*3.4. Stiffness Degradation*

The deflection of the beams increased with the increase of the unloading–reloading cycles under the same applied load, which was called stiffness degradation. According to JGJ/T 101-2015 [37], the stiffness of a beam is expressed by secant stiffness *Kij*, which can be calculated by the following equation.

$$\mathcal{K}\_{ij} = \frac{|+F\_{ij}| + \left| -F'\_{ij} \right|}{|+\Delta\_{ij}| + \left| -\Delta'\_{ij} \right|} \tag{4}$$

where *Fij* represents the peak load of the *jth* cycle under a displacement of *ith*; Δ*ij* represents the largest displacement of the *jth* cycle under a displacement of *ith*; *Fij*' represents the minimum load of the *jth* cycle under a displacement of *ith*; Δ*ij*' represents the residual deflection of the *jth* cycle under a displacement of *ith*; *j* is the number of cycles under a displacement of *ith*, where *j* is less than or equal to 3 in this study.

As the loading mode was cyclic in this research, *Fij*" = 0, Δ*ij* = 0. Therefore, the secant stiffness *Kij* can be simplified as the following equation:

$$K\_{ij} = \frac{|+F\_{ij}|}{|+\Delta\_{ij}|} \tag{5}$$

Figure 11 depicts the stiffness–displacement curves of all beams. The stiffness of the beams decreased with the increase of displacement, and the stiffness degradation rate decreased with the increase of displacement. In particular, the stiffness degradation rate was the highest from the initiation of cracking to an actuator displacement of 6 mm. The stiffness of the beams remained unchanged before cracking. After cracking to an actuator displacement of 6 mm, the crack width and height increased rapidly, and the effective section of a beam decreased accordingly, leading to a higher rate of stiffness degradation. When the actuator displacement reached 6 mm, the crack height of a beam changed little, and the stiffness degradation was small. Noticeably, increasing the number of unloading–reloading cycles decreased the stiffness under the same deflection, but the stiffness degradation rate of beams decreased. The stiffness of the beams in the second cycle was 4.00% lower than in the first cycle under the same deflection, and their stiffness in the third cycle was 1.59% lower than in the second cycle. The main reason for this was that after the first cycle, new cracks appeared, and old cracks further developed, leading to rapid stiffness degradation. The peak load of the second cycle decreased under the same displacement, and no new cracks appeared, which had little effect on the stiffness of the beams.

**Figure 11.** *Cont*.

**Figure 11.** Stiffness–displacement curves: (**a**) B0.56C60V1.0S3; (**b**) B0.77C60V1.0S3; (**c**) B1.15C60V1.0S3; (**d**) B1.65C60V1.0S3; (**e**) B1.15C60; (**f**) B1.15C60V0.5S3; (**g**) B1.15C60V1.5S3; (**h**) B1.15C30V1.0S3; (**i**) B1.15C60V1.0S4; (**j**) B1.15C60V1.0S5.

Figure 12 illustrates stiffness–displacement curves of the beams in the first cycle under different variables. The stiffness had increased with the increase of the BFRP reinforcement ratio, but the stiffness degradation rate decreased. After cracking, the restraint force of the beams with a higher reinforcement ratio on crack width expansion was higher than that of the beams with a lower reinforcement ratio, so the stiffness degradation rate of beams with a higher reinforcement ratio was reduced. The increase of steel-fiber volume fraction and number of hook-ends helped to enhance the stiffness. From Figure 12b,c, it can be found that steel-fiber volume fraction and the number of hook-ends had a significant effect on the stiffness–displacement curve of the beams in the early stage of loading, but the effect became less significant in the later stage. Increasing volume fraction and number of hook-ends of steel fibers was beneficial for improving the tensile strength of concrete, and the random distribution of fibers helped to hinder the further development of cracks, thus reducing the deflection of the beams in the early stage of loading. The effects of fibers on deflection were reduced at the later stage of loading because most steel fibers in the tensile zone were pulled out at the ultimate failure. 5D steel fibers had higher tensile strength and more hook-ends than the 3D and 4D steel fibers. Therefore, the bond strength between concrete and fibers was higher than other steel fibers, which made the stiffness of beam B1.15C60V1.0S5 higher than that of beams B1.15C60V1.0S3 and B1.15C60V1.0S4. The effect of concrete strength on beam stiffness–displacement curves are depicted in Figure 12d. Increasing concrete strength can increase the stiffness of the beams, but it has little effect on the stiffness degradation rate.

**Figure 12.** *Cont*.

**Figure 12.** Stiffness–displacement curves of beams at the first loading–unloading cycle with respect to (**a**) reinforcement ratio; (**b**) steel-fiber volume fraction; (**c**) steel fiber shape; (**d**) concrete strength.

#### **4. Experimental Results versus Model Prediction**

FRP-RC beams usually possess larger deflection than RC beams due to the FRP bars having a lower Young's modulus. In this regard, the serviceability limit states usually control the structural design of FRP-RC beams. Controlling the deformation of FRP-RC beams under cyclic loading is particularly important for design. At present, most studies and design codes use the effective moment of inertia method to evaluate the deflection of FRP-RC beams under static loading, which is also used in this paper to predict and evaluate the deflections of the BFRP-RC beams with steel fibers under cyclic loading. Results from various analytical models/empirical equations were compared with experimental results from this research, through which the analytical model/empirical equations were evaluated for their appropriateness for calculating the deflection of the BFRP-RC beams with steel fibers under cyclic loading. Table 7 summarizes the comparisons between the experimental and theoretical results of the deflection of beams tested at a crack width of 0.5 mm.
