and RC slab is developed to examine the structural behavior of the air-backed RC slab due to UWCEs.At first, the reliability of the numerical method is validated by comparing the numerical results of an UWCE test with experimental data.Then, the difference in dynamic behavior of air-backed and water-backed RC slabs due to UWCEs is explored with the calibrated model.The results indicate that the blast response of the air-backed slab induced by UWCE is fiercer than that of water-backed slab with equal charge mass.In addition, parametric studies are also conducted to explore the effects of the charge mass, standoff distance, reinforcement spacing, concrete compression strength, and boundary condition on the blast performance of the air-backed RC slab.

1.Introduction

RC material is common in engineering.However, impact and blast loads induced by terrorist attacks or unexpected incidents may absolutely destroy structures,resulting in huge casualties and property damage.In recent years,the dynamic response and failure mechanism analysis of RC structures induced by blast loads has been widely investigated[1-9].In this work,we intend to explore the resistance of air-backed RC slabs against UWCEs.

As a common component in engineering, most of slabs are exposed to the air and threatened by air explosive loads.But some(submerged pipes, drilling platforms, sluices and submarine tunnels, and so on) are embedded in water, could suffer submerged blast loads.Therefore, it is very meaningful to explore the resistance of RC slabs against different explosion scenarios.At present,there are numerous studies about the behaviors of RC components against air explosions available in open literatures.Zhao [10] and Wang [11] et al.investigated the damage mechanism and mode of square RC slabs induced by air close-in blast loading.Shi et al.[12]discussed the local damage and fragment characteristics of RC slabs subjected to close-in explosions.Dua [13-15] studied the blast behaviors of slabs against contact explosions.They found that concrete slabs exhibited two failure patterns.Ichino et al.[16]examined the influence factors on the performance of two-stage plate induced by contact explosions, and discussed the prediction equations for the local failure depth.Li et al.[17,18]experimentally investigated the anti-explosion performance of UHPC slabs.Yu et al.[19] evaluated the strengthening effectiveness of BFRP bar on concrete slabs.Although the above literatures focused on the blast resistance of RC slabs against air explosions, no thorough investigation has been conducted on the effects of submerged blast loads.

For underwater explosions,there are many achievements on the dynamic response of ships, submarines and propellers [20-25].Zhang et al.[26] applied SPH method to investigate the shapedcharge jet penetration due to submerged explosions.Results indicated that the air-backed plate was damaged more seriously than water-backed plate.Schiffer and Tagarielli [27,28] performed investigations on the blast behaviors of water-backed and air-backed metal panels induced by submerged explosions.For submarine RC structures against blast loading, Zhao [29,30] and Yang [31] et al.used blast tests and numerical methods to research the blast performance of slabs under air blast loads and UWCEs.Hai and Ren[32] investigated the resistance of air-backed RC slabs against submerged non-contact explosions.Zhuang[33]and Yan[34]et al.investigated the damage modes of RC columns induced by underwater explosions.However, with regard to the structural response and failure patterns of air-backed RC slabs due to UWCEs,there has no past research effort to the best knowledge of authors.

In this study, a numerical approach is devised to explore the blast resistance of an air-backed RC slab against UWCEs.At first,the load characteristics induced by underwater explosion, and the response of air-backed and water-backed slabs induced by uniform impact loads are reviewed.Then,a complicated numerical model is established, and the verification of the numerical method is performed through comparing the numerical results with experimental data.With the validated model, numerical simulations of air-backed and water-backed RC slabs against UWCEs are then carried out.Besides, a comprehensive investigation on the blast resistance of the air-backed RC slab affected by various parameters(e.g., charge weight, standoff distance, reinforcement spacing,concrete strength and boundary condition) subjected to underwater explosions is further considered.

2.Theoretical background

The underwater explosion loads mainly consists of shock wave and bubble, as shown in Fig.1.

Cole has done much research in underwater explosions, and devised empirical formulas to predict the shock wave and the maximum bubble radius[35].

where Pmis the peak pressure, θ is the decay constant, W is the explosive weight,kg.R is the distance.Rmaxis the maximum bubble radius.KRis constant for explosive charge, and H is the depth of charge.For TNT explosive, k1=52.12 MPa, k2= 0.0895, α1=1.18,α2= -0.185, l = 5760, β = 0.89, KR= 3.38.It is greatly difficult to accurately measure the pressure in the vicinity the explosives due to the extremely high pressure.Hence, the above empirical equations for pressure are feasibility to the distance over six charge radii.However, the pressure in the near-field is crucial for structures when subjected to close-in explosions.

