Annals of Nuclear Energy 68 (2014) 262–275

Contents lists available at ScienceDirect

Annals of Nuclear Energy

journal homepage: www.elsevier .com/locate /anucene

Local impact effects on concrete target due to missile: An empiricaland numerical approach

http://dx.doi.org/10.1016/j.anucene.2014.01.0150306-4549/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Address: Flat Number-11, Satpura, Anushaktinagar,Mumbai 400094, India. Tel.: +91 9757052920; fax: +91 022 25995143.

E-mail addresses: rajivranjan_barc@yahoo.com (R. Ranjan), sauvik@iitb.ac.in(S. Banerjee), rksingh175@yahoo.co.in (R.K. Singh), pbanerji@iitr.ac.in (P. Banerji).

Rajiv Ranjan a,⇑, Sauvik Banerjee a, R.K. Singh b, Pradipta Banerji c

a Dept. of Civil Engg, Indian Institute of Technology, Mumbai 400076, Indiab Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai 400085, Indiac Indian Institute of Technology, Roorkee, Uttarakhand 247667, India

a r t i c l e i n f o

Article history:Received 25 July 2013Received in revised form 28 November 2013Accepted 8 January 2014Available online 21 February 2014

Keywords:Local impact effectReinforced concrete targetEmpiricalAnalyticalNumerical simulation

a b s t r a c t

Concrete containment walls and internal concrete barrier walls of a Nuclear Power Plant safety relatedstructures are often required to be designed for externally and internally generated missiles. Potentialmissiles include external extreme wind generated missiles, aircraft crash and internal accident generatedmissiles such as impact due to turbine blade failure and steel pipe missiles resulting from pipe break. Theobjective of the present paper is to compare local missile impact effects on reinforced concrete targetusing available empirical formulations with those obtained using LS-DYNA numerical simulation. Theuse of numerical simulations for capturing the transient structural response has become increasinglyused for structural design against impact loads. They overcome the limits of applicability of the empiricalformulae and also provide information on stress and deformation fields, which may be used to improvethe resistance of the concrete. Finite element (FE) analyses of an experimental impact problem reportedby Kojima (1991) are carried out that are able to capture the missile impact effects; in terms of local andglobal damage. The continuous surface cap model has been used for modelling concrete behaviour. Arange of missile velocity has been considered to simulate local missile impact phenomenon and modesof failure and to capture the concrete response from elastic to plastic fracture. A comparison is then madebetween the empirical formulations, numerical simulation results, and available experimental results ofslab impact tests. While the numerical simulation is able to capture the experimental trend and results, acomparison of penetration depth and scabbing and perforation limits as per different empirical formula-tion shows substantial divergence.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Concrete has been widely used over many years by structuralengineers in the design and construction of protective structuresto resist impact loads. Impact loads are of sub-second durationand of magnitude many times higher than any other loads thatact during the design life of the structure. Concrete containmentwalls and internal concrete barrier walls of a Nuclear Power Plantsafety related structures must be analyzed to ensure requisite mar-gin for missile impact. Potential missiles include external extremewind generated missiles, aircraft crash, and internal accident gen-erated missiles (turbine blade, and steel pipe missiles resultingfrom pipe break). Impacting missiles can be classified as either

‘hard’ or ‘soft’ depending upon whether the missile deformabilityis small or large relative to the target deformability.

The three important methods of studying local effects on a con-crete target arising from missile impact are experimental, analyticaland numerical simulation. Empirical formulae based on experimen-tal data are especially important in this field due to the complexityof the phenomena. These empirical formulations are defined interms of impact parameters such as mass, velocity and shape ofmissile, its rigidity, relative stiffness, and mass ratio of missileand target, mechanical properties of the target, reinforcementamount, size and area of impact. Simple analytical models are avail-able which offers more efficient and economic way of predicting thelocal missile impact effects and help to continue the experimentallybased empirical formulae and often extend the range of parametersfor which it could be valid. However, with the developments ofcomputational tools, computational mechanics and material con-stitutive models, the numerical simulation of missile impact effectshave become more exact and reliable. In addition, with exhaustive

R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275 263

numerical data generated through computation, carefully con-trolled limited number of experiments can be performed.

Islam et al. (2011) conducted numerical simulation of highvelocity ogive-shaped projectile penetration of concrete targetand emphasized the importance of element erosion in the lagrang-ian based FE method. Jose et al. (2010) has indicated the impor-tance of effect of reinforcement and erosion criteria for betterrepresentation of local missile impact effect on concrete slab. Sangiand May (2009) has reported numerical simulation of an experi-mental drop tower test using two different mesh sizes and two dif-ferent concrete model and was able to capture the kinematicresponse under low velocity impact. Tahmasebina (2008) carriedout numerical simulation of slab impact test and has found diver-gence in terms of mesh convergence and failure modes. Murrayet al. (2007) carried out drop tower impact test on plain, under-reinforced and over-reinforced concrete beams and has showngood agreement in terms of deflection time histories and damagemode under low strain-rate loading. Cotovos et al. (2008) studiedthe localized impact loading on reinforced beams under highstrain-rate loading such as blast and ballistic loads. Liu et al.(2011) have studied the oblique penetration of missile into RCCslab using a dynamic constitutive model based on composite dam-age formulations for tension and compression failure and showedgood agreement with experimental data. Suanno et al. (2003) car-ried out impact analysis of reactor containment for impact loadand has discussed the importance of constitutive modelling of im-pact related parameters to avoid error in numerical simulation.

This paper presents an overview on the empirical formulationof studying the local missile impact effect and computation of pen-etration depth, scabbing and perforation limits for a typical Rein-forced Cement Concrete (RCC) slab panel to assess the localimpact resistance. The paper attempts to quantify the factor ofsafety that is in-built in these formulations as most of them areconservative in nature. Seldom structural adequacy check requiresexact determination of local missile impact effects as the marginsavailable are less and it becomes pertinent to quantify the exactstructural response for qualification against such loads.

A benchmark problem has been used to determine the applica-bility of these empirical formulations in predicting local missile im-pact effects. A numerical simulation using finite element (FE) codeLS-DYNA has been carried out to capture the different modes of fail-ure and capture and concrete response from elastic to plastic failurethat will occur for low velocity impact to high velocity impact. Thecontinuous surface cap model has been used for modelling concretebehaviour. This model includes isotropic constitutive equations forthree stress-invariant shear surfaces with translation for prepeakhardening, a hardening cap that expands and contracts, damage-based softening with erosion and rate effects for high strain rateapplications. The reinforcement bars have been modelled as trusselement using plastic kinematic material model using strain rate ef-fect of Symonds–Cowper model.

Numerical simulation for studying impact problem on concretesuffers from mesh sensitivity. The present paper addresses this is-sue while simulating impact problem within the framework ofcontinuum damage modelling and highlights its effect on thestructural response. Mesh sensitivity study has been carried outusing three different mesh sizes to arrive at a FE model that is ableto predict the concrete response using the concrete material mod-el. Segment based penalty formulation has been used to model thecontact interface between missile and concrete. Nodes to surfacecontact algorithm has been used to model contact between missileand rebar, if any, which will occur for high velocity impact. A rangeof missile velocity has been considered to simulate local missileimpact phenomenon and modes of failure and to capture the con-crete response from elastic to plastic fracture. A comparison is thenmade between the empirical formulations, numerical simulation

results and available experimental results of slab impact tests re-ported by Kojima (1991).

