Interface Fatigue Test of Hybrid-Bonded Fiber-Reinforced Plastic-Reinforced Concrete Specimen

Abstract

This research produced five hybrid-bonded fiber-reinforced plastic (HB-FRP)-reinforced beam specimens, and different fatigue load amplitudes were used as parameter variables for the fatigue performance tests of the FRP–concrete interface. The results show that FRP sliding with the number of cycles can be roughly divided into fatigue initiation stage, fatigue development stage, and fatigue damage stage, and finally, because the load is too large and friction, pin, and bonding cannot provide a greater inhibition effect, FRP adhesive length is not enough to withstand the stripping load and failure. FRP slip increases with increasing fatigue load amplitude for the same number of fatigue cycles; for the specimens with fatigue damage, the interfacial stiffness of FRP–concrete decreases with the increase in the number of cyclic, and the rate of stiffness damage at the FRP–concrete interface accelerates with increasing fatigue load amplitude. For fatigue load magnitudes higher than 0.6, the slopes of fatigue bond–slip curves decrease with the increase in the number of cycles. When the fatigue load magnitude is lower than 0.4, the slope of the fatigue bond–slip curve increases with the number of cycles and is close to the slope of the monotonically loaded curve. Due to the difference in load class, the bond strength of HB-F-1 continues to decrease with the increase in fatigue times, decreasing by 26% after 100 fatigue cycles and decreasing to 7.33 MPa after 5000 fatigue cycles. The bond strength of the sample HB-F-2 first increased and then decreased with the increase in fatigue times. After 10,000 fatigue cycles, the bond strength decreased by 8%, and at 11,133,300 fatigue breaks, the bond strength of the sample HB-F-3 continued to increase with the increase in fatigue times. At 2 million fatigue load cycles, the bond strength increased to 7 MPa, far from reaching the peak strength. The empirical formulas for the fatigue life curve of HB-FRP-reinforced specimens under single steel fasteners are proposed.

1. Introduction

FRP (fiber-reinforced plastic), with its lightweight, high-strength, corrosion-resistant properties, is widely used in the repair and reinforcement of railroad bridges. These structures that have been damaged require reinforcement to increase the capacity of the member beams and slabs. Additionally, these structures must be sufficiently durable to prevent fatigue failure of the members. Repeated cyclic loading can cause fatigue damage to a structure, even if the load amplitude is significantly lower than the ultimate load capacity of the structure. This fact is often overlooked during the analysis and design of FRP-reinforced concrete structures.

The types of fiber composites used in engineering reinforcement can be divided into AFRP, BFRP, CFRP, and GFRP, and the fatigue properties of these composites have been tested. Hollaway [1] studied the fatigue properties of GFRP, AFRP, and CFRP materials through experiments. The results show that the fatigue properties of the three fiber-reinforced composites are CFRP, AFRP, and GFRP from high to low, and the fatigue properties of AFRP are similar to those of GFRP.

For AFRP, Zhang et al. [2] experimentally studied the flexural fatigue performance of reinforced concrete beams reinforced by AFRP and explored the development law of strain hysteresis and stiffness degradation law of reinforcement and concrete in reinforced specimens. The research shows that the fatigue cracking resistance of the component can be improved, and the fatigue life of the component can be prolonged by using aramid fiber reinforcement. Due to its own characteristics, there are few studies on the fatigue performance of reinforced concrete members of AFRP.

For BFRP, based on the four-point flexural loading test, Xie et al. [3] explored the fatigue failure mode at the interface of BFRP- and CFRP-reinforced concretes, the expansion law of fatigue cracks at the interface, and the variation law of mid-span deflection and FRP strain of members with the number of loading cycles. The results show that the static load stripping capacity of CFRP–concrete interface is about 50% higher than that of BFRP concrete interface. Tian [4] studied the influence of fatigue load on the interface properties of BFRP–concrete double-shear specimens under multiple variables, gave the law of interface stripping failure, and established the empirical formula for the three-stage degradation of interface stiffness. Considering the impact load or continuous loading in actual service, Zhang [5] and Yuan [6] studied the influence of different loading rates on the interface properties of BFRP-reinforced specimens and found that the increase in loading rate would have a negative effect on the maximum bearing capacity and effective bond length of specimens, and the lower bearing capacity and effective bond length would also decrease when the specimens were damaged.

