Yi Fn ,Li Chen ,Heng-bo Xing ,Qin Fng ,Fng-yu Hn
a State Key Laboratory of Disaster Prevention & Mitigation of Explosion & Impact, Army Engineering University of PLA, Nanjing, Jiangsu, 210007, China
b Engineering Research Center of Safety and Protection of Explosion&Impact of Ministry of Education,Southeast University,Nanjing,Jiangsu,211189,China
c State Key Laboratory of High Performance Civil Engineering Materials, Nanjing, Jiangsu, 211103, China
Keywords: Ultra-high-performance concrete Reinforced concrete slabs Explosion Fragment velocity Blast resistance
ABSTRACT When an explosion occurs close to or partially within the face of a concrete structure,fragments are rapidly launched from the opposite face of the structure owing to concrete spalling,posing a significant risk to nearby personnel and equipment.To study the lead fragment velocity of ultra-high-performance concrete (UHPC),partially embedded explosion experiments were performed on UHPC slabs of limited thickness using a cylindrical trinitrotoluene charge.The launch angles and velocities of the resulting fragments were the determined using images collected by high-speed camera to document the concrete spalling and fragment launching process.The results showed that UHPC slabs without fiber reinforcement had a fragment velocity distribution of 0-118.3 m/s,which are largely identical to that for a normal-strength concrete (NSC) slab.In addition,the fragment velocity was negatively correlated to the angle between the velocity vector and vertical direction.An empirical Eq.for the lead spall velocity of UHPC and NSC slabs was then proposed based on a large volume of existing experimental data.
When an explosive charge is detonated in the close-in range of a concrete structure,the compressive stress waves generated by the explosion pass through the concrete structure and reflect off the free surface on the opposite side,converting the compressive waves into tensile waves that cause concrete spalling.High-speed fragments released through this process can cause serious harm to nearby personnel,equipment,or structures.Therefore,examination of the fragmentation of ultra-high-performance concrete(UHPC)and estimation of its fragment size and velocity distribution are important in the design of protective structures for fortification against blast loads.The explosions involved in the study of concrete slab spalling typically employ near,contact,or embedded charges,each with a different spatial relationship to the concrete structure.
A significant body of research exists on the fragment size distribution of concrete structures subjected to various explosion types.Wu et al.[1]discovered that the concrete fragment sizes generated by near-charge explosions exhibited both a Weibull and a Rosin-Rammler-Sperling-Bennet distribution.Shi et al.[2]collected the reinforced concrete slab fragments generated by nearcharge blasts to analyze their mass and size distribution,finding that the total number of fragments and percentage of small fragments increased with the mass of the explosive charge.Li et al.[3]performed contact-charge explosion experiments on seven different concrete slabs,including two normal-strength concrete(NSC) and five UHPC slabs.The size distributions of the NSC and UHPC slab fragments were best described by the Weibull distribution and log-normal distribution,respectively.
Moreover,a significant quantity of experimental data has been collected on the fragment velocities generated by concrete structures subjected to blast loads.McVay [4]and the United Facilities Criteria(UFC)[5]studied the fragmentation of concrete slabs at the opposite end of near-charge explosions and measured the fragment velocity distribution in a small number of conditions.Shi et al.[2]measured the distance travelled by fragments from a vertical concrete slab to determine their ejection velocity in the direction normal to the slab.However,these velocities likely contained significant errors because the possibility of continued fragment rolling owing to their inertia was not considered.Furthermore,it was impossible to precisely determine the initial height and velocity direction of the fragments.Dörr et al.[6]measured the debris launch velocities (hereafter referred to as ‘spall velocities’) of concrete structures subjected to internal explosive loads,and on this basis proposed an empirical Eq.for predicting spall velocities.Lu and Xu [7,8]applied the conservation of energy approach to propose spall velocity Eq.s for the concrete fragments generated by internal (near-charge) explosions;the results closely agreed with the experimental data of Dörr et al.[6].