When the slab is subjected to underwater explosion, the medium behind the structure may be air (air-backed slab, such as underwater traffic tunnel,sluice gate)or water(water-backed slab,such as submarine pipeline, deep gate).The response of a flexible slab under impact loads can be estimated using Taylor's plate theory [36].When a plane wave Pi(t) impacts an infinite slab, a reflected wave Pr(t) will happen and depart from the slab, as displayed in Fig.2.

The pressure acting on the front of the slab is

According to the law of momentum conservation

where Pf(t) is pressure on the front surface;ρ is the density of the medium on the incident side (water), and c denotes the wave velocity.u(t)signifies the position of the slab.Combine Eqs.(6)and(7)

According to Newton's second law

where ρsis the slab density; hsis the slab thickness; Pb(t) signifies the back pressure.

The incident pressure Pi(t)induced by an underwater explosion can be expressed as Eq.(1).

For the air-backed slab,the Pb(t)can be regarded as 0,then the motion equation can be described as

Fig.2.Sketch of air-backed or water-backed slab under impact loads.

For the water-backed slab, pb（t） = ρc ˙u（t）, then the motion equation:

From Eqs.(10) and (11), it can be seen that the air-backed slab will experience more violent response, although the impact loads are the same.Generally, for structures with inerratic geometries under far-field blast effects, the blast loads can be predicted by empirical formulas, and structural response can be estimated.The single degree of freedom (SDOF) system which simplifies the motion of the continuum structure into the one-dimensional motion of the feature direction is brought into the design of anti-blast structures.This method can also quickly present the maximum deflection, speed and acceleration of the component caused by uniform blast loads.But for more complex situations, such as contact or close-in explosions, the high frequencies and short duration dynamic loads exhibit strong spatial and time variations,leading to sharp stress gradients in the structures and a varying strain rate for the duration of the analysis.Theoretical analysis on the local damage of concrete is quite complex.In addition, local concrete damage is dependent on the stress wave shapes, magnitudes, angles of incidence, and the dynamic properties of the concrete material under high strain rate [3].Accurately evaluating the dynamic behaviors of structures under contact or close-in explosions is very complicated.Hence,the existing research results of structural response induced by contact or close-in explosions mainly depended on numerical simulations and experimental tests[1-3,13,37,38].

3.Description of the material model

3.1.Model and parameter for concrete

The properties of concrete material are related to the loading rate.At a high strain rate,the strength of concrete can be improved significantly.There are many material models, such as K&C (Karagozian and Case), CSCM (Continuous Surface Cap Model), JHC(Johnson Holmquist Concrete), and RHT (Riedel, Hiermaier and Thoma)to simulate the concrete under blast loading.In this work,the RHT material model[39]is used,and the suitability of the RHT model in modelling dynamic behaviors of concrete material has been proved in previous studies [40-42].

Strain rate effects of concrete under impact loads are reflected:

where α and δ denote the factor of strain rate.

It should be noted that the moisture content of dry and damp concrete is different.Previous results [43-45] found that the strength of the concrete increased with the increasing moisture content under dynamic loading.Zhao et al.[30]concluded that the explosion loads played the main role on the dynamic response of the slabs instead of dry or damp concrete.The material parameters in the RHT model were calibrated according to the experimental results of RC slab subjected to underwater explosion.However,further studies of the moisture content effects on the behaviors of concrete against blast loads are deemed necessary.All the material configurations of the concrete applied in this study are shown in Table 1.

3.2.Model and parameter for steel reinforcement

Johnson-Cook material model is selected to represent the reinforcement under explosion loads.Yield stress Y can be expressed[46].

where A denotes yield stress, 400 MPa.B and n are constants reflecting strain hardening, 510 MPa and 0.26.C and m are 0.014 and 1.03.?psignifies effective plastic strain.T is material temperature.

3.3.Model and parameter for TNT

TNT, as a reference for measuring the energy released when explosives explode,is widely used in the performance evaluation of structures under blast loads.Other explosives, such as RDX and HMX can be equivalent to TNT based on the principle of energy equality[47-49].TNT explosive is described by JWL equation[46].

where p denotes hydrostatic pressure.A, B and ω are the pressure coefficients, and V denotes specific volume.R1is the principal eigenvalue, and R2is the secondary eigenvalue.The parameters of TNT charge are displayed in Table 2.

Table 1Material parameters in the RHT model.

Table 2Material parameters for TNT charge.

3.4.Model and parameter for air

Because the acoustic impedance of the concrete is much larger than that of the air, the pressure of the shock wave transmitted through the slab into the air may be very small.Hence, using an ideal gas state equation may have negligible influence on the blast response of air-backed slab subjected to underwater explosions.Pressure for perfect gas is

where γ denotes gas adiabatic exponent,1.4.ρ represents the current density.ρ0is the initial density, 1.292 × 10-3g/cm3.E0= 0.25 MPa is initial internal energy.

