Debris velocity of concrete structures subjected to explosive loading

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2006; 30:917–926Published online 15 February 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.504

Debris velocity of concrete structures subjectedto explosive loading

Kai Xun,y and Yong Lu

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

SUMMARY

When designing above-ground ammunition storage facilities, one has to take into account the debrishazard resulting from accidental explosions. The purpose of this paper is to develop a predictive methodfor debris dispersion around an ammunition storage site in case of an accidental detonation in a reinforcedconcrete storage structure. The concrete slabs/walls break up into debris when it is overloaded by theinternal blast. The debris velocity is one of the important parameters to describe the debris dispersion. Theparameters that affect the debris velocity are complex. This study adopts the energy approach to simplifythe formulation. The failure process in a relatively thin concrete slab/wall is treated using the concept ofexpansion. Based on energy conservation, a general formula is derived for the debris launch velocity in acubicle structure subjected to internal blast loading. The dynamic strength of concrete and reinforcementare considered in the fracture process. The analytical results are found to be consistent with the relevantexperimental results. Copyright # 2006 John Wiley & Sons, Ltd.

KEY WORDS: concrete; reinforcement; explosion; dynamic strength; debris velocity

1. INTRODUCTION

When designing an ammunition storage magazine, one has to take into account the possibilityof accidental explosions. Of particular concern regarding the harmful physical effects of anaccident is the concrete debris throw, which can be a governing factor for the determination ofthe safe sitting distance of above-ground ammunition magazines and other explosives storagefacilities. Experimental and analytical studies in recent years have led to the development ofsome conceptual and empirical approaches for the prediction of initial debris parameters anddebris throw distance.

Under blast loading, the concrete damage and fragmentation result from both impulsiveloading by stress waves and gas-driven fracture propagation. Studies [1–3] tend to support theview that stress waves generated by the detonation of an explosive charge are responsible for the

Received 19 June 2005Revised 28 September 2005Accepted 3 January 2006Copyright # 2006 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: Kai Xu, School of Civil and Environmental Engineering, Nanyang Technological University,Singapore 639798, Singapore.

development of a damage zone in the concrete material and the subsequent fragment sizedistribution, while the explosion gases are important in the separation of a crack pattern thathas already been formed during the passage of the stress wave, and in the subsequent launch ofthe fragments. Fragment formation is deemed to be a much slower process than the propagationof the stress wave; however, there is evidence that the fragment size is determined immediatelyafter the passage of the stress wave and long before the gas penetration and throw movementtake place [3,4]. The findings by Wilson [5] suggest that the information about fragment sizedistribution during shock loading is important since a majority of the fragments are formed inthis stage. In other words, the fragment formation is under very high loading rate; hence, thestrain-rate effect on material properties plays an important role in the break-up andfragmentation process.

The purpose of this study is to develop a predictive method for debris dispersion aroundammunition storage sites in case of an accidental detonation in a cubicle structure. The concretestructure breaks up into debris when it is overloaded by an internal high explosive detonation.The debris velocity is one of the important parameters to describe the debris dispersion. Thispaper derives the formulation based on energy considerations. The concrete slabs/walls areassumed to have relatively small thickness, such that the deformation is considered as primarilyexpansion in the energy formulation. The dynamic strength of concrete and reinforcement areconsidered in the fracture process. Equations are derived for calculating fragment velocity of theconcrete slabs/walls subjected to explosive loading. The analytical results are compared withrelevant experimental data.

2. DETERMINATION OF DEBRIS VELOCITY

Closed-space reinforced concrete structures are considered in this paper. Under an internalblast, the structure is damaged and breaks up due to expansion under blast shock waves andexplosive gas. The explosive energy is consumed in the fragmentation of structural componentsand the debris launch. The explosion pressure works along the path from the explosive charge tothe fragment units.

Generally speaking, the total energy, U, for a brittle structure undergoing crack developmentand debris launching can be expressed as

U ¼ Uc þUk ð1Þ

where Uc is the energy consumed by the damage and fragmentation of the concrete structure,and Uk is the kinetic energy in the system due to fragment motions.

