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Computers and Structures 84 (2006) 1729–1742
Characterization of structural effects from above-groundexplosion using coupled numerical simulation
Yong Lu a,*, Zhongqi Wang b
a School of Civil and Environmental Engineering, Nanyang Technological University, Nanyang 639798, Singaporeb The State Key Laboratory of Explosion Science and Technology, Beijing 100081, China
Received 29 August 2005; accepted 22 May 2006Available online 28 August 2006
Abstract
The response of a building structure to a nearby explosion is complicated by the drastic spatial and time variation of the blast load.Existing studies on the structural responses to explosion effects often employ simplified structural model with assumed loading patterns,such as element-based (beam-column, slab) models, single degree of freedom or lumped mass systems. The validity of a simplifiedapproach depends on whether the governing response and failure mechanisms are well represented in the simplified scheme. For suchvalidation more sophisticated models are required. This paper presents a numerical simulation study aiming to characterize the variousstructural effects of above-ground explosions. A coupled numerical approach with combined Lagrangian and Eulerian methods isadopted to allow for the incorporation of the essential processes, namely the explosion, shock wave propagation, shock wave-structureinteraction and structural response, in the same model. The computational domain extends to the soil around the base of the structure,allowing also for an evaluation of the significance of the ground vibration effect. Results show that for a typical above-ground explosionscenario, the critical structural damage is dominated by air shock loading, while the ground shock induces only some additional vibrationwhose structural effect is minor. The distribution of structural damage tends to be governed by member level effects on the front face ofthe structure, whereas the global dynamic response of the system appears to be insignificant. Similar modeling approach may be appliedto explore other blast-induced complex response phenomena.� 2006 Elsevier Ltd. All rights reserved.
Keywords: Above-ground explosion; Structural effects; Numerical simulation; Coupled model; Air shock wave; Ground vibration
1. Introduction
The potential threat to building structures of explosivedetonations is a subject of extensive studies worldwide inrecent years. A major portion of the study is devoted tothe structural effects under such extreme loading conditionsfor concerns over the damage and potential collapse of thestructural system. As a matter of fact, the response ofbuilding structures to nearby explosions is complicatedby the drastic spatial and time variation of the blastloads. Typical analysis of the structural responses to theexplosions often employs simplified structural model with
0045-7949/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruc.2006.05.002
* Corresponding author.E-mail address: [email protected] (Y. Lu).
assumed loading patterns. The simplification of the struc-tural model is usually based on the understanding of theprimary dynamic phenomenon that may govern the criticalresponse concerned. Typical choices of such models includeelement (beam-column or slab) based dynamic models [1–4], single degree of freedom systems [5,6], and lumped massmodels [7]. Different models address the problem fromdifferent angles, and their appropriate use requires a com-prehensive understanding of the general response charac-teristics, for which more sophisticated models withrealistic structural and loading representation are neces-sary. Numerical simulation is often called upon for suchpurposes.
The commonly used numerical simulation method forthe response of structures subjected to explosion adopts
1730 Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742
the so-called decoupling approach, in which the structure isusually modeled using finite element method, while theloading is simplified as a time history of pressure or veloc-ity to be applied at selected points on the structure. Theloading curves are obtained from separate numerical simu-lation of the explosion in air or using empirical formulae(e.g., [8–10]). A typical loading function is a triangle shape.Although such simplification on the loading may be appro-priate for certain engineering applications, many importantphenomena are ignored. The interaction between the shockwave and the structure, particularly for structure with com-plex geometry, the ground reflection, as well as the blast-induced ground vibration effect, are difficult to be modeledin a realistic manner with an uncoupled model [9,11].
The present study adopts the fully coupled numericalsimulation approach. The computational domain encom-passes the structure, the surrounding air, the charge deto-nation, as well as a sufficient layer of soil medium so thatthe explosion-induced ground vibration effect can also beincluded for evaluation. The main objective is to character-ize the structural effects for above-ground explosions, withfocus on the relative importance of localized damage andthe global dynamic response on the overall behaviour ofa structural system, as well as the significance of the groundvibration in the entire process. The models for the constit-uent materials are carefully selected and validated, and anewly developed three-phase soil model for high dynamicloading is employed. A generic multi-storey reinforced con-crete frame is used as an example to illustrate the character-istics of the structural effects and the dominant mechanismsgoverning the damage to the structural system in a typicalabove-ground explosion scenario.
