The materials properties of modeled beams are shown in Figure 1 . Due to the symmetry of geometry, loading, boundary conditions, and material properties, a quarter FE model was built and analyzed. The use of a quarter model significantly reduces computational time. In ANSYS terminology, the term model generation usually takes on the narrower meaning of generating the nodes and elements that represent the spatial volume and connectivity of the actual system [25] .

Thus, model generation in this discussion will mean the process of defining the geometric configuration of the model’s nodes and elements. From the available element library in ANSYS, the elements used in this work as fallow:

An eight-node solid element, Solid65, was used to model the concrete. The solid element has eight nodes with three degrees of freedom at each node –translations in the nodal x, y, and z directions. The element is capable of plastic deformation, cracking in three orthogonal directions, and crushing. The value of the shear transfer coefficient (βt) ranges from 0.0 to 1.0, with 0.0 representing a smooth crack and 1.0 representing a rough crack. In this paper, a shear transfer coefficient of the open crack of βt is 0.25, and a shear transfer coefficient of closed crack βc is 0.8 are used. The modulus of elasticity ( $E_c$), and the modulus of rupture ( $f_r$) for concrete both are calculated in terms of the concrete compressive strength ( $f_c^{/}$)

$E_c=4700\sqrt{f_c^{/}}$ (1)

$f_r=0.62\sqrt{f_c^{/}}$ (2)

The Poisson’s ratio for concrete is usually taken as 0.2 and the stress strain relationship can be obtained from following equations.

$f=E_c\epsilon /\left(1+{\left(\epsilon /{\epsilon }_0\right)}^2\right)$ (3)

${\epsilon }_0=\frac{2f_c^{/}}{E_c}$ (4)

$E_c=\frac{f}{\epsilon }$ (5)

Where $f$ strees at any strain (ε), ε is strain at stress $\left(f\right)$, ε₀: strain at the ultimate compressive strength $f_c^{/}$.

nk-8 element was used to model steel reinforcement. Two nodes are required for this element. Each node has three degrees of freedom, – translations in the nodal x, y, and z directions. The element is also capable of plastic deformation. The geometry and node locations for this element type.

An eight-node solid element, Solid45, was used for the steel plates at the supports in the beam models. The element is defined with eight nodes having three degrees of freedom at each node – translations in the nodal x, y, and z directions.

To study the contact between two bodies, the surface of one body is conventionally taken as a contact surface and the surface of the other body as a target surface. ARGE170 is used to represent various 3D target surfaces for the associated contact elements (CONTA175). CONTA175 may be used to represent contact and sliding between two surfaces (or between a node and a surface, or between a line and a surface) in 2D or 3D. The element is applicable to 2D or 3D structural contact analyses. Here in this study, concrete is considered as contact and the FRP as target element [24] . For concrete, ANSYS computer program requires input data for material properties such as Mishing, materials contact and target, boundary condition and the uniaxial stress-strain relationship for concrete in compression.

For a FE model of RC beams, failure was considered when a solution for a 10 N load increment could not reach a convergence. The FE models of the beams typically failed when shear steel reinforcement yielded followed by severe cracking of concrete. This in turn caused the FE simulation to terminate due to a divergence. Divergence in the FE solution coincided with a considerably large deflection, exceeding the displacement limitation of the ANSYS software.

Four reinforced concrete beams were tested, one beam without strengthening and considered as reference beam, one beam was strengthened with FRP without pre-damage and two other beams were strengthened after subjected to pre-damage load. Each beam possessed a rectangular section with dimensions of 1200 mm × 100 mm × 200 mm. The beams were reinforced with stirrups of 6-mm diameter spaced 120-mm center to center. The longitudinal tensile reinforcement with a diameter of 10 mm was provided in the top and bottom of the beam.

When the beam BM1 was considered as control beam without strengthening, BM2 and BM3 were exposed to an elastic load by cracking up to 50% and 75% of the control beam’s load capacity, respectively, before being strengthened with CFRP. BM4 was strengthened and tested to failure load without pre-loading.

