MOJ eISSN: 2572-8520 MOJCE

Civil Engineering
Research Article
Volume 4 Issue 4

Parametric study on local impact damage of concrete members
Hyeon-Jong Hwang
College of Civil Engineering, Hunan University, China
Received: August 09, 2018 | Published: August 30, 2018

Correspondence: Hyeon-Jong Hwang, College of Civil Engineering, Hunan University, Changsha, Hunan, China, Tel +86 1357 4826 119, Email

Citation: Hwang HJ. Parametric study on local impact damage of concrete members. MOJ Civil Eng. 2018;4(4):369‒372. DOI: 10.15406/mojce.2018.04.00130

Abstract

To predict the local damage of a concrete target subjected to impact load, various empirical and theoretical equations were proposed because the local failure mechanism of concrete on collision is complicated. In the present study, using an empirical model and an energy based model, a parametric study was performed to investigate the effects of design parameters on the penetration depth. All design parameters including concrete strength, rebar ratio, concrete density, aggregate size, steel fiber volume ratio, impact velocity, projectile mass, projectile diameter, and concrete target configuration were considered. The results showed that the energy based model is useful to predict the penetration depth of the local impact-damaged concrete target under various design conditions.

Keywords: impact load, penetration, kinetic energy, resistant energy, energy-based penetration model

Introduction

High-velocity impact causes local failure in reinforced concrete structures.1,2 For civilian and military structures under impact load, the impact load resistance needs to be accurately evaluated. However, due to the complicated mechanism of concrete subjected to impact load, the impact resistance of concrete members has been studied by a number of tests and empirical methods. In order to predict the penetration depth of concrete targets, Petry et al.2 proposed a method based on a simplified equation of motion for the first time, considering the projectile mass, impact velocity, sectional area of the projectile, and factor related to concrete penetrability. Army Corps of Engineers (ACE)3 evaluated the allowable concrete target thickness to restrain significant local failure, and National Defense Research Committee (NDRC)4 and Kennedy5 additionally considered the nose shape effect of the projectile in ACE model.4 The effects of maximum aggregate size, elastic modulus of the projectile, strain-rate of concrete, thickness of the concrete target, concrete density, configuration of the projectile, and hybrid-fibers on the penetration depth of the concrete target have been studied.6-15 Unlike empirical models, Hwang et al.16 developed an energy-based model for the penetration depth and residual velocity of a projectile. Comparing the kinetic energy of a projectile with the resistant energy of a concrete target directly, the penetration depth of the concrete target can be estimated. Although the energy-based model predicts well the penetration depth of the concrete target, it is difficult to understand the effect of each design parameter on the penetration depth intuitionally in initial design stage. Thus, to investigate the impact damage of the concrete target under high velocity impact, a parametric study was performed. The modified NDRC model5 and energy-based model16 were used, and the penetration depth of the local impact-damaged concrete target under various design conditions was examined. The analysis results can provide an insight to researchers and practical engineers.

Existing methods

Modified NDRC

The modified NDRC method5 was developed based on the test results of plain concrete. Although various design conditions cannot be considered, the modified NDRC method has been widely used for simple calculation. The penetration depth x of concrete targets is defined as follows.

x=2d G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaajugibi aadIhacqGH9aqpcaaIYaGaamizaKqbaoaakaaakeaajugibiaadEea aSqabaaaaa@3DF9@ for G 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadEeacqGHKjYOcaqGGaGaaGymaaaa@3C04@ (1a)

x=( G+1 )d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaajugibi aadIhacqGH9aqpjuaGdaqadaGcbaqcLbsacaWGhbGaey4kaSIaaGym aaGccaGLOaGaayzkaaqcLbsacaWGKbaaaa@40E1@ for G> 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadEeacqGH+aGpcaqGGaGaaGymaaaa@3B57@ (1b)

G=3.8× 10 5 Nm d f c ( V i d ) 1.8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq =Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0x fr=xfr=xb9adbaqaceGaciGaciaabiqaamGabiabaaGcbaGaam4rai abg2da9iaaiodacaGGUaGaaGioaiabgEna0kaaigdacaaIWaWaaWba aSqabeaacqGHsislcaaI1aaaaOWaaSaaaeaacaWGobGaamyBaaqaai aadsgadaGcaaqaaiqadAgagaqbamaaBaaaleaacaWGJbaabeaaaeqa aaaakmaabmaabaWaaSaaaeaacaWGwbWaaSbaaSqaaiaadMgaaeqaaa GcbaGaamizaaaaaiaawIcacaGLPaaalmaaCaaabeqaaKqzadGaaGym aiaac6cacaaI4aaaaaaa@4F2C@ (in kg, MPa, m/s, and m) (1c)

Where d is the projectile diameter; N is the nose shape factor (= 0.72 for flat nose, 0.84 for hemispherical nose, 1.0 for blunt nose, and 1.13 for sharp nose); m is the projectile mass; f’c is the concrete compressive strength; and Vi is the initial velocity.

Energy-based penetration model

In the study of Hwang et al.,16 the penetration depth x of concrete targets is estimated by comparing the kinetic energy EK of projectile with the resistant energy ER of the concrete target. The kinetic energy EK is defined as a function of the projectile mass m, initial velocity Vi, and residual velocity Vr.