Lönnqvist[9],Yang et al.[10],and Kuenzel et al.[11]used highspeed cameras to measure the lead spall velocity of fragments launched from the back of contact-charge explosion loaded concrete slabs.These experiments generally demonstrated that spall velocities increased with decreasing thickness for each equivalent trinitrotoluene (TNT) mass;that is,given the same slab thickness,the spall velocities increased with the explosive mass.Yang et al.[10]performed a dimensional analysis to fit the experimentally measured spall velocities and thereby obtain an empirical Eq.describing them.However,the applicability of this Eq.was limited owing to the small volume of data used in their analysis.
Erik et al.[12]measured the shape,velocity magnitude and angle,and cloud density of the fragments produced by the explosion of a fully embedded charge in concrete,then analyzed the factors affecting the initial debris velocity using their experimental data.Haberacker et al.[13]performed experiments to measure the launch velocities,launch angles,masses,and shapes of the fragments ejected from the backs of brick walls subjected to near,contact,and embedded charges.However,no experimental data could be extracted from these reports,as their contents are confidential.
Numerical simulations can be used to acquire information that is difficult to obtain experimentally;this is particularly useful in the study of explosions.For example,Zhou and Hao[14]constructed a two-dimensional axially symmetrical model of a concrete material with randomly distributed high-strength coarse aggregates and a low-strength mortar matrix using AUTODYN to study the failure and fracturing of concrete slabs under contact blast loading.They found a significant correlation between the size distribution of the fragments and the aggregate size,and determined that the smaller the fragments,the higher their launch velocities.Wu et al.[15]used LS-DYNA with a zero-thickness cohesive element,which sidesteps the mass loss associated with conventional element erosion algorithms,to simulate the failure of concrete structures owing to internal blast loads created by charges contained in chambers.The explosive charge was placed in a rigid cubic chamber with a concrete slab cover to analyze the size distribution,launch velocities,launch angles,and projectile distances of the fragments generated by various internal blast loads.It was shown that an increase in the blast load decreased the number,average volume,and maximum characteristic length of large debris,and nonlinearly increased the total number of fragments.The fragment launch velocities exhibited a normal distribution in which the average velocity increased with the blast load.Furthermore,60%-70% of the debris was launched at angles between 80°and 90°,i.e.almost vertical to the surface.
The size distribution of the fragments from blast-loaded concrete structures has been studied by multiple researchers,likely owing to the relative ease of collecting and analyzing blastgenerated fragments.However,there are limited studies on fragment launch velocities (magnitudes and angles) and little publicly available data describing the fragment launch velocities of UHPC structures.Furthermore,only a few Eq.s can be used to calculate fragment velocities.This scarcity of fragment velocity studies can mainly be ascribed to the difficulty of measuring the positional changes,velocities,launch angles,masses,and sizes of randomly generated fragments in a three-dimensional space.In most cases,it was only possible to measure one type of these data in a given experiment,e.g.lead spall velocity.In addition,although numerical simulations are a convenient way to obtain blast-related data,there remain many disparities between the results obtained by numerical simulations and real-world experiments.
To investigate the distribution of the spall velocities from a UHPC structure subjected to blast loads in this study,a novel experimental device was specially constructed to perform partially embedded explosion experiments on two UHPC slabs and one NSC slab.The mass,size,and spatial distributions of the fragments were examined,and their launch velocities (magnitudes and angles)were obtained by analyzing high-speed camera footage of the concrete spalling process.Finally,an empirical Eq.for the lead spall velocity was derived from the experimental data collected in this and previous studies.
A cross-sectional schematic of the slab specimens employed in this study is shown in Fig.1.The square slabs had a side length of 3000 mm and thickness of 560 mm or 620 mm according to the thickness required for the explosion to perforate the slab.Three slabs were tested:one made of NSC and two made of UHPC.The test program is described in Table 1.
Table 1 Test program.
Fig.1.Cross-sectional schematic of the experimental concrete slab specimen.
The concrete slabs were reinforced with 8 mm diameter HRB400 rebars with a 400 MPa yield strength(Fig.1).These rebars were spaced at 100 mm in the horizontal directions and arranged in three layers with a 30 mm protective cover.The vertical rebar ties were arranged in a 200 × 200 mm2quincunx pattern.