For GFRP, Bizindavyi et al. [7] studied the fatigue properties of the GFRP–concrete interface by using a single shear test. Considering the influence of bond length and bond width on the interface bond stress under fatigue load, the S–N curve of the interface under fatigue load was obtained. Zeng et al. [8] conducted a fatigue loading test on six GFRP-reinforced concrete beams. The results show that the cracking moment of the reinforced beam is 30~40% higher than that of the unreinforced beam. In addition, the ultimate bearing capacity increased by more than 30%.

For CFRP, Oudah et al. [9] and Christian et al. [10] demonstrated through tests that concrete beams reinforced with FRP exhibit superior fatigue performance. Sobuz et al. [11] discovered that the flexural stiffness of reinforced beams can be effectively improved by using CFRP sheet reinforcement. This also reduces crack spacing and width while shortening the extension of cracks along the height of the section. Dong et al. [12] conducted tests and found that the fatigue failure modes of reinforced beams mainly include steel bar fracture failure and CFRP debonding failure. As the upper limit of fatigue load increases, the failure mode shifts from steel bar fatigue fracture to CFRP debonding failure, resulting in a significant decrease in the life of reinforced beams. Hunebum et al. [13] conducted a double-shear fatigue test to examine the impact of FRP types and the number of adhesive layers on the fatigue bond performance of the FRP–concrete interface. The results showed that an increase in FRP stiffness and the number of adhesive layers led to an increase in bearing capacity during fatigue debonding failure, while the relative slip of the interface decreased. Diab et al. [14] conducted a double-shear fatigue test and found that the rate of interface stripping decreases as the number of cycles increases. They proposed that the upper limit of the interface bonding capacity under fatigue load should be 30% of that under static load to avoid fatigue stripping failure. They also established an interface bonding slip model and crack propagation model for predicting fatigue life. Li [15] conducted beam tests in addition to single or double shear tests to better reflect the actual interface of flexural members and study the fatigue performance of the CFRP–concrete interface. The results showed that the interface fatigue life decreases with an increase in stress level or CFRP to concrete width ratio and increases with an increase in concrete strength or bond length, and they proposed an empirical formula to predict the fatigue life of the CFRP–concrete interface based on experimental studies. The formula considers the effects of multiple factors. In addition, scholars have carried out a large number of experimental studies, and the overall fatigue performance of CFRP-reinforced concrete members has obtained more comprehensive results.

In recent years, researchers have studied and reviewed different types of FRP-reinforced concrete members and their interface bonding properties under fatigue load and proposed new FRP anchoring methods, details of which can be seen in reference [16,17,18]. Hybrid-bonded fiber-reinforced plastic (HB-FRP) is a novel reinforcement method that uses steel fasteners and chemical bolts to mechanically anchor concrete, based on the traditional FRP-bonded concrete. Its strengthening principle comprises the dotting effect of chemical bolts, the bonding effect between FRP and the reinforced concrete, and the frictional bite effect between HB-FRP and the concrete interface. This technology strengthens the bond between FRP and the surface it is applied to, preventing premature stripping and increasing the utilization rate of FRP [19,20,21]. Currently, research on HB-FRP reinforcement technology primarily focuses on enhancing the load-carrying capacity of reinforced members, particularly reinforced concrete beams. Additionally, research has been conducted on the bond slip constitutive of FRP–concrete and the corresponding numerical simulation. However, there has been less research on the working performance under fatigue load [22,23]. In this study, five HB-FRP-reinforced specimens were fabricated to investigate the fatigue bond properties of the HB-FRP–concrete interface using the fatigue load amplitude as the test parameter.

