**1. Introduction**

The corrosion of steel bars in RC structures shortens the service life of RC structures and significantly increases maintenance costs. Over the past decades, fibre-reinforced polymer (FRP) bars have been used extensively in the construction industry as a new type of reinforcement that replaces steel bars to solve corrosion problems [1,2]. Compared with steel bars, FRP bars are corrosion-free, magnetically transparent, and lighter but with higher tensile strength [3]. These important features enable FRP-RC structures to withstand various complex and corrosive environments with desirable performances. Based on the raw materials used for manufacturing FRPs, FRPs are divided into four categories, including basalt-fiber-reinforced plastic (BFRP), aramid-fiber-reinforced plastic (AFRP), glass-fiber-reinforced plastic (GFRP) and carbon-fiber-reinforced plastic (CFRP). CFRP has a higher Young's modulus and tensile strength than any other FRP, but its high price hinders its wider applications in construction. Although GFRP and AFRP are less expensive, their alkali resistance is poor, leading to a large degree of strength degradation when reinforcing concrete with alkalinity of pH 12~13. In this regard, BFRP is now used in more

**Citation:** Zhu, H.; Li, Z.; Chen, Q.; Cheng, S.; Li, C.; Zhou, X. A New Analytical Model for Deflection of Concrete Beams Reinforced by BFRP Bars and Steel Fibres under Cyclic Loading. *Polymers* **2022**, *14*, 1797. https://doi.org/10.3390/ polym14091797

Academic Editors: Wei Wu, Hao-Yang Mi, Chongxing Huang, Hui Zhao and Tao Liu

Received: 13 April 2022 Accepted: 26 April 2022 Published: 28 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

applications in construction due to its relatively low cost, good thermal resistance, and excellent freeze/thaw resistance [4]. At the same time, the bond performance between BFRP bars and concrete is also better than that between GFRP bars and concrete. More importantly, BFRP is made of volcanic basalt, which is widely recognized as a type of green construction material with high sustainability credentials; therefore, BFRP has greater application prospects [5]. However, the Young's modulus of FRP bars is lower than that of steel bars, which leads to FRP-RC structures usually possessing larger crack widths and deflection than RC structures [6–8]. In addition, FRP-RC structures are prone to brittle failure because FRP bars are usually brittle while steel bars are ductile.

To solve the above problems, researchers have proposed various ways to improve the structural performance of FRP-RC beams, which include composite reinforcement composed of steel core and FRP to reinforce RC beams [9], increasing the transverse reinforcement ratio for FRP-RC beams [10], grouting FRP bars in corrugated sleeves to reinforce RC beams [11] and adding fibers as additional reinforcement into FRP-RC beams [12]. Previous studies have demonstrated that fibers can improve ductility and strain-hardening of cementitious composite mortars and grouts [13]. Adding discrete fibers into a concrete matrix is the most effective way to improve the serviceability performances of FRP-RC structures. More importantly, the bridging effect of fibers leads to the pseudo-ductile behavior of FRP-RC structures [14–29]. Chellapandian et al. [14] investigated the effects of adding macro-synthetic fibers into concrete on the cracking, stiffness, and deformability of GFRP-RC beams. They concluded that the fibers improved the post-cracking behavior of GFRP-RC beams with their deformation largely enhanced by adding only 1% by volume of steel fibers, which also transformed the GFRP-RC beams from brittle flexure–shear failure to ductile flexural failure with higher pseudo-ductility. Filipe et al. [15] found that the use of fibers enhanced the stiffness of FRP-RC members and helped to reduce crack spacing and width. The same findings were also obtained by other researchers [16,17]. Ibrahim and Eswari [18] investigated the strength and ductility of GFRP = laminated RC beams incorporated with various amounts of discrete steel fibers. Their study revealed that incorporating steel fibers can effectively improve the strength and ductility of FRP-RC beams. Issa et al. [19] concluded that polypropylene fibers, glass fibers, and steel fibers all improved the ductility of FRP-RC beams, in particular, in the case of steel fibers, which increased the ductility of the beams by 277.8%. Short discrete fibers not only improved the tensile properties of concrete but also improved the shear capacity of concrete beams [20–23]. Zhu et al. [24] studied the effects of partially steel fibers reinforced concrete (SFRC) on the flexural behavior of FRP-RC beams. The results suggest that compared with full section SFRC beams, partially reinforced SFRC beams cannot provide a better performance, and steel fibers helped to reduce the deflection of the FRP-RC structures. Similar findings were also reported by other researchers. In summary, previous studies have systematically studied the flexural behaviors of FRC beams strengthened with FRP bars under static loading, including crack behaviors, ductility, deflection, and ultimate moment capacity.

However, for practical purposes, RC beams always bear cyclic loading rather than static loading [30]. The flexural behaviors of RC beams under static loading are different from those under cyclic loading, so it is imperative to investigate the effects of the deterioration of concrete and FRP bars on the flexural performances of FRP-RC beams under cyclic loading. Zhu et al. [27] studied the influence of steel fiber on the bearing capacity of the BFRP-RC beams, and the results showed that steel fiber can help to increase the ultimate compressive strain of concrete so as to increase the bearing capacity of the BFRP reinforced concrete beams, and the calculation method of flexural capacity of the BFRP bars and steel-fiber-reinforced concrete beams was established. Li et al. [28,29] analyzed the influence of steel fibers on crack width and ductility of the BFRP-RC beams; the results showed that steel fibers were beneficial in reducing crack width and increasing the ductility of the BFRP-RC beams. The calculation method of maximum crack width and the evaluation method of ductility were put forward. However, the research on the dRC beams with steel fibers under cyclic load is still rare in literature.

The purpose of this research is to study the volume fraction and type of steel fibers on the deformation and flexural behavior of the BFRP-RC beams. In addition, the effects of concrete strength and BFRP reinforcement ratio on the deformation and flexural behavior of the BFRP-RC beams were also investigated. Ten beams were prepared and loaded via four-point bending under cyclic loading until failure, which included nine beams reinforced by both BFRP bars and steel fibers and one BFRP-RC beam without steel fibers as a reference case. The responses of the beams under cyclic loading were compared and analyzed from the aspects of failure mode, ultimate moment capacity, service load moment, load-deflection relation, envelop curves, residual deflection, and stiffness degradation. In addition, a new analytical model for the deformation of the BFRP-RC beams with steel fibers was proposed. Compared with other models, the newly proposed model in this paper accords better to experimental results.
