Strengthening of RC Beams with CFC Panels for Improving Impact Resistance

Abstract

In this study, continuous fiber composite (CFC) panels were used as a strengthening material to improve the impact resistance of reinforced concrete (RC). Both experimental tests and numerical analyses were carried out to investigate the impact resistance of RC beams strengthened with CFC panels. The experiments involved repeated drop-weight impact tests at constant speed. The experimental results confirm that the strengthening of RC beams with CFC panels improves the impact resistance, thereby increasing the number of repeated impacts that can be allowed before a specified residual displacement is reached. In addition, a virtual particle model based on the conventional smoothed particle hydrodynamics (SPH) method, which takes into account the mechanical properties of the adhesive, was introduced as an analytical method to simulate the impact fracture behavior of RC beams strengthened with CFC panels. The analysis results show that the improved SPH method proposed in this study can accurately reproduce the impact behavior of RC beams strengthened with CFC panels and predict the allowable number of repeated impacts. Furthermore, a parametric study was carried out using a validated analytical approach to compare the load-bearing capacity and discuss the impact performance of RC beams with three types of CFC panel reinforcement.

1. Introduction

Reinforced concrete (RC) is a versatile composite material widely used in structures owing to its variety of applications and properties. Owing to their widespread utilization, RC structures can be subjected to different types of loads during their service, including impact loads. During the last few decades, many researchers have investigated the structural component behavior when subjected to impact loads, such as RC beams, and this has been a major concern in the case of RC beams in civil engineering structures [1,2].

Many researchers have conducted studies on structural members subjected to impact loads. RC beams suffer from two main failure modes: flexural failure [3,4] and shear failure [5,6]. The failure mode of RC beams with stirrups under low-to-high impact velocities was investigated by Fujikake et al. [7] and Cheng and Wen [8]. The authors investigated the relationship between the impact velocity and the failure mode of RC beams, and the results demonstrate that the failure mode of RC beams tends to change from bending to shear when the impact velocity increases. This failure mode phenomenon was also demonstrated by adopting a numerical analysis [9]. Satadru Das Adhikary et al. [10] conducted a drop-weight impact test and provided a numerical method that mainly focused on the failure mode of an RC beam under low-velocity impact. The results confirm the literature that the RC beam encountered the flexural failure mode while subjected to low-impact velocity. Besides that, concrete crushing was observed in the impact region and tended to increase while the impact velocity increased.

To improve the impact resistance of RC beams in order to guarantee safety and durability, external strengthening materials and methods were studied and introduced. Bars or sheets made of fiber-reinforced polymers (FRP) [11,12,13] and carbon fiber-reinforced polymers (CFRP) [14,15,16] have been commonly used to improve the load-bearing performance of RC structural members. The results prove that the external strengthening of the FRP and CFRP improved the stiffness and load resistance of the conventional RC beams’ either flexure or shear, indicating that a larger load capacity and lower displacement were acquired, as well as the crack propagation was controlled. The improvement of the shear capacity was clearly proved by wrapping CFRP around the RC beams without the stirrup; the beams were advantageous for shear strengthening [17]. In addition, aramid fiber-reinforced polymer (AFRP) [18] sheets and ultra-high-performance concrete (UHPC) [19] have been employed to strengthen RC beams against impact loads. The impact resistance capacity of the RC beam was significantly increased, but it still suffered from either sheet debonding or rupturing, which is likely to occur because of insufficient adhesive strength, particularly when impact loads are applied.

Although many previous studies have been conducted on FRP-reinforced RC beams using either sheets or rods to improve the impact resistance of RC beams, they might practically suffer from unsuitable workability and long-term behavior when exposed to a different and severe environment [20]. Therefore, further research is required to precisely evaluate the retrofitting effectiveness of the reinforcing materials on the impact response of RC beams. The CFC panels were fabricated as a sandwich structure comprising a carbon fiber core dovetailed by flexible boards that allow for easy installation on existing concrete structures, whereas the dovetail layer provides weather-resistance properties. The load-bearing capacity of RC beams reinforced with CFC panels, which TAISEI Corporation or TAISEI Construction Company, Ltd., has already adopted for seismic reinforcement of RC columns, was studied for impact loads and the applicability as reinforcement material for impact resistance in the current research.