2. Local missile impact effects

The impact load effects can be broadly classified into two cate-gories: local and global. The local effects relate to the local structuralresponse in terms of spalling, scabbing, perforation, and penetra-tion. The global effect includes the structural response in terms ofglobal displacement, concrete cracking, yielding of steel and stres-ses in concrete and steel members for identifying critical zones.

Displacement of missile into the target with possible formationof inlet funnel (facing spalling) without passing through it is calledpenetration. Spalling is the ejection of target material from theproximal/front face of the target by reflection of tensile waves fromstructure behind surface. Scabbing is the ejection of structuralmaterial from the distal or back face of the target due to impacton the proximal side. Perforation is defined as the complete passageof the missile through the target with or without a residual velocity.

The impacting missile can be classified as ‘hard’ or ‘soft’depending upon the relative deformability of the missile and thetarget. If the deformability of the missile is negligible comparedto the target, the missile can be considered as hard. However, ifthe missile deformability is moderate or high compared to the tar-get, the missile can be considered to be soft. This paper basicallyconcerns with the hard missile.

3. Empirical formulations

The problem of missile impacting a concrete target is an extre-mely complicated phenomenon and hence empirical formulas areespecially important in this field. The empirical formulations forpredicting local missile impact effects for hard missiles are basedon the regression analysis of experimental test results conductedby striking projectiles/missiles on reinforced concrete target slabs.Large numbers of impact experiments involving reinforced concreteslabs are reported in literature (Kennedy, 1976; Sliter, 1980; Li et al.,2005). Various methods such as free fall, air guns, cannons and actu-ators have been used to obtain high velocity impact. Some of theempirical formulations also possess a partial theoretical basis.

Each of the empirical formulations have their range of applica-tion in terms of mass, velocity and shape and size of the missileowing limitations due to experimental setup and are based onthe assumption of rigid mass impacting a flexible target. Most ofthese formulations are unit-dependent which poses difficultywhile comparing different experimental results. The missile noseshape factors used in many of the formulae are ambiguous (Liet al., 2003) and thus introduce uncertainty in the determinationof the local missile impact effects.

The empirical formulae mentioned in this paper have been col-lected on the basis of publications and review works of Kennedy(1976), Bangash (1993) and Li et al. (2005). The penetration depth(x), perforation limit (e) and scabbing limit (hs) for some of the for-mulations in FPS (mass in lb, strength and elasticity in psi, length ininch, velocity in ft/s) and SI (mass in kg, strength and elasticity inPa, length in meter, velocity in m/s) units are re-produced.

3.1. Petry formula (1910)

The Petry penetration formula (Bangash, 1993; Li et al., 2005;Amirikian, 1950) is the oldest penetration formula and was oneof the most common formulas in the United States (US) for predict-ing the penetration depth for infinitely thick concrete target. Theformula was derived from the solution of equation of motion inwhich the instantaneous resisting force is expressed by a constant

264 R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275

term and a drag resisting component depending upon the square ofthe impact velocity V0:

x ¼ 12KpApLog10 1þ V20

215; 000

!FPS units

xd¼ k0

m

d3 Log10 1þ V20

19;974

!SI units

where Ap is missile section pressure expressed as its mass per unitarea. In the original Petry formula, the value of the concrete pene-trability factor Kp was independent of the strength of concrete. Inthe Modified Petry I formula, the concrete penetrability factor wassuggested as 0.00799 for plain concrete, 0.00426 for normal rein-forced concrete and 0.00284 for special reinforced concrete inwhich near and rear face reinforcement are laced with special tiesfor FPS units. In SI units, the concrete penetrability factor was sug-gested as 6.36E�04 for plain concrete, 3.39E�04 for normal rein-forced concrete and 2.26E�04 for special reinforced concrete.

In the Modified Petry II formula, (Amirikian, 1950) modifiedconcrete penetrability factor as per a function of concrete strengthand suggested perforation (e) and scabbing (hs) limits based onpenetration depth calculated from the above formula:

ed¼ 2

xd

andhs

d¼ 2:2

xd

3.2. Ballistic Research Laboratory’s (BRL) formula (1941)

The BRL formula (Kennedy, 1976; Li et al., 2005; Adeli andAmin, 1985) gives the penetration depth of rigid missile in an infi-nitely thick target as:

xd¼ 427ffiffiffiffi

fc

p m

d3

� �d0:2 V0

1000

� �1:33

FPS units

xd¼ 1:33 � 10�3ffiffiffiffi

fc

p m

d3

� �d0:2V1:33

0 SI units

where m and d are the mass and the diameter of the missile, respec-tively and fs in the unconfined compression strength of concrete.Based on the above penetration depth, Chelapati et al. (1972) sug-gested the perforation as:ed¼ 1:3

xd

The modified BRL formula for scabbing limit is (Kennedy, 1976):

hs

d¼ 2:0

xd

3.3. Army corps of engineer formula (ACE) (1946)

The ACE formula (Kennedy, 1976; Li et al., 2005; Chelapati et al.,1972) is based on experimental test results prior to 1943 from theOrdnance department of US Army and Ballistic Research Labora-tory and predicts the penetration depth of rigid missile in concretetarget. The penetration formulation gives a non-zero penetrationdepth when the velocity is zero.

xd¼ 282:6ffiffiffiffi

fc

p m

d3

� �d0:215 V0

1000

� �1:5

þ 0:5 FPS units

xd¼ 3:5 � 10�4ffiffiffiffi

fc

p m

d3

� �d0:215V1:5

0 þ 0:5 SI units

Based on regression analysis of experimental data from ballistictests performed in 1943 on 37 mm, 75 mm, 76.2 mm and 155 mmsteel cylinders, the ACE formula for perforation and scabbing limitswere given as:

ed¼ 1:32þ 1:24

xd

for 1:35 <xd< 13:5 or 3:0 <

ed< 18:0

hs

d¼ 2:12þ 1:36

xd

for 0:65 <xd< 11:75 or 3:0 <

hs

d< 18:0

In 1944, additional test data for 0.5 caliber bullets (12.7 mm indiameter) were obtained and the perforation and scabbing limitsfor the same range of validity were modified as:

ed¼ 1:23þ 1:07

xd

andhs

d¼ 2:28þ 1:13

xd

3.4. Modified NDRC formula (1946)

The NDRC formula (Kennedy, 1976, 1966; Bangash, 1993) wasput forward by the US National Defense Research Committee basedon the ACE formula, further testing and a penetration model whereit was assumed that the contact force increased linearly to a con-stant maximum value when the penetration depth is small. TheNDRC formula was proposed based on an approximate theory ofpenetration for a non-deformable missile penetrating a massiveconcrete target which offered a good approximation with theexperimental results available at that time. The penetration depthwas defined as a function of G-function given as:

G ¼ KNmd

V0

1000d

� �1:8

FPS units

xd¼ 2G0:5 for G � 1;

xd¼ Gþ 1 for G < 1

where K is the concrete penetrability factor considered same asModified Petry formulae and N is nose shape co-efficient given as0.72 for flat, 0.84 for hemispherical, 1.0 for blunt and 1.14 for sharpnose missile. In 1946, the study was stopped without completelydefining the K factor due to reduced interest in missile penetration.Later Kennedy (1966) suggested the K factor in terms of concretecylinder compressive strength by fitting experimental data avail-able for larger missile diameter. The modified NDRC penetrationformula defined by a new G-function and the penetration depth is:

