Failure behavior of reinforced concrete slabs subjected to moderate-velocity impact by a steel projectile

Abstract

This study investigates the failure characteristics of reinforced concrete slabs subjected to moderate-velocity impacts by conducting impact tests and numerical simulations. In a series of tests, a spherical steel projectile with a mass of 8.3 kg and a diameter of 80 mm is collided with an reinforced concrete slab at an impact velocity of 65–90 m/s. To investigate the failure characteristics of the reinforced concrete slab, impact motion of the projectile, reaction force, and strain–time history on the back surface and reinforcing bars of the reinforced concrete slab were measured. Failure modes obtained experimentally were compared with the Central Research Institute of Electric Power Industry formula proposed for the local damage of reinforced concrete slabs. Test results revealed that a circular scabbing crack on the back surface of the reinforced concrete slab was completed while there is a sharp increase in the reaction force. Numerical simulations using a high-fidelity concrete model reasonably reproduced the failure characteristics of an reinforced concrete slab. Numerical results demonstrated that the scabbing failure of an reinforced concrete slab subjected to a moderate-velocity impact was initiated by the penetration of the projectile and was completed during the reaction force response.

Keywords

Moderate-velocity impact reinforced concrete slab failure characteristic local damage

Introduction

In recent years, tornado events and volcanic eruptions have increased owing to climate change and crustal movement. For instance, 24 fatalities and 387 injuries were caused by the Moore tornado in Oklahoma, United States in 2013 (National Institute of Standards and Technology, 2013). A volcanic eruption at Mt. Ontake in Japan in 2014 (Cabinet Office Japan, 2015b) left 58 humans dead, while several were injured by the volcanic cinders in addition to the pyroclastic and debris flow. In such incidents, structural damage caused by the missiles generated by wind pressure (called tornado missiles) and volcanic cinders (i.e. volcanic missiles) was reported. To protect humans and structures from the impact loads, a reliable design of shelters for protection from the tornado and volcanic missiles should be promptly established. With regard to designing buildings providing protection from collisions with tornado and volcanic missiles in Japan, guidelines have been reported that have assigned a design impact velocity of 40 and 150 m/s for a tornado and volcanic missile, respectively (Cabinet Office Japan, 2015a; Nuclear Regulation Authority Japan, 2013). However, details of the impact condition or failure mechanism of the structure were not examined sufficiently.

Numerous studies on the failure behavior of reinforced concrete (RC) slabs, subjected to impact load over a wide range of impact velocities, have been conducted in the past few years. Miyamoto et al. (1991, 1994) proposed a safety verification method for an RC railing of a bridge exposed to a vehicle impact, based on the concepts of load-carrying and energy capacity, by conducting scaled vehicle collision tests and finite element analysis. Other studies on the impact behavior of RC slabs subjected to low-velocity impact loads involved experiments and numerical programming (Othman and Marzouk, 2016; Xiao et al., 2016; Zineddin and Krauthammer, 2007). In these studies, primarily, the overall response of RC slabs (flexure and punching shear failures) was investigated at an impact velocity of less than 10 m/s. The local damage of RC slabs subjected to a high-velocity impact of 150–1000 m/s was investigated for bullet and aircraft collisions (Beppu et al., 2008; Chang, 1981; Chen et al., 2008; Gomez and Shukla, 2001; Huang et al., 2005; Hughes, 1984; Ito et al., 1995; Kennedy, 1976; Li and Chen, 2003; Li et al., 2005, 2007; Werner et al., 2013). These studies reported that a local failure occurred in the vicinity of the collided part because of stress wave interaction and local deformation. That is, scabbing is caused by the propagation of cracks due to the tensile stress wave transformed from the incident compressive stress wave at the free surface. Chen et al. (2008) proposed formulae predicting the penetration depth of concrete impacted by a hard projectile based on the dimension analysis using a theoretical penetration model. Some studies have been devoted to the investigation of the impact resistant behavior of RC structures as a function of moderate velocities in the range of 10–150 m/s (Ito et al., 1995; Li et al., 2007; Yankelevsky, 1997). However, few formulae have been proposed based on the detailed failure mechanism of RC slabs subjected to the projectile impact on a basis of measured test and numerical analysis.

