Experimental and Theoretical Study on Anchorage Loss of Prestressed CFRP-Reinforced Concrete Beams

Abstract

To investigate the anchorage loss mechanism of externally prestressed CFRP tendons in concrete beams, this study introduces a novel theoretical calculation system (TCS) developed through both the finite element method (FEM) and experimental validation. Firstly, the FEM and the proposed TCS were employed based on the mechanism of anchorage loss to compute the deformation of each part of the prestressed tendon–main beam connection system, ensuring result accuracy through mutual validation. Subsequently, field tests, designed according to FEM guidelines, measured the anchorage loss in externally prestressed CFRP tendons, with long-term monitoring included. Finally, experimental data were then used to refine the TCS. The results indicate that deformation at the connecting screw and the front end of the steel reaction frame constitutes approximately 95% of the total deformation, with theoretical calculations aligning closely with the FEM results. The field tests revealed that the anchorage loss of a 12 m long CFRP tendon under 950 MPa prestress accounted for about 35% of the total prestress loss. The discrepancy in deformation compared with the theoretical results was due to a gap of approximately 0.4 mm between the two threaded connections, which can be minimized by improving construction techniques. After correction, the calculation error was reduced to about 5%. Control variable studies confirmed that anchorage loss is influenced by the prestress level, the dimensions of the steel reaction frame front end, the connecting screw length, and the number of thread gaps. This study provides a comprehensive approach for accurately predicting and mitigating anchorage loss in externally prestressed CFRP tendons, with significant implications for future engineering applications.

1. Introduction

Carbon fiber-reinforced polymer (CFRP) tendons exhibit high modulus, light weight, fatigue resistance, and corrosion resistance. The development of external prestressing CFRP tendon reinforcement technology has revealed significant advantages over other reinforcement techniques. It enhances bridge durability without substantially increasing structural weight. Additionally, it provides crack closure, combined flexural and shear reinforcement, and high efficiency. Therefore, it has been widely used in flexural reinforcement in bridge engineering [1,2,3,4,5]. To ensure effective reinforcement, prestress loss has become a key research focus, with anchorage loss being a significant component of the total prestress loss. Anchorage loss occurs due to the deformation of components within the prestressed tendon–main beam connection system. This deformation reduces the prestress in the tendons.

In bridge structures, the calculation method for anchorage loss of externally prestressed CFRP tendons typically follows the method used for steel bars and strands [6]. However, significant differences in practical anchoring methods prevent simple interchangeability [7]. The anchorage loss of externally prestressed CFRP tendons primarily originates from anchoring deformation, which depends on the type of anchoring device and tensioning method. These anchoring devices include the clamp type, the bonding type, and the combination type [8,9,10,11,12,13,14]. Zhou et al. [15] designed a small-sized variable-stiffness wedge (VSW) for BFRP tendons. This anchorage resulted in a deformation of 0.6 mm due to short-term losses, causing approximately 29 MPa of prestress loss. The prestress loss structure of externally prestressed CFRP tendons, similar to that of steel strands, is divided into short-term and long-term losses [10]. Páez, P. M. [16] proposed a simplified method to evaluate prestress loss, improving design efficiency and accuracy. This method includes both long-term and short-term losses, with short-term losses involving factors such as elastic shortening, concrete shrinkage, and initial losses in prestressing steel. Kim et al. [17] found that in the Near-Surface Mounted (NSM) post-tensioning method, CFRP relaxation, a long-term loss, is the primary source of prestress loss, while short-term loss is mainly due to the anchoring set. Wang et al. [18] highlighted the similarity in the loss structure between CFRP tendons and steel strands. Comparing test and calculation values of anchoring loss for clamp-bonding anchoring devices revealed that the difference does not exceed 15%. Xu et al. [19] introduced the main components of prestress loss in CFRP tendon-reinforced concrete beams. Tao et al. [20] studied CFRP tendon reinforcement, finding that anchoring loss accounts for 45.9% of the total loss, while concrete elastic shrinkage loss is negligible. A report by the Research Committee on Continuous Fiber Reinforcing Materials [21] indicates that under long-term prestressing, CFRP tendons relax, with a creep fracture stress coefficient of 0.85 after 100 years. Additionally, some scholars [22,23] researched the prestress loss of carbon fiber cloth and sheets, concluding that early loss is rapid, but prestress stabilizes over time. Wang et al. [24] reinforced concrete beams with CFRP and studied the impact of anchoring deformation on prestress loss. The test results showed that anchor deformation and slippage between the anchor and CFRP fabric cause an instantaneous prestress loss rate of approximately 12.6% to 18.2%. Kim et al. [25] established a mechanical model, examined the anchoring effect of CFRP plates, and predicted the magnitude of short-term loss in their anchoring system, suggesting that short-term prestress loss should be 10% of the applied prestress. Jun et al. [26] found that prestressing CFRP-reinforced plates on RC beams of different sizes affects both short-term and long-term prestress losses, with smaller sizes resulting in higher losses. Shi et al. [27] developed a BFRP composite wedge anchor with 85% efficiency. At an axial force of 65 kN, no slippage occurred. In concrete beam reinforcement tests, a pre-tensioning load 1.05 times the predetermined value was applied to compensate for prestress losses.

