Experimental study on bond failure characteristics of ribbed CFRP bars embedded in ultra-high-performance concrete

This study investigates the bond failure mechanism between carbon fiber-reinforced polymer (CFRP) ribs and ultra-high-performance concrete (UHPC) in detail. Strain gauges were externally applied to the surface of CFRP bars in long anchor specimens to analyze in detail the variation law of bond stresses and slips with anchorage position, as well as the distribution differences of anchorage-related variables along the anchorage length under different failure modes. Twelve experimental specimens were subjected to pull-out tests, and the test observations were evaluated based on a theoretical analysis. The results indicate that the failure modes of the specimens mainly included pull-out failure and tensile-fracture failure of the CFRP bars, and the ultimate load was closely related to the anchorage length. The anchorage length of the CFRP bars in UHPC should be 20–30 days for 10 mm bars and 10–20 days for 12 mm bars. The anchorage performance of UHPC was comparable to that of steel-tube anchors, and it demonstrated excellent long-term performance. The bond-stress distribution along the anchorage length was nonuniform. As the anchorage length increased, the high-stress region contracted and the average bond strength decreased. Moreover, the relative slip decreased nonlinearly along the anchorage length, and the slip difference between the loading end and free end increased progressively with increasing load.

1 Introduction

Reinforced-concrete structures have been widely used in engineering since the late 19th century (Achillides and Pilakoutas, 2004). When exposed to harsh environments such as humidity, heat, and corrosion can lead to severe performance degradation of structures, such as underground space engineering (Chen et al., 2022; Ding et al., 2022; Park et al., 2022). In recent years, with the increasing use of compact and durable structures and components, the demand for ultra-high-performance concrete (UHPC) (Ke et al., 2025; Shokrgozar et al., 2024; Wang et al., 2019) and fiber-reinforced polymer (FRP) bars (Mazaheripour et al., 2013b; Shan et al., 2021) has grown in the infrastructure-engineering sector. UHPC is a new type of cementitious material characterized by ultra-high strength, excellent fatigue resistance, and superior tensile and corrosion resistance (Yoo and Yoon, 2017; Zou et al., 2023), making it an ideal alternative for flexural structures. The high strength-to-weight ratio and good fatigue resistance of FRP bars make them promising substitutes for conventional steel bars. Among FRP bars, carbon fiber-reinforced polymer (CFRP) bars offer high strength and elastic modulus, thereby enabling a more efficient structural performance. The combination of CFRP bars and UHPC is particularly significant for the application in underground space engineering. Specifically, both materials are extremely durable and well-suited for use in tough underground conditions, offering great potential for engineering durability longelity (Yang et al., 2021; Pan and Yin, 2025).

Adequate bonding between FRP bars and concrete is fundamental to achieving composite action and ensuring the structural performance of FRP-reinforced concrete systems (Yoo et al., 2015). Extensive research has been conducted on the bond behavior between FRP bars and normal concrete. Typical FRP bars with ribbed, sand-coated, indented (Abbass et al., 2013), or fiber-wound surface finishes can effectively enhance bond strength with concrete. Among them, ribbed FRP bars provide the most significant enhancement (Ke et al., 2023b). The combination of CFRP bars and UHPC is highly innovative, offering excellent mechanical properties and good durability; however, related research and applications are currently limited. From a design and analysis perspective, the relative slip between CFRP bars and UHPC influences the cracking behavior of CFRP UHPC structures, which is a critical issue for application (Li et al., 2024). Therefore, the bond–slip relationship must be accurately predicted. Existing design codes for normal and high-strength concrete structures are not applicable to those utilizing UHPC (Ke et al., 2024), and models developed for normal-strength fiber-reinforced concrete cannot be extended to UHPC (Ahmad et al., 2011). Moreover, the bond behavior of CFRP bars in conventional concrete has been extensively investigated, research into the fundamental bond mechanisms governing CFRP bars embedded in UHPC remains notably limited (Solyom and Balázs, 2020). Therefore, it is necessary to analyze in detail the distribution of bond stress and the evolution of relative slip along the entire bond length.