This study aims at investigating experimentally and numerically the fundamental failure characteristics of RC slabs subjected to moderate-velocity impacts. To this end, impact tests were conducted with a steel projectile. Based on the motion of the projectile during the impact response of an RC slab, the failure process of RC slabs, subjected to moderate-velocity impacts, was discussed experimentally. Numerical simulations of impact tests were conducted to examine the reproducibility of projectile motion and to study the failure mechanism of an RC slab.

Moderate-velocity impact tests and failure characteristics of RC slab

Moderate-impact test apparatus

A moderate-impact velocity test machine was used to conduct impact tests. This machine can launch a projectile of 8.3 kg at a velocity of 20–90 m/s by adjusting air pressure. A schematic of the projectile launching test and the experimental setup are shown in Figure 1 and Photo 1, respectively. A pair of laser velocity sensors is fixed at the muzzle of an acceleration tube to measure the average velocity at intervals of 50 cm. The muzzle of the acceleration tube was approximately at a distance of 0.5 m from an RC slab because of attachment of a projectile catcher which holds the projectile after rebounding. Consequently, the velocity of a projectile is slightly increased after it passes through the acceleration tube. Hence, the velocity of the projectile was obtained by differentiating the displacement time histories from captured images by a high-speed camera (Resolution: 1280 × 152; frame rate: 32,000 fps).

Figure.1.

Schematic of launching test machine.

Photo 1.

A view of experimental setup.

Steel projectile and RC slab

Photo 2 shows the steel projectile used in the tests. The 8.3-kg mass steel projectile (JIS: SKS93) had a hemispherical nose with a diameter of 80 mm. The mass of 8.3 kg corresponds to the lightest tornado missile made of a steel pipe (8.4 kg) designated in “Assessment guide for tornado effect on nuclear power plants of Japan” (Nuclear Regulation Authority Japan, 2013). Targets, for tracking using the high-speed camera, were set on the lateral side of the projectile. Figure 2 displays the dimensions of the RC slabs and arrangement of the reinforcing bars. RC slabs are 1225 mm in height and width, with thicknesses of 100, 200, and 300 mm. The specimens were reinforced with reinforcing bars with a reinforcing ratio of 0.00%–1.43% in both directions on the top and bottom sides. To investigate the effects of the reinforcement ratio on the failure state of RC slabs, three different 200-mm-thick reinforced specimens including a plain concrete slab were cast. To ensure the specimens are intact during transportation and replacement of an RC slab, safety-reinforcing bars (broken lines in Figure 2) were set along the four sides of the specimens. The average compressive strength of the concrete is 44.1 MPa using cylindrical specimens with a diameter of 100 mm and a length of 200 mm.

Figure 2.

Dimensions of RC slabs and arrangement of rebar (----- safety rebar): (a) t = 100 mm, r_s = 0.57%; (b) t = 200 mm, r_s = 0.51%; (c) t = 300 mm, r_s = 0.53%; (d) t = 200 mm, r_s = 0.0%; (e) t = 200 mm, r_s = 0.99%; and (f) t = 200 mm, r_s = 1.43%.

Photo 2.

A steel projectile.

Measurement items and test case

The displacement time history of the projectile was obtained by analyzing the images captured by a high-speed camera after the collision. The velocity and acceleration of the projectile were calculated by differentiating the displacement–time history, while the impact load of the projectile was indirectly obtained by multiplying the mass of the projectile with the acceleration. The reaction force of RC slabs was measured by eight piezo-type electric load-cells (response frequency of 30 kHz) set on a reaction wall as shown in Figure 3.

Figure 3.

Setup of an impact test: (a) impact surface, (b) back surface, and (c) side view.

