*3.2. Load-Deflection Curve and Envelope Curve*

Load-deflection curves of beams under cyclic loading were commonly used for examining their flexural behaviours, from which the envelope curve, energy dissipation, residual deflection, stiffness, etc., was able to be derived. The envelope curve was the curve connecting the peak load of all cycles in the load-deflection curve of a beam under cyclic loading. The enclosed area in the load-deflection curve after the unloading–reloading cycle represented the energy dissipation of the beam under this unloading–reloading cycle. The residual deflection was defined as the irrecoverable deflection of a beam after the load was unloaded to 0. The load degradation coefficient meant the reduction coefficient of the peak load at the same displacement in different unloading–reloading cycles. Figure 7 presents the load-deflection curves of all beams tested in this research. The red curves, blue curves, and pink curves indicate the first cycle, the second cycle, and the third cycle envelope curves, respectively, of the load-deflection curves. The envelope curve was also an important index for studying the flexural performance of a beam under cyclic loading. From Figure 7, it is obvious that the load-deflection curves of all beams demonstrate the identical characteristics, i.e., all load-deflection curves increased linearly after cracking; the residual deflection increased with unloading–reloading cycles, especially at larger deflection; the peak load and energy consumption of the beam under the same deflection decreased gradually with the increase of loading–unloading cycles. Figure 8 reproduces the first cycle envelope curves for all beams.

**Figure 7.** *Cont*.

**Figure 7.** Load-deflection curves for all beams tested: (**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**) B1.15C60V1.0S4; (**i**) B1.15C60V1.0S5; (**j**) B1.15C30V1.0S3.

**Figure 8.** Envelope curve (first cycle) for beams with different: (**a**) BFRP reinforcement ratio; (**b**) steelfiber volume fraction; (**c**) steel fiber shape; and (**d**) concrete strength.

3.2.1. Number of Unloading–Reloading Cycles

From Figure 7, it can be found that unloading–reloading cycles at the same stroke displacement significantly reduced the peak load of a beam. However, the peak load reduction rate decreased with the increase of unloading–reloading cycles. According to the experimental results, the average peak load of the second cycle was 3%~12% lower than that of the first cycle, while the average peak load of the third cycle was only 1%~5.38% lower than that of the second cycle. This was due to greater damage to the beams caused by the increase in loading in the first cycle, leading to the increase in crack width and height, the decrease of the effective area of concrete, and hence the reduction of stiffness. The peak load of the second and third unloading–reloading cycles was lower than that of the first unloading–reloading cycle. But the width and height of cracks after the second and the third unloading–reloading cycles were comparable to those after the first unloading– reloading cycle. The decrease in stiffness was only related to the internal damage and bond between concrete and BFRP bars. Therefore, the reduction rate of peak load was decreased with the increase of unloading–reloading cycles. For example, for beam B1.15C60V1.0S3 with a stroke displacement of 6 mm, the peak load degradation coefficient after the second and the third unloading–reloading cycles were 7.12% and 1.08%, respectively.

More importantly, the deflection of all beams increased with the increase of the number of unloading–reloading cycles under the same applied load. Table 8 shows the deflections of the beams at the first cycle and the deflections after three unloading–reloading cycles under the same applied load. It can be seen from Table 8 that after three loading and unloading cycles of the first stage under the same applied load, the deflections of the beams increased by 11% on average, but after three loading and unloading cycles of the second and third stages under the same applied load, the deflections of the beams increased by only 8% on average. The reason was that the skeleton curves of the beams were bilinear; due to the lower elastic modulus of the BFRP bars, the stress of the BFRP bars increased

rapidly after concrete cracking, resulting in a large increase of the deflection after three loading and unloading cycles of the first stage. Therefore, the deflections of the beams under cyclic loading can be calculated by the following formula.

$$
\Delta' = \Delta \times \left(1 + 11\% \right) \times \left(1 + 8\% \right)^{n-1} \tag{3}
$$

where Δ*<sup>n</sup>* is the deflection of a beam after three cycles under cyclic loading, Δ is the deflection of the beam under static loading, and *n* is the loading grade under cyclic loading (*n* ≥ 1).

**Table 8.** Beam's deflections at the first cycle and after three loading–unloading cycles under the same applied load.


Note: Δ*<sup>m</sup>* is the deflection of a beam at the first cycle under the mth loading stage, and Δ*m* is the deflection of the beam after the third unloading–reloading cycle under the mth loading stage.