E K = m 2 ( V i 2 V r 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq =Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0x fr=xfr=xb9adbaqaceGaciGaciaabiqaamGabiabaaGcbaGaamyram aaBaaaleaacaWGlbaabeaakiabg2da9maalaaabaGaamyBaaqaaiaa ikdaaaWaaeWaaeaacaWGwbWaa0baaSqaaiaadMgaaeaacaaIYaaaaO GaeyOeI0IaamOvamaaDaaaleaacaWGYbaabaGaaGOmaaaaaOGaayjk aiaawMcaaaaa@4557@ (2)

Where Vr is the residual velocity of the projectile after perforation (i.e., Vr = 0 when perforation failure does not occur).

Figure 1 shows local failure mode of concrete targets. Due to deformation of the projectile and concrete target on collision, the kinetic energy Ek is absorbed by the deformed energy EDP of the projectile and deformed energy EDC of the concrete target (Figure 1). On the basis of the idealized concrete cone and failure surface in the tunneling zone, the spalling resistant energy ES, tunneling resistant energy ET, and scabbing resistant energy EC are generated to absorb the kinetic energy Ek. The resistant energy Er of the concrete target is determined from the sum of EDP, EDC, ES, ET, and EC.

Figure 1 Local failure mode of concrete target under impact load.

E R = E DP + E DC + E S + E T + E C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq =Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0x fr=xfr=xb9adbiqaceGaciGaciaabiqaamGabiabaaGcbaGaamyram aaBaaaleaacaWGsbaabeaakiabg2da9iaadweadaWgaaWcbaGaamir aiaadcfaaeqaaOGaey4kaSIaamyramaaBaaaleaacaWGebGaam4qaa qabaGccqGHRaWkcaWGfbWaaSbaaSqaaiaadofaaeqaaOGaey4kaSIa amyramaaBaaaleaacaWGubaabeaakiabgUcaRiaadwealmaaBaaaba qcLbmacaWGdbaaleqaaaaa@4B0A@ (3a)

E DP + E DC =[ A p L 2 E p + A p 2 b 3 24 E cd I g ] ( ρ p V i 2 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq =Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0x fr=xfr=xb9adbaqaceGaciGaciaabiqaamGabiabaaGcbaGaamyram aaBaaaleaacaWGebGaamiuaaqabaGccqGHRaWkcaWGfbWaaSbaaSqa aiaadseacaWGdbaabeaakiabg2da9maadmaabaWaaSaaaeaacaWGbb WaaSbaaSqaaiaadchaaeqaaOGaamitaaqaaiaaikdacaWGfbWaaSba aSqaaiaadchaaeqaaaaakiabgUcaRmaalaaabaGaamyqamaaDaaale aacaWGWbaabaGaaGOmaaaakiaadkgadaahaaWcbeqaaiaaiodaaaaa keaacaaIYaGaaGinaiaadweadaWgaaWcbaGaam4yaiaadsgaaeqaaO GaamysamaaBaaaleaacaWGNbaabeaaaaaakiaawUfacaGLDbaadaqa daqaamaalaaabaGaeqyWdi3aaSbaaSqaaiaadchaaeqaaOGaamOvam aaDaaaleaacaWGPbaabaGaaGOmaaaaaOqaaiaaikdaaaaacaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@5BF3@ (3b)

E S ( x )=( k s k bs )[ π f td 12 + A s f y cos θ s sin θ s 3 ( d+2xtan θ s ) 2 ]×[ 4 x 3 tan 2 θ s +6d x 2 tan θ s +3 d 2 x ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPq=BgrFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq =Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0x fr=xfr=xb9adbaqaceGaciGaciaabiqaamGabiabaaGcbaGaamyram aaBaaaleaacaWGtbaabeaakmaabmaabaGaamiEaaGaayjkaiaawMca aiabg2da9maabmaabaGaam4AamaaBaaaleaacaWGZbaabeaakiaadU gadaWgaaWcbaGaamOyaiaadohaaeqaaaGccaGLOaGaayzkaaWaamWa aeaadaWcaaqaaiabec8aWjaadAgadaWgaaWcbaGaamiDaiaadsgaae qaaaGcbaGaaGymaiaaikdaaaGaey4kaSYaaSaaaeaacaWGbbWaaSba aSqaaiaadohaaeqaaOGaamOzamaaBaaaleaacaWG5baabeaakiaabo gacaqGVbGaae4CaiabeI7aXnaaBaaaleaacaWGZbaabeaakiaaboha caqGPbGaaeOBaiabeI7aXnaaBaaaleaacaWGZbaabeaaaOqaaiaaio dadaqadaqaaiaadsgacqGHRaWkcaaIYaGaamiEaiaabshacaqGHbGa aeOBaiabeI7aXnaaBaaaleaacaWGZbaabeaaaOGaayjkaiaawMcaam aaCaaaleqabaGaaGOmaaaaaaaakiaawUfacaGLDbaacqGHxdaTdaWa daqaaiaaisdacaWG4bWaa0baaSqaaaqaaiaaiodaaaGccaqG0bGaae yyaiaab6gadaahaaWcbeqaaiaaikdaaaGccqaH4oqCdaWgaaWcbaGa am4CaaqabaGccqGHRaWkcaaI2aGaamizaiaadIhadaqhaaWcbaaaba GaaGOmaaaakiaabshacaqGHbGaaeOBaiabeI7aXnaaBaaaleaacaWG ZbaabeaakiabgUcaRiaaiodacaWGKbWaaWbaaSqabeaacaaIYaaaaO GaamiEaaGaay5waiaaw2faaaaa@8610@

