## 3 Numerical Modeling  
The simulations were based on the experimental work by [PERSON] et al. [31]. The commercial FE software package ANSYS [32] was used to develop FE models. Thus, the geometries of the FE models were designed to be the same as the geometries of the specimens used in the previous work [31]. The representative geometry and FE meshes used for the simulations are shown in Figure 3. Half of the geometry of the RC beam reinforced by the Fe-SMAs and rebars was modeled based on the symmetry of the geometry, i.e., the \(y\)-\(z\) plane of the FE model was considered as the plane of symmetry where \(x\)-direction movements of the FE nodes were constrained during the simulations. The averaged FE mesh size was 25 mm. The bending deformation of the models was simulated by applying \(z\)-axis displacement on the top of the model where pressure was applied experimentally, while the \(z\)-axis movement of the supported position was fixed. Two types of FE elements of SOLID65 and SOLID185 of ANSYS were used in the simulations. The SOLID65 elements [32] used for the concrete material are the 8-noded brick elements with three translational degrees of freedom per node with capability of cracking in tensional load and crushing under compression. The Fe-SMA bars and the rebars were modeled using the 3D structural 8-noded elements of the SOLID185. Perfect bonding between the concrete and the Fe-SMA bars as well as between the concrete and the rebars were assumed throughout the FE simulations. Figure 2: Photos of (**a**) Fe-SMA bars used for the experiments, (**b**) Fe-SMA and stirrup assembly, (**c**) molding for concrete casting and (**d**) concrete casting process. In order to simulate the cracking and non-linear response of the concrete, the constitutive model was based on the theory of [PERSON] and [PERSON] [33]. The concrete model used in this study requires five input parameters of uniaxial tensile strength (\(f_{t}\)), uniaxial compressive strength (\(f^{\prime}_{c}\)), compressive strength for a state of biaxial compression superimposed on hydrostatic stress state (\(f_{2}\)) and biaxial compressive strength (\(f_{cb}\)). In this study, the concrete was modeled with \(f^{\prime}_{c}\) and \(f_{t}\) of 46.2 and 4.28 MPa, respectively, based on the experiments. The other three parameters of \(f_{cb},f_{1}\) and \(f_{2}\) were set based on the original model by [PERSON] and [PERSON] [34] as \(1.2f^{\prime}_{c}\), \(1.45f^{\prime}_{c}\) and \(1.725f^{\prime}_{c}\), respectively. Moreover, the values for the open and closed shear crack coefficients of 0.3 and 0.5, respectively, were used for the simulations. The non-linear compressive stress-strain response of the concrete under uniaxial compression was modeled using the relation proposed by [PERSON] et al. [35], as in Equation (1) and shown in Figure 3a. \[f_{c}={f^{\prime}}_{c}\left[2\left(\frac{\varepsilon}{\varepsilon_{0}}\right) -\left(\frac{\varepsilon}{\varepsilon_{0}}\right)^{2}\right], \tag{1}\]  
where \(f_{c}\) is the compressive stress in the concrete at the corresponding strain of \(\varepsilon\). \(\varepsilon_{0}\) of Equation (1), which is given as follows:  
\[\varepsilon_{0}=\frac{2f_{c}^{\prime}}{E_{c}}, \tag{2}\]  
where \(E_{c}\) is the elastic modulus of the concrete, which was set to 32 GPa in the FE simulations. The behavior of the concrete in tension was modeled as linear elastic up to the tensile strength of concrete. When the tensile stress of the concrete element reached its tensile strength, the stress relaxation by the cracking was represented by a stepwise drop in the concrete tensile stress by 40%. Then, the curve descended linearly to zero tensile stress at a strain value 6 times higher than the strain where the tensile crack started to appear, similar to a previous work done by [PERSON] [35]. For the Fe-SMA bars, an experimentally determined elastic modulus of 150 GPa [31] with Poisson's ratio of 0.33 was used.