Research ArticleExperimental Investigation on the Deformation and Fracture ofSteel Square Tubes under Double-Explosion Loadings
Chong Ji ,1 Yang Yu,1 You Zhou,2 Fuyin Gao ,1 Xingbo Xie,1 and Jianyu Wu1
1College of Field Engineering, Army Engineering University of PLA, Nanjing 210007, China2Northwest Institute of Nuclear Technology, Xi’an 710024, Shanxi, China
+e experimental investigations on the square tubes with various stand-off distances and wall thickness were helpful to un-derstand the dynamic response of metal shell under single and double explosion. +erefore, the effect of stand-off distance, wallthickness, and amounts of explosion on the deformation and damage of the square tubes and the regular pattern of the fracturedevelopment was analyzed by dimensions of local plastic deformation, volume of the depression area, and crack type. +e resultreveals that the deformation and fracture mode of steel square tubes gradually transform from local deformation to rupture withthe decrease of wall thickness and stand-off distance. Besides, the failure degrees of square tubes under a double explosion wererelatively higher than those of square tubes under a single explosion. In addition, the experiment indicates that the side corners ofthe square tube are very vulnerable, and they are damaged easily by the stress concentration and shear effect. +e conclusionprovides an important scientific basis for the structural design of square-tube structures and calculation of engineering protection.
1. Introduction
With the frequent occurrence of worldwide terrorist explo-sions and industrial explosions, research on the antiexplosioncapability of engineering structures is attracting increasingattention from international experts. Metal thin-walled hol-low members (tubes) have high stiffness excellent energyabsorption characteristics, are light weight and easy to pro-cess, and are widely used in ocean platforms, gas and pe-troleum pipelines, construction structures, aerospaceindustry, and other military or civilian fields. Engineeringstructures are often damaged by a single or continuous ex-plosion, whichmeans that the structure of a metal thin-walledhollow member may be in danger of multiple explosions. Asone of the important forms of thin-walled hollow structures,a square tube will produce local or global plastic deformationor rupture when subjected to an explosion impact load,resulting in the structure losing its original function. In thiscondition, a more serious damage may occur if it is subjectedto explosion impact again. +erefore, the study of the dy-namic response mechanism of the square-tube structure
under multiple explosion loads is of great engineering ap-plication value for predicting the deformation characteristicsof the structure and improving its antiexplosion capability.
In recent years, numerous studies on the dynamic re-sponse of thin-walled structures such as cylindrical shellssubjected to explosion and impact loading have been con-ducted. Yuen et al. [1] used a series of tests and numericalanalysis to obtain the dynamic response of a cylindrical tubeunder lateral blast load detonated at very close proximity.+e experimental results indicated that for a constant stand-off distance, the permanent midpoint deflection increased asthe mass of the explosive increased. Larger load diametersappear to cause more damage for the same charge mass. Gaoet al. [2] studied the damage of the water-filled cylindricalshell subjected to explosion impact through experiments andnumerical simulations.+e results show that the existence ofwater provides a “foundation” pressure for deformationresistance. Song et al. [3] built a mathematical model toanalyse the deflection and the deformation angle of the steelcircular tubes subjected to lateral blast loads. +e calculationresults agree well with experiment observations when the
HindawiShock and VibrationVolume 2018, Article ID 1074340, 15 pageshttps://doi.org/10.1155/2018/1074340
deformation is relatively small. Clubley [4] investigated theinfluence of long-duration blast loads on the structuralresponse of aluminium cylindrical shell structures. Pre-liminary modelling predicted a global sway and localisedplate buckling, and subsequent experimental testing showeda crushing failure of the shell before any translationalmovement occurred. Kim et al. [5] investigated energyabsorption capability and bending collapse behaviour of analuminium (Al)/carbon fiber reinforced plastic (CFRP) shortsquare hollow section (SHS) beam under transverse qua-sistatic loading. Abedi et al. [6] found the pipe displacementand peak particle velocity under a blast wave equivalentdynamic load by establishing a mathematical model. Wuet al. [7] conducted experimental and numerical studies onthe dynamic response of metal cylindrical shells under thecombined effects of fragments and shock waves and ob-tained three failure modes. +e effects of preformed holes,hole-spacing, and TNTcharge at a certain stand-off distanceon the deformation/failure of a cylindrical shell were ana-lyzed. Rajabiehfard et al. [8] investigated axisymmetriccircular cylindrical shells subjected to axial impact, and twotypes of loading were analyzed. Rushton et al. [9] carried outan internal charge explosion test for a seamless steel tubewith diameter of 324mm and wall thickness of 9.5mm.Because of the small amount of explosive used (0.8 kg of PE4explosive), only the bulge phenomenon of the tube wall wasobserved. With respect to the square tube, Wegener andMartin [10] investigated the permanent deformation ofa simply supported thin-walled square tube under the ex-plosive load of a flake explosive. Bambach [11, 12] tested thelocal and global deformation of an aluminium alloy squaretube subjected to the impact load of an explosion and ob-tained an empirical formula for calculating its deformationaccording to the experimental results. Jama et al. [13, 14]theoretically and experimentally studied the dynamic re-sponses of three different sizes of square tubes under uni-formly distributed explosion loads and used numericalsimulation to gain an insight into the temporal distributionof the global and local deformation and the adiabatictemperature rise in the beams as a result of impulsiveloading. Karagiozova et al. [15, 16] developed a model ofdeformation of a metal hollow section beam under a uni-form blast loading, in order to reveal the characteristicfeatures of deformation and energy absorption of hollowsection beams under such loading. Jones [17] considered thethin-walled square tube as an ideal rigid-plastic structureand conducted experimental and theoretical studies on itsplastic deformation under impact loadings.