In various studies, the behavior of the structural members was investigated through experimental tests and numerical analyses. In the experiments, the drop-weight impact test is the most common methodology [5,7,21] to measure the impact force between a falling weight and a beam to observe the failure mode and determine the dynamic response. On the other hand, the experiments could lead to numerical analyses for the validation and improvement of numerical models, ultimately resulting in more accurate and dependable simulations. Therefore, along with experimental approaches, most of the previous studies were especially intended to propose a numerical analysis due to its cost-effectiveness, rapid prototyping, reproducibility, and ability to predict and explore a wide range of scenarios, which are limitations of the experiments. Most researchers applied the finite element method (FEM) to perform the impact response analysis of RC beams [22,23]. In addition, the smoothed particle hydrodynamics (SPH) method developed by Gingold and Monaghan, and Lucy in 1977 [24,25], is a numerical technique and belongs to the category of mesh-free discretization techniques that could be used in continuum mechanics to simulate solid dynamics, including elasticity and plasticity [26]. Considering the advantage of SPH over FEM that it is easy to analyze the final fracture conditions accurately and realistically by taking advantage of the mesh-free method [27], SPH is often employed to analyze the impact behavior of structures [26,28].

In the current study, the investigation of beam behaviors was conducted through experimental tests and numerical analyses to investigate the impact resistance of reinforced concrete (RC) beams strengthened with CFC panels. In the experiment, the drop-weight tests were performed to compare the impact resistance of conventional RC beams with that of specimens strengthened with different strengthening patterns of CFC panels, such as those attached either to the bottom or to the sides and bottom. Additionally, the study also presents an impact analysis using the SPH method to simulate the impact fracture behavior of RC beams. Considering the mechanical properties of adhesives to the SPH method, virtual particles were proposed and attempted to analytically reproduce the peeling between concrete and the CFC panels. Furthermore, a parametric study was conducted using a self-coded Java program that verified the accuracy by impact experiment results, and the fracture behavior of each test object under the impact load was analyzed.

2. Experimental Program

2.1. Material Properties and Specimen Preparation

As shown in Figure 1, the CFC panel used in this experiment had a structure consisting of a 1.0 mm thick carbon fiber core sandwiched between two 3.0 mm thick flexible boards that are made from cement and reinforcing fibers.

The RC beams had a simple rectangular cross-section 120 mm in height and 100 mm in width with a concrete cover of 30 mm. The specimens had a length of 1200 mm and a span of 1000 mm between the supports. The reinforcements were arranged as shown in Figure 2, with 2Ø10 mm longitudinal rebar at the bottom, 2Ø6 mm compression reinforcing bars at the top, and Ø6 mm stirrups spaced 100 mm apart. The purpose of the tests was to investigate the effect of using the CFC panels on the response of RC beams subjected to impact loading. Four different types of beam specimens were used: (a) conventional RC as a standard for comparison (RC), (b) RC fully reinforced with CFC panels at the bottom (RC_FB_CFC), (c) RC partially reinforced with CFC panels at the bottom and sides (RC_P3S_CFC), and (d) RC fully reinforced with CFC panels at the bottom and sides (RC_F3S_CFC), as shown in Figure 3.

The CFC panels were installed using the following procedures, as shown in Figure 4: starting with preparation of the concrete surface by grinding, dusting, and priming the surface; placing the CFC panels and sealing the edge gaps between the concrete and the CFC panel with epoxy resin, named “CFC putty”; finishing with the epoxy resin injection and final polishing. The strengthened CFC panels were allowed to cure at room temperature for 1–2 days before starting the drop-weight impact tests.

The constants for each material are given in

Table 1. Material properties in this study, provided by manufacturer.

Description	Concrete	Reinforcement	CFC 1		Adhesive
			FB 2	CFS 3
Density [kg/m3]	2350.0	7853.2	1600.0	1818.0	1170.0
Young’s modulus [N/mm2]	30,300.0	188,000.0	13,000.0	245,000.0	1500.0
Poisson’s ratio	0.23	0.30	0.15	0.10	0.35
Compressive strength [N/mm2]	45.5		47.2		85.1
Tensile strength [N/mm2]	2.60	491.0	18.5		55.6
Yield strength [N/mm2]		358.0		3400.0

¹ Continuous fiber composite panel. ² Flexible board. ³ Carbon fiber sheets.

. On the 28th day, the concrete was subjected to a material test according to JIS [29,30], and its Young’s modulus, mean compressive strength, and mean tensile strength are listed in this table. The SD295A rebar was subjected to a uniaxial tensile test in accordance with JIS [31], and its average yield strength and tensile strength were measured. For the mechanical properties of the CFC panel components, the material constants were also obtained by various tests according to JIS [32,33] and JSCE [34] standards, but nominal values were adopted for CF injection.

