Analytical Model for Concrete Cover Separation in Prestressed Near-Surface-Mounted Carbon Fiber-Reinforced Polymer-Strengthened Reinforced Concrete Beams

Abstract

This study presents a novel method for calculating the stress and bending moment in reinforced concrete (RC) beams strengthened with prestressed near-surface-mounted (NSM) carbon fiber-reinforced polymer (CFRP), focusing specifically on separation failure at the end concrete cover. By characterizing the geometry of the failure body and integrating it with the concrete tooth model, a comprehensive theoretical framework has been developed. This framework utilizes sectional strain distribution characteristics to establish the bending moment–curvature relationship and the load–deflection curve during loading. Comparisons with experimental data confirm the accuracy and applicability of this analytical model. The results demonstrate that the model is capable of accurately predicting the load–deflection behavior of the strengthened beam. Additionally, this study underscores the substantial impact of the CFRP prestress level on the concrete cover separation failure, showing that optimizing prestress settings can effectively enhance the ductility and bearing capacity of the strengthened beam.

1. Introduction

Fiber-reinforced polymer (FRP), with its superior characteristics, such as being lightweight, high-strength, and corrosion-resistant, has been widely applied in the structural strengthening of existing concrete structures. Typically, FRP materials are externally bonded (EB) to the surface of the structure and prestressed using anchors [1,2]. However, the effectiveness of this strengthening method is constrained by several factors, including bonding area, cost, and anchor durability. In recent years, researchers have explored the near-surface-mounted (NSM) technique [3,4,5], which embeds FRP within the concrete cover, creating multi-sided bonding. This technique significantly enhances bond performance and eliminates the need for permanent anchors. Nevertheless, under combined prestressing and external loads, the stress concentration at the FRP ends can lead to the premature separation failure of the concrete cover [6,7,8]. Unlike midspan concrete crushing or FRP tensile fracture, whose ultimate loads can be calculated using the plane section assumption, determining the ultimate strength during concrete cover separation (CCS) failure is more complex. Currently, limited research focuses on developing analysis models for CCS failure in prestressed FRP-strengthened structures.

Most of the existing analysis models for CCS failure are extensions of the original EB FRP strengthening analysis model, primarily encompassing a shear capacity-based strength model and a concrete tooth model. However, significant differences exist in terms of bonding performance between the NSM and EB techniques [9,10]. In NSM-strengthened beams, failures at the CFRP–concrete bonding interface are relatively rare. Consequently, the definitions and analyses of failure modes in some models diverge from the actual conditions, leading to unsatisfactory prediction outcomes. The following section briefly outlines the current calculation models for the bearing capacity of CCS failure.

In the shear capacity model, it is proposed that the debonding strength of strengthened beams is primarily determined by the shear capacity of concrete. A correlation coefficient derived from the experimental data is used to quantify this relationship. However, the contributions of stirrup and bending moment are either neglected or only partially considered, leading to a relatively simplistic formula. Based on the existing experimental data and models, Oehlers [11], Smith and Teng [12,13], and Yao and Teng [14,15] developed a debonding failure analysis model that incorporates the interaction between bending moment and shear force. This model initially predicts the failure loads for two extreme scenarios: flexural debonding at the plate end occurring in the pure bending region and pure shear bond debonding at the plate end under high shear stress and low bending moment (e.g., near the support of a simply supported beam). Subsequently, an interaction equation is formulated to predict bond failure under any combination of bending moment and shear force.

The concept of the concrete tooth model was first introduced by Zhang [16]. This theory suggests that prior to separation, the shear stress between the strengthening material and concrete is uniformly distributed. Once the crack enters the stable propagation phase, the concrete cover between adjacent cracks can be regarded as a “tooth”. Under the influence of CFRP at the bottom of the beam, the stress condition of this “tooth” resembles that of a cantilever member. When the stress concentration at the tooth exceeds the tensile strength of the interface, a horizontal branch crack initiates along the bottom edge of the tensile reinforcement, leading to the failure of the separation. De Lorenzis and Al-Mahmoud et al. [17,18] extended the concrete tooth model to evaluate the ultimate strength of CCS failure in NSM FRP-strengthened beams. They refined the crack spacing based on the specific characteristics of NSM strengthening and derived the principal stress of CFRP at failure by incorporating the stress equilibrium relationship. Zhang et al. [19,20] developed a 2D nonlinear finite element analysis model to investigate the CCS failure at the end of the strengthened beams. This model specifically examined the behavior of the RC beam between two cracks near the critical end of the FRP strip. By adjusting the external load, they simulated the bending moments on the cracked sections. Cohesive elements were utilized to connect the tensile reinforcement and concrete, accounting for the radial stress effect. The failure bending moment was determined by evaluating the FRP strain at the point of failure. Through numerical regression, they proposed a predictive model for FRP strain at the onset of separation failure.

