Numerical analysis of reinforced concrete beams subjected to impact loads

1 Introduction

Concrete structures are usually subject to a variety of impact loads, such as falling rocks, terrorist attacks, and unexpected collisions, during their serviceability. After the reinforced concrete member is exposed to impact loading, it is likely to experience significant local deformation; as a result of the debris, there is subsequent damage and a loss of inner strength. To avoid this damage, there has been a growing desire for current reinforce concrete (RC) members to be strengthened with impact resistance. Fiber-reinforced polymer (FRP) has been progressively used in civil engineering infrastructure strengthening and rehabilitation since the 1990s because of the rising stiffness-to-weight ratio, high strength to weight, excellent durability, and ease of usage in situ applications [1]. However, currently, there is not very limited research focusing on carbon fiber-reinforced polymer (CFRP) strengthened reinforced concrete members under impact although an increasing number of field applications of FRP-strengthened reinforced concrete members are vulnerable to impact loading. As CFRPs have commonly been employed efficiently to strengthen reinforced concrete members against static and fatigue loads, examining the CFRP strengthening technique to improve the performance of the structural concrete members under varied impact loads is the main focus of this study. As a result, this study explains how to use a numerical modeling technique to study the behavior of RC beam that has been subjected to impact loads.

Numerical solutions, such as the finite element (FE) method, might be deemed more suitable for determining the behavior of RC member subjected to impact loads in order to solve the entire impact problems and to provide information regarding the behavior of RC member. A three-dimensional FE model permits users to analyze the different components of any difficult problem, such as the load and boundary conditions acting on the structures. For the instance of impact loading, RC member and impactor can also be properly modeled, in addition to nonlinearities caused by material behavior, and can always be considered. The method of numerically simulating RC beams exposed to impact loading is a difficult process. As a result, an accurate simulation necessitates the employment of a concrete model and a contact method that can be used for nonlinear explicit finite element analysis (FEA). In this portion of the investigation, some preceding papers associated with numerical modeling of reinforced concrete members subjected to high, low, and explosive velocity of impact loading are addressed.

Mokhatar and Abdullah [2] examined the adequacy for numerical modeling techniques of predicting the behavior of RC slab and steel reinforcement failure mechanisms when exposed to impact loads. To obtain a good knowledge for the impacted slab behavior, an FE model was adopted using ABAQUS and the result was validated using the data of Chen and May [3]. The concrete, steel reinforcement, steel support, and steel projectile were all split into four segments in the model. The steel support was modeled with undeformable discrete rigid elements (R3D4), whereas eight-node (C3D8R) continuum solid elements have been used to simulate the concrete zone. Following the assembly of all elements, steel supports were connected to the slab by the tie contact method. Surface-to-surface contact technique was employed to describe the connection between the concrete region and the steel impactor. For mechanical restriction formulation, the kinematic contact technique with a friction coefficient of 0.2 has been used. With the rate of 1.5, the increase in compression and tension strengths of concrete owing to impact load was addressed. The FE modeling was confirmed using experimental data, which included the impact force–time history and a final fracture pattern. Mesh sensitivity has been used to choose a sufficiently refined mesh that yielded reasonably accurate results. After that, the impact load of three concrete modeling was computed: the concrete damage plasticity (CDP) model has brittle-cracking behaviors, while the cap plasticity and Drucker–Prager models have ductile behaviors. The findings demonstrate that ductility models, rather than brittle-cracking models, may accurately predict the behavior of RC slab under dynamic impact loading and accord well with experimental results. However, the final crack pattern derived using the CDP model agrees well with experimental data, but there was no influence of spallation, which is not the genuine slab failure mode.