The existing method for tensioning externally prestressed CFRP tendons in concrete beams involves tensioning the anchoring device and then fixing it with a nut, which differs from the tensioning method of steel strands. This method results in much smaller tendon shrinkage values compared with steel strands. However, the anchoring deformation values used for calculating prestressed CFRP tendon anchorage loss mostly adopt those for prestressed steel bars or steel strand clamp-type anchors, deviating significantly from actual conditions. There is also a lack of testing data on the anchorage deformation of prestressed CFRP tendons, leading to less accurate existing calculation methods for CFRP tendon anchorage loss. Therefore, experimental measurements of the anchorage deformation of prestressed CFRP tendons and establishing an anchorage loss assessment method aligned with actual engineering conditions are necessary for more accurate evaluations.

To investigate the influencing factors of anchorage loss and the magnitude of anchoring deformation, finite element method (FEM) models and a theoretical calculation system (TCS) were developed based on the loss mechanism. Figure 1 illustrates the overall research process, including the development of FEM models, the establishment of the TCS, and the subsequent steps involved. These simulations and theoretical calculations determined the magnitude of anchorage deformation. Subsequently, practical bridge tests were conducted according to the established model to obtain actual anchorage loss values, total deformation, and the proportion of anchorage loss in total loss under long-term loads. Control variable studies were conducted on the influencing factors to understand their impact on anchorage loss. Based on the experimental results, corrections were made to the errors in the theoretical calculations, providing references for the calculation of anchorage loss in different practical engineering scenarios.

2. FEM Calculation and Theoretical Analysis

2.1. Mechanism of Anchorage Loss

Figure 2 illustrates the externally prestressed CFRP tendon reinforcement system for concrete beams. The reinforcement system is fixed to the bottom of the beam via chemical anchors and mechanical connections. During reinforcement, prestress is applied to the CFRP tendons to actively close any cracks that develop in the bottom span of the beam during normal use, thereby extending the bridge’s service life.

The reinforcement system consists of CFRP tendons and the CFRP tendon–main beam connection system. As shown in Figure 3, the CFRP tendon–main beam connection system includes a CFRP tendon anchorage, a connecting screw, a fixed nut, and a steel reaction frame. Prestressing is applied to the CFRP tendons by using a jack, achieving active reinforcement.

As shown in Figure 4, when applying prestress to the CFRP tendons, the tensioning device must be placed at the end of the steel reaction frame, i.e., the tensioning end. Subsequently, the hydraulic jack moves the CFRP tendons and the front steel plate towards the front end of the steel reaction frame via a screw. This elongates the CFRP tendon, generating prestress to resist deformation. Once the desired prestress level is reached, the fixation nuts are tightened to complete the prestressing process of the CFRP tendons, and the strengthening procedure is finished as illustrated in steps 1 to 3 in Figure 4.

Before completing CFRP tendon reinforcement, the main causes of prestress loss are anchorage loss, CFRP tendon friction loss, and concrete elastic compression loss. After reinforcement is completed, under the long-term action of prestress, there occur CFRP tendon relaxation loss, CFRP tendon temperature difference loss, and concrete shrinkage creep loss.

This research study examines the phenomenon of anchoring loss, which occurs when the stressed region at the tensioning end changes before and after unloading. Due to the alteration in the stressed region (illustrated in the red area in Figure 5), before unloading, the anchorage and the rear end of the steel reaction frame are under compression. After unloading, the anchorage, the connection screw, and the front section of the steel reaction frame experience tensile stress, whereas the fixed nut undergoes compressive stress. This modification causes distortion in the components, which is then transmitted to the CFRP tendon, resulting in relaxation and loss of prestress, commonly referred to as anchorage loss. To calculate anchorage loss, some assumptions are taken into account.