From existing research, current analysis models predominantly focus on non-prestressed strengthened beams and neglect the stress concentration at the end of the prestressed CFRP. This oversight has resulted in an incomplete understanding of CCS failure mode. Directly applying these models to prestressed strengthened beams would likely result in an overestimation of their load-bearing capacity. Furthermore, in prestressed NSM CFRP-strengthened structures, the risk of CCS failure increases significantly with higher levels of prestress. Currently, there is a notable lack of analysis models that specifically address varying levels of prestress in CFRP.

The objective of this study is to develop an analytical model to predict the bending performance of prestressed NSM CFRP-strengthened beams. The bearing capacity at the onset of CCS is assessed using a concrete tooth model that incorporates the morphology of the failure surface. This value is then compared with the loads within which the compression region crushes or the CFRP ruptures, thereby identifying potential failure modes. Subsequently, the midspan deflection is determined using the moment–curvature relationship. Finally, the proposed analytical model’s predictive accuracy is validated by comparing its results with the experimental data.

2. Analytical Framework

Based on the mechanical behavior and material characteristics of the prestressed strengthened beam at various stages during the loading process, the relationship between load and midspan deflection is described by trilinear behavior: pre-cracking, post-cracking, and post-yielding stages. This behavior is defined by four key points: initial camber (ic), concrete cracking (cr), tensile reinforcement yielding (y), and ultimate failure (u). Additionally, the release of prestress induces negative camber in the strengthened beams, resulting in compressive strains in the concrete and tensile reinforcement within the tension zone. During loading, the initial compression deformation at the bottom of the concrete and the tensile reinforcement gradually diminish to zero and subsequently evolve into tensile strains. The points where strains transition from compressive to zero are designated as the concrete decompression (cd) and the steel decompression (sd), respectively.

In accordance with the principles of strain compatibility and static equilibrium [21,22], the following assumptions are made in the analysis:

(1)

The strains in the reinforcement, CFRP strips, and concrete vary linearly with their respective distances from the neutral axis.

(2)

A perfect bond is presumed between the reinforcement, CFRP, and surrounding concrete, ensuring that no relative slip occurs.

(3)

The ultimate compressive strain of concrete is taken as 0.0033.

In this study, the NSM technique demonstrates superior bonding performance for the CFRP–concrete interface, exhibiting minimal deterioration under normal use conditions. However, the developed analytical model does not account for the descending section following the peak load of the strengthened beam. Despite these simplifying assumptions, they are unlikely to significantly impact the predictive accuracy of the model.

Figure 1 illustrates a simplified depiction of the characteristic points along with their corresponding cross-sectional strain distributions. In the analytical model, three failure modes are evaluated: concrete crushing, CFRP rupture, and concrete cover separation. The ultimate failure mode is identified by comparing the ultimate bearing capacities associated with each respective failure mode.

Figure 2a illustrates the geometric configuration and loading details of the four-point bending simply supported beam. As shown in Figure 2b, the concrete’s compressive response is linear until the tensile reinforcement yields. After yielding, the compressive contribution is modeled using a rectangular stress distribution, as specified by ACI-440.2R-17. The tensile behavior of concrete remains linear until cracking initiates. Figure 2c,d show that an idealized elastic–plastic model is employed to model the mechanical response of the tensile reinforcement, whereas CFRP demonstrates linear elastic properties up to ultimate strength.

The load–deflection (L–D) relationship for the strengthened beam is predicted based on its moment–curvature response, with a focus on midspan deflection. The analysis assumed that the curvature of the beam section varies linearly across different stages. Figure 3 illustrates the segmentation of the curvature change area, which is segmented into three regions along its length: pre-cracking, post-cracking, and post-yielding of tensile reinforcement. The midspan deflection is determined by aggregating the deflections of the region within the half-span length; deflection is obtained by integrating the corresponding curvature function, as described in Equation (1).

$${\delta }_{i(L_i-L_{i+1})}={\displaystyle \underset{L_i}{\overset{L_{i+1}}{\int }}{\chi }_i(x)\cdot }x\cdot d(x)$$

(1)

where L_i and L_i+1 represent the distances from the edges of the selected section to the support, while x denotes the length variable along the region.