Figure 4 illustrates the arrangement of strain gauges. To investigate the impact response of an RC slab, four 5-mm-long strain gauges were set on reinforcing bars at distances of 125 and 375 mm from the center of the RC slab, while 60-mm-long strain gauges were set at 100 mm intervals from the center of the back surface. To measure the collision time, a 60-mm-long strain gauge was set at the center of the impact surface as shown in Figure 4(c).

Figure 4.

Arrangement of the strain gauges ( strain gauge): (a) rebar, (b) back surface, and (c) strain gauge measuring the collision time (impact surface).

Table 1 presents the test cases and their parameters including the impact velocity, thickness of an RC slab, and reinforcing ratio. To investigate the effect of the impact velocity on the transition of the failure state, an impact velocity in the range of 65–90 m/s was employed for five specimens with a thickness of 200 mm and reinforcing ratio of 0.51%. Effects of the slab thickness on the failure state were investigated using the 100- (Case 1) and 300-mm (Cases 10–12)-thick specimens with a reinforcing ratio of 0.53%–0.57%. Finally, the influence of reinforcing ratios between 0.00% and 1.43% on the failure state of the 200-mm-thick specimens was examined at an impact velocity of 65 m/s.

Table 1.

Test cases and results.

Case	Thickness (mm)	Steel ratio (%)	Velocity (m/s)	Failure mode	Penetration depth (mm)	Maximum of impact load (kN)	Maximum of reaction force (kN)
1	100	0.57	69.7 (estimated)	Perforation	–	Non-measured	138
2	200	0.51	65.0	Spalling	39	779	375
3	200	0.51	74.8	Scabbing limit	43	778	Failed
4	200	0.51	74.8	Scabbing limit	41	876	368
5	200	0.51	81.2	Scabbing	53	906	257
6	200	0.51	89.7	Scabbing	52	840	167
7	200	0.00	66.0	Spalling	39	820	339
8	200	0.99	64.4	Spalling	38	702	328
9	200	1.43	64.1	Spalling	38	870	387
10	300	0.53	74.8	Spalling	49	992	394
11	300	0.53	81.9	Spalling	48	1,055	305
12	300	0.53	85.7	Spalling	48	993	281

Failure state

Definition of failure mode

Table 1 summarizes the test results and impact velocities calculated via image analysis. The impact velocity of test Case1 was measured only by the velocity sensor at the muzzle of the acceleration tube because of the protection of the high-speed camera. Because the failure mechanism of an RC slab subjected to a moderate-velocity impact was not clearly categorized, for instance, into local damage and global failure, local failure mode was applied when referring to previous studies, which includes spalling (fracture in the impact surface), scabbing (failure on the back surface without a perforation hole), perforation (perforation hole with projectile passing), and scabbing limit (Kennedy, 1976).

Failure mode and penetration depth

The failure modes of RC slabs can be understood in terms of the relation between the thickness of an RC slab and velocity of the projectile as shown in Figure 5. Figure 5 also shows Central Research Institute of Electric Power Industry (CRIEPI) formula (Ito et al., 1995; Li et al., 2005) which was proposed for assessment of the scabbing and perforation limits as follows

Figure 5.

Failure mode in terms of the relation between the thickness of the specimen and impact velocity of the projectile.

t_{s} = α_{s} \times {(\frac{V_{0}}{V})}^{0.13} \frac{{(M V^{2})}^{0.4}}{d^{0.2} {f'}_{c}^{0.4}}

(1a)

t_{p} = α_{p} \times {(\frac{V_{0}}{V})}^{0.25} {(\frac{M V^{2}}{d \cdot {f^{'}}_{c}})}^{1 / 2}

(1b)

where t_s is the scabbing limit, t_p is the perforation limit, V₀ is the control velocity (60.96 m/s), V is the impact velocity (m/s), M is the mass of the projectile (kg), d is the diameter of the projectile (m), ${f^{'}}_{c}$ is the compressive strength of concrete (MPa), and α_s and α_p are constants equal to 6.96 × 10⁻³ and 8.96 × 10⁻⁴, respectively.

The figure exhibits that the test results of the perforation limit show a good correspondence with the CRIEPI formula, but the CRIEPI formula slightly overestimates the scabbing limit of the 200-mm-thick RC slab by approximately 15%.

