Numerical studies of the combined effects of blast and fragment loadingUlrika Nystro m*, Kent GylltoftDepartment of Civil and Environmental Engineering, Chalmers University of Technology, Sven Hultins gata 8, SE-412 96 Goteborg, Swedenarti cle i nfoArticle history:Received 22 July 2008Received in revised form20 February 2009Accepted 24 February 2009Available online 6 March 2009Keywords:Numerical simulationBlast loadFragment impactCombined loadingConcreteabstractThe well-known synergetic effect of blast and fragment loading, observed in numerous experiments, isoften pointed out in design manuals for protective structures. However, since this synergetic effect is notwell understood it is often not taken into account, or is treated in a very simplied manner in the designprocess itself. A numerical-simulation tool has been used to further study the combined blast and frag-ment loading effects on a reinforced concrete wall. Simulations of the response of a wall strip subjected toblast loading, fragment loading, and combined blast and fragment loading were conducted and the resultswere compared. Most damage caused by the impact of fragments occurred within the rst 0.2 ms afterfragments arrival, and in the case of fragment loading (both alone and combined with blast) the number ofexural cracks formed was larger than in the case of blast loading alone. The overall damage of the wallstrip subjected to combined loading was more severe than if adding the damages caused by blast andfragment loading treated separately, which also indicates the synergetic effect of the combined loading. 2009 Elsevier Ltd. All rights reserved.1. IntroductionThe combined loading of blast and fragments, caused byexplosions, isconsideredtobesynergeticinthesensethat thecombinedloading results indamage greater thanthe sumofdamagecausedbytheblastandfragmentloadingtreatedsepa-rately [1]. This is a well-knownphenomenon pointed out in some oftheliteratureanddesignmanualswithintheareaof protectivedesign[2]. However, duetothecomplexnatureof theeffectofcombined loading, its high parameter dependence and the limitednumber of documentations and comparable experiments, thedesignmanuals oftendisregardtheeffect or treat it inaverysimplied manner.In order to increase the understanding of the combined effectsof blastandfragmentloading, numerical simulationswerecon-ducted. Thesimulationsconsistofawallstripsubjectedtoblastand fragment loading, applied both separately and simultaneously.To be able to drawgeneral conclusions about the effect ofcombined loading the complexities of the structure and theappliedloadswerereduced. Thecasesstudiedinthispaperarepurely academic, where both the structure and the loads are basedon a real case, i.e. requirements of protective capacity stated intheSwedishShelterRegulations[3], butarestrictlyidealisedinorder to give useful results from which conclusions can be drawn.Usinganumerical-simulationtool ismotivatedbye.g. thehighcost of undertaking tests, and the possibility to better follow andunderstand the principal phenomena related to this kind ofloading.This work is a substudy within a project with the long-term aimto study and increase the knowledge of blast and fragment impacts,and the synergy effect of these loads, onreinforced concretestructures. Theresearchprojectisacollaborationofmanyyearsduration between Chalmers University of Technology and theSwedishRescue Services Agency. Inearlier studies withintheframework ofthisproject, theeffect ofblastwavesinreinforcedconcrete structures, fragment impacts on plain concrete, anddesignwith regard to explosions and concrete, reinforced and bre-reinforced, subjected to projectile impact were studied by Johans-son [4], Leppa nen [5] and Nystro m [6,7], respectively.2. Theoretical framework2.1. Weapon load characteristicsAs detonation of the explosive ller in a cased bomb is initiated,the inside temperature and pressure will increase rapidly and thecasingwill expanduntil it breaksupinfragments. Theenergyremaining after swelling and fragmenting the casing, and impart-ing velocity to the fragments, expands into the surrounding air andthus creates a blast wave. Thereby, the structures around a bombdetonationwill beexposedtobothblast andfragmentloading,*Corresponding author Tel.: 46 031 772 22 48; fax: 46 031 772 22 60.E-mail address: ulrika.nystrom@chalmers.se (U. Nystro m).Contents lists available at ScienceDirectInternational Journal of Impact Engineeringj ournal homepage: www. el sevi er. com/ l ocat e/ i j i mpeng0734-743X/$ see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijimpeng.2009.02.008International Journal of Impact Engineering 36 (2009) 9951005whichmeansthatatleastthreetypesofloading effectsmustbeconsidered:impulse load from blast waveimpulse load from striking fragmentsimpact load from striking fragmentswhere impulse is considered to give a global response and impacta local response caused by the penetration of the fragments.Therearemanydifferenttypesofweapons, designedtohavea specic effect on the surroundings. In design of protective struc-turesathreat-determinationmethodology, basedonprobabilityaspects, must be usedto decide what loadconditions the structure istobedesignedfor. Therearemethodologiesforcalculatingthecharacteristics of the blast and fragment loads caused by explosion,whicharewell acceptedinthedesignof protectivestructures.However, even though the blast load characteristics for a bare high-explosivedetonationcanbeestimatedwithgreat accuracy, theloads from a cased bomb cannot be determined as accurately [2].Due to the complexity of not only the blast itself but also the frag-mentation of the casing, these load estimations are more uncertain.Since the properties of the bomb (geometry, casing material andthickness, type of explosive ller, etc.) and its position relative tothe target, as well as the surrounding environment, have inuenceon the loading conditions, all these parameters must be consideredduringanalysisof theloadingeffect. Alsothedistancefromthedetonation (stand-off) will greatly inuence the loading properties.This is mainly due to the change in peak pressure for the blast waveand the change in velocity of the fragments, which both decreasewith increasing distance. The retardation of the blast wave is largerthan that of the fragments, leading to a difference in arrival time. Inthe range closest to the bomb, i.e. within a few metres, the blastwavewill reachthetargetbeforethefragments, whileatlargerdistances the fragments will arrive before the blast wave. For a 250-kg general-purpose bomb (GP-bomb), with 50 weight per cent TNT,the blast front and the fragments will strike the target at the sametime at an approximate distance of 5 m; according to Fig. 1, wherethe time of arrival of the blast load and the fragments are calculatedby means of ConWep [8] (based on equations of Kingery and Bul-mash [9]) and Janzon [10], respectively.2.1.1. Blast loadingThe blast load resulting from a detonation of an uncased chargein free air, i.e. distant from the nearest reecting surface, is wellknown and often idealised as shown in Fig. 2. The detonation takesplace at time t 0 and arrives at the point studied at time ta. As theblast wave arrives, the pressure increases from the ambient pres-sure, P0, to P0Ps, where Psis the incident overpressure causedby the detonation. As time goes on, the overpressure decays and attime T after the time of arrival the pressure is again equal to theambient pressure P0 and the positive phase is over. Due to a partialvacuum formed behind the blast front [11] a negative pressure Ps(relative to the ambient pressure) appears and the negative phase isentered. Thedurationof thenegativephaseislonger thanthepositive phase, but the amplitude of the negative pressure is limitedbytheambientpressure, P0, andisoftensmallcomparedtothepeakoverpressure, Ps. However, indesign withregardtoexplo-sionsthenegativephaseisconsideredlessimportant thanthepositive phase and is therefore often disregarded.As the blast wave strikes a surface, e.g. a wall, it is reected andits behaviour changes. The so-called normal reection, taking placeas the blast wave is reected against a perpendicular surface, maylead to signicantly enhanced pressures, where the reected peakoverpressure Prwill be between 2 and 8 [2], and according to Refs.[11,12] as much as 20, times higher than the incident overpressurePs. According to Ref. [13] the shape of the reected pressure has thesame general shape as the incident pressure, as shown in Fig. 3.For cased charges the blast load characteristics depend not onlyon the type and amount of explosive and the stand-off distance, butalsoonthe properties (geometrical andmaterial) of the casing. Sincethere is less knowledge about howthe casing affects the blast wave,there are also less generic expressions describing this. In Ref. [14] anexpression for calculating an equivalent uncased charge weight isgiven as a function of the ratio between the casing weight and theactual charge weight. However, the reduced blast pressure due tothe energy consumed during casing break-up is often not taken intoaccount in the design manuals [2], which also is used in this study.2.1.2. Fragment loadingAs mentionedearlier, thecasingof abombwill swell afterinitiationoftheexplosivellerduetothehighpressure. During02468100 2 4 6 8 10Distance [m]FragmentsBlast waveArrival time [ms]Fig. 1. Timeofarrivalforblastwave andfragmentsas functionsofthestand-offfora 250-kg GP-bomb with 50 weight per cent TNT, from Ref. [5].PT+T - tP0+Ps+P0P0-Ps-taFig. 2. Incident blast wave idealisation for 125 kg TNTat 5 mstand-off, based on Ref. [4].0100020003000400050000 1 2 3 4 5Time [ms]Pressure, P [kPa]Incident pressureReflected pressureFig. 3. Positive phase of reected and incident blast wave for 125 kg TNT at 5 m stand-off, calculated according to Ref. [13].U. Nystrom, K. Gylltoft / International Journal of Impact Engineering 36 (2009) 9951005 996swelling, cracks will form and propagate in the casing; and as thecracks meet or reach a free border, fragments are formed [15]. Thenoseandthetailsectionofthebomb willbreakupinasmallernumber of massive fragments and the body will fracture into manysmall fragments.Derivationof theoretical expressionsdescribingthefragmen-tation processanditscharacteristicsforcasedbombsisdifcult.This is partly due to the complexity of the phenomenon itself andpartly due to the great variation of bomb properties, which highlyinuences the fragmentation process. However, there are expres-sionsforestimatingthemassdistributionandvelocitiesof thefragments that are based on theoretical considerations andconrmedwithalargenumber oftests [16]. Inthederivationofthese expressions, the bomb casing is normally idealised asacylinderwithevenlydistributedexplosives, meaningthat themethods apply especially to items that can reasonably be approx-imated as either cylindrical items or as a series of cylindrical items[16]. The more an item deviates from this ideal, the less reliable arethe estimations made using these methodologies.In order to estimate the fragment mass distribution, a relation-ship developed by Mott [17,18] (presented in e.g. [16,19]) is oftenused. For design purposes a design fragment is used. The mass ofthe design fragment is often determined by specifying a condencelevel giving the probability that the weight of the fragment is thelargest fragment produced. However, this method of determiningthe design fragment is justied in design where the damage causedby the individual fragments is of interest as the hazardous case. Inthe case of design against the fragment cluster, another approachmay be more desirable where the combined effect of the fragmentimpact andimpulseis of interest. This is discussedfurther inSection 4.2.The initial velocity of the fragments can be estimated from theGurney equation [20] (presented in e.g. [16,19]), which also derivesfrom an assumption of a cylindrical casing. Since this equation isbasedonenergybalancewithintheexplosive andmetal casesystem, without takingintoaccount theloss