As illustrated in Figure 4, the potential failure modes of the strengthened beam under static loading mainly include CCS failure, concrete crushing, and CFRP rupture (collectively referred to as traditional bending failure). In the prestressed NSM CFRP-strengthened structure, the combined effects of prestress and applied loads result in a significant stress concentration at the CFRP ends. This region experiences substantial shear forces, normal tensile forces, and bending moments, leading to the horizontal branching of flexural-shear cracks at the height of the tensile reinforcement near the CFRP ends. Consequently, this increases the risk of separation failure in the end concrete cover. To identify the most probable failure mode, the ultimate bearing capacities for each scenario are calculated and compared. Specifically, the ultimate loads for ductile failure, characterized by concrete crushing and CFRP rupture, are assessed using the plane section assumption based on material strengths. The method for calculating the ultimate load due to brittle failure (CCS) will be detailed in the subsequent section.

3. Calculation Model for CCS Failure

This study proposes a method where separation is presumed to initiate while the principal tensile stress, transferred from the CFRP strip end to the adjacent concrete cover, reaches the tensile strength of the concrete. A semi-quadrilateral pyramid model is utilized to analyze the fractured concrete section, as shown in Figure 5, with its geometric dimensions determined by factors such as beam width, concrete cover thickness, and CFRP dimensions. The analysis neglects any mutual slip between the CFRP and concrete. By incorporating the concrete tensile strength (f_ct), the resistance force (F_cf) exerted by the concrete fracture body corresponding to each CFRP strip can be calculated.

To evaluate the impact of several factors on concrete crack in CCS failure, including the weak surface of the concrete beneath the tensile reinforcement, along with the relative positioning between the tensile reinforcement and CFRP, the quantity of the CFRP, and the spacing between the adjacent CFRPs. It is assumed that the dimensions of the concrete fracture body at the edge of the prestressed NSM CFRP are constrained by specific geometric conditions [21,22,23], with its height and width being restricted by the thickness of the concrete cover and the width of the beam, respectively, calculated by Equations (2) and (3).

$$s_{\mathrm{cz}}=c_{\mathrm{c}}$$

(2)

$$s_{\mathrm{cy}}=\min (s_{\mathrm{f}}/2, s_{\mathrm{f}}^{\prime }, b_{\mathrm{g}}{/\tan 30}^{\circ }+a_{\mathrm{g}}/2)$$

(3)

where s_cy represents the width of the fractured body, s_f denotes the spacing between adjacent CFRP strips, $s_{\mathrm{f}}^{\prime }$ indicates the distance from the CFRP to the edge of the beam, and a_g and b_g are the width and depth of the groove, respectively. s_cz is the height of the fractured body, and c_c represents the thickness of the concrete cover. When a single CFRP strip is used for strengthening, s_f/2 should be disregarded; when more than two CFRP strips are utilized for strengthening, $s_{\mathrm{f}}^{\prime }$ for the middle CFRP strip should be ignored.

When formulating assumptions regarding the geometric configuration of the fractured body, the angles between the principal baseline of the semi-vertebral section and the directions of the length x, width y, and height z of the reinforced beam are denoted as α₁, α₂, and α₃, respectively. Based on prior research [23,24], the typical range for α₂ is between 28.5° and 35°, and the length of the fractured body, referred to as the resistance bond length (l_rb), can be calculated based on the thickness of c_c using Equation (4).

$$l_{\mathrm{rb}}=s_{\mathrm{cz}}/\tan {\alpha }_2$$

(4)

As shown in Figure 6, the vertical eccentricity y_c of the CFRP axial tensile stress F_f relative to the centroid of the fracture section induces an active moment M_f, which contributes to the fracture initiation at the CFRP end. Based on the geometric configuration of the fracture body, the value of the vertical eccentricity y_c is calculated using Equations (5)–(7).

