Study on Flexural Performance of Reinforced Concrete Beams Strengthened with FRP Grid–PCM Composite Reinforcement

Abstract

To study the flexural performance of fiber-reinforced polymer (FRP) grid–polymer cement mortar (PCM)-composite-strengthened RC beams, the finite element numerical simulation of FRP grid–PCM composite RC beams is carried out using ABAQUS to analyze the effects of the amount of FRP grid reinforcement, the type of FRP grid material, and the geometry of FRP grid on the flexural performance of reinforced concrete beams and to establish the flexural capacity calculation formula of FRP grid–PCM-reinforced RC beams in case of debonding failure, based on analysis of the influencing factors. The results show that increasing the reinforcement of the FRP grid and increasing the stiffness of the FRP grid can improve the flexural bearing capacity of RC beams, and the change of FRP grid geometry has little effect on the flexural bearing capacity of RC beams. The established formula for calculating the flexural bearing capacity of FRP grid-reinforced concrete beams can better predict the flexural capacity of reinforced concrete beams under peeling failure.

1. Introduction

Reinforced concrete beams are subject to deterioration or damage due to environmental, seismic, or other hazards and need to be strengthened and repaired [1]. The FRP grid–PCM reinforcement method lays FRP grids on the concrete surface and utilizes PCM as the bonding material and protection so that the three become a single unit that is subjected to joint stresses, thus improving the structural capacity, stiffness, and durability of the structure [2]. In this method of reinforcement, layer thickness is small, has low environmental dependence, and construction is simple, with high construction efficiency [3]. It overcomes the shortcomings of FRP reinforcement, such as easy aging of epoxy resin, poor resistance to high temperature and fire, and poor adhesion to concrete in humid environments [4].

Few scholars have systematically studied the flexural performance of reinforced concrete beams strengthened with FRP grid–PCM composite. Dai JG et al. [5] studied the flexural performance of reinforced concrete beams reinforced with different inorganic mortars composited with FRP grids; it was found that the polymer cement mortar has a relatively dense microstructure, which can form an integral whole with the FRP grids well. Rashid K et al. [6] found that the externally applied FRP grid–PCM composite layer could effectively inhibit the formation and development of concrete cracks, and the FRP grids could effectively improve the flexural stiffness and ultimate load capacity of the test beams. Koutas L et al. [7] studied the influence of mesh geometry on the anti-bending effect. It was found that doubling the spacing of meridional carbon fiber grids reduces the contribution of CFRP grids to the overall flexural bearing capacity of RC beams, while the zonal grid spacing has no substantial negative effect on the flexural strength of reinforced concrete beams. Pan Yi et al. [8] found that the stiffness and flexural capacity of the reinforced beams were greater when a greater unit reinforcement amount of CFRP grids was used; at the same time, the failure mode of the reinforced beams was changed to the debonding failure of the reinforcement layer. Yang et al. [9] carried out bending tests on RC beams strengthened with CFRP grids. The prefabrication and cast-in-place installation of the reinforced layer are considered in the study. When bonding CFRP grids with ECC or epoxy mortar, reliable end anchoring is required to prevent peeling failure. The higher the stiffness of CFRP, the more prone to debonding failure. Raoof SM et al. [10] confirmed that the damage of FRP grid–PCM-reinforced beams is often due to the debonding of the composite layer from the substrate. Therefore, it is necessary to quantify and predict the effectiveness of the FRP grid–PCM composite layer. Katsuyuki et al. [11] found that the relationship between bending moment and curvature of the CFRP grid–PCM-strengthened beam members were in good agreement with the theoretical values, so it can be assumed that the CFRP grid-strengthened beam members still satisfy the flat cross-section assumption, and the formula for calculating the flexural capacity of the strengthened beams can be deduced accordingly. He Weidong [12] analyzed the mechanical properties of FRP grid–PCM-strengthened RC beams undergoing concrete crushing and FRP grid rupture and proposed the formulae for predicting the flexural capacity in these two states, but the prediction of specimens in which the composite layer was debonded from the concrete substrate was poor.

Currently, the research on FRP grid–PCM-composite-strengthened RC beams is mostly based on experimental studies, but the actual experimental process is cumbersome, and the experimental conditions are easily restricted, while finite element simulation can overcome such difficulties. Liu Wei et al. [13] verified the reliability of a concrete damage plasticity model in the nonlinear analysis of concrete structures. Hu Wenhao et al. [14] conducted numerical simulations and analyses of the flexural performance of FRP grid–PCM-composite-reinforced concrete beams, thereby confirming the validity of the bond–slip cohesion modeling method introduced at the interface between the composite layer and the concrete during the modeling process.

