*4.1. Theoretical Calculation of Deflection of FRP-RC Beams*

To simplify the analysis, the following assumptions were taken when evaluating the deflection of FRP-RC beams.

(1) A beam section is homogeneous before concrete cracking, and the contribution of the BFRP bars to the total moment of inertia of a beam section is neglected. Therefore, the total moment of inertia (*Ig*) can be obtained by the following equation.

$$I\_{\mathcal{S}} = \frac{bh^3}{12} \tag{6}$$

(2) After a crack is initiated in concrete, the contribution of the concrete in the tension zone is neglected. Therefore, the moment of inertia (*Icr*) of the cracked beam section can be obtained by the following equation.

$$I\_{cr} = \frac{b}{3} d^3 k^3 + n\_f A\_f d^2 (1 - k)^2 \tag{7}$$

$$k = \sqrt{2\rho\_f n\_f + \left(\rho\_f n\_f\right)^2} - \rho\_f n\_f \tag{8}$$

$$m\_f = \frac{E\_f}{E\_c} \tag{9}$$

where *d* is the effective depth of the beam section, *k* is the ratio of the depth of the neutral axis to the depth of reinforcement bars, *nf* is the ratio of Young's modulus of FRP bars to the modulus of elasticity of concrete, *Ec* is the Young's modulus of concrete, *Ef* is the Young's modulus of FRP bars, and *ρ<sup>f</sup>* is the FRP reinforcement ratio.

Currently, there are various calculation models for the effective moment of inertia (*Ie*) of an FRP-RC beam section. Bischoff [38,39] recommended that the effective moment of inertia (*Ie,bischoff*) of an FRP-RC beam section can be obtained by Equation (10).

$$I\_{\varepsilon,b\text{ishoff}} = \frac{I\_{\varepsilon r}}{1 - \left(1 - \frac{I\_{\varepsilon r}}{I\_{\varepsilon}}\right) \left(\frac{M\_{\text{cr}}}{M\_{\text{cr}}}\right)^2} \tag{10}$$

where *Mcr* is the cracking moment, and *Ma* is the applied moment.

According to Benmokrane et al. [40], the effective moment of inertia (*Ie,benmokrane*) of an FRP-RC beam section can be evaluated by Equation (11).

$$I\_{c, \text{hemuokrante}} = \left(\frac{M\_{cr}}{M\_a}\right)^3 \frac{I\_\mathcal{g}}{7} + 0.84 \left[1 - \left(\frac{M\_{cr}}{M\_a}\right)^3\right] I\_{cr} \le I\_\mathcal{g} \tag{11}$$

Alsayed et al. [41] proposed that the effective moment of inertia (*Ie,alsayed*) of an FRP-RC beam section can be evaluated by Equation (12).

$$\begin{aligned} I\_{\mathfrak{c},\text{slayed}} &= \left( 1.4 - \frac{2}{15} \left( \frac{M\_{\mathfrak{s}}}{M\_{\text{cr}}} \right) \right) I\_{\text{cr}} & \quad for \quad 1 < \frac{M\_{\mathfrak{s}}}{M\_{\text{cr}}} < 3\\ I\_{\mathfrak{c},\text{slayed}} &= I\_{\text{cr}} & \quad for \quad 3 < \frac{M\_{\mathfrak{s}}}{M\_{\text{cr}}} \end{aligned} \tag{12}$$

Canadian ISIS [42] code recommends that the effective moment of inertia (IISIS) of an FRP-RC beam section can be evaluated by Equation (13).

$$I\_{\xi} = \frac{I\_{\xi} I\_{cr}}{I\_{cr} + \left[1 - 0.5 \left(\frac{M\_{cr}}{M\_d}\right)^2\right] \left(I\_{\xi} - I\_{cr}\right)}\tag{13}$$

Combined with classical beam theory and the effective moment of inertia method, the mid-span deflection (Δ) of an FRP-RC beam can be obtained by Equation (14).

$$
\Delta = \frac{pl\_a}{48E\_cI\_c} (3l\_o^2 - 4l\_a^2) \tag{14}
$$

where *p* is the applied load, *la* is the shear span, and *lo* is the clear span.

Using the various effective moment of inertia formulas in literature and design codes summarized above, combined with the deflection calculation method from the classical beam theory, the deflection of a BFRP-RC beam with steel fibers at 0.5 mm crack width can be obtained. However, none of the above analytical models for the effective moment of inertia considers the positive contribution of steel fibers to the moment of inertia of the section of a beam strengthened with BFRP bars. Indeed, for a BFRP-RC beam with steel fibers, the contribution of steel fibers in the concrete tensile zone cannot be neglected, because steel fibers reinforced concrete can bear large tensile stress after concrete cracking.