Figure 6 illustrates the penetration depths observed in the tests as compared with the values obtained from the Hughes formula given below (Hughes, 1984; Li et al., 2005)

\frac{x}{d} = 0.19 \frac{N I}{S}

(2a)

I = \frac{M V^{2}}{f_{t} d^{3}}

(2b)

S = 1 + 12.3 \ln (1 + 0.03 I)

(2c)

where x is the penetration depth (m), N is the nose shape factor (hemisphere: 1.12), I is an impact parameter, S is the strain-rate factor, and f_t is the tensile strength of concrete (N/m²).

Figure 6.

Penetration depth.

It is seen in Figure 6 that the observed penetration depth increases with increasing impact velocity. By comparing the two sets of penetration depths, we can infer that the experimentally obtained values are approximately 20% lower than those calculated from the Hughes formula in this test condition.

Failure state

Photos 3 to 5 show the failure states of the RC slabs categorized with respect to the reinforcing ratio and slab thickness. In all test cases, the steel projectile was not deformed after the tests. In the photos, cracks are emphatically depicted by black and white lines. Photo 3 shows the failure states of a 100-mm-thick RC slab with a 0.57% reinforcing ratio and of 200-mm-thick RC slabs with a 0.51% reinforcing ratio; the impact velocity was between 65 and 90 m/s. The 100-mm-thick RC slab at a velocity of 69.7 m/s (Case 1) shows perforation in which a perforation hole is clearly generated in the impact area. The failure mode in Case 2 (thickness of 200 mm and impact velocity of 65.0 m/s) is spalling with circular cracks on the back surface of the specimen, but no diagonal crack is observed in the cross section. In Case 3 and Case 4 (thickness of 200 mm and impact velocity of 74.8 m/s), diagonal cracks are clearly generated in the cross section, but scabbing does not occur. In Case 5, at an impact velocity of 81.2 m/s, scabbing is considered the failure mode because a part of the diagonal shear crack reaches the back surface, and scabbing is partially generated. Note that a crack in the direction parallel to the back surface connecting the diagonal cracks is initiated (white broken line in Photo 3) in the cross section of specimens in Cases 4 to 6. The crack appears to be a “spalling fracture” owing to the stress wave interaction (Chang, 1981). In Case 6, scabbing is eventually observed at an impact velocity of 89.7 m/s. Photo 4 shows the failure state of 300-mm-thick RC slabs at an impact velocity of 75–85 m/s. In all test cases, the failure mode is spalling, and no visible crack is created in the cross section and on the back surface of the specimen.

Photo 3.

Failure states of specimens (100 and 200 mm thickness and steel ratio 0.51%).

Photo 4.

Failure states of specimens (300 mm thickness and steel ratio 0.53%).

Photo 5 shows a comparison of the failure states of the 200-mm-thick RC slabs with the different reinforcing ratios at an impact velocity of 65 m/s. The resistant effect by the reinforcement is not apparently recognized, as the diagonal cracks are not initiated in the cross section in all the cases. However, the number of cracks on the back surface slightly decreases as the reinforcing ratio increases.

Photo 5.

Failure states of specimens (200 mm thickness and steel ratio 0.00%–1.43%).

Motion of projectile and impact response of RC slab

In this section, the failure mechanism is discussed by comparing the projectile motion before and after collision with the impact response of RC slabs in Cases 2, 4, and 6.

Motion of projectile

Figure 7 shows the displacement–time histories of the projectile in Cases 2, 4, and 6, where the origin indicates the time of collision. In Figure 7, the displacement indicates position of the projectile after the impact. In Cases 2 and 6, time duration of measurement is 3 ms because the tracing target in image analysis was obstructed due to fragments caused by the spalling. In Cases 2 and 4 (failure modes are spalling and scabbing limit, respectively), the displacement increases after the collision yields a maximum displacement of 30–40 mm in 1.0 ms. However, in Case 6, the displacement continues to increase in 3 ms due to the scabbing failure.