2.2. Experimental Method and Condition

The experimental program consisted of drop-weight impact tests for four specimens, which are mechanical tests in which a defined weight was dropped onto the specimen from a specified height.

Figure 5 shows the schematic diagram of the test setup. A 100 kg steel weight was used as an impactor, with lateral movement prevented by guide rails. A fixture was applied to all the specimens to prevent rebound during impact loading. All the specimens were subjected to the same loading condition at mid-span, which consisted of repeatedly dropping a 100 kg from a height of 459 mm (equivalent to an impact velocity of 3 m/s) until the cumulative residual displacement reached 1% of the span, which was considered to be the failure condition of all the specimens. During the test, the impact force was measured using a CLP-500KNB load cell manufactured by Tokyo Sokki Kenkyujo, the mid-span deflection was recorded using a laser displacement meter (Sensor Head: LB-300, amplifier unit: LB-1200), and the final crack propagation was recorded using a digital or high-speed camera.

3. Experimental Results

3.1. Failure Mode and Crack Propagation

Figure 6 shows the distribution of cracks in each specimen for each number of impacts. This figure is an enlarged view of the central part of the beam span. First, the RC beam specimen, which serves as the reference, was severely flexurally fractured in the first impact and reached the end. On the other hand, in the case of RC_FB_CFC, whose bottom was reinforced with a CFC panel, cracks occurred during the first impact but did not reach the end, and the second impact resulted in peeling failure. Similarly, in the case of RC_F3S_CFC reinforced on three sides, there was almost no damage with the first impact, but cracks with small widths were observed on the sides of the CFC panel after the second impact, and the specimen reached its final state after the third impact. In other words, it was confirmed that the reinforcement effect on the impact resistance of the CFC panel is sufficient; however, as for the RC_P3S_CFC specimen, it reached the end after the first impact, which is presumed to be due to the problem of adhesiveness.

Each type of specimen underwent between one and three impacts until failure occurred. As shown in Figure 7, flexural cracks were observed as the final crack pattern, and the impact contact area was crushed in all the specimens. A similar phenomenon was observed in Refs. [8,9,35]. The crack propagation of CFC panel-strengthened RC beams seems to spread widely in comparison with conventional RC beams, which demonstrates that the crack reduction of the midspan was achieved by the CFC panel-strengthened RC beams. Among the strengthened specimens, separation of the sandwiched structure of the CFC panel was not observed, and there was only interfacial debonding between the CFC panel and concrete.

3.2. Impact Response

Figure 8a presents the displacement–time history of each specimen under the first impact. The unstrengthened RC beam specimen (RC) experienced a larger midspan displacement than all the CFC-reinforced RC beams, demonstrating that reinforcing RC beams with CFC panels can improve their impact resistance. The maximum displacement of RC was 16.95 mm. Among the CFC-strengthened specimens, RC_F3S_CFC was the most effective and achieved the highest impact resistance, as shown as the lowest maximum displacement of 8.15 mm, followed by RC_FB_CFC 11.2 mm and RC_P3S_CFC 13.75 mm. Similarly, the residual displacement tended to decrease because of the strengthening of the CFC panel. According to Ref. [36], the authors determined the restitution coefficient of displacement, which was calculated as 1 minus the residual displacement divided by the maximum displacement, in order to illustrate the recovery performance. The conventional RC beam exhibited the lowest restitution coefficient of displacement (0.253), whereas the restitution coefficients of RC_P3S_CFC, RC_FB_CFC, and RC_F3S were 0.335, 0.635, and 0.822, respectively. Regarding the restitution coefficient of each specimen, the strengthened specimens exhibited a better recovery performance.

Figure 8b presents the measurement results of the impact force–time history of each specimen under the first impact. RC_F3S_CFC clearly earned the highest maximum impact force of 144.11 kN in 0.0113 s, which is the shortest duration of the impact process, whereas the conventional RC beam earned the lowest impact force of 117.95 kN in 0.0193 s, which is the longest duration of the impact process. The maximum impact force of RC_FB_CFC and RC_P3S_CFC were 123.78 kN and 119.17 kN within the duration of the impact process of 0.0195 s and 0.0153 s, respectively.