Cast TNT charges with a density of 1.55 g/cm3were used in the experiments.The diameter,height,aspect ratio,and average mass of the cylindrical TNT charges were 122 mm,561 mm,4.6,and 10,165 g,respectively (Fig.1).The TNT charges were detonated using an electric igniter at an initiation point located on the axis atop the TNT charge.In each concrete slab,the cylindrical TNT charge was placed at the geometric center of the slab"s surface in a prefabricated hole,150 mm in diameter and 280 mm deep(half the height of the cylindrical charge).
Commercial Grade C60 concrete (as defined in the GB/T50107-2010 Chinese Standard) was used as the NSC.Standard tests were conducted to determine that the average compressive strength,elastic modulus,tensile strength,and Poisson"s ratio of the NSC were 51.7 MPa,36.0 GPa,2.85 MPa,and 0.2,respectively.The UHPC was provided by Sobute New Materials Co.Ltd.[16],and had a design bulk density of 2480 kg/m3.The UHPC had a water/cement ratio of 0.19,sand percentage of 44% by mass,gravel with particle sizes of 5-16 mm and a compressive strength ≥120 MPa,additive(PCA-I) content of 2.0% by mass,and a total cementitious material content of 680 kg/m3(75%Onoda P II 525 cement and 25%Sobute New Materials HDC(V) UHPC admixture).The standard tests determined the UHPC to have an average compressive strength of 104.0 MPa,an elastic modulus of 43.44 GPa,a tensile strength of 5.0 MPa,and a Poisson"s ratio of 0.13-0.16.
An explosion produces intense flaring and copious amounts of dust that will wrap around the slab and thereby interfere with the observation of fragments being launched from the back of the specimen slab.To address this problem,a blast-resistant chamber was specially designed and constructed as shown in Fig.2.The base(Fig.2(a))was filled with quartz sand to buffer the shockwave and impact load from the fragments.A window sealed with optical glass was installed on one side of the chamber to allow the use of a highspeed camera to photograph the inner chamber from a close distance.A sheet of coordinate paper (comprising alternating black and white blocks with a side length of 0.1 m) was installed on the opposite side of the chamber to measure the speed of the ejected fragments.
Fig.2.Structural diagrams of the experimental apparatus:(a)Base;(b)Upper surface.
The top of the experimental apparatus is shown in Fig.2(b).Three removable supporting plates were placed on top of the base,one of which supported the slab in the middle of the base on ledges running parallel along each side of the viewing axis through the optical glass window.The length of the overlap between these ledges on supporting plate and specimen was 0.2 m.The area of this overlap was covered with a～5 cm thick quartz sand layer.The other two sides of the specimen were unsupported.A FASTCAM SA-Z high-speed camera was used in this experiment (Fig.3),protected in a wooden box,and triggered simultaneously with the detonation of the cylindrical charge.
Fig.3.High-speed camera and its protective box.
A schematic of the fragment velocity measurement system,illustrating the relative positions of the high-speed camera,fragments,and coordinate paper,is shown in Fig.4.The center of the camera lens was approximately level with the upper surface of the sand cushioning layer.The optical axis of the lens was aimed upward along the center of the specimen slab"s bottom surface in the same plane as the vertical centerline of the coordinate paper.In the horizontal plane,the center of the specimen slab was 4.7 m from the camera lens and 4.2 m from the coordinate paper.The fragments generated were expected to spread out in an approximately axisymmetric manner.To calculate the displacement of a fragment between consecutive video frames,it was necessary to establish the relationship between the fragment and coordinate paper in space,and account for the visual biases resulting from the angle of the camera with respect to the coordinate paper.
Fig.4.Fragment velocity measurement system.
