Fundamental concepts in blast resistanceevaluation of structures1G. Razaqpur, Waleed Mekky, and S. FooAbstract: This study criticallydiscusses the fundamentalconcepts used for evaluating the flexural and axial resistance ofstructures under blast. Simplified methods based on single degree of freedom are emphasized. The paper begins with howto estimatethe blast parameters for a given charge size and standoff distance. These parameters include side-on and re-flected pressures, positive phase duration, and side-on and reflectedimpulses. Subsequently, blast damage criteriaare de-fined in accordance with prevailing guidelines and some of their short comings are discussed. To assess the impact ofblast on the flexural safety and performance of structures, some simple methods are presented. The methods are either em-pirical or are based on the principles of energy and momentum conservation. The analyticalresults are in closed-form orin the form of pressureimpulse (PI) diagrams. The effect of strain rate on both blast-induced flexural deflection andstrength of structures, with particularemphasis on reinforced concrete structures, is discussed.Key words: blast, beam, column, concrete, pressure, impulse, strain rate.Resume: Cette etude analyse les concepts fondamentaux utilises pour evaluer la resistance en flexion et axiale des struc-tures soumises a`un souffle dune explosion. Laccentest mis sur des methodes simplifiees basees sur un seul degrede li-berte. Cet article debute en expliquant comment estimer les parame`tres du souffle pour une charge et une distance delexplosion donnees. Ces parame`tres comprennent les pressions laterales et reflechies, la duree de la phase positive ainsique les impulsions laterales et reflechies. Par la suite les crite`res de dommages causes par le souffle sont definis selon deslignes directricesapplicables et quelques inconvenients sont abordes. Certaines methodes simples sont presentees afindevaluer limpact du souffle sur la securiteet le comportement en flexion des structures. Les methodes sont soit empiri-ques ou basees sur les principes de conservation de lenergie et du moment. Les resultats sont sous forme analytique ousous forme de diagrammes pression-impulsion (PI). Leffet du taux de contraintes sur la flexion induite par le souffle etsur la resistance des structures est aborde, avec une attentionspeciale aux structures en beton arme.Mots-cles : souffle, poutre, colonne, beton, pression, impulsion, taux de contraintes.[Traduit par la Redaction]IntroductionRecent eventshavecreatedconcernabout thevulnerabil-ityofbuildingsandother structurestoblast loads. Theef-fect of blast on a building depends on the amount andlocationof theexplosivechargefromthebuildingandonthestrengthandgeometryof thebuildingstructure.Detona-tionof highexplosivematerials mayproducesevereover-pressures, primary and secondary fragments, fire, heat,groundshock, vibrations, etc. Theexplosionmayoccur in-sideoroutsideabuildingandmaycausedamagetostruc-tural andnonstructural elements, tothe buildingcontents,suchas equipment, services, andfacilities andmaycausehuman fatalities inside or outside the building. Althoughfromthesecuritypoint ofview, knowledgeandassessmentof each of these effects are important, the scope of thepresent study is limitedto the structure.Thepurposeof thisstudyistopresent relativelysimplemethods for assessing the effect of external blast on buildingstructures and their components. These methods are intendedfor individualswithanappreciationfor theirscopeandlim-itationsandknowledgeof buildingdesign andstructuralen-gineering. Thestudyfocusesprincipallyontheflexural andaxialbehaviourof structuralelementsand is concernedwithbendingfailureonly. Althoughinsomecasesother modesof failuremaybedominant, e.g., theones associatedwithshear, torsion, andbuckling, ormayberelatedtolocal fail-ure mechanisms such as spalling, breaching, debonding,etc., thesearenotaddressedinthispaper. Withthepreced-ingcaveat inmind, theresultscouldbeusedfor assessingReceived 11 November 2007. Revision accepted16 February 2009. Published on the NRC Research Press Web site at cjce.nrc.caon22 August 2009.G. Razaqpur2and W. Mekky. Centre for Effective Design of Structures, McMaster University, Department of Civil Engineering JHE-301, 1280 Main Street West, Hamilton, ON L8S 4L7, Canada.S. Foo. Public Works and Government Services, Gatineau, QC K1A 0S5, Canada.Written discussion of this article is welcomed and will be received by the Editor until 31 December 2009.1This article is one of a selection of papers published in the Special Issue on Blast Engineering.2Corresponding author (e-mail:razaqpu@mcmaster.ca).1292Can. J. Civ. Eng. 36: 12921304 (2009) doi:10.1139/L09-032 Published by NRC Research Presstheeffectsof member damageonthesafetyof theoverallstructure. Finally, although the emphasis is on reinforcedconcrete structures, the presentedmethods canbe adaptedfor other types of materials.Determining blast load characteristicsThereisawidevarietyof highexplosivesavailableandeach has its own destructive power, depending upon itsmass specific energy. Somecommonexplosives are TNT,RDX, ANFO, Dynamite, HMX, and PETN. We will useTNTasthereferenceexplosiveandall other chemical ex-plosives canbeconvertedtotheir TNTequivalent (Zukasand Walters 1997).Theexplosioncreatesaspherical shockwaveorfront (alayerofhighlycompressedair)associatedwithhightransi-ent pressurethat decaysrapidlyinthecaseofaveragesizecharges(i.e.,2000 kg or less).Thepeakvalueandthe posi-tivephasedurationoftheblastpressuretimecurvecharac-terizethepotential destructiveeffects of anexplosionandare of primary interest to structural designers. AtypicalpressuretimecurveisillustratedinFig. 