E S ( x )=( k s k bs )[ π f td 12 + A s f y cos θ s sin θ s 3 ( d+2xtan θ s ) 2 ]×[ 4 x 3 tan 2 θ s +6d x 2 tan θ s +3 d 2 x ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPq=BgrFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq =Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0x fr=xfr=xb9adbaqaceGaciGaciaabiqaamGabiabaaGcbaGaamyram aaBaaaleaacaWGtbaabeaakmaabmaabaGaamiEaaGaayjkaiaawMca aiabg2da9maabmaabaGaam4AamaaBaaaleaacaWGZbaabeaakiaadU gadaWgaaWcbaGaamOyaiaadohaaeqaaaGccaGLOaGaayzkaaWaamWa aeaadaWcaaqaaiabec8aWjaadAgadaWgaaWcbaGaamiDaiaadsgaae qaaaGcbaGaaGymaiaaikdaaaGaey4kaSYaaSaaaeaacaWGbbWaaSba aSqaaiaadohaaeqaaOGaamOzamaaBaaaleaacaWG5baabeaakiaabo gacaqGVbGaae4CaiabeI7aXnaaBaaaleaacaWGZbaabeaakiaaboha caqGPbGaaeOBaiabeI7aXnaaBaaaleaacaWGZbaabeaaaOqaaiaaio dadaqadaqaaiaadsgacqGHRaWkcaaIYaGaamiEaiaabshacaqGHbGa aeOBaiabeI7aXnaaBaaaleaacaWGZbaabeaaaOGaayjkaiaawMcaam aaCaaaleqabaGaaGOmaaaaaaaakiaawUfacaGLDbaacqGHxdaTdaWa daqaaiaaisdacaWG4bWaa0baaSqaaaqaaiaaiodaaaGccaqG0bGaae yyaiaab6gadaahaaWcbeqaaiaaikdaaaGccqaH4oqCdaWgaaWcbaGa am4CaaqabaGccqGHRaWkcaaI2aGaamizaiaadIhadaqhaaWcbaaaba GaaGOmaaaakiaabshacaqGHbGaaeOBaiabeI7aXnaaBaaaleaacaWG ZbaabeaakiabgUcaRiaaiodacaWGKbWaaWbaaSqabeaacaaIYaaaaO GaamiEaaGaay5waiaaw2faaaaa@8610@

for 0x t s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaajugibi aaicdacqGHKjYOcaWG4bGaeyizImQaamiDaKqbaoaaBaaaleaajugW aiaadohaaSqabaaaaa@4148@ (3c)

E T ( x )= 4m ρ p d ( 2.2ψ f cd )( x t s )0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq =Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0x fr=xfr=xb9adbaqaceGaciGaciaabiqaamGabiabaaGcbaGaamyram aaBaaaleaacaWGubaabeaakmaabmaabaGaamiEaaGaayjkaiaawMca aiabg2da9maalaaabaGaaGinaiaad2gaaeaacqaHbpGCdaWgaaWcba GaamiCaaqabaGccaWGKbaaamaabmaabaGaaGOmaiaac6cacaaIYaGa eqiYdK3aaOaaaeaaceWGMbGbauaadaWgaaWcbaGaam4yaiaadsgaae qaaaqabaaakiaawIcacaGLPaaadaqadaqaaiaadIhacqGHsislcaWG 0bWaaSbaaSqaaiaadohaaeqaaaGccaGLOaGaayzkaaGaeyyzImRaaG imaaaa@546E@ for t s xh t c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaacaWG0b WaaSbaaSqaaiaadohaaeqaaOGaeyizImQaamiEaiabgsMiJkaadIga cqGHsislcaWG0bWaaSbaaSqaaiaadogaaeqaaaaa@4229@ (3d)

E C ( x )=( k s k bc )[ π f td 12 + 2 A s f y 15 { d+4( xh+2 t c ) } 2 ]×[ 16 ( xh+2 t c ) 3 +12d ( xh+2 t c ) 2 +3 d 2 ( xh+2 t c ) ]0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacPi=BMqFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq =Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0x fr=xfr=xb9adbaqaceGaciGaciaabiqaamGabiabaaGcbaGaamyram aaBaaaleaacaWGdbaabeaakmaabmaabaGaamiEaaGaayjkaiaawMca aiabg2da9maabmaabaGaam4AamaaBaaaleaacaWGZbaabeaakiaadU gadaWgaaWcbaGaamOyaiaadogaaeqaaaGccaGLOaGaayzkaaWaamWa aeaadaWcaaqaaiabec8aWjaadAgadaWgaaWcbaGaamiDaiaadsgaae qaaaGcbaGaaGymaiaaikdaaaGaey4kaSYaaSaaaeaacaaIYaGaamyq amaaBaaaleaacaWGZbaabeaakiaadAgadaWgaaWcbaGaamyEaaqaba aakeaacaaIXaGaaGynamaacmaabaGaamizaiabgUcaRiaaisdadaqa daqaaiaadIhacqGHsislcaWGObGaey4kaSIaaGOmaiaadshadaWgaa WcbaGaam4yaaqabaaakiaawIcacaGLPaaaaiaawUhacaGL9baadaah aaWcbeqaaiaaikdaaaaaaaGccaGLBbGaayzxaaGaey41aq7aamWaae aacaaIXaGaaGOnamaabmaabaGaamiEaiabgkHiTiaadIgacqGHRaWk caaIYaGaamiDamaaBaaaleaacaWGJbaabeaaaOGaayjkaiaawMcaam aaDaaaleaaaeaacaaIZaaaaOGaey4kaSIaaGymaiaaikdacaWGKbWa aeWaaeaacaWG4bGaeyOeI0IaamiAaiabgUcaRiaaikdacaWG0bWaaS baaSqaaiaadogaaeqaaaGccaGLOaGaayzkaaWaa0baaSqaaaqaaiaa ikdaaaGccqGHRaWkcaaIZaGaamizamaaCaaaleqabaGaaGOmaaaakm aabmaabaGaamiEaiabgkHiTiaadIgacqGHRaWkcaaIYaGaamiDamaa BaaaleaacaWGJbaabeaaaOGaayjkaiaawMcaaaGaay5waiaaw2faai abgwMiZkaaicdaaaa@8AD9@