Figure 7.

Displacement–time histories of projectile: (a) Case 2 (velocity = 65.0 m/s), (b) Case 4 (velocity = 74.8 m/s), and (c) Case 6 (velocity = 89.7 m/s).

Figure 8 illustrates the impact velocity–time histories. The impact velocity curve in black presents the smoothened data with the moving average method of seven words. In all the cases, impact velocity decreases to zero or becomes negative (called rebounding velocity) between 1.0 and 1.5 ms. In Case 6, the impact velocity reaches zero at 3 ms.

Figure 8.

Velocity–time histories of projectile: (a) Case 2 (velocity = 65.0 m/s), (b) Case 4 (velocity = 74.8 m/s), and (c) Case 6 (velocity = 89.7 m/s).

Figure 9 displays the impact load–time histories obtained by multiplying the mass and acceleration of the projectile. As a high-frequency wave oscillates owing to time differentiation, the impact load–time history is smoothened with the moving average method of three words (black line in Figure 9). A comparison between the impulses integrated by the impact load–time history and the initial momentum calculated by multiplying the mass and impact velocity of the projectile is shown in Figure 10. It can be seen that the impulse is in good accordance with the momentum, which implies that the results of image analysis are valid and reliable. The duration of the impact load and maximum impact load in all the cases are 1.0–1.5 ms and 1000–1100 kN, respectively.

Figure 9.

Impact load–time histories: (a) Case 2 (velocity = 65.0 m/s), (b) Case 4 (velocity = 74.8 m/s), and (c) Case 6 (velocity = 89.7 m/s).

Figure 10.

Relation between impulse and momentum of projectile.

Impact response and reaction force of RC slabs

Figure 11 shows the strain response on the back surface of the RC slabs. In all cases, the strain gauge B1 on the center of the back surface exhibits extremely high positive (tensile) values because of the rupturing of the strain gauge by the stress wave generated by the collision with the projectile. In Case 2, involving spalling failure, compressive strain is initially generated for strain gauges of B2–B4, which later converges to zero. While strain gauges B2–B4 in Cases 4 and 6 initially show compressive strains of 500–1000 μ, strain gauges B2 and B3 exhibit a tensile strain at 0.5–1.0 ms because the strain gauges were ruptured by the scabbing cracks on the back surface as shown in Figure 12.

Figure 11.

Strain–time histories of back surface: (a) Case 2 (velocity = 65.0 m/s), (b) Case 4 (velocity = 74.8 m/s), and (c) Case 6 (velocity = 89.7 m/s).

Figure 12.

Reference between location of strain gauge and scabbing.

Figure 13 displays the strain–time histories of the tensile reinforcing bars. In Cases 2 and 4, strain gauges experience a maximum tensile strain of 1200–1800 μ. Strain gauges in Case 6 are under a maximum tensile strain of 5000 μ, which is significantly greater than for the other cases because the failure mode in Case 6 is scabbing. All the test cases show that the tensile strains sharply increase at 0.5–1.0 ms and show maximum at 1.5 ms.

Figure 13.

Strain–time histories of lower rebar: (a) Case 2 (velocity = 65.0 m/s), (b) Case 4 (velocity = 74.8 m/s), and (c) Case 6 (velocity = 89.7 m/s).

Figure 14 illustrates the reaction force–time histories of the RC slabs. In all the cases, the reaction force starts arising at 0.5 ms and sharply increases after 1.0 ms. By comparing the reaction force with the impact load–time histories as shown in Figure 9, the reaction force is generated at 0–0.2 ms after the time required for achieving the maximum impact load.

Figure 14.

Reaction force–time histories: (a) Case 2 (velocity = 65.0 m/s), (b) Case 4 (velocity = 74.8 m/s), and (c) Case 6 (velocity = 89.7 m/s).