of energyduringrupture of the casing, it is an upper bound estimation.As the fragments travel throughthe air their velocity willdecreaseduetothedragforces. Smaller, lighterfragmentswillretardfasterthanlarger, heavierfragments. Equationsdescribingthis behaviour exist as well, e.g. [16,19].2.2. Concrete behaviour under static and dynamic loadingIt is well known that the two most pronounced disadvantages ofconcreteareitslowtensilestrengthandbrittlebehaviour. Thetensile strength of normal-strength concrete is less than one tenthof the compressive strength, and after fracture initiation, i.e. afterthe tensile strengthis reached, the ability totransfer stressesthrough the material decreases rapidly. For high-strength concretethebrittlebehaviour canalsobeseeninthecaseof uni-axialcompression, but the post-fracture ductility in compressionincreases, with a decreasing compressive strength.Inmulti-axial loadingconditions the behaviour of concretediffersfromthebehaviourunderuni-axialloading. Theductility,stiffness and strength incompression increase withincreasedconnement, and for very high lateral pressures the compressivestrengthmaybemorethan15timeshigherthantheuni-axialcompressivestrength[5]. Suchhighlateral pressuresmayoccurduring impact and perforation of e.g. projectiles and fragments.High dynamic loading, giving a high strain-rate in the material,also affects the strength and ductility of the concrete. In the case ofhigh-ratetensileloading, theultimateuni-axial tensilestrengthmay be as much as 57 times higher than the static tensile strength[21], andeventhoughthe effect onthe ultimate compressivestrength is less pronounced it may still be more than doubled [22].It has recently also been indicated that the fracture energy is strain-rate-dependent [2325].3. MethodTests have beenconductedaroundthe worldto study thecombined effects of blast and fragment loading, but these are oftennot suitablefordrawinggeneral conclusionsabout thelocal orglobal structural behaviour. Thisisduetothegreatvariationofparameters involved, e.g. load characteristics and stand-off, whichaffects the results. Numerical simulations are often used to inves-tigate the effect of blast andfragments, andmake it possible tostudythe inuence of different parameters stand-off distance, fragmentsize, materials, etc. whichis costly inexperimental testing.Nevertheless, thenumerical simulations cannot fullysupersedeexperiments, but should be used in combination, and experimentsare needed to verify the numerical models used in the simulations.The reinforced concrete structure used in the study presentedhere is based on a wall strip in a civil defence shelter, fullling therequirements of protective capacity related to conventional bombsin the Swedish Shelter Regulations [3]. The loads applied, i.e. theblast wave and fragment loading, are also based on the load de-nitions in Ref. [3].As no suitable experiments, with combined blast and fragmentloading, werefoundforthisstudy, twoseparateexperimentsonblast load and single fragment impact were used to verify and cali-brate the numerical model. The validation and calibration processwere done within a preliminary study and are only briey describedin this paper. Conclusions from the preliminary study were used tobuild upthe numerical model of the wall strip subjected to blast andfragment impacts usedinthe mainstudy. Single-degree-of-freedomanalyses wereusedtondwhat loadcombinationcausedthe largestdeection: simultaneous arrival of the two loads, blast load arrivingrst, or fragments arriving rst. The results from the SDOF analyseswere used to decide the arrival times for the loads in the numericalsimulation of combined loading. The numerical results of the wallstripresponsewerecomparedandanalysedinordertoseetheeffects of combined loading.For further information about this study the reader is referred toRef. [26] where a detailed description of the load characterisationand the preliminary study is presented.4. Wall element and load characteristicsThe Swedish Shelter Regulations [3] govern the design of civildefence shelters in Sweden, and contain the requirements speciedfor these protective structures. Here only the criteria for protectivecapacity related to conventional bombs are specied, but it shouldbepointedout that civil defenceshelters alsoaredesignedtowithstand e.g. radioactive radiation, chemical and biologicalwarfare, and explosive gases.According to Ref. [3], a civil defence shelter should be designed towithstandtheeffect of a pressurewave correspondingtothatproduced by a 250-kg GP-bomb with 50 weight per cent TNT, whichbursts freely outside at a distance of 5.0 m from the shelter duringfree pressure release. Further, the shelter must also be able to with-stand the effect of fragments from a burst as described above. In thecaseof fragment loadingit is thefragment cluster that is meant, whilelarger individual fragments may damage and penetrate the shelter.4.1. Wall elementIn the Swedish Shelter Regulations [3], the civil defence shelteris conceived as a reinforced, solid concrete structure. For a shelterU. Nystrom, K. Gylltoft / International Journal of Impact Engineering 36 (2009) 9951005 997without backlling the minimum thicknesses of the roof, walls andoor are specied as 350, 350 and 200 mm, respectively, and theconcrete should full a requirement of at least C25/30, according toRef. [27] (corresponds to mean cylindrical compressive strength of25 MPa). Hot-rolledreinforcementbarswithaspeciedrequire-ment of strain hardening must be used. The reinforcement must beplacedintwo perpendicular alignments inbothedges of thestructural element and the minimumand maximumreinforcementcontent is 0.14 and 1.10%, respectively. A minimum reinforcement-bar diameter of 10 mm and maximum bar spacing of 200 mm arerequired, with a maximum concrete cover of 50 mm.The wall studied has a total height of 3 m and is simplied to besimply supported with a span length of 2.7 m, as seen in Fig. 4. Therough simplication of the support conditions was not made in anattempt to imitate the real behaviour of the wall.InRef. [3], equivalent staticloads, representingtheweaponeffect, are used in the design process. A static load of 50 kN/m2isused to calculate the required amount of reinforcement in the walls,giving reinforcement bars 10 s170 (which corresponds to a rein-forcementareaof465 mm2/mineachfaceofthewallelement).Deformedreinforcementbars(B500BT), withayieldstrengthof500 MPawereassumedandthedistancefromconcreteedgetocentre of reinforcement bars was chosen as 35 mm. The concretewas assumed to have a concrete strength of 35 MPa.4.2. Load characteristicsInFig. 