As mentioned for the failure mode and crack propagation, specimens RC and RC_P3S_CFC were considered as failed specimens after the first impact because their cumulative residual displacement reached 1% of the span; hence, no further impacts were applied. In contrast, RC_FB_CFC and RC_F3S_CFC were subjected to additional impacts until their residual displacement reached 1% of the span or delamination of the CFC panel was observed.

The response of each type of specimen that underwent the repeated impacts test is shown in Figure 9. The displacement tended to gradually increase as the number of repeated impacts increased. In Figure 9a,c, due to the second repeated impact, RC_FB_CFC reached 15.57 mm. of maximum displacement with a restitution coefficient of 0.247, although the maximum displacement of RC_F3S_CFC was only 9.14 mm. and the residual displacement increased slightly owing to a restitution coefficient of 0.821, which were slightly decreased when compared to the first impact that was applied. With the third repeated impact, the higher displacement of RC_F3S_CFC was generated for 12.84 mm with 0.708 as the restitution coefficient. The impact forces are shown in Figure 9b,d. The impact force continuously decreased under the repeated impact load, demonstrating that the impact force tended to decrease as the impact number increased while increasing the duration of the impact process. The relationship between the cumulative residual displacement and the number of repeated impacts was plotted for all the specimens, as shown in Figure 10. The RC_FB_CFC and RC_F3S_CFC specimens withstood an increasing number of repeated impacts before reaching the specified residual displacement. RC_FB_CFC failed at the second impact owing to CFC panel delamination, whereas RC_F3S_CFC failed at the third impact.

4. Analytical Program

4.1. Geometric Model

In this study, SPH was used to simulate the response of beams to impact. SPH is a numerical technique and belongs to the category of mesh-free discretization techniques that could be used in continuum mechanics to simulate solid dynamics, including elasticity and plasticity. The way the spatial domain is discretized in SPH approaches is different from the FEM. A collection of points, known as particles, are used in the SPH approach to discretize the domain. The advantage of SPH over the FEM is that SPH can predict similar response behavior with more accurate and realistic failure patterns [27].

The SPH analytical model is presented in Figure 11. The analytical model using a self-coded Java program comprised two parts: the 100 kg falling weight and the RC beam. According to the objective of acquiring a detailed crack pattern, the RC beam in the analytical models was discretized with a particle size of 6.0 mm. According to the desktop capacity and to minimize the analysis time, only half of the beam was included in the analysis area, taking the center of the span as a plane of symmetry that was controlled by preventing movement in the x-direction. Furthermore, the RC beam was constrained using top and bottom hinges at the fulcrum position, similar to the experimental conditions, to prevent the beam from bouncing or z-direction movement.

In the analytical model of the strengthened RC beams, different materials were defined, namely, concrete, reinforcing steel, CFC panels, and the adhesive between the CFC panels and concrete. The mechanical properties of these four materials are listed in Table 1.

Figure 12 shows the adhesion model used in this analysis. Virtual adhesive particles were placed between the concrete and CFC particles. In the analysis, the strains of the concrete and CFC panel particles were calculated based on the conventional SPH method, and their average value was used as the strain of the adhesive particles. The stress of the adhesive particles was then calculated using the constitutive law of the adhesive material and controlling failure through the von Mises equivalent failure criterion. When the adhesive particles reached the failure state, the interaction between the adjacent CFC and concrete particles was excluded from the SPH analysis (excluded from the mutual influence domain) to account for the delamination of the CFC panel.

4.2. Material Models

Reinforcement: The steel of the reinforcing bars was modeled as an elastoplastic material that follows the von Mises yield condition and the simple bilinear stress–strain relationship, shown in Figure 13, for both compression and tension. The hardening stiffness after yielding was set to 1/100 of the initial stiffness.

Concrete: The concrete was modeled as an elastoplastic model based on the linear pressure-dependent Drucker–Prager yield criterion, which is expressed as:

$$f\left(J_2,I_1\right)=\sqrt{J_2}+\alpha I_1-k=0,$$

(1)

where $J_2$ is the second invariant of the deviatoric stress, $I_1$ is the first invariant of the stress ($I_1={\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}$), $\alpha =\frac{{\sigma }_c-{\sigma }_t}{\sqrt{3}({\sigma }_c+{\sigma }_t)}$ and $k=\frac{{2\sigma }_c{\sigma }_t}{\sqrt{3}({\sigma }_c+{\sigma }_t)}$ are material constants, and ${\sigma }_c$ and ${\sigma }_t$ are the compressive and tensile strengths, respectively.

