Experimental Research and Analysis on Fatigue Life of Carbon Fiber Reinforced Polymer (CFRP) Tendons

The fatigue life of carbon fiber reinforced polymer (CFRP) tendons was studied in this paper. A new wedge-type anchorage system was applied to the fatigue test of CFRP tendons and demonstrated an excellent fatigue resistance. In the test and analytical data, the fatigue stress ranged from 200 MPa to 800 MPa, and maximum stresses from 0.37 to 1.0 fu (fu = ultimate tensile strength of CFRP tendons) were determined. The main work and results were that the stress range and stress level (maximum stress) were two key parameters affecting the fatigue life of CFRP tendons. A bilinear equation and a linear equation considering the fatigue life of CFRP tendons jointly affected by the stress range and the maximum stress were established. The error of predicted results and test results was 0.038 and 0.083, respectively, both representing good prediction accuracy. The predicted results of Whitney’s method showed that, at a 95% confidence level, when the stress range was 200 MPa, 400 MPa, and 600 MPa, the maximum stress limit of CFRP tendons, which were not broken in a fatigue test of 2 million times, was 63.9% fu, 53.0% fu, and 36.8% fu, respectively.


Introduction
Fiber-reinforced polymer (FRP) composites have the characteristics of light weight, high strength, corrosion resistance, and electromagnetic insulation, which can replace steel for engineering construction in particular scenarios [1][2][3]. On the basis of the fiber variety, FRP can be classified into glass fiber-reinforced polymer (GFRP), carbon fiber-reinforced polymer (CFRP), basalt fiber-reinforced polymer (BFRP), and aramid fiber-reinforced polymer (CFRP), among others. Among all FRPs, CFRP has the properties of superior fatigue resistance [4][5][6][7], high-creep rupture limit [8][9][10], low-creep performance [11], and other benefits, which has deemed it as the most ideal cable material to replace steel, thereby realizing the long life of major engineering structures. Cable structures bear repetitive loads during their service life. Fatigue rapture may bring disastrous consequences due to its burstiness. Therefore, a study on the high-precision prediction method of fatigue life of CFRP cables is required.
Stable and reliable anchorages are key to applying FRP cables to practical application. Due to low transverse strength and the delicate surface of FRP tendons, a dedicated anchorage system for FRP cables needs to be developed. The authors' research team developed a new wedge-type anchorage system and in [11] applied this new device to the study of long-term creep performance of CFRP tendons. This paper applied this anchorage system to the fatigue performance test of the pre-stressed CFRP tendons. The fatigue stability of the anchorage system was also tested. the back of the anchorage where the tensile stresses are lower. However, the stability, reliability, and economic performance of these mechanical anchorages still need to be improved [21]. The development of a stable and efficient anchorage for FRP tendons is an important task in the application study of CFRP cables.
On the basis of the new wedge-type anchorage of FRP tendons developed by the research team, this paper conducted an experimental study on the fatigue life of CFRP tendons. In the test, the stress range ∆σ was 600 Mpa and 800 Mpa, and the maximum stress ranged from 0.375 to 0.843 f u . On the basis of the test data of this paper and that in [12], a bilinear model, which considers the fatigue life of CFRP tendons jointly affected by stress range and maximum stress, was established. Meanwhile, the reliability of the fatigue life of CFRP tendons was analyzed.

