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Computers and Structures 84 (2006) 431–438

Numerical simulation study of spallation in reinforced concreteplates subjected to blast loading

Kai Xu, Yong Lu *

School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798, Singapore

Received 12 March 2004; accepted 29 September 2005Available online 21 November 2005

Abstract

In the design of protective structures, concrete walls are often used to provide effective protection against blast from incidental events.With a reasonable configuration and proper reinforcement, the protective structure could sustain a specified level of blast without globalfailure. However, the concrete wall may generate spallation on the back side of the wall, posing threats to the personnel and equipmentinside the structure. For this concern, it is important to establish appropriate concrete spallation criteria in the protective design. Earlieranalytical studies have been based on simplified one-dimensional wave theory, which does not consider the complex three-dimensionalstress conditions in the case of close-in explosion and the structural effects. This paper presents a numerical simulation study on the con-crete spallation under various blast loading and structural conditions. A sophisticated concrete material model is employed, taking intoaccount the strain rate effect. The erosion technique is adopted to model the spallation process. Based on the numerical results, the spall-ation criteria are established for different levels of spallation. Comparison of the analytical results with experimental data shows a favor-able agreement. It is also shown that the structural effects can become significant for relatively large charge weight and longer distancescenarios.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Reinforced concrete; Plate; Blast loading; Spallation; Numerical simulation

1. Introduction

The behavior, analysis and design of hardened struc-tures for protection against short duration dynamic load-ing effects, such as those induced by air blast, is a subjectof extensive studies in the last few decades. Short durationhigh magnitude loading conditions significantly influencestructural response. The frequencies of explosive loadscan be much higher than conventional loads. Furthermore,short duration dynamic loads often exhibit strong spatialand time variations, resulting in sharp stress gradients inthe structures. This also results in a varying strain ratefor the duration of the analysis. Consequently, the analysisof structures made of brittle materials such as concrete sub-jected to blast loading becomes a very complex issue.

0045-7949/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2005.09.029

* Corresponding author. Fax: +65 67910676.E-mail address: cylu@ntu.edu.sg (Y. Lu).

Reinforced concrete walls are often used as barricade inthe protective design against potential blast and fragmenta-tion. Generally speaking, when a concrete wall or plate issubjected to blast detonation, explosive charges will causevarious regions of distinctive stress and failure situations.As a result, regional limited material failure can beobserved. Near the surface facing the explosive charge,the concrete experiences triaxial compression and the con-crete material can fail due to high compression. This crush-ing failure results in a crater. The equation of state of thecurve relating the volumetric deformation to the hydro-static stress exhibits an increase due to increase of volumet-ric deformation. On the opposite side of the plate, due tothe interaction between the compressive wave generatedby the detonation and the free surface, the shock wave isreflected and converted into a tensile wave which couldproduce tensile cracking if the material strength is reached.Because of the low resistance of concrete to tension, one

432 K. Xu, Y. Lu / Computers and Structures 84 (2006) 431–438

can observe a tensile spalling. Between these regions, theconcrete is subjected to compression with a moderateamount of confinement and responses with couplingbetween the volumetric and deviatoric stress components.It is generally accepted that the failure strength of concreteis rate dependent although the exact physical mecha-nisms responsible for the rate effect are not clearly knownat the moment. Therefore, it is important that the strainrate effect is adequately incorporated in the analyticalmodel.

Spalling of the concrete wall on the inner side of theprotective structure poses threat to the personnel andequipment inside the structure even though the structureand the wall itself do not suffer general failure. For eco-nomical reasons, most damage curves for reinforced con-crete walls are generated from small-scale experiments.An extensive review of concrete spallation processes andcompiled experimental data is presented in McVay [1].Nash et al. [2] studied and defined thresholds for spallationof concrete based on an approximate one-dimensionalnumerical model. Until recently, no procedure has beenavailable for predicting concrete spall damage and spallvelocities considering the true three-dimensional concreteresponses.