The tensile softening response was assumed as defined by the bilinear function developed by Hillerborg [38], in which the tensile stress ${\sigma }_t$ decreases after cracking and tends to be zero at a total crack width $w$ of 3.6 times the first crack opening, as shown in Figure 14. The fracture energy $G_f$ of the concrete was defined using an existing experimental formula based on the compressive strength. Furthermore, the anisotropic damage law was used to compute the tensile cracking of concrete. The tensile damage in each principal direction at each time step was calculated using the relationship between the damage and accumulated plastic strain, as shown in Figure 15.

The accumulated plastic strain in the current principal direction was calculated by summing the plastic strain increments determined at each step along the global coordinates. According to the tensile softening, the principal accumulated plastic strain and direction must be calculated to evaluate the tensile damage in each principal direction $d_{t\_pr}$ by applying the following equation:

$$d_{t\_pr}=d_{t\_lim}\left(e^{a-e^{\left(b\left(c-\left({\epsilon }^p-\frac{{\epsilon }_{max}^p}{2}\right)\right)\right)}}\right),$$

(2)

where$d_{t\_lim}$ is the tensile damage limit and ${\epsilon }^p$ is the accumulated plastic strain. The constant values that were applied in this study are as follows: $a=0.02,b=230,$and $c=0.02$, and are the Gompertz parameters, and ${\epsilon }_{max}^p=0.025$ is the maximum accumulated plastic strain.

The damage in each principal direction is transformed into a global coordinate system as follows:

$$\left\{\begin{array}{c}{D}_1\\ D_2\\ D_3\end{array}\right\}=\left|d_{1,t\_pr}\left\{\begin{array}{c}{x}_1\\ y_1\\ z_1\end{array}\right\}\right|+\left|d_{2,t\_pr}\left\{\begin{array}{c}{x}_2\\ y_2\\ z_2\end{array}\right\}\right|+\left|d_{3,t\_pr}\left\{\begin{array}{c}{x}_3\\ y_3\\ z_3\end{array}\right\}\right|,e_{i,t\_pr}={\left\{x_i,y_i,z_i\right\}}^t,$$

(3)

$$D_i={\sum }_{i=1}^3\left|d_{i,t\_pr}{e}_{i,t\_pr}\right|,$$

(4)

where $D_i$ is the damage index in the global coordinates, and $e_{i,t\_pr}$ is the principal direction vector ($i=\mathrm{1,2},3$).

To consider the effect of tensile damage in this analysis, the elastic stiffness was multiplied by a reduction factor as follows:

$${\Phi }_{ij}=\sqrt{(1-D_i)(1-D_j)},$$

(5)

$$E^e=\left(\begin{array}{c}(\lambda +2\mu ){\Phi }_{11}\\ \begin{array}{c}\lambda {\Phi }_{12}\\ \lambda {\Phi }_{13}\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}\end{array}\begin{array}{c}\lambda {\Phi }_{12}\\ \begin{array}{c}(\lambda +2\mu ){\Phi }_{22}\\ \lambda {\Phi }_{23}\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}\end{array}\begin{array}{c}\lambda {\Phi }_{13}\\ \begin{array}{c}\lambda {\Phi }_{23}\\ (\lambda +2\mu ){\Phi }_{33}\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}\end{array}\begin{array}{c}0\\ \begin{array}{c}0\\ 0\\ 2\mu {\Phi }_{12}\end{array}\\ \begin{array}{c}0\\ 0\end{array}\end{array}\begin{array}{c}0\\ \begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}2\mu {\Phi }_{23}\\ 0\end{array}\end{array}\begin{array}{c}0\\ \begin{array}{c}0\\ 0\\ 0\end{array}\\ \begin{array}{c}0\\ 2\mu {\Phi }_{13}\end{array}\end{array}\right),$$

(6)

where ${\Phi }_{ij}$ is the reduction rate of the elastic stiffness, $\lambda =\frac{E\nu }{(1+\mu )(1-2\mu )}$ and $\mu =\frac{E}{2(1+\nu )}$ are Lamé’s constants, $E$ is the Young’s modulus, and $\nu$ is the Poisson’s ratio.