+e above studies mainly focused on the single-explosion load of thin-walled shell structures, and fewstudies have been conducted on the blast resistance char-acteristics of protective structures under multiple explosionloads. With the increasing awareness of security problems inpractical applications, the problem of multiple explosions isreceiving growing attention. Zhang et al. [18] reduced themultiple blast loads to a series of pulsed loads that were longenough apart.+e experiment shows that under the multipleloads, as the number of loads increases, the damage ofstructural materials gradually increases, and the bearing
capacity of the structure decreases. Kumar et al. [19] studiedin detail the influence of stand-off distance, explosive amount,strain rate, and provision of stiffeners on the dynamic be-haviour of semiburied structure under soil-structure in-teraction and multiple explosion conditions. Henchie et al.[20] and Yuen et al. [21] studied the impact tests of five similaruniform blast loads on plates, and the impact, deformation,and material changes of plates under multiple blast loads areanalyzed. At the same time, ABAQUS is used to simulate theexperiment. +e numerical simulation results are comparedwith the experimental results, and the influence of repeatedblast loading on the material properties of the test plate isfurther analyzed. Zhou et al. [22] obtained five differentfailure modes by studying the double explosions of metalcylindrical shells under different explosive conditions. +eresults show that the deformed cylindrical shell subjected tothe first explosive shock absorbs more energy than the un-structured shell under a given explosive load.
In the above studies, where the scholars mainly focus onthe dynamic response of engineering structures undera single explosion and its calculation method, the failuremodes and dynamic responses of engineering structuresunder multiple explosions are rarely studied. Owing to thespecial structure of the square tube, there are still no pub-lished studies on the deformation and damage of steel squaretubes subjected to multiple explosions. +erefore, it is ofgreat theoretical significance and application value to studythe antiexplosion characteristics of structures under mul-tiple explosive loads.
Field experiments were carried out to investigate the in-fluences of initial conditions on the damage effects of the steelsquare tube on the side of the local explosion load. Squaretubes (with wall thicknesses of 3.0, 3.5, and 4.0mm) weresubjected to a single explosion (of 160 g of TNT charge) anda double explosion (where the first explosion is of 100 g ofTNT charge and the second explosion is of 160 g of TNTcharge) with different stand-off distances. +e failure modes,deflection of the centre of the tube, the radial width, axiallength, and volume of the depressed area were analyzed.+rough a comparison and analysis of the experimental re-sults, the key factors that affect the deformation of thestructure and the influence rules were obtained so as toprovide a reference for the antiexplosion capability analysisand prediction of deformation failuremode of the square tube.
2. Experimental Research
2.1. Experimental Setup. +e square tubes under the single-and double-explosion tests were Q235 steel square tubeswith outer diameter of 100mm × 100mm; wall thicknessesof 3.0mm, 3.5mm, and 4.0mm; and length of 100 cm. +echemical composition of the specimen is listed in Table 1.+e ordinary mechanical properties are listed in Table 2.+eexperiment uses cylindrical press-fitting 100 g and 160 gTNT charge columns as explosion sources with dimensionsof Φ 48mm × 34mm and Φ 48mm × 54mm.
+e charge was vertically installed above the square tubeand aligned with a line passing through its centre point. +ecentre of the upper surface of the column charge was
2 Shock and Vibration
initiated by an electric detonator. �e TNT density is1.63 g/cm3, and the detonation velocity is 6950m/s.
Sketches of the double-explosion experiment setup areshown in Figure 1. �e PVC tubes were thin and of lowstrength and thus had little e�ect on the experimental re-sults. �e square tubes were simply supported on a bracket[2, 6–8]. �e contact area between the square tube and thebracket was kept small enough to ensure that this contactarea had little in�uence on the experimental results.