Numerical simulations

Numerical models

Numerical simulations were conducted using the hydro code ANSYS AUTODYN (version 15.0). Figure 15 shows the numerical models. For saving the cost of simulations, quarter models were constructed because of the structural symmetry. An RC slab and the projectile were modeled using Lagrangian solid elements, and the initial condition of the impact velocity was applied to all the nodes of the projectile model. The numbers of concrete and projectile elements were 605,160 and 1969, respectively. The element size of concrete was 5 mm × 5 mm × 5 mm. The reinforcing bar was modeled with beam elements embedded in the concrete model with an element size of 5 mm. The steel support was modeled by shell elements with thickness of 25 mm. The nodes in the hatched area of the steel support in Figure 15 were fixed. For modeling of contact between the RC slab and the projectile models, the slide-line method was adapted.

Figure 15.

Finite element model of the test calculation.

Material models

The CAPROUS constitutive model (Itoh et al., 2016) considering the nonlinear characteristics of concrete such as pressure dependency, strain-rate dependency, strain hardening, and softening is applied for the concrete material. In the CAPROUS constitutive model, the pressure is calculated with the porous equation of state proposed by Morishita and Asonuma (2005) as shown in equation (3). The yield criterion with the pressure-dependent strain hardening and softening as shown in equations (4) to (7) and Figure 16 (Han and Chen, 1985) was employed

p = {f^{'}}_{c} (a μ + b μ^{2} + c μ^{3})

(3)

f = σ_{y} - s (h, p) σ_{f, c}

(4)

where σ_y and σ_f,c are the equivalent and yield stress, s is the shape factor, and h is the hardening parameter.

Figure 16.

Nonlinear yield surface of CAPROUS (Han and Chen, 1985).

The initial yield surface means the elastic limit under small pressure, which expands due to the hardening behavior of concrete. The yield stress σ_f,_c is calculated by equation (5)

σ_{f, c} = {\begin{matrix} α [\frac{{f^{'}}_{c} - {f^{'}}_{t}}{{f^{'}}_{c} + {f^{'}}_{t}} (3 p + {f^{'}}_{t}) + {f^{'}}_{t}] & (p \geq {f^{'}}_{c} / 3) \\ α [\frac{{f^{'}}_{b c} - {f^{'}}_{c}}{2 {f^{'}}_{b c} - {f^{'}}_{t}} (3 p - {f^{'}}_{c}) + {f^{'}}_{c}] & (p \leq {f^{'}}_{c} / 3) \end{matrix}

(5)

where ${f^{'}}_{c}$ , ${f^{'}}_{b c}$ , and f_t are the uniaxial compressive strength, biaxial compressive strength, and uniaxial tensile strength of concrete, respectively; μ (= ρ/ρ₀ – 1) is the volume change rate; and ρ₀ is the initial density. The constants are given as follows: a = 3.72 × 10², b = −1.47 × 10⁴, and c = 2.48 × 10⁵.

The shape factor s is a function of the hardening parameter h and the pressure p as shown in equation (6)

s = {\begin{matrix} 1 & (p < p_{t}) \\ 1 + \frac{(1 - h) [- p_{t} (2 p_{c} + p) - 2 p_{c} p + p^{2}]}{{(p_{c} - p_{t})}^{2}} & (p_{t} \leq p \leq p_{c}) \\ \frac{h (\bar{p} - p) (\bar{p} + p - 2 p_{c})}{{(\bar{p} - p_{c})}^{2}} & (p_{c} \leq p \leq \bar{p}) \\ 0 & (p > \bar{p}) \end{matrix}

(6)

where $p_{t} = - {f^{'}}_{t}$ and $p_{c} = {f^{'}}_{c} / 3$ .

The shape factor in the range $p_{c} \leq p \leq \bar{p}$ gives the cap part of the yield surface. The variable $\bar{p}$ is expressed as follows

\bar{p} = {\begin{matrix} \frac{w}{1 - h} & (p < {\bar{p}}_{0}) \\ \frac{w}{1 - h} {(\frac{p}{{\bar{p}}_{0}})}^{k} & (p \geq {\bar{p}}_{0}) \end{matrix}

(7)

where

w = \frac{2}{3} (1 - h_{0}) {f^{'}}_{c} {\bar{p}}_{0} = \frac{2}{3} {f^{'}}_{c}

(8)

where h₀ and k are material properties.