$$y_{\mathrm{cp}}={\displaystyle {\int }_0^{lrb}\left(A_{(x)}\cdot y_{cp(x)}/V_{cp}\right)\cdot dx}={\displaystyle {\int }_0^{lrb}\left[\left(2\cdot s_{cy(x)}\cdot s_{cz(x)}\cdot \frac{1}{2}\right)\cdot \left(s_{cz(x)}/3\right)/V_{cp}\right]\cdot dx=}{\displaystyle \frac{c_c}{4}}$$

(5)

in which,

$$s_{\mathrm{cy}(x)}={\displaystyle \frac{s_{\mathrm{cy}}\cdot x}{l_{\mathrm{rb}}}},s_{\mathrm{cz}(x)}={\displaystyle \frac{c_{\mathrm{c}}\cdot x}{l_{\mathrm{rb}}}}$$

(6)

$$V_{\mathrm{cp}}={\displaystyle {\int }_0^{lrb}{A}_{(x)}\cdot dx}={\displaystyle {\int }_0^{lrb}\left(s_{cy(x)}\cdot s_{cz(x)}\right)\cdot dx=}{\displaystyle \frac{s_{\mathrm{cy}}\cdot c_{\mathrm{c}}\cdot l_{\mathrm{rb}}}{3}}$$

(7)

Therefore, the eccentricity between the centroid of the fractured body and the centroid of the CFRP is calculated using Equation (8):

$$y_{\mathrm{c}}=l_{\mathrm{f}}-y_{\mathrm{cp}}$$

(8)

where l_f is the distance between the centroid of the CFRP strip and the tension surface of the concrete.

As shown in Figure 6, the tensile stress transferred by CFRP to the fracture of the surrounding concrete is resisted by the bending moment M_r, which is generated by the tensile stress F_ctv on the top surface of the fracture body. Assuming a linear distribution of tensile stress, CCS failure initiates at the CFRP end and reaches its maximum extent at this location, gradually decreasing to zero at the cross-section in relation to the bond length l_rb. The tensile strength of concrete is employed to quantify the tensile stress at various cross-sections, and the calculation formula was derived from the principle of stress equilibrium, as shown in Equation (9).

$$f_{\mathrm{ct}(x)}=f_{\mathrm{ct}}-({\displaystyle \frac{f_{\mathrm{ct}}\cdot x}{l_{\mathrm{t}}}})$$

(9)

where l_t represents the length of the top surface of the fractured body, in which l_t = l_rb/cosα₂.

The tensile strength is derived by integrating the length l_t and area a_(x) of the top surface of the fractured body, as calculated using Equation (10).

$$F_{\mathrm{ct}}={\displaystyle {\int }_0^{l\mathrm{t}}{f}_{\mathrm{ct}(x)}\cdot a_{(x)}}\cdot dx={\displaystyle \frac{f_{\mathrm{ct}}\cdot l_{\mathrm{t}}\cdot s_{\mathrm{cy}}}{6}}$$

(10)

in which: $a_{(x)}={\displaystyle \frac{s_{\mathrm{cy}}\cdot x}{l_{\mathrm{t}}}}$.

F_ct is projected onto the vertical direction, and Equation (11) is used to calculate the contribution of concrete to the resisting bending moment M_r within the bond length l_rb.

$$M_{\mathrm{r}}=F_{\mathrm{ctv}}\cdot (l_{\mathrm{rb}}-x_{\mathrm{t}}\cdot \cos {\alpha }_2)$$

(11)

in which: $F_{\mathrm{ctv}}=F_{\mathrm{ct}}\cdot \cos {\alpha }_2$.

The distance x_t is measured from the action point of F_ct, as illustrated in Figure 5, to the CFRP end, and it is calculated using Equation (12).

$$x_{\mathrm{t}}={\displaystyle \frac{\left({\displaystyle {\int }_0^{l_{\mathrm{t}}}{f}_{\mathrm{ct}(x)}\cdot a_{(x)}\cdot x\cdot dx}\right)}{F_{\mathrm{ct}}}}={\displaystyle \frac{l_{\mathrm{t}}}{2}}$$

(12)