3.5.Model and parameter for water

Two-phase expansion EOS is used to model the water [46]

When μ > 0

where μ=ρ/ρ0-1,ρ0denotes the initial density.The term e signifies internal energy.A1, A2and A3are 2.2, 9.54 and 14.57 GPa, respectively.B0, B1, T1and T2are 0.28, 0.28, 2.2 and 0, respectively.

3.6.Coupled Lagrange-euler numerical approach

Coupled Lagrange and Euler(CLE)method[50],which combines the advantages of Euler and Lagrange methods, can effectively model the fluid-structure interaction and large deformation issues.The slabs are modeled by Lagrange element,in which the numerical mesh moves and distorts with the physical material.The air,water and TNT are modeled by Euler method, in which the numerical mesh is fixed in space and the physical materials flow through the mesh.The CLE method is implemented using a special solution strategy: the Lagrange mesh can invade the Euler mesh, and the Euler region can exert pressure on the Lagrange surface.In other words,the Lagrange element provides displacement boundaries for the Euler element,and the Euler region exerts pressure boundaries on the Lagrange region.

3.7.Erosion

Large deformation or high-velocity distortion of the elements will cause the false calculation or low efficiency.Erosion is a numerical mechanism for the automatic deletion of distorted elements once the criteria are met.The erosion technique has been widely employed in simulating the fracture, cutting and breach of concrete structures under impact or blast loadings for avoiding the extreme distortions of Lagrange meshes.However, the deletion of the cells will lead to the reduction of mass and energy.Ignoring the influence of the erased grids is not in conformity with the physics of the problem.Hence, erosion criteria must be selected cautiously.The common erosion criteria are based on principle strain [10,51],effective strain [52], tensile strength [53,54], geometric strain[40,55],and so on.In the present study,geometric strain of 0.5[40]is used as the erosion criterion.

4.Model validation

Field test is an essential approach to explore the blast loads and its influence on structures.But it is impractical to carry out substantial tests to study the anti-explosion performance of structures because of the high requirements in cost, time, equipment, safety and repeatability.With advanced computer techniques, it is advantageous to explore the blast resistance of structures using numerical method.Once properly validated, the numerical method can be used as a replacement for costly blast experiments.

There are two major loads in underwater explosions, namely shock wave and bubble.However,the development of the bubble is restricted by structure in contact or very close-in underwater explosions because the distance between the structure and the explosive is smaller than the maximum diameter of the bubble.The effects of the shock wave and bubble on the RC slab are considered based on the CLE method.The non-linear finite element code,AUTODYN is employed for the numerical investigations in this work, which has been widely utilized in structural engineering to simulate the impact and explosion-related issues, and has the ability to conduct advanced dynamics analysis with complex material models.The numerical method,after verification with testing data,is used in an extensive research on the resistance of air-backed RC slab subjected to submerged explosions.

4.1.Descriptions of field blast test

To validate the accuracy and reliability of the numerical model developed for simulating blast resistance of air-backed RC slab against underwater contact explosion, a field test of water-backed RC slab subjected to underwater contact explosion was used to verify the RC model and the CLE method.The RC slab was immersed in water.Ideally testing data on air-backed slab subjected to underwater blast loads should be used to calibrate the numerical model since the model will be principally used to simulate the behaviors of air-backed slab under submarine explosions.Unfortunately, testing data of air-backed RC slabs subjected to underwater explosions are not available in the open literature.However,a calibrated model that can reliably predict water-backed RC slab damage induced by underwater explosion should also be able to predict air-backed RC slab response.

The RC slab had dimensions of 500 mm × 500 mm with a thickness of 60 mm was employed to verify the analytical method.The geometry and reinforced details of the slab are presented in Fig.3(a).The TNT explosive was positioned on the slab center,and the explosive mass was 6 g.The concrete had the compressive strength of 38 MPa,and a cover thickness of 20 mm.The diameter of the rebar was 8 mm.The yield strength and Young's modulus of the reinforcement were 400 MPa and 200 GPa, respectively.The slab was simply fixed on two opposite sides.Two PCB138A01 piezotronics pressure sensors were placed in the positions shown in Fig.3(b) to capture the explosion-induced pressure-time histories in the water.The PCB138A01 pressure sensor is characterized by built-in integrated amplification circuit, which can convert high impedance charge signal into low impedance voltage output.The measurement range of the pressure sensor is 68,950 kPa with a resolution of 0.14 kPa, and the resonant frequency is larger than 1000 kHz.The measuring devices were connected to a Blast-PRO multichannel data acquisition system that sampled pressure data at a rate of 1000 kHz.The pressure sensor and Blast-PRO acquisition system are displayed in Fig.3(c).