1a. Inthisfigure,Psisthepeakoverpressure, whichistheamount inexcessof the atmospheric pressure andis alsoknownas side-onpressure. It decays tozerointimetdafter whichnegativepressure(suction)develops. Whentheshockfront strikesasurfaceat anangle, thepressurewaveisreflected, causingan increase in pressure onthe reflecting surface. The re-flectedpressure is denotedbyPr. The shockfront is fol-lowedbythe movingcompressedair, whichcreates blastwind anddynamicpressurePd, resultingin drag forces. Thedynamicpressureisrelatedtotheairparticlesvelocityandtheresultingwindpressureisrelativelysmall andisoftenignored.Althoughtheactual pressurevariationwithtimeisrela-tivelycomplex, it is commonpractice(Biggs1964; Bakeret al. 1983; U.S. Army 1990) to assume for the positivephaseof thepressurealinear pressuretimerelation, withzerorise time,as indicatedinFig. 1a. Inanalytical work,sometimeanexponential pressuretimevariationisalsoas-sumed, but in practiceit is common to assume a linear pres-sure pulse.For assessing blast damage potential,one must first deter-mine the magnitude of the peakside-onpressure, Ps, thepeakreflectedpressure, Pr, andthepositivephasedurationof the pressure, td. Thesequantitiescanbe calculatedby us-ing theoretical expressions as given by Biggs (1964), thecharts givenbyBaker et al. (1983) andTM5-1300(U.S.Army1990), specializedcomputer programs, suchasCon-Wep (1990), or computational fluid dynamics and blastphysics. Thelattertwoapproachesareoftenintheresearchrealmrather thancommonpractice. Figure1bfromTM5-1300(U.S.Army1990)showsatypical chart. Observethattheabscissaof thechart isthescaleddistanceZ, whichisdefined as1 Z RW1=3where Ris the standoff distance (m) and Wis the TNTequivalentchargeweight(kg).ThequantityZisreferredtoasthescaleddistance, whichreflectsthecombinedeffectsof chargeweight andstandoff distance. UsingFig. 1b, theside-onoverpressure Ps, the normal reflected pressure Pr,the scaled incident impulse Is/W1/3, the scaled normal re-flectedimpulseIr/W1/3, andthescaledpositivephasedura-tion td/W1/3of the blast can be determined. Equation [1]indicates that thevariationof blast pressureonabuildingwill depend onthe distance of the various points onthebuildingfromtheexplosionsource.ItisthescaleddistanceZthat determines theintensityof the pressureandscaledimpulse at a point rather than the charge size or the standoffdistancealone.Statistical analyses showthat the different approachesusedtoestimate blast parameters includingsoftware, suchas CONWEP, can predictthe reflectedpressure and impulserelativelywell. Figure 2acompares experimental reflectedpressurevaluesfortypicalscaleddistanceswiththeircorre-spondingpredictedvaluescalculatedbyanumberofavail-able software (Bogosian et al. 2002). While most of thepredictedvaluesarereasonable, CONWEPpredictsthere-sultsbetter thantheotheravailablesoftware. Togaugetherelative accuracyof CONWEPpredictions, Fig. 2bshowstypical measuredreflectedpressurevaluesasafunctionofscaledistance andthemeanandplusminus twostandarddeviation(twosigma)ofthepredictedvaluesofCONWEP.Itisclearthat CONWEPpredictspressurevaluesrelativelywell. Ahigherdegreeofaccuracymaybeunwarrantedbe-causeinrealityfor thesamescaleddistance, thereflectedpressure can vary significantly, depending upon the sur-roundinggroundfeatures includingthe presence of build-ings, trees, etc.Response of structures to blast loadsDepending on the time of maximum response of the struc-turetmandthepositivephasedurationtdoftheblast pres-sure, (or as anapproximate alternative depending on utd,where uis the natural circular frequencyof the structuremodeledas asingledegreeof freedom(SDOF) oscillator)thestructureisassumedtobesubjectedtooneofthethreeloadingregimes; namely, impulsive, dynamic, or pressure.The time of maximumresponse depends on whether thestructure behaves elasticallyor plasticallywhenmaximumresponse is reached. The time tomaximumresponse is afunctionoftheratioofthepositivephasedurationofpres-sure to the fundamental period of the structure (td/T), asgiven by Biggs (1964).For a response that is governed by impulse, utd 0.4, theresponsedoesnot dependoneither themaximumpressureor the shape of the pressuretime curve. Therefore, the mag-nitude of the pressure, which may be many times higherthananystaticpressurethat astructurecansafelyresist, isbasicallyirrelevant becausethestructuredoesnot havead-equate time todeformunder this transient pressure. Fromthe basic mechanics point of view, for the pressuretobefeltbythestructure,thestructuremusthavethetimetode-form under the applied load, and the work done by the pres-surewouldbeconvertedtostrainenergy.Butifthetimeofapplication of the loadis tooshort, incomparisontothefundamental periodofthestructure, thestructurewouldnothaveadequate time todeformandthe pressurecannot doanywork. Hence, inthiscasethestructurestrengthislessRazaqpur et al. 1293Published by NRC Research Pressimportant thanitsductilitytoresist theblast energy. Struc-turesthatrespondinthiswayhaverelativelylongvibrationperiod compared with the positive phase duration of theblast, whichmaybeafewmilliseconds. Ontheotherhand,structures withshort periodof vibrationsubjectedtolongduration pressure profiles, i.e., utd> 40, respond in thequasi-staticorpressureregime, whichmeansthattheymustbe able to resist the maximumblast pressure. Structures thatfallbetweenthesetwolimitsrespondtoadynamicloadre-gime, whichmeansthatboththemaximumpressureanditstime variationaffectthe response of the structure.Although responses due to impulsive and quasi-staticloads can be estimated by relatively simple procedures basedon conservation of energy and momentum, the determinationof response within the dynamic load regime requires full dy-namic analysis. For single degree of freedom systems(SDOF)it isnot difficult toperformfull dynamicanalysis,but for systems withmanydegrees of freedomit is moreFig. 