This study systematically investigated the distribution patterns of bond stress and relative slip in CFRP bars embedded in UHPC under different failure modes by surface-mounting strain gauges on long-anchorage specimens. The fundamental anchorage length range for CFRP bars was clarified. Through normalization procedures, a position function was proposed to preliminarily characterize bond stress distributions under selected experimental parameters (including anchorage lengths and concrete cover thicknesses). However, the proposed position function remains applicable only within the parametric range of this test series and cannot yet represent a universal bond-slip constitutive law for broader experimental conditions or other interfacial scenarios. Future work will refine and extend the applicability of this functional model via large-scale, multi-condition experiments and numerical inversion studies. These findings can serve as references for CFRP-UHPC structural design and safety assessment, yet their practical generalization requires further validation and optimization through subsequent research.

2 Materials and methods

2.1 Materials

2.1.1 CFRP bars

You may insert up to 5 heading levels into your manuscript as can be seen in “Styles” tab of this template. These formatting styles are meant as a guide, as long as the heading levels are clear, Frontiers style will be applied during typesetting. Typical FRP bars with ribbed, sand-coated, indented, or fiber-wound surface finishes can effectively enhance their bond strength with concrete. Among them, ribbed FRP bars offer the most effective improvement in bond strength. The CFRP bars for our tests were threaded ribbed CFRP bars manufactured domestically. Ribbed CFRP bars of three different diameters (8 mm, 10 mm, and 12 mm) were adopted. CFRP rib parameters of 8 mm, 10 mm, and 12 mm diameters: the elastic modulus increases slowly with the increase of diameter (119.9–124.3 GPa), the tensile strength of 8 mm rib is the highest (2,590.4 MPa), and the tensile strength of 10 mm and 12 mm is about 2000 MPa. Geometric parameters (rib radius, width, etc.) and CLR, Rr ratios increase significantly with diameter, and the surface structure of 12 mm ribs is more complex, which can be used to enhance anchoring performance (Ke et al., 2023a). Figure 1 presents both the photographs and schematic diagrams of the bars.

Figure 1

Figure 1. Ribbed CFRP bars used in this study.

CFRP bars with different surface finishes were designed to enhance their interaction with concrete. Variations in surface characteristics significantly influenced the bond performance between FRP bars and concrete. When analyzing results across different manufacturers or for different diameters of the same type of FRP bar, the influence of the surface geometric ratios must be considered. Previous studies have shown that the bond behaviour between FRP bars and concrete improves with increasing rib height, decreasing rib spacing, and a higher ratio of rib-projection area to rib-shear area.

2.1.2 UHPC

The mix proportion of the UHPC used in this study is listed in Table 1. The aggregates, cementitious materials, and steel fibers are shown in Figure 2. Refined quartz sand with a particle size of 0.2–1.25 mm and fineness modulus of 3.22 was employed as the aggregate. P·O 52.5 cement (ordinary Portland cement with a strength grade of 52.5 MPa) produced by Guangxi Yufeng Group Co., Ltd., along with semi-condensed silica fume, served as the cementitious materials. The use of a water-reducing agent was essential for achieving a low water-to-binder ratio, high strength, and excellent workability. A polycarboxylic acid-based water-reducing agent with a 35% water reduction rate and 23% solid content was added at 3% of the binder mass. The geometrical and mechanical properties of the steel fibers are summarised in Table 2. The addition of short, straight, copper-coated steel fibers with an aspect ratio of 65 (length: 13 mm, diameter: 0.2 mm) enhanced the ductility and mechanical properties of the material. The volume fraction of the steel fibers was 2%, consistent with that used in commercially available UHPC. The water-to-binder ratio for all specimens was 0.144.

Table 1

Table 1. Mix proportion of UHPC (kg/m³).

Figure 2

Figure 2. (a) Quartz sand (16–26 mesh); (b) quartz sand (26–40 mesh); (c) quartz sand (40–70 mesh); (d) ordinary Portland cement 52.5; (e) silica fume; (f) steel fiber (length = 13 mm, diameter = 0.2 mm).

Table 2

Table 2. Physical and mechanical properties of steel fibers.