In this study, when the square tube is subjected to theexplosion load for the �rst time (that is, the �rst time underthe explosive load of the 100 g TNTcharge), it is de�ned as the�rst explosion of the double explosion. When the square tubeis subjected to explosion loading again (that is, the secondtime under the explosive load of the 160 g TNTcharge), this isde�ned as the second explosion of the double explosion. �edeformed square tubes under the blast load may be subjectedto the explosion impact again anywhere. �e �rst and secondexplosions of the double-explosion tests were set in the samedirection and considering the worst conditions. �e stand-o�distance was de�ned as the distance between the bottom of thecharge and the midpoint of the front zone of the square tube.When the deformed square tube was subjected to the secondexplosion, the stand-o� distance became the distance from thebottom of the charge to the centre of the depression zone. Forthe purpose of comparison, the stand-o� distances in the �rstand second explosion tests of the double-explosion experi-ments were the same. Figure 2 describes in detail the stand-o�distance of the �rst and second explosion tests.
2.2. Experimental Results and Discussion. �irty-six testswere conducted in this study. �e experiment numbers werede�ned as TZ1 to TZ36. Table 3 lists the �nal test conditionsand results, which include the wall thickness a of the squaretube, stand-o� distance R, charge mass m, deformation pa-rameters, and failure modes for each test. lx and ly are the localplastic deformation values of the oval concave in the axial andradius direction, respectively. δlocal and δglobal represent thevertical distance from the midpoint and the lowest point ofthe back zone to the highest point of the square tube, re-spectively (indicated in Figure 3). V is the volume of thedepressed area measured by the �lling method, and themeasurement method is shown in Figure 4(a). �e front zonerepresents the upper half of the square tube facing the ex-plosion, and the back zone represents the lower half of thesquare tube against the explosion. When partial cracks aregenerated, the value of d1 was recorded as “−.”�e description
of the plastic hinge is shown in Figure 4(b). �e bulge zone islocated on the side wall, and the crimp zone is located on theside corners of the square tube. A is the blasting midpoint, Jand K are in the axial plastic hinge, and G andH are located atthe side corners of the square tube. TZn is the test number.For example, TZn-1 and TZn-2 express the �rst and secondexplosion test of the double-explosion test TZn, respectively.�e charge mass “100 g + 160 g” represents the two explosiontests of the square tubes subjected to the �rst and secondexplosions of the double explosion.
According to the typical beam deformationmodes studiedby Menkes and Opat [23], Wu et al. [7] studied the damage ofcylindrical shells under the combined e�ect of fragments andshock waves and divided the cylindrical shell deformation intothree modes, namely, Mode I, Mode II, and Mode III. Basedon the experiment results and previous ones, the deformationof a square tube subjected to double-explosion loadings isdivided into four failure modes in this report. Owing to thein�uence of the machining accuracy and measurement error,when δglobal is less than 1.05 timesΦ, no global deformation isconsidered to have occurred in the square tube:
Mode Ia: local plastic deformation at the front zone ofsquare tube, without global deformation (δglobal ≤1.05Φ)Mode Ib: local plastic deformation at the front zone ofsquare tube, with small global deformation (1.05Φ <δglobal ≤ 1.4Φ)Mode II: coupling of local cracking deformation andglobal deformation, and the crack presents a radialH-type crackMode III: coupling of local cracking deformation andglobal deformation, and the crack presents an axialH-type crack
3. Discussion
3.1. Analysis of Typical Failure Modes. Figure 5 shows thetwo typical deformation modes of a square tube underdouble-explosion loads, namely, Mode Ia and Mode Ib. It is
Table 1: �e chemical compositions of steel Q235.
Elements C (%) Si (%) Mn (%) P (%) S (%)Contents 0.15 0.3 0.35 0.045 0.05
Table 2: Mechanical properties.
Elongationεf (%)
Elastic modulusE (GPa)
Yield strengthσs (MPa)
Tensile strengthσb (MPa)
26 208 239 430
Fixed pole
PVC tubeDetonator
TNT chargeSteel square tube
Bracket
Figure 1: Sketches of experiment setup.
Shock and Vibration 3
recorded as Mode Ia andMode Ib when the square tube onlysubject to local plastic deformation at the front zone.
+e shock wave and detonation product first act on theblasting midpoint and then gradually expand to the zone near
the blasting midpoint, when the stress received at the blastingmidpoint exceeds its yield strength. Local plastic deformationoccurs when the stress generated at the blastingmidpoint by theexplosion is greater than the yield strength of the square tube.
Initiation point
Midpoint
Initial square tube
100g charge
R
(a)
Initiation point
Midpoint
Deformed square tube after the first explosion
160g charge
R
(b)
Figure 2: Stand-off distance description of the first and second explosion test. (a)+e first explosion of double-explosion test. (b)+e secondexplosion of double-explosion test.
Table 3: Detailed experimental data of the single- and double-explosion tests.