1. (a) Typical high explosive blast pressure profile and (b) surface blast parameters as function of scaled distance (modified from U.S.Army 1990).1294 Can. J. Civ. Eng. Vol. 36, 2009Published by NRC Research Presscomplex, particularlywhenoneneedstoconsiderinelastic-ity and strain rateeffect.In practice,for preliminaryevalua-tionone couldapproximate the dynamic responsewithoutrecoursetofull dynamic analysis, as shownlater, but onemust be aware of the limitations of such methods.Determining the blast resistance of amemberThekeytoaccuratedeterminationoftheblast resistanceof structural members is the response of their constituentmaterialstothehighstrainratesimposedbyablastload. Itis generally known that materials exhibit significantly higherstrength than theirstaticstrengthwhen they aresubjectedtohigher strain rates. Bischoff and Perry (1991) introduced ap-proximaterangesofstrainratesfordifferent loadingcondi-tions. It was stated that ordinary static strain rate rangesfrom106to105s1,whileblastimposedstrainraterangesfrom102to 104s1. These ranges agree with the valuesspecified in the U.S. Defence Special Weapons Agency(DSWA) report (Malvar and Ross 1998).Properties of concrete under high strain rateAccordingtoFuet al. (1991), theearliest dynamictestson concrete in compression were conducted by Abrams(1917) to investigate the effect of strain rate on the compres-sivestrengthofplainconcrete. Themainconclusionofthispioneer workwasthat thehigher therateof straining, theFig. 2. (a) Measured and predicted reflectedpositive pressure. (b) Measured and predicted reflectedpositive pressure (modified from Bo-gosian et al. 2002).Razaqpur et al. 1295Published by NRC Research Presshigherthecompressivestrengthofconcrete. Figure3illus-tratesthestrainrateeffecton the stressstrain curve of con-crete incompression(Malvar andRoss 1998), where onecanobserveasignificant increase inbothits strengthandenergyabsorptioncapacitywithhigherstrainrate.Notethatthestrainratedoesnothaveanoticeableeffectontheelas-tic modulus, but the maximum or failure strain, Fig. 3, is al-most an order of magnitude larger than the typical staticfailurestrainof0.0035. BasedonFig. 3, forstrainratesof500s1or higher, it maybe reasonable touse at least astrainof0.02atfailure, avaluethatisalmostsixtimestheconventionalstatic failurestrain.Scott et al. (1982)conductedanextensiveseriesoftestsonconcrete samples subjectedtoconcentric andeccentricloadsat strainratesvaryingfrom3.3 106to0.0167s1.Test resultsindicatedanincreaseofabout 25%inboththestress andthe strain at failure due tothe increase inthestrain rate.Also, the shape of the stressstrain curve of con-fined concrete was found to be strongly affected by thechange inthe loadingrate. As a result, andas shownbyothers(Dilgeretal. 1984;Soroushianetal. 1986), boththesecant andthe rupture moduli significantlyincreasedwithincreasing loading rate.The U.S. Defence Special Weapons Agency (DSWA)sponsored a recent numerical study to investigate the re-sponseofreinforcedconcretestructuresundertheeffect ofhighstrainratesrangingfrom10to103s1, resultingfrominternal explosion. Complete results of the study were not re-leased for publication, but some of these results are discussedbyMalvarandRoss(1998), inwhichtheapparent strengthremarkably increased, by more than 50% for reinforcing steeland by more than100% and600% for concreteincompres-sion and tension, respectively. Consequently, the latter inves-tigatorsproposedthefollowingexpressionsfortheeffectofhigh strain rates on the tensile strength of concrete,2 DIF ftdfts_ 3_ 3s d; _ 31 s13 DIF ftdftsb_ 3_ 3s 13; _ 3> 1 s14 logb 6d 25 d 11 8 f0c=f0cowhereftdandftsarethedynamicandstatictensilestrengthsof concrete, respectively, DIF is the dynamic increase factorfortensilestrength, _ 3and_ 3sarethehighstrainrate(upto104s1)andstaticstrainrate(106*105s1)andf 0cois thefractionofthecompressivestrength, f 0c, ofconcrete( f 0ccanbe assumed 10 MPa).It was concluded that under strain rate greater than 200 s1,the dynamic increase factor could reach up to 6. The magni-tudeofdynamicincreaseisdependent uponseveral factorsincludingstaticstrengthofmaterialunderconsiderationandrate of applied load. In general, the higher the static strengthofamaterial thelowertheincreaseinitsdynamicstrength(lower strain rate sensitivity). Figures 4a and 4b illustrate therelationshipbetweenstrainrateandtheDIFforconcreteintension and compression, respectively. In the figures, experi-mental data collected by Bischoff and Perry (1991) frommanydifferentauthors arecomparedwith predictionsof theCEB(1985) expressions andthe Malvar andRoss (1998)modifiedform of that expression. In additionto the increasein the tensile and compressive strength of concrete, the straincorrespondingtothepeakstrengthintensionandcompres-sion and the concrete ultimate strain are believed to beshiftedtohighervalueswiththeincreaseinstrainrate. Te-desco and Ross (1998) discuss and suggest specific relationsthat can be used to quantifythe extentof the increasein thevalues of the above parameters.Properties of steelBecuaseof the isotropicproperties of steel,its elastic andinelastic response todynamic loadingcanbe more easilymonitored and assessed (Scott et al. 1982). Norris et al.