This paper will use ABAQUS 2022 finite element software to numerically simulate FRP grid–PCM-composite-strengthened RC beams to study the effects of the amount of FRP grid reinforcement, the type of FRP grid materials, and the geometry of the FRP grids on the flexural performance of the reinforced concrete beams and to carry out regression analysis on the simulated beams where the debonding failure of the composite layer occurs, as well as to determine the formula of the peeling strain of the FRP grid to propose the FRP grid-reinforced RC beams’ flexural load capacity calculation formula.

2. Finite Element Modeling and Validation

2.1. Experimental Introduction

To verify the correctness of the modeling approach, this paper selects the test beams in the literature [15] for numerical simulation analysis, and four working conditions are selected: one unstrengthened RC beam and three BFRP grid–PCM-composite-strengthened RC beams. The test beams were loaded by four-point bending, and the dimensions of the beams as well as the reinforcement arrangement are shown in Figure 1. The FRP grid–PCM composite layer was attached to the bottom of the beams, and the spacing of the BFRP grid was 50 mm × 50 mm, the width of the composite layer was 150 mm, and the length was 1400 mm. The main test variables of the test beams are listed in

Table 1. Test variables of the test beams.

Beam ID	BFRP Grid Strengthening Amount
	Thickness (mm)	The Cross-Sectional Area of Single Limb (mm2)
B0
BG-L-PCM	3	45
BG-M-PCM	4	60
BG-H-PCM	5	75

, and the mechanical properties of each material are shown in

Table 2. Mechanical properties of materials.

Material		Compressive Strength (MPa)	Tensile Strength (MPa)	Yield Strength (MPa)	Elastic Modulus (GPa)
Concrete		30			30
PCM		70	12		23
Steel bar	$\varphi$6			440	200
	$\varphi$10			440	200
	$\varphi$13			440	200
BFRP grid	3 mm		455.6		17.4
	4 mm		410.0		16.3
	5 mm		382.7		15.4

.

2.2. Numerical Simulation Model

In this paper, the finite element software ABAQUS is used to numerically simulate the test beam described in Section 2.1.

2.2.1. Material Constitutive Model

In the analysis, the concrete is modeled by the uniaxial stress–strain curve model of concrete given in the Code for the Design of Concrete Structures (GB50010-2010) [16]; the constitutive relationship curves are shown in Figure 2. Since both PCM and concrete belong to cementitious composites with similar stress performance, the same constitutive model as that of concrete is used.

The constitutive relation of concrete under uniaxial tension can be determined by Equations (1)–(4):

$${\sigma }_t=(1-d_t)E_c{\epsilon }_t$$

(1)

$$d_t=\left\{\begin{array}{cc}1-{\rho }_t[1.2-0.2x^5]\hfill & \hfill x\le 1\\ 1-{\displaystyle \frac{{\rho }_t}{{\alpha }_t(x-1{)}^{1.7}+x}}\hfill & \hfill x>1\end{array}\right.$$

(2)

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_{t,r}}}$$

(3)

$${\rho }_t={\displaystyle \frac{f_t}{E_c{\epsilon }_{t,r}}}$$

(4)

where $E_c$ is the elastic modulus of concrete, ${\rho }_t$ is the parameter values of the descending section of the stress–strain curve under uniaxial tension, $f_t$ is the uniaxial tensile strength of concrete, ${\epsilon }_{t,r}$ is the peak tensile strain of concrete corresponding to the peak stress faut, ${\alpha }_t$ is the parameter values of the descending part of the tensile stress–strain curve, and $d_t$ is the damage evolution parameters of concrete under uniaxial tension.

The constitutive relation under uniaxial compression can be determined by Equations (5)–(9):

$${\sigma }_c=\left(1-d_c\right)E_c{\epsilon }_c$$

(5)

$$d_c=\left\{\begin{array}{cc}1-{\displaystyle \frac{{\rho }_{c}n}{n-1+x^n}}\hfill & \hfill x\le 1\\ 1-{\displaystyle \frac{{\rho }_c}{{\alpha }_c{\left(x-1\right)}^2+x}}\hfill & \hfill x>1\end{array}\right.$$

(6)

$${\rho }_c={\displaystyle \frac{f_c}{E_c{\epsilon }_{c,r}}}$$

(7)

$$n={\displaystyle \frac{E_c{\epsilon }_{c,r}}{E_c{\epsilon }_{c,r}-f_c}}$$

(8)

$$x={\displaystyle \frac{\epsilon }{{\epsilon }_{c,r}}}$$

(9)

where ${\sigma }_c$ is the compressive stress of concrete when the compressive stress of concrete becomes ${\epsilon }_{c0}$, $f_c$ is the design value of the axial compressive strength of concrete, and ${\epsilon }_{c,r}$ is the compressive strain of concrete when the compressive stress reaches fc. When the calculated ${\epsilon }_{c,r}$ value is less than 0.002, it is taken as 0.002. ${\epsilon }_{cu}$ is the ultimate compressive strain of concrete. $f_c$ is the uniaxial compressive strength of concrete. n is the coefficient. When the calculated value is greater than 2.0, it is 2.0.