Based on the relationship between the embedment depth of CFRP and the centroid of the fracture body illustrated in Figure 7, when the active bending moment M_f induced by the stress in the CFRP exceeds the resistant bending moment M_r of the surrounding concrete, it can be inferred that the separation of the concrete cover begins. Consequently, before CCS failure occurs within the effective bond length l_rb, the maximum axial stress F_f in the CFRP is determined by Equation (13):

$$M_{\mathrm{f}}=M_{\mathrm{r}}\to F_{\mathrm{f}\left(\max \right)}\cdot y_{\mathrm{c}}=M_{\mathrm{r}}\to F_{\mathrm{f}\left(\max \right)}={\displaystyle \frac{M_{\mathrm{r}}}{y_{\mathrm{c}}}}$$

(13)

According to Equation (13), as the embedded position of CFRP moves farther from the tensile surface (with a larger l_f), the induced active bending moment M_f decreases, thereby reducing the possibility of end CCS failure in the strengthened beam. However, it is essential to acknowledge that the significant impact of the effect of the moment arm of the axial force on eccentricity y_c is not the sole determinant of separation failure. Concrete may also experience tensile fractures due to the axial stress F_f exerted by the CFRP. In this scenario, the effective stress F_fe on the CFRP strip is the sum of the peak stresses across all individual strips, as shown in Equation (14).

$$F_{\mathrm{fe}}={\displaystyle \sum _{i=1}^N{F}_{f(\max )}}$$

(14)

where N represents the quantity of the NSM CFRP strips.

Considering the effective stress F_fe of the CFRP strip, the bending behavior of the end section within the bond length l_rb can be determined based on the distribution of sectional strain, as given by Equation (15). At this stage, the tensile reinforcement of the strengthened beam has already yielded. According to ACI-440, a rectangular stress block is employed to model the compressive contribution of the concrete, while the stresses in both the compressive and tensile reinforcements are limited by their respective yield strengths (f_sy = ε_sy·E_s).

$$\begin{array}{cc}{M}_{\mathrm{CCS}}^{l_{\mathrm{rb}}}\hfill & ={\alpha }_1\cdot {\beta }_1\cdot \left(1-{\displaystyle \frac{{\beta }_1}{2}}\right)\cdot {\epsilon }_{\mathrm{cc},\mathrm{CCS}}\cdot E_{\mathrm{c}}\cdot b\cdot c_{\mathrm{CCS}}^2+{\epsilon }_{s,\mathrm{CCS}}^{\prime }\cdot E_{\mathrm{s}}\cdot A_s^{\prime }\cdot (c_{\mathrm{CCS}}-d_s^{\prime })\hfill \\ & +{\epsilon }_{s,\mathrm{CCS}}\cdot Es\cdot A_s\cdot (d_s-c_{\mathrm{CCS}})+F_{\mathrm{fe}}\cdot (d_{\mathrm{f}}-c_{\mathrm{CCS}})\hfill \end{array}$$

(15)

where E_c and E_s represent the elastic modulus of concrete and steel reinforcements, respectively. d_s’, d_s, and d_f denote the distances from the centroids of the compressive, tensile reinforcement, and CFRP to the compressive surface of the concrete, respectively. A_s’ and A_s indicate the cross-sectional areas of the compressive and tensile reinforcement, respectively, while ε_sy denotes the yield strain of the tensile reinforcement. Moreover, based on the plane section assumption and principles of static equilibrium, the strains in other materials are obtained from the strains of the CFRP, as described in Equations (16)–(19).

$${\displaystyle \frac{1}{2}}{\epsilon }_{\mathrm{cc},\mathrm{CCS}}\cdot E_{\mathrm{c}}\cdot c_{\mathrm{CCS}}\cdot b+A_s^{\prime }\cdot {\epsilon }_{s,\mathrm{CCS}}^{\prime }\cdot E_{\mathrm{s}}-A_s\cdot {\epsilon }_{s,\mathrm{CCS}}\cdot E_{\mathrm{s}}-A_{\mathrm{f}}\cdot {\epsilon }_{\mathrm{fe}}\cdot E_{\mathrm{f}}=0$$

(16)

in which:

$${\epsilon }_{\mathrm{cc},CCS}={\displaystyle \frac{{\epsilon }_{\mathrm{fe}}\cdot c_{\mathrm{CCS}}}{\left(d_{\mathrm{f}}-c_{\mathrm{CCS}}\right)}}$$