With further action of the shock wave, the depressionzone of the square tube expands to the axial and radial di-rections, and the depth and area of the depression zone alsoincreased. Owing to the limitation of the cross-sectional shapeof the square tube, the axial velocity of the depression zone ishigher than that in the radial direction. When the depressionzone is developed in the radial direction, the side corners ofthe square tube are subjected to the pull of the centre of thesquare. �e side corners develop to the centre in the radialdirection and the axially inclined deformation develops toboth sides, and ultimately an hourglass deformation is formed.When the square tube deformation of Mode Ia occurs, theenergy absorbed by the square tube is mainly transformed intothe plastic work that produces the local plastic deformation.For Mode Ib, the energy absorbed by the square tube is alsoconverted into the energy of the global deformation.
Figure 6 shows the typical deformation of the square tubeunder double-explosion loadings inMode II.With the decreaseof the stand-o� distance or wall thickness, the deformation anddegree of damage of the square tube increase gradually. Whenthe stress received at the midpoint exceeds the ultimate rupturestress and the square tube begins to crack and fail, then plasticdeformation and crack growth occur simultaneously. FromFigure 6(a), we can see that the crack of critical rupture of thetube develops along the axial direction from the midpointunder double-explosion loadings. With the decrease in stand-o� distance, the explosive load acting on the square tube in-creases. �e energy of the explosive load has residual energy inaddition to a similar axial crack, as depicted in Figure 6(a), andthe crack that develops in axial direction is converted into radialdirection and �nally forms anH-type crackwith radial crackingin Figure 6(b). In Mode II, the energy generated by the ex-plosion on the square tube �rst causes local plastic deformationand small global deformation of the steel tube, and a part of theenergy causes the square tube to crack when the stress exceedsthe yield strength of the square tube.
Figure 7 shows the failure mode of Mode III under thedouble-explosive loadings. As we can see from Figure 7(b),the front zone of the square tube is completely ruptured inthe centre of the blasting surface under the action of the highexplosive load. �e resulting high-speed fragments hit theback zone to produce a certain amount of bulge, and thefragments accumulate in the bulge. �e front zone hasa broken gap with a size of 141.7mm × 91.7mm and thesquare tube has a larger global deformation.�e side cornersof the square tube facing the blasting surface have longercracks in the axial direction, and the fragments generated atthe same time impact the back zone with high speed,resulting in a long crack in the axial direction of the sidecorners of the back zone. Finally, the axial H-type crackingappears. Besides, the explosive load acts on the tube whenthe side corners in addition to the explosive load havea downward force, while the side walls of the square tubegenerate an upward support force on the side corners, that is,the side corners are subjected to a pair of strong shear forces.�erefore, the side corners are easily broken by shear failure,and this rupture is easy to develop rapidly along the axialdirection of the side corners. Figures 7(a) and 7(b) con�rmthis statement; there is no obvious thinning on the sidecorners, and the rupture is neat and in a 45° angle, which isthe typical characteristic of a shear failure.
As a contrast, Mode III presents a completely di�erentcrack development path from that in Mode II, and the crackof Mode II is more localised on the front zone. Owing to thein�uence of the side corners of the square tube, the cracks ofMode III are along the side corners, and there are radialcracks and bulging on the side walls of the tube. �e sidecorners of the tube are the junction of the blasting surfaceand the side walls, and thus, it is easy to produce stressconcentration there. In addition, the side corners of the frontzone will be subjected to the larger tension of the blastingmidpoint and to the intermediate movement under the
δlocal δglobal
lly
d
lylx
Figure 3: Description of the parameters of the deformation.
Filler
(a)
KJGA
H
Bulge zone
Crimp zoneSide corners
(b)
Figure 4: Description of the measurement methods and plastic hinge of square tube deformation. (a) Measurement methods. (b) Plastichinge.
Shock and Vibration 5
action of the explosive load. Besides, the side corners of thetube can also be crimped during the movement process,resulting in a more severe stress concentration.
3.2. Deformation and Damage Comparison of Square Tubesunder Single and Double Explosion. +e predamage on thesurface of square tube will decrease the strength of the tubeand cause stress concentration thereby making the tubevulnerable to impact load. +e final deformation andfracture of the thin-walled square tube is closely related to
the interaction between these two factors. +e effects ofsingle- and double-explosion loads on the dynamic responseof thin-walled cylindrical shells are discussed below.
Figure 8 shows the deformation and fracture comparisonof square tubes with different wall thicknesses under a singleand double explosion. It can be seen that the deformationand damage of the square tube under the action of thedouble explosion was more severe than that under the singleexplosion. Obviously, the deformation modes of the squaretubes under a single-explosion load areMode Ia andMode Ib,while the deformation modes under double-explosion
Planform view
Cross sectionSide view
(a)
Planform view
Cross sectionSide view
(b)
Figure 5: Typical failure modes of cylindrical shell in Mode Ia and Mode Ib. (a) Mode Ia: local plastic deformation without globaldeformation (a � 3.0mm, R � 24 cm, m � 100 g + 160 g, TZ1). (b) Mode Ib: local plastic deformation with small global deformation (a �
3.0mm, R � 20 cm, m � 100 g + 160 g, TZ5).