G ¼ 180Nm

dffiffiffiffifc

p V0

1000d

� �1:8

FPS units

G ¼ 3:8 � 10�5 Nm

dffiffiffiffifc

p V0

d

� �1:8

SI units

The penetration depth is given as:

xd¼ 2G0:5 for G � 1;

xd¼ Gþ 1 for G < 1

The perforation and scabbing limits are predicted by extending theACE formula to thin targets and are given as:

ed¼ 1:32þ 1:24

xd

for 1:35 <xd

� �< 13:5 or 3 <

ed� 18;

ed¼ 3:19

xd

� �� 0:718

xd

� �2for

xd

� �� 1:35 or

ed� 3

hs

d¼ 2:12þ 1:36

xd

for 0:65 <xd

� �< 13:5 or 3 <

hsd� 18;

hs

d¼ 7:91

xd

� �� 5:06

xd

� �2for

xd

� �� 0:65 or

hsd� 3

The primary advantage of the NDRC formula is that since it is basedupon a theory of penetration it can be extrapolated beyond therange of available test data with a greater confidence. The theoryof penetration enables not only to calculate the total depth of pen-etration, but also to calculate the impact force–time history andpenetration-depth time history.

R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275 265

3.5. Ammann and Whitney formula (1976)

The Amman and Whitney formula (Kennedy, 1976; Li et al.,2005) was proposed to predict the penetration of explosively gen-erated small fragments at relatively high velocities over 300 m/s.The formula is expressed as:

xd¼ 282ffiffiffiffi

fc

p Nm

d3

� �d0:2 V0

1000

� �1:8

FPS units

xd¼ 6 � 10�4ffiffiffiffi

fc

p Nm

d3

� �d0:2V1:8

0 SI units

The formula is similar to the form given by ACE and NDRC formula.N is the nose shape factor defined in NDRC formula. The formulawas intended to predict the penetration of small explosively gener-ated fragments travelling at over 300 m/s.

3.6. Whiffen formula (1943)

The Whiffen formula (Bangash, 1993; Li et al., 2005) was givenby British Road Research Laboratory based on extensive range ofwartime data in the United Kingdom (UK) from penetration studiesof fragments from many types of bombs into reinforced concreteand extended investigations involving larger ranges of missilediameter and concrete aggregate size (a).

xd¼ 870ffiffiffiffi

fc

p !

m

d3

� �da

� �0:1 V0

1750

� �n

with n ¼ 10:70ffiffiffiffifc

0:25p FPS units

xd¼ 2:61ffiffiffiffi

fc

p !

m

d3

� �da

� �0:1 V0

533:4

� �n

with n ¼ 97:51ffiffiffiffifc

0:25p SI units

The range of application for the Whiffen formula is: (i) 0.5 < d < 38 in.,(ii) 0.3 < m < 22,000 lb, (iii) 800 < fc < 10,000 psi, (iv) 0 < V0 < 1750 ft/s and (v) 0.5 < d/a < 50 in FPS units or (i) 12.7 < d < 965.2 mm, (ii)0.136 < m < 9979.2 kg, (iii) 5.52 < fc < 68.95 MPa, (iv)0 < V0 < 1127.8 m/s and (v) 0.5 < d/a < 50 in SI units.

3.7. Kar formula (1978)

Kar (1978) revised the NDRC formula following regression anal-ysis to account for the difference in rigidity of the missile and tar-get material and is given as:

G ¼ 180Nm

dffiffiffiffifc

p V0

1000d

� �1:8 EEs

� �1:25

FPS units

G ¼ 3:8 � 10�5Nm

dffiffiffiffifc

p V0

d

� �1:8 EEs

� �1:25

SI units

wherexd¼ 2G0:5 for G � 1 and

xd¼ Gþ 1 for G < 1

where E and Es are Young’s modules of the missile and steel, respec-tively. The perforation and scabbing limits account for both the size ofaggregates, a and the Young’s modulus of the missile and is given as:

e� a ¼ 1:32dþ 1:24x for 1:35 <xd� 13:5;

e� ad¼ 3:19

xd

� �� 0:718

xd

� �2for

xd� 1:35

ðhs � aÞb ¼ 2:12dþ 1:36x for 0:65 <xd� 11:75;

hs � ad

b ¼ 7:19xd

� �� 5:06

xd

� �2for

xd� 0:65

where b ¼ EEs

� �0:2. If the material of the missile is steel, the penetra-

tion depth prediction formula is identical to the modified NDRCformula.

3.8. CEA–EDF perforation formula (1974)

Based on extensive study on protection of Nuclear Power Plantstructures, CEA–EDF (Commissariat à l’Énergie Atomique – Electricitde France) (Berriaud et al., 1978) developed a perforation limit for-mula based on the ballistic performance of reinforced concretestructure under missile impact. The formulation was based on aseries of drop-weight and air gun tests directed towards low veloc-ity impact.

ed¼ 0:82

m0:5V0:750

q0:125c f 0:375

c d1:5 SI units

where qc is the density of concrete and H0 is the slab thickness. Theballistic limit defined as minimum impact velocity to perforate thetarget was defined as:

VP ¼ 1:3q1=6c f 1=2

cdH2

0

m

!2=3

SI units

The range of application for the CEA–EDF perforation formula is: (i)0.5 < d/H0 < 1.5, (ii) 0:5 < m=ðqH3

0Þ < 1:5 and (iii) 20 < V0 < 200 m/s.

3.9. UKAEA formula (1990)

The United Kingdom Atomic Energy Authority (UKAEA) formulaLi et al., 2005; Barr, 1990 originated in UK based on the extensivestudies of the protection of Nuclear Power Plants in United King-dom. The UKAEA formula is further modification to NDRC formula,directed towards the low impact velocities which are more signif-icant to nuclear industry.

G ¼ 180Nm

dffiffiffiffifc

p V0

1000d

� �1:8

FPS units

G ¼ 3:8� 10�5 Nm

dffiffiffiffifc

p V0

d

� �1:8

SI units

The relation between G and penetration is given as:

xd¼ 0:275� ½0:0756� G�2 for G � 0:0726;

xd

¼ ½4G� 0:242�0:5 for 0:0726 < G < 1:065xd¼ Gþ 0:9395 for G � 1:065

The relation between G and penetration is suggested by Barr (1990)and is given as:

hs

d¼ 5:3G0:33

The perforation velocity Vp is further modified according to theCEA–EDF perforation formula and (Fullard, 1991) for a flat nosemissile and is given as:

VP ¼ Va for Va � 70m=s; VP ¼ Va½1þVa

500

� �2

� for Va > 70m=s

where

VP ¼ 1:3q16ck

12c

pH20

pm

!23

ðr þ 0:3Þ1=2 1:2� 0:6cr

H0

� �� �

where p and cr are the perimeter of the missile cross-section and re-bar spacing, respectively. qc is the density of concrete in kg/m3,kc = fc for fc < 37 MPa, and kc = 37 MPa for fc > 37 MPa.