5theblast loadcausedbytheGP-bombspeciedinSection 4 with a stand-off of 5.0 m, calculated with ConWep [8], isshown together with a simplied relationship. It should be kept inmind that design codes do often not take into account the energyconsumed for swelling and fragmenting the casing of bombs. As anapproximationthisenergylossisalsoneglectedinthepresentstudy even though it would be more accurate to reduce the pres-sureoftheblastloadinordertoimitatetherealbehaviour. Theblast load is assumed to be uniformover the wall, whichisreasonably accurate forthisstand-off[4]. Theimpulsedensityofthe blast load is, according to Ref. [8], 2795 Ns/m2.Since the geometry and casing material of the bomb used in thedesign of civil defence shelters are not specied, the size and massdistribution cannot be calculated without making certainassumptions. In this study, the American GP-bomb Mk82 was usedas a reference when estimating the mass distribution of the bombspecied in the Swedish Shelter Regulations. According to ConWep[8] the Mk82 has a nominal weight of 500 lb (226.8 kg) andcontains 192.0 lb (87.09 kg) of the high-explosive H-6, corre-spondingto242.9 lb(110.2 kg)equivalentweightof TNT, andisthereforerelativelyclosetothebombspeciedintheSwedishShelterRegulations[3]. Themassdistributionwasestimatedbyscaling the inner casing diameter andthe casing thickness tocorrespondto thesomewhatincreased volume ofexplosivellercompared to the Mk82; for more details see Ref. [26].All fragments usedinthesimulations wereassumedtobespherical and of the same size, corresponding to a design fragment.It was further assumed that the fragments were uniformlydistributed over the wall. Even though these idealizations are roughand differ from the real fragment loading caused by a bomb, wherethe mass, shape andvelocity of the fragments differ andthedistributionof thefragmentsoverthewall isnotuniform, theywerenecessaryinorder toreducethecomplexityof boththenumerical model and the resultsproduced. It can be pointed outthat a more realistic fragment mass distribution would give a morenon-uniformdamageoverthewall, wherethelargerfragmentswould give a larger local damage than the smaller fragments.As mentioned in Section 2.1.2, the design fragment, calculatedwith a condence level (often taken as 95%), is used for design withregard to fragment impact. However, this design fragment and thecorrespondingeffectonthetargetarenotrepresentativeof thefragmentimpulseloadwhichisimportanttocapturetheglobalresponse caused by the fragments and not only their local effect.Thismeansthatanother approachmustbeusedtondarepre-sentative fragment size in this study and it was decided to use theimpulse caused by the real fragmentload on the wall, causedbya vertically placed bomb, to dene a representative weight of thefragments. From estimations of the mass and velocity distributionamong the fragments, the corresponding fragment impulse distri-butionwas calculated, andarepresentativefragment sizewasdeterminedasthefragmentmassgivingtheaverageimpulseonthe structure, for details see Ref. [26]. This resulted in a fragmentmassof 21.9 gandafragment diameterof 17.5 mm. Theinitial5.0 m3.0 m0.2 m0.35 m0.35 m0.352.7 mmFig. 4. Civil defence shelter and simplied model of one of its walls.0100020003000400050000 1 2 3 4 5Time [ms]Pressure, P [kPa]Simplified relationConWepTime[ms]Pressure[kPa]0.00.30.61.01.52.03.08.9745 0062 9301 713836340137220Fig. 5. The reected pressure load as function of time for 125 kg TNT at a distance of 5.0 m, according to ConWep [8], and the simplied relationship for this which is used in thisstudy, based on Ref. [4].U. Nystrom, K. Gylltoft / International Journal of Impact Engineering 36 (2009) 9951005 998fragmentvelocitywascalculatedtoapproximately1890 m/s(byuseof theGurneyequation), andat thedistanceof 5.0 mthevelocity is decreased to 1760 m/s. The fragment density isapproximately 0.65 kg/m2and the corresponding impulse intensitycaused by the fragments is 1125 Ns/m2.4.3. SDOF estimationsThe single-degree-of-freedommethod (SDOF method) was usedin order to nd what combination of arrival times of the blast andfragment load that resulted in the maximumdeection. Thesimplied relation of the blast load, presented in Section 4.2, wasused for the blast load and a triangular load was assumed for thefragment loading. The duration of the fragment loading wasassumed to 0.1 ms, which is the approximate time it takes for thefragment to penetrate the concrete, and its impulse intensity was asdened in Section 4.2, giving a peak pressure of 22.5 MPa. It shouldbepointedoutthatonlytheimpulseloadofthefragmentswastakenintoconsiderationintheSDOFanalysespresentedinthispaper, since the penetration by the fragments and the subsequentdamage were not considered. An ideal-plastic material response ofthe SDOF system was used and the maximum value of the internaldynamic resistance Rmof the wall strip was calculated to be 275 kN;for details see Appendix A and Ref. [26].In Fig. 6 the results are shown for ve different cases ofcombined loading:1. loads arrive at the same time (simultaneous loading)2. blastwaverst, fragmentsarriveatmaximumwall velocitycaused by the blast3. blast wave rst, fragments arrive at maximum wall deectioncaused by the blast4. fragmentsrst, blastwavearrivesatmaximum wallvelocitycaused by the fragments5. fragments rst, blast wave arrives at maximum wall deectioncaused by the fragmentsAs seen, the caseof simultaneous loadingcauses themostsevere deection (equalling 139.2 mm at time 42.2 ms). For furtherinformation about the SDOF method the reader is referred to Refs.