2. Materials and Methods

2.1. Test Specimens

The dimensions of the concrete specimens used in this study are presented in Figure 1. The concrete is poured in layers from the bottom up, making it necessary to allocate longitudinal and hoop reinforcements inside the concrete to prevent cracking during transport. The arrangement of the reinforcement is also shown in Figure 1. The hoop and longitudinal reinforcements are made of Q235 steel (jinan, China) with a diameter of 6 mm. The spacing between each hoop reinforcement is 10 cm.

This study aims to simulate the influence of a crack in the middle on the bonding performance of FRP. To achieve this, a steel hinge (jinan, China) is connected in the middle of two concrete pieces, allowing them to rotate freely around the top of the gap. Holes are designed on both sides of the plates where the hinge is in contact with the concrete, and a tight connection is used to ensure the hinge and concrete are securely connected. The dimensions and physical drawings of the steel hinge are shown in Figure 2 and Figure 3.

During specimen fabrication, counterpoise screws were threaded into the formwork beforehand and secured with bolts to ensure the anchoring effect of the steel hinges on the concrete. To prevent the FRP plate from debonding before the experiment due to steel hinge rotation, a steel plate was affixed to the lower formwork to prevent rotation. The fixation diagram of the steel hinge and steel plate is shown in Figure 4. After completing the aforementioned steps, concrete was poured. Three cubic specimens of 150 mm side length were poured for each different grade of concrete, and standard curing was carried out for 28 days to prepare for measuring the compressive strength of the concrete specimens.

After the completion of concrete curing, it is necessary to polish the concrete. This includes grinding the contact surface between FRP and concrete to smooth the concrete surface and expose the new concrete layer, ensuring high-quality adhesion between FRP and concrete. The corners at the gap between the two concrete blocks at the bottom are rounded to an arc with a radius of at least 20 mm. This is to prevent FRP fabric from experiencing brittle fracture at that location. After the grinding is completed, the corresponding position of the concrete will be cleaned up.

After completing the grinding process, proceed to create holes and mark the positions for bolt implantation. Use an electric hammer to drill holes that are at least 7 cm deep. Once the holes are drilled, remove any dust using a steel brush and an air pump. After cleaning is complete, proceed with the bolt implantation process.

FRP is bonded, and steel fasteners are arranged. To ensure that the bond damage occurred on the other side of the measured area, four steel fasteners were arranged on one side of the non-measured area. Additionally, carbon plate adhesive was applied to all FRP plates to observe the debonding of FRP. The arrangement of steel fasteners and FRP is illustrated in Figure 5, and the dimensions of steel fasteners are presented in Figure 6.

2.2. Material Properties

The concrete cube test block of 150 × 150 × 150 mm was poured at the same time as pouring the specimen, and the compressive strength test of the cube was carried out. The final average compressive strength was 45.32 MPa. The tensile strength of the FRP (jinan, China) plate used in the test is 2400 MPa, and the elastic modulus is 160 GPa.

2.3. Arrangement of Measuring Points

In order for the debonding to occur only on one side of the single steel fastener, multiple steel fasteners are arranged on the other side to inhibit the debonding of the FRP. To determine the change rule of surface strain on CFRP plates, strain gauges were continuously arranged along the center line of the plate on both monotonically and repeatedly loaded specimens. For the specimen with a gap of 140 mm from the steel fasteners, there are 11 strain gauges in the bonding measurement area, which are densely arranged close to the center of the span. Additionally, strain gauges are bonded to the surface of the CFRP plate in the non-bonding area of the span to measure the maximum strain of the CFRP plate. During monotonic loading tests, a multi-channel static strain recorder was used to collect strain data. During repetitive loading tests, strain data were collected using a multi-channel dynamic strain recorder with the strain gauge arrangement illustrated in Figure 7. To determine the horizontal relative displacement at the bottom of the span of the two concrete specimens, a linear variable displacement transducer was placed on each side of the bottom of the span of the specimen, as shown in Figure 8.