Steel bars were modeled using a bilinear stress–strain relationship with post-yield strain hardening. The expression and the relationship curve are shown in Figure 3a. FRP grids were modeled using a linear elastic stress–strain relationship. The expression and the relationship curve are shown in Figure 3b.

2.2.2. Bond–Slip Model at PCM–Concrete Interface

It can be seen from the test that the composite layer and concrete slip. Some specimens will be peeled off because of the large slip, so in the finite element modeling, the interaction between the concrete and composite layer is modeled by the bond–slip model proposed by Lu Xinzheng et al. [17]; the detailed calculation formula is shown in Equations (10)–(15); the relation curve is shown in Figure 4. In ABAQUS, cohesion contact is set on the interface between composite layer and concrete to represent the cohesion model. Interface shear stiffness $K_0$, interface maximum shear stress ${\tau }_{max}$, and interface limit slip $S_f$ are the key parameters of the bond contact setting. After several trial calculations, this paper inputs $K_0$ = 46 MPa, ${\tau }_{max}$ = 1.4 MPa, and $S_f$ = 0.354 mm.

$$\tau =\left\{\begin{array}{cc}{\tau }_{max}s/s_0\hfill & \hfill (0<s\le s_0)\\ {\tau }_{max}(s_f-s)/(s_f-s_0)\hfill & \hfill (s_0<s\le s_f)\\ 0\hfill & \hfill {(s}_f<s)\end{array}\right.$$

(10)

$${\tau }_{max}={\alpha }_1{\beta }_w{f}_t$$

(11)

$$S_0=0.0195{\beta }_w{f}_t$$

(12)

$$s_f=2G_f/{\tau }_{max}$$

(13)

$$G_f=0.308{\beta }_w^2\sqrt{f_t}$$

(14)

$${\beta }_w=\sqrt{{\displaystyle \frac{2-b_f/b_c}{1+b_f/b_c}}}$$

(15)

where $\tau$ and $s$ are local interfacial shear stress and relative slip, respectively; ${\tau }_{max}$ and $s_0$ are bond strength and their corresponding slip, respectively; $s_f$ is corresponding slip when interfacial shear stress is 0; $G_f$ is interfacial failure energy (fracture energy); $f_t$ is tensile strength of concrete; ${\alpha }_1$ is the finite element method to calculate the regression parameters; ${\beta }_w$ is taken as the width influence coefficient; and $b_f$ and $b_c$ are the width of the FRP grid and the concrete section, respectively.

2.2.3. Additional Details of the Numerical Simulation Model

Concrete and PCM are simulated using a continuous solid element with the cell type of a C3D8R element and a concrete damage plasticity model; the steel bars are used as truss elements with the cell type of T3D2 elements; and the FRP grid is used as shell elements with the cell type of S4R elements. To prevent stress concentration, rigid pads are placed at the support and the loading point.

According to the force characteristics of the simply supported beam, the left end bearing boundary conditions were set as (U1 = 0, U2 = 0, U3 = 0, UR1 = 0, UR2 = 0), and the right end support boundary conditions as were set as (U2 = 0, U3 = 0, UR1 = 0, UR2 = 0).

This simulation adopts the displacement loading method of centralized force loading, setting the reference point above the mat as the displacement loading point, and setting the coupling constraint between the loading point and the mat to realize the transfer of centralized force to the mat.

The remaining components are constrained as follows: steel bars are embedded in the concrete; the FRP grid is also embedded in the PCM; the rigid pads are constrained with concrete using the tie constraints, and the coupling constraints are used between the rigid pads and the loading points. After analysis and comparison, and finally combined with the size of the model, the mesh of the model is divided by 25 mm. Additionally, the mesh quality of the model beam was checked. The model beam had good accuracy while avoiding longer computation time. The finite element model of the strengthened beam is shown in Figure 5.

2.3. Simulation Results and Correctness Verification

To verify the reliability of the above finite element model, the strain cloud and load–midspan deflection curves of RC beams obtained from the finite element analysis are compared with the corresponding tests in this section, which are shown in Figure 6 and Figure 7, respectively. The comparisons of the characteristic load and midspan deflection values of the strengthened beams with the specific values of the experimental results are shown in

Table 3. Comparison of simulation results and experimental results.

Beam ID	Yield Load (kN)		Error (%)	Ultimate Load (kN)		Error (%)	Deflection Value (mm)		Error (%)
	Experimental Value	Simulation Value		Experimental Value	Simulation Value		Experimental Value	Simulation Value
B0	120.3	129.8	7.89	139.6	139.8	−0.14	39.15	39.84	1.76
BG-L-PCM	159.2	164.6	3.39	223.9	211.8	−5.40	22.00	21.14	−3.90
BG-M-PCM	160.2	165.5	3.31	228.4	213.3	−6.61	18.48	15.90	−13.96
BG-H-PCM	163.2	166.3	1.90	219.5	217.1	−1.09	13.85	14.63	5.63

Note: Error value = (simulated value − experimental value)/experimental value × 100%; experimental values in the table are from the literature [15].