for h2 t c xh t c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaacaWGOb GaeyOeI0IaaGOmaiaadshadaWgaaWcbaGaam4yaaqabaGccqGHKjYO caWG4bGaeyizImQaamiAaiabgkHiTiaadshadaWgaaWcbaGaam4yaa qabaaaaa@44AF@ (3e)

Where Ap is the cross-sectional area of projectile; L is the projectile length; Ep is the elastic modulus of projectile (= 205000 MPa for steel); b is the concrete target width; Ecd is the elastic modulus of concrete targets under impact load; Ig is the gross moment of inertia of concrete target; ρp is the projectile density (= 7850 kg/m3 for steel); ks is the size effect factor (= ( 300/h ) 0.25 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaadaqada qaamaalyaabaGaaG4maiaaicdacaaIWaaabaGaamiAaaaaaiaawIca caGLPaaadaahaaWcbeqaaiaaicdacaGGUaGaaGOmaiaaiwdaaaGccq GHKjYOcaaIXaaaaa@41FF@ , where h is in mm); h is the concrete target thickness; kbs and kbc is the stress concentration effect factors (= 4 h / π( d+ t s tan θ s ) 1.25 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaajuaGda WcgaGcbaqcLbsacaaI0aqcfa4aaOaaaOqaaKqzGeGaamiAaaWcbeaa aOqaaKqbaoaakaaakeaajugibiabec8aWLqbaoaabmaakeaajugibi aadsgacqGHRaWkcaWG0bWcdaWgaaqaaKqzadGaam4CaaWcbeaajugi biaabshacaqGHbGaaeOBaiabeI7aXLqbaoaaBaaaleaajugWaiaado haaSqabaaakiaawIcacaGLPaaaaSqabaaaaKqzGeGaeyizImQaaGym aiaac6cacaaIYaGaaGynaaaa@53EA@ ); tan θ s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaacaqG0b Gaaeyyaiaab6gacqaH4oqCdaWgaaWcbaGaam4Caaqabaaaaa@3D5A@ = 2.0, 1.9, 1.55, and 0.9 for flat, round, ogive, and sharp nose shape, respectively; ftd is the concrete tensile strength increased by strain rate effect; As is the sum of effective reinforcing bar area in the concrete cone in horizontal and vertical directions; fy is the rebar yield strength; f′cd is the concrete compressive strength increased by strain rate effect; and = 1.0, 0.9, 0.7, and 0.2 for flat, round, ogive, and sharp nose shape, respectively.

In Eqs 3c–3e, the allowable maximum spalling and scabbing depths ts and tc are affected by geometric and material properties of the concrete target, and it is defined as follows.

t s = t c = k 1 k 2 k 3 k 4 d0.5h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaacaWG0b WaaSbaaSqaaiaadohaaeqaaOGaeyypa0JaamiDamaaBaaaleaacaWG Jbaabeaakiaab2dacGaJao4Aamacmc4gaaWcbGaJakacmciIXaaabK aJacGccaWGRbWaaSbaaSqaaiaaikdaaeqaaOGaam4AamaaBaaaleaa caaIZaaabeaakiaadUgadaWgaaWcbaGaaGinaaqabaGccaWGKbGaey izImQaaGimaiaac6cacaaI1aGaamiAaaaa@505C@ (4a)

k 1 =2.1 ( h d ) 0.3 1.750 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaacaWGRb WaaSbaaSqaaiaaigdaaeqaaOGaaeypaiaaikdacaGGUaGaaGymamaa bmaabaWaaSaaaeaacaWGObaabaGaamizaaaaaiaawIcacaGLPaaada ahaaWcbeqaaiaaicdacaGGUaGaaG4maaaakiabgkHiTiaaigdacaGG UaGaaG4naiaaiwdacqGHLjYScaaIWaaaaa@48A7@ (4b)

k 2 =10.025 V f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaacaWGRb WaaSbaaSqaaiaaikdaaeqaaOGaaeypaiaaigdacqGHsislcaaIWaGa aiOlaiaaicdacaaIYaGaaGynaiaadAfadaWgaaWcbaGaamOzaaqaba aaaa@4191@ (in %) (4c)