2.4. Testing Procedures

The loading method of monotonic loading and repeated loading specimens is to take four-point bending loading on the specimen beam through the distribution beam, and the loading method is shown in Figure 9. The monotonic loading test adopts the hydraulic servo loading system, and the loading speed is 0.005 mm/s until the specimen undergoes the damage of the CFRP debonding, fatigue test using hydraulic pulsation fatigue test monotonic loading using displacement control machine. Repeated loading of the specimen before the first monotonic loading test, loaded from 0 to the upper limit value of the repeated load, and then unloaded to 0. Repeated load changes according to the sinusoidal law. The frequency of repetitive loading in this test was chosen to be 3 Hz, the loading mode and parameters of each specimen are shown in the

Table 1. Test parameters.

Specimen ID	Testing Method	Pmax/Pu	Pmin/Pmax	Concrete Strength	Loading Frequency
HB-M-1	Static	1	-	C50
HB-M-2	Static	1	-	C50
HB-F-1	Fatigue	0.75	0.1	C50	3
HB-F-2	Fatigue	0.6	0.1	C50	3
HB-F-3	Fatigue	0.4	0.1	C50	3

.

3. Results

3.1. Load–FRP Slip Relationship

3.1.1. Monotonic

The FRP slip in the region below the steel fasteners can be calculated by the following equation for monotonic loading and repeated loading tests:

$$s_i=s_{i-1}+{\displaystyle \frac{\left({\epsilon }_i+{\epsilon }_{i-1}\right)}{2}}\cdot \left(x_i-x_{i-1}\right)$$

(1)

In the formula, $s_i$ represents the slip of the i section; ${\epsilon }_i$ represents the measured strain value of the i node; $x_i$ represents the distance from the i node to the free end; and the detailed layout is shown in Figure 10. Calculations were carried out according to the above equation, and the load–FRP slip curves of the specimens were plotted based on the calculation results.

The load–FRP slip curves under monotonic loads are shown in Figure 11. The results indicate that when the load is between 0 and 30 kN, the FRP slip increases linearly with the increase in load, and the specimen is in an elastic working state. At this point, the interface between the FRP and the concrete has not debonded. However, when the load exceeds 30 kN, local debonding starts to occur at the bond end of the FRP in the span, and the slope of the load–FRP slip curve of the specimen starts to decrease, and the amount of FRP slip increases. After localized debonding, the interfacial debonding gradually extended towards the bearing end. When the FRP debonding reached the position of the steel fasteners, it was inhibited due to the friction effect and pinning action provided by the steel fasteners and chemical bolts. When the remaining bond length of FRP was insufficient to withstand the external load, the load ceased to increase, and the specimen was damaged. The ultimate loads of specimens HB-M-1 and HB-M-2 were 84.5 kN and 88.5 kN, respectively. Therefore, the average ultimate load of the two monotonically loaded specimens, 86.5 kN, was defined as P_u.