.

In the CDP model, the development of cracks in RC beams is represented by tensile strains, and the tensile strain clouds of four specimens are shown in Figure 6. It can be seen from the figure that the crack distribution of each model is in good agreement with the test, and the plastic tensile strain of the unstrengthened beam B0 is greater and concentrated in the middle of the span, which is consistent with the test results that show that the cracks are concentrated in the middle of the span. Compared with the unstrengthened beam B0, the tensile strain of the FRP grid–PCM-composite-reinforced beam is reduced from 0.02287 to 0.02066, indicating that the maximum crack width of the concrete beam is significantly reduced, and the external FRP grid–PCM composite layer can effectively inhibit the formation and expansion of the cracks in reinforced concrete beams.

In the CDP model, the crack development of reinforced concrete beams is represented by equivalent plasticity (PEEQ), and the equivalent plastic strain cloud diagrams of the four specimens are shown in Figure 6. It can be seen from the figures that the crack distribution of each model is consistent with the test results, and the plastic tensile strain of the unreinforced beam B0 is greater and concentrated in the middle of the span, which is consistent with the test results. Compared with the unstrengthened beam B0, the equivalent strain of the FRP grid–PCM-composite-strengthened beam decreases from 0.09211 to 0.04228, indicating that the maximum crack width of the concrete beam decreases significantly, and the external FRP grid–PCM composite layer can effectively restrain the crack formation and propagation of the reinforced concrete beam.

As shown in Figure 7, the load-deflection of the beams can be divided into three stages (pre-cracking, cracking, and post-yielding stages). In the pre-cracked stage, the test beams show linear elastic behavior. When the test beams crack, they experience a subsequent loss of bending stiffness. In the post-yielding stage, there are some small fluctuations in the load-deflection response of the strengthened beam, which is due to the local slip between the composite layer and the concrete surface of the strengthened beam. The development trend of the four groups of comparison curves is basically the same, indicating that the simulation results of the four beams are in good agreement with the experimental results.

As shown in Table 3, there are some errors between the simulated and tested values of the characteristic loads and midspan deflections. The possible sources of errors include: (1) The boundary conditions of the reinforced beams are treated as ideally simply supported in the numerical analysis, whereas it is difficult to ensure that the beams are in an ideally simply supported state in the tests. (2) There is a certain gap between the intrinsic relationship of the concrete, FRP grids, and other materials used in the model and the actual stress state of each material in the experiments. (3) In the simulation of strengthened beams, the bond–slip model used in this paper is with reference to the bond–slip model of RC beams strengthened with FRP sheets, which is in error from the actual situation.

On the whole, the simulation results of the strengthened beams are basically consistent with the variation trend of the experimental results, and the errors of the characteristic load and deflection values are controlled within 15%, so the modeling method adopted in this paper is feasible, and the analysis of the influencing factors can be further carried out.

3. Analysis of Factors Affecting Flexural Performance of Reinforced Beams

The study shows that [16] the damage of FRCM flexural reinforced RC beams often occurs due to the loss of bond between the composite material and the matrix, which depends on: (i) the mechanical properties of the FRP grid material; (ii) the fiber-matrix bonding properties; and (iii) the bond between the cement matrix and the concrete matrix. These three situations are greatly influenced by the type of FRP grid material, the degree of FRP grid immersion in cement mortar, and the geometric arrangement of the grid. As a result, this paper will investigate the effects of FRP grid material type, the amount of FRP grid reinforcement, and FRP grid geometry on the flexural performance of FRP grid–PCM-composite-strengthened reinforced concrete beams.

To facilitate the subsequent analysis, the beams are numbered using the WXYZ format. W represents the type of FRP grid material (C for carbon fiber, B for basalt fiber, and G for glass fiber); X represents the thickness of the grid (3, 4, and 5 indicate that the thickness of the grid is 3 mm, 4 mm, and 5 mm, respectively); Y represents the number of layers of the grid (O and T represent the 1- and 2-layer FRP grids, respectively); and Z represents the spacing of the grid (50-50 indicates that the grid warp and weft spacing are 50 mm and 50 mm respectively, 50-100 indicates that the grid warp and weft spacing are 50 mm and 100 mm respectively, and 100-50 indicates that the grid warp and weft spacing are 100 mm and 50 mm respectively).

3.1. Effect of the Amount of FRP Grid Reinforcement on the Flexural Performance of Reinforced Beams

To study the effect of the amount of FRP grid reinforcement on the flexural performance of the reinforced beams, BG-L-PCM is used as the reference beam. The rest of the parameters remain unchanged; the grid thickness and the number of grid layers are adjusted; the grid thicknesses are taken as 3, 4, and 5 mm; and the number of grid layers are taken as 1 and 2 layers.