The UHPC mix was added in the following sequence: fine aggregate, cement, silica fume, the mixture of water and the water-reducing agent, and steel fibers. To ensure a more uniform material distribution, the dry components were pre-mixed for 2 min. Then, the mixture of water and the water-reducing agent was added and stirred for 5 min to achieve better workability. Finally, while mixing, steel fibers with a volume fraction of 2.0% were gradually introduced through a screen into a horizontal shaft-forced mixer. Mixing continued for a further 3 min after the addition of the steel fibers. The workability was measured using a slump cone test, as specified in the Standard for test method of performance on ordinary fresh concrete (GB/T 50,080-2016). The average slump flow exceeded 630 mm (Figure 3), and no segregation of the steel fibers from the UHPC matrix was observed. The UHPC was then cast into plastic moulds and sealed with a waterproof membrane. After curing at room temperature for 16 h, the specimens were demoulded and subsequently cured in hot water at 90 °C ± 2 °C for 72 h. Following thermal curing, the specimens were stored at room temperature until the day of testing.

Figure 3

Figure 3. Slump-flow test.

Specimens were fabricated for the cube compressive strength test, the elastic modulus test and the tensile strength test. The specimen used for the compressive strength test is 100 mm × 100 mm × 100 mm. In accordance with the recommendations in Fundamental characteristics and test methods of ultra-high-performance concrete (T/CBMF 37-2018), prism specimens measuring 100 mm × 100 mm × 300 mm were used to determine both the prism compressive strength and elastic modulus, as shown in Figure 4b. The uniaxial tensile stress–strain behavior of UHPC was measured using dog bone-shaped specimens, and the tensile test setup is illustrated in Figure 4b. The tensile stress–strain relationship exhibited strain-hardening behavior, with a tensile-strain capacity of 2,800 με and tensile strength of 8.2 MPa. The average value of the mechanical properties of the material: cubic compressive strength 160.3 MPa, prismatic compressive strength 130.5 MPa, tensile strength 8.2 MPa, elastic modulus 46.8 GPa. The data show that the compressive strength of the material is significantly higher than the tensile strength, and the elastic modulus is at a medium to high level, indicating that the material may have good performance in the compressive structure.

Figure 4

Figure 4. (a) Compressive strength test for cubic specimens; (b) tensile strength test; (c) compressive strength test for prism specimens (elastic modulus test).

2.2 Specimen design and fabrication

2.2.1 Specimen design

Unlike ordinary steel bars, CFRP bars are composite bars fabricated through specific molds via drawing and extrusion processes. Splitting and grooving them at the center like steel bars would be impractical as they are anisotropic materials, and the CFRP bars used in this study have small diameters. Therefore, strain gauges were adhered directly to the surface of the CFRP bars, as shown in Figure 5.

Figure 5

Figure 5. Specimens design.

Four sets of long anchor specimens were designed to further investigate the distribution of bond stresses and slips between the CFRP bars and UHPC. According to the recommendations of CSA S806-12. (2012) Design and Construction of Building Structures with FibreReinforced Polymers, 2012), ACI 440.3R-04 (ACI Committee 440, 2004; ASTM D7913/D7913M-14, 2020; Huang et al., 2020), an anchorage length of 5 days was used to evaluate the bond characteristics between the FRP steel bars and concrete. The bond strengths derived from specimens with a 5 days anchorage length were used to extrapolate the critical anchorage length. As 10 mm and 12 mm ribbed CFRP bars tend to achieve higher bond strength, the anchorage lengths for 10 mm bars were set at 200 mm (20 days) and 300 mm (30 days), and for 12 mm bars at 120 mm (10 days) and 240 mm (20 days). Strain gauges were externally applied to the surface of the CFRP bars, and three specimens were cast for each configuration. The specimen details are provided in Table 3.

Table 3

Table 3. Details of specimens.

2.3 Investigated parameters

2.3.1 CFRP bar diameter

Ribbed CFRP bars with diameters of 10 mm, and 12 mm were used in this study to investigate the effect of bar diameter on bond performance. The CFRP bars were manufactured using a tension-pultrusion process, resulting in a threaded ribbed surface geometry.

2.3.2 Anchorage length

Anchorage length is a key factor in bond-performance studies. For CFRP The anchorage lengths with bars with diameters of 10 mm and 12 mm, the anchorage lengths were set to 20 days, 30 days and 10 days, 20 days in this study, respectively.

2.3.3 Thickness of cover layer

An adequate cover layer is essential for fully mobilising the mechanical properties of the reinforcement and ensuring effective composite action with the concrete. Concrete-splitting failure occurs when tensile stress exceeds the tensile strength of the concrete. Given the excellent mechanical properties of UHPC, the critical thickness of the cover layer may be lower than in conventional concrete. Therefore, the thickness of the cover layer was set to 6.3d ays and 7.5 days to investigate the effect of the cover layer thickness on bond performance in this study.