The range of application of the UKAEA penetration formula is:(i) 25 < V0 < 300 m/s, (ii) 22 < fc < 44 MPa and (iii) 5000 < m/d3 < 200,000 kg/m3. The range of application of the UKAEA

266 R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275

scabbing formula is: (i) 3000 < m/d3 < 222,200 kg/m3, (ii)29 < V0 < 38 m/s and (iii) 26 < fc < 44 MPa. The range of applicationof the UKAEA perforation formula is: (i) 11 < V0 < 300 m/s, (ii)22 < fc < 52 MPa, (iii) 0.0 < r < 0.75% (% reinforcement each wayeach face), (iv) 0.33 < H0/(p/p) < 5.0, (v) 150 < m/p2H0 < 104 kg/m3

and (vi) 0.12 < cr/H0 < 0.49. If cr/H0 > 0.49, the CEA–EDF formulafor perforation is to be used. Barr (1990) concluded that the perfo-ration velocity for a hemispherical nose shape missile is higher by30% compared to that of a flat nose missile.

3.10. Bechtel formula (1976)

Bechtel Power Corporation (Sliter, 1980; Bangash, 1993; Liet al., 2005) developed the scabbing limit formula based on testdata applicable to Nuclear Power Plant structures. The scabbinglimit formula is applicable for hard missile such as solid steel rod.

hs

d¼ 15:5m0:4V0:5

0

f 0:5c d1:2 FPS units

hs

d¼ 38:98m0:4V0:5

0

f 0:5c d1:2 SI units

According to Bangash (1993) and Sliter (1980), Bechtel formula forsteel pipe missile is nearly 35% lesser compared to solid steel rod.

3.11. Stone and Webster formula (1976)

Stone and Webster (Sliter, 1980; Li et al., 2005) proposed thefollowing non-dimensional scabbing limit formula:

hs

d¼ mV2

0

cd3

!0:333

where c is a non-dimensional factor having a linear relationshipwith H0/d and varies from 0.35 to 0.37 when H0/d varies from 1.5to 3.0. The non-dimensional factor c is given as;

c ¼ 0:013H0

dþ 0:33

The range of application of the scabbing limit formula is: (i)20.7 < fc < 31 MPa and (ii) 1.5 < hs/d < 3.0. The scabbing limit agreeswith the test data of Sliter (1980).

3.12. Haldar–Hamieh formula (1984)

Haldar and Hamieh (Li et al., 2005; Haldar and Hamieh, 1998)suggested the use of an impact factor I given as:

I ¼ NmV20

d3fc

where N is the nose shape factor as defined in NDRC formula and I isa dimensionless impact factor. The penetration depth is given as:

xd¼ �0:0308þ 0:225I when 0:3 � I � 4:0;

xd¼ 0:6740þ 0:0567I when 4 < I � 21

xd¼ 1:1875þ 0:0299I when 21 < I � 455

Based on the above formula for penetration depth, it was suggestedthat perforation depth (e) could be calculated as per NDRC formula.The scabbing limit as given by NDRC can be used if I < 21. If I exceedthis value, the following formula is recommended:

hs

d¼ 3:3437þ 0:0342I for 21 < I < 385

3.13. Adeli–Amin formula (1985)

The impact factor I defined by Haldar and Hamieh was adoptedby Adeli and Amin (1985) to fit data collections on penetration,perforation and scabbing.

xd¼ :0416þ :1698I � :0045 I2 for 0:3 � I < 4;

xd¼ :0123þ :1960I � :008I2 þ :0001I3 for 4 � I < 21

ed¼ 1:8685þ :4035I � :0114 I2 for 0:3 < I < 21;

hs

d¼ 0:9060þ 0:3214I � 0:0106I2 for 0:3 � I < 21

The range of application for Adeli–Amin formula is: (i) 0.7 < H0/d < 18, (ii) 0.11 < m < 343 kg, (iii) 27 < V0 < 312 m/s and (iv) x/d 6 2.0.

3.14. Hughes formula (1984)

Hughes (1984) assumed that the penetration resistance first in-creased linearly (same as the assumption used in the NDRC formu-lae) and then decreased parabolically with the penetration depth,and suggested the following formulation for the penetration depth:

xd¼ 0:19

NhIh

S

where Nh is a missile nose shape coefficient, which is 1.0, 1.12, 1.26and 1.39 for flat, blunt, spherical and very sharp noses, respectively.They were obtained by fitting above equation to the predictions ofthe NDRC penetration formula for a given nose shape over its wholerange of applicability. Ih is a non-dimensional impact factor definedas:

Ih ¼mV2

d3ft

Hughes employed the tensile strength of the concrete instead of itscompressive strength as used in other formulations. Hughes ac-counted for the influence of the strain rate on the tensile strengthof concrete by introducing a Dynamic Increase Factor (DIF) S so thatthe tensile strength ft was replaced by Sft. The dynamic tensilestrength was then converted into the dynamic compressivestrength by multiplying by a constant coefficient. Hughes obtainedS through an empirical calibration with penetration results,

S ¼ 1:0þ 12:3 lnð1:0þ 0:03IhÞ

The perforation and scabbing limit is predicted as:

ed¼ 3:6

xd

forxd< 0:7;

ed¼ 1:58

xdþ 1:4

forxd� 0:7

hs

d¼ 5:0

xd

forxd< 0:7;

hs

d¼ 1:74

xdþ 2:3 for

xd� 0:7

4. Numerical approach

The numerical simulation of reinforced concrete structure sub-jected to impact load is a complex phenomenon. For reinforcedconcrete structure subjected to impact load, the material model-ling becomes more complex as it is a composite material and be-haves differently under tension and compression, involvingfactors such as: nonlinearity, strain rate effect, tensile stiffening,strain softening, and material failure under multiaxial stress state.Development of crack in concrete and the post failure behaviour isa major source of material non-linearity in reinforced concrete

R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275 267

structure and has to be accounted in the material model. In addi-tion to the conventional requirement of a material model to be ableto represent the above complex material behaviour, the other real-istic requirement is its ability to represent accumulation of dam-age, residual strength and contact interface modelling. For softmissile impact it should be also able to take care of self-contact.

The response of concrete to high strain rate loading such as im-pact from a missile and impulse from blast loading has been ofinterest due to continuous use of this material in the design of pro-tective barriers that safeguards safety related structures, systemsand components. A dynamic strength increase in concrete was firstobserved by Abrams (1917) and since then it has been generallyaccepted that concrete and concrete-like material are strain-ratesensitive and the constitutive models of such material must in-clude strain-rate effects. The strain-rate effects on the tensilestrength of concrete are stronger than those on the compressivestrength of concrete (CEB, 1993; Malvar and Ross, 1998; Yamaguchiand Fujitima, 1989).

During the last three decades, extensive work has been carriedout to study the structural response of concrete structures sub-jected to impact loading. With stringent requirements of safetyand enhanced testing capabilities, a large number of impact testshas been conducted, much of which can be categorized into pene-tration and perforation testing. Penetration and perforation testshave been carried out on concrete scaled and prototype modelswith different objectives. In penetration tests the focus has beenon the performance study of the target and missile in terms of pen-etration depth and missile penetrability. In perforation tests the fo-cus has been on the residual velocity of the missile. However, thestructural response of concrete and its resistance remains theimportant parameters in the overall performance study. These testsprovide valuable observations on material responses that are usedto validate different modelling techniques related to impact.

The numerical simulation overrides the limitations of empiricalformulations which works within a range of limitations of experi-mental parameters and can be used as a design application. Forthe engineering practice, most of the empirical formulae did not in-clude the effect of reinforcement bars or metallic liner on the sur-face. Only in some of the empirical formulations such as UKAEA,the effect of reinforcing bars has been taken into account in the per-foration formulae, which was based on fitting experimental data.The influence of amount of reinforcement and its location on thefront and rear faces, the effect of missile deformability are still areasof research where extensive studies are required to be done.

