**5. Simplified Interface Model**

### *5.1. Simplified Mechanical Model*

According to the analysis of the interface bond mechanism between the CFRP grid– PCM reinforcing layer and concrete, the tensile stress is balanced by the anchorage action of the CFRP grid in the PCM and the interface bond action between the CFRP grid–PCM reinforcing layer and the concrete. Therefore, a simplified model based on a nonlinear spring system was proposed from the perspective of mechanical analysis, as shown in Figure 9. It is clear that the resistant action of the horizontal grid plays a dominant role in the anchorage action of the CFRP grid in the PCM, so the tensile springs and shear-resistant springs are used to simulate the resistant action of the CFRP grid in the PCM and the interface bond action between the reinforcing layer and the concrete, respectively. In order to reduce the complexity of the model, the two categories of springs are equivalent to the same category of spring in terms of stiffness.

**Figure 9.** Schematic diagram of the specimen interface.

In view of the brittle failure characteristics of specimens, a nonlinear spring was used for the equivalent simulation. Figure 10 shows the load–deflection curve of the nonlinear spring. The load–deflection curve can be divided into two stages: In the first stage, the resistant action of the spring increased linearly with the increase in deflection until the spring deflection reached the critical deflection *d*t. When the spring deflection was equal to the critical deflection *d*t, the resistant action of the spring reached the maximum value *R*max. In the second stage, the spring deflection increased rapidly and resistant action gradually decreased after the spring deflection exceeded the critical deflection *d*t. When the spring deflection reached the maximum value *d*max, the resistant action of the spring was equal to zero. At this point, the spring failed, and the stress was transferred to the adjacent springs until all springs failed, and the spring system lost its bearing capacity.

**Figure 10.** Stiffness curve of the nonlinear spring.

The critical deflection *d*<sup>t</sup> and maximum deflection *d*max of the nonlinear spring were determined by the concrete material test, and the maximum resistant action *R*max was determined by the interface shear strength between the PCM and concrete, along with the tensile strength of the PCM. Each spring represents the local interface shear area and tensile area of the PCM. The bearing capacity of the nonlinear spring can be calculated by Equation (1):

$$\begin{aligned} R\_{\text{max}} &= (R\_{\text{s}} + R\_{\text{t}}) \times \beta \\ R\_{\text{s}} &= h \times s \times \sigma\_{\text{s}} \\ R\_{\text{t}} &= t \times s \times \sigma\_{\text{t}} \end{aligned} \tag{1}$$

where *β* is the adjustment coefficient; *s* is the spring spacing; *t* is the PCM thickness; *h* is the horizontal grid interval; *σ*<sup>s</sup> is the interface shear strength between the PCM and the concrete; and *σ*<sup>t</sup> is the tensile strength of the PCM.

#### *5.2. Finite Element Analysis*

In order to conduct a further investigation of the interface bond mechanism and tensile stress transfer mode between the CFRP grid–PCM reinforcing layer and the concrete, the FEM analysis was carried out on the basis of pull-out tests. At first, the effectiveness of the simplified model was verified via the numerical simulation method. Then, the stress variation of the CFRP grid was explored based on the results of the FEM analysis.

#### 5.2.1. Establishment of the Model

Based on the simplified mechanical model, the two-dimensional FEM model was established using the FEM software ABAQUS. According to the stress behavior of the CFRP grid in the PCM, when the vertical CFRP grid was elongated along the axial direction under the tensile load, the horizontal CFRP grid would generate resistant action and cause apparent lateral deformation. Therefore, beam elements and truss elements were used to simulate the horizontal grid and vertical grid, respectively. Considering that the grid points of the CFRP grid have a certain rigidity, the crossing points of the horizontal and vertical grids were simulated by rigid joints. The nonlinear springs were arranged along the axis of the horizontal grids, and a nonlinear spring was arranged on the two sides of

the grid points to constrain the CFRP grid. Taking the specimen C8D100, for example, the FEM model is shown in Figure 11. The spring stiffness was calculated by Equation (1), where the interface shear strength was obtained via the direct shear tests between the PCM and the concrete [26], and the tensile strength of the PCM was obtained from the material property tests (see Table 4). The critical deformation *d*t and maximum deformation *d*max of the nonlinear spring were 0.1 mm and 0.3 mm, respectively. The tensile strength *σ*<sup>t</sup> of the PCM and interface shear strength *σ*<sup>s</sup> between the PCM and the concrete were 3.21 N/mm2 and 4.69 N/mm2, respectively. Spring spacing was 2.5 mm. It was assumed that the stress distribution in the area around each spring gradually decreased, and that only half of the area around each spring plays a role in bearing stress. Thus, the adjustment coefficient was set as 0.5. The bearing capacity of springs at different locations can be obtained by substituting these parameters into Equation (1), as shown in Table 7. It can be seen that the maximum resistant actions *R*max of the bottom springs were lower than those of the non-bottom springs, because the support area of the CFRP grid near the bottom loaded edge was smaller than that of the CFRP grid far away from the bottom loaded edge.

**Figure 11.** FEM analysis model.

**Table 7.** The maximum force of the spring.


### 5.2.2. Model Verification

Figure 12 shows the deformation of the CFRP grid for the FEM model based on the specimen C8D100. It can be seen that the deformation of the CFRP grid near the loaded edge was larger than that of the CFRP grid far away from the loaded edge, while the tensile stress was gradually transferred from the loaded edge to the free edge. Figure 13 shows the comparison of the maximum tensile loads obtained via the pull-out tests and FEM calculations. It can be seen that the ratios of the test values to the simulation values were close to 1, indicating that the simplified interface model with a nonlinear spring system could effectively reflect the mechanical behavior of the concrete specimens with a CFRP grid–PCM reinforcing layer. There was a certain error between the ultimate load of the FEM results and the test values. The main reason for these differences was that the simplified model considered the interface interaction between the CFRP grid–PCM and the concrete, as well as the interaction between the CFRP grid and the PCM from the overall point of view, which is an equivalent simplification of the interaction of the various parts. In addition, the nodes of the CFRP grids were regarded as rigid connections, which caused the inaccurate expression of the working performance of the actual nodes under loading.