2.4 Test procedure

To minimise the influence of the strain gauges on the interfacial bond, the smallest possible strain gauges were selected. We employed strain gauges measuring 1 mm × 1 mm. The positions for adhering the strain gauges were first marked on the surface of the CFRP bars. A small flat area was then ground using a file, followed by cleaning with alcohol to ensure proper adhesion. The strain gauges were affixed using adhesive. Once the adhesive had cured, sealant was applied over the gauges to provide insulation and protect against moisture. Polyvinyl chloride (PVC) pipes were placed at both ends of the bond region between the CFRP bar and UHPC to control the bond length. UHPC was cast vertically into horizontally placed moulds, and a rubber mallet was used to gently tap the side walls during casting to prevent the strain-gauge wires from being compressed against the bar surface, which could compromise interfacial bonding. Strain gauges were applied in a staggered arrangement on opposite sides of the bar surface, with a spacing of either 20 mm or 40 mm. To further reduce the influence of the gauges on the interface, a 20 mm spacing was used near the loading end, whereas a 40 mm spacing was adopted near the free end. The detailed arrangement is shown in Figure 6.

Figure 6

Figure 6. Arrangement of strain gauges for CFRP bars: (a) specimen C-12-10d-6.3d; (b) specimen C-10-20d-7.5d; (c) specimen C-12-20d-6.3d; (d) specimen C-10-30d-7.5d.

Due to the limited size of the specimens, the pull-out tests could not be conducted directly on a conventional testing machine. Instead, a 300 kN through-centre jack was employed for loading, as shown in Figure 7. A welded steel support was placed between the base of the jack and top surface of the vertically oriented specimen to facilitate the placement of displacement gauges at the loading end.

Figure 7

Figure 7. Pull-out test setup.

The loading method affects bond performance. According to Ref (Huang et al., 2020), the use of a conventional compression-type pull-out setup without a PVC pipe at the loading end influences the test results. Introducing a PVC pipe at the loading end reduces the confinement effect of the setup on the concrete under compression, and this reduction becomes more pronounced as the PVC-pipe length increases. Based on these findings, the influence of the loading device on bond performance was not considered in this study. Two linear variable-displacement transducers were installed at the loading and free ends of the CFRP bars to measure the respective slippage. The load was read from the force display of the jack, with an accuracy of 0.01 kN. The relative slip between the CFRP bar and UHPC was captured using a DH3816N data-acquisition system, with load and displacement recorded synchronously. In accordance with CAN/CSA S806-02 (Ahmad et al., 2011), the test was terminated when any of the following conditions were met: (1) a through-crack formed in the UHPC; (2) the CFRP bar fractured in tension; or (3) the displacement at the free end of the CFRP bar exceeded 10 mm while the load remained constant.

3 Test results and analysis

3.1 Test results

The results of the pull-out specimens with externally applied strain gauges are presented in Table 4. The table summarises the outcomes of the pull-out tests, including the ultimate external loads, slip values at the loading end, and failure modes.

Table 4

Table 4. Results of specimens with externally applied strain gauges.

3.2 Failure characteristics

The failure modes observed in the ribbed CFRP bars bonded to UHPC included pull-out and tensile-fracture failures, as shown in Figure 8. In specimens C-10-20d-7.5d and C-12-10d-6.3d, shear pull-out failure occurred at the interface between the CFRP bars and UHPC. The thickness of the cover layer was sufficient, and no cracks appeared on the UHPC surface. In contrast, specimens C-10-30d-7.5d and C-12-20d-6.3d experienced bar fracture near the end of the embedded steel tube when the peak bond stress was reached. The failure was abrupt, the load decreased to zero immediately, and the specimens lost their load-bearing capacity. It can be concluded that the cover thicknesses of 7.5days and 6.3days are sufficient for 10 mm and 12 mm diameter CFRP bars because no splitting failure of UHPC was observed.

Figure 8

Figure 8. Failure modes of long anchor specimens: (a) pull-out failure; (b) FRP fracture failure.