Many commercial available numerical analyses codes have dif-ferent constitutive models for concrete that are able to capture theresponse for both the missile and the target. For the present study,LS-DYNA has been used in simulating the impact phenomenon. Thecurrent study is based on simulation of a missile impacting againsta reinforced concrete slab and validation of the modelling tech-

Fig. 1. FE model for concrete slab

niques for its capability to capture experimental results of slab im-pact tests reported by Kojima (1991). The present simulation isbased on the data available and can be made more accurate if bet-ter data is available.

5. Problem description

The local missile impact effect on a reinforced concrete slab isnumerically simulated and compared with those obtained fromthe empirical formulations. The problem description has been ta-ken from the experimental work done by Kojima (1991) who per-formed a series of comparative impact tests to study the localbehaviour on reinforced concrete slabs subjected to high speed im-pact. The steel missiles were projected at high speed from a powerdriven launcher which collided with the reinforced concrete target.

The target was a square reinforced concrete slab with sides1.2 m in length and 0.12 m thick. The reinforcement ratio of the tar-get slab was 0.6% for both the directions. The compressive strengthof the concrete was 28 MPa. Deformed bars of 10 mm diameter andyield strength of 428 MPa were used for the main reinforcement. Ahard-nosed missile made up of solid steel and of weight 2 kg wasused as a missile. The cylindrical missile of 60 mm diameter and ahemispherical nose is 100 mm in length. The FE mesh of the rein-forced concrete slab and missile are shown in Fig. 1. The missilevelocities measured during the experimental tests of Kojima(1991) have been used for the numerical simulation. Few lowerrange missile velocities have also been used so that the concrete re-sponse from elastic to inelastic and subsequent fracture can be sim-ulated. LS-DYNA which is a general-purpose multiphysicssimulation software package developed by the Livermore SoftwareTechnology Corporation (LSTC) has been used for the present study.

5.1. FE modelling of concrete

A 3-D finite element model is developed to simulate the rein-forced concrete target with its full dimensions 1.2 � 1.2 � 0.12(m) using 2, 30, 400 solid elements. There are 16 solid elementof size 10 � 10 7.5 (mm) across the thickness of the slab. The meshsize is chosen based on a comprehensive mesh discretization sen-sitivity analysis using three different mesh sizes. The concrete slabis modelled as eight node hexahedron solid element with constantstress formulation using one-point volume integration usingGaussian quadrature. To avoid the problem of hourglassing, Flana-gan–Belytschko stiffness form with exact volume integration withhourglass control has been used in the analysis.

The continuous surface cap model (CSCM) which was developedfor DYNA 3D analysis tools for roadside safety applications II(2007) program by the US Department of Transportation (FederalHighway Administration, 2007) has been used to represent theconcrete. This model represents a recent development trend in

, missile and reinforcement.

268 R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275

continuum damage models for concrete. The cap model is based onthe idea that continued hydrostatic loading will cause yielding.CSCM employs a coupled deviatoric and volumetric failure surfaceto describe concrete yield and flow. The model considers effects ofdamage, strain rate and triaxial stress on the concrete strength.

The CSCM strength model uses an ultimate failure surfacewhich combines both shear failure and cap hardening. Thestrength model includes an ability to describe the pre-peak nonlin-earity in concrete stress–strain curves involving low confinementlevels. Rubin scaling function is incorporated in the strength modelto represent the effects of the third invariant of the deviatoricstress tensor, which is directly proportional to the angle b in thedeviatoric plane. The Rubin function is used to scale the materialstrength according to the triaxial state of stress. While other mod-els often decouple the volumetric and deviatoric responses, CSCMincorporates the two into the failure surface through the use ofthe cap surface. The formulation allows for the cap to expand inplastic compaction and contract in plastic dilation. The motion ofthe cap is determined using the concrete pressure–volumetricstrain response or equation of state. CSCM incorporates damageapplied in two forms: strain softening and modulus reduction byapplying a scalar damage parameter, d, varying from zero to one,to the undamaged stress tensor.

Damage initiates and accumulates through the use of brittle andductile strain-based energy terms and their corresponding thresh-olds. The ductile damage algorithm limits the possible damageaccumulation based on the amount of confining pressure, allowingmore ductile behaviour to be represented for concrete in higherconfinement. CSCM involves a strength failure surface, damageaccumulation, and strain rate effects. The model considers manyof the responses of concrete in a range of pressure levels and triax-ial stress states.

The material model, with an unconfined compressive strengthof 28 MPa and aggregate size of 10 mm has been used in the pres-ent study. The suite of material properties has been primarily ob-tained through Comité Euro-International du Béton – FédérationInternationale de la Précontrainte (CEB-FIP) CEB, 1993 supple-mented through correlations with impact tests. The strain-rate ef-fect is included in the analysis. An erosion criterion based on theconcrete model damage parameter d and maximum principalstrain has been used to model failure in terms of physically sepa-rating the eroded elements from the rest of the mesh. The erosionmodel represents a numerical remedy to distortion, which cancause excessive and unrealistic deformation of the mesh. In thepresent analysis, element erodes when the maximum principalstrain exceeds 20%. This value is based on the damage patternand results from the experimental works of Kojima (1991). Theconsequence of possible discrepancy in the erosion specified is lim-ited as the damage level of the concrete material is basically gov-erned by the material model itself.

0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5

Forc

e (k

N)

Time (msec)

Fig. 2. Triangular impulse.

5.2. FE modelling for reinforcement

Reinforcements are modelled using 3, 600 truss elements shar-ing the common nodes of the solid element. Full bond between thetruss element and concrete is assumed. The reinforcement is mod-elled using plastic material model with isotropic hardening. Thematerial has a density of 7800 kg/m3, modulus of 210 GPa, Poisson’ratio of 0.3, yield stress of 428 MPa, tangent modules of 1.043 GPaand failure strain of 15%. Cowper–Symonds strain rate model(Jones, 1989) has been used to model the strain rate effect. A valueof 40.4 and 5.0 has been used to model the strain rate parametersof the Symonds–Cowper model. The 10 mm diameter reinforce-ment bars have yield strength of 428 MPa and ultimate tensilestrength of 581 MPa.

5.3. FE modelling for missile

A 3-D finite element model is developed for the solid cylindricalmissile. The model includes 8, 500 hexahedron solid elements withconstant stress formulation. The missile is modelled as using plasticmaterial model with isotropic hardening. The material has a densityof 7830 kg/m3, modulus of 200 GPa, Poisson’ ratio of 0.3, yield stressof 350 MPa, tangent modules of 2.0 GPa and failure strain of 25%.Cowper–Symonds strain rate model has been used to model thestrain rate effect. A value of 40.4 and 5.0 has been used to modelthe strain rate parameters of the Symonds–Cowper model.

All the four bottom corners of the slab are restrained in the z-vertical translational degree of freedom by applying constraintsfor four nodes at each corner. The FE model for the slab, missile,and reinforcement is given in Fig. 1.

6. Mesh sensitivity analysis

One of the important and desirable requirements of non-linearconcrete material model is convergence of the FE solution withreasonable mesh refinement. FE solutions are known to have con-vergence problems if the material being modelled contains soften-ing formulation. With softening formulation, there is a tendency ofthe greatest amount of damage to accumulate in the smallest ele-ment. This is the result of modelling smaller fracture energy in thesmaller elements. In the CSCM model there is an inbuilt regulationfor mesh size dependency as the fracture energy is independent ofmesh size. The concrete model maintains constant fracture energyregardless of the element size. The model calculates the damageparameters in terms of element characteristics length (cube rootof element volume), the fracture energy, Gf and initial damagethresholds. The fracture energy is calculated separately from fiveuser inputs, fracture energy in tension, Gft, compression, Gfc andshear Gfs, shear to compression transition parameter and shear totension transition parameter.