2. Description of the coupled model and computational
domain
2.1. Overview of coupled Eulerian–Lagrangian numerical
approach
Due to computational constraints, much of the existingworks on the numerical analysis of structures under blastloading are based on either the ‘‘uncoupled method’’ orthe ‘‘partial coupled method’’. In the ‘‘uncoupled method’’the main physical process is divided into several consecu-tive phases, and the output of one phase is used as the inputof the next phase. For example, in the analysis of a struc-ture under explosion in air, the problem may be dividedinto three phases: (1) the detonation of charge and forma-tion of air shock wave; (2) the propagation of air shockwave, including the ground reflection or other boundaryeffects; and (3) the response of structure under air shockwave. In the ‘‘partial coupled method’’, the above threephases are reduced into two phases, combining either thefirst two or the last two phases. In these methods a funda-mental question is the adequacy of defining the loads onthe structures. Usually the time histories of stress or veloc-ities are applied on the structure as boundary conditions in
the ‘‘uncoupled method’’ or on the computational bound-ary in the ‘‘partial coupled method’’. Whereas it may beconsidered appropriate to define the blast load for rela-tively simple and symmetric situations, it becomes ques-tionable in cases where the shape of structure is notregular or when other influencing factors may be signifi-cantly involved [9,11], such as the ground reflection, thefree surface effect, and so on. In these situations a fullcoupled approach including the explosion source is desired.
To incorporate the various physical processes into asingle model, many complex phenomena should be consid-ered, such as large deformation, the fluid-structure interac-tion, etc. This is beyond the capacity of common structuralanalysis codes because these codes usually do not includethe consideration of energy conservation. The hydrocodes(or ‘‘wave codes’’) are suited for such purpose.
In typical hydrocodes there are two primary ways ofdescribing the continuum media; one is the Euleriandescription, the other is the Lagrangian description. Inthe Lagrangian method, the numerical mesh moves anddeforms with the physical material. Fig. 1a depicts themovement of the materials and the mesh at two instants.In this method, free surfaces and material interfaces arelocated at element boundaries and hence can be maintainedthroughout the calculation. As the materials are containedin their original cells, history dependent material propertiescan be well described. Thus, it is well suited for solid mate-rials. On the other hand, in the Eulerian method thenumerical mesh is fixed in space and the physical materialflows through the mesh. Fig. 1b depicts the movement ofthe materials and the mesh at two instants. As the meshis fixed, there is no mesh distortion problem when largedeformations or flow occur. The material motions aretracked by numerical techniques, and extra computationalwork may be required in order to maintain interfaces andlimit numerical diffusion. The Eulerian method is typicallyused for describing fluids and gases.
A combination of the advantages of both descriptionswill facilitate the modeling of a problem involving fluid(air)-structure interactions, through appropriate couplingof the two methods. This has been made possible in varioushydrocodes, e.g., LS-DYNA [12], AUTODYN [13], amongothers. Several algorithms for coupling the Eulerian andLagrangian descriptions have been put forward, such asthe arbitrary Lagrangian–Eulerian method (ALE) and thecoupled Lagrangian–Eulerian method [14]. The ALEmethod usually involves a complicated rezoning technique.Some recent studies on fluid-structure interactions also usedan ALE formulation that does not require remeshing[15,16]. In the present study, the coupled Lagrangian–Eule-rian method is adopted. In explicit hydrocodes such asAUTODYN [13] used in the present study, this coupling isachieved by a strategy such that, a Lagrangian interfacecan ‘‘cut’’ through the fixed Eulerian mesh in an arbitrarymanner, whereas the Eulerian cells interacted by theLagrangian interface defines a stress (pressure) profile forthe Lagrangian boundary vertices. In return, the Lagrangian
Material 1 Material 2
Interface t = 0 t = t1
Material 1 Material 2Interface
t = 0 t = t1
Lagrangian mesh
P V
V
P
V
Eulerian mesh P
V
V = velocity boundary conduction
P = pressure boundary condition
Fig. 1. Schematic of Lagrangian/Eulerian methods and a coupling approach. (a) Lagrangian method; (b) Eulerian method; (c) Coupling Lagrangian–Eulerian meshes.
R
D
H
Air
Soil L
Charge (W)
Eulerian method for explosion products and air
Lagrangian method for structure and soil
Fig. 2. Configuration of a generic scenario and coupled numerical model.(a) Configuration of a generic scenario. (b) Coupled model for numericalsimulation.
Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742 1731
interface defines a geometric constraints (velocity boundaryconditions) to the flow of material in the Eulerian mesh.Fig. 1c shows schematically a typical coupled situation.
2.2. General model setting
Fig. 2a shows a general configuration of a buildingstructure subjected to an above-ground explosion, whereR is the standoff distance of the charge from the front faceof the building, D is the detonation height above theground surface, and H, L are the height and width of thestructure, respectively.
In the numerical model as shown in Fig. 2b, the coupledLagrangian and Eulerian approach is adopted. The explo-sion products and the air are modeled with Eulerian mesh,while the structure and the soil are modeled with Lagrang-ian mesh. The coupled Eulerian–Lagrangian algorithm isemployed to facilitate the interaction between the air shockwave and the structure or the ground. Along the truncatedboundary for the entire computational domain, the trans-mission boundary conditions [13] are applied to allow forfree passage of shock/stress waves.