Consider a nominal fragment unit, i. The energy consumed in the failure process can beexpressed as

Uci ¼ZVi

u dV ð2Þ

where u is the density of the structure failure energy. The expression of the failure energy will bediscussed in detail later in the application.

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:917–926

K. XU AND Y. LU918

When the loading rate is high, the mechanical response of a material is generally different fromthat at a low loading rate. Such rate dependence is observed for nearly all brittle materials [6].Concrete also exhibits an enigmatic phenomenon of increased resistance when it is loaded at a veryhigh rate. The dynamic strength of the structure increases with the strain rate.

The rapid stain rate effect on the tensile strength of the reinforcing steel can be modelled as [7]

fd

fs¼

’e10�4

� �a

ð3Þ

where, for the yield stress,

a ¼ 0:074� 0:04fy

414ð4Þ

and, for the ultimate stress,

a ¼ 0:019� 0:009fy

414ð5Þ

where fy is the static yield strength of the reinforcing steel.

10-8 10-6 10-4 10-2 100 1020

1

2

3

4

5

6

7

8

Present model

Tedesco et al. (1993)

Reinhardt et al. (1990)

Weerheijm (1992)

John et al. (1987)

Strain rate ε

Dyn

amic

/sta

tic s

tren

gth

Figure 1. Normalized concrete tensile strength vs strain rate.


DEBRIS VELOCITY OF CONCRETE STRUCTURES 919

In this study, the concrete dynamic strength change with strain rate is modelled based on atheoretical model [8] in conjunction with a collection of experimental results [9–12], as shown inFigure 1. The function can be approximated as

fcd

fcs¼ 1þ 1:505’e0:295 ð6Þ

where fcd and fcs are the dynamic and static tensile strength of the concrete, respectively.There has not been consistent experimental evidence to show the strain-rate effect on the

dynamic ultimate strain for concrete and reinforcing steel [7,13]. For this reason, it is assumedthat the dynamic ultimate strain of concrete and reinforcement remain unchanged at differentstrain rates.

After the dynamic stress reaches the dynamic material strength, the fragments are formed.Driven by explosion gas, fragments continuously accelerate. When the fragment velocity reachesthe expansion velocity of the explosion gas, the driving action of the expansion gas ceases; at thispoint, the fragment velocity reaches its maximum. This velocity is referred to as the maximumfragment velocity. After this moment, the fragments go on flying due to their inertia forces,while decelerating due to the air drag. In fact, during the acceleration of the fragments, theexplosion gas can push its way through the interstices between the fragments, and surpass them;this quickly increases the air pressure ahead of the fragment, and thus shortens the accelerationprocess. The kinetic energy of the fragments when they reach the maximum fragment velocitycan be expressed as

Uki ¼1

2Virv2i ð7Þ

where Vi is the volume of the fragment, r is the density of the material and vi is the velocity ofthe fragment.

In the process of the change of the concrete structure from the original un-deformed state toexpansion, fragment formation and fragment acceleration, the work is done by the explosiveloading. The explosion gas pressure works along the path from the explosive charge to thefragment of the structure. For a given fragment, when the acceleration process ends, the workdone by the explosive charge is given by

Wig ¼Z re

r0

SiPg dr ð8Þ

where Wig is the work done by the explosion gas pressure in the fragment acceleration process,r0 is the original distance of the fragment to the blast centre, re is the distance of the fragment tothe blast centre at the moment when the fragment reaches its maximum velocity, Pg is theinstantaneous pressure of explosion gas, and Si is the front area of the fragment in contact withthe blast wave.

According to Sun [14], the overpressure of the blast in a closed-space structure may beexpressed as

Pg ¼ 0:1692m

SL

� �1=3þ0:0269

m

SL

� �2=3þ2:031

m

SL

� �ð9Þ

where Pg is the overpressure, m is the mass of the explosive charge, S is the internal cross sectionof the closed-space structure and L is the length of the internal space.