k 3 =5.94 ( 2.1 ρ c )/ 1000 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaacaWGRb WaaSbaaSqaaiaaiodaaeqaaOGaaeypaiaaiwdacaGGUaGaaGyoaiaa isdacqGHsisldaWcgaqaamaabmaabaGaaGOmaiaac6cacaaIXaGaeq yWdi3aaSbaaSqaaiaadogaaeqaaaGccaGLOaGaayzkaaaabaGaaGym aiaaicdacaaIWaGaaGimaaaacqGHKjYOcaaIXaaaaa@4A35@ (in kg/m3) (4d)

k 4 = 0 .23s a /d +0.771 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaacaWGRb WaaSbaaSqaaiaaisdaaeqaaOGaaeypamaalyaabaGaaeimaiaab6ca caqGYaGaae4maiaabohadaWgaaWcbaGaamyyaaqabaaakeaacaWGKb aaaiabgUcaRiaaicdacaGGUaGaaG4naiaaiEdacqGHLjYScaaIXaaa aa@4689@ (4e)

where k1 is the coefficient related to the concrete target thickness; k2 is the coefficient related to the steel fiber volume ratio; k3 is the coefficient related to the concrete density ; ρ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaacqaHbp GCdaWgaaWcbaGaam4yaaqabaaaaa@3A88@ and k4 is the coefficient related to the maximum size sa of coarse aggregates.

In Eqs. 3b–3e, impact force increases the elastic modulus Ecd, compressive strength f′cd, and tensile strength ftd of concrete due to strain rate effect, and it is defined as follows.

ε . c =2 ε p h E p ρ p = V i 2 h ρ p E p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWfGaqaaGGaaKqzGeGae8xTdugaleqabaqcLbsacaGGUaaaaSWa aSbaaWqaaiaadogaaeqaaKqzGeGaeyypa0JaaGOmaOWaaSaaaeaaju gibiab=v7aLTWaaSbaaWqaaiaadchaaeqaaaGcbaqcLbsacaWGObaa aOWaaOaaaeaadaWcaaqaaKqzGeGaamyraSWaaSbaaWqaaiaadchaae qaaaGcbaqcLbsacqWFbpGCkmaaBaaaleaajugWaiaadchaaSqabaaa aaqabaqcLbsacqGH9aqpkmaalaaabaqcLbsacaWGwbWcdaWgaaadba GaamyAaaqabaWcdaahaaadbeqaaiaaikdaaaaakeaajugibiaadIga aaGcdaGcaaqaamaalaaabaqcLbsacqWFbpGClmaaBaaameaacaWGWb aabeaaaOqaaKqzGeGaamyraSWaaSbaaWqaaiaadchaaeqaaaaaaSqa baaaaa@5828@ (5a)

E cd = E cs ( 105 ε c . ) 0.026 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadwealmaaBaaameaacaWGJbGaamizaaqabaqcLbsacqGH 9aqpcaWGfbWcdaWgaaadbaGaam4yaiaadohaaeqaaOWaaeWaaeaaju gibiaaigdacaaIWaGaaGynaOWaaCbiaeaaiiaajugibiab=v7aLPWa aSbaaWqaaKqzadGaam4yaaadbeaaaSqabeaajugibiaac6caaaaaki aawIcacaGLPaaadaahaaWcbeqaaKqzadGaaGimaiaac6cacaaIWaGa aGOmaiaaiAdaaaaaaa@4EF4@ (5b)

f td =f ( 106 ε c . ) 0.018 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadAgalmaaBaaameaacaWG0bGaamizaaqabaqcLbsacqGH 9aqpcaWGMbGcdaqadaqaaKqzGeGaaGymaiaaicdacaaI2aGcdaWfGa qaaGGaaKqzGeGae8xTdu2cdaWgaaadbaGaam4yaaqabaaaleqabaqc LbsacaGGUaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaajugWaiaaic dacaGGUaGaaGimaiaaigdacaaI4aaaaaaa@4BF8@ for ε . c <10/s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWfGaqaaGGaaKqzGeGae8xTdugaleqabaqcLbsacaGGUaaaaOWa aSbaaSqaaKqzadGaam4yaaWcbeaajugibiab=Xda8iab=fdaXiab=b daWiab=9caViaadohaaaa@42E5@ (5c)

f td =0.0062 f t ( 10 6 ε c . ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb WcdaWgaaqcfayaaKqzadGaamiDaiaadsgaaKqbagqaaKqzGeGaeyyp a0JaaGimaiaac6cacaaIWaGaaGimaiaaiAdacaaIYaGaamOzaOWaaS baaKqbagaajugWaiaadshaaKqbagqaaOWaaeWaaKqbagaajugibiaa igdacaaIWaWcdaahaaqcfayabeaajugWaiaaiAdaaaGcdaWfGaqcfa yaaKqzGeGaeqyTduMcdaWgaaqcfayaaKqzadGaam4yaaqcfayabaaa beqaaKqzGeGaaiOlaaaaaKqbakaawIcacaGLPaaalmaaCaaajuaGbe qaaKqzadGaaGymaiaac+cacaaIZaaaaaaa@5AAC@ for ε . c 10/s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWfGaqaaGGaaKqzGeGae8xTdugaleqabaqcLbsacaGGUaaaaOWa aSbaaSqaaKqzadGaam4yaaWcbeaajugibiab=vMiZkab=fdaXiab=b daWiab=9caViaadohaaaa@43A7@ (5d)