Characteristics and Anchorages of CFRP Tendons
Smooth and round CFRP tendons with a diameter of 8 mm produced in Jiangsu Hengshen (Zhenjiang, China) were employed in this study. The tendons were made from 12k, Type T700 continuous CFRP filaments (Hengshen, Zhenjiang, China), which were immersed in high toughness epoxy resin and then solidified, having a volume content of 65% and a density of 1.6 g/cm 3 .
A new anchorage system developed by the research team is shown in Figure 1, which mainly included the steel wedge, barrel, and nut. To reduce the friction between the wedge and the barrel, the wedge's outer surface was polished and the taper angle was made 1:20. The wedge was manufactured with a flexible thin layer on its inner surface. In this manner, the placement of an additionally flexible sleeve, which is commonly used at present [24][25][26], was omitted, such that the assembly was simplified. The incorporation of the flexible thin layer not only increased the friction between the wedge and the CFRP tendons, but also avoided the premature failure of FRP tendons due to the concentrated stress induced by the clamp load [27]. The nut was connected with the barrel. In practical applications, the nut was placed against the anchorage plates fixed on the end of the concrete member to counteract the load caused by prestressing the tendons. angle design and curved angle design, respectively, which can transfer the compressive stresses to the back of the anchorage where the tensile stresses are lower. However, the stability, reliability, and economic performance of these mechanical anchorages still need to be improved [21]. The development of a stable and efficient anchorage for FRP tendons is an important task in the application study of CFRP cables. On the basis of the new wedge-type anchorage of FRP tendons developed by the research team, this paper conducted an experimental study on the fatigue life of CFRP tendons. In the test, the stress range Δσ was 600 Mpa and 800 Mpa, and the maximum stress ranged from 0.375 to 0.843 . On the basis of the test data of this paper and that in [12], a bilinear model, which considers the fatigue life of CFRP tendons jointly affected by stress range and maximum stress, was established. Meanwhile, the reliability of the fatigue life of CFRP tendons was analyzed.

Characteristics and Anchorages of CFRP Tendons
Smooth and round CFRP tendons with a diameter of 8 mm produced in Jiangsu Hengshen (Zhenjiang, China) were employed in this study. The tendons were made from 12k, Type T700 continuous CFRP filaments (Hengshen, Zhenjiang, China), which were immersed in high toughness epoxy resin and then solidified, having a volume content of 65% and a density of 1.6 g/cm³.
A new anchorage system developed by the research team is shown in Figure 1, which mainly included the steel wedge, barrel, and nut. To reduce the friction between the wedge and the barrel, the wedge's outer surface was polished and the taper angle was made 1:20. The wedge was manufactured with a flexible thin layer on its inner surface. In this manner, the placement of an additionally flexible sleeve, which is commonly used at present [24][25][26], was omitted, such that the assembly was simplified. The incorporation of the flexible thin layer not only increased the friction between the wedge and the CFRP tendons, but also avoided the premature failure of FRP tendons due to the concentrated stress induced by the clamp load [27]. The nut was connected with the barrel. In practical applications, the nut was placed against the anchorage plates fixed on the end of the concrete member to counteract the load caused by prestressing the tendons.

Static Test Setup
The static tensile strength test system was used to test the static strength and static elastic modulus of CFRP tendons under static conditions. As shown in Figure 2, the system mainly consisted of a hollow jack, a load cell, and a loading brace. The loading brace was employed to observe the rupture of CFRP tendons. A strain gauge was placed in the middle of the CFRP tendons to record the tendon strain during the loading process. The load cell was used to record the external load, thereby

Static Test Setup
The static tensile strength test system was used to test the static strength and static elastic modulus of CFRP tendons under static conditions. As shown in Figure 2, the system mainly consisted of a hollow jack, a load cell, and a loading brace. The loading brace was employed to observe the rupture of CFRP tendons. A strain gauge was placed in the middle of the CFRP tendons to record the tendon strain during the loading process. The load cell was used to record the external load, thereby determining

Fatigue Test Setup
The fatigue performance test of CFRP tendons was performed using the MTS-810 fatigue testing machine. The test loading was controlled by load, with the accuracy of 0.1 kN. The specialized wedge of the testing machine directly clamped the cup in the anchorage to exert fatigue load, as shown in Figure 3. The free length of CFRP tendons was 500 mm. A mark was set on the surface of CFRP tendons near the interior of the anchorage area to observe the slippage between CFRP tendons and the wedge during fatigue loading. A strain gauge was installed in the middle of the tendons to record the tendon strain.

Static Test Loading Procedure
The loading rate was set as 200 MPa/min in the static strength test, and the load and tendon strain were recorded by TDS-530 static data acquisition machine (Tokyo institute of instrumentation, Tokyo, Japan) at the frequency of 1 Hz. The static elastic modulus of CFRP tendons was obtained from the data points of ultimate tensile strength at 20% and 50%.