The typical damage suffered on the top and bottom surfaces of a slab specimen owing to the detonation of a cylindrical charge is shown in Fig.5.All of the slabs in this study were fully penetrated by the explosion.The fragments launched from the slab impacted the quartz sand cushion below the slab,forming a debris pile,as shown in Fig.6.The impact formed a crater on the sand cushioning layer,causing the quartz sand around the debris pile to bulge upwards.The conical debris piles were predominantly located directly below the hole in the slab,while a few fragments of varying size were scattered around the perimeter of the debris pile.These scattered fragments may have bounced off the debris pile or rolled out of it;alternatively,they may have been directly launched from the back of the specimen.
Fig.5.Damage on the top and bottom surfaces of a specimen slab: (a) Top surface;(b) Bottom surface.
Fig.6.Top-down view of the debris piles.
After detonation,GPa-level shockwaves were formed at the locations where the concrete slab was in contact with the explosive,compressing the adjacent material into powder and forming a cavity.At a certain distance from the explosive charge,the concrete broke into smaller fragments.At a further distance,radial and circumferential cracks were formed in the concrete.The blast compression wave eventually reflected off the free surface on the bottom of the specimen slab,transforming it into tensile waves which then caused the spalling of the slab.The intensity of the reflected tensile waves increased toward the center of the slab specimens,whereas the size of the released fragments decreased[17,18].As the blast compression waves travelled farther from the explosive charge,their intensity was weakened by the repeated reflection between the old and new (spalled) free surfaces.This process primarily causes tensile failure in the concrete,leading to the generation of large fragments and explaining the considerable disparities between the fragment volumes shown in Fig.6.
In the experiment,the finer fragments were usually found in the lower layers of the debris pile with the larger fragments on top.These larger fragments always had one flat surface,indicating that they were ejected from either the top or bottom surfaces of the slab.As shown in Fig.5,several larger fragments remained at the edges of the hole and spalling crater instead of falling off the slab;this can be attributed to mechanical interlocking between the fragments[19],as well as the presence of the rebar.Consequently,it can be concluded that the average launch velocity of the smaller fragments was greater than that of the larger fragments,which is consistent with the findings of [2,14].
Fig.7 shows a selection of fragments from Test 1.The fragment volumes exhibited a wide distribution.The larger fragments exhibited a flattened shape with relatively sharp edges and were significantly thicker in their middle portions than at their sides.The size of a fragment was defined as the maximum distance between two points on its profile.The smallest fragments only weighed a few grams or were dust-like.Conversely,the largest fragments were always larger than 200 mm and weighed over 2 kg.The largest fragment was obtained from Test 1(260 mm),whereas the heaviest fragment was obtained from Test 2(2.8 kg).In this experiment,there was no significant correlation between the size and mass of the largest fragment,thickness of the slab,and concrete strength.
Fig.7.Shapes of a sampling of fragments from Test 1.
As there was a gap between the specimen slab and supporting plates,the fire and dust generated by the explosion reached the back of the slab through this gap,thus obstructing the high-speed camera.Consequently,only two sets of images were useable for the image analysis.
3.2.1.Analysis of the fragmentflight process
The raw images collected by the high-speed camera during Test 2 are shown in Fig.8.These images were cropped to pixels 394-723 along the vertical axis before increasing their brightness.The highspeed camera was configured to take 1000 frames per second,that is,1.0 ms between frames.An analysis of the images determined that each pixel showing the coordinate paper represented a physical length of 6.13 mm;each pixel thus represented a distance of 3.24 mm on the vertical plane passing through the center of the slab specimen,parallel to the coordinate paper.
Fig.8.Photographs of spall fragments taken using the high-speed camera in Test 2.
In Test 2,no significant flaring or spalling was observed in the 39th frame,whereas flaring and some spalling were observed in the 40th frame,suggesting that the detonation occurred between these two frames.In Fig.8,fragmentation first occurred at the center of the slab,where the fragments were launched with high velocities and large displacements.The fragments in the middle of the slab covered half the distance between the slab and ground by the 43rd frame,and reached the surface of the sand cushion in the 44th frame.In the 45th frame,some of the sand on the ground can be observed to have been raised by the impact of fragments on the sand cushion.Moreover,it can be observed that the camera was slightly shaking during this frame.However,this will not affect the results because the shaking did not occur until five frames after the explosion;sufficient data was collected in this time to determine fragment velocity and angle.