f t =0.3 f c ' 2/3 ( 1+ 2 3 l f d f V f ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb GcdaWgaaqcfayaaKqzadGaamiDaaqcfayabaqcLbsacqGH9aqpcaaI WaGaaiOlaiaaiodacaWGMbGcdaqhaaqcfayaaKqzadGaam4yaaqcfa yaaKqzadGaai4jaaaakmaaCaaajuaGbeqaaKqzadGaaGOmaiaac+ca caaIZaaaaOWaaeWaaKqbagaajugibiaaigdacqGHRaWkkmaalaaaju aGbaqcLbsacaaIYaaajuaGbaqcLbsacaaIZaaaaOWaaSaaaKqbagaa jugibiaadYgakmaaBaaajuaGbaqcLbmacaWGMbaajuaGbeaaaeaaju gibiaadsgakmaaBaaajuaGbaqcLbmacaWGMbaajuaGbeaaaaqcLbsa caWGwbGcdaWgaaqcfayaaKqzadGaamOzaaqcfayabaaacaGLOaGaay zkaaGcdaahaaqcfayabeaajugWaiaaigdacaGGVaGaaG4maaaaaaa@65A9@ for f c <50 MPa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaajugibi qadAgagaqbaSWaaSbaaeaajugWaiaadogaaSqabaqcLbsacqGH8aap caaI1aGaaGimaiaabccacaqGnbGaaeiuaiaabggaaaa@41BD@ (5e)

f t =2.12ln[ 1+0.1( f c +8 ) ]( 1+ 2 3 l f d f V f ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb GcdaWgaaqcfayaaKqzadGaamiDaaqcfayabaqcLbsacqGH9aqpcaaI YaGaaiOlaiaaigdacaaIYaGaaeiBaiaab6gakmaadmaajuaGbaqcLb sacaaIXaGaey4kaSIaaGimaiaac6cacaaIXaGcdaqadaqcfayaaKqz GeGabmOzayaafaWcdaWgaaqcfayaaKqzadGaam4yaaqcfayabaqcLb sacqGHRaWkcaaI4aaajuaGcaGLOaGaayzkaaaacaGLBbGaayzxaaGc daqadaqcfayaaKqzGeGaaGymaiabgUcaROWaaSaaaKqbagaajugibi aaikdaaKqbagaajugibiaaiodaaaGcdaWcaaqcfayaaKqzGeGaamiB aOWaaSbaaKqbagaajugWaiaadAgaaKqbagqaaaqaaKqzGeGaamizaO WaaSbaaKqbagaajugWaiaadAgaaKqbagqaaaaajugibiaadAfakmaa BaaajuaGbaqcLbmacaWGMbaajuaGbeaaaiaawIcacaGLPaaaaaa@6A9F@ for f c 50 MPa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaajugibi qadAgagaqbaSWaaSbaaeaajugWaiaadogaaSqabaqcLbsacqGHLjYS caaI1aGaaGimaiaabccacaqGnbGaaeiuaiaabggaaaa@427F@ (5f)

f cd = f c ( 10 5 ε . c /3 ) 0.014 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGMb GbauaalmaaBaaajuaGbaqcLbmacaWGJbGaamizaaqcfayabaqcLbsa cqGH9aqpceWGMbGbauaalmaaBaaajuaGbaqcLbmacaWGJbaajuaGbe aakmaabmaaleaajugibiaaigdacaaIWaGcdaahaaadbeqaaKqzadGa aGynaaaakmaaxacaleaajugibiabew7aLbadbeqaaKqzadGaaiOlaa aakmaaBaaameaajugWaiaadogaaWqabaqcLbsacaGGVaGaaG4maaWc caGLOaGaayzkaaGcdaahaaadbeqaaKqzadGaaGimaiaac6cacaaIWa GaaGymaiaaisdaaaaaaa@56D1@ for ε . c <30/s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWfGaqaaGGaaKqzGeGae8xTdugaleqabaqcLbsacaGGUaaaaOWa aSbaaSqaaKqzadGaam4yaaWcbeaajugibiab=Xda8iab=ndaZiab=b daWiab=9caViaadohaaaa@42E9@ C (5g)

f cd =0.012 f c ( 10 5 ε . c /3 ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWGMb GbauaalmaaBaaajuaGbaqcLbmacaWGJbGaamizaaqcfayabaqcLbsa cqGH9aqpcaaIWaGaaiOlaiaaicdacaaIXaGaaGOmaiqadAgagaqbaS WaaSbaaKqbagaajugWaiaadogaaKqbagqaaOWaaeWaaSqaaKqzGeGa aGymaiaaicdakmaaCaaameqabaqcLbmacaaI1aaaaOWaaCbiaSqaaK qzGeGaeqyTdugameqabaqcLbmacaGGUaaaaOWaaSbaaWqaaKqzadGa am4yaaadbeaajugibiaac+cacaaIZaaaliaawIcacaGLPaaakmaaCa aameqabaGaaGymaiaac+cacaaIZaaaaaaa@57CC@ for ε . c 30/s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWfGaqaaGGaaKqzGeGae8xTdugaleqabaqcLbsacaGGUaaaaOWa aSbaaSqaaKqzadGaam4yaaWcbeaajugibiab=vMiZkab=ndaZiab=b daWiab=9caViaadohaaaa@43AB@ (5h)