1. The stress conditions between the CFRP tendon anchor and the front plate only impact the outward deformation of the anchor, not the loss of prestress in the CFRP tendon, both before and after unloading. The effect is disregarded in later computations.

2. Since only a little amount of deformation is sent to the anchors at both ends, it can be disregarded. Thus, it is inferred that all the deformation is transferred to the CFRP tendon.

To summarize, the loss of anchorage in the CFRP tendon may be categorized into four components: deformation at the connecting screw, the front end of the steel reaction frame, the fixed nut, and the rear end of the anchor.

2.2. Description FEM Model

Based on the above loss mechanism, a FEM model for the prestressed CFRP tendon tensioning end was established, as shown in Figure 6. The deformation of each component (steel reaction frame, fixed nut, connecting screw, anchor, and CFRP tendon) was determined by using FEM analysis. The following assumptions and limitations were made for the model:

1. “Tie” constraints are applied between the CFRP tendons and the anchor, assuming that no relative slippage occurs between them.

2. “Hard contact” is used among the anchorage, the connecting screw, the fixed nut, and the steel reaction frame.

3. The steel reaction frame is anchored to the bottom of the reinforced concrete beam via chemical anchors, and the connection between them is treated as a rigid body.

4. The boundary conditions dictate that displacements and rotations in all three directions of the beam are restricted to zero. During the prestressing process, the CFRP tendon experiences displacement solely along the z-axis, while the anchoring remains stationary along the x- and y-axes.

5. A load of 950 MPa is applied to the surface of the CFRP tendon. Furthermore, the simulation study of the tensioning end deformation is not influenced by the length of the CFRP tendon. Therefore, the length of the CFRP tendon is adjusted to align with the anchor.

The steel reaction frame is made of a 16 mm thick steel plate, with some dimensions as shown in Figure 7 below. The embedment depth of the anchor screw is 170 mm, and the diameter of the connecting screw is taken as 16.6 mm. The material parameters are shown in

Table 1. Corresponding relationship between components and materials in deformation area.

Component	Material	Elastic Modulus (MPa)	Poisson Ratio
Steel reaction frame	Q235	210,000	0.2
Fixed nut	Grade 8.8	210,000	0.3
Connecting screw	Grade 8.8	210,000	0.3
Anchor	Q235	210,000	0.3
CFRP tendon	CFRP	160,000	0.31

.

2.3. Analysis of FEM Results

The analysis results of the model are subdivided into the following six regions within the tensioning end, as illustrated in Figure 8: front end of the steel reaction frame (A), fixed nut (B), connecting screw at the location of the fixed nut (C), middle part of the connecting screw (D), connecting screw at the rear part of the anchor (E), and rear part of the anchor (F).

The stress contour plot in Figure 9 illustrates the distribution of stress in the simulation results, indicating that the highest stress concentration arises at the points where the fixed nut engages with the connecting screw threads, with a peak value of 640 MPa. Additionally, significant stress is observed at the connection points of the connecting screw at the left and right ends, approximately three turns of threads away from the ends. In the case of the anchor screws, discernibly elevated stress levels are evident on the two screws proximal to the CFRP tendon in comparison to those positioned at the distant end. Consequently, it is deduced that the predominant loss of anchorage originates from the vicinity encompassing the fixed nut, the connecting screw, and the rear extremity of the anchor. The designated nodes within these regions are denoted as a–f (Figure 9), correlating with the corresponding deformation areas A–F (Figure 8).

Table 2. Deformation amount of each deformation area.

Component		Deformation (mm)	Proportion of Total Deformation (%)
Front end of the steel reaction frame		0.270822	41.09
Fixed nut		0.001575	0.24
Connecting screws	(a)	0.027201	54.87
	(b)	0.309988
	(c)	0.024496
Rear of anchor		0.025051	3.8
Total deformation		0.659133	/

presents the deformation magnitude at the tensioning end, where the connecting screw includes the contact part between the connecting screw and the fixed nut (a), the middle section of the connecting screw (b), and the contact part between the connecting screw and the back of the anchor (c). The analysis of the deformation percentages reveals that the primary contributors to deformation at the tensioning end are the steel reaction frame and the connecting screw. Deformation at the fixed nut and the rear section of the anchor screw accounts for only 4.04% of the total, rendering it negligible for subsequent calculations. Furthermore, it is noted that the deformations of the two connecting screws, situated at the fixed nut and the rear section of the anchor screw, are approximately equal, each measuring around 0.025 mm. This deformation predominantly manifests along the three threads of the stressed section, allowing for simplification in subsequent calculations.