However, mesh sensitivity study has been carried out to achieveconvergence by studying the response of the slab around the areaof impact. The impact problem has been decoupled into an impulseone so that mesh sensitivity can be studied for the concrete mate-rial model. A triangular impulse has been applied at the centre ofthe RCC slab using different mesh sizes. The triangular impulseof peak impulse of 600 kN acts normal to area of 1600 mm2 forduration of 0.25 ms and then reduces to zero at 0.5 ms. Four meshsizes using 8, 12 and 16 element across the depth of the slab havebeen studied. For all the three models the material parameters,geometrical parameters, element formulations have been keptthe same. The triangular impulse used in the simulation is shownin Fig. 2. The model details of the four cases are given in Table 1.

The analysis has been carried out for duration of 5.5 ms and theoutput parameters studied and compared are support reaction, dis-placement of the central node at top and bottom of slab, pressure

Table 1FE model details.

Parameters for (1.2 m � 1.2 m � 0.12 m slab) Case A (20 � 20 � 15) Case B (10 � 10 � 10) Case C (10 � 10 � 7.5)

Element size (mm) 20 � 20 � 15 10 � 10 � 10 10 � 10 � 7.5Number of element across depth 8 12 16Number of nodes 33,489 2,04,974 2,48,897Number of beam element 1800 3600 3600Number of solid element 28,800 1,87,200 2,30,400

Table 2Results of numerical simulation for all the three mesh sizes.

Parameters Case A (20 � 20 � 15) Case B (10 � 10 � 10) Case C (10 � 10 � 7.5)

Support reaction (kN) 303.97 187.46 187.71Displacement of central top node (mm) �2.13 �2.42 �2.42Displacement of central bottom node (mm) �2.08 �2.18 �2.20Velocity of central top node (m/s) �8.39 �10.55 �10.46Pressure of central top element (GPa) 0.135 0.152 0.162Z-stress of central top element (GPa) (-compressive) �0.215 �0.245 �0.265Maximum shear stress of central top element (GPa) 0.061 0.0716 0.0774Maximum principal stress of central top element (GPa) (-compressive) �0.096 �0.1026 �0.1103

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Dis

plac

emen

t (m

m)

Time (msec)

Case A

Case B

Case C

0 1 2 3 4 5 6

Fig. 3. Displacement of top central node of slab.

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Dis

plac

emen

t (m

m)

Time (msec)

Case A

Case B

Case C

0 1 2 3 4 5 6

Fig. 4. Displacement of bottom central node of slab.

0 1 2 3 4 5 6

Pres

sure

(G

Pa)

Time (msec)

Case A

Case B

Case C

Fig. 6. Pressure of element around area of impulse.

-200

-100

0

100

200

300

400

Supp

ort r

eact

ion

(kN

)

Time (msec)

Case A

Case B

Case C

0 1 2 3 4 5 6

Fig. 5. Total support reaction.

R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275 269

in the element close to the top central node and external workdone by the impulsive force. The results of the simulation for thethree cases are given in Table 2.

The displacement profile of top and bottom mid-span node ofthe slab is shown in Figs. 3 and 4, respectively. The displacementprofile and peak displacement converges for Case-C. The total sup-port reactions for the three cases are shown in Fig. 5. The pressuredue to the impulsive force on the top central node is shown inFig. 6. From the mesh sensitivity study, it is inferred that the FEmesh of Case-C is adequate enough to capture the response andhence this mesh has been used for further coupled impact analysis.

7. Contact algorithm

The modelling of contact along the interfaces of missile andconcrete forms an important aspect in the numerical simulation.Two types of contact have been used to model the interactionbetween the missile and the reinforced concrete slab. Eroding sur-face to surface algorithm has been used to model contact betweenthe missile and concrete. This contact is based on penalty methodwith a segment based approach for contact detection to avoid

Supp

ort r

eact

ion

(kN

)

Time (msec)

Fig. 8. Support reaction-time history.

270 R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275

penetration. In the segment based approach, in addition to themaster and slave contact stiffness, an additional mass and timestep based penalty stiffness invokes a segment based contact algo-rithm which has its origin in Pinball contact developed by Bely-tschko and his co-workers (Barr, 1990). Eroding node to surfacealgorithm has been used to model contact between missile and re-bar. This contact deals with the interaction between reinforcementnodes and missiles segments, if any. This situation arises for highvelocity impact.

The friction between concrete and missile is modelled using adynamic decay exponential function. The static and dynamic coef-ficient of friction between concrete and missile has been assumedas 0.2 and 0.1, respectively. The static and dynamic friction be-tween missile and reinforcement has been assumed as 0.57.

Fig. 9. Deformed shape.

8. Coupled impact analysis

Numerical simulation for coupled impact analysis has been car-ried out by giving the missile an initial velocity that makes contactwith the slab panel at its centre. Simulation has been carried outfor a range of velocities so that the structural response; from elasticto inelastic and subsequent fracture and erosion can be studied.The missile velocities considered are 1 m/s, 50 m/s, 95 m/s,164 m/s and 215 m/s; the later three velocities corresponding tothe experimental work of Kojima (1991). The time step used inthe calculation is 5E�6 s. The penetration depth, missile velocity:rebound or residual, contact forces, support reaction and energycontent for all the five velocity impacts are extracted and studied.

9. Numerical results

9.1. Missile velocity 1 m/s

The response of the concrete slab is elastic with few elementsaround the area of impact going into the inelastic range. No spall-ing or scabbing is observed. No erosion of internal energy is ob-served and the contact force observed is similar to the supportreaction. The contact force and support reaction time historiesare shown in Figs. 7 and 8, respectively.

Fig. 10. Damage fringe for bottom of slab.

Fig. 11. Deformed shape.

9.2. Missile velocity 50 m/s

The deformed shape of the slab with the penetration of missileat �11.59 mm is shown in Fig. 9. The missile rebounds with avelocity of 11.13 m/s. The peak contact force is 277.56 kN andthe support reaction is 104.9 kN. Spalling of concrete around areaof impact is observed; with no scabbing at the rear face as shownin Fig. 10. The damage plot is of effective plastic strain which isplotted as maximum of brittle and ductile damage.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.1 0.2 0.3 0.4

Con

tact

for

ce (

kN)

Time (msec)

Fig. 7. Contact force–time history.

9.3. Missile velocity 95 m/s

The deformed shape of the slab with the penetration of missileis shown in Fig. 11. The missile penetrates to �40.54 mm (Fig. 12)and then rebounds with a velocity of 6.89 m/s (Fig. 13). The time atwhich missile rebounds represents the time at which it loses con-tact with the concrete. The peak contact force is 334.25 kN (Fig. 14)and the support reaction is 149.14 kN (Fig. 15). It is observed thatthere is a phase lag between the time instant of peak force and thepeak support reaction. Scabbing of concrete is observed (Fig. 16).

9.4. Missile velocity 164 m/s

The entire concrete gets scabbed but the missile will be unableto perforate the target. The deformed shape of the slab is shown in

Mis

sile

pen

etra

tion

(mm

)

Time (sec)

Fig. 12. Penetration of missile into concrete slab.