Fatigue Test Loading Procedure
The whole process of this fatigue test adopted the loading method of constant amplitude sinusoid. The loading frequency in this paper was 5 Hz. The study [28] showed that the temperature rise inside the materials can be ignored when the loading frequency is less than 10 Hz. The fatigue loading was divided into the following stages: static pre-loading, twice-static loading, followed by

Fatigue Test Setup
The fatigue performance test of CFRP tendons was performed using the MTS-810 fatigue testing machine. The test loading was controlled by load, with the accuracy of 0.1 kN. The specialized wedge of the testing machine directly clamped the cup in the anchorage to exert fatigue load, as shown in Figure 3. The free length of CFRP tendons was 500 mm. A mark was set on the surface of CFRP tendons near the interior of the anchorage area to observe the slippage between CFRP tendons and the wedge during fatigue loading. A strain gauge was installed in the middle of the tendons to record the tendon strain.
anchorage area to observe the slippage of tendons.

Fatigue Test Setup
The fatigue performance test of CFRP tendons was performed using the MTS-810 fatigue testing machine. The test loading was controlled by load, with the accuracy of 0.1 kN. The specialized wedge of the testing machine directly clamped the cup in the anchorage to exert fatigue load, as shown in Figure 3. The free length of CFRP tendons was 500 mm. A mark was set on the surface of CFRP tendons near the interior of the anchorage area to observe the slippage between CFRP tendons and the wedge during fatigue loading. A strain gauge was installed in the middle of the tendons to record the tendon strain.

Static Test Loading Procedure
The loading rate was set as 200 MPa/min in the static strength test, and the load and tendon strain were recorded by TDS-530 static data acquisition machine (Tokyo institute of instrumentation, Tokyo, Japan) at the frequency of 1 Hz. The static elastic modulus of CFRP tendons was obtained from the data points of ultimate tensile strength at 20% and 50%.

Fatigue Test Loading Procedure
The whole process of this fatigue test adopted the loading method of constant amplitude sinusoid. The loading frequency in this paper was 5 Hz. The study [28] showed that the temperature rise inside the materials can be ignored when the loading frequency is less than 10 Hz. The fatigue loading was divided into the following stages: static pre-loading, twice-static loading, followed by

Static Test Loading Procedure
The loading rate was set as 200 MPa/min in the static strength test, and the load and tendon strain were recorded by TDS-530 static data acquisition machine (Tokyo institute of instrumentation, Tokyo, Japan) at the frequency of 1 Hz. The static elastic modulus of CFRP tendons was obtained from the data points of ultimate tensile strength at 20% and 50%.

Fatigue Test Loading Procedure
The whole process of this fatigue test adopted the loading method of constant amplitude sinusoid. The loading frequency in this paper was 5 Hz. The study [28] showed that the temperature rise inside the materials can be ignored when the loading frequency is less than 10 Hz. The fatigue loading was Materials 2019, 12, 3383 5 of 12 divided into the following stages: static pre-loading, twice-static loading, followed by fatigue loading. The upper limit of the fatigue exerted by the pre-loading was 20% of the load, with the purpose of removing the spaces and poor contact between the wedges. The twice-static loading stage was done to record the initial elastic modulus of CFRP tendons. The formal fatigue loading conducted the static loading test during shutdown at fixed times to obtain the residual fatigue elastic modulus. The loading mode was shown in Figure 4. This paper focused on the fatigue life of CFRP tendons and, hence, the residual elastic modulus was not included. When tendons were ruptured with the fatigue cycle being less than 2 million, the cycles at rupture were recorded. When tendons were not ruptured with the fatigue cycle reaching 2 million, the static loading continued until rupture occurred and the strength of tendons after fatigue loading was tested.
Materials 2019, 12, 3383 5 of 12 fatigue loading. The upper limit of the fatigue exerted by the pre-loading was 20% of the load, with the purpose of removing the spaces and poor contact between the wedges. The twice-static loading stage was done to record the initial elastic modulus of CFRP tendons. The formal fatigue loading conducted the static loading test during shutdown at fixed times to obtain the residual fatigue elastic modulus. The loading mode was shown in Figure 4. This paper focused on the fatigue life of CFRP tendons and, hence, the residual elastic modulus was not included. When tendons were ruptured with the fatigue cycle being less than 2 million, the cycles at rupture were recorded. When tendons were not ruptured with the fatigue cycle reaching 2 million, the static loading continued until rupture occurred and the strength of tendons after fatigue loading was tested.