(1959)testedtwotypesofsteelwithstaticyieldstrengthof330 and278MPa undertensionatstrainratesrangingfrom105to 0.1 s1. Strength increase of 9%*21%and10%*23%wasobservedforthetwosteel types, respec-tively. DowlingandHarding(1967) conductedtensile ex-periments using the tensile version of Split-HopkintonPressure Bar (SHPB) on mild steel using strain rates varyingbetween103and2000s1. It wasconcludedfromthistestseriesthat materialsofbody-centredcubic(BCC)structure,suchasmildsteel, showthegreatest strainratesensitivity,theirlowertensileyieldstrengthalmost doubled, theirulti-mate tensile stress increased by about 50%, their upper yieldtensile strength considerably increased, and the ultimateten-silestraindecreasedbydifferent percentagesdependingonthestrainrate. Figures5aand5billustratethestressstrainrelationships of mildsteel under the effect of strainratesvarying from 103to 1750 s1.Malvar(1998)alsostudiedstrengthenhancement ofsteelFig. 3. Strain rate effect on the stressstrain curve of concrete incompression (Yong and Xu 2004).1296 Can. J. Civ. Eng. Vol. 36, 2009Published by NRC Research Pressreinforcingbars under theeffect of highstrainrates. Thiswas again described in terms of the DIF, which can be eval-uated for different steel grades and for yield stresses fy,ranging from 290 to 710 MPa as6 DIF _ 3 104awhere for calculatingyield stressa =ay and7 ay 0:074 0:04fy=414For ultimatestrength calculations, a =au, and8 au 0:019 0:009fy=414For example, for fy=400MPa anda relatively lowerstrain rate of 100 s1, eq. [6] gives DIF = 1.63.Section responseToillustratetheinfluenceofthestrainrateonamemberresponse to fast dynamically applied axial load and (or)bendingmoment, let us consider atypical reinforcedcon-crete column and examine its response under differentstrainrates. Assumea500 500mmsquarecolumnwithrein-forcement ratioof 3.2%, f 0c=35MPa, andfy=400MPa.For the sake of simplicity, the effect of confinement isignored and the longitudinal steel is assumed to be uni-formly distributed around the column. Also, perfect bondbetween reinforcingbars and concreteis assumed.Momentcurvature relationThe response of a column under an axial load and a bend-ing moment can be conveniently represented by its momentcurvature diagram. Using first principles, based on straincompatibility and equilibriumrequirements, the momentcurvaturediagramsinFig. 6areconstructed. Eachdiagramisconstructedbyassumingacertainstrainprofile(extremefibers strain) anddeterminingthe stresses inconcrete andsteel, using their stressstrain relations, corresponding tothat profile. Next theforceresultant of thestresses, whicharegenerallyanaxial forceandamoment, aredetermined.Meanwhile, thecurvatureisgivenbytheslopeofthestrainprofile.This process yields one point on the momentcurva-ture diagram. By assuming anotherstrain profileand repeat-Fig. 4. Relationship between strain rate and the dynamic increasefactor (DIF) for concrete under (a) tension and (b) compression.Fig. 5. Stress-strain relationships of mild steel under the effect ofdifferent strain rates (Dowling and Harding 1967).Razaqpur et al. 1297Published by NRC Research Pressing the same process, another point is found and this processis continueduntil the complete range of straininthe ex-tremefibresiscovered. Intheconstructionofthediagramsin Fig. 6, stressstrain relationships similar to those inFigs. 3and5wereused. Figures6ato6cshowthetypicalbehaviour of acolumnunder aconstant axial loadbut in-creasingmoment. For thesakeof simplicity, thediagramsinFig. 6donot includethetension-stiffeningeffect, i.e., itisassumedthat crackedconcretecarries zerotension. Thestrainratevaluesareassumedtobeconstant overthecrosssection for any momentcurvature diagram, ranging from107to 1000 s1, where the slowest rate corresponds toquasi-staticconditions, and the moment is assumed to be in-ducedbyloadsappliedatdifferent strainrates. Theappliedaxial loadisassumedtovaryfrom5%to60%ofthestaticaxial load capacityof the column.Asexpected, themaximummoment capacityof thecol-umnincreases withstrainrate. For instance, for acolumnsubjectedto an axialload that is 40% of its static axial loadcapacity, its ultimate moment capacity increases almost250%and its energy absorptioncapacity350%whenthestrainrateincreases from1 107to1000s1. Similarly,itsductilityanddeformationcapacityalsoincreasesnotice-ably. Theincreaseindeformationcapacityis important tomentionbecausethelevel of blast damageinamember isoftendefinedinterms of its deformationcapacity, as wewill seeinthefollowingsections. Theresponseofthecol-umninFig. 6ais dominatedbytheaxial loadduetothelargemagnitudeoftheappliedaxial load, whilethat ofthecolumninFig.6disdominatedbythemomentbecausetheaxial load is small. In both cases, substantial increase inbothstrengthandenergyabsorptioncanbeobserved. It isalso important to observe that in each case under the 100 s1orhigher strainrate, thecolumnbehavesessentiallyelasti-callyuptoamoment equal toat least twiceitsstaticulti-mate moment capacity. It is noticed in Fig. 6d that thecrackingmoment of thecolumnsubjectedtostrainrateof100s1orhigherismuchgreaterthantheultimatestrengthof the same column under static conditions.Based on Fig. 6, the current practice of assuming the con-creteandsteel dynamicincreasefactor tobe1.25, assug-gestedbytheU.S. ArmyManual TM5-1300(U.S. Army1990), seemshighlyconservative. Thislevel of increaseisachievedunder strainrateof 1.0s1, whichis well belowthestrain-raterangespecifiedforblast (BischoffandPerry1991).Thisanalysismayindicatesubstantialincreaseinre-Fig. 6. Momentcurvature response of a typical column subjected to an axial load and a dynamic moment applied at four different strainrates: (a) quasi-static, (b) 1 s1, (c) 10 s1, and (d) 1000 s1).1298 Can. J. Civ. Eng. Vol. 36, 2009Published by NRC Research Presssistance of a member under blast loads over its staticstrength, and this may diminish the need for retrofit insome cases.Interaction diagramInteraction diagrams are commonly used to assess thestrength of a column under combinationsof an appliedaxialload and a moment. For the square column described earlier,consideringthe effect of strainrate onconcrete andsteelpropertiesand using first principles,interactiondiagrams,asillustratedinFig. 