2.3. Material model for concrete
The response of the concrete under shock loading is acomplex non-linear and rate-dependent process. A varietyof constitutive models for the dynamic and static response
Uniaxial Compression
Failure Surface
Elastic Limit Surface
Residual Surface
Uniaxial Tension
P
Tensile Elastic Strength
Compressive Elastic Strength
fc
ft
Y
Fig. 3. Concrete strength model.
1732 Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742
of concrete have been proposed over the years. The RHT(Riedel–Thoma–Hiermaier) model is adopted in the pres-ent study [17]. The RHT model contains many pertinentfeatures for describing a brittle material, namely pressurehardening, strain hardening, strain rate hardening, thirdinvariant dependence for compressive and tensile meridi-ans, and cumulative damage (strain softening). It can beused in conjunction with the existing tensile crack softeningalgorithm.
As shown in Fig. 3, the material model uses threestrength surfaces: an elastic limit surface, a failure surfaceand a remaining strength surface for the crushed material.The failure surface Y is defined as a function of pressure p,the lode angle h and strain rate _e,
Y fail ¼ Y TXCðpÞ � R3ðhÞ � F RATEð_eÞ ð1Þwhere Y TXC ¼ fc½Aðp� � p�spallF RATEð_eÞÞN �, with fc being thecompressive strength, A the failure surface constant, N
the failure surface exponent, p* the pressure normalizedby fc, p�spall ¼ p�ðft=fcÞ. F RATEð_eÞ is the strain rate function.R3(h) defines the third invariant dependency of the modelas a function of the second and third stress invariantsand a meridian ratio Q2.
The elastic limit surface is scaled from the failure surface,
Y elastic ¼ Y fail � F elastic � F CAPðpÞ ð2Þwhere Felastic is the ratio of the elastic strength to failurestrength, FCAP(p) is a function that limits the elastic devia-toric stresses under hydrostatic compression, and it variesin the range of (0,1) for pressure between initial compac-tion pressure and the solid compaction pressure. Linearhardening is used prior to the peak load. During harden-ing, the current yield surface (Y*) is scaled between theelastic limit surface and the failure surface via
Y � ¼ Y elastic þepl
eplðpre-softeningÞðY fail � Y elasticÞ ð3Þ
where epl, epl(pre-softening) are the current and pre-softeningplastic strain.
A residual (frictional) failure surface is defined as
Y �resid ¼ B � p�M ð4Þwhere B is the residual failure surface constant, M is theresidual failure surface exponent.
As for the damage in concrete, additional plastic strain-ing of the material leads to damage and strength reductionfollowing the hardening phase. Damage is accumulated via
D ¼X Depl
efailurep
ð5Þ
efailurep ¼ D1ðp� � p�spallÞ
D2 P eminf ð6Þ
where D1 and D2 are damage constants, eminf is the mini-
mum strain to reach failure.The post-damage failure surface is then interpolated via
Y �fracture ¼ ð1� DÞY �failure þ DY �residual ð7Þand the post-damage shear modulus is interpolated via
Gfracture ¼ ð1� DÞGinitial þ DGresidual ð8Þwhere Ginitial, Gresidual, Gfracture are the shear moduli.
For the volumetric compaction under hydrostatic pres-sure, the RHT model adopts Herrmann’s P–a equationof state [18], which depicts correctly the behavior at highstresses and provides also a reasonably detailed descriptionof the compaction process at low stress levels.
The RHT model for concrete has been evaluated favour-ably in the modeling of concrete perforation under shockloading, and systematic parameters have been obtainedfor several kinds of concrete [19].
2.4. Elastic-strain hardening plastic model for steel
Under blast loading, the reinforcing steel may be subjectto strain hardening, strain rate hardening and heat soften-ing effects. In this study, the John-Cook model [20], whichis a rate-dependent elastic–plastic model, is adopted tomodel the response of the steel bars in the concrete. Themodel defines the yield stress Y as
Y ¼ ½Y 0 þ Benp�½1þ C log e�p�½1� T m
H� ð9Þ
where Y0 is the initial yield strength, ep is the effective plasticstrain, e�p is the normalized effective plastic strain rate, B, C,n, m are material constants. TH is homologous temperature,TH = (T � Troom)/(Tmelt � Troom), with Tmelt being themelting temperature and Troom the ambient temperature.
