# Analytical Study of Reinforced Concrete Beams Tested under Quasi-Static and Impact Loadings
## 3 Finite Element Model (FEM)  
ABAQUS FEA software, a commercially available finite-element analysis program, was used to model the RC beams with and without GFRP strengthening schemes. For concrete modeling, 3D eight-node linear isoparametric elements with reduced integration (C3D8R) were utilized. For longitudinal and transverse bars, truss elements (T3D2) were applied. FRP were modeled using shell elements (S4R). Embedded region coupling was utilized to simulate the bond between longitudinal and transverse reinforcements with concrete. This coupling enables the user to identify one region as the host, and another one as embedded. In this paper, reinforcements represented the embedded region, and concrete was the host region [25]. In order to prevent the scattering of results, coupling restraint was applied. This coupling restraint is formed through a reference point (RP) situated at the center of the support on the bottom of the beam. For impact, the loading is exerted via a hammer at the mid-span of the beam, and a face-to-face constraint was utilized. Therefore, one could obtain force displacement results at one point with minimal errors. Since the beams were simply supported, U1, U2, and U3 degrees of freedom were set to zero. The arrangement of reinforcing bars, RC beams, supports, and hammer in the software are presented in Figure 5. It is noteworthy that the analyses conducted for these models were quasi-static and dynamic. The BS and BS-GFRP samples shown in Table 5 and all samples mentioned in Table 4 were modeled using quasi-static loading, and BI samples in Table 5 were modeled using dynamic/impact analyses. Dynamic explicit analysis was employed for the impact tests, while dynamic implicit analysis was used for quasi-static analyses. To simulate the behavior of steel reinforcing bars in ABAQUS under a high strain rate, a linear kinematic hardening model was used [25]. In order to model the behavior of confined concrete, a concrete damage plasticity (CDP) model was utilized based on the work of [PERSON] et al. [26]. laboratory and software results are negligible except for BI-400. It can be inferred that a minimum drop height of around 500 mm is required to make the RC beam fail, which was not met for BI-400 as its drop height was only 400 mm. This is the most probable cause of the 20% difference. Comparing the results, the beam BS-GFRP that is strengthened with GFRP and tested under quasi-static loading showed a 29.3% increase in load-carrying capacity in the laboratory experiment and a 30.0% increase in the software model. Figures 14-22 present the load vs. mid-span displacement curves under quasi-static loading based on Table 4. Results from the FEM analysis are in very good agreement with the experimental results, with a maximum difference of 5%. In effect, the FEM analysis is seen to accurately predict the rate of load-carrying capacity vs. displacement before and after the maximum load. The load-displacement curve can be idealized by a bilinear graph similar to Figure 23. The stiffness of the beam can then be calculated using Equation (1): [27]  
\[\text{Stiffness}=\text{V}_{\text{y}}/\Delta_{\text{y}}. \tag{1}\]  
The load-displacement curve of C-NS beam is idealized in Figure 24, and the stiffness was calculated. Following the same procedure, the stiffness of all beams was calculated, and the values are illustrated in Figures 25 and 26. Figure 21: Load vs. mid-span deflection for B2-6B-S-1. Figure 23: Load vs. displacement bilinear graph [27]. Figure 22: Load vs. mid-span deflection for B3-S-2. As shown in Figure 25, it is apparent that the stiffness of the beam under impact loading increased with respect to its stiffness under quasi-static loading. Also, increasing the impact velocity reduced the stiffness (the stiffness increased from BS to BI-400 and then decreased from BI-400 to BI-2000). The same trend was observed in FEM as well as experimental results. Fabric GFRP increased the stiffness of the beam under both quasi-static and impact loadings.