In recent years, many efforts have been devoted to thedevelopment of reliable methods and algorithms for a morerealistic analysis of structures and structural componentssubjected to high dynamic loading. Some techniques, suchas specialized contact algorithms and adaptive meshing orerosive techniques exist nowadays in computer codes suchas the transient dynamic code LS-DYNA [3] used in thepresent study. With such techniques, it is possible to carryout three-dimensional computations of concrete structureconcerning a realistic reproduction of the weapon effects.On this basis, numerical results may be generated to sup-plement experimental studies for developing appropriatedesign guidelines. For example, the criteria for damagesuch as breaching, heavy damage, moderate damage, andslight damage can be constructed in terms of the scaledstandoff distance (actual distance divided by the cube rootof the explosive weight) and the target thickness based onthe numerical calculations.

The objective of this paper is to study the generalbehavior of concrete plates subjected to air blast loadingusing numerical simulation approach, with particularfocus on the spall damage. The dynamic material strengthsare incorporated and a realistic explosive loading is con-sidered in the simulation. The dynamic fracture criterionfor concrete is employed in the implementation of theerosion technique to capture the fracture and materialseparation process. The fracture location and spall veloci-ties are predicted in the computation, while the effect ofthe plate dimension is investigated. The numerical resultsare compared with experimental data and the data gene-rated using a simplified method. Empirical spallation crite-ria considering the three-dimensional concrete responseare proposed.

2. 3-D Numerical model

2.1. Basic considerations

Nash et al. [2] predicted concrete wall spallation usingone-dimensional model. In that model, constant strainbar elements were used for stress wave propagation. Theconcrete wall was represented as a prismatic bar withdimensions based on the thickness and the size of theexplosive charge. Therefore, the model could be suitableonly for modeling plane wave propagation. Besides, initia-tion of the spallation by the planar waves was consideredto be independent of the wall thickness for the exponen-tially decaying pressure time histories used. The attenua-tion of the stress intensity through the wall thickness wasnot considered.

In real three-dimensional situations, spatial distributionof the loading across the face of a wall or divergence of thestress wave through the wall thickness can significantlyalter the stress wave from a planar to a non-planar front.This diverging stress waves can attenuate their intensitythrough the wall thickness. Hence, the results from theone-dimensional model are expected to be on the conserva-tive side. To more realistically simulate and predict the con-crete spallation under various charge weights and standoffdistances, it is necessary to resort to the general 3-D numer-ical simulation. A 3-D numerical model will also allow aninvestigation into the global structural effect on the localconcrete spallation.

2.2. Concrete model in 3-D numerical simulation

In the present study, the ‘‘pseudo-tensor concrete/geo-logical model’’ [3] is employed to model the concrete. Thismodel provides the necessary elements needed to describethe concrete behavior under high dynamic and complexstress conditions. In this model, the stress tensor is sepa-rated into the hydrostatic tensor and the deviatoric stresstensor. The hydrostatic stress tensor changes the concretevolume and the deviatoric stress tensor conjugates theshape deformation. A strain rate multiplier is used to mod-ify the dynamic yield strength of the concrete material.

For hydrostatic tensor, the compaction model is a multi-linear approximation in internal energy. Pressure is definedby

p ¼ CðevÞ þ cT ðevÞE; ð1Þ

where E is the internal energy per initial volume, c is theratio of specific heats. The volumetric strain, ev, is givenby the natural logarithm of the relative volume. As shownin Fig. 1, the model contains an elastic path from thehydrostatic tension cutoff to the point T of elastic limit.When tension stress exceeds the hydrostatic tension cutoff,tensile failure occurs, which corresponds to the bulk failureregion. When the volumetric strain is greater than the pointT, compaction occurs and gradually the concrete convertsinto a granular kind of material. Then the volumetric strain

Pres

sure

TVolumetric Strain

Tension cutoff

Fig. 1. Pressure versus volumetric strain curve.

Point 3

Point 1

Point 2 Maximum

Yield

Residual

p -ft

(a)

Point 1Yield

Point 2 Maximum

Point 3 Residual

(b)

∆σ

σ

Fig. 2. Strength model for concrete material [3]: (a) two-curves strengthmodel and the yield surface; (b) typical resulting stress–strain curve.