From the preceding analysis, it is deduced that the deformations within the anchorage area primarily stem from four components: (1) the front end of the steel reaction frame; (2) the fixed nut; (3) the connecting screw; (4) the rear section of the anchor screw. However, the deformations associated with components (2) and (4), specifically the fixed nut and the rear section of the anchor screw, are deemed negligible in comparison with those of the front end of the steel reaction frame and the connecting screw. Consequently, for optimization purposes, these negligible deformations are omitted from subsequent calculations. As such, the crux of the theoretical calculation revolves around establishing the computational framework for analyzing the deformations of the front end of the steel reaction frame and the connecting screw.

2.4. Theoretical Calculation Analysis

The FEM results indicate that significant deformation primarily occurs at the front end of the steel reaction frame and the connecting bolt. To address the deformation of these components, a TCS is proposed.

The deformation at the front end of the steel reaction frame primarily arises from the interaction between this section and the fixed nut, simplifying the analysis to the deformation of the contact area. As depicted in Figure 10, this contact area consists of the vertical plate welded to the side plate of the steel reaction frame. Recognizing that the side plate also experiences deformation, we model the support at both ends as simply supported. Consequently, calculating the deformation of the steel reaction frame simplifies to determining the deflection of a simply supported beam under a concentrated force at its midspan. There are two simplifications to consider in the calculation.

The formula for calculating the deformation ${\mathsf{\omega }}_1$ of the front end of the steel reaction frame is as follows:

$${\mathsf{\omega }}_1={\displaystyle \frac{\mathrm{F}{\mathrm{x}}^3}{48{\mathrm{E}}_1\mathrm{I}}},$$

(1)

where $\mathrm{F}$ is the concentrated force, $\mathrm{x}$ is the distance between the two simply supported positions, ${\mathrm{E}}_1$ is the modulus of elasticity of Q235, and $\mathrm{I}$ is the moment of inertia of the cross-section.

The deformation of the connecting screw can be conceptualized as the deformation of a cylinder experiencing axial tensile force, as depicted in Figure 11. Given that the stress primarily accumulates at the three turns of threads in the contact portion of the connecting screw and the resulting deformation is relatively minor, we simplify the analysis by treating the connecting screws at these locations as part of the middle section for uniform deformation calculation.

According to the stress–strain relationship, the formula for calculating the deformation ${\mathsf{\omega }}_2$ of the connecting screw under the action of prestress can be obtained as follows:

$${\mathsf{\omega }}_2={\displaystyle \frac{4\mathrm{F}\left({\mathrm{y}}_1+{\mathrm{y}}_2+{\mathrm{y}}_3\right)}{\mathsf{\pi }{{\mathrm{d}}_1}^2{\mathrm{E}}_2}},$$

(2)

where ${\mathrm{y}}_1$ represents the length of the middle section of the connecting screw, ${\mathrm{y}}_2$ denotes the length of the stressed section of the connecting screw situated at the fixing, ${\mathrm{y}}_3$ denotes the length of the stressed section of the connecting screw positioned at the rear part of the anchor, ${\mathrm{d}}_1$ represents the effective diameter of the cross-section of the connecting screw, and ${\mathrm{E}}_2$ represents the modulus of elasticity of steel (Grade 8.8).

By combining Equations (1) and (2), we can simplify them to obtain the TCS of the total deformation (a) at the tensioning end. From this, we can derive the formula for calculating the anchorage loss, as shown in the specific equation below:

$$\mathrm{a}={\mathsf{\omega }}_1+{\mathsf{\omega }}_2,$$

(3)

$${\mathsf{\sigma }}_{\mathrm{l}}={\displaystyle \frac{\mathrm{a}{\times \mathrm{E}}_{\mathrm{c}\mathrm{f}\mathrm{r}\mathrm{p}}}{\mathrm{L}}},$$

(4)

where ${\mathsf{\omega }}_1$ is the deformation of the steel reaction frame under the action of prestress, ${\mathsf{\omega }}_2$ represents the deformation of the connecting screw under the action of prestress, $\mathrm{L}$ denotes the length of the CFRP tendon, and ${\mathrm{E}}_{\mathrm{c}\mathrm{f}\mathrm{r}\mathrm{p}}$ is the modulus of elasticity of the CFRP tendon.