Jahami et al. [4] studied numerically the use of CFRPs as a strengthening method to enhance the structural behavior of post-tensioned one-way slabs subjected to direct impact from falling objects (e.g., rocks). The volumetric FE modeling program, ABAQUS CAE, was used for their research. The proposed study consisted of a 300 cm × 600 cm × 25 cm PT slab subjected to a direct impact using a 500 kg concrete block. A supporting system was provided by steel cylinders linked to the slab by welded steel rebar. The studied parameters were the covered area and the number of CFRP sheets. As for the falling object, it was dropped at the slab mid-span in order to study the damage at critical zones. A linear brick element C3D8R with eight nodes was used. Each node has three degrees of freedom, and the steel bars and tendons were modeled using a three-dimensional truss element with two modes. It was considered that all steel reinforcement was fully embedded in the concrete.

The CDP model was used to characterize the nonlinear relationships between stress and strain in concrete using ABAQUS software. This model is considered to be suitable for dynamic and impact load analysis. A concrete compressive strength of 30 MPa was used in the analysis. In terms of steel reinforcement, a yield strength of 420 MPa with elastic-perfectly plastic behavior was considered. The yield strength of prestressing tendons was determined to be 1,860 MPa, with bi-linear stress–strain curve. The sheets of CFRP were modeled as shell elements. The governed mode of failure was punching shear mode, and by using more layers of CFRP sheets, the damage was reduced and more energy was dissipated, but for a certain limit since the main cause of failure was the concrete crushing under high compressive and tensile stresses.

2 Selected beams for the numerical analysis

Kishi et al. [5] used a set of simply supported reinforced concrete beams strengthened with CFRP to evaluate their response under impact load. The samples also included a unstrengthened reference sample. Two samples have been simulated and analyzed in the present research. These specimens included a unstrengthening beam (the reference for beams unstrengthening beam by carbon fiber [NI]) plus another strengthening beam strengthened beam by carbon fiber (CI), which was strengthened with one bottom layer of the CFRP sheet in the longitudinal direction. The tested beams were designated as 200 mm in width and 250 mm in height, with 3,000 mm clear span. Two longitudinal rebars with 19 mm diameter have been cast at the top and the bottom fibers of the beam, and the rebars have been welded to 9 mm-thick steel plates located at the beam’s ends to provide full anchorage of the beam. Steel stirrups of 10 mm diameters were placed at 100 mm spacing. The CFRP sheet was attached to a tension-bottom surface with a gap of 50 mm between the support point and the sheet’s end. Beam (CI) was strengthened by bonding one CFRP sheet with a mass of 600 g/m². The properties of FRP sheets are specified by the manufacturers (Fibex 2014; Toray 2019). These properties were obtained by an experimental test based on JIS K 7165 (JIS 2008). The deformed rebar had yield strengths of 403 MPa for beam (CI) and 382 MPa for beam (NI). A cylindrical impact having a mass weight of 300 kg and a drop-weight height of 2.5 and 3 m (impact speed of 7 and 7.7 m/s) was applied to the tested beams (NI and CI), respectively. The concrete had a compressive strength of 32 MPa for beam (NI) and 33 MPa for beam (CI).

3 FEA

In order to achieve reliable results for analyzing the response of structural elements under impact loads with ABAQUS/Explicit, the proper geometric and material modeling parameters must be carefully selected. The modeling methods used in this work are described in the following sections.

3.1 Selection of element types

For numerical simulation of reinforced concrete structural members, an accurate model of the structural elements and, constituently, the members operating as a composite consisting of concrete and steel are necessary. C3D8R is an eight-node brick element having three translation degrees of freedom for each node as shown in Figure 1a and was chosen from a large library of solid elements each with different capability accuracies and efficiencies to model the concrete beam, ABAQUS user manual [6]. ABAQUS/Explicit adopts a reduced integration technique to integrate various response outputs over this type of element. The steel reinforcing bars can be modeled in ABAQUS using various procedures. In this work, a truss element (T3D2) with two-node linear displacement is used to simulate steel bars as shown in Figure 1b. A four-node S4R element having six degrees of freedom at each node, three displacements and three rotations in each direction, has been chosen to represent the CFRP material as shown in Figure 1c. Cohesive elements were employed to represent the contact zone for CFRP plus concrete. Each node of cohesive element (COH3D8) has six degrees of freedom. In similar examples, such as Al-Zubaidy et al., this kind of element has been widely utilized to modeling this type of contact area [7]. ABAQUS/Explicit also employs a reduced integration technique, linear interpolation, and hourglass control over this element. To model the steel support of the beams, the discrete rigid element (R3D4) was chosen. Finally, the impactor has been modeled as a separate rigid body with a reference point for determining the impactor’s mass.