2.5. Three-phase model for soil medium
There exist many soil models for the static and dynamicresponse of soils [21]. But in the case of explosion, therange of variation of stress in soils is much larger than whatis usually encountered in the common soil dynamics. Assoil is a multi-phase mixture composed of solid mineralparticles, water and air, the deformation mechanism andthe contribution of different phases vary with drasticchange of the stress condition. The authors recently devel-oped a numerical three-phase soil model for simulatingblast wave propagation in soils [22,23]. This model extendsfrom the conceptual model described in [9], and the soil isconsidered as an assemblage of solid particles with differentsizes and shapes that form a skeleton and their void are
Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742 1733
filled with water and air (Fig. 4a). The solid particles,water, air as well as the skeleton formed by the solid parti-cles deform under different laws when external load acts onthe soil mass. The model formulation can be roughlydivided into two main parts: the equation of state andthe strength model. The volumetric ratios of the solid,water and air phases are assumed to be a1, a2, a3, respec-tively. An overview of the three-phase soil model formula-tion is given in what follows.
2.5.1. The equation of state (EOS)
To satisfy the continuity requirements, the total volumechange of the multi-phase soil system must be equal to thesum of volume changes associated with each phase, i.e.
DVV 0
¼ DV w
V 0
þ DV g
V 0
þ DV s
V 0
ð10Þ
where V is the volume of a soil element, V0 is the initialtotal volume of the element, Vw is the volume of water,Vg and Vs are volumes of air and soil particles, respectively.Denote the volume of voids as Vp, Vp = Vg + Vw, andhence V = Vs + Vp.
The pressure load causes deformation in each phase, aswell as friction between the solid particles and deformationof the bond between the solid particles. The friction forceand the force due to the bond are all exerted on the solidphase. Satisfying the equilibrium leads to
dp � dV � oV s
opdp
� �oV g
opb
þ oV w
opb
� ��1
þ opa
oV p
þ opc
oV p
" #¼ 0
ð11Þ
0.5
10-2
10-1
100
101
102
Numerical Resul Empirical (ExperP
eak
Pre
ssur
e (M
Pa)
Scaled D
Solidparticles
Voids
Bondlinkage
Fig. 4. Three-phase soil model and typical verification with peak pressure aComparison with experiment.
where p is the total hydrostatic pressure, ps is the pressureexerted on the solid phase, pa is the pressure borne by thefriction between the solid particles, pb is the pressure borneby the water and gas, or the ‘‘pore pressure’’, pc is the pres-sure borne by the bond between the solid particles, and pe isthe pressure carried by the soil skeleton which is equal tothe sum of pa and pc.
Eq. (11) describes the volumetric deformation underthe hydrostatic pressure, in which oV s
op ,oV g
opb, oV w
opb, opa
oV p, opc
oV p
can be obtained from their independent equations of stateor stress–strain relationship [9,24]. In particular, thenormal stress in the skeleton of soil can be assumed asproportional to the deformation of the soil skeleton.Hence,
pa ¼ fKpDV p ð12Þ
where f is the friction coefficient of the solid particles, Kp isthe coefficient of proportionality, DVp is the incrementalvolume of voids in the soil, DVp = Vp � Vp0, with Vp,Vp0 being the current and initial volume of voids, respec-tively. And for the bonds between the solid particles, itcan be represented by a series of elastic brittle filaments.The resisting stress in each filament obeys the Hooke’slaw until the filament breaks. Introducing a damage vari-able D, it follows
pc ¼ E0ð1� DÞDV p=V p ð13Þ
where E0 is the initial modulus of the bonds.
5
tsimental) Results
α3 =0.15
α3 =0.01
α3 =0.04
α3 =0.00
istance (m kg-1/3)
P
Solid particlesA
B
CD
E
Air
Elastobrittle linkage
Water
Friction betweenblocks
ttenuation. (a) Concept of three-phase soil model for shock loading. (b)
1734 Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742
The continuum damage model is applied to describe thedamage of the soil skeleton. Based on the filament breakingmodel, the damage can be defined as
D ¼ 1� exp � 1
gðbeeffÞg
� �ð14Þ
where B, g are constants related to the properties of thesoil, b is a constant, eeff is the effective strain.
With the above definitions and the initial conditionp(V0) = p0, the pressure p at any time instant can beobtained from Eq. (11).
2.5.2. The strength model for soils
In the soil model, the viscosity of the water and air isneglected, so the total shear stress is borne by the soil skel-eton formed by the solid particles. To include the effect ofhydrostatic stress on the shearing resistance of the soil, themodified von Mises’ yield criterion [25] is adopted and fur-ther modified to take into account the strain rate effect, as
f ¼ffiffiffiffiffiJ 2
p� ðaI1 � kÞ 1þ b ln
_eeff
_e0
� �¼ 0 ð15Þ
where a and k are material constants related to the fric-tional and cohesive strengths of the material, respectively;I1, J2 are the first and deviatoric stress invariants; _e0 isthe reference effective strain rate; b is a strain rate enhance-ment factor, and _eeff is the effective strain rate defined as
_eeff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
3d _eijd _eij
rð16Þ
The plastic potential function employed in this model is thePrandtl–Reuss type:
QffiffiffiffiffiJ 2
p� �¼
ffiffiffiffiffiJ 2
p� Y ¼ 0 ð17Þ
where Y is the yield limit defined by the yield function.The above soil model has been calibrated against avail-
able experimental and empirical data [22,23]. Fig. 4b illus-trates a typical calibration against experimental resultsdescribed in [9].