[6] and [19].5. Numerical modelHydrocodes are used for highly time-dependent dynamicproblem-solving by use of nite difference, nite volume and niteelement techniques. The differential equations for conservation ofmass, momentumandenergy, together withmaterial models,describingthebehaviourof thematerialsinvolved, andasetofboundary conditions give the solution of the problem. Thenumerical hydrocode AUTODYN 2D and 3D [28]was used in thisstudy, and the Lagrangian solver technique was employed.5.1. Calibration and validation of numerical modelExperiments and ndings fromnumerical simulations ofexperiments described in the literature were used in the calibrationand validation process for the numerical model used in this study.Thenumerical simulationsforthecalibrationprocesswereper-formed in2Dand 3D. The use of 2Dwas to prefer since it reduces thecomputational time, but since beam elements (used to simulate thereinforcing bars) could not be used in the 2D simulations, 3D wasusedwhenreinforcedconcretewassimulated. However, 3Dwasused with the width of one element and use of boundary conditionsto emulate a 2D simulation. Below is a brief summary of the cali-bration and validation process; for further description see Ref. [26].MagnussonandHansson[29]describedexperimentsonrein-forced concrete beams, of length 1.72 m, subjected to blast loading,and thereafter used AUTODYN 3D to simulate the beam response.They concluded that it was possible to simulate the beam responsewiththeRHTmaterial model providedthat theprincipal-stresstensile-failure model with a fully associated ow rule (i.e. the owrule was associated in both the deviatoric and meridian planes) wasusedinthesimulations, togetherwithcracksoftening. Thiswasalso found by means of numerical simulations of the same exper-iment conducted within the calibration process made in the studypresented here. In this process it was also found that an elementlength of 12 mm gave approximately the same beam response asa ner mesh of 6-mm elements; hence, the coarser mesh of 12-mmelements should be accurate enough to simulate the beamresponsewhen subjected to blast loading.Leppa nen [30] performed and described experiments witha single fragment impacting a concrete block, with size750 350 500 mm. An AUTODYN 2D model with axial symmetryand different element sizes (12 mm) was used in the calibrationprocess, andit was concludedthat thenumerical model gaveaccurate results. However, the size of the fragment used in Ref. [30]differedfromthefragment sizeusedinthis study, andhence,additional 2D simulations were conducted to investigate the effectof the element size. It was concluded that the resulting crater in thesimulations with an element size of 6 mm was somewhat differentfrom the crater in simulations where smaller elements were used,but still an acceptable approximation of the damage caused by thefragmentimpact. Thus, itwasnotconsideredworththegreatlyincreased computational time to use a ner mesh in the main study.5.2. Material modelsThestandardmaterial model for concretewithcompressivestrength of 35 MPa in the material library of AUTODYN was used todescribe the behaviour of concrete. This material model wasdevelopedbyRiedel, Hiermayer andThoma(therefore calledtheRHT model) [31], and consists of three pressure-dependentsurfaces in the stress space. The RHT model also takes into accountpressure hardening, strain hardening and strain-rate hardening aswell as the third invariance in the deviatoric plane. However, thepreliminary study, i.e. the calibration and validation process0204060801001201401600 10 20 30 40 50Time [ms]Deflection [mm]Fragments arrive at maximum velocityFragments arrive at maximum deflectionBlast arrives at maximum velocityBlast arrives at maximum deflectionSimultaneous loadingFig. 6. Mid-point deection of wall strip subjected to blast load and fragment impulseload, calculatedwithSDOFmethod. Sincetheresponse ofsimultaneousloadingandloading where fragments arrive rst and blast at time of maximum velocity are almostidentical these lines are seen as one in the gure.U. Nystrom, K. Gylltoft / International Journal of Impact Engineering 36 (2009) 9951005 999describedinSection5.1, showedthatit wasnecessarytomakesomemodicationswithinthemodelto get accurateresults. Forexample, it wasconcludedthat aprincipal-stresstensile-failuremodel was necessary to describe the behaviour of the wall strip inthecaseof blast loading, insteadof thehydrodynamictensile-failuremodel usedasdefault intheRHTmodel. Thechangetoaprincipal-stresstensile-failuremodel leadstoacut-off of thestrain-rate dependence of the ultimate tensile strength.To describe the behaviour of the reinforcing steel, a piecewiselinear JohnsonCookmaterial model wasused, includingstrainhardening but not strain-rate and thermal effects. A linear elasticsteel material model, with a shear modulus of 81.1 GPa, was used forthesupports, andavonMises material model, simplifyingthematerial behaviour to linear-elasticideal-plastic with yieldstrength of 800 MPa, was used for the fragments. A linear equationof state (EOS) was usedfor the reinforcing steel, the supports andthefragments, while the nonlinear P-a EOS was used for the concrete.The input in the different material models used in the simulations ispresented in Tables 13; for further information about the materialmodels used, the reader is referred to Ref. [26,31,32].5.3. MeshSince fragment penetration is a local effect, requiring relativelysmall elements, a numerical model of even a 1.0-mwide strip of theTable 1Employed material data for concrete, input to the RHT model.Equation of state P alphaReference density 2.75000E 00 (g/cm3)Porous density 2.31400E 00 (g/cm3)Porous soundspeed 2.92000E 03 (m/s)Initial compaction pressure 2.33000E 04 (kPa)Solid compaction pressure 6.00000E 06 (kPa)Compaction exponent 3.00000E 00 ()Solid