3.1.2. Fatigue

The load–FRP slip curves under fatigue loading are shown in Figure 12. It is seen that except for specimen HB-F-3, for which the first cycle data were not measured, the area enclosed by the load–FRP slip curves of specimens HB-F-1 and HB-F-2 in the first cycle of loading is the largest, indicating that a large amount of fracture energy was consumed during the first cycle and there was a significant decrease in stiffness during the first cycle, and in the subsequent cycle of loading, the FRP slip varies with loading. The FRP slip varies approximately linearly with loading in the subsequent cycles. As the number of fatigue cycles increases, the slope of the curve gradually decreases, and the bond stiffness of FRP continues to decrease. Eventually, specimen HB-F-1 with load level P_max/P_u = 0.75 experiences fatigue damage after 5000 cycles, causing the FRP slip to reach 0.42 mm, increasing the hysteresis curve area, and resulting in complete debonding of the FRP. Similarly, specimen HB-F-2 with load level P_max/P_u = 0.6 also experiences fatigue damage. HB-F-2 experienced fatigue damage at 1,113,300 cycles with a load level of P_max/P_u = 0.6, causing the FRP slip to reach 0.51 mm. Meanwhile, specimen HB-F-3 with a load level of P_max/P_u = 0.4 was loaded up to 2 million times without experiencing any fatigue damage, although the FRP slip did reach 0.15 mm. It is evident that, under the same number of fatigue cycles, the amount of FRP slip increases with the load amplitude.

3.2. Interface Stiffness Damage

While stiffness generally refers to the ability of a material to resist deformation, the stiffness of an interface refers to the ability to resist slip. The stiffness of the interface decreases with fatigue loading. In this study, the stiffness of the interface is defined as K_d, and the stiffness of the interface is roughly calculated by the upper fatigue limit value and the corresponding FRP slip at the fastener position with the following formula:

$$K_d=P_{max}/S_f$$

(2)

In the formula, P_max is the upper fatigue load limit value, and S_f is the FRP slip at the fastener position corresponding to the upper fatigue load limit value.

From the calculation results in Figure 13, it can be seen that for specimens with fatigue damage, the interfacial stiffness of FRP–concrete all decreases with the increase in the number of cyclic loads, and the larger the fatigue load amplitude is, the faster the FRP–concrete interfacial stiffness is damaged.

3.3. FRP Slip–Cycle Count Relationship

The relationship between FRP slip and the number of cycles under repeated loading is plotted as a curve in Figure 14. It can be seen that the change in FRP slip with the number of cycles can be roughly divided into three stages: The first stage is the fatigue initiation stage (about 0–10% of the fatigue life), in which the FRP slip increases rapidly with the increase in fatigue number, which is due to the continuous deterioration of FRP–concrete bond performance and the formation of the initial interfacial cracks in this stage, and the development of the initial crack is faster, which leads to the development of the FRP slip more quickly. The second stage is the fatigue development stage (about 10% to 90% of the fatigue life); FRP slip with the increase in the fatigue rate slows down; this is due to the friction and pinning role in the participation of the interfacial crack expansion being slow, so that the FRP slip increases slowly. The third stage is the fatigue damage stage (about 90% to 100% of the fatigue life); in this stage, FRP slip increases with the increase of fatigue number. The third stage is the fatigue damage stage (about 90~100% of the fatigue life), in which the FRP slip increases with the number of fatigue cycles and the acceleration rate accelerates. This is due to the fact that the friction, pinning, and bonding cannot provide greater inhibition because of the excessive load, and the retained adhesion length of the FRP is not long enough to withstand the debonding load.

3.4. Development of FRP Strain Distribution

3.4.1. Monotonic

The distribution of FRP strain along the bond length of each monotonically loaded specimen during the loading process is shown in Figure 15. The curves show the distribution of FRP strain along the bond length under different loading levels. It can be seen that when the load is small, the FRP strain decays rapidly from the bonded end position in the span to the free end; with the increase in load, the bond between FRP and concrete near the end region gradually deteriorates, the ability to transfer shear stress decreases, and the decay of FRP strain in the end range slows down, and the initial cracks begin to appear at the interface between the concrete and FRP to form the initial debonding; with the continual increase in load, the debonding region continues to expand and gradually moves to the free end. continues to expand, and gradually to the free end of the development, the effective force transfer region is also gradually moved to the end, until the stripping developed to the fastener position. The friction effect of fasteners, chemical bolts, and pinning bolts inhibits the transfer of shear force, which is manifested as a large strain difference. When FRP debonding occurred at the fastener position, the strain difference between the two sides decreased, and the FRP debonding continued to develop toward the free end until the FRP was completely debonding.