K. Xu, Y. Lu / Computers and Structures 84 (2006) 431–438 433

almost does not increase any further. Unloading occursalong the unloading bulk modulus to the tension cutoff.Reloading always follows the unloading path to the pointwhere unloading began, and continues on the loading pathshown in Fig. 1. Unloading and reloading follow the samepath, which is interpolated between the elastic slope andgranular material state.

A two-curve model is used to analyze the deviatoricstress tensor, as shown in Fig. 2(a), where the upper curverepresents the maximum yield strength curve and the lowercurve is the failed material residual strength curve.

The function Dr which limits the deviatoric stresses isdefined as a linear combination of the two fixed strengthcurves which are functions of pressure as

Dr ¼ gDrm þ ð1� gÞDrr; ð2Þwhere

Dr ¼ffiffiffiffiffiffiffi3J 2

p;

Drm ¼ a0 þp

a1 þ a2pðmaximum strength curveÞ;

Drr ¼ a0f þp

a1f þ a2pðresidual strength curveÞ;

and p = �(rxx + ryy + rzz)/3 is the pressure (stresses arepositive in tension, pressure is positive in compression).J 2 ¼ ðs21 þ s22 þ s23Þ=2 is the second invariant of the devia-toric stress tensor. s1 is the first principal deviatoric stress,s1 = max[(r1 � p), (r2 � p), (r3 � p)]. s2 and s3 are secondand third principal deviatoric stress, respectively.

The parameter g is a modified effective plastic strainmeasure k. The function g(k) is intended to first increasefrom some initial value up to unity, then decrease to zerorepresenting softening. The effective plastic strain arisesfrom the physical mechanisms such as internal cracking,and the extent of this cracking is affected by the hydrostaticpressure when the cracking occurs. This mechanism givesrise to the confinement effect on concrete behavior. Thefunction to account for this phenomenon is given in theform

k ¼Z ep

0

1þ prcut

� ��b1

dep; ð3Þ

where the effective plastic strain increment is given by

dep ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2=3Þdepijdepij

q.

The yield surface is actually migrating between Drr(residual strength) and Drm (maximum strength). The ini-tial yield surface is given by

Dry ¼ gyDrm þ ð1� gyÞDrr; ð4Þwhere gy = g(0) is the initial value of g before any plasticityhas occurred. A typical value of this coefficient for concreteis gy = 0.309 [3]. The subsequent initial yield surface isdepicted in Fig. 2(a).

If Dr is greater than Dry, plastic flow occurs. The well-known radial return algorithm pushes the stress vectorback to the yield surface perpendicular to the hydrostaticaxis, the corresponding strain increment is calculated bythe difference between the actual effective and yield stress.A typical resulting stress and strain relationship is shownin Fig. 2(b).

2.3. Strain rate effect and mixture model for reinforced

concrete

A theoretical model for concrete dynamic strengthchange with strain rate has been formulated by the authors

434 K. Xu, Y. Lu / Computers and Structures 84 (2006) 431–438

based on continuum fracture mechanics and experimentaldata. The model stems from micro-crack nucleation,growth and coalescence to formulate the evolution of dam-age. The concrete is assumed as homogeneous continuumwith pre-existing micro-cracks. Damage to concrete isdefined as the probability of fracture at a given crack den-sity, obtained by integrating a crack density function overtime. Based on the damage function, the stress response ata particular time, and hence the dynamic stress–strain rela-tionship, is established under a given strain rate. Therequired material constants representing initial crack prop-erties are derived from material dynamic strength tests.Details of the model derivation can be found in [4].Fig. 3(a) and (b) illustrates the modeling results for con-crete under tension and compression as compared to someexperimental results [5–10].

