*4.2. A New Model for the Deflection of the BFRP-RC Beams with Steel Fibers*

As elaborated above, when calculating the deflections of the BFRP-RC beams with steel fibers, the effect of steel fibers on the deflection should be considered. However, the influences of steel fibers on the deflection of a BFRP-RC beam with steel fibers should be considered after concrete cracking [19]. The distance from the center of the mass of a fiber to the neutral axis of the beam cannot be calculated, resulting in the inability to obtain its area and moment of inertia. Rather, some scholars believe that the steel fibers in a beam section can be taken as a whole, which can obtain its area and moment of inertia [19]. The distribution and orientation of steel fibers dictate the concrete's performance, especially at the post-cracking stage. Generally, the distribution of steel fibers is described by the non-uniformity coefficient *ηv*, while the orientation is by the orientation coefficient *η*0. Zhu [19] recommends that the total area of steel fibers in an SFRC beam is obtained by the following equations:

$$A\_{sf} = \eta\_0 \eta\_v bh \rho\_{sf} = \eta bh \rho\_{sf} \tag{15}$$

where *η* is the steel fiber effective coefficient (in this research, *η* was taken as 0.16).

Figure 13a depicts the gross section and transformed an uncracked section of a beam. Since the area moments of the compression and tension zones of the beam are equal, Equations (16) and (17) can be derived as follows:

$$\frac{h\mathbf{x}\_0^2}{2} + \frac{(n\_{sf} - 1)b\mathbf{x}\_0^2 A\_{sf}}{2h} = \frac{b(h - \mathbf{x}\_0)^2}{2} + (n\_{sf} - 1)A\_{sf}\frac{(h - \mathbf{x}\_0)^2}{2h} + (n\_f - 1)A\_f(d - \mathbf{x}\_0) \tag{16}$$

$$\alpha\_0 = \frac{\frac{bh^2}{2} + (n\_f - 1)A\_f d + \frac{(n\_{sf} - 1)A\_{sf}h}{2h}}{bh + (n\_f - 1)A\_f + (n\_{sf} - 1)A\_{sf}} \tag{17}$$

$$m\_{sf} = \frac{E\_{sf}}{E\_c} \tag{18}$$

The moment of inertia of the gross section (*Ig*) of a BFRP-RC beam with steel fibers is calculated by the following equation.

$$I\_{\mathcal{S}} = \frac{b}{3} \left[ \mathbf{x}\_0^3 + (h - \mathbf{x}\_0)^3 \right] + (n\_f - 1) A\_f (d - \mathbf{x}\_0)^2 + \frac{(n\_{sf} - 1) A\_{sf}}{3h} \left[ \mathbf{x}\_0^3 + (h - \mathbf{x}\_0)^3 \right] \tag{19}$$

Figure 13b describes the cracked and transformed cracked sections of a beam. Since the area moments of the compression and tension zones of the beam are equal, Equations (20) and (21) can be derived as follows:

$$\frac{n\_{\rm xcr}^2}{2} + \frac{(n\_{sf} - 1)bx\_{cr}^2 A\_{sf}}{2h} = n\_{sf}A\_{sf}(d - x\_{cr}) + \frac{n\_{sf}(h - x\_{cr})^2 A\_{sf}}{2h} \tag{20}$$

$$\mathbf{x}\_{\text{cf}} = \frac{-\left(\mathbf{n}\_{sf}A\_{sf} + \mathbf{n}\_{f}A\_{f}\right) + \sqrt{\left(\mathbf{n}\_{sf}A\_{sf} + \mathbf{n}\_{f}A\_{f}\right)^{2} + 2\left(\mathbf{b} - \frac{\mathbf{A}\_{sf}}{\hbar}\right)\left(\frac{\mathbf{n}\_{sf}}{2}\hbar \mathbf{A}\_{sf} + \mathbf{n}\_{f}A\_{f}d\right)}{\mathbf{b} - \frac{\mathbf{A}\_{sf}}{\hbar}} \tag{21}$$

The moment of inertia of the cracked section (*Icr*) of a BFRP-RC beam with steel fibers is calculated by the following equation.

$$I\_{cr} = \frac{b}{3} \mathbf{x}\_{cr}^3 + n\_f A\_f (d - \mathbf{x}\_{cr})^2 + \frac{n\_{sf} A\_{sf}}{3h} (h - \mathbf{x}\_{cr})^3 \tag{22}$$

After the moment of inertia of the gross and cracked section of a BFRP-RC beam with steel fibers was obtained, the deflection of the beam can be calculated by introducing Equations (13) and (14).

**Figure 13.** Sectional parameters of the gross and cracked sections: (**a**) gross section; (**b**) cracked section. "筋" = steel.

According to the loading regime adopted in this research, the deformation of the beam can be checked according to the static loading before the stroke of the actuator reached 6 mm, but the cyclic loading effect shall be considered after the actuator's stroke reached 6 mm. Table 9 lists the deflections of all beams investigated in this research when the stroke of the actuator reached 6 mm from experiment and calculation from various models in literature and design codes, as well as the new analytical model established in this study. From Table 9, the calculated deflections from the Bischoff, Benmokrane, Alsayed, and Canadian ISIS models are 13%~49% higher than the experimental value. While the average deflections calculated by the proposed analytical model in this paper are only 9% higher than the experimental ones, and the coefficient of variation is only 0.22, which is lower than the coefficient of variation of any other existing model investigated. Therefore, the analytical model proposed in this paper is more reliable and accurate for evaluating the deflection of the BFRP-RC beams with steel fibers.

**Table 9.** Deflections of the beams from the experiment and calculated from the proposed analytical momodelhen the stroke of the actuator reached 6 mm.


Note: *F*<sup>1</sup> is the applied load on the beams when the stroke of the actuator reached 6 mm; Δ<sup>1</sup> is the deflection of the beams when the stroke of the actuator reached 6 mm; Δ*<sup>c</sup>* is the deflection of the beams calculated from the analytical model established in this paper; Δ*Benm.* is the deflection of the beams calculated using Benmokrane's model; Δ*Bisch.* is the deflection of the beams calculated using Bischoff's model; Δ*Alsa* is the deflection of the beams calculated from Alsayed's model; Δ*ISIS* is the deflection of the beams calculated using the Canadian ISIS model.

According to the loading regime adopted in this study, the cyclic loading effect shall be considered after the stroke of the actuator reached 6 mm. In sum, the deflection of a beam under a certain load can be calculated by using Equations (13), (14), (19) and (22). By introducing the calculated results into Equation (1), the deflection of the beam under cyclic loading can be calculated. Table 10 compares the calculated deflection and the actual deflection of the beams after three loading and unloading cycles. From Table 10, it can be seen that the ratio between the calculated value from the model to the counterpart from the experiment was 0.99, and the coefficient of variation was 0.16, suggesting that the analytical model proposed in this paper can accurately evaluate the deflection of the BFRP-RC beams with steel fibers under cyclic loading.


**Table 10.** Deflections of the beams were obtained from the experiment and calculated from the proposed analytical model after three loading and unloading cycles.

Note: Δ*cn* is the deflection of the beams calculated from the analytical model established in this paper after three loading and unloading cycles.