2.6. Modeling of charge detonation and air
For the explosive detonation, the Jones–Wilkens–Lee(JWL) equation of state [26] is adopted:
P ¼ C1 1� xR1v
� �expð�r1vÞ
þ C2 1� xR2v
� �expð�r2vÞ þ xe
vð18Þ
where v is the specific volume, e is specific energy. The val-ues of constants C1, R1, C2, R2, x for many common explo-sives have been determined from dynamic experiments.
The following equation of state is used for air:
P ¼ ðc� 1Þ qq0
E ð19Þ
where E is the specific energy, and c is taken equal to 1.4.
3. Numerical investigation
A multi-storey reinforced concrete frame is chosen forthe analysis to characterize the structural effects from anabove-ground explosion. Fig. 5a and b shows the geometryand the dimensions. The frame has seven storeys in total,with six stories above the ground and one story embeddedin the soil. The explosion scenario is chosen such that bothlocal and entire structural system response can be repre-sented to allow for a comprehensive evaluation of theresponse characteristics. Thus, a charge of 1000 kg TNTequivalent is selected, representing a major incident. Thestand-off distance is chosen to be 30 m measuring fromthe front face of the structure, giving a scaled standoff dis-tance equal to 3 m/kg1/3. Based on trial analysis, a nearerstandoff distance would generally result in more localizeddamage. The detonation centre is assumed to be 1 m abovethe ground surface, representing the kind of vehicle carriedexplosives.
The dimensions of the frame structure are typical, with atotal above-ground height of 20 m, total width of 10 m, anda basement height of 3 m. The height of the first storey ispurposely made larger than the remaining storey, as is usu-ally seen in multi-storey residual and office buildings. Thereinforced concrete beam and columns are assumed to havea uniform cross-sectional depth of 400 mm. The concretehas the following properties: compressive strength =30 MPa, tensile strength = 3 MPa, Young’s modulus =26 GPa. A gross reinforcement volumetric ratio of 2% isassumed. The reinforcing steel has a yield strength of460 MPa, and Young’s modulus of 200 GPa.
The numerical calculation is carried out using hydro-code AUTODYN [13] with necessary user defined subrou-tines. Fig. 5c shows the numerical model configuration andthe mesh. In order to study the propagation of the blastwave in the air and soil medium towards to the structure,as well as the distribution of the air pressure loading(reflected overpressure) on the structure, a number of tar-get points are arranged in the numerical model to recordthe pressure histories.
Fig. 6 depicts the contours of the pressure and velocityat several time instants. These graphs show clearly thepropagation of the shock wave. The incident and reflectedwaves can also be clearly observed.
3.1. Propagation of blast wave in air, soil and pressureload on structure
Fig. 7 shows the blast overpressure on the front face ofthe structure. As expected, the peak overpressure tends todecrease gradually at higher positions on the front face.The peak pressure near the top level (Target 64) is about50% of that at the first storey level. The arrival time ofthe shock front gradually delays as the height increases.The shape of the pressure curves and main pulse durationdo not exhibit sensible difference among different targetpoints.
30m
1m
20m
Air
Soil 10m
Charge W=1000kg
10m
5m
0.4m
0.4m
3m
Charge
Air
Structure
SoilTransmission
boundary
Transmission boundary
Eulerian mesh
Lagrangian mesh
Fig. 5. Example case configuration and the numerical model setting. (a) Scenario configuration. (b) Dimensions of structure. (c) Numerical model andmesh.
Fig. 6. Computed pressure and velocity fields. (a) Pressure field. (b) Velocity field.
Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742 1735
Fig. 8 shows the overpressure histories at selected tar-gets along the straight path from the charge centre to thestructure. Two significant observations may be made fromthese pressure history plots; (1) the attenuation of the peakpressure is rapid (exponential) with increase of the distancefrom the charge centre, and (2) with the target points closerto the building structure, the reflected wave becomes moreand more significant. At a target point very close to the
structure face (point 72), the peak reflected overpressureappears to be higher than the incident wave overpressure,which is reasonable.
In order to measure the stress wave in the soil, severaltarget points are arranged inside the soil layer, and typicalrecorded soil pressure curves are plotted in Fig. 9. As canbe seen, the pressure in the soil also attenuates rapidly withthe increase of the distance from the charge location. Near
64
63
62
61 605958575655 54
43 0 100 200 300 400
0.0
0.2
0.4
0.6
0.8
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 55
0 100 200 300 400
0.0
0.2
0.4
0.6
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 57
0 100 200 300 400
0.0
0.2
0.4
0.6
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 58
0 100 200 300 400
0.0
0.2
0.4
0.6
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 59
0 100 200 300 400
0.0
0.2
0.4
0.6
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 61
0 100 200 300 400
0.0
0.2
0.4
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 62
0 100 200 300 400
0.0
0.1
0.2
0.3
0.4
0.5
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 63
0 100 200 300 400
0.0
0.1
0.2
0.3
0.4
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 64
Fig. 7. Blast pressure load on the front face of structure.