By substituting the FEM data into the TCS, with $\mathrm{F}$ set to 107.4 $\mathrm{k}\mathrm{N}$, $\mathrm{x}$ set to 79 $\mathrm{m}\mathrm{m}$, ${\mathrm{E}}_1$ set to $210$,$000$ $\mathrm{M}\mathrm{P}\mathrm{a}$, $\mathrm{I}$ set to $\mathrm{19,660.8}\mathrm{m}{\mathrm{m}}^4$, ${\mathrm{y}}_1$ set to 139 $\mathrm{m}\mathrm{m}$, ${\mathrm{y}}_2$ set to 7.5 $\mathrm{m}\mathrm{m}$, ${\mathrm{d}}_1$ set to 16.6 $\mathrm{m}\mathrm{m}$, and ${\mathrm{E}}_1$ set to $210$,$000$ $\mathrm{M}\mathrm{P}\mathrm{a}$, the results are as follows:

$${\mathsf{\omega }}_1=0.267192 \mathrm{m}\mathrm{m}\hspace{1em}{\mathsf{\omega }}_2=0.363915 \mathrm{m}\mathrm{m}\hspace{1em}\mathrm{a}=0.631107 \mathrm{m}\mathrm{m}\hspace{1em}{\mathsf{\sigma }}_{\mathrm{l}}=8.70 \mathrm{M}\mathrm{P}\mathrm{a}.$$

Figure 12 illustrates a comparison between the results of the FEM and the TCS. It is evident that the deformation at the front end of the steel reaction frame differs by only 1.34%, while the deformation of the connecting screw differs by 0.61%. This comparison highlights the applicability of the theoretical formulas discussed previously. The combined deformations obtained from the TCS for both components amount to 0.631107 mm, deviating from the value obtained from the FEM simulation results (0.659133 mm) by only 4.3%.

3. Experimental Study

3.1. Overview of Project

To study the anchorage loss before and after tensioning, to continuously monitor the long-term prestress loss, and to analyze the proportion of anchorage loss in the total prestress loss, this paper conducted an experimental study on the prestressed reinforcement of concrete beams. Figure 13 shows the prestressed CFRP tendon–main beam connection system: the anchor and the steel reaction frame were connected through the internal thread at the end of the anchor and the connecting screw rod, and the steel reaction frame was connected to the bottom of the beam through anchoring screws (embedded depth of 170 mm). The steel reaction frame served two different purposes at its ends: one end was used as the fixed end, where a through-type pressure sensor was installed to monitor the magnitude of prestress and prestress loss; the other end was used as the tensioning end, where a hydraulic jack was placed to apply prestress to the CFRP tendons. Note that the CFRP tendons used in this reinforcement system had a smooth surface, a diameter of 12 mm, and an elastic modulus of 160 GPa, and the anchorages were physical connection anchorages with high anchorage efficiency. Subsequent experiments disregarded the relative slippage between the anchorage and the CFRP tendons.

To validate the reliability of the FEM model and the TCS mentioned above, seven CFRP tendons were selected as test objects, consisting of one control group and six variable groups. The experimental conditions were modified by adjusting factors such as the dimensions of the front steel plate, prestress magnitude, length of the connecting screws, and torque applied to the fixed nuts, as detailed in

Table 3. Test conditions.

Group	CFRP Tendon Length $\mathbf{L}$ (mm)	Prestress (MPa)	Connecting Screw Length ${\mathbf{y}}_1$ (mm)	Side Panel Spacing (mm)	Fixed Nut Torque (N·m)
1	12,000	950	139	79	/
2	12,000	950	135	59	400
3	12,000	950	137	59	/
4	12,000	475	374	59	400
5	12,000	475	370	59	/
6	12,000	950	374	59	400
7	12,000	950	376	59	/

. Each condition was tested twice, and the average result was recorded.