In the numerical model, the strain rate effect is incorpo-rated as follows. At any given pressure, the failure surfacesare expanded by a rate enhancement factor which dependson the effective deviatoric strain rate. The rate enhance-ment is expressed as follows:

(a)

(b)

Fig. 3. Dynamic strength enhancement versus strain rate under uniaxialstress: (a) dynamic tensile strength ratios; (b) dynamic compressivestrength ratios.

rc ¼3a1f 0

crf þ a2f 02c r

2f

3a0a1 þ ð1þ a0a2Þf 0crf

; ð5Þ

where rc is rate enhancement factor at fixed pressure, rf isthe rate enhancement factor from an unconfined uniaxialstrength (tension and compression) model. In this paper,rf follows the proposed model [4] as depicted in Fig. 3. f 0

c

is compressive strength, and ai are parameters definingthe maximum stress curve.

The strain rate is calculated by the following equation:

_ep ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2=3Þ_epij _e

pij

q. ð6Þ

The reinforcement dynamic strength change with strainrate is obtained from the equation given by Liu and Owen[11],

rd

rs

¼ klog10_ed_es

� �þ 1; ð7Þ

where rd and rs are the dynamic and static yield strength;_ed and _es are the corresponding strain rate; the parameter kis 0.03 and _es is assumed to be approximately 10�2/s.

Tables 1 and 2 list the tabulated strain rate effect on con-crete and reinforcement strengths used in the numericalsimulations.

In the numerical model, the effect of reinforcement onthe reinforced concrete behavior is simulated by a mixturemodel. In this mixture model, a reinforcement fraction, fr,is defined along with properties of the reinforcement mate-rial. The bulk modulus, shear modulus, and yield strengthare then calculated from the mixture rule, e.g., for the bulkmodulus the rule gives

K ¼ ð1� frÞKc þ frKr; ð8Þwhere Kc and Kr are the bulk modulus for the concrete andthe reinforcing steel, respectively. The modulus is updatedas damage in the concrete (and steel) accumulates. The rel-ative importance of concrete in the mixture decreases asdamage increases.

Table 2Reinforcement dynamic strength change with strain rate

Strain rate (s�1) 1E�3 1E�2 1E�1 1E0

Strength (MPa) 530 530 545.9 561.8

Strain rate (s�1) 1E+1 1E+2 1E+3 1E+4Strength (MPa) 577.7 593.6 609.5 625.4

Table 1Concrete dynamic strength change with strain rate

Strain rate (s�1) 1E�7 1E�6 1E�5 1E�4 1E�3

Strength (MPa) 3.85 3.881083 3.902165 3.92325 3.94433

Strain rate (s�1) 1E�2 0.33241 1.5 2 5Strength (MPa) 3.9654 7.364031 9.86365 10.46515 12.40048

Strain rate (s�1) 10 20 50 150 500Strength (MPa) 14.1134 16.2069 19.95419 27.58541 46.7145

Fig. 4. Typical concrete break-up simulated using pseudo-tensor concrete model and the erosion algorithm.

K. Xu, Y. Lu / Computers and Structures 84 (2006) 431–438 435

2.4. Simulation of fracture by erosion algorithm

In order to simulate the physical fracture or crushing ofconcrete in the numerical model, the so-called erosion algo-rithm is implemented. When the material response in anelement reaches certain critical value, the element is imme-diately deleted. The deletion process is irreversible, whichmeans when the applied load is reversed, the deleted mate-rial will not be able to offer further resistance. This tech-nique can be employed to capture the physical fractureprocess if no significant reverse loading occurs to the frac-tured elements. Fig. 4 shows an example calculation ofbreak-up of a concrete plate subjected to blast using theerosion algorithm. The details of fracture are revealed fromthe simulation in a rather realistic manner.