(17)

$${\epsilon }_{\mathrm{s},CCS}^{\prime }={\displaystyle \frac{{\epsilon }_{\mathrm{fe}}\cdot \left(c_{\mathrm{CCS}}-d_s^{\prime }\right)}{\left(d_{\mathrm{f}}-c_{\mathrm{CCS}}\right)}}\le {\epsilon }_{\mathrm{sy}}$$

(18)

$${\epsilon }_{\mathrm{s},CCS}={\displaystyle \frac{{\epsilon }_{\mathrm{fe}}\cdot \left(d_{\mathrm{s}}-c_{\mathrm{CCS}}\right)}{\left(d_{\mathrm{f}}-c_{\mathrm{CCS}}\right)}}\le {\epsilon }_{\mathrm{sy}}$$

(19)

by integrating the equations, Equation (20) can be derived:

$$\begin{array}{r}\left(E_{\mathrm{c}}\cdot b\right)\cdot c_{\mathrm{CCS}}^2+2\cdot \left(E_{\mathrm{s}}\cdot A_{\mathrm{s}}\prime +E_{\mathrm{s}}\cdot A_{\mathrm{s}}+N\cdot E_{\mathrm{f}}\cdot a_{\mathrm{f}}\cdot b_{\mathrm{f}}\right)\cdot c_{\mathrm{CCS}}\\ -2\cdot \left(E_{\mathrm{s}}\cdot A_{\mathrm{s}}\prime \cdot d_{\mathrm{s}}\prime +E_{\mathrm{s}}\cdot A_{\mathrm{s}}\cdot d_{\mathrm{s}}+N\cdot E_{\mathrm{f}}\cdot a_{\mathrm{f}}\cdot b_{\mathrm{f}}\cdot d_{\mathrm{f}}\right)=0\end{array}$$

(20)

Finally, with failures attributed to CCS failure, the ultimate flexural capacity of the strengthened beam can be determined using Equation (21). This equation is derived from the distribution of bending moment along the beam:

$$M_{\mathrm{CCS}}^{\mathrm{u}}={\displaystyle \frac{b_{\mathrm{s}}\cdot M_{\mathrm{CCS}}^{l_{\mathrm{rb}}}}{(l_{\mathrm{rb}}+l_{ub})}}$$

(21)

where b_s represents the spacing from the support to the closest point of the load application, also referred to as the shear span length. l_ub denotes the spacing from the support to the CFRP termination, corresponding to the length of the unstrengthened part of the shear span. These distances are illustrated in Figure 8.

4. Validation of the Analytical Model

To evaluate and analyze the flexural performance of strengthened beams where the concrete cover at the CFRP end undergoes separation failure, this study integrates test data from our own experiments, previous studies within our research group [3,25,26], and external references [24,27] for validation. The dataset includes non-prestressed, fully prestressed, and gradually prestressed NSM CFRP-strengthened beams. The geometric configurations and strengthening details are summarized in

Table 1. Geometric configuration and reinforcement particulars of the strengthened beam.

Specimen	lb (mm)	lub (mm)	bs (mm)	b (mm)	h (mm)	ds’ (mm)	ds (mm)	df (mm)	As’ (mm2)	As (mm2)	Af (mm2)	sf (mm)	sf’’ (mm)
PRS2900 [3]	2900	200	1150	160	350	40	310	340	402	402	64	65	47.5
PRS2900-G300-II [23]	2900	200	1150	160	350	40	310	340	402	402	64	65	47.5
PRS900-L250 [24]	2000	250	1000	150	250	35	215	240	402	402	64	60	47.5
PRS900-L1900 + 2 × 200 [24]	2000	250	1000	150	250	35	215	240	402	402	64	60	47.5
LB2C1 [25]	2000	200	800	160	280	40	240	272	101	226	101	80	40
S2V2R2 [22]	1400	50	500	100	177	25	152	171	101	85	28	30	35

Note: The symbols used in this table are consistent with those defined in the preceding sections. Specifically, l_b represents the bond length of the CFRP, whereas l_b’ denotes the spacing from the CFRP end to the support. sf and s_f’ indicate the spacing between adjacent CFRP strips and the distance from the outermost CFRP sheet to the edge of the beam, respectively.

, while the material properties are provided in

Table 2. Material properties of the strengthened beam.