Where ε . c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWfGaqaaGGaaKqzGeGae8xTdugaleqabaqcLbsacaGGUaaaaOWa aSbaaSqaaKqzadGaam4yaaWcbeaaaaa@3DB2@ is the strain rate of concrete; E cs MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=Mj0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadweak8aadaWgaaWcbaqcLbmapeGaam4yaiaadohaaSWd aeqaaaaa@3C6C@ is the elastic modulus of concrete (= 4700 f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeGabiqaciGaciGaaeGabaWaceGaeaaakeaajugibi aaisdacaaI3aGaaGimaiaaicdajuaGdaGcaaGcbaqcLbsaceWGMbGb auaalmaaBaaabaqcLbmacaWGJbaaleqaaaqabaaaaa@3FB3@ in MPa) under static load; ft is the concrete tensile strength; lf is the steel fiber length; df is the steel fiber diameter; and Vf is the volume ratio of steel fibers. The compressive and tensile strength increased by strain rate effect are specified in Model Code 2010.17 In Eq. 5e and 5f, the effect of steel fiber on the tensile strength of concrete is considered on the basis of the test results of Musmar.18

Figure 2 shows the relationship between the penetration depth x and resistant energy ER corresponding to the kinetic energy EK. When the kinetic energy EK is less than the resistant energy E1, E2, or E3, the penetration depth is linearly interpolated between 0 and x1, x1 and x2, or x2 and x3, respectively.

Figure 2 Resistant energy-penetration depth relationship.

Parametric analysis

To investigate the effects of design parameters on the penetration depth, a parametric study was performed for a concrete target under impact load. A prototype concrete target has the width b = 1000 mm; thickness h = 300 mm; concrete density ρ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaajugibi abeg8aYTWaaSbaaeaajugWaiaadogaaSqabaaaaa@3C50@ = 2350 kg/m3; coarse aggregate size Sa MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0=Mr0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb qcLbmacaWGHbaaaa@3B13@ = 20 mm; steel fiber volume ratio Vf MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0=Mr0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGwb qcLbmacaWGMbaaaa@3B1B@ = 0%; and effective reinforcing bar area As = 0 mm2. A prototype steel projectile has the diameter d = 20 mm; mass m = 100 g; initial velocity Vi = 300 m/s; length L = 40 mm; and round nose shape. Design parameters are as follows: the concrete strength f c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0=Mr0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaqaaaaa aaaaWdbiaadAgajuaGpaWaaSbaaSqaaKqzadWdbiaadogaaSWdaeqa aKqzadWdbiabgkdiIcaa@3EF9@ ranges from 20 to 120 MPa; rebar area As MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0=Mr0xXdbba91rFfpec8Eeeu0xXdbba9frFj0=OqFf ea0dXdd9vqai=hGuQ8kuc9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb qcLbmacaWGZbaaaa@3B13@ from 0 to 0.48 % (yield strength fy = 400 MPa); coarse aggregate size sa from 5 to 30 mm; concrete density ρ c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8qqaqFr0x c9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8fr Fve9Fve9Ff0dmeaabiqaciGaciGaaeGabaWaceGaeaaakeaajugibi abeg8aYTWaaSbaaeaajugWaiaadogaaSqabaaaaa@3C50@ from 2350 to 2500 kg/m3; steel fiber volume ratio Vf from 0 to 2.0% (fiber shape ratio lf / df = 60); projectile initial velocity Vi from 100 to 1100 m/s; projectile mass m from 100 to 1100 g; projectile diameter d from 10 to 60 mm; and concrete target type (single target and dual target).

Figure 3 compares the penetration predictions of the modified NDRC model5 and energy based model16 according to the concrete target parameters. As concrete strength increases, the penetration depth decreases (Figure 3). In general, the prediction of the modified NDRC model5 is greater than that of the energy based model.16 The penetration depth of the energy based model16 is significantly increased at the concrete targets using low strength concrete under impact of Vi = 600 m/s. This is because the tunneling resistance is additionally generated to absorb the kinetic energy in the concrete target using low strength concrete. Reinforcing bars improve the spalling and scabbing resistance, which decreases the penetration depth and damage (Figure 3B). The use of small aggregate size and concrete with large density increase the penetration depth because the impact force of the projectile easily passes through small coarse aggregate in the concrete target (Figure 3C). The use of steel fibers increases the flexural tensile strength of concrete, which decreases the penetration depth (Figure 3D).

Figure 3 Penetration depth according to design parameters of concrete target.

Figure 4 compares the penetration predictions of the modified NDRC model5 and energy based model16 according to the projectile parameters. As the impact velocity increases, the penetration depth increases. Particularly, the prediction of the energy based model16 is significantly increased at Vi = 600 ~ 700 m/s because the tunneling resistance is activated (Figure 4A). Projectile mass increases the kinetic energy, which increases the penetration depth, but the increase ratio is reduced at a certain level (Figure 4B). Thus, although both the impact velocity and projectile mass increases the impact energy, the impact damage of the concrete target is more vulnerable to the high-velocity impact than the heavy-mass impact. Large projectile diameter increases the resisting area of the concrete target, which decreases the penetration depth.

Figure 4 Penetration depth according to design parameters of projectile.