3.3 Effect of anchorage length on ultimate load

The relationship between the ultimate load and anchorage length is shown in Figure 9. It was noted that the test results for anchorage length 5 days (50 mm for 10 mm diameter CFRP bar and 60 mm for 12 mm diameter CFRP bar) were derived from our previous study (Ke et al., 2023b). It can ben seen that, for a bar diameter of 10 mm and anchorage length of 30 days, the ultimate load reached 142.99 kN, which was close to the theoretical fracture load. When the bar diameter was 12 mm and anchorage length was 20 days, the ultimate load reached 181.41 kN. In this case, the specimen failed due to CFRP-bar fracture near the end of the anchorage zone. The actual fracture load was slightly lower than the theoretical value, possibly because the CFRP bar was not perfectly horizontally embedded in the anchorage. Because the setup lacked a spherical hinge, a small initial eccentricity angle likely developed during loading, placing the CFRP bar under a combined tensile–bending–shear state rather than pure axial tension. Given the low shear strength of CFRP bars, this reduced the force-utilisation efficiency. The CFRP bars with diameters of 10 mm and 12 mm fractured within the unbonded zones near the loading ends, meeting the fundamental requirements of the anchorage system. As seen from Table 4, CFRP bar fracture occurred at anchorage lengths of 30 days for the 10 mm bars and 20 days for the 12 mm bars. Considering the limited step intervals in the test design, the anchorage length of the CFRP bars in UHPC should be 20 days−30 days for 10 mm bars and 10 days–20 days for 12 mm bars. The steel tubes used in the anchorage system made of steel pipe and expansive cement had a length of 250 mm, which is comparable to the bond lengths in the aforementioned specimens. This indicates that UHPC can offer an anchorage performance equivalent to that of steel-tube anchors, with the added advantage of superior long-term performance. These experimental results provide a valuable reference for determining the required anchorage length of CFRP bars in structural applications where UHPC serves as the anchorage material and CFRP bars as the load-resisting reinforcement. During anchorage, the tensile-force direction of the FRP bars must be aligned with the axial direction or the force-utilisation efficiency factor must be considered.

Figure 9

Figure 9. Relationship between ultimate load and anchorage length.

3.4 Distribution of bond stresses along the anchorage length

The local bond stress between the CFRP bars and UHPC cannot be directly measured during the test. Instead, the bond stress is assumed to be uniformly distributed over short bond lengths. The stress in the CFRP bars is inferred from the strains recorded by strain gauges applied to the bar surface, and the bond stress at the interface is indirectly calculated using the equilibrium of sectional forces, as shown in Equation 1.

${\tau }_i=\frac{d_p}{\pi d·d_x}=\frac{A\left({\sigma }_{i+1}-{\sigma }_i\right)}{\pi d·d_x}=\frac{E\left({\epsilon }_{i+1}-{\epsilon }_i\right)}{\pi d·d_x}(1)$

where τ_i is the bond stress between the ith and i+1-th measurement points; d_p is the change in tensile force in the CFRP bar over the interval d_x; d is the diameter of the CFRP bar; d_x is the distance between the ith and i+1-th measurement points; A is the cross-sectional area of the CFRP bar; ε_i+1 and ε_i are the strains at the i+1-th and ith measurement points, respectively; and E_f is the elastic modulus of the CFRP bar.

The distribution of bond stress along the anchorage length was analysed for specimens exhibiting two different failure modes. Specimen C-10-30 days-6.3 demonstrated a higher force-utilisation efficiency factor for the ultimate tensile load of the CFRP bar. Therefore, the bond-stress distribution was examined in detail for specimens C-12-10d-6.3d (pull-out failure) and C-10-30d-6.3d (tensile-fracture failure). During testing, strain gauges near the loading end registered strain early on, whereas those near the free end recorded zero strain. At this stage, the load had not yet been transmitted to the free end, and no relative slip was observed between the CFRP bar and UHPC. The interfacial bond was mainly governed by chemical adhesion. As the applied load increased, the force was gradually transmitted toward the free end. At different stress levels, the distribution of interfacial-bond stresses along the anchorage length varied. As mentioned above, assuming a uniform bond-stress distribution between adjacent strain gauges, the stress in the bar was calculated from the measured strain, and the interfacial-bond stress was then derived. The midpoint between each pair of strain gauges was taken as the representative location for the bond stress in that segment.