3.4.2. Fatigue

The distribution of peak strain of FRP along the bond length under repeated loading is shown in Figure 16. It can be seen that the FRP end strain of specimen HB-F-1 reached 4000 με at the first loading, and the FRP debonding had developed to the fastener position. At this time, due to the existence of friction at the fastener, the pinning bolt action inhibited the development of FRP debonding; when the repeated loading reached 5000 times, the FRP at the fastener position was completely debonded, and the debonding then rapidly developed to the end of the fatigue damage. The strain at the end of specimen HB-F-2 reached 2700 µε during the first loading and increased to 2900 µε after 1000 repetitions, and the FRP debonding developed to the position of the fastener, and then the FRP strain gradually decreased with the increase in the number of cycles. When the number of repetitive loadings reaches 1 million times, the FRP at the fastener position is completely debonded, and finally, fatigue damage occurs in the specimen after 1,113,300 cycles of loading. Different from the previous two, specimen HB-F-3 was loaded to 1000 times; the strain at the end of the FRP was only 1750 µε, and with the increase in the number of cycles, the strain increased until the FRP was debonded only 90 mm (from the end) when loaded to 2 million times, and the strain reached 2500 µε, and fatigue damage did not occur. The FRP debonding mechanism is shown in Figure 17.

3.5. Bond Stress–Slip Responses

The interfacial shear stress of FRP at the fastener location can be referred to in Figure 10 and derived from the formula proposed by Liu [14]:

$${\tau }_{m,i}={\displaystyle \frac{\left({\epsilon }_{i+1}-{\epsilon }_{i-1}\right)E_f{t}_f}{b_s}}$$

(3)

In the formula, $E_f$ is the modulus of elasticity of the FRP plate; $t_f$ is the thickness of the FRP plate; and $b_s$ is the width of the fastener.

After calculation, the FRP bond–slip curves of each specimen fastener position were plotted as shown in Figure 18. As can be seen from the figure, the peak shear stress of specimen HB-F-1 reaches 13.17 MPa at the first loading, and with the increase in the number of cycles and slip, the maximum shear stress of specimen HB-F-1 shows a gradual decreasing tendency and finally drops to 7.33 MPa at 5000 times of cyclic loading, and fatigue damage to the specimen occurs. The maximum bond stress of specimen HB-F-2 shows an increasing trend before cyclic loading for 1000 times, and the shear stress reaches a peak value of 13.9 MPa at 1000 times and then continues to decrease with cyclic loading, and fatigue damage occurs eventually. On the other hand, the shear stress of specimen HB-F-3 reaches only 7.03 MPa when the cyclic loading reaches 2 million times, and it shows a continuous increasing trend.

The slopes of the bond–slip curves under different fatigue loads are plotted as shown in Figure 19. It can be seen that the slopes of the curves for specimens HB-F-1 and HB-F-2 have a tendency to decrease gradually when the load levels are 0.75 and 0.6, and the decreasing slopes indicate that the interfaces are beginning to have damage and that the energy dissipated by the damage will not be recovered. When the load level is 0.4, the slope of the bond–slip curve of specimen HB-F-3 increases with the increase in the number of cycles, which indicates that the interface is still in the elastic stage without damage, and the interface shear stress increases with the increase in the number of cycles, and it still does not reach the peak when the number of load cycles reaches 2 million times. It can be seen that, when the fatigue load amplitude is higher than 0.6, the slope of the fatigue bond–slip curve decreases with the increase in the number of cycles; when the fatigue load amplitude is lower than 0.4, the slope of the fatigue bond–slip curve increases with the increase in the number of cycles and is close to the slope of the curve of monotonic loading.