Figure 5 compares the effect of the number of concrete target on the penetration depth. Single concrete target has a thickness of 300 mm, and dual concrete target consists of two concrete targets with a thickness of 150 mm. Under low-velocity impact, the single concrete target exhibits better penetration resistance than that of the dual concrete target. However, the penetration depth of the single concrete target is significantly increased at Vi = 600~700 m/s due to tunneling failure. In the dual concrete target, perforation failure occurs in the first target at Vi = 700 m/s. As a result, larger impact energy can be dissipated by additional scabbing resistance of the first concrete target at the same penetration depth. Ultimately, the penetration depth can be reduced in the dual concrete target under large impact load. Reinforcing bars in the dual concrete target decreases the penetration depth effectively.

Figure 5 Penetration depths according to the number of concrete target.

Figure 6 shows the minimum thickness of concrete target to avoid perforation failure (i.e., EK = ER) according to concrete strength. Although the same kinetic energy EK is considered, the penetration depth is more vulnerable to high-velocity impact. As concrete strength increases, the concrete target thickness decreases. In general, the concrete target thickness is decreased by about 30% at every two times concrete strength. When reinforcing bars are used, the concrete target thickness is decreased by only 5%. The use of steel fiber with 2.0% volume ratio decreases about 18% of the concrete target thickness. Thus, high strength fiber-reinforced concrete is recommended to improve the impact resistance of concrete targets.

Figure 6 Minimum thickness of concrete target without perforation failure.

Conclusion

To investigate the effects of design parameters on the local impact damage of concrete members, a parametric study was performed by the modified NDRC model and energy-based model, addressing various design conditions of a concrete member and projectile. The parametric study results showed that the use of higher strength concrete, greater reinforcement ratio, larger aggregate size, lower concrete density, and larger steel fiber volume ratio in concrete target design was desirable to decrease the impact damage and penetration depth. The projectile with high-velocity, heavy mass and small diameter increased the penetration depth. Particularly, compared to the single concrete target, dual concrete target system was more efficient to absorb the large impact energy. To avoid perforation failure effectively in concrete targets, high strength fiber-reinforced concrete should be used. This study can be used to guide the direction of future experimental and theoretical studies.

Acknowledgement

This research was financially supported by National Key Research Program of China (2016YFC0701400) and National Natural Science Foun­dation of China (Grant No. 51650110500). The authors are grateful to the authorities for their supports.

Conflicts of interest

The author declares there is no conflict of interest.

References

  1. Li J, Wu C, Hao H, et al. Blast Resistance of Concrete Slab Reinforced with High Performance Fibre Material. Journal of Structural Integrity and Maintenance. 2016;1(2):51–59.
  2. Kennedy RP. A Review of Procedures for the Analysis and Design of Concrete Structures to Resist Missile Impact Effects. Nuclear Engineering and Design.1976;37(2):183–203.
  3. ACE. Fundamentals of Protective Structures. Report AT120 AT1207821. Army Corps of Engineers, Office of the Chief of Engineers, USA; 1946.
  4. NDRC. Effects of Impact and Explosion. Summary Technical Report of Division 2, V.1, National Defense Research Committee, Washington, DC, USA; 1946.
  5. Kennedy RP. Effects of an aircraft crash into a concrete reactor containment building. Anaheim, CA: Holmes & Narver Inc., USA; 1966.
  6. Whiffen P. UK Road Research Laboratory. Note No. MOS/311, UK; 1943.
  7. Kar AK. Local Effects of Tornado Generated Missiles. Journal of the Structural Division. 1978;104(5):809–816.
  8. Hughes G. Hard Missile Impact on Reinforced Concrete. Nuclear Engineering and Design. 1984;77(1):23–35.
  9. Kojima I. An Experimental Study on Local Behaviour of Reinforced Concrete Slabs to Missile Impact. Nuclear Engineering and Design. 1991;130(2):121–132.
  10. Reid SR, Wen HM. Predicting Penetration, Cone Cracking. Scabbing and Perforation of Reinforced Concrete Targets Struck by Flat–Faced Projectiles. UMIST Report ME/AM/02.01/TE/G/018507/Z, UK; 2001.
  11. Forrestal MJ, Altman BS, Cargile JD, et al. An Empirical Equation for Penetration Depth of Ogive–Nose Projectiles into Concrete Targets. International Journal of Impact Engineering. 1994;15(4):395–405.
  12. Li QM, Chen XW. Dimensionless Formulae for Penetration Depth of Concrete Target Impacted by A Non–Deformable Projectile. International Journal of Impact Engineering. 2003;28(1):93–116.
  13. Guirgis S, Guirguis E. An Energy Approach Study of the Penetration of Concrete by Rigid Missiles. Nuclear Engineering and Design. 2009;4:819–829.
  14. Almusallam TH, Siddiqui N, Iqbal RA, et al. Response of Hybrid–Fiber Reinforced Concrete Slabs to Hard Projectile Impact. International Journal of Impact Engineering. 2013;58:17–30.
  15. Peng Y, Wu H, Fang Q, et al. A Note on the Deep Penetration and Perforation of Hard Projectiles into Thick Targets. International Journal of Impact Engineering. 2015;85:37–44.
  16. Hwang HJ, Kim S, Kang THK. Energy–Based Penetration Model for Local Impact–Damaged Concrete Members. ACI Structural Journal. 2017;114(5):1189–1200.
  17. Fib. fib Model Code for Concrete Structures 2010. Fib, Ernst & Sohn, Germany. 2010:420.
  18. Musmar M. Tensile Strength of Steel Fiber Reinforced Concrete. Contemporary Engineering Sciences. 2013;6(5):225–237.
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