Fig.3.Experimental setup: (a) Dimensions and reinforcement arrangements; (b) Test devices, (c) Pressure sensors and acquisition system, (d) Numerical model.(Unit: mm).

A 3D model which involving TNT explosive,water,precast steel frame and RC slab is created to reproduce the blast test,as displayed in Fig.3(d).The slab and precast steel support are discretized with Lagrange elements while the steel reinforcement is described using beam meshes.The concrete and reinforcement bar have a strong bond without any slips.The element size is 5 mm for the concrete and steel reinforcement.Euler method is used to describe the explosive and water.The water domain is 600 mm×500 mm×600 mm.The fine size is approximately 3 mm for the TNT charge,and gradually increasing mesh sizes are used for Euler domain.Outflow boundaries are implemented on the external surfaces of Euler domain.Y-direction restrictions are imposed on the support bottom.

4.2.Numerical and experimental results

Fig.4 compares the damage from the blast test and numerical method.There was a small crushing area at the slab center.Several compressive cracks and a large number of tensile cracks distributed on the front and back surfaces,respectively.The damage degree of the back was worse than that of front surface, and more cracks could be found on the back.The damage status from numerical results is consistent well with the blast test.In the numerical simulation,several cracks appear on the front,and extensive cracks on the back propagate from the center to the corners.

Fig.4.Comparison of damage status: (a) Front; (b) Back.

Comparison of measured and simulated pressure-time histories is given in Fig.5, the peak values and the global change trend of simulated pressure data show good agreement with the testing data except for some minor errors.The errors may be caused by the special environment in the blast test.Because the test was conducted underwater,the positions of the sensors might slightly shift when the sample and the sensors ware placed into the water.In addition,the explosion caused water fluctuation,thus affecting the stability of the pressure sensors.The peak values of the calculation and test results at measuring point 1 are 38 and 40 MPa.The peak value at measuring point 2 is 16 MPa,and the computed results give a prediction of 14.5 MPa.From Figs.4 and 5,it can be seen that the calibrated model successfully predicts the blast pressure in water and describes the dynamic behaviors for RC slab induced by UWCEs.

5.Numerical simulation

Fig.5.Comparison of pressure-time history curves: (a) Measuring point 1; (b) Measuring point 2.

The calibrated model based on the experimental data is believed reliable for simulating the resistance of air-backed RC slabs against submerged explosions because the fluid-structure coupling method and material constitutive models are the same.

5.1.Finite element models

The geometry of an air-backed RC slab is given in Fig.6.The length,width,and thickness of the slab are 1000,500,and 60 mm,respectively.The concrete has the compressive strength of 40 MPa.The diameters of the reinforcement bars are 8 mm, and have the yield strength of 400 MPa.Two opposite sides of the air-backed slab are defined as fully fixed supports, and the other two sides are assumed as the free boundaries, as shown in Fig.6(b).10 g TNT is detonated at the top center.Both the sizes of the concrete and reinforcement elements are 5 mm.Nodes of the steel layers are attached to the concrete nodes one to one at the intersections preventing the two materials from sliding.The dimensions of the Euler domain are 1200 mm × 700 mm ×700 mm.The fine size is 3 mm for the TNT charge, and gradually increasing mesh sizes are used for Euler domain.240,000 Lagrange elements and 1,080,000 Euler elements are used in the model.

5.2.Blast response of air-backed and water-backed slab

When RC slabs suffer underwater explosions,the common fluids behind the structures are air and water.Compared with the air,water has greater density and can be considered an incompressible fluid, resulting in the dynamic behaviors of the air-backed and water-backed slabs may be different.Therefore, the difference in dynamic behaviors of air-backed and water-backed RC slabs subjected to UWCEs is explored.The numerical model of the waterbacked slab is same as that of the air-backed slab, as displayed in Fig.6(b).Only difference is that the air is replaced by water.

Fig.6.Numerical model: (a) Geometry of RC slab, (b) Details of model; (c) Finite element model.(Unit: mm).

Fig.7.Propagation process of blast wave: (a) Water-backed RC slab; (b) Air-backed RC slab.

Fig.8.Comparison of pressure time histories between air-backed and water-backed slabs: (a) Point A; (b) Point B.