The goal of the comparison between the FE model by using ANSYS14.5 and the experimental results was to ensure that the material properties, elements, and convergence criteria are adequate to model the response of the member and make sure that the simulation process is correct. Therefore, in this study, the beams which conducted in the previous experimental test were simulated for verification study. Figure 2 shows the geometry and tested beams. The comparison between experimental and FE results by using ANSYS were illustrated in Figure 3 . From Figure 3 , it is evident that there was a good correlation between the numerical and experimental load-deflection curves at all loading stages. The FE models were able to predict accurately the load capacities for the simulated RC beams. This confirmed the validity of the developed FE models and reliability of the FE simulation. Table 1 showed the numerical and experimental cracking and failure loads for the calibrated beams.

As shown from Table 1 that the results obtained from FE analysis are in accordance with those from experimental works with variation of not more than 20%.

For parametric study, a rectangular reinforced concrete beam was conducted by using ANSYS finite element model. The model program includes instrumentation, and four RC beam specimens with 400 mm depth, 150 mm width, 2300 mm length and 2000 mm clear span. In these beam specimens, the same steel reinforcement layout was provided, where two tensile steel bars with 12-mm diameter were arranged to the bottom of beam, two tensile steel bars with 10-mm diameter were arranged to the top of beam, shear reinforcement spacing 200 mm with 8 mm diameter stirrups were used throughout the entire beam length. The thickness of the concrete cover layer was 25 mm at the lateral and upper faces of the beam and 35 mm at the bottom side which have 360 Mpa yield strength for basic iron and 240 Mpa for links. The dimensions and details of the modeling beams are presented in Figure 4 . The used CFRP strips have a width of 150 mm and a thickness of 1.1 mm per layer.

The obtained vertical displacement distributions of reinforced concrete beam for four different cases are presented in Figure 5 , the maximum deflections are shown in Table 2 . As shown from the results, deformation distribution features are almost same by comparing these subfigures, but the value of maximum deflection is very distinct. Especially, the maximum deflection decreases with the increase of CFRP layers. The Misses stress of reinforcing bars beam for three different cases are presented in Figure 6 , where the maximum stress of reinforcing bars with different layers of FRP is presented in Table 3 . The Misses stress and maximum stress distributions of FRP with different layers are presented in Figure 7 and Table 4 , respectively. The horizontal stress distribution and evolution of concrete beam for four different cases are presented in Figure 8 , Figure 9 , Figure 10 and Figure 11 , respectively. The results of the load-deflection curve are shown in Figure 12 .

From Table 2 , we find that the maximum deflection for the case of without FRP is the minimum, while the maximum deflection of other cases with FRP is much larger than that of the case without FRP. For comparison, the deflection increased in strengthened beams when the number of layers increased, which resulted in greater capacity compared with control beam. The deflection increased by 464%, 290%, and 260% for one, two, and three layers, respectively. This is because the brittle fracture happened easily in the non-strengthened beam, causing the RC beam without FRP strengthening fail earlier than the strengthened one. Furthermore, when reinforced using FRP at bottom side of the beam, the stiffness is improved especially for two layers case compared with one layer. However, the improvement effect is not obvious for three layers case. Therefore, from the point of deformation control, the optimized reinforcement scheme should be the two layers FRP case.

From Figure 6 and Table 3 , the maximum stress in reinforcing bar for the case of without FRP is minimum, only 154.575 MPa, which is much less than that of the other cases with FRP. This due to the whole brittle fracturing happened earlier for the non-strengthened beam. In addition, when reinforced using FRP at bottom of RC beam, the carried part of external force for reinforcing bar is almost same for different layers of FRP due to most of the external force is passed on the CFRP at the bottom of RC beam. All the maximum stress in reinforcing bars happens in the middle of the beam span as shown in Figure 7 and Table 4 .

As shown from Figure 7 and Table 4 , the maximum stress in one-layer FRP is 1847.55MPa, which is more than that of the two FRP layers. This is because most of the stress is carried by the first layer before the contribution of two layers start. Moreover, the maximum value of Misses stress also occurs at the center of the CFRP for all the four cases.

As shown from Figures 8-11 , the tension stress is distributed at the bottom of beam, while the compression stress is distributed at the top of beam.