Several measurement lines were defined to help characterize the fragment velocity distribution,and the angle between the velocity of a fragment and the vertical direction was defined as α.As shown in Fig.9,a white dotted line-labelled line 1-was drawn at the horizontal center of the fragment (approximately corresponding to a pixel coordinate of 550).The velocity of each fragment was calculated by recording the vertical coordinate where its lower boundary intersected with this line.Additional lines were then drawn 50 pixels away on the left and right sides of line 1,labelled line 2 and line 3,respectively.By analyzing the images,the fragments between measurement line 2 and line 3 always had α values less than 15°,whereas beyond these lines,α ranged between 15°and 27°.Thus,the α value of a fragment generally increased with distance from the central point at line 1.
Fig.9.Illustration of the measurement lines.
To account for the effects of α in the measurement of fragment speed,the four red lines labelled a-d in Fig.9 were drawn at an angle of 16.9°to the white lines.As the angle between the velocity of a fragment and lines a-d was always less than 15°,the difference between the measured and actual fragment velocities was generally less than 3.4%.
For further analysis,the Beyond Compare software was used to perform a pixel-wise comparison between each frame to accurately determine the location of a fragment by identifying the inter-frame differences.The white,blue,and red regions of the resulting comparative image represent fully identical,similar,and different pixels,respectively.Fig.10 shows the pixel-wise comparisons between frames 39/40 and 41/42.As these frames are close to each other in the temporal dimension,their differences are dominated by the fragments moving at high speeds.However,it is clearly necessary to remove interference owing to flare and dust from the explosion using the raw images as guides.By analyzing the effects of different tolerances on the image resolution,a tolerance of 2 was found to be optimal in removing such interference.
Fig.10.Pixel-wise comparisons performed using the Beyond Compare software: (a) Comparison between the 39th and 40th frames;(b) Comparison between the 41st and 42nd frames.
3.2.2.Analysis of fragment velocities
Fig.11 shows the calculated fragment velocities for Test 2.The horizontal and vertical axes of Fig.11(a) depict the frame number and pixel coordinates,respectively.The similar slopes for line 1,line 2,and line 3 indicate similar fragment velocities,which were determined to be 118.3 m/s,113.4 m/s,and 111.5 m/s,respectively,with coefficients of determination (R2) of 0.998,0.997,and 0.989,respectively,using the linear least square method.Based on the specimen geometry,the vertical pixel coordinate of any fragment as it separates from the specimen slab"s lower surface(between line 2 and line 3) should range between 524 and 528.Thus,as the detonation occurred between the 39th and 40th frames,the fitted lines should pass through the grey rectangle in Fig.11(a).As all three fitted lines meet this requirement,the credibility of the calculated velocities is confirmed.
Fig.11.Calculation of fragment velocities in Test 2: (a) Measurement line 1-line 3;(b) Measurement line a-line d.
In Fig.11(b),the horizontal coordinate is time,with 0 being the start time of the 41st frame,and the vertical coordinate corresponds to the relative displacements from the pixel coordinates in the 40th frame;thus,the velocity of a fragment is defined as the slope of its line in this Fig.11(b).The fragment velocities at inner lines b and c were 98.5 m/s and 83.0 m/s (R2=0.996 and 0.995),respectively,whereas that at outer lines a and d were 44.4 m/s and 49.5 m/s (R2=0.979 and 0.992),respectively.
Note that the initial kinetic energies of ejected fragments are produced by the tensile waves in the concrete slab,which are formed by the reflection of the compressive wave off the free surface.These fragments are then accelerated by the blast wave penetrating the concrete slab.As the explosion occurs over a short time,the fragments quickly reach their maximum velocities.Therefore,the fragment velocities measured in this experiment can be assumed to be constant within the experimental time frame.