3.6. Bond Stress–Fatigue Life Responses

The relationship between bond strength and fatigue life of each specimen is plotted as shown in Figure 20. As can be seen from the figure, except for HB-F-3, which did not reach the peak bond strength, both HB-F-1 and HB-F-2 experienced a decrease in bond strength after reaching the peak value. Due to the difference in load ratings, the bond strength of HB-F-1 decreased by 26% after 100 cycles of fatigue and that of HB-F-2 decreased by 8% after 10,000 cycles of fatigue, while the bond strength of the former decreased to 7.33 MPa at 5000 cycles of fatigue and broke, and the bond strength of the latter decreased to 1.3 MPa at 11,133,300 cycles of fatigue and broke, while the bond strength of specimen HB-F-3 reached only 7 MPa at 2 million fatigue load cycles, which is far from reaching the peak strength. It can be seen that when the interfacial bond strength is similar, the specimens with large load amplitude are damaged earlier, and the specimens with fatigue damage all reach the peak bond strength during the loading process.

3.7. S–N Curve Calculation

The S–N curve (fatigue life curve) is a curve that represents the relationship between the level of applied stress on a material and its fatigue life. The curve is the basis for fatigue life prediction and fatigue-resistant design of members. The HB-FRP specimens in this test were compared with the EB-FRP fatigue test by Li [15], and the parameters of each specimen are shown in

Table 2. Parameter information.

Specimen	FRP Width/mm	FRP Thickness/mm	FRP Bonding Length/mm	Concrete Strength/MPa
EB-FRP	50	0.222	160	62.2
HB-FRP	50	2	360	55

and

Table 3. EB-FRP specimen fatigue life.

Specimen ID	Pu	Pmin/Pu	Pmax/Pu	Fatigue Life
E-M-0	29.17	-	1	1
E-F-0	-	0.15	0.8	2600
E-F-1	-	0.15	0.7	32,000
E-F-2	-	0.15	0.65	168,900
E-F-3	-	0.15	0.55	1,550,000
E-F-4	-	0.15	0.54	2,380,000
E-F-5	-	0.15	0.45	>2,000,000

.

The main well-established expressions for approximate S–N curves are the power function formula (double logarithmic formula) and the exponential function formula (single logarithmic formula). In this study, the single logarithmic formula is used to express the S–N curve of the FRP–concrete interface as shown in the following equation:

S = a − blgN

(4)

In the formula, a and b are material constants related to material properties, loading mode, and environmental factors. Where the stress level S is calculated with respect to the magnitude of the stress amplitude and the average stress amplitude:

$$\Delta S={\displaystyle \frac{{\sigma }_{\max }-{\sigma }_{\min }}{{\sigma }_{\mathrm{u}}}}={\displaystyle \frac{P_{\max }-P_{\min }}{P_{\mathrm{u}}}}$$

(5)

$$S_{\mathrm{a}}={\displaystyle \frac{{\sigma }_{\max }+{\sigma }_{\min }}{2{\sigma }_{\mathrm{u}}}}={\displaystyle \frac{P_{\max }+P_{\min }}{2P_{\mathrm{u}}}}$$

(6)

In the formula, ∆S is the normalized CFRP stress amplitude; $S_a$ is the normalized CFRP mean stress amplitude; ${\sigma }_{max}$ is the peak stress of CFRP under repeated loads; and ${\sigma }_{min}$ is the valley stress of CFRP under repetitive loading.

The fatigue design of FRP-reinforced concrete beams is carried out by a Goodman–Smith diagram defining the stress level as follows:

$$S={\displaystyle \frac{\Delta S}{1-S_{\mathrm{a}}}}$$

(7)

The fatigue test data for the two reinforced specimens are shown in

Table 4. Calculation of stress level for each specimen.