Mis

sile

vel

ocity

(m

m/s

)(E

+03

)

Time (sec)

Fig. 13. Velocity profile of missile during penetration.

Con

tact

for

ce (

N)

(E+

06)

Time (sec)

Fig. 14. Contact force–time history.

Supp

ort r

eact

ion

(N)

(E+

06)

Time (sec)

Fig. 15. Support reaction-time history.

Fig. 16. Damge fringe for bottom of slab.

Fig. 17. Deformed shape.

Fig. 18. Damage fringe for bottom of slab.

Ene

rgy

(N-m

m)

(E+

06)

Time (sec)

Slave Energy

Master Energy

Fig. 19. Contact energy: missile and concrete.

R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275 271

Fig. 17. The missile penetrates till the reinforcement level;scabbing the concrete in the process. The damage of the bottomof slab is shown in Fig. 18. The transfer of contact energy betweenthe slave and master segments for missile-concrete is shown inFig. 19. There is a small amount of variation between the master

and the slave energy transfer between missile and concrete dueto dissipation of energy on account of friction. However, the fric-tional energy loss due to friction is very small and is in line withthose reported in the literature. The energy balance, includingthe hourglass energy is shown in Fig. 20. The peak contact forceis 805.2 kN and the support reaction is 116.21 kN.

Kinetic Energy

Internal Energy

Total Energy

Hourglass Energy

Fig. 20. Energy balance.

Fig. 21. Perforation of the slab.

Mis

sile

vel

ocity

(m

m/s

)(E

+03

)

Time (sec)

Fig. 22. Velocity profile of missile during perforation.

Fig. 23. Damage fringe of top surface of slab.

Fig. 24. Damage fringe of bottom surface of slab.

Ene

rgy

(N-m

m)

(E+

06)

Time (sec)

Kinetic Energy

Internal Energy

Total Energy

Hourglass Energy

Fig. 25. Energy balance.

Fig. 26. Axial force in rebars.

272 R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275

9.5. Missile velocity 215 m/s

The missile perforates the concrete slab (Fig. 21) and ejects witha residual velocity of �55.20 m/s (Fig. 22). During this process, themissile penetrates till the bottom reinforcement level and makes

contact with the rebar. The rebar now deforms with a subsequentreduction in the velocity of the penetrating missile leading to achange in the change in the slope of the displacement–time curve.A radial cone type failure is simulated with complete failure ofrebar which is perpendicular to the travel path of the missile.Two bars get ruptured during the perforation process. The plot ofdamage fringe for top and bottom surface of the slab is shown inFigs. 23 and 24, respectively. The energy balance and the axial forcein the rebars are shown in Figs. 25 and 26, respectively.

The summary of the observed parameters from all the fivenumerical simulations are given in Table 3.

Table 3Results of numerical simulation.

Parameters Missile velocity

1 m/s 50 m/s 95 m/s 164 m/s 215 m/s

Penetration depth, x (mm) �0.13 �11.59 �40.54 �104.94 –a

Residual velocity, Vrb 0.80 11.13 6.89 8.71 �55.21

Ratio of hourglass/internal energy 0.149 0.149 0.149 0.067 0.069Total energy (kN-mm) 0.998 2478.5 8964 26,667 45,832Contact force between missile and concrete (kN) 13.56 277.56 334.25 805.20 1348.4Contact force between missile and rebar (kN) –c –c –c 30.00 165.74Support reaction (kN) 13.35 104.9 149.14 116.21 93.44Remarks No damage Spalling Scabbing Scabbing Perforation

a Missile perforates.b ‘�’ denotes residual and ‘+’ denotes rebound.c Missile does not penetrate up to rebar depth.

Table 4Comparison of numerical simulation with experimental results.

Parameters VP = 95 m/s VP = 164 m/s VP = 215 m/s

Numerical Experimental Numerical Experimental Numerical Experimental

Penetration Depth, x (mm) �40.54 �44.0 �104.94 �100.0 – –Maximum reaction (kN) 149.14 145.0 116.21 118.0 93.44 106Numbers of rebars ruptured 0 0 0 1 2 3Damage Scabbing Scabbing Scabbing Scabbing Perforation Perforation

Pene

trat

ion

x (

mm

)

Velocity (m/sec)

Mod Petry 2nd BRL ACE

NDRC Ammann and Whitney Whiffen

Kar Adeli-Amin Hughes

UKAEA Numerical Experiment

Experimental x = 44 mm

Numerical x = 40.54 mm

Fig. 27. Penetration depth vs. velocity as per different emperical formulations.

R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275 273

10. Comparison with experimental results

The comparison between the numerical simulation and experi-mental results are given in Table 4. It is observed that the damagemode of the slab from the numerical simulation matches closelywith that of the tests. Close match is observed in the peak reactionfor all the three cases. For the missile velocity of 95 m/s, the ob-served penetration is �40.54 mm whereas the measured was�44.0 mm. Similarly, for the missile velocity of 164 m/s, the ob-served penetration is �104.94 mm whereas the measured fromthe test was �100.0 mm. The cause for such discrepancies can beassumptions such as effects of failure strain for concrete and steel,effects of some of the material input parameters that have been as-sumed and erosion criteria for concrete.

11. Comparison with empirical formulations

The penetration depth, scabbing and perforation limits as pervarious empirical formulations have been obtained and the resultshave been compared with the numerical simulations. The plot ofpenetration depth verses velocity is given in Fig. 27. The plot showsthe variation in the prediction of the penetration depth as per dif-ferent formulations. A comparison of the penetration depth as ob-tained from the numerical experiment shows different behaviourcompared to the empirical trend. The test results for a velocity of95 m/s shows that ACE, NDRC and Kar formulations over predictsthe penetration depth. However, ACE and Kar gives the closest re-sults as compared to the test results. The missile diameter is60 mm and is very close to the ballistic test data of steel cylinders

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300

Sca

bbin

g de

pth,

hs

(mm

)

Velocity m/s

Mod Petry 2nd BRL ACE NDRC

Kar UKAEA Bechtel Hughes

Experimental x = 100.0 mm

Numericalx = 104.94 mm

Fig. 28. Scabbing depth vs. velocity as per different emperical formulations.

0

50

100

150

200

250

300

0 50 100 150 200 250 300

Perf

orat

ion

limit,

e (

mm

)

Velocity m/sec

Mod Petry 2nd BRL ACE

NDRC Kar CEA-EDF

Hughes

Slab Thickness=120 mm

Fig. 29. Perforation depth vs. velocity for different emperical formulations.

274 R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275

on which ACE formulation was based and hence close match isobserved between empirical and numerical results.

The plot of scabbing limit vs. velocity as per different empericalformulations is given in Fig. 28. The plot shows the variation in theprediction of the scabbing depth as per different formulations. Ex-cept the Modified Petry formula, all the formulas predict that theslab will be scabbed for a missile velocity of 164 m/s and are con-servative in predicting the scabbing limit.

The slab will perforate the slab for a velocity of 215 m/s as perthe empirical formulations. Only the Modified Petry formula pre-dicts a perforation limit of 116 mm. All other formulations are con-servative in predicting the perforation limit. The plot of perforationlimit verses velocity is given in Fig. 29.