Planform view
Cracked axially Cross sectionSide view
(a)
Planform view
Cross sectionSide view
(b)
Figure 6: Typical failure modes of square tube in Mode II. (a) +e square tube is just cracking (a � 4.0mm, R � 13 cm, m � 100 g + 160 g,TZ33). (b) +e crack presents a radial H-type crack (a � 3.0mm, R � 18 cm, m � 100 g + 160 g, TZ7).
6 Shock and Vibration
loadings may be Mode Ia, Mode Ib, Mode II, and Mode III.From the comparison between Figures 8(a) and 8(c), it can beseen that the deformation and damage of the square tubes underthe single and double explosion are more obvious when thestand-off distance is smaller or the wall thickness is thinner. InFigure 8(a), under the condition of a � 4.0mm and R � 20 cm,the local plastic deformation of the blasting point of the tubeunder the single explosion is 18.5mm, while the deformationunder the double explosion is 25.8mm, and both of them areMode Ia. However, in Figure 8(c), under the condition of a �
3.0mm and R � 12 cm, the local plastic deformation of theblasting point of the tube under the single explosion is 63.1mm,while the deformation under the double explosion is 141.5mm.+e deformationmode of the former is Mode Ib and that of thelatter is Mode III, and the difference between the deformationparameters and the deformation model is very large.
Figure 9 illustrates the failure grade comparison of squaretubes under single explosion and double explosions. +efailure grades from 1 to 4 corresponded to the failure modesfromMode Ia toMode III. A high-failure grade value resultedin severe damage of the square tubes. +erefore, the failuregrades of square tubes under double explosions were relativelyhigher than those of square tubes under single explosion.When the stand-off distance or the thickness was small, thefailure grade difference was remarkably obvious.
In order to study quantitatively the relationship betweenthe single-explosion and double-explosion effects of V in thedepression area, the gain coefficientw of the depression areaV
is introduced. Among them, the gain coefficientw is defined as
w �Vd
Vs, (1)
where Vd is the volume in the depression area with doubleexplosions and Vs is the volume in the depression area witha single explosion. According to the definition in this studyand the experimental data in Table 3, the calculation resultsof the single-explosion and the double-explosion intervalsare compared with that shown in Figure 10.
As shown in Figure 10, the gain coefficient w between thesingle explosion and the double explosion is 1.26–1.5. +eenergy absorbed by the square tube under the double ex-plosion is much larger than that absorbed under the singleexplosion, resulting in greater deformation damage.
When the square tube is subjected to the first explosion ofthe double explosion, there is a local depression in the frontzone of the blasting surface and the square tube is subjected tolocal plastic deformation and small global deformation.Whenthe tube is subjected to the second explosion of the doubleexplosion, it easily leads to stress concentration in the de-formed depression zone, resulting in the decrease of thebearing capacity of the square tube and the reduction of itsantiexplosion capability. In addition, owing to the tensileplastic deformation of the front zone caused by the firstexplosion, the wall thickness of the front zone thins and theultimate tensile strength is reduced. +erefore, under thesame explosion loading, the degree of deformation anddamage under a double explosion was evidently higher thanthat under a single explosion, namely, the predamaged squaretube is more susceptible to damage than an undamaged tubewhen it undergoes the explosion loading again.
3.3. Influence of the Stand-Off Distance on the Dynamic Re-sponse of the Square Tubes Subjected to Double Explosion.Previous work on free field explosions indicates that the peakoverpressure of the blast increases with decreasing stand-off
Planform view
Cross sectionSide view
(a)
Planform view
Back crack Cross sectionSide view
(b)
Figure 7: Typical failure modes of square tube inMode III. (a)+e crack presents an axial H-type crack (back without penetration) (a � 3.0mm,R � 16 cm,m � 100 g + 160 g, TZ9). (b)+e crack presents an axial H-type crack (back penetration) (a � 3.0mm, R � 12 cm,m � 100 g + 160 g,TZ11).
Shock and Vibration 7
distance [21, 22, 24, 25]. Figures 11–14 depict various de-formations of square tubes with wall thicknesses of 3.0, 3.5,and 4.0mm under double-explosion loadings. +e chargestand-off distances in these experiments were set to 24, 22,20, 18, 16, 14, 13, and 12 cm. +e results indicate that thedegree of plastic deformation zone and crack length ofsquare tubes become gradually more severe as the stand-offdistance decreases.
In Figure 12, the stand-off distances in these six double-explosion experiments are 24, 22, 20, 18, 16, and 12 cm.When the stand-off distances are 24, 22, and 20 cm, thedeformation mode of the square tube belongs to eitherMode Ia or Mode Ib, and the dimensions of their localplastic deformation (lx × ly) are 357.3mm × 73.3mm,360.9mm × 73.1mm, and 365.8mm × 69.2mm, re-spectively. Local plastic deformation occurred on the tube
due to the blasting centre is far from the front zone of thesteel tube.+is mechanism belongs to deformationMode Iaand Mode Ib.