The initial propagation process of blast waves is displayed in Fig.7.The peak pressure impacting on the structure exceeds 600 MPa(t=10 μs),which will crush the concrete.The compressive waves propagating through the slab are reflected.The superposition stresses of incident compressive and reflected tensile stresses exceed ultimate tensile strength, which will cause the concrete spall (t = 50 μs).Meanwhile, it can also be found that stress wave reflection happens near the free edges(t=100 μs).As displayed in Figs.6(a) and 6(b), two target points are arranged on the front and back to record the pressure time histories during the underwater contact explosions, respectively.Fig.8(a) shows pressure-time histories in the water near the front surface of the slabs (Point A), it can be found that the peak and the variation trends of the pressure time histories in the water-backed and airbacked conditions are highly similar.However, the air-backed slab has no obstruction of back water and more easily deforms backwards, resulting in the pressure acting on the front surface of the air-backed slab is smaller than that of the water-backed slab after the peak.Transmitted pressure time histories in the waterbacked and air-backed conditions are shown in Fig.8(b).Compared with the pressure transmitted into water (waterbacked),the pressure in the air(air-backed)is negligible.The peaks of the pressure at Point B in the water-baked and air-backed conditions are 15 and 0.01 MPa, respectively.

Figs.7 and 8 demonstrate that the propagation process of blast waves is significantly different when the fluids behind the slab are water and air.Because the wave impedance of water is greater than that of air,transmitted waves from the back surface of the RC slab in the water can be observed clearly,but the transmitted waves in the air behind the slab is insignificant.Greater reflected tensile stress happens on the back of the air-backed slab,which will cause more serious spall damage.

Fig.9(a)exhibits the damage evolution of water-backed RC slab caused by the UWCE of 10 g TNT,the development of the explosive products is also captured.The development of explosive products is restricted because of the existence of the structure.At t = 0.3 ms,the effect of the shock wave is weaker, and the spall on the back almost stops.No obvious global deformation is observed, the damage is believed caused by the wave propagation and reflection(Fig.9(a), t = 0.3 ms).However, the concrete damage further develops due to the effects of the explosive products with high pressure.There is a compressive zone in the vicinity of the detonation point, which causing minor cracks close to the free boundaries and the global bending deformation(Fig.9(a),t=1.0 ms).Due to the restriction of reinforcement,the concrete on the front of the slab fails across the transversal direction.The final failure mode of the water-backed slab exhibits a breach failure with global bending deformation (Fig.9(a), t = 10 ms).

Fig.9(b)displays the damage process of the air-backed slab.The diameter of the spall on the back is larger than that in the waterbacked slab.This is because that the wave impedance of air is smaller than water,resulting in a stronger reflected tensile wave on the back of the air-backed slab(Fig.9(b),t=0.3 ms).More serious compressive zone happens under the effects of the explosive products due to the lack of water resistance (Fig.9(b), t = 1.0 ms).The final damage of the air-backed slab induced by the UWCE is obviously larger than that of water-backed slab, although the charge weight is same.The average diameters of the breach obtained from the water-backed and air-backed slabs are 13 and 18 cm.Meanwhile,more cracks occur on the back of the air-backed slab.In addition,it can be seen that obvious cracking appears along the two fixed support edges (Fig.9(b), t = 10 ms).

Fig.9.Damage of RC slab induced by UWCE: (a) Water-backed RC slab; (b) Air-backed RC slab.

Fig.10 compares the kinetic energy and deformation of the water-backed and air-backed slabs due to UWCEs.As can be seen from Fig.10(a), kinetic energy time histories of the water-backed slab are similar with that of the air-backed slab, but the peak of the water-backed slab is obviously smaller.The peak kinetic energy of the air-backed slab is 2300 J,which is about 2.5 times that of the water-backed slab.Fig.10(b) presents the peak deflection of the slab along the X-direction with air and water behind it.Because the elements in the vicinity of the explosive experience much fiercer blast loads,and the erosion criterion is met,resulting in the data of those elements is interrupted.But it can still be seen that the global deformation of the water-backed slab is significantly smaller than that of the water-backed slab.This is because that the water behind the RC slab will impede the backward deformation of the structure.

In Section 2, the theoretical analysis demonstrates that the airbacked slab will exhibit a greater response under uniformly distributed impact loads,compared with that of water-backed slab.In this section, the difference in structural response between the air-backed and the water-backed slabs induced by UWCEs is investigated by numerical method.Similarly, air-backed slab will absorb more energy from an UWCE, and suffer more serious damage.

Fig.10.Difference in dynamic response of water-backed and air-backed slabs: (a) Kinetic energy time history curves, (b) Peak displacement along X-direction.

5.3.Parametric study

Numerous studies were conducted to explore the behaviors of RC slabs under blast loads, but few achievements have been reported to compare the influence factors on the resistance of airbacked RC slab against UWCEs.The air-backed RC slab described in Section 5.1 is used as the basic model to explore the influence of charge weight, standoff distance, reinforcement spacing, concrete strength and boundary condition on the resistance against UWCEs.The material properties and reinforcement arrangements are same as those in Sections 5.1 and 5.2.

