# Analytical Study of Reinforced Concrete Beams Tested under Quasi-Static and Impact Loadings
## 1 Introduction  
Concrete structures are subjected to a variety of different loading conditions in day-to-day situations, with the two most ubiquitous of these conditions being quasi-static and impact loadings. With regard to the former, there exists an abundance of research studying the behavior of reinforced concrete (RC) elements under quasi-static load. Conversely, structures such as high-rise buildings are often exposed to damaging impact forces, including the incidental impact of objects, explosions, sudden collapse of cranes, and unregulated motion of heavy machinery. As such, it is also imperative to study the behavior of RC beams under these dynamic loading conditions. To this end, there have been numerous studies conducted on impact loading and its effects on RC structures, with the first known dynamic test being conducted on concrete in 1917 [1]. After years of inactivity, there has been a resurgence in various impact loading experiments performed on concrete over the past 50 years. Researchers such as [PERSON] and [PERSON] [2], [PERSON] et al. [3], [PERSON] et al. [4], [PERSON] et al. [5], and [PERSON] et al. [6] studied the behavior of RC elements under dynamic conditions and discovered that with incremental increases in loading rate, the ultimate stress and strain of concrete increases by about 25%. Moreover, with respect to the compressive strength of RC elements, experiments carried out by [PERSON] [7], and [PERSON] and [PERSON] [8] suggest that the compressive strength of concrete under dynamic loads increased by 85-100%. However, the existing literature on the bending capacity of RC beams in response to impact loading remains comparatively scarce. [PERSON] et al. conducted experiments on simply supported RC beams [9] and noted that the flexural capacity and rigidity ofRC beams show an incremental increase at high rates of strain. Similarly, [PERSON] et al. [10] ran dynamic experiments on beams under strain rates of 0.001 s\({}^{-1}\) and discovered that the loading capacity of beams increased by 30% in high strain rates. Comparable experiments have also been carried out by [PERSON] et al. [11] to experimentally and analytically evaluate the effect of hammer height and longitudinal reinforcement on the behavior of RC beams, using a mass-spring damper system to simulate the impact response. Furthermore, [PERSON] and [PERSON] [12] performed an accurate numerical simulation (less than 12% average absolute error) using an artificial neural network to predict the response of RC beams to impact loading, with concrete compressive strength, hammer velocity, and mass of hammer being considered as variables. Another important variable--namely, contact stiffness--was studied experimentally and numerically by [PERSON] et al. [13], with results indicating that this variable can significantly influence the peak of load--displacement diagram. Moreover, contact stiffness significantly influenced the bending moment and shear force of the RC beams. Given the advancing age of RC structures, especially those in developed countries, the importance of using advanced materials--such as fiber-reinforced polymers (FRP)--for rehabilitation purposes encompassing both quasi-static and impact loadings should be emphasized. The use of externally bonded FRP as a strengthening technique for RC structures has gained popularity over the past three decades, with a notable surge in its use being observed in recent years. As such, it is necessary to study the behavior of FRP-strengthened RC elements under the aforementioned loading conditions. With respect to quasi-static loading, there has been an increasing trend in the number of studies conducted in recent years [14; 15], establishing the role of externally bonded FRP in augmenting the load-carrying capacity of RC elements.