The yield stress was calculated by multiplying the shape factor. The dynamic increase factor (DIF) was calculated by the following equations proposed by Yamaguchi et al. (1989) as shown in equations (9a) and (9b)

γ_{t} = 0.8267 + 0.02987 \log_{10} {\dot{ε}}_{o c t} + 0.04379 {(\log_{10} {\dot{ε}}_{o c t})}^{2}

(9a)

γ_{c} = 1.021 - 0.05076 \log_{10} {\dot{ε}}_{o c t} + 0.02583 {(\log_{10} {\dot{ε}}_{o c t})}^{2}

(9b)

where γ_t and γ_c are the DIF of the compressive and tensile strength, respectively, and ${\dot{ε}}_{o c t}$ is the octahedral shearing strain rate.

For the tensile fracture in the CAPROUS model, the spall failure criterion that the element is fractured when the pressure of an element reached the prescribed critical pressure p_spall is used (Itoh et al., 2016). The spall pressure p_spall is given by equation (10)

p_{s p a l l} = - (\frac{2}{3}) \frac{{f^{'}}_{c}^{d} {f^{'}}_{t}^{d}}{{f^{'}}_{c}^{d} - {f^{'}}_{t}^{d}}

(10)

where ${f^{'}}_{c}^{d}$ and ${f^{'}}_{t}^{d}$ are the dynamic compressive and tensile strength, respectively.

After reaching the spalling pressure, the softening gradient K_soft in the pressure–volume change rate relation was given according to the fracture energy as shown in equation (11) and Figure 17 (Itoh et al., 2016)

K_{s o f t} = - \frac{L_{0} p_{s p a l l}^{2}}{2 G_{f}}

(11)

where L₀ is the representative length and G_f is the fracture energy.

Figure 17.

Loading path after spall failure (Itoh et al., 2016).

In the CAPROUS model, the parameters were calculated based on the uniaxial compressive strength (Itoh et al., 2016), and basic parameters used were shown in Table 2, citing other parameters in the literature Itoh et al. (2016).

Table 2.

Material properties of concrete (Itoh et al., 2016).

Materials	Symbol	Concrete (CAPROUS)
Compressive strength (MPa)	f′_c	44.1
Tensile strength (MPa)	${f^{'}}_{t}$	$0.08 {f^{'}}_{c}$
Biaxial compressive strength	${f^{'}}_{b c}$	$1.15 {f^{'}}_{c}$
Failure surface constant	a	0.95
Young’s modulus (GPa)	E	25
Poisson ratio	ν	0.2
Density (g/cm³)	ρ	2.3

The projectile was modeled as elastic, and the steel material of the reinforcing bar was modeled with the Johnson–Cook yield criterion (Johnson and Cook, 1983; Schwer, 2007)

σ_{f, s} = [A + B {(ε_{e f f}^{p})}^{N}] (1 + C \ln \dot{ε}) [1 - {(T_{H})}^{M}]

(12)

where σ_f,s is the yield stress, $ε_{e f f}^{p}$ is the effective plastic strain, $\dot{ε}$ is the strain rate, T_H = {(T − T_r)/(T_m − T_r)} is the homologous temperature, T_m is the melt temperature, and T_r is the room temperature. The parameters of the Johnson–Cook model used are listed in Table 3 (Schwer, 2007).

Table 3.

Material properties of steel for rebar (Schwer, 2007).