CFC panel and adhesive: As shown in Figure 1, the CFC panels have a composite structure consisting of a 1.0 mm thick carbon fiber core sandwiched between two 3.0 mm thick flexible boards. In the numerical analyses, the CFC panels were modeled as an elastic–plastic material that conforms with the von Mises criterion. Figure 16 and Figure 17 present the stress–strain relationships that dictate the behavior of the carbon fiber and flexible boards, which were defined using the results of the material characterization tests.

Furthermore, a multilinear elastic–plastic stress–strain relationship was used to model the composite response of the CFC panel structure. The stress–strain relationship shown in Figure 18 assumes that the carbon fiber and flexible board that consist of the CFC panel yield or break separately according to their individual stress–strain characteristics during the process of uniform deformation, and the overall stiffness of this panel is obtained as the sum of these materials’ stiffnesses. Specifically, the response of the CFC panel structure is divided into four strain regions: ${\epsilon }_{f1}<{\epsilon }_{c1}<{\epsilon }_{f2}<{\epsilon }_{c2}$, where the Young’s modulus and stress in each region are calculated using the following equations:

Elastic range $\left(\epsilon \le {\epsilon }_{f1}\right)$

$$E_1=\frac{{E_{f1}A}_f+{E_{c1}A}_c}{A_f+A_c},$$

(7)

$${\sigma }_{y1}=E_{f1}{\epsilon }_{f1}{A}_f+{E_{c1}{\epsilon }_{f1}A}_c,$$

(8)

Plastic stage 1 ${(\epsilon }_{f1}\le \epsilon \le$ ${\epsilon }_{c1})$

$$E_2=\frac{{E_{f2}A}_f+{E_{c1}A}_c}{A_f+A_c},$$

(9)

$${\sigma }_{y2}={\sigma }_{y1}+E_2({\epsilon }_{c1}-{\epsilon }_{f1}),$$

(10)

Plastic stage 2 ${(\epsilon }_{c1}\le \epsilon \le {\epsilon }_{f2})$

$$E_3=\frac{{E_{f2}A}_f+{E_{c2}A}_c}{A_f+A_c},$$

(11)

$${\sigma }_{y3}={\sigma }_{y2}+E_2({\epsilon }_{f2}-{\epsilon }_{c1}),$$

(12)

Plastic stage 3 ${(\epsilon }_{f2}\le \epsilon \le {\epsilon }_{c2})$

$$E_4=\frac{{E_{c2}A}_c}{A_f+A_c},$$

(13)

$${\sigma }_{y4}={\sigma }_{y3}+E_4({\epsilon }_{c2}-{\epsilon }_{f2}),$$

(14)

where $E_1,E_2,E_3,\mathrm{a}\mathrm{n}\mathrm{d} E_4$ are the Young’s modulus, ${\sigma }_{y1},{\sigma }_{y2},{\sigma }_{y3},{\mathrm{a}\mathrm{n}\mathrm{d} \sigma }_{y4}$ are the stresses, $A_f$ is the area of the carbon fiber, and $A_c$ is the area of the flexible board.

The virtual adhesive particles simulate the boning between the concrete and CFC panel particles. In this study, peeling of the adhesive layer was assumed to occur when the stress reached the adhesive strength given by the von Mises yield criterion:

$$\left(J_2\right)=\sqrt{J_2}-k=0,$$

(15)

where $J_2=\frac{1}{6}[{\left({\sigma }_{11}-{\sigma }_{22}\right)}^2+{\left({\sigma }_{22}-{\sigma }_{33}\right)}^2+{\left({\sigma }_{33}-{\sigma }_{11}\right)}^2+6\left({\sigma }_{12}^2+{\sigma }_{23}^2+{\sigma }_{31}^2\right)]$ is the second invariant of the deviatoric stress, and $k=\frac{{\sigma }_y}{\sqrt{3}}$ is a positive material constant.

Based on the assumptions above, the CFC panel particles separated from the concrete particles when either the adhesive particles reached the yielding condition, or the strain exceeded the strain limit of the CFC panel.

5. Analytical Program Verification

5.1. Failure Mode and Crack Propagation

The analytical final crack propagation of each specimen and analytical crack propagation at the middle range of the specimen subjected to each impact were compared with the experiment, as shown in Figure 19.