There may be a variety of criteria governing the ‘‘ero-sion’’ of the material. Typically, the material fracture andfailure under tension and compression may be defined bythe magnitude of the effective plastic strain and volumetrictensile strain, respectively. In the present study, the failurecriteria are defined on an empirical basis. Typical concretestrain at peak tensile stress under static loading is around0.0002 (one-tenth of that at peak compressive stress). Con-sidering the softening phase, the concrete at fracture withpractically complete loss of tensile strength may beassumed as 5 · 0.0002 = 0.001. For the explosion casesunder consideration, the maximum strain rate is generallyon the order of 10–100 s�1 (for example, the maximumeffective strain rate from the numerical simulation for 5and 30 kg charge at standoff distance of 1 m is about 25and 140 s�1, respectively). For this magnitude of strainrate, the corresponding dynamic strength enhancement fac-tor can reach 5.0 or above. Furthermore, the confinementeffect from the reinforcement, which is not reflected inthe mixture concrete model, will also contribute in delayingthe fracture of concrete. Taking all these influences intoaccount and in conjunction with trial parametric analysis,it is found that the dynamic tensile fracture strain shouldbe around 0.01 for spallation with the mixture RC mate-rial. Thus, the principal tensile strain reaching 0.01 isadopted as the primary criterion in the implementation ofthe erosion algorithm in the numerical simulation.

It is worth pointing out that, although the above selec-tion of the dynamic strain of concrete for erosion is primar-ily based on judgment rather than solid experimentalevidences, which are lacked, the consequence of possiblediscrepancy in this quantity is limited. This is because thedamage status of the concrete material is essentially gov-erned by the material model itself, which is independentof the selection of the fracture strain. The material wouldhave been rendered practically ineffective due to damagebefore reaching the erosion level defined above.

As will be seen later, the simulation results using the ero-sion criterion mentioned above for concrete spallationshow a consistent comparison with the relevant experimen-tal observations, and this also indicates that the adoptionof the simple criterion is effective in the spallation simula-tion for RC plates. Of course, more robust criteria for ero-sion may result in more accurate simulation results, andthis is a subject requiring further study.

3. Numerical simulation study

According to McVay [1], the spall damage of concreteplates can be divided into the following three categories(i) no damage: from initial state to a few barely visiblecracks. The critical state for this category is referred as‘‘light cracking’’ in this paper. (ii) Threshold for spall: froma few cracks and a hollow sound to a large bulge in the con-crete with a few small pieces of spall on the surface. (iii)Medium spall: from a very shallow spall to spall penetra-tion up to one third of the plate thickness.

For the numerical simulation, a generic square rein-forced concrete plate of L (width) · L (length) · t (thick-ness) with four fixed sides is considered. The thickness t

is assumed to be 300 mm constant, while three differentlengths of L = 1.5, 3.0, and 4.5 m are considered forobserving different degree of structural effect on the con-crete spallation. This gives rise to three length-to-thicknessratios of 5, 10 and 15, respectively. Assuming the detona-tion is along the central normal direction, a quarter ofthe plate is included in the numerical model consideringsymmetry. The total number of solid elements in the modelwas respectively 19,200, 76,800, and 172,800 for the three

0.1 1 10 1000.01

0.1

1

10

Stan

doff

(m

)

Explosive weight (kg)

Fit function of simulation resultsExperimental results, (McVay, 1988)L/t=5.0L/t=10L/t=151D model assuming spall velocityVelocity=15.24 m/s (Nash et al. 1995)Velocity=30.48 m/s (Nash et al. 1995)

A

B

C

Fig. 6. Critical charge weight–standoff distance for no-damage (up to lightcracking) category.

436 K. Xu, Y. Lu / Computers and Structures 84 (2006) 431–438

plate dimensions with a uniform element size of 18.75 ·18.75 · 25 mm.

The concrete material considered in the present study isof 40 MPa class. The properties of the concrete material areas follows: static Young�s modulus = 27 GPa, mass den-sity = 2400 kg/m3, shear modulus = 10.8 GPa, Poisson�sratio = 0.20. The pressure cutoff level for tensile fractureis taken to be the same as the quasi-static tensile strengthof the concrete equal to 3.85 MPa. The reinforcement ratioin each direction of the plate is 2%. The reinforcement hasyield strength of 530 MPa. Its elastic modulus is 200 GPaand the Poisson�s ratio is 0.3.

