## 1 Introduction  
The utilization of concrete structures is of crucial significance in modern architectural and infrastructural projects. The factors related to flexural and shear performance have always been considered critical criteria that require particular focus throughout the design and construction phases. Concrete beams that are either simply supported or continuous are frequently used as structural components. These beams are subjected to cyclic loads or are exposed to corrosive conditions over time, which ultimately results in the accumulation of damage and a decline in performance [1]. Extensive research spanning decades has demonstrated that the application of external prestressing techniques not only enhances the structural flexural and shear capacity but also contributes to an improvement in rigidity [2; 3; 4]. Researchers have initiated the evaluation and application of fiber-reinforced polymer (FRP) tendons as a viable and environmentally friendly substitute for traditional steel reinforcement, owing to the notable progress in FRP tendon manufacturing technology [5; 6; 7; 8]. In comparison with traditional reinforcing materials, FRP composites provide a linear elastic response without a distinct yield point. Additionally, these composites offer the advantage of flexible arrangement, possess high specific strength, exhibit exceptional resistance to corrosion, and are non-magnetic [9; 10; 11]. The utilization of carbon fiber-reinforced polymer (CFRP) materials has been extensively employed for the purpose of flexural and shear reinforcement in reinforced concrete beams [12; 13]. In recent decades, researchers have extensively explored the use of CFRP materials for the externally prestressed flexural or shear reinforcement of concrete beams. [PERSON] et al. [14] and [PERSON] et al. [15] investigated the flexural performance of concrete beams reinforced with externally prestressed CFRP tendons. In their study, they compared the results with beams that were strengthened with externally prestressed steel-stranded wire. The modulus of elasticity associated with CFRP materials is comparatively lower than that of steel-stranded wire. This discrepancy results in a discernible reduction in rigidity and an increase in deflection after the occurrence of cracking, thus emphasizing the importance of mid-span deflection in concrete beams that are reinforced with CFRP materials under normal operating conditions. [PERSON] et al. [16] used CFRP strips to provide separated reinforcement to the positive and negative moment regions of continuous beams. Their experimental findings suggest that the beams reinforced in both positive and negative moment regions exhibit a higher ultimate load capacity compared to the beams reinforced only in the negative moment zone. [PERSON] et al. [17] and [PERSON] et al. [18] conducted experimental testing on concrete beams reinforced with CFRP materials, with a focus on investigating the influence of the initial damage level on the test results. Their findings indicate that the initial level of damage had a limited impact on both the ultimate load-bearing capacity and the ductility of concrete beams reinforced through external prestressing. However, a positive correlation was observed between the initial damage level and the mid-span deflection during the loading process. [PERSON] et al. [19] and [PERSON] et al. [20; 21] demonstrated that augmenting the extent of internal flexural reinforcement in concrete beams resulted in a corresponding enhancement of their shear capacity. Hence, it is anticipated that the augmentation of shear capacity through the use of externally prestressed CFRP tendons will have a comparable effect to that achieved by internal reinforcement. [PERSON] et al. [22] and [PERSON] et al. [23] conducted an empirical investigation on the strengthening of reinforced concrete beams in both flexural and flexural-shear conditions with FRP sheets.