Static Tensile Properties
In the static loading test, the five specimens demonstrated the same failure mode, namely, burst fiber rupture in the middle, as shown in Figure 5. No tendon slippage or premature failure was detected in the anchorage area, which indicated the efficiency of the anchorage. The static test data of CFRP tendons are shown in Table 1, where the average static strength was 2136 MPa (Coefficient of variation, COV, 3.2%), and the average static elastic modulus was 150.2 GPa (COV 2.1%). For design consideration, the ultimate tensile strength of CFRP tendons was determined to be 2024 MPa, with a 95% confidence level calculated using the Equation (1): where , , σ are the strength standard value, average value, and variance of CFRP tendons, respectively.

Static Tensile Properties
In the static loading test, the five specimens demonstrated the same failure mode, namely, burst fiber rupture in the middle, as shown in Figure 5. No tendon slippage or premature failure was detected in the anchorage area, which indicated the efficiency of the anchorage. The static test data of CFRP tendons are shown in Table 1, where the average static strength was 2136 MPa (Coefficient of variation, COV, 3.2%), and the average static elastic modulus was 150.2 GPa (COV 2.1%). For design consideration, the ultimate tensile strength of CFRP tendons was determined to be 2024 MPa, with a 95% confidence level calculated using the Equation (1): where f k , µ f , σ f are the strength standard value, average value, and variance of CFRP tendons, respectively.
Materials 2019, 12, 3383 5 of 12 fatigue loading. The upper limit of the fatigue exerted by the pre-loading was 20% of the load, with the purpose of removing the spaces and poor contact between the wedges. The twice-static loading stage was done to record the initial elastic modulus of CFRP tendons. The formal fatigue loading conducted the static loading test during shutdown at fixed times to obtain the residual fatigue elastic modulus. The loading mode was shown in Figure 4. This paper focused on the fatigue life of CFRP tendons and, hence, the residual elastic modulus was not included. When tendons were ruptured with the fatigue cycle being less than 2 million, the cycles at rupture were recorded. When tendons were not ruptured with the fatigue cycle reaching 2 million, the static loading continued until rupture occurred and the strength of tendons after fatigue loading was tested.

Static Tensile Properties
In the static loading test, the five specimens demonstrated the same failure mode, namely, burst fiber rupture in the middle, as shown in Figure 5. No tendon slippage or premature failure was detected in the anchorage area, which indicated the efficiency of the anchorage. The static test data of CFRP tendons are shown in Table 1, where the average static strength was 2136 MPa (Coefficient of variation, COV, 3.2%), and the average static elastic modulus was 150.2 GPa (COV 2.1%). For design consideration, the ultimate tensile strength of CFRP tendons was determined to be 2024 MPa, with a 95% confidence level calculated using the Equation (1): where , , σ are the strength standard value, average value, and variance of CFRP tendons, respectively.

Fatigue Tensile Properties
The failure mode of CFRP tendons is shown in Figure 6. All of the specimens exhibited a similar failure mode in the middle portion. There was no slip and advance failure in the anchorage area, which indicated that the anchorage had good fatigue stability. The fatigue life data of each specimen are shown in Table 2.

Fatigue Tensile Properties
The failure mode of CFRP tendons is shown in Figure 6. All of the specimens exhibited a similar failure mode in the middle portion. There was no slip and advance failure in the anchorage area, which indicated that the anchorage had good fatigue stability. The fatigue life data of each specimen are shown in Table 2.  As seen from the data in Table 2, at the same stress range, the fatigue life of CFRP tendons increased with decreasing maximum stress. At the stress range of 600 MPa, the average fatigue life  As seen from the data in Table 2, at the same stress range, the fatigue life of CFRP tendons increased with decreasing maximum stress. At the stress range of 600 MPa, the average fatigue life of CFRP tendons was 541 cycles with the maximum stress of 1800 MPa, whereas the fatigue life of CFRP tendons exceeded 2 million cycles with the maximum stress reduced to 800 MPa. After 2 million cycles of fatigue loading for specimen 16, the static strength of CFRP tendons was 2081 MPa, about 97.4% of the average static strength of CFRP tendons. At the same maximum stress, the fatigue life of CFRP tendons with the stress range of 800 MPa was 11.7-41.0% of that with the stress range of 600 MPa. Evidently, the stress range and the maximum stress jointly affected the fatigue life of CFRP tendons.