Figure 1

(a) Modeling concrete, (b) modeling of bars, and (c) modeling of CFRP.

3.2 Material modeling

When performing any type of nonlinear FEA, material properties can play an important part in the analysis. For the concrete simulation, the concrete damage plasticity model has been chosen to provide a general capacity for analyzing concrete structures in monotonic, periodic, and dynamic loading. It is an isotropic damaged pattern with tension cracking and compression crushing modes. The concrete modeling has been done in two steps. In a first step, the modulus of elasticity and Poisson’s ratio have been determined, while in a second step, the model of damage plasticity has been determined, which included the nonlinear part of a stress–strain curve for concrete as seen in Figure 2 [8].

Figure 2

Behavior of concrete. (a) Uniaxial compression and (b) uniaxial tension (ABAQUS user manual) [8].

Table 1 lists the input parameters used to define the constitutive concrete model.

Table 1

Damage plasticity model parameters

Dilation angle	Eccentricity	fbo/fco	K	Viscosity parameter
51	0.1	1.16	0.667	0

Steel is a little easier to describe, as demonstrated in Table 2, because of its precise depiction under the concepts of elastic and plastic. The steel longitudinal bars and stirrups are modeled as elastic–perfect plastic materials by specifying yield stress and elastic modulus for bars.

Table 2

Steel reinforcement material properties

Beam designate	Young’s modulus (MPa)	Poisson’s ratio	Density (kg/m3)	Yield stress (MPa)
CI	200,000	0.3	7,800	403
NI	200,000	0.3	7,800	382

A lamina concept that is necessary to characterize the linear elastic response of the CFRP sheet was used to determine the modulus of elasticity of shear in two different directions plus Poisson’s ratio. ABAQUS/Explicit provides Hashin’s failure criteria [9], which are widely used to characterize damage in the composite materials, to generate the damage of the CFRP sheet within that model. The selected model takes into account the longitudinal, transverse tensile, and compressive strengths of CFRP sheets, as well as the shear strengths in longitudinal plus transverse directions. The listed values in Table 3 are obtained from the experimental study carried out by Kishi et al. [5], and other properties are obtained from available data presented in the literature [10].

Table 3

Material properties of the CFRP sheet*

Property	Value
Density	1,820 kg/m3
E 1 Elastic modulus in the longitudinal direction	245 GPa
E 2 Elastic modulus in the transverse direction	17 GPa
G 12 In-plane shear modulus	4.5 GPa
Longitudinal tensile strength	3,900 MPa
Longitudinal compressive strength	2,400 MPa
Transverse tensile strength	111 MPa
Transverse compressive strength	290 MPa
Longitudinal shear strength	120 MPa

*Material properties are obtained from Dolce [10].

4 Finite element model validation

To gain full knowledge of the dynamic behavior of the CFRP sheet-strengthened beam, a numerical model’s capacity to reflect the actual behavior for the investigated event must be examined from the start. The numerical modeling is validated by comparing the impact force–time history, maximum impact force, displacement–time history, and the maximum displacement against experimental data. Comparisons of the impact force–time history obtained through experimental data and numerical analysis show that numerical modeling is capable of accurately capturing the force–time history of the analyzed beams. The impact time produced by the numerical modeling differs slightly from it found in experimental tests, as shown in Figure 3. The variation could be due to changes in fixity between the tests and simulations. In previous work, when the same test approach was used, a similar difference in the impact duration value acquired by experimental data and numerical modeling was reported. For example, Zeinoddini et al. [13] discovered that the highest variance between numerical and experimental impact duration is approximately 26%. A graphical comparison of impact force–time history of the selected samples is shown in Figure 3.