**Figure 12.** Diagram of specimen deformation. (**a**) diagram of specimens C6D50 and C8D50 deformation; (**b**) diagram of specimens C6D50 and C8D50 deformation.

**Figure 13.** Comparison between the FEM and corresponding experimental results.

The FEM analysis results were compared with the test results, as shown in Figure 14. It can be observed that the pull-out test results and the FEM analysis results were almost identical in the strain distribution patterns of the tensile vertical grids; that is, the tensile strains gradually deceased from the loaded edge to the free edge. Therefore, it is reasonable to reflect the stress transfer change by the equivalent springs in the simplified mechanical model. In addition, the FEM strain values at some grid points were inconsistent with the test strain values. As shown in Figure 14a,c, the FEM strain value of specimens C6D50 and C8D50 at nodes GP1 and GP2 were essentially consistent with the test strain value; however, a discrepancy occurred at GP3. The test strain value increased suddenly at GP3, indicating that the specimen bore a large load and the force transfer form changed, which was probably related to the cracking of the mortar in the test process that was unable to be completed in the simplified simulation. In conclusion, since the simplified model considering the equivalent simplification based on reasonable analysis was the inaccurate simulation of the stress of the original specimens, the stress state of the vertical or transverse reinforcement was different from that in the test, and a certain error occurred between

the FEM value of the node strain and the test value. Further investigations using a fine mechanical model considering grid point failure should be conducted in the follow-up study.

**Figure 14.** Comparison of strain distribution for vertical grids. (**a**) comparison of strain distribution for vertical grids in specimen C6D50; (**b**) comparison of strain distribution for vertical grids in specimen C6D100; (**c**) comparison of strain distribution for vertical grids in specimen C8D50; (**d**) comparison of strain distribution for vertical grids in specimen C8D100.

### 5.2.3. Strain Analysis of CFRP Grids

Figure 15 shows the strain distribution in the FEM results to explain the stress transmission change. The results show that the trend of strain distribution on vertical grid points was almost identical for all pull-out specimens during the loading process. There were three stages in the variation of strain distribution with load: In the initial stage, the strain of the vertical grids was concentrated at the loaded edge, indicating that the load was mainly undertaken by GP1 and GP2, and the strain at GP1 was larger than at GP2, owing to the transmission mode of the vertical grid load. With the load increasing, the strain growth tendency from GP1 to GP5 (GP3) decreased, indicating that the load was gradually transferred from the loaded edge to the free edge. In the final stage, the load of specimens was close to its maximum, the spring failure expanded, and the strain at the grid points increased. With all springs failed, the specimens completely lost their carrying capacity. In addition, as shown in Figure 15a,b, comparing the strain distribution at the grid points at a certain load for C6D50 and C6D100 (i.e., 2 kN and 6 kN), it can be seen that with the upward transfer of stress, GP3 in C6D50 is less than GP2 (at the same height) in

C6D100, indicating that more grid points can share the stress with small grid spacing at the identical cross-sectional area. As shown in Figure 15a,c, for specimen C6D50, at a load of 2.0 kN, the grid strain of GP4 and GP5 is close to 0. When the load reaches 10.0 kN, the strain of GP1 to GP5 decreases; due to the large cross-sectional area, the grid point strain of C6D50 is greater than that of specimen C8D50, which is consistent with the results of carrying capacity, indicating that the upper grid point of the vertical grid can effectively bear the load and improves the carrying capacity of the specimens.

**Figure 15.** Strain distribution for CFRP grids. (**a**) strain distribution for CFRP grids in specimen C6D50; (**b**) strain distribution for CFRP grids in specimen C6D100; (**c**) strain distribution for CFRP grids in specimen C8D50; (**d**) strain distribution for CFRP grids in specimen C8D100.

#### 5.2.4. Strain Difference Analysis

The variation in strain difference on both sides of vertical grids in different specimens is shown in Figure 16, wherein the load–strain difference curve shows an upward tendency with the increase in pull-out load for all specimens, and the linear relationship mostly occurs in the initial stage. The farther away from the loaded edge, the smaller the strain difference, indicating that the resistance of the transverse reinforcement is smaller. As can be seen from Figure 16a,c, the strain difference at GP1 is close to that of GP2, which shows that the force is mainly undertaken by these two grid points. As the load reached a certain value (specimen C6D50 in 11.8 kN and specimen C8D50 in 17.9 kN), the strain difference of GP1 changed with the gradual failure of some springs at the grid point after the softening stage, and the load was borne by the spring at the next adjacent position. Then, with the

load increasing, the force was continuously transmitted upward, and the resistance of the transverse grids further reduced until the ultimate load. As can be seen from Figure 16b,d, the overall strain difference variations of specimens C6D100 and C8D100 were effectively consistent with the others. However, the strain difference of GP1 was smaller than that of GP2 throughout the whole loaded process, due to the concentrated load at GP1 for the larger grid interval of CFRP grids, resulting in the premature failure of some springs, after which the load was transmitted to GP2, which bore the main load, even if the elastic spring near GP1 did not completely quit working.

**Figure 16.** Strain difference distribution for CFRP grids. **(a**) C6D50; (**b**) C6D100; (**c**) C8D50; (**d**) C8D100.