Discussion
The fatigue life data of CFRP tendons with the stress ranges of 200 MPa, 400 MPa, 600 MPa, and 800 MPa are shown in Figure 7, where the stress ranges of 200 MPa and 400 MPa are from [12], and 600 MPa and 800 MPa are the test results of this paper. In [12], leadline tendons were fabricated with PAN-type carbon fibers (fiber volume fraction = 65%) embedded in an epoxy matrix. The ultimate strength was 1999.2 MPa (COV 1.6%). The static elastic modulus was 149.6 GPa (COV 8.3%). The material properties of the tendons reported by [12], including the fiber type, fiber volume fraction, and the ultimate strength, were similar to the tendons of this paper, which makes the experimental results of these two groups comparable. The data points in Figure 8 represent the average value of several test results at the same level. As can be seen from these data, the stress range and the stress level (the maximum stress) jointly affected the fatigue life of CFRP tendons. At the same maximum stress, the greater the stress range, the shorter the fatigue life of CFRP tendons. At the same stress range, the fatigue life of CFRP tendons increased with the lowering of the maximum stress. The changing speed of fatigue life varied at different stages, which was slow with greater maximum stress.  As shown in Figure 8, the effect of the stress range on the slope K of S-curves was evident. In addition, the slope K varied with the maximum stress. Most of the available methods for predicting the S-curve ignore the effect of the stress range. To take into account both the effect of the stress range and the varying slopes of the S-curve with the maximum stress, a bilinear equation, which  As shown in Figure 8, the effect of the stress range on the slope K of S-curves was evident. In addition, the slope K varied with the maximum stress. Most of the available methods for predicting the S-curve ignore the effect of the stress range. To take into account both the effect of the stress range and the varying slopes of the S-curve with the maximum stress, a bilinear equation, which has the simplest form for practical application, is proposed for predicting the fatigue life of CFRP Coefficients K 1 and K 2 were adopted in this paper to separately describe the changing rate of fatigue life for the case where σ max was greater than 1400 MPa and those where σ max was smaller than 1400 MPa. Equation (2) is shown below: where σ max1 and σ max2 represent the maximum and minimum σ max on the same stress range curve. At the stress ranges of 200 MPa, 400 MPa, 600 MPa, and 800 MPa, σ max1 was 2000 MPa, 2000 MPa, 1800 MPa, 1800 MPa, respectively, and σ max2 was 1400 MPa, 1200 MPa, 1000 MPa, and 800 MPa, respectively; σ 1400 = 1400 MPa; N 1 and N 2 are the fatigue life at σ max1 and σ max2 ; N 1400 is the fatigue life when σ max = 1400 MPa.
The calculation results of K 1 and K 2 at different stress ranges are shown in Figure 8 (Supplementary material Table S1). It can be seen that K 2 was considerably less than K 1 at the same stress range, and the difference between K 1 and K 2 was smaller with increasing stress range. At the stress range of 400 MPa, S-N curve of the maximum stress σ max and fatigue life (log N) presented the form of double broken lines. With increasing stress range, S-N curve tended to display a linear relationship. This figure reflects that different stress ranges brought different changing rates of fatigue life.
As shown in Figure 8, the effect of the stress range on the slope K of S-N curves was evident. In addition, the slope K varied with the maximum stress. Most of the available methods for predicting the S-N curve ignore the effect of the stress range. To take into account both the effect of the stress range and the varying slopes of the S-N curve with the maximum stress, a bilinear equation, which has the simplest form for practical application, is proposed for predicting the fatigue life of CFRP tendons in Equation (3): There are four parameters in this model, namely, a, b, c, d, among which parameters b and d are the effects of stress range ∆σ on the fatigue life and its changing speed. At the same time, two stages were divided at σ max = 1400 Mpa. The values of the four parameters were determined by fitting the experimental results. If σ max was no less than 1400, the optimal values of a, b, c, d were 2931, −0.005, −151, −0.47, respectively. If σ max was less than 1400, the optimal values of a, b, c, d were 1661, 0.42, −7.5, −0.25, respectively.
The comparison of the calculation results of prediction in Equation (3) with test results is shown in Figure 9 (Supplementary material Table S2). The test data were mainly derived from the results of [12] and the test in this paper, and the test values corresponding to the data points in the figure are the average values of test results at the same level. The error e of the prediction results was 0.040, and prediction Equation (3) has high prediction accuracy. In Figure 9, n test is test data and n cal is the calculated value of Equation (3).
Prediction Equation (3) has higher accuracy in predicting the fatigue life of CFRP tendons, but with slightly complex forms. The stress range ∆σ and the maximum stress σ max were the main factors influencing the fatigue life of CFRP tendons. A simplified equation to calculate the fatigue life is given in the paper, as shown in Equation (4): There are three parameters in this model, namely, a 1 , b 1 , c 1 . According to the test results, the optimal values of a 1 , b 1 , c 1 were 3031, −1.16, and −231, respectively.