7a, are constructed. Incompliance withcurrent practicefor staticloads, when constructing such dia-grams, once thesectionis cracked, the tensile strengthofconcrete is neglected. However, recalling the high strainratesensitivityofconcreteintension,itmaybeappropriateto consider the contribution of tensile stresses in the un-crackedtensile zoneinthe caseof veryhighstrainrates.Ignoringconfinement effect, thepureaxial compressionca-pacityof acolumndependsonitsreinforcement ratioandyield stress and on the compressive strength of concrete.Under highstrainrates, theyieldstressandconcretecom-pressive strength increase, as discussed earlier, causing a tre-mendous increaseinthecompressiveaxial capacityof thecolumn. In the case of the column tensile capacity, it ismainlyafunctionof thereinforcement yieldstress, whichhas a limiting increase factor of only two. The balanced fail-ure point is characterized by initial yielding inreinforce-ment. As the yield stress increases with strain rate, thecorresponding yielding strain also increases, allowing themaximummoment capacityof thecolumntoincreaseandcausing the failure envelop to almost triple in size. It is clearthat substantial increaseinstrainrateincreasesthecolumncapacityseveralfolds. Consequently,theprevailingpracticeof increasingconcreteandsteel strengths by25%maybegrosslyconservativedependingonthestrainrate. Newandappropriate methods of design need to be developed thatguaranteestructural safetywithinthesamelimits as thoseunderothertypesofextremeloads, suchashighwindandseismic loads.Let us studythe variationof the balancedmoment andbalancedaxial loadwithstrainratefor theprecedingcol-umn. Figure 7b shows the variationof the balancedmomentandbalancedaxialloadwithstrainratenormalizedbytheirquasistaticvalues. Aconsistent increaseinthesequantitiesis noticed with the normalized axial load and moment reach-ingvaluesof2.4and2.67atstrainrateof1000s1. Chartsofthiskindcanbedevelopedforarangeofcolumnsizes,reinforcement ratios, andgeometriessimilartoaxial loadmomentinteractiondiagramsgiveninexistingdesignhand-books (CAC 2006). Once chartsareavailable, designers canestimatethestrainrateandthenfindthedynamicincreasefactor for the moment and axial load to more realistically as-sess the strength of beams and columns.Blast damage classification and assessmentAkeystepinthedamagevulnerabilityassessment of abuildingisestablishment ofdamagelimit states. Toreducethe risk of an event to an acceptable level, it is useful to de-fine performance-based damage criteria, or limit states,whichcorrespondtocertainlevelsofprotection. Inthecaseof a building, the followingmaybe some of the damagelimit states: protection of building occupants fromsevere injury ordeath; avoidanceof damagetoimportant equipment andfacil-ities within the building; avoidanceof damagetoother buildingcontents, for ex-ample, documents, furniture, etc.; avoidance of damage tononstructural elements, for ex-ample doors, walls, and partitions; avoidanceof damage to structural elements; avoidanceof building collapse;and avoidanceofharmtopeopleanddamagetootherstruc-tures in the vicinity of the target structure.Itisclearthattheabovelimit statescanbeavoidedbyanumber of measures, without necessarily strengthening or al-teringthebuildingstructure. Themost effectivewayofre-ducingblast damagetostructureistoincreasethestandoffdistance. Theoretically, a1kgbombat 1mfromatargetyields the same peak side-on overpressure as a 1000 kgFig. 7. (a) Interaction diagram for a typical column under differentstrain rates and (b) balanced moment and axial load variationin acolumn with strain rate (normalizedto balanced moment and axialload staticallyapplied).Razaqpur et al. 1299Published by NRC Research Pressbombat 10mfromthesametarget. This clearlydemon-strates the advantage of increased standoff distance. As aquickandhighlyconservativeguideadaptedfromtheU.S.ArmyManual TM5-1300(U.S. Army1990), Table1pro-vides someguidelinesfor the levelof protectionthatcanbeaffordedbyconventionallydesignedbuildings against cer-tain chargesizes at specifiedstandoff distances. The variousprotectioncategoriesin Table 1 are defined as Nrepresentsnoprotection:possiblecollapseofbuilding,majordamagetoequipment andfacilities, anddeathandsevere injuries; Lrepresentslowlevel of protection: moderatetoheavydamagetobuilding, 50%to75%ofthewallsbeingse-verelydamaged,occupants of exposed structuremay suf-fer temporary hearing loss andinjuryfromblast waveand flying debris, and equipmentwill get damaged; Mrepresents mediumlevel of protection: buildingwillsustainlight tomoderate damage, includingdestructionof theroofand loss ofan externalwall,occupantsof ex-posed structure may suffer minor injuries fromflyingdebris, andequipment maysuffer light damage due toflying debris; H represents high level of protection: building may suffercosmeticdamage, includingbrokenglass, cracking, anddamage to building face and damaged partition walls,brokenjoistsandstuds, occupantsmaysuffersuperficialinjuries, andequipment maybescratchedanddentedbydebris. Bakeret al. (1983)assumethat theseprotectioncategories correspondtocomplete collapse, partial col-lapse, major structural damage, andminor structural