Once the maximum tension stress of middle span reaches the tensile strength of concrete material, the concrete beam will fracture along top direction from bottom at the middle span gradually, where the maximum tension stress is suddenly dropped while the compression stress at the top of beam is suddenly increased much more.

From the load-deflection curves shown as Figure 12 , the results show that the FRP layer and thickness has greater effect for increasing the load caring capacity of beams with increasing of 60%, 40%, and 30% for three, two, and one layer, respectively, compared with those without strengthening. Accordingly, the deflection increases with an increase of load applied at mid span of the beam, while in the yield and fracturing stage of concrete it decreases with the increase of CFRP layer number. In the elastic stage, the deformation is almost the same for the different cases with and without CFRP. Once reaching the initial bearing capacity of RC beam, the plastic flow happens and holds for some seconds until the bearing capacity is regained due to the bearing action of CFRP.

By setting the failure parameters of concrete material, the generated cracks can be simulated. The parameters and coefficients used in FE simulation are: open shear transfer coefficient ${\beta }_t=0.5$, the closed shear transfer coefficient ${\beta }_c=0.9$, uniaxial cracking stress $f_t=2.5MPa$, uniaxial crushing stress $f_c=25MPa$ and biaxial crushing stress $f_b=30MPa$. And when the hydrostatic pressure is ${\sigma }_m=20MPa$, the hydro biaxial crushing stress $f{\text{'}}_b=40MPa$, the hydro uniaxial crushing stress $f{\text{'}}_c=35MPa$. The tensile crack factor $R^t=0.5$. The fracturing results with the increase of load or time are as Figure 13 .

From crack distribution occurred, the cracks first happened in the middle of span and gradually developed along two sides. As load increases, the cracks number and width are increased and distributed along the bottom of the beams which lead to failure.

The first crack at the integration points is as presented in Figure 14 . From this figure, the first cracks developed from the bottom of mid-span diagonally to the top end, then gradually continued to the beam sides.

The first cracks at integration points for beam strengthened with two layers FRP are presented in Figure 15 .

The first cracks at integration points for beam strengthened with three layers FRP are presented in Figure 16 .

According to the results blotted in the above figures, the first cracks appeared at mid-span in all specimens and then propagated to other beam chords according to load increasing and strengthening. During the first loading steps, no cracks appeared at a small force, and the initial flexural cracks were distributed at a length of 15 mm right and left sides from beam loading point during step 20,142 s (20.142 kN) in specimens with one-layer CFRP. In addition, beams with two and three layers CFRP have the same crack width at initial load step 20,642 s. The maximum displacements of these beams in the initial steps donated were less than the yield displacements, which resulted in small cracks. Moreover, the beams that had been strengthened with CFRPs exhibited elastic behavior under applying loads. The entire beam failed due to the flexural of the longitudinally reinforcing. As shown from control beam, when the applied load increased, the crack had increased and covered about 85% of the spacemen length with corresponding load of 57.039 kN. However, the beams strengthened with two and three layers of CFRP show more crack distribution with high capacity compared with control beam, this due high contribution of CFRP. Furthermore, the beams with three layers of CFRP have greater capacity compared to other specimens, as shown in Figure 16 which the crack was covers about 92.5% of beam length with load step 62 kN prior to failure. In addition, this beam has failed at a load of 71.719 kN with crack distributed on 97.5 of beam sections. According to the measurements made during the simulation, the diagonal cracks in control beam, which was reinforced with ordinary steel, appeared at top of beam at load step 57.039 kN. In addition, the diagonal cracks appeared at top of beams strengthened with one, two and three layers of CFRP measured at load step 55.486 kN, 61.356 kN and 65.257 kN respectively. The crack damage of all the specimens at the end of the simulations occurred due to concrete cover spalled off and longitudinal reinforced steel was yielded and CFRP was ruptured without debonding because it modeled contacted with concrete as full bond. In conclusion, the bearing capacity improved with the increasing of CFRP layers, and the deflection decreased with the increasing of CFRP layers. As shown from Figures 13-15 , the cracks appeared early in beams with low strengthening. Finaly, it concluded that the presence of FRP has a greater effect on beam capacity and the CFRP layer and thickness play an important issue for failure load of strengthened beams compared with control beam.