Figure 3

Comparison between the experimental data and numerical result: (a) impact force–time history of the beam (NI) and (b) impact force–time history of the beam (CI).

The numerical results indicated extremely good agreement with the relevant experimental data in terms of displacement–time history; the analysis was done for 0.05 s to reduce the time and additional effort of the analysis, as shown in Figure 4.

Figure 4

Comparison between the experimental data and numerical results: (a) displacement–time history of the beam (NI) and (b) displacement–time history of the beam (CI).

Concerning peak impact force, a comparison of numerical and experimental findings shows that the numerical predictions are quite close to the experimental values reported in Table 4. This table shows that the numerical findings have a maximum deviation of less than 10% when compared to the equivalent experimental data. Also, for displacement–time histories, as shown in Table 4, the difference is less than 5%. As a result, a numerical modeling showed capability to forecast the deflection time history and peak impact force of the impacted beam with a good degree of accuracy.

5 Parametric study on the performance of beam under impact loading

In the experimental tests, only a small number of samples can be evaluated; however, by using the validated numerical model, more parameters can be studied to gain a better knowledge of the performance of CFRP-strengthened beam under impact loads. These parameters are explained in Table 5, and beams were analyzed by ABAQUS (2017) software program.

where m represents mass (kg) and v represents velocity (m/s).

6 Numerical result of the parametric study

In addition to the original velocity of 7.7 m/s (height of impact mass = 3 m and impact mass = 300 kg) used in the experimental work, three more velocities were chosen: 8.3 m/s (H = 3.5 m), 8.85 m/s (H = 4 m), and 9.4 m/s (H = 4.5 m), to study the influence of impact velocity on the effectiveness of the CFRP strengthened beam (CI), and the numerical results were compared to those obtained for beam (NI). Various impact velocities are examined as shown in Figure 5. In general, as shown in Figure 6, the effectiveness of the CFRP sheets used for the CI beam was clear by reducing the displacement as compared to the NI beam, but with the higher impact velocity, the effectiveness of the CFRP was not significantly increased. The reason for this response is that the CFRP is an insensitive strain rate material.

Figure 5

Displacement–time history of the beams NI and CI for different impact velocities. (a) For the NI beam and (b) for the CI beam.

Figure 6

Maximum displacement of the beams NI and CI against the impact velocity.

During the experimental program, all samples were tested under the impact mass of 300 kg. In this section, three additional masses were suggested, 348, 396, and 447 kg, to examine the influence of mass changes on the efficiency of a strengthening beam (CI) and the obtained results were compared with those of unstrengthen beam (NI). It is important to mention here that the impact energy values in this study were 10333.5, 11748.4, and 13254 J, which equals the impact energy used in the preceding study concerning the influence of impact velocity, to investigate the effect of the same impact energy resulting from two different components (velocity and mass). It can be seen in Figure 7 (impact mass = 447 kg) that the difference in the impact duration was quite high compared with that of the previous study, shown in Figure 5 (for the impact velocity of 9.4 m/s), even when the same energy of the impact was used. In addition, the difference in the displacement shown in the two figures is significant. This difference can be attributed to the variation in the strain rate between the two considered cases. For instance, the average strain rate (measured under impact load) for a beam impacted with a 300 kg mass impactor at a velocity of 9.4 m/s was about 1.23 times the value of that corresponds to a beam impacted with the equal impact energy but with different components (mass = 447 kg and velocity = 7.7 m/s). Since the flow of stress in concrete increases as the strain rate increases, the displacement value for a beam with a high strain rate is anticipated to be smaller than that of a beam with a lower strain rate. Also, it was observed that for the strengthened beam CI with CFRP sheets, the maximum displacement was reduced when the results have been compared to those obtained for unstrengthening beam NI, as shown in Figure 8.

Figure 7

Displacement–time history for different impact masses. (a) For the NI beam and (b) for the CI beam.

Figure 8

Maximum displacement of beams NI and CI versus the impact mass.