da-mage scenarios, respectively.While as a quick guide Table 1 may be useful, it is highlyempiricalin nature becauseitassumes thatallbuildings, re-gardless of theirform and materialsof construction,providethe same level of protection. Infact, that assumptionhasbeenmadetoestablishdamagecriteriaandprovidesimpledesignguidelines. Morerecently, this deficiencyhas beenaddressedinsomepublications (TaskCommittee onBlastResistant Design 1997), where different response criteriaare given for several reinforced concrete, reinforced ma-sonry, and steel structures. Nevertheless, as most of thestated criteria are based on static or lowspeed dynamictests, theyneedtobe verifiedbyfieldor laboratorytestsunderblast conditions. Formoreimprovedassessment, onemustperformproperstructural analysisandevaluationwithdueconsiderationof the dynamic nature of the blast loadand the response of structures to high strain rate.Because of the ductile response of steel andreinforcedconcrete members, damage to them can be assessed in termsoftheirdeformations. Thedeformationcriterionmaybeex-pressedeither interms of maximumdeflection, maximumsupportrotation,maximumstrain, ormaximumductilityra-tio. Thesequantitiesareinterrelatedandinmanycasesonecanderive closed-formexpressions torelate themtoeachother.Thedamageinsteel memberscanbeclassified, asgiveninTable2(U.S. Army1990), andisexpressedintermsofsupportrotation, q,ordisplacement ratio, m,wheremistheratio of the maximumdisplacementexperienced to the max-imumelastic displacement. The table lists commonstruc-tural members and their governing damage criteria. Theprotectioncategories1to4correspondtohighertonopro-tectionlimits as defined earlier.Thesecategories wereoriginallydefinedfor militaryfa-cilities, but theywereinterpretedbythewriterstosuit thepurposeof thisstudy. Whitneyet al. (1989) giveother setof criteriafor steel beams. Todetermine mfor amember,its momentcurvature relation can be established in thesamemanner asfor staticloads, but onemust includetheeffect of strainrate onthe yieldandultimate strengthofsteel, as discussed earlier. For reinforced concrete structures,damage criteria are defined in the document TM5-1300(U.S. Army1990) intermsof themember endrotation, q,similarlytothesteel members. For minor damage q 48.Notice that reinforced concrete members that are designedaccording to conventional design methods, e.g., based onCSA standard A23.3 (CSA 2004), are assumed to reach gen-erallythelimit of their utilityandwill collapseif q>48.However, greater rotations can be achieved in structureswith special reinforcement (laced diagonal reinforcement)designedspecificallytoresist blast loads or structures de-signed against progressive collapse. Continuous memberswith adequate and properly detailed reinforcement in thepositive and negative moment regions may undergo substan-tial deformations andresist largeloads throughmembraneaction (Nielsen 1984).Anumberofissuesrelatedtotheblastresistanceofrein-forcedconcretemembersarenotclearatthisjuncture. Thisincludesthemethodofcalculationofthememberendrota-tions, which depend on the length of the plastic hinge,bond-sliprelationof thereinforcingbarsunder highstrainrates, tension stiffening, and the momentcurvature responseof the member. Unfortunately, none of the above recom-mendations explicitlystate howthe structure deformationsmust be calculated. As we noticed in Fig. 6, strain rategreatlyaffects boththestrengthandenergyabsorptionca-pacity of a member. Furthermore, as in Fig. 6a and 6bTable 1. Effect of charge size and standoff distance on level of protection attainableinconventionallyconstructed buildings.Charge size(kg) Standoff distance (m)25 028 (N) 2835 (L) 3547 (M) 4779 (H)100 045 (N) 4555 (L) 4573 (M) 7397 (H)25 058 (N) 5873 (L) 7397 (M) 97128 (H)450 0 73 (N) 7391 (L) 91122 (M) 122152 (H)Note: N, no protection; L, low level of protection; M, medium level of protection; H, high level ofprotection.1300 Can. J. Civ. Eng. Vol. 36, 2009Published by NRC Research Pressstrainratecanalsosignificantlyaffect theultimate curva-ture of a member. Since bothmember joint rotations andmaximumdeflectionarefunctionsofitsmomentcurvatureresponse, it isimportant toestablishproper proceduresforcalculatingtheactual deformationexperiencedbyamem-ber.Thereislittleexperimental datacurrentlyavailableintheopenliteraturetoverifythevalidityof thecurrent recom-mendationssuchasthoseinTM5-1300(U.S.Army1990).Theoretical and empirical analyses, similar to those illus-tratedinFigs. 7and8, respectively, indicatethat structuresunder blast can resist, depending on the strain rate, veryhigh forces and can undergo large deformations withoutcompleteloss of strength and sudden collapse.Blast damage evaluation methodsSeveral methods exist for quantitativelyevaluatingblastdamage. In addition to refined methods of dynamic analysis,suchasthoseinLS-DYNA(LivermoreSoftwareTechnol-ogy Corp. 2007), which are not discussed here, blast damagecanbeevaluatedusingempiricalmethods,simpleanalyticalmethods basedonenergyandmomentumconservation, orresponse spectrumanalysis of single degree of freedom(SDOF) systems. Alternatively, the so-called pressureim-pulse(PI)diagramscanbeused, whicharebasedoncon-servationof energyandmomentum. Wewill discusssomeofthesimpleapproachesinthefollowingsections. Fortheresponsespectrummethod, referencecanbemadetoBiggs(1964).Evaluating damage using pressureimpulse diagramsThelevel ofdamagetolow-risewood, unreinforcedma-sonry, and light industrial structures can be estimated bymeans of empirical pressure impulse (PI) diagrams, asgiveninFig.8,plottedbyBakeretal.