5.3.1.Influence of explosive mass

To study the effect of explosive mass on dynamic behaviors of the air-backed RC slab, TNT explosive of four different weights (5,10,15,and 20 g)detonated on the top center of slab are taken into consideration.Results show that the TNT mass has a significantly influence on the blast response of the air-backed RC slab.Fig.11(a)shows the deformation of the slab along the X direction.Because of the constraints on the two fixed edges, the deflections around the two fixed edges are negligible.With the increase of the charge mass,the deformation of the slab increases significantly.When the charge mass is 20 g, the air-backed slab collapses and loses its carrying capacity.The relationship between peak velocity and charge mass is summarized in Fig.11(b).It can be found that the peak velocity increases approximately linearly with increasing charge mass.For the air-backed slab under the explosions of 5,10,15 and 20 g TNT, the peak values of velocity are -37, -54, -76 and-86 m/s,respectively.Compared with the peak velocity caused by 5 g TNT, the peak velocity is increased 46, 105 and 132%,respectively.

Fig.12 displays the failure modes of the air-backed RC slab caused by contact explosions with charge weights of 5,10,15 and 20 g.It can be seen that the center zone of the slab has been penetrated, massive minor cracks appear on the back surface, and global bending deformation can be observed.With the increasing charge mass, the dimensions of breach and spall increase.The increasing volume of flying debris due to the spalling and crushing damage will pose a great threat to personnel and instruments shielded by the structure.In addition,shear failure happens at the ends of the slab with the increase of the charge mass.When the charge mass is 20 g, the puncture area covers the entire width of the slab,and the air-backed slab collapses,as shown in Fig.11(d).In conclusion, the increasing explosive charge could increase the deflection and cause worse damage.

Fig.11.Influence of charge mass on dynamic response of air-backed slab: (a) Peak displacement along X-direction; (b) Relationship between peak velocity and charge weight.

Fig.12.Influence of charge mass on damage of air-backed slab: (a) 5 g; (b) 10 g; (c) 15 g; (d) 20 g.

5.3.2.Influence of standoff distance

Four different standoff distances are investigated by varying the initiation distance from 0, 5,10-20 cm, and the explosive mass is 10 g.In general,increasing the standoff distance can greatly reduce the dynamic response of the air-backed RC slab.As shown in Fig.13(a), the maximum deflection can be reduced significantly.The peak velocity decays approximately exponentially with the increasing standoff distance,as displayed in Fig.13(b).For example,the peak velocity decreases from-54, -27, -15 to-8.1 m/s when the standoff distance increases from 0, 5,10-20 cm, respectively.Compared with the UWCE,the peak velocity can be reduced by 85%when the standoff distance is increased by 20 cm.

Fig.14 exhibits the damage distribution of the air-backed RC slab with various standoff distances.Fierce blast loads cause breach failure with flexural deformation in the slab when the standoff distance is smaller than 5 cm,as displayed in Figs.14(a)and 14(b).When the standoff distance is 10 cm, the slab exhibits local spall and global flexural failure, as shown in Fig.14(c).There is no obvious spall damage due to the smaller reflected tensile stress when the initiation distance exceeds 20 cm.The air-backed slab presents a flexural failure mode, and the spall is insignificant, as displayed in Fig.14(d).It can be concluded the failure mode changes from breach failure with flexural deformation, to flexural failure with spall damage, and flexural failure as the standoff distance increases.

5.3.3.Influence of reinforcement spacing

The reinforcement of the air-backed slab may also affect its dynamic behaviors because reinforcement can provide restriction to the concrete and improve the stiffness of the structure.Hence,the influence of reinforcement spacing on the blast resistance of the air-backed RC slab is also explored.Four different reinforcement schemes are considered in this section.The fundamental characteristics of the four reinforcement details are summarized as follows.

Case A: control specimen slab, the reinforcement scheme(Fig.15(a)) is same as that described in Section 5.1.

Fig.13.Influence of standoff distance on dynamic response of air-backed slab: (a) Peak displacement along X-direction; (b) Relationship between peak velocity and standoff distance.

Case B: reducing the transverse reinforcement spacing, as displayed in Fig.15(b).

Case C: reducing the longitudinal reinforcement spacing, as displayed in Fig.15(c).

Case D:reducing the transverse and longitudinal reinforcement spacing, as displayed in Fig.15(d).