12. Discussion

The numerical simulation shows that the target response isdependent of the relative kinetic energy of the missile and theinternal energy absorbing capacity of the target. At very low veloc-ity, the entire kinetic energy is absorbed as internal energy of theconcrete slab with no loss of energy due to material erosion andthe response is elastic. As kinetic energy increases, and exceedsthe elastic strain energy absorbing capacity of the concrete slab,material undergoes inelastic straining and subsequently gets frac-tured in the form of spalling. When the kinetic energy of the mis-sile exceeds the total strain energy absorbing capacity of theconcrete slab around area of impact, failure occurs in the form ofscabbing and with further increase in the velocity in the form ofperforation.

There is an increase in the contact force as the velocity is in-creased from 1 m/s to 215 m/s which are as expected. The fasterthe missile, the greater is its kinetic energy and the greater contactforce it will experience. However, the trend of the support reactionshows different behaviour. There is an increase in the support reac-tion from 1 m/s to 95 m/s and thereafter it reduces. It is observedthat in the range of 95 m/s to 215 m/s, the greater the missilevelocity, the lower the peak reaction force. This is basically dueto greater amount of damage accumulation at greater velocitywherein larger amount of internal energy gets eroded. However,at low velocity range from 1 m/s to 95 m/s, where the damageare less associated, the increase in velocity results in an increasein the support reaction. This behaviour was noted in the experi-mental works and is observed in the numerical simulation as well.

The loss of frictional energy due to sliding at the interface ofmissile-concrete is insignificant and is observed in the energy bal-ance of the system. There is a phase lag between the time instant ofthe peak support reaction and contact force. This is due to the tra-vel path of the wave to travel from the impact location to theboundary nodes that are nearly 560 mm away from the contactarea. As the velocity is increased the failure behaviour graduallyshifts from global to local.

13. Conclusion

Numerical simulations of dynamic response have becomeincreasingly important for the structural design of Nuclear PowerPlant structures against impact loads. In contrast to empirical for-mulae, they are not bound by the limits of applicability and also

R. Ranjan et al. / Annals of Nuclear Energy 68 (2014) 262–275 275

provide information on stress and deformation fields, which maybe used to improve the resistance of the concrete. A set of numer-ical simulations have been carried out using the explicit FEM codeLS-DYNA, and the simulation results compared with the experi-mental results and empirical formulations. The result showsclearly that the missile velocity is, probably, the main factor indefining the target damage. The local and global behaviour of theconcrete slab subjected to missile impact for all the three experi-mental tests could be predicted closely by the FE analyses usingLS-DYNA. The cause for the observed discrepancies can be assump-tions such as effects of failure strain for concrete and steel, effectsof some of the material input parameters that have been assumedand erosion criteria for concrete. Sensitivity study on the effect ofmaterial parameters and particularly erosion criteria will lead toan identification of the variables affecting the results and can bebetter optimized to fit the experimental data.

References

Abrams, D.A., 1917. Effect of rate of application of load on compressive strength ofconcrete. Proc. ASTM 17, 364–369.

Adeli, H., Amin, A.M., 1985. Local effects of impactors on concrete structures. Nucl.Eng. Des. 88, 301–317.

Amirikian, A., 1950. Design of protective structures. Report NT-3726, Bureau ofYards and Docks, Department of the Navy.

Bangash, M.Y.H., 1993. Impact and Explosion: Structural Analysis and Design. CRCPress, Boca Raton, FL.

Barr, P., 1990. Guidelines for the design and assessment of concrete structuressubjected to impact. Report, UK Atomic Energy Authority, Safety and ReliabilityDirectorate, HMSO, London.

Berriaud, C., Sokolovsky, Gueraud, R., Dulac, J., Labrot, R., 1978. Local behaviour ofreinforced concrete slabs under missile impact. Nucl. Eng. Des. 45, 457–469.

CEB, 1993. CEB-FIP model code 1990. Trowbridge, Wiltshire UK: Comite Euro-International Du Beton, Redwood Books.

Chelapati, C.V., Kennedy, R.P., Wall, I.B., 1972. Probabilistic assessment of hazard fornuclear structures. Nucl. Eng. Des. 19, 333–364.

Cotovos et al., 2008. Behaviour of RC beams subjected to high rates of concentratedloading. J. Struct. Eng. 134 (12), 1839–1851.

Federal Highway Administration, USA, 2007. User Manual for LS-DYNA ConcreteMaterial May 2007.

Fullard, K., 1991. Interpretation of low velocity damage tests on concrete slabs. In:Proceedings of the international conference of earthquake, blast and impact,Manchester, UK, pp. 126–133.

Haldar, A., Hamieh, H., 1998. Local effect of solid missile on concrete structures.ASCE J. Struct. Div. 110 (5), 948–960.

Hughes, G., 1984. Hard missile impact on reinforced concrete. Nucl. Eng. Des. 37,23–35.

Islam et al., 2011. Numerical study of concrete penetration/perforation under highvelocity impact by ogive-nose steel projectile. Comput. Struct. 8 (1), 111–123.

Jones, N., 1989. Structural Impact. Cambridge University Press.Jose, A. et al., 2010. Benchmarking finite element simulation of hard impact missiles

on reinforced concrete slabs. ACI Fall 2010 Convention, Pittsburgh, PA.Kar, A.K., 1978. Local effects of tornado generated missiles. ASCE J. Struct. Div. 104

(ST5), 809–816.Kennedy, R.P., 1966. Effects of an Aircraft Crash into a Concrete Reactor

Containment Building. Holmes & Narver Inc., Anaheim, CA.Kennedy, R.P., 1976. A review of procedures for the analysis and design of concrete

structures to resist missile impact effects. Nucl. Eng. Des. 37, 183–203.Kojima, Isao, 1991. An experimental study of local behaviour of reinforced concrete

slabs to missile impact. Nucl. Eng. Des. 8, 121–132.Li, Q.M. et al., 2003. Dimensionless formulae for penetration depth of concrete

target impacted by a non-deformable projectile. Int. J. Impact Eng. 28, 93–116.Li, Q.M. et al., 2005. Local impact effect of hard missile on concrete target. Int. J.

Impact Eng. 32, 224–284.Liu et al., 2011. Numerical simulation of oblique-angle penetration into concrete

targets. Comput. Math. Appl. 6 (2011), 2168–2171.LS-DYNA: LS-DYNA keyword User’s Manual Version 971, Lawrence Livermore

Software Technology Corporation (CA), USA.Malvar, L.J., Ross, C.A., 1998. Review of strain rate effects for concrete in tension. ACI

Mater. J. 95 (6), 735–739.Murray, Y.D. et al., 2007. Evaluation of LS-DYNA concrete material model 159.

McLean, VA: US Dept. of transportation, FHA.NDRC, 1946. Effects of impact and explosion. Summary Technical Report of

Division-2.vol., National Defense Research Committee, Washington DC.Sangi, A.J., May, I.M., 2009. High-mass, low velocity impacts on reinforced concrete

slabs. In: 7th European LS-DYNA conference.Sliter, G.E., 1980. Assessment of empirical concrete formulas. ASCE J. Struct. Div. 106

(ST5), 1023–1045.Suanno, et al., 2003. Non-linear analysis of reinforced concrete shell subjected to

impact load. SMiRT 17, Prague, Czech Republic, August 17–23, 2003.Tahmasebina, F., 2008. Finite element simulations of reinforced concrete structures

under impact accident. Struct. Surv. 26 (5), 445–454.Yamaguchi, H., Fujitima, K., 1989. Strain rate effect on the dynamic response of

reinforced concrete slab under impact loading. Trans. Architect. Inst. Jpn..