The reinforcement effect is modeled using the mixturemodel described before. The loading is applied above themiddle point of the plate with varying standoff distanceH (Fig. 5(a)). The loading at different points on the frontface of the slab for a given charge is computed by the pro-gram based on CONWEP blast model, which relates thereflected overpressure to the scaled distance and alsoaccount for the angle of incidence of the blast wave [12].At each standoff distance, a series of calculations with dif-ferent charge weights are carried out to observe theresponse of the RC plate and the damage state, and thecritical charge weights corresponding to the three spallingdamage states are recorded.

Fig. 6 shows the critical charge weight–standoff distancediagram corresponding to the state of ‘‘light cracking’’.Coordinates above the critical curve produces no concretedamage, while the coordinates below the curve will producelight cracking. Shown in the figure are also the experimen-tal data points at this limit state.

From the numerical results in Fig. 6, three distinctiveregions can be identified. The first region is from point Aup to point B, where the critical conditions are basicallyindependent of the plate dimension (span length in partic-ular). This is because in such cases the charge is very closeto the plate, so that the effect is primarily localized andhence is not significantly affected by the boundaryconditions. From point B (6 kg@0.5 m) to point C

Fig. 5. Reinforced concrete plate configuratio

(100 kg@3 m) appears to be a dimension sensitive region,and it is featured by the local damage combined with par-tial global failure along the fixed sides. The plate withshorter span (L/t = 5) begins to show the boundary effectearlier (starting from 6 kg@0.5 m), followed by the casewith L/t = 10 (starting from 30 kg@1 m) and then L/t =15 (starting from 70 kg@1.3 m). Beyond point C, the dam-age will involve complete boundary failure with brokenedges; as a result, the spall damage becomes less depen-dent on the span length. Of course, under such a failuremode the spall damage may no longer be of a majorconcern.

The numerical results compare reasonably well with theexperimental data. The approximate one-dimensional pre-diction results [2] appear to be on the conservative side inthe small charge-standoff distance range, and this isbelieved to be due to the ignorance of the non-planar stresswave.

The best fit curve of the numerical results for the lightcracking state for plates with larger L/t ratios (L/tP 10)can be generally expressed as

y ¼ �0:2018þ 0:4051w0:311 ð0:2 kg 6 w 6 100 kgÞ; ð9Þ

ns (4 sides fixed): (a) overview; (b) mesh.

Fig. 8. Typical threshold spall scenarios (L/t = 10): (a) 1 kg char-ge@0.13 m; (b) 10 kg charge@0.56 m; (c) 50 kg charge@1.56 m.

0.01 0.1 1 10 1000.01

0.1

1

10

Stan

doff

(m

)

Fit function of simulation resultsExperimental results (McVay, 1988)L/t=5.0L/t=10L/t=151D model assuming spall velocityVelocity=15.24 m/s (Nash et al. 1995)Velocity=30.28 m/s (Nash et al. 1995)

K. Xu, Y. Lu / Computers and Structures 84 (2006) 431–438 437

where y is the charge standoff distance in meters, w is thecharge weight in kg. The curve is capped at w = 100 kgwhich signifies the dominance of global failure of the plate.

Fig. 7 depicts the critical state corresponding to thethreshold for spalling category. The numerical results agreefavorably with the experimental data. The effect of theboundary conditions for threshold spalling is similar tothe light cracking criterion; however, the dimension-sensi-tive region appears to shift leftwards to start at approxi-mately 5 kg@0.4 m and end at around 100 kg@2 m.When the explosive loading is about 5 kg, the thresholdcurve for the case with L/t = 5 begins to divert from theother curves, signifying the influence of the boundaryeffect. When the charge weight reaches 100 kg with a stand-off distance of 2 m, almost complete boundary failurewould occur. For the cases with L/t = 10, and L/t = 15,the boundary effect becomes significant from about15 kg@0.7 m and 75 kg@1 m, respectively. Beyond thecharge weight of about 100 kg, the three curves cometogether due to boundary failure.

