IntroductionRC structures play a primary role in protecting lives and property against external hazards, such as blasts, impacts, and attacks (Carrasquillo et al. 1981; Ibrahim Ary and Kang 2012; Kang and Ibrahim Ary 2012). Since 2006, public buildings and infrastructures have been damaged by more than 200,000 terrorist activities (GTD 2018). These blasts and impacts can be catastrophic. Thus, major buildings and infrastructures likely deemed to be targets should consider protection performance. High-strength concrete and ultrahigh performance concrete (UHPC) offer many advantages in the field of RC structure protection and fortification. When the same protection performance is necessary, high-strength concrete and UHPC are advantageous in that they reduce the cross-section of the RC member compared to the same functionality in normal strength concrete (Kang et al. 2019). In the case of UHPC or higher strength concrete mixed with steel fibers, they possess a high level of energy absorption and an increasing level of impact performance on the rear face, which is opposite of the point of collision. Therefore, impact performance and applications of high-strength concrete and UHPC have been the subject of research (Ngo et al. 2013; Riedel et al. 2010).Various studies have been performed to evaluate the local impact-resistant performance of RC walls subjected to external attacks and collisions. In 1910, Petry first proposed an equation to predict the damage level of concrete under impact loading. Numerous studies and research programs were subsequently conducted in the 1940s and 1980s (Li et al. 2005; Hwang et al. 2017; Kim et al. 2017). Those conducted in the 1940s were primarily geared toward reduction in structural damage associated with World War II. Mainstream research done in the 1980s was tied to improving the safety of nuclear power plants. Equations derived during these two eras to determine local impact damage were empirically derived from test results conducted on normal-strength concrete.With the development of high-strength concrete and UHPC (Beppu et al. 2008; Drdlová et al. 2015; Kong et al. 2017), additional experimental and analytical studies on high-velocity impact loading have been conducted. Based on experimental data, prediction equations for penetration depth, scabbing limit thickness, and perforation limit thickness have been developed. Existing equations, although not all-inclusive, consider design parameters, such as concrete compressive strength and thickness of the concrete target, along with the velocity, mass, nose shape, and diameter of the projectile (Kennedy 1976; Kojima 1991; Li et al. 2005; Kim et al. 2017).Nose shape factors proposed by previous researchers have been used in the development of local impact damage equations. In particular, values proposed in the modified National Defense Research Committee (NDRC) equation (NDRC 1946) are used in the Kar equation (Kar 1978), Haldar and Hamieh equation (Haldar and Hamieh 1984), and Ammann and Whitney equation (Ammann and Whitney 1963) to determine local impact damage. Given that the NDRC equation was developed in 1946, and plausible impact resistance differences from that of normal-strength concrete, it is vital to confirm whether or not these nose shape factors and existing equations used to determine local impact damage may be applied to high-strength concrete and UHPC.To evaluate the validity of existing equations for penetration depth due to high-velocity impact loading, relatively rigid projectile impact loading tests were performed on concrete targets in this study: (1) using normal- to high-strength concrete, including UHPC, and (2) with projectiles having two kinds of nose shapes (conical versus hemispherical).Design compressive strength for normal-strength concrete, high-strength concrete, and UHPC were 35, 80, and 150 MPa, respectively. The high-strength concrete with coarse aggregates used in this study had enhanced fluidity, while UHPC had characteristics of very high fluidity without coarse aggregates, enhanced ductility and toughness provided by steel fibers, and very high strength (measured above 150 MPa).Penetration depth and nose shape impacts were evaluated based on measured concrete strength. Application of the existing equations for the determination of the penetration depth was compared with the test results. The potential with respect to the performance-based impact design of buildings and infrastructures is also discussed.Review of Existing Equations for