Specimen	Concrete		Reinforcement		CFRP
	fc (MPa)	Ec (GPa)	fsy (MPa)	Es (GPa)	ffu (MPa)	Ef (GPa)
PRS2900 [3]	42.47	32.5	440	200	2550	142
PRS2900-G300-II [23]	46.82	32.5	440	200	2550	142
PRS900-L250 [24]	47.69	32.5	440	200	2068	131
PRS900-L1900 + 2 × 200 [24]	42.61	32.5	440	200	2068	131
LB2C1 [25]	32	30	545	205	2350	170
S2V2R2 [22]	46	32.5	730	200	2740	159

Note: The material performance parameters listed herein are sourced from various references [3,24,25,26,27].

.

4.1. L–D Relationship

Figure 9 illustrates the prediction of the analytical model for the L–D behavior of the strengthened beams experiencing CCS failure at the CFRP end. The results show a good agreement between the analytical predictions and experimental data, with the three-line relationship accurately depicting the failure process. The initial negative camber caused by the application of prestress leads to a slight overestimation of flexural stiffness in the early loading stage. However, as loading progresses, the discrepancy diminishes.

The failure at the CFRP end due to CCS is an early failure mode, especially when excessive prestress is applied to CFRP. For example, beams PRS2900 and PRS900-L250 are fully prestressed with prestress levels of 1000 MPa and 900 MPa, respectively. Significant stress concentrations occur at the ends of these strengthened beams as the tensile reinforcement nears its yield stress. This leads to a substantial increase in beam deflection, accelerating crack propagation at the CFRP ends and their horizontal branches. Under the combined effects of high shear stress from prestress and external loads, the concrete cover at the CFRP ends may separate prematurely.

4.2. Comparison of Characteristic Loads

As illustrated in Figure 9a, the analysis results consistently overestimated the characteristic loads at all levels. Specifically, the cracking load was overestimated by 6.51% to 15.60%, the yield load by −0.18% to 5.51%, and the failure load by 3.55% to 12.39%. These discrepancies were more pronounced compared to those observed when predicting the behavior of strengthened beams subjected to bending failure. Considering that CCS is a brittle failure mode with no clear warning signs prior to failure, these deviations are deemed acceptable within the context of this study. From the comparison of characteristic deflections presented in Figure 9b, the analysis model slightly underestimated the cracking deflection and yield deflection, with deviations ranging from 8.33% to 21.59% and 3.16% to 15.01%, respectively. The prediction results for the failure deflection of the strengthened beams exhibited significant variability, with differences ranging from −15.01% to 10.29%. This variability can be attributed to the substantial influence of the end crack spacing (the length of concrete tooth) on the failure load calculations for the prestressed strengthened beams. In the absence of precise data from other research, such deviations are expected and considered reasonable.

The prediction results of the analysis model for the bending behavior of the strengthened beams are outlined in

Table 3. Comparison of main results.

Specimen	α1 (°)	α2 (°)	α3 (°)	lrb (mm)	Ffe (kN)	Pccu (kN)	Pccsu (kN)	Pccsu(exp) (kN)	Pccsu
									/Pccsu(exp)
PRS2900 [3]	35.5	30	33.7	51.96	40.28	178.4	162.3	155	1.05
PRS2900-G300-II [23]	35.5	30	33.7	51.96	27.17	197.9	194.7	188	1.04
PRS900-L250 [24]	36.9	33.9	30	44.64	24.8	125.3	99.1	95	1.04
PRS900-L1900 + 2 × 200 [24]	36.9	33.9	30	44.64	19.92	155	143.1	135	1.06
LB2C1 [25]	35	35	35	57.12	20.4	164.3	118.8	116.1	1.02
S2V2R2 [22]	35	35	35	35.7	5.69	88.1	84.46	78.5	1.13

and illustrated in Figure 10. The data and figure demonstrate a high degree of consistency between the predicted results and experimental observations. However, the model underestimated the ultimate deflection of the prestressed NSM CFRP-strengthened beams. This discrepancy is likely due to the confinement effect of the concrete cover in the shear span section, which restricted the displacement and rotation of the end concrete tooth. Consequently, even when the axial stress in the CFRP reached the calculated threshold, failure did not occur. Moreover, the model overestimated the failure deflection of the non-prestressed beams. This discrepancy can be attributed to the increased crack width in the flexural-shear section prior to failure, which diminished the confinement effect of the concrete and resulted in significant deformation and shear dislocation at the ends of the concrete teeth. These factors were not adequately considered by the analytical model.