(1983)basedonJar-rett(1968)damagecriterion. EachPIcurveisthelocusofpressureandimpulse combinations that produce adefinedlevel of damage ina member. Thus, PI curves are alsocalledisodamagecurves. TouseaPI diagram, onemustfirst determinethepressureandimpulseforagivenchargesize and standoff distance, e.g., using charts similar toFig. 1b. Next, one must plotthe calculatedpressure and im-pulse in Fig. 8 and determinethe level of damage caused bythe particularcombinationof pressure and impulse based onthepositionoftheplottedpointwithrespecttotheisodam-age curves. The PI diagrams can be theoretically con-structedfor varioustypesof members and for various limitsstates as described later.Since Jarretts methodis empirical andis derivedfrombombdamagetolow-risewoodandmasonryconstruction,it may not be suitable for application to modern ductile steeland reinforced concrete structures. In the case of these struc-tures, the basic principles of structural dynamics can beused,inconjunctionwiththeirforcedeformationbehaviourtoestimatetheirmaximumresponse. Tworelativelysimpleanalytical methodsareavailablefor thispurpose. Baker etal. (1983)statethat theestimatedresponsesbasedonthesemethodsarereasonablyaccurateandareinfair agreementwith experimental data from actualblast tests.Beforewedescribethesemethods,letusfirstgenericallyTable 2. Deformation criteriafor steel structures.Maximum deformation(whichever governs)Member typeHighest level ofprotection* Additional consideration q(8)Ductilityratiom = Xm/XyBeams, spandrels, 1 None 2 10girts or purlins 2 None 12 20Frame structures 1 None 2{(H/25)/Xy{Cold-formed steel floorand wall panels1 Without tensile membrane action 1.25 1.75With tensile membrane action 4 6Open-web joists 1 Not controlled by maximum end reaction 2 4Controlled by maximum end reaction 1 1Plates 1 None 2 102 None 12 20*Protection categories as defined in this section.{Applies to members only.{Applies to structure with height H.q is the maximum member end rotation measured from chord joining the member endsFig. 8. Empiricalpressureimpulse (PI) diagram (modified fromBaker et al. 1983).Razaqpur et al. 1301Published by NRC Research Pressdescribe the forcedeformation relationship of an elasticplastic member. Let Rm, K and Xy denote the ultimate resist-ance, elasticstiffness, andtheelasticdeformationlimit ofthemember. ThedeformationXymaycorrespondtothede-formationat yield. Theresistancemaybeintermsofaxialresistance, Nr, flexural resistance, Mr, etc. withthe corre-spondingdeformations beingaxial elongation/ shortening,curvature, etc. Inpreliminaryevaluation, wenormallydealwith axial and flexural deformations. To check possibleshear failure, one needs to calculate the maximum shear cor-respondingtothemaximummoment andthencomparethecalculatedvaluewiththeshearcapacityofthemember. Forsteel structures, lateral-torsional interaction and bucklingmay also be important.Basedonfundamental mechanics, themaximumdeflec-tionofthestructureXmaxdependsonthestructuremassM,stiffness K, ultimate resistance Rm, durationtd, andmaxi-mumvalueoftheappliedpressureload, Fm. Let Xcbethecharacteristicdeflectionassociatedwithaspecifieddamagelevel orlimit state. ThedeflectionXccanberelated(Bakeret al. 1983) to two dimensionless quantities given by the fol-lowing;9 p FmXcK10 i IXcM=Kpwhere i is a normalizedimpulseandI is the actualimpulse.Thus, thePI diagramcanbeplottedinaCartesianplanewitheq. [9] as the abscissa andeq. [10] as the ordinate.TheimpulseI for any pulse shape can be determined by in-tegrating the area under the pressuretime curve.ThePIdiagramscanbeconstructedforasingledegreeof freedomspring-masssystemfor specifieddisplacementscorresponding to each limit state. As seen in Fig. 8, thecompleterangeofpressureandimpulsethatastructurecanbe subjected to is covered by the PI plane.To constructaPI diagram,givenapulseshape, we startbyassumingaveryhighvalueforimpulsetoapproximatetheasymptoticorlowest pressurecorrespondingtothisim-pulsethat wouldcausefailure. Usingtheassumedimpulseandthemaximumpressure, theSDOFsystemis analyzeddynamically to calculateits maximum deflection. The calcu-lateddeflectionis comparedwiththe specifieddeflection,andifitisfoundtobedifferent, thepressureisrevisedandtheanalysisisrepeateduntilthecalculateddeflectionisap-proximatelyequal tothespecifieddeflection. Thecombina-tionofpressureandimpulsethatproduceadeflectionequaltothespecifieddeflectionconstituteonepoint onthePIdiagram. This process is continued by assuming anothervalue of I andrepeating the preceding procedure. Inthismanner, the completePI diagram is established.Since the response of the SDOF spring-mass system is de-pendentonthestiffnessandstrengthofthespring, itisim-portanttoclarifyhowthesequantitiesarecalculated. Inthecase of columns or beams, one could establish the axialloaddeformation or momentcurvature response of themember, asillustratedinFig. 6. AccordingtoFig. 6, boththestiffnessandthestrengthof themember arefunctionsof strainrate, thereforetheP-I diagrams arealsofunctionofthestrainrate. Consequently, unlessthestrainrateises-tablishedfor a specifieddeflection, one cannot produce aunique PI diagram for a structuralmember.Using the aforementioned procedure, the PI diagram cor-responding to different levels of damage are plotted inFig. 