Figures 10, 11 present the distributions of strain and bond stress along the anchorage length of the CFRP bars under different loading levels. Strain decreased progressively from the loading end to the free end, which is associated with the gradual transfer of force along the bar. As shown in Figure 11, the distribution of the interfacial-bond stress was clearly nonuniform, and the maximum bond stress occurred near the loading end. During initial loading, the bond-stress distribution curve exhibited significant fluctuations. Subsequently, bond stress increased with increasing load, and the shape of the distribution curve stabilised. At a load level of 20 kN, most of the external load was resisted by the front half of the anchorage zone near the loading end. As a result, the bond stresses were relatively high near the loading end, and stresses near the free end were relatively small. As the external load increased, the bond stresses at all measured points tended to increase. At higher load levels, the force extended to the free end, and the bond stress in the segment approximately 100 mm from the loading end also reached a high level. This behaviour is mainly attributed to the ribbed surface of the CFRP bars and relatively short anchorage length employed. When the displacement at the free end increased, the previously unbonded section of the CFRP bar at the free end began to engage in bonding. However, due to the large rib height (12 mm diameter) and high shear strength of UHPC, the bar segment in the previously unbonded region could not fully enter the bonded region, causing stress concentrations in this area. With shorter anchorage lengths, the nonuniformity of the bond-stress distribution tended to weaken.

Figure 10

Figure 10. Distribution of bond stress and CFRP bar strain along the anchorage length of specimen C-12-10d-6.3d (pull-out failure): (a) CFRP bar-strain distribution; (b) Bond-stress distribution.

Figure 11

Figure 11. Distribution of bond stress and CFRP bar strain along the anchorage length of specimen C-10-30d-7.5d (tensile-fracture failure): (a) CFRP bar-strain distribution; (b) Bond-stress distribution.

As shown in Figure 11, the peak bond stress for the specimen that experienced tensile-fracture failure was also located near the loading end. When the failure mode transitioned from pull-out to tensile fracture, the proportion of the external load transferred to a specific point (220 mm) decreased significantly. The nonuniformity in the bond-stress distribution along the anchorage length also became more pronounced. The bond stress near the free end decreased to a low level, indicating that the anchorage length approached the critical effective anchorage length. Although the ultimate load increased with longer anchorage lengths, the average bond strength displayed a decreasing trend. This is because longer anchorage lengths cause a more uneven stress distribution along the anchorage direction, with a relatively small high-stress region and, consequently, a lower average bond strength. Conversely, for shorter anchorage lengths, the high-stress zone is more extensive and stress distribution is more uniform, leading to a relatively higher average bond strength.

3.5 Distribution of slips along the anchorage length

In pull-out tests, localised slip along the anchorage length cannot be measured directly. It can generally be calculated indirectly from the strain recorded by strain gauges and slip values measured by displacement gauges installed at the loading and free ends. The elongation of the CFRP bars in each bonded segment is calculated from the stress values of the CFRP bars in that segment. Then, based on the local force equilibrium, the deformation at the measurement point caused by the compressive stress resulting from the interaction between UHPC and the CFRP bars is determined. By incrementally superimposing these two deformation components and the displacement at the free end, the relative slip at each measurement point can be obtained. The specific calculation procedure is as follows.

The rate of change of slip between the CFRP bar and UHPC in a locally bonded segment corresponds to the strain difference between the CFRP bar and UHPC, as expressed by the following equation:

$\frac{\mathrm{d}s}{\mathrm{d}x}={\epsilon }_f-{\gamma }_c{\epsilon }_c(2)$

where ε_f is the strain in the CFRP bar; ε_c is the strain in the concrete; γ_c is the correction coefficient that accounts for the nonuniform deformation of the concrete, taken as 1.0 (ASTM, 2014); and dx is the spacing between two adjacent strain gauges in the measurement segment.

Based on the local equilibrium of bonding forces in a differential segment, the following equation can be established:

$A_{f}d{\sigma }_f+A_{c}d{\sigma }_c=0(3)$

where A_c is the cross-sectional area of the concrete (mm²); A_f is the cross-sectional area of the CFRP bar (mm²); and σ_c and σ_f are the stresses in the concrete and CFRP bar, respectively (MPa).