Fig. 8 illustrates some typical threshold spall scenarios.The boundary effect can be clearly observed fromFig. 8(c). The critical curve based on the numerical resultsfor plates with L/t P 10 can be generally approximated bythe following expression:

y ¼ �0:445þ 0:5763w0:2314 ð0:35 kg 6 w 6 100 kgÞ.ð10Þ

The critical curve for medium spall damage, defined asspall penetration to around 1/3 plate thickness, is plottedin Fig. 9. The numerical results compare well with theexperimental data. It is interesting to note that the numer-ical simulation results do not exhibit remarkable differencefor different span length-to-thickness ratios for this state ofdamage. This may be explained by the fact that the relativeamount of charge is increased to generate the mediumspallation as compared to the light cracking and thresholdspall states. Consequently, boundary failure occurs in thepotential dimension-sensitive distance range as observedin the light cracking and threshold spall cases, rendering

0.1 1 10 1000.01

0.1

1

10

Stan

doff

(m

)

Explosive weight (kg)

Fit function of simulation resultsExperimental results (McVay, 1988)L/t=5.0L/t=10L/t=151D model assuming spall velocityVelocity=15.24 m/s (Nash et al. 1995)Velocity=30.48 m/s (Nash et al. 1995)

Fig. 7. Critical charge weight–standoff distance for no damage forthreshold spall category.

Explosive weight (kg)

Fig. 9. Critical charge weight–standoff distance for no damage formedium spall category.

the boundary effect to become insignificant in this range.The boundary failure can be observed from the damagecontours shown in Fig. 10(c) and (d).

It is noted in Fig. 10(a) that a crater (dent) is producedwith small charge at very close distance.

The fit curve for the medium spall criterion based on thenumerical simulation results can be expressed as

y ¼ �0:0328þ 0:1476w0:4827 ð0:5 kg 6 w 6 100 kgÞ.ð11Þ

From Fig. 9, it can also be observed that the predictionusing the simplified one-dimensional wave approach tends

Fig. 10. Typical medium spall scenarios (L/t = 15): (a) 1 kg char-ge@0.12 m; (b) 5 kg charge@0.5 m; (c) 50 kg charge@0.87 m (boundaryfailure visible); (d) 100 kg charge@ 1.27 m (boundary failure visible).

438 K. Xu, Y. Lu / Computers and Structures 84 (2006) 431–438

to overestimate the blast effect in causing spalling in thesmall charge range and hence remains on the conservativeside as for the other categories of spallation.

4. Conclusions

The spalling of reinforced concrete plates subjected toexplosive loading is one of the most important problemsfor safety analysis in fortification engineering. The previousstudies on this topic are usually based on material sampletesting and simplified approximate one-dimensional waveanalysis. Whereas these approaches provide useful bench-mark results in understanding the problem, they cannotreproduce the three-dimensional effect and the contributionof the global structural response in the spalling process.

In this study, a three-dimensional numerical model is setup for the simulation of spallation of RC plates. Thedynamic enhancement factor of concrete strength isobtained from a separate study based on continuum frac-ture mechanics and relevant experimental data. Threeconcrete damage categories under blast loading are investi-

gated, namely no-damage (light cracking) category, thres-hold spall damage category and medium spall damagecategory. The respective limit criteria in the form of criticalcharge weight–standoff distance relationships are obtainedfrom the parametric calculations using the numericalmodel. Comparison of the curves based on the numericalresults with existing experimental data shows a favorableagreement. In particular, the numerical results reveal threequantity-distance regimes. In the ‘‘small’’ quantity regime,the critical standoff distance is also small, subsequently theeffect is a localized phenomenon and hence the spallation israther independent of the boundary conditions. However,non-planar wave effect appears to be significant in suchcases; and because of this, the simplified one-dimensionalwave model tends to overestimate the spallation effect ofthe blast. In the ‘‘intermediate’’ quantity regime, the spall-ation appears to be dimension sensitive. This is becausepartial failure occurs along the fixed boundary before spall-ing takes place around the middle of the plate; and subse-quently, the limit state curve will change course due to thealtered boundary condition. The ‘‘large’’ regime is domi-nated by the global failure along the boundary lines.Empirical equations are proposed as criteria for no-dam-age (up to light cracking), threshold spall and medium spallcategories.

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