There are three parameters in this model, namely, , , . According to the test results, the optimal values of , , were 3031, −1.16, and −231, respectively. The comparison of the prediction results of Equation (4) with the test results is shown in Figure  10 (Supplementary material Table S3). The error of prediction results was 0.085 and the simplified calculation Equation (4) also had better prediction accuracy. In Figure 10, is test data and is the calculated value of Equation (4).  As shown in Table 2, at the same stress level, the fatigue life of CFRP presented relatively great discreteness. A certain reliability should be considered to predict the fatigue life of CFRP tendons during the design. Therefore, on the basis of the test data (from this paper and [12]), the fatigue life reliability of CFRP tendons was analyzed. The normal lifetime distribution (NLD) [16] method and Whitney's method [16] were used to predict the fatigue life of CFRP tendons.
NLD method is a simplified function model based on probability distribution. It was assumed that the fatigue life of each specimen was in normal distribution and the variation coefficient was 15%. On the basis of the assumption above, the safety guarantee rate was taken as 95%, and the fatigue life at each load level was calculated in Equation (5): where refers to the number of specimens at the ith load level, and refers to the average fatigue life at the ith load level.
Through fitting, the S-curve between the load level with 95% guarantee rate and the fatigue life can be obtained, and the curve can be expressed as Equation (6):  Table S3). The error of prediction results was 0.085 and the simplified calculation Equation (4) also had better prediction accuracy. In Figure 10, n test is test data and n cal is the calculated value of Equation (4).
There are three parameters in this model, namely, , , . According to the test results, the optimal values of , , were 3031, −1.16, and −231, respectively. The comparison of the prediction results of Equation (4) with the test results is shown in Figure  10 (Supplementary material Table S3). The error of prediction results was 0.085 and the simplified calculation Equation (4) also had better prediction accuracy. In Figure 10, is test data and is the calculated value of Equation (4).  As shown in Table 2, at the same stress level, the fatigue life of CFRP presented relatively great discreteness. A certain reliability should be considered to predict the fatigue life of CFRP tendons during the design. Therefore, on the basis of the test data (from this paper and [12]), the fatigue life reliability of CFRP tendons was analyzed. The normal lifetime distribution (NLD) [16] method and Whitney's method [16] were used to predict the fatigue life of CFRP tendons.
NLD method is a simplified function model based on probability distribution. It was assumed that the fatigue life of each specimen was in normal distribution and the variation coefficient was 15%. On the basis of the assumption above, the safety guarantee rate was taken as 95%, and the fatigue life at each load level was calculated in Equation (5) where refers to the number of specimens at the ith load level, and refers to the average fatigue life at the ith load level.
Through fitting, the S-curve between the load level with 95% guarantee rate and the fatigue life can be obtained, and the curve can be expressed as Equation (6): As shown in Table 2, at the same stress level, the fatigue life of CFRP presented relatively great discreteness. A certain reliability should be considered to predict the fatigue life of CFRP tendons during the design. Therefore, on the basis of the test data (from this paper and [12]), the fatigue life reliability of CFRP tendons was analyzed. The normal lifetime distribution (NLD) [16] method and Whitney's method [16] were used to predict the fatigue life of CFRP tendons.
NLD method is a simplified function model based on probability distribution. It was assumed that the fatigue life of each specimen was in normal distribution and the variation coefficient was 15%. On the basis of the assumption above, the safety guarantee rate was taken as 95%, and the fatigue life at each load level was calculated in Equation (5): where m i refers to the number of specimens at the ith load level, and N i refers to the average fatigue life at the ith load level. Through fitting, the S-N curve between the load level with 95% guarantee rate and the fatigue life can be obtained, and the curve can be expressed as Equation (6): where σ refers to the stress level, referring to the maximum stress in this paper; R k refers to the characteristic value of fatigue life; and k are fitting parameters.
Whitney's method is based on two assumptions: (1) the S-N curve conforms to the power function, i.e., Equation (6); (2) the fatigue life conforms to the two-parameter Weibull distribution, as Equation (7): where N i is the cycle number at the ith stress level; N i and α f i are the scale parameter and the shape parameter for the Weibull distribution, respectively; and P s (N i ) is probability of survival after N i cycles. The values of σ 0 and k are determined by linearly fitting the experimental log(σ i ) − log N i . α f is solved using the maximum likelihood estimators equations [16]. Thus, the S-N curves with different confidence levels are calculated with Equation (8): The calculated results with 95% confidence levels are listed in Table 3. It can be seen from Table 3 for the stress ranges of 200 MPa and 400 MPa, the results calculated using NLD method and Whitney's method were quite close. For higher stress ranges (i.e., 600 MPa and 800 MPa), the predicted results using NLD method were slightly larger. Despite the relative simplicity of NLD method, it assumed 15% COV is typically lower than the experimental results for CFRP, making the predictions less confident. In Whitney's method, the COV is determined on the basis of the experiments and was thus more suitable for reflecting the variability of the material properties of FRP material [16].