By using the tensioning method described in Section 2.1, the prestress ${\mathsf{\sigma }}_{\mathrm{a}}$ is recorded before tightening the fixed nuts, and after removing the jack, the prestress ${\mathsf{\sigma }}_{\mathrm{b}}$ is recorded. The difference between ${\mathsf{\sigma }}_{\mathrm{a}}$ and ${\mathsf{\sigma }}_{\mathrm{b}}$ represents the anchorage loss of the prestressed CFRP tendons ${\mathsf{\sigma }}_{\mathrm{l}}$, which is essential for subsequent data analysis. Two points require attention.

1. Accurately controlling the length of the connecting rods is challenging in engineering. To examine the impact of varying connecting rod lengths on anchorage loss, pre-tensioning force was applied before the official tensioning of CFRP tendons. The length of the connecting screws at this stage was then measured for subsequent calculations.

2. To study the effect of thread clearances on prestress loss, groups 3, 5, and 7 required additional locking with a pipe wrench after normal tightening to minimize gaps. During fixing, all groups simulated actual construction workers tightening the nuts. For groups 3, 5, and 7, a specific torque was applied with an electric torque wrench as a control.

3.2. Results and Discussion

The measured data show that jack rebounding during the experimental (EXP) process made it difficult to precisely control the set tension value. By substituting the data into Equation (4), with $\mathrm{L}$ set to 11,600 mm and ${\mathrm{E}}_{\mathrm{c}\mathrm{f}\mathrm{r}\mathrm{p}}$ to 160,000 MPa, the deformation magnitude was calculated, as shown in

Table 4. Anchorage loss of prestressed CFRP tendon.

Group	${\mathsf{\sigma }}_{\mathbf{a}}$ (MPa)	${\mathsf{\sigma }}_{\mathbf{b}}$ (MPa)	${\mathsf{\sigma }}_{\mathbf{l}}$ (MPa)	Deformation (a/mm)
1	948.51	933.88	14.63	1.061
2	951.52	941.65	9.87	0.716
3	947.98	935.71	12.27	0.890
4	486.86	476.28	10.58	0.784
5	485.58	472.43	13.16	0.954
6	975.28	957.20	18.08	1.311
7	962.36	941.14	21.22	1.539

.

The EXP results indicate a relationship between anchorage loss and factors such as prestress magnitude, dimensions of the front end of the steel reaction frame, connecting rod dimensions, and clearance in threaded connections. For instance, in the control group, anchorage loss constituted about 1.54% of the total applied prestress, resulting in a total deformation of approximately 1 mm. By using EXP data from seven conditions and the TCS, the anchorage loss was calculated, as shown in

Table 5. Theoretical calculation value of anchorage loss.

Group	Steel Reaction Frame	Connection Screw	Deformation	${\mathsf{\sigma }}_{\mathbf{l}}$ (MPa)
1	0.267	0.364	0.631	8.70
2	0.112	0.355	0.467	9.196
3	0.111	0.359	0.470	11.995
4	0.057	0.471	0.528	10.046
5	0.057	0.465	0.522	12.719
6	0.118	0.978	1.097	17.888
7	0.113	0.936	1.049	19.988

. Data from both sources are summarized in Figure 14.

Comparing the data revealed significant differences in the seven deformation values (a) obtained from the experiments. The analysis indicated that the discrepancies were due to gaps between the anchorage and the connecting screw and between the fixed nut and the connecting screw during the experiments. These gaps increased actual deformation (a), causing a greater decrease in the prestress of the CFRP tendon. The magnitude of prestress, thread length, and screw diameter do not directly affect gap size; rather, they relate to the torque applied to the nut and the number of threaded connections. To address this, a correction was applied. Based on experimental data and standard construction processes, a gap value of 0.2 mm was assumed for each threaded connection. For groups 2, 4, and 6, where special treatments were applied, a gap value of 0.1 mm was used. We integrated the conclusions of the above analysis into the TCS and added a calculation section for the gap values caused by threaded connections in the TCS, resulting in a modified theoretical calculation system (MTCS).