The analytical model was able to efficiently simulate the flexural cracks on the specimens by using the anisotropic constitutive law in the tension region. In addition, the use of adhesive particles with limited adhesive strength in the numerical analyses made it possible to analytically reproduce the delamination between the concrete and CFC panels, as well as the interfacial delamination of the latter. Furthermore, the analysis results confirm that the specimens reinforced with CFC panels exhibited a better impact resistance and a wider area of bending cracks. The delamination of the CFC particles from the concrete particles was effectively simulated; however, the interfacial debonding of the bottom CFC panel was not clearly captured due to the underlying SPH assumptions. In the analysis, the coupling condition holds as long as the bottom and side CFC particles remain within the same influence domain; thus, to predict the exact failure mechanism of the RC_F3S_CFC specimen, additional improvements are required in the SPH modelling approach.

5.2. Impact Response

Figure 20 presents the analytical results against the experimental results, in terms of the displacement and impact force time histories. The results demonstrate that the proposed numerical method can adequately reproduce the impact response of CFC-strengthened RC beams.

The repeated impacts of RC_FB_CFC and RC_F3S_CFC are shown in Figure 21. As the figure demonstrates, except for RC_F3S_CFC, the displacement responses of all the specimens under repeated impact loading were accurately reproduced. This confirms that the proposed analysis method can predict the impact response and failure process of RC beams strengthened with bottom panels, such as the RC_FB_CFC specimen; however, the proposed method could not adequately simulate the displacement response of the RC_F3S_CFC during the third impact. This may be attributed to the simplified implementation of an elastic–plastic relationship to model the flexural–tensile behavior of the CFC panels. Therefore, to accurately reproduce the impact response, it is necessary to use a more advanced mechanical model, in which the inside of the panel reaches the elastic limit before the side of the CFC panel peels off.

Figure 22 shows the relationship between the cumulative residual displacements and number of repeated impacts for each of the four types of beams. In this study, a residual displacement of 10 mm (1/100 of the span length) was set as the design limit. As shown in Figure 22, the proposed analysis method can accurately predict the number of repeated impacts that each beam type can resist before reaching the design limit, although there is a slight difference between the residual displacement at the time of the second and third impacts estimated by the numerical analysis of the RC_F3S_CFC types. The simulation confirmed that the number of repeated impacts until each type of residual displacement reached the design limit could be roughly predicted.

5.3. Strain Energy Distribution

Figure 23 shows the strain energy distribution of all the specimens after releasing a single impact of 3.0 m/s impact velocity, which can indicate each specimen’s ability to absorb and dissipate energy. Among the strengthened specimens, CFC panels can help to protect the RC beam from damage or crack propagation during impact events. The strain energy of the strengthened specimens was not concentrated around the steel reinforcement, but was dissipated in the tension region, whereas the strain energy was differently distributed in the conventional RC beam, and it clearly showed that steel reinforcement played an important role in absorbing energy.

In RC_F3S_CFC, the full bottom and side surface strengthening achieved the best performance in impact response, followed by the full bottom surface strengthening; however, in RC_P3_CFC, the partial bottom and side surfaces’ strengthening performed an interesting strain energy distribution. The specimen had the ability to absorb energy in a wide area, which can minimize the damage or crack width that especially propagates in the tension region. Unfortunately, the midspan displacement of RC_P3S_CFC seemed to be higher than that of the other strengthened specimen, which was significantly caused by the interfacial debonding of the CFC panels. In order to observe the full performance of the strengthened specimens, the adhesion methods needed to be improved to prevent the CFC panels from debonding.

6. Parametric Study

The impact resistance performance of RC beams and RC beams reinforced with CFC panels were verified through falling-weight impact experiments and SPH analysis simulations. Tests were conducted until the residual displacement reached the design limit. In this study, the impact behavior of each specimen was examined up to the ultimate limit through a parametric analysis. Specifically, fracture simulations were performed for all the specimens subjected to three repeated impacts at a constant impact velocity of 3.0 m/s, and the relationship between the number of impacts and the cumulative residual displacement was investigated. Additionally, the fracture behavior of each specimen was considered when the impact velocity was increased to 4.0 m/s and 5.0 m/s.

Table 2. Parametric study. * is same as in Figure 24.

Specimen ID	Parameter of Parametric Study
	Impact Velocity	Simulation Type
RC *	3.0 m/s	Three repeated impacts
RC_FB_CFC
RC_P3S_CFC
RC_F3S_CFC
RC_4 m/s	4.0 m/s	Single impact
RC_FB_CFC_4 m/s
RC_P3S_CFC_4 m/s
RC_F3S_CFC_4 m/s
RC_5 m/s	5.0 m/s	Single impact
RC_FB_CFC_5 m/s
RC_P3S_CFC_5 m/s
RC_F3S_CFC_5 m/s

lists the specific analytical conditions used.