A summary of the measured fragment velocity distribution in Test 2 is shown in Fig.12.The fragments at the center exhibited the highest velocities with an α of 0°(the vertical direction).The fragment α and launch velocity decreased with increasing distance from the slab center,as the fragments at the center were located directly below the explosive charge.This confirms that the blast stress wave,which will be reflected normally(at 0°)off the back of the concrete slab,was most intense at this point.As this point was also closest to the hole formed by the explosion,a fragment at this point gained the highest energy and acceleration from the blast wave.At locations farther away from the center,the shockwave will be obliquely reflected off the back of the slab,with the angle of reflection increasing with increasing distance from the center.Furthermore,the acceleration provided by the blast wave decreased with increasing distance from the hole formed by the explosion.
Fig.12.Distribution of the fragment velocities in Test 2.
The fragment velocities calculated for Test 3 are shown in Fig.13.Owing to the severity of the obstruction caused by the dust,it was only possible to measure the fragment velocity at the central point,which was 81.6 m/s (R2=0.990).Note that because the smoke completely obscured the view of the fragments in Test 1,the fragment velocity could not be measured.
Fig.13.Fragment velocity calculations for Test 3.
Fragment velocity calculations are important when evaluating the threat posed by spalling concrete debris and designing structural fortifications.If a concrete slab is sufficiently thick,a contact explosion will not cause concrete spalling at its back.As the slab in question becomes thinner,spalling begins to occur but the launch velocities of the resulting debris usually remain below 1.5 m/s [5].As the thickness of the concrete slab is further reduced,the explosion penetrates the slab,accelerating the fragments to velocities greater than 100 m/s.The energy from the detonation induces severe compression and spalling in the concrete structure before dissipating into the air.Furthermore,the part of the blast wave propagating to the back of the slab will accelerate its spalling fragments.
As fragment velocity depends on the material characteristics of the explosive and concrete,shape and mass of the explosive charge,size of the concrete slab,and type of reinforcement,it is extremely difficult to derive an analytical solution for fragment velocity.Therefore,this study attempted to derive an empirical equatioin for fragment velocity based on publicly available data and the experimental data from this work.
In the UFC standard [5],initial fragment velocity is typically calculated using the Gurney method[20],where the initial velocity is a function of the shape,type,and weight of the explosive charge as well as the mass ratio between the explosive charge and its metallic casing,as follows:
Note that Eq.(1) is applicable to rectangular explosives in contact with a metal plate of equal surface area.As the thickness-tosurface area ratio is small in this scenario,the velocity of the fragment is always perpendicular to the plate.The forms of Eq.s applying to other scenarios are generally similar to that of Eq.(1).Indeed,a term similar to theC/Mcasused in Eq.(1) can be used to calculate concrete fragment velocity when the explosive charge is contained inside a chamber [6-8]as follows:
whereMconis the mass of the entire concrete slab(kg);γ is the ratio of the mass of the explosive to the volume of the chamber(kg/m3);Lis the ratio of the volume of the chamber to cross-sectional area of the concrete slab (m);andMais the mass per unit area of the concrete slab (kg/m2).
In a contact explosion on a concrete slab,fragment velocity can be calculated using Eq.(1)only if the concrete slab is infinitesimally thin and the explosion-induced changes in the physical properties of the materials are negligible.IfMcasis infinitesimally small,the maximum velocity isHowever,though concrete spalling is induced by the reflection of the tension wave off of the back free surface of the slab,the basic concept underlying the Gurney method is energy conservation with a one-dimensional assumption.As a result,the Gurney method cannot be directly used to analyze the concrete fragment velocity in this scenario.Thus,an Eq.for the velocity of the concrete fragments generated when an explosive charge is detonated on the surface of a concrete slab of a certain thickness can be derived using the form of Eq.(1)to fit the experimental results as follows:
However,a contact explosion is a localized interaction between the charge and the plate;therefore,it is obviously unreasonable to use the mass of the entire concrete slab(Mcon)in Eq.(3).When the concrete slab is sufficiently wide,the effects of its width on the damage induced within and the associated fragment velocities will be negligible.Therefore,C/Mconin Eq.(3)can be replaced withwhereHis the thickness of the concrete slab and ρconis the density of the concrete.As the concrete density can be considered constant,Eq.(3) can be revised as
Thus,a new empirical Eq.for predicting the lead spall velocity of a concrete slab is ultimately obtained.