Charge6671 70 6972 68 67
0 10 20 30 40 50
0
4
8
12
16
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 66
0 10 20 30 40 50
0
1
2
3
4
5
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 67
0 20 40 60 80 100
0.0
0.5
1.0
1.5
2.0
2.5
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 68
0 30 60 90 120 150 180
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 70
0 100 200 300 400
0.0
0.1
0.2
0.3
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 71
0 100 200 300 400
0.000.050.100.150.200.250.30
Ove
rpre
ssur
e /M
Pa
Time /ms
Target 72
Fig. 8. Overpressure histories at targets in air.
1736 Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742
the charge the pressure exhibits extremely high value. Infact, as can be seen from the contours shown in Fig. 6, a
half-spherical crater of radius about 2 m is created beneaththe point of detonation.
Charge
4752 51 5053 49 48
0 10 20 30 40 50
0
10
20
30
40
Pre
ssur
e /M
Pa
Time /ms
Target 47
0 10 20 30 40 50
0
1
2
3
4
5
6
Pre
ssur
e /M
Pa
Time /ms
Target 48
0 20 40 60 80 100-0.20.00.20.40.60.81.01.2
Pre
ssur
e /M
Pa
Time /ms
Target 50
0 50 100 150 200-0.10.00.10.20.30.40.50.6
Pre
ssur
e /M
Pa
Time /ms
Target 51
0 50 100 150 200-0.05
0.00
0.05
0.10
0.15
0.20
0.25
Pre
ssur
e /M
Pa
Time /ms
Target 52
0 50 100 150 200
0.00
0.05
0.10
0.15
0.20
0.25
Pre
ssur
e /M
Pa
Time /ms
Target 53
Fig. 9. Stress wave propagation inside soil.
Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742 1737
3.2. Structural responses
3.2.1. General response featuresFig. 10 shows the damage contours of the entire struc-
ture and a close-up of the damage in the first two storeys.Severe damage is observed to occur mainly in the ele-
ments on the front face of the structure, due apparentlyto the strike of direct air shock wave. The remaining partof the structure exhibit more or less a uniform distributionof damage among different elements. This may beexplained by the fact that the elements not subjected tothe direct shock are involved through the global dynamicsystem (with interconnection through beam-column joints).As the degree of damage to these components is generally
Fig. 10. Damage contours of the structure. (a) Ov
minor with a damage index below 0.2, it can be inferredthat the global dynamic response plays a relatively insignif-icant role as compared to the direct shock effect on thefront face of the structure. This can be confirmed by theglobal vibration results as will be discussed later.
The damage to the front face elements shows a distinc-tive pattern. Within each storey, the severest damage isconcentrated at both ends of the element on the outer sideand around the middle portion on the inner side. This indi-cates that these front face elements respond to the shockload more like individual elements with fixed ends. Thisobservation is significant in the sense that the analysis ofthe response in critical elements may be simplified into indi-vidual elements with a somewhat fixed end condition. For
erall; (b) Close-in at first and second storeys.
29
28
115
1442
2120191817
Z
R7
0 100 200 300 400
-0.1
0.0
0.1
0.2
0.3Target 2
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-0.3-0.2-0.10.00.10.20.3 Target 3
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400
0123456 Target 5
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6 Target 7
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-1.0
-0.5
0.0
0.5
1.0
1.5
2.0Target 8
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-0.6-0.4-0.20.00.20.40.60.8
Target 9
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400
-0.4
-0.2
0.0
0.2
0.4
0.6 Target 14
R-V
eloc
ity /m
/s
Time /ms0 100 200 300 400
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6 Target 21
R-V
eloc
ity /m
/s
Time /ms
Fig. 11. Horizontal velocity time histories at selected targets on structure.
0 100 200 300 400-15
-10
-5
0
5
10
15 Target 2
R-A
ccel
erat
ion
/g
Time /ms0 100 200 300 400
-20-15-10
-505
10152025
Target 3
R-A
ccel
erat
ion
/g
Time /ms
0 100 200 300 400
-200
20406080
100120
Target 5
R-A
ccel
erat
ion
/g
Time /ms
0 100 200 300 400
-15-10
-505
101520
Target 7
R-A
ccel
erat
ion
/g
Time /ms0 100 200 300 400
-60-40-20
0204060 Target 8
R-A
ccel
erat
ion
/g
Time /ms
0 100 200 300 400-20
-10
0
10
20 Target 9
R-A
ccel
erat
ion
/g
Time /ms
0 100 200 300 400
-20-10
010203040
Target 14
R-A
ccel
erat
ion
/g
Time /ms
0 100 200 300 400
-10
0
10
20 Target 21
R-A
ccel
erat
ion
/g
Time /ms
Fig. 12. Horizontal acceleration time histories at selected targets on structure.