K. XU AND Y. LU920

The maximum overpressure at the shock wave front, Ps, can be written as [15]

Ps ¼14:0717

%Rþ

5:5397%R2�

0:3572%R3þ

0:00625%R4

� �� 0:098 MPa 0:054 %R40:3

Ps ¼6:1938

%R�

0:3262%R2þ

2:1324%R3

� �� 0:098 MPa 0:34 %R41

Ps ¼0:662%Rþ

4:05%R2þ

3:288%R3

� �� 0:098 MPa 14 %R410

%R ¼ r=ffiffiffiffim3

pm=kg1=3

ð10Þ

where %R is the scaled distance, r is the actual distance of the point considered from the chargecentre and m is the mass of the charge.

The duration of the shock wave overpressure t can be obtained according to Henrych[15] as

t ¼ 1:5� 10�3ffiffir

p ffiffiffiffim6

pðsÞ ð11Þ

where r is the distance from the blast centre.The energy delivered to the fragment i by blast shock overpressure can be calculated as

Wis ¼I2i2Mi

¼1

2Mi

ZPrsðtÞ dt

� �Si

� �2ð12Þ

where Ii is the shock wave impulse, Mi is the mass of the fragment i, Si is the front area of thefragment in contact with the shock wave, t is the duration of the overpressure, and Prs is thereflection overpressure of the shock wave.

Because of the relatively small thickness, it can be assumed that during the passage of theshock stress wave, there is no attenuation of the stress intensity across the wall thickness. Thestrain rate in the concrete due to the shock overpressure can then be approximated as

’ecd ¼Prs

Ect=2ð13Þ

where ’ecd is the strain rate, Ec is Young’s modulus of concrete, and the reflection overpressurePrs is usually between 2 and 8 times the incident shock wave overpressure Ps [15]. For the close-in range, the reflection overpressure factor is on the higher end. The present study concernsrelatively high loading density and near-range cases, so the reflection overpressure is chosen tobe 5 times the shock wave overpressure. The strain rate of reinforcement is assumed to be thesame as that of the surrounding concrete. According to energy conservation, the kinetic energyof the fragment can be expressed as

Uki ¼Ui �Uci ¼Wig þWis �Uci

¼Z re

r0

SiPg drþI2i2Mi

�1

2

ZVi

u dV ð14Þ



For the fragment i under consideration, the ejection velocity can then be obtained by thefollowing equation:

vi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

Z re

r0

SiPg drþI2i2Mi

�1

2

ZVi

u dV

� �.Mi

sð15Þ

3. APPLICATION AND COMPARISON WITH EXPERIMENTAL OBSERVATIONS

3.1. Experimental arrangement

The debris throw from an internal detonation in a concrete structure is a complex process. Anexperimental study of the debris launch velocity for clamped reinforced concrete slabs wasconducted by Dorr et al. [16]. The fragmentation process was recorded with a high-speed videocamera and a photo pole. The frame rate of the high-speed video camera is 200 framesper second. The camera speed was appropriate to show the overall effects of the debris motion.

The test arrangement was a subsurface cubicle chamber of 1 m� 1 m� 1 m dimensions andan inner volume V ¼ 1 m3 (Figure 2). The reinforced concrete slab was mounted on top of thecubicle chamber. The test slab was 100mm thick with two layers of reinforcement. The concretewas B35 class (35MPa compressive strength), with Young’s modulus Ec ¼ 27 GPa; Poisson’sratio nc ¼ 0:2; mass density r ¼ 2550 kg=m3; and static tensile strength fs ¼ 4 MPa: Thematerial properties of the reinforcement bar are mass density 7800 kg=m3; yield strength337MPa, yield strain 0.002, Young’s modulus 200GPa, and Poisson’s ratio nr ¼ 0:3: Thediameter of the reinforcement bars is 8 mm and the reinforcement ratio is 0.6%. The spacing ofthe reinforcement grids was 100mm. Hemispherical HE Pentolite charges (Plastic Nitropenta)were detonated at the centre of the chamber. The loading density varied in the range of0.125–4 kg/m3.

Reinforced Concrete

Steel Box

0.5 m

0.5 m

Charge

0.5 m 0.5 m

Figure 2. Configuration of the test box for clamped concrete slab tests.