There is a significant body of data demonstrating the close relationship between the damage suffered by concrete slabs in close-in explosions and the value ofH/[4,9,22].Consequently,several methods based onH/have been proposed to predict whether spalling or penetration will occur in a concrete slab during contact explosions.As a rule,the damage suffered by a concrete slab increases with decreasingH/.In this study,v/was therefore plotted againstH/to analyze the effects ofH/on the fragment velocity (Fig.14).Experimental fragment velocity data from the literature were also included in this plot,as shown in Table 2.Though in a contact explosion,His the thickness of the slab,as this study employed partially embedded explosives,Hwas defined as the distance between the bottom of the explosive charge and back of the concrete slab.As the charge was embedded a certain depth inside the slab,the sides of the charge were obstructed by the concrete,increasing the damage compared to that caused by a contact explosion.However,this effect may be diminished if the explosive has an excessively large aspect ratio.The equivalent TNT masses in this study (whoseH/values are indicated by the red arrows in Fig.14) were therefore calculated by accounting for the effects of the embedment and aspect ratio using the methods of Haas and Rinehart [23].
Fig.14.Comparison of the fragment velocities from concrete spalling.
Note that if the explosive and slab are separated by a distance(as in a near-charge explosion),the intensity of the shockwave entering the concrete slab will be attenuated by the air between them.However,as this scenario differs completely from a contact explosion,it was not included in the discussion and analysis in this study.
In Fig.14,increasing the value ofH/decreases the damage in the concrete slab and fragment velocity.The fragment velocity is at its maximumis zero;that is,when the concrete slab is infinitesimally thin.In multiple studies,concrete spalling was not observed to occur at the back of a sufficiently thick slab,i.e.the fragment velocity fell to zero.Morishita [22]experimentally determined the threshold for concrete spalling to be,whereas Zheng [26]obtained a threshold of 0.342.The experimental data of Zhang [25](Fig.14) showed that the fragment velocity falls to zero after
An empirical equation for fragment velocity based on the International System of Units was therefore obtained by fitting the experimental data in Fig.14 using the least squares method(R2=0.999).In this equation.,the concrete spalling threshold was conservatively set toH/=0.36 as follows:
Fig.14 shows the curve corresponding to Eq.(5) as well as two other curves corresponding to a 20% increase and decrease in the exponent (25).All the measured fragment velocities are also plotted on the graph.The empirical equation and experimental data agree well with one another.Although the maximum velocity was unaffected by the changes to the exponent,an increase to the absolute exponent value decreased the fragment velocity andthreshold,corresponding to the point at whichvfalls to zero.Note that the correct value of the exponent may depend on factors not otherwise included in Eq.(5),such as the compressive and tensile strengths of the concrete,explosive shape and mode of detonation,or inclusion of rebar/steel fibers.
In Fig.14,the experimentally measured fragment velocities obtained by Yang et al.[10]are shown to be generally smaller than those of Lönnqvist [9],whose measurements are in turn smaller than those of Kuenzel et al.[11].Consequently,the presence of steel reinforcing bars (Table 2) may significantly affect fragment velocities.As the reinforcement ratio depends on the diameter and spacing of the rebar as well as the slab thickness,and theH/term already accounts for the effects of slab thickness,it is only necessary to define a parameter that independently characterizes the effects of the rebar content on fragment velocity.As the concrete fragments are located on the opposite face from the explosive(bottom of the slab),the rebar on the bottom of the slab will exert a greater effect on the fragment velocity than that on the top.Thus,the reinforcement content β is defined as a new dimensionless parameter describing the volumetric steel content of a 1000 × 1000 × 20 mm3rectangle that passes through the bottom rebar of the concrete slab,with its height (20 mm)oriented in the same direction as the slab thickness.As these dimensions are larger than the diameters of most rebars,it can be generalized to a wide range of experimental settings.The expression for β is given by
whereDis the rebar diameter and δ is the rebar spacing (mm).