As shown in Fig.16(a), the deformation of the slab decreases with the increasing ratio of the reinforcement.Enhancing the longitudinal reinforcement can result in the smaller bending deformation.However, it can be found that decreasing the transverse reinforcement spacing,the reductions of the peak deflections are inconspicuous.Comparing with Case A, the average reduction rates of peak deflections along the length direction with schemes of Cases B, C and D are 2.9, 13.7 and 17.9%, respectively.The main reason is that the slab bends due to the underwater explosion,increasing the longitudinal reinforcement can improve the bending stiffness and reduce the overall deformation.Fig.16(b) shows the kinetic energy time histories of the air-backed slabs subjected to the UWCE of 10 g TNT.From the kinetic energy time histories,the peak kinetic energy and decay trends of the slabs with various reinforcement schemes are very similar.The peak values of the airbacked slabs reinforced with Cases A, B, C and D are 2327, 2270,2300 and 2235 J,respectively.Compared with the control specimen slab Case A, the peak values have a decrease of 2.4,1.2 and 4.0%,respectively.Because the volume of steel bar is very small compared with the concrete (reinforcement ratio), the proportion of kinetic energy of reinforcement in the RC slab is relatively small.Hence, decreasing reinforcement spacing can lead to smaller kinetic energy of the RC slab, but the reductions are insignificant.

The failure modes of the air-backed RC slab with different reinforcement schemes are shown in Fig.17.It can be found that decreasing the reinforcement spacing can mitigate the damage of the slab, but the reduction is disappointing.The volume of fragments can be reduced by decreasing the transverse reinforcement spacing because restraint effects enhance the strength and ductility of the concrete, as displayed in Fig.17(b).Decreasing the longitudinal reinforcement spacing can improve the flexural capacity but can hardly mitigate the breach failure of the slab.Conversely, a larger spall area of concrete on the back happens due to pulling effects of the longitudinal steel bars, as shown in Fig.17(c).Decreasing both the transverse and longitudinal reinforcement spacing can provide stronger restraint to the concrete thus slightly alleviate the spall damage.In addition,the concrete cracking at the fixed supports can be mitigated,as shown in Fig.17(d).The average diameters of the breach reinforced with Cases A,B,C and D are 18,16,18 and 16 cm, respectively.The results indicate that the reinforcement spacing has limited influence on the local dynamic response (breach and spall) of the slab, but decreasing the reinforcement spacing can improve the flexural rigidity of the slab.

Fig.15.Details of the reinforcement schemes: (a) Case A; (b) Case B; (c) Case C; (d) Case D.

Fig.16.Influence of reinforcement spacing on dynamic response of air-backed slab: (a) Peak displacement along X-direction; (b) Kinetic energy time histories.

Fig.17.Influence of reinforcement spacing on damage distribution: (a) Case A; (b) Case B; (c) Case C; (d) Case D.

5.3.4.Influence of concrete strength

In this section, numerical simulations are conducted to discuss the effect of concrete strength on the blast resistance of the airbacked slab subjected to the UWCE of 10 g TNT.The compressive strength of the concrete is 40, 70,100 and 140 MPa, respectively.The reinforcement details are same as that in Section 5.1(Fig.6(a)).

From Fig.18(a), it can be seen that the peak displacement exhibits a significant mitigation with the increase of concrete strength.The reason is that the flexural rigidity and shear strength are improved by the increasing strength of concrete.Fig.18(b)shows the peak strains of the longitudinal rebar measured at the center of the slab.It can be found that improving the compressive strength of the concrete causes the larger peak strains of the reinforcement.For example, the peak strain at the mid-span induced by the contact explosion of 10 g TNT increases from 0.0059, 0.0076, 0.0115 to 0.0175 when the compressive strength increases from 40, 70,100-140 MPa, respectively.This is because that the higher-strength concrete can better resist the explosive loads and prolong the duration of the loads.In addition,it should be noted that all the peak values are larger than the yield strain of the reinforcement.

The damage modes of the air-backed slab with various compressive strengths are displayed in Fig.19.The blast resistance of the slab is stronger as the compressive strength increases.When subjected to the UWCE, the slab will directly suffer violent shock waves, resulting in the breach failure.The results indicate improving the strength of concrete can not only reduce the flexural deformation, but also effectively decrease and alleviate the spall and breach of concrete.The lengths of the spall damage for slabs with compressive strength of 40,70,100 and 140 MPa are 56,36,32 and 15 cm,respectively.The failure mode changes from breach and flexural failure, to breach failure as the compressive strength increases, as shown in Figs.18(c) and 18(d).In conclusion, the compressive strength has significant effect on the survivability of air-backed slabs when subjected to UWCEs.

Fig.18.Influence of compressive strength on dynamic response of air-backed slab:(a)Peak displacement along X-direction;(b)Relationship between peak strain and compressive strength.

Fig.19.Influence of compressive strength on damage of air-backed slab: (a) 40 MPa; (b) 70 MPa; (c) 100 MPa; (d) 140 MPa.