EOS PolynomialBulk modulus A1 3.52700E 07 (kPa)Parameter A2 3.95800E 07 (kPa)Parameter A3 9.04000E 06 (kPa)Parameter B0 1.22000E 00 ()Parameter B1 1.22000E 00 ()Parameter T1 3.52700E 07 (kPa)Parameter T2 0.00000E 00 (kPa)Reference temperature 3.00000E 02 (K)Specic heat 6.54000E 02 (J/kgK)Compaction curve StandardStrength RHT concreteShear modulus 1.67000E 07 (kPa)Compressive strength (fc) 3.50000E 04 (kPa)Tensile strength (ft/fc) 1.00000E 01 ()Shear strength (fs/fc) 1.80000E 01 ()Intact failure surface constant A 1.60000E 00 ()Intact failure surface exponent N 6.10000E 01 ()Tens./Comp. meridian ratio (Q) 6.80500E 01 ()Brittle to ductile transition 1.05000E 02 ()G (elastic)/(elastic-plastic) 2.00000E 00 ()Elastic strength/ft7.00000E 01 ()Elastic strength/fc5.30000E 01 ()Fractured strength constant B 1.60000E 00 ()Fractured strength exponent M 6.10000E 01 ()Compressive strain-rate exponenta 3.20000E 02 ()Tensile strain-rate exponentd 3.60000E 02 ()Max. fracture strength ratio 1.00000E 20 ()Use CAP on elastic surface? YesFailure RHT concreteDamage constant D1 4.00000E 02 ()Damage constant D2 1.00000E 00 ()Minimum strain to failure 1.00000E 02 ()Residual shear modulus fraction 1.30000E 01 ()Tensile failure Principal stressPrincipal tensile failure stress 3.50000E 03 (kPa)Max. principal stress difference/2 1.01000E 20 (kPa)Crack softening YesFracture energy, GF1.20000E 02 (J/m2)Flow rule Bulking (Associative)Stochastic failure NoErosion Geometric strainErosion strain 2.00000E 00 ()Type of geometric strain InstantaneousTable 2Employed material data for reinforcing steel.Equation of state LinearReference density 7.83000E 00 (g/cm3)Bulk modulus 1.59000E 08 (kPa)Reference temperature 3.00000E 02 (K)Specic heat 4.77000E 00 (J/kgK)Thermal conductivity 0.00000E 00 (J/mKs)Strength Piecewise JCShear modulus 8.18000E 07 (kPa)Yield stress (zero plastic strain) 5.49330E 05 (kPa)Eff. Plastic strain #1 6.70000E 03 ()Eff. Plastic strain #2 1.62000E 02 ()Eff. Plastic strain #3 2.86000E 02 ()Eff. Plastic strain #4 4.57000E 02 ()Eff. Plastic strain #5 6.45000E 02 ()Eff. Plastic strain #6 9.21000E 02 ()Eff. Plastic strain #7 1.27800E 01 ()Eff. Plastic strain #8 1.79200E 01 ()Eff. Plastic strain #9 1.79201E 01 ()Eff. Plastic strain #10 1.00000E 01 ()Yield stress #1 5.62000E 05 (kPa)Yield stress #2 5.68000E 05 (kPa)Yield stress #3 6.27000E 05 (kPa)Yield stress #4 6.78000E 05 (kPa)Yield stress #5 7.15000E 05 (kPa)Yield stress #6 7.46000E 05 (kPa)Yield stress #7 7.76000E 05 (kPa)Yield stress #8 7.95000E 05 (kPa)Yield stress #9 7.95000E 05 (kPa)Yield stress #10 7.95000E 05 (kPa)Strain-rate constant C 0.00000E 00 ()Thermal softening exponent m 0.00000E 00 ()Melting temperature 0.00000E 00 (K)Ref. strain-rate (1/s) 1.00000E 00 ()Failure NoneErosion NoneTable 3Employed material data for supports and fragments.Equation of state LinearReference density 7.83000E 00 (g/cm3)Bulk modulus 1.59000E 08 (kPa)Reference temperature 3.00000E 02 (K)Specic heat 4.77000E 00 (J/kgK)Thermal conductivity 0.00000E 00 (J/mKs)Strength ElasticShear modulus 8.18000E 07 (kPa)Failure NoneErosion NoneEquation of state LinearReference density 7.83000E 00 (g/cm3)Bulk modulus 1.59000E 08 (kPa)Reference temperature 3.00000E 02 (K)Specic heat 4.77000E 00 (J/kgK)Thermal conductivity 0.00000E 00 (J/mKs)Strength von MisesShear modulus 8.18000E 07 (kPa)Yield stress 8.00000E 05 (kPa)Failure NoneErosion NoneU. Nystrom, K. Gylltoft / International Journal of Impact Engineering 36 (2009) 9951005 1000shelterwall wouldhavebeenverylargeandrequiredextensivecomputational time. By use of symmetries and plane-strain-boundaryconditions themodel was limitedtoa 84 1512 350 mmpart of the wall, representing 4.25% of a metre-wide wall strip whichreduces thetotal number of elements usedinthemodel fromapproximately 2.3 million to 98106. Even though the width of thewall stripis only9.6times thefragment dimensionandthesize of thefront face crater caused by the fragment impact (measured in the 2Dsimulations used for calibration described in Section 5.1) approxi-mates the width of the wall strip included in the model the use ofplane-strain-boundary conditions are applicable. The plane-strain-boundary condition takes into account the next row of fragmentswhichstrikes thewall 168 mmfromtherowof fragments includedinthe model. Including a wider part of the wall in the model, where thechoice of width of course has to be done with respect to the loadingcase, would only lead to negligible changes in the results.Due to the varying need of element sizes when simulating theeffectsofblastandfragmentimpact, anermeshofLagrangianelements (size 6 6 6 mm) was used on the front face of the wallstrip, and a coarser mesh of elements (size 12 12 6 mm) of thesame element type was used on the rear side of the wall strip (seeFig. 7), resultinginatotal numberof 97020concreteelements,where 59976 of those were in the front zone where the ner meshwas usedand37044intherear zonewiththecoarser mesh. The total168134484 102 8+240[mm]Transversereinforcement Border between fine and coarse meshxy204204192204204192204FragmentLongitudinalreinforcementFig. 7. Numerical mesh of wall strip used in simulations, also showing reinforcement in the modelled wall strip.Fig. 8. Response of wall strip subjected to blast load at time of maximum mid-point deection.U. Nystrom, K. Gylltoft / International Journal of Impact Engineering 36 (2009) 9951005 1001number ofelementsalongthedepth(x-direction), theheight(y-direction) inthe front- andrear- zones, andthewidth(z-direction) ofthe concrete structure included in the model were, 38, 252, 126 and14, respectively.The wallstripwassupportedby two semicylindricalsupportswitharadiusof 84 mmtoavoidlocal crushingof theelementsaround the supports. The nodes of the support were joined togetherwith the interfacing concrete nodes. In order to allow for rotationaroundthesupportsonlythelineof backnodeswaspreventedfrommoving in the x-direction. The supports were modelled with 4elements along the radius of the half-cylinders, resulting