1738 Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742
the particular example structure herein, the first storey col-umn experiences disproportionately severer damage thancolumns in other storeys on the front face, due apparentlyto the relatively weaker resistance of the column because ofits longer length, as well as the relatively higher loading atthe first storey level.
3.2.2. Vibration response of the structure
To understand the dynamic response of the structuralsystem more precisely, the vibration histories at targetpoints all over the structure are recorded. The arrangementof the target points can be seen from the top of Fig. 11. Thevelocity histories were recorded during the numerical calcu-
Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742 1739
lation. The displacements and acceleration can be calcu-lated from the velocities by integration or differentiationoperations.
Fig. 11 plots the horizontal velocity histories at typicaltarget points. As can be seen, the highest velocity of about6 m/s occurs at the first storey column on the front face(represented by target 5), followed by the second storey col-umn (about 1.7 m/s, at target 8). The maximum velocity at
-4
-1
2
5
8
11
14
17
20
-50050100
Floor level
Mid-column
(m)
(mm) -0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0
Dis
p(m
)
Fig. 13. Front face displacement profile and time histories. (a) Front face distargets 4–14.
Fig. 14. Numerical models for isolated air pressure load/ground vibration andground vibration effect. (c) Damage under air shock. (d) Damage under groun
the joint locations is generally on a lower order of about0.5 m/s, including that at the interior joints (e.g., target21). The trend of significantly higher response on the frontface columns as compared to the joint response which rep-resents the global dynamic response can also be observedfrom the accelerations shown in Fig. 12.
Fig. 13a depicts the displacement profile of the frontface elements at 50 ms when the displacements throughout
.10.1 0.2 0.3 0.4
Time (s)
t5t6
t4
t7-14
placement profile at 50 ms; (b) Displacement time histories for front face
corresponding damage. (a) Model for air shock wave effect. (b) Model ford motion.
0 100 200 300 400
0123456 Target 5
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400
0
2
4
6 Target 5
R-V
eloc
ity /m
/s
Time /ms0 100 200 300 400
-0.2
-0.1
0.0
0.1
0.2
0.3Target 5
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6 Target 7
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6 Target 7
R-V
eloc
ity /m
/s
Time /ms0 100 200 300 400
-0.05
0.00
0.05
0.10 Target 7
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-1.0
-0.5
0.0
0.5
1.0
1.5
2.0Target 8
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400
-0.5
0.0
0.5
1.0
1.5
2.0Target 8
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400
-0.10
-0.05
0.00
0.05
0.10
0.15 Target 8
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-0.6-0.4-0.20.00.20.40.60.8
Target 9
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-0.6-0.4-0.20.00.20.40.60.8
Target 9
R-V
eloc
ity /m
/s
Time /ms0 100 200 300 400
-0.10
-0.05
0.00
0.05
0.10 Target 9
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400
-0.4
-0.2
0.0
0.2
0.4
0.6 Target 14
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400
-0.4
-0.2
0.0
0.2
0.4 Target 14
R-V
eloc
ity /m
/s
Time /ms0 100 200 300 400
-0.05
0.00
0.05
0.10Target 14
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-0.4-0.3-0.2-0.10.00.10.20.30.4 Target 18
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400
-0.4-0.3-0.2-0.10.00.10.20.3
Target 18
R-V
eloc
ity /m
/s
Time /ms0 100 200 300 400
-0.05
0.00
0.05
0.10
0.15
0.20Target 18
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6 Target 19
R-V
eloc
ity /m
/s
Time /ms
0 100 200 300 400
-0.4
-0.2
0.0
0.2
0.4
0.6 Target 19
R-V
eloc
ity /m
/s
Time /ms0 100 200 300 400
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15 Target 19
R-V
eloc
ity /m
/s
Time /ms
Fig. 15. Comparison of response velocities from combined and separate air shock/ground vibration effects (left = combined; middle = air shock only;right = ground vibration only).
1740 Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742
the entire structure reaches about the maximum value(except the first storey column where plastic deformation
still carries on to a later time). The displacement time his-tories are plotted in Fig. 13b. The strong concentration of
-0.02
-0.01
0.00
0.01
0.02
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time (s)
Dis
p (m
)
Fig. 16. Horizontal displacement at targets 1–14 under ground shockonly.
Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742 1741
the damage in the first storey column, as well as significantelement (column) level responses within each storey as evi-denced in the damage contours shown in Fig. 10, is clearlyobserved. On the contrary, the global response as repre-sented by the drifts at the floor (or joint) locations is rela-tively insignificant. As a matter of fact, the maximum totalhorizontal displacement at the floor (or joint) locations isonly on the order of 20 mm, while the relative drift betweenadjacent floors is no more than 10 mm. In terms of the driftratio (inter-storey drift divided by the storey height), it isup to 0.3–0.4%. This amount of drift ratio is lower thanthe general threshold for sensible structural damageaccording to practical standards for reinforced concreteframes [27,28].
In summary, the observed response features demonstrateclearly that the global system response only plays an insig-nificant role in the development of damage to the structureunder a nearby explosion like the case under consideration.If the standoff distance gets nearer, it can be expected thatthe localized damage could be even severer; whereas for far-ther away explosions, the relative contribution of the globaleffect in the overall response may somehow increase, but itdoes not seem to cause concerns since its damaging effectwould reduce further from what is mentioned above.
4. Discussion on the significance of ground shock effect
In the above-mentioned coupled numerical model theresponse of the structure is a combined effect of the airshock wave and the ground vibration generated by theexplosion. Although it is generally understood that forthe explosion scenarios under consideration (structuressubjected to nearby above-ground explosions) the damageto the structure is primarily caused by the air shock wave, itis not clear as to what extent the explosion induced groundvibration can affect the damage and vibration response ofthe structure.
In order to examine the relative significance of theexplosion-induced ground vibration on the structuralresponse, two additional analyses are performed using thecoupled model for separate air overpressure and groundvibration effects, respectively. In the analysis for the airshock effect only, an artificial rigid barrier is placed in frontof the basement in the soil to block the propagation of thestress wave in the soil towards the structure base (Fig. 14a).Likewise, for the other case a rigid barrier is placed in frontof the main structure to block the propagation of the airshock wave towards the structure so that only the groundvibration is effective (Fig. 14b).
Fig. 14c and d shows a comparison of the damage con-tours for the structure separately under the air shock waveand ground shock effect. From the figure and comparing tothe damage under the combined loads shown in Fig. 10, itcan be clearly observed that the damage of the structure isalmost entirely attributable to the air shock wave effect,while the damage due to the ground vibration effect is prac-tically negligible.
The above trend is also evident from the comparison ofthe velocity responses at typical target points as shown inFig. 15. While the velocity response of the frame appearsto be almost identical under combined loading or air shockonly, the velocity response under the ground vibrationalone is generally of one order of magnitude lower.
Fig. 16 shows typical displacement responses at frontface target points when the frame is subjected to theground vibration alone. The maximum displacement is lessthan 15 mm, and the inter-storey drift ratio is only about0.2%. This is consistent with the low damage level as indi-cated by the contour shown in Fig. 14d.
5. Conclusions
A coupled numerical model is used to study the charac-teristics of the complicated effects of an above-groundexplosion on a building frame. The model encompassesthe charge detonation, propagation of shock wave in air,propagation of stress wave in the ground, as well as thedynamic response of the structure, through a coupled Eule-rian–Lagrangian computational scheme. Thus, it allows fora comprehensive simulation of the entire process in a morerealistic manner. A typical explosion scenario involving amulti-storey frame and a typical charge detonation is ana-lyzed. Based on the results, the following conclusions maybe drawn:
(1) The damage to the building frame primarily occurs inthe elements on the front face, due primarily to theeffect of the direct air shock wave. The mode of dam-age to the front face of the structure appears to begoverned by the element level response, with concen-tration of damage at the element ends and around themid-span. The global dynamic response of the systemis relatively insignificant concerning the structuraldamage. As a result, considerably less severe damageis observed in the interior elements and the structuralelements on the back side of the structure.
(2) Based on the above observations, for practical appli-cations it is deemed appropriate to consider a simpli-fied element based model for the damage analysis atpotential critical regions on the front face of thestructure.
1742 Y. Lu, Z. Wang / Computers and Structures 84 (2006) 1729–1742
(3) For the typical explosion scenario under consideration(1000 kg with a standoff distance equal to 30 m, or ascaled distance of 3 m/kg1/3), the structural damageand the critical vibration responses are dominated bythe air shock loading, while the explosion-inducedground vibration appears to only enhance to a limitedextent the vibrations. The ground motion alone doesnot cause sensible damage to the structure. It can beexpected that with further increase of the standoff dis-tance, the relative importance of the ground motion inthe overall structural response may increase because ofthe slower attenuation of the stress wave in soil as com-pared to the shock wave in air. However, as the abso-lute magnitude of the ground vibration decreases, theground vibration effect should not become a majorproblem as far as structural damage is concerned.
Similar modeling approach may be applied to exploreother blast-induced complex response phenomena. Itshould also be pointed out that the scope of this study islimited to the direct effect of the explosion. The post-blastanalysis concerning progressive collapse deals with a differ-ent process which will obviously involve the entire struc-tural system. The post-blast analysis may be carried outon the basis of the residual state of the structure as a resultof the direct blast effect.
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