The experimental data in Fig.14 were re-plotted to analyze the effect of β as shown in Fig.15.The horizontal and vertical axes are unchanged,but the β value of each data point is indicated using colors.The black data points (β=0) always exhibit the highest fragment velocity at each value offollowed by the blue data points (β=0.64),and finally the red data points (β ＞3.9).Consequently,it can be concluded that the fragment velocity decreases with increasing β.However,more research is required to quantify the effects of β on the fragment velocity.
Fig.15.Effects of β on fragment velocity.
Fig.14 includes the fragment velocity data from Test 2 and Test 3,which used NSC and UHPC slabs with the same reinforcement content.As the compressive strength,reinforcement content,and Gurney constant of the NSC slab used in this study were identical to those of the NSC slab used by Yang et al.[10],the fragment velocities for the NSC slab in this study were similar to those determined by Yang et al.However,owing its relatively higher reinforcement content,the fragment velocities for the NSC slab in this study were lower than those observed by Lönnqvist [9].
The compressive and tensile strengths of the UHPC employed in this study were approximately twice those of the NSC used by Yang et al.[10],who used commercial Grade C40 concrete.However,the fragment velocities for the UHPC slabs obtained in this study were identical to those obtained by Yang et al.[10]whenH/was the same,as shown in Fig.14.As the UHPC slabs in this study otherwise had a similar reinforcement content to the NSC slabs used by Yang et al.[10],it can be concluded that concrete strength does not significantly affect the fragment velocity.This observation can be explained by analyzing the effects of concrete strength on the explosive damage suffered by concrete slabs.Morishita et al.[22]performed contact explosion experiments on high-strength concrete slabs with a compressive strength of 84.8 MPa(approximately twice that of NSC) with no steel fiber reinforcement.They found that concrete strength did not have a significant effect on concrete spalling.Hence,it follows that concrete strength would not significantly affect fragment velocity.
Finally,note that as the addition of steel fibers has been found to significantly reduce the damage suffered by a concrete slab (especially UHPC slabs) during a contact explosion [3,27,28]and thus reduce fragment velocities,the proposed Eq.is not applicable to UHPC slabs that have been reinforced with steel fibers.
In this work,partially embedded explosion experiments were performed on reinforced UHPC and NSC slabs.The characteristics and launch velocities of the fragments ejected from the backs of these concrete slabs were then quantified and used to obtain an equation for lead fragment velocity.The following conclusions were drawn from the results.
(1) The fragment velocities for the UHPC slab ranged from 0 to 118.3 m/s with a maximum angle of 27°from vertical.The fragment velocities decreased as this angle increased.Wide ranges of fragment volumes were generated by the partially embedded explosions.Smaller fragments exhibited a higher mean velocity than their larger counterparts.
(2) Fragment velocity was observed to be negatively correlated withH/;that is,the greater the damage suffered by the concrete,the higher the fragment velocity.The fragment velocity was also negatively correlated with the reinforcement content.However,there was no significant correlation observed between the fragment velocity and concrete strength because the fragment velocities of a UHPC slab showed no significant differences compared to those of an equivalent NSC slab.It was not possible to analyze or quantify the relationship between angle and velocity at this time owing to the limitations of the data collected in this study.
(3) An equation was proposed to describe the lead spall velocity of fragments generated by a contact explosion on a concrete slab based on a theoretical Eq.for the fragment velocities and experimental data collected in this and previous studies.However,further research is required to confirm the effectiveness of the equivalence method used to calculate the effects of an embedded-charge explosion based on a contactcharge explosion.
Funding
This work was supported by the National Natural Science Foundation of China [No.51978166].The funding source was not involved in the study design;in the collection,analysis,and interpretation of data;in the writing of the report;and in the decision to submit the article for publication.
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.
Acknowledgements
We would like to thank Editage [www.editage.cn]for English language editing.
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