Local Impact Load ResistanceNose Shape FactorAlthough many studies have been performed on the local impact load resistance of concrete, limited research literature pertaining to the projectile nose shape coefficient is available (Li et al. 2005; Kennedy 1976; Haldar and Hamieh 1984; Zhang et al. 2017). Within the modified NDRC (Kennedy 1976), Hughes (1984), University of Manchester Institute of Science and Technology (UMIST) (Reid and Wen 2001), and Hwang et al. (2017) equations, the nose of the projectile is classified as either flat, blunt, hemispherical, or sharp (Table 1).Table 1. Nose shape factors of existing equationsTable 1. Nose shape factors of existing equationsEquationsParametersModified NDRC (Kennedy 1976)0.72, 0.84, 1.0, and 1.14 for flat, blunt, hemispherical, and sharp, respectivelyHughes (1984)1.0, 1.12, 1.26, and 1.39 for flat, blunt, hemispherical, and sharp, respectivelyYoung (Sandia) (1997)0.56+0.18Rn/dp for ogive,0.56+0.25ln/dp for conical, and0.56+0.09(ln+Rn)dp for bluntUMIST (Reid and Wen 2001)0.72, 0.84, 1.0, and 1.13 for flat, blunt, hemispherical, and sharp, respectivelyLi and Chen (2003)13(Rn/dp)−124(Rn/dp)2 for ogive nose1−18(Rn/dp)2 for blunt/spherical nose11+4(ln/dp)2 for conical noseHwang et al. (2017)2.0, 1.9, 1.55, and 0.9 for flat, hemispherical, ogive, and sharp, respectively, and these values are used to calculate the spalling energy1.0, 0.9, 0.7, and 0.2 for flat, hemispherical, ogive, and sharp, respectively, and these values are used to calculate the tunneling energyLi and Chen (Chen and Li 2002; Li and Chen 2003) proposed a nose shape factor dependent upon the geometric configuration of the nose (Fig. 1). However, Young (1997) (called the Sandia equation) advocated a nose shape factor based upon the geometric properties of the nose and soil penetration data under low-velocity impact.Existing Equations for Penetration DepthThe modified NDRC nose shape factors are defined as 0.72, 0.84, 1.0, and 1.14 for flat, hemispherical, blunt, and very sharp noses, respectively. These nose shape factors were adopted in the Kar, Haldar and Hamieh, United Kingdom Atomic Energy Authority (UKAEA), and Ammann and Whitney equations (Li et al. 2005) (1) xpe=2dpGfor  xpe/dp≤2(2) xpe=(G+1)dpfor  xpe/dp>2(3) G=1.751×10−5KNpMpdp(Vimpdp)1.8where xpe = penetration depth (m); Mp = mass of the projectile (kg); K = penetrability of concrete (=2.17/fc′); fc′ = compressive strength of concrete (Pa); Np = nose shape factor; dp = diameter of the projectile (m); Vimp = impact velocity of the projectile (m/s); and G = impact function (dimensionless) of the modified NDRC equation.Hughes (1984) proposed a penetration equation [Eqs. (4)–(6)] using the nondimensional impact factor (I) defined as a function of mass, velocity, and diameter of the projectile, along with the modulus of rupture of concrete (fr). The dynamic increase factor (S) was used to calculate the penetration depth. Nose shape factors were defined as 1, 1.12, 1.26, and 1.39 for flat, blunt, spherical, and sharp noses, respectively (4) (5) (6) where I = impact factor; S = dynamic increase factor; and fr = modulus of rupture of concrete (Pa).As part of the research conducted on behalf of the United States Department of Energy on Earth Penetrating Weapons (EPW), Young (1997), at Sandia National Laboratories (SNL), studied impact resistance of natural earth materials (soil, rocks, ice, and frozen soil) along with concrete. In 1967, Young proposed a penetration equation, which has been updated several times (Young 1997). The equation [Eqs. (7)–(11)] included nose shape factors for ogive, conical, and blunt noses depending upon the geometric properties of the projectile (Table 1). These nose shape factors were originally presented based on soil impact test data under low-velocity impact and applied later to other materials (7) xpe=0.0008×Kh×Y×Np(Mp/Ap)0.7ln(1+2.15Vimp210−4)for  Vimp<61  m/s(8) xpe=0.000018×Kh×Y×Np(Mp/Ap)0.7ln(Vimp−30.5)for  Vimp≥61  m/s(9) Kh=0.46(Mp)0.15for  Mp≤182  kg, elseKh=1.0(10) Y=0.085Ke(11−Prebar)⁢(tcTc)−0.06(35/fc′)0.3(default value of Y is 0.9)(11) where fc′ = compressive strength of concrete (MPa); Tc = ratio of the concrete target thickness to projectile diameter (if Tc is over 6, set to be 6); Kh = correction factor for lightweight projectile; Ke = factor to consider the edge effect on the concrete target; Freinf = coefficient related to reinforcement (20 for reinforced concrete or 30 for plain concrete); W1 = ratio of the target width to the penetrator diameter; Prebar = volume ratio of rebar (%); and tc = concrete curing time (if tc is over 1 year, set to be 1).As part of the General Nuclear Safety Research Program, impact loading tests were conducted to evaluate the impact load resistance of reinforced concrete walls in 1985 (Reid and Wen 2001; Li et al. 2005). The penetration depth equation [Eqs. (12) and (13)] was proposed to evaluate kinetic energy and residual energy once the missile punctures the target. New nose shape factors were presented as 0.72, 0.84, 1.0, and 1.13 for flat, hemispherical, blunt, and sharp noses, respectively (12) xpdp=(2π)Np0.72MVimp2σtdp3(13) σt=4.2fc′+135×106+(0.014fc′+0.45×106)Vimpwhere fc′ = compressive strength of concrete (MPa); and σt = rate-dependent strength of concrete.Based on the dynamic cavity expansion model, Luk and Forrestal (1987) proposed an equation to estimate the depth of penetration. Forrestal et al. (1994) calibrated the factors using 6 sets of impact test data for an ogive nose projectile. Li and Chen (2003) further developed the penetration depth model [Eqs. (14)–(19)] with modified S factors for concrete strength and studied the impact factor N* (Chen and Li 2002; Li and Chen 2003). The factor N* is defined as a function of the nose shape and sectional pressure, which is composed of the mass, density, and diameter of the projectile. The nose shape factor (Np) was calculated based on the nose geometry of the projectile (Table 1) (14) xpdp=(1+(kπ/4N*)(1+I/N*)4kπIfor  xpdp≤k(15) xpdp=2πNln[1+I/N*1+(kπ/4N*)]+kfor  xpdp>k(16) (17) I=1S(MpVimp2dp3fc′)=172fc−0.5(MpVimp2dp3fc′)(18) k=(0.707+hdp)for  xdp<5, k=2 for  xdp≥5(19) xpedp=1.628(1+kπ/4N1+I/N4kπI)1.395for  xpedp≤0.5where fc′ = compressive strength of concrete (Pa); h = thickness of target (m); ρc = density of concrete (kg/m3); k = factor related to the target thickness and the diameter of the projectile; and I = impact factor.Hwang et al. (2017) compared impact energy with resistant energy to estimate the penetration depth and residual velocity of the projectile. Using the law of conservation of energy, an equation for the penetration depth of thick and thin concrete targets was proposed and verified by test results. The impact mechanism was divided into three stages: spalling, tunneling, and scabbing. Spalling refers to the truncated dome or bowl shape failure at the face of the concrete target. Tunneling refers to the projectile penetrating into the concrete target and making a tunnel without significant failure besides passage of the projectile. Scabbing refers to concrete cone failure at the rear face of the concrete target. The nose shape factors were presented separately in accordance with spalling or tunneling (Table 1). The shape of the spalled concrete cone was considered related to the shape of the nose of the projectile, and the nose shape factor for spalling was adopted as tanθs. It is noted that Eqs. (20)–(36) present the penetration depth of the concrete target under spalling failure (20) xpe=tsEk−EDP−EDCES(ts)(for  spalling failure)(21) (22) k1=2.1(hdp)0.3−1.75≥0(23) k2=1−0.025Vf(in  %)(24) k3=5.94−(2.1ρc)/1,000≤1(in kg/m3)(25) (26) EDP=πdp28lpEp(ρpVimp22)2(27) EDC=b396EcdIe(ρpVimp22πdp24)2(28) (29) (30) ES=[ftd+AsfycosθssinθsAsp]Vsckskbs(31) ftd=ft(106ε˙c)0.018for  ε˙c<10/s(32) ftd=0.0062ft(106ε˙c)1/3for  ε˙c≥10/s(33) ft=0.3fc′2/3for  fc′<50  MPa (in MPa)(34) ft=2.12ln[1+0.1(fc′+8)]for  fc′≥50  MPa (in MPa)(35) (36) Vsc=πts12[4ts2tan2θs+6dptstanθs+3dp2]where dp, Lp, ρp, Ep, and Vimp = diameter, length, density, elastic modulus, and impact velocity of the projectile, respectively; Asp and Vsc = projected area and volume of the idealized concrete cone, respectively; ks = size effect factor of the concrete target [=(300/h) 0.25≤1  (mm)]; kbs = stress concentration effect factor of the concrete target [=4/√(bs/h)≤1.25]; bs = average perimeter of the concrete cone [=π(dp+tstanθ)]; As = sum of the effective reinforcing bar area in the concrete cone in horizontal and vertical directions; fy = yield strength of the reinforcing bar; ft = concrete tensile strength; Ecd and ftd = elastic modulus and tensile strength of concrete affected by the strain rate, respectively; b = concrete target width; Ie = effective moment of inertia of the concrete target; Ec = elastic modulus of concrete; EK = kinetic energy of the projectile; EDP = deformed energy of projectile; EDC and ES = deformed energy and spalling-resistant energy of the concrete target, respectively; h = thickness of the concrete target; k1 = coefficient related to h; k2 = coefficient related to the steel fiber volume ratio; k3 = coefficient related to the concrete density; and k4 = coefficient related to the maximum size of coarse aggregates.Experimental ProgramTest Plan and SpecimensThe experiment was planned to investigate the prediction accuracy of existing impact equations for high-strength concrete and UHPC, as well as the validity of nose shape factors in existing impact equations (Kim 2018). Test parameters included concrete strength and two projectile nose shapes: hemispherical and conical. A design compressive strength of 35 MPa was set as the reference strength, and test targets with design compressive strengths of 80, 100, 120, and 150 MPa were prepared (Table 2). To simulate field conditions, high-strength concrete specimens in the range of 80–120  MPa were developed without high-temperature curing, and steel fibers were not included. The maximum coarse aggregates size was limited to 20 mm. However, UHPC test targets having a design compressive strength of 150 MPa were developed using laboratory mix proportions without coarse aggregates and were steam cured. Because it is not known whether the effect of steel fibers on concrete penetration resistance is insignificant (Kim et al. 2017, 2018), steel fiber having a volume of 1.5% was also used.Table 2. Concrete mix proportion and material propertiesTable 2. Concrete mix proportion and material propertiesMaterialfc,design′ (MPa)3580100120150W/B (%)4218.814.913.520S/a (%)47403635—Water (kg/m3)160150140130196.9Cement (kg/m3)2104004704321,287.8aFine aggregates (kg/m3)826578484473866.4Coarse aggregates (kg/m3) [maximum size (mm)]942 [25]882 [20]876 [20]892 [20]—ZrSF (kg/m3)—120141144—Fly ash (kg/m3)76————Slag powder (kg/m3)95280329384—Superplasticizer (kg/m3)—1011.751224.4Air-entraining agent (kg/m3)3.05———0.9Steel fibers (kg/m3) 16 mm and 20 mm————39 and 78fc,meas′ (MPa)411038196162fr,meas (MPa)5.58.467.848.37—fct,meas (MPa)3.467.095.734.4314.23fdt,meas for 150 MPaDensity (kg/m3)2,3302,4212,4642,4262,406High-strength concrete for the 80–120  MPa test targets was designed such that compressive strength would develop after 120 days without steam curing. Zirconia silica fume (ZrSF) was added to maintain concrete strength and improve fluidity (Joh et al. 2012; Koh et al. 2012). Concrete mix proportions were considered for actual construction practical application. After material testing, ready mixed concrete was used for the concrete targets. Six specimens with the dimensions of 200×200×100  mm were manufactured for each target compressive strength.According to the Hughes (1984) study, the penetration depth is overestimated when the ratio of the concrete target thickness to projectile diameter is less than 5. Therefore, in this study, the thickness of the concrete target was designed to be 100 mm so that the thickness-to-diameter ratio satisfies the value of 5, and the scabbing failure would not occur on the rear face of the specimen. Using the Hughes equation (Hughes 1984), a minimum thickness of 82 mm would be required when a hemispherical projectile with a diameter of 20 mm (weight of about 90 g) collides with the 35 MPa concrete target at an impact velocity of 140  m/s. However, the modified NDRC equation (Kennedy 1976) predicted a need for a minimum thickness of 67 mm under the same collision conditions.Two straight steel fibers having a nominal tensile strength of 2,000  MPa with a diameter of 0.2 mm were used for the UHPC targets. Lengths of 16 mm (volume ratio of 0.5%) and 20 mm (volume ratio of 1.0%) were used respectively. In order to suppress the fire-balling and poor orientation when manufacturing UHPC mixed with two types of steel fibers, these steel fibers were gradually inputted into a fan-mixer machine.The UHPC targets were made using a premixed binder with zirconium, filler, and ground granulated blast furnace slag (GGBS). To address the scabbing limit thickness and overestimation of the penetration equation when the ratio of thickness to the diameter of the projectile is over 5, the size of the UHPC targets were designed to be 200×200×50  mm. Previous test results showed that the penetration depth of the UHPC target was about 3–4  mm when subject to a 20 mm sphere projectile at 200  m/s (Kang et al. 2016). The modified NDRC equation (Kennedy 1976) predicted the need for a minimum thickness of 58 mm to ensure against scabbing failure.For impact loading tests, two types of projectiles were prepared (Fig. 2): conical and