5. Influence of Prestress Level at the CFRP End

The end prestress level of CFRP significantly influences the bending behavior of strengthened beams. As illustrated in Figure 11, integrating a non-prestressed section into the existing fully prestressed section can substantially increase the load-bearing capacity of the tensile reinforcement after yielding [26]. Specifically, adding a 100 mm non-prestressed section at both ends of the strengthened beam increases the ultimate bearing capacity by 6.67%, while the corresponding midspan deflection rises by 13.32%. Extending this section to 200 mm results in more substantial improvements: a 26.67% increase in ultimate bearing capacity and an 83.59% rise in midspan deflection. However, extending the non-prestressed section to 300 mm only yields marginal gains of 5% in ultimate bearing capacity and 6.03% in midspan deflection compared to the 200 mm configuration. Therefore, when the non-prestressed section length is too short, its impact on performance enhancement is limited; conversely, when the total bonded length exceeds 90% of the clear span, increasing the non-prestressed section length offers relatively limited benefits.

When the CFRP length is limited, the fully prestressed section experiences immediate CCS failureonece the tensile reinforcement reaches its yield strength. By reducing the length of the fully prestressed section and implementing a gradient prestress distribution [25], both the ductility and failure load of the strengthened beam can be significantly enhanced. As illustrated in Figure 12a, applying a single-stage prestress gradient increases the load and deflection at yield and failure as the gradient length grows. However, once the gradient length exceeds 300 mm, the ultimate load begins to decrease while the beam exhibits significant ductility characteristics. As shown in Figure 12b, for a two-stage prestress gradient, the ultimate load and corresponding deflection of the beam increase markedly, peaking at a gradient length of 400 mm before gradually decreasing. The data also indicate that as the gradient length increases, the ductility of the beam becomes more pronounced, reducing the probability of CCS failure. Therefore, it is recommended that the single-stage gradient length should not exceed 300 mm, and the two-stage gradient length should not exceed 400 mm. An excessively short gradient prestressing length is inadequate to alleviate the stress concentration at the CFRP end, resulting in limited enhancement of the load-bearing performance of the strengthened beam and maintaining CCS as the predominant failure mode. Conversely, an overly long gradient prestressing length can diminish the flexural rigidity of the strengthened beam, leading to excessive deformation under the same external load and potentially altering the failure mode to a ductile one. The ultimate bearing capacity can only be accurately assessed by comparing the ultimate loads under different failure modes. Therefore, during the design process, it is crucial to achieve an optimal balance and avoid both excessively short and overly long gradient prestressing lengths.

6. Conclusions

In this study, an innovative method was developed to calculate the stress and bending moment in CFRP by modeling the geometry of the failure body and integrating it with a concrete tooth model. A comprehensive theoretical framework was further established that transitions from bending moment–curvature relationships to L–D characteristics. The results highlight the significant impact of the end prestress level of CFRP under the CCS failure mechanism, providing specific recommendations for optimizing CFRP strengthening design in practical engineering applications. The key conclusions are summarized as follows:

(1)

A comprehensive analysis model was developed, incorporating strengthening details and assuming the geometric configuration of the CCS failure body. This model utilized a concrete tooth model to facilitate the accurate determination of CFRP stress values and the corresponding bending moment at failure.

(2)

An analytical framework was developed to investigate the loading process. By analyzing the strain distribution across sections of the strengthened beam under load, the moment–curvature response can be derived. Integrating the curvature over different sections allows for an accurate determination of the L–D relationship during the loading process.

(3)

The calculation results of the analysis model were compared with the experimental data to validate the model’s accuracy. The model slightly overestimates the characteristic load and underestimates the midspan deflection in strengthened beams. Specifically, the deviation for the failure load ranges from 3.55% to 12.39%, while the deviation for the failure deflection ranges from −15.01% to 10.29%.

(4)

The prestress level at the CFRP end has a substantial impact on the failure mode of CCS. Reducing the prestress level or increasing the length of the low-prestressed region enhances the beam’s ductility and decreases the probability of CCS failure. It is recommended that when employing a non-prestressed bonding section, the total CFRP strengthening length should not exceed 90% of the clear span. For single-stage gradient prestressed sections, this length should be limited to 300 mm, while for two-stage gradient prestressed sections, it should be limited to 400 mm.