K. XU AND Y. LU922

3.2. Analytical results

For brittle materials like concrete, the fragments moving a distance along the slab/wall thicknesscan be considered as completely fractured, so that the explosion gas begins to surpass thefragment movement through the interstices between the fragments. Therefore, the locationswhere the fragment starts and ceases to accelerate may be considered as r0 and r0 þ t;respectively, where t is the thickness of the slab/wall. Thus, the work done during the fragmentacceleration process by the explosive gas pressure can be calculated as

Wg ¼Z r0þt

ro

SPg dr

¼ 0:1692m

SðLþ 0:5tÞ

� �1=3

tþ 0:0269m

SðLþ 0:5tÞ

� �2=3

tþ 2:031m

SðLþ 0:5tÞ

� �t

¼ ð0:0166m1=3 þ 0:0026m2=3 þ 0:1934mÞ � 106 ð16Þ

where m is the mass of the explosive charge.The energy delivered by the shock wave can be evaluated by considering Equations (10)–(12),

as well as the reflection factor, given the time function for the shock overpressure. It can also becalculated directly based on the impulse formula. Using the impulse formula from Henrych [15]and by regression for the curve, it has

Ws ¼I2

2M¼

1

2MS

ZPrsðtÞ dt

� �2

¼ 1920 m2 ð17Þ

When the equivalent stress and equivalent strain reach the tensile dynamic strength and failurestrain during the stress wave passage, the material is considered to be at the failure state. Theenergy consumed in the failure process can be expressed as

Uc ¼ZV

u dV ¼ZV

Z %e

0

s de dV ¼1

2

ZV

½fyd %eyd þ fcd %ecd� dV

¼1

2St½fyd %eyd Rr þ fcd %ecd Rc� ð18Þ

where %eyd is the reinforcement dynamic yield strain, %ecd is the dynamic failure strain ofconcrete, fyd is the reinforcement dynamic yield strength and fcd is the concrete dynamicstrength and Rc and Rr are the volumetric ratio of concrete and reinforcement in the slab,respectively. It is worth pointing out that Equation (18) applies to a nominal fragment unit. Itdoes not directly involve the energy for cracking and fragment formation, and this is based onthe consideration that the release of the cumulated elastic strain energy in Equation (18) will beresponsible for cracking and fragmentation, so it does not require additional work from theblast pressure.

Herein, Rr ¼ 0:006 and Rc ¼ 0:994: The yield strain of the reinforcement is approximately

%eyd ¼ 0:002: For concrete, the tensile strain at peak stress is 0.00015 [17]. The ultimate strain canbe chosen as %ecd ¼ 0:0003: After substituting the material parameters into the equation, thefollowing approximate equation can be obtained:

Uc ¼ 375:68þ 9:72 ln m ð19Þ



The mean velocity of the debris can be calculated as

v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðWg þWs �UcÞ=M

qðm=sÞ ð20Þ

where M is the total mass of the plate and m is the mass of explosive charge.For the present case, the final debris velocity can be obtained after substituting Equations

(16), (17) and (19) into Equation (20) as

v ¼ 650

ffiffiffiffiffiffiffigLMa

rðm=sÞ ð21Þ

where g is the charge loading density in kg/m3, Ma is the specific mass per slab/wall area inkg/m2 and L is the chamber volume divided by the cross-section area.

3.3. Comparison with experimental results

The average debris launch velocity for the test cases is obtained using Equation (21). Figure 3plots the analytical results compared with the clamped frame test results, as well as the valuesdetermined with the empirical formula derived from the test data. The empirical formula iswritten as follows [18]:

v ¼ 525

ffiffiffiffiffiffiffigLMa

rðm=sÞ ð22Þ

0.1 1 1010

100

Present theory results

Empirical results Dorr et al. (2002)

Small frame Dorr et al. (2002)

Big frame Dorr et al. (2002)

Vel

ocity

(m

/s)

Loading density (kg/m3)

Figure 3. Variation of debris launch velocity with charge loading density.


K. XU AND Y. LU924

where v is debris velocity, g is the charge loading density, L is the cubicle volume divided by thecross-section area, and Ma is the mass per plate area.

From Figure 3 it can be observed that the present analytical results agree well with theexperimental data. On the other hand, the prediction by the empirical formula of Equation (22)is lower than the experimental results.