Solving Equation 3 yields the strain in the UHPC, ε_c, as follows:

${\epsilon }_c=-A_f{E}_f{\epsilon }_f/\left(A_c{E}_c\right)(4)$

where E_c and E_f are the elastic moduli (MPa) of the concrete and CFRP bars, respectively. Substituting Equation 4 into Equation 2 yields the relative slip between the CFRP bar and UHPC at a given measurement point:

$s\left(x\right)={\displaystyle \sum _0^x}{\int }_i^{i+1}{\epsilon }_f\left(1+{\gamma }_c{A}_f{E}_f/\left(A_c{E}_c\right)\right)\mathrm{d}x(5)$

Accordingly, the slip values s_l(x) at various measurement points and the slip value sf at the free end are related as follows:

${\epsilon }_l\left(x\right)={\epsilon }_f+\epsilon \left(x\right)(6)$

The distribution of relative slip under different load levels is shown in Figure 12. The relative slip was nonuniformly distributed along the anchorage length and underwent a nonlinear reduction from the loading end to the free end. At the early stage of loading, slip at the free end was negligible. As the load increased, the force was progressively transferred towards the free end, and the relative slip near the free end increased accordingly. Simultaneously, the difference in relative slip between the loading and free ends became more pronounced, which is attributed to the deformation characteristics of the CFRP bars. In the later stages of loading (loading from 60 kN to 100 kN and form 100 kN–148 kN), as the bond stress approached its peak value at the interface, the slip increased more rapidly. For specimens experiencing pull-out failure, the initial slip load at the free end was relatively low, whereas the displacement at the free end was larger than that observed in specimens that experienced tensile-fracture failure. For specimens experiencing tensile-fracture failure, due to the sufficiently long anchorage length and extended bond-transfer path, even in the later loading stage, the bond stress in the anchorage zone near the free end remained relatively low. As a result, the deformation and loading in this region contributed less to the overall bond performance of the anchorage zone. The specimen failed due to CFRP-bar fracture when a small amount of slip occurred at the free end.

Figure 12

Figure 12. Distribution of relative slip: (a) specimen C-12-10d-6.3d; (b) specimen C-10-30d-7.5d.

The recommended setting of the anchorage length to 20–30 times the diameter of the steel bar (20 days - 30 days) is for the following key reasons, including the mechanical properties of the CFRP bars, the bonding characteristics between the CFRP bars and UHPC, and the behavior of the anchorage system under load. First, the ribbed surface of CFRP bars plays a significant role in enhancing the bond strength with UHPC, but as the anchorage length increases, the bond stress distribution becomes less uniform, leading to a reduction in average bond strength. The nonuniformity of bond stresses is due to the variation in the interaction between the ribs and the UHPC matrix along the anchorage length. At shorter anchorage lengths, the bond stress is concentrated in a smaller region, resulting in higher local bond strength and a more uniform stress distribution. As the anchorage length increases, the high-stress region contracts, and the bond stress is spread over a larger area, reducing the overall bond strength. This results in a decreasing trend in average bond strength with increasing anchorage length. Additionally, the slip between the CFRP bar and UHPC increases nonlinearly with the load, and the relative slip is larger at the free end of the anchorage zone, which further supports the observed optimal anchorage length range of 20–30 days.

3.6 Position function

The position function f(x) is a dimensionless empirical relationship established to describe the interfacial shear stress distribution along the anchorage length in bonded anchor systems (Mazaheripour et al., 2013a). The specific function development procedure is as follows: First, experimentally obtained local interfacial shear stresses are correlated with discrete locations (distance from the loading end) within the anchorage zone. Normalized data processing yields values of local relative slip and interfacial shear stress. Finally, polynomial regression via least squares fitting establishes the position function. This function explicitly characterizes the statistical distribution pattern of normalized shear stress versus normalized anchorage length for specific anchorage system.

This study takes specimen C-12-10d-6.3d and specimen C-12-10d-6.3d as examples to establish corresponding position functions, as shown in Figure 13. The detailed position functions are shown in Equations 7, 8. The coefficients of determination R2 are 0.993 and 0.961 for specimens C-12-10d-6.3d and C-10-30d-7.5d, respectively, indicating that the simplified model can effectively reflect the stress attenuation characteristics under the selected slip level. In addition, the fitted curves exhibit high consistency across varying slip levels, demonstrating the proposed position function’s robust stability in characterizing spatial decay features of bond stress. Specimen C-12-10d-6.3d and Specimen C-10-30d-7.5d reached peak stress at approximately x/l_b = 0.25 and 0.10, respectively, subsequently exhibiting monotonically decaying stress distributions along the anchorage length.