The MTCS for the total deformation ${\mathsf{\sigma }}_{\mathrm{l}}$ at the tensioning end was derived from this analysis, yielding the formula for anchorage loss shown below:

$$\mathrm{a}={\mathsf{\omega }}_1+{\mathsf{\omega }}_2+{\mathrm{n}\mathsf{\omega }}_3,$$

(5)

$${\mathsf{\sigma }}_{\mathrm{l}}={\displaystyle \frac{\mathrm{a}{\times \mathrm{E}}_{\mathrm{c}\mathrm{f}\mathrm{r}\mathrm{p}}}{\mathrm{L}}},$$

(6)

where ${\mathsf{\omega }}_1$ is the deformation caused by prestress on the steel reaction frames, ${\mathsf{\omega }}_2$ is the deformation caused by prestress on the connecting screw, ${\mathsf{\omega }}_3$ is the gap value present at each threaded connection point, $\mathrm{n}$ is the number of threaded connections, $\mathrm{l}$ is the length of the CFRP tendon, and ${\mathrm{E}}_{\mathrm{c}\mathrm{f}\mathrm{r}\mathrm{p}}$ is the modulus of elasticity of the CFRP tendon.

The adjusted comparative data are shown in Figure 15. The results of EXP and MTCS are basically consistent, with an error controlled within approximately 5%, demonstrating high accuracy. These results can be used for the validation of experimental data and theoretical calculations in subsequent sections. This validates that applying external force to tighten the fixed nut and using a wrench for the threaded connections effectively reduce the gap.

By monitoring the prestress changes in CFRP tendons over 200 days, the prestress variation curve shown in Figure 16 was obtained. Anchorage loss occurred immediately after removing the jack, leading to an instantaneous decrease in prestress, followed primarily by relaxation and temperature-induced losses. The total prestress loss varied from 3% to 5%, with anchorage loss representing a significant portion. For a 12 m CFRP bar with a prestress of 950 MPa, anchorage loss comprised around 35% of the total loss, reaching up to 44% under the most adverse conditions. The difference in proportions was due to the smaller prestress under this condition, resulting in a smaller relaxation loss and thus an increased proportion of anchorage loss in the total loss.

3.3. Theoretical Validation

The experimental data were theoretically verified by using Formulas (5) and (4) obtained above, resulting in the data shown in Figure 17. The group analysis showed the following: By comparing (a) and (b), increasing the screw length by 176% resulted in a 79% increase in total deformation. This suggests that screw length significantly affects anchorage loss. In (c) and (d), a 100% increase in prestressing force led to a 66% increase in total deformation. The differing increments further confirm the presence of gaps. In (e), a 33.9% increase in the spacing between the side plates resulted in a 19.2% increase in total deformation. This indicates that increasing the size of the steel reaction frame front end increases anchorage loss.

The proposed theoretical framework shows high accuracy in reinforcing concrete beams with externally prestressed CFRP tendons. Anchorage loss forms a significant part of the total prestress loss and is categorized as a short-term loss. The primary factors influencing it are the dimensions of the steel reaction frame front end, the size of the connecting screws, and the gap values in threaded connections. Properly sizing these components in the reinforcement design can effectively reduce anchorage loss.

4. Conclusions

The anchorage loss mechanism of strengthening reinforced concrete beams with externally prestressed CFRP tendons was studied. FEM simulations were conducted, and the TCS was established to calculate the anchorage loss of the prestressed tendon–main beam connection system. Field tests on actual bridges were then performed to obtain the actual anchorage loss and the proportion of the total prestress loss, validating the reliability of the FEM simulations and correcting the TCS. The influencing factors of anchorage loss were also clarified. The conclusions are as follows:

• The FEM simulations revealed that the deformation of the connecting screw and the front end of the steel reaction frame accounts for approximately 95% of the total anchorage loss in a CFRP tendon–main beam connection system. The TCS was established based on the stress forms of these components, and the calculated results are consistent with the FEM analysis, indicating a high level of reliability.
• In the control group experiment, a CFRP tendon with a length of 12 m and a diameter of 12 mm, subjected to a prestress of 950 MPa, showed an anchorage loss of approximately 1.5% of the set prestress value, corresponding to an anchorage deformation of about 1 mm. Further studies indicated that the deformation magnitude is influenced by the prestress level, the dimensions of the steel reaction frame’s front end, the connecting screw length, and the thread gap values.
• The experimental results show relative discrepancies with both the FEM and theoretical results, attributed to gaps values ${\mathsf{\omega }}_3$ between the threaded connections. A 2 mm gap at each threaded connection was identified, and this can be reduced by tightening the threads with external force. However, the influencing pattern of these gaps requires further investigation. After adjusting the calculation formulas, the error was controlled within approximately 5%, demonstrating higher precision.
• This study provides a comprehensive framework for accurately predicting and mitigating anchorage loss in externally prestressed CFRP tendons, with significant implications for future engineering applications.