In this subsection, a total of six FRP grid-reinforced beams with different amounts of reinforcement are subjected to finite element analysis, and the load–midspan deflection curves obtained are shown in Figure 8. The characteristic loads, deflection values, and damage modes are shown in

Table 4. Effect of FRP grid reinforcement amount on reinforced beams.

Beam ID	Cross-Sectional Area of the Grid ${\mathit{A}}_{\mathit{f}} \left({\mathbf{mm}}^2\right)$	Yield Load (kN)	Ultimate Load (kN)	Ultimate Load Increase Rate (%)	Deflection Value (mm)	FRP Grid Strain	Failure Mode
B3O50-50	135	164.4	211.8	51.5	21.14	0.0163	D + FR
B4O50-50	180	165.5	213.3	52.6	15.90	0.0114	D + CC
B5O50-50	225	166.3	217.1	55.3	14.63	0.0098	D + CC
B3T50-50	270	173.8	218.5	56.3	13.40	0.0120	D
B4T50-50	360	175.3	216.1	54.6	11.01	0.0084	D
B5T50-50	450	183.9	215.5	54.1	9.79	0.0066	D

Note: D = debonding, denoting composite layer stripping damage; CC = concrete crush, denoting concrete crushing damage; FR = fibers rupture, denoting FRP grid fracture, the same below; FRP grid strain is the ultimate load corresponding to the grid strain in the span, the same below; the same as below.

. As can be seen from Figure 7, the simulated beams show a clear three-stage variation. In the first two stages, the curves of the six beams are in good agreement, and in the third stage it is found that the stiffness of beam B5T50-50 is greater than that of beam B3O50-50; the stiffness of the beam decreases with the decrease in the amount of FRP grid reinforcement. As can be seen from Table 4, the ultimate load of the reinforced beams increases nonlinearly. With the increase of the reinforcement amount by 51.5–56.3%, while the deflection of the beams decreases from 21.14 mm to 9.79 mm, the strain of the FRP grid decreases from 0.0163 mm to 0.0066 mm. A total of three damage modes occur in the simulated beams: beam B3O50-50 undergoes the damage of the composite layer peeling off after the FRP grid fracture; beam B4O50-50 and beam B5O50-50 both undergo damage after the composite layer peeling damage, when the concrete in the compression zone was crushed; and beam B3T50-50, beam B4T50-50, and beam B5T50-50 all undergo damage only after the composite layer peeling damage. In conclusion, as the amount of FRP grid reinforcement increases, its flexural stiffness and ultimate load are improved to different degrees, and the midspan deflection of the beams is also reduced.

3.2. Effect of FRP Grid Material Type on Flexural Performance of Reinforced Beams

To investigate the effect of FRP grid material type on the flexural performance of the reinforced beams, the FRP grid material was changed to CFRP, BFRP, and GFRP, respectively, using BG-L-PCM as the reference beam with the rest of the parameters unchanged.

In this subsection, a total of three reinforced beams with different FRP grids are subjected to finite element analysis, and the load–midspan deflection curves of the reinforced beams are obtained, as shown in Figure 8. The characteristic loads, deflection values, and damage modes are shown in

Table 5. Effect of FRP grid material types on reinforced beams.

Beam ID	FRP Grid		Yield Load (kN)	Ultimate Load (kN)	Ultimate Load Increase Rate (%)	Deflection Value (mm)	FRP Grid Strain	Failure Mode
	$\mathbf{Elastic}\mathbf{Modulus}\left(\mathbf{G}\mathbf{P}\mathbf{a}\right)$	Tensile Strength (MPa)
B3O50-50	17.4	455.6	164.4	211.8	51.5	21.14	0.0163	D + FR
C3O50-50	41	492	174.7	214.9	53.7	11.65	0.0073	D
G3O50-50	21	501	165.2	216.9	55.2	18.57	0.0203	D + FR

. From Figure 9, it can be seen that the simulated beams show a clear three-stage variation, where the curves of the three beams match well during the first two stages, and in the third stage, it is found that due to the higher elastic modulus of CFRP grid, beam C3O50-50 shows the highest improvement in bending stiffness, followed by beam B3O50-50. As seen in Table 5, the ultimate loads of beam G3O50-50, beam C3O50-50, and beam B3O50-50 increased by 55.2%, 53.7%, and 51.5%, respectively; the deflections of beam G3O50-50, beam C3O50-50, and beam B3O50-50 were 18.57 mm, 11.65 mm, and 18.57 mm, respectively. A total of two damage modes were experienced by the simulated beams: beam B3O50-50 and beam G3O50-50 were damaged by composite layer peeling and then the FRP grid was pulled off; beam C3O50-50 was damaged by composite layer peeling. In summary, although the CFRP grid can improve the flexural stiffness of the reinforced beams, it is not ideal to improve the load carrying capacity; the modulus of elasticity of the BFRP grid is small, but it is ideal to improve the load carrying capacity of the reinforced beams as well as the deflection.