in a totalnumber of 336 elements in one support.Inthesimulationsincludingimpactingfragments, theseweremodelledwithtwoelementsalongtheirradius, resultingin16elementsperhalf-fragmentandatotalnumberof120fragmentelements in the model. Embedded beam elements with the samelengthas thesurroundingconcreteelements andwithcircularcross-sectionwereusedtomodelthereinforcementbars, givinga total number of 630 beam elements in the model.6. ResultsTheresponses of thewall stripestimatedinthenumericalsimulations for blast loading, fragment loading, and combined blastandfragmentloading(simultaneousloading)arepresentedanddiscussed below. As the damage differs at different locations withinthe wall strip, the damage is shown in three views for each case:a top view, a side view at the section of reinforcement, and one inthe middle of the wall strip (the section where the fragments strikethewall strip). Inthegureswiththewall stripresponses, thecolour red indicates fully damaged concrete.6.1. Blast loadingIn the case of blast loading, the maximum deection is 65.2 mmand takes place 29.0 ms after the arrival. In Fig. 8 the damage in thewall strip is shown at time of maximum deection, where it can beseen that cracks have formed at the rear side of the wall strip andhave propagated towards the front face. The damage is localised toFig. 9. Response of wall strip subjected to fragment impacts at time of maximum mid-point deection.Fig. 10. Response of wall strip subjected to fragment impacts at time (a) 0.25 ms, (b) 0.6 ms and (c) 9 ms after time of fragment arrival, seen at section of reinforcement.U. Nystrom, K. Gylltoft / International Journal of Impact Engineering 36 (2009) 9951005 1002relatively few cracks, even though it can be seen that crack initia-tionhastaken placerather densely alongthelengthofthestrip.Damaged concrete can also be seen along the reinforcement closeto the fully developed cracks; at these locations the reinforcementbars were yielding.Whenstudyingthecrackdevelopment, it wasseenthat thelocalisedcrackclosest tothesupport wasformedalreadyafter1 ms, while no damage of the concrete was observed in the middleof the beam at this time. This indicates a direct shear crack due tothe inertia effects, i.e. internal momentum, related to severedynamic loading. After approximately 2 ms, also the localisedcracks in the middle of the beam have formed, and these have thecharacter of exural cracks.6.2. Fragment loadingIn Fig. 9 the wall strip subjected to fragment loading is shown attimeof maximumdeection. This is reached13.3 ms after thefragments strike the wall, and amounts to 11.0 mm. As can be seen,thesimulateddamagecausedbythemulti-fragment impact ismore complex than in the case of blast loading. The total damageconsistsof local damageonthefront face, i.e. craters, scabbingcracks at the rear of the wall strip, direct shear cracks close to thesupports, and bending cracks in the more central parts of the beam.When comparing Figs. 8 and 9, it can be seen that there are morebending cracks formed in the case of fragment impact than for blastloading, resulting in an increased energy-absorbing capacity sincethe reinforcement bars can yield at more locations. This means thatalsotheload-bearingcapacitymayincrease. However, theload-bearing capacity will at the same time be reduced by the decreasedeffectiveheightduetothedamageonthefrontfaceofthewallstrip.To better distinguish the modes of damage and to betterunderstand their evolution, the beamresponse is shown atdifferent times, i.e. after 0.25, 0.6 and 9 ms, in Fig. 10. After 0.25 ms(Fig. 10a)thefragmentimpactshavecausedcratersonthefrontface, and the reected stress wave has caused scabbing cracks at therear of the wall strip. The scabbing effect was not expected in thesimulations, but 2Dsimulationsof fragment impact, takingthemultiple simultaneous impact of fragments and also the strain-ratedependence of the tensile strengthinto account, conrmthisbehaviour; seeRef. [26]. However, inrealitythetwoscabbingcracks probably represent one crack which appears at the level oftensilereinforcementandnot in betweenthetwo reinforcementlayers, as in this case.Approximately 0.6 ms after the arrival of fragments, crackspropagate at the rear side of the wall strip, close to the supports, seeFig. 10b. These are probably direct shear cracks, as also observed inblast loading; see Section 6.1.Attime9 ms, exuralcrackshavestartedtopropagateinthewall strip, as seen in Fig. 10c. These cracks form at the rear face ofthetarget, but alsoat thelevel of thescabbingcracks, whichindicatesthat thewall striphasstartedtoact astwoseparatestructures with sliding between the two separate planes formed bythe horizontal scabbing cracks.6.3. Combination of blast and fragment loadingThemaximummid-point deectionincaseof simultaneousloading of blast and fragment is 85.7 mm and occurs after 33.4 ms.TheresponseofthewallstripattimeofmaximumdeectionisFig. 11. Response of wall strip subjected to combined blast and fragment loading at time of maximum mid-point deection.blastblastfragmentfragmentConcrete surfacePenetrating fragmentFig. 12. Schematically shown connement effects from blast loading, sblast, on concreteelement compressed bysfragment due to fragment penetration.0204060801000 5 10 15 20 25 30 35Time [ms]Deflection [m/s]Combined loadingFragment loadingBlast loadingFig. 13. Mid-point deection of wall strip subjected to combined, blast and fragmentloading from numerical simulations.U. Nystrom, K. Gylltoft / International Journal of Impact Engineering 36 (2009) 9951005 1003shown in Fig. 11. As the damage caused by the fragment impact, i.e.the front face craters and the scabbing cracks at the rear of the strip,appears very early (at less than 0.25 ms, as seen in Section 6.2) thedamageinthecaseofcombined, simultaneousloadingisrathersimilar to the case of fragment loading alone. Due to the blast load,the deection is larger and the damage in the concrete surroundingthe reinforcement bars is more severe than in the case of fragmentimpact alone.Further, thediametersofthefrontface craters arereduced incase of combined loading compared to fragment loading alone. Thiscan probablybeexplainedbyincreasedconnementeffects. Theblast wave causes pressure on the front face, acting perpendicularto the concrete surface, and gives a lateral pressure to the materialcompressed by the fragment penetration; schematically shown inFig. 