9abyassumingaconstant strainrate. It canbeseenthat theblast resistanceof amember is greatlydependentonitsductility. EachPIdiagraminFig.9acorrespondstoa certainductility ratio.Theeffectof ductilityon the abilityof amember toresist bothhigher pressureandimpulseisclear.To investigatethe effectof strain rateon the PI diagramof amember,Fig. 9b illustratesthePI diagram of aSDOFsystem corresponding to different amounts of specified max-imumdisplacement andtwostrainrates. Itcanbeobservedin this figure, that if the member is to remain elastic, even a10-fold increase in the strain rate does not have a significanteffecton its blast resistance. On the other hand, if the mem-ber isallowedtoundergoplasticdeformation, e.g., Xmax=4Xy or whenm = 4, the increase of strain rate nearly doublesthe resistance of the member in the pressure regime. Ofcourse, higherstrainratethanthoseinFig. 9aareencoun-teredunder blast loads, andtheeffect ofstrainrateontheFig. 9. Effect of (a) ductilityand (b) strain rate on a pressure-im-pulse (PI) diagram.1302 Can. J. Civ. Eng. Vol. 36, 2009Published by NRC Research PressPIdiagramathigherstrainrateswouldbeevenmoredra-matic. Tolba (2001) measured average strain rates in thesteel reinforcement inconcreteslabs, whichvariedfromaslow as 2 s1to as high as 120 s1.Evaluating damage using analytical expressionsFor elasticplasticsystems, wecanusethelawsof con-servationofenergyandmomentumtodeterminearelation-shipbetweentheenergyabsorbedbythestructureandtheenergyor impulseimpartedtoit bytheblast load. Inthiscase, full dynamicanalysis is avoidedandtheresponseisapproximatedasbeingeither intheimpulseorquasi staticregime. Usingtheseconcepts, wecanshowthat for aonedegreeof freedomsystemthefollowingrelationships hold(Biggs 1964).For amember withamuchsmaller fundamental periodthanthe positivephasedurationof the pressure utd4andwithaspecifieddisplacement ratiom, itsrequiredulti-mate capacityRr is11 Rr Fm2m2m 1 Alternatively, for a member with available capacity Ra, itsrequired maximumdisplacementratiomr is given by12 mr 121 Fm=Rawhereineq.[11]Fmisthemaximumorpeakappliedblastload, misthespecifieddisplacementratio,andmris there-quireddisplacement ratio. Note that mcannot exceed theductility ratio of a member.It is evident fromthe above equations that Ramust begreater thanFm, regardlessof theductilityratio, otherwisethe member willfailimmediatelyupon application of Fm. Ifthe member is to remain elastic under the blast load, its resist-ance must be equal or greater than twice the maximum blastforce. Clearly, amember remainingelasticwill not experi-ence appreciable damage. Alternatively, eq. [11] indicatesthat amemberwiththeoreticallyinfiniteductilitymust stillhave a resistance exceeding the maximum applied blast load,if collapse is tobe prevented. For preliminaryevaluation,under blast loads, a 25%increase in steel and concretestrengths may be conservatively assumed, but, as pointed ear-lier, the member strength may be significantly higher becauseof the actual strain rate. According to eq. [11], for maximumdisplacementratioof 5, the required resistanceis 1.11 timesthe maximum applied load. Equations [11] and [12] apply tomembers with elasticplastic behaviour that are responding inpressure regime. Equation [11] is useful for design purposes,while eq. [12] is suitable for analysis or evaluation purposes.For elasticplasticmembers with relatively long period ofvibrationthanthepositivephasedurationoftheblast pres-sure, (utd>40), whichrespondintheimpulseregime, therequiredstrengthandductilityare givenbyeqs. [13] and[14], respectively.13 Rr Iu2m 1p14 m 12I2u2R2a 1 Intheseequations uisthenatural circular frequencyofthe structure and I is the total impulse impartedto the struc-ture. Once again, eq. [14] is appropriate for evaluation,while eq. [13] is suitable for design. Observe that inthiscasenotonlytheimpulse, butalsothestructurenaturalfre-quencyandultimatestrengthaffect themaximumdeforma-tionofthestructure. Forapurelyelasticresponse, i.e., m=1, the required ultimate strength of the member must equal Iu, while for a more ductile response, that is, m>1thestrength can be less than Iu. For instance, for available duc-tilityratioof 5, therequiredresistancewouldbe0.33I u.Thelatter twoequationsshowthat it isnot themagnitudeofthemaximumpressurebut rathertheimpulsethat deter-mines the potential severityof the damage ina structure.Hence, insuchcases,blastpressuresseveralordersofmag-nitude larger than the static pressure capacity can be resistedby a structure without experiencingfailure.Summary and conclusionsBasedontheresultsof thisstudy, thefollowingconclu-sions can be reached:(1) The blast loadparameters for agivencharge size andstandoff distance can be calculatedwith a reasonable de-gree of accuracy using existing methods and design aids.(2) Thestructureresponsetoablast event cannot bedeter-mined accurately using single degree of freedom models.The structure response is a function of a number of inter-acting parameters, includingthe stiffness and ductilityofthestructureandblastwaveparameters. Sincestructuralnonlinearitiesandstrainratecanalterthestructurestiff-ness, strength, andductilitydramatically, theresults ofsingle degree of freedommodels that donot considerthese parametershave to be used cautiously.(3) Member strength and energy absorption capacity in-creases several foldunder veryhighstrainrates com-paredwithits correspondingstaticstrengthandenergyabsorption capacity.(4) Currently,nosimplemethod isavailabletoestablishthestrain rates experienced by a structure during a blastevent. Thedevelopment ofsuchamethodwill improvegreatly existing blast evaluationmethods.(5) ThePI diagramisaconvenient tool for assessingtheblast resistanceof a tool corresponding to specifiedlimitstates of damage. Suchdiagrams canbedevelopedfordifferent structural members made of concrete, steel,etc., but realistic diagrams must includethestrainrateeffect.(6) Thereisneedfordetailedexperimental datatovalidatebothcurrent simplifiedandadvancedmethods of blastanalysis. 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