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: [email protected]
(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)