## 3 Finite Element Simulation  
### Finite Element Model  
Generally, a FE model to simulate the nonlinear behavior of RC structures consists of a three-dimensional solid element and a nonlinear material model for the in-elasticity of  
Figure 1: Details of experimental specimen: (**a**) SL-CTRL; (**b**) SL-30S-4; (**c**) SL-60S-4; (**d**) SL-90S-4 [18].  
Figure 2: Stress-strain curve of Fe-SMA with recovery stress [22].  
the constituent materials. As mentioned before, an FE model using the 3D solid element takes a longer time to build lots of nodes and elements, and requires high computational time for the complex analysis. To reduce this time, [PERSON] et al. [23] proposed a beam element FE model using 2D fiber elements. FE models using 2D fiber elements are used by many researchers because they can accurately simulate the nonlinear flexural behavior of the RC member, while reducing running time by building the model with the beam element [24; 25; 26].  
Therefore, in this study, the nonlinear analyses of the RC beams reinforced by Fe-SMA strips were modeled with the 2D fiber element and the non-linear beam-column element, and were simulated with OpenSees, a general computational simulation program for fiber elements.  
Figure 3 shows an FE model for predicting the flexural behavior of the RC beams strengthened with Fe-SMA strips using the NSM method. As shown in Figure 3, the FE model consists of a total of 6 nodes (white letters) and 5 elements (red letters). Each element was modeled with a fiber cross section and nonlinear beam-column element. The mesh size of the cross-section was 10 mm for accurate analysis. In addition, 5 integration points were set and the Gauss-Lobatto integration was used to improve the accuracy of the analysis.  
As shown in Figure 2(a), the cross-section of element 1 was modeled with a concrete and steel reinforcement because the Fe-SMA strip was not embedded on this region. As mentioned before, 90% of the net-span of test specimens was strengthened with the Fe-SMA strip. Elements 2 to 5 were modeled with the concrete, steel reinforcement, and Fe-SMA strip, as shown in Figure 2(b)-d depending on the amount of Fe-SMA strip. [PERSON] et al. [18] suggested that the Fe-SMA strip should be bonded over 600 mm as the effective bond length. As shown in Figure 1, the bond length of Fe-SMA was 1100 mm, which exceeded the effective bond length. Therefore, the FE model assumed that perfect bonding exists between the concrete and the Fe-SMA strip and between the concrete and the tensile reinforcement. The geometry of the test specimens and their loads were symmetric, as shown in Figure 1. To simplify the FE analysis and reduce the running time, only a half-length of the test specimen was modeled, as shown in Figure 3.  
For the simulation of the flexural behavior of the RC beams strengthened with Fe-SMA strip using the NSM method, a displacement control analysis employing the Newton-Raphson method was conducted, to calculate the reaction by applying a vertical displacement to the loading point (node 5 in Figure 3). In the displacement control analysis, the increment of displacement was 0.01 mm per step, and the analyses were performed until the strain of the extreme compression fiber of the concrete reached 0.003 according to ACI 318 [27]. Fundamentally, a shear span-to-effective depth ratio (a/d) greater than 6 for an RC beam means sufficient strength for shear failure [28]. The FE models had a/d values of 6.47, which indicates all of the simulated beams can be expected to fail in the compression-controlled section or the tension-controlled section. Since the Fe-SMA strip is based on steel material, an increase in the amount of Fe-SMA strip can change the failure section from the tension-controlled to the compression-controlled.  
Figure 3: A half of the FE model for RC beams strengthened by Fe-SMA strips: (a) SL-CTRL; (b) SL-30S-4; (c) SL-60S-4; (d) SL-90S-4.