Specimen ID	lgN	$\mathit{\Delta }\mathit{S}$	Sa	S
E-M-0	0	1	0.5	2
E-F-0	3.414	0.648	0.476	1.237
E-F-1	4.505	0.553	0.423	0.959
E-F-2	5.227	0.500	0.399	0.833
E-F-3	6.190	0.401	0.349	0.616
E-F-4	6.376	0.388	0.345	0.594
E-F-5	>6.301	0.301	0.299	0.430
HB-M-1	0	0.961	0.487	1.846
HB-M-2	0	1.032	0.515	2.123
HB-F-1	3.694	0.614	0.411	1
HB-F-2	6.042	0.425	0.305	0.571
HB-F-3	>6.301	0.289	0.260	0.322

.

The relationship between stress level S and fatigue life N of the specimen plotted from the calculations summarized in Table 4 is shown in Figure 21. It can be seen that the fatigue life N decreases with the increase in the stress level S. The E-F-5 and HB-F-3 specimens did not experience fatigue damage after 2 million repetitive loadings (no debonding damage of FRP). From the above test results, the empirical equations for the S–N curves under this condition were fitted.

EB-FRP specimen:

$$\left\{\begin{array}{c}\mathrm{S}=-0.2237\mathrm{l}\mathrm{g}\mathrm{N}+2,\mathrm{S}>0.3/0.7\\ \mathrm{N}>2,000,000,\mathrm{S}\le 0.3/0.7\end{array}\right.$$

(8)

HB-FRP specimen:

$$\left\{\begin{array}{c}\mathrm{S}=-0.2216\mathrm{l}\mathrm{g}\mathrm{N}+2,\mathrm{S}>0.29/0.74\\ \mathrm{N}>2,000,000,\mathrm{S}\le 0.29/0.74\end{array}\right.$$

(9)

From the curve data in Figure 21, it can be seen that the fatigue life curves of EB-FRP and HB-FRP-reinforced specimens with single fasteners have basically the same trend, which may be caused by the fact that the number of fasteners is too small and the inhibition of FRP debonding is not obvious.

The fatigue life versus load level was shown with the aid of Goodman–Smith plots, as shown in Figure 22. It can be seen that the upper limit dividing line is determined according to Equations (2)–(8), and the lower limit dividing line is a straight line passing through the origin and the point (1, 1). The area between the upper limit demarcation line and the lower limit demarcation line is the area where the fatigue life is greater than 2 million times; the area above the upper limit demarcation line is the area where the fatigue life is less than 2 million times; and the point falling on the upper limit demarcation line is the point where the fatigue life is 2 million times. It can be seen that the stress amplitude corresponding to a fatigue life of 2 million times is related to the minimum load level (σ_min/σ_u), i.e., the stress amplitude corresponding to a fatigue life of 2 million times decreases with the increase of σ_min/σ_u.

4. Conclusions

In this study, the experimental study on the fatigue bonding performance of mechanically anchored FRP–concrete interface was carried out by designing different load level parameters, and the specific conclusions are as follows:

(1) Under the same number of fatigue cycles, the larger the load amplitude, the larger the slip of FRP. For the specimens with fatigue damage, the interfacial stiffness of FRP–concrete decreases with the increase in the number of cyclic, and the larger the fatigue load amplitude, the faster the FRP–concrete interface stiffness is damaged.

(2) When the fatigue load amplitude is higher than 0.6, the slope of the fatigue bond–slip curve decreases with the increase in the number of cycles; when the fatigue load amplitude is lower than 0.4, the slope of the fatigue bond–slip curve increases with the increase in the number of cycles and is close to the slope of the curve with monotonic loading.

(3) The fatigue life curves of EB-FRP- and HB-FRP-reinforced specimens with single fasteners have basically the same trend, which may be caused by the small number of fasteners, which does not have a significant inhibitory effect on FRP debonding.

(4) The stress amplitude corresponding to a fatigue life of 2 million times is related to the minimum load level (σ_min/σ_u), and the stress amplitude corresponding to a fatigue life of 2 million times decreases with the increase of σ_min/σ_u.