When the stand-off distance is decreased from 20 cm to18 cm, the deformation mode of the square tube istransformed from Mode Ib to Mode II, and their localplastic deformation dimensions (lx × ly) are 365.8mm ×
69.2mm and 339.5mm × 76.4mm, respectively. +e im-pact loads in the front zone of the tube increased as thestand-off distance decreased, which causes the tube to crackand a certain global deformation. However, the plasticdeformation dimension decreases, even though the stand-off distance decreases. +is is because the energy of squaretube will not only cause the local plastic deformation of thetube, but also the most energy will be transformed into thefracture energy.
Cross section(single)
Cross section(double)
Planform view
Side view
Double explosion
Single explosion
Single explosion
Double explosion
(a)
Cross section(single)
Cross section(double)
Planform view
Side view
Double explosion
Single explosion
Single explosion
Double explosion
(b)
Cross section(single)
Cross section(double)
Planform view
Side view
Double explosion
Single explosion
Single explosion
Double explosion
(c)
Figure 8:+e deformation and damage comparison of square tubes subjected to single and double explosions. (a) a � 4.0mm, R � 20 cm. (b)a � 3.5mm, R � 14 cm. (c) a � 3.0mm, R � 12 cm.
8 Shock and Vibration
When the stand-o� distance is decreased from 18 cm to16 cm, the deformation mode of the square tube is trans-formed fromMode II to Mode III, and the axial deformationvalue lx of the square tube is almost invariable, and the radialdeformation value ly of the square tube is obviously in-creased. G and H move towards the blasting midpoint of Aunder the pull e�ect; besides, they will move downwardunder the action of the explosive load and also havea bulging motion on the side walls. �erefore, the squaretube at the G and H points appears crimped, and the bulgeon the side walls is enlarged. When the stand-o� distancecontinues to decrease from 16 cm to 12 cm and the failure
mode of the square tube transforms from Mode II to ModeIII, the rupture damage and the plastic deformation zoneincrease; correspondingly, the axial plastic deformationvalue lx decreases from 340mm to 304.6mm, and the radialplastic deformation value ly increases from 80.3mm to112.9mm, respectively. �is result indicates that as thestand-o� distance is further reduced, the shock wave on thesquare tube becomes further strengthened and the blastplastic deformation zone becomes enlarged once again.
In order to quantitatively study the damage and failuree�ect of the stand-o� distance R and the in�uence ofV in thedepression area, taking the wall thickness a as 4.0mm, forexample, the gain coe®cient u of the depression area V withthe stand-o� distance R is introduced. �e gain coe®cient uis de�ned as
Experimental results5
4
3
2
Failu
re g
rade
Stand-off distance (cm) Wall thickness (
mm)
1
5
4
3
2
1
2.5
3.0
3.5
4.0
4.52624
2220
18161412
10
(a)
Experimental results5
4
3
2
Failu
re g
rade
Stand-off distance (cm) Wall thickness (
mm)
1
5
4
3
2
1
2.5
3.0
3.5
4.0
4.52624
2220
18161412
10
(b)
Figure 9: �e failure grade of square tubes under single and double explosions. (a) Failure grade under single explosion. (b) Failure gradeunder double explosion.
3
2
24 22 20 18 16 14 13 12
w
R (cm)
1
0
α = 3.0mmα = 3.5mmα = 4.0mm
Figure 10: �e volume relation between the single and doubleexplosions.
1.33.8 4.8
11.4
22.8
0
5
10
15
20
25
30
20–18 18–16 16–14 14-13 13-12
u
R (cm)
Figure 11: Depression area volume changes with the stand-o�distance.
Shock and Vibration 9
u �V2 −V1
R2 −R1, (2)
where the stand-off distances R1 are 12, 13, 14, 16, and18 cm, respectively, and V1 is the volume of the depressionarea corresponding to the stand-off distance R1. +e stand-off distances R2 are 13, 14, 16, 18, and 20 cm respectively, andV2 is the volume of the depression area corresponding to thestand-off distance R2. According to the definition in thisstudy and the experimental data in Table 3, the calculationresults of the stand-off distance in a 12–20 cm interval arecompared with those shown in Figure 11.
As shown in Figure 11, the volume growth rate of thedepressed area increases when the stand-off distance R isreduced from 20 cm to 12 cm (u from 1.3 to 22.8). It can be
seen that with the decrease in stand-off distance, the damageof the explosion load to the square tube is greater and thevolume of the depression increases.