3.3. Effect of FRP Grid Geometry on Flexural Performance of Reinforced Beams

To investigate the effect of FRP grid geometry on the flexural performance of the reinforced beams, BG-L-PCM is used as the reference beam, and the rest of the parameters remain unchanged. Controlling the total meridional cross-sectional area $A_f$ = ${135\mathrm{m}\mathrm{m}}^2$ and the total cross-sectional area in the warp direction $A_f$ = 1.260 ${\mathrm{m}\mathrm{m}}^2$, the grid spacing and single limb width are adjusted, respectively, and the thickness of all grids is 3 mm. Two different grid spacings are added; its warp and weft direction spacing are 50 × 100 and 100 × 50 mm, respectively, as shown in Figure 10.

In this subsection, a total of three reinforced beams with different geometries are subjected to finite element analysis, and the load–midspan deflection curves of the reinforced beams are obtained, as shown in Figure 11. The characteristic loads, midspan deflection values, and damage modes are shown in Table 5. As can be seen from Figure 11, the simulated beams show a clear three-stage variation, and the curves of the three beams match well in the first two stages. In the third stage, it is found that the deflection values corresponding to the ultimate load of beams B3O50-100 are the greatest, those of beams B3O100-50 are the second greatest, and those of beams B3O50-50 are the lowest. From

Table 6. Effect of FRP grid geometry on reinforced beams.

Beam ID	Grid Warp × Weft Spacing/mm	Yield Load (kN)	Ultimate Load (kN)	Ultimate Load Increase Rate (%)	Deflection Value (mm)	FRP Grid Strain	Failure Mode
B3O50-50	50 × 50	164.6	211.8	51.5	21.14	0.0163	D + FR
B3O50-100	50 × 100	165.1	211.6	51.4	23.06	0.0230	CC
B3O100-50	100 × 50	165.5	214.3	53.2	22.37	0.0222	CC

, it can be seen that when the warp spacing of the FRP grid was increased from 50 mm to 100 mm, the ultimate load of beam B3O50-100 decreased by 0.2 kN compared to that of beam B3O50-50, but the deflection increased by 3.79 mm. When the latitudinal spacing of the FRP grid was increased from 50 mm to 100 mm, the ultimate load of beam B3O100-50 compared to that of beam B3O50-50 increased by 2.5 kN. Two damage modes occurred in the simulated beams: the FRP grid was pulled off after the composite layer peeling damage occurred in beam B3O50-50, and the concrete crushing damage occurred in the compression zone between beam B3O50-100 and beam B3O100-50. This indicates that when the grid spacing is increased, the bonding area between the FRP grid and the PCM increases, the composite layer has better integrity, and the flexural reinforcement effect is somewhat improved.

4. Calculation Formula of Flexural Bearing Capacity of RC Beams Strengthened with FRP Grid and PCM Composite

From the numerical simulation results of FRP grid–PCM-composite-reinforced concrete beams in Section 3, it can be seen that the simulated beams have a total of three damage modes: (1) concrete crushing damage in the compression zone; (2) FRP grid pulling damage; (3) FRP grid–PCM composite layer peeling damage. The research on the flexural capacity of the first two damage modes is more systematic and comprehensive, but the research on the flexural capacity of the third damage mode is less systematic and comprehensive, so research on the calculation model of the bending capacity of composite layer peeling will be carried out in this section.

4.1. Modeling of Bending Load Capacity Calculation

The results of the finite element analysis above found that the type of FRP grid material and the amount of FRP grid reinforcement are the main influencing factors when the simulated beam undergoes peeling damage, so these two factors will be taken into account to propose a formula for calculating the peeling strain of the FRP grid when the simulated beam undergoes peeling damage. Considering that beams with different FRP grid materials selected for reinforcement have different cross-sectional areas, this paper takes the axial stiffness, i.e., the product of the modulus of elasticity of the FRP grid material and the total cross-sectional area of the FRP grid in the latitudinal direction in the fitting $E_f{A}_f$ to represent the difference between the FRP grid material type and the amount of reinforcement. After several attempts, the following relational assumptions are presented in the form of power functions to ensure fitting accuracy:

$${\epsilon }_{deb}=a{\left(E_f{A}_f\right)}^b$$

(16)

In Equation (16), $a\mathrm{and}b$ is the coefficient to be found; ${\epsilon }_{deb}$ is the FRP grid peeling strain; $E_f$ is the modulus of elasticity of the FRP grid material; and $A_f$ is the total cross-sectional area of the FRP grid in the latitudinal direction.

The relationship between FRP grid debonding strain and axial stiffness was plotted in the form of scatter points, and is fitted with Equation (10). The fitting result is shown in Figure 12.