4. CONCLUSIONS

On the basis of the physical process of the explosion action, a simple approach to calculate thefragment velocity is formulated in this paper. The energy conservation on the reinforcedconcrete fragmentation and debris launch is used to derive the general equations of debrislaunch velocity. The results obtained for a series of test cases are compared with theexperimental data and the relevant empirical equation. The analytical results are found to be infavourable agreement with the test results. This method can be applied for a rapid prediction ofthe debris launch velocity when engineering the safety design of explosives storage facilities.

It is worth pointing out that the blast effect and concrete fragmentation are very complexphenomena, and various uncertainties exist. For this reason, the semi-theoretical method asadopted in this paper is deemed an appropriate approach for engineering applications.However, further improvement is possible with more refined assumptions on the energydissipation mechanisms, distribution of blast load, fragmentation process, etc. so that therobustness and the applicability of the prediction can be further enhanced.

REFERENCES

1. Simha KRY, Fourney WL, Dick RD. An investigation of the usefulness of stemming in crater blasting. Proceedingsof the Second International Symposium on Rock Fragmentation by Blasting, Colorado, Keystone, 1987; 591–599.

2. Hommert PJ, Kuszmaul JS, Parrish RL. Computational and experimental studies of the role of stemming incratering. Proceedings of the Second International Symposium on Rock Fragmentation by Blasting, Colorado,Keystone, 1987; 550–562.

3. Brinkman JR. Separating shock wave and gas expansion breakage mechanisms. Proceedings of the SecondInternational Symposium on Rock Fragmentation by Blasting, Colorado, Keystone, 1987; 6–15.

4. Zhao Y, Huang J, Wang R. Fractal characteristics of mesofractures in compressed rock specimens. InternationalJournal of Rock Mechanics and Mining Science & Geomechanics Abstracts 1993; 30:877–882.

5. Wilson WH. An experimental and theoretical analysis of stress wave and gas pressure effects in bench-blasting.Ph.D. Dissertation, University of Maryland, 1987.

6. Clifton RJ. Response of materials under dynamic loading. International Journal of Solids and Structures 2000;37:105–113.

7. Malvar LJ. Review of static and dynamic properties of steel reinforcing bars. ACI Material Journal 1998; 95(5):609–616.

8. Lu Y, Xu K. Modelling of dynamic behaviour of concrete material under blast loading. International Journal ofSolids and Structures 2004; 41:131–143.

9. Tedesco JW, Ross CA, Kuennen ST. Experimental and numerical analysis of high strain rate splitting tensile tests.ACI Material Journal 1993; 90:162–169.

10. Reinhardt HW, Rossi P, Mier van JGM. Joint investigation of concrete at high rates of loading. Material andStructure Research Testing 1990; 23:213–216.

11. Weerheijm J. Concrete under impact tensile loading and lateral compression. Doctoral Thesis, TNO Prins MauritsLaboratory, 1992.

12. John R, Shah SP, Jeng YS. A Fracture Mechanics Model to Predict the Rate Sensitivity of Mode I Fracture ofConcrete. Northwestern University: Evanston, 1987.

13. Park SW, Xiao Q, Zhou M. Dynamic behaviour of concrete at high strain rates and pressure. Part II: numericalsimulation. International Journal of Impact Engineering 2001; 25:887–910.

14. Sun YB. Explosion Action and Explosive Infilling Design. Defence Industry Publication 1987; 235–238.



15. Henrych J. The Dynamics of Explosion and Its Use. Elsevier: New York, 1979.16. Dorr A, Michael K, Gurke G. Experimental investigations of the debris launch velocity from internally overloaded

concrete structures. Final Report DLV 4-2002, Institut Kurzzeitdynamik Ernst-Mach-Institut, 2002.17. Malvar LJ, Ross CA. Review of strain rate effects for concrete in tension. ACI Material Journal 1998; 95(6):735–739.18. Grundler J, Guerke G. Debris launch velocity}a new approximation formula. 29th DDESB Seminar, New Orleans,

Louisiana, 2000.


K. XU AND Y. LU926

Documents

Debris velocity of concrete structures subjected to explosive loading