Similar laws are also found in Figures 13 and 14.+rough the above law, it can be seen that the degree ofdeformation and damage of the square tubes becomegradually more severe as the stand-off distance decreases.A part of the energy absorbed by the square tube isconverted to rupture damage energy and global de-formation of the tube. Figures 12(d) and 12(e) show thedeformation of the square tube more clearly by the de-scription of the cross section and axial deflection. Whenthe stand-off distance decreases from 24 cm to 18 cm, theglobal deformation value, δglobal, increases by 4.5mm,namely, from 103.5mm to 108mm, compared with the
R = 24 cm
R = 20 cm
R = 18 cm
R = 16 cm
R = 12 cm
R = 22 cm
(a)
R = 24 cm
R = 20 cm
R = 18 cm
R = 16 cm
R = 12 cm
R = 22 cm
(b)
R = 24 cm R = 22 cm R = 20 cm R = 18 cm R = 16 cm R = 12 cm
(c)
–100 –80 –60 –40 –20 0 20 40 60 80 100 120
0
20
40
60
80
100
Radius from impact midpoint (mm)
Def
lect
ion
(mm
)
24 cm22 cm20 cm
18 cm16 cm12 cm
(d)
Distance from impact midpoint (mm)
Def
lect
ion
(mm
)
24 cm22 cm20 cm
18 cm16 cm12 cm
0 20 40 60 80 100 120 140 160 180 200 220
120
100
80
60
40
20
(e)
Figure 12: Damage of square tubes with a thickness of 3.0mm at different stand-off distances. (a) Planform view. (b) Side view. (c) Crosssection. (d) Deformation curve on cross section. (e) Deflection in axial direction.
10 Shock and Vibration
case where the stand-off distance is reduced from 18 cm to12 cm, in which it increases by 33.1 mm, namely, from108mm to 141.1mm. +is result demonstrates that thedeformation and damage of the steel tube is quite large oreven the fracture deformation occurs, although the chargedistance decreases the same size. +is difference is at-tributed to the fact that the shock wave pressure is ex-ponentially attenuated as the propagation distanceincreases and the attenuation amplitude is larger in thearea near the explosion for the same propagation distance.Besides, the explosion load on the blasting surface of thesquare tube is the common effect of the explosion products
and the shock wave in the near distance from the blastingcentre, while the explosion load is mainly due to the effectof the shock wave at the larger stand-off distance.
3.4. Influence of Wall :ickness on the Dynamic Response ofSquare Tubes Subjected to Double Explosion. Figures 15–17depict various deformations of square tubes with stand-offdistance of 18, 16, and 12 cm under double-explosionloadings. +ree sizes of wall thickness were selected as 1,2, and 3 in this test. +e results indicate that the degree ofplastic deformation zone and crack length of square tubes
R = 22cm
R = 18cm
R = 16cm
R = 14cm
R = 12cm
R = 20cm
(a)
R = 22cm
R = 18cm
R = 16cm
R = 14cm
R = 12cm
R = 20cm
(b)
R = 22cm R = 20cm R = 18cm R = 16cm R = 14cm R = 12cm
(c)
Figure 13: Damage of square tubes with a thickness of 3.5mm at different stand-off distances. (a) Planform view. (b) Side view. (c) Crosssection.
R = 20 cm
R = 16 cm
R = 14 cm
R = 13 cm
R = 12 cm
R = 18 cm
(a)
R = 20 cm
R = 16 cm
R = 14 cm
R = 13 cm
R = 12 cm
R = 18 cm
(b)
R = 20cm R = 18cm R = 16cm R = 14cm R = 13cm R = 12cm
(c)
Figure 14: Damage of square tubes with a thickness of 4.0mm at different stand-off distances. (a) Planform view. (b) Side view. (c) Crosssection.
Shock and Vibration 11
a = 3.0mm
a = 3.5mm
a = 4.0mm
(a)
a = 3.0mm
a = 3.5mm
a = 4.0mm
(b)
a = 3.0mm a = 3.5mm a = 4.0mm
(c)
0
20
40
60
80
100
Defl
ectio
n (m
m)
–40 –20 0 20 40 60–60Distance from impact midpoint (mm)
Figure 15: Damage of square tubes with di�erent thicknesses at a stand-o� distance of 18 cm. (a) Planform view. (b) Side view. (c) Crosssection. (d) Deformation curve on cross section. (e) De�ection in axial direction.
12 Shock and Vibration
become gradually more severe as the wall thicknessdecreases.
Figure 15 shows that the wall thickness has an im-portant influence on the deformation mode of the squaretube subjected to a double explosion. When the stand-offdistance is 12 cm and the charge mass is 100 g + 160 g, thedeformation mode of the square tube is Mode II for thetubes with thicknesses of 3.5mm or 4.0mm and Mode IIIfor the tube with a thickness of 3.0mm. Although theultimate rupture strength of the square tube for the dif-ferent wall thicknesses is equal, the deformation anddamage extent of the tube with a thickness of 3.0mm arethe largest compared with those of the tubes with thick-nesses of 3.5mm and 4.0mm, for a given charge mass andstand-off distance. Besides, when the wall thickness de-creases from 4mm to 3.5mm, the global deformationvalue, δglobal, increases by 9.1mm, namely, from 107.8mmto 116.9mm, compared with the case where the wallthickness changes from 3.5mm to 3.0mm, in which itincreases by 24.6mm, namely, from 116.9mm to141.5mm. From this result, it is obvious that the de-formation in the latter case is considerably larger than thatin the former one, although the decrease in wall thick-nesses is equal in both cases. Figure 15(d) shows the de-flection on the cross section of the square tubes for threevalues of wall thickness. +e cross section of the squaretube becomes wider as the wall thickness decreases, which,remarkably, weakens the ability of the square tube to resistbending. Figure 15(e) illustrates the axial deformationcurve of the square tubes. +e deformation of the squaretube is still localised in spite of the rupture of the square-tube structure.