Three repeated impact simulations were performed for three types of CFC-reinforced test specimens, but the ordinary RC beam was excluded from the study because a large residual displacement occurred with one impact.

As shown in Figure 24, it is recognized that when the number of repeated impacts increases, the residual displacement correspondingly increases as the damaged area increases. When subjected to three repeated impacts, RC_F3S_CFC exhibits the best performance among the strengthened specimens. During the first two impacts, the effectiveness of the three strengthening methods were ranked as follows: RC_F3S_CFC < RC_FB_CFC < RC_P3S_CFC; however, after the third impact, RC_P3S_CFC outperformed RC_FB_CFC. Thus, the relative superiority of RC_P3S_CFC and RC_FB_CFC depend on the adhesive strength of the CFC panels or the reinforcement area of the RC_P3S_CFC specimen. However, in general, it is expected that safety against ultimate failure is higher for RC_P3S_CFC, which reinforces the bottom and side surfaces.

Figure 24. Results of parametric study on three repeated impacts prediction: (a) displacement at midspan; (b) impact force at midspan; (c) relationship between cumulative residual displacement and number of impacts. Note: *—Cannot demonstrate residual displacement.

Figure 24. Results of parametric study on three repeated impacts prediction: (a) displacement at midspan; (b) impact force at midspan; (c) relationship between cumulative residual displacement and number of impacts. Note: *—Cannot demonstrate residual displacement.

As shown in Figure 25, when subjected to an impact velocity of 5.0 m/s, all the specimens except RC_F3S_CFC experienced significant fractures with residual displacements of 30 mm or more after a single impact; however, RC_F3S_CFC displayed a residual displacement of 10.35 mm, demonstrating remarkable impact resistance.

In addition, under relatively severe impact conditions, such as impact velocities of 4.0 m/s or 5.0 m/s, RC_P3S_CFC exhibited superior impact resistance compared to RC_FB_CFC after a single impact. This indicates that RC_P3S_CFC offers greater safety than RC_FB_CFC.

7. Conclusions

In this study, both experimental and numerical analyses were carried out to investigate the reinforcement effect of applying CFC panels to improve the impact resistance of RC beams subjected to impact loading. For the analytical study, an analysis program using virtual particles that takes into account the characteristics of the adhesive in the SPH method was developed, and its accuracy was verified by comparing it with the results of the impact experiments. Subsequently, a parametric study was conducted using a validated analytical program to examine the relationship between the method of reinforcement with CFC panels and impact resistance performance. The main results of this study are summarized as follows.

In the impact test, flexural cracks were observed in all the specimens; however, it was confirmed that reinforcement with the CFC panels significantly improved the impact resistance of the beams and suppressed the propagation of cracks throughout the beams. The strengthened specimens performed better in terms of recovery capability compared to the unstrengthened specimens, which was apparent after the first impact. The full bottom strengthening and the partial bottom and side surfaces strengthening acquired higher restitution coefficients compared to the unstrengthened RC beams by 151% and 32%, respectively. In particular, the specimens reinforced with CFC panels on the entire bottom and side surfaces showed the greatest improvement in impact resistance, as demonstrated by a higher restitution coefficient compared to the conventional RC beam for 225% after the first impact and being able to withstand up to three repeated impacts before reaching the failure condition. In terms of energy dissipation, the strengthened specimens had better performance compared to the conventional RC beam. The partial bottom and side surface strengthening was able to absorb energy over a wide area, which helped to reduce crack propagation. It was also found that the CFC panels were prone to interface delamination due to the fracture of the adhesive layer. An anchorage system is suggested to be applied to improve the interfacial bonding between the concrete and CFC panels, as well as the three layers of CFC panels.

In the analytical study using the SPH method, a simple linear elastoplastic model, which combines the individual stress–strain relationships of carbon fiber and flexible board for the CFC panels, was employed, and it was confirmed that the model can easily and accurately reproduce the impact behavior of RC beams reinforced with CFC panels. It was also confirmed that the interfacial delamination of the CFC panels could be reproduced by placing adhesive particles between the CFC panels and concrete that bond the two and by setting the conditions for their failure.

From the parametric analysis results for the increasing impact velocity and the number of repeated impacts, it can be concluded that the ultimate performance to complete failure is improved by integrally reinforcing the bottom and sides of the beam rather than reinforcing only the bottom.