## 2 Finite Element Modeling and Validation
### Establishment of the Finite Element Model
#### 2.2.3 Modeling Details and Parameter Settings  
During modeling, it is assumed that no bond slip occurs at the steel-concrete interface. Rigid pads are set at the supports and loading points. The nodes at the mid-lines of the bottoms of the rigid pads at the lower parts of the left and right sides of the beam are restrained in 5 degrees of freedom (U\({}_{1}=0\), U\({}_{2}=0\), U\({}_{3}=0\), U\({}_{\rm R1}=0\), U\({}_{\rm R2}=0\)) and 4 degrees of freedom (U\({}_{2}=0\), U\({}_{3}=0\), U\({}_{\rm R1}=0\), U\({}_{\rm R2}=0\)), respectively, and the deflection-controlled loading mode is adopted. An embedded constraint is used between the steel bars and the concrete; a Tie constraint is used between the pads and the concrete. The rigid pads and the concrete are divided at intervals of 30 mm. The PCM with relatively small dimensions and the bonding layer are divided at intervals of 15 mm. To more clearly demonstrate the stress of the FRP grid, it is divided at intervals of 10 mm. The grid checking function built into ABAQUS shows that the meshing of the simulated beam is reasonable and there are no incorrect elements. The finite element model of the strengthened beam is shown in Figure 5. The concrete, FRP grid, PCM, and pads are all simulated by three-dimensional solid elements with the element type C3D8R; steel bars are simulated by truss elements with the element type T3D2; and a Cohesive Element is used between the concrete and the strengthening layer with the element type COH3D8 (an eight-node three-dimensional bonding element). Figure 5: Finite element model of the reinforced beam. Both the test beams and the simulated beams suffered shear-compression failure. As can be seen from Figure 6, the equivalent plastic strain of the simulated beam is in good agreement with the distribution and propagation of cracks on the beam surface. Compared with the unstrengthened beam C20, the equivalent plastic tensile strain area in the shear span zone of the strengthened beam is more concentrated and smaller in area. This is because the presence of the FRP grid increases the number of cracks and reduces the crack width and extension height. Therefore, the externally bonded FRP grid-PCM composite layer can effectively inhibit the formation and propagation of cracks during the shear process of RC beams. The strengthened beams with smaller shear span ratios have higher shear resistance and deformation capacity, and there are more diagonal cracks in the shear span zone. Therefore, the corresponding equivalent plastic strain values in the figure are larger and more concentrated in range. Before the final failure of the beam, diagonal shear cracks appear and develop, the equivalent plastic strain of the concrete in the shear span zone increases, and the bonding capacity with the strengthening layer is weakened. These factors jointly lead to the descending section of the load-midspan deflection curve of the strengthened beam. It can also be seen from Figure 7 that the curve shapes of the four beams are generally consistent with the test results. Compared with the unstrengthened beam, the shear behavior capacities of the strengthened beams are increased to varying degrees, indicating that pasting the FRP grid-PCM composite strengthening layer in the shear span zone can effectively improve the capacity of RC beams. Table 2 shows the comparison between the test results and the finite element simulation results. As the shear span ratio increases, the  
Figure 7: Load–midspan deflection curve. increases in the ultimate loads of beams S16-P-G-O, S20-P-G-O, and S24-P-G-O compared with that of C20 gradually decrease. This is because as the distance from the support to the loading point increases, the internal force arm of the bending moment increases, the stress state on the diagonal section of the shear span zone changes, and the compressive stress of the concrete in the shear span zone decreases while the tensile stress increases, resulting in a decrease in the bearing capacity of the beam.