In order to quantitatively analyse the damage and dy-namic response characteristics of the square tube with dif-ferent wall thicknesses subjected to explosion loading, thestand-off distance R � 20 cm is taken as an example. +e gaincoefficient v of the depression area V with the wall thicknessW is introduced. +e gain coefficient v is defined as
v �V3.0 −V3.5
V3.5 −V4.0, (3)
where V3.0, V3.5, and V4.0 are the volumes of the depressionarea when the wall thicknesses are 3, 3.5, and 4.0mm,respectively.
According to the definition and experimental data inTable 3, when the thickness of a decreases from 4.0mm to3.5mm and from 3.5mm to 3.0mm, the volume change ofthe depression area increases by 6.85 times (v � 6.85). It canbe seen that the change in wall thickness has a considerableinfluence on the change in volume of the depression areaunder the effect of the explosion load.
As can be seen from the above analysis, with the decreasein the wall thickness of a square tube, the correspondingdegrees of deformation and damage are increased. Fur-thermore, when the wall thickness of the tube is reduced forthe same interval (4.0mm to 3.5mm and 3.5mm to 3.0mm),the antiexplosion capability of the tube rapidly decreases. Inpractical engineering applications, the wall thickness ofa square tube should be increased to strengthen the re-sistance capability of the square-tube structure undera double explosion. Moreover, the construction cost andefficiency of the project should be considered.
4. Conclusions
+e main purpose of this study was to investigate the de-formation and fracture of steel square tubes under double-explosion loadings. To achieve this purpose, square tubeswith different wall thicknesses were subjected to a singleexplosion (with 160 g of TNTcharge) and a double explosion(the first explosion with 100 g of TNTcharge and the secondexplosion with 160 g of TNTcharge) with different stand-offdistances.
+e results obtained from this study are summarised asfollows:
a = 3.0mm
a = 3.5mm
a = 4.0mm
(a)
a = 3.0mm
a = 3.5mm
a = 4.0mm
(b)
a = 3.0mm a = 3.5mm a = 4.0mm
(c)
FIGURE 16: Damage of square tubes with different thicknesses at a stand-off distance of 16 cm. (a) Planform view. (b) Side view. (c) Crosssection.
Shock and Vibration 13
(1) Four major failure modes were recorded during theexplosion tests, namely, Mode Ia, Mode Ib, Mode II,and Mode III. Mode Ia denotes local plastic de-formation; Mode Ib denotes local plastic deformationwith a small global deformation, in which the energyabsorbed by the square tube is mainly converted toplastic work. Mode II andMode III represent differentfracture types at the front zone of the square tube withvarious degrees of deformation, and the energyabsorbed by the square tube was mainly converted toplastic work and fracture energy.
(2) As the failure mode of the square tube is transformedfromMode II to Mode III, the fracture pattern changesfrom the radial H-type crack to axial H-type crack,respectively.+e cracks in the radial direction no longerincrease owing to the presence of the side corners of thesquare tube, while the length of the cracks in the sidecorners of the square tube increases in the axial di-rection. Mode III presents a completely different crackdevelopment path from that of Mode II, and the crackof Mode II is more localised on the front zone.
(3) +e failure degrees of square tubes under a doubleexplosion were relatively higher than those of squaretubes under a single explosion, namely, the pre-damaged square tube is more susceptible to damagethan an undamaged tube when it undergoes theexplosion loading once again. When the stand-offdistance or the thickness was small, the difference infailure degree was remarkably obvious.
(4) +e experiment indicates that the side corners of thesquare tube are very vulnerable, and they easilysustain damage owing to the stress concentrationand shear effect. +is phenomenon is very unfav-ourable for the stability of square-tube structures inengineering applications.
Future research work will be focused on the numericalsimulation of the dynamic responses and damages of squaretubes subjected to multiple blast loads.
Data Availability
+e data used to support the findings of this study areavailable from the corresponding author upon request.
Conflicts of Interest
+e authors declare that they have no conflicts of interest.
Acknowledgments
+is research was supported by the National Natural ScienceFoundation of China (nos. 51678567 and 11102233) and theChina Postdoctoral Science Foundation (nos. 2015M582791and 2016T90998). +e authors would like to gratefully ac-knowledge this support.
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(a)
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14 Shock and Vibration
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