12. This reduction offrontface damage may lead to ahigherload-bearing capacity thaninthe case withfragment impactsalone, but since the effective height of the wall strip is reduced, theload-bearingcapacityisstillaffected. However, asinthecaseoffragmentloadingalone, thenumberofexuralcracksformedislargerthaninthecaseofblastloadingalone, allowingtherein-forcement bars to yield at more locations, which may improve theload-bearing capacity.7. Comparison of mid-point deections and velocitiesInFig. 13themid-point deections of thethreewall stripssubjectedtoblast, fragmentandcombinedloading, respectively,are shown. As seen, the mid-point deection in the case ofcombined loading is larger than the sum of the deections causedby blast and fragment loading separately, which indicates a synergyeffect.InFig. 14themid-pointvelocitiesfromthesimulationswithblast, fragment and combined loading are shown. The velocity forcombinedloading isrstinuenced by thefragmentimpact, butalready after a fraction of a millisecond the velocity seems close tothe velocity of the wall strip subjected to blast loading alone. Afterapproximately2 ms, thevelocityforcombinedloadingincreasesand exceeds the velocity for blast loading.In Table 4 the mid-point deections estimated in the numericalanalyses are presented together with results from SDOF analyses.Input parameters for the SDOF analyses are shown in Appendix A.In the case of blast loading the estimations of the deection madeinSDOFandnumerical simulations agreewell. Inthe caseof fragmentloading the difference is larger, which probably can be explained bythelimitationsintheSDOF analysestotakeintoaccounte.g. theenergyconsumedduringpenetrationandsubsequent crushingof theconcrete, the formation of many exural cracks, and inertia effects,which may increase the load-bearing capacity. The results from thenumerical simulation and the SDOF analysis differ also for combinedloading. The difference is even larger than in the case of fragmentloadingandmaybe explained bymagnication ofthelimitationsalready used as explanation for the case of fragment loading.8. Summary and conclusionsIn blast loading, the elongation of the rear face of the wall strip islocalised to a few cracks where yielding of the reinforcement takesplace. In fragment loading, the exural cracks to which the elon-gation of the reinforcement is localised are numerous, the energy-absorbing capacity of the wall strip may thus be increased.In the simulations involving fragment loading, scabbing cracksformed due to the reected stress wave. The appearance of thesecrackswasunexpected, butwasconrmedwitha2Dsimulationstudy, indicating that the case of multi-fragment impact may leadto scabbing also when the single fragment impact does not. It maytherefore be necessary to take this effect into account in design ofprotective structures. However, it is questionable whether thelocation and size of the scabbing cracks simulated are realistic.Most damagecausedbythefragment impact occurs within0.2 ms after arrival, which is short compared to the response timeof the element, indicating that in the case of combined loading thebearing capacity and the mid-point deection of the wall strip maybehighlyinuencedbythefragmentimpactsincethestructurethus loses part of its effective height.Thelargermid-pointdeectionofthewall stripsubjectedtoblast loading, compared to the deection in the case of fragmentloading, was expected since the impulse from the blast was almost2.5 times the impulse caused by the fragments.The damage caused by combined loading is more severe than ifaddingthe damages causedby the blast andfragment loading treatedseparately. The size of the front face craters, though, is an exceptionsince these are larger inthe caseof fragment loadingthanincombined loading. It can be concluded that the mid-point deectionin combined loading (85.7 mm) is larger than the sum of mid-point012345Time [ms]Velocity [m/s]0123450 0,5 1 1,5 2 2,5 3 0 5 10 15 20 25 30 35Time [ms]Velocity [m/s]Combined loadingFragment loadingBlast loadingCombined loadingFragment loadingBlast loadingFig. 14. Mid-point velocity of wall strip subjected to blast and fragment loading from numerical simulations.Table 4Mid-point deections.Load uAUTODYN [mm] uSDOF [mm]Blast 65.2 64.0Fragment 11.0 13.9Combined 85.7 139U. Nystrom, K. Gylltoft / International Journal of Impact Engineering 36 (2009) 9951005 1004deections for blast and fragment loading treated separately (intotal76.2 mm), indicating a synergy effect in combined loading.AcknowledgementThe work presentedinthispaperisdone withintheresearchproject Concrete structures subjected to blast and fragmentimpacts: dynamicbehaviour of reinforcedconcrete, nanciallysupportedbytheSwedishRescueServicesAgency. Theauthorswouldliketothankthemembersofthereferencegroupfortheproject: Bjo rnEkengren, M.Sc., at theSwedishRescueServicesAgency, MorganJohansson, Ph.D., at ReinertsenAB, andJoosefLeppa nen, Ph.D., at FB Engineering AB.Appendix ASince an ideal-plastic material behaviour was assumed for theinternal resistance of the SDOF system the equation of motion usedto describe the movement of the mid-point in a simply supportedbeam, with mass, M, and length, L, subjectedto a uniformlydistributed load, q(t), can be simplied to:23M u R qt,L (1)where u is the mid-point acceleration of the beam [6].When assuming an ideal-plastic material behaviour, the internalresistance, R, in Eq. (1) equals the maximum value of the load thatthe beam(or wall strip) can bear, i.e. R Rmgiven that thedisplacement u s0. Before any displacement occurs (u 0), if theexternal load is smaller than the maximum load-bearing capacity(P(t)