As previously stated, different CFRP thicknesses have been investigated in this parametric study in order to gain a thorough understanding of the effect of thickness for CFRP on the impact performance of a strengthened beam, including the thickness of 0.66, 0.99, and 1.32 mm in addition to the reference simulated thickness of 0.33 mm. All analyzed beams of the selected three thicknesses were modeled with the same boundary conditions as well as kinetic energy. The simulation results indicated that the thickness of CFRP sheet had a considerable influence on the obtained displacement, as shown in Figure 9. The displacement histories clearly show that the impact duration was significantly reduced as a result of increasing the CFRP sheet thickness. For example, when the CFRP thickness was increased to 0.66 mm, the reduction in maximum displacement was increased by about 18.8% compared with the CFRP thickness of 0.33 mm. However, when the CFRP thickness was increased to 1.32 mm, the reduction in maximum displacement was increased about 11% compared with the CFRP thickness of 0.66 mm, as shown in Figure 9. This response occurred as a result of the stiffness imbalance between the adherents (CFRP plus concrete), since by increasing the CFRP thickness, the interfacial shear stress in adhesive material was increased. Also, the results appear to be in line with the previous study, which found that the thicknesses and elastic modulus of CFRP sheets influence the interfacial shear stress in the adhesive material (Wu et al. [14] and Yuan et al. [15]).

Figure 9

Displacement–time histories for various CFRP sheet thicknesses.

Figure 10 shows the effect of the compressive strength of concrete on the displacement behavior of a unstrengthen beam that was tested (NI). For compressive strength of 42 and 52 MPa, the decrease in the maximum displacement was 6.4 and 11%, respectively, as compared with the corresponding displacement obtained for compressive strength 32 MPa. As the compressive strength of concrete increases, modulus elasticity of concrete increases, and the rigidity of the beam increases [16].

Figure 10

Displacement–time histories for various compressive strengths of concrete.

7 Conclusions

The following conclusions have been drawn from analyzing the considered beams:

1. The simulation results showed that increasing thickness of the CFRP layer had a significant effect on the displacement of all analyzed strengthened beams. When the thickness was increased to 0.66, 0.99, and 1.32 mm, the reduction in maximum displacement increased by about 18.8, 24, and 28% compared with the beam strengthened by CFRP thickness of 0.33 mm.

2. The effectiveness of the CFRP was found to be dependent on the applied impact energy and is also dependent on the components of the impact energy. It was observed that even though the same energy of the impact loading was used, the difference in the impact duration was quite high for the impact mass of 396 kg compared to the impact velocity of 8.85 m/s, for example.

3. Generally, it has been found that by using a CFRP sheet for strengthening reinforced concrete beams, they greatly improve the members’ impact behavior by improving stiffness as well as increasing load capacities. The enhanced performance characteristics of strengthening beams under impact loads correlate with the applied kinetic energy and CFRP thicknesses.

4. For beams with high compressive strength, the deflection values have been reduced because of the increase in stiffness.

5. The CFRP effectiveness was found to be dependent on the kinetic energy and significantly sensitive to the components of kinetic energy (i.e., velocity and mass).

6. The increase in the thickness of the CFRP sheet had a significant influence on the displacement of all analyzed beams. The increase in the carbon fiber sheet thickness has a certain effect, after which the reduction ratio begins to decrease, because the increase in the thickness of CFRP sheet led to increase in the interfacial of shear stress in adhesive materials because of the stiffness imbalance among the adherent materials (CFRP and concrete).

Notations

CDP

concrete damage plasticity

CFRP

carbon fiber-reinforced polymer

E ₁

modulus of elasticity for longitudinal direction

E ₂

modulus of elasticity for transverse direction

f ′ c

concrete compressive strength

FE

finite element

FE

finite element analysis

FRP

fiber-reinforced polymer

g

ground acceleration = 9.81 m/s²

G ₁₂

modulus of shear in plane

H

height of the drop, m

m

mass, kg

RC

reinforce concrete

v

velocity, m/s