5.3.5.Influence of boundary condition

Boundary condition may affect the deflection of explosive loaded structures.Hence, numerical investigations are conducted to examine the influence of two different boundary conditions on the failure mechanism of the air-backed slab subjected to the UWCE of 10 g TNT:(1)fully fixed supports are applied on the two edges of the slab,as displayed in Fig.6(b);(2)normal constraints(fixed in Y direction) are applied on the two edges of the slab.

Fig.20 exhibits the blast behaviors of the air-backed slab with different boundary conditions.The slab can absorb the blast energy and transform it into internal energy through distortion and cracking.From the internal energy histories, the peaks of internal energy with fully fixed supports and normal constraints are extremely close, but the decay rate with fully fixed supports is smaller than that with normal constraints, as displayed in Fig.20(a).The reason is that the peak internal energy is induced by the shock waves,and the structural response is extremely localized in the initial stages of the explosion, which is not affected by the boundary conditions.The residual internal energy of the slab with fully fixed supports is greater than that of the normal constraints.This demonstrates that the fully fixed supports can absorb more external energy and have batter blast resistance.

The strain time histories of the longitudinal rebar measured at the center of the slab with different boundaries are extracted and compared,as displayed in Fig.20(b).The initial phase of the strain time histories with fully fixed supports and normal constrains is consistent, which also demonstrates that the boundary condition has insignificant influence on the local response of the structure induced by shock waves.However, the constraint condition has great effect on the global structural response due to the effects of high-pressure detonation products.The flexural rigidity of the slab with fully fixed constrains is stronger than that with normal constrains,resulting in smaller deflection.The maximum strains of the longitudinal rebar with fully fixed supports and normal constrains are 0.0095 and 0.0059, respectively.

The damage modes of the air-backed RC slab with different boundaries are shown in Fig.21.It can be seen that the slabs with fully fixed supports and normal constrains suffer equally local damage.However, the global deformation of the slab with normal constrains is larger than that with fully fixed supports, and more cracks occur on the back surface.Besides, penetrating cracks covered the whole width of the slab with normal constrains, and the slab collapses,as shown in Fig.21(b).These results also indicate that fully fixed supports are beneficial for improving the resistance to blast loading by increasing the flexural capacities of slab, but have negligible influence on the characteristic of the breach.

Fig.20.Influence of boundary condition on dynamic response of air-backed slab: (a) Internal energy time history curves; (b) Strain time history curves.

Fig.21.Influence of boundary condition on damage of air-backed slab: (a) Fully fixed supports; (b) Normal constraints.

6.Conclusions

In this work, a 3D analytical model is developed for the simulation of the dynamic behaviors of an air-backed RC slab subjected to UWCEs.The reliability of the numerical method is validated against testing data.Then, the difference in the blast resistance of air-backed and water-backed RC slabs against UWCEs is explored.Besides, the effects of charge mass, standoff distance, reinforcement spacing, compressive strength of concrete, and boundary condition are parametrically analyzed on the blast performance of the air-backed RC slab.Conclusions can be made through this research:

(1) The fluid types behind RC slabs have great effect on the structural response induced by UWCEs.The blast response of the air-backed slab is significantly greater than that of the water-backed slab.Not only a smaller area of local damage(breach and spall) is found, but also a smaller global deformation happens when water behind the slab.

(2) Increasing the charge mass could increase the deflection and cause worse damage.The increasing volume of the flying fragments generated from the spall and crush of the slab will bring great threat to personnel and instruments.In addition,shear failure will happen at the ends of the slab with the increasing charge mass.When the charge mass is 20 g, the area of the puncture covered the whole width of the slab,and the air-backed slab collapses.

(3) The failure mode of the air-backed RC slab changes from breach with flexural failure, to flexural failure with spall damage, and flexural failure as the standoff distance increases.Decreasing the reinforcement spacing can mitigate the damage degree and deformation of the slab, but the reductions are disappointing.

(4) Concrete strength has strong influence on the blast resistance of the air-backed slab.The failure mode of the slab changes from breach with flexural failure to breach failure as the compressive strength increases.Higher compressive strength can reduce the threat of high-speed fragments.

(5) Constraint condition can significantly affect the deflection of the air-backed RC slab due to UWCEs.Fully fixed supports at the two opposite edges are beneficial for improving the flexural capacities of the slab, but have negligible influence on the local damage, such as breach and spall.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The supports from the Natural Science Research of Jiangsu Higher Education Institutions of China(21KJB580001),the National Natural Science Foundation of China (Grant No.52209162,51979152), Educational Commission of Hubei Province of China(T2020005),Young Top-notch Talent Cultivation Program of Hubei Province, and Jiangxi Provincial Natural Science Foundation(20212BAB214044).