Figure 13

Figure 13. Distribution of normalised bond stress along the normalised anchorage position: (a) specimen C-12-10d-6.3d; (b) specimen C-10-30d-7.5d.

It should be noted that this position function serves only as a simplified representation and does not account for interface frictional dissipation, concrete–CFRP micro-roughness, or multi-level load (slip) interactions in this study. Consequently, its applicability is limited to the current testing scenario. Future work will involve larger-scale, multi-slip-level experiments combined with high-fidelity numerical inversion to systematically refine the function form and parameters toward a more general bond–slip constitutive model.

For specimen C-12-10d-6.3d, the position function is:

$f\left(x\right)=12.71\left(\frac{x}{l_b}\right)-37.57{\left(\frac{x}{l_b}\right)}^2+42.92{\left(\frac{x}{l_b}\right)}^3-18.05{\left(\frac{x}{l_b}\right)}^4(7)$

For specimen C-10-30d-7.5d, the position function is:

$f\left(x\right)=22.16\left(\frac{x}{l_b}\right)-81.84{\left(\frac{x}{l_b}\right)}^2+104.78{\left(\frac{x}{l_b}\right)}^3-45.31{\left(\frac{x}{l_b}\right)}^4(8)$

4 Conclusion

The distribution patterns of bond stress and relative slip along the anchorage length were investigated for long anchor specimens instrumented with externally applied strain gauges on the surface of ribbed CFRP bars. Differences in bond stress and relative slip under various failure modes were identified, and the basic anchorage-length range of CFRP bars in UHPC was determined. In addition, a position function for the bond–slip relationship was established. The main conclusions are as follows:

1. The failure modes observed in the specimens included pull-out failure and the tensile-fracture failure of the CFRP bars. For a bar diameter of 10 mm with an anchorage length of 30 times the bar diameter, the ultimate load reached 142.99 kN; for a 12 mm diameter bar with an anchorage length of 20 bar diameters, the ultimate load reached 181.41 kN. Both failures were attributed to CFRP bar fracture. It can be concluded that the cover thicknesses of 7.5 days and 6.3 days are sufficient for 10 mm and 12 mm diameter CFRP bars because no splitting failure of UHPC was observed.

2. The anchorage length of the CFRP bars in UHPC should be 20–30 days for 10 mm bars and 10–20 days for 12 mm bars. The anchorage performance of UHPC was comparable to that of the steel-tube anchors used in this study, suggesting that UHPC is a viable anchorage material for CFRP bars and can offer superior long-term performance. These findings provide a reference for determining the required anchorage length in anchorage systems utilising UHPC with CFRP bars with diameters of 10 mm and 12 mm as load-bearing reinforcing bars. During anchorage, the direction of force application on the FRP bar should align with the longitudinal axis, or a force-utilisation efficiency factor should be considered.

3. In both pull-out and tensile-fracture failure specimens, local peak bond stresses were concentrated near the loading end. Bond stress was nonuniformly distributed along the anchorage length. When the failure mode shifted from pull-out to tensile fracture, the nonuniformity of bond-stress distribution became more pronounced. Longer anchorage lengths resulted in a smaller high-stress region and lower average bond strength, whereas shorter anchorage lengths produced a larger high-stress region, more uniform stress distribution, and higher average bond strength.

4. Relative slip was nonuniformly distributed along the anchorage length, showing a nonlinear decrease from the loading end to the free end. The slip difference between the loading and free ends increased with increasing load, which is attributable to the deformation characteristics of the CFRP bars. As the ultimate load was approached, the slope of the bond–slip curve decreased and the rate of increase in relative slip accelerated. A position function for the bond–slip relationship curve was established to enable its correction.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

JZ: Conceptualization, Formal Analysis, Methodology, Writing – original draft. CY: Data curation, Funding acquisition, Project administration, Writing – review and editing. ZA: Investigation, Writing – review and editing. ZZ: Visualization, Writing – review and editing. LK: Writing – review and editing. XW: Formal Analysis, Writing – review and editing.

Funding

The authors declare that financial support was received for the research and/or publication of this article. This research was funded by Guangxi Key Research and Development Program, grant number AB24010039, and Guangxi Science and Technology Major Program, grant number AA23073017.

Conflict of interest

Authors JZ, CY, and XW were employed by Guangxi Construction Testing Center Co., Ltd.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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