From Figure 12, the FRP grid peeling strain relation is given as follows:

$${\epsilon }_{deb}=2.8174{\left(E_f{A}_f\right)}^{-0.6652}$$

(17)

The coefficients are obtained after retaining three significant figures:

$${\epsilon }_{deb}=2.82{\left(E_f{A}_f\right)}^{-0.665}$$

(18)

The stress–strain distribution in the midspan cross-section of the reinforced beam when debonding failure occurs is shown in Figure 12. For further analysis, the following assumptions are made: (i) during the stressing process, the strains of concrete, reinforcement, and FRP grid in the cross-section of the test beam satisfy the assumption of flat section; (ii) the bond between concrete and reinforcement as well as between the polymer-cement mortar and the FRP grid is reliable, and there is no relative slippage; (iii) the effects of the concrete in the tension zone and polymer-cement mortar on flexural capacity are neglected; (iv) after the concrete cracking, the concrete in the tension zone does not participate in the structural force; (v) the FRP constitutive model is linear before the maximum strain; (vi) the material constitutive relationship between the concrete in the compression zone and the stressed longitudinal reinforcement follows the literature [16]. At this time, the stress and strain distribution of the section is shown in Figure 13, and the mechanical expression of the section is shown in the Formulas (19)–(23).

$$C_c={\sigma }_c\left({\epsilon }_c\right)b_c\beta x_0$$

(19)

$$T_c={\epsilon }_{sc}{A}_{sc}{E}_s$$

(20)

$${\epsilon }_{sc}={\epsilon }_{deb}{\displaystyle \frac{x_0-a_s^{\prime }}{h-x_0}}$$

(21)

$$T_t=f_{y,st}{A}_{st}$$

(22)

$$T_{f,deb}={\epsilon }_{deb}{E}_f{A}_f$$

(23)

in the Formulas (19)–(23), $C_c$ and $T_c$ are the pressures on concrete and compression bars in the compression zone, respectively; $T_t$ and $T_{f,deb}$ are the tensile forces provided by the tensile reinforcement and the FRP grid, respectively; $b_c$ and $h$ are the width and height of the RC beam section, respectively; $\beta$ is the equivalent rectangular stress pattern coefficient of the concrete in the compression zone, and is generally taken as 0.8; $x_0$ is the height of the concrete in the compression zone; $A_{sc}$ and $A_{st}$ are the total cross-sectional area of compression and tension bars, respectively; $f_{y,st}$ is the yield strength of the tension bar; $a_s^{\prime }$ is the distance from the compression bar to the upper edge of the concrete beam; ${\epsilon }_{sc}$ is the strain of the compression bar; ${\epsilon }_c$ is the strain at the upper edge of the concrete in the compression zone; and ${\sigma }_c\left({\epsilon }_c\right)$ denotes the concrete strain ${\epsilon }_c$ corresponding to the stress in the principal curve.

Balanced by internal forces:

$$C_c+T_c=T_t+T_{f,deb}$$

(24)

$$M_{u,deb}=M_{C_c}+M_{T_c}+M_{T_{f,deb}}$$

(25)

where $M_{u,deb}$ is the value of the section bending moment at the time of stripping damage, and $M_{C_c}$, $M_{T_c}$, and $M_{T_{f,deb}}$ are the values of the bending moment provided by the concrete, compression longitudinal bars, and FRP grid in the compression zone, respectively.

Thus, the flexural capacity of the FRP grid flexural reinforced RC beam is obtained as follows:

$$P_{u,deb}={\displaystyle \frac{2M_{u,deb}}{a}}={\displaystyle \frac{2}{a}}\left[C_c\left(h_0-{\displaystyle \frac{x}{2}}\right)+T_c\left(h_0-a_s^{\prime }\right)+T_{f,deb}{a}_s\right]$$

(26)

where $P_{u,deb}$ is the calculated value of bending capacity; $a_s$ is the distance from the tension reinforcement to the lower edge of the concrete beam; $a$ is the length of shear span section; and $x$ is the height of the compression zone of the equivalent rectangular stress diagram.

4.2. Validation of the Bending Load Capacity Formula

To verify the correctness of the formula for calculating the flexural bearing capacity of reinforced concrete beams strengthened with FRP grids established in this paper, the test data in the literature [9,18,19,20,21,22] were collected. All the test beams in the table were rectangular reinforced concrete simply supported beams, which were strengthened with FRP grids, and peeling failure occurred. The FRP grid debonding strain and ultimate load of the test beam are compared with the values calculated by Formulas (18) and (26). The results are shown in

Table 7. Validation of Formulas (18) and (26) for FRP grid debonding strain and flexural capacity of reinforced beams.