Prediction
Results Equation ( According to the data in Table 3, the equation describing the relationship between the stress range and the maximum stress at 2 million times of fatigue life of CFRP tendons was established in this paper, as shown in Equation (9): ∆σ = 1350 − σ max .
On the basis of either one of the stress ranges and the maximum stress, the other one can be calculated through Equation (9). When the actual loading value was less than the calculated value, the fatigue life of CFRP tendons could be more than 2 million times.

Conclusions
The fatigue life of CFRP tendons was studied through fatigue test and analysis, and the fatigue performance of the new wedge-type anchorage was verified in this paper. The main conclusions are shown as follows: (1) In the fatigue cycle, CFRP tendons exhibited burst rupture in the middle portion. The new wedge-type anchorage presented excellent fatigue resistance without tendon slippage or tendon failure at anchorage. (2) The stress range ∆σ and the maximum stress σ max were two key parameters affecting the fatigue life N of CFRP tendons. At the same stress range, the greater the maximum stress, the shorter the fatigue life of CFRP tendons. At the same maximum stress, the greater the stress range, the shorter the fatigue life of CFRP tendons. (3) The bilinear equation and simplified equation for predicting the fatigue life of CFRP tendons established in this paper considered the effects of the stress range ∆σ and the maximum stress σ max . Both were able to provide accurate predictions. The bilinear equation had a higher level of accuracy, whereas the simplified equation was simpler and easy to use. (4) The predictions obtained using the Whitney's method suggested that at the 95% confidence level, the CFRP tendons were able to be subjected to 2 million cyclic loads without fatigue failure when the maximum stresses were 63.9%, 53.0%, and 36.8% f u for the stress range of 200 MPa, 400 MPa, 600 MPa, respectively.
In particular, the application scope for the conclusions and equations of this paper were that the stress range was 200-800 MPa and the maximum stress was 0.37-1.0 f u for the fatigue life of CFRP tendons.