Materials	Symbol	Rebar (Johnson–Cook)
Yield strength (MPa)	σ_y	286
Young’s modulus (GPa)	E	200
Initial yield stress (MPa)	A	286
Hardening constant (MPa)	B	500
Hardening exponent	N	0.23
Strain-rate constant	C	0.017
Thermal softening exponent	M	0.92
Melting temperature (K)	T_M	1773
Reference strain rate	${\dot{ε}}_{0}$	1.0

Results and discussion

For instance, the numerical result of Cases 4 and 6, whose failure mode is scabbing limit and scabbing (200 mm thickness and impact velocity of 74.8 and 89.6 m/s), is exhibited as follows. Figure 18 shows the comparison of failure state between numerical and test results. In the numerical simulation, radial cracks were generated on the impact and back surfaces, and diagonal cracks clearly opened in the cross section similar to the test results. Figure 19 shows the displacement, velocity, and impact load–time histories of the projectile and reaction force–time history. The velocity and acceleration–time histories of the projectile were numerically obtained by differentiating the average displacement–time history of the projectile model. Although the maximum displacement is 12% smaller than that of the test result, the velocity of the projectile is well reproduced in terms of the rapid decrease from the initial velocity, followed by convergence to zero at 3.0 ms. The impact load–time history is reproduced reasonably well showing a sharp increase to the peak load and the duration of 3 ms.

Figure 18.

Failure state of RC specimens: (a) Case 4 (velocity = 74.8 m/s) and (b) Case 6 (velocity = 89.7 m/s).

Figure 19.

Displacement, velocity, impact load of projectile, and reaction force–time histories: (a) Case 4 (velocity = 74.8 m/s) and (b) Case 6 (velocity = 89.7 m/s).

The maximum impact load reaches approximately 1000 kN, which is identical to the test results. In the reaction force–time history, the maximum reaction force is 1.5–3.0 times greater than the test result, because the damage intensity caused by the collision of the projectile in the numerical simulation decreased as compared to the test result.

Figure 20 depicts the failure state of an RC slab obtained via numerical simulations. As the penetration depth progresses, the diagonal cracks in the cross section occur at 0.6 ms, and they develop to the back surface at 1.0 ms. After that, the development of the cracks was completed at 1.0–1.5 ms, which is coincident with the time when the strain gauge attached on the circular crack in the test showed the extremely large tensile figure.

Figure 20.

Numerical results of failure state on the cross section of RC slab (Case 6).

Figure 21 shows the pressure distribution in the model of an RC slab. Expansive pressure is generated at 0.13 ms after the collision owing to the reflection of the incident compressive pressure at the back surface. However, Figure 20 revealed that spalling failure did not occur by reflected pressure wave. A high compressive pressure is generated in the impact region because of the penetration of the projectile during 0.1–1.0 ms. Although the pressure begins to disperse after 1.0 ms, expansive pressure is generated on the place where diagonal cracks were generated in 0.6–1.0 ms which corresponds to the tensile strain on the back surface after 1.0 ms in the test. The numerical result suggests that the diagonal crack observed in the test was reproduced during 1.0 ms. Hence, the failure on the back surface of an RC slab subjected to moderate velocity seems to be completed after the initial local damage by the penetration of projectile.

Figure 21.

Numerical results of pressure distribution on the cross section of RC slab (Case 6).

Conclusion

This study aimed at investigating the failure characteristics of RC slabs experimentally and numerically, subjected to a moderate-velocity impact. The main conclusions were obtained as follows.

To investigate the impact response of an RC slab subjected to a moderate-velocity impact, impact tests were conducted. As the impact velocity increased, the penetration depth increased, and the failure mode of an RC slab developed from spalling to scabbing. The failure mode estimated by the CRIEPI formula and the penetration depth estimated by the Hughes formula slightly overestimate the test results. Failure status of the RC slab was not affected by a reinforcing ratio of 0.00%–1.43%.

By comparing the impact load calculated using a high-speed camera image with the reaction force and the strain response of the RC slab, it was found that a circular crack of scabbing on the back surface of the RC slab appeared before the reaction force sharply increased.

Numerical simulation of the moderate-velocity impact test was conducted. Although there were some discrepancies between the test and numerical results, the numerical model reasonably reproduced the test results. From the comparison between the test and numerical results, the failure of the RC slab subjected to a moderate-velocity impact was the local failure caused by the initial local damage and completed while there is a sharp increase in the reaction force.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by JSPS KAKENHI (Grant Nos 15K06203 and 5289139).

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