Beam ID	${\mathit{\epsilon }}_{\mathit{d}\mathit{e}\mathit{b},\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}$	${\mathit{\epsilon }}_{\mathit{d}\mathit{e}\mathit{b},\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{r}\mathit{i}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{a}\mathit{l}}$	${\mathit{\epsilon }}_{\mathit{d}\mathit{e}\mathit{b},\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{r}\mathit{i}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{a}\mathit{l}}/{\mathit{\epsilon }}_{\mathit{d}\mathit{e}\mathit{b},\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}$	${\mathit{P}}_{\mathit{u},\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}$ (kN)	${\mathit{P}}_{\mathit{u},\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{r}\mathit{i}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{a}\mathit{l}}$ (kN)	${\mathit{P}}_{\mathit{u},\mathit{e}\mathit{x}\mathit{p}\mathit{e}\mathit{r}\mathit{i}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathit{a}\mathit{l}}$$/{\mathit{P}}_{\mathit{u},\mathit{c}\mathit{a}\mathit{l}\mathit{c}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}$
II-12 [9]	0.00976	0.00901	0.92	95.4	104.8	1.10
V12 [9]	0.00754	0.00812	1.08	120.1	117.6	0.98
ROH [18]	0.00573	0.00539	0.94	223.6	204.0	0.91
MTF2 [19]	0.00904	0.01002	1.11	169.3	155.2	0.92
A2 [20]	0.00740	0.00720	0.97	205.2	190.3	0.93
S2-T1-P2-2 [21]	0.01207	0.01150	0.95	71.7	66	0.92
H_4_X [22]	0.00744	0.00800	1.08	106.3	96.8	0.91
Average value			1.00			0.96
Standard deviation			0.076			0.067
Coefficient of variation			0.076			0.067

.

As can be seen from Table 7, the mean, standard deviation, and coefficient of variation of the ratio of the test value of the debonding strain of the reinforced beam to the calculated value are 1.00, 0.076, and 0.076, and the mean, standard deviation, and coefficient of variation of the ratio of the test value of the flexural capacity to the calculated value are 0.96, 0.067, and 0. 067. The overall agreement is better, but there is still a certain degree of error, which is considered by the author to be the reason for the error. The author believes that the reasons for the error are as follows:

(1)

The bending capacity calculation formula of FRP grid–PCM-composite-strengthened reinforced concrete beams established in this paper is based on certain assumptions, so there is a certain error between the calculated value and the actual bending capacity of the reinforced beams.

(2)

When establishing the FRP grid peeling strain formula, only the case of peeling at the interface between the composite layer and the concrete is considered, which will be different from the interface when peeling damage occurs in the actual test reinforced beam.

In summary, the formula established in this paper for calculating the peeling strain of FRP grid-reinforced RC beams with the occurrence of composite layer peeling damage has certain feasibility and can provide a certain theoretical basis for calculating or evaluating the flexural load capacity of FRP grid-reinforced concrete beams, but the applicability of the formula still needs more FRP grid-reinforced research and experimental data to be optimized.

5. Conclusions

In this paper, FRP grid–PCM-composite-reinforced RC beams are modeled and analyzed by the finite element software ABAQUS, and the effects of the main influencing factors on the flexural performance of FRP grid–PCM-composite-reinforced RC beams are investigated. The following conclusions are obtained:

(1)

The finite element model established based on bond–slip cohesive contact is feasible and effective. It can fit the test phenomenon and the magnitude of bending load capacity better and can be used to simulate the bending specimens of FRP grid–PCM-composite-reinforced RC beams.

(2)

An FRP grid can effectively improve the flexural stiffness and bearing capacity of reinforced concrete beams and has a non-linear relationship with the amount of FRP grid reinforcement and the elastic modulus of the grid.

(3)

When debonding failure occurs in the reinforced beams subjected to bending, the FRP grid strain does not change significantly with the grid shape, but it is more affected by the amount of grid reinforcement and the type of grid material, and the greater the amount of grid reinforcement and the grid elasticity modulus, the lower the grid strain when debonding occurs.

(4)

The formula for calculating the debonding strain and the formula for calculating the flexural capacity of FRP grid-strengthened reinforced concrete beams with interfacial debonding failure were established. Based on the experimental data collected, the formulas were verified, and the results show that the peeling strain formula and flexural capacity formula proposed in this paper can provide a theoretical basis for the calculation or assessment of the flexural capacity of FRP grid-strengthened reinforced concrete beams.

6. Recommendations

Based on the research in this paper, a brief suggestion is made to provide a further outline for further research in the future.

(1)

In the existing analysis, the discussion on the interface of reinforced concrete beams strengthened with FRP grid–PCM is not detailed enough, and the change law of interface stress cannot be explained systematically and comprehensively. In addition, in view of the complexity of the interface of FRP grid–PCM-strengthened concrete structures, and given that the simulation theory of the interface is still in its infancy, it is of great significance to explore a reasonable and appropriate interface simulation method for in-depth analysis of the mechanical behavior of the interface between FRP grid–PCM and concrete.

(2)

Among the existing formulas for calculating the flexural capacity of reinforced concrete beams strengthened with FRP grids, more influence factors can be given for the formula with FRP grid peeling so as to further improve the accuracy of the formula.