hemispherical. The conical projectile had a diameter of 20 mm, length of 40 mm, and nose length of 10 mm at 45°. The hemispherical projectile had the same configuration except that the radius of the nose was 10 mm. To increase the strength of the projectiles, the projectiles were subjected to heat treatment. The steel type projectile was SCM 440, and the featuring tensile strength and hardness were rated at 100–130 ksi (700–1,000  MPa) and 18–22 HRc (Rockwell hardness).TestingMaterial Curing and TestsThe material properties for the 35–120  MPa concrete targets were ascertained using five cylinder molds of 100×200  mm and five prism molds of 100×100×400  mm. The specimens for material tests were cured under the same conditions—air-dry curing for 24 h, and then sealed curing for 28 days—like that of the concrete target specimens for high-velocity impact tests. The compressive strength of concrete was measured according to KS F 2405/ISO 4012:1978 (KATS 2010; ISO 1978a). To measure the concrete strain, a strain gauge was attached. Bending strength tests were carried out according to KS F 2408/ISO 4013:1978 (KATS 2016; ISO 1978b), and from the five test results, an average value was obtained.For the UHPC concrete targets having a specified strength of 150 MPa, five 100×200  mm and four dog-bone specimens were used to measure the compressive and direct tensile strength, respectively. The specimens were cured at room temperature after 48 h of high-temperature wet curing at 90°C±5°C. For the dog-bone specimens, testing for direct tensile strength was conducted according to KCI (2012) structural design guidelines. In addition, the material properties of UHPC were measured by independent testing institutes.Impact TestsAn impact test device (Fig. 3) was designed by the authors (Kim et al. 2019), whereby a projectile would collide with a concrete target specimen at the velocity of 100–120  m/s using an air compressor. The air compressor machine compresses air into the air tank, and the discharged compressed air pushes a projectile at the starting point of the barrel as soon as the valve is opened. The projectile moves along the barrel and hits the fixed specimen in the chamber, where its velocity is measured by the velocity measurement device installed in the chamber. In general, for impact tests, high velocity is defined as over 100  m/s. The minimum impact velocity of the projectile was set to 100  m/s in this experiment (Xiao et al. 2017). The front and back sides of the concrete target specimen were firmly fixed with steel frames. To measure the collision speed of each projectile, a velocity measurement device was installed in front of the concrete target specimen.Test Results and AnalysisMaterial Test ResultsThe average compressive strength for the target specimens whose design compressive strengths were 35, 80, 100, 120, and 150 MPa were measured at 41, 103, 81, 96, and 162 MPa, respectively (Table 2 and Fig. 4).For the high-strength concrete target specimens, measured concrete strength for that having design compressive strengths of 100 and 120 MPa was 81 and 96 MPa, respectively, about 20% less. The reason for the difference in concrete strength may be attributed to unknown cause(s), such as segregation during the moving or mix process and/or the curing temperature being relatively low in developing the target strength because the 100 and 120 MPa concrete specimens were poured in October (average temperature of 13.5°C). Although the specified concrete strengths of 100 and 120 MPa were not reached, their measured strengths were sufficient to perform the impact test for the purpose of this study. Meanwhile, the design concrete of 80 MPa reached an effective strength of 103 MPa, having been poured during September (with an average temperature of 22°C and a peak temperature of 29°C, which was 6°C higher than past years). This was an unexpectedly strong concrete specimen as a result.The measured strains at peaks of 35, 80, 100, and 120 MPa ranged from 0.025 to 0.028 and that for UHPC was 0.0038.For the concrete specimens, the average measured flexural strengths were 5.5, 8.5, 7.8, and 8.4 MPa, and the splitting tensile strengths were 3.46, 7.09, 5.73, and 4.43 MPa for the 35, 80, 100, and 120 MPa, respectively. The average direct tensile strength was 14.23 MPa for the 150 MPa.Penetration DepthTable 3 and Fig. 5 show the relationship between the measured penetration depth (xtest) and the concrete compressive strength in the target specimens. In the context of collision experiments, the results are often significantly varied, depending on the fluctuating conditions. Thus, it may become difficult to analyze the experimental results. It could be considered a good experiment when a researcher can successfully obtain consistent results under the same conditions. Hwang et al. (2017) provide a coefficient of variation (COV) of 0.293 across 13 test programs focused on various projectile sizes [see Tables 3 and 4 in the study by Hwang et al. (2017)]. Meanwhile, the COV in this experiment was 0.116., which is indicative of this program having a strong plan and obtaining consistent results. For the conical and hemispherical projectiles, their average xtest values were 13.22, 10.10, 9.63, 9.01, and 6.97 mm for the 41, 81, 96, 103, and 162 MPa concrete target specimens, respectively. The trends and observations made were that as the concrete strength increased, the penetration depth decreased. In short, the use of high-strength concrete and UHPC can significantly improve local impact load resistance when compared to normal-strength concrete. The decrease in high-strength concrete and UHPC target specimens was linear.Table 3. Impact test results and testing of predicted penetration depthsTable 3. Impact test results and testing of predicted penetration depthsSpecimenVimp (m/s)xtest (mm)xtestEK (10−2  mm/J)xtestxNDRCxtestxHughesxtestxLi-ChenxtestxHwangxtestxUMISTxtestxYoung41C-1C11512.572.320.780.680.351.001.980.5641C-2C11413.072.430.810.710.371.052.080.5941C-3C11313.912.660.880.760.391.202.260.6441C-4H11914.072.210.860.790.371.102.570.5941C-5H11812.51.990.770.710.330.992.300.5313.182.470.820.720.371.082.100.6013.292.100.820.750.351.042.430.5613.222.320.820.730.361.072.240.5881C-1C1169.331.670.680.560.300.792.200.5081C-2C13510.021.340.640.540.280.701.840.4581C-3C13110.71.510.700.590.310.782.060.4981C-4H1209.41.430.670.590.280.752.560.4781C-5H13311.411.440.750.670.310.812.640.5081C-6H1239.731.410.680.600.280.762.540.4810.021.510.670.570.300.762.030.4810.181.430.700.620.290.772.580.5010.101.470.690.590.290.772.310.4896C-1C1119.491.870.650.550.290.822.120.4796C-2C1268.211.270.570.480.250.721.870.4196C-3C12910.081.480.670.560.290.842.100.4896C-4H1239.61.410.650.580.270.802.500.4596C-5H11710.441.680.780.670.330.973.190.5596C-6H1129.961.750.720.630.300.902.930.519.261.540.630.530.291.052.360.4910.001.610.720.630.300.892.870.489.631.360.670.580.290.842.450.48103C-1C1147.831.470.620.500.271.022.310.47103C-2C1239.391.500.690.560.301.092.410.51103C-3H1259.41.400.710.610.301.013.020.50103C-4H1218.841.460.700.590.——0.690.590.330.952.640.48COV———0.1160.1530.1620.1560.2080.161Table 4. Contact force prediction by Hertz contact theoryTable 4. Contact force prediction by Hertz contact theoryProjectilefc,meas′ (MPa)Ec* (MPa)xtest (mm)F (kN)Conical projectile4144,40413.1812,4518162,31110.0210,1159667,7759.269,44210370,2208.618,45716289,4967.397,984Hemisphere projectile4144,40413.297,8468162,31110.186,9039667,77510.007,14510370,2209.286,56416289,4966.424,506In Table 3, the nose shape of the projectile is used to classify the values of xtest. For the conical projectile, the average penetration depths were 13.18, 10.02, 9.26, 8.61, and 7.39 mm for the 41, 81, 96, 103, and 162 MPa concrete specimens, respectively. Meanwhile, the average penetration depth for the hemispherical projectile was 13.29, 10.18, 10.0, 9.28, and 6.42 mm for the 41, 81, 96, 103, and 162 MPa concrete specimens, respectively. In the 41 to 103 MPa concrete specimens, the hemispherical projectile achieved greater penetration depth. However, greater penetration depth in the 162 MPa concrete specimens was achieved by the conical projectile.To address the different impact velocities and masses of the projectile in each test, the penetration depth (xtest) normalized by the kinetic energy (EK) for the specimen is depicted in Fig. 6. The kinetic energies of the hemispherical and conical projectiles differed by 40 J and were 630 J and 590 J, respectively.The penetration depth according to the nose shape is depicted in Table 3. The value difference results indicate that a deeper penetration depth is achieved by the conical projectile results when compared to the hemispherical projectile for normal-strength concrete. However, the value discrepancy was not significant in the high-strength concrete and UHPC target specimens.The ratios of (xtest/EK) of a conical projectile to that of a hemispherical projectile were 1.18 (=0.025/0.021), 1.06 (=0.015/0.014), 0.95 (=0.0154/0.016), 1.05 (=0.0148/0.0142), and 1.05 (=0.0131/0.0124) in the 41, 81, 96, 103, and 162 MPa concrete specimens, respectively. The value for normal-strength concrete was significantly higher than that for high-strength concrete and UHPC. It can be derived that the penetration depth of normal-strength concrete targets is sensitive to the nose shape of a projectile. However, the effect of the nose shape on the penetration depth is insignificant in high-strength concrete and UHPC targets.Nose Shape Factors in Existing EquationsTable 3 compares test results of the penetration depth with predictions of existing equations. Although the nose shape of projectiles varies in reality, impact equations except for those by Li and Chen (2003) and Young (1997) are classified into four types with rather unclear criteria.For this study, conical nose shape factors of 1.14, 1.39, 0.685, 1.13, 0.5, and 1.8 were used in the modified NDRC (1976), Hughes (1984), Young (1997), UMIST (Reid and Wen 2001), Li and Chen (2003), and Hwang et al. (2017) equations. The Hwang et al. (2017) equation recommended 0.9 for a sharp nose shape on the basis of existing test data using very sharp projectiles, in which the ratio of ln to dp is large (i.e., 1.0 to 3.2, which are the same as 63°–81°). But in this study, the conical nose shape was at 45°. Given the nose shape factor of 1.9 for hemispherical projectiles with the ratio of Rn to dp=0.5, a nose shape factor of 1.8 was applied to the conical nose shape with the ratio of ln to dp=0.5 in the Hwang et al. (2017) equation.For hemispherical projectiles, the nose shape factors suggested by each equation were used. In the existing equations, the actual compressive and tensile strengths of concrete were used to predict the penetration depth. For the Hwang et al. (2017) equation, concrete tensile strength was estimated from the splitting tensile strength [i.e., 0.9fct,meas in Eurocode 2, (CEN 2004)] or direct tensile strength test. It is noted that due to low reliability pertaining to a few of the splitting test results, the larger of the test result and tensile strength in Eqs. (33) and (34) provided by the fib model code (fib 2010) were used.In view of Fig. 7, the Hwang et al. (2017) equation predicted the test results better than the other equations posed, with the average ratio of test results to predictions being 0.95 and a coefficient of variation (COV) of 0.156. As concrete strength increased, both the modified NDRC (1976) and Hughes (1984) equations overestimated the penetration depth. In general, the prediction accuracy summarized in Table 3 using existing impact equations according to the nose shape, with the exception of the Hwang et al. (2017) equation, decreased as concrete strength increased.The prediction accuracy of the modified NDRC (1976) equation according to the nose shape was determined to be equal to 0.82. The test-to-prediction ratio of the penetration depth due to a hemispherical nose was slightly larger than that of a conical projectile in the concrete targets for compressive strengths of 81 through 103 MPa. The Hughes (1984) equation showed a similar trend. For these two equations, the results indicate that the penetration depth associated with a conical projectile is relatively overestimated compared to that of a hemispherical projectile in high-strength concrete.The Young (1997) equation significantly overestimated the penetration depth, showing the average ratio of 0.48, in which inaccurate results had been already recognized; the Young (1997) equation is questionable when the penetration depth is less than about 3dp, and the allowable weight range of the projectile is at least 4.5 kg.However, the UMIST equation (Reid and Wen 2001) underestimated the penetration depth, showing an average ratio of 2.64. Underestimation is attributed to impact test conditions deviating significantly from the allowable application range in which the projectile’s diameter, mass, velocity, and the ratio of the penetration depth prediction to dp are 50–600  mm, 35–2,500  kg, 3–66  m/s, and 0–2.5, respectively.The Li and Chen (2003) equation overestimated the penetration depth with an average ratio of 0.33, in which overestimation is attributed to the impact factor (I) being less than 0.5 in Eq. (17).Penetration Analysis with Hertzian Contact TheoryThe Hertzian contact theory defines the relationship between the contact force and elastic penetration depth when two elastic bodies collide (Hannor et al. 2015). According to the Hertzian contact theory, the contact force (Fhemi) related to the indentation depth (d) (Fig. 8) due to a hemispherical indenter can be determined from Eqs. (37) and (38) (Hannor et al. 2015), and the contact force (Fcone) due to a conical projectile can be calculated from Eq. (39) (Seddon 1956) (37) (38) (39) Fcone=4E2cotθπ(1−v2)d2where Fhemi and Fcone = applied forces to the hemispherical and conical projectiles, respectively; E* = effective elastic modulus of the target with respect to the elastic modulus of the indenter (projectile); E1 and E2 = elastic moduli of the indenter and target, respectively; v1 and v2 = Poisson’s ratios of the indenter and target, respectively; R = radius of the hemispherical nose; d = indentation depth (Fig. 8); and θ = angle between the plane and the side surface of the conical nose.Although the Hertzian contact theory considers the elastic condition, the indentation depth (d) was assumed to be equal to the penetration depth for the sake of analysis in this experimental study to evaluate the tendency of the impact force (the applied force in the Hertzian contact theory). The sectional diameter, elastic modulus, density, and Poisson’s ratio of the projectile were assumed to be 20 mm, 200,000  MPa, 7,850  kg/m3, and 0.3, respectively. The elastic modulus of steel was also considered not affected by the strain rate (CEB 1988).The elastic modulus of the 41, 81, 96, 103, and 162 MPa concrete specimens were calculated in accordance with ACI 318-14 (ACI 2014) to be 30,095, 42,300, 46,050, 47,700, and 59,821  MPa, respectively. The Poisson’s ratio of all specimens was assumed to be 0.21 (Carrasquillo et al. 1981). In a high-velocity impact test, the elastic modulus of concrete is affected by the strain rate, which can be estimated from Eq. (29) (Hwang et al. 2017) and Eq. (40) (fib 2010). The angle (θ) in Eq. (39) was assumed to be equal to the angle of the conical projectile (i.e., 45° in this study), and the thickness in Eq. (37) was the measured value of each test specimen (40) Ec*Ec=(ε˙c30×10−6)0.026Table 4 lists impact force predictions according to the concrete compressive strength and nose shape. Impact forces (i.e., the contact force in the Hertzian contact theory) of conical and hemispherical projectiles per the equation decrease as the concrete compressive strength increases. It should be noted that the impact force of the conical projectile on the 41 MPa concrete target specimens was 59% (=12,451/7,846) greater than that of the hemispherical projectile.In Table 1, the ratios of the nose shape factors between conical and hemispherical noses in the modified NDRC (1976), Hughes (1984), Hwang et al. (2017), UMIST (Reid and Wen 2001), and Young (1997) equations were 1.14 (=1.14/1.0), 1.1 (=1.39/1.26), 1.06 (=1.9/1.8; inverted value for resistant energy), 1.13 (=1.13/1.0), and 1.05 (=0.685/0.65), respectively. The ratio predicted by the Li and Chen (2003) equation was 1.00 for the conical projectile at an angle of 45°. In other words, all the equations except for the Li and Chen (2003) equation consider the differences in the impact force between the hemispherical nose and conical nose at an angle of 45°. Actual differences in the impact force between the conical and hemispherical projectiles were 1.47 (=10,115/6,903), 1.32 (=9,442/7,145), 1.29 (=8,457/6,564), and 1.77 (=7,984/4,506) for 81, 96, 103, and 162 MPa for the concrete specimens, respectively.The ratios of the actual differences, except for that of 162 MPa, decreased as the concrete strength increased. The average impact velocity of the hemispherical projectile in tests for 162 MPa was found to be 10  m/s slower than that of the conical projectile. However, the gap between the impact velocities of the two projectiles for the 41 through 103 MPa concrete test targets was less than 5  m/s. A small penetration depth may have caused errors in the calculation of impact forces. Thus, if the impact velocity of two projectile types in tests for 162 MPa concrete were similar, a decreasing trend would have been maintained or be similar to that found for the other high-strength concrete specimens.In the high-strength concrete and UHPC target specimens, a decreasing trend in contact force can be deduced from the relationship between the elastic moduli of the concrete and the projectile. The elastic modulus ratio of steel to normal-strength concrete is 4.5 (=200,000/44,404), but the elastic modulus